Stability and Bifurcation Analyses of Reduced-Order Models of Forced ...
STABILITY AND BIFURCATION ANALYSES OF RECUDED-ORDER MODELS OF FORCED AND NATURAL CIRCULATION BWRS
BY QUAN ZHOU B.E., Tsinghua University, 1995 M.E. Tsinghua University, 2000
DISSERTATION
Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Nuclear Engineering in the Graduate College of the University of Illinois at Urbana-Champaign, 2006
Urbana, Illinois
© Copyright by Quan Zhou, 2006
Stability and Bifurcation Analyses of Reduced-Order Models of Forced and Natural Circulation BWRs Abstract Quan Zhou, Ph.D. Department of Nuclear, Plasma and Radiological Engineering University of Illinois at Urbana-Champaign, 2006 Rizwan-uddin, Advisor
Current design of boiling water reactors (BWRs) relies on recirculation pumps to remove fission heat from cores. The forced circulation BWRs faced nuclear-coupled thermal hydraulic stability problems in an island in the power-flow plane—under high power and low flow conditions. Previous studies of the stability problem were focused on simulations using large scale and high fidelity commercial codes. Even the existing studies using the reduced order model focused on linear stability of the BWR systems. Few analyses have been made to review phase and bifurcation characteristics of power and flow oscillations in BWR core. New designs of BWRs relying on natural circulation to extract core heat, such as SBWR and ESBWR, have been completed, and they are promising candidates to replace the current forced-circulation BWRs. In addition to the complicated nuclear-coupled density-wave oscillations, natural circulation systems with their long risers must also address the issue of flashing and associated nuclear-coupled thermal hydraulic stability. In this case, small perturbation of flow rate can be dramatically intensified by vaporization due to pressure changes. Previous studies of the nuclearcoupled instabilities and flashing-induced instabilities of natural circulation BWRs were limited, partly due to difficulties in developing models that fully respect pressure
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dependence of water and steam thermodynamic properties under low system pressure condition. In this dissertation, stabilities of the forced and natural circulation BWRs were studied separately. In the first part of the dissertation, a reduced-order model was modified and used for stability and bifurcation analyses of forced-circulation BWRs. The model was comprised of Ordinary Differential Equations (ODEs) with respect to thermal hydraulics, fuel dynamics, and neutronics. An extra parameter considering inhomogeneity of core loading was introduced. Stability and characteristics of oscillations were studied based on linear stability and Poincare-Andronov-Hopf Bifurcation (PAH-B) theorems of a general dynamic system. Results show that different bifurcations (supercritical or subcritical) may occur along the stability boundary in a parameter space. Moreover, it was found that although the conjugate pair of eigenvalue with the largest real part determines linear stability of the system, the other pair with the second largest real part is also important defining phase characteristics of oscillations. In the second part of this dissertation, a new reduced-order model was developed for the natural circulation BWRs. Water saturation enthalpy in this model depends on local pressure, allowing analysis of the flashing phenomenon. ODEs of flow enthalpy, water saturation enthalpy, steam quality and flow velocities were derived node by node with classic weighted residual approach. The nodalization schemes of the natural circulation loop are based on modeling the boiling boundary. Moving boundary scheme differentiates cases of the boiling boundary, but the fixed boundary scheme treats all cases of boiling the same. Both the schemes have advantages and disadvantages, and they are implemented for different applications. The model was validated by comparing with
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experiments performed on loop SIRIUS and natural circulation reactor Dodewaard. Stability and bifurcation analyses revealed complexity of dynamics of the natural circulation system. It is expected the studies in this dissertation are of value helping improve safety of BWR systems.
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ACKNOWLEDGEMENTS
First, I would like to thank my advisor, Professor Rizwan-uddin for his continuous guidance and encouragement throughout my years of study at University of Illinois. I would like to thank him also for his support and care in my life and job search. I would also like to thank Professor Barclay Jones, Professor Magdi Ragheb and Professor S Balachandar for serving on my final defense committee. I wish to extend special recognition to financial support provided by the U.S. Department of Energy under a Nuclear Engineering Education Research (NEER) Grant, number DE-FG07-00ID13923. I would also like to acknowledge support under the Computational Science and Engineering Fellowship program at University of Illinois at Urbana Champaign. I would like to thank my parents for the encouragement given to me since my childhood. Without their support, I could not have come to USA for my Ph.D. studies. I would also like to thank my officemates Daniel Rock, Federico Teruel, Jianwei Hu, Yizhou Yan, and Yuxiang Gu for making Room 251 NEL a pleasant working environment.
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Table of Contents NOMENCLATURE .................................................................................................. xi LIST OF TABLES..................................................................................................... xiii LIST OF FIGURES ................................................................................................... xiv PART I. STABILITY AND BIFURCATION ANALYSES OF FORCED CIRCULATION BWRS ............................................................................................ 1 CHAPTER 1. INTRODUCTION TO INSTABILITIES OF FORCED CIRCULATION BWRS ............................................................................................ 1 1.1. Instability Incidents and Experiments in BWRs.......................................... 1 1.2. Types of Instabilities.................................................................................... 7 1.3. Physical Mechanisms................................................................................... 10 1.3.1. Thermal Hydraulics Mechanism: Density Wave Oscillations (DWO) ........................................................................................................................ 10 1.3.2. Neutron Kinetics Mechanisms........................................................... 12 1.4. Previous Work ............................................................................................. 15 1.4.1. Detailed Models ................................................................................. 17 1.4.2. Reduced-order Models....................................................................... 18 1.5. Tools to Analyze Dynamical Systems ......................................................... 20 1.5.1. Frequency Domain Methods (Classical Stability Analysis) .............. 20 1.5.2. Bifurcation Analyses.......................................................................... 21 1.5.3. Numerical Integration ........................................................................ 22 1.6. Scope of Part I of This Thesis...................................................................... 22 CHAPTER 2. A REDUCED-ORDER MODEL OF FORCED-CIRCULATION BWR AND REVIEW OF BIFURCATION ANALYSIS METHODS .................................................................................................................................... 25 2.1. Forced Circulation BWR Model.................................................................. 25 2.1.1. Neutron Kinetics ................................................................................ 25 2.1.2. Consideration of In-Homogeneity ..................................................... 28 2.1.3. Heated Channel Thermal Hydraulics................................................. 30 2.1.4. Heated Conduction Model ................................................................. 33 2.2. Stability and Bifurcation Analysis Method.................................................. 34 2.2.1. Neutron Kinetics ................................................................................ 35 2.2.2. Poincaré-Andronov-Hopf Bifurcation (PAH-B)................................ 37 2.2.3. Semi-Analytical Method and BIFDD ................................................ 42 CHAPTER 3. STABILITY AND BIFURCATION ANALYSES OF FORCED CIRCULATION BWR MODEL ............................................................................... 3.1. Eigenvalues and Eigenvectors of the BWR Model...................................... 3.2. Homogeneous Core (F = 1) ......................................................................... 3.2.1. Results in Nsub—ρext Space.................................................................
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46 46 54 54
3.2.2. Results in ρext—∆Pext and Nsub— ∆Pext Space.................................... 3.3. Non-Homogeneous Core (F > 1) ................................................................. 3.3.1. Results in Nsub— ∆Pext Space............................................................. 3.3.2. Results in F—∆Pext Space ................................................................. 3.4. Discussion ....................................................................................................
58 61 61 66 68
CHPATER 4. NUMERICAL SIMULATIONS OF FORCED-CIRCULATION BWR MODEL ..................................................................................................................... 70 4.1. Numerical Simulations of Homogeneous Core (F=1) ................................. 70 4.2. Numerical Simulations of Non-Homogeneous Core (F>1)......................... 76 CHPATER 5. SUMMARY, CONCLUSIONS AND FUTURE WORKS OF PART I .................................................................................................................................... 92 5.1. Summary of the Model and Analysis Method ............................................. 92 5.2. Conclusions of Results................................................................................. 94 5.3. Future Work ................................................................................................. 97 PART II. STABILITY AND BIFURCATION ANALYSES OF NATURAL CIRCULATION BWRS ............................................................................................ 98 CHAPTER 6. INTRODUCTION TO INSTABILITIES OF NATURAL CIRCULATION BWR .......................................................................................................................... 98 6.1. Instabilities of Natural Circulation BWRs under Low Pressure.................. 98 6.2. Physical Mechanisms................................................................................... 104 6.2.1. Geysering ........................................................................................... 104 6.2.2. Flashing.............................................................................................. 106 6.3. Previous Work ............................................................................................. 109 6.4. Scope of Part II of This Thesis .................................................................... 112 CHAPTER 7. REDUCED-ORDER MODEL OF NATURAL-CIRCULATION BWRS .................................................................................................................................... 115 7.1. Nodalization of Natural Circulation System................................................ 115 7.2. Differential Equations.................................................................................. 121 7.2.1. Dimensional and Non-Dimensional Partial Differential Equations (PDEs)............................................................................................................ 121 7.2.2. Ordinary Differential Equations (ODEs) in A Node ......................... 125 7.2.3. Node with Small Length .................................................................... 129 7.2.4. Phase Variables across Boundaries.................................................... 130 7.3. Closure of Boiling Boundary and Its Time-Dependent Derivatives............ 134 CHPATER 8. STABILITY AND BIRFURCATION ANALYSES OF NATURAL CIRCULATION BWR MODEL ............................................................................... 141 8.1. Validations and Comparisons ...................................................................... 141 8.1.1. Convergence ...................................................................................... 141 8.1.2. Model Validations and Comparison with Experiments ..................... 147
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8.2. Stability and Bifurcation Analyses of the Thermal Hydraulics Model........ 8.3. Stability and Bifurcation Analyses of the Nuclear-Coupled Model ............ 8.3.1. Effect of Nodalization Schemes......................................................... 8.3.2. In-phase and Out-of-phase Oscillations............................................. 8.3.3. Bifurcation Analyses.......................................................................... 8.3.4. Sensitivity Analyses...........................................................................
153 164 164 166 172 176
CHPATER 9. NUMERICAL SIMULATIONS OF NATURAL CIRCULATION MODEL ..................................................................................................................... 180 9.1. Thermal-Hydraulic Model ........................................................................... 180 9.2. Nuclear-Coupled Thermal-Hydraulics Model of A Natural Circulation BWR .................................................................................................................... 188 CHPATER 10. SUMMARY, CONCLUSIONS AND FUTURE WORK OF PART II .................................................................................................................................... 204 10.1. Summary of Natural Circulation BWR Model .......................................... 204 10.2. Summary of Results................................................................................... 205 10.3. Future Work ............................................................................................... 209 APPENDIX A NON-DIMENSIONAL VARIABLES AND PARAMETERS IN THE FORCED CIRCULATION BWR MODEL .............................................................. 210 A.1. Thermal-Hydraulic Variables and Parameters............................................ 210 A.2. Fuel Dynamic Variables and Parameters .................................................... 210 A.3. Neutronic Variables and Parameters........................................................... 211 APPENDIX B IMPACT OF THE ERROR OF INTERMEDIATE VARIABLE I6(t) .................................................................................................................................... 212 APPENDIX C TYPICAL DIMENSIONAL PARAMETERS OF FORCEDCIRCULATION BWR .............................................................................................. 217 APPENDIX D DIFFERENT DEFINITIONS OF SUBCOOLING AND PHASE CHANGE NUMBERS FOR FORCED AND NATURAL CIRCULATION MODEL .................................................................................................................................... 219 APPENDIX E NON-DIMENSIONAL VARIABLES AND PARAMETERS IN THE NATURAL CIRCULATION BWR MODEL ........................................................... 221 APPENDIX F ODES OF THERMAL HYDRAULICS OF NATURAL CIRCULATION LOOP ......................................................................................................................... 223 APPENDIX G EXPANSION PARAMETERS WHEN NODE LENGTH IS CLOSE TO ZERO ......................................................................................................................... 227 APPENDIX H TYPICAL DIMENSIONAL PARAMETERS OF SIRIUS FACILITY .................................................................................................................................... 230
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APPENDIX I TYPICAL DIMENSIONAL DODEWAARD REACTOR................ 231 I.1. Thermal-Hydraulic Parameters..................................................................... 231 I.2. Fuel Dynamic and Neutronic Parameters..................................................... 232 BIBLIOGRAPHY...................................................................................................... 233
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Nomenclature A: Bi : C (r , t ) : D(r , t ) : Fr J0
cross sectional flow area Biot number averaged delayed neutron precursor group concentration neutron diffusion coefficient Froud number Bessel function of first kind of order 0
J1 K L Ni N exp
Bessel function of first kind of order 1 pressure loss coefficient flow channel length number of node in region i (=1, 2,3) area number Boussinesq expansion number
Nf
friction number
N flash
flashing number
N pch
phase change number
Na
N (r , t ) N sub P Psys R T a af
reactor core radius temperature expansion parameter of flow enthalpy expansion parameter of water saturation enthalpy
cα cD cc e f h hf
void reactivity coefficient Doppler reactivity coefficient clad specific heat eigenvector of Jacobian matrix friction factor enthalpy or heat transfer coefficient water saturation enthalpy
hinlet hm k
coolant inlet enthalpy flow enthalpy pressure loss coefficient
n q' s
neutron number density linear heat flux expansion parameter of steam quality
neutron number density subcooling number pressure system pressure of natural circulation loop
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t v x z ∆Pdrv ∆Pext ∆h fg Λ Σa
Σf α
β λ µ ρ ω ik
Subscripts 1φ 2φ avg c dc exit fric grav i in or inlet l loc lim m o out or outlet r sat
time neutron or cooland velocity steam quality axial spatial coordinate static water pressure along channel and riser external pressure drop of the forced-circulation model vapor-liquid enthalpy difference neutron generation time macroscopic absorption cross section macroscopic fission cross section void fraction or thermal diffusivity delayed neutron fraction precursor decay constant boiling boundary reactivity or density modal eigenvalues
µ
single-phase two-phase average channel downcomer riser exit frictional gravitational node number channel or riser inlet subcooled water, pressure loss due to local restriction smallest value mixture reference channel or riser outlet riser saturation boiling boundary
Superscripts ~ *
steady-state value dimensional quantity
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List of Tables Table 1.1:
Instability events during BWR operations and during stability tests..... 3
Table 2.1:
Properties of fixed points and periodic solutions close to the stability boundary ................................................................................................ 41
Table 3.1:
Elements corresponding to n0 and n1 in the eigenvector corresponding to e1 ........................................................................................................ 52 Elements corresponding to n0 and n1 in the eigenvector corresponding to e2 ........................................................................................................ 53
Table 3.2:
Table 4.1:
Perturbations for phase variables for F = 4.6 and 5.0. Operating parameter set ( N sub , ρ ext , ∆Pext ) corresponds to (1.0, 0.0, 8.5) ............................... 83
Table 8.1:
Critical eigenvalues E imag and elements EV n0 and EV n1 (corresponding to n0 (t ) and n1 (t ) ) respectively in the eigenvector at four operating points shown in Figures 8.13 and 8.14. ............................................................ 171
Table 9.1:
Operating points and characteristics of oscillations and PAH-B predicted by BIFDD............................................................................................... 195
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List of Figures Figure 1.1: Figure 1.2: Figure 1.3: Figure 1.4: Figure 1.5:
Figure 1.6: Figure 2.1: Figure 2.2: Figure 3.1: Figure 3.2:
Figure 3.3:
Figure 3.4:
Diagram of Swiss boiling water reactor KKL ....................................... 2 Relative power and flow rate during instability event of LaSalle unit 2 ................................................................................................................ 5 Flow patterns and void distributions of different types of oscillations.. 9 (a) In-phase oscillations ......................................................................... 9 (b) Out-of-phase oscillations.................................................................. 9 Illustration of pressure drops and inlet flow along a uniformly heated channel with constant flow cross-section .............................................. 11 Shapes and variations of the fundamental and first azimuthal modes ... 14 (a) 3-dimensional shape of the fundamental mode of a homogeneously loaded core ............................................................................................. 14 (b) 3-dimensional shape of the first azimuthal mode of a homogeneously loaded core ............................................................................................. 14 (c) fundamental mode during an oscillation .......................................... 14 (d) first azimuthal mode during an oscillation....................................... 14 Power-flow map of KKL plant (Switzerland) ....................................... 16 Examples of stable and unstable periodic solution ................................ 39 (a) stable periodic solution..................................................................... 39 (b) unstable periodic solution................................................................. 39 Examples of stability boundary and oscillation curves calculated by BIFDD ................................................................................................................ 45 Critical values of ∆Pext , when N sub = 1.0, ρ ext = 0.0 and F = 1.0, 2.0, 3.0, 4.6, and 5.0 respectively ........................................................................ 49 Eight rightmost eigenvlaues corresponding to the critical values of ∆Pext for the five cases shown in Figure 3.1 ................................................... 50 (a) F = 1.0 .............................................................................................. 50 (b) F = 2.0 .............................................................................................. 50 (c) F = 3.5 .............................................................................................. 50 (d) F = 4.6 .............................................................................................. 50 (e) F = 5.0 .............................................................................................. 50 ................................................................................................................ 56 (a) Stability boundary in N sub - ρ ext space ............................................. 56 (b) Fifteen percent oscillation amplitude ( ε = 0.15) curves in N sub (ρ ext − ρ ext ,critical ) space ........................................................................... 56 ................................................................................................................ 59 (a) Stability boundaries in ρ ext - ∆Pext space .......................................... 59 (b) Fifteen percent oscillation amplitude ( ε = 0.15) curves in ρ ext -
(∆P
ext
Figure 3.5:
− ∆Pext ,critical ) space ....................................................................... 59
................................................................................................................ 60
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(a) Stability boundary in N sub - ∆Pext space ........................................... 60 (b) Fifteen percent oscillation amplitude ( ε = 0.15) curves in N sub -
(∆P
ext
Figure 3.6:
Figure 3.7:
Figure 3.8:
Figure 4.1:
Figure 4.2:
Figure 4.3: Figure 4.4: Figure 4.5: Figure 4.6: Figure 4.7: Figure 4.8:
Figure 4.9:
− ∆Pext ,critical ) space ....................................................................... 60
................................................................................................................ 62 (a) Stability boundaries in N sub — ∆Pext space for F = 1.0, 2.0, 3.5, 4.6 and 5.0........................................................................................................... 62 (b) 5% oscillation curves in N sub — (∆Pext − ∆Pext ,critical ) space for F = 1.0, 2.0, 3.5, 4.6 and 5.0................................................................................ 62 ................................................................................................................ 64 (a) Stability boundary and the boundary along which the real part of the second largest pair of eigenvalue is zero for F = 5.0 ............................. 64 (b) 5% oscillation amplitude curves corresponding to the eigenvalue pairs with the largest and second largest real parts......................................... 64 Stability boundaries in F— ∆Pext space for N sub = (a) 4.50; (b) 3.25; (c) 1.90; (d) 1.40; (e) 1.30; (f) 1.18 ............................................................. 67 Results of numerical integration for parameter values corresponding to point 1 in Figure 3.3. Operating parameters ( N sub , ρ ext , ∆Pext ) are (9.0, 0.03005, 8.0).. ........................................................................................ 72 Results of numerical integration for parameter values corresponding to point 2 in Figure 3.3. Operating parameters ( N sub , ρ ext , ∆Pext ) are (3.0, 0.01320, 8.0). ......................................................................................... 74 Results of numerical integration at point 3 in Figure 3.3. Operating parameters ( N sub , ρ ext , ∆Pext ) at the point are (3.0, 0.01, 8.0). .............. 75 Results of numerical integration of the governing set of nonlinear ODEs for F = 2.0. Operating parameters ( N sub , ρ ext , ∆Pext ) are (1.0, 0.0, 8.5)...... 77 Results of numerical integration for F = 3.0. Operating parameters ( N sub , ρ ext , ∆Pext ) at the point are (1.0, 0.0, 8.5). ................................... 78 Results of numerical integration for F = 4.6. Operating parameters ( N sub , ρ ext , ∆Pext ) at the point are (1.0, 0.0, 8.5). ................................... 79 Results of numerical integration for F = 5.0. Operating parameters ( N sub , ρ ext , ∆Pext ) at the point are (1.0, 0.0, 8.5). ................................... 80 Perturbations of n0 and n1 for F = 4.6. Operating parameters ( N sub , ρ ext , ∆Pext ) at the point are (1.0, 0.0, 8.5). Perturbations of timedependent variables are introduced as ξv , where ξ =0.01 is a small number, v is the eigenvector corresponding to the pair of eigenvalues with the largest real part................................................................................. 84 Perturbations of n0 and n1 for F = 5.0. Operating parameters ( N sub , ρ ext , ∆Pext ) at the point are (1.0, 0.0, 8.5). Perturbations of timedependent variables are introduced as ξv , where ξ =0.01 is a small
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Figure 4.10: Figure 4.11: Figure 4.12: Figure 4.13:
Figure 6.1: Figure 6.2: Figure 6.3:
Figure 6.4: Figure 6.5:
Figure 7.1: Figure 7.2:
Figure 7.3: Figure 7.4: Figure 7.5: Figure 8.1:
number, v is the eigenvector corresponding to the pair of eigenvalues with the largest real part................................................................................. 85 Results of numerical integration for F = 5.0. Operating parameters ( N sub , ρ ext , ∆Pext ) at the point are (2.2303, 0.0, 3.9). ............................. 88 Results of numerical integration for F = 5.0. Operating parameters ( N sub , ρ ext , ∆Pext ) at the point are (2.2303, 0.0, 4.1).. ............................ 89 Results of numerical integration for F = 5.0. Operating parameters ( N sub , ρ ext , ∆Pext ) at the point are (0.8, 0.0, 7.85).. ................................ 90 Results of numerical integration for F = 5.0. Operating parameters ( N sub , ρ ext , ∆Pext ) at the point are (0.8, 0.0, 8.2). ................................... 91 Diagram of natural circulation BWR design and schematic flow path. 99 Typical geysering flow signals in two parallel heated channels............ 102 (a) Flow signal in channel 1, u(1) .......................................................... 102 (b) Flow signal in channel 2, u(2).......................................................... 102 Typical flow signals recorded in natural circulation SIRIUS loop. System pressure is P = 0.2 MPa, heat flux is q’’ = 53 kW/m2.. ......................... 103 (a) single phase natural circulation flow, subcooling number Nsub = 29.8.............................................................................................. 103 (b) intermittent wave due to flashing, Nsub = 23.6 ................................. 103 (c) sinusoidal oscillations, Nsub = -0.08.................................................. 103 (d) steady two-phase flow, Nsub = -2.17................................................. 103 Flow regimes in a channel corresponding to four points of stage I and II of a geysering cycle in the experiment by Aritomi et al.. .......................... 105 Flow pattern and enthalpy profiles during a cycle of intermittent wave caused by flashing.................................................................................. 107 (a) Both channel and riser are single phase ........................................... 107 (b) The hot fluid reaches the upper part of the riser .............................. 107 (c) Flow is accelerated ........................................................................... 107 (d) Flashing is suppressed when the cold fluid reaches the upper part of the riser ........................................................................................................ 107 Schematic plot of natural circulation system... ...................................... 116 Cases based on position of boiling boundary... ..................................... 117 (a) Boiling boundary is out of the riser outlet........................................ 117 (b) Boiling boundary is in the riser ........................................................ 117 (c) Boiling boundary stays at the connection of channel and riser ........ 117 (d) Boiling boundary is in the channel................................................... 117 Nodalization schemes for the case 2 (boiling boundary in the riser)..... 120 Conditions across a boundary.... ............................................................ 132 Intermediate unknowns for ODE evaluations. Boiling boundary is in the channel in this example.......................................................................... 136 Steady state boiling boundary and inlet flow velocity calculated with fixed boundary schemes. Total number of nodes are N = 3, 5, and 9...... ...... 143
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Figure 8.2:
Figure 8.3:
(a) boiling boundary µ .......................................................................... 143 (b) inlet flow velocity vinlet .................................................................... 143 Stability boundaries in inlet subcooling—heat flux parameter space for two different nodalization schemes....... ....................................................... 145 (a) moving boundary scheme................................................................. 145 (b) fixed boundary scheme..................................................................... 145 (c) comparison of moving and fixed boundary schemes ....................... 145 Comparison of steady state flow rates between experiments and analyses. System pressure is Psys = 0.2 MPa......................................................... 148 (a) heat flux q ' = 1.7647 (kW/m) ........................................................... 148 (b) q ' = 2.9412 (kW/m) .......................................................................... 148 (c) q ' = 4.1177 (kW/m) .......................................................................... 148
Figure 8.4:
Figure 8.5:
Figure 8.6: Figure 8.7:
Figure 8.8: Figure 8.9:
(d) q ' = 5.2941 (kW/m) .......................................................................... 148 Comparisons of experimental data with the stability boundaries evaluated using the model developed in chapter 7. System pressure is Psys = 0.1 MPa........ ................................................................................................ 150 (a) Comparison of stability boundary .................................................... 150 (b) Comparison of no-boiling boundary for different Boussinesq expansion parameter values .................................................................................... 150 (c) Comparison of both stability boundary and no-boiling boundary.... 150 Comparison of experimental data points with stability boundaries evaluated using the model developed in Chapter 7. System pressure is Psys = 0.2 MPa........ ................................................................................................ 152 (a) Comparison of stability boundary .................................................... 152 (b) Comparison of no-boiling boundary for different Boussinesq expansion parameter values .................................................................................... 152 (c) Comparison of both stability boundary and no-boiling boundary.... 152 Stability boundary and the boundary on which the second rightmost pair of * eigenvalues has zero real part. System pressure Psys = 0.1 MPa........... 156 7.5% oscillation curves for different segments of the SB shown in Figure 8.6........................................................................................................... 157 (a) segment A-T ..................................................................................... 157 (b) segment T-C ..................................................................................... 157 (c) segment D-E ..................................................................................... 157 (d) segment E-F...................................................................................... 158 (e) segment F-G ..................................................................................... 158 Stability boundary and the boundary of the eigenvalues with the second * largest real part. System pressure Psys = 0.2 MPa........ ......................... 159 7.5% oscillation curves for different segments of the SB shown in Figure 8.8........................................................................................................... 160 (a) segment A-T ..................................................................................... 160 (b) segment T-C ..................................................................................... 160 (c) segment C-D..................................................................................... 160 xvii
Figure 8.10: Figure 8.11:
Figure 8.12:
(d) segment E-F...................................................................................... 160 Stability boundary calculated for system parameters corresponding to the * Dodewaard reactor. System pressure Psys = 1.07 MPa......... ................ 162 7.5% oscillation curves for the segments of the SB shown in Figure 8.10......................................................................................................... (a) segment A-B..................................................................................... (b) segment C-D..................................................................................... (c) segment D-E ..................................................................................... * SBs for different size nodes in ∆Tinlet — ρ ext parameter space for........
163 163 163 163 165
*
(a) Psys = 7.0 MPa.................................................................................. 165 *
(b) Psys = 0.4 MPa ................................................................................. 165 *
*
Figure 8.13:
Boundaries in ∆Tinlet — ρ ext space for Psys = 7.0 MPa. The solid line is the
Figure 8.14:
stability boundary, and the dotted line is the boundary on which the second largest real part is zero........ ................................................................... 168 * * Boundaries in ∆Tinlet — ρ ext space for Psys = 0.4 MPa. The solid line is the
Figure 8.15:
Figure 8.16: Figure 8.17:
Figure 9.1:
stability boundary (SB), and the dotted line is the boundary on which the second largest real part is zero........ ....................................................... 169 7.5% oscillation curve along the SB shown in Figure 8.13......... .......... 173 (a) segment A-X..................................................................................... 173 (b) segment X-B..................................................................................... 173 (c) segment B-C ..................................................................................... 173 7.5% oscillation curve along the SB shown in Figure 8.14......... .......... 174 (a) segment D-E ..................................................................................... 174 (b) segment E-F...................................................................................... 174 Sensitivity analysis results for different operating parameters......... ..... 177 (a) System pressure ................................................................................ 177 (b) Riser length ...................................................................................... 177 (c) Pressure loss coefficient at riser exit ................................................ 177 (d) Pressure loss coefficient at channel exit........................................... 178 (e) Pressure loss coefficient at channel inlet.......................................... 178 Results of numerical simulation for operating parameters corresponding to * point P1 in Figure 8.1(a). For this point, Psys = 0.2 MPa, q ' = 6.0 kW/m, *
Figure 9.2:
and ∆Tinlet = 35.0 (K)............................................................................ 182 (a) Non-dimensional boiling boundary.................................................. 182 (b) Non-dimensional inlet velocity ........................................................ 182 Results of numerical simulation for operating parameters corresponding to * point P2 in Figure 8.1(a). For this point, Psys = 0.2 MPa, q ' = 6.0 kW/m, *
and ∆Tinlet = 31.496 (K)........................................................................ 183 (a) Non-dimensional boiling boundary.................................................. 183 (b) Non-dimensional inlet velocity ........................................................ 183
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Figure 9.3:
Results of numerical simulation for operating parameters corresponding to * point P3 in Figure 8.1(a). For this point, Psys = 0.2 MPa, q ' = 6.0 kW/m, *
Figure 9.4:
and ∆Tinlet = 15.0 (K)............................................................................ 184 (a) Non-dimensional boiling boundary.................................................. 184 (b) Non-dimensional inlet velocity ........................................................ 184 Results of numerical simulation for operating parameters corresponding to * point P4 in Figure 8.1(a). For this point, Psys = 0.2 MPa, q ' = 6.0 kW/m, *
Figure 9.5:
and ∆Tinlet = 8.8 (K).............................................................................. 185 (a) Non-dimensional boiling boundary.................................................. 185 (b) Non-dimensional inlet velocity ........................................................ 185 Results of numerical simulation for operating parameters corresponding to * point P5 in Figure 8.1(a). For this point, Psys = 0.2 MPa, q ' = 6.0 kW/m, *
Figure 9.6:
and ∆Tinlet = 5.0 (K).............................................................................. 186 (a) Non-dimensional boiling boundary.................................................. 186 (b) Non-dimensional inlet velocity ........................................................ 186 Results of numerical simulation for operating parameters corresponding to * point P1 in Figure 8.10. For this point, Psys = 1.07 MPa, q ' = 379.0 kW/m, *
Figure 9.7:
and ∆Tinlet = 4.0 (K).............................................................................. 189 (a) Non-dimensional boiling boundary.................................................. 189 (b) Non-dimensional inlet velocity ........................................................ 189 Results of numerical simulation for operating parameters corresponding to * point P2 in Figure 8.10. For this point, Psys = 1.07 MPa, q ' = 379.9 kW/m, *
Figure 9.8:
and ∆Tinlet = 4.0 (K).............................................................................. 190 (a) Non-dimensional boiling boundary.................................................. 190 (b) Non-dimensional inlet velocity ........................................................ 190 Results of numerical simulation for operating parameters corresponding to * point P3 in Figure 8.10. For this point, Psys = 1.07 MPa, q ' = 256.95 *
Figure 9.9:
kW/m, and ∆Tinlet = 8.5 (K).................................................................. 191 (a) Non-dimensional boiling boundary.................................................. 191 (b) Non-dimensional inlet velocity ........................................................ 191 Operating points in the inlet subcooling—reactivity parameter space for numerical simulations of the nuclear-coupled model.......... .................. 194 * (a) Psys = 7.0 (MPa) .............................................................................. 194 *
(b) Psys = 0.4 (MPa).............................................................................. 194 Figure 9.10:
Results of numerical integration for point a in Figure 9.9(a). Perturbation at t = 0 is given by ( n1 : n1, ss → n1, ss + 0.0001).......... ................................ 196 (a) time evolution of n 0 (t ) .................................................................... 196 (b) time evolution of n1 (t ) ..................................................................... 196
xix
Figure 9.11:
Figure 9.12:
Figure 9.13:
Results of numerical integration for point b in Figure 9.9(a). Perturbations at t = 0 are given by ( n0 : n0, ss → n0, ss + 0.0001, n1 : n1, ss → n1, ss + 0.0001).......... ......................................................................................... 197 (a) time evolution of n 0 (t ) .................................................................... 197 (b) time evolution of n1 (t ) ..................................................................... 197 Results of numerical integration for point c in Figure 9.9(a). Perturbations at t = 0 are given by ( n0 : n0, ss → n0, ss + 0.003, n1 : n1, ss → n1, ss + 0.001).......... ........................................................................................... 198 (a) time evolution of n 0 (t ) .................................................................... 198 (b) time evolution of n1 (t ) ..................................................................... 198 Results of numerical integration for point d in Figure 9.9(a). Perturbations at t = 0 are given by ( n0 : n0, ss → n0, ss + 0.001, n1 : n1, ss → n1, ss + 0.001).......... ........................................................................................... 199 (a) time evolution of n 0 (t ) .................................................................... 199
Figure 9.14:
(b) time evolution of n1 (t ) ..................................................................... 199 Results of numerical integration for point e in Figure 9.9(a). Perturbation at t = 0 is given by ( n0 : n0, ss → n0, ss + 0.0001)......................................... 200 (a) time evolution of n 0 (t ) .................................................................... 200
Figure 9.15:
Figure 9.16:
(b) time evolution of n1 (t ) ..................................................................... 200 Results of numerical integration for point f in Figure 9.9(b). Perturbation at t = 0 is given by ( n0 : n0, ss → n0, ss + 0.002)........................................... 201 (a) time evolution of n 0 (t ) .................................................................... 201 (b) time evolution of n1 (t ) ..................................................................... 201 Results of numerical integration for point g in Figure 9.9(b). Perturbations at t = 0 are given by ( n0 : n0, ss → n0, ss + 0.0001, n1 : n1, ss → n1, ss + 0.0001).......... ......................................................................................... 202 (a) time evolution of n 0 (t ) .................................................................... 202
Figure 9.17:
(b) time evolution of n1 (t ) ..................................................................... 202 Results of numerical integration for point h in Figure 9.9(b). Perturbations at t = 0 are given by ( n0 : n0, ss → n0, ss + 0.001, n1 : n1, ss → n1, ss + 0.001).......... ........................................................................................... 203 (a) time evolution of n 0 (t ) .................................................................... 203 (b) time evolution of n1 (t ) ..................................................................... 203
Figure B.1: Figure B.2:
Comparisons of I6(t) evaluated with expressions (B.1), (B.2), and (B.3).......... ............................................................................................. 215 Comparisons of numerical simulations for operating point 6 on page 118 of reference [30]. Oscillations are evaluated with...................................... 216 (a) expression (B.2)................................................................................ 216 (b) expression (B.3) ............................................................................... 216
xx
PART I Stability and Bifurcation Analyses of Forced Circulation BWRs
21
Chapter1 Introduction to Instabilities of Forced Circulation BWRs
1.1 . Instability Incidents and Experiments in BWRs Boiling water reactors (BWRs) are characterized by a single-loop. That is, unlike pressurized water reactors (PWRs) that have a primary and a secondary loop, the coolant in current design of BWRs that flows through the core is the fluid that drives the turbine. Figure 1.1 shows a diagram of the Swiss KernKraftwerk power plant at Leibstadt (KKL) – a forcedcirculation BWR [1]. Subcooled water enters the cylindrical core from the bottom and starts boiling at a certain height along the reactor core. Typical exit quality and void fraction at the top of the core are about 0.15 and 0.65, respectively. Steam is separated from water by the steam separators at the top of the upper plenum and is sent to drive the turbine. The separated water, mixed with feed water, is forced by recirculation pumps, and flows back to the bottom of the core through the downcomer region. Under certain operating conditions, this BWR system becomes unstable and small perturbations, naturally present in the system, do not die out, leading to power and flow oscillations. Such power and flow instability phenomena have been observed for over two decades, and the BWR in almost all such cases had to be automatically or manually scrammed. Table 1.1 lists reports of instabilities found in literature over the last two decades. A specific example of such an instability is the event that occurred in TVO unit 1 (ABB design, 710 MWe power) on February 23, 1987. Before the event, power and core flow rate were about 60% and 30% respectively [2], and the axial power distribution had two peaks (a double-humped
1
Figure 1.1: Diagram of Swiss boiling water reactor KKL [1].
2
Table 1.1: Instability events during BWR operations and during stability tests Time
Country
BWR
Condition
1980
Finland
TVO-II
Test
1981
USA
Vermont Yankee
Test
1982
Italy
Caorso-1
Operation
1983
Italy
Caorso-1
Test
1984
Italy
Caorso-1
Operation
1987
Sweden
Forsmark-1
Test
1987
Finland
TVO-I
Operation
1988
USA
Lasalle-2
Operation
1989
Sweden
Forsmark-1
Operation
1989
Sweden
Ringhals
Test
1990
Switzerland
KKL-1
Test
1991
Germany
Isar-1
Operation
1991
Spain
Confrentes
Operation
1992
USA
WNP-2
Operation
1995
Mexico
Laguna Verde-1
Operation
1996
Sweden
Forsmark-1
Operation
1998
Sweden
Oskarshamn-3
Operation
1999
Sweden
Oskarshamn-2
Operation
3
distribution). This combination of power and flow rate along with the axial power distribution is very unfavorable to the stability of the reactor. Under these conditions, a power oscillation with peak-to-peak amplitude of about 12% was indicated by Average Power Range Monitors (APRMs). However, these signals were not noticed by the operators. What made the conditions worse was that under these conditions a routine test of the feedwater heating system was performed. During the test, a group of bypass valves failed to turn back to their normal positions, which caused a dramatic decrease in feedwater temperature. Combination of the adverse operating conditions and valve malfunction resulted in large amplitude oscillations of power and flow rate at once. A few second later, the scram limit under this flow rate, 90% power level, was reached, and the reactor was automatically shut down. On March 9, 1988, another instability event occurred in LaSalle unit 2 (GE design, BWR5, 1,130 MWe) [3], which is also the first such instability event in the U.S. The event was caused by technicians who inadvertently opened a valve during a functional test. The transient process following this mistake caused trips of both circulation pumps, and rapid flow reduction from 76% to 29%. Due to inadequate moderation of neutrons, reactor power also decreased rapidly from 84% to approximately 40%. Following power reduction, feed water heater was automatically isolated and led to a reduction in feed water temperature. A positive reactivity was thus induced into the core through void reactivity feedback. As a result, the power began to increase. Approximate 5 minutes into the event, a power oscillation was detected through APRM readings between 25% and 50% of full power with 2 to 3 s period. In addition to global power indicated by APRM signal, Local Power Range Monitor (LPRM) signals revealed occurrence of regional oscillations with different phases. The power and flow oscillations finally resulted in automatic scram. Figure 1.2 shows the power and flow rate of the actual event over a 9 minute window.
4
Figure 1.2 Relative power and flow rate during instability event of LaSalle unit 2 [3]
5
Another event took place at Laguna-verde, unit 1 (BWR/5, 1,308 MWe) in Mexico on January 24, 1995 [4]. The instability was excited by flow control valve movement at about 37% power and 37.8% flow rate. In order to shift recirculation pump to high speed, the flow control valves were closed to their minimum positions. Traces of power and flow oscillations were noticeable after 40 seconds from the beginning of valve closure. At 2 min 10 sec, APRM signal showed an oscillation with growth rate of about 1.02. At 3 min 10 sec, the valves were closed to their minimum positions, but the oscillations persisted with the same growth rate. At that point, for safety reasons the reactor was manually scrammed. The magnitude of oscillation was less than 10% of rated power during the whole instability period. Fortunately, until now safety features have always functioned as expected and no serious accidents have been caused by these instabilities. However, potential of serious failure of fuel elements still exists. Following the instability events [5-10] observed during operations, many stability tests [10-13] have been conducted in BWR plants to study behaviors of reactors under adverse operating conditions (high power and low flow rate). Some of the tests reveal that BWRs are capable of complicated large amplitude regional power and flow oscillations that may not be noticeable from APRM readings, and consequently reactors may not be scrammed in-time [14]. Because of this safety concern, extensive research interests have been focused on the stability problem of forced-circulation BWR design, widely used in commercial power plants. In addition to stability concerns for the current generation of BWRs, similar concerns also exist for next generation of BWR designs. These designs have features similar to current generation of BWRs that would necessitate their stability analyses. Moreover, some of these designs have additional features – like natural circulation for normal operation or during accident scenarios – that require additional modeling and experiments. Natural circulation is, for example,
6
achieved by the addition of long risers on top of the heated core. These long risers not only impact the core and system dynamics, they lead to the possibility of flashing (during startup, under low pressure operations) induced instabilities that may result due to flashing in the unheated riser section. Goal of this work is to address stability issues in both, forced as well as natural circulation BWR designs. There are significant differences in their modeling that the thesis is essentially divided into two parts: stability and bifurcation analyses of, 1) forced circulation; and 2) natural circulation BWRs. The first five chapters are focused on forced circulation BWRs. However, several sections in Chapter 1 are relevant to forced as well as natural circulation systems – such as discussion of dynamical systems. Chapters 6 through 10 are focused on modeling and analysis of natural circulation BWR systems. Types of instabilities for forced circulation BWRs and their physical mechanisms are described in the next two subsections in this chapter. Previous work is described in Section 1.4. A brief review of dynamical systems is given in Section 1.5, and the scope of the first part of this thesis is outlined in the last section of this chapter, Section 1.6.
1.2. Types of Instabilities Event classification process has led to identification of two different kinds of oscillations according to phase of oscillations in different regions [8]. The first one is called in-phase or corewide oscillation, which is characterized by global and uniform (across the core) power and flow changes. Figure 1.3(a) shows a diagram of flow pattern of in-phase oscillations taken from reference [15], where arrow thickness is proportional to flow intensity, and density of bubbles is proportional to void fraction. Flow rate and void fraction across any horizontal core cross section 7
is identical; i.e., the left and right half of the core experience similar power and flow characteristics at any given time. Dynamics is primarily determined by momentum dynamics of the recirculation loop and the heat transfer conditions in the core. The global property of in-phase oscillations make it easier to be detected through monitoring Average Power Range Monitor (APRM) signals. The second one is the so-called out-of-phase or regional oscillations, where two halves of the core in a horizontal plane may experience out-of-phase oscillitations. Figure 1.3(b) shows a schematic diagram of flow pattern in out-of-phase oscillations. At the beginning of a cycle (left panel, t = 0), the flow rate and void fraction in (parallel) assemblies in the left half of the core (0 < θ < π) are respectively, smaller and larger than corresponding quantities in the assemblies in the right half of the core (π < θ < 2 π). After half of the oscillation period, flow rate in the left half of the core is increased and void fraction is decreased, while the changes in the right half are opposite. Overall, the power level and total flow rate in the core as a whole stay stationary or vary with much smaller amplitude. The possibility of power and flow in half of the core oscillating 180 degrees out of phase with the oscillations in the other half poses a bigger threat to safety of BWRs than in-phase oscillations, because it is difficult to detect them from monitoring APRM signals only. Another distinction between oscillatory instabilities can be made based on the characteristics of the amplitude of oscillations. In most self-sustained oscillation cases, signals recorded indicate ever-increasing amplitudes until scram. However, in several cases, amplitudes of oscillations are found to increase at the beginning and then saturate at a constant value. The oscillations with constant periods and amplitudes are usually called limit-cycles. Existence of stable limit-cycles in some cases is considered to be a clear indication of non-linear nature of BWR dynamics.
8
(a)
(b) Figure 1.3: Flow patterns and void distributions of different types of oscillations (a) in-phase oscillations. In this case, the flow rates and void distributions in different areas in the core vary with almost the same phase; (b) out-of-phase oscillations. In this case, flow rate and void distributions in the left half of the core oscillate out-of-phase from those in the right half of the core. [15]
9
1.3. Physical Mechanisms Root causes of various instabilities have been topics of studies for two decades. It appears that some of the instability events are induced by inappropriately designed control systems. They can be fixed relatively easily by adjusting the gain of control systems. Causes of the others, however, are inner feedback mechanisms existing in BWR physics. [These can only be avoided by staying away from regions in parameter space where these instabilities are likely to occur.] Since water in BWRs acts as both coolant and moderator of neutrons, feedback mechanisms come first from thermal-hydraulics of two-phase flow, and then additionally from coupled neutron kinetics. The thermal hydraulics and neutronics mechanisms that lead to oscillatory behavior will be discussed in this section.
1.3.1. Thermal-Hydraulics Mechanism: Density Wave Oscillations (DWOs) Small perturbations in inlet flow rate in two-phase heated channel flow can cause changes in local vapor generation rate and therefore in two-phase mixture density. Transmission of the perturbation along the channel produces a perturbation in local pressure drop, which may in-turn intensify the flow rate perturbation under certain operating conditions. It has long been understood that such a self-amplified thermal-hydraulic feedback process can produce so-called density wave oscillations (DWOs) [8]. Mechanism of DWOs can be illustrated by an example. Consider a heated channel with single phase (upstream) and two-phase (downstream) flow [8]. Flow is due to a constant imposed pressure drop across the channel (see Figure 1.4). A perturbation in inlet flow, as it travels downstream, may cause local density changes in the two-phase region. Specifically, larger value of inlet flow rate causes lower steam quality and higher mixture density. Changes in density,
10
Figure 1.4: Illustration of pressure drops and inlet flow along a uniformly heated channel with constant flow cross-section [8]. The density wave oscillations occur when the phases of total pressure drop and inlet flow rate are 180± (half cycle) apart.
11
in turn, lead to perturbation in velocity. If the acoustic effect, which is indeed very small, is ignored in the channel, this density perturbation will be transmitted upward with the flow speed. Since the local frictional pressure drop depends upon density and velocity of two-phase flow, the local pressure drop along the channel is perturbed with a delay that depends upon the propagation of the perturbation. The impact of the perturbation on the total pressure drop, which is the sum of local pressure drops along the channel, will also be delayed. Since the total pressure drop must remain constant (boundary condition), perturbation in local pressure drops are compensated by changes in inlet flow rate. This leads to another round of perturbations flowing downstream. This mechanism of DWO can be used to explain why the periods of oscillations observed in BWRs are about 2~3 seconds, which is the transmission time of density wave from the bottom to the top of the channel. Other studies that suggest a lower role of “traveling” density waves have also been proposed [16].
1.3.2. Neutron Kinetics Mechanisms In light water reactors (LWRs), neutrons are moderated by water. Since level of moderation depends upon the amount of water in the core, fission rate, and hence power, in a BWR is coupled with flow conditions through void (and fuel temperature) reactivity feedback coefficients. The design principles of BWRs require void and fuel temperature feedback coefficients to be negative. While two-phase thermal-hydraulics is the primary mechanism behind the oscillation in BWR, flow rate oscillations force the power to oscillate as well. Hence, an accurate modeling of the instabilities and the oscillatory phenomenon requires a coupled analysis of reactor thermal hydraulics and reactor neutron kinetics. One approach to include neutron
12
kinetics is via a modal decomposition of the the neutron flux which can provide a clear understanding of the in-phase and out-of-phase power oscillations in BWRs. Under steady-state operating conditions, the neutron flux in a homogeneously loaded cylindrical core is given by [17] Ψ 0 (r ,θ , z ) = J 0 (2.4048r / R ) Sin (π z / L )
(1.1)
This is known as the fundamental mode. Note that power generated in a core is proportional to neutron flux. The first azimuthal mode is given by
Ψ1 (r , θ , z ) = J1 (3.8317 r / R ) Sin (π z / L ) Sin (θ )
(1.2)
where, J 0 and J 1 are Bessel functions. Shapes of the fundamental and the first azimuthal modes are shown in Figures 1.5 (a) and (b). It is well known that at steady state, higher order modes die out and only the fundamental mode exists. During a transient process, however, power distribution depends on a combination of all modes. Figure 1.5 (c) and (d) show schematic plots of variations of fundamental and azimuthal modes during a transient. If amplitude of the fundamental mode is large, power oscillations will be in-phase; whereas if the amplitude of the azimuthal mode is large, the power oscillations will be out-of-phase. In general, small cores tend to have large amplitude of the fundamental mode, and hence in-phase oscillations. However, cores of modern commercial BWRs are very large. For example, the BWR of KKL-1 (Switzerland) has 648 assemblies [1]. This leads to potential decoupling of different parts of the core. Consequently, the oscillation of neutron flux (power) in different parts of the core may have quite different characteristics, and amplitude of the azimuthal mode can be large under certain operating conditions. When such a decoupling occurs, it may lead to out-of-phase oscillations.
13
(a)
(b) ψ1(r)
ψ0(r) t2 t1 t1
t0
r
t0
r
(c)
(d)
Figure 1.5: Shapes and variations of the fundamental and first azimuthal modes, (a) 3-dimensional shape of the fundamental mode of a homogeneously loaded core; (b) 3-dimensional shape of the first azimuthal mode of a homogeneously loaded core; (c) fundamental mode during an oscillation; (d) first azimuthal mode during an oscillation.
14
Instabilities in forced-circulation BWRs have become regulatory concerns. Power-flow map of BWR is the most important operational indicator which represents coupling of power and flow rate due to void and Doppler feedbacks. Power-flow maps of BWRs have to be validated for each core loading to avoid the potential regions of instabilities. Figure 1.6 shows a power-flow map of KKL power plant (Switzerland) [18], on which three regions are identified. BWR can be normally operated on points in region I of the power-flow map, where decay ratios of the coupled system are less then 0.8. Operating conditions can only enter and pass briefly through region II, where decay ratios are less than 1.0 but greater than 0.8. Region III is the exclusion zone, where decay ratios are expected to be greater than 1.0. This region must be avoided during the operation. The exclusion region for the KKL power plant is over the range of 40~70% of the rated power and 30~45% of the rated flow [18].
1.4. Previous Work A number of simulations and analyses have been carried out to better understand the mechanisms of these instabilities. These studies, in general, followed two approaches. The first approach considers BWR to be a complicated, coupled, thermal-hydraulic, heat conduction, and neutron kinetics system, and emphasizes fidelity of simulations. Models in this approach are usually comprised of complete set of partial differential equations in space and time. Large scale multi-purpose system codes like RAMONA, RETRAN, and RELAP are used in these studies to accurately reproduce the actual instability events in computer simulations. The approach is helpful in understanding specific instability events, but the system codes are cumbersome, timeconsuming, and of limited value to explore large operating parameter spaces. The second approach, on the other hand, focuses more on qualitative aspects and parametric trends of
15
130 120
3515 MWt MEOD
Thermal Power (%)
110
%100 P=3138 MWt
100 90 80
3138 MWt MEOD
70 60
Natural Circulation
40 30
I
II
III
50
80% Rod Line
20 10 0 0
10
20
30
40
50
60
70
80
90
100 110 120
Core Flow (%)
Figure 1.6: Power-flow map of KKL plant (Switzerland) [18]. Region II is the monitoring region and region III is the exclusion zone.
16
different operating conditions. Models used in these studies, usually called reduced order models (or ROMs), are obtained by reducing the set of PDEs to a set of non-linear ordinary differential equations (ODEs) via nodalization and standard reduction methods. Rich analytical techniques in frequency and time domains are available to analyze these reduced order models. Even when numerically integrated, these models yield results much faster than the detailed models used in the system codes, and hence allow exploration of very large portions of the operating parameter space. These features of reduced order models make them good tools to understand not only specific events but also to improve general knowledge of BWRs as dynamical systems. Previous works carried out using both of these approaches will be discussed in this section.
1.4.1 Detailed Models In 1987, Takigawa et al. [19] studied out-of-phase oscillations observed during stability tests in Caorso unit 1, employing a three-dimensional neutronics and a multi-channel thermalhydraulic code, TOSDYN-2. Two cases were considered. The first one includes only one unstable channel, and the second one includes 10 unstable channels in each half of the core. The out-ofphase oscillations were successfully reproduced by the second case. In 1988, Cheng et al. [20] analyzed the instability event in LaSalle-2 using Brookhaven National Laboratory (BNL) boiling water reactor analysis code (BPA). The code employs one-group point reactor model to predict fission power in the core, and the two-phase flow is modeled using the non-equilibrium drift flux model. Their simulations of APRM signals show a "remarkable resemblance to the actual APRM traces" [20]. Some sensitivity studies are also carried out. Araya et al. [21] simulated the same event using thermal-hydraulic transient analysis code RETRAN. Their calculations considered not only the point in the power flow map at which the instability event happened, but also determined
17
the stability boundary in the subcooling-number—power-to-flow-ratio space. Results agree qualitatively with the measurements. The instability event in Forsmark-1 was studied by Analytis et al. [9] using a three-dimensional BWR transient simulation code RAMONA 3-12. The code includes a three-dimensional neutron kinetics model, which solves two-group time-dependent diffusion equations using analytical nodal method. Each assembly is considered a heated channel and other components of BWR—including lower and upper plenum, recirculation pumps and downcomers—are considered. The analysis is based on the finding that large amplitude local oscillations were found in two bundles in the core, which were suspected to be improperly seated. In these calculations, inlet pressure loss coefficients of the channels corresponding to the two positions were reduced to simulate the effects of unseated bundles. Oscillations similar to the measured data were obtained. Several other studies are discussed in reference [10].
1.4.2. Reduced-order Models Reduced order models have been used to carry out (linear) stability analysis and bifurcation analysis, as well as for numerical integration. In 1986, stability analyses were carried out by March-Leuba et al. [15] [22]. They used a very simple, one-channel model, which contains one ODE for heat transfer, two ODEs for thermal-hydraulics to account for void reactivity feedbacks, and two ODEs for point reactor kinetics – one for power and one for delayed neutron precursor concentration. The system of 5 ODEs was integrated numerically for given operating parameters and initial conditions. Typical linear and nonlinear system behaviors, such as diverging and converging oscillations and period doubling, were observed. This analysis was, later, extended to a two-channel model with fundamental as well as first azimuthal modes for neutronics analysis [17]. The modification in the neutron kinetics model was integrated in the
18
frequency domain code LAPUR [23]. This analysis showed that different combinations of stable and unstable fundamental and first azimuthal modes may occur in different regions of power— flow map. A more detailed, and consequently complicated, reduced order model was developed by Karve et al. in 1997 [24]. This model includes detailed ω-mode neutron kinetics, fuel rod heat conduction model, and single- and two-phase channel thermal-hydraulics. Stability analysis was carried out, and numerical integrations were performed to determine the nature of oscillations. Neutron kinetics in the reduced order model developed by Muñoz-Cobo et al. [25], [26] is based on the λ-mode. Lumped equations were used to account for fuel rod heat transfer model and for channel thermal-hydraulics. Stability analyses of their model were conducted by numerical simulations, and out-of-phase oscillations were excited when the void feedback coefficients was gradually increased. Tsuji et al. [27] developed a reduced order model, which includes point reactor kinetics, a conduction heat transfer model and heated channel thermal-hydraulics. Stability boundary is determined for different external pressure drops and for external reactivity in the power-flow map. Sensitivity analysis of stability boundaries for power distributions and several operating parameters were presented. They also carried out a bifurcation analysis. The nature of PoincareAndronov-Hopf bifurcation is studied using the bifurcation code, BIFOR2. Numerical simulations were carried out to validate results of BIFOR2. In another study, asymptotic expansion method was used by Muñoz-Cobo and Verdu [28] for bifurcation studies. Complexity of this method increases dramatically for more detailed models. Rizwan-uddin [29] used another bifurcation code, BIFDD, to study the dynamics of the simple BWR model proposed by March-leuba et al. [15]. Stability boundaries in several parameter spaces as well as the nature of bifurcation were determined. Nonlinearity of the system was also
19
studied using numerical simulations. Period doubling and existence of a turning point in the bifurcation diagram were confirmed. Dokhane et al. [18] improved the model developed by Karve et al. by incorporating draft flux model in two-phase flow modeling and using more realistic reactivity feedback in nuclear coupling. Stability and bifurcation properties of the model are analyzed using code BIFDD. It is shown that both the void distribution parameter C0 and the drift velocity Vgj have stabilizing effects.
1.5. Tools to Analyze Dynamical Systems As mentioned in the previous section, objectives of BWR stability analyses include:
• determine stability margins of BWRs under normal and during transient conditions • predict dynamical behavior of the reactors in instability events • help BWR designers and operators limit occurrence and impact of instabilities Moreover, rather than using the detailed system codes, this study is based on the reduced order model approach discussed above. To achieve the goals itemized above, reduced order models of BWRs can be analyzed using different mathematical techniques in frequency and time domains. We here briefly review these mathematical techniques.
1.5.1 Frequency Domain Methods (Classical Stability Analysis) The non-linear, time-dependent ODEs of the models can be linearized and transferred to frequency domain through Laplace transform. Open and close-loop transfer functions can be deduced through the Laplace transform and control system block diagram. Usually, the form of the close-loop transfer function is given by a polynomial of order n, divided by another polynomial of order m (> n). Root-locus method calculates all m roots (poles) of the denominator 20
of the close-loop transfer function. Trajectories of the poles are plotted in the complex plane. If the trajectories stay to the left of the imaginary axis, the system is stable (because poles with negative real parts correspond to solutions in the time domain with negative exponents). Nyquistdiagram technique is another technique to determine stability of the system in frequency domain. It uses open-loop transfer function instead of closed-loop transfer function. The Nyquist diagram is actually a trajectory in complex plane of the open-loop function G (i w) , where w is frequency varying from − ∞ to + ∞ . Two rules to determine stability of the system are: if the open-loop transfer function is stable, the condition of stability of the close-loop system is that the trajectory of G (i w) does not encircle the point (-1, 0); if the open-loop transfer function is unstable and has k poles with positive real parts, the condition of stability of the close-loop function is that the trajectory of G (i w) (counter-clockwise) encircles the point (-1, 0) k times. Classical stability analysis divides the parameter space into stable and unstable regions. Infinitesimal perturbations grow when operating in the unstable region and decay when operating in the stable region. Results are not applicable to large amplitude perturbations, nor for large time. That is, in case of unstable behavior, for example resulting in oscillations with growing amplitude, classical stability analysis sheds no light on the long term amplitude of the resulting oscillations – they may continue to grow or may saturate at a certain value.
1.5.2. Bifurcation Analyses While classical stability analysis divides the parameter space into stable and unstable regions, bifurcation analyses go a step further, and shed some light on the nonlinear characteristics of the system. [Note that in very limited cases results of bifurcation analyses may be global. Most results are local in nature – obtained via perturbation methods – and hence valid in region close to
21
the point about which perturbations were carried out.] For example, bifurcation analysis may provide results for perturbations that are larger than “infinitesimal”. It is possible for a “steadystate” (fixed point) to be stable for “infinitesimal” perturbations (according to classical (linear) stability analysis) but unstable if perturbations larger than the specified amplitude are applied. Bifurcation analysis may also identify regions in parameter space (close to the stability boundary) where the steady-state is unstable but there exist stable periodic solutions. Since the classical stability and bifurcation analysis methods will be used as the primary mathematical tools in this thesis, they will be discussed in detail in the second chapter.
1.5.3. Numerical Integration Reduced order models are generally based on a set of nonlinear, ordinary differential equations (ODEs). Classical numerical integration methods, such as Runge-Kutta, Gear or Adams-Bashforth method, can be applied to solve the ODEs. Since, in many cases, the ODEs are stiff, it generally requires A-stable or α-stable versions of the methods. The numerical integration of the dynamical system does not rely on any assumptions on the perturbation amplitude – as do methods to determine stability and bifurcation characteristics of the system – and hence it is very useful in testing the validity of the results obtained using other methods. It is also the only means to determine system dynamics under conditions where other analytical methods are invalid – such as in regions away from the stability boundary where the results of the local bifurcation analysis are not applicable.
1.6. Scope of Part I of This Thesis
22
A number of reduced order models have been developed and analyzed to reveal mechanisms and parametric trends of instabilities of forced-circulation BWRs. However, there are several limitations in these studies: • Some reduced order models are overly simplified. For example, the pioneering work by
March-Leuba et al. is based on a model that has only 5 ODEs to represent thermal-hydraulics, heat conduction and neutron kinetics of BWRs. Realizing that the focus of that study was “nonlinear dynamics,” and the physics of BWRs was captured using empirically fitted constants to an actual transfer function, the over-simplification in the model may lead to loss of some key details of BWR dynamics. It is therefore necessary to study BWR dynamics using a model that better balances simplicity and accuracy. • Most studies, even those carried out using reduced order models, have focused on numerical
simulations and classical stability analyses, and relatively few models have been analyzed for their bifurcation characteristics. This is due to the fact that complexity of bifurcation analysis increases dramatically with number of equations in reduced order models. Therefore, only rather simple reduced order models have been used for studies, such as those by Tsuji et al. [27], Muñoz-Cobo et al. [28] and Rizwan-uddin [29]. Moreover, even when carried out, bifurcation characteristics were determined for only one parameter, consequently limiting the scope of these analyses to a small region of the rather large parameter space. Because of the approaches usually followed for the bifurcation analyses, it is not trivial to repeat them for different system parameters as bifurcation parameter. • In-phase oscillations have attracted more attention than out-of-phase oscillations. Instability
events and numerical simulations have shown transition from in-phase to out-of-phase oscillations as operating parameters change, and it is premised that this transition has close relationship with
23
the radial in-homogeneity of the core. However, to the author’s knowledge, the question of how the transition occurs and the mechanism through which inhomogeneity helps excite the out-ofphase oscillations have not been adequately answered. Purpose of the study presented in part I of this thesis is to tackle the above-mentioned limitations of existing studies. Specifically, a detailed model developed by Karve et al. [24] is used for steady-state, stability and bifurcation analyses. Model is also numerically integrated. This model is chosen because it represents a healthy balance between the level of detail, yet simple enough to be amenable to the tools of analyses available. Components of this model are briefly reviewed in the next chapter. A modification introduced in the model to artificially model radial core inhomogeneity is discussed in more detail. After the model, a brief introduction to bifurcation theory and the semi-analytic approach that makes it possible to perform bifurcation analysis without massive algebra are presented. In chapter three, results of stability and bifurcation analyses are presented and discussed. These are presented first for the homogeneous core, and then for the inhomogeneous core. Discussion of the complex relationship between inhomogeneity and in-phase and out-of-phase oscillations, as well as insight gained in understanding BWR dynamics from the results of the bifurcation analysis are particular focus of this chapter. Results of numerical integration performed for operating points in relevant parameter spaces are presented in Chapter 4. These results are used to verify predictions of bifurcation analyses. Chapter five summarizes stability and bifurcation analyses of forcedcirculation BWRs.
24
Chapter 2 A Reduced-Order Model of Forced-Circulation BWR and Review of Bifurcation Analysis Methods
Karve et al. [30] developed a reduced order model to study dynamics of forced-circulation BWRs. Although the model is simple, it is much more detailed than the 5-equation model used by March-Leuba et al [31] and retains several key details of BWR dynamics. The balance between simplicity and accuracy of this model make it a good candidate for stability and bifurcation studies. In this chapter, a brief review of the model is given in the first section. Emphasis is placed on the neutron kinetics part of the model, where a modification is introduced. The second section of the chapter is dedicated to a review and discussion of Poincare-Andronov-Hopf bifurcation theorem. This is a key theorem used in this work to identify regions of parameter space (close to stability boundaries) where stable or unstable limit cycles exist.
2.1. Forced-Circulation BWR Model The complete model can be divided into three components: neutron kinetics, fuel rod heat conduction and channel thermal-hydraulics. Complete and detailed derivation of the model can be found in [30]. Only those components of the model that are relevant to the analyses and discussion in the latter part of the thesis are reviewed below.
2.1.1. Neutron Kinetics
25
Governing equations of the neutron kinetics are derived from the matrix form of the onegroup neutron diffusion and one delayed neutron precursor equations
dY =M Y dt
(2.1)
where v∇ ⋅ D (r , t )∇ − vΣ a (r , t ) + (1 − β )vνΣ f (r , t ) λ M = − λ vβν Σ f
Y (r , t ) = [N (r , t ), C (r , t )]
T
(2.2)
(2.3)
where N (r , t ) and C (r , t ) are neutron and precursor densities, v is neutron velocity, ν is average neutrons released per fission reaction. These equations and reduction to their modal counterparts can be found in textbooks [15]. A one-group treatment for neutron kinetics is sufficient because of the time-scale of oscillations that are being modeled here [15]. Steady-state eigenvalue ω ik and eigenfunction φ ik of M satisfy
M 0 (r )φ ik (r ) = ω ik φik (r ) ,
(2.4)
1 ψ k ; i = 0, 1 ; k = 0,1 vβν Σ f (r ) (ω ik + λ )
φik (r ) = (2.5)
where ψ k , k = 0, 1 are fundamental and azimuthal modes of the steady-state matrix M 0 . For homogeneously loaded core, functions and shapes of the modes are shown in section 1.3.2 of chapter 1. The space and time dependent vector Y (r , t ) , thus, can be expanded in the eigenfunctions
Y (r , t ) = u 00 (t )φ 00 (r ) + u 01 (t )φ 01 (r ) + u10 (t )φ10 (r ) + u11 (t )φ11 (r )
26
(2.6)
where, u ik (t ) , i = 0, 1, k = 0, 1 are time dependent expansion parameters. The ω -mode based equations are developed by plugging the expansion (2.6) into equation +
(2.1) and taking inner product of both sides using the adjoint eigenfunction φik (r ) . This leads to a set of two ODEs for each mode k = 0, 1 1 ρ dn k (t ) = ω ik n k (t ) + (ω1k − ω 0 k )u k (t ) + ∑ km n m (t ) dt m =0 Λ k
(2.7)
1 ρ du k (t ) = ω1k u k (t ) + ∑ km n m (t ) , dt m = 0 Λ 1k
(2.8)
where nk (t ) = u0 k (t ) + u1k (t )
(2.9)
u k (t ) = u1k (t ) Λ ik =
(2.10)
φ ik+ (r ), φ ik (r )
V
ψ k (r ), vυΣ f (r )ψ k (r )
(2.11) V
n k , k = 0, 1 are time-dependent neutron densities of the fundamental and first azimuthal modes; u k , k = 0, 1 are time-dependent precursor densities of the fundamental and first azimuthal modes; Λ ik , i = 0, 1, k = 0, 1 are effective neutron generation time, and
V
denotes the inner product
over the volume of the core. The governing ODEs for neutron kinetics given above are coupled with fuel rod heat conduction and channel thermal-hydraulics through reactivity feedback ρ km (t ) =
1
∑ρ
ext , kml
− cα ,kml (α l (t ) − α 0,l ) − c D , kml (Tavg ,l (t ) − Tavg ,0,l )
l =0
(2.12)
27
where ρ ext is the control rod induced external reactivity, cα ,kml and c D ,kml are void coefficient and Doppler feedback coefficients for different modes and channels. α l and Tavg ,l are average void fraction and temperature of fuel pellet for different channels. Here the subscript l represents the lth channel. The core must be divided into at least two channels (l = 0, 1) to capture the feedback between modal neutronics with fundamental and first azimuthal modes, and core thermal hydraulics. α l and Tavg ,l are determined as intermediate variables in the solution of the heated channel thermal-hydraulics and the heat conduction parts.
2.1.2. Consideration of In-Homogeneity Parameters in the governing equations of neutron kinetics, such as cα ,kml , c D ,kml , ω km and Λ ik , depend on inner products of the modes and their adjoint functions. These parameters were evaluated by Karve et al. [30] using the fundamental and azimuthal modes given by equations (1.1) and (1.2). They are simply the Bessel’s functions in radial direction, which, however, are valid only for cylindrical cores that are not far from homogeneous configurations. There are several factors that can lead to in-homogeneous cores, which in-turn will lead to modes that may be quite different from the Bessel functions. These factors include: non-uniform burn-up, asymmetric control rod insertion, and variations in core inlet temperature between different (lumped) channels. Analysis of operational data from reactors that experienced instability events suggests that occurrence of out-of-phase oscillations may be closely related to the in-homogeneity of the core. For example, Miró et al. [12] found that radial power distributions that roughly correspond to the shape of Bessel functions, generally tend to generate in-phase oscillations; while bowl-shaped radial power distributions, with a local minimum at the center, tend to produce outof-phase oscillations. Hence, accurate evaluation of neutron kinetics parameters for different 28
loadings – including the inhomogeneity effects – is necessary to correctly predict characteristics of oscillations. Quantitative representation of different loadings in in-homogeneous cores requires a modal decomposition approach [32]. In this approach, loading pattern, burn-up history and control rod insertion of the core are specified, and 2-D or 3-D neutronics codes are used to solve the core configuration for eigenvalues and eigenfunctions. The evaluation process is fairly complicated and time-consuming. Moreover, it is case-specific, which eliminates the biggest strength of the reduced-order models. An alternate way to account for in-homogeneity of different core loadings is to modify system parameters to parametrically simulate the effects of inhomogeneous core loading. Recognizing that the major impact of core in-homogeneity on the dynamics of the core is to change the relative likelihood of in-phase or out-of-phase oscillations, reactivity feedbacks of fundamental and first azimuthal modes are the most appropriate parameters to be modified to incorporate the effects of inhomogeneous core on core dynamics. It is therefore appropriate that these feedback coefficients be varied, for example by multiplying by a suitable factor, to reflect relative strength of reactivity feedbacks for different cores. Observations and simulations have already confirmed that most oscillations for roughly homogeneous cores are in-phase. It indicates that reactivity feedback of azimuthal mode is much smaller than that of the fundamental mode in this case. On the other hand, occurrence of out-ofphase oscillations in in-homogeneous cores depends on how much the in-homogeneity can increase the feedback to the azimuthal mode. Consideration of in-homogeneity, therefore, leads to the introduction of an amplification factor F to modify reactivity feedback coefficients of the azimuthal mode,
cα ,1ml ,nu = F ⋅ cα ,1ml ,u
29
(2.13)
c D ,1ml ,nu = F ⋅ c D ,1ml ,u
(2.14)
where, cα ,1ml ,nu and c D ,1ml ,nu are feedback coefficients of an in-homogeneous (or non-uniform, hence the subscript nu) core; and cα ,1ml ,u and c D ,1ml ,u are feedback coefficients of a homogeneous (or uniform, hence the subscript u) core. Although there are other parameters that are affected by the in-homogeneity, their impact on dynamics of the modes are assumed to be minimal, and their values are held constant for different core configurations.
2.1.3. Heated Channel Thermal Hydraulics Heated channel thermal hydraulics model presented in this section and the fuel rod heat conduction model presented in the next section are taken from reference [30]. Except for one mistake found in reference [30] in the derivation of the reduced order model – identified and corrected below – these two models used here are identical to those in reference [30]. Most of the details in deriving these models are hence omitted. They can be found in reference [30]. A one-dimensional model is used to simulate water flowing through a heated channel in a BWR core. The channel consists of single-phase and two-phase regions. Ordinary differential equations (ODEs) describing dynamics of two key variables – water enthalpy in the single-phase region and steam quality in the two-phase region – are developed through conservation equations. Non-dimensional energy conservation equation (for enthalpy) in single phase region for a channel with spatially uniform but time-dependent heat flux is given by ∂h ( z , t ) ∂hm ( z , t ) = N ρ N r N pch ,1φ (t ) + vinlet (t ) m ∂z ∂t
(2.15)
where hm ( z , t ) is water enthalpy, and N pch ,1φ is the phase change number in the single-phase region, which is proportional to input heat flux. Definitions of dimensionless variables and
30
parameters are given in Appendix A. A quadratic polynomial in space, with time dependent coefficients, is assumed to approximate the enthalpy hm ( z , t ) = hm (0, t ) + a1 (t ) z + a 2 (t ) z 2 . ODEs for the time-dependent expansion parameters a1 (t ) and a2 (t ) are derived using classical weighted residual approach. Their forms are given by
[
]
da1 (t ) 6 = N ρ N r N pch ,1φ (t ) − vinet (t )a1 (t ) − 2vinlet (t )a 2 (t ) dt µ (t )
[
da 2 (t ) 6 =− 2 N ρ N r N pch ,1φ (t ) − vinlet (t )a1 (t ) dt µ (t )
]
(2.16)
(2.17)
where µ (t ) is the location of the boiling boundary, which separates the single-phase and the twophase regions. Above equations are applicable over 0 < z < µ(t). A similar approach is followed to derive equations for steam quality in the two-phase region; µ(t) < z < 1. For the homogeneous equilibrium model of the two-phase flow, the nondimensional continuity equation is [30]
∂ρ m ( z , t ) ∂ρ ( z , t ) + v m ( z, t ) m = − N pch, 2φ (t ) ρ m ( z , t ) ∂t ∂z
(2.18)
where, vm ( z , t ) = vinlet + N pch ,2φ ( z − µ (t )) is fluid mixture velocity. The mixture density ρ m ( z , t ) is expanded as
ρm ( z, t ) =
1 1 + s1 ( t ) ( z − µ (t )) + s2 ( t ) ( z − µ (t )) 2
(2.19)
where s1 (t ) and s 2 (t ) are the time-dependent expansion parameters for the steam quality. By applying the weighted residual method to the two-phase continuity equation, the ODEs for the expansion parameters s1 (t ) and s2 (t ) are determined as ds1 (t ) 1 = ( f 3 (t ) f 1 (t ) + f 4 (t )) dt f 2 (t )
31
(2.20)
ds 2 (t ) 1 = ( f 5 (t ) f 1 (t ) + f 6 (t )) dt f 2 (t )
(2.21)
where f1(t), f2(t), f3(t), f4(t), f5(t) and f6(t) are complicated functions of a1 (t ) , a2 (t ) , s1 (t ) , s2 (t ) and vinlet (t ) . Details and complete expressions of fi(t), i = 1, 2, …, 6 can be found in reference [30]. There is however one mistake in reference [30]. The expression for I6(t) given in Appendix E of the reference [30], which is an intermediate function in the expressions of fi(t), i = 1, 2, …, 6, has an error. The error occurred due to the fact that not enough terms were kept in expansions necessary for evaluation of certain expressions. The correct expression is δ 13 (t ) δ 1 2 (t )[3 − δ 1 (t )δ 3 (t )] δ e (t ) + + log 1 − 1 − δ 1 (t )δ 3 (t ) δ 1 (t ) δ 1 (t ) − δ e (t ) 2 δ e (t ) − δ 2 (t ) + 2δ (t )δ (t ) + 1 e 1 2 1 I 6 (t ) = 2 3 s1 (t ) − 4 s 2 (t ) 2δ e (t ) 2 2 3 + δ e (t )δ 1 (t ) + 2δ e (t )δ 1 (t )δ 3 (t ) + 3δ 4 (t ) 2δ 3 (t )δ (t ) 2 2 3 2 1 e + e + δ e (t )δ 1 (t ) + 2δ e (t )δ 1 (t )δ 3 (t ) 3 4
(2.22)
Impact of this error on the results of the stability analysis reported in reference [30] are minimal, and are discussed in Appendix B ODE for inlet flow velocity vinlet (t ) is obtained by integrating the momentum conservation equations over both the single-phase and two-phase regions. Constant pressure drop boundary condition across the length of the channel is imposed [30]. Form of the ODE is given by dvinlet (t ) 1 ( f8 (t ) + f9 (t ) f1 (t ) + f10 (t ) f11 (t ) ) = dt f 7 (t ) where fi(t), i = 7, 8, … 11 are complicated functions of a1 (t ) , a 2 (t ) , s1 (t ) , s 2 (t ) , and vinlet .
32
(2.23)
2.1.4. Heat Conduction Model For fuel rod heat conduction model, non-dimensional energy equation for temperature distribution in cylindrical fuel pellet, assuming azimuthal symmetry, is given by 2 1 ∂θ p (r , t ) ∂ θ p (r , t ) 1 ∂θ p (r , t ) = + + c q (n0 (t ) − n~0 ) 2 ∂t r ∂r αp ∂r
(2.24)
A gap conductance is considered between the fuel pellet and cladding. Because the conductivity of the cladding is much higher than that of the fuel pellet, to simplify heat conduction in the gap and cladding region the transient temperature distribution in the cladding, θ c (r , t ) , can be approximated by its steady-state form. Thus, the temperature distribution in the cladding and gap conductance is lumped into boundary condition for pellet temperature θ p (r , t ) . Due to large temperature gradient in the fuel pellet, a single quadratic profile over the entire pellet was found to not be adequate [30]. Hence, the pellet temperature distribution,
θ p (r , t ) , is represented using quadratic profiles over two segments, given by T1 (t )v1a (r ) + T2 (t )v1b (r ) T1 (t )v2 a (r ) + T2 (t )v2b (r )
θ p (r , t ) =
0 < r < rd rd < r < rp
(2.25)
where rp is radius of the fuel pellet, and rd = 0.83r p is an empirically determined value [30]. v1a (r ) , v1b ( r ) , v 2 a (r ) and v 2b (r ) are quadratic polynomials which are determined by imposing the boundary and interface conditions. To derive the ODEs for T1 (t ) and T2 (t ) in the single-phase and the two-phase regions, variational principle is applied. The unknown expansion parameters T1 (t ) and T2 (t ) that minimize the given variational [30] satisfy the ODEs dT1 (t ) = l1,1T1 (t ) + l 2,1T2 (t ) + l3,1c q (n0 − n~0 ) dt
33
(2.26)
dT2 (t ) = l1, 2T1 (t ) + l 2, 2T2 (t ) + l 3, 2 c q (n0 − n~0 ) dt
(2.27)
where l1,1, l2,1, l3,1, l1,2, l2,2 and l3,2 are constants that depend upon the operating parameters.
BWR as a dynamical system is hence represented by the combination of the three components: two ODEs for neutron kinetics for each mode (equations (2.7) and (2.8)), five ODEs for channel thermal hydraulics for each half of the core (equations (2.16), (2.17), (2.20), (2.21) and (2.23)); and two ODEs for the heat conduction model in the pellet for single phase and two phase regions (equations (2.26) and (2.27)). The evaluation of the right hand sides of these ODEs requires the evaluation of the steadys1 depend on v~inlet and n~0 , and all state values of the variables first. At the steady-state, a~1 and ~
other variables are 0. v~inlet and n~0 can be evaluated by imposing balance of reactivity and pressure drop at steady-state. [Tilde (~) here represents a steady-state quantity.] The model was tested by Karve [30] to validate the assumptions of quadratic representations of water enthalpy profile in the single-phase region and steam quality profile in the twophase region. Details can be found in Appendix C of reference [30].
2.2. Stability and Bifurcation Analysis Method Stability analysis (frequency domain), bifurcation analysis, or numerical integration (time domain) methods, as stated in the section 1.5, can be used to study dynamics of the BWR model. They are major mathematical tools for study of the reduced-order model in this dissertation. In this section, the stability and bifurcation analysis methods are discussed in more details than the section 1.5. Then a numeric-analytic approach to reduce complexity of evaluation is introduced.
34
This discussion will serve as background information to understand results presented in the next chapter. The numerical integration (time domain) method, although is important for analysis, is quite straightforward, and not repeated here.
2.2.1. Stability analysis of a BWR as a dynamical system The reduced order BWR model is formulated and represented by a set of nonlinear ODEs. It can be considered a dynamical system, whose behavior can be analyzed through mathematical tools developed for this purpose. General form of autonomous dynamical system can be written as dF = G (F ; λ ) dt
(2.28)
where, F = [F1
F2 … Fn ]
T
λ = [λ1 λ 2
[
G ( F , λ ) = G1 ( F ; λ ) G2 ( F ; λ )
(2.29)
λ m ]T Gn ( F ; λ )
(2.30)
]
T
(2.31)
F1, F2, … Fn are a set of time-dependent variables (also called phase-variables); λ1, λ2, … λm are
operating parameters; and G1, G2, … Gn are non-linear functions of F and λ . Steady-state solutions (or fixed points) of the dynamical system, F (t ) = F0 (λ ) , satisfy the condition
G ( F0 ; λ ) = 0
(2.32)
Note that there may be more than one fixed points corresponding to a given set of parameter values. Fixed points are simply the steady-state operating points of the dynamical system. Fixed points may be stable or unstable. Hence, safe operation of systems such as BWRs require that
35
stability of their steady-state operating points under different operating conditions be determined. Stability of a fixed point is governed by the eigenvalues of the Jacobian matrix. Jacobian of the dynamical system, A-double-bar, is a matrix of derivatives of Gi, i = 1, 2, …, n with respect to Fj, j = 1, 2, …, n. It can be written as
∂G1 ∂F 1 ∂G2 A = ∂F 1 ∂G n ∂F1
∂G1 ∂F2 ∂G2 ∂F2 ∂Gn ∂F2
∂G1 ∂Fn ∂G2 … ∂Fn ∂Gn … ∂Fn λ …
(2.33)
It is well known that stability of a fixed point depends on eigenvalues of the Jacobian matrix evaluated at that fixed point. Specifically: a) If all eigenvalues of the Jacobian matrix have negative real parts, the fixed point is stable. Infinitesimal perturbations will die away and the trajectory in phase space will converge to
the fixed point. b) If one or more of the eigenvalues has positive real parts, the fixed point is unstable. Infinitesimal perturbations will grow and will lead to a trajectory in phase space that diverges away from the fixed point. c) If the rightmost eigenvalue has a zero real part (lies on the imaginary axis), the fixed point is neutral. Stability of the fixed point is then determined by higher order analysis. Clearly, since the fixed points depend on the operating parameters λ , the parameter space can be divided into stable and unstable regions, divided by a stability boundary (SB). A stability boundary which separates stable regions from unstable regions can be calculated by simply tracking the sets of (critical) operating parameters for which the rightmost eigenvalue, i.e., the eigenvalue with the largest real part, lies on the imaginary axis.
36
It is necessary to note that the results of (linear) stability analysis are only valid for infinitesimal perturbations. Results of local (linear) stability analysis are not necessarily applicable
to perturbations that are larger than infinitesimal. The results of stability analyses also do not shed any light on the system behavior as time increases and amplitudes of phase variables become finite. Specifically, the experimentally observed stable limit cycle oscillations cannot be predicted by stability analysis alone. System dynamics following large perturbations and the determination of amplitudes of oscillations when oscillations grow into a stable limit cycle is discussed in the next sub-section.
2.2.2. Poincaré-Andronov-Hopf Bifurcation (PAH-B)
Poincaré-Andronov-Hopf bifurcation (PAH-B) theorem concerns about dynamical properties of another type of special solution of the system: periodic solution, which satisfy F1 (t ; λ ) = F1 (t + T ; λ )
(2.34)
From any point on this solution, the system will come back to the same condition after a fixed period T. The periodic solution clearly corresponds to stable amplitude oscillation (limit cycle) of BWR. The theorem predicts existence of periodic solutions, branching from fixed points, when λ is close to its critical value and the following two conditions [33] are satisfied: a) a pair of pure imaginary eigenvalues ( ± iω c ) of the Jacobian of the system exists at
λ = λcritical and the real parts of all the other eigenvalues are negative. b) the real part of the complex conjugate eigenvalues with zero real part crosses the imaginary axis with non-zero speed as λ is varied, i.e.
d (real part ) dλ
The periodic solutions, which are indexed by a constant ε , are given by
37
≠ 0. λ = λcritical
F1 (t ; λ (ε )) = F0 (λc ) + εv e iω (ε ) t + O(ε 2 )
(2.35)
where, F0 (λc ) is the fixed point of the system. v is the eigenvector corresponding to the pure imaginary eigenvalue iω c normalized such that its first non-zero element is equal to one,. v = [v1
vn ]
T
v2
(2.36)
Constant ε is the so-called “index constant” which specifies amplitude of the periodic solution.
λ (ε ) is the set of operating parameter which corresponds to the periodic solution of ε . ω (ε ) is frequency of the periodic solution. Similar to the fixed point, periodic solution can be stable or unstable. Figure 2.1 shows examples of stable and unstable periodic solutions in two-dimensional phase variable space. The x-axis F1(t) and y-axis F2(t) are two phase variables of the system. For stable periodic solution (Figure 2.1(a)), if the system starts at a condition perturbed from the periodic solution, trajectory of the time-dependent variables will be attracted to the periodic solution as time evolves. For unstable periodic solution (Figure 2.1(b)), if initial condition of the system is at a point only slightly deviating from the periodic solution, trajectory of states will be expelled by the periodic solution. Stability of periodic solutions, whose existence is guaranteed by the PAH bifurcation theorem, is determined in a way similar to that of fixed points, i.e. by introducing a perturbation to the periodic solution, and then analyzing whether the perturbation grows or decays. Difference between them is that grows or decays of perturbations for the periodic solutions are analyzed through Poincaré map [34]. Suppose an arbitrary perturbation is introduced to a periodic solution at t = 0
F (t = 0; λ (ε )) = F1 (t = 0; λ (ε )) + I 0 e
38
(2.37)
3
0
0
y
y
3
-3
-3 -3
0
3
-3
0
x
3
x
(a)
(b)
Figure 2.1: Examples of stable and unstable periodic solution. ODEs and parameters of the system can be found in page 148 of reference [34]. (a) stable periodic solution. Orbits close to the periodic solution is attracted when time evolves. (b) unstable periodic solution. Orbits close to the periodic solution is expelled.
39
where I 0 is an n × n identity matrix, e is a vector of arbitrary perturbation. At t = T, the perturbation becomes F (t = T ; λ (ε )) = F1 (t = T ; λ (ε )) + I T e
(2.38)
where, I T is a n × n matrix called monodromy matrix, which maps the initial condition I 0 to the matrix at t = T. Since F1 (t = 0; λ (ε )) = F1 (t = T ; λ (ε )) , grow or decay of the perturbation after each circle of period T can be determined by analyzing properties of the monodromy matrix. Suppose eigenvalues of the monodromy matrix are given by
ρi = eβ T , i
i = 1, n
(2.39)
where β i , i = 1, n are called Floquet exponents. The perturbation decays if all the Floquet exponents have negative real parts. The periodic solution is stable and the corresponding bifurcation is called supercritical PAH bifurcation. Otherwise the perturbation grows in magnitude. The periodic solution is unstable leading to what is called subcritical PAH bifurcation. For the supercritical bifurcation, stable periodic solutions can only co-exist with unstable fixed points. For the case of subcritical bifurcation, unstable periodic solutions can only co-exist with stable fixed points. The reason is easy to understand: if there were stable periodic solution coe-existing with stable fixed point for a specific set of operating parameters, they will both attract trajectories and there must be another unstable periodic solution repelling trajectories to separate them. Similarly, if there were unstable periodic solutions co-existing with unstable fixed points, they will both repel trajectories and there must be another stable periodic solution attracting trajectories to separate them. Table 2.1 lists all possible combinations of properties of fixed points and periodic solutions for a specific set of operating parameters close to stability boundary. For supercritical bifurcation, if the system is operated on a set of operating parameters
40
Table 2.1. Properties of fixed points and periodic solutions close to the stability boundary
Supercritical PAH-B
Subcritical PAH-B
Stable Region
Unstable Region
Fixed point is stable; Periodic
Fixed point is unstable;
solution does not exist
Periodic solution is stable
Fixed point is stable; Periodic
Fixed point is unstable;
is unstable
Periodic solution does not exist
41
of stable fixed points, any perturbation from the fixed point will decay. If it is operated on parameters of unstable fixed point, small perturbation from the fixed point will grow. However, the amplitude of the perturbation will finally saturate because of existence of a stable periodic solution in this case. For subcritical bifurcation, if the system is (linearly) stable, small perturbation not crossing bond of periodic solution will decay, but large perturbation crossing the bond will grow because of existence of the unstable periodic solution. If the system is (linearly) unstable, i.e. operated on parameters of unstable fixed points, even small perturbation will grow un-bonded. From safety point of view, operating conditions corresponding to the supercritical bifurcation is clearly favored because of the inherent bonded nature of the bifurcation.. Operating conditions to the subcritical bifurcation is un-favored because of the inherent un-bonded nature of the bifurcation.
2.2.3. Semi-Analytical Method and BIFDD
Quantitative evaluations of the periodic solutions requires calculations of the vector of operating parameter λ (ε ) , frequency ω (ε ) , and the Floquet exponent with the large real part
β (ε ) as functions of the index constant ε . A pure analytical method employing LindstedtPoincaré asymptotic expansion [33], [34] can be applied for the evaluation. The unknown parameters are expanded to polynomials of ε
1 ω (ε ) = 1 ω c + τ 2 ε 2 + O(ε 4 )
(2.40)
λ (ε ) = λc + µ 2 ε 2 + O(ε 4 )
(2.41)
β (ε ) = 0 + β 2ε 2 + O(ε 4 )
(2.42)
Substitute the expansions with the parameters in periodic solutions. A set of equations can be obtained by equating coefficients of like powers of ε on left and right hand sides of the ODEs. 42
Expansion parameters, then, can be acquired through solving the equations. It is noted that expansion forms (2.40), (2.41), and (2.42) ignore the first and third order terms of the ε . This is because the solutions of the equations of like powers of ε will lead to expansion parameters of the first order and third order terms equal to zero. Bifurcation analysis requires evaluation of µ 2 , τ 2 and β 2 . Analytical evaluation of µ 2 ,
τ 2 and β 2 , even for reduced order simple models, is a very tedious task and quickly becomes almost impossible with increasing number of differential equations in the model. Another drawback of the analytical bifurcation studies is that each is specific to a bifurcation parameter, and must be repeated if the impact of a different parameter is to be studied. Numerical integration (time domain method) can be used to determine bifurcation properties of the system, but is also time consuming and leads to results valid for specific set of parameter values. Due to the limitations on the two approaches (numerical integration and analytical bifurcation studies), there has been limited investigation of the large parameter space even for simple models of BWRs. Analytical bifurcation analysis carried out numerically is an attractive alternative to the two approaches described above [29]. In this approach the governing set of nonlinear equations are neither integrated numerically in time nor treated entirely analytically. Rather, the forms of ODEs and Jacobian are evaluated analytically, yet the complicated process of Lindstedt-Poincaré expansion is carried out numerically. This approach, which will henceforth be called analytic-
numeric approach, allows accurate and efficient evaluation of the entire parameter space of interest. General-purpose bifurcation codes such as BIFDD [35] have been developed to perform analytic-numeric bifurcation analysis of set of ODEs and ODEs with delays. For a given set of nonlinear ODEs (or ODEs with delays) and the corresponding Jacobian matrix, the code determines the critical value of the bifurcation parameter, nature of the bifurcation, and the
43
oscillation amplitude when operating close to the critical value. BIFDD allows any parameter of choice in the model to be selected as the bifurcation parameter, and hence the entire parameter space can easily be spanned. Bifurcation analysis when carried out analytically or via an analytic-numeric approach, yields—for all other parameters fixed—the critical value of the bifurcation parameter λ = λcritical , frequency of oscillation ω c corresponding to the critical value, and parameters µ 2 , τ 2 and β 2 . A negative (positive) value of β 2 indicates a supercritical (subcritical) PAH bifurcation, i.e., stable (unstable) limit cycle oscillations as the bifurcation parameter is varied across the critical value.
τ 2 is a correction factor to the oscillation frequency, and µ 2 relates the oscillation amplitude to the value of the bifurcation parameter through equation (2.41). Figure 2.2 shows an example of BIFDD calculation in a two-dimensional parameter space. The solid line is the stability boundary (SB), which separates stable and unstable regions. The three doted lines are curves corresponding to periodic solutions of index constant ε = 0.05, 0.1 and 0.15, respectively. Since amplitudes the periodic solutions are given by multiplication of ε and steady-state eigenvector v (equation (2.35)), the curves are also called 5%, 10% and 15% oscillation curves, which indicate amplitudes of oscillations on these curves are 5%, 10% and 15% to the magnitudes of steady-state eigenvectors, respectively. It is also shown in the Figure 2.2 that the oscillation curves are in the unstable regions above the intersection point P. Unstable fixed points co-exist with stable periodic solutions in this case. According to PAH-B theorem, property of bifurcation is thus supercritical. The oscillations curves are in the stable regions below the point P, indicating a co-existence of stable fixed points and unstable periodic solutions. Property of bifurcation below the point P is subcritical.
44
Supercritical
5
ε = 0.05
Stability Boundary
ε = 0.10 ε = 0.15
P 3
Unstable
Stable
Subcritical
Parameter 2
4
2
1
0 5
6
7
8
9
10
11
12
Parameter 1
Figure 2.2. Examples of stability boundary and oscillation curves calculated by BIFDD. The oscillations curves are in the unstable region above the point P, indicating supercritical PAH-B; they are in the stable region below the point P, indicating subcritical PAH-B.
45
Chapter 3 Stability and Bifurcation Analyses of Forced-Circulation BWR Model
In this chapter, results obtained using stability analysis and using the analytic-numeric methods are presented and discussed. A set of typical BWR operating parameters are used for these studies. Dimensional values of these parameters can be found in Appendix C. Results of standard stability analysis—carried out using the eigenvalues and eigenvectors of the Jacobian of the BWR model—are presented first. Focus is on the relationship of the eigenvalues and eigenvectors of the Jacobian to characteristics of BWR dynamics. In the second and third sections, stability boundaries (linear analysis) and constant amplitude oscillation curves (bifurcation analysis) for homogeneous core and non-homogeneous cores are presented in multiple parameter spaces. The operating parameters of interest include: amplification factor F, non-dimensional subcooling number Nsub, non-dimensional pressure drop along the core ∆Pext, control rod induced reactivity ρext, void reactivity feedback coefficient Cα, fuel temperature feedback coefficient CD, gap conductance coefficient hg, pressure loss coefficient at channel outlet
kexit, and pressure loss coefficient at channel inlet kinlet.
3.1. Eigenvalues and Eigenvectors of the BWR Model Eigenvalues and eigenvectors of the Jacobian matrix play important roles in determining the stability properties of a system of ODEs. As is well known (and discussed in section 2.2 of chapter 2), the eigenvalue with the largest real part determines the stability of the system. For the
46
case of density-wave oscillations or nuclear-coupled density-wave oscillations, the eigenvalue with the largest real part is usually a complex conjugate pair. The instability that results when this pair crosses the imaginary axis leads to oscillatory behavior. Elements of the corresponding eigenvector determine the relative magnitude of different phase variables in the resulting oscillations. Since in the analysis of most models that include only the fundamental mode, it is the
same pair of complex conjugate eigenvalues that crosses the imaginary axis across different sections of the stability boundary, the frequency and amplitude of the resulting oscillations— dictated by the elements in the corresponding eigenvector—have a smooth and continuous dependence on parameter values along the stability boundary. However, this is not necessarily the case when fundamental and first azimuthal modes are included in the modal neutronics analyses and reactivity feedback parameters are varied. That is, it is possible that different pairs of complex conjugate eigenvalues may cause the system to become unstable on different segments of the stability boundary. Since, the evolution of different phase variables depend upon the elements of the eigenvector that corresponds to the eigenvalues with the largest real part, the characteristic oscillations that results along different segments of the stability boundary may be qualitatively very different. In practice, one must track two (rightmost) pairs of complex conjugate eigenvalues to study the stability of BWR thermalhydraulics coupled with fundamental and first azimuthal modal neutronics. Therefore, to determine the parametric dependence of different qualitative behavior along different segments of the stability boundary, the bifurcation code BIFDD was modified to evaluate all the eigenvalues of the Jacobian matrix (the original code only tracked the eigenvalues(s) with the largest real part). The modified code was used to perform stability and bifurcation analyses of the BWR model presented in Chapter 2.
47
Figure 3.1 shows critical values of the total pressure drop ∆Pext for N sub = 1.0, ρ ext = 0.0 and amplification factor F = 1.0, 2.0, 3.0, 4.6, and 5.0 respectively. That is, the system at each one of these points has one pair of complex conjugate eigenvalues with zero real part. All the other operating parameters are fixed at their typical values. As F increases, critical ∆Pext at which the real part of the rightmost eigenvalue is zero, does not change significantly for F ≤ 4.6, but experiences a jump for F > 4.6. The dramatic change in the value of the critical ∆Pext indicates a possible change in the qualitative behavior of the system as F is increased above 4.6. Figure 3.2 (a-e) shows eight rightmost eigenvalues for critical parameter values for each of the five cases shown in Figure 3.1.[Rest of the eigenvalues are farther to the left and are not shown here.] There are two pairs of complex conjugate eigenvalues and four real eigenvalues. Pairs of complex conjugate eigenvalues are denoted by e1 and e2, and represented by squares and stars. Because the operating points are critical, by definition, real part of the rightmost pair of eigenvalues is always zero. For the F = 1 case, the pair e1 is on the imaginary axis and the pair e2 is far from the imaginary axis. The real eigenvalues lie between the pair e1 and e2. As F increases, the pair e2 approaches the imaginary axis. For F = 4.6, the real part of e2 is larger than all of the real eigenvalues and the complex conjugate pair lies just slightly to the left of the imaginary axis. Real part of e2 in this case is close to zero, and hence both e1 and e2 will almost equally influence the evolution of a BWR transient. As F is increased to 5.0, the pair e2 moves to the right of the pair e1, and becomes the rightmost pair of eigenvalues. Characteristic differences in the dynamical behavior of systems that become unstable when the eigenvalue e2 is on the imaginary axis and those systems that become unstable when the eigenvalue e1 is on the imaginary axis, can be identified by studying the elements in the eigenvectors associated with the two pairs of eigenvalues. Eigenvectors associated with e1 and e2
48
8.55 8.50 8.45
∆Pext
8.40 8.35 8.30 8.25 8.20 8.15 1
2
3
4
5
F
Figure 3.1: Critical values of ∆Pext , when N sub = 1.0, ρ ext = 0.0 and F = 1.0, 2.0, 3.0, 4.6, and 5.0 respectively.
49
Imaginary
Imaginary
-1.0
-0.8
-0.6
-0.4
-0.2
4
4
2
2
0 0.0 -2
0.2 Real
-1.0
-0.8
-0.6
-0.4
-2
-4
(b) Imaginary
Imaginary
-0.8
-0.6
-0.4
0.2 Real
-4
(a)
-1.0
0 0.0
-0.2
-0.2
4
4
2
2
0 0.0 -2
0.2 Real
-1.0
-0.8
-0.6
-0.4
0 0.0
-0.2
-2
-4
0.2 Real
-4
(c)
(d) Imaginary 4
Key:
2
-1.0
-0.8
-0.6
-0.4
-0.2
0 0.0 -2
complex eigenvalue e1 complex eigenvalue e2 real eigenvalues
0.2 Real
-4
(e)
Figure 3.2: Eight rightmost eigenvlaues corresponding to the critical values of ∆Pext for the five cases shown in Figure 3.1. F = (a) 1.0; (b) 2.0; (c) 3.5; (d) 4.6; (e) 5.0
50
for different values of F are listed in Tables 3.1 and 3.2, respectively. For clarity, only the elements corresponding to the fundamental mode expansion parameter n0 and azimuthal mode expansion parameter n1 are shown in the tables. In the eigenvector corresponding to e1, shown in Table 3.1, the magnitudes of the elements of n0 are much larger than those of elements of n1 . However, in the eigenvector corresponding to e2 (Table 3.2), the trend is reversed, and the magnitudes of the elements of n1 are much larger than those of elements of n0 . As F is increased, the ratio of the magnitude of elements of n0 to n1 in the eigenvector corresponding to eigenvalue
e1, and the ratio of the magnitude of elements of n1 to n0 in the eigenvector corresponding to eigenvalue e2 decrease. However, they are still of order of 10 even for F = 4.6. As discussed in chapter 2 (equation (2.35) in section 2.3.2), oscillation amplitude of time-dependent variables depend both, on index constant ε and on magnitude of the corresponding element in the eigenvector v . Since the index constant remains the same for all the variables, the magnitudes of the elements of the eigenvector indicate relative amplitudes of the corresponding variables. Hence, the oscillation amplitude of n0 is much larger than that of n1 , if the oscillations are produced by the pair e1. In this case the oscillations are dominated by the in-phase component. The oscillation amplitude of n1 will be much larger than that of n0 in oscillations produced by the pair of eigenvalue e2, and hence the oscillations will be primarily out-of-phase. It is clear that the two pairs of complex conjugate eigenvalues — with the largest and second largest real part — are responsible for the in-phase and out-of-phase (fundamental and the first azimuthal mode) oscillations. Either one of these—depending upon the parameter values— may be the dominant eigenvalue. To avoid confusion, the pair of eigenvalues for which magnitude of the element of n0 in the corresponding eigenvector is much larger than that of the element of
51
Table 3.1: Elements corresponding to n0 and n1 in the eigenvector corresponding to e1 n0
n1
n0
n1
F
Real
Imaginary
Magnitude
Real
Imaginary
Magnitude
1.0
1.3656
4.8667
5.0546
0.004590
0.01369
0.01444
350.04
2.0
1.3615
4.8533
5.0407
0.01231
0.03922
0.04111
122.61
3.5
1.3482
4.7732
4.9600
0.03740
0.1920
0.1956
25.36
4.6
1.6890
5.1131
5.3848
-0.6087
-0.4666
0.7670
7.02
52
Table 3.2: Elements corresponding to n0 and n1 in the eigenvector corresponding to e2 n0
n1
n1
n0
F
Real
Imaginary
Magnitude
Real
Imaginary
Magnitude
1.0
-0.009630
-0.01334
0.01645
2.7592
3.9449
4.8140
292.64
2.0
-0.01475
-0.03072
0.03407
2.8850
6.3208
6.9481
203.94
3.5
-0.02896
-0.1192
0.1227
2.6818
8.5689
8.9788
73.18
4.6
0.2584
0.2299
0.3459
1.8988
8.9610
9.1600
26.48
53
n1 , will be called the fundamental mode eigenvalue pair ( a f + i w f ), and the other pair will be
called the azimuthal mode eigenvalue ( a a + i wa ). It is found that among all the system parameters, the amplification factor F has the largest impact on the relative order of magnitude of the fundamental mode eigenvalues and the azimuthal mode eigenvalues. This is not very surprising since the value of F directly reflects the relative strength of the reactivity feedbacks from the two modes. For this reason, results of stability and bifurcation analyses in the next two sections will be presented separately, for homogeneous core, where F = 1, and for nonhomogeneous cores, where F can be much greater than 1.
3.2. Homogeneous Core (F = 1) For homogeneous cores, azimuthal mode eigenvalues are far from the imaginary axis. For the range of values studied for other parameters in the system—parameters other than F—the dominant eigenvalues for homogeneous cores remain the fundamental mode eigenvalue pair and the azimuthal mode eigenvalue pair was never found to be close to the imaginary axis. Stability boundaries and oscillation curves are hence, only associated with the fundamental mode eigenvalues. Therefore, oscillations, when the system first becomes unstable, will be predominantly in-phase.
3.2.1. Results in N sub - ρ ext Space
Constant pressure drop ( ∆Pext ) boundary condition is used to carry out the stability and bifurcation analyses. If the imposed pressure drop is treated as a parameter—fixed for the entire plain in which SB is presented—then the inlet flow rate at steady-state at each point on the SB would in general be different. This often makes the analysis and interpretation of the effect of the 54
bifurcation parameter more difficult because different points on the SB not only have different values for the bifurcation parameter, but also correspond to different flow rates. Hence, SBs, along which the steady-state inlet flow rate is constant—and consequently each point on the SB in general corresponds to a different value of imposed pressure drop—are often desirable. The latter are in fact easier to evaluate, since total steady-state inlet flow velocity v~inlet is the same for all points on the SB, and if needed, ∆Pext can be calculated for each point on the SB after the SB has been evaluated by simply solving for ∆Pext explicitly from the pressure balance
∆Pext = ∆Pinlet + ∆Pexit + ∆Pfric ,1 + ∆Pgrav ,1 + ∆Pacc , 2 + ∆Pfric , 2 + ∆Pgrav , 2 where, ∆Pinlet and ∆Pexit are local pressure losses at channel inlet and outlet, ∆Pfric is pressure loss due to friction term in the momentum equation, ∆Pgrav is pressure loss due to gravity, ∆Pacc is the loss due to expansion in two-phase flow. Subscripts 1 and 2 denote single- and two-phase, respectively. [In the former case (when ∆Pext is fixed) v~inlet is an unknown, and hence the above equation must be solved simultaneously with the eigenvalue equations.] Stability boundaries (SBs) for the two cases are compared in Figure 3.3(a). SBs are shown for fixed ∆Pext = 8, 25, and 50, and for fixed v~inlet = 1. For comparison, ∆Pext along the SB for the v~inlet = 1 case varies between 45.0 and 175.0. Types of bifurcation along the SBs in Figure 3.3 (a) are shown in Figure 3.3 (b). Fifteen percent oscillation curves in N sub - (ρ ext − ρ ext ,critical ) space are shown for each SB, that is, the index constant ε along the results of the bifurcation analysis is 0.15. Note that (ρ ext − ρ ext ,critical ) = 0 corresponds to the SB. Hence, the periodic solutions are stable when the curves are on the unstable side of the SB ( (ρ ext − ρ ext ,critical ) > 0) and the bifurcation is supercritical. Otherwise, the
55
12 ∆Pext=47.0
10
∆Pext=8.0 ∆Pext=25.0 ∆Pext=50.0
Nsub
8 6
vinlet=1.0
4
0 -0.05
∆Pext=175.0
∆Pext=58.0
2
0.00
0.05
0.10
0.15
0.20
0.25
0.30
ρext (a) 12
Supercritical 10
∆Pext=8.0 ∆Pext=25.0 ∆Pext=50.0
Nsub
8
vinlet=1.0
6 4
Subcritical 2 0 -0.004
-0.002
0.000
0.002
0.004
0.006
0.008
ρext-ρext,critical (b) Figure 3.3: (a) Stability boundary in N sub - ρ ext space; (b) Fifteen percent oscillation amplitude ( ε = 0.15) curves in N sub - (ρ ext − ρ ext ,critical ) space.
56
periodic solutions are unstable and the bifurcation is subcritical. For example, for ∆Pext = 8.0, in the upper part of the stability boundary for N sub > 4.2, where the oscillation curve is in the stable region, the bifurcation is subcritical. Thus, the small perturbations will decay, but “large” perturbations, greater than the amplitude of the periodic solution, will grow. For 0.6 < N sub < 4.2, the oscillation curve is in the unstable region and the bifurcation is supercritical, the periodic solution is stable, and any finite but small perturbation will evolve to this stable periodic solution. For even lower values of N sub (< 0.6), the 15% oscillations curve returns to the stable region, and the bifurcation is subcritical. The low value of N sub where the transition between sub- and supercritical bifurcation occurs is close to the typical value of N sub for forced-circulation BWRs. This change from sub- to supercritical bifurcation might explain the fact that stable amplitude oscillations are observed in some BWR instability cases, and growing amplitude oscillations (until scram) in others. The same back and forth transitions between supercritical and subcritical PAH-B can be observed for ∆Pext = 25, 50, and v~inlet = 1 cases. Comparing the three 15% oscillation amplitude curves for the three fixed ∆Pext cases, it can be seen that for intermediate value of N sub (supercritical case), higher the imposed pressure drop, faster will be the rise in stable oscillation amplitudes as the SB is crossed and the operating point is moved into the unstable region. [Recall that when operating at the ε = 0.15 curve, the theory predicts the oscillation amplitude to be about 15% of the steady-state value.] Also, comparison between fixed v~inlet ( = 1.0 ) case and fixed ∆Pext ( = 50.0 ) case shows that though the SBs for the two cases were fairly close to each other, the rise in oscillation amplitude is faster for the constant ∆Pext case than for the constant v~inlet case.
57
3.2.2 Results in ρ ext - ∆Pext and N sub - ∆Pext Space
The SBs in ρ ext - ∆Pext space are shown in Figure 3.4 (a). Four SBs for N sub = 0.6, 1.5, 3.0, and 10.0 are plotted. Clearly N sub very strongly impacts the SB. The difference is evident especially for large values of ρ ext . An interesting, and well known, trend is that very small or large N sub cases have larger stable region than the intermediate N sub cases. The SBs for the case
N sub = 0.6 and 10.0 are very close but relatively far from those for the case N sub = 1.5 and 3.0. Corresponding results of the bifurcation analysis are shown in Figure 3.4 (b). For N sub = 0.6, bifurcation changes from subcritical to supercritical and then back to subcritical as ρ ext is increased. The supercritical region (for intermediate values of ρ ext ) increases in size as N sub is increased, but then diminishes as N sub approaches 10.0. At the same time, the amplitude of stable oscillation rises the slowest with (ρ ext − ρ ext ,critical ) for N sub ≈ 1.5. Also note that although the SBs for cases of very small (0.6) and large (10.0) N sub are close, nature of bifurcation along these SBs are quite different. There is a supercritical region for the case of N sub = 0.6, but only subcritical bifurcation is found for the case of N sub = 10.0. This implies quite different behaviors of oscillations for the cases of very small and large N sub values. [Since the operating value of N sub for BWR is about 0.6, the cases of large value of N sub are generally rare.] SBs for ρ ext = 0.00, 0.03, 0.08 and 0.13 is shown in Figure 3.5(a) in N sub - ∆Pext space. Results of the bifurcation analysis are shown in Figure 3.5(b). Once again as N sub is decreased along the SB, bifurcation changes from subcritical to supercritical, and back to subcritical. The effect of large ρ ext is to shrink the stable regions in the N sub - ∆Pext space. The supercritical region
58
0.14 0.12
Unstable
0.10 0.08
ρext
0.06
Nsub=0.6 Nsub=1.5 Nsub=3.0 Nsub=10.0
0.04 0.02
Stable
0.00 -0.02 -0.04 -0.06 0
20
40
60
80
100
120
140
160
180
200
∆Pext
(a)
0.14
Subcritical
Nsub=0.6 Nsub=1.5 Nsub=3.0 Nsub=10.0
0.12 0.10 0.08
ρext
0.06 0.04 0.02 0.00 -0.02
Supercritical
-0.04 -0.06 -4
-3
-2
-1
0
1
2
∆Pext-∆Pext,cr
(b) Figure 3.4: (a) Stability boundaries in ρ ext - ∆Pext space; (b) Fifteen percent oscillation amplitude ( ε = 0.15) curves in ρ ext - (∆Pext − ∆Pext ,critical ) space. 59
12
ρext=0.0 ρext=0.03
10
ρext=0.08 ρext=0.13
Nsub
8
6
Unstable
Stable
4
2
0 -40
-20
0
20
40
60
80
100
120
140
160
180
200
∆Pext
(a)
12
Supercritical
Subcritical
10
ρext=0.0 ρext=0.03
Nsub
8
ρext=0.08 ρext=0.13
6
4
2
0 -4
-3
-2
-1
0
1
2
∆Pext-∆Pext.cr
(b) Figure 3.5: (a) Stability boundary in N sub - ∆Pext space; (b) Fifteen percent oscillation amplitude ( ε = 0.15) curves in N sub - (∆Pext − ∆Pext ,critical ) space 60
in the N sub - (∆Pext − ∆Pext ,critical ) space at low value of N sub increases with increasing ρ ext .
3.3. Non-Homogeneous Core (F > 1) For non-homogeneous core, the amplification factor is greater than 1.0. For nonhomogeneous core cases, the azimuthal mode eigenvalues in the complex plane may cross the fundamental mode eigenvalues, and nature of oscillation may change from in-phase to out-ofphase. Results of the crossing and impact on the dynamical properties of the system are discussed in N sub - ∆Pext and F- ∆Pext spaces, respectively.
3.3.1. Results in N sub - ∆Pext Space
Figures 3.6 (a) and (b) show SBs and five percent oscillation curves ( ε = 0.05 ) for F = 1.0, 2.0, 3.5, 4.6 and 5.0. The SBs for F = 1.0, 2.0, 3.5 and 4.6 cannot be distinguished from each other. This shows that, for the set of parameter values used here, the factor F has very little impact on the location of the stability boundary for F < 4.6. However, SB for F = 5.0 when N sub < 1.177 differs significantly from corresponding SBs for lower values of F. The five percent oscillation curves for the first four cases, shown in Figure 3.6 (b), are also almost indistinguishable, but there is a jump in the five percent oscillation curve for F = 5.0 case at N sub = 1.177. The difference in SBs and jump in the oscillation curves are caused by an exchange of the dominant eigenvalue from the fundamental mode eigenvalues pair ( a f + i w f ) to the azimuthal mode eigenvalues pair ( a a + i wa ), as shown in the section 3.1. As mentioned in chapter 2, the code BIFDD is modified to determine boundaries corresponding to the fundamental and first azimuthal mode eigenvalues. Real part of the
61
4.5
F=1.0 F=2.0 F=3.5 F=4.6 F=5.0
4.0 3.5 3.0
Nsub
2.5
Stable
2.0
Unstable
1.5 1.0
*
0.5 0.0
1
2
3
4
5
6
7
8
9
∆Pext
(a)
4.5
F = 1.0 F = 2.0 F = 3.5 F = 4.6 F = 5.0
4.0 3.5 3.0
Nsub
2.5 2.0 1.5 1.0 0.5 0.0
-4
-3
-2
-1
0
1
∆Pext - ∆Pext,critical
(b) Figure 3.6: (a) Stability boundaries in
(∆P
ext
N sub — ∆Pext
space; (b) 5% oscillation curves in
− ∆Pext ,critical ) space for F = 1.0, 2.0, 3.5, 4.6 and 5.0.
62
N sub —
fundamental mode eigenvalues is zero along what will be called the fundamental mode boundary, and real part of the azimuthal mode eigenvalues is zero along what will be called the azimuthal mode boundary. For F = 5.0 case, the SB (solid line) in Figure 3.7(a), on which the real part of the rightmost (dominant) eigenvalue pair is zero, is comprised of two curves that meet at the point T ( N sub = 1.177, ∆Pext = 7.618). Above this point, the eigenvalues with the largest real part along the SB are the fundamental mode eigenvalues, that is, a f = 0 and a a < 0. However, for N sub < 1.177 (below the point T), the eigenvalues with the largest real part along the SB are the azimuthal mode eigenvalues ( a a = 0 and a f < 0). The dotted line in Figure 3.7(a) denotes the boundary along which the real part of the second rightmost pair of complex conjugate eigenvalue is zero. (This means that the rightmost pair of complex conjugate eigenvalue along the dotted line has positive real part). The dotted curve is also comprised of two branches. Along the branch above the point T, the real parts of the azimuthal mode eigenvalues are zero ( a a = 0 and a f > 0). However, along the lower branch, the real parts of the fundamental mode eigenvalues are zero ( a f = 0 and a a > 0). Hence, the parameter space is divided into four parts, denoted by letters A, B, C and D. Regions A and C are respectively the unstable and stable regions, where both real parts a f and a a are respectively positive and negative. In region B, a f is positive while a a is negative. Hence, the system is unstable. However, since the element of the eigenvector corresponding to the fundamental mode is much larger than the element corresponding to the azimuthal mode, growing oscillations in this region will be dominated by the fundamental mode. If both the fundamental mode and azimuthal mode are perturbed, the azimuthal mode in the beginning may actually decrease in amplitude since its amplitude is initially dominated by the (fast decaying) component
63
5 Stability boundary Re(second largest eigenvalue) = 0
4 B 3
Stable
Nsub
Unstable Point 1
2
C
** Point 2
Nsub,T
T 1
Point 3
A
* *
Point 4 D
0 2
3
4
5
6
7
∆Pext
8
9
∆Pext,T
10
(a)
5 Subcritical 4 ε = 0.05 curve 3
Nsub
ε = 0.05 curve of the 2nd largest eigenvalue
2 1 Supercritical 0 -5
-4
-3
-2
-1
0
1
∆Pext-∆Pext,critical
(b) Figure 3.7: (a) Stability boundary and the boundary along which the real part of the second largest pair of eigenvalue is zero for F = 5.0; (b) 5% oscillation amplitude curves corresponding to the eigenvalue pairs with the largest and second largest real parts. 64
that corresponds to the first azimuthal mode eigenvalue pair. Once the decaying part has died out, even the azimuthal mode will start to grow (due to the slowly growing component that corresponds to the fundamental mode eigenvalue). In region D, a a is positive, but a f is negative. Similar to the region B, the system is unstable. However, since the element of the eigenvector corresponding to the azimuthal mode is much larger than the element corresponding to the fundamental mode, growing oscillations in this region will be dominated by the azimuthal mode. If both modes are perturbed, the fundamental mode in the beginning may actually decrease in amplitude, since its amplitude is initially dominated by the (fast decaying) component that corresponds to the fundamental mode eigenvalue pair. Once the decaying part has died out, even the fundamental mode will start to grow (due to the slowly growing component that corresponds to the azimuthal mode eigenvalue). These results are conceptually similar to those obtained by March-Leuba et al. [17] using a simple model. However, in-phase and out-of-phase oscillations in their studies were obtained using different sets of boundary conditions in the model. Results reported here are obtained always using the same boundary condition of constant pressure drop along the channel. Results of the bifurcation analyses for F = 5 case are presented in Figure 3.7(b) in the form of ten percent oscillation amplitude curve in N sub — (∆Pext − ∆Pext ,critical ) space. The solid line corresponds to the SB (solid line) in Figure 3.7(a), while the dotted curve corresponds to the dotted curve in Figure 3.7(a) (i.e., to the second rightmost eigenvalue pair crossing the imaginary axis). [The bifurcation is subcritical for N sub > 1.874.] In Figure 3.7(b), a jump in the oscillation curve (solid line) occurs at the value of N sub corresponding to point T. The discontinuity exists because of the degeneracy at this value of N sub , where both the fundamental and first azimuthal mode eigenvalues have zero real parts. The oscillation curve for N sub > 1.177 is determined using 65
the fundamental mode eigenvalues, while for N sub < 1.177 it must be determined using the first azimuthal mode eigenvalues. Note that the oscillation curve for each pair of fundamental and azimuthal mode eigenvalues is continuous over the entire range of N sub . However, only a segment (the solid part) of each is relevant. Though the PAH-B theorem is strictly applicable only along the solid lines in Figure 3.7(a), results of the numerical simulations (reported below) show that the conclusions regarding the stable and unstable periodic solutions may be roughly applicable even to the second pair of complex conjugate eigenvalues that cross the imaginary axis
right behind the first pair.
3.3.2. Results in F- ∆Pext Space
To even more explicitly identify the role of F in determining the nature (in-phase or outof-phase) oscillations, Figures 3.8 (a-f) show SBs in the F— ∆Pext parameter space for N sub = 4.50, 3.25, 1.90, 1.40, 1.30 and 1.18. The solid and dashed lines are respectively the boundaries along which the real parts of the fundamental and first azimuthal mode eigenvalue pairs are zero. System stability is determined by the boundary on the right. For all values of N sub the fundamental mode boundary is independent of F. The first azimuthal mode boundary does vary with F; much more strongly for smaller values of N sub than for larges values. For N sub = 4.50 and
F = 1.00, the system becomes unstable due to the fundamental mode eigenvalue pair. However, as F is increased, for 1.50 < F < 3.53, it is the first azimuthal mode eigenvalue pair that crosses the imaginary axis as the stability boundary is crossed. Boundaries for the two modes in this region are close and hence both types of oscillations are likely to be manifested as the SB is crossed. For
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Figure 3.8. Stability boundaries in F— ∆Pext space for N sub = (a) 4.50; (b) 3.25; (c) 1.90; (d) 1.40; (e) 1.30; (f) 1.18. The fundamental mode eigenvalue pair is on the imaginary axis along the solid line, while the first azimuthal mode eigenvalue pair is on the imaginary axis along the dashed line. System stability boundary is given by the rightmost segments of the solid and dashed curves. 67
F > 3.53, once again it is the fundamental mode that determines the stability of the system. Similar behavior is observed for N sub = 3.25 case (Figure 3.8 (b)). Hence for 2.32 < F < 3.20 the two boundaries almost overlap leading to the degenerate case. For F < 2.32 and F > 3.20, stability is determined by the fundamental mode. As N sub is decreased to 1.90 (Figure 3.8(c)) and 1.40 (Figure 3.8(d)), the azimuthal mode boundary stays to the left of the fundamental mode boundary for all values of F, and hence it is only the fundamental mode that becomes unstable in the vicinity of the SB. However, as N sub is decreased further to 1.30 (Figure 3.8(e)) and then to 1.18 ( Figure 3.8 (f)), the boundary for the first azimuthal mode shifts to the right. For these two values of the subcooling number, for some intermediate range of F (5.48 < F < 7.37 for N sub = 1.30, and 5.03 < F < 10.29 for N sub = 1.18) it is the first azimuthal mode that determines the stability of the system.
3.4 Discussion Results of stability and bifurcation analyses presented above show the complexity of BWR dynamics, even when modeled by a simple, two-channel, reduced order model with relatively small number of operating parameters. For relatively homogeneous cores with F ~ 1.0, oscillations are primarily of in-phase type, and the complexity is primarily manifested in the nonlinearity of oscillations (bifurcation type) and whether the periodic solution in the vicinity of the SB is stable or unstable. Parameters such as inlet subcooling, external reactivity and pressure drop have large impact on the bifurcation characteristics of the system. The subcritical bifurcation, which may occur under certain combination of operating parameters, specifically is of significance.
68
As the in-homogeneity is increased, in addition to the variation in the bifurcation type, the system may experience in-phase or out-of-phase oscillations. The factor F has no-doubt the largest impact on whether the oscillations are in-phase or out-of-phase. However, other parameters, as shown in section 3.3.2, are also important, and the complete picture of BWR dynamics can be painted only after taking into account the effects of different parameters. The results presented in this chapter help provide insights into the nonlinear dynamics of BWRs. However, stability and even bifurcation analyses carried out here are local in nature, not valid for very large perturbations, not valid for operating points very far from the SBs and for large times. They provide useful insight as to what must be studied using, say, numerical simulation studies. In the next chapter the governing set of nonlinear ordinary differential equations are integrated numerically to test the predictions of the stability and bifurcation analyses presented here as well as to study the effects of large perturbations and dynamical behavior over long period of time.
69
Chapter 4 Numerical Simulations of Forced-Circulation BWR Model
In this chapter, the force-circulation BWR model presented in chapter 2 and studied for stability and bifurcation analyses in chapter 3 is numerically integrated to study the impact of large perturbation amplitude as well as to ascertain the system behavior for large time. Steadystate (or fixed points) for a given set of parameter values is first determined using the NLEQ2; a computer code to solve set of non-linear algebraic equations [36] . It uses “damped affine invariant Newton method with rank- strategy” to solve potentially highly nonlinear equations. Perturbations at the fixed points are introduced and time evolutions of the phase variables are numerically calculated using the ODE integration code DMEDBF [37]. The code DMEDBF is based on “Modified Extended Backward Differentiation Formulae”, which includes explicit prediction and implicit correction schemes. The method is A-stable for scheme of orders less than 4 and α-stable for orders up to 9. Since the BWR model can be potentially stiff for some set of operating parameter values, the code DMEBDF was found to be very helpful in reducing numerical problems when the system is evaluated at these operating points. Results of the numerical simulations for homogeneous and non-homogenous cores are given in Sections 4.1 and 4.2, respectively.
4.1. Numerical Simulations for Homogeneous Core (F = 1) To investigate the time evolution, and validate the results obtained using BIFDD, three points are selected in N sub - ρ ext parameter space for numerical simulations. These are shown in
70
Figure 3.3. ( N sub , ρ ext , ∆Pext ) for point 1 is (9.0, 0.03005, 8.0). The critical value of ρ ext for N sub = 9.0 and ∆Pext = 8.0 is 0.030053, therefore, point 1 is in the stable region but very close to the stability boundary. According to the bifurcation results obtained using the code BIFDD and presented in Fig. 3.3 (b), property of PAH-B along this part of the SB is subcritical. As expected, small perturbations introduced in the phase-variables, die out, and the phase variables slowly converge to the steady-state values. However, since the bifurcation at this point is subcritical, large perturbations may grow. Figure 4.1 shows time evolutions of one such case of a rather large perturbation for point 1. In this case, three phase variables were given arbitrary but large perturbations at t = 0 (a1,0: 0.8593 → 1.0593; s1,0: 15.9232 → 16.1232; vinlet,0: 0.3231 → 0.5231). Other phase variables are kept at their steady-state values. Despite the fact that point 1 is in the stable region and small perturbations in phase variables do indeed decay down to zero, these large perturbations lead to growing amplitude oscillations of the phase variables. Figure 4.1(a) shows the time evolution of the fundamental mode expansion parameter n0. It is clear that there is an unstable periodic solution repelling the trajectory in phase space away to ever larger amplitude. Figure 4.1 (b) shows the time-evolution of the azimuthal mode expansion parameter n1. Amplitude of n1 is much smaller than of n0, hence characteristic of power oscillations of the system will appear as in-phase. Note that even the azimuthal mode expansion parameter starts to grow (not shown here) after the initial drop in its amplitude. Figure 4.1 (c) shows the time evolutions of inlet flow velocities of two parallel channels that comprise the core, vinlet,0 and vinlet,1, Figure 4.1 (d) shows the same data (two inlet velocities) but only over the first 50 (nondimensional) seconds. It is clear that after the initial transients the two channels oscillate with the same phase, confirming the in-phase nature of the oscillations predicted in chapter 3.
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2.4
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vinlet,0 vinlet,1
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v
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40
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(c)
(d)
Figure 4.1: Results of numerical integration for parameter values corresponding to point 1 in Figure 3.3. Operating parameters ( N sub , ρ ext , ∆Pext ) are (9.0, 0.03005, 8.0). Phase variables a1,0, s1,0 and vinlet,0 are perturbed as follows—a1,0: 0.8593 → 1.0593; s1,0: 15.9232 → 16.1232; vinlet,0: 0.3231 → 0.5231). The phase variables shown are: (a) n0 ; (b) n1 ; (c) and (d) vinlet , 0 and vinlet ,1 .
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50
Points 2 and 3 in Fig. 3.3 are also close to the SB for ∆Pext = 8.0. Operating parameters ( N sub ,
ρ ext , ∆Pext ) for them are (3.0, 0.01320, 8.0) and (3.0, 0.01, 8.0), respectively. Critical value of ρ ext for N sub = 3.0 and ∆Pext = 8.0 is 0.01317, therefore point 2 is in the unstable region and point 3 is in the stable region. As shown in Fig. 3.3 (b), PAH-B along this part of the SB is supercritical. For parameter values corresponding to point 2, very large amplitude perturbations are introduced in three phase variables at t = 0 (a1,0: 0.4517 → 0.5517; s1,0: 8.3693 → 8.4693; vinlet,0: 0: 0.3597 → 0.4587). Remaining phase variables are kept at their steady-state value. Results of the numerical simulations are shown in Figure 4.2. Despite the fact that operating point 2 is in the unstable region, even the very large perturbations do not cause the phase variables to grow indefinitely, but instead, as predicted by the supercritical PAH-B, phase variables evolve to stable limit cycles. Figure 4.2 (a) shows the time evolution of the amplitude of n0. Note that because the operating point is very close the SB, evolution to the stable limit cycle is very slow. The oscillation amplitude does indeed saturate for t > 1000 and oscillates between 0.75 and 1.06. Thus, while point 1 is in the stable region, because of the subcritical PAH-B there, a large amplitude perturbation at that point may be more dangerous than a large amplitude perturbation at point 1— despite the fact that point 1 is in the unstable region. As for the in-phase vs out-of-phase oscillations, time evolutions of n1, vinlet,0, and vinlet,1, shown in Figure 4.2 (b), (c) and (d) indicate that in-phase oscillations similar to those at point 1 result at point 2. Perturbations introduced in phase variables for system operating at point 3 in Fig. 3.3 are given by (vinlet,0: 0.3616 → 0.4616; vinlet,1: 0.3616 → 0.2616). Because point 3 is in the stable region and the bifurcation there is supercritical, even these large perturbations, as expected, decay to zero, and the system, as shown in Figure 4.3, returns to its steady-state condition. Thus, results of the numerical integrations performed for parameter values corresponding
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1.2
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Figure 4.2: Results of numerical integration for parameter values corresponding to point 2 in Figure 3.3. Operating parameters ( N sub , ρ ext , ∆Pext ) are (3.0, 0.01320, 8.0). Perturbations of timedependent variables are introduced as (a1,0: 0.4517 → 0.5517; s1,0: 8.3693 → 8.4693; vinlet,0: 0.3597 → 0.4597); The signals show time evolutions of (a) n0 ; (b) n1 ; (c) and (d) vinlet ,0 and
vinlet ,1 .
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0.025
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Figure 4.3: Results of numerical integration at point 3 in Figure 3.3. Operating parameters ( N sub , ρ ext , ∆Pext ) at the point are (3.0, 0.01, 8.0). Perturbations of time-dependent variables are introduced as (vinlet,0: 0.3616 → 0.4616; vinlet,1: 0.3616 → 0.2616); The signals show time evolutions of (a) n0 ; (b) n1 ; (c) and (d) vinlet , 0 and vinlet ,1 .
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to points 1, 2 and 3 confirm the predictions of the semi-analytic bifurcation studies carried out using the bifurcation code BIFDD. Moreover, results of these numerical integrations can be used to estimate the perturbation amplitude necessary to lead a system operating in a stable region away from the stability boundary but close to a subcritical PAH-B point to “become unstable.” Similarly, results of these numerical integrations can also be used to estimate the oscillation amplitude of stable limit cycles after supercritical PAH-B.
4.2. Numerical Simulations of Non-homogeneous Core (F > 1) Results of numerical simulations for amplification factor F = 2.0, 3.5, 4.6 and 5.0 are shown in Figures 4.4 through 4.7. The operating point ( N sub = 1.0, ρ ext = 0.0, and ∆Pext = 8.5) is shown in the Figure 3.6 (a). For the set of operating parameter ( N sub = 1.0, ρ ext = 0.0), the critical value of external pressure along the channel is ∆Pext ,critical = 8.2085 for F = 2.0, 3.5, 4.6, and
∆Pext ,critical = 8.5362 for F = 5.0. Therefore, the operating point is stable for the cases F = 2.0, 3.5, 4.6, and unstable for the case F = 5.0. Phase variables n0 and n1 are perturbed at t = 0 as follows (n0: n0,ss → n0,ss + 0.05; n1: n1,ss → n1,ss + 0.05), where n0,ss is the steady state value of n0 and is about the same (0.5827) for different values of F; and n1,ss = 0.0 is the steady state value of n1. There is no significant change in the decay ratio of n0 as F is increased from 2 to 5 (Figures 4.4 to 4.7). However, dramatic change in the decay ratio of n1 can be seen. For F = 2.0, oscillations in
n1 , shown in Fig. 4.4, very quickly decay to zero. As F is increased from 2 toward 5, oscillations in n1 take increasingly longer time to decay down to zero (see Figures 4.5-4.6). In fact, for F = 5.0, decay ratio of n1 is greater than 1.0, implying an unstable first azimuthal mode, and hence a switch in oscillations from in-phase to out-of-phase. Evolutions of velocities also show similar
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Figure 4.4: Results of numerical integration of the governing set of nonlinear ODEs for F = 2.0. Operating parameters ( N sub , ρ ext , ∆Pext ) are (1.0, 0.0, 8.5). Perturbations are introduced in the phase variables n0 and n1 (n0: 0.5827 → 0.6327; n1: 0.0 → 0.05). The figures show time evolution of: (a) n0 ; (b) n1 ; (c) and (d) vinlet , 0 and vinlet ,1 .
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Figure 4.5: Results of numerical integration for F = 3.0. Operating parameters ( N sub , ρ ext , ∆Pext ) at the point are (1.0, 0.0, 8.5). Perturbations of time-dependent variables are introduced as (n0: 0.5827 → 0.6327; n1: 0.0 → 0.05); The signals show time evolutions of (a) n0 ; (b) n1 ; (c) and (d) vinlet , 0 and vinlet ,1 .
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Figure 4.6: Results of numerical integration for F = 4.6. Operating parameters ( N sub , ρ ext , ∆Pext ) at the point are (1.0, 0.0, 8.5). Perturbations of time-dependent variables are introduced as (n0: 0.5827 → 0.6327; n1: 0.0 → 0.05); The signals show time evolutions of (a) n0 ; (b) n1 ; (c) and (d) vinlet , 0 and vinlet ,1 .
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Figure 4.7: Results of numerical integration for F = 5.0. Operating parameters ( N sub , ρ ext , ∆Pext ) at the point are (1.0, 0.0, 8.5). Perturbations of time-dependent variables are introduced as (n0: 0.5827 → 0.6327; n1: 0.0 → 0.05); The signals show time evolutions of (a) n0 ; (b) n1 ; (c) and (d)
vinlet , 0 and vinlet ,1 .
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trend. For small values of F, phase and amplitude of inlet velocities of the two channels get close quickly after the initial perturbations. For F = 2.0 and 3.0, time evolutions of the two velocities in the last 40 seconds of integration can not even be distinguished (Figures 4.4 (d) and 4.5 (d)). When F is increased even further, difference in the phases and amplitudes of the two velocities becomes more evident. For F = 4.6, it takes much longer for the two channel velocities to get in phase. As F is increased even further, the nature of oscillations changes to out-of-phase. This can be observed in the time evolution of the two velocities during the last 40 seconds of the evolutions for F = 5.0 (Figure 4.7 (d)). Moreover, the time evolutions of the velocities for F = 5.0 show
beating phenomena. It is well known that beating is caused by summation of two signals with very close frequencies. This fact suggests that the oscillations for F = 5.0 case are composed of two slightly different frequencies (resulting from the two pair of complex conjugate eigenvalues very close to the imaginary axis). For F = 5.0 case, frequencies for the fundamental and the first azimuthal mode eigenvalue pairs for the operating point on the stability boundary are 4.2319 and 4.3837, respectively. The ratio of the two frequencies suggest a beating period of ~80 seconds, which is a little higher than that observed in the Figure 4.7 (c) and (d) (about 55 seconds). The overestimate is caused by the frequency difference between the operating point on the stability boundary and the point of actual numerical simulation, which is, although close to, still not on the boundary. Characteristic changes from F = 4.6 to F = 5.0 can be shown clearer, when new initial perturbations are selected to eliminate effect of the second frequency. The perturbation is given by a small number ξ = 0.01 multiplying the eigenvectors corresponding to the pair of eigenvalues with the largest real parts. Table 4.1 lists perturbations for phase variables at t = 0.0 for F = 4.6 and 5.0. Evolutions of perturbations for n0 and n1 are compared for F = 4.6 and 5.0 in the Figure
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4.8 and 4.9 respectively. It is clear that amplitude of n0 are much larger than that of n1 for the case
F = 4.6, hence in-phase oscillation is predominant. For the case F = 5.0, however, amplitude of n1 is much larger than that of n0, and characteristics of oscillations becomes dominant out-of-phase. Results of numerical simulations for the non-homogeneous cores agree with the results, presented in chapter 3, of the stability analysis and the expected change in instability from predominantly in-phase to pre-dominantly out-of-phase oscillations with an exchange in predominant eigenvalue pair. For small values of F, when the pair of complex conjugate eigenvalues for the fundamental mode is on the imaginary axis, the pair of complex conjugate eigenvalues for the azimuthal mode is far to the left of the imaginary axis. Oscillations are thus dominated by the fundamental mode. As F is increased, i.e. for operation at SB for larger values of F, the azimuthal mode eigenvalues pair moves closer and closer to the imaginary axis. For F = 5.0, the pair of eigenvalues for the first azimuthal mode crosses the pair for the fundamental mode, thus the oscillations are dominated by the first azimuthal model. [In fact, for some value of 4.7 < F < 5.0, the two pairs are simultaneously on the imaginary axis, leading to a degenerate case.] Additional numerical simulations were carried out for the F = 5.0 case to analyze the switch from in-phase to out-of-phase oscillations. These were carried out for operating points corresponding to points 1-4 shown in Fig. 3.7 (a). Parameter values ( N sub , ρ ext , ∆Pext ) at points 14 in Figure 3.7 (a) are (2.2303, 0.0, 3.9000), (2.2303, 0.0, 4.1000), (0.8000, 0.0, 7.8500), and (0.8, 0.0, 8.2000), respectively. Points 1 and 2 are above the separation point T, while points 3 and 4 are below the separation point T. Recall that point T divides the SB into an “in-phase stability boundary” above it from the “out-of-phase stability boundary” below it. Time evolutions of n0
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Table 4.1: Perturbations for phase variables for F = 4.6 and 5.0. Operating parameter set ( N sub , ρ ext , ∆Pext ) corresponds to (1.0, 0.0, 8.5)
Phase Variables a1,0 a2,0 s1,0 s2,0 vinlet,0 T1,1,0 T2,1,0 T1,2,0 T2,2,0 n0 u0 a1,1 a2,1 s1,1 s2,1 vinlet,1 T1,1,1 T2,1,1 T1,2,1 T2,2,1 n1 u1
Perturbation for F = 4.6 -1.401ä10-3 2.997ä10-3 -8.157ä10-4 1.717ä10-2 1.289ä10-3 -3.165ä10-4 2.528ä10-3 -3.181ä10-4 2.979ä10-3 -1.014ä10-3 -7.967ä10-4 -1.607ä10-3 3.772ä10-3 1.143ä10-3 1.738ä10-2 1.521ä10-3 -3.515ä10-4 2.944ä10-3 -3.528ä10-4 3.479ä10-3 7.589ä10-4 7.522ä10-4
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Perturbation for F = 5.0 1.644ä10-3 -3.709ä10-3 3.643ä10-5 -1.739ä10-2 -1.499ä10-3 3.752ä10-4 -2.977ä10-3 3.770ä10-4 -3.509ä10-3 1.119ä10-4 1.048ä10-4 -1.542ä10-3 3.393ä10-3 -5.222ä10-4 1.687ä10-2 1.394ä10-3 -3.544ä10-4 2.778ä10-3 -3.563ä10-4 3.272ä10-3 6.931ä10-4 5.668ä10-4
0.010
n0 - n0,ss n1 - n1,ss
ni - ni,ss, i = 0, 1
0.005
0.000
-0.005
-0.010 0
20
40
60
80
100
time
Figure 4.8: Perturbations of n0 and n1 for F = 4.6. Operating parameters ( N sub , ρ ext , ∆Pext ) at the point are (1.0, 0.0, 8.5). Perturbations of time-dependent variables are introduced as ξv , where ξ =0.01 is a small number, v is the eigenvector corresponding to the pair of eigenvalues with the largest real part. Details of initial perturbation can be seen in the Table 4.1.
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n0 - n0,ss n1 - n1,ss
0.03
ni - ni,ss, i = 0, 1
0.02
0.01
0.00
-0.01
-0.02 0
20
40
60
80
100
time
Figure 4.9: Perturbations of n0 and n1 for F = 5.0. Operating parameters ( N sub , ρ ext , ∆Pext ) at the point are (1.0, 0.0, 8.5). Perturbations of time-dependent variables are introduced as ξv , where ξ =0.01 is a small number, v is the eigenvector corresponding to the pair of eigenvalues with the largest real part. Details of initial perturbation can be seen in the Table 4.1.
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and n1 following small perturbations are shown in Figures 4.10 through 4.13. At point 1 (Figure 4.10), in-phase oscillation results to increasing amplitude of n0 and an initially decreasing amplitude of n1 . Note that oscillation amplitude of n1 as well starts to increase at large values of t. Point 2, however, is in the stable region C. Therefore, small perturbations produce decreasing amplitude oscillations of both n0 and n1 (Figure 4.11). However, n0 decreases much slower than
n1 , causing oscillations that are primarily in-phase. At points 3 and 4, the oscillations are predominantly out-of-phase because as was shown in chapter 3, the magnitude of the element corresponding to the out-of-phase mode in the eigenvector corresponding to the dominant eigenvalues is much larger than the magnitude of the element corresponding to the in-phase mode. For point 3, real parts of both eigenvalues are positive, therefore amplitudes of n0 and n1 increas fast (Figure 4.12). At point 4, predominantly out-of-phase oscillations occur, because the real part of the fundamental mode eigenvalue is negative while the real part of the first azimuthal mode eigenvalue is positive and the magnitude of the elements of the eigenvector corresponds to the dominant eigenvalue (Figure 4.13). Clearly the dominant oscillations in the high and low N sub regions are characteristically different. In the high N sub region, the in-phase oscillations are dominant because the critical eigenvalues on the SB correspond to the fundamental mode. However, it should be noted that this characteristic will be evident for parameter values that correspond to a stable system, and for parameter values in a very narrow region close to the SB in the unstable region (region B in the Figure 3.7(a)) where the fundamental mode eigenvalues has positive real part while the first azimuthal mode eigenvalues has negative real part. In a vast majority of the unstable region, both modes are actually unstable and hence both n0 and n1 are expected to grow rapidly.
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In the low N sub region (below point T), however, the out-of-phase oscillations are dominant because the critical eigenvalues correspond to the azimuthal mode. Once again, this characteristic will be evident for parameter values that correspond to a stable system and a narrow region in the unstable side (region D in the Figure 3.7(a)), where the first azimuthal mode eigenvalues has positive real part while the fundamental mode eigenvalues has negative real part. For the rest part of unstable region, since both eigenvalues have positive real parts, amplitudes of
n0 and n1 both grow rapidly. In the region close to the separation point T, effects of both, the fundamental and azimuthal mode eigenvalues, are evident, and oscillations in this region are a blend of in-phase and out-phase. Thus, numerical simulations confirm the findings of the numeric-analytic bifurcation analysis.
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0.336
0.10
0.334 0.08
0.332
0.06
0.330
n1
n0
0.328
0.04
0.326 0.02
0.324 0.00
0.322 0.320
-0.02
0.318 -20
0
20
40
60
80
100
120
140
-20
160
0
20
40
60
80
100
120
140
160
time
time
(b)
(a) vinlet,0 vinlet,1
0.162
vinlet,0 vinlet,1
0.162
0.160
0.158
0.158
v
v
0.160
0.156
0.156
0.154
0.154
0.152
0.152
-20
0
20
40
60
80
100
120
140
120
160
125
130
135
140
145
150
time
time
(d)
(c)
Figure 4.10: Results of numerical integration for F = 5.0. Operating parameters ( N sub , ρ ext , ∆Pext ) at the point are (2.2303, 0.0, 3.9). Perturbations of time-dependent variables are introduced as (n1: 0.0 → 0.1); The signals show time evolutions of (a) n0 ; (b) n1 ; (c) and (d) vinlet , 0 and vinlet ,1 .
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0.10
0.39
0.08
0.38
0.06
0.37
0.04
n1
n0
0.40
0.36
0.02
0.35
0.00
0.34
-0.02 0
20
40
60
80
100
0
20
40
time
60
80
100
time
(a)
(b) 0.182
0.182
vinlet,0 vinlet,1
0.180 0.178
vinlet,0 vinlet,1
0.180 0.178 0.176
0.174
0.174
0.172
0.172
v
v
0.176
0.170
0.170
0.168
0.168
0.166
0.166
0.164
0.164 0.162
0.162 0
20
40
60
80
80
100
85
90
95
100
time
time
(d)
(c)
Figure 4.11: Results of numerical integration for F = 5.0. Operating parameters ( N sub , ρ ext , ∆Pext ) at the point are (2.2303, 0.0, 4.1). Perturbations of time-dependent variables are introduced as (n0: 0.3523 → 0.4; n1: 0.0 → 0.1); The signals show time evolutions of (a) n0 ; (b) n1 ; (c) and (d)
vinlet , 0 and vinlet ,1 .
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0.565 0.2 0.560 0.555
0.1
0.550
n1
n0
0.545
0.0
0.540 0.535
-0.1
0.530 0.525
-0.2
0.520 0
20
40
60
80
100
0
20
40
time
(a) 0.41
80
100
(b) 0.42
vinlet,0 vinlet,1
0.40
60
time
vinlet,0 vinlet,1
0.41
0.40 0.39
v
v
0.39 0.38
0.38
0.37
0.37
0.36 0.36 0.35 0
20
40
60
80
100
80
85
90
time
95
100
time
(c)
(d)
Figure 4.12: Results of numerical integration for F = 5.0. Operating parameters ( N sub , ρ ext , ∆Pext ) at the point are (0.8, 0.0, 7.85). Perturbations of time-dependent variables are introduced as (n1: 0.0 → 0.05); The signals show time evolutions of (a) n0 ; (b) n1 ; (c) and (d) vinlet , 0 and vinlet ,1 .
90
0.020
0.60
0.015
0.58
0.010 0.005
0.56
n1
n0
0.000 -0.005
0.54
-0.010
0.52
-0.015 -0.020
0.50
-0.025
-20
0
20
40
60
80
100
120
140
160
-20
0
20
40
60
100
120
140
160
(b)
(a) vinlet,0 vinlet,1
0.406 0.404
vinlet,0 vinlet,1
0.406 0.404
0.402
0.402
0.400
0.400
0.398
v
v
80
time
time
0.396
0.398 0.396
0.394
0.394
0.392
0.392
0.390
0.390 -20
0
20
40
60
80
100
120
140
160
130
time
135
140
145
150
time
(c)
(d)
Figure 4.13: Results of numerical integration for F = 5.0. Operating parameters ( N sub , ρ ext , ∆Pext ) at the point are (0.8, 0.0, 8.2). Perturbations of time-dependent variables are introduced as (n0: 0.5573 → 0.5); The signals show time evolutions of (a) n0 ; (b) n1 ; (c) and (d) vinlet , 0 and vinlet ,1 .
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Chapter5 Summary, Conclusions and Future Works of Part I
This chapter summarizes the model and analysis method, concludes the stability and bifurcation results, and suggests future works for studies of force-circulation BWRs.
5.1. Summary of the Model and Analysis Method Stability and bifurcation analyses are carried out for a reduced order model of BWR. The model, which is originally developed by Karve et al., includes three parts: neutron kinetics, fuel rod heat conduction and channel thermal-hydraulics. In the part of neutron kinetics, neutron density and precursor density are assumed an expansion of both the fundamental and first azimuthal mode. The ODEs of the time-dependent parameters of the expansion are derived from one-group time-dependent diffusion equation and the precursor equation. Operating parameters presented in the ODEs, including void reactivity feedback coefficient, Doppler reactivity feedback coefficient, and neutron generation time, are given by forms of inner products of functions and adjoints of the modes. The functions and adjoints of the two modes, however, depend on core loadings and other factors which affect power distributions. The original model by Karve et al. evaluates the functions and adjoints with an assumption that the core is homogeneously loaded, thus failed to generate any out-of-phase oscillations. To account for effects of different loadings, an extra parameter, amplification factor F is introduced. The factor is multiplied to the void and Doppler feedback coefficients of the azimuthal mode, so that the out-of-phase oscillations can be generated for large value of F.
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In the part of heated channel thermal hydraulics, the spatial enthalpy distribution in the single phase region, and the steam quality in the two phase region in the BWR core are approximated by quadratic expansion profiles. Two phase flow is studied with the homogeneous equilibrium model (HEM). The ODEs of time-dependent expansion parameters of the enthalpy and steam quality are derived through weighted-residual approach applied to conservation equations of HEM. Similar to the heated channel thermal-hydraulics, in the part of fuel rod heat conduction, temperature distribution in fuel pellet is represented by piecewise quadratic profiles. The ODEs of the expansion parameters of the quadratic profiles are derived through variational principle. The three parts of the reduced-order model are coupled by reactivity feedbacks and heat transfer correlations. Dynamic properties of the model are studied through stability analysis, bifurcation analysis, and numerical integration. Linear stability of the model is determined by the largest real part of the Jacobian matrix of the dynamical system. Stability boundary, which separates linearly stable and unstable regions, can be plotted in operating parameter space, by tracking the operating points with the zero largest real parts. Properties of oscillations, or periodic solutions, are determined by Poincare-Andronov-Hopf bifurcation (PAH-B) theorem. Stated by the theorem, stable or unstable periodic solutions, called supercritical PAH-B or subcritical PAH-B, is determined by Floquet exponent. The supercritical PAH-B corresponds to oscillations with saturated amplitudes and the subcritical PAH-B corresponds to those with unlimited amplitudes when operating condition cross the stability boundary from stable to unstable side. Relative amplitudes of the different time-dependent variables are determined by magnitudes of the corresponding element of the eigenvector. Large ratio of the magnitude of n0 (the expansion
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parameter of the fundamental mode) to the magnitude of n1 (the expansion parameter of the azimuthal mode), corresponds to in-phase oscillations; large ratio of the magnitude of the n1 to the magnitude of the n0 corresponds to out-of-phase oscillations. The properties of supercritical and subcritical PAH-B and in-phase and out-of-phase oscillations can be studied using a mathematical code BIFDD, which is modified from its original version to evaluate the entire spectrum of eigenvalues—rather than only the one with the largest real part. Given the analytical forms of ODEs, the code can automatically calculate fixed point, determine eigenvalues of the Jacobian matrix at the fixed point and track the operating points of the stability boundary with zero real parts of the eigenvalues. It then numerically evaluates all higher order derivatives necessary for calculations of the properties of the bifurcation and oscillation. The properties of the ODE system can also be determined through observations of numerical integrations at certain operating points. This provides an alternative means to validate and verify stability and bifurcation results by BIFDD.
5.2. Conclusions of Results For azimuthally symmetric flux shape (F = 1), only in-phase oscillations are found in all regions of parameter space studied. The stability boundaries calculated with BIFDD are found identical with those calculated by Karve et al. using different method. Both sub- and supercritical bifurcations along the stability boundaries are found capable. For high or low values of inlet subcooling, the properties of bifurcation are subcritical. For intermediate values of the inlet subcooling, the properties of bifurcation are supercritical. The value of inlet subcooling number separating the lower subcritical region and the supercritical region is close to the typical nominal
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value of BWR. The back and forth change from supercritical to subcritical PAH-B close to the typical nominal inlet subcooling might explain the fact that stable amplitude oscillations are observed in some BWRs while growing amplitude oscillations in others. As the amplification factor F is increased, spectrum of eigenvalues calculated by BIFDD shows that real part of the second pair of complex conjugate eigenvalues increases and approaches the imaginary axis. Eigenvectors of the two pairs of complex conjugate eigenvalues suggest characteristically different oscillations. The oscillations produced by one pair of complex conjugate eigenvalues are mainly in-phase, and those produced by the other pair of complex conjugate eigenvalues are mainly out-of-phase. For this reason, the former is called fundamental mode (in-phase) eigenvalue, and the latter is called azimuthal mode (out-of-phase) eigenvalue. As
F is increased, it is found the azimuthal mode eigenvalues, which is the original pair with the second largest real part, may cross the fundamental mode eigenvalues and become the pair with the largest real part. The change of the pair of eigenvalues with the largest real part (fundamental mode or azimuthal mode eigenvalues) represents a characteristic transition of the system dynamics. For F = 5, boundaries and bifurcation curves of the eigenvalues with the largest and second largest real parts in N sub — ∆Pext parameter space show that above certain value of inlet subcooling, the fundamental mode eigenvalue reaches the imaginary axis (as the imposed pressure drop is increased) before the first azimuthal mode eigenvalue. For these high values of inlet subcooling in-phase oscillations dominate. However, for smaller values of inlet subcooling out-ofphase oscillations dominate due to the azimuthal mode eigenvalue that reaches the imaginary axis first as the bifurcation parameters is varied. Hence, the parameter space is divided into four regions separated by boundaries of the two pairs of eigenvalues. Operating points in different
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regions may experience oscillations of different characteristics: in the narrow unstable region B above the transition point T (see Figure 3.7(a)), the in-phase oscillations will dominate; in the narrow unstable region D below the transition point T, the out-of-phase will dominate; in the unstable region A or stable region C, the oscillations are combinations of both in-phase and outof-phase. Bifurcation results show that both sub- and super-critical bifurcation can occur along the stability boundaries for in-phase as well as out-of-phase oscillations. Stability boundaries in F— ∆Pext parameter space show that for large or small values of
N sub , azimuthal mode eigenvalues can cross the fundamental mode eigenvalues. In this case, the system becomes unstable due to fundamental mode when F is large or small. The system becomes unstable, however, due to the first azimuthal mode for intermediate values of F. Therefore, oscillations will change from in-phase to out-of-phase and back to in-phase as F is increased. For intermediate values of N sub , azimuthal mode eigenvalues do not cross the fundamental mode eigenvalues as F is varied. Hence, dominant oscillations are only in-phase. Numerical integrations are performed for both small and large values of F. For small value of F (=1.0), It is found that for subcritical PAH-B, the large perturbation to operating points in stable region but close to the stability boundary may cause oscillations with unlimited amplitudes. For supercritical PAH-B, perturbations to operating points in unstable region may cause oscillations with saturated amplitudes. For large value of F( = 5.0), numerical integration results show dramatic change of characteristics of oscillations from in-phase to out-of-phase or vice versa, when inlet subcooling crosses a critical value. Beating phenomena when operating point is close to the transition point reveal the fact that the oscillations are composed by two frequencies, one corresponding to the
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fundamental mode eigenvalues and another corresponding to the azimuthal mode eigenvalues. These results agree with the predictions of the stability and bifurcation analyses.
5.3. Future Work In the reduced-order model, the amplification factor F is introduced to roughly account for effects of different core loadings. The value of F is actually given arbitrarily to reveal qualitative properties of oscillations. We suggest improvement can be made in the future to reduce the arbitrariness caused by the factor F. This can be done by employing the modal decomposition approach mentioned in the section 2.1.2 of the Chapter 2. Two- or three-dimensional diffusion equation codes can be used to evaluate a number of typical core loadings during normal or abnormal operating conditions. The functions and adjoints of the fundamental and azimuthal modes, thus, can be numerically determined, and operating parameters presented in the ODEs of neutron kinetics can be calculated. It is of importance to learn what specific core loadings and operating conditions may lead to out-of-phase oscillations.
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PART II Stability and Bifurcation Analyses of Natural Circulation BWRs
98
Chapter 6 Introduction to Instabilities of Natural Circulation BWRs
Second part of this dissertation, starting from this chapter, is devoted to modeling and instability analyses of natural circulation BWRs during start-up. Unlike forced circulation BWRs that rely on recirculation pumps for flow through the core, natural circulation BWR designs rely on natural circulation caused by density difference for flow through the core and for heat extraction purposes.
6.1. Instabilities of Natural-Circulation BWRs under Low Pressure Safety is one of the most important concerns in nuclear engineering. To ensure safety, current techniques rely on “defense in depth” strategy based on multiple and different active components. These lead to increased capitol cost and extend construction period, consequently limiting competitiveness of nuclear energy to fossil fuels. To improve both safety and economy of next generation light water reactors (LWRs), passive safety concepts are introduced in designs. The essence of the passive safety is to reduce active components as much as possible and ensure reactor safety by non-active (passive) components and physical laws. New designs of BWRs that include passive safety concepts, such as SBWR and ESBWR by GE Energy, rely on natural circulation to remove fission heat from cores. They are promising candidates to replace the current forced-circulation BWRs. Figure 6.1 shows a diagram of the natural circulation design ESBWR [38] [39] and a schematic plot of flow path. In natural
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Dryer
Steam separator
Riser
DownBoiling
~12 m
Riser
~25 m
comer
Boundary
Downcomer
Channel
Single Phase Region
Core
~3 m
(a)
(b)
Figure 6.1: Diagram of (a) natural circulation BWR design and (b) schematic flow path. Parameters of lengths of core, riser and containment are, in particular, for ESBWR design. They may differ from other natural circulation BWR designs [38]
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circulation design, there are no pumps and associated pipes. Therefore, a number of severe accidents, like pump trip and LOCA due to recirculation pipes, are completely avoided. Flow path of the natural circulation is comprised of heated channels in the core, adiabatic risers above the channels and downcome. Cold water flows into the core and is boiling at certain height along the channel and riser. Steam is separated at the top of riser. The separated water and feed water are mixed and flow back through the downcomer to the entrance of the core. Driving force for flow through the core is, instead, provided by gravity force due to density difference between the coreriser and the downcomer. The long riser, about 12 m in the case of ESBWR [39], ensures enough density difference between the core-riser and downcomer for adequate flow through the core. Designs of short core length and low pressure loss steam separator also help reduce friction loss and increases flow rate. By these means, power density of ESBWR can reach 48.5 kw/l [40], which is lower than the corresponding value for forced circulation system, but economically acceptable. Nominal system pressure of the ESBWR is 7.0 MPa, about the same as that for forced-circulation BWR. However, the system pressure can be significantly lower than its nominal value during start-up, even as low as the atmosphere pressure. The natural circulation BWR may experience nuclear-coupled density wave oscillations (DWOs) under certain adverse operating conditions similar to those of forced-circulation BWRs, when the system pressure is about 7.0 MPa. However, an even larger concern for stability of the natural circulation system is when the system pressure is much lower during start-up. It may experience more complicated instabilities other than DWOs. Types of these instabilities depend on many factors. For example, if the length of the riser is short, geysering can be dominant in a natural circulation loop, which is caused by formation and condensation of large bubbles. Figure 6.2 shows typical flow signals recorded in geysering experiments by Aritomi et al [41] in a natural
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circulation heated channel laboratory system. The experimental apparatus is comprised of two parallel heated channels and adiabatic risers. A geysering cycle in one of two channels includes four stages (Figure 6.2(a)). Slow increase of velocity from reverse flow in stage I is followed by a rapid increase in stage II. In stage III, flow rate decreases slowly. Then in stage IV, it decreases rapidly leading to reverse flow. Figure 6.2 (a) and (b) show that flow rate oscillations in the two channels are out-of-phase. This leads to an almost invariant total flow rate. If the riser is long, a situation for most natural circulation BWR designs, another type of instability, flashing, may be dominant. Flashing here means boiling in the adiabatic riser not due to heating but due to pressure drop along the riser. Figure 6.3 shows results of flashing experiments in a natural circulation loop with long riser performed by Inada et al. [42]. [Note that definition of the subcooling number in this case of natural circulation is different from that used in Part I (forced circulation system) of this thesis. The two definitions are explained and compared in Appendix D (also explain the fact that Nsub can be negative in this case in the appendix)]. When inlet subcooling is large (Figure 6.3(a)), flow rate in the natural circulation loop is small because there is no boiling. Driving force of recirculation, in this case, is provided by density difference between hot water in the channel-riser and cold water in the downcomer, and the flow shown in the Figure 6.3(a) is stable. As inlet subcooling is decreased, flashing occurs in the upper part of the riser and intermittent peaks of the flow rate with period of about ~100 seconds appear (Figure 6.3(b)). If the inlet subcooling is even lower, the intermittent peaks are wider and interconnected, and flow pattern evolves to a sinusoidal wave (Figure 6.3(c)). Typical periods of sinusoidal oscillations range from several seconds to several tens of seconds. For very small inlet subcooling, amplitude of the sinusoidal wave is very small and the natural circulation flow evolves to stable two-phase flow (Figure 6.3(d)).
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II
III
I
IV
(a)
(b)
Figure 6.2: Typical geysering flow signals in two parallel heated channels [41].(a) Flow signal in channel 1, u(1); (b) Flow signal in channel 2, u(2). Geysering cycle comprises of four stages. Flow oscillations in the two channels are out-of-phase.
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(a)
(b)
(c)
(d)
Figure 6.3: Typical flow signals recorded in natural circulation SIRIUS loop [42]. System pressure is P = 0.2 MPa, heat flux is q’’ = 53 kW/m2. (a) single phase natural circulation flow, subcooling number Nsub = 29.8; (b) intermittent wave due to flashing, Nsub = 23.6; (c) sinusoidal oscillations, Nsub = -0.08; (d) steady two-phase flow, Nsub = -2.17.
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The potential of more complicated instabilities during start-up poses a threat to safety of natural circulation BWR systems. For geysering-type instabilities, the dramatic change of void fractions in the core during a cycle may cause large perturbations in reactivity. For flashing-type instabilities, creation and condensation of steam is quite dramatic, thus flow rate changes rapidly for intermittent and sinusoidal waves. Moreover, the flashing front (boiling boundary) may fall below the riser into the channel, causing perturbations in reactivity. Several start-up procedures for different natural circulation designs have already been proposed. The focus of the start-up procedures is to avoid potential hazardous instabilities by maintaining single phase state of the natural circulation flow when the system pressure is low. However, possibility of boiling and instabilities during start-up, caused by off-normal conditions, still can not be completely ruled out. Therefore more extensive studies of the mechanisms of the boiling related instabilities are necessary.
6.2. Physical Mechanisms
6.2.1. Geysering
Figure 6.4 shows mechanisms of geysering type instability postulated by Aritomi et al. [41]. Flow patterns corresponding to four points in stage I and II of a geysering cycle are plotted. According to Chiang et al., geysering in two heated parallel channels occurs at very low flow speed ( < 0.2m/s) [43]. Under this flow rate, large slug bubbles can be created at specific position of the channel and their lengths grow as they move upward. Since driving force of natural circulation depends on the density difference between the channel-riser and the downcomer, the flow is slowly accelerated. When these slug bubbles reach the channel exit, sudden condensation
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Figure 6.4: Flow regimes in a channel corresponding to four points of stage I and II of a geysering cycle in the experiment by Aritomi et al. [41]
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decreases pressure in the channel, and the flow is accelerated very rapidly. These processes lead to the first two stages (slow and rapid increase) of the geysering cycle. Since the experimental system is comprised of two parallel channels, and total flow rate can not change as quickly as that in a single channel due to inertia, the flow velocity increase in one channel due to slug movement and condensation causes decrease in the other channel. Therefore, the flow velocities in the two channels are out-of-phase. The last two stages (slow and rapid decrease) in the geysering cycle in one channel are actually caused by the first two stages in the other channel.
6.2.2. Flashing
Different from geysering, flashing-induced instabilities are not caused by production and condensation of slugs. It is due to bulk boiling and associated changes of natural circulation driving force. For long riser and low system pressure, pressure drop can be large along the path of the channel and riser. Boiling may occur in the adiabatic riser simply due to local pressure drop (flashing). When flashing occurs in the adiabatic riser, small perturbation of flow can be dramatically intensified by feedbacks due to vaporization and local pressure variation. Inada et al. [42] [44] reported the results of flashing experiments. When inlet subcooling is large, both, core and riser are filled with single phase water and the flow is stable. As the inlet subcooling is decreased, Figure 6.5 shows sequence of changes in the flow regimes and enthalpies along the channel and the riser. The flow enthalpy is generally increasing along the channel due to heat input. Since the pressure along the channel and the riser is generally decreasing, the saturation enthalpy also decreases. Therefore, flow is the least subcooled at the top of the riser. In Figure 6.5(a), the flow enthalpy along the channel and riser is lower than the local saturation enthalpy. Flow rate in this situation is small because there is no boiling. A small perturbation in
106
z
z
Saturation Enthalpy
Saturation Enthalpy
Flow Enthalpy
q’
Flow Enthalpy
q’
enthalpy
enthalpy
(a)
(b)
z
z
Saturation Enthalpy
Saturation Enthalpy
Flow Enthalpy
Flow Enthalpy
q’
q’
enthalpy
enthalpy
(c)
(d)
Figure 6.5: Flow pattern and enthalpy profiles during a cycle of intermittent wave caused by flashing. (a) Both channel and riser are single phase. A mass of hot water travels upward; (b) The hot fluid reaches the upper part of the riser. Flashing starts in the riser and “travels” towards riser inlet and outlet; (c) Flow is accelerated. Cold water enters the channel and travels upward; (d) Flashing is suppressed when the cold fluid reaches the upper part of the riser. Flow is decelerated and another cycle starts.
107
inlet water temperature at the bottom of the heated channel leads to a “bump” in water temperature that travels towards outlet of the riser. When it reaches the upper part of the riser (not necessary the top of the riser) where the saturation enthalpy is low (Figure 6.5(b)), flow enthalpy can be higher than the saturation enthalpy, leading to flashing. Under low pressure condition, void fraction varies strongly with steam quality [45]. The small amount of steam produced in the upper part of the riser may cause rapid increase of the void fraction of two phase flow. Because friction in two-phase flow is much higher than that in single phase flow, it may decrease local pressure at which the flashing occurs even further. This will in-turn cause rapid vaporization and increase of void fraction in the riser. If flashing starts in the middle section of the riser, there may be two flashing fronts—one traveling upward toward the riser outlet and one traveling downward toward the riser inlet. If flashing occurs at the top of the riser, only one flashing front will travel downward toward the riser inlet. In natural circulation loop, large void fraction in the channelriser means large density difference between the channel-riser and the downcomer, hence large gravity driving force. Therefore, the flow is accelerated (Figure 6.5(c)). The higher flow velocity then will decrease the flow enthalpy at the entrance of the channel. A bulk of cold water will travel along the channel and riser. When the cold water reaches upper part of the riser, it suppresses flashing and the flow returns to initial single phase state (Figure 6.3 (d)). The flow is then decelerated and this process repeats, producing another intermittent wave. If inlet subcooling is decreased or heat input is increased, the intermittent wave evolves to a sinusoidal wave. In this case, small inlet subcooling or large heat input keeps the flashing front below the top of the riser. The movement of flashing front is thus confined to the riser and the channel, and the condition in which both the channel and riser are covered with single phase water may not occur. This leads to
108
smooth changes in pressure at the flashing front and the smooth shape of sinusoidal wave. Further drop in inlet subcooling or increase of heat input can produce stable two-phase flow.
6.3. Previous Work For natural circulation systems, a series of stability experiments and tests have been conducted in laboratories and in a natural circulation driven BWR. A large volume of data has been accumulated. Experiments demonstrating the geysering phenomenon are described first, followed by those carried out to study flashing and flashing induced instabilities. Aritomi et al. [46] conducted experiments in a loop comprised of two parallel channels, which are partly heated by electric heaters in the center. The unheated sections act as risers. Length of the heated part is 1.0 m, and the length of the adjustable riser is from 0.25 to 0.75 m. The system pressure is close to the atmosphere pressure. Driving mechanisms of geysering in parallel channels under natural and forced-circulation conditions are identified. It is found that formation and condensation of large bubbles are essential in geysering. Increase of flow rate and inlet subcooling can help suppress geysering. Weissler et al. [47] first studied dynamics of natural circulation in a loop under low pressure. Their experiments showed that flashing-induced instabilities were possible in natural circulation systems. Jiang et al. [48] performed experiments in an apparatus which is comprised of two parallel heated channels and long risers. They observed flashing-induced instabilities in their experiments (“the maximum void fraction caused by flashing is about 5% under a system pressure of 1.5 MPa, and about 80% under 0.1 MPa.” [49]). Five stages of instabilities in the test loop were identified [48]. When inlet subcooling is decreased, flow pattern in the loop changes sequentially from stable single phase flow, unstable subcooled boiling flow, intermittent flow, unstable
109
flashing flow, to stable flashing flow. Based on experimental results, a start-up strategy was suggested to use nitrogen gas to increase pressure of the system to 0.3 MPa. Flashing-induced instabilities can be suppressed effectively by this means. Furuya et al. [44] performed experiments on another natural circulation loop (SIRIUS), and identified similar flow pattern transitions when inlet subcooling is decreased or thermal power is increased. They reported decay ratios, frequencies and other oscillation characteristics for different operating points in a parameter space comprised of channel inlet subcooling and channel heat power. Also, they found that increasing inlet restriction can stabilize the natural circulation flow. Manera et al. [39] carried out studies of flashing-induced instabilities in the CIRCUS natural circulation loop in Delft, the Netherlands. Through visual observation, they found complicated mode of flashing-induced instabilities. For small inlet subcooling, boiling was found to start at some point in the riser close to the top, and the flashing fronts traveled in both upward and downward directions. As the flow is accelerated, flashing is suppressed, but repeats in the same way with a period of about ~100 seconds. The phenomenon was explained to be caused by non-equilibrium character of flashing. They also suggested that decreasing compressible volume and increasing system pressure can stabilize the system. Van Bragt [50] summarized stability tests performed in a natural circulation BWR (Dodewaard, the Netherlands). Power and flow signals were measured during start-up of cycle 2426. Decay ratios for all cases but one were less than 1.0, indicating that the system was very stable. For the only exception, clear reasons are still unknown, but carry-under effect in the downcomer was suggested to be the likely cause of instability. Ishii et al. designed a natural circulation loop applying scaling analysis for simulation of natural circulation reactor [51]. A number of experiments of flashing-induced instabilities in the natural circulation loop were carried out and recorded [52].
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Computer simulations and theoretical analyses of various natural circulation instabilities are limited when compared to experiments, partly due to difficulties in developing models that fully respect pressure dependence of water and steam thermodynamic properties. This coupling is the key for the simulation of flashing and other instabilities under low pressure. Analyses can be divided into those carried out using large scale system codes, and those performed using reduced order models. Paniagua et al. [53] modified the system code RAMONA-4B to take the pressure dependence into consideration. Original RAMONA code is designed for transient simulations of forced-circulation BWRs. The thermal-hydraulics model inside is a 4-equation drift flux model. The modification includes a new option in the code to evaluate all thermodynamic properties of water and steam based on local pressure field. Tests against experiments by Wang et al. [54] showed noticeable improvements predicting amplitudes and frequencies of geysering instabilities. Yet the under-prediction of the amplitude is still high (about 19%). Anderson et al. [55] simulated flashing-induced instabilities with system code TRACG, which is a General Electric version of the TRAC code. It contains a two-fluid model for two-phase flow and tested correlations. Qualitative agreements with experiments were found. Manera et al. [56] simulated the experiments conducted in the CIRCUS facility—that showed flashing—with a 4-equation drift flux code, FLOCAL. Generally good agreements of amplitudes, frequencies and stability maps were found. A reduced order model of low-pressure natural circulation system was developed by Inada and Ohkawa [57]. In their model, thermal-hydraulics in a vertical heated channel and adiabatic riser above the channel is considered. Homogeneous equilibrium model (HEM) was used in the two-phase region, and flashing effect was included by assuming linear dependence of saturation
111
enthalpy on local pressure. They performed a frequency domain analysis. The stability boundaries follow the experimental results qualitatively, and better agreement was found under lower system pressure condition. They also found the system to be more stable if water is boiling in the heated channel rather than in the adiabatic riser. Van Bragt et al. [58] [59] extended Inada's method adding neutronics and heat transfer models. The two-phase region in the riser is discretized in a number of nodes with equal sizes. In order to simplify the complexity associated with the derivation of the reduced order model, linear variations of the steam quality and saturation enthalpy in the nodes were assumed. The inertial term in the single phase momentum equation and the time derivative term of saturation enthalpy in the energy equation were ignored. They analyzed the reduced order model in both time and frequency domains. The results qualitatively agree with the experimental data. Jiang and Emendorfer [49] considered subcooled boiling and flashing in their reduced order model of a natural circulation system. They discretized the riser into several nodes. Three fronts—the subcooled boiling, saturation boiling and flashing—were modeled. The Clausius-Clapeyron equation was used to determine the flashing front. Only hydrostatic pressure drop along the riser, relevant to the drop in saturation enthalpy, was included. Comparisons between simulation results and experimental data were presented for relatively high system pressure (1.5MPa) and low pressure (0.1 MP). Only steady-state results were reported. Better agreements were observed for high pressure conditions. Kuran et al. developed a simple one-dimensional model based on kinematic wave theory and drift flux formulations [60]. The model shows capability of developing limit cycle oscillations and more complicated perioddoubling oscillations.
6.4. Scope of Part II of This Thesis
112
It is not surprising that most existing system codes are incapable of analyzing natural circulation systems under low pressure conditions because they are originally developed for forced-circulation BWRs, for which the system pressures are always high. Usually the thermodynamic properties of water and steam in these codes are evaluated at the average system pressure, and consideration of their variation with local pressure is unnecessary. This leads to their inability to capture the essence of geysering and flashing phenomena when the system pressure is low. Other than the few exceptions—such as TRACG, FLOCAL and the modified RAMONA—calculations using system codes suffer from long running time and complicated data analyses. These limitations restrict their use to numerical simulations of specific cases. They are not suitable for parametric studies or for bifurcation studies like those presented in part I of this thesis for forced circulation systems. Therefore, reduced order models are needed for efficient stability and bifurcation analyses over large region of parameter spaces. However, most existing reduced order models are too simplistic. Some of them do not consider pressure dependence of thermodynamic properties, while others rely on several simplifying assumptions and simplifications to reduce the difficulties in the derivation of the reduced order model. [While all such assumptions and simplifications cannot be avoided, they certainly can be reduced in the derivation of an improved model.] Moreover, only one reduced order model (van Bragt et al. [50]), to author’s knowledge, has considered nuclear-coupling. Equally importantly, only a few reduced order models have been analyzed as a dynamical system to study bifurcation characteristics of natural circulation boiling water systems . Due to the limitations outlined above, the second part of this thesis (following four chapters) focuses on the development and analysis of a new reduced-order model to study
113
flashing-induced instabilities in natural circulation BWRs under low pressure. Although geysering instabilities may be important under certain conditions (high heat flux, short risers, etc…), modeling of the geysering instabilities necessitates modeling of a single large bubble and its dynamics. It can not be modeled using the simple HEM or even using the more detailed drift flux model (DFM). Therefore, it is beyond the scope of this dissertation and is recommended as future work. A new reduced order model is developed in chapter 7. Focus will be on the nodalization scheme, reduced order model equations, and the iteration process. After a model validation exercise, stability and bifurcation analyses of the model developed in chapter 7 are carried out in chapter 8. Numerical simulations for different operating conditions are presented and discussed in chapter 9. The last chapter summarizes the work on dynamics of natural circulation BWR under low pressure, and suggests future work.
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Chapter7 Reduced-Order Model of Natural-Circulation BWRs
A reduced order model of natural-circulation BWRs is derived in this chapter, which also includes neutron kinetics, fuel rod conduction and thermal-hydraulics parts. However, the Neutron kinetics and fuel rod heat conduction parts do not differ from those of the forced and natural circulation models described in the Chapter 2. Details of them will not be presented in this chapter. Only the derivation of thermal-hydraulics part of the model is presented.
7.1. Nodalization of Natural Circulation System Figure 7.1 shows a schematic plot of the natural circulation loop considered in the model. It is comprised of two heated channels (each of which represents a half of the core), two separate *
*
risers above the channels and a common down comer. Length of them are Lc , Lr and Ldc
*
respectively, where Ldc = Lc + Lr . The channels are heated by time-dependent heat flux q l (t ) , *
*
*
'
l = 0, 1. The value k c ,in ,l , k c ,out ,l , k r ,out ,l and k dc ,in are local pressure loss coefficients due to shape
changes of flow path or restrictions at the channel inlet, channel outlet, riser outlet, and *
*
*
downcomer inlet respectively. The inlet subcooling is denoted by ∆Tinlet = Tsat ,inlet − Tinlet , which is the difference between inlet flow temperature and the local saturation temperature. A system *
pressure is imposed by setting the pressure at outlet of the risers, Pexit , to a fixed value. Different cases, shown in the Figure 7.2, based on the location of the boiling boundary, are considered separately. For case 1, inlet subcooling is large or heat flux is small. Boiling does not occur in the channel-riser. The whole system is characterized by single phase flow. For case 2, 115
Pexit
kr,out
Pexit
kr,out
Lr kc,out Lc
q0
kc,out q1
∆Tinlet
kc,in
∆Tinlet
kc,in
Figure 7.1: Schematic plot of natural circulation system
116
Ldc
Boiling Boundary
Boiling Boundary
Riser
q’
Riser
q’
Channel
Channel
vinlet
v inlet
(a)
(b)
Riser
Riser Boiling Boundary
Boiling Boundary
q’
q’
Channel
Channel
vinlet
vinlet
(d)
(c)
Figure 7.2: Cases based on position of boiling boundary. (a) Boiling boundary is out of the riser outlet; (b) Boiling boundary is in the riser; (c) Boiling boundary stays at the connection of channel and riser; (d) Boiling boundary is in the channel
117
water enthalpy at the core exit is less than the saturation enthalpy, and flashing occurs in the riser. The case 3 is a degenerated case. For low inlet subcooling or high heat flux, because of the pressure loss between the channel outlet and riser inlet, water enthalpy can be less than the saturation value at the riser outlet, but greater at the the riser inlet. Therefore, for case 3, the boiling boundary may stay at the interface of channel outlet and riser inlet. For case 4, very small inlet subcooling or large heat input causes the boiling boundary falling into the channel. Nodes are required to separate the channels, risers, and dowcomer into smaller parts, so that approximations of spatial profiles of enthalpies, qualities and other variables of the systems can be made to each node. The nodalization for the downcomer is relatively easy, since it is comprised of subcooled single-phase water. Only one node for the whole downcomer is enough. For the channel-riser, cases of boiling add complex to the nodalization schemes. For cases 2 and 4, both include three regions starting from the channel inlet to the riser outlet. The first region is single-phase region from channel inlet to either boiling boundary or channel outlet. The middle region is between the boiling boundary and channel outlet. It can be either single-phase (for case 2) or two-phase (for case 4). The last region is two-phase region from either channel outlet or boiling boundary to the riser outlet. For the case 1 and 3, only two regions exist from the channel inlet to riser outlet. The first region is single phase from channel inlet to channel outlet, and the second region is from the channel outlet to riser outlet, which is either single phase (for case 1) or two-phase (for case 3). One possible scheme of nodalization is to consider different cases separately and allow multiple moving node boundaries. Figure 7.3 (a) shows an example of this kind of nodalization for the case 2 (boiling in the riser). The first region, single phase region in the channel, is discretized by two nodes, each of which is a half size of the channel. The boundary S2 between
118
the two nodes is fixed. The second region, single phase region in the lower part of the riser, is also discretized by two nodes with equal length. However, since the upper boundary of this region is the boiling boundary which is moving during transient process, the boundary S4 between the two nodes is assumed moving proportionally with the boiling boundary, and the length of each node is always the half of this region. Similarly, the last region, two phase region at the upper part of the riser, is dicretized to two nodes. The boundary S6 is assumed moving proportionally too, and length of each node is the half of the region 3. One advantage of this nodalization scheme is that it can significantly reduce occurrence of the node with zero length. If the boiling boundary does not move across the riser outlet and channel outlet, the node length is always non-zero, which helps avoid numerical difficulties during calculation. However, it also unnecessarily limits movement of the boiling boundary. If the boiling boundary moves back and forth between the riser and channel, or it moves in and out of the riser during flashing-induced instabilities, the case of boiling will correspondingly changed back and forth between 2 and 3, or between 1 and 2. The node with zero length during numerical simulations will be inevitable in these situations. Figure 7.3 (b) shows the example of an alternative scheme of nodalization. First, the channel and riser are discretized to two nodes respectively. The node lengths in the channel and riser are half of the channel and riser respectively. The boundaries (S1, S2, S3, S4, S6) of these nodes are fixed. Then, the boiling boundary, which is the boundary S5, is allowed to move in the fixed nodes freely and splits one of them into two nodes (in this case, it is the last node in the riser getting split). Therefore, for the node 1, 2, 3, upper and lower boundaries of them are fixed. For the node 4, the lower boundary is fixed, but upper boundary is moving. For the node 5 the lower boundary is moving, but upper boundary is fixed. When the boiling boundary moves across any of the fixed boundaries, a node with zero length will be created. This degenerated situation (the node
119
S7
6
Boiling Boundary
S6
5
S6
Boiling Boundary
5
S5
S5
4
4 S4
S4
3
3 S3
S3 2
2 q’
q’
S2
S2 1
1
S1
S1 (b)
(a)
Figure 7.3: Two nodalization schemes for the case 2 (boiling boundary in the riser). (a) Two nodes are discretized for each region. Node surface S4 and S6 move proportionally with the boiling boundary S5; (b) Two nodes are discretized for channel and riser respectively first. Boiling boundary S5 splits the last node in the riser into two.
120
of zero length) will create numerical difficulties and need to be taken special care. Both the two schemes are programmed and performances of them are evaluated for different application. For convenience, in the rest of the thesis, the first scheme with multiple moving boundaries will be called moving boundary scheme, and the second on with only one moving boundary (the boiling boundary) will be called fixed boundary scheme.
7.2. Differential Equations
7.2.1. Dimensional and Non-Dimensional Partial Differential Equations (PDEs)
The thermal-hydraulic model is based on the following two physical and modelling assumptions:
• Water saturation enthalpy varies linearly with local pressure, and single phase water density varies linearly with flow enthalpy. Other thermo properties of water and steam are constant and their values are evaluated with the system pressure (pressure at the riser outlet).
• Two phase flow can be modeled by Homogeneous Equilibrium Model (HEM) Although better assumptions such as quadratic pressure dependence and non-equilibrium twophase flow models can be applied, they will greatly increase complexity and decrease computational efficiency of the model. Dynamical properties of the natural circulation BWRs can be solved through conservation equations for mass, energy and momentum. It is difficult and unnecessary to solve the full threedimensional versions of the equations, because the aspect ratio of flow channels in BWR reactors, which is equal to the ratio of the length in axial dimension to the length in radial dimension, is
121
generally very large. Moreover, cross mixing of flow is generally small in BWRs, because there are cases surrounding fuel assemblies. One dimensional, cross-sectionally averaged equations are generally enough to describe dynamics of the system. For single phase flow, one-dimensional mass equation can be simplified to *
*
v m (t ) = vinlet (t ) Ainlet *
*
A*
(7.1) *
where, vinlet (t ) is the flow velocity at the channel inlet, Ainlet is the flow area at the channel inlet and A* is the local flow area. Conservation equation of energy is given by *
*
∂hm ( z * , t * ) ∂hm ( z * , t * ) q '' (t ) * + v m (t ) = * ∂t * ∂t * A ρf
(7.2)
where, q '' (t ) is time-dependent heat flux into the channel, water density ρ f is the saturation value evaluated with the system pressure. Conservation equation of momentum equation can be written as
dv * (t ) ∆ρ f ( z * , t * ) f 1φ (v m * (t )) 2 m + + + g 1 + ∂P ( z , t ) * 2 ρ D dt ρ − = f h f ∂z * K (v * (t )) 2 δ ( z * − z * ) m K *
*
*
(7.3)
where ∆ρ f ( z * , t * ) is water density expansion due to temperature (enthalpy) change, K is local *
pressure loss coefficient and z K is the position on which the pressure loss occurs. From the first assumption, water saturation enthalpy depends on the local pressure *
h f ( z , t ) = E1 P * ( z * , t ) + E 2
(7.4)
The constants E1 and E 2 can be determined if two pressures and corresponding saturation *
*
enthalpies are known. Here, one of the pressures used is the system pressure Plow = Psys , which is the fixed pressure at the riser outlet. The other one is given by the system pressure plus static
122
(
)
water head of the downcomer Phigh = Psys + Lc + Lr gρ f , which is roughly the steady state *
*
*
*
pressure at channel inlet if friction and local pressure loss are ignored in the downcomer. *
*
*
*
Saturation enthalpies h f ,low and h f ,high corresponding to the two pressures Plow and Phigh are evaluated with ASME water thermo-property codes. The value of E1 and E 2 , therefore, can be *
*
*
*
calculated easily by plugging the h f ,low , h f ,high , Plow and Phigh into the equation (7.4). The left hand side of equation (7.3), thus, can be replaced by term − E1 ∂h f
*
∂z .
The density expansion term ∆ρ f ( z * , t * ) is assumed depending on local flow enthalpy
(
((
)
)
)
∆ρ f z * , t * = α h h z * , t * − h f ,high + β h *
(7.5)
where α h is the Bousinesque density expansion parameter related to flow enthalpy, β h is a constant to modify the density expansion for better accuracy. Their values are determined to agree with experiments. For two phase flow, the dimensional form of mass equation for HEM is given by
(
)
∂ρ m ( z * , t * ) ∂ * * + * vm ( z * , t * ) ρ m ( z * , t * ) = 0 * ∂z ∂t *
(7.6)
where flow density ρ m is a function of steam quality given by *
x(z * , t * ) (1 − x(z * , t * )) + ρ m * ( z * , t * ) = 1 ρ ρ g f
(7.7)
Therefore, the equation can be transformed to
(
)
∂x z * , t * ρ g = + x z* ,t* * ∂t ∆ρ fg
(
(
)
(
* * ∂v m * z * , t * * * * ∂x z , t v ( z , t ) − m ∂z * ∂z *
)
)
Dimensional form of conservation equation of energy for two phase flow is given by
123
(7.8)
∂ ∂ q '' (t ) * * * * * * * * * * * * * z t h z t + v z t h z t ρ , , ρ , , = m m m m m ∂t * ∂z * A*
( (
))
[
( )(
(
) (
(
)]
) (
(7.9)
*
where, the hm is the flow enthalpy *
(
)
*
))
(
*
hm z * , t * = h f z * , t 1 − x z * , t * + hg x z * , t *
)
(7.10)
Differential equation of the water saturation enthalpy can be obtained from equations (7.6), (7.9) and (7.10).
*
(
∂h f z , t ∂t
*
*
*
*
)
(
)
(
q '' (t ) ρ f ρ g ∆h fg * z * , t * ∂v * z * , t * m * − ∆ρ fg ∂z * A = * ∂h f z * , t * * * v m z , t ∂z *
(
(
)
*
)
)
(
)
x z* ,t* 1 + − ρ g 1 − x z * , t * ρ f
(
(
))
(7.11)
*
where, ∆h fg ( z * , t * ) = hg − h f ( z * , t * ) is the specific latent enthalpy. The conservation equation of momentum for two phase flow can be simplified by the mass equation. The form is given by
(
* ∂h * z * , t * ∂v m f = − E1 ∂t * ∂z *
)
ρ m + vm *
*
* f 2φ v m ∂v m +g+ * 2 Dh ∂z
*2
*2 * + Kv m δ ( z * − z K )
(7.12)
note the ∂P * ∂z * term (the first term) on the right hand side of the classic momentum equation has already been replaced by E1∂h f
*
∂z * term with the pressure dependence assumption.
Using the non-dimensional variables and numbers defined in Appendix E, conservation equations of mass, energy and momentum for single phase flow can be converted into the nondimensional forms v m (t ) = N a vinlet (t ) N pch (t ) ∂hm ( z , t ) ∂h( z , t ) = − + v m (t ) N ∂z ∂t flash
124
(7.13) (7.14)
∂h f ( z , t ) ∂z
=
1 ∂v m (1 + N exp,h hm ( z , t ) + N dev ,h ) 2 2 + + N 2φ v m + Kv m δ ( z − z K ) (7.15) ∆Pdrv ∂t Fr
Similary, non-dimensional forms of the equation (7.8), (7.11) and (7.12) for two-phase flow are given by ∂v ( z , t ) ∂x( z , t ) ∂x( z , t ) = ( N ρ N r + x( z , t )) m − vm ( z, t ) ∂z ∂z ∂t
(7.16)
∂v m ( z , t ) (hg − h f ( z , t ) )( N ρ N r + x ( z , t ) ) + (1 − x ( z , t ) ) ∂z ∂h f ( z , t ) N pch ( t ) ( N ρ N r + x ( z , t ) ) =− + N flash N ρ N r (1 − x ( z , t ) ) ∂t ∂h f ( z , t ) vm ( z , t ) ∂z
(7.17)
N ρ N r + x( z , t ) ∂h f ( z , t ) ∂v m ( z , t ) − = ∆Pdrv Nρ Nr ∂z ∂t ∂v ( z , t ) + N f ,2 vm ( z, t ) 2 vm ( z, t ) m ∂z 1 2 + Kv m ( z , t ) δ ( z − z K ) Fr
+
(7.18)
7.2.2. Ordinary Differential Equations (ODEs) in A Node
Phase variables, which represent thermal-hydraulic status of the system include flow enthalpy ( h( z , t ) ), saturation enthalpy ( h f ( z , t ) ), steam quality ( x( z, t ) ), and flow velocity ( v m ( z , t ) ). All of them are functions of both spatial coordinate z and time t. Although the partial differential equations (PDEs) of the phase variables can be solved directly through the multiple numerical schemes discretizing both spatial and time coordinates, such as up-wind, Leap-Frog, and Crank-Nicolson schemes, these schemes, however, generally requires large number of nodes
125
to achieve satisfactory accuracy, and small time step to avoid numerical instabilities. It is also quite difficult to study the system with stability and bifurcation analysis techniques with these pure numerical schemes. An alternative approach is to reduce the set of PDEs to a set of ODEs through spatial discretization, and weighted residue approach can be employed for the reduction. There are two options available to carry out this step. The phase variable can be expanded to either high order polynomial profiles, or linear profiles with time-dependent expansion parameters in each of the nodes. The advantage of higher order polynomial expansion is that less number of nodes, hence less time-dependent parameters, is needed to represent the phase variables. However, the derivation of the ODEs for higher order polynomials is complicated and generally requires iterative calculations. The piece-wise linear expansion route, though, may require more variables, is much easier to develop and may achieve higher efficiency of computation without the need of complicated iterative calculations. Therefore, the piece-wise linear expansion scheme is used in this work. In a node of z i −1 (t ) ≤ z ≤ z i (t ) , where z i −1 (t ) and z i (t ) are lower and upper boundaries of the node respectively. The profiles of the phase variables in a node are given by h f ( z , t ) = h f ,i −1 (t ) + a f ,i (t )( z − z i −1 (t ))
(7.19)
hm ( z , t ) = hm ,i −1 (t ) + a m ,i (t )( z − z i −1 (t ))
(7.20)
x( z , t ) = xi −1 (t ) + N ρ N r si (t )(z − z i −1 (t ) )
(7.21)
v m ( z , t ) = v m , i −1 (t ) + vi (t )( z − z i −1 (t ))
(7.22)
where a f ,i (t ) , a m,i (t ) , si (t ) , and vi (t ) are time-dependent parameters of the profiles of the phase variables. h f ,i −1 (t ) , hm ,i −1 (t ) , xi −1 (t ) , and v m ,i −1 (t ) are values of the phase variables at the lower boundary of the node. Values of the phase variables at the upper boundary are given by
126
h f ,i (t ) = h f ,i −1 (t ) + a f ,i (t )∆z i (t )
(7.23)
hm ,i (t ) = hm ,i −1 (t ) + a m ,i (t )∆z i (t )
(7.24)
xi (t ) = xi −1 (t ) + N ρ N r si (t )∆z i (t )
(7.25)
v m ,i (t ) = v m ,i −1 (t ) + vi (t )∆z i (t )
(7.26)
where, ∆z i (t ) = z i (t ) − z i −1 (t ) is the node length. For single phase flow, since the water is assumed incompressible and subcooled, the expansion parameters v(t) and s(t) are equal to zero. Expansion parameter af(t) can be obtained from the equations (7.15) dv m ,i (t ) (1 + N exp,h hm ,i (t ) + N dev ,h ) + + 1 dt F a f ,i (t ) = r ∆Pdrv 2 2 N f 1v m ,i (t ) + Kv m (t ) δ ( z − z K )
(7.27)
where
v m,i (t ) = N a ,i vinlet (t )
(7.28)
hm ,i (t ) = (hm ,i −1 (t ) + hm ,i (t )) 2
(7.29)
N a ,i is the node area number which is ratio between flow areas of the node and the channel inlet. vinlet (t ) is the flow rate at the channel inlet. ODE for the time-dependent expansion parameters am(t) is derived by plugging equation (7.20) into equation (7.14) and integrating the governing PDE along the node. It is given by da m ,i (t ) dt
=
Am ,1 (a m , a f , s, v, ∆z (t )
)
+ Am , 2 (a m , a f , s, v,
Form of the nonlinear functions Am ,1 and Am , 2 can be found in Appendix F.
127
)
(7.30)
For two phase flow, the flow enthalpy hm ( z , t ) is a function of steam quality x( z , t ) , water saturation enthalpy h f ( z , t ) and steam saturation enthalpy hg (equation (7.10)). Therefore, the expansion parameter am(t) can be explicitly determined from equation (7.10) and (7.20) a m ,i (t ) = a f ,i (t )(1 − xi −1 (t )) + N ρ N r si (t )(hg − h f ,i −1 (t )) − N ρ N r a f ,i (t )si (t )∆z (t ) (7.31) ODEs for the expansion parameters af(t), s(t), and v(t) can be is derived through integrating the governing PDE along the node. Their generic forms are given by da f (t ) dt
=
A f ,1 (a m , a f , v, s,...) ∆z
+ A f , 2 (a m , a f , v, s,...)
(7.32)
ds (t ) S1 (a m , a f , v, s,...) = + S 2 (a m , a f , v, s,...) dt ∆z
(7.33)
dv(t ) V1 (a m , a f , v, s,...) = + V2 (a m , a f , v, s,...) dt ∆z
(7.34)
where A f ,1 , A f , 2 , S1 , S 2 , V1 , and V2 are nonlinear functions of the expansion parameters. Their forms are presented in Appendix F. ODEs for the phase variables at the upper boundary of the node are calculated by the linear representations of the profiles from equations (7.19-7.22). dh f ,i (t ) dt dhm ,i (t ) dt
=
=
dh f ,i −1 (t ) dt dhm ,i −1 (t ) dt
+
+
da f ,i (t ) dt da m ,i (t ) dt
∆z (t ) + a f ,i (t )
d∆z (t ) dt
(7.35)
∆z (t ) + a m ,i (t )
d∆z (t ) dt
(7.36)
dxi (t ) dxi −1 (t ) 1 dsi (t ) d∆z (t ) = + ∆z (t ) + si (t ) dt dt N ρ N r dt dt dv m ,i (t ) dt
=
dv m ,i −1 (t ) dt
+
dvi (t ) d∆z (t ) ∆z (t ) + vi (t ) dt dt
128
(7.37)
(7.38)
7.2.3. Node with Small Length In the fixed boundary nodalization scheme where all node lengths are fixed and timeinvariant ( ∆z ) except for the last single-phase and the first two-phase nodes—as these nodes are divided by a moving boiling boundary, and their lengths should add up to ∆z —the length of the last single-phase node or the first two-phase node can be very small. The first terms of right hand sides (RHSs) of Equations (7.30), (7.32), (7.33) and (7.34) are of forms of nonlinear functions divided by the node length. As the length goes to zero, small changes of the numerator Am,1 , A f ,1 , S1 and V1 cause large changes of the values of ODEs. Thus, time scale for the node with very
small length can be significantly larger than those of the other nodes with large lengths. This will lead to a stiff ODE system with significantly different time scales. It is well known that numerical integration schemes are stable for stiff system only when the time step is very small. Classic numerical integration codes usually use schemes of adaptive time steps to solve stiff problem, where stiffness of the system is detected continuously during the integration and time steps are adjusted accordingly. However, for the problem of flashing, decreasing time steps of the numerical integration can not solve the stiffness caused by small node length, instead, it may exacerbate it. For smaller time step, the calculation of the boiling boundary is finer and it can move closer to a fixed boundary. This will lead to even smaller node length and larger stiffness. The time step then is forced to decrease even further again and again, until the numerical integration code crashes. To solve this difficulty, a limit to the node length ∆z lim is set. When the node length is smaller than this limit, evaluations of ODEs of equations (7.30), (7.32), (7.33), and (7.34) are stopped. Values of a f ,i , a m ,i , si and vi in the node i, whose length is very small, are explicitly solved from large stiffness conditions
129
A f ,1 (a m , a f , v, s,...) = 0
(7.39)
Am ,1 (a m , a f , v, s,...) = 0
(7.40)
S1 (a m , a f , v, s,...) = 0
(7.41)
V1 (a m , a f , v, s,...) = 0
(7.42)
These conditions are valid because ODEs of equations (7.30), (7.32), (7.33) and (7.34) are still finite as the node length goes to zero. Only the zero numerators in the first term of the right hand sides can avoid occurrence of infinite numbers when the denominator is zero. Forms of a f ,i , a m ,i , si and vi are given by functions of operating parameters and values at the lower boundary of the
node. Details can be found in the Appendix G.
7.2.4. Phase Variables across Boundaries For the system considered in the section 7.1, phase variables and ODEs may experience jumps across node boundaries, i.e. the values at the upper boundary of a lower node may not be the same at the lower boundary of an upper node. Two types of boundaries have to be identified. The first are those boundaries due to nodalization, such as those of boundaries S2, S4, and S6 in the Figure 7.3(a), S2 and S4 in the Figure 7.3(b). There is no pressure loss and changes of physical conditions, such as differences of heat flux, flow area, and flow patterns across the boundaries. Therefore, phase variables and ODEs are not supposed to change across the boundary. The second are those boundaries, across which there are local pressure loss and changes of physical conditions, such as the boundaries S3 and S5 in the Figure 7.3(a) and Figure 7.3(b). The boundary S3 is the interface between the channel and riser, and there is a local pressure loss coefficient K c ,out at the boundary, and heat flux is changed from q ' (t ) to zero from the channel to
130
the riser. The boundary S5 is the boiling boundary, thus flow patterns are changed from single phase to two-phase. In these cases, phase variables and/or ODEs are going to jump across the boundaries. Figure 7.4 shows conditions across a boundary with jump of the phase variables. Suppose the saturation enthalpy, flow enthalpy, steam quality, and flow velocity at the exit of a lower node Sur1, which is a surface below but infinitely close to the boundary between the nodes, are h f ,1 , hm ,1 , x1 , and v m ,1 , and those at the entrance of an upper node Sur2, which is a surface above but
also infinitely close to the boundary, are h f , 2 , hm , 2 , x 2 , and v m , 2 . There are a local pressure loss coefficient K at the boundary and a flow area change across the boundary. Area number at the surface Sur1 and Sur2 are N a ,1 and N a , 2 respectively. The flow enthalpy hm is not subject to change from Sur1 to Sur2 due to conservation of energy.
hm ,1 = hm , 2
(7.43)
Because of pressure loss, the saturation enthalpy h f will take a jump, however. h f , 2 (t ) = h f ,1 (t ) +
Nρ Nr K 2 v m ,1 (t ) ∆Pdrv N ρ N r + x1 (t )
(7.44)
The changes of flow quality and velocities across a boundary can be derived through conservation of flow enthalpy (equation 7.43) and conservation of mass flow rate. Combining the equation (7.10) and equation (7.43), we will find x 2 (t ) =
hx ,1 (t ) − hx , 2 (t ) hg − hx , 2 (t )
+
hg − hx ,1 (t )
hg − hx , 2 (t )
131
x1 (t )
(7.45)
Na,2
hf,2, hm,2, x2, vm,2
Sur2 K
hf,1, hm,1, x1, vm,1
Sur1 Na,1
Figure 7.4: Conditions across a boundary. Sur1 is a surface below but infinitely close to the boundary. Sur2 is the surface above but infinitely close to the boundary. Conditions on Sur1 include hf,1, hm,1, x1, and vm,1. Area number of the Sur1 is Na,1. Conditions on Sur2 include hf,2, hm,2, x2, vm,2. Area number of the Sur2 is Na,2. There is a pressure loss coefficient K on the boundary.
132
where, hx ,1 (t ) and hx , 2 (t ) denote enthalpies of water phase in the lower and upper nodes. If a node is single phase, hx ,i (t ) = hm ,i (t ) ; other wise, hx ,i (t ) = h f ,i (t ) . Non-dimensional density of flow is given by
ρ m (t ) =
Nρ Nr
(7.46)
N ρ N r + x(t )
Through conservation of mass flow rate across the boundary
ρ m,1 (t )v m,1 (t ) N a ,1 = ρ m, 2 (t )v m, 2 (t ) N a , 2
(7.47)
find
v m , 2 (t ) =
N a , 2 N ρ N r + x 2 (t ) N a ,1 N ρ N r + x1 (t )
v m ,1 (t )
(7.48)
Time derivatives of equations (7.43), (7.44), (7.45), and (7.48) show ODE changes across the boundary.
dhm , 2 (t ) dt
=
dhm ,1 (t ) dt
Nρ Nr 2 dx1 (t ) − v (t ) 2 m ,1 dt dh f , 2 (t ) dh f ,1 (t ) K (N ρ N r + x1 (t )) = − ∆Pdrv dt dt 2N ρ N r dv m ,1 (t ) v m ,1 (t ) dt N ρ N r + x1 (t ) 1 − x (t ) dhx ,1 (t ) 1 − hg − hx , 2 (t ) dt (h − h (t ) ) dhx , 2 (t ) dx 2 (t ) g x ,1 (1 − x1 (t ) ) = + (hg − hx , 2 (t ) )2 dt dt hg − hx ,1 (t ) dx1 (t ) hg − hx , 2 (t ) dt
133
(7.49)
(7.50)
(7.51)
dv m , 2 (t ) dt
=−
N a,2 N a ,1
N ρ N r + x 2 (t ) dx1 (t ) − N N x t dt + ( ) ρ 1 r dx 2 (t ) v m ,1 (t ) − N ρ N r + x1 (t ) dt N ρ N r + x 2 (t ) dv m ,1 (t ) v m ,1 (t ) dt
(7.52)
7.3. Closure of Boiling Boundary and Its Time-Dependent Derivative For a set of the expansion parameters, the ODEs given by (7.30), (7.32), (7.33) and (7.34) depend on evaluations of intermediate unknowns. These include the boiling boundary µ (t ) , the saturation enthalpy at the boiling boundary h f , µ (t ) , the inlet saturation enthalpy h f ,inlet (t ) , time derivative of inlet flow velocity
dvinlet (t ) dµ (t ) , time derivatives of the boiling boundary and dt dt
time derivative of the inlet saturation enthalpy
dh f ,inlet (t ) dt
. If these unknowns are determined as
functions of the expansion parameters, ODEs of the expansion parameters, thus, can be evaluated sequentially from the lowest node at the channel inlet to the highest one at the riser outlet. However, the unknowns do not depend on local expansion parameters, but all the expansion parameters. Analytical forms of the functions therefore are fairly complicated and derivations can be tedious processes. Alternative numerical evaluations of the unknowns require iterative processes. In this section, the numerical evaluation method is discussed. The pressure at the riser outlet is fixed as system pressure. Non-dimensional process and dependence of saturation enthalpy and local pressure lead to the saturation enthalpy at riser outlet fixed
h f ,r ,out (t ) = 1
134
(7.53)
For a set of expansion parameters a f ,i , a m ,i , si and vi , values of the unknowns have to get the boundary condition (7.53) satisfied.
µ (t ) , h f , µ (t ) , h f ,inlet (t ) and
dvinlet (t ) are inter-related. If any one of them is chosen as the dt
iterative variable, the rest three can be determined as functions of the chosen one by pressure balance along the single phase regions. Figure 7.5 shows an example of the case “boiling in the channel”. The number of nodes for the three regions are N 1 , N 2 and N 3 respectively, and the total number of nodes N = N 1 + N 2 + N 3 . Node lengths in the three regions from the channel inlet to the riser outlet are given by
∆z1 (t ) = µ (t ) N 1 ,
(7.54)
∆z 2 (t ) = (Lc − µ (t )) N 2
(7.55)
∆z 3 = Lr N 3
(7.56)
From the assumption 1 in the section 7.2.1, flow is at equilibrium state at the boiling boundary, thus flow enthalpy hm , N1 +1 is equal to the saturation enthalpy h f , µ (t ) . N1
h f , µ = hm , N1 +1 = hm ,inlet + ∑ a m ,i (t )∆z1 (t )
(7.57)
i =1
where hm ,inlet =
N sub dvinlet (t ) is the flow enthalpy at the channel inlet. The value of can be N flash dt
determined through integration of the single phase momentum equation (7.15) along the dashed line in the Figure 7.5 from the riser outlet to the boiling boundary.
dvinlet (t ) 1 = ∆Pext − ∆Pgrav − ∆Pfric − ∆Ploc dt Mass
[
where
135
]
(7.58)
Kr,out
h,f, r,out=1
Kdc,in
N3
Kc,out
Ldc hf, µ
N2 N1 µ
Kc,in
vinlet, dvinlet/dt
hf,inlet
Figure 7.5: Intermediate unknowns for ODE evaluations. Boiling boundary is in the channel in this example.
136
Mass = N a ,dc (Lc + Lr ) + N a ,c µ (t )
(7.59)
∆Pext = (h f , µ (t ) − 1)∆Pdrv
(7.60)
∆Pgrav
N1 (1 + N exp,h ,c hm ,i −1 (t ) + N dev ,h ,c ) ∑ ∆z1 (t ) + Fr i =1 N1 2 N exp,h ,c a m ,i (t ) ∆z1 (t ) = ∑ − i =1 Fr 2 (Lc + Lr ) F r
∆Pfric = N f ,1 (N a ,c vinlet (t )) µ (t ) 2
(
(7.62)
)
∆Ploc = k c ,in N a ,c + k dc ,in N a ,dc vinlet (t ) 2
2
(7.61)
2
(7.63)
Mass is non-dimensional mass along the dashed line, ∆Pext is the pressure between the boiling
boundary and riser outlet, ∆Pgrav is the pressure loss (and gain) due to gravity, ∆Pfric is the pressure loss due to friction, ∆Ploc is the pressure loss due to restrictions along the integration line. The value of inlet saturation enthalpy h f ,inlet (t ) is given by integration along the dashed line from riser outlet to the channel inlet
h f ,inlet (t ) =
1 ∆Pdrv
dvinlet (t ) Mass dc dt + ∆Pgrav ,dc + ∆Pfric ,dc + ∆Ploc ,dc + 1
(7.64)
where, Mass dc = N a ,dc (Lc + Lr )
(7.65)
∆Pext = (h f ,inlet (t ) − 1)∆Pdrv
(7.66)
∆Pgrav = −
(Lc + Lr ) Fr
∆Pfric = 0
(7.67) (7.68)
137
∆Ploc = k dc ,in (N a ,dc vinlet (t ))
2
(7.69)
The subscript “dc” denote the values in the downcomer region. For the moving boundary scheme, different cases of boiling are modeled differently, thus the approximate position of the boiling boundary is known at least roughly (out of the riser, in the riser, at the connection of the channel and riser, or in the channel). It is convenient to choose the boiling boundary as the iterative variable and other three unknowns are determined through equations (7.54 – 7.69) as functions of the boiling boundary. For the fixed boundary scheme, however, there is no previous knowledge about the position of boiling boundary µ (t ) . It can be in the channel, in the riser, at the connection between the channel and riser, or more specifically, out of the riser. A bad initial guess of the boiling boundary may lead to non-convergence of the iterative process. To avoid this difficulty, the saturation enthalpy at the channel inlet h f ,inlet (t ) , instead of the boiling boundary µ (t ) , is chosen as the a better candidate of the iteration variable. The
dvinlet (t ) , thus, is represented as a function dt
of h f ,inlet (t ) (equation (7.64)). Initial guess of the boiling boundary µ (t ) is at the riser outlet ( µ (t ) = Lc + Lr ). The flow enthalpy hm (t ) and saturation enthalpy h f (t ) are evaluated sequentially from the lowest node. Since the boiling boundary is initially assumed at the exit of the highest node, virtually all the nodes evaluated are single-phase. If in a specific node i, the flow enthalpy hm,i (t ) are larger than the saturation enthalpy h f ,i (t ) at the node exit, it indicates the boiling boundary is actually located at somewhere in the middle of the node i ( z i −1 < µ (t ) < z i ). The value of µ (t ) has to get h f , µ (t ) = hm , µ (t ) satisfied. From equation (7.19), (7.20) and (7.27), the value of µ (t ) and h f , µ (t ) is given by
138
µ (t ) = z i −1 −
2 E3
(7.70)
2
E 2 + E 2 − 4 E1 E3
h f , µ (t ) = hm , µ (t ) = hm ,i −1 (t ) + a m ,i (µ (t ) − z i −1 )
(7.71)
where, E1 =
E2 =
N exp,h
(7.72)
2 Fr ∆Pdrv 1 ∆Pdrv
dv m ,i (t ) (1 + N exp,h hm ,i −1 + N dev ,h ) 2 + + N f ,1v m ,i − a m ,i (7.73) Fr dt
E3 = h f ,i −1 − hm ,i −1
(7.74)
Once the boiling boundary is determined (either as iterative variable for the moving boundary scheme, or as function of the h f ,inlet (t ) for the fixed boundary scheme), saturation enthalpies above the boiling boundaries h f ,i , i = N µ ,
N + 1 can be determined sequentially
along the nodes from the boiling boundary to the riser outlet from equations (7.19-7.22) and (7.44). The saturation enthalpy at the riser outlet h f ,r ,out = h f , N +1 , therefore, is a single value function of the iterative variable y
h f ,r ,out (t ) = h f , N (t ) = g ( y (t ))
(7.75)
for the moving boundary scheme y (t ) = µ (t ) ; for the fixed boundary scheme y (t ) = h f ,inlet (t ) . A simple Newton’s method can be used to find the iterative variable that satisfies the equation ()
(
(
µ (t )(n +1) = µ (t )(n ) + 1 − g y (t )(n )
)) DG
(7.76)
where DG = dg ( y (n ) ) dy (n ) . It is found that the iteration of Newton’s method converges rather quickly. In most cases, less than 4 iterations are enough.
139
The afro-mentioned process can be applied similarly to evaluate the time derivatives of the boiling boundary
dh f ,inlet (t ) dµ (t ) and inlet saturation enthalpy . If there is no considerations of dt dt
the zero-length node along the channel and riser, it is shown that the time derivative of the saturation enthalpy at the riser outlet
dh f ,r ,out (t ) dt
is a linear function of either
dh f ,inlet (t ) dµ (t ) or . dt dt
Therefore only one Newton’s iteration is enough for the solution of time derivative of the boundary condition (7.53). If there is node with almost zero length, the solutions of (7.39-7.42) cause a nonlinear function of the
dh f ,r ,out (t ) dt
to the
method are required.
140
dh f ,inlet (t ) dt
. Multiple iterations of Newton’s
Chapter 8 Stability and Bifurcation Analyses of Natural-Circulation BWR Model
In this chapter, the stability and bifurcation analyses of the thermal-hydraulics and nuclear coupled thermal hydraulics models of the natural circulation BWR are carried out. The methodology is the same as that in section 2.2. For the thermal-hydraulics model, two sets of operating parameters listed in Appendix H and I are used for these studies. One corresponds to the natural circulation loop SIRIUS [44] [61] and the other corresponds to the natural circulation BWR Dodewaard reactor (the Netherlands) [45] [59] [62] [63]. Results obtained using BIFDD are first validated and compared with experiments carried out on the SIRIUS facility. Stability, bifurcation and sensitivity analyses are then carried out using the two sets of data. For the nuclear coupled model, only the set of operating parameters corresponding to the Dodewaard reactor are used. Characteristic of bifurcations and regions of in-phase/out-of-phase oscillations are determined and analyzed in different operating parameter spaces.
8.1. Validations and Comparisons
8.1.1 Convergence The set of parameters for SIRIUS facilities are used for testing convergence of different nodalization schemes. Figure 8.1 shows steady state non-dimensional boiling boundary ( µ ) and *
inlet flow ( vinlet ) as a function of inlet subcooling temperature ( ∆Tinlet ) calculated using the fixed
141
boundary scheme. The system pressure is Psys = 0.2 MPa, and heat flux in the channel is q ' = 6.0 (kW/m). Total numbers of nodes are N = 3, 5, and 9 respectively (the nodes with fixed boundaries in the channel and riser are equal, and N 1 = N 2 = 1, 2, and 4). The boiling boundary is affected little by the number of nodes. For the flow velocity, the results appear to converge for N = 9. As the subcooling increases, Figure 8.1(a) shows that the steady-state boiling boundary *
moves upward. For ∆Tinlet > 10.6 (K), the boiling boundary is in the riser and flashing induced *
boiling due to pressure drop, instead of heating, begins. For ∆Tinlet > 34.8 (K), the boiling boundary reaches the riser outlet, and no boiling at steady state occurs in the channel or the riser. It is noted that the value of subcooling for the no-boiling case strongly depends on the Boussinesq [64] expansion parameters N exp h and constant N devh (Appendix E). In this calculation, the parameters N exp h and N devh are assumed constant for all single phase nodes in the channel and *
riser, and they are evaluated using thermodynamic properties of water at pressure Phigh (section 7.2.1). The parameters N exp h and N devh in the downcomer are assumed to be zero. However, thermodynamic properties along the channel and riser may be quite different due to changes in the local pressure. Consequently, the values of these two parameters, N exp h and N devh , will be parametrically changed later in this chapter. Figure 8.1(b) shows the steady state inlet flow velocity. It is noted that the inlet flow velocity is increasing with inlet subcooling at low values of inlet subcooling due to lower friction as the boiling boundary moves upward. Inlet flow velocity starts to decrease with inlet subcooling as the boiling boundary reaches into the riser—because of lower density difference between the channel-riser and the downcomer, and thus a drop in the natural circulation driving force.
142
5
P2 P1
4
3
µ
P3
N=3 N=5 N=9 N=17
2
P4
1
P5 0
0
10
20
30
40
*
∆Tinlet (K)
(a)
7
P4
N=3 N=5 N=9 N=17
6
P3
P5
vinlet
5
4
3
P2 P1 2
1 0
10
20
30
40
*
∆Tinlet (K)
(b) Figure 8.1: Steady state boiling boundary and inlet flow velocity calculated with fixed boundary schemes. Total number of nodes are N = 3, 5, and 9. (a) boiling boundary µ ; (b) inlet flow velocity vinlet .
143
Figure 8.2 shows the stability results in the form of stability boundaries (SBs), determined using BIFDD, in the inlet subcooling temperature—heat flux parameter space. The system pressure is Psys = 0.2 MPa. Figure 8.2 (a) shows the SBs calculated with the moving boundary schemes. Total numbers of nodes are N = 3, 6, 12, and 24 (number of nodes in the three regions are equal). For small values of N , the stable region is larger, indicating insufficient nodalization may lead to non-conservative results. As N is increased, the stable area is decreased. The stable area for N = 12 and 24 are very close, and the SBs appear to have converged. Converged boundary is discontinuous at one point, and has a discontinuity in slope at another point. The discontinuity in slope is due to the fact that there are two different complex conjugate pairs, and as shown in chapter 3 for forced circulation systems, a different pair is responsible for the instability on the two sides of the discontinuity. This is explained in much more detail later in this chapter. Reason for the discontinuity in the SB along the separation boundary is discussed below. In addition to the SB, another boundary represented by the short dash-dot line is shown in Fig. 8.2, which separates the parameter space into two regions. The steady state boiling boundary in region in parameter space above this separation boundary is in the riser; while the boiling boundary in the region below the separation boundary is in the channel. The SB is thus divided (by this dash-dot separation boundary) into a segment along which the boiling boundary is in the riser and a segment along which the boiling boundary is in the channel. The SB is discontinuous where this transition occurs. (The discontinuity is exacerbated by an insufficient number of nodes.) The discontinuity can be explained by the fact that there is a qualitative change in the system when the boiling boundary moves from the channel into the riser. [These qualitative changes include a concentrated pressure drop at the channel exit; change in area; and no heat flux in the riser. Moreover, boiling (or flashing) in the riser is due to drop in pressure, unlike in the
144
15
15
N=3 N=6 N=12 N=24
14
Unstable
13
Unstable
12
*
11 10 9 8
11 10 9 8
Riser
7
N=3 N=5 N=9 N=17 N=33
13
∆Tinlet (K)
*
∆Tinlet (K)
12
14
Riser
Stable
7
Stable
6
6
channel 5
channel
5 0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.001
q' (kW/m)
0.002
0.003
0.004
0.005
0.006
0.007
q' (kW/m)
(a)
(b)
15 14
N=24, Moving Bound. N=33, Fixed Bound.
13
*
∆Tinlet (K)
12
Unstable
11 10 9 8
Riser
7
Stable
6
Channel
5 0.001
0.002
0.003
0.004
0.005
0.006
0.007
q' (kW/m)
(c)
Figure 8.2: Stability boundaries in inlet subcooling—heat flux parameter space for two different nodalization schemes. (a) moving boundary scheme; (b) fixed boundary scheme; (c) comparison of moving and fixed boundary schemes.
145
channel where heat addition leads to boiling.] The SB only indicates stability in response to an infinitesimal perturbation about a steady state. For this infinitesimal perturbation, it is possible that the system is stable for a steady state with a boiling boundary in the channel but infinitesimally close to the channel exit, but unstable for another steady state with the boiling boundary in the riser but infinitesimally close to the channel exit. Part of the reason for this discontinuity is the failure of the homogeneous equilibrium model used here to represent the two phase flow. With the inclusion of a subcooled boiling model and thermodynamics nonequilibrium, the discontinuity is likely to disappear. Figure 8.2(b) shows results for the fixed boundary scheme. Total numbers of nodes are N = 3, 5, 9, 17, and 33. Here the converged SB is discontinuous not only on the boundary that
separates the region where the boiling boundary is in the channel from the region where it is in the riser, but also at locations where the boiling boundary is in the channel or in the riser. These jumps in the SBs are more pronounced for smaller numbers of nodes. As the number of nodes is increased, the discontinuities are smoothed out, and the SBs converge to a limiting boundary. These discontinuities result due to the movement of the boiling boundary across fixed nodes. Every time when the boiling boundary moves across a fixed boundary, a node will change from single phase to a two phase or vice versa. This leads to changes in the dynamical system and dynamics of the system accordingly, thus, creating a jump in the SB. For the fixed boundary scheme, the SB is always assumed to be a single valued function of heat flux. This condition is valid when the heat flux is small and large. In these cases, SB segment exists either in region above the separation boundary (boiling in the riser) or below (boiling in the channel). However, for the intermediate value of heat flux, segments of SB can exist in both the two regions. To maintain the single valued condition, segments of the SB above the separation
146
boundary in Figure 8.2(b) are tailed off. The part of the SB segment where the boiling boundary is in the riser in the Figure 8.2 (b) does not smoothly approach the separation boundary. It instead suddenly jumps from the riser region to the channel region. Figure 8.2(c) compares the converged SBs for the moving and fixed boundary schemes. They agree with each other satisfactorily (except the part that is tailed off by the fixed boundary scheme). However, it takes more nodes in the fixed boundary scheme than the moving boundary scheme to reach a smooth and converged boundary. The results suggest that the moving boundary scheme is superior to the fixed boundary scheme for the purpose of stability analysis.
8.1.2. Model Validations and Comparisons with Experiments Preliminary evaluation of the model carried out by comparing steady state and stability results predicted by the model with experimental data is presented in this section. More detailed stability and bifurcation analyses are presented in the next section. Figure 8.3 compares converged steady state flow velocities with experimental data by Inada et al [61]. The system pressure for the experiments is Psys = 0.2 MPa. Heat flux in the channel is q ' = 1.7647, 2.9412, 4.1177, and 5.2941 (kW/m), respectively. Good agreement between the model based on HEM and experimental data is found when heat flux and subcooling are high (Figure 8.3 (c) and (d)). The model, however, over-estimates flow rates for low heat flux and inlet subcooling (Figure 8.3(a) and (b)). This over-estimation is (most likely) due to the lack of subcooled boiling model as well as due to the use of HEM. Figure 8.4 compares stability boundaries calculated using the model developed here with experimental results by Inada et al [61], in the inlet subcooling—heat flux parameter space. The system pressure is Psys = 0.1 MPa. The open squires represent the operating points with stable
147
1.5
1.5
Experiment Analysis
Experiment Analysis
*
vinlet (m/s)
1.0
*
vinlet (m/s)
1.0
0.5
0.5
0.0
0.0 5
10
15
20
5
10
15
*
20
25
30
*
∆Tinlet (K)
∆Tinlet (K)
(a)
(b) 1.5
1.5
Experiment Analysis
Experiment Analysis 1.0
*
*
vinlet (m/s)
vinlet (m/s)
1.0
0.5
0.5
0.0
0.0 10
15
20
25
30
10
15
20
25
30
35
*
*
∆Tinlet (K)
∆Tinlet (K)
(d)
(c)
Figure 8.3: Comparison of steady state flow rates between experiments and analyses. System pressure is Psys = 0.2 MPa. (a) heat flux q ' = 1.7647 (kW/m); (b) q ' = 2.9412 (kW/m); (c) q ' = 4.1177 (kW/m); (d) q ' = 5.2941 (kW/m)
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40
flow rates found in the experiments. The solid circles represent those with unstable flow rates in the experiments. As shown in Figure 8.4, when the inlet subcooling is large, no boiling occurs in the natural circulation system. The flow is single phase and stable. As the subcooling is decreased, flashing starts in the riser and unstable intermittent waves occur. As the subcooling is decreased further, the flow is still unstable, but it evolves from the intermittent waves to the sinusoidal waves. For small inlet subcooling, amplitude of the sinusoidal wave is very small, and the flow is again stable. Figure 8.4(a) shows comparison of the stability boundary—which separates the stable twophase flow from the unstable sinusoidal wave flow—calculated using BIFDD with the experimental data,. Only the segment in the riser is plotted. Good agreement is found in the range 2.0 (kW/m) < q ' < 4.66 (kW/m). For the heat flux less than 2.0 (kW/m), the analytical stability results slightly over-estimate the stable region. This over estimation is evidently because of the lack of a subcooled boiling model. At point T (heat flux is 4.66 (kW/m)), the stability boundary (solid line) turns sharply back toward lower heat flux region. This result indicates that operating points with heat fluxes greater than 4.66 (kW/m) are unstable. The experimental data however shows at least one more case of stable flow for heat flux greater than 4.66 kW/m. The discrepancy between the analytical result and the experimental data beyond the point T might be explained by a mechanism similar to the crossing of eigenvalues in section 3.1 of Chapter 3. As it will be shown in the next section, the SB turns back because the pair of eigenvalues with the second largest real part (along the stability boundary shown) crosses the eigenvalues with the largest real part. Figure 8.4(a) also shows the continuation of the SB (dotted line) on which the real part of the second rightmost eqignvalue pair is equal to zero. The combined segments of the SB, A-T and the dotted line T-B capture the division between the stable and unstable experimental data points
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50
50
Stable Unstable
*
∆Tinlet (K)
40
30
*
∆Tinlet (K)
40
Stable Unstable
E=1.0 E=1.5 E=2.0
30
20
20
B
A T 10 0.000
0.002
0.004
10 0.000
0.006
q' (MW/m)
0.002
0.004
0.006
q' (MW/m)
(a)
(b)
50
Stable Unstable
*
∆Tinlet (K)
40
E=1.5
30
20
B
A T 10 0.000
0.002
0.004
0.006
q' (MW/m)
(c) Figure 8.4: Comparisons of experimental data with the stability boundaries evaluated using the model developed in chapter 7. System pressure is Psys = 0.1 MPa. (a) Comparison of stability boundary; (b) Comparison of no-boiling boundary for different Boussinesq expansion parameter values; (c) Comparison of both stability boundary and no-boiling boundary.
150
satisfactorily. More detailed discussion of these issues will be presented in the next section. Figure 8.4(b) shows comparison of analytical no-boiling boundary (boundary in parameter space above which no boiling occurs in the system; flow is all single phase and stable) with the experimental data, which separates the stable single phase flow from the unstable intermittent wave flows. The analytical boundary is calculated by tracking the operating points on which boiling occurs exactly at the riser outlet. Above the boundary, the model predicts no-boiling in the natural circulation system. As described at the beginning of this section, the no-boiling condition strongly depends on the Boussinesq expansion parameter N exp h and N devh . Values of the *
parameters are evaluated at the system pressure Phigh , which is roughly the pressure at the channel inlet, and assumed constant in the entire single phase region. However, due to pressure drop, rate of the Boussinesq expansion is different along nodes in the channel and riser. To account for this effect, the expansion parameter is parametrically modified by multiplying N exp h and N devh by a factor E , varying E between 1 and 2. Figure 8.4 (b) shows the no-boiling boundaries for E = 1.0, 1.5 and 2.0. The boundary for E = 1.5 case is found to best match the experimental boundary separating the stable experimental data points from the unstable ones. Figure 8.4 (c) combines the results of Figures 8.4 (a) and (b). The segments A-T and T-B in Figure 8.4(a) and the no-boiling boundary in Figure 8.4(b) are plotted on the same figure. Comparison with the experimental data show good agreement, especially for large heat flux values. *
Similar comparisons are made for the system pressure Psys = 0.2 MPa (Figure 8.5). In Figure 8.5(a), the SB calculated by BIFDD is compared with the experimental data. When heat flux q ' is less (greater) than 4.0 (kW/m), boiling along the operating points on the SB occurs in
151
50
50
Stable Unstable
40
40
E=1.0 E=1.5 E=2.0
30
*
∆Tinlet (K)
*
∆Tinlet (K)
30
Stable Unstable
20
20
10
10
In Riser 0 0.000
0.002
In Channel
0.004
0 0.000
0.006
q' (MW/m)
0.002
0.004
0.006
q' (MW/m)
(a)
(b)
50
40
Stable Unstable
*
∆Tinlet (K)
E=1.5 30
20
10
In Riser 0 0.000
0.002
In Channel
0.004
0.006
q' (MW/m)
(c) Figure 8.5: Comparison of experimental data points with stability boundaries evaluated using the model developed in Chapter 7. System pressure is Psys = 0.2 MPa. (a) Comparison of stability boundary; (b) Comparison of no-boiling boundary for different values of the Boussinesq expansion parameter; (c) Comparison of both, stability boundary and no-boiling boundary with the experimental data.
152
the riser (channel). The SB separates the stable experimental data points from the unstable ones satisfactorily. Figure 8.5(b) plots the no-boiling boundaries for E = 1.0, 1.5 and 2.0. The boundary for E = 1.5 is again found to best match with the experimental data. The combined results are presented in the Figure 8.5(c).
8.2. Stability and Bifurcation Analyses of the Thermal-Hydraulics Model In this section, natural circulation thermal-hydraulics is studied as a dynamical system. Stability boundaries and oscillation curves are calculated and analyzed for the SIRIUS facility *
[44] under low pressure conditions; Psys = 0.1 and 0.2 MPa. Behavior of natural circulation *
system under higher pressure condition ( Psys = 1.07 MPa) is studied for the Dodewaard reactor system (without neutronics). Description and dimensions and parameters of the SIRIUS facility are given in reference [44] [61] and Appendix H. Figure 8.6 shows, in the inlet subcooling—heat flux parameter space, the stability boundary (solid line) and the boundary on which the second rightmost eigenvalue pair has zero real part (the dotted line). This second boundary is given only for the case when the boiling boundary is in the riser. [The second boundary has no significance for the case when the boiling boundary is in the channel. That is, when the boiling boundary is in the channel (and no neutronics), the other pairs of eigenvalues of Jacobian matrix are all far to the left of the pair with *
the largest real part.] System pressure is Psys = 0.1 MPa. Unlike the SB shown in Figure 8.4, here
both branches of the SB—that is, segment on which boiling starts in the riser and segment on which boiling starts in the channel—are plotted. They are separated by the separation boundary (short dash-dot line). The two branches do not exactly meet at the same point on the separation
153
boundary, D and E.. The discrepancy is again due to the sudden jump in system characteristics as the boiling boundary, constrained by a HEM, jumps from the channel into the riser. On the branch with boiling in the riser, the two pairs of complex conjugate eigenvalues *
cross at the point T ( q ' = 4.66 (kW/m), ∆Tinlet = 15.33 (K)). Along the segment of the SB from point A to T, real part of the right most pair of eigenvalues e1 is zero. As one moves from A toward
T, the second rightmost pair of eigenvalues e2 approaches the imaginary axis. The pair e2 crosses the pair e1 at the point T. Beyond the point T, the real part of e1 on the dotted line segment T-B is still zero, but the pair e1 is now the second rightmost pair. Real part of the rightmost pair of eigenvalue e2 is greater than zero and the system is unstable. The SB segment C-T is determined by the pair e2. Along the SB segment C-T and the dotted line segment T-C’, the real part of e2 is zero, but the segment T-C’ is in the unstable region. Similar to the crossing of eigenvalues described in the section 3.1, this change of eigenvalue pair responsible for the instability represents characteristic change in the type of oscillations that result as the SB is crossed. Moreover, this also explains the jump in slope in the SB in Fig. 8.2. The most significant difference is in the period of oscillations produced by the pair e1 and those due to e2. The imaginary part of the rightmost pair of eigenvalue determines the frequency of oscillations for the operating points close to the SB. In this case, the imaginary part for the pair
e1 is 16.01; while for the pair e2, the imaginary part is 3.32. Therefore, the frequency that results due to the pair e1 is about 5 times larger than that produced by the pair e2. This fact may help explain the slight discrepancy between the SB calculated and the experimental data (for one experimental data point) in section 8.1.2. In the region of the parameter space below the segment T-B and to the right of the segment C-T, the system is unstable, because the real part of the rightmost pair e2 is greater than zero. However, the real part of the second rightmost pair e1 is less
154
than zero. Low frequency oscillation produced by the pair e2 will hence grow, but high frequency oscillations produced by the pair e1 will initially decay. Since the high frequency oscillations are the easiest to observe and record, when operating point crosses the segment T-B, the experimental observation of high frequency (initially) decaying oscillations might have suggested a stable system, and the experiment might not have been continued long enough to observe the unstable low frequency oscillations. This might have led to the characterization of the data point for the rightmost value of heat flux studied as “stable.” Figure 8.7 shows 7.5% oscillation curves (OCs) calculated using BIFDD for different *
*
segments along the SB. The y-axis is ∆Tinlet − ∆Tinlet ,critical , where subscript “critical” denotes the value on the SB. For the SB branch on which boiling is in the riser, the oscillation curve for the segment A-T is in the unstable region, and the bifurcation is supercritical. On the other hand, the OC for the segment T-C is in the stable region, and the bifurcation is subcritical. For the SB branch on which boiling is in the channel, the bifurcation is subcritical along the segment D-E and along a part of the segment E-F. Bifurcation turns supercritical below a certain value of subcooling near the middle of the segment E-F, and stays supercritical for the entire segment F-G. *
Figure 8.8 shows results of the stability analysis for the system pressure Psys = 0.2 MPa. *
Similar to the case of Psys = 0.1 MPa, two pairs of complex conjugate eigenvalues cross in the complex plane along the branch of SB when the boiling boundary is in the riser. At the point T ( q ' *
= 3.62 (kW/m), ∆Tinlet = 11.04 (K)), both pairs of complex conjugate eigenvalues have zero real part. The imaginary parts (frequencies) of e1 and e2 at the point T are 14.60 and 3.68, respectively. Along the segment A-T, e1 is the dominant pair, while e2 is the dominant pair along T-D . Figure 8.9 shows 7.5% oscillation curves for different segments along the SB in Figure 8.8. The
155
25
Stability Boundary Boundary of the 2nd eigenvalue 20
C' B
T
15
*
∆Tinlet (K)
A
Stable 10
5
Riser
C
E
D Unstable
Channel
F G
0 0.000
0.002
0.004
0.006
q' (MW/m)
Figure 8.6: Stability boundary and the boundary on which the second rightmost pair of *
eigenvalues has zero real part. System pressure Psys = 0.1 MPa. The SB is divided into segments along which boiling is in the riser, and the other one along which boiling is in the channel.
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0.010
T ∆Tinlet -∆Tinlet,critical (K)
0.008
T
*
0.008
0.006
0.004
0.002
0.000 0.000
0.006
*
*
*
∆Tinlet -∆Tinlet,critical (K)
0.010
Supercritical
0.004
C 0.002
Subcritical
A 0.002
0.004
0.000 0.000
0.006
0.002
0.004
0.006
q' (MW/m)
q' (MW/m)
(b)
(a)
E Subcritical
-0.05
*
*
∆Tinlet -∆Tinlet,critical (K)
0.00
-0.10
-0.15
-0.20
D
-0.25 0.000
0.002
0.004
0.006
q' (MW/m)
(c)
Figure 8.7: 7.5% oscillation curves for different segments of the SB shown in Figure 8.6. (a) segment A-T; (b) segment T-C; (c) segment D-E; (d) segment E-F; (e) segment F-G
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0.012
0.008
*
∆Tinlet -∆Tinlet,critical (K)
E
0.004
*
Subcritical 0.000
Supercitical -0.004
F 0.000
0.002
0.004
0.006
q' (MW/m)
(d) 0.007
F
*
*
∆Tinlet -∆Tinlet,critical (K)
0.006
0.005
0.004
0.003
0.002
G
0.001
0.000 0.000
0.002
0.004
Supercritical
0.006
q' (MW/m)
(e)
Figure 8.7 (continue): 7.5% oscillation curves for different segments of the SB shown in Figure 8.6. (d) segment E-F; (e) segement F-G
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16
Stability Boundary Boundary of 2nd Eigenvalue
14
Unstable
12
T D'
*
∆Tinlet (K)
B
A
10
8
Stable Riser
6
C D E
F
Channel
0.000
0.002
0.004
0.006
q' (MW/m)
Figure 8.8: Stability boundary and the boundary of the eigenvalues with the second largest real *
part. System pressure Psys = 0.2 MPa. The SB is divided by the separation boundary that divides the parameter space into a region where the boiling boundary is in the riser (above) and a region where the boiling boundary is in the channel (below).
159
0.006
0.000
*
∆Tinlet -∆Tinlet,critical (K)
T
*
∆Tinlet -∆Tinlet,critical (K)
T 0.005
0.004
Subcritical
-0.010 -0.015 -0.020
*
*
0.003
-0.005
0.002
A
-0.025 -0.030
0.001
Supercritical
0.000 0.000
0.002
0.004
-0.035 -0.040 0.000
0.006
C 0.002
C ∆Tinlet -∆Tinlet,critical (K)
0.0001
Supercritical
*
*
∆Tinlet -∆Tinlet,critical (K)
0.0002
D
0.12 0.10
*
0.08
0.0000
-0.0001
Subcritical
*
0.06 0.04
Subcritical
0.02 0.00 0.00407
0.006
(b)
(a)
0.14
0.004
q' (MW/m)
q' (MW/m)
0.00408
0.00409
0.00410
0.00411
-0.0002
-0.0003
-0.0004 0.000
0.00412
q' (MW/m)
E 0.002
0.004
F 0.006
q' (MW/m)
(c)
(d)
Figure 8.9: 7.5% oscillation curves for different segments of the SB shown in Figure 8.8. (a) segment A-T; (b) segment T-C; (c) segment C-D; (d) segment E-F.
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bifurcation is supercritical for the segment A-T, but subcritical for the segment T-C and C-D. The bifurcation along the SB branch along which the boiling boundary is in the channel (segment E-F) moves back and forth from the subcritical to the supercritical case. Dynamics of a natural circulation loop under high pressure is studied using system parameters corresponding to the Dodewaard natural circulation reactor (the Netherlands). Note that neutronics is not included at this stage. Coupled system is studied in the next section. System *
pressure in Figure 8.10 is Psys = 1.07 MPa. Pressure loss coefficient K c ,out at the connection of the riser and channel in the data of Dodewaard reactor is not zero (1.095). Therefore, the branches of the SB corresponding to the cases of boiling boundary in the riser and boiling boundary in the channel are separated by not just one dashed line but two lines. The branch to the left of the dashed lines corresponds to boiling in riser, and the branch to the right of the dashed lines corresponds to boiling in channel. The region in the middle of the two dashed lines corresponds to the special case where the boiling boundary stays at the connection. Since the system pressure is relatively high, flashing in the riser is relatively less intense. Region in the parameter space where boiling (flashing) may start in the riser (to the left of the dashed lines) is much smaller than that of the region where boiling starts in the channel (to the right of the dashed lines). Most of the SB (segments C-D-E) is in the second region. Figure 8.11 shows 7.5% oscillation curves for different *
segments of the SB in ∆Tinlet —( q' - q' critical ) parameter space. Along the segment A-B, which is the SB branch along which boiling starts in the riser, the oscillation curve is in the stable region *
*
for ∆Tinlet > 3.04 (K) and ∆Tinlet < 1.88 (K), and the bifurcation is subcritical. It is in the unstable *
region for 1.88 (K) < ∆Tinlet < 3.04 (K), and the bifurcation is supercritical. For the segment C-D, characteristics of bifurcation rapidly changes from supercritical to subcritical as subcooling is
161
12
At Connection D
10
In Riser
In Channel
* P3
*
∆Tinlet (K)
8
6
P1
4
B 2
Unstable
Stable
A
P2
**
C
E
0 0.0
0.1
0.2
0.3
0.4
0.5
q' (MW/m)
Figure 8.10: Stability boundary calculated for system parameters corresponding to the Dodewaard *
reactor. System pressure Psys = 1.07 MPa. The SB is separated into a branch corresponding to boiling in the riser, and the other one corresponding to boiling in the channel.
162
12
Supercritical Subcritical
Supercritical Subcritical
10
10
8
8
∆Tinlet (K)
∆Tinlet (K)
12
6
4
C 2
A
0 -0.002
0 -0.0010
-0.0008
-0.0006
-0.0004
-0.0002
0.0000
0.0002
0.000
0.001
0.002
(b)
(a)
10
-0.001
q'-q'critical (MW/m)
q'-q'critical (MW/m)
12
6
4
B
2
D
Subcritical
Supercritical
D
∆Tinlet (K)
8
6
4
E
2
0 -0.0003
-0.0002
-0.0001
0.0000
0.0001
0.0002
q'-q'critical (MW/m)
(c)
Figure 8.11: 7.5% oscillation curves for the segments of the SB shown in Figure 8.10. (a) segment A-B; (b) segment C-D; (c) segment D-E.
163
increased. It turns back to supercritical again along the segment D-E and stays supercritical even for very low inlet subcooling. The analyses of stability and bifurcation properties under low and high pressure conditions reveal complexities of the dynamics of the natural circulation boiling/flashing systems.
8.3. Stability and Bifurcation Analyses of the Nuclear-Coupled Model In this section the nuclear coupled model is studied using the moving boundary scheme.
8.3.1. Effect of Nodalization Schemes Coupling among thermal-hydraulics, fuel dynamics and neutronics introduces extra feedbacks, which are crucial to the dynamics of the system. However, accurate evaluations of the feedback signals, the void fraction in the channel, and fuel temperature rely on nodalization scheme. The higher the number of nodes in each single-phase or two-phase region, the more accurate is the estimate of the void fraction and fuel temperature. Large number of nodes on the other hand dramatically increase the number of phase variables, leading to computational inefficiency. Therefore, an analysis of the impact of the number of nodes on the results of the stability analysis is carried out first. Stability boundaries (SBs) are calculated, for two different system pressures, using BIFDD for different nodalization schemes in the so-called inletsubcooling—reactivity parameter space (Figures 8.12) for parameter values that roughly correspond to the Dodewaard natural circulation *
*
BWR [45] [62]. System pressure for these two SBs are Psys = 7.0 MPa and Psys = 0.4 MPa. Here, the x axis, ρ ext , is the external reactivity introduced by control rod movement. The y axis, *
∆Tinlet , is the inlet subcooling (saturation temperature minus inlet temperature). Different regions
164
40
N=1 N=2 N=4 N=8 N = 16
4.5 4.0 3.5
30
∆Tinlet (K)
∆Tinlet (K)
5.0
N=1 N=2 N=4 N=8 N = 16
50
Unstable
20
Stable
10
3.0
Unstable
2.5 2.0 1.5
Stable
1.0 0.5
0
0.0
-0.04
-0.03
-0.02
-0.01
0.00
0.01
0.02
-0.04 -0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 0.04
0.03
ρext
ρext
(b)
(a)
Figure 8.12: SBs for different size nodes in ∆Tinlet — ρ ext parameter space for (a) Psys = 7.0 MPa, *
*
*
(b) Psys = 0.4 MPa. The boiling boundary is in the channel, and different regions are discretized to the same number of nodes characterized by N.
165
along the channel inlet to the riser outlet are discretized using identical number of nodes. Different nodalization schemes are characterized by N, which is the number of nodes in each (single phase and two-phase) region. Figure 8.12(a) for high system pressure case, shows that as the number of nodes increases, the SBs converge to a limiting SB. The results point out that insufficient nodalization may cause large deviation in both the area of stable regions and in the shapes of SBs from those obtained using sufficiently large number of nodes. Compared to the SB of N = 16 case, the one with N = 1 significantly over-estimates the area of the stable region. As N is increased, the SBs tend to converge. Good convergence of SB is achieved in regions of low and high ρ ext for the N = 8 case, but the SB may be over-estimated by as much as 10% in the middle range of ρ ext . For the low system pressure case (0.4 MPa), Figure 8.12(b) shows a much better convergence for relatively smaller number of nodes compared to the high system pressure case. Even the SB for the N = 4 case is very close to that of the converged SB. SBs for the N = 8 and N = 16 cases are too close for most values of ρ ext to be distinguished from each other. Results presented in the rest of the paper were obtained using N = 8.
8.3.2. In-phase and Out-of-phase Oscillations *
Figures 8.13 and 8.14 show SBs (solid lines) in ∆Tinlet — ρ ext space for system pressure of 7.0 MPa and 0.4 MPa, respectively. Nature of instability that results—in-phase or out of phase— as these SBs are crossed, may however vary significantly along different parts of the SB. As shown in part I of this dissertation for forced circulation BWR system, the two rightmost pairs of eigenvalues can be associated with in-phase and out-of-phase oscillations (section 3.1). The relative magnitude of the elements in the corresponding eigenvectors indicate as to which mode—
166
fundamental or first azimuthal—will be dominant in the unstable system as the corresponding eigenvalue crosses the imaginary axis. It was shown that the eigenvalue pair that corresponds to the first azimuthal mode may cross the imaginary axis before the eigenvalue pair that corresponds to the fundamental mode leading to predominantly out-of-phase oscillations. Hence, even for the natural circulation system it is important that boundaries associated with the eigenvalues with the largest and second largest real part be taken into stability consideration. Relative magnitude of the elements in the eigenvector can then be used to identify whether it is the in-phase or the out-ofphase mode that will be dominant in the oscillations when the system becomes unstable as the SB is crossed. Figures 8.13 and 8.14 show two boundaries associated with the two pairs of complex conjugate eigenvalues with largest real parts. One of these pairs has zero real part along the boundary with squares, while the second pair has zero real part along the boundary with triangles. For low system pressure (Figure 8.14) the two boundaries do not intersect, and hence the entire SB is composed of the boundary associated with one pair of eigenvalue—in this case the one with *
squares. However, Figure 8.13, for Psys = 7.0 MPa case, shows that the two boundaries intersect *
at point X ( ρ ext = -0.0335, ∆Tinlet = 8.65 K). Hence, the SB (the solid line) is composed of a segment of the boundary with squares and another segment with triangles. Characteristic difference of oscillations associated with the two rightmost pairs of complex conjugate eigenvalues is further illustrated by analyzing the eigenvalues and the corresponding *
eigenvectors at four points on the two boundaries. Point 1 ( ρ ext = -0.0278, ∆Tinlet = 20.0 K) and *
*
point 2 ( ρ ext = -0.0306, ∆Tinlet = 20.0 K) are on the stability map of the system with Psys = 7.0 *
*
MPa; point 3 ( ρ ext = 0.0124, ∆Tinlet = 2.0 K) and point 4 ( ρ ext = 0.0179, ∆Tinlet = 2.0 K) are on
167
55
Stability Boundary Boundary of 2nd largest real part
50
14
45
35
Unstable
B
∆Tinlet (K)
∆Tinlet (K)
40 IV
30 25 2 1
20
II
Stable
* *
I
10
See inset
X
10 8
C -0.02
I
6
III
2
A -0.03
Stable
IV
A
III
0 -0.04
X
Unstable
4
15
5
II
12
-0.01
0.00
0.01
0 -0.040
0.02
-0.038
-0.036
-0.034
-0.032
-0.030
ρext
ρext
(b)
(a)
*
*
Figure 8.13: Boundaries in ∆Tinlet — ρ ext space for Psys = 7.0 MPa. The solid line is the stability boundary, and the dotted line is the boundary on which the second largest real part is zero. The line with open squares is the boundary on which the real part of the in-phase eigenvalue is zero, and the line with open triangles is the boundary on which the real part of the out-of-phase eigenvalue is zero.
168
7
Stability boundary Boundary of the 2nd largest real part
6 II
∆Tinlet (K)
5
E
4
Unstable III
D 3
Stable
3
I
2
*
*
4
1 F 0 -0.04
-0.03
-0.02
-0.01
0.00
0.01
0.02
0.03
ρext *
*
Figure 8.14: Boundaries in ∆Tinlet — ρ ext space for Psys = 0.4 MPa. The solid line is the stability boundary (SB), and the dotted line is the boundary on which the second largest real part is zero. The two boundaries do not cross. The SB is the boundary on which the real part of the in-phase eigenvalue is zero, and the dotted line is the boundary on which the real part of the out-of-phase eigenvalue is zero.
169
*
the stability map of the system with Psys = 0.4 MPa. Table 8.1 lists the pure imaginary eigenvalues of the Jacobian E imag and magnitudes of the elements corresponding to n 0 (t ) and
n1 (t ) , EV n0 and EV n1 , in the corresponding eigenvectors at these points. Points 1 and 3 (2 and 4) are chosen to analyze the nature of oscillations associated with the eigenvalues that have a zero real part on the boundary with squares (triangles). For points 1 and 3, the magnitude of the element of the eigenvector corresponding to
n 0 (t ) is much smaller than that for n1 (t ) , suggesting large amplitude oscillations in n 0 (t ) , but much smaller amplitude oscillations for n1 (t ) at these two points. The pure imaginary eigenvalues at these two points and along the boundary with squares, thus, are called in-phase or fundamental mode eigenvlaues. For points 2 and 4, the magnitude of the element of the eigenvector corresponding to n1 (t ) is large, while that of n0 (t ) is very small. These eigenvalues, and those corresponding to the boundaries with triangles, are hence called out-of-phase or azimuthal mode eigenvlaues. Therefore, a characteristic change in the nature of oscillation along the SB (solid line) must be expected due to eigenvalue crossing at point X (Figure 8.13). To the left of point X, segment A-X is the SB due to out-of-phase oscillations. To the right of point X, segment X-B-C is the SB due to in-phase oscillations. Parameter space in Figure 8.13 is divided by these boundaries of fundamental and first azimuthal mode eigenvalues into four regions. In region I, all eigenvalues have negative real parts and hence the system is stable. Small perturbation will cause decreasing amplitude oscillations. In region II, the fundamental mode eigenvalue has positive real part and the first azimuthal mode eigenvalue has negative real part. The system is hence unstable, and small perturbations will lead to oscillations dominated by in-phase component. In region III, the first azimuthal mode
170
Table 8.1: Critical eigenvalues E imag and elements EV n0 and EV n1 (corresponding to n0 (t ) and
n1 (t ) ) respectively in the eigenvector at four operating points shown in Figures 8.13 and 8.14.
Point 1
Point 2
Point 3
Point 4
Eimag
±2.306 i
±1.709 i
±3.030 i
±4.079 i
EVn0
0.156
0.170×10-11
0.00972
0.110×10-13
EVn1
0.943×10-12
0.0595
0.324×10-13
0.00530
171
eigenvalue has positive real part and the fundamental mode eigenvalue has negative real part. The system is hence unstable, and small perturbations will lead to oscillations dominated by iout-ofphase component. In the last region (IV), both eigenvalues have positive real part, and hence the system is unstable and the oscillations will therefore be a combination of in-phase and out-ofphase modes. For low system pressure, instead of four regions, three regions of the parameter space are identified. The system is stable in region I. Growing oscillations in region II will be dominated by the fundamental mode; and in unstable region III both modes will contribute about equally to the growing oscillations (Figure 8.14).
8.3.3 Bifurcation Analyses In addition to in-phase and out-of-phase, the oscillations in a natural circulation BWR can also be characterized by whether they evolve to stable periodic oscillations or to unstable growing amplitude oscillations. As discussed in section 2.3, PAH bifurcation theorem provides us with conditions that lead to one or the other. According to PAH-B theorem, if the periodic solution (oscillations) exists in the unstable region, it is stable and the bifurcation is supercritical; if it exists in the stable region, the periodic solution is unstable and the bifurcation is subcritical. Nature of PAH-B (supercritical or subcritical) along the SB will again be represented by fixed amplitude oscillation curves. Figures 8.15 and 8.16 show 7.5% oscillation curves for different *
segments of the SBs shown in Figures 8.13 and 8.14 in ∆Tinlet —( ρ ext – ρext,critical ) parameter space. Here
ρext,critical
is the reactivity on the SB. The amplitude of the limit cycle along these curves is
about 7.5% of the magnitude of the corresponding elements in the eigenvector. In Figure 8.15(a), for the out-of-phase segment A-X, the oscillation curve is in the unstable region, indicating a
172
Subcritical
Supercritical
30
30
∆Tinlet (K)
∆Tinlet (K)
B
25
25 20 15 10
Subcritical
Supercritical
35
35
X
20 15 10
X
5
5 A
0 -0.0012
-0.0008
0 -0.0004
-0.03
0.0000
-0.02
-0.01
0.00
0.01
ρext - ρext,critical
ρext - ρext,critical
(b)
(a) Subcritical Supercritical 35 B
30
∆Tinlet (K)
25 20 15 10 5 C 0 0.000
0.005
0.010
0.015
0.020
0.025
0.030
ρext - ρext,critical
(c)
Figure 8.15: 7.5% oscillation curve along the SB shown in Figure 8.13. (a) segment A-X. (b) segment X-B. (c) segment B-C.
173
0.02
Supercrtical 4.0
Subcritical
Subcritical
3.0
∆Tinlet (K)
3.0
∆Tinlet (K)
E
3.5
D
3.5
2.5 2.0 1.5
2.5 2.0 1.5
1.0
1.0
0.5
0.5
0.0 -0.020
Supercritical
4.0
E
-0.015
-0.010
-0.005
0.000
0.005
0.0 -0.01
F
0.00
0.01
ρext - ρext,critical
ρext - ρext,critical
(a)
(b)
0.02
Figure 8.16: 7.5% oscillation curve along the SB shown in Figure 8.14. (a) segment D-E. (b) segment E-F.
174
0.03
supercritical PAH-B along the SB. From the out-of-phase segment (A-X) of the SB to in-phase *
segment X-B, oscillation curve has a jump from the unstable side to the stable side. For ∆Tinlet < 28.91 K, the oscillation curve of segment X-B is in the stable region, and type of PAH-B is *
subcritical. For 28.91 < ∆Tinlet < 31.42 K (point B), the oscillation curve returns to the unstable region, and consequently type of PAH-B along this segment of the SB is supercritical. For segment B-C, Figure 8.15(c) shows that oscillation curve along the B-C branch is always in the unstable region. Type of PAH-B along this segment is hence supercritical. Similar back-and-forth transitions from sub- to supercritical bifurcations are observed for the low system pressure case. In Figure 8.16(a), the type of bifurcation is supercritical along the D-E branch of the SB. Along the *
E-F branch of the SB (Figure 8.16(b)), when ∆Tinlet > 2.762 K, the bifurcation is supercritical; *
while for ∆Tinlet < 2.762 K, it is subcritical. *
For the out-of-phase segment A-X (Figure 8.15(a)) and low ∆Tinlet region of the in-phase segment B-C (Figure 8.15(c)), deviations of ρ ext from its critical value required for the 7.5% *
amplitude, are much smaller than those for the segment X-B and high ∆Tinlet region of the segment B-C, indicating that much larger amplitude oscillations (in-phase or out-of-phase) may result for the same deviation along the SB in the low subcooling region than those along the SB in the high subcooling region. Large amplitude oscillations at lower inlet subcooling are due to longer two-phase regions in the channel and riser. Similar trend can be found in the low system pressure case (Figure 8.16). The change in characteristic of bifurcation (supercritical or subcritical) and change in characteristic of oscillations (in-phase or out-of-phase) along the SB, thus, reveal complexity of the dynamic behavior of the coupled natural circulation BWR system.
175
8.3.4 Sensitivity Analyses Effect of operating parameters other than ∆Tinlet and ρ ext on the SBs is investigated. Since *
behavior of two-phase natural circulation systems under high system pressure conditions (close to 7.0 MPa) is relatively better understood, sensitivity analysis of operating parameters is focused on relatively low system pressure. To study the effect of system pressure, SBs are plotted in ∆Tinlet — ρ ext parameter space *
*
*
for five different values of Psys (Figure 8.17(a)). Impact of a change in Psys on systems with *
relatively large pressure ( Psys = 1.5, 1.0 MPa) is different from that on systems at low pressure (
Psys
*
= 0.4, 0.3, 0.2 MPa). Stable region decreases as
Psys
*
decreases if the system pressure is *
P relatively high, especially for small values of ρ ext . Thus, a drop in sys has a destabilizing effect under these conditions. If the system pressure is relatively low, however, a drop in
Psys
*
causes the
stable region to increase. Therefore, instead of destabilizing, it has a stabilizing effect on the dynamics of the system. Comparing these results with those of pure thermal-hydraulics model, where a drop in system pressure is a destabilizing factor, this new stabilizing effect can be attributed to the coupling between system thermal-hydraulics and neutron kinetics. At low system pressure, without neutronics coupling, small perturbation can quickly grow due to two-phase flow instability. In the coupled system, however, void fraction feedback acts as a stabilizing force. Perturbations in void fraction lead to negative feedback of reactivity, which suppresses thermalhydraulic perturbations. *
Figure 8.17(b) shows the effect of riser length Lr , for Psys = 0.4 MPa case. In natural
176
∆Tinlet (K)
OP
Lr = 0.30 m Lr = 3.06 m Lr = 5.00 m
OP
6 5
∆Tinlet (K)
7 6
7
Pexit = 0.2 MPa Pexit = 0.3 MPa Pexit = 0.4 MPa Pexit = 1.0 MPa Pexit = 1.5 MPa
8
5 4 3
3 2
2
1
1
0
0 -0.05 -0.04 -0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 0.04
Unstable
4
Stable
-0.04
-0.03
-0.02
-0.01
0.02
0.03
(b)
(a)
kr,out = 0.00 kr,out = 0.50 kr,out = 10.0
5.0 OP
4.0 3.5
∆Tinlet (K)
0.01
ρext
ρext
4.5
0.00
Unstable
3.0 2.5
Stable
2.0 1.5 1.0 0.5 -0.04
-0.03
-0.02
-0.01
0.00
0.01
0.02
0.03
ρext
(c) Figure 8.17: Sensitivity analysis results for different operating parameters. Out-of-phase oscillation SBs are indicated by OP. A change in slope of the SB indicates transition to in-phase. (a) System pressure; (b) Riser length; (c) Pressure loss coefficient at riser exit
177
kc,out = 0.00 kc,out = 1.10 kc,out = 10.0
5.0 4.5 4.0
∆Tinlet (K)
3.5 3.0
Unstable
2.5 2.0
Stable
1.5 1.0 0.5 0.0
-0.04 -0.03 -0.02 -0.01
0.00
0.01
0.02
0.03
ρext
(d)
kc,in = 0.00 kc,in = 1.73 kc,in = 10.0
6.0 5.5 5.0
OP
∆Tinlet (K)
4.5 4.0
Unstable
3.5 3.0 2.5 2.0 1.5
Stable
1.0 0.5 0.0 -0.04 -0.03 -0.02 -0.01 0.00
0.01
0.02
0.03
0.04
ρext
(e)
Figure 8.17 (continue): Sensitivity analysis results for different operating parameters. Out-ofphase oscillation SBs are indicated by OP. A change in slope of the SB indicates transition to inphase. (d) Pressure loss coefficient at channel exit; (e) Pressure loss coefficient at channel inlet
178
circulation system, increasing Lr increases the driving force leading to higher flow rate. Therefore, it has a strong stabilizing effect. However, Figure 8.17(b) also shows that out-of-phase oscillations are possible at low reactivity for long risers. Effect of pressure loss coefficient at riser outlet K r ,out is not as strong as those for Psys
*
and Lr (Figure 8.17(c)), but SBs indicate the possibility of out-of-phase oscillations for low reactivity if K r ,out is high. Figure 8.17(d) shows the effect of an increase in pressure loss at the outlet of the core. Increasing K c ,out generally tends to cause a loss of stability. The last operating parameter studied is the pressure loss coefficient at the core inlet, K c ,in . As shown in Figure 8.17(e), an increase in
K c ,in has a strong stabilizing effect.
179
Chapter 9 Numerical Simulations of Natural-Circulation Model
In this chapter, results of numerical simulations for a number of representative operating points in the parameter space are presented. These results are compared with the results of stability and bifurcation analyses presented in Chapter 8. Results for only the thermal hydraulic model are presented first in Section 9.1, followed by those for the nuclear-coupled thermal hydraulic system in Section 9.2.
9.1. Thermal-Hydraulic Model Simulation (numerical integration) of the complete nonlinear thermal hydraulic system of a natural circulation two-phase flow loop under low pressure is carried out first. The parameters correspond to the operating points P1 through P5 in Figure 8.1 (a). System pressure and heat *
fluxes for the operating points are fixed at Psys = 0.2 (MPa) and q ' = 6.0 (kW/m). A perturbation ( vinlet : vinlet , ss + 0.001) is introduced in the system at t = 0, where vinlet , ss is the steady-state value of the inlet velocity. Results are presented in Figures 9.1 through 9.5. [Realizing that the perturbations introduced are very small, the effect of larger amplitude perturbations are also studied, and those results are presented after the results for small amplitude perturbations.] *
For the point P1, the inlet subcooling is ∆Tinlet = 35.0 (K), which is high enough to prevent boiling in the channel and the riser. Figure 9.1(a) shows the straight line at the riser exit location, indicating that the boiling boundary is not in the heated channel or the riser, and hence that there is no boiling in this case. Figure 9.1(b) shows the time evolution of the inlet velocity. The single
180
phase flow corresponding to this operating point is seen to be quite stable, and the initial *
perturbation decays rapidly. Figure 9.2, which is for inlet subcooling ∆Tinlet = 31.496 (K) case (operating point P2 in Figure 8.1(a)), shows an example of the intermittent wave. The corresponding steady state boiling boundary and inlet velocity are µ ss = 4.367 and vinlet = 2.157. The boiling boundary is very close to the riser outlet (4.38). Figure 9.2(a) showing the evolution of the boiling boundary shows that the system at this operating point is unstable, and the boiling boundary oscillation amplitude increases rapidly in the beginning. At time t = 8.522, it reaches the riser outlet (4.38), and the flow in the channel and riser stays single-phase for a while before boiling starts again, starting at the top of the riser. The boiling boundary then moves downward. This leads to a series of intermittent waves in which boiling starts at the riser exit, followed by a downward movement of the boiling boundary. After reaching a minimum, the boiling boundary again travels upward and reaches the riser exit. For a brief interval, the flow is single phase in both the channel and the riser. The non-dimensional period of the intermittent oscillations is about 2.2, which is close to the time it takes for the fluid to travel from the channel inlet to the riser exit. Figure 9.2(b) shows the corresponding intermittent oscillations in inlet velocity. The inlet velocity and the boiling boundary oscillate in-phase. As the inlet subcooling is further decreased, the intermittent waves turn into sinusoidal *
waves. Figure 9.3 shows growing amplitude sinusoidal oscillations for ∆Tinlet = 15.0 K case (corresponding to point P3 in Figure 8.1(a)). The amplitudes of the boiling boundary and inlet velocity increase, clearly indicating that the operating point is unstable. However, before a limit cycle is reached—if there is one—the numerical algorithm fails to converge at about t = 5.0. These are high frequency oscillations where the period of the oscillations is about 0.5. Clearly, the compressibility effect is playing a role in these oscillations.
181
Boiling Boundary
5.5
5.0
4.5
4.0
3.5
3.0 0
2
4
6
8
10
12
14
16
18
20
Time
(a) 2.160
2.158
vinlet
2.156
2.154
2.152
2.150 0
2
4
6
8
10
12
14
16
18
20
Time
(b) Figure 9.1: Results of numerical simulation for operating parameters corresponding to point P1 in *
*
Figure 8.1(a). For this point, Psys = 0.2 MPa, q ' = 6.0 kW/m, and ∆Tinlet = 35.0 (K). (a) Nondimensional boiling boundary; (b) Non-dimensional inlet velocity.
182
Boiling Boundary
4.38
4.37
4.36
4.35
4.34
4.33
4.32 0
10
20
30
40
50
Time
(a) 2.170
2.165
vinlet
2.160
2.155
2.150
2.145 0
10
20
30
40
50
Time
(b) Figure 9.2: Results of numerical simulation for operating parameters corresponding to point P2 in *
*
Figure 8.1(a). For this point, Psys = 0.2 MPa, q ' = 6.0 kW/m, and ∆Tinlet = 31.496 (K). (a) Nondimensional boiling boundary; (b) Non-dimensional inlet velocity
183
Boiling Boundary
2.485
2.480
2.475
2.470
2.465 0
1
2
3
4
5
Time
(a) 5.52
5.50
vinlet
5.48
5.46
5.44
5.42
5.40 0
1
2
3
4
5
Time
(b) Figure 9.3: Results of numerical simulation for operating parameters corresponding to point P3 in *
*
Figure 8.1(a). For this point, Psys = 0.2 MPa, q ' = 6.0 kW/m, and ∆Tinlet = 15.0 (K). (a) Nondimensional boiling boundary; (b) Non-dimensional inlet velocity. 184
Boiling Boundary
0.80162
0.80160
0.80158
0.80156
0.80154
0.80152
0.80150 0
10
20
30
40
50
40
50
Time
(a) 6.5266 6.5264 6.5262 6.5260
vinlet
6.5258 6.5256 6.5254 6.5252 6.5250 6.5248 6.5246 0
10
20
30
Time
(b) Figure 9.4: Results of numerical simulation for operating parameters corresponding to point P4 in *
*
Figure 8.1(a). For this point, Psys = 0.2 MPa, q ' = 6.0 kW/m, and ∆Tinlet = 8.8 (K). (a) Nondimensional boiling boundary; (b) Non-dimensional inlet velocity.
185
Boiling Boundary
0.38036
0.38034
0.38032
0.38030
0.38028 0
10
20
30
40
50
40
50
Time
(a) 5.2568 5.2566 5.2564
vinlet
5.2562 5.2560 5.2558 5.2556 5.2554 5.2552 0
10
20
30
Time
(b) Figure 9.5: Results of numerical simulation for operating parameters corresponding to point P5 in *
*
Figure 8.1(a). For this point, Psys = 0.2 MPa, q ' = 6.0 kW/m, and ∆Tinlet = 5.0 (K). (a) Nondimensional boiling boundary; (b) Non-dimensional inlet velocity. 186
Results for numerical simulation for operating parameters corresponding to Point P4 (inlet *
subcooling, ∆Tinlet = 8.8(K)) are shown in Figure 9.4. Inlet subcooling is now so small that the boiling boundary at steady state ( µ ss = 0.802) is in the channel for this case. Operating point P4 is clearly unstable, but the rise in amplitude per period is much smaller than that for point P3. Figure 9.5 shows results of simulations for operating parameters corresponding to point P5 (inlet *
subcooling, ∆Tinlet = 5.0 (K)). The steady state boiling boundary is for this case even lower in the channel ( µ ss = 0.380), and the simulations show the operating point to be stable with single and two-phase natural circulation flows. The evolutions of the boiling boundary and inlet velocity oscillations as the inlet subcooling is decreased qualitatively agree with the experimental data obtained by Inada et al. [42]. Results of numerical simulations reported above are obtained following a rather small perturbation in the inlet velocity (about the steady state value). Additional results obtained by numerically integrating the set of governing ODEs for different set of parameter values and using different perturbation amplitudes are reported below. They are also compared with the results of stability analysis. Figures 9.6 through 9.8 show evolution of the boiling boundary and inlet velocity for three *
operating points—P1, P2, and P3—in the ∆Tinlet — q ' parameter space shown in Figure 8.10. The set of operating parameters correspond to those of the Dodewaard reactor, and system pressure is *
Psys = 1.07 (MPa). [Note that neutronics is not included at this stage.] Inlet subcooling and heat *
flux ( ∆Tinlet , q ' ) for points P1, P2 and P3 are given by (4.0, 379.0), (4.0, 379.9), and (8.5, 256.95), respectively. Points P1 and P2 are respectively on the stable and unstable side of the SB in a region where the bifurcation is supercritical. For point P1, a perturbation is introduced at t = 0
187
( vinlet : vinlet , ss → vinlet , ss + 0.01). Figures 9.6(a) and (b) show time evolutions of the boiling boundary and inlet velocity. As expected, the system is stable at the point P1. For point P2, the same initial perturbation is introduced. Figure 9.7 shows that the system evolves to a stable limit cycle with saturated oscillation amplitude. Point P3 is on the stable side of the SB in a region where bifurcation is subcritical. A quite large perturbation ( vinlet : vinlet , ss → vinlet , ss + 0.2, a m ,i :
a m ,i , ss + 0.2, i = 1,…,8) is introduced. Figure 9.8 shows the time evolutions for parameter values corresponding to point P3. Even though point P3 is in the stable region, since the perturbation given is large and the bifurcation at point 3 is subcritical, the perturbations grow, being repelled by the unstable limit cycle around the stable fixed point.
9.2. Nuclear-Coupled Thermal-Hydraulics Model of A Natural Circulation BWR Numerical simulations are carried out to gain further insight into the dynamics of the natural circulation, nuclear-coupled thermal hydraulic system as well as to compare the results obtained using numerical simulations with those obtained earlier—for stability and bifurcation analysis—using BIFDD. Results of numerical simulations are given here for eight points on the stability maps of high and low pressure systems. These points—labeled a through h—are shown in Figures 9.9(a) and (b). Table 2 lists operating parameters as well as characteristics of bifurcation (sub- or supercritical) and the nature of oscillations (in-phase or out-of-phase) predicted by BIFDD for these operating points. Points a and b are close to the out-of-phase segment of the stability boundary (Figure 10a).
188
Boiling Boundary
0.2070 0.2068 0.2066 0.2064 0.2062 0.2060 0.2058 0.2056 0
50
100
150
200
150
200
Time
(a) 1.920
vinlet
1.915
1.910
1.905
1.900 0
50
100
Time
(b) Figure 9.6: Results of numerical simulation for operating parameters corresponding to point P1 in *
*
Figure 8.10. For this point, Psys = 1.07 MPa, q ' = 379.0 kW/m, and ∆Tinlet = 4.0 (K). (a) Nondimensional boiling boundary; (b) Non-dimensional inlet velocity.
189
0.2070
Boiling Boundary
0.2065
0.2060
0.2055
0.2050
0.2045
0.2040 0
20
40
60
80
100
80
100
Time
(a) 1.920
1.915
vinlet
1.910
1.905
1.900
1.895
1.890 0
20
40
60
Time
(b) Figure 9.7: Results of numerical simulation for operating parameters corresponding to point P2 in *
*
Figure 8.10. For this point, Psys = 1.07 MPa, q ' = 379.9 kW/m, and ∆Tinlet = 4.0 (K). (a) Nondimensional boiling boundary; (b) Non-dimensional inlet velocity.
190
Boiling Boundary
0.80
0.78
0.76
0.74
0.72
0.70 0
20
40
60
80
100
120
140
160
180
200
220
140
160
180
200
220
Time
(a) 2.50 2.45 2.40
vinlet
2.35 2.30 2.25 2.20 2.15 2.10 0
20
40
60
80
100
120
Time
(b) Figure 9.8: Results of numerical simulation for operating parameters corresponding to point P3 in *
*
Figure 8.10. For this point, Psys = 1.07 MPa, q ' = 256.95 kW/m, and ∆Tinlet = 8.5 (K). (a) Nondimensional boiling boundary; (b) Non-dimensional inlet velocity. 191
Point a is in the unstable region, while point b is in the stable region. Bifurcation along this segment of the SB is supercritical. For point a, initial perturbation is given by ( n1 : n1, ss → n1, ss + 0.0001). Figure 9.10 shows results of numerical simulations for the expansion parameters of the fundamental and first azimuthal modes n0 (t ) and n1 (t ) . As expected, stable amplitude limit cycle of n1 (t ) and very small amplitude of n 0 (t ) result. For point b, the perturbations of n0 (t ) and n1 (t ) are given by ( n0 : n0, ss → n0, ss + 0.0001, n1 : n1, ss → n1, ss + 0.0001). Figure 9.11 shows decreasing amplitude oscillations for both n 0 (t ) and n1 (t ) . Points c, d and e are close to the in-phase segment of the SB. Bifurcation at point c is subcritical; while those for points d and e are supercritical. Perturbations for the point c is ( n0 : n0, ss → n0, ss + 0.003, n1 : n1, ss → n1, ss + 0.001). Figure 9.12, corresponding to point c, shows increasing amplitude of n 0 (t ) (and initially decaying n1 (t ) ) for large perturbation even though point c is in the stable region, consistent with the subcritical nature of the bifurcation. For point d (in the stable region) and e (in the unstable region), perturbations lead to decreasing amplitude oscillations (point d) and stable limit cycle oscillations (point e) for
n 0 (t ) while initially decreasing amplitude oscillations result for n1 (t ) in both cases (Figures 9.13 and 9.14). For the case of low system pressure, Figures 9.15, 9.16, and 9.17 show that oscillations at points f, g, and h are in-phase as suggested by the stability analysis. Large perturbation at point f, which is in the stable region, causes increasing amplitude oscillation consistent with the subcritical bifurcation there. Perturbations at point g (unstable region), and h (stable region) cause limit cycle and decreasing amplitude oscillations, respectively, for n 0 (t ) . Numerical simulations of systems with high and low system pressures agree with predictions of BIFDD satisfactorily.
192
The numerical simulations of nuclear coupled natural circulation BWR model show characteristic complexities of oscillations similar to the forced circulation BWR model. Depending on operating conditions, they both can generate in-phase or out-of-phase oscillations as well as supercritical or subcritical bifurcations. However, two differences between the forced and natural circulation systems can be pointed out. First; in the forced circulation case, the out-ofphase oscillations were observed (for the range of parameter values studied) only when the inhomogeneity of core loading is considered. If the core loading is homogeneous, the oscillations are in-phase for typical BWR operating parameters. However, for the natural circulation model, even if the core loading is homogeneous, for the set of operating parameters corresponding to the Dodewaard reactor, out-of-phase oscillations can still be generated in certain areas of the operating parameter space. This may suggest higher tendency towards out-of-phase instability for the natural circulation BWRs. Second; system pressure for forced circulation BWR is generally very high. Therefore it shows only trivial effects on the dynamics of the system. However, pressure is found to play a very important role in determining the characteristics of oscillations for natural circulation BWRs. The system pressure impacts decay ratios, amplitudes, frequencies, and phase and bifurcation properties. The differences are of importance in safety considerations for the natural circulation BWR designs.
193
35
Pexit = 7.0 MPa
30
∆Tinlet (K)
25
c
d e
**
*
20
Unstable
15
Stable
10
a b
**
5
OP
0 -0.04
-0.03
-0.02
-0.01
0.00
0.01
0.02
ρext (a)
Pexit = 0.4 MPa
g h
4.0
**
3.5
∆Tinlet (K)
3.0
Unstable
2.5
f
Stable
2.0
*
1.5 1.0 0.5 -0.04
-0.03
-0.02
-0.01
0.00
0.01
0.02
0.03
ρext (b) Figure 9.9: Operating points in the inlet subcooling—reactivity parameter space for numerical *
*
simulations of the nuclear-coupled model. (a) Psys = 7.0 (MPa); (b) Psys = 0.4 (MPa).
194
Table 9.1: Operating points and characteristics of oscillations and PAH-B predicted by BIFDD *
Psys (MPa)
ρ ext
*
∆Tinlet (K)
Characteristics Characteristics of oscillations
of PAH-B
Point a
7.0
-0.03413
6.0
Out-of-phase
Supercritical
Point b
7.0
-0.03400
6.0
Out-of-phase
Supercritical
Point c
7.0
-0.02781
20.0
In-phase
Subcritical
Point d
7.0
-0.00840
20.0
In-phase
Supercritical
Point e
7.0
-0.00823
20.0
In-phase
Supercritical
Point f
0.4
0.01237
2.0
In-phase
Subcritical
Point g
0.4
-0.02570
3.8
In-phase
Supercritical
Point h
0.4
-0.02540
3.8
In-phase
Supercritical
195
n0-n0,ss
3.0x10
-6
2.0x10
-6
1.0x10
-6
0.0 -1.0x10
-6
-2.0x10
-6
-3.0x10
-6
0
20
40
60
80
100
120
140
160
100
120
140
160
Time
n1-n1,ss
(a)
1.0x10
-4
8.0x10
-5
6.0x10
-5
4.0x10
-5
2.0x10
-5
0.0 -2.0x10
-5
0
20
40
60
80
Time
(b)
Figure 9.10: Results of numerical integration for point a in Figure 9.9(a). Perturbation at t = 0 is given by ( n1 : n1, ss → n1, ss + 0.0001). (a) time evolution of n 0 (t ) ; (b) time evolution of n1 (t ) .
196
n0-n0,ss
1.0x10
-4
8.0x10
-5
6.0x10
-5
4.0x10
-5
2.0x10
-5
0.0 0
50
100
150
200
250
300
200
250
300
Time
n1-n1,ss
(a)
1.0x10
-4
8.0x10
-5
6.0x10
-5
4.0x10
-5
2.0x10
-5
0.0 -2.0x10
-5
0
50
100
150
Time
(b)
Figure 9.11: Results of numerical integration for point b in Figure 9.9(a). Perturbations at t = 0 are given by ( n0 : n0, ss → n0, ss + 0.0001, n1 : n1, ss → n1, ss + 0.0001). (a) time evolution of n 0 (t ) ; (b) time evolution of n1 (t ) .
197
n0-n0,ss
4.0x10
-3
3.0x10
-3
2.0x10
-3
1.0x10
-3
0.0 -1.0x10
-3
-2.0x10
-3
-3.0x10
-3
0
100
200
300
400
300
400
Time
n1-n1,ss
(a)
1.0x10
-3
8.0x10
-4
6.0x10
-4
4.0x10
-4
2.0x10
-4
0.0 0
100
200
Time
(b)
Figure 9.12: Results of numerical integration for point c in Figure 9.9(a). Perturbations at t = 0 are given by ( n0 : n0, ss → n0, ss + 0.003, n1 : n1, ss → n1, ss + 0.001). (a) time evolution of n 0 (t ) ; (b) time evolution of n1 (t ) .
198
n0-n0,ss
1.2x10
-3
1.0x10
-3
8.0x10
-4
6.0x10
-4
4.0x10
-4
2.0x10
-4
0.0 -2.0x10
-4
-4.0x10
-4
-6.0x10
-4
0
100
200
300
400
300
400
Time
n1-n1,ss
(a)
1.0x10
-3
8.0x10
-4
6.0x10
-4
4.0x10
-4
2.0x10
-4
0.0 0
100
200
Time
(b)
Figure 9.13: Results of numerical integration for point d in Figure 9.9(a). Perturbations at t = 0 are given by ( n0 : n0, ss → n0, ss + 0.001, n1 : n1, ss → n1, ss + 0.001). (a) time evolution of n 0 (t ) ; (b) time evolution of n1 (t ) .
199
n0-n0,ss
1.2x10
-4
1.0x10
-4
8.0x10
-5
6.0x10
-5
4.0x10
-5
2.0x10
-5
0.0 -2.0x10
-5
-4.0x10
-5
-6.0x10
-5
-8.0x10
-5
0
100
200
300
400
300
400
Time
(a) 1.0x10
-9
-10
8.0x10
-10
6.0x10
-10
4.0x10
n1-n1,ss
-10
2.0x10
0.0 -10
-2.0x10
-10
-4.0x10
-10
-6.0x10
-10
-8.0x10
-1.0x10
-9
0
100
200
Time
(b) Figure 9.14: Results of numerical integration for point e in Figure 9.9(a). Perturbation at t = 0 is given by ( n0 : n0, ss → n0, ss + 0.0001). (a) time evolution of n 0 (t ) ; (b) time evolution of n1 (t ) .
200
n0-n0,ss
3.0x10
-2
2.5x10
-2
2.0x10
-2
1.5x10
-2
1.0x10
-2
5.0x10
-3
0.0 -5.0x10
-3
-1.0x10
-2
0
20
40
60
80
100
60
80
100
Time
n1-n1,ss
(a) 1.0x10
-10
8.0x10
-11
6.0x10
-11
4.0x10
-11
2.0x10
-11
0.0 -2.0x10
-11
-4.0x10
-11
-6.0x10
-11
-8.0x10
-11
-1.0x10
-10
0
20
40
Time
(b) Figure 9.15: Results of numerical integration for point f in Figure 9.9(b). Perturbation at t = 0 is given by ( n0 : n0, ss → n0, ss + 0.002). (a) time evolution of n 0 (t ) ; (b) time evolution of n1 (t ) .
201
n0-n0,ss
1.2x10
-4
1.0x10
-4
8.0x10
-5
6.0x10
-5
4.0x10
-5
2.0x10
-5
0.0 -2.0x10
-5
-4.0x10
-5
-6.0x10
-5
0
50
100
150
200
150
200
Time
n1-n1,ss
(a)
1.0x10
-4
8.0x10
-5
6.0x10
-5
4.0x10
-5
2.0x10
-5
0.0 0
50
100
Time
(b) Figure 9.16: Results of numerical integration for point g in Figure 9.9(b). Perturbations at t = 0 are given by ( n0 : n0, ss → n0, ss + 0.0001, n1 : n1, ss → n1, ss + 0.0001). (a) time evolution of n 0 (t ) ; (b) time evolution of n1 (t ) .
202
n0-n0,ss
1.2x10
-3
1.0x10
-3
8.0x10
-4
6.0x10
-4
4.0x10
-4
2.0x10
-4
0.0 -2.0x10
-4
-4.0x10
-4
0
100
200
300
400
500
400
500
Time
(a)
-3
1.0x10
-4
n1-n1,ss
8.0x10
-4
6.0x10
-4
4.0x10
-4
2.0x10
0.0 0
100
200
300
Time
(b) Figure 9.17: Results of numerical integration for point h in Figure 9.9(b). Perturbations at t = 0 are given by ( n0 : n0, ss → n0, ss + 0.001, n1 : n1, ss → n1, ss + 0.001). (a) time evolution of n 0 (t ) ; (b) time evolution of n1 (t ) .
203
Chapter 10 Summary, Conclusions and Future Work of Part II
This chapter summarizes development of the model for natural circulation BWR and discusses results of stability and bifurcation analyses and numerical simulations. Potential areas for future work are also discussed.
10.1. Summary of Natural Circulation BWR Model A new nuclear-coupled reduced order model is developed to simulate instabilities in natural circulation BWRs under either low or high system pressures. Thermal-hydraulic system of natural circulation BWR is simplified to two, parallel, heated channels with common inlet (lower plenum), two risers above the channels with common outlet (upper plenum), and a downcomer. Fuel temperature dynamics and neutronics of the model do not differ from those of the forcedcirculation model [30]. A key to catch the essence of flashing-induced instabilities when system pressure is low is to appropriately model dependence of water saturation enthalpy and local pressure. The model assumes a linear relation between the two. Homogeneous Equilibrium Model (HEM) is chosen to model two-phase flow dynamics due to its simplicity and computational efficiency. Two different nodalization schemes are applied. In the first, “moving boundaries,” that separate nodes,, move in proportion to the movement of the boiling boundary. This nodalization scheme has the advantage of simplicity for programming purposes, and is shown to be necessary for applications such as stability and bifurcation analyses. However, this nodalization scheme fails to simulate the cases in which the boiling boundary may cross from the channel to the riser, or 204
move out of the riser. In these special cases, the difficulty arises because of node(s) with near zero length. The second nodalization scheme uses “fixed boundaries” for each node and the only moving boundary in this case is the boiling boundary. The only zero length node in this case occurs when the boiling boundary approaches a fixed node boundary. This causes numerical difficulties during simulations, and it is addressed properly in the model development. Weighted residual approach is applied to mass, energy and momentum conservation PDEs for each node to reduce them into a set of ODEs. Time-dependent coefficients of step-wise linear spatial profiles of flow enthalpy, water saturation enthalpy (linearly varying with pressure), steam quality and flow velocity are chosen as phase variables of the set of ODEs. Boiling boundary is determined iteratively using Newton’s method. For the special case when node length is close to zero, the set of ODEs are very stiff. Two approaches are used to avoid numerical difficulty to integrate a set of stiff ODEs. First, a numerical integration code with better numerical stability is used. Second, evaluation of the specific ODEs for the node is stopped if the length is less than a given small number. Phase variables in that tiny node are assumed to be given by the values (flow enthalpy, saturation enthalpy, quality and velocity) at the inlet of the node.
10.2. Summary of Results The thermal-hydraulic model is validated for its convergence and accuracy first. The set of parameters for the validation exercise correspond to those of the SIRIUS facility [44]. As number of nodes increases, steady state velocities and boiling boundaries converge rapidly. Good agreement is found between model results and experimental data for steady state flow rate for large heat flux. The modle over-estimates the flow rate for lower values of heat flux. This is likely due to the fact that model ignores subcooled boiling effect. Convergence of the stability
205
boundaries for the “moving boundary scheme,” however, is found to be much faster than that for the “fixed boundary scheme.” Stability boundaries are found to be consistent with the experiments for lower heat flux values. For larger values of the linear heat flux, the stability boundary for the system pressure Psys = 0.1 MPa under-estimates the stable area. It is presumed that the disagreement is caused by the exchange of two pairs of complex conjugate eigenvalues with the largest real parts. No-boiling boundaries, which separate areas in the operating parameter space where boiling appears in the riser or channel from those where no boiling exists at the steadystate, are studied for different Boussinesq expansion parameters. Good agreement with experimental data is found for adjusted expansion parameters (E = 1.5). Stability and bifurcation for the thermal hydraulic model are analyzed for the set of parameters corresponding to the SIRIUS facility under low pressure conditions (0.1 and 0.2 MPa), and the Dodewaard reactor under higher pressure condition (1.07 MPa). Results of the SIRIUS facility indicate that the stability boundary is separated into two segments; one in which the boiling boundary is in the riser, and second in which the boiling boundary is in the channel. For the segment corresponding to boiling in the riser, similar to the case of forced-circulation BWR model with large in-homogeneity, two conjugate pairs of eigenvalues with the largest and second largest real parts play an important role in defining the stability boundary. Imaginary parts suggest very different frequencies of oscillations for the two pairs. For lower values of the heating power, the pair corresponding to the higher frequency is the one with the largest real part. When the heating power is increased, the second pair, corresponding to the lower frequency, approaches the imaginary axis. As the heating power crosses a threshold value, the second eigenvalue pair crosses the first and becomes the one with the largest real part. It also represents a transition of oscillations from high frequency to low frequency. Similar to the approach followed for studying
206
the forced-circulation BWR model, the stability boundary and the boundary corresponding to the second pair of eigenvalues when plotted together,help explain the slight discrepancy between the analytical stability boundary and the experimental data. For the stability boundary segment corresponding to the “boiling boundary in the channel” case, the pair with the second largest real part is always far to the left of the pair with the largest real part. Results of thermal hydraulics simulations for the Dodewaard reactor (without neutronics) indicate less intense flashing in the riser under higher system pressure conditions. Moreover, stability boundary segment corresponding to the “boiling boundary in the riser” case is much shorter than that corresponding to the “boiling boundary in the channel” case. Bifurcation analyses of the thermal hydraulic model show both supercritical and subcritical bifurcation along different segment of the stability bondary. The natural circulation system generally follows the same trend as the forced-circulation system. The supercritical bifurcation occurs in regions of parameter space with intermediate inlet subcooling values. The subcritical bifurcation, however, occurs in regions with large and small inlet subcooling. Finite amplitude oscillations curves also indicate that much less deviation from the stability boundary is required for the same amplitude of oscillation if the boiling boundary is in the channel. This implies much more intense oscillations are expected to occur as the stability boundary is crossed if the boiling boundary is in the channel. Dynamics of the nuclear coupled natural circulation BWR model are studied for the set of operating parameters corresponding to the Dodewaard reactor. Nominal system pressure Psys = 7.0 MPa and a much lower pressure Psys = 0.4 MPa are chosen to represent typical operating conditions a natural circulation BWR may encounter. Core loading is assumed to be homogeneous. For the high system pressure case, characteristic of oscillation is in-phase along
207
most of the stability boundary. However, in a small region of low reactivity, the azimuthal mode eigenvalue pair crosses the fundamental mode eigenvalue pair, and the oscillations in this region is hence expected to be out-of-phase. For the low system pressure, characteristics of oscillations are found to be always in-phase along the entire stability boundary. Bifurcation results show both supercritical and subcritical PAH-B along the stability boundaries for low as well as high system pressure operations. Sensitivity analysis of various operating parameters is also carried out. It is shown that increasing system pressure may lead to a more stable or a less stable system depending upon the value of the absolute system pressure. Increasing pressure loss coefficient at the outlet of core or riser has a destabilizing effect, while increase of pressure loss coefficient at the core inlet has a stabilizing effect. Numerical simulations of various operating points for thermal hydraulics system and nuclear coupled thermal hydraulic system are presented in Chapter 9. Results of numerical simulations of the thermal hydraulic model (without nuclear coupling) show characteristic similarity with experimental data. As inlet subcooling is decreased, inlet velocity and boiling boundary of the system experience, sequentially, stable single phase flow, intermittent wave, unstable sinuous wave, and stable two-phase flow. Non-dimensional period of the intermittent wave is about 2.2, which is close to the time it takes for the fluid to travel from the channel inlet to the riser exit. Results of numerical simulations of the nuclear coupled system show increasing or saturated amplitudes of oscillations, consistent with the bifurcation properties of the operating points (subcritical or supercritical, respectively). Numerical simulations also show large amplitude fundamental or azimuthal mode oscillations consistent with the predictions of the stability analysis results (in-phase or out-of-phase, respectively). In general, numerical simulation results
208
confirmed the characteristics predicted by the stability and bifurcation analyses. In addition, they also showed mixed oscillation characteristics with larger amplitude, that were beyond the scope of stability and bifurcation analyses.
10.3. Future Work In the thermal hydraulics model of the natural circulation system, the two phase flow is simulated using the Homogeneous Equilibrium Model (HEM). The HEM does not consider different velocities between the steam phase and water phase and assumes thermal equilibrium between the two phases. The assumptions of HEM are not valid for certain applications, such as the low flow rate and void fraction, and when there is a large subcooled boiling region. In future work, the HEM can be replaced by the more accurate drift flux model which takes into account the relative velocity between the two phases and the radially non-uniform distribution of the vapor phase. In addition, a subcooled boiling model would also help in achieving better agreement with the experimental data for lower values of the heating power. Validation of the model is currently limited due to limited access to experimental data. Only two sets of data (SIRIUS and Dodewaard) are used in the validation exercise. If more sets of experimental data and details of the facilities, such as those of PANDA and PUMA, become available in the future, additional validation exercises can be carried out.
209
Appendix A Non-dimensional Variables and Parameters in the Forced Circulation BWR Model
The non-dimensional variables and parameters were defined in the forced circulation BWR model by Karve et al. [30]. The following definitions are reproduced from Appendix A in the reference [30].
A.1. Thermal-Hydraulic Variables and Parameters h = h * ∆h fg v = v * vo
t = t * vo
N sub
*
z = z * L*
L*
P = P * ρ f vo*
*
Nρ = ρg
(
ρf*
*
(h =
ρ = ρ* ρ f *
*
* f , sat
Nr = ρ f
)
− hinlet ∆ρ fg *
∆h fg ρ g *
(
N f ,1 = f 1φ L* 2 Dh
*
)
*
*
*
N pch =
Fr = vo*
*
∆ρ fg
) *
∆ρ fg
*
A* vo ∆h fg , sat ρ f ρ g *
2
r = r * L*
210
2
L* q '
A.2. Fuel Dynamic Variables and Parameters T = T * To
*
(gL ) *
*
*
*
v = v * vo
z = z * L*
*
(
α p ,c = α p ,c * L* vo * *
Bic , g , p =
h∞ rc , g , p k c,g , p
*
)
*
*
c q = c q no L*
*
2
(k
N pch
Λ = Λ* vo *
Σ f ,a = Σ f ,a L*
*
) *
*
R = R * L*
To
∆ρ fg L* q ' = * * * * * A vo ∆h fg , sat ρ f ρ g
A.3. Neutronic Variables and Parameters n = n * no
* p
*
L*
ω ik = ω ik * L* vo *
211
Appendix B Impact of the Error of Intermediate Variable I6(t)
In the reduced order model by Karve et al. [30], the profile of single phase enthalpy and two phase steam quality are represented by quadratic profiles. Weighted residual approach is applied to reduce the conservation PDEs to a set of ODEs of the expansion parameters of the profiles. As the weighted residual procedure is applied to the two phase continuity and energy equations, a number of intermediate variables which are used in the final forms of ODEs are defined. These are included δ 1 (t ) , δ 3 (t ) , δ e (t ) , ( I i (t ) , i = 1, …6), and ( M i (t ) , i = 1, ..3). The original form of I6(i) is given by
δ 13 (t ) 1 + 2 + δ 1 (t ) − δ e (t ) δ 3 (t )[1 − δ e (t )δ 3 (t )] 2( ) δ 1 t [3 − δ 1 (t )δ 3 (t )] log 1 − δ e (t ) − 1 − δ 1 (t )δ 3 (t ) 1 δ 1 (t ) I 6 (t ) = 2 2 2 s1 (t ) − 4s 2 (t ) 1 − δ 1 (t )δ 3 (t ) + δ 1 (t )δ 3 (t ) − δ 13 (t )δ 3 3 (t ) δ 3 2 (t )[1 − δ 1 (t )δ 3 (t )] [1 − 3δ 1 (t )δ 3 (t )]log[1 − δ e (t )δ 3 (t )] δ 3 2 (t )[1 − δ 1 (t )δ 3 (t )]
[
]
+
(B.1)
~ Because steady-state δ 3 = 0 , it will cause both numerators and denominators are zero for terms
on the right-hand-side of (B.1). Taylor’s expansions are used for the certain trouble terms to avoid numerical difficulties. However, it is found the expanded expression given in the reference [30] (equation (E.28) in Appendix E) has a mistake. The wrong form of I6(t) in the reference [30] is given by
212
δ 13 (t ) δ 1 2 (t )[3 − δ 1 (t )δ 3 (t )] δ e (t ) log 1 − + + 1 − δ 1 (t )δ 3 (t ) δ 1 (t ) δ 1 (t ) − δ e (t ) 2 δ e (t ) − δ 2 (t ) + 2δ (t )δ (t ) + 1 1 e 2 2δ e 3 (t ) 1 3 2 2 + δ 1 (t ) − δ e (t )δ 1 (t ) + 2δ e (t )δ 1 (t )δ 3 (t ) + I 6 (t ) = 2 (B.2) s1 (t ) − 4 s 2 (t ) 3 4 2 3δ e (t ) − δ 4 (t ) + 2δ e (t )δ 1 (t ) + δ 2 (t )δ 2 (t ) + 2δ (t )δ 3 (t )δ 2 (t ) + 3 1 e 1 e 1 4 3 O δ 3 3 (t ) The correct form is given by
[
]
δ 13 (t ) δ 1 2 (t )[3 − δ 1 (t )δ 3 (t )] δ e (t ) log 1 − + + 1 − δ 1 (t )δ 3 (t ) δ 1 (t ) δ 1 (t ) − δ e (t ) 2 δ e (t ) − δ 2 (t ) + 2δ (t )δ (t ) + 1 1 e 2 3 1 2δ e (t ) 2 2 + δ e (t )δ 1 (t ) + 2δ e (t )δ 1 (t )δ 3 (t ) + I 6 (t ) = 2 s1 (t ) − 4 s 2 (t ) 3 4 3 3δ e (t ) + 2δ e (t )δ 1 (t ) + δ 2 (t )δ 2 (t ) + 2δ (t )δ 3 (t )δ 2 (t ) + 3 e 1 e 1 4 3 O δ 3 3 (t )
[
(B.3)
]
Figure B.1 compared the expressions (B.1), (B.2) and (B.3) when δ 3 (t ) is changed. Values for (s1, s2, δ1, δe) are fixed at (13.23, 0.0, -0.076, 0.50). The expression (B.3) has much closer agreement with the un-expanded expression (B.1) then the expression (B.2). It is noted that the mistake does not affect stability analysis results in the reference [30]. The reason is Jacobian matrices are always evaluated at steady states for the stability analysis, and they are the same value at the steady-states evaluated with the expression (B.2) or (B.3). Therefore, there is no difference of the eigenvalues of the Jacobian Matrices for the two
213
expressions, which determine linear stabilities of the system. However, higher order derivatives are required for bifurcation analyses and they are different for the two expressions. The mistake thus has a significant impact to characteristics of bifurcation of the system. For example, Figure B.2 shows comparisons of numerical simulations for the operating point 6 on the page 118 of reference [30]. The oscillation calculated using expression (B.2) indicates an increasing amplitude for large perturbation. Therefore, the system has subcritical PAH bifurcation close to this operating point. The one calculated using expression (B.3), however, indicates a decreasing amplitude for the same perturbation. The system should have supercritical PAH bifurcation. It is crucial to correct this mistake for appropriate results of bifurcation analyses.
214
0.0008
(B.1) (B.2) (B.3)
I6(t)
0.0007
0.0006
0.0005
0.0004 0.0
0.1
0.2
0.3
0.4
δ3(t)
Figure B.1: Comparisons of I6(t) evaluated with expressions (B.1), (B.2), and (B.3)
215
0.70
0.65
n0
0.60
0.55
0.50
0.45 0
20
40
60
80
100
120
140
100
120
140
time (s)
(a) 0.70
0.65
n0
0.60
0.55
0.50
0.45 0
20
40
60
80
time (s)
(b) Figure B.2: Comparisons of numerical simulations for operating point 6 on page 118 of reference [30]. Operating parameters ( N sub , ρ ext , ∆Pext ) are (1.123, 0.0, 8.2). Phase variables n0, a1,0, s1,0 and vinlet,0 are perturbed as follows—n0: 0.5800 → 0.5916; a1,0: 0.26608 → 0.25610; s1,0: 4.9306 → 4.7367; vinlet,0: 0.39400 → 0.39383). Oscillations are evaluated with (a) expression (B.2); (b) expression (B.3) 216
Appendix C Typical Dimensional Parameters of Forced-Circulation BWR
Typical dimensional parameters of forced-circulation BWR are listed in this appendix. They are reproduced from the Appendix H of reference [30]. A* = 1.442 × 10 −4 m 2
c p = 325.0 J kg −1 K −1
D * = 1.424 cm
c q = 4.6901 × 10 −6 W cm −3
L* = 3.81 m
f 1φ = 0.01467
R * = 2.32 m
f 2φ = 0.016357
*
Tinlet ,o = 551 K *
To = 561 K *
*
*
g = 9.81 m s −2 *
hg = 5678.2 W m −2 K −1 *
Tsat = 561 K
k c = 17.0 W m −1 K −1
c1,0 = −0.101
k exit = 5.74 × 10 −1 W m −1 K −1
c1,1 = −0.100
k inlet = 19
*
k p = 2.7 W m −1 K −1
*
n0 = 3.6245 × 10 7 cm −3
c 2, 0 = −4.3 × 10 −5 K −1 c 2,1 = −4.3 × 10 −5 K −1 *
c c = 330.0 J kg −1 K −1 *
c f = 5.307 × 10 3 J kg −1 K −1
*
*
P * = 7.2 × 10 6 N m −2 *
pc = 16.2 × 10 −3 m
217
*
ρ c * = 6.5 × 10 3 kg m −3
*
ρ f * = 736.49 kg m −3
*
ρ g * = 37.71 kg m −3
rc = 6.135 × 10 −3 m
rg = 5.322 × 10 −3 m rp = 5.2 × 10 −3 m
(
)
v * = 1 / νΣ f Λ* = 1.48987 × 10 4 m s −1 *
*
v o = 2.67 m s −1
λ* = 0.08 s −1 µ f * = 9.693 × 10 −5 N m −2 s
∆h fg = 1494.2 × 10 3 J kg −1
νΣ f * = 0.01678 cm −1
Λ* = 4.0 × 10 −5 s
ω 00 * = 0 s −1
*
Σ a = 0.016527585 cm −1
ω10 * = −140.08 s −1
α c * = 7.925 × 10 −6 m 2 s −1
ω 01* = −0.0571756 s −1
α p * = 0.797 × 10 −6 m 2 s −1
ω11* = −490.78 s −1
β = 0.0056
ξ h * = 3.855 × 10 −2 m
*
218
Appendix D Different Definitions of Subcooling and Phase Change Numbers for Forced and Natural Circulation Models
Multiple definitions of subcooling and phase change numbers ( N sub and N pch ) are present in this dissertation. In the forced-circulation BWR model, pressure along the flow channel is assumed constant. Definitions of the two numbers are given by
(h
N sub =
* f , sat
*
∆h fg ρ g *
(D.1)
*
∆ρ fg
L* q '
N pch =
)
− hinlet ∆ρ
A vo ∆h fg , sat ρ f ρ g *
*
*
(D.2)
where, thermo properties ρ f , ρ g , h f ,sat , hg ,sat and hinlet are evaluated at the constant system *
*
*
pressure. In the natural circulation model, however, pressure is varying from the channel inlet to the riser outlet. It is necessary to specify what pressure the thermo properties in the expressions of the dimensionless numbers are evaluated at. The definitions of subcooling and phase change numbers are given by h f , sat ,high − hinlet ∆ρ fg *
N sub =
N pch =
∆h fg , sat ,high L* q '
*
*
ρg ∆ρ fg
* * A* v o ∆h fg , sat ,high ρ f ρ g
where the subscript “high” and “low” specify the high and low reference pressures. 219
(D.3)
(D.4)
In the section 6.1 of chapter 6, results of natural circulation experiment performed by Inada et al. [42] are discussed. Another definition of the subcooling number is used in the reference [42], which is given by h f , sat ,low − hinlet ∆ρ fg *
N sub =
∆h fg , sat ,low
*
*
(D.5)
ρg
Because the water saturation enthalpy in the nominator is evaluated at the low reference pressure *
(system pressure), it can be less than the inlet flow enthalpy hinlet . The subcooling number can be negative for the cases of low inlet subcooling.
220
Appendix E Non-dimensional Variables and Parameters in the Natural Circulation BWR Model
The non-dimensional thermal hydraulic variables and parameters defined in the natural circulation BWR model are listed below. The asterisks indicate the original dimensional quantities. Non-dimensional fuel dynamic and neutronic variables and parameters do not differ from those defined for the forced-circulation BWR model. They can be found in Appendix A.
*
t = t * vo / Lc v = v * / vo
*
z = z * / Lc
*2
*
P = P * /(ρ f vo )
*
*
ρm = ρm / ρ f
Nρ =
*
*
h = (h * − h f , sat ,high ) /( h f , sat ,low − h f , sat ,high )
ρg*
Nr =
ρf*
ρf* ∆ρ fg
*
N f ,1 = f 1φ L* /(2 Dh )
*2
Fr = vo /( gL* ) N f , 2 = f 2φ L* /(2 Dh )
N pch =
h f , sat ,high − h f ,sat ,low ∆ρ fg *
N flash =
*
∆h fg , sat ,high
*
*
ρg*
L* q '
∆ρ fg
* * A* v o ∆h fg , sat ,high ρ f ρ g
*
(v )
* 2
N exp =
221
(h
o
* f , sat , high
− h f , sat ,low
*
)
h f , sat ,high − hinlet ∆ρ fg *
N sub =
N exp,h
*
∆h fg , sat ,high
(h =
*
f , sat ,low
*
*
*
N a = Ainlet / Alocal
ρg*
)
− h * f , sat ,high ∂ρ l
ρf*
*
∂hl
*
N dev ,h = −
* hinlet *
222
N exp,h N sub N flash
Appendix F ODEs of Thermal Hydraulics of Natural Circulation Loop In chapter 7, profiles of water saturation enthalpy, flow enthalpy, steam quality and flow velocity are assumed linear equations in a node. Their forms are given by h f ( z , t ) = h f ,i −1 (t ) + a f ,i (t )( z − z i −1 (t ))
(F.1)
hm ( z , t ) = hm ,i −1 (t ) + a m ,i (t )( z − z i −1 (t ))
(F.2)
x( z , t ) = xi −1 (t ) + N ρ N r si (t )(z − z i −1 (t ) )
(F.3)
v m ( z , t ) = v m , i −1 (t ) + vi (t )( z − z i −1 (t ))
(F.4)
ODEs of the expansion parameters can be derived through weighted residual approach.
Details of arriving at ODEs for a m,i (t )
ODE of flow enthalpy expansion parameter a m,i (t ) can be derived from the single phase energy conservation equation (7.14). Plugging the equation (F.2) into the (7.14) and integrating the governing PDE along the node, we will arrive at da m ,i (t ) dt
=
Am ,1 (a m , a f , s, v, ∆z (t )
)
+ Am , 2 (a m , a f , s, v,
)
(F.5)
where
Am ,1
N pch (t ) + N a ,local vinlet (t )ai (t ) + N = −2 flash dhi −1 ( z i −1 (t ), t ) − a (t ) dz i −1 (t ) i dt dt
223
(F.6)
Am , 2 = 0
(F.7)
Details of arriving at ODEs for a f ,i (t ) Two phase energy conservation PDE (7.17) can be used to derive ODE of water saturation enthalpy expansion parameter a f ,i (t ) . Form of the ODE is given by da f (t ) dt
=
A f ,1 (a m , a f , v, s,...) ∆z
+ A f , 2 (a m , a f , v, s,...)
(F.8)
where A f ,1 and A f , 2 are nonlinear functions defined by the weighted residual procedure. They are given by
A f ,1
v m ,i −1 (t )a f ,i (t ) + x t ( ) 1 − i vi (t ) N ρ N r (hg − h f ,i −1 (t ) )1 + N N (1 + xi −1 (t )) + r ρ = −2 N pch (t ) ( ) x t i 1 − (1 + xi −1 (t )) + 1 + N ρ N r N flash dh (t ) f ,i −1 − a (t ) dz i −1 (t ) f ,i dt dt
A f ,2
vi (t )a f ,i (t ) + x ( t ) i −1 vi (t ) N ρ N r (hg − h f ,i −1 (t ) )1 + N N I 1 (t ) + r ρ (hg − h f ,i −1 (t ) )si (t ) − vi (t ) N ρ N r I (t ) − 1 + xi −1 (t ) a f ,i (t ) 2 = − N N r ρ vi (t ) N ρ N r a f ,i (t ) si (t ) I 3 (t ) + N pch (t ) 1 + xi −1 (t ) I (t ) + 1 N flash N N ρ r N pch (t ) si (t ) I 2 (t ) N flash 224
(F.9)
(F.10)
where intermediate variables I 1 , I 2 and I 3 are defined by integration I 1 (t ) =
z 2 1 i 1 1 − dz (F.11) ∆z (t ) ∆z (t ) z∫i −1 1 − xi −1 (t ) − N ρ N r s i (t )( z − z i −1 (t )) 1 − xi −1 (t )
I 2 (t ) =
I 3 (t ) =
2 ∆z (t ) 2 2 ∆z (t ) 2
zi
( z − z i −1 (t )) dz i −1 (t ) − N ρ N r s i (t )( z − z i −1 (t ))
(F.12)
( z − z i −1 (t )) 2 ∫ 1 − xi −1 (t ) − N ρ N r si (t )( z − z i−1 (t )) dz zi −1
(F.13)
∫ 1− x
zi −1 zi
They can be expanded by Taylor’s expansion formula ∞
I 1 (t ) = ∑ TI 1,n (t )
(F.14)
n =1 ∞
I 2 (t ) = ∑ TI 2,n (t )
(F.15)
n =1 ∞
I 3 (t ) = ∑ TI 3, n (t )
(F.16)
n =1
where, 2(N ρ N r si (t ) ) ∆z (t ) n −1 n
TI 1, n (t ) =
(n + 1) ⋅ (1 − xi −1 (t ) )n +1
2(N ρ N r si (t ) ) ∆z (t ) n −1
(F.17)
n −1
TI 2,n (t ) =
(n + 1) ⋅ (1 − xi −1 (t ) )n
2(N ρ N r si (t ) ) ∆z (t ) n
(F.18)
n −1
TI 3,n (t ) =
(n + 2) ⋅ (1 − xi −1 (t ) )n
Details of arriving at ODEs for si (t ) 225
(F.19)
Mass conservation equations (7.16) is integrated along the node to obtain ODE of the quality expansion parameter si (t ) . It is given by ds (t ) S1 (a m, a f , v, s,...) = + S 2 (a m, a f , v, s,...) dt ∆z
(F.20)
where nonlinear functions x (z (t ), t ) 1 + i −1 i −1 vi (t ) − Nρ Nr S1 = 2 v m ,i −1 ( z i −1 (t ), t )si (t ) − 1 dxi −1 ( z i −1 (t ), t ) − s (t ) dz i −1 (t ) i N ρ N r dt dt
(F.21)
S2 = 0
(F.22)
Details of arriving at ODEs for vi (t ) ODE of vi (t ) comes from governing equation momentum conservation (7.18). Similarly, it has expression dv(t ) V1 (a m , a f , v, s,...) = + V2 (a m , a f , v, s,...) dt ∆z
(F.23)
where
x (t ) 1 2 − ∆Pdrv 1 + i −1 a f ,i (t ) − vi (t )v m ,i −1 (t ) − N f , 2 v m ,i −1 (t ) − Nρ Nr Fr V1 = 2 dz i −1 (t ) dv m ,i −1 ( z i −1 (t ), t ) − vi (t ) dt dt
(F.24)
2 V2 = ∆Pdrv si (t )a f ,i (t ) − vi (t ) 2 − N f , 2 2v m ,i −1 (t )vi (t ) + vi (t ) 2 ∆z (t ) 3
(F.25)
226
Appendix G Expansion Parameters When Node Length Is Close to Zero
Numerical difficulties will arise, if ODEs of the expansion parameters are evaluated for the node with very small length. In this case, the ODE system will be too stiff to integrate because of the first terms on the right hand sides of equations (7.30), (7.32), (7.33) and (7.34) (the terms with ∆z (t ) as denominators). To avoid this difficulty, a smallest node length is set up in the code. If the node length is smaller than this value, ODEs are not evaluated in the specific node. Instead, explicit expressions for the expansion parameters are used assuming the nominators of the first terms equal to zero, i.e. A f ,1 (a m , a f , v, s,...) = 0
(G.1)
Am ,1 (a m , a f , v, s,...) = 0
(G.2)
S1 (a m , a f , v, s,...) = 0
(G.3)
V1 (a m , a f , v, s,...) = 0
(G.4)
Replace the left hand side of (G.2) with the the equation (F.6). Explicit form of the expansion parameter ai (t ) is given by N pch dhi −1 (t ) N a ,local vinlet (t ) − dz i −1 (t ) ai (t ) = − + N dt dt flash The expansion parameters a f ,i (t ) , si (t ) and vi (t ) can be obtained by solving the equations (G.1), (G.3) and (G.4) From equation (G.1) and (F.9), we have 227
(G.5)
m1 (t )a f ,i (t ) + n1 (t )vi (t ) = e1 (t )
(G.6)
where, m1 (t ) = v m ,i −1 (t ) − n1 (t ) = N ρ N r
(h
dz i −1 (t ) dt
− h f ,i −1 (t ) ) 1 + xi −1 (t ) N ρ N r 1 − xi −1 (t )
g
N pch (t ) 1 1 + xi −1 (t ) + N N ρ N r 1 − xi −1 (t ) e1 (t ) = − flash dh f ,i −1 (t ) dt
(G.7)
(G.8)
(G.9)
From the equations (G.4) and (F.24), we have m2 (t )a f ,i (t ) + n2 (t )vi (t ) = e2 (t )
(G.10)
where, x (t ) m2 (t ) = ∆Pdrv 1 + i −1 N ρ N r
(G.11)
dz i −1 (t ) dt
(G.12)
n2 (t ) = −v m ,i −1 (t ) +
2 + N v t ( ) f , 2 m ,i −1 1 e2 (t ) = − (1 + N exp,h h f ,i −1 (t ) + N dev ,h ) + Fr dv (t ) m ,i −1 dt
(G.13)
Therefore, a f ,i (t ) and vi (t ) are given by a f ,i (t ) =
e1 (t )n2 (t ) − e2 (t )n1 (t ) m1 (t )n2 (t ) − m2 (t )n1 (t )
228
(G.14)
e2 (t )m1 (t ) − e1 (t )m2 (t ) m1 (t )n2 (t ) − m2 (t )n1 (t )
(G.15)
m3 (t )vi (t ) + n3 (t )si (t ) = e3 (t )
(G.16)
vi (t ) = Similarly, from (G.3) and (F.21), we have
where, m3 (t ) = 1 +
xi −1 (t ) Nρ Nr
n3 (t ) = −v m ,i −1 (t ) +
dz i −1 (t ) dt
(G.17)
(G.18)
e3 (t ) =
1 dxi −1 (t ) N ρ N r dt
(G.19)
si (t ) =
e3 (t ) − m3 (t )vi (t ) n3 (t )
(G.20)
si (t ) is given by
229
Appendix H Typical Dimensional Parameters of SIRIUS Facility The following dimensional parameters of SIRIUS facility [44] [61] are used in Chapter 8 and 9 in this dissertation. Note: water and steam densities ρ f and ρ g are evaluated at the *
*
(
given pressure at the riser outlet Plow = Psys . The pressure Phigh = ρ f g Lc + Lr *
*
*
*
*
*
) is the static
pressure at the inlet of channel. Thermo properties of water and steam related to the pressures *
*
Plow and Phigh are evaluated using ASME steam table routines.
*
*
Lr = 5.746 m
*
Ar = 5.925 × 10 −4 m 2
*
Dr = 2.747 × 10 −2 m
Lc = 1.7 m Ac = 4.798 × 10 −4 m 2
Dc = 9.2 × 10 −3 m *
*
*
*
Ad ,i = 2.663 × 10 −4 m 2
Ad ,o = 2.663 × 10 −4 m 2
K c ,in = 15.5
K c ,out = 0.0
K d ,in = 0.0
K r ,out = 10.0
f 1,c = 0.0309
f 1,r = 0.0309
g * = 9.81 m / s 2
Pr ,out = 0.2 MPa
q ' = 5.294 × 10 −3 MW / m
v o = 1.126 × 10 −1 m / s
*
*
*
Tsub = 18.739 K 230
Appendix I Typical Dimensional Parameters of Dodewaard Reactor I.1. Thermal-Hydraulic Parameters The following dimensional parameters of Dodewaard reactor [62] [63] are used in Chapter 8 and 9 in this dissertation. Note: water and steam densities ρ f and ρ g are evaluated *
*
(
at the given pressure at the riser outlet Plow = Psys . The pressure Phigh = ρ f g Lc + Lr *
*
*
*
*
*
) is the
static pressure at the inlet of channel. Thermo properties of water and steam related to the *
*
pressures Plow and Phigh are evaluated using ASME steam table routines.
*
*
Lr = 3.06 m
*
Ar = 1.573 × 10 −2 m 2
*
Dr = 1.415 × 10 −1 m
Lc = 1.794 m
Ac = 6.981 × 10 −3 m 2 Dc = 1.419 × 10 −2 m *
*
*
*
Ad ,i = 2.146 × 10 −2 m 2
Ad ,o = 1.598 × 10 −2 m 2
K c ,in = 1.73
K c ,out = 1.095
K d ,in = 0.5
K r ,out = 0.5
f1,c = 0.0164
f1,r = 0.0164
g * = 9.81 m / s 2
Pr ,out = 1.07 MPa
q ' = 8.497 × 10 −3 MW / m
vo = 7.233 × 10 −1 m / s
*
*
*
Tsub = 1.8 K 231
I.2. Fuel Dynamics and Neutronics Parameters The following fuel dynamics and neutronics parameters of Dodewaard reactorare used in Chapter 8 and 9 in this dissertation *
α o = 0.3849
*
Tavg ,o = 832.5 K
*
β = 0.0061
*
λ* = 0.084 s −1
*
Λ* = 5.0 × 10 −5 s
hg = 5678.2 W m −2 K −1
*
k ∞ = 1.01
α p * = 0.797 × 10 −6 m 2 s −1
L2 = 8.43 × 10 −4 m 2
α c * = 7.925 × 10 −6 m 2 s −1
νΣ f * = 0.01678 cm −1
rc = 6.135 × 10 −3 m
rg = 5.322 × 10 −3 m rp = 5.2 × 10 −3 m k c = 17.0 W m −1 K −1 k p = 2.7 W m −1 K −1
*
cc = 330.0 J kg −1 K −1
cα = −0.078
R * = 0.796 m
c D = −2.0 × 10 −5 K −1
*
*
To = 564.5 K
*
g = 9.81 m s −2
*
c q = 2.034 × 10 9 W cm −3
232
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BY QUAN ZHOU B.E., Tsinghua University, 1995 M.E. Tsinghua University, 2000
DISSERTATION
Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Nuclear Engineering in the Graduate College of the University of Illinois at Urbana-Champaign, 2006
Urbana, Illinois
© Copyright by Quan Zhou, 2006
Stability and Bifurcation Analyses of Reduced-Order Models of Forced and Natural Circulation BWRs Abstract Quan Zhou, Ph.D. Department of Nuclear, Plasma and Radiological Engineering University of Illinois at Urbana-Champaign, 2006 Rizwan-uddin, Advisor
Current design of boiling water reactors (BWRs) relies on recirculation pumps to remove fission heat from cores. The forced circulation BWRs faced nuclear-coupled thermal hydraulic stability problems in an island in the power-flow plane—under high power and low flow conditions. Previous studies of the stability problem were focused on simulations using large scale and high fidelity commercial codes. Even the existing studies using the reduced order model focused on linear stability of the BWR systems. Few analyses have been made to review phase and bifurcation characteristics of power and flow oscillations in BWR core. New designs of BWRs relying on natural circulation to extract core heat, such as SBWR and ESBWR, have been completed, and they are promising candidates to replace the current forced-circulation BWRs. In addition to the complicated nuclear-coupled density-wave oscillations, natural circulation systems with their long risers must also address the issue of flashing and associated nuclear-coupled thermal hydraulic stability. In this case, small perturbation of flow rate can be dramatically intensified by vaporization due to pressure changes. Previous studies of the nuclearcoupled instabilities and flashing-induced instabilities of natural circulation BWRs were limited, partly due to difficulties in developing models that fully respect pressure
iii
dependence of water and steam thermodynamic properties under low system pressure condition. In this dissertation, stabilities of the forced and natural circulation BWRs were studied separately. In the first part of the dissertation, a reduced-order model was modified and used for stability and bifurcation analyses of forced-circulation BWRs. The model was comprised of Ordinary Differential Equations (ODEs) with respect to thermal hydraulics, fuel dynamics, and neutronics. An extra parameter considering inhomogeneity of core loading was introduced. Stability and characteristics of oscillations were studied based on linear stability and Poincare-Andronov-Hopf Bifurcation (PAH-B) theorems of a general dynamic system. Results show that different bifurcations (supercritical or subcritical) may occur along the stability boundary in a parameter space. Moreover, it was found that although the conjugate pair of eigenvalue with the largest real part determines linear stability of the system, the other pair with the second largest real part is also important defining phase characteristics of oscillations. In the second part of this dissertation, a new reduced-order model was developed for the natural circulation BWRs. Water saturation enthalpy in this model depends on local pressure, allowing analysis of the flashing phenomenon. ODEs of flow enthalpy, water saturation enthalpy, steam quality and flow velocities were derived node by node with classic weighted residual approach. The nodalization schemes of the natural circulation loop are based on modeling the boiling boundary. Moving boundary scheme differentiates cases of the boiling boundary, but the fixed boundary scheme treats all cases of boiling the same. Both the schemes have advantages and disadvantages, and they are implemented for different applications. The model was validated by comparing with
iv
experiments performed on loop SIRIUS and natural circulation reactor Dodewaard. Stability and bifurcation analyses revealed complexity of dynamics of the natural circulation system. It is expected the studies in this dissertation are of value helping improve safety of BWR systems.
v
ACKNOWLEDGEMENTS
First, I would like to thank my advisor, Professor Rizwan-uddin for his continuous guidance and encouragement throughout my years of study at University of Illinois. I would like to thank him also for his support and care in my life and job search. I would also like to thank Professor Barclay Jones, Professor Magdi Ragheb and Professor S Balachandar for serving on my final defense committee. I wish to extend special recognition to financial support provided by the U.S. Department of Energy under a Nuclear Engineering Education Research (NEER) Grant, number DE-FG07-00ID13923. I would also like to acknowledge support under the Computational Science and Engineering Fellowship program at University of Illinois at Urbana Champaign. I would like to thank my parents for the encouragement given to me since my childhood. Without their support, I could not have come to USA for my Ph.D. studies. I would also like to thank my officemates Daniel Rock, Federico Teruel, Jianwei Hu, Yizhou Yan, and Yuxiang Gu for making Room 251 NEL a pleasant working environment.
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Table of Contents NOMENCLATURE .................................................................................................. xi LIST OF TABLES..................................................................................................... xiii LIST OF FIGURES ................................................................................................... xiv PART I. STABILITY AND BIFURCATION ANALYSES OF FORCED CIRCULATION BWRS ............................................................................................ 1 CHAPTER 1. INTRODUCTION TO INSTABILITIES OF FORCED CIRCULATION BWRS ............................................................................................ 1 1.1. Instability Incidents and Experiments in BWRs.......................................... 1 1.2. Types of Instabilities.................................................................................... 7 1.3. Physical Mechanisms................................................................................... 10 1.3.1. Thermal Hydraulics Mechanism: Density Wave Oscillations (DWO) ........................................................................................................................ 10 1.3.2. Neutron Kinetics Mechanisms........................................................... 12 1.4. Previous Work ............................................................................................. 15 1.4.1. Detailed Models ................................................................................. 17 1.4.2. Reduced-order Models....................................................................... 18 1.5. Tools to Analyze Dynamical Systems ......................................................... 20 1.5.1. Frequency Domain Methods (Classical Stability Analysis) .............. 20 1.5.2. Bifurcation Analyses.......................................................................... 21 1.5.3. Numerical Integration ........................................................................ 22 1.6. Scope of Part I of This Thesis...................................................................... 22 CHAPTER 2. A REDUCED-ORDER MODEL OF FORCED-CIRCULATION BWR AND REVIEW OF BIFURCATION ANALYSIS METHODS .................................................................................................................................... 25 2.1. Forced Circulation BWR Model.................................................................. 25 2.1.1. Neutron Kinetics ................................................................................ 25 2.1.2. Consideration of In-Homogeneity ..................................................... 28 2.1.3. Heated Channel Thermal Hydraulics................................................. 30 2.1.4. Heated Conduction Model ................................................................. 33 2.2. Stability and Bifurcation Analysis Method.................................................. 34 2.2.1. Neutron Kinetics ................................................................................ 35 2.2.2. Poincaré-Andronov-Hopf Bifurcation (PAH-B)................................ 37 2.2.3. Semi-Analytical Method and BIFDD ................................................ 42 CHAPTER 3. STABILITY AND BIFURCATION ANALYSES OF FORCED CIRCULATION BWR MODEL ............................................................................... 3.1. Eigenvalues and Eigenvectors of the BWR Model...................................... 3.2. Homogeneous Core (F = 1) ......................................................................... 3.2.1. Results in Nsub—ρext Space.................................................................
vii
46 46 54 54
3.2.2. Results in ρext—∆Pext and Nsub— ∆Pext Space.................................... 3.3. Non-Homogeneous Core (F > 1) ................................................................. 3.3.1. Results in Nsub— ∆Pext Space............................................................. 3.3.2. Results in F—∆Pext Space ................................................................. 3.4. Discussion ....................................................................................................
58 61 61 66 68
CHPATER 4. NUMERICAL SIMULATIONS OF FORCED-CIRCULATION BWR MODEL ..................................................................................................................... 70 4.1. Numerical Simulations of Homogeneous Core (F=1) ................................. 70 4.2. Numerical Simulations of Non-Homogeneous Core (F>1)......................... 76 CHPATER 5. SUMMARY, CONCLUSIONS AND FUTURE WORKS OF PART I .................................................................................................................................... 92 5.1. Summary of the Model and Analysis Method ............................................. 92 5.2. Conclusions of Results................................................................................. 94 5.3. Future Work ................................................................................................. 97 PART II. STABILITY AND BIFURCATION ANALYSES OF NATURAL CIRCULATION BWRS ............................................................................................ 98 CHAPTER 6. INTRODUCTION TO INSTABILITIES OF NATURAL CIRCULATION BWR .......................................................................................................................... 98 6.1. Instabilities of Natural Circulation BWRs under Low Pressure.................. 98 6.2. Physical Mechanisms................................................................................... 104 6.2.1. Geysering ........................................................................................... 104 6.2.2. Flashing.............................................................................................. 106 6.3. Previous Work ............................................................................................. 109 6.4. Scope of Part II of This Thesis .................................................................... 112 CHAPTER 7. REDUCED-ORDER MODEL OF NATURAL-CIRCULATION BWRS .................................................................................................................................... 115 7.1. Nodalization of Natural Circulation System................................................ 115 7.2. Differential Equations.................................................................................. 121 7.2.1. Dimensional and Non-Dimensional Partial Differential Equations (PDEs)............................................................................................................ 121 7.2.2. Ordinary Differential Equations (ODEs) in A Node ......................... 125 7.2.3. Node with Small Length .................................................................... 129 7.2.4. Phase Variables across Boundaries.................................................... 130 7.3. Closure of Boiling Boundary and Its Time-Dependent Derivatives............ 134 CHPATER 8. STABILITY AND BIRFURCATION ANALYSES OF NATURAL CIRCULATION BWR MODEL ............................................................................... 141 8.1. Validations and Comparisons ...................................................................... 141 8.1.1. Convergence ...................................................................................... 141 8.1.2. Model Validations and Comparison with Experiments ..................... 147
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8.2. Stability and Bifurcation Analyses of the Thermal Hydraulics Model........ 8.3. Stability and Bifurcation Analyses of the Nuclear-Coupled Model ............ 8.3.1. Effect of Nodalization Schemes......................................................... 8.3.2. In-phase and Out-of-phase Oscillations............................................. 8.3.3. Bifurcation Analyses.......................................................................... 8.3.4. Sensitivity Analyses...........................................................................
153 164 164 166 172 176
CHPATER 9. NUMERICAL SIMULATIONS OF NATURAL CIRCULATION MODEL ..................................................................................................................... 180 9.1. Thermal-Hydraulic Model ........................................................................... 180 9.2. Nuclear-Coupled Thermal-Hydraulics Model of A Natural Circulation BWR .................................................................................................................... 188 CHPATER 10. SUMMARY, CONCLUSIONS AND FUTURE WORK OF PART II .................................................................................................................................... 204 10.1. Summary of Natural Circulation BWR Model .......................................... 204 10.2. Summary of Results................................................................................... 205 10.3. Future Work ............................................................................................... 209 APPENDIX A NON-DIMENSIONAL VARIABLES AND PARAMETERS IN THE FORCED CIRCULATION BWR MODEL .............................................................. 210 A.1. Thermal-Hydraulic Variables and Parameters............................................ 210 A.2. Fuel Dynamic Variables and Parameters .................................................... 210 A.3. Neutronic Variables and Parameters........................................................... 211 APPENDIX B IMPACT OF THE ERROR OF INTERMEDIATE VARIABLE I6(t) .................................................................................................................................... 212 APPENDIX C TYPICAL DIMENSIONAL PARAMETERS OF FORCEDCIRCULATION BWR .............................................................................................. 217 APPENDIX D DIFFERENT DEFINITIONS OF SUBCOOLING AND PHASE CHANGE NUMBERS FOR FORCED AND NATURAL CIRCULATION MODEL .................................................................................................................................... 219 APPENDIX E NON-DIMENSIONAL VARIABLES AND PARAMETERS IN THE NATURAL CIRCULATION BWR MODEL ........................................................... 221 APPENDIX F ODES OF THERMAL HYDRAULICS OF NATURAL CIRCULATION LOOP ......................................................................................................................... 223 APPENDIX G EXPANSION PARAMETERS WHEN NODE LENGTH IS CLOSE TO ZERO ......................................................................................................................... 227 APPENDIX H TYPICAL DIMENSIONAL PARAMETERS OF SIRIUS FACILITY .................................................................................................................................... 230
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APPENDIX I TYPICAL DIMENSIONAL DODEWAARD REACTOR................ 231 I.1. Thermal-Hydraulic Parameters..................................................................... 231 I.2. Fuel Dynamic and Neutronic Parameters..................................................... 232 BIBLIOGRAPHY...................................................................................................... 233
x
Nomenclature A: Bi : C (r , t ) : D(r , t ) : Fr J0
cross sectional flow area Biot number averaged delayed neutron precursor group concentration neutron diffusion coefficient Froud number Bessel function of first kind of order 0
J1 K L Ni N exp
Bessel function of first kind of order 1 pressure loss coefficient flow channel length number of node in region i (=1, 2,3) area number Boussinesq expansion number
Nf
friction number
N flash
flashing number
N pch
phase change number
Na
N (r , t ) N sub P Psys R T a af
reactor core radius temperature expansion parameter of flow enthalpy expansion parameter of water saturation enthalpy
cα cD cc e f h hf
void reactivity coefficient Doppler reactivity coefficient clad specific heat eigenvector of Jacobian matrix friction factor enthalpy or heat transfer coefficient water saturation enthalpy
hinlet hm k
coolant inlet enthalpy flow enthalpy pressure loss coefficient
n q' s
neutron number density linear heat flux expansion parameter of steam quality
neutron number density subcooling number pressure system pressure of natural circulation loop
xi
t v x z ∆Pdrv ∆Pext ∆h fg Λ Σa
Σf α
β λ µ ρ ω ik
Subscripts 1φ 2φ avg c dc exit fric grav i in or inlet l loc lim m o out or outlet r sat
time neutron or cooland velocity steam quality axial spatial coordinate static water pressure along channel and riser external pressure drop of the forced-circulation model vapor-liquid enthalpy difference neutron generation time macroscopic absorption cross section macroscopic fission cross section void fraction or thermal diffusivity delayed neutron fraction precursor decay constant boiling boundary reactivity or density modal eigenvalues
µ
single-phase two-phase average channel downcomer riser exit frictional gravitational node number channel or riser inlet subcooled water, pressure loss due to local restriction smallest value mixture reference channel or riser outlet riser saturation boiling boundary
Superscripts ~ *
steady-state value dimensional quantity
xii
List of Tables Table 1.1:
Instability events during BWR operations and during stability tests..... 3
Table 2.1:
Properties of fixed points and periodic solutions close to the stability boundary ................................................................................................ 41
Table 3.1:
Elements corresponding to n0 and n1 in the eigenvector corresponding to e1 ........................................................................................................ 52 Elements corresponding to n0 and n1 in the eigenvector corresponding to e2 ........................................................................................................ 53
Table 3.2:
Table 4.1:
Perturbations for phase variables for F = 4.6 and 5.0. Operating parameter set ( N sub , ρ ext , ∆Pext ) corresponds to (1.0, 0.0, 8.5) ............................... 83
Table 8.1:
Critical eigenvalues E imag and elements EV n0 and EV n1 (corresponding to n0 (t ) and n1 (t ) ) respectively in the eigenvector at four operating points shown in Figures 8.13 and 8.14. ............................................................ 171
Table 9.1:
Operating points and characteristics of oscillations and PAH-B predicted by BIFDD............................................................................................... 195
xiii
List of Figures Figure 1.1: Figure 1.2: Figure 1.3: Figure 1.4: Figure 1.5:
Figure 1.6: Figure 2.1: Figure 2.2: Figure 3.1: Figure 3.2:
Figure 3.3:
Figure 3.4:
Diagram of Swiss boiling water reactor KKL ....................................... 2 Relative power and flow rate during instability event of LaSalle unit 2 ................................................................................................................ 5 Flow patterns and void distributions of different types of oscillations.. 9 (a) In-phase oscillations ......................................................................... 9 (b) Out-of-phase oscillations.................................................................. 9 Illustration of pressure drops and inlet flow along a uniformly heated channel with constant flow cross-section .............................................. 11 Shapes and variations of the fundamental and first azimuthal modes ... 14 (a) 3-dimensional shape of the fundamental mode of a homogeneously loaded core ............................................................................................. 14 (b) 3-dimensional shape of the first azimuthal mode of a homogeneously loaded core ............................................................................................. 14 (c) fundamental mode during an oscillation .......................................... 14 (d) first azimuthal mode during an oscillation....................................... 14 Power-flow map of KKL plant (Switzerland) ....................................... 16 Examples of stable and unstable periodic solution ................................ 39 (a) stable periodic solution..................................................................... 39 (b) unstable periodic solution................................................................. 39 Examples of stability boundary and oscillation curves calculated by BIFDD ................................................................................................................ 45 Critical values of ∆Pext , when N sub = 1.0, ρ ext = 0.0 and F = 1.0, 2.0, 3.0, 4.6, and 5.0 respectively ........................................................................ 49 Eight rightmost eigenvlaues corresponding to the critical values of ∆Pext for the five cases shown in Figure 3.1 ................................................... 50 (a) F = 1.0 .............................................................................................. 50 (b) F = 2.0 .............................................................................................. 50 (c) F = 3.5 .............................................................................................. 50 (d) F = 4.6 .............................................................................................. 50 (e) F = 5.0 .............................................................................................. 50 ................................................................................................................ 56 (a) Stability boundary in N sub - ρ ext space ............................................. 56 (b) Fifteen percent oscillation amplitude ( ε = 0.15) curves in N sub (ρ ext − ρ ext ,critical ) space ........................................................................... 56 ................................................................................................................ 59 (a) Stability boundaries in ρ ext - ∆Pext space .......................................... 59 (b) Fifteen percent oscillation amplitude ( ε = 0.15) curves in ρ ext -
(∆P
ext
Figure 3.5:
− ∆Pext ,critical ) space ....................................................................... 59
................................................................................................................ 60
xiv
(a) Stability boundary in N sub - ∆Pext space ........................................... 60 (b) Fifteen percent oscillation amplitude ( ε = 0.15) curves in N sub -
(∆P
ext
Figure 3.6:
Figure 3.7:
Figure 3.8:
Figure 4.1:
Figure 4.2:
Figure 4.3: Figure 4.4: Figure 4.5: Figure 4.6: Figure 4.7: Figure 4.8:
Figure 4.9:
− ∆Pext ,critical ) space ....................................................................... 60
................................................................................................................ 62 (a) Stability boundaries in N sub — ∆Pext space for F = 1.0, 2.0, 3.5, 4.6 and 5.0........................................................................................................... 62 (b) 5% oscillation curves in N sub — (∆Pext − ∆Pext ,critical ) space for F = 1.0, 2.0, 3.5, 4.6 and 5.0................................................................................ 62 ................................................................................................................ 64 (a) Stability boundary and the boundary along which the real part of the second largest pair of eigenvalue is zero for F = 5.0 ............................. 64 (b) 5% oscillation amplitude curves corresponding to the eigenvalue pairs with the largest and second largest real parts......................................... 64 Stability boundaries in F— ∆Pext space for N sub = (a) 4.50; (b) 3.25; (c) 1.90; (d) 1.40; (e) 1.30; (f) 1.18 ............................................................. 67 Results of numerical integration for parameter values corresponding to point 1 in Figure 3.3. Operating parameters ( N sub , ρ ext , ∆Pext ) are (9.0, 0.03005, 8.0).. ........................................................................................ 72 Results of numerical integration for parameter values corresponding to point 2 in Figure 3.3. Operating parameters ( N sub , ρ ext , ∆Pext ) are (3.0, 0.01320, 8.0). ......................................................................................... 74 Results of numerical integration at point 3 in Figure 3.3. Operating parameters ( N sub , ρ ext , ∆Pext ) at the point are (3.0, 0.01, 8.0). .............. 75 Results of numerical integration of the governing set of nonlinear ODEs for F = 2.0. Operating parameters ( N sub , ρ ext , ∆Pext ) are (1.0, 0.0, 8.5)...... 77 Results of numerical integration for F = 3.0. Operating parameters ( N sub , ρ ext , ∆Pext ) at the point are (1.0, 0.0, 8.5). ................................... 78 Results of numerical integration for F = 4.6. Operating parameters ( N sub , ρ ext , ∆Pext ) at the point are (1.0, 0.0, 8.5). ................................... 79 Results of numerical integration for F = 5.0. Operating parameters ( N sub , ρ ext , ∆Pext ) at the point are (1.0, 0.0, 8.5). ................................... 80 Perturbations of n0 and n1 for F = 4.6. Operating parameters ( N sub , ρ ext , ∆Pext ) at the point are (1.0, 0.0, 8.5). Perturbations of timedependent variables are introduced as ξv , where ξ =0.01 is a small number, v is the eigenvector corresponding to the pair of eigenvalues with the largest real part................................................................................. 84 Perturbations of n0 and n1 for F = 5.0. Operating parameters ( N sub , ρ ext , ∆Pext ) at the point are (1.0, 0.0, 8.5). Perturbations of timedependent variables are introduced as ξv , where ξ =0.01 is a small
xv
Figure 4.10: Figure 4.11: Figure 4.12: Figure 4.13:
Figure 6.1: Figure 6.2: Figure 6.3:
Figure 6.4: Figure 6.5:
Figure 7.1: Figure 7.2:
Figure 7.3: Figure 7.4: Figure 7.5: Figure 8.1:
number, v is the eigenvector corresponding to the pair of eigenvalues with the largest real part................................................................................. 85 Results of numerical integration for F = 5.0. Operating parameters ( N sub , ρ ext , ∆Pext ) at the point are (2.2303, 0.0, 3.9). ............................. 88 Results of numerical integration for F = 5.0. Operating parameters ( N sub , ρ ext , ∆Pext ) at the point are (2.2303, 0.0, 4.1).. ............................ 89 Results of numerical integration for F = 5.0. Operating parameters ( N sub , ρ ext , ∆Pext ) at the point are (0.8, 0.0, 7.85).. ................................ 90 Results of numerical integration for F = 5.0. Operating parameters ( N sub , ρ ext , ∆Pext ) at the point are (0.8, 0.0, 8.2). ................................... 91 Diagram of natural circulation BWR design and schematic flow path. 99 Typical geysering flow signals in two parallel heated channels............ 102 (a) Flow signal in channel 1, u(1) .......................................................... 102 (b) Flow signal in channel 2, u(2).......................................................... 102 Typical flow signals recorded in natural circulation SIRIUS loop. System pressure is P = 0.2 MPa, heat flux is q’’ = 53 kW/m2.. ......................... 103 (a) single phase natural circulation flow, subcooling number Nsub = 29.8.............................................................................................. 103 (b) intermittent wave due to flashing, Nsub = 23.6 ................................. 103 (c) sinusoidal oscillations, Nsub = -0.08.................................................. 103 (d) steady two-phase flow, Nsub = -2.17................................................. 103 Flow regimes in a channel corresponding to four points of stage I and II of a geysering cycle in the experiment by Aritomi et al.. .......................... 105 Flow pattern and enthalpy profiles during a cycle of intermittent wave caused by flashing.................................................................................. 107 (a) Both channel and riser are single phase ........................................... 107 (b) The hot fluid reaches the upper part of the riser .............................. 107 (c) Flow is accelerated ........................................................................... 107 (d) Flashing is suppressed when the cold fluid reaches the upper part of the riser ........................................................................................................ 107 Schematic plot of natural circulation system... ...................................... 116 Cases based on position of boiling boundary... ..................................... 117 (a) Boiling boundary is out of the riser outlet........................................ 117 (b) Boiling boundary is in the riser ........................................................ 117 (c) Boiling boundary stays at the connection of channel and riser ........ 117 (d) Boiling boundary is in the channel................................................... 117 Nodalization schemes for the case 2 (boiling boundary in the riser)..... 120 Conditions across a boundary.... ............................................................ 132 Intermediate unknowns for ODE evaluations. Boiling boundary is in the channel in this example.......................................................................... 136 Steady state boiling boundary and inlet flow velocity calculated with fixed boundary schemes. Total number of nodes are N = 3, 5, and 9...... ...... 143
xvi
Figure 8.2:
Figure 8.3:
(a) boiling boundary µ .......................................................................... 143 (b) inlet flow velocity vinlet .................................................................... 143 Stability boundaries in inlet subcooling—heat flux parameter space for two different nodalization schemes....... ....................................................... 145 (a) moving boundary scheme................................................................. 145 (b) fixed boundary scheme..................................................................... 145 (c) comparison of moving and fixed boundary schemes ....................... 145 Comparison of steady state flow rates between experiments and analyses. System pressure is Psys = 0.2 MPa......................................................... 148 (a) heat flux q ' = 1.7647 (kW/m) ........................................................... 148 (b) q ' = 2.9412 (kW/m) .......................................................................... 148 (c) q ' = 4.1177 (kW/m) .......................................................................... 148
Figure 8.4:
Figure 8.5:
Figure 8.6: Figure 8.7:
Figure 8.8: Figure 8.9:
(d) q ' = 5.2941 (kW/m) .......................................................................... 148 Comparisons of experimental data with the stability boundaries evaluated using the model developed in chapter 7. System pressure is Psys = 0.1 MPa........ ................................................................................................ 150 (a) Comparison of stability boundary .................................................... 150 (b) Comparison of no-boiling boundary for different Boussinesq expansion parameter values .................................................................................... 150 (c) Comparison of both stability boundary and no-boiling boundary.... 150 Comparison of experimental data points with stability boundaries evaluated using the model developed in Chapter 7. System pressure is Psys = 0.2 MPa........ ................................................................................................ 152 (a) Comparison of stability boundary .................................................... 152 (b) Comparison of no-boiling boundary for different Boussinesq expansion parameter values .................................................................................... 152 (c) Comparison of both stability boundary and no-boiling boundary.... 152 Stability boundary and the boundary on which the second rightmost pair of * eigenvalues has zero real part. System pressure Psys = 0.1 MPa........... 156 7.5% oscillation curves for different segments of the SB shown in Figure 8.6........................................................................................................... 157 (a) segment A-T ..................................................................................... 157 (b) segment T-C ..................................................................................... 157 (c) segment D-E ..................................................................................... 157 (d) segment E-F...................................................................................... 158 (e) segment F-G ..................................................................................... 158 Stability boundary and the boundary of the eigenvalues with the second * largest real part. System pressure Psys = 0.2 MPa........ ......................... 159 7.5% oscillation curves for different segments of the SB shown in Figure 8.8........................................................................................................... 160 (a) segment A-T ..................................................................................... 160 (b) segment T-C ..................................................................................... 160 (c) segment C-D..................................................................................... 160 xvii
Figure 8.10: Figure 8.11:
Figure 8.12:
(d) segment E-F...................................................................................... 160 Stability boundary calculated for system parameters corresponding to the * Dodewaard reactor. System pressure Psys = 1.07 MPa......... ................ 162 7.5% oscillation curves for the segments of the SB shown in Figure 8.10......................................................................................................... (a) segment A-B..................................................................................... (b) segment C-D..................................................................................... (c) segment D-E ..................................................................................... * SBs for different size nodes in ∆Tinlet — ρ ext parameter space for........
163 163 163 163 165
*
(a) Psys = 7.0 MPa.................................................................................. 165 *
(b) Psys = 0.4 MPa ................................................................................. 165 *
*
Figure 8.13:
Boundaries in ∆Tinlet — ρ ext space for Psys = 7.0 MPa. The solid line is the
Figure 8.14:
stability boundary, and the dotted line is the boundary on which the second largest real part is zero........ ................................................................... 168 * * Boundaries in ∆Tinlet — ρ ext space for Psys = 0.4 MPa. The solid line is the
Figure 8.15:
Figure 8.16: Figure 8.17:
Figure 9.1:
stability boundary (SB), and the dotted line is the boundary on which the second largest real part is zero........ ....................................................... 169 7.5% oscillation curve along the SB shown in Figure 8.13......... .......... 173 (a) segment A-X..................................................................................... 173 (b) segment X-B..................................................................................... 173 (c) segment B-C ..................................................................................... 173 7.5% oscillation curve along the SB shown in Figure 8.14......... .......... 174 (a) segment D-E ..................................................................................... 174 (b) segment E-F...................................................................................... 174 Sensitivity analysis results for different operating parameters......... ..... 177 (a) System pressure ................................................................................ 177 (b) Riser length ...................................................................................... 177 (c) Pressure loss coefficient at riser exit ................................................ 177 (d) Pressure loss coefficient at channel exit........................................... 178 (e) Pressure loss coefficient at channel inlet.......................................... 178 Results of numerical simulation for operating parameters corresponding to * point P1 in Figure 8.1(a). For this point, Psys = 0.2 MPa, q ' = 6.0 kW/m, *
Figure 9.2:
and ∆Tinlet = 35.0 (K)............................................................................ 182 (a) Non-dimensional boiling boundary.................................................. 182 (b) Non-dimensional inlet velocity ........................................................ 182 Results of numerical simulation for operating parameters corresponding to * point P2 in Figure 8.1(a). For this point, Psys = 0.2 MPa, q ' = 6.0 kW/m, *
and ∆Tinlet = 31.496 (K)........................................................................ 183 (a) Non-dimensional boiling boundary.................................................. 183 (b) Non-dimensional inlet velocity ........................................................ 183
xviii
Figure 9.3:
Results of numerical simulation for operating parameters corresponding to * point P3 in Figure 8.1(a). For this point, Psys = 0.2 MPa, q ' = 6.0 kW/m, *
Figure 9.4:
and ∆Tinlet = 15.0 (K)............................................................................ 184 (a) Non-dimensional boiling boundary.................................................. 184 (b) Non-dimensional inlet velocity ........................................................ 184 Results of numerical simulation for operating parameters corresponding to * point P4 in Figure 8.1(a). For this point, Psys = 0.2 MPa, q ' = 6.0 kW/m, *
Figure 9.5:
and ∆Tinlet = 8.8 (K).............................................................................. 185 (a) Non-dimensional boiling boundary.................................................. 185 (b) Non-dimensional inlet velocity ........................................................ 185 Results of numerical simulation for operating parameters corresponding to * point P5 in Figure 8.1(a). For this point, Psys = 0.2 MPa, q ' = 6.0 kW/m, *
Figure 9.6:
and ∆Tinlet = 5.0 (K).............................................................................. 186 (a) Non-dimensional boiling boundary.................................................. 186 (b) Non-dimensional inlet velocity ........................................................ 186 Results of numerical simulation for operating parameters corresponding to * point P1 in Figure 8.10. For this point, Psys = 1.07 MPa, q ' = 379.0 kW/m, *
Figure 9.7:
and ∆Tinlet = 4.0 (K).............................................................................. 189 (a) Non-dimensional boiling boundary.................................................. 189 (b) Non-dimensional inlet velocity ........................................................ 189 Results of numerical simulation for operating parameters corresponding to * point P2 in Figure 8.10. For this point, Psys = 1.07 MPa, q ' = 379.9 kW/m, *
Figure 9.8:
and ∆Tinlet = 4.0 (K).............................................................................. 190 (a) Non-dimensional boiling boundary.................................................. 190 (b) Non-dimensional inlet velocity ........................................................ 190 Results of numerical simulation for operating parameters corresponding to * point P3 in Figure 8.10. For this point, Psys = 1.07 MPa, q ' = 256.95 *
Figure 9.9:
kW/m, and ∆Tinlet = 8.5 (K).................................................................. 191 (a) Non-dimensional boiling boundary.................................................. 191 (b) Non-dimensional inlet velocity ........................................................ 191 Operating points in the inlet subcooling—reactivity parameter space for numerical simulations of the nuclear-coupled model.......... .................. 194 * (a) Psys = 7.0 (MPa) .............................................................................. 194 *
(b) Psys = 0.4 (MPa).............................................................................. 194 Figure 9.10:
Results of numerical integration for point a in Figure 9.9(a). Perturbation at t = 0 is given by ( n1 : n1, ss → n1, ss + 0.0001).......... ................................ 196 (a) time evolution of n 0 (t ) .................................................................... 196 (b) time evolution of n1 (t ) ..................................................................... 196
xix
Figure 9.11:
Figure 9.12:
Figure 9.13:
Results of numerical integration for point b in Figure 9.9(a). Perturbations at t = 0 are given by ( n0 : n0, ss → n0, ss + 0.0001, n1 : n1, ss → n1, ss + 0.0001).......... ......................................................................................... 197 (a) time evolution of n 0 (t ) .................................................................... 197 (b) time evolution of n1 (t ) ..................................................................... 197 Results of numerical integration for point c in Figure 9.9(a). Perturbations at t = 0 are given by ( n0 : n0, ss → n0, ss + 0.003, n1 : n1, ss → n1, ss + 0.001).......... ........................................................................................... 198 (a) time evolution of n 0 (t ) .................................................................... 198 (b) time evolution of n1 (t ) ..................................................................... 198 Results of numerical integration for point d in Figure 9.9(a). Perturbations at t = 0 are given by ( n0 : n0, ss → n0, ss + 0.001, n1 : n1, ss → n1, ss + 0.001).......... ........................................................................................... 199 (a) time evolution of n 0 (t ) .................................................................... 199
Figure 9.14:
(b) time evolution of n1 (t ) ..................................................................... 199 Results of numerical integration for point e in Figure 9.9(a). Perturbation at t = 0 is given by ( n0 : n0, ss → n0, ss + 0.0001)......................................... 200 (a) time evolution of n 0 (t ) .................................................................... 200
Figure 9.15:
Figure 9.16:
(b) time evolution of n1 (t ) ..................................................................... 200 Results of numerical integration for point f in Figure 9.9(b). Perturbation at t = 0 is given by ( n0 : n0, ss → n0, ss + 0.002)........................................... 201 (a) time evolution of n 0 (t ) .................................................................... 201 (b) time evolution of n1 (t ) ..................................................................... 201 Results of numerical integration for point g in Figure 9.9(b). Perturbations at t = 0 are given by ( n0 : n0, ss → n0, ss + 0.0001, n1 : n1, ss → n1, ss + 0.0001).......... ......................................................................................... 202 (a) time evolution of n 0 (t ) .................................................................... 202
Figure 9.17:
(b) time evolution of n1 (t ) ..................................................................... 202 Results of numerical integration for point h in Figure 9.9(b). Perturbations at t = 0 are given by ( n0 : n0, ss → n0, ss + 0.001, n1 : n1, ss → n1, ss + 0.001).......... ........................................................................................... 203 (a) time evolution of n 0 (t ) .................................................................... 203 (b) time evolution of n1 (t ) ..................................................................... 203
Figure B.1: Figure B.2:
Comparisons of I6(t) evaluated with expressions (B.1), (B.2), and (B.3).......... ............................................................................................. 215 Comparisons of numerical simulations for operating point 6 on page 118 of reference [30]. Oscillations are evaluated with...................................... 216 (a) expression (B.2)................................................................................ 216 (b) expression (B.3) ............................................................................... 216
xx
PART I Stability and Bifurcation Analyses of Forced Circulation BWRs
21
Chapter1 Introduction to Instabilities of Forced Circulation BWRs
1.1 . Instability Incidents and Experiments in BWRs Boiling water reactors (BWRs) are characterized by a single-loop. That is, unlike pressurized water reactors (PWRs) that have a primary and a secondary loop, the coolant in current design of BWRs that flows through the core is the fluid that drives the turbine. Figure 1.1 shows a diagram of the Swiss KernKraftwerk power plant at Leibstadt (KKL) – a forcedcirculation BWR [1]. Subcooled water enters the cylindrical core from the bottom and starts boiling at a certain height along the reactor core. Typical exit quality and void fraction at the top of the core are about 0.15 and 0.65, respectively. Steam is separated from water by the steam separators at the top of the upper plenum and is sent to drive the turbine. The separated water, mixed with feed water, is forced by recirculation pumps, and flows back to the bottom of the core through the downcomer region. Under certain operating conditions, this BWR system becomes unstable and small perturbations, naturally present in the system, do not die out, leading to power and flow oscillations. Such power and flow instability phenomena have been observed for over two decades, and the BWR in almost all such cases had to be automatically or manually scrammed. Table 1.1 lists reports of instabilities found in literature over the last two decades. A specific example of such an instability is the event that occurred in TVO unit 1 (ABB design, 710 MWe power) on February 23, 1987. Before the event, power and core flow rate were about 60% and 30% respectively [2], and the axial power distribution had two peaks (a double-humped
1
Figure 1.1: Diagram of Swiss boiling water reactor KKL [1].
2
Table 1.1: Instability events during BWR operations and during stability tests Time
Country
BWR
Condition
1980
Finland
TVO-II
Test
1981
USA
Vermont Yankee
Test
1982
Italy
Caorso-1
Operation
1983
Italy
Caorso-1
Test
1984
Italy
Caorso-1
Operation
1987
Sweden
Forsmark-1
Test
1987
Finland
TVO-I
Operation
1988
USA
Lasalle-2
Operation
1989
Sweden
Forsmark-1
Operation
1989
Sweden
Ringhals
Test
1990
Switzerland
KKL-1
Test
1991
Germany
Isar-1
Operation
1991
Spain
Confrentes
Operation
1992
USA
WNP-2
Operation
1995
Mexico
Laguna Verde-1
Operation
1996
Sweden
Forsmark-1
Operation
1998
Sweden
Oskarshamn-3
Operation
1999
Sweden
Oskarshamn-2
Operation
3
distribution). This combination of power and flow rate along with the axial power distribution is very unfavorable to the stability of the reactor. Under these conditions, a power oscillation with peak-to-peak amplitude of about 12% was indicated by Average Power Range Monitors (APRMs). However, these signals were not noticed by the operators. What made the conditions worse was that under these conditions a routine test of the feedwater heating system was performed. During the test, a group of bypass valves failed to turn back to their normal positions, which caused a dramatic decrease in feedwater temperature. Combination of the adverse operating conditions and valve malfunction resulted in large amplitude oscillations of power and flow rate at once. A few second later, the scram limit under this flow rate, 90% power level, was reached, and the reactor was automatically shut down. On March 9, 1988, another instability event occurred in LaSalle unit 2 (GE design, BWR5, 1,130 MWe) [3], which is also the first such instability event in the U.S. The event was caused by technicians who inadvertently opened a valve during a functional test. The transient process following this mistake caused trips of both circulation pumps, and rapid flow reduction from 76% to 29%. Due to inadequate moderation of neutrons, reactor power also decreased rapidly from 84% to approximately 40%. Following power reduction, feed water heater was automatically isolated and led to a reduction in feed water temperature. A positive reactivity was thus induced into the core through void reactivity feedback. As a result, the power began to increase. Approximate 5 minutes into the event, a power oscillation was detected through APRM readings between 25% and 50% of full power with 2 to 3 s period. In addition to global power indicated by APRM signal, Local Power Range Monitor (LPRM) signals revealed occurrence of regional oscillations with different phases. The power and flow oscillations finally resulted in automatic scram. Figure 1.2 shows the power and flow rate of the actual event over a 9 minute window.
4
Figure 1.2 Relative power and flow rate during instability event of LaSalle unit 2 [3]
5
Another event took place at Laguna-verde, unit 1 (BWR/5, 1,308 MWe) in Mexico on January 24, 1995 [4]. The instability was excited by flow control valve movement at about 37% power and 37.8% flow rate. In order to shift recirculation pump to high speed, the flow control valves were closed to their minimum positions. Traces of power and flow oscillations were noticeable after 40 seconds from the beginning of valve closure. At 2 min 10 sec, APRM signal showed an oscillation with growth rate of about 1.02. At 3 min 10 sec, the valves were closed to their minimum positions, but the oscillations persisted with the same growth rate. At that point, for safety reasons the reactor was manually scrammed. The magnitude of oscillation was less than 10% of rated power during the whole instability period. Fortunately, until now safety features have always functioned as expected and no serious accidents have been caused by these instabilities. However, potential of serious failure of fuel elements still exists. Following the instability events [5-10] observed during operations, many stability tests [10-13] have been conducted in BWR plants to study behaviors of reactors under adverse operating conditions (high power and low flow rate). Some of the tests reveal that BWRs are capable of complicated large amplitude regional power and flow oscillations that may not be noticeable from APRM readings, and consequently reactors may not be scrammed in-time [14]. Because of this safety concern, extensive research interests have been focused on the stability problem of forced-circulation BWR design, widely used in commercial power plants. In addition to stability concerns for the current generation of BWRs, similar concerns also exist for next generation of BWR designs. These designs have features similar to current generation of BWRs that would necessitate their stability analyses. Moreover, some of these designs have additional features – like natural circulation for normal operation or during accident scenarios – that require additional modeling and experiments. Natural circulation is, for example,
6
achieved by the addition of long risers on top of the heated core. These long risers not only impact the core and system dynamics, they lead to the possibility of flashing (during startup, under low pressure operations) induced instabilities that may result due to flashing in the unheated riser section. Goal of this work is to address stability issues in both, forced as well as natural circulation BWR designs. There are significant differences in their modeling that the thesis is essentially divided into two parts: stability and bifurcation analyses of, 1) forced circulation; and 2) natural circulation BWRs. The first five chapters are focused on forced circulation BWRs. However, several sections in Chapter 1 are relevant to forced as well as natural circulation systems – such as discussion of dynamical systems. Chapters 6 through 10 are focused on modeling and analysis of natural circulation BWR systems. Types of instabilities for forced circulation BWRs and their physical mechanisms are described in the next two subsections in this chapter. Previous work is described in Section 1.4. A brief review of dynamical systems is given in Section 1.5, and the scope of the first part of this thesis is outlined in the last section of this chapter, Section 1.6.
1.2. Types of Instabilities Event classification process has led to identification of two different kinds of oscillations according to phase of oscillations in different regions [8]. The first one is called in-phase or corewide oscillation, which is characterized by global and uniform (across the core) power and flow changes. Figure 1.3(a) shows a diagram of flow pattern of in-phase oscillations taken from reference [15], where arrow thickness is proportional to flow intensity, and density of bubbles is proportional to void fraction. Flow rate and void fraction across any horizontal core cross section 7
is identical; i.e., the left and right half of the core experience similar power and flow characteristics at any given time. Dynamics is primarily determined by momentum dynamics of the recirculation loop and the heat transfer conditions in the core. The global property of in-phase oscillations make it easier to be detected through monitoring Average Power Range Monitor (APRM) signals. The second one is the so-called out-of-phase or regional oscillations, where two halves of the core in a horizontal plane may experience out-of-phase oscillitations. Figure 1.3(b) shows a schematic diagram of flow pattern in out-of-phase oscillations. At the beginning of a cycle (left panel, t = 0), the flow rate and void fraction in (parallel) assemblies in the left half of the core (0 < θ < π) are respectively, smaller and larger than corresponding quantities in the assemblies in the right half of the core (π < θ < 2 π). After half of the oscillation period, flow rate in the left half of the core is increased and void fraction is decreased, while the changes in the right half are opposite. Overall, the power level and total flow rate in the core as a whole stay stationary or vary with much smaller amplitude. The possibility of power and flow in half of the core oscillating 180 degrees out of phase with the oscillations in the other half poses a bigger threat to safety of BWRs than in-phase oscillations, because it is difficult to detect them from monitoring APRM signals only. Another distinction between oscillatory instabilities can be made based on the characteristics of the amplitude of oscillations. In most self-sustained oscillation cases, signals recorded indicate ever-increasing amplitudes until scram. However, in several cases, amplitudes of oscillations are found to increase at the beginning and then saturate at a constant value. The oscillations with constant periods and amplitudes are usually called limit-cycles. Existence of stable limit-cycles in some cases is considered to be a clear indication of non-linear nature of BWR dynamics.
8
(a)
(b) Figure 1.3: Flow patterns and void distributions of different types of oscillations (a) in-phase oscillations. In this case, the flow rates and void distributions in different areas in the core vary with almost the same phase; (b) out-of-phase oscillations. In this case, flow rate and void distributions in the left half of the core oscillate out-of-phase from those in the right half of the core. [15]
9
1.3. Physical Mechanisms Root causes of various instabilities have been topics of studies for two decades. It appears that some of the instability events are induced by inappropriately designed control systems. They can be fixed relatively easily by adjusting the gain of control systems. Causes of the others, however, are inner feedback mechanisms existing in BWR physics. [These can only be avoided by staying away from regions in parameter space where these instabilities are likely to occur.] Since water in BWRs acts as both coolant and moderator of neutrons, feedback mechanisms come first from thermal-hydraulics of two-phase flow, and then additionally from coupled neutron kinetics. The thermal hydraulics and neutronics mechanisms that lead to oscillatory behavior will be discussed in this section.
1.3.1. Thermal-Hydraulics Mechanism: Density Wave Oscillations (DWOs) Small perturbations in inlet flow rate in two-phase heated channel flow can cause changes in local vapor generation rate and therefore in two-phase mixture density. Transmission of the perturbation along the channel produces a perturbation in local pressure drop, which may in-turn intensify the flow rate perturbation under certain operating conditions. It has long been understood that such a self-amplified thermal-hydraulic feedback process can produce so-called density wave oscillations (DWOs) [8]. Mechanism of DWOs can be illustrated by an example. Consider a heated channel with single phase (upstream) and two-phase (downstream) flow [8]. Flow is due to a constant imposed pressure drop across the channel (see Figure 1.4). A perturbation in inlet flow, as it travels downstream, may cause local density changes in the two-phase region. Specifically, larger value of inlet flow rate causes lower steam quality and higher mixture density. Changes in density,
10
Figure 1.4: Illustration of pressure drops and inlet flow along a uniformly heated channel with constant flow cross-section [8]. The density wave oscillations occur when the phases of total pressure drop and inlet flow rate are 180± (half cycle) apart.
11
in turn, lead to perturbation in velocity. If the acoustic effect, which is indeed very small, is ignored in the channel, this density perturbation will be transmitted upward with the flow speed. Since the local frictional pressure drop depends upon density and velocity of two-phase flow, the local pressure drop along the channel is perturbed with a delay that depends upon the propagation of the perturbation. The impact of the perturbation on the total pressure drop, which is the sum of local pressure drops along the channel, will also be delayed. Since the total pressure drop must remain constant (boundary condition), perturbation in local pressure drops are compensated by changes in inlet flow rate. This leads to another round of perturbations flowing downstream. This mechanism of DWO can be used to explain why the periods of oscillations observed in BWRs are about 2~3 seconds, which is the transmission time of density wave from the bottom to the top of the channel. Other studies that suggest a lower role of “traveling” density waves have also been proposed [16].
1.3.2. Neutron Kinetics Mechanisms In light water reactors (LWRs), neutrons are moderated by water. Since level of moderation depends upon the amount of water in the core, fission rate, and hence power, in a BWR is coupled with flow conditions through void (and fuel temperature) reactivity feedback coefficients. The design principles of BWRs require void and fuel temperature feedback coefficients to be negative. While two-phase thermal-hydraulics is the primary mechanism behind the oscillation in BWR, flow rate oscillations force the power to oscillate as well. Hence, an accurate modeling of the instabilities and the oscillatory phenomenon requires a coupled analysis of reactor thermal hydraulics and reactor neutron kinetics. One approach to include neutron
12
kinetics is via a modal decomposition of the the neutron flux which can provide a clear understanding of the in-phase and out-of-phase power oscillations in BWRs. Under steady-state operating conditions, the neutron flux in a homogeneously loaded cylindrical core is given by [17] Ψ 0 (r ,θ , z ) = J 0 (2.4048r / R ) Sin (π z / L )
(1.1)
This is known as the fundamental mode. Note that power generated in a core is proportional to neutron flux. The first azimuthal mode is given by
Ψ1 (r , θ , z ) = J1 (3.8317 r / R ) Sin (π z / L ) Sin (θ )
(1.2)
where, J 0 and J 1 are Bessel functions. Shapes of the fundamental and the first azimuthal modes are shown in Figures 1.5 (a) and (b). It is well known that at steady state, higher order modes die out and only the fundamental mode exists. During a transient process, however, power distribution depends on a combination of all modes. Figure 1.5 (c) and (d) show schematic plots of variations of fundamental and azimuthal modes during a transient. If amplitude of the fundamental mode is large, power oscillations will be in-phase; whereas if the amplitude of the azimuthal mode is large, the power oscillations will be out-of-phase. In general, small cores tend to have large amplitude of the fundamental mode, and hence in-phase oscillations. However, cores of modern commercial BWRs are very large. For example, the BWR of KKL-1 (Switzerland) has 648 assemblies [1]. This leads to potential decoupling of different parts of the core. Consequently, the oscillation of neutron flux (power) in different parts of the core may have quite different characteristics, and amplitude of the azimuthal mode can be large under certain operating conditions. When such a decoupling occurs, it may lead to out-of-phase oscillations.
13
(a)
(b) ψ1(r)
ψ0(r) t2 t1 t1
t0
r
t0
r
(c)
(d)
Figure 1.5: Shapes and variations of the fundamental and first azimuthal modes, (a) 3-dimensional shape of the fundamental mode of a homogeneously loaded core; (b) 3-dimensional shape of the first azimuthal mode of a homogeneously loaded core; (c) fundamental mode during an oscillation; (d) first azimuthal mode during an oscillation.
14
Instabilities in forced-circulation BWRs have become regulatory concerns. Power-flow map of BWR is the most important operational indicator which represents coupling of power and flow rate due to void and Doppler feedbacks. Power-flow maps of BWRs have to be validated for each core loading to avoid the potential regions of instabilities. Figure 1.6 shows a power-flow map of KKL power plant (Switzerland) [18], on which three regions are identified. BWR can be normally operated on points in region I of the power-flow map, where decay ratios of the coupled system are less then 0.8. Operating conditions can only enter and pass briefly through region II, where decay ratios are less than 1.0 but greater than 0.8. Region III is the exclusion zone, where decay ratios are expected to be greater than 1.0. This region must be avoided during the operation. The exclusion region for the KKL power plant is over the range of 40~70% of the rated power and 30~45% of the rated flow [18].
1.4. Previous Work A number of simulations and analyses have been carried out to better understand the mechanisms of these instabilities. These studies, in general, followed two approaches. The first approach considers BWR to be a complicated, coupled, thermal-hydraulic, heat conduction, and neutron kinetics system, and emphasizes fidelity of simulations. Models in this approach are usually comprised of complete set of partial differential equations in space and time. Large scale multi-purpose system codes like RAMONA, RETRAN, and RELAP are used in these studies to accurately reproduce the actual instability events in computer simulations. The approach is helpful in understanding specific instability events, but the system codes are cumbersome, timeconsuming, and of limited value to explore large operating parameter spaces. The second approach, on the other hand, focuses more on qualitative aspects and parametric trends of
15
130 120
3515 MWt MEOD
Thermal Power (%)
110
%100 P=3138 MWt
100 90 80
3138 MWt MEOD
70 60
Natural Circulation
40 30
I
II
III
50
80% Rod Line
20 10 0 0
10
20
30
40
50
60
70
80
90
100 110 120
Core Flow (%)
Figure 1.6: Power-flow map of KKL plant (Switzerland) [18]. Region II is the monitoring region and region III is the exclusion zone.
16
different operating conditions. Models used in these studies, usually called reduced order models (or ROMs), are obtained by reducing the set of PDEs to a set of non-linear ordinary differential equations (ODEs) via nodalization and standard reduction methods. Rich analytical techniques in frequency and time domains are available to analyze these reduced order models. Even when numerically integrated, these models yield results much faster than the detailed models used in the system codes, and hence allow exploration of very large portions of the operating parameter space. These features of reduced order models make them good tools to understand not only specific events but also to improve general knowledge of BWRs as dynamical systems. Previous works carried out using both of these approaches will be discussed in this section.
1.4.1 Detailed Models In 1987, Takigawa et al. [19] studied out-of-phase oscillations observed during stability tests in Caorso unit 1, employing a three-dimensional neutronics and a multi-channel thermalhydraulic code, TOSDYN-2. Two cases were considered. The first one includes only one unstable channel, and the second one includes 10 unstable channels in each half of the core. The out-ofphase oscillations were successfully reproduced by the second case. In 1988, Cheng et al. [20] analyzed the instability event in LaSalle-2 using Brookhaven National Laboratory (BNL) boiling water reactor analysis code (BPA). The code employs one-group point reactor model to predict fission power in the core, and the two-phase flow is modeled using the non-equilibrium drift flux model. Their simulations of APRM signals show a "remarkable resemblance to the actual APRM traces" [20]. Some sensitivity studies are also carried out. Araya et al. [21] simulated the same event using thermal-hydraulic transient analysis code RETRAN. Their calculations considered not only the point in the power flow map at which the instability event happened, but also determined
17
the stability boundary in the subcooling-number—power-to-flow-ratio space. Results agree qualitatively with the measurements. The instability event in Forsmark-1 was studied by Analytis et al. [9] using a three-dimensional BWR transient simulation code RAMONA 3-12. The code includes a three-dimensional neutron kinetics model, which solves two-group time-dependent diffusion equations using analytical nodal method. Each assembly is considered a heated channel and other components of BWR—including lower and upper plenum, recirculation pumps and downcomers—are considered. The analysis is based on the finding that large amplitude local oscillations were found in two bundles in the core, which were suspected to be improperly seated. In these calculations, inlet pressure loss coefficients of the channels corresponding to the two positions were reduced to simulate the effects of unseated bundles. Oscillations similar to the measured data were obtained. Several other studies are discussed in reference [10].
1.4.2. Reduced-order Models Reduced order models have been used to carry out (linear) stability analysis and bifurcation analysis, as well as for numerical integration. In 1986, stability analyses were carried out by March-Leuba et al. [15] [22]. They used a very simple, one-channel model, which contains one ODE for heat transfer, two ODEs for thermal-hydraulics to account for void reactivity feedbacks, and two ODEs for point reactor kinetics – one for power and one for delayed neutron precursor concentration. The system of 5 ODEs was integrated numerically for given operating parameters and initial conditions. Typical linear and nonlinear system behaviors, such as diverging and converging oscillations and period doubling, were observed. This analysis was, later, extended to a two-channel model with fundamental as well as first azimuthal modes for neutronics analysis [17]. The modification in the neutron kinetics model was integrated in the
18
frequency domain code LAPUR [23]. This analysis showed that different combinations of stable and unstable fundamental and first azimuthal modes may occur in different regions of power— flow map. A more detailed, and consequently complicated, reduced order model was developed by Karve et al. in 1997 [24]. This model includes detailed ω-mode neutron kinetics, fuel rod heat conduction model, and single- and two-phase channel thermal-hydraulics. Stability analysis was carried out, and numerical integrations were performed to determine the nature of oscillations. Neutron kinetics in the reduced order model developed by Muñoz-Cobo et al. [25], [26] is based on the λ-mode. Lumped equations were used to account for fuel rod heat transfer model and for channel thermal-hydraulics. Stability analyses of their model were conducted by numerical simulations, and out-of-phase oscillations were excited when the void feedback coefficients was gradually increased. Tsuji et al. [27] developed a reduced order model, which includes point reactor kinetics, a conduction heat transfer model and heated channel thermal-hydraulics. Stability boundary is determined for different external pressure drops and for external reactivity in the power-flow map. Sensitivity analysis of stability boundaries for power distributions and several operating parameters were presented. They also carried out a bifurcation analysis. The nature of PoincareAndronov-Hopf bifurcation is studied using the bifurcation code, BIFOR2. Numerical simulations were carried out to validate results of BIFOR2. In another study, asymptotic expansion method was used by Muñoz-Cobo and Verdu [28] for bifurcation studies. Complexity of this method increases dramatically for more detailed models. Rizwan-uddin [29] used another bifurcation code, BIFDD, to study the dynamics of the simple BWR model proposed by March-leuba et al. [15]. Stability boundaries in several parameter spaces as well as the nature of bifurcation were determined. Nonlinearity of the system was also
19
studied using numerical simulations. Period doubling and existence of a turning point in the bifurcation diagram were confirmed. Dokhane et al. [18] improved the model developed by Karve et al. by incorporating draft flux model in two-phase flow modeling and using more realistic reactivity feedback in nuclear coupling. Stability and bifurcation properties of the model are analyzed using code BIFDD. It is shown that both the void distribution parameter C0 and the drift velocity Vgj have stabilizing effects.
1.5. Tools to Analyze Dynamical Systems As mentioned in the previous section, objectives of BWR stability analyses include:
• determine stability margins of BWRs under normal and during transient conditions • predict dynamical behavior of the reactors in instability events • help BWR designers and operators limit occurrence and impact of instabilities Moreover, rather than using the detailed system codes, this study is based on the reduced order model approach discussed above. To achieve the goals itemized above, reduced order models of BWRs can be analyzed using different mathematical techniques in frequency and time domains. We here briefly review these mathematical techniques.
1.5.1 Frequency Domain Methods (Classical Stability Analysis) The non-linear, time-dependent ODEs of the models can be linearized and transferred to frequency domain through Laplace transform. Open and close-loop transfer functions can be deduced through the Laplace transform and control system block diagram. Usually, the form of the close-loop transfer function is given by a polynomial of order n, divided by another polynomial of order m (> n). Root-locus method calculates all m roots (poles) of the denominator 20
of the close-loop transfer function. Trajectories of the poles are plotted in the complex plane. If the trajectories stay to the left of the imaginary axis, the system is stable (because poles with negative real parts correspond to solutions in the time domain with negative exponents). Nyquistdiagram technique is another technique to determine stability of the system in frequency domain. It uses open-loop transfer function instead of closed-loop transfer function. The Nyquist diagram is actually a trajectory in complex plane of the open-loop function G (i w) , where w is frequency varying from − ∞ to + ∞ . Two rules to determine stability of the system are: if the open-loop transfer function is stable, the condition of stability of the close-loop system is that the trajectory of G (i w) does not encircle the point (-1, 0); if the open-loop transfer function is unstable and has k poles with positive real parts, the condition of stability of the close-loop function is that the trajectory of G (i w) (counter-clockwise) encircles the point (-1, 0) k times. Classical stability analysis divides the parameter space into stable and unstable regions. Infinitesimal perturbations grow when operating in the unstable region and decay when operating in the stable region. Results are not applicable to large amplitude perturbations, nor for large time. That is, in case of unstable behavior, for example resulting in oscillations with growing amplitude, classical stability analysis sheds no light on the long term amplitude of the resulting oscillations – they may continue to grow or may saturate at a certain value.
1.5.2. Bifurcation Analyses While classical stability analysis divides the parameter space into stable and unstable regions, bifurcation analyses go a step further, and shed some light on the nonlinear characteristics of the system. [Note that in very limited cases results of bifurcation analyses may be global. Most results are local in nature – obtained via perturbation methods – and hence valid in region close to
21
the point about which perturbations were carried out.] For example, bifurcation analysis may provide results for perturbations that are larger than “infinitesimal”. It is possible for a “steadystate” (fixed point) to be stable for “infinitesimal” perturbations (according to classical (linear) stability analysis) but unstable if perturbations larger than the specified amplitude are applied. Bifurcation analysis may also identify regions in parameter space (close to the stability boundary) where the steady-state is unstable but there exist stable periodic solutions. Since the classical stability and bifurcation analysis methods will be used as the primary mathematical tools in this thesis, they will be discussed in detail in the second chapter.
1.5.3. Numerical Integration Reduced order models are generally based on a set of nonlinear, ordinary differential equations (ODEs). Classical numerical integration methods, such as Runge-Kutta, Gear or Adams-Bashforth method, can be applied to solve the ODEs. Since, in many cases, the ODEs are stiff, it generally requires A-stable or α-stable versions of the methods. The numerical integration of the dynamical system does not rely on any assumptions on the perturbation amplitude – as do methods to determine stability and bifurcation characteristics of the system – and hence it is very useful in testing the validity of the results obtained using other methods. It is also the only means to determine system dynamics under conditions where other analytical methods are invalid – such as in regions away from the stability boundary where the results of the local bifurcation analysis are not applicable.
1.6. Scope of Part I of This Thesis
22
A number of reduced order models have been developed and analyzed to reveal mechanisms and parametric trends of instabilities of forced-circulation BWRs. However, there are several limitations in these studies: • Some reduced order models are overly simplified. For example, the pioneering work by
March-Leuba et al. is based on a model that has only 5 ODEs to represent thermal-hydraulics, heat conduction and neutron kinetics of BWRs. Realizing that the focus of that study was “nonlinear dynamics,” and the physics of BWRs was captured using empirically fitted constants to an actual transfer function, the over-simplification in the model may lead to loss of some key details of BWR dynamics. It is therefore necessary to study BWR dynamics using a model that better balances simplicity and accuracy. • Most studies, even those carried out using reduced order models, have focused on numerical
simulations and classical stability analyses, and relatively few models have been analyzed for their bifurcation characteristics. This is due to the fact that complexity of bifurcation analysis increases dramatically with number of equations in reduced order models. Therefore, only rather simple reduced order models have been used for studies, such as those by Tsuji et al. [27], Muñoz-Cobo et al. [28] and Rizwan-uddin [29]. Moreover, even when carried out, bifurcation characteristics were determined for only one parameter, consequently limiting the scope of these analyses to a small region of the rather large parameter space. Because of the approaches usually followed for the bifurcation analyses, it is not trivial to repeat them for different system parameters as bifurcation parameter. • In-phase oscillations have attracted more attention than out-of-phase oscillations. Instability
events and numerical simulations have shown transition from in-phase to out-of-phase oscillations as operating parameters change, and it is premised that this transition has close relationship with
23
the radial in-homogeneity of the core. However, to the author’s knowledge, the question of how the transition occurs and the mechanism through which inhomogeneity helps excite the out-ofphase oscillations have not been adequately answered. Purpose of the study presented in part I of this thesis is to tackle the above-mentioned limitations of existing studies. Specifically, a detailed model developed by Karve et al. [24] is used for steady-state, stability and bifurcation analyses. Model is also numerically integrated. This model is chosen because it represents a healthy balance between the level of detail, yet simple enough to be amenable to the tools of analyses available. Components of this model are briefly reviewed in the next chapter. A modification introduced in the model to artificially model radial core inhomogeneity is discussed in more detail. After the model, a brief introduction to bifurcation theory and the semi-analytic approach that makes it possible to perform bifurcation analysis without massive algebra are presented. In chapter three, results of stability and bifurcation analyses are presented and discussed. These are presented first for the homogeneous core, and then for the inhomogeneous core. Discussion of the complex relationship between inhomogeneity and in-phase and out-of-phase oscillations, as well as insight gained in understanding BWR dynamics from the results of the bifurcation analysis are particular focus of this chapter. Results of numerical integration performed for operating points in relevant parameter spaces are presented in Chapter 4. These results are used to verify predictions of bifurcation analyses. Chapter five summarizes stability and bifurcation analyses of forcedcirculation BWRs.
24
Chapter 2 A Reduced-Order Model of Forced-Circulation BWR and Review of Bifurcation Analysis Methods
Karve et al. [30] developed a reduced order model to study dynamics of forced-circulation BWRs. Although the model is simple, it is much more detailed than the 5-equation model used by March-Leuba et al [31] and retains several key details of BWR dynamics. The balance between simplicity and accuracy of this model make it a good candidate for stability and bifurcation studies. In this chapter, a brief review of the model is given in the first section. Emphasis is placed on the neutron kinetics part of the model, where a modification is introduced. The second section of the chapter is dedicated to a review and discussion of Poincare-Andronov-Hopf bifurcation theorem. This is a key theorem used in this work to identify regions of parameter space (close to stability boundaries) where stable or unstable limit cycles exist.
2.1. Forced-Circulation BWR Model The complete model can be divided into three components: neutron kinetics, fuel rod heat conduction and channel thermal-hydraulics. Complete and detailed derivation of the model can be found in [30]. Only those components of the model that are relevant to the analyses and discussion in the latter part of the thesis are reviewed below.
2.1.1. Neutron Kinetics
25
Governing equations of the neutron kinetics are derived from the matrix form of the onegroup neutron diffusion and one delayed neutron precursor equations
dY =M Y dt
(2.1)
where v∇ ⋅ D (r , t )∇ − vΣ a (r , t ) + (1 − β )vνΣ f (r , t ) λ M = − λ vβν Σ f
Y (r , t ) = [N (r , t ), C (r , t )]
T
(2.2)
(2.3)
where N (r , t ) and C (r , t ) are neutron and precursor densities, v is neutron velocity, ν is average neutrons released per fission reaction. These equations and reduction to their modal counterparts can be found in textbooks [15]. A one-group treatment for neutron kinetics is sufficient because of the time-scale of oscillations that are being modeled here [15]. Steady-state eigenvalue ω ik and eigenfunction φ ik of M satisfy
M 0 (r )φ ik (r ) = ω ik φik (r ) ,
(2.4)
1 ψ k ; i = 0, 1 ; k = 0,1 vβν Σ f (r ) (ω ik + λ )
φik (r ) = (2.5)
where ψ k , k = 0, 1 are fundamental and azimuthal modes of the steady-state matrix M 0 . For homogeneously loaded core, functions and shapes of the modes are shown in section 1.3.2 of chapter 1. The space and time dependent vector Y (r , t ) , thus, can be expanded in the eigenfunctions
Y (r , t ) = u 00 (t )φ 00 (r ) + u 01 (t )φ 01 (r ) + u10 (t )φ10 (r ) + u11 (t )φ11 (r )
26
(2.6)
where, u ik (t ) , i = 0, 1, k = 0, 1 are time dependent expansion parameters. The ω -mode based equations are developed by plugging the expansion (2.6) into equation +
(2.1) and taking inner product of both sides using the adjoint eigenfunction φik (r ) . This leads to a set of two ODEs for each mode k = 0, 1 1 ρ dn k (t ) = ω ik n k (t ) + (ω1k − ω 0 k )u k (t ) + ∑ km n m (t ) dt m =0 Λ k
(2.7)
1 ρ du k (t ) = ω1k u k (t ) + ∑ km n m (t ) , dt m = 0 Λ 1k
(2.8)
where nk (t ) = u0 k (t ) + u1k (t )
(2.9)
u k (t ) = u1k (t ) Λ ik =
(2.10)
φ ik+ (r ), φ ik (r )
V
ψ k (r ), vυΣ f (r )ψ k (r )
(2.11) V
n k , k = 0, 1 are time-dependent neutron densities of the fundamental and first azimuthal modes; u k , k = 0, 1 are time-dependent precursor densities of the fundamental and first azimuthal modes; Λ ik , i = 0, 1, k = 0, 1 are effective neutron generation time, and
V
denotes the inner product
over the volume of the core. The governing ODEs for neutron kinetics given above are coupled with fuel rod heat conduction and channel thermal-hydraulics through reactivity feedback ρ km (t ) =
1
∑ρ
ext , kml
− cα ,kml (α l (t ) − α 0,l ) − c D , kml (Tavg ,l (t ) − Tavg ,0,l )
l =0
(2.12)
27
where ρ ext is the control rod induced external reactivity, cα ,kml and c D ,kml are void coefficient and Doppler feedback coefficients for different modes and channels. α l and Tavg ,l are average void fraction and temperature of fuel pellet for different channels. Here the subscript l represents the lth channel. The core must be divided into at least two channels (l = 0, 1) to capture the feedback between modal neutronics with fundamental and first azimuthal modes, and core thermal hydraulics. α l and Tavg ,l are determined as intermediate variables in the solution of the heated channel thermal-hydraulics and the heat conduction parts.
2.1.2. Consideration of In-Homogeneity Parameters in the governing equations of neutron kinetics, such as cα ,kml , c D ,kml , ω km and Λ ik , depend on inner products of the modes and their adjoint functions. These parameters were evaluated by Karve et al. [30] using the fundamental and azimuthal modes given by equations (1.1) and (1.2). They are simply the Bessel’s functions in radial direction, which, however, are valid only for cylindrical cores that are not far from homogeneous configurations. There are several factors that can lead to in-homogeneous cores, which in-turn will lead to modes that may be quite different from the Bessel functions. These factors include: non-uniform burn-up, asymmetric control rod insertion, and variations in core inlet temperature between different (lumped) channels. Analysis of operational data from reactors that experienced instability events suggests that occurrence of out-of-phase oscillations may be closely related to the in-homogeneity of the core. For example, Miró et al. [12] found that radial power distributions that roughly correspond to the shape of Bessel functions, generally tend to generate in-phase oscillations; while bowl-shaped radial power distributions, with a local minimum at the center, tend to produce outof-phase oscillations. Hence, accurate evaluation of neutron kinetics parameters for different 28
loadings – including the inhomogeneity effects – is necessary to correctly predict characteristics of oscillations. Quantitative representation of different loadings in in-homogeneous cores requires a modal decomposition approach [32]. In this approach, loading pattern, burn-up history and control rod insertion of the core are specified, and 2-D or 3-D neutronics codes are used to solve the core configuration for eigenvalues and eigenfunctions. The evaluation process is fairly complicated and time-consuming. Moreover, it is case-specific, which eliminates the biggest strength of the reduced-order models. An alternate way to account for in-homogeneity of different core loadings is to modify system parameters to parametrically simulate the effects of inhomogeneous core loading. Recognizing that the major impact of core in-homogeneity on the dynamics of the core is to change the relative likelihood of in-phase or out-of-phase oscillations, reactivity feedbacks of fundamental and first azimuthal modes are the most appropriate parameters to be modified to incorporate the effects of inhomogeneous core on core dynamics. It is therefore appropriate that these feedback coefficients be varied, for example by multiplying by a suitable factor, to reflect relative strength of reactivity feedbacks for different cores. Observations and simulations have already confirmed that most oscillations for roughly homogeneous cores are in-phase. It indicates that reactivity feedback of azimuthal mode is much smaller than that of the fundamental mode in this case. On the other hand, occurrence of out-ofphase oscillations in in-homogeneous cores depends on how much the in-homogeneity can increase the feedback to the azimuthal mode. Consideration of in-homogeneity, therefore, leads to the introduction of an amplification factor F to modify reactivity feedback coefficients of the azimuthal mode,
cα ,1ml ,nu = F ⋅ cα ,1ml ,u
29
(2.13)
c D ,1ml ,nu = F ⋅ c D ,1ml ,u
(2.14)
where, cα ,1ml ,nu and c D ,1ml ,nu are feedback coefficients of an in-homogeneous (or non-uniform, hence the subscript nu) core; and cα ,1ml ,u and c D ,1ml ,u are feedback coefficients of a homogeneous (or uniform, hence the subscript u) core. Although there are other parameters that are affected by the in-homogeneity, their impact on dynamics of the modes are assumed to be minimal, and their values are held constant for different core configurations.
2.1.3. Heated Channel Thermal Hydraulics Heated channel thermal hydraulics model presented in this section and the fuel rod heat conduction model presented in the next section are taken from reference [30]. Except for one mistake found in reference [30] in the derivation of the reduced order model – identified and corrected below – these two models used here are identical to those in reference [30]. Most of the details in deriving these models are hence omitted. They can be found in reference [30]. A one-dimensional model is used to simulate water flowing through a heated channel in a BWR core. The channel consists of single-phase and two-phase regions. Ordinary differential equations (ODEs) describing dynamics of two key variables – water enthalpy in the single-phase region and steam quality in the two-phase region – are developed through conservation equations. Non-dimensional energy conservation equation (for enthalpy) in single phase region for a channel with spatially uniform but time-dependent heat flux is given by ∂h ( z , t ) ∂hm ( z , t ) = N ρ N r N pch ,1φ (t ) + vinlet (t ) m ∂z ∂t
(2.15)
where hm ( z , t ) is water enthalpy, and N pch ,1φ is the phase change number in the single-phase region, which is proportional to input heat flux. Definitions of dimensionless variables and
30
parameters are given in Appendix A. A quadratic polynomial in space, with time dependent coefficients, is assumed to approximate the enthalpy hm ( z , t ) = hm (0, t ) + a1 (t ) z + a 2 (t ) z 2 . ODEs for the time-dependent expansion parameters a1 (t ) and a2 (t ) are derived using classical weighted residual approach. Their forms are given by
[
]
da1 (t ) 6 = N ρ N r N pch ,1φ (t ) − vinet (t )a1 (t ) − 2vinlet (t )a 2 (t ) dt µ (t )
[
da 2 (t ) 6 =− 2 N ρ N r N pch ,1φ (t ) − vinlet (t )a1 (t ) dt µ (t )
]
(2.16)
(2.17)
where µ (t ) is the location of the boiling boundary, which separates the single-phase and the twophase regions. Above equations are applicable over 0 < z < µ(t). A similar approach is followed to derive equations for steam quality in the two-phase region; µ(t) < z < 1. For the homogeneous equilibrium model of the two-phase flow, the nondimensional continuity equation is [30]
∂ρ m ( z , t ) ∂ρ ( z , t ) + v m ( z, t ) m = − N pch, 2φ (t ) ρ m ( z , t ) ∂t ∂z
(2.18)
where, vm ( z , t ) = vinlet + N pch ,2φ ( z − µ (t )) is fluid mixture velocity. The mixture density ρ m ( z , t ) is expanded as
ρm ( z, t ) =
1 1 + s1 ( t ) ( z − µ (t )) + s2 ( t ) ( z − µ (t )) 2
(2.19)
where s1 (t ) and s 2 (t ) are the time-dependent expansion parameters for the steam quality. By applying the weighted residual method to the two-phase continuity equation, the ODEs for the expansion parameters s1 (t ) and s2 (t ) are determined as ds1 (t ) 1 = ( f 3 (t ) f 1 (t ) + f 4 (t )) dt f 2 (t )
31
(2.20)
ds 2 (t ) 1 = ( f 5 (t ) f 1 (t ) + f 6 (t )) dt f 2 (t )
(2.21)
where f1(t), f2(t), f3(t), f4(t), f5(t) and f6(t) are complicated functions of a1 (t ) , a2 (t ) , s1 (t ) , s2 (t ) and vinlet (t ) . Details and complete expressions of fi(t), i = 1, 2, …, 6 can be found in reference [30]. There is however one mistake in reference [30]. The expression for I6(t) given in Appendix E of the reference [30], which is an intermediate function in the expressions of fi(t), i = 1, 2, …, 6, has an error. The error occurred due to the fact that not enough terms were kept in expansions necessary for evaluation of certain expressions. The correct expression is δ 13 (t ) δ 1 2 (t )[3 − δ 1 (t )δ 3 (t )] δ e (t ) + + log 1 − 1 − δ 1 (t )δ 3 (t ) δ 1 (t ) δ 1 (t ) − δ e (t ) 2 δ e (t ) − δ 2 (t ) + 2δ (t )δ (t ) + 1 e 1 2 1 I 6 (t ) = 2 3 s1 (t ) − 4 s 2 (t ) 2δ e (t ) 2 2 3 + δ e (t )δ 1 (t ) + 2δ e (t )δ 1 (t )δ 3 (t ) + 3δ 4 (t ) 2δ 3 (t )δ (t ) 2 2 3 2 1 e + e + δ e (t )δ 1 (t ) + 2δ e (t )δ 1 (t )δ 3 (t ) 3 4
(2.22)
Impact of this error on the results of the stability analysis reported in reference [30] are minimal, and are discussed in Appendix B ODE for inlet flow velocity vinlet (t ) is obtained by integrating the momentum conservation equations over both the single-phase and two-phase regions. Constant pressure drop boundary condition across the length of the channel is imposed [30]. Form of the ODE is given by dvinlet (t ) 1 ( f8 (t ) + f9 (t ) f1 (t ) + f10 (t ) f11 (t ) ) = dt f 7 (t ) where fi(t), i = 7, 8, … 11 are complicated functions of a1 (t ) , a 2 (t ) , s1 (t ) , s 2 (t ) , and vinlet .
32
(2.23)
2.1.4. Heat Conduction Model For fuel rod heat conduction model, non-dimensional energy equation for temperature distribution in cylindrical fuel pellet, assuming azimuthal symmetry, is given by 2 1 ∂θ p (r , t ) ∂ θ p (r , t ) 1 ∂θ p (r , t ) = + + c q (n0 (t ) − n~0 ) 2 ∂t r ∂r αp ∂r
(2.24)
A gap conductance is considered between the fuel pellet and cladding. Because the conductivity of the cladding is much higher than that of the fuel pellet, to simplify heat conduction in the gap and cladding region the transient temperature distribution in the cladding, θ c (r , t ) , can be approximated by its steady-state form. Thus, the temperature distribution in the cladding and gap conductance is lumped into boundary condition for pellet temperature θ p (r , t ) . Due to large temperature gradient in the fuel pellet, a single quadratic profile over the entire pellet was found to not be adequate [30]. Hence, the pellet temperature distribution,
θ p (r , t ) , is represented using quadratic profiles over two segments, given by T1 (t )v1a (r ) + T2 (t )v1b (r ) T1 (t )v2 a (r ) + T2 (t )v2b (r )
θ p (r , t ) =
0 < r < rd rd < r < rp
(2.25)
where rp is radius of the fuel pellet, and rd = 0.83r p is an empirically determined value [30]. v1a (r ) , v1b ( r ) , v 2 a (r ) and v 2b (r ) are quadratic polynomials which are determined by imposing the boundary and interface conditions. To derive the ODEs for T1 (t ) and T2 (t ) in the single-phase and the two-phase regions, variational principle is applied. The unknown expansion parameters T1 (t ) and T2 (t ) that minimize the given variational [30] satisfy the ODEs dT1 (t ) = l1,1T1 (t ) + l 2,1T2 (t ) + l3,1c q (n0 − n~0 ) dt
33
(2.26)
dT2 (t ) = l1, 2T1 (t ) + l 2, 2T2 (t ) + l 3, 2 c q (n0 − n~0 ) dt
(2.27)
where l1,1, l2,1, l3,1, l1,2, l2,2 and l3,2 are constants that depend upon the operating parameters.
BWR as a dynamical system is hence represented by the combination of the three components: two ODEs for neutron kinetics for each mode (equations (2.7) and (2.8)), five ODEs for channel thermal hydraulics for each half of the core (equations (2.16), (2.17), (2.20), (2.21) and (2.23)); and two ODEs for the heat conduction model in the pellet for single phase and two phase regions (equations (2.26) and (2.27)). The evaluation of the right hand sides of these ODEs requires the evaluation of the steadys1 depend on v~inlet and n~0 , and all state values of the variables first. At the steady-state, a~1 and ~
other variables are 0. v~inlet and n~0 can be evaluated by imposing balance of reactivity and pressure drop at steady-state. [Tilde (~) here represents a steady-state quantity.] The model was tested by Karve [30] to validate the assumptions of quadratic representations of water enthalpy profile in the single-phase region and steam quality profile in the twophase region. Details can be found in Appendix C of reference [30].
2.2. Stability and Bifurcation Analysis Method Stability analysis (frequency domain), bifurcation analysis, or numerical integration (time domain) methods, as stated in the section 1.5, can be used to study dynamics of the BWR model. They are major mathematical tools for study of the reduced-order model in this dissertation. In this section, the stability and bifurcation analysis methods are discussed in more details than the section 1.5. Then a numeric-analytic approach to reduce complexity of evaluation is introduced.
34
This discussion will serve as background information to understand results presented in the next chapter. The numerical integration (time domain) method, although is important for analysis, is quite straightforward, and not repeated here.
2.2.1. Stability analysis of a BWR as a dynamical system The reduced order BWR model is formulated and represented by a set of nonlinear ODEs. It can be considered a dynamical system, whose behavior can be analyzed through mathematical tools developed for this purpose. General form of autonomous dynamical system can be written as dF = G (F ; λ ) dt
(2.28)
where, F = [F1
F2 … Fn ]
T
λ = [λ1 λ 2
[
G ( F , λ ) = G1 ( F ; λ ) G2 ( F ; λ )
(2.29)
λ m ]T Gn ( F ; λ )
(2.30)
]
T
(2.31)
F1, F2, … Fn are a set of time-dependent variables (also called phase-variables); λ1, λ2, … λm are
operating parameters; and G1, G2, … Gn are non-linear functions of F and λ . Steady-state solutions (or fixed points) of the dynamical system, F (t ) = F0 (λ ) , satisfy the condition
G ( F0 ; λ ) = 0
(2.32)
Note that there may be more than one fixed points corresponding to a given set of parameter values. Fixed points are simply the steady-state operating points of the dynamical system. Fixed points may be stable or unstable. Hence, safe operation of systems such as BWRs require that
35
stability of their steady-state operating points under different operating conditions be determined. Stability of a fixed point is governed by the eigenvalues of the Jacobian matrix. Jacobian of the dynamical system, A-double-bar, is a matrix of derivatives of Gi, i = 1, 2, …, n with respect to Fj, j = 1, 2, …, n. It can be written as
∂G1 ∂F 1 ∂G2 A = ∂F 1 ∂G n ∂F1
∂G1 ∂F2 ∂G2 ∂F2 ∂Gn ∂F2
∂G1 ∂Fn ∂G2 … ∂Fn ∂Gn … ∂Fn λ …
(2.33)
It is well known that stability of a fixed point depends on eigenvalues of the Jacobian matrix evaluated at that fixed point. Specifically: a) If all eigenvalues of the Jacobian matrix have negative real parts, the fixed point is stable. Infinitesimal perturbations will die away and the trajectory in phase space will converge to
the fixed point. b) If one or more of the eigenvalues has positive real parts, the fixed point is unstable. Infinitesimal perturbations will grow and will lead to a trajectory in phase space that diverges away from the fixed point. c) If the rightmost eigenvalue has a zero real part (lies on the imaginary axis), the fixed point is neutral. Stability of the fixed point is then determined by higher order analysis. Clearly, since the fixed points depend on the operating parameters λ , the parameter space can be divided into stable and unstable regions, divided by a stability boundary (SB). A stability boundary which separates stable regions from unstable regions can be calculated by simply tracking the sets of (critical) operating parameters for which the rightmost eigenvalue, i.e., the eigenvalue with the largest real part, lies on the imaginary axis.
36
It is necessary to note that the results of (linear) stability analysis are only valid for infinitesimal perturbations. Results of local (linear) stability analysis are not necessarily applicable
to perturbations that are larger than infinitesimal. The results of stability analyses also do not shed any light on the system behavior as time increases and amplitudes of phase variables become finite. Specifically, the experimentally observed stable limit cycle oscillations cannot be predicted by stability analysis alone. System dynamics following large perturbations and the determination of amplitudes of oscillations when oscillations grow into a stable limit cycle is discussed in the next sub-section.
2.2.2. Poincaré-Andronov-Hopf Bifurcation (PAH-B)
Poincaré-Andronov-Hopf bifurcation (PAH-B) theorem concerns about dynamical properties of another type of special solution of the system: periodic solution, which satisfy F1 (t ; λ ) = F1 (t + T ; λ )
(2.34)
From any point on this solution, the system will come back to the same condition after a fixed period T. The periodic solution clearly corresponds to stable amplitude oscillation (limit cycle) of BWR. The theorem predicts existence of periodic solutions, branching from fixed points, when λ is close to its critical value and the following two conditions [33] are satisfied: a) a pair of pure imaginary eigenvalues ( ± iω c ) of the Jacobian of the system exists at
λ = λcritical and the real parts of all the other eigenvalues are negative. b) the real part of the complex conjugate eigenvalues with zero real part crosses the imaginary axis with non-zero speed as λ is varied, i.e.
d (real part ) dλ
The periodic solutions, which are indexed by a constant ε , are given by
37
≠ 0. λ = λcritical
F1 (t ; λ (ε )) = F0 (λc ) + εv e iω (ε ) t + O(ε 2 )
(2.35)
where, F0 (λc ) is the fixed point of the system. v is the eigenvector corresponding to the pure imaginary eigenvalue iω c normalized such that its first non-zero element is equal to one,. v = [v1
vn ]
T
v2
(2.36)
Constant ε is the so-called “index constant” which specifies amplitude of the periodic solution.
λ (ε ) is the set of operating parameter which corresponds to the periodic solution of ε . ω (ε ) is frequency of the periodic solution. Similar to the fixed point, periodic solution can be stable or unstable. Figure 2.1 shows examples of stable and unstable periodic solutions in two-dimensional phase variable space. The x-axis F1(t) and y-axis F2(t) are two phase variables of the system. For stable periodic solution (Figure 2.1(a)), if the system starts at a condition perturbed from the periodic solution, trajectory of the time-dependent variables will be attracted to the periodic solution as time evolves. For unstable periodic solution (Figure 2.1(b)), if initial condition of the system is at a point only slightly deviating from the periodic solution, trajectory of states will be expelled by the periodic solution. Stability of periodic solutions, whose existence is guaranteed by the PAH bifurcation theorem, is determined in a way similar to that of fixed points, i.e. by introducing a perturbation to the periodic solution, and then analyzing whether the perturbation grows or decays. Difference between them is that grows or decays of perturbations for the periodic solutions are analyzed through Poincaré map [34]. Suppose an arbitrary perturbation is introduced to a periodic solution at t = 0
F (t = 0; λ (ε )) = F1 (t = 0; λ (ε )) + I 0 e
38
(2.37)
3
0
0
y
y
3
-3
-3 -3
0
3
-3
0
x
3
x
(a)
(b)
Figure 2.1: Examples of stable and unstable periodic solution. ODEs and parameters of the system can be found in page 148 of reference [34]. (a) stable periodic solution. Orbits close to the periodic solution is attracted when time evolves. (b) unstable periodic solution. Orbits close to the periodic solution is expelled.
39
where I 0 is an n × n identity matrix, e is a vector of arbitrary perturbation. At t = T, the perturbation becomes F (t = T ; λ (ε )) = F1 (t = T ; λ (ε )) + I T e
(2.38)
where, I T is a n × n matrix called monodromy matrix, which maps the initial condition I 0 to the matrix at t = T. Since F1 (t = 0; λ (ε )) = F1 (t = T ; λ (ε )) , grow or decay of the perturbation after each circle of period T can be determined by analyzing properties of the monodromy matrix. Suppose eigenvalues of the monodromy matrix are given by
ρi = eβ T , i
i = 1, n
(2.39)
where β i , i = 1, n are called Floquet exponents. The perturbation decays if all the Floquet exponents have negative real parts. The periodic solution is stable and the corresponding bifurcation is called supercritical PAH bifurcation. Otherwise the perturbation grows in magnitude. The periodic solution is unstable leading to what is called subcritical PAH bifurcation. For the supercritical bifurcation, stable periodic solutions can only co-exist with unstable fixed points. For the case of subcritical bifurcation, unstable periodic solutions can only co-exist with stable fixed points. The reason is easy to understand: if there were stable periodic solution coe-existing with stable fixed point for a specific set of operating parameters, they will both attract trajectories and there must be another unstable periodic solution repelling trajectories to separate them. Similarly, if there were unstable periodic solutions co-existing with unstable fixed points, they will both repel trajectories and there must be another stable periodic solution attracting trajectories to separate them. Table 2.1 lists all possible combinations of properties of fixed points and periodic solutions for a specific set of operating parameters close to stability boundary. For supercritical bifurcation, if the system is operated on a set of operating parameters
40
Table 2.1. Properties of fixed points and periodic solutions close to the stability boundary
Supercritical PAH-B
Subcritical PAH-B
Stable Region
Unstable Region
Fixed point is stable; Periodic
Fixed point is unstable;
solution does not exist
Periodic solution is stable
Fixed point is stable; Periodic
Fixed point is unstable;
is unstable
Periodic solution does not exist
41
of stable fixed points, any perturbation from the fixed point will decay. If it is operated on parameters of unstable fixed point, small perturbation from the fixed point will grow. However, the amplitude of the perturbation will finally saturate because of existence of a stable periodic solution in this case. For subcritical bifurcation, if the system is (linearly) stable, small perturbation not crossing bond of periodic solution will decay, but large perturbation crossing the bond will grow because of existence of the unstable periodic solution. If the system is (linearly) unstable, i.e. operated on parameters of unstable fixed points, even small perturbation will grow un-bonded. From safety point of view, operating conditions corresponding to the supercritical bifurcation is clearly favored because of the inherent bonded nature of the bifurcation.. Operating conditions to the subcritical bifurcation is un-favored because of the inherent un-bonded nature of the bifurcation.
2.2.3. Semi-Analytical Method and BIFDD
Quantitative evaluations of the periodic solutions requires calculations of the vector of operating parameter λ (ε ) , frequency ω (ε ) , and the Floquet exponent with the large real part
β (ε ) as functions of the index constant ε . A pure analytical method employing LindstedtPoincaré asymptotic expansion [33], [34] can be applied for the evaluation. The unknown parameters are expanded to polynomials of ε
1 ω (ε ) = 1 ω c + τ 2 ε 2 + O(ε 4 )
(2.40)
λ (ε ) = λc + µ 2 ε 2 + O(ε 4 )
(2.41)
β (ε ) = 0 + β 2ε 2 + O(ε 4 )
(2.42)
Substitute the expansions with the parameters in periodic solutions. A set of equations can be obtained by equating coefficients of like powers of ε on left and right hand sides of the ODEs. 42
Expansion parameters, then, can be acquired through solving the equations. It is noted that expansion forms (2.40), (2.41), and (2.42) ignore the first and third order terms of the ε . This is because the solutions of the equations of like powers of ε will lead to expansion parameters of the first order and third order terms equal to zero. Bifurcation analysis requires evaluation of µ 2 , τ 2 and β 2 . Analytical evaluation of µ 2 ,
τ 2 and β 2 , even for reduced order simple models, is a very tedious task and quickly becomes almost impossible with increasing number of differential equations in the model. Another drawback of the analytical bifurcation studies is that each is specific to a bifurcation parameter, and must be repeated if the impact of a different parameter is to be studied. Numerical integration (time domain method) can be used to determine bifurcation properties of the system, but is also time consuming and leads to results valid for specific set of parameter values. Due to the limitations on the two approaches (numerical integration and analytical bifurcation studies), there has been limited investigation of the large parameter space even for simple models of BWRs. Analytical bifurcation analysis carried out numerically is an attractive alternative to the two approaches described above [29]. In this approach the governing set of nonlinear equations are neither integrated numerically in time nor treated entirely analytically. Rather, the forms of ODEs and Jacobian are evaluated analytically, yet the complicated process of Lindstedt-Poincaré expansion is carried out numerically. This approach, which will henceforth be called analytic-
numeric approach, allows accurate and efficient evaluation of the entire parameter space of interest. General-purpose bifurcation codes such as BIFDD [35] have been developed to perform analytic-numeric bifurcation analysis of set of ODEs and ODEs with delays. For a given set of nonlinear ODEs (or ODEs with delays) and the corresponding Jacobian matrix, the code determines the critical value of the bifurcation parameter, nature of the bifurcation, and the
43
oscillation amplitude when operating close to the critical value. BIFDD allows any parameter of choice in the model to be selected as the bifurcation parameter, and hence the entire parameter space can easily be spanned. Bifurcation analysis when carried out analytically or via an analytic-numeric approach, yields—for all other parameters fixed—the critical value of the bifurcation parameter λ = λcritical , frequency of oscillation ω c corresponding to the critical value, and parameters µ 2 , τ 2 and β 2 . A negative (positive) value of β 2 indicates a supercritical (subcritical) PAH bifurcation, i.e., stable (unstable) limit cycle oscillations as the bifurcation parameter is varied across the critical value.
τ 2 is a correction factor to the oscillation frequency, and µ 2 relates the oscillation amplitude to the value of the bifurcation parameter through equation (2.41). Figure 2.2 shows an example of BIFDD calculation in a two-dimensional parameter space. The solid line is the stability boundary (SB), which separates stable and unstable regions. The three doted lines are curves corresponding to periodic solutions of index constant ε = 0.05, 0.1 and 0.15, respectively. Since amplitudes the periodic solutions are given by multiplication of ε and steady-state eigenvector v (equation (2.35)), the curves are also called 5%, 10% and 15% oscillation curves, which indicate amplitudes of oscillations on these curves are 5%, 10% and 15% to the magnitudes of steady-state eigenvectors, respectively. It is also shown in the Figure 2.2 that the oscillation curves are in the unstable regions above the intersection point P. Unstable fixed points co-exist with stable periodic solutions in this case. According to PAH-B theorem, property of bifurcation is thus supercritical. The oscillations curves are in the stable regions below the point P, indicating a co-existence of stable fixed points and unstable periodic solutions. Property of bifurcation below the point P is subcritical.
44
Supercritical
5
ε = 0.05
Stability Boundary
ε = 0.10 ε = 0.15
P 3
Unstable
Stable
Subcritical
Parameter 2
4
2
1
0 5
6
7
8
9
10
11
12
Parameter 1
Figure 2.2. Examples of stability boundary and oscillation curves calculated by BIFDD. The oscillations curves are in the unstable region above the point P, indicating supercritical PAH-B; they are in the stable region below the point P, indicating subcritical PAH-B.
45
Chapter 3 Stability and Bifurcation Analyses of Forced-Circulation BWR Model
In this chapter, results obtained using stability analysis and using the analytic-numeric methods are presented and discussed. A set of typical BWR operating parameters are used for these studies. Dimensional values of these parameters can be found in Appendix C. Results of standard stability analysis—carried out using the eigenvalues and eigenvectors of the Jacobian of the BWR model—are presented first. Focus is on the relationship of the eigenvalues and eigenvectors of the Jacobian to characteristics of BWR dynamics. In the second and third sections, stability boundaries (linear analysis) and constant amplitude oscillation curves (bifurcation analysis) for homogeneous core and non-homogeneous cores are presented in multiple parameter spaces. The operating parameters of interest include: amplification factor F, non-dimensional subcooling number Nsub, non-dimensional pressure drop along the core ∆Pext, control rod induced reactivity ρext, void reactivity feedback coefficient Cα, fuel temperature feedback coefficient CD, gap conductance coefficient hg, pressure loss coefficient at channel outlet
kexit, and pressure loss coefficient at channel inlet kinlet.
3.1. Eigenvalues and Eigenvectors of the BWR Model Eigenvalues and eigenvectors of the Jacobian matrix play important roles in determining the stability properties of a system of ODEs. As is well known (and discussed in section 2.2 of chapter 2), the eigenvalue with the largest real part determines the stability of the system. For the
46
case of density-wave oscillations or nuclear-coupled density-wave oscillations, the eigenvalue with the largest real part is usually a complex conjugate pair. The instability that results when this pair crosses the imaginary axis leads to oscillatory behavior. Elements of the corresponding eigenvector determine the relative magnitude of different phase variables in the resulting oscillations. Since in the analysis of most models that include only the fundamental mode, it is the
same pair of complex conjugate eigenvalues that crosses the imaginary axis across different sections of the stability boundary, the frequency and amplitude of the resulting oscillations— dictated by the elements in the corresponding eigenvector—have a smooth and continuous dependence on parameter values along the stability boundary. However, this is not necessarily the case when fundamental and first azimuthal modes are included in the modal neutronics analyses and reactivity feedback parameters are varied. That is, it is possible that different pairs of complex conjugate eigenvalues may cause the system to become unstable on different segments of the stability boundary. Since, the evolution of different phase variables depend upon the elements of the eigenvector that corresponds to the eigenvalues with the largest real part, the characteristic oscillations that results along different segments of the stability boundary may be qualitatively very different. In practice, one must track two (rightmost) pairs of complex conjugate eigenvalues to study the stability of BWR thermalhydraulics coupled with fundamental and first azimuthal modal neutronics. Therefore, to determine the parametric dependence of different qualitative behavior along different segments of the stability boundary, the bifurcation code BIFDD was modified to evaluate all the eigenvalues of the Jacobian matrix (the original code only tracked the eigenvalues(s) with the largest real part). The modified code was used to perform stability and bifurcation analyses of the BWR model presented in Chapter 2.
47
Figure 3.1 shows critical values of the total pressure drop ∆Pext for N sub = 1.0, ρ ext = 0.0 and amplification factor F = 1.0, 2.0, 3.0, 4.6, and 5.0 respectively. That is, the system at each one of these points has one pair of complex conjugate eigenvalues with zero real part. All the other operating parameters are fixed at their typical values. As F increases, critical ∆Pext at which the real part of the rightmost eigenvalue is zero, does not change significantly for F ≤ 4.6, but experiences a jump for F > 4.6. The dramatic change in the value of the critical ∆Pext indicates a possible change in the qualitative behavior of the system as F is increased above 4.6. Figure 3.2 (a-e) shows eight rightmost eigenvalues for critical parameter values for each of the five cases shown in Figure 3.1.[Rest of the eigenvalues are farther to the left and are not shown here.] There are two pairs of complex conjugate eigenvalues and four real eigenvalues. Pairs of complex conjugate eigenvalues are denoted by e1 and e2, and represented by squares and stars. Because the operating points are critical, by definition, real part of the rightmost pair of eigenvalues is always zero. For the F = 1 case, the pair e1 is on the imaginary axis and the pair e2 is far from the imaginary axis. The real eigenvalues lie between the pair e1 and e2. As F increases, the pair e2 approaches the imaginary axis. For F = 4.6, the real part of e2 is larger than all of the real eigenvalues and the complex conjugate pair lies just slightly to the left of the imaginary axis. Real part of e2 in this case is close to zero, and hence both e1 and e2 will almost equally influence the evolution of a BWR transient. As F is increased to 5.0, the pair e2 moves to the right of the pair e1, and becomes the rightmost pair of eigenvalues. Characteristic differences in the dynamical behavior of systems that become unstable when the eigenvalue e2 is on the imaginary axis and those systems that become unstable when the eigenvalue e1 is on the imaginary axis, can be identified by studying the elements in the eigenvectors associated with the two pairs of eigenvalues. Eigenvectors associated with e1 and e2
48
8.55 8.50 8.45
∆Pext
8.40 8.35 8.30 8.25 8.20 8.15 1
2
3
4
5
F
Figure 3.1: Critical values of ∆Pext , when N sub = 1.0, ρ ext = 0.0 and F = 1.0, 2.0, 3.0, 4.6, and 5.0 respectively.
49
Imaginary
Imaginary
-1.0
-0.8
-0.6
-0.4
-0.2
4
4
2
2
0 0.0 -2
0.2 Real
-1.0
-0.8
-0.6
-0.4
-2
-4
(b) Imaginary
Imaginary
-0.8
-0.6
-0.4
0.2 Real
-4
(a)
-1.0
0 0.0
-0.2
-0.2
4
4
2
2
0 0.0 -2
0.2 Real
-1.0
-0.8
-0.6
-0.4
0 0.0
-0.2
-2
-4
0.2 Real
-4
(c)
(d) Imaginary 4
Key:
2
-1.0
-0.8
-0.6
-0.4
-0.2
0 0.0 -2
complex eigenvalue e1 complex eigenvalue e2 real eigenvalues
0.2 Real
-4
(e)
Figure 3.2: Eight rightmost eigenvlaues corresponding to the critical values of ∆Pext for the five cases shown in Figure 3.1. F = (a) 1.0; (b) 2.0; (c) 3.5; (d) 4.6; (e) 5.0
50
for different values of F are listed in Tables 3.1 and 3.2, respectively. For clarity, only the elements corresponding to the fundamental mode expansion parameter n0 and azimuthal mode expansion parameter n1 are shown in the tables. In the eigenvector corresponding to e1, shown in Table 3.1, the magnitudes of the elements of n0 are much larger than those of elements of n1 . However, in the eigenvector corresponding to e2 (Table 3.2), the trend is reversed, and the magnitudes of the elements of n1 are much larger than those of elements of n0 . As F is increased, the ratio of the magnitude of elements of n0 to n1 in the eigenvector corresponding to eigenvalue
e1, and the ratio of the magnitude of elements of n1 to n0 in the eigenvector corresponding to eigenvalue e2 decrease. However, they are still of order of 10 even for F = 4.6. As discussed in chapter 2 (equation (2.35) in section 2.3.2), oscillation amplitude of time-dependent variables depend both, on index constant ε and on magnitude of the corresponding element in the eigenvector v . Since the index constant remains the same for all the variables, the magnitudes of the elements of the eigenvector indicate relative amplitudes of the corresponding variables. Hence, the oscillation amplitude of n0 is much larger than that of n1 , if the oscillations are produced by the pair e1. In this case the oscillations are dominated by the in-phase component. The oscillation amplitude of n1 will be much larger than that of n0 in oscillations produced by the pair of eigenvalue e2, and hence the oscillations will be primarily out-of-phase. It is clear that the two pairs of complex conjugate eigenvalues — with the largest and second largest real part — are responsible for the in-phase and out-of-phase (fundamental and the first azimuthal mode) oscillations. Either one of these—depending upon the parameter values— may be the dominant eigenvalue. To avoid confusion, the pair of eigenvalues for which magnitude of the element of n0 in the corresponding eigenvector is much larger than that of the element of
51
Table 3.1: Elements corresponding to n0 and n1 in the eigenvector corresponding to e1 n0
n1
n0
n1
F
Real
Imaginary
Magnitude
Real
Imaginary
Magnitude
1.0
1.3656
4.8667
5.0546
0.004590
0.01369
0.01444
350.04
2.0
1.3615
4.8533
5.0407
0.01231
0.03922
0.04111
122.61
3.5
1.3482
4.7732
4.9600
0.03740
0.1920
0.1956
25.36
4.6
1.6890
5.1131
5.3848
-0.6087
-0.4666
0.7670
7.02
52
Table 3.2: Elements corresponding to n0 and n1 in the eigenvector corresponding to e2 n0
n1
n1
n0
F
Real
Imaginary
Magnitude
Real
Imaginary
Magnitude
1.0
-0.009630
-0.01334
0.01645
2.7592
3.9449
4.8140
292.64
2.0
-0.01475
-0.03072
0.03407
2.8850
6.3208
6.9481
203.94
3.5
-0.02896
-0.1192
0.1227
2.6818
8.5689
8.9788
73.18
4.6
0.2584
0.2299
0.3459
1.8988
8.9610
9.1600
26.48
53
n1 , will be called the fundamental mode eigenvalue pair ( a f + i w f ), and the other pair will be
called the azimuthal mode eigenvalue ( a a + i wa ). It is found that among all the system parameters, the amplification factor F has the largest impact on the relative order of magnitude of the fundamental mode eigenvalues and the azimuthal mode eigenvalues. This is not very surprising since the value of F directly reflects the relative strength of the reactivity feedbacks from the two modes. For this reason, results of stability and bifurcation analyses in the next two sections will be presented separately, for homogeneous core, where F = 1, and for nonhomogeneous cores, where F can be much greater than 1.
3.2. Homogeneous Core (F = 1) For homogeneous cores, azimuthal mode eigenvalues are far from the imaginary axis. For the range of values studied for other parameters in the system—parameters other than F—the dominant eigenvalues for homogeneous cores remain the fundamental mode eigenvalue pair and the azimuthal mode eigenvalue pair was never found to be close to the imaginary axis. Stability boundaries and oscillation curves are hence, only associated with the fundamental mode eigenvalues. Therefore, oscillations, when the system first becomes unstable, will be predominantly in-phase.
3.2.1. Results in N sub - ρ ext Space
Constant pressure drop ( ∆Pext ) boundary condition is used to carry out the stability and bifurcation analyses. If the imposed pressure drop is treated as a parameter—fixed for the entire plain in which SB is presented—then the inlet flow rate at steady-state at each point on the SB would in general be different. This often makes the analysis and interpretation of the effect of the 54
bifurcation parameter more difficult because different points on the SB not only have different values for the bifurcation parameter, but also correspond to different flow rates. Hence, SBs, along which the steady-state inlet flow rate is constant—and consequently each point on the SB in general corresponds to a different value of imposed pressure drop—are often desirable. The latter are in fact easier to evaluate, since total steady-state inlet flow velocity v~inlet is the same for all points on the SB, and if needed, ∆Pext can be calculated for each point on the SB after the SB has been evaluated by simply solving for ∆Pext explicitly from the pressure balance
∆Pext = ∆Pinlet + ∆Pexit + ∆Pfric ,1 + ∆Pgrav ,1 + ∆Pacc , 2 + ∆Pfric , 2 + ∆Pgrav , 2 where, ∆Pinlet and ∆Pexit are local pressure losses at channel inlet and outlet, ∆Pfric is pressure loss due to friction term in the momentum equation, ∆Pgrav is pressure loss due to gravity, ∆Pacc is the loss due to expansion in two-phase flow. Subscripts 1 and 2 denote single- and two-phase, respectively. [In the former case (when ∆Pext is fixed) v~inlet is an unknown, and hence the above equation must be solved simultaneously with the eigenvalue equations.] Stability boundaries (SBs) for the two cases are compared in Figure 3.3(a). SBs are shown for fixed ∆Pext = 8, 25, and 50, and for fixed v~inlet = 1. For comparison, ∆Pext along the SB for the v~inlet = 1 case varies between 45.0 and 175.0. Types of bifurcation along the SBs in Figure 3.3 (a) are shown in Figure 3.3 (b). Fifteen percent oscillation curves in N sub - (ρ ext − ρ ext ,critical ) space are shown for each SB, that is, the index constant ε along the results of the bifurcation analysis is 0.15. Note that (ρ ext − ρ ext ,critical ) = 0 corresponds to the SB. Hence, the periodic solutions are stable when the curves are on the unstable side of the SB ( (ρ ext − ρ ext ,critical ) > 0) and the bifurcation is supercritical. Otherwise, the
55
12 ∆Pext=47.0
10
∆Pext=8.0 ∆Pext=25.0 ∆Pext=50.0
Nsub
8 6
vinlet=1.0
4
0 -0.05
∆Pext=175.0
∆Pext=58.0
2
0.00
0.05
0.10
0.15
0.20
0.25
0.30
ρext (a) 12
Supercritical 10
∆Pext=8.0 ∆Pext=25.0 ∆Pext=50.0
Nsub
8
vinlet=1.0
6 4
Subcritical 2 0 -0.004
-0.002
0.000
0.002
0.004
0.006
0.008
ρext-ρext,critical (b) Figure 3.3: (a) Stability boundary in N sub - ρ ext space; (b) Fifteen percent oscillation amplitude ( ε = 0.15) curves in N sub - (ρ ext − ρ ext ,critical ) space.
56
periodic solutions are unstable and the bifurcation is subcritical. For example, for ∆Pext = 8.0, in the upper part of the stability boundary for N sub > 4.2, where the oscillation curve is in the stable region, the bifurcation is subcritical. Thus, the small perturbations will decay, but “large” perturbations, greater than the amplitude of the periodic solution, will grow. For 0.6 < N sub < 4.2, the oscillation curve is in the unstable region and the bifurcation is supercritical, the periodic solution is stable, and any finite but small perturbation will evolve to this stable periodic solution. For even lower values of N sub (< 0.6), the 15% oscillations curve returns to the stable region, and the bifurcation is subcritical. The low value of N sub where the transition between sub- and supercritical bifurcation occurs is close to the typical value of N sub for forced-circulation BWRs. This change from sub- to supercritical bifurcation might explain the fact that stable amplitude oscillations are observed in some BWR instability cases, and growing amplitude oscillations (until scram) in others. The same back and forth transitions between supercritical and subcritical PAH-B can be observed for ∆Pext = 25, 50, and v~inlet = 1 cases. Comparing the three 15% oscillation amplitude curves for the three fixed ∆Pext cases, it can be seen that for intermediate value of N sub (supercritical case), higher the imposed pressure drop, faster will be the rise in stable oscillation amplitudes as the SB is crossed and the operating point is moved into the unstable region. [Recall that when operating at the ε = 0.15 curve, the theory predicts the oscillation amplitude to be about 15% of the steady-state value.] Also, comparison between fixed v~inlet ( = 1.0 ) case and fixed ∆Pext ( = 50.0 ) case shows that though the SBs for the two cases were fairly close to each other, the rise in oscillation amplitude is faster for the constant ∆Pext case than for the constant v~inlet case.
57
3.2.2 Results in ρ ext - ∆Pext and N sub - ∆Pext Space
The SBs in ρ ext - ∆Pext space are shown in Figure 3.4 (a). Four SBs for N sub = 0.6, 1.5, 3.0, and 10.0 are plotted. Clearly N sub very strongly impacts the SB. The difference is evident especially for large values of ρ ext . An interesting, and well known, trend is that very small or large N sub cases have larger stable region than the intermediate N sub cases. The SBs for the case
N sub = 0.6 and 10.0 are very close but relatively far from those for the case N sub = 1.5 and 3.0. Corresponding results of the bifurcation analysis are shown in Figure 3.4 (b). For N sub = 0.6, bifurcation changes from subcritical to supercritical and then back to subcritical as ρ ext is increased. The supercritical region (for intermediate values of ρ ext ) increases in size as N sub is increased, but then diminishes as N sub approaches 10.0. At the same time, the amplitude of stable oscillation rises the slowest with (ρ ext − ρ ext ,critical ) for N sub ≈ 1.5. Also note that although the SBs for cases of very small (0.6) and large (10.0) N sub are close, nature of bifurcation along these SBs are quite different. There is a supercritical region for the case of N sub = 0.6, but only subcritical bifurcation is found for the case of N sub = 10.0. This implies quite different behaviors of oscillations for the cases of very small and large N sub values. [Since the operating value of N sub for BWR is about 0.6, the cases of large value of N sub are generally rare.] SBs for ρ ext = 0.00, 0.03, 0.08 and 0.13 is shown in Figure 3.5(a) in N sub - ∆Pext space. Results of the bifurcation analysis are shown in Figure 3.5(b). Once again as N sub is decreased along the SB, bifurcation changes from subcritical to supercritical, and back to subcritical. The effect of large ρ ext is to shrink the stable regions in the N sub - ∆Pext space. The supercritical region
58
0.14 0.12
Unstable
0.10 0.08
ρext
0.06
Nsub=0.6 Nsub=1.5 Nsub=3.0 Nsub=10.0
0.04 0.02
Stable
0.00 -0.02 -0.04 -0.06 0
20
40
60
80
100
120
140
160
180
200
∆Pext
(a)
0.14
Subcritical
Nsub=0.6 Nsub=1.5 Nsub=3.0 Nsub=10.0
0.12 0.10 0.08
ρext
0.06 0.04 0.02 0.00 -0.02
Supercritical
-0.04 -0.06 -4
-3
-2
-1
0
1
2
∆Pext-∆Pext,cr
(b) Figure 3.4: (a) Stability boundaries in ρ ext - ∆Pext space; (b) Fifteen percent oscillation amplitude ( ε = 0.15) curves in ρ ext - (∆Pext − ∆Pext ,critical ) space. 59
12
ρext=0.0 ρext=0.03
10
ρext=0.08 ρext=0.13
Nsub
8
6
Unstable
Stable
4
2
0 -40
-20
0
20
40
60
80
100
120
140
160
180
200
∆Pext
(a)
12
Supercritical
Subcritical
10
ρext=0.0 ρext=0.03
Nsub
8
ρext=0.08 ρext=0.13
6
4
2
0 -4
-3
-2
-1
0
1
2
∆Pext-∆Pext.cr
(b) Figure 3.5: (a) Stability boundary in N sub - ∆Pext space; (b) Fifteen percent oscillation amplitude ( ε = 0.15) curves in N sub - (∆Pext − ∆Pext ,critical ) space 60
in the N sub - (∆Pext − ∆Pext ,critical ) space at low value of N sub increases with increasing ρ ext .
3.3. Non-Homogeneous Core (F > 1) For non-homogeneous core, the amplification factor is greater than 1.0. For nonhomogeneous core cases, the azimuthal mode eigenvalues in the complex plane may cross the fundamental mode eigenvalues, and nature of oscillation may change from in-phase to out-ofphase. Results of the crossing and impact on the dynamical properties of the system are discussed in N sub - ∆Pext and F- ∆Pext spaces, respectively.
3.3.1. Results in N sub - ∆Pext Space
Figures 3.6 (a) and (b) show SBs and five percent oscillation curves ( ε = 0.05 ) for F = 1.0, 2.0, 3.5, 4.6 and 5.0. The SBs for F = 1.0, 2.0, 3.5 and 4.6 cannot be distinguished from each other. This shows that, for the set of parameter values used here, the factor F has very little impact on the location of the stability boundary for F < 4.6. However, SB for F = 5.0 when N sub < 1.177 differs significantly from corresponding SBs for lower values of F. The five percent oscillation curves for the first four cases, shown in Figure 3.6 (b), are also almost indistinguishable, but there is a jump in the five percent oscillation curve for F = 5.0 case at N sub = 1.177. The difference in SBs and jump in the oscillation curves are caused by an exchange of the dominant eigenvalue from the fundamental mode eigenvalues pair ( a f + i w f ) to the azimuthal mode eigenvalues pair ( a a + i wa ), as shown in the section 3.1. As mentioned in chapter 2, the code BIFDD is modified to determine boundaries corresponding to the fundamental and first azimuthal mode eigenvalues. Real part of the
61
4.5
F=1.0 F=2.0 F=3.5 F=4.6 F=5.0
4.0 3.5 3.0
Nsub
2.5
Stable
2.0
Unstable
1.5 1.0
*
0.5 0.0
1
2
3
4
5
6
7
8
9
∆Pext
(a)
4.5
F = 1.0 F = 2.0 F = 3.5 F = 4.6 F = 5.0
4.0 3.5 3.0
Nsub
2.5 2.0 1.5 1.0 0.5 0.0
-4
-3
-2
-1
0
1
∆Pext - ∆Pext,critical
(b) Figure 3.6: (a) Stability boundaries in
(∆P
ext
N sub — ∆Pext
space; (b) 5% oscillation curves in
− ∆Pext ,critical ) space for F = 1.0, 2.0, 3.5, 4.6 and 5.0.
62
N sub —
fundamental mode eigenvalues is zero along what will be called the fundamental mode boundary, and real part of the azimuthal mode eigenvalues is zero along what will be called the azimuthal mode boundary. For F = 5.0 case, the SB (solid line) in Figure 3.7(a), on which the real part of the rightmost (dominant) eigenvalue pair is zero, is comprised of two curves that meet at the point T ( N sub = 1.177, ∆Pext = 7.618). Above this point, the eigenvalues with the largest real part along the SB are the fundamental mode eigenvalues, that is, a f = 0 and a a < 0. However, for N sub < 1.177 (below the point T), the eigenvalues with the largest real part along the SB are the azimuthal mode eigenvalues ( a a = 0 and a f < 0). The dotted line in Figure 3.7(a) denotes the boundary along which the real part of the second rightmost pair of complex conjugate eigenvalue is zero. (This means that the rightmost pair of complex conjugate eigenvalue along the dotted line has positive real part). The dotted curve is also comprised of two branches. Along the branch above the point T, the real parts of the azimuthal mode eigenvalues are zero ( a a = 0 and a f > 0). However, along the lower branch, the real parts of the fundamental mode eigenvalues are zero ( a f = 0 and a a > 0). Hence, the parameter space is divided into four parts, denoted by letters A, B, C and D. Regions A and C are respectively the unstable and stable regions, where both real parts a f and a a are respectively positive and negative. In region B, a f is positive while a a is negative. Hence, the system is unstable. However, since the element of the eigenvector corresponding to the fundamental mode is much larger than the element corresponding to the azimuthal mode, growing oscillations in this region will be dominated by the fundamental mode. If both the fundamental mode and azimuthal mode are perturbed, the azimuthal mode in the beginning may actually decrease in amplitude since its amplitude is initially dominated by the (fast decaying) component
63
5 Stability boundary Re(second largest eigenvalue) = 0
4 B 3
Stable
Nsub
Unstable Point 1
2
C
** Point 2
Nsub,T
T 1
Point 3
A
* *
Point 4 D
0 2
3
4
5
6
7
∆Pext
8
9
∆Pext,T
10
(a)
5 Subcritical 4 ε = 0.05 curve 3
Nsub
ε = 0.05 curve of the 2nd largest eigenvalue
2 1 Supercritical 0 -5
-4
-3
-2
-1
0
1
∆Pext-∆Pext,critical
(b) Figure 3.7: (a) Stability boundary and the boundary along which the real part of the second largest pair of eigenvalue is zero for F = 5.0; (b) 5% oscillation amplitude curves corresponding to the eigenvalue pairs with the largest and second largest real parts. 64
that corresponds to the first azimuthal mode eigenvalue pair. Once the decaying part has died out, even the azimuthal mode will start to grow (due to the slowly growing component that corresponds to the fundamental mode eigenvalue). In region D, a a is positive, but a f is negative. Similar to the region B, the system is unstable. However, since the element of the eigenvector corresponding to the azimuthal mode is much larger than the element corresponding to the fundamental mode, growing oscillations in this region will be dominated by the azimuthal mode. If both modes are perturbed, the fundamental mode in the beginning may actually decrease in amplitude, since its amplitude is initially dominated by the (fast decaying) component that corresponds to the fundamental mode eigenvalue pair. Once the decaying part has died out, even the fundamental mode will start to grow (due to the slowly growing component that corresponds to the azimuthal mode eigenvalue). These results are conceptually similar to those obtained by March-Leuba et al. [17] using a simple model. However, in-phase and out-of-phase oscillations in their studies were obtained using different sets of boundary conditions in the model. Results reported here are obtained always using the same boundary condition of constant pressure drop along the channel. Results of the bifurcation analyses for F = 5 case are presented in Figure 3.7(b) in the form of ten percent oscillation amplitude curve in N sub — (∆Pext − ∆Pext ,critical ) space. The solid line corresponds to the SB (solid line) in Figure 3.7(a), while the dotted curve corresponds to the dotted curve in Figure 3.7(a) (i.e., to the second rightmost eigenvalue pair crossing the imaginary axis). [The bifurcation is subcritical for N sub > 1.874.] In Figure 3.7(b), a jump in the oscillation curve (solid line) occurs at the value of N sub corresponding to point T. The discontinuity exists because of the degeneracy at this value of N sub , where both the fundamental and first azimuthal mode eigenvalues have zero real parts. The oscillation curve for N sub > 1.177 is determined using 65
the fundamental mode eigenvalues, while for N sub < 1.177 it must be determined using the first azimuthal mode eigenvalues. Note that the oscillation curve for each pair of fundamental and azimuthal mode eigenvalues is continuous over the entire range of N sub . However, only a segment (the solid part) of each is relevant. Though the PAH-B theorem is strictly applicable only along the solid lines in Figure 3.7(a), results of the numerical simulations (reported below) show that the conclusions regarding the stable and unstable periodic solutions may be roughly applicable even to the second pair of complex conjugate eigenvalues that cross the imaginary axis
right behind the first pair.
3.3.2. Results in F- ∆Pext Space
To even more explicitly identify the role of F in determining the nature (in-phase or outof-phase) oscillations, Figures 3.8 (a-f) show SBs in the F— ∆Pext parameter space for N sub = 4.50, 3.25, 1.90, 1.40, 1.30 and 1.18. The solid and dashed lines are respectively the boundaries along which the real parts of the fundamental and first azimuthal mode eigenvalue pairs are zero. System stability is determined by the boundary on the right. For all values of N sub the fundamental mode boundary is independent of F. The first azimuthal mode boundary does vary with F; much more strongly for smaller values of N sub than for larges values. For N sub = 4.50 and
F = 1.00, the system becomes unstable due to the fundamental mode eigenvalue pair. However, as F is increased, for 1.50 < F < 3.53, it is the first azimuthal mode eigenvalue pair that crosses the imaginary axis as the stability boundary is crossed. Boundaries for the two modes in this region are close and hence both types of oscillations are likely to be manifested as the SB is crossed. For
66
12
10
10
8
8
6
Unstable
Stable
F
F
12
4
4
2
2
0 1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
0 1.5
6.0
3.5
4.0
(b)
10
10
4.5
5.0
5.5
6.0
8
6
F
Stable
Unstable
4
2
2
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
0 3.5
6.0
5.5
6.0
(d)
10
8
8
Unstable
Stable
4
2
2
6.0
6.5
7.0
7.5
8.0
0 5.0
8.5
6.5
7.0
5.5
6.0
6.5
7.5
8.0
Stable
Unstable
6
4
5.5
5.0
(c)
10
5.0
4.5
∆Pext
12
4.5
4.0
∆Pext
12
6
Stable
Unstable
6
4
F
F
3.0
(a)
8
F
2.5
∆Pext
12
0 4.0
2.0
∆Pext
12
0 1.5
Stable
Unstable
6
7.0
7.5
∆Pext
∆Pext
(e)
(f)
8.0
8.5
9.0
9.5
Figure 3.8. Stability boundaries in F— ∆Pext space for N sub = (a) 4.50; (b) 3.25; (c) 1.90; (d) 1.40; (e) 1.30; (f) 1.18. The fundamental mode eigenvalue pair is on the imaginary axis along the solid line, while the first azimuthal mode eigenvalue pair is on the imaginary axis along the dashed line. System stability boundary is given by the rightmost segments of the solid and dashed curves. 67
F > 3.53, once again it is the fundamental mode that determines the stability of the system. Similar behavior is observed for N sub = 3.25 case (Figure 3.8 (b)). Hence for 2.32 < F < 3.20 the two boundaries almost overlap leading to the degenerate case. For F < 2.32 and F > 3.20, stability is determined by the fundamental mode. As N sub is decreased to 1.90 (Figure 3.8(c)) and 1.40 (Figure 3.8(d)), the azimuthal mode boundary stays to the left of the fundamental mode boundary for all values of F, and hence it is only the fundamental mode that becomes unstable in the vicinity of the SB. However, as N sub is decreased further to 1.30 (Figure 3.8(e)) and then to 1.18 ( Figure 3.8 (f)), the boundary for the first azimuthal mode shifts to the right. For these two values of the subcooling number, for some intermediate range of F (5.48 < F < 7.37 for N sub = 1.30, and 5.03 < F < 10.29 for N sub = 1.18) it is the first azimuthal mode that determines the stability of the system.
3.4 Discussion Results of stability and bifurcation analyses presented above show the complexity of BWR dynamics, even when modeled by a simple, two-channel, reduced order model with relatively small number of operating parameters. For relatively homogeneous cores with F ~ 1.0, oscillations are primarily of in-phase type, and the complexity is primarily manifested in the nonlinearity of oscillations (bifurcation type) and whether the periodic solution in the vicinity of the SB is stable or unstable. Parameters such as inlet subcooling, external reactivity and pressure drop have large impact on the bifurcation characteristics of the system. The subcritical bifurcation, which may occur under certain combination of operating parameters, specifically is of significance.
68
As the in-homogeneity is increased, in addition to the variation in the bifurcation type, the system may experience in-phase or out-of-phase oscillations. The factor F has no-doubt the largest impact on whether the oscillations are in-phase or out-of-phase. However, other parameters, as shown in section 3.3.2, are also important, and the complete picture of BWR dynamics can be painted only after taking into account the effects of different parameters. The results presented in this chapter help provide insights into the nonlinear dynamics of BWRs. However, stability and even bifurcation analyses carried out here are local in nature, not valid for very large perturbations, not valid for operating points very far from the SBs and for large times. They provide useful insight as to what must be studied using, say, numerical simulation studies. In the next chapter the governing set of nonlinear ordinary differential equations are integrated numerically to test the predictions of the stability and bifurcation analyses presented here as well as to study the effects of large perturbations and dynamical behavior over long period of time.
69
Chapter 4 Numerical Simulations of Forced-Circulation BWR Model
In this chapter, the force-circulation BWR model presented in chapter 2 and studied for stability and bifurcation analyses in chapter 3 is numerically integrated to study the impact of large perturbation amplitude as well as to ascertain the system behavior for large time. Steadystate (or fixed points) for a given set of parameter values is first determined using the NLEQ2; a computer code to solve set of non-linear algebraic equations [36] . It uses “damped affine invariant Newton method with rank- strategy” to solve potentially highly nonlinear equations. Perturbations at the fixed points are introduced and time evolutions of the phase variables are numerically calculated using the ODE integration code DMEDBF [37]. The code DMEDBF is based on “Modified Extended Backward Differentiation Formulae”, which includes explicit prediction and implicit correction schemes. The method is A-stable for scheme of orders less than 4 and α-stable for orders up to 9. Since the BWR model can be potentially stiff for some set of operating parameter values, the code DMEBDF was found to be very helpful in reducing numerical problems when the system is evaluated at these operating points. Results of the numerical simulations for homogeneous and non-homogenous cores are given in Sections 4.1 and 4.2, respectively.
4.1. Numerical Simulations for Homogeneous Core (F = 1) To investigate the time evolution, and validate the results obtained using BIFDD, three points are selected in N sub - ρ ext parameter space for numerical simulations. These are shown in
70
Figure 3.3. ( N sub , ρ ext , ∆Pext ) for point 1 is (9.0, 0.03005, 8.0). The critical value of ρ ext for N sub = 9.0 and ∆Pext = 8.0 is 0.030053, therefore, point 1 is in the stable region but very close to the stability boundary. According to the bifurcation results obtained using the code BIFDD and presented in Fig. 3.3 (b), property of PAH-B along this part of the SB is subcritical. As expected, small perturbations introduced in the phase-variables, die out, and the phase variables slowly converge to the steady-state values. However, since the bifurcation at this point is subcritical, large perturbations may grow. Figure 4.1 shows time evolutions of one such case of a rather large perturbation for point 1. In this case, three phase variables were given arbitrary but large perturbations at t = 0 (a1,0: 0.8593 → 1.0593; s1,0: 15.9232 → 16.1232; vinlet,0: 0.3231 → 0.5231). Other phase variables are kept at their steady-state values. Despite the fact that point 1 is in the stable region and small perturbations in phase variables do indeed decay down to zero, these large perturbations lead to growing amplitude oscillations of the phase variables. Figure 4.1(a) shows the time evolution of the fundamental mode expansion parameter n0. It is clear that there is an unstable periodic solution repelling the trajectory in phase space away to ever larger amplitude. Figure 4.1 (b) shows the time-evolution of the azimuthal mode expansion parameter n1. Amplitude of n1 is much smaller than of n0, hence characteristic of power oscillations of the system will appear as in-phase. Note that even the azimuthal mode expansion parameter starts to grow (not shown here) after the initial drop in its amplitude. Figure 4.1 (c) shows the time evolutions of inlet flow velocities of two parallel channels that comprise the core, vinlet,0 and vinlet,1, Figure 4.1 (d) shows the same data (two inlet velocities) but only over the first 50 (nondimensional) seconds. It is clear that after the initial transients the two channels oscillate with the same phase, confirming the in-phase nature of the oscillations predicted in chapter 3.
71
2.4
0.3
2.2
0.2 2.0
0.1
1.6
n1
n0
1.8
0.0
1.4
-0.1 1.2
-0.2
1.0 0.8
-0.3 -20
0
20
40
60
80
100
120
140
160
-20
0
20
40
60
time
(a)
100
120
140
160
(b)
0.55
0.55
vinlet,0 vinlet,1
0.50 0.45
0.45
0.40
0.40
0.35
0.35
0.30
0.30
0.25
0.25
0.20
0.20 -20
0
20
40
60
80
100
120
140
vinlet,0 vinlet,1
0.50
v
v
80
time
160
0
10
20
time
30
40
time
(c)
(d)
Figure 4.1: Results of numerical integration for parameter values corresponding to point 1 in Figure 3.3. Operating parameters ( N sub , ρ ext , ∆Pext ) are (9.0, 0.03005, 8.0). Phase variables a1,0, s1,0 and vinlet,0 are perturbed as follows—a1,0: 0.8593 → 1.0593; s1,0: 15.9232 → 16.1232; vinlet,0: 0.3231 → 0.5231). The phase variables shown are: (a) n0 ; (b) n1 ; (c) and (d) vinlet , 0 and vinlet ,1 .
72
50
Points 2 and 3 in Fig. 3.3 are also close to the SB for ∆Pext = 8.0. Operating parameters ( N sub ,
ρ ext , ∆Pext ) for them are (3.0, 0.01320, 8.0) and (3.0, 0.01, 8.0), respectively. Critical value of ρ ext for N sub = 3.0 and ∆Pext = 8.0 is 0.01317, therefore point 2 is in the unstable region and point 3 is in the stable region. As shown in Fig. 3.3 (b), PAH-B along this part of the SB is supercritical. For parameter values corresponding to point 2, very large amplitude perturbations are introduced in three phase variables at t = 0 (a1,0: 0.4517 → 0.5517; s1,0: 8.3693 → 8.4693; vinlet,0: 0: 0.3597 → 0.4587). Remaining phase variables are kept at their steady-state value. Results of the numerical simulations are shown in Figure 4.2. Despite the fact that operating point 2 is in the unstable region, even the very large perturbations do not cause the phase variables to grow indefinitely, but instead, as predicted by the supercritical PAH-B, phase variables evolve to stable limit cycles. Figure 4.2 (a) shows the time evolution of the amplitude of n0. Note that because the operating point is very close the SB, evolution to the stable limit cycle is very slow. The oscillation amplitude does indeed saturate for t > 1000 and oscillates between 0.75 and 1.06. Thus, while point 1 is in the stable region, because of the subcritical PAH-B there, a large amplitude perturbation at that point may be more dangerous than a large amplitude perturbation at point 1— despite the fact that point 1 is in the unstable region. As for the in-phase vs out-of-phase oscillations, time evolutions of n1, vinlet,0, and vinlet,1, shown in Figure 4.2 (b), (c) and (d) indicate that in-phase oscillations similar to those at point 1 result at point 2. Perturbations introduced in phase variables for system operating at point 3 in Fig. 3.3 are given by (vinlet,0: 0.3616 → 0.4616; vinlet,1: 0.3616 → 0.2616). Because point 3 is in the stable region and the bifurcation there is supercritical, even these large perturbations, as expected, decay to zero, and the system, as shown in Figure 4.3, returns to its steady-state condition. Thus, results of the numerical integrations performed for parameter values corresponding
73
1.2
0.15
1.1
0.10
0.05
n0
n1
1.0
0.00
0.9 -0.05 0.8 -0.10 0.7 -0.15 0
100
1000
1100
0
1200
100
1000
1100
1200
time
time
(b)
(a) 0.48 0.45
vinlet,0 vinlet,1
0.46 0.44 0.42
vinlet,0 vinlet,1
0.40
v
v
0.40 0.38
0.35
0.36 0.34 0.32
0.30
0.30 0
100
1000
1100
0
1200
2
4
6
8
10
12
14
16
18
20
time
time
(d)
(c)
Figure 4.2: Results of numerical integration for parameter values corresponding to point 2 in Figure 3.3. Operating parameters ( N sub , ρ ext , ∆Pext ) are (3.0, 0.01320, 8.0). Perturbations of timedependent variables are introduced as (a1,0: 0.4517 → 0.5517; s1,0: 8.3693 → 8.4693; vinlet,0: 0.3597 → 0.4597); The signals show time evolutions of (a) n0 ; (b) n1 ; (c) and (d) vinlet ,0 and
vinlet ,1 .
74
0.025
0.846
0.020 0.845
0.015 0.010
0.844
n0
n1
0.005 0.000
0.843 -0.005 -0.010
0.842
-0.015 -0.020
0.841 -10
0
10
20
30
40
50
60
70
80
90
-10
100 110
0
10
20
30
40
50
60
70
80
90
100
time
time
(b)
(a) 0.40
vinlet,0 vinlet,1
0.45
0.39
vinlet,0 vinlet,1
0.38 0.37
0.40
0.35
v
v
0.36 0.35 0.34 0.33 0.30 0.32 0.31 0.25 -10
0
10
20
30
40
50
60
70
80
90
0.30
100 110
0
time
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15
time
(c)
(d)
Figure 4.3: Results of numerical integration at point 3 in Figure 3.3. Operating parameters ( N sub , ρ ext , ∆Pext ) at the point are (3.0, 0.01, 8.0). Perturbations of time-dependent variables are introduced as (vinlet,0: 0.3616 → 0.4616; vinlet,1: 0.3616 → 0.2616); The signals show time evolutions of (a) n0 ; (b) n1 ; (c) and (d) vinlet , 0 and vinlet ,1 .
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to points 1, 2 and 3 confirm the predictions of the semi-analytic bifurcation studies carried out using the bifurcation code BIFDD. Moreover, results of these numerical integrations can be used to estimate the perturbation amplitude necessary to lead a system operating in a stable region away from the stability boundary but close to a subcritical PAH-B point to “become unstable.” Similarly, results of these numerical integrations can also be used to estimate the oscillation amplitude of stable limit cycles after supercritical PAH-B.
4.2. Numerical Simulations of Non-homogeneous Core (F > 1) Results of numerical simulations for amplification factor F = 2.0, 3.5, 4.6 and 5.0 are shown in Figures 4.4 through 4.7. The operating point ( N sub = 1.0, ρ ext = 0.0, and ∆Pext = 8.5) is shown in the Figure 3.6 (a). For the set of operating parameter ( N sub = 1.0, ρ ext = 0.0), the critical value of external pressure along the channel is ∆Pext ,critical = 8.2085 for F = 2.0, 3.5, 4.6, and
∆Pext ,critical = 8.5362 for F = 5.0. Therefore, the operating point is stable for the cases F = 2.0, 3.5, 4.6, and unstable for the case F = 5.0. Phase variables n0 and n1 are perturbed at t = 0 as follows (n0: n0,ss → n0,ss + 0.05; n1: n1,ss → n1,ss + 0.05), where n0,ss is the steady state value of n0 and is about the same (0.5827) for different values of F; and n1,ss = 0.0 is the steady state value of n1. There is no significant change in the decay ratio of n0 as F is increased from 2 to 5 (Figures 4.4 to 4.7). However, dramatic change in the decay ratio of n1 can be seen. For F = 2.0, oscillations in
n1 , shown in Fig. 4.4, very quickly decay to zero. As F is increased from 2 toward 5, oscillations in n1 take increasingly longer time to decay down to zero (see Figures 4.5-4.6). In fact, for F = 5.0, decay ratio of n1 is greater than 1.0, implying an unstable first azimuthal mode, and hence a switch in oscillations from in-phase to out-of-phase. Evolutions of velocities also show similar
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0.64
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n1
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65
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time
80
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time
(c)
(d)
Figure 4.4: Results of numerical integration of the governing set of nonlinear ODEs for F = 2.0. Operating parameters ( N sub , ρ ext , ∆Pext ) are (1.0, 0.0, 8.5). Perturbations are introduced in the phase variables n0 and n1 (n0: 0.5827 → 0.6327; n1: 0.0 → 0.05). The figures show time evolution of: (a) n0 ; (b) n1 ; (c) and (d) vinlet , 0 and vinlet ,1 .
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v
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vinlet,0 vinlet,1
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80
0.396
100
60
65
70
75
t
80
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t
(c)
(d)
Figure 4.5: Results of numerical integration for F = 3.0. Operating parameters ( N sub , ρ ext , ∆Pext ) at the point are (1.0, 0.0, 8.5). Perturbations of time-dependent variables are introduced as (n0: 0.5827 → 0.6327; n1: 0.0 → 0.05); The signals show time evolutions of (a) n0 ; (b) n1 ; (c) and (d) vinlet , 0 and vinlet ,1 .
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0.64
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n0
0.60 0.59
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vinlet,0 vinlet,1
v
v
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65
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75
t
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t
(c)
(c)
Figure 4.6: Results of numerical integration for F = 4.6. Operating parameters ( N sub , ρ ext , ∆Pext ) at the point are (1.0, 0.0, 8.5). Perturbations of time-dependent variables are introduced as (n0: 0.5827 → 0.6327; n1: 0.0 → 0.05); The signals show time evolutions of (a) n0 ; (b) n1 ; (c) and (d) vinlet , 0 and vinlet ,1 .
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100
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v
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65
70
75
t
80
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t
(c)
(d)
Figure 4.7: Results of numerical integration for F = 5.0. Operating parameters ( N sub , ρ ext , ∆Pext ) at the point are (1.0, 0.0, 8.5). Perturbations of time-dependent variables are introduced as (n0: 0.5827 → 0.6327; n1: 0.0 → 0.05); The signals show time evolutions of (a) n0 ; (b) n1 ; (c) and (d)
vinlet , 0 and vinlet ,1 .
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trend. For small values of F, phase and amplitude of inlet velocities of the two channels get close quickly after the initial perturbations. For F = 2.0 and 3.0, time evolutions of the two velocities in the last 40 seconds of integration can not even be distinguished (Figures 4.4 (d) and 4.5 (d)). When F is increased even further, difference in the phases and amplitudes of the two velocities becomes more evident. For F = 4.6, it takes much longer for the two channel velocities to get in phase. As F is increased even further, the nature of oscillations changes to out-of-phase. This can be observed in the time evolution of the two velocities during the last 40 seconds of the evolutions for F = 5.0 (Figure 4.7 (d)). Moreover, the time evolutions of the velocities for F = 5.0 show
beating phenomena. It is well known that beating is caused by summation of two signals with very close frequencies. This fact suggests that the oscillations for F = 5.0 case are composed of two slightly different frequencies (resulting from the two pair of complex conjugate eigenvalues very close to the imaginary axis). For F = 5.0 case, frequencies for the fundamental and the first azimuthal mode eigenvalue pairs for the operating point on the stability boundary are 4.2319 and 4.3837, respectively. The ratio of the two frequencies suggest a beating period of ~80 seconds, which is a little higher than that observed in the Figure 4.7 (c) and (d) (about 55 seconds). The overestimate is caused by the frequency difference between the operating point on the stability boundary and the point of actual numerical simulation, which is, although close to, still not on the boundary. Characteristic changes from F = 4.6 to F = 5.0 can be shown clearer, when new initial perturbations are selected to eliminate effect of the second frequency. The perturbation is given by a small number ξ = 0.01 multiplying the eigenvectors corresponding to the pair of eigenvalues with the largest real parts. Table 4.1 lists perturbations for phase variables at t = 0.0 for F = 4.6 and 5.0. Evolutions of perturbations for n0 and n1 are compared for F = 4.6 and 5.0 in the Figure
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4.8 and 4.9 respectively. It is clear that amplitude of n0 are much larger than that of n1 for the case
F = 4.6, hence in-phase oscillation is predominant. For the case F = 5.0, however, amplitude of n1 is much larger than that of n0, and characteristics of oscillations becomes dominant out-of-phase. Results of numerical simulations for the non-homogeneous cores agree with the results, presented in chapter 3, of the stability analysis and the expected change in instability from predominantly in-phase to pre-dominantly out-of-phase oscillations with an exchange in predominant eigenvalue pair. For small values of F, when the pair of complex conjugate eigenvalues for the fundamental mode is on the imaginary axis, the pair of complex conjugate eigenvalues for the azimuthal mode is far to the left of the imaginary axis. Oscillations are thus dominated by the fundamental mode. As F is increased, i.e. for operation at SB for larger values of F, the azimuthal mode eigenvalues pair moves closer and closer to the imaginary axis. For F = 5.0, the pair of eigenvalues for the first azimuthal mode crosses the pair for the fundamental mode, thus the oscillations are dominated by the first azimuthal model. [In fact, for some value of 4.7 < F < 5.0, the two pairs are simultaneously on the imaginary axis, leading to a degenerate case.] Additional numerical simulations were carried out for the F = 5.0 case to analyze the switch from in-phase to out-of-phase oscillations. These were carried out for operating points corresponding to points 1-4 shown in Fig. 3.7 (a). Parameter values ( N sub , ρ ext , ∆Pext ) at points 14 in Figure 3.7 (a) are (2.2303, 0.0, 3.9000), (2.2303, 0.0, 4.1000), (0.8000, 0.0, 7.8500), and (0.8, 0.0, 8.2000), respectively. Points 1 and 2 are above the separation point T, while points 3 and 4 are below the separation point T. Recall that point T divides the SB into an “in-phase stability boundary” above it from the “out-of-phase stability boundary” below it. Time evolutions of n0
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Table 4.1: Perturbations for phase variables for F = 4.6 and 5.0. Operating parameter set ( N sub , ρ ext , ∆Pext ) corresponds to (1.0, 0.0, 8.5)
Phase Variables a1,0 a2,0 s1,0 s2,0 vinlet,0 T1,1,0 T2,1,0 T1,2,0 T2,2,0 n0 u0 a1,1 a2,1 s1,1 s2,1 vinlet,1 T1,1,1 T2,1,1 T1,2,1 T2,2,1 n1 u1
Perturbation for F = 4.6 -1.401ä10-3 2.997ä10-3 -8.157ä10-4 1.717ä10-2 1.289ä10-3 -3.165ä10-4 2.528ä10-3 -3.181ä10-4 2.979ä10-3 -1.014ä10-3 -7.967ä10-4 -1.607ä10-3 3.772ä10-3 1.143ä10-3 1.738ä10-2 1.521ä10-3 -3.515ä10-4 2.944ä10-3 -3.528ä10-4 3.479ä10-3 7.589ä10-4 7.522ä10-4
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Perturbation for F = 5.0 1.644ä10-3 -3.709ä10-3 3.643ä10-5 -1.739ä10-2 -1.499ä10-3 3.752ä10-4 -2.977ä10-3 3.770ä10-4 -3.509ä10-3 1.119ä10-4 1.048ä10-4 -1.542ä10-3 3.393ä10-3 -5.222ä10-4 1.687ä10-2 1.394ä10-3 -3.544ä10-4 2.778ä10-3 -3.563ä10-4 3.272ä10-3 6.931ä10-4 5.668ä10-4
0.010
n0 - n0,ss n1 - n1,ss
ni - ni,ss, i = 0, 1
0.005
0.000
-0.005
-0.010 0
20
40
60
80
100
time
Figure 4.8: Perturbations of n0 and n1 for F = 4.6. Operating parameters ( N sub , ρ ext , ∆Pext ) at the point are (1.0, 0.0, 8.5). Perturbations of time-dependent variables are introduced as ξv , where ξ =0.01 is a small number, v is the eigenvector corresponding to the pair of eigenvalues with the largest real part. Details of initial perturbation can be seen in the Table 4.1.
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n0 - n0,ss n1 - n1,ss
0.03
ni - ni,ss, i = 0, 1
0.02
0.01
0.00
-0.01
-0.02 0
20
40
60
80
100
time
Figure 4.9: Perturbations of n0 and n1 for F = 5.0. Operating parameters ( N sub , ρ ext , ∆Pext ) at the point are (1.0, 0.0, 8.5). Perturbations of time-dependent variables are introduced as ξv , where ξ =0.01 is a small number, v is the eigenvector corresponding to the pair of eigenvalues with the largest real part. Details of initial perturbation can be seen in the Table 4.1.
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and n1 following small perturbations are shown in Figures 4.10 through 4.13. At point 1 (Figure 4.10), in-phase oscillation results to increasing amplitude of n0 and an initially decreasing amplitude of n1 . Note that oscillation amplitude of n1 as well starts to increase at large values of t. Point 2, however, is in the stable region C. Therefore, small perturbations produce decreasing amplitude oscillations of both n0 and n1 (Figure 4.11). However, n0 decreases much slower than
n1 , causing oscillations that are primarily in-phase. At points 3 and 4, the oscillations are predominantly out-of-phase because as was shown in chapter 3, the magnitude of the element corresponding to the out-of-phase mode in the eigenvector corresponding to the dominant eigenvalues is much larger than the magnitude of the element corresponding to the in-phase mode. For point 3, real parts of both eigenvalues are positive, therefore amplitudes of n0 and n1 increas fast (Figure 4.12). At point 4, predominantly out-of-phase oscillations occur, because the real part of the fundamental mode eigenvalue is negative while the real part of the first azimuthal mode eigenvalue is positive and the magnitude of the elements of the eigenvector corresponds to the dominant eigenvalue (Figure 4.13). Clearly the dominant oscillations in the high and low N sub regions are characteristically different. In the high N sub region, the in-phase oscillations are dominant because the critical eigenvalues on the SB correspond to the fundamental mode. However, it should be noted that this characteristic will be evident for parameter values that correspond to a stable system, and for parameter values in a very narrow region close to the SB in the unstable region (region B in the Figure 3.7(a)) where the fundamental mode eigenvalues has positive real part while the first azimuthal mode eigenvalues has negative real part. In a vast majority of the unstable region, both modes are actually unstable and hence both n0 and n1 are expected to grow rapidly.
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In the low N sub region (below point T), however, the out-of-phase oscillations are dominant because the critical eigenvalues correspond to the azimuthal mode. Once again, this characteristic will be evident for parameter values that correspond to a stable system and a narrow region in the unstable side (region D in the Figure 3.7(a)), where the first azimuthal mode eigenvalues has positive real part while the fundamental mode eigenvalues has negative real part. For the rest part of unstable region, since both eigenvalues have positive real parts, amplitudes of
n0 and n1 both grow rapidly. In the region close to the separation point T, effects of both, the fundamental and azimuthal mode eigenvalues, are evident, and oscillations in this region are a blend of in-phase and out-phase. Thus, numerical simulations confirm the findings of the numeric-analytic bifurcation analysis.
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0.336
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Figure 4.10: Results of numerical integration for F = 5.0. Operating parameters ( N sub , ρ ext , ∆Pext ) at the point are (2.2303, 0.0, 3.9). Perturbations of time-dependent variables are introduced as (n1: 0.0 → 0.1); The signals show time evolutions of (a) n0 ; (b) n1 ; (c) and (d) vinlet , 0 and vinlet ,1 .
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0.10
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Figure 4.11: Results of numerical integration for F = 5.0. Operating parameters ( N sub , ρ ext , ∆Pext ) at the point are (2.2303, 0.0, 4.1). Perturbations of time-dependent variables are introduced as (n0: 0.3523 → 0.4; n1: 0.0 → 0.1); The signals show time evolutions of (a) n0 ; (b) n1 ; (c) and (d)
vinlet , 0 and vinlet ,1 .
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0.565 0.2 0.560 0.555
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(c)
(d)
Figure 4.12: Results of numerical integration for F = 5.0. Operating parameters ( N sub , ρ ext , ∆Pext ) at the point are (0.8, 0.0, 7.85). Perturbations of time-dependent variables are introduced as (n1: 0.0 → 0.05); The signals show time evolutions of (a) n0 ; (b) n1 ; (c) and (d) vinlet , 0 and vinlet ,1 .
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0.020
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0.010 0.005
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n1
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0.000 -0.005
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0.406 0.404
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v
v
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time
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130
time
135
140
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(c)
(d)
Figure 4.13: Results of numerical integration for F = 5.0. Operating parameters ( N sub , ρ ext , ∆Pext ) at the point are (0.8, 0.0, 8.2). Perturbations of time-dependent variables are introduced as (n0: 0.5573 → 0.5); The signals show time evolutions of (a) n0 ; (b) n1 ; (c) and (d) vinlet , 0 and vinlet ,1 .
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Chapter5 Summary, Conclusions and Future Works of Part I
This chapter summarizes the model and analysis method, concludes the stability and bifurcation results, and suggests future works for studies of force-circulation BWRs.
5.1. Summary of the Model and Analysis Method Stability and bifurcation analyses are carried out for a reduced order model of BWR. The model, which is originally developed by Karve et al., includes three parts: neutron kinetics, fuel rod heat conduction and channel thermal-hydraulics. In the part of neutron kinetics, neutron density and precursor density are assumed an expansion of both the fundamental and first azimuthal mode. The ODEs of the time-dependent parameters of the expansion are derived from one-group time-dependent diffusion equation and the precursor equation. Operating parameters presented in the ODEs, including void reactivity feedback coefficient, Doppler reactivity feedback coefficient, and neutron generation time, are given by forms of inner products of functions and adjoints of the modes. The functions and adjoints of the two modes, however, depend on core loadings and other factors which affect power distributions. The original model by Karve et al. evaluates the functions and adjoints with an assumption that the core is homogeneously loaded, thus failed to generate any out-of-phase oscillations. To account for effects of different loadings, an extra parameter, amplification factor F is introduced. The factor is multiplied to the void and Doppler feedback coefficients of the azimuthal mode, so that the out-of-phase oscillations can be generated for large value of F.
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In the part of heated channel thermal hydraulics, the spatial enthalpy distribution in the single phase region, and the steam quality in the two phase region in the BWR core are approximated by quadratic expansion profiles. Two phase flow is studied with the homogeneous equilibrium model (HEM). The ODEs of time-dependent expansion parameters of the enthalpy and steam quality are derived through weighted-residual approach applied to conservation equations of HEM. Similar to the heated channel thermal-hydraulics, in the part of fuel rod heat conduction, temperature distribution in fuel pellet is represented by piecewise quadratic profiles. The ODEs of the expansion parameters of the quadratic profiles are derived through variational principle. The three parts of the reduced-order model are coupled by reactivity feedbacks and heat transfer correlations. Dynamic properties of the model are studied through stability analysis, bifurcation analysis, and numerical integration. Linear stability of the model is determined by the largest real part of the Jacobian matrix of the dynamical system. Stability boundary, which separates linearly stable and unstable regions, can be plotted in operating parameter space, by tracking the operating points with the zero largest real parts. Properties of oscillations, or periodic solutions, are determined by Poincare-Andronov-Hopf bifurcation (PAH-B) theorem. Stated by the theorem, stable or unstable periodic solutions, called supercritical PAH-B or subcritical PAH-B, is determined by Floquet exponent. The supercritical PAH-B corresponds to oscillations with saturated amplitudes and the subcritical PAH-B corresponds to those with unlimited amplitudes when operating condition cross the stability boundary from stable to unstable side. Relative amplitudes of the different time-dependent variables are determined by magnitudes of the corresponding element of the eigenvector. Large ratio of the magnitude of n0 (the expansion
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parameter of the fundamental mode) to the magnitude of n1 (the expansion parameter of the azimuthal mode), corresponds to in-phase oscillations; large ratio of the magnitude of the n1 to the magnitude of the n0 corresponds to out-of-phase oscillations. The properties of supercritical and subcritical PAH-B and in-phase and out-of-phase oscillations can be studied using a mathematical code BIFDD, which is modified from its original version to evaluate the entire spectrum of eigenvalues—rather than only the one with the largest real part. Given the analytical forms of ODEs, the code can automatically calculate fixed point, determine eigenvalues of the Jacobian matrix at the fixed point and track the operating points of the stability boundary with zero real parts of the eigenvalues. It then numerically evaluates all higher order derivatives necessary for calculations of the properties of the bifurcation and oscillation. The properties of the ODE system can also be determined through observations of numerical integrations at certain operating points. This provides an alternative means to validate and verify stability and bifurcation results by BIFDD.
5.2. Conclusions of Results For azimuthally symmetric flux shape (F = 1), only in-phase oscillations are found in all regions of parameter space studied. The stability boundaries calculated with BIFDD are found identical with those calculated by Karve et al. using different method. Both sub- and supercritical bifurcations along the stability boundaries are found capable. For high or low values of inlet subcooling, the properties of bifurcation are subcritical. For intermediate values of the inlet subcooling, the properties of bifurcation are supercritical. The value of inlet subcooling number separating the lower subcritical region and the supercritical region is close to the typical nominal
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value of BWR. The back and forth change from supercritical to subcritical PAH-B close to the typical nominal inlet subcooling might explain the fact that stable amplitude oscillations are observed in some BWRs while growing amplitude oscillations in others. As the amplification factor F is increased, spectrum of eigenvalues calculated by BIFDD shows that real part of the second pair of complex conjugate eigenvalues increases and approaches the imaginary axis. Eigenvectors of the two pairs of complex conjugate eigenvalues suggest characteristically different oscillations. The oscillations produced by one pair of complex conjugate eigenvalues are mainly in-phase, and those produced by the other pair of complex conjugate eigenvalues are mainly out-of-phase. For this reason, the former is called fundamental mode (in-phase) eigenvalue, and the latter is called azimuthal mode (out-of-phase) eigenvalue. As
F is increased, it is found the azimuthal mode eigenvalues, which is the original pair with the second largest real part, may cross the fundamental mode eigenvalues and become the pair with the largest real part. The change of the pair of eigenvalues with the largest real part (fundamental mode or azimuthal mode eigenvalues) represents a characteristic transition of the system dynamics. For F = 5, boundaries and bifurcation curves of the eigenvalues with the largest and second largest real parts in N sub — ∆Pext parameter space show that above certain value of inlet subcooling, the fundamental mode eigenvalue reaches the imaginary axis (as the imposed pressure drop is increased) before the first azimuthal mode eigenvalue. For these high values of inlet subcooling in-phase oscillations dominate. However, for smaller values of inlet subcooling out-ofphase oscillations dominate due to the azimuthal mode eigenvalue that reaches the imaginary axis first as the bifurcation parameters is varied. Hence, the parameter space is divided into four regions separated by boundaries of the two pairs of eigenvalues. Operating points in different
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regions may experience oscillations of different characteristics: in the narrow unstable region B above the transition point T (see Figure 3.7(a)), the in-phase oscillations will dominate; in the narrow unstable region D below the transition point T, the out-of-phase will dominate; in the unstable region A or stable region C, the oscillations are combinations of both in-phase and outof-phase. Bifurcation results show that both sub- and super-critical bifurcation can occur along the stability boundaries for in-phase as well as out-of-phase oscillations. Stability boundaries in F— ∆Pext parameter space show that for large or small values of
N sub , azimuthal mode eigenvalues can cross the fundamental mode eigenvalues. In this case, the system becomes unstable due to fundamental mode when F is large or small. The system becomes unstable, however, due to the first azimuthal mode for intermediate values of F. Therefore, oscillations will change from in-phase to out-of-phase and back to in-phase as F is increased. For intermediate values of N sub , azimuthal mode eigenvalues do not cross the fundamental mode eigenvalues as F is varied. Hence, dominant oscillations are only in-phase. Numerical integrations are performed for both small and large values of F. For small value of F (=1.0), It is found that for subcritical PAH-B, the large perturbation to operating points in stable region but close to the stability boundary may cause oscillations with unlimited amplitudes. For supercritical PAH-B, perturbations to operating points in unstable region may cause oscillations with saturated amplitudes. For large value of F( = 5.0), numerical integration results show dramatic change of characteristics of oscillations from in-phase to out-of-phase or vice versa, when inlet subcooling crosses a critical value. Beating phenomena when operating point is close to the transition point reveal the fact that the oscillations are composed by two frequencies, one corresponding to the
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fundamental mode eigenvalues and another corresponding to the azimuthal mode eigenvalues. These results agree with the predictions of the stability and bifurcation analyses.
5.3. Future Work In the reduced-order model, the amplification factor F is introduced to roughly account for effects of different core loadings. The value of F is actually given arbitrarily to reveal qualitative properties of oscillations. We suggest improvement can be made in the future to reduce the arbitrariness caused by the factor F. This can be done by employing the modal decomposition approach mentioned in the section 2.1.2 of the Chapter 2. Two- or three-dimensional diffusion equation codes can be used to evaluate a number of typical core loadings during normal or abnormal operating conditions. The functions and adjoints of the fundamental and azimuthal modes, thus, can be numerically determined, and operating parameters presented in the ODEs of neutron kinetics can be calculated. It is of importance to learn what specific core loadings and operating conditions may lead to out-of-phase oscillations.
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PART II Stability and Bifurcation Analyses of Natural Circulation BWRs
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Chapter 6 Introduction to Instabilities of Natural Circulation BWRs
Second part of this dissertation, starting from this chapter, is devoted to modeling and instability analyses of natural circulation BWRs during start-up. Unlike forced circulation BWRs that rely on recirculation pumps for flow through the core, natural circulation BWR designs rely on natural circulation caused by density difference for flow through the core and for heat extraction purposes.
6.1. Instabilities of Natural-Circulation BWRs under Low Pressure Safety is one of the most important concerns in nuclear engineering. To ensure safety, current techniques rely on “defense in depth” strategy based on multiple and different active components. These lead to increased capitol cost and extend construction period, consequently limiting competitiveness of nuclear energy to fossil fuels. To improve both safety and economy of next generation light water reactors (LWRs), passive safety concepts are introduced in designs. The essence of the passive safety is to reduce active components as much as possible and ensure reactor safety by non-active (passive) components and physical laws. New designs of BWRs that include passive safety concepts, such as SBWR and ESBWR by GE Energy, rely on natural circulation to remove fission heat from cores. They are promising candidates to replace the current forced-circulation BWRs. Figure 6.1 shows a diagram of the natural circulation design ESBWR [38] [39] and a schematic plot of flow path. In natural
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Dryer
Steam separator
Riser
DownBoiling
~12 m
Riser
~25 m
comer
Boundary
Downcomer
Channel
Single Phase Region
Core
~3 m
(a)
(b)
Figure 6.1: Diagram of (a) natural circulation BWR design and (b) schematic flow path. Parameters of lengths of core, riser and containment are, in particular, for ESBWR design. They may differ from other natural circulation BWR designs [38]
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circulation design, there are no pumps and associated pipes. Therefore, a number of severe accidents, like pump trip and LOCA due to recirculation pipes, are completely avoided. Flow path of the natural circulation is comprised of heated channels in the core, adiabatic risers above the channels and downcome. Cold water flows into the core and is boiling at certain height along the channel and riser. Steam is separated at the top of riser. The separated water and feed water are mixed and flow back through the downcomer to the entrance of the core. Driving force for flow through the core is, instead, provided by gravity force due to density difference between the coreriser and the downcomer. The long riser, about 12 m in the case of ESBWR [39], ensures enough density difference between the core-riser and downcomer for adequate flow through the core. Designs of short core length and low pressure loss steam separator also help reduce friction loss and increases flow rate. By these means, power density of ESBWR can reach 48.5 kw/l [40], which is lower than the corresponding value for forced circulation system, but economically acceptable. Nominal system pressure of the ESBWR is 7.0 MPa, about the same as that for forced-circulation BWR. However, the system pressure can be significantly lower than its nominal value during start-up, even as low as the atmosphere pressure. The natural circulation BWR may experience nuclear-coupled density wave oscillations (DWOs) under certain adverse operating conditions similar to those of forced-circulation BWRs, when the system pressure is about 7.0 MPa. However, an even larger concern for stability of the natural circulation system is when the system pressure is much lower during start-up. It may experience more complicated instabilities other than DWOs. Types of these instabilities depend on many factors. For example, if the length of the riser is short, geysering can be dominant in a natural circulation loop, which is caused by formation and condensation of large bubbles. Figure 6.2 shows typical flow signals recorded in geysering experiments by Aritomi et al [41] in a natural
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circulation heated channel laboratory system. The experimental apparatus is comprised of two parallel heated channels and adiabatic risers. A geysering cycle in one of two channels includes four stages (Figure 6.2(a)). Slow increase of velocity from reverse flow in stage I is followed by a rapid increase in stage II. In stage III, flow rate decreases slowly. Then in stage IV, it decreases rapidly leading to reverse flow. Figure 6.2 (a) and (b) show that flow rate oscillations in the two channels are out-of-phase. This leads to an almost invariant total flow rate. If the riser is long, a situation for most natural circulation BWR designs, another type of instability, flashing, may be dominant. Flashing here means boiling in the adiabatic riser not due to heating but due to pressure drop along the riser. Figure 6.3 shows results of flashing experiments in a natural circulation loop with long riser performed by Inada et al. [42]. [Note that definition of the subcooling number in this case of natural circulation is different from that used in Part I (forced circulation system) of this thesis. The two definitions are explained and compared in Appendix D (also explain the fact that Nsub can be negative in this case in the appendix)]. When inlet subcooling is large (Figure 6.3(a)), flow rate in the natural circulation loop is small because there is no boiling. Driving force of recirculation, in this case, is provided by density difference between hot water in the channel-riser and cold water in the downcomer, and the flow shown in the Figure 6.3(a) is stable. As inlet subcooling is decreased, flashing occurs in the upper part of the riser and intermittent peaks of the flow rate with period of about ~100 seconds appear (Figure 6.3(b)). If the inlet subcooling is even lower, the intermittent peaks are wider and interconnected, and flow pattern evolves to a sinusoidal wave (Figure 6.3(c)). Typical periods of sinusoidal oscillations range from several seconds to several tens of seconds. For very small inlet subcooling, amplitude of the sinusoidal wave is very small and the natural circulation flow evolves to stable two-phase flow (Figure 6.3(d)).
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II
III
I
IV
(a)
(b)
Figure 6.2: Typical geysering flow signals in two parallel heated channels [41].(a) Flow signal in channel 1, u(1); (b) Flow signal in channel 2, u(2). Geysering cycle comprises of four stages. Flow oscillations in the two channels are out-of-phase.
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(a)
(b)
(c)
(d)
Figure 6.3: Typical flow signals recorded in natural circulation SIRIUS loop [42]. System pressure is P = 0.2 MPa, heat flux is q’’ = 53 kW/m2. (a) single phase natural circulation flow, subcooling number Nsub = 29.8; (b) intermittent wave due to flashing, Nsub = 23.6; (c) sinusoidal oscillations, Nsub = -0.08; (d) steady two-phase flow, Nsub = -2.17.
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The potential of more complicated instabilities during start-up poses a threat to safety of natural circulation BWR systems. For geysering-type instabilities, the dramatic change of void fractions in the core during a cycle may cause large perturbations in reactivity. For flashing-type instabilities, creation and condensation of steam is quite dramatic, thus flow rate changes rapidly for intermittent and sinusoidal waves. Moreover, the flashing front (boiling boundary) may fall below the riser into the channel, causing perturbations in reactivity. Several start-up procedures for different natural circulation designs have already been proposed. The focus of the start-up procedures is to avoid potential hazardous instabilities by maintaining single phase state of the natural circulation flow when the system pressure is low. However, possibility of boiling and instabilities during start-up, caused by off-normal conditions, still can not be completely ruled out. Therefore more extensive studies of the mechanisms of the boiling related instabilities are necessary.
6.2. Physical Mechanisms
6.2.1. Geysering
Figure 6.4 shows mechanisms of geysering type instability postulated by Aritomi et al. [41]. Flow patterns corresponding to four points in stage I and II of a geysering cycle are plotted. According to Chiang et al., geysering in two heated parallel channels occurs at very low flow speed ( < 0.2m/s) [43]. Under this flow rate, large slug bubbles can be created at specific position of the channel and their lengths grow as they move upward. Since driving force of natural circulation depends on the density difference between the channel-riser and the downcomer, the flow is slowly accelerated. When these slug bubbles reach the channel exit, sudden condensation
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Figure 6.4: Flow regimes in a channel corresponding to four points of stage I and II of a geysering cycle in the experiment by Aritomi et al. [41]
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decreases pressure in the channel, and the flow is accelerated very rapidly. These processes lead to the first two stages (slow and rapid increase) of the geysering cycle. Since the experimental system is comprised of two parallel channels, and total flow rate can not change as quickly as that in a single channel due to inertia, the flow velocity increase in one channel due to slug movement and condensation causes decrease in the other channel. Therefore, the flow velocities in the two channels are out-of-phase. The last two stages (slow and rapid decrease) in the geysering cycle in one channel are actually caused by the first two stages in the other channel.
6.2.2. Flashing
Different from geysering, flashing-induced instabilities are not caused by production and condensation of slugs. It is due to bulk boiling and associated changes of natural circulation driving force. For long riser and low system pressure, pressure drop can be large along the path of the channel and riser. Boiling may occur in the adiabatic riser simply due to local pressure drop (flashing). When flashing occurs in the adiabatic riser, small perturbation of flow can be dramatically intensified by feedbacks due to vaporization and local pressure variation. Inada et al. [42] [44] reported the results of flashing experiments. When inlet subcooling is large, both, core and riser are filled with single phase water and the flow is stable. As the inlet subcooling is decreased, Figure 6.5 shows sequence of changes in the flow regimes and enthalpies along the channel and the riser. The flow enthalpy is generally increasing along the channel due to heat input. Since the pressure along the channel and the riser is generally decreasing, the saturation enthalpy also decreases. Therefore, flow is the least subcooled at the top of the riser. In Figure 6.5(a), the flow enthalpy along the channel and riser is lower than the local saturation enthalpy. Flow rate in this situation is small because there is no boiling. A small perturbation in
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z
z
Saturation Enthalpy
Saturation Enthalpy
Flow Enthalpy
q’
Flow Enthalpy
q’
enthalpy
enthalpy
(a)
(b)
z
z
Saturation Enthalpy
Saturation Enthalpy
Flow Enthalpy
Flow Enthalpy
q’
q’
enthalpy
enthalpy
(c)
(d)
Figure 6.5: Flow pattern and enthalpy profiles during a cycle of intermittent wave caused by flashing. (a) Both channel and riser are single phase. A mass of hot water travels upward; (b) The hot fluid reaches the upper part of the riser. Flashing starts in the riser and “travels” towards riser inlet and outlet; (c) Flow is accelerated. Cold water enters the channel and travels upward; (d) Flashing is suppressed when the cold fluid reaches the upper part of the riser. Flow is decelerated and another cycle starts.
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inlet water temperature at the bottom of the heated channel leads to a “bump” in water temperature that travels towards outlet of the riser. When it reaches the upper part of the riser (not necessary the top of the riser) where the saturation enthalpy is low (Figure 6.5(b)), flow enthalpy can be higher than the saturation enthalpy, leading to flashing. Under low pressure condition, void fraction varies strongly with steam quality [45]. The small amount of steam produced in the upper part of the riser may cause rapid increase of the void fraction of two phase flow. Because friction in two-phase flow is much higher than that in single phase flow, it may decrease local pressure at which the flashing occurs even further. This will in-turn cause rapid vaporization and increase of void fraction in the riser. If flashing starts in the middle section of the riser, there may be two flashing fronts—one traveling upward toward the riser outlet and one traveling downward toward the riser inlet. If flashing occurs at the top of the riser, only one flashing front will travel downward toward the riser inlet. In natural circulation loop, large void fraction in the channelriser means large density difference between the channel-riser and the downcomer, hence large gravity driving force. Therefore, the flow is accelerated (Figure 6.5(c)). The higher flow velocity then will decrease the flow enthalpy at the entrance of the channel. A bulk of cold water will travel along the channel and riser. When the cold water reaches upper part of the riser, it suppresses flashing and the flow returns to initial single phase state (Figure 6.3 (d)). The flow is then decelerated and this process repeats, producing another intermittent wave. If inlet subcooling is decreased or heat input is increased, the intermittent wave evolves to a sinusoidal wave. In this case, small inlet subcooling or large heat input keeps the flashing front below the top of the riser. The movement of flashing front is thus confined to the riser and the channel, and the condition in which both the channel and riser are covered with single phase water may not occur. This leads to
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smooth changes in pressure at the flashing front and the smooth shape of sinusoidal wave. Further drop in inlet subcooling or increase of heat input can produce stable two-phase flow.
6.3. Previous Work For natural circulation systems, a series of stability experiments and tests have been conducted in laboratories and in a natural circulation driven BWR. A large volume of data has been accumulated. Experiments demonstrating the geysering phenomenon are described first, followed by those carried out to study flashing and flashing induced instabilities. Aritomi et al. [46] conducted experiments in a loop comprised of two parallel channels, which are partly heated by electric heaters in the center. The unheated sections act as risers. Length of the heated part is 1.0 m, and the length of the adjustable riser is from 0.25 to 0.75 m. The system pressure is close to the atmosphere pressure. Driving mechanisms of geysering in parallel channels under natural and forced-circulation conditions are identified. It is found that formation and condensation of large bubbles are essential in geysering. Increase of flow rate and inlet subcooling can help suppress geysering. Weissler et al. [47] first studied dynamics of natural circulation in a loop under low pressure. Their experiments showed that flashing-induced instabilities were possible in natural circulation systems. Jiang et al. [48] performed experiments in an apparatus which is comprised of two parallel heated channels and long risers. They observed flashing-induced instabilities in their experiments (“the maximum void fraction caused by flashing is about 5% under a system pressure of 1.5 MPa, and about 80% under 0.1 MPa.” [49]). Five stages of instabilities in the test loop were identified [48]. When inlet subcooling is decreased, flow pattern in the loop changes sequentially from stable single phase flow, unstable subcooled boiling flow, intermittent flow, unstable
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flashing flow, to stable flashing flow. Based on experimental results, a start-up strategy was suggested to use nitrogen gas to increase pressure of the system to 0.3 MPa. Flashing-induced instabilities can be suppressed effectively by this means. Furuya et al. [44] performed experiments on another natural circulation loop (SIRIUS), and identified similar flow pattern transitions when inlet subcooling is decreased or thermal power is increased. They reported decay ratios, frequencies and other oscillation characteristics for different operating points in a parameter space comprised of channel inlet subcooling and channel heat power. Also, they found that increasing inlet restriction can stabilize the natural circulation flow. Manera et al. [39] carried out studies of flashing-induced instabilities in the CIRCUS natural circulation loop in Delft, the Netherlands. Through visual observation, they found complicated mode of flashing-induced instabilities. For small inlet subcooling, boiling was found to start at some point in the riser close to the top, and the flashing fronts traveled in both upward and downward directions. As the flow is accelerated, flashing is suppressed, but repeats in the same way with a period of about ~100 seconds. The phenomenon was explained to be caused by non-equilibrium character of flashing. They also suggested that decreasing compressible volume and increasing system pressure can stabilize the system. Van Bragt [50] summarized stability tests performed in a natural circulation BWR (Dodewaard, the Netherlands). Power and flow signals were measured during start-up of cycle 2426. Decay ratios for all cases but one were less than 1.0, indicating that the system was very stable. For the only exception, clear reasons are still unknown, but carry-under effect in the downcomer was suggested to be the likely cause of instability. Ishii et al. designed a natural circulation loop applying scaling analysis for simulation of natural circulation reactor [51]. A number of experiments of flashing-induced instabilities in the natural circulation loop were carried out and recorded [52].
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Computer simulations and theoretical analyses of various natural circulation instabilities are limited when compared to experiments, partly due to difficulties in developing models that fully respect pressure dependence of water and steam thermodynamic properties. This coupling is the key for the simulation of flashing and other instabilities under low pressure. Analyses can be divided into those carried out using large scale system codes, and those performed using reduced order models. Paniagua et al. [53] modified the system code RAMONA-4B to take the pressure dependence into consideration. Original RAMONA code is designed for transient simulations of forced-circulation BWRs. The thermal-hydraulics model inside is a 4-equation drift flux model. The modification includes a new option in the code to evaluate all thermodynamic properties of water and steam based on local pressure field. Tests against experiments by Wang et al. [54] showed noticeable improvements predicting amplitudes and frequencies of geysering instabilities. Yet the under-prediction of the amplitude is still high (about 19%). Anderson et al. [55] simulated flashing-induced instabilities with system code TRACG, which is a General Electric version of the TRAC code. It contains a two-fluid model for two-phase flow and tested correlations. Qualitative agreements with experiments were found. Manera et al. [56] simulated the experiments conducted in the CIRCUS facility—that showed flashing—with a 4-equation drift flux code, FLOCAL. Generally good agreements of amplitudes, frequencies and stability maps were found. A reduced order model of low-pressure natural circulation system was developed by Inada and Ohkawa [57]. In their model, thermal-hydraulics in a vertical heated channel and adiabatic riser above the channel is considered. Homogeneous equilibrium model (HEM) was used in the two-phase region, and flashing effect was included by assuming linear dependence of saturation
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enthalpy on local pressure. They performed a frequency domain analysis. The stability boundaries follow the experimental results qualitatively, and better agreement was found under lower system pressure condition. They also found the system to be more stable if water is boiling in the heated channel rather than in the adiabatic riser. Van Bragt et al. [58] [59] extended Inada's method adding neutronics and heat transfer models. The two-phase region in the riser is discretized in a number of nodes with equal sizes. In order to simplify the complexity associated with the derivation of the reduced order model, linear variations of the steam quality and saturation enthalpy in the nodes were assumed. The inertial term in the single phase momentum equation and the time derivative term of saturation enthalpy in the energy equation were ignored. They analyzed the reduced order model in both time and frequency domains. The results qualitatively agree with the experimental data. Jiang and Emendorfer [49] considered subcooled boiling and flashing in their reduced order model of a natural circulation system. They discretized the riser into several nodes. Three fronts—the subcooled boiling, saturation boiling and flashing—were modeled. The Clausius-Clapeyron equation was used to determine the flashing front. Only hydrostatic pressure drop along the riser, relevant to the drop in saturation enthalpy, was included. Comparisons between simulation results and experimental data were presented for relatively high system pressure (1.5MPa) and low pressure (0.1 MP). Only steady-state results were reported. Better agreements were observed for high pressure conditions. Kuran et al. developed a simple one-dimensional model based on kinematic wave theory and drift flux formulations [60]. The model shows capability of developing limit cycle oscillations and more complicated perioddoubling oscillations.
6.4. Scope of Part II of This Thesis
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It is not surprising that most existing system codes are incapable of analyzing natural circulation systems under low pressure conditions because they are originally developed for forced-circulation BWRs, for which the system pressures are always high. Usually the thermodynamic properties of water and steam in these codes are evaluated at the average system pressure, and consideration of their variation with local pressure is unnecessary. This leads to their inability to capture the essence of geysering and flashing phenomena when the system pressure is low. Other than the few exceptions—such as TRACG, FLOCAL and the modified RAMONA—calculations using system codes suffer from long running time and complicated data analyses. These limitations restrict their use to numerical simulations of specific cases. They are not suitable for parametric studies or for bifurcation studies like those presented in part I of this thesis for forced circulation systems. Therefore, reduced order models are needed for efficient stability and bifurcation analyses over large region of parameter spaces. However, most existing reduced order models are too simplistic. Some of them do not consider pressure dependence of thermodynamic properties, while others rely on several simplifying assumptions and simplifications to reduce the difficulties in the derivation of the reduced order model. [While all such assumptions and simplifications cannot be avoided, they certainly can be reduced in the derivation of an improved model.] Moreover, only one reduced order model (van Bragt et al. [50]), to author’s knowledge, has considered nuclear-coupling. Equally importantly, only a few reduced order models have been analyzed as a dynamical system to study bifurcation characteristics of natural circulation boiling water systems . Due to the limitations outlined above, the second part of this thesis (following four chapters) focuses on the development and analysis of a new reduced-order model to study
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flashing-induced instabilities in natural circulation BWRs under low pressure. Although geysering instabilities may be important under certain conditions (high heat flux, short risers, etc…), modeling of the geysering instabilities necessitates modeling of a single large bubble and its dynamics. It can not be modeled using the simple HEM or even using the more detailed drift flux model (DFM). Therefore, it is beyond the scope of this dissertation and is recommended as future work. A new reduced order model is developed in chapter 7. Focus will be on the nodalization scheme, reduced order model equations, and the iteration process. After a model validation exercise, stability and bifurcation analyses of the model developed in chapter 7 are carried out in chapter 8. Numerical simulations for different operating conditions are presented and discussed in chapter 9. The last chapter summarizes the work on dynamics of natural circulation BWR under low pressure, and suggests future work.
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Chapter7 Reduced-Order Model of Natural-Circulation BWRs
A reduced order model of natural-circulation BWRs is derived in this chapter, which also includes neutron kinetics, fuel rod conduction and thermal-hydraulics parts. However, the Neutron kinetics and fuel rod heat conduction parts do not differ from those of the forced and natural circulation models described in the Chapter 2. Details of them will not be presented in this chapter. Only the derivation of thermal-hydraulics part of the model is presented.
7.1. Nodalization of Natural Circulation System Figure 7.1 shows a schematic plot of the natural circulation loop considered in the model. It is comprised of two heated channels (each of which represents a half of the core), two separate *
*
risers above the channels and a common down comer. Length of them are Lc , Lr and Ldc
*
respectively, where Ldc = Lc + Lr . The channels are heated by time-dependent heat flux q l (t ) , *
*
*
'
l = 0, 1. The value k c ,in ,l , k c ,out ,l , k r ,out ,l and k dc ,in are local pressure loss coefficients due to shape
changes of flow path or restrictions at the channel inlet, channel outlet, riser outlet, and *
*
*
downcomer inlet respectively. The inlet subcooling is denoted by ∆Tinlet = Tsat ,inlet − Tinlet , which is the difference between inlet flow temperature and the local saturation temperature. A system *
pressure is imposed by setting the pressure at outlet of the risers, Pexit , to a fixed value. Different cases, shown in the Figure 7.2, based on the location of the boiling boundary, are considered separately. For case 1, inlet subcooling is large or heat flux is small. Boiling does not occur in the channel-riser. The whole system is characterized by single phase flow. For case 2, 115
Pexit
kr,out
Pexit
kr,out
Lr kc,out Lc
q0
kc,out q1
∆Tinlet
kc,in
∆Tinlet
kc,in
Figure 7.1: Schematic plot of natural circulation system
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Ldc
Boiling Boundary
Boiling Boundary
Riser
q’
Riser
q’
Channel
Channel
vinlet
v inlet
(a)
(b)
Riser
Riser Boiling Boundary
Boiling Boundary
q’
q’
Channel
Channel
vinlet
vinlet
(d)
(c)
Figure 7.2: Cases based on position of boiling boundary. (a) Boiling boundary is out of the riser outlet; (b) Boiling boundary is in the riser; (c) Boiling boundary stays at the connection of channel and riser; (d) Boiling boundary is in the channel
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water enthalpy at the core exit is less than the saturation enthalpy, and flashing occurs in the riser. The case 3 is a degenerated case. For low inlet subcooling or high heat flux, because of the pressure loss between the channel outlet and riser inlet, water enthalpy can be less than the saturation value at the riser outlet, but greater at the the riser inlet. Therefore, for case 3, the boiling boundary may stay at the interface of channel outlet and riser inlet. For case 4, very small inlet subcooling or large heat input causes the boiling boundary falling into the channel. Nodes are required to separate the channels, risers, and dowcomer into smaller parts, so that approximations of spatial profiles of enthalpies, qualities and other variables of the systems can be made to each node. The nodalization for the downcomer is relatively easy, since it is comprised of subcooled single-phase water. Only one node for the whole downcomer is enough. For the channel-riser, cases of boiling add complex to the nodalization schemes. For cases 2 and 4, both include three regions starting from the channel inlet to the riser outlet. The first region is single-phase region from channel inlet to either boiling boundary or channel outlet. The middle region is between the boiling boundary and channel outlet. It can be either single-phase (for case 2) or two-phase (for case 4). The last region is two-phase region from either channel outlet or boiling boundary to the riser outlet. For the case 1 and 3, only two regions exist from the channel inlet to riser outlet. The first region is single phase from channel inlet to channel outlet, and the second region is from the channel outlet to riser outlet, which is either single phase (for case 1) or two-phase (for case 3). One possible scheme of nodalization is to consider different cases separately and allow multiple moving node boundaries. Figure 7.3 (a) shows an example of this kind of nodalization for the case 2 (boiling in the riser). The first region, single phase region in the channel, is discretized by two nodes, each of which is a half size of the channel. The boundary S2 between
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the two nodes is fixed. The second region, single phase region in the lower part of the riser, is also discretized by two nodes with equal length. However, since the upper boundary of this region is the boiling boundary which is moving during transient process, the boundary S4 between the two nodes is assumed moving proportionally with the boiling boundary, and the length of each node is always the half of this region. Similarly, the last region, two phase region at the upper part of the riser, is dicretized to two nodes. The boundary S6 is assumed moving proportionally too, and length of each node is the half of the region 3. One advantage of this nodalization scheme is that it can significantly reduce occurrence of the node with zero length. If the boiling boundary does not move across the riser outlet and channel outlet, the node length is always non-zero, which helps avoid numerical difficulties during calculation. However, it also unnecessarily limits movement of the boiling boundary. If the boiling boundary moves back and forth between the riser and channel, or it moves in and out of the riser during flashing-induced instabilities, the case of boiling will correspondingly changed back and forth between 2 and 3, or between 1 and 2. The node with zero length during numerical simulations will be inevitable in these situations. Figure 7.3 (b) shows the example of an alternative scheme of nodalization. First, the channel and riser are discretized to two nodes respectively. The node lengths in the channel and riser are half of the channel and riser respectively. The boundaries (S1, S2, S3, S4, S6) of these nodes are fixed. Then, the boiling boundary, which is the boundary S5, is allowed to move in the fixed nodes freely and splits one of them into two nodes (in this case, it is the last node in the riser getting split). Therefore, for the node 1, 2, 3, upper and lower boundaries of them are fixed. For the node 4, the lower boundary is fixed, but upper boundary is moving. For the node 5 the lower boundary is moving, but upper boundary is fixed. When the boiling boundary moves across any of the fixed boundaries, a node with zero length will be created. This degenerated situation (the node
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S7
6
Boiling Boundary
S6
5
S6
Boiling Boundary
5
S5
S5
4
4 S4
S4
3
3 S3
S3 2
2 q’
q’
S2
S2 1
1
S1
S1 (b)
(a)
Figure 7.3: Two nodalization schemes for the case 2 (boiling boundary in the riser). (a) Two nodes are discretized for each region. Node surface S4 and S6 move proportionally with the boiling boundary S5; (b) Two nodes are discretized for channel and riser respectively first. Boiling boundary S5 splits the last node in the riser into two.
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of zero length) will create numerical difficulties and need to be taken special care. Both the two schemes are programmed and performances of them are evaluated for different application. For convenience, in the rest of the thesis, the first scheme with multiple moving boundaries will be called moving boundary scheme, and the second on with only one moving boundary (the boiling boundary) will be called fixed boundary scheme.
7.2. Differential Equations
7.2.1. Dimensional and Non-Dimensional Partial Differential Equations (PDEs)
The thermal-hydraulic model is based on the following two physical and modelling assumptions:
• Water saturation enthalpy varies linearly with local pressure, and single phase water density varies linearly with flow enthalpy. Other thermo properties of water and steam are constant and their values are evaluated with the system pressure (pressure at the riser outlet).
• Two phase flow can be modeled by Homogeneous Equilibrium Model (HEM) Although better assumptions such as quadratic pressure dependence and non-equilibrium twophase flow models can be applied, they will greatly increase complexity and decrease computational efficiency of the model. Dynamical properties of the natural circulation BWRs can be solved through conservation equations for mass, energy and momentum. It is difficult and unnecessary to solve the full threedimensional versions of the equations, because the aspect ratio of flow channels in BWR reactors, which is equal to the ratio of the length in axial dimension to the length in radial dimension, is
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generally very large. Moreover, cross mixing of flow is generally small in BWRs, because there are cases surrounding fuel assemblies. One dimensional, cross-sectionally averaged equations are generally enough to describe dynamics of the system. For single phase flow, one-dimensional mass equation can be simplified to *
*
v m (t ) = vinlet (t ) Ainlet *
*
A*
(7.1) *
where, vinlet (t ) is the flow velocity at the channel inlet, Ainlet is the flow area at the channel inlet and A* is the local flow area. Conservation equation of energy is given by *
*
∂hm ( z * , t * ) ∂hm ( z * , t * ) q '' (t ) * + v m (t ) = * ∂t * ∂t * A ρf
(7.2)
where, q '' (t ) is time-dependent heat flux into the channel, water density ρ f is the saturation value evaluated with the system pressure. Conservation equation of momentum equation can be written as
dv * (t ) ∆ρ f ( z * , t * ) f 1φ (v m * (t )) 2 m + + + g 1 + ∂P ( z , t ) * 2 ρ D dt ρ − = f h f ∂z * K (v * (t )) 2 δ ( z * − z * ) m K *
*
*
(7.3)
where ∆ρ f ( z * , t * ) is water density expansion due to temperature (enthalpy) change, K is local *
pressure loss coefficient and z K is the position on which the pressure loss occurs. From the first assumption, water saturation enthalpy depends on the local pressure *
h f ( z , t ) = E1 P * ( z * , t ) + E 2
(7.4)
The constants E1 and E 2 can be determined if two pressures and corresponding saturation *
*
enthalpies are known. Here, one of the pressures used is the system pressure Plow = Psys , which is the fixed pressure at the riser outlet. The other one is given by the system pressure plus static
122
(
)
water head of the downcomer Phigh = Psys + Lc + Lr gρ f , which is roughly the steady state *
*
*
*
pressure at channel inlet if friction and local pressure loss are ignored in the downcomer. *
*
*
*
Saturation enthalpies h f ,low and h f ,high corresponding to the two pressures Plow and Phigh are evaluated with ASME water thermo-property codes. The value of E1 and E 2 , therefore, can be *
*
*
*
calculated easily by plugging the h f ,low , h f ,high , Plow and Phigh into the equation (7.4). The left hand side of equation (7.3), thus, can be replaced by term − E1 ∂h f
*
∂z .
The density expansion term ∆ρ f ( z * , t * ) is assumed depending on local flow enthalpy
(
((
)
)
)
∆ρ f z * , t * = α h h z * , t * − h f ,high + β h *
(7.5)
where α h is the Bousinesque density expansion parameter related to flow enthalpy, β h is a constant to modify the density expansion for better accuracy. Their values are determined to agree with experiments. For two phase flow, the dimensional form of mass equation for HEM is given by
(
)
∂ρ m ( z * , t * ) ∂ * * + * vm ( z * , t * ) ρ m ( z * , t * ) = 0 * ∂z ∂t *
(7.6)
where flow density ρ m is a function of steam quality given by *
x(z * , t * ) (1 − x(z * , t * )) + ρ m * ( z * , t * ) = 1 ρ ρ g f
(7.7)
Therefore, the equation can be transformed to
(
)
∂x z * , t * ρ g = + x z* ,t* * ∂t ∆ρ fg
(
(
)
(
* * ∂v m * z * , t * * * * ∂x z , t v ( z , t ) − m ∂z * ∂z *
)
)
Dimensional form of conservation equation of energy for two phase flow is given by
123
(7.8)
∂ ∂ q '' (t ) * * * * * * * * * * * * * z t h z t + v z t h z t ρ , , ρ , , = m m m m m ∂t * ∂z * A*
( (
))
[
( )(
(
) (
(
)]
) (
(7.9)
*
where, the hm is the flow enthalpy *
(
)
*
))
(
*
hm z * , t * = h f z * , t 1 − x z * , t * + hg x z * , t *
)
(7.10)
Differential equation of the water saturation enthalpy can be obtained from equations (7.6), (7.9) and (7.10).
*
(
∂h f z , t ∂t
*
*
*
*
)
(
)
(
q '' (t ) ρ f ρ g ∆h fg * z * , t * ∂v * z * , t * m * − ∆ρ fg ∂z * A = * ∂h f z * , t * * * v m z , t ∂z *
(
(
)
*
)
)
(
)
x z* ,t* 1 + − ρ g 1 − x z * , t * ρ f
(
(
))
(7.11)
*
where, ∆h fg ( z * , t * ) = hg − h f ( z * , t * ) is the specific latent enthalpy. The conservation equation of momentum for two phase flow can be simplified by the mass equation. The form is given by
(
* ∂h * z * , t * ∂v m f = − E1 ∂t * ∂z *
)
ρ m + vm *
*
* f 2φ v m ∂v m +g+ * 2 Dh ∂z
*2
*2 * + Kv m δ ( z * − z K )
(7.12)
note the ∂P * ∂z * term (the first term) on the right hand side of the classic momentum equation has already been replaced by E1∂h f
*
∂z * term with the pressure dependence assumption.
Using the non-dimensional variables and numbers defined in Appendix E, conservation equations of mass, energy and momentum for single phase flow can be converted into the nondimensional forms v m (t ) = N a vinlet (t ) N pch (t ) ∂hm ( z , t ) ∂h( z , t ) = − + v m (t ) N ∂z ∂t flash
124
(7.13) (7.14)
∂h f ( z , t ) ∂z
=
1 ∂v m (1 + N exp,h hm ( z , t ) + N dev ,h ) 2 2 + + N 2φ v m + Kv m δ ( z − z K ) (7.15) ∆Pdrv ∂t Fr
Similary, non-dimensional forms of the equation (7.8), (7.11) and (7.12) for two-phase flow are given by ∂v ( z , t ) ∂x( z , t ) ∂x( z , t ) = ( N ρ N r + x( z , t )) m − vm ( z, t ) ∂z ∂z ∂t
(7.16)
∂v m ( z , t ) (hg − h f ( z , t ) )( N ρ N r + x ( z , t ) ) + (1 − x ( z , t ) ) ∂z ∂h f ( z , t ) N pch ( t ) ( N ρ N r + x ( z , t ) ) =− + N flash N ρ N r (1 − x ( z , t ) ) ∂t ∂h f ( z , t ) vm ( z , t ) ∂z
(7.17)
N ρ N r + x( z , t ) ∂h f ( z , t ) ∂v m ( z , t ) − = ∆Pdrv Nρ Nr ∂z ∂t ∂v ( z , t ) + N f ,2 vm ( z, t ) 2 vm ( z, t ) m ∂z 1 2 + Kv m ( z , t ) δ ( z − z K ) Fr
+
(7.18)
7.2.2. Ordinary Differential Equations (ODEs) in A Node
Phase variables, which represent thermal-hydraulic status of the system include flow enthalpy ( h( z , t ) ), saturation enthalpy ( h f ( z , t ) ), steam quality ( x( z, t ) ), and flow velocity ( v m ( z , t ) ). All of them are functions of both spatial coordinate z and time t. Although the partial differential equations (PDEs) of the phase variables can be solved directly through the multiple numerical schemes discretizing both spatial and time coordinates, such as up-wind, Leap-Frog, and Crank-Nicolson schemes, these schemes, however, generally requires large number of nodes
125
to achieve satisfactory accuracy, and small time step to avoid numerical instabilities. It is also quite difficult to study the system with stability and bifurcation analysis techniques with these pure numerical schemes. An alternative approach is to reduce the set of PDEs to a set of ODEs through spatial discretization, and weighted residue approach can be employed for the reduction. There are two options available to carry out this step. The phase variable can be expanded to either high order polynomial profiles, or linear profiles with time-dependent expansion parameters in each of the nodes. The advantage of higher order polynomial expansion is that less number of nodes, hence less time-dependent parameters, is needed to represent the phase variables. However, the derivation of the ODEs for higher order polynomials is complicated and generally requires iterative calculations. The piece-wise linear expansion route, though, may require more variables, is much easier to develop and may achieve higher efficiency of computation without the need of complicated iterative calculations. Therefore, the piece-wise linear expansion scheme is used in this work. In a node of z i −1 (t ) ≤ z ≤ z i (t ) , where z i −1 (t ) and z i (t ) are lower and upper boundaries of the node respectively. The profiles of the phase variables in a node are given by h f ( z , t ) = h f ,i −1 (t ) + a f ,i (t )( z − z i −1 (t ))
(7.19)
hm ( z , t ) = hm ,i −1 (t ) + a m ,i (t )( z − z i −1 (t ))
(7.20)
x( z , t ) = xi −1 (t ) + N ρ N r si (t )(z − z i −1 (t ) )
(7.21)
v m ( z , t ) = v m , i −1 (t ) + vi (t )( z − z i −1 (t ))
(7.22)
where a f ,i (t ) , a m,i (t ) , si (t ) , and vi (t ) are time-dependent parameters of the profiles of the phase variables. h f ,i −1 (t ) , hm ,i −1 (t ) , xi −1 (t ) , and v m ,i −1 (t ) are values of the phase variables at the lower boundary of the node. Values of the phase variables at the upper boundary are given by
126
h f ,i (t ) = h f ,i −1 (t ) + a f ,i (t )∆z i (t )
(7.23)
hm ,i (t ) = hm ,i −1 (t ) + a m ,i (t )∆z i (t )
(7.24)
xi (t ) = xi −1 (t ) + N ρ N r si (t )∆z i (t )
(7.25)
v m ,i (t ) = v m ,i −1 (t ) + vi (t )∆z i (t )
(7.26)
where, ∆z i (t ) = z i (t ) − z i −1 (t ) is the node length. For single phase flow, since the water is assumed incompressible and subcooled, the expansion parameters v(t) and s(t) are equal to zero. Expansion parameter af(t) can be obtained from the equations (7.15) dv m ,i (t ) (1 + N exp,h hm ,i (t ) + N dev ,h ) + + 1 dt F a f ,i (t ) = r ∆Pdrv 2 2 N f 1v m ,i (t ) + Kv m (t ) δ ( z − z K )
(7.27)
where
v m,i (t ) = N a ,i vinlet (t )
(7.28)
hm ,i (t ) = (hm ,i −1 (t ) + hm ,i (t )) 2
(7.29)
N a ,i is the node area number which is ratio between flow areas of the node and the channel inlet. vinlet (t ) is the flow rate at the channel inlet. ODE for the time-dependent expansion parameters am(t) is derived by plugging equation (7.20) into equation (7.14) and integrating the governing PDE along the node. It is given by da m ,i (t ) dt
=
Am ,1 (a m , a f , s, v, ∆z (t )
)
+ Am , 2 (a m , a f , s, v,
Form of the nonlinear functions Am ,1 and Am , 2 can be found in Appendix F.
127
)
(7.30)
For two phase flow, the flow enthalpy hm ( z , t ) is a function of steam quality x( z , t ) , water saturation enthalpy h f ( z , t ) and steam saturation enthalpy hg (equation (7.10)). Therefore, the expansion parameter am(t) can be explicitly determined from equation (7.10) and (7.20) a m ,i (t ) = a f ,i (t )(1 − xi −1 (t )) + N ρ N r si (t )(hg − h f ,i −1 (t )) − N ρ N r a f ,i (t )si (t )∆z (t ) (7.31) ODEs for the expansion parameters af(t), s(t), and v(t) can be is derived through integrating the governing PDE along the node. Their generic forms are given by da f (t ) dt
=
A f ,1 (a m , a f , v, s,...) ∆z
+ A f , 2 (a m , a f , v, s,...)
(7.32)
ds (t ) S1 (a m , a f , v, s,...) = + S 2 (a m , a f , v, s,...) dt ∆z
(7.33)
dv(t ) V1 (a m , a f , v, s,...) = + V2 (a m , a f , v, s,...) dt ∆z
(7.34)
where A f ,1 , A f , 2 , S1 , S 2 , V1 , and V2 are nonlinear functions of the expansion parameters. Their forms are presented in Appendix F. ODEs for the phase variables at the upper boundary of the node are calculated by the linear representations of the profiles from equations (7.19-7.22). dh f ,i (t ) dt dhm ,i (t ) dt
=
=
dh f ,i −1 (t ) dt dhm ,i −1 (t ) dt
+
+
da f ,i (t ) dt da m ,i (t ) dt
∆z (t ) + a f ,i (t )
d∆z (t ) dt
(7.35)
∆z (t ) + a m ,i (t )
d∆z (t ) dt
(7.36)
dxi (t ) dxi −1 (t ) 1 dsi (t ) d∆z (t ) = + ∆z (t ) + si (t ) dt dt N ρ N r dt dt dv m ,i (t ) dt
=
dv m ,i −1 (t ) dt
+
dvi (t ) d∆z (t ) ∆z (t ) + vi (t ) dt dt
128
(7.37)
(7.38)
7.2.3. Node with Small Length In the fixed boundary nodalization scheme where all node lengths are fixed and timeinvariant ( ∆z ) except for the last single-phase and the first two-phase nodes—as these nodes are divided by a moving boiling boundary, and their lengths should add up to ∆z —the length of the last single-phase node or the first two-phase node can be very small. The first terms of right hand sides (RHSs) of Equations (7.30), (7.32), (7.33) and (7.34) are of forms of nonlinear functions divided by the node length. As the length goes to zero, small changes of the numerator Am,1 , A f ,1 , S1 and V1 cause large changes of the values of ODEs. Thus, time scale for the node with very
small length can be significantly larger than those of the other nodes with large lengths. This will lead to a stiff ODE system with significantly different time scales. It is well known that numerical integration schemes are stable for stiff system only when the time step is very small. Classic numerical integration codes usually use schemes of adaptive time steps to solve stiff problem, where stiffness of the system is detected continuously during the integration and time steps are adjusted accordingly. However, for the problem of flashing, decreasing time steps of the numerical integration can not solve the stiffness caused by small node length, instead, it may exacerbate it. For smaller time step, the calculation of the boiling boundary is finer and it can move closer to a fixed boundary. This will lead to even smaller node length and larger stiffness. The time step then is forced to decrease even further again and again, until the numerical integration code crashes. To solve this difficulty, a limit to the node length ∆z lim is set. When the node length is smaller than this limit, evaluations of ODEs of equations (7.30), (7.32), (7.33), and (7.34) are stopped. Values of a f ,i , a m ,i , si and vi in the node i, whose length is very small, are explicitly solved from large stiffness conditions
129
A f ,1 (a m , a f , v, s,...) = 0
(7.39)
Am ,1 (a m , a f , v, s,...) = 0
(7.40)
S1 (a m , a f , v, s,...) = 0
(7.41)
V1 (a m , a f , v, s,...) = 0
(7.42)
These conditions are valid because ODEs of equations (7.30), (7.32), (7.33) and (7.34) are still finite as the node length goes to zero. Only the zero numerators in the first term of the right hand sides can avoid occurrence of infinite numbers when the denominator is zero. Forms of a f ,i , a m ,i , si and vi are given by functions of operating parameters and values at the lower boundary of the
node. Details can be found in the Appendix G.
7.2.4. Phase Variables across Boundaries For the system considered in the section 7.1, phase variables and ODEs may experience jumps across node boundaries, i.e. the values at the upper boundary of a lower node may not be the same at the lower boundary of an upper node. Two types of boundaries have to be identified. The first are those boundaries due to nodalization, such as those of boundaries S2, S4, and S6 in the Figure 7.3(a), S2 and S4 in the Figure 7.3(b). There is no pressure loss and changes of physical conditions, such as differences of heat flux, flow area, and flow patterns across the boundaries. Therefore, phase variables and ODEs are not supposed to change across the boundary. The second are those boundaries, across which there are local pressure loss and changes of physical conditions, such as the boundaries S3 and S5 in the Figure 7.3(a) and Figure 7.3(b). The boundary S3 is the interface between the channel and riser, and there is a local pressure loss coefficient K c ,out at the boundary, and heat flux is changed from q ' (t ) to zero from the channel to
130
the riser. The boundary S5 is the boiling boundary, thus flow patterns are changed from single phase to two-phase. In these cases, phase variables and/or ODEs are going to jump across the boundaries. Figure 7.4 shows conditions across a boundary with jump of the phase variables. Suppose the saturation enthalpy, flow enthalpy, steam quality, and flow velocity at the exit of a lower node Sur1, which is a surface below but infinitely close to the boundary between the nodes, are h f ,1 , hm ,1 , x1 , and v m ,1 , and those at the entrance of an upper node Sur2, which is a surface above but
also infinitely close to the boundary, are h f , 2 , hm , 2 , x 2 , and v m , 2 . There are a local pressure loss coefficient K at the boundary and a flow area change across the boundary. Area number at the surface Sur1 and Sur2 are N a ,1 and N a , 2 respectively. The flow enthalpy hm is not subject to change from Sur1 to Sur2 due to conservation of energy.
hm ,1 = hm , 2
(7.43)
Because of pressure loss, the saturation enthalpy h f will take a jump, however. h f , 2 (t ) = h f ,1 (t ) +
Nρ Nr K 2 v m ,1 (t ) ∆Pdrv N ρ N r + x1 (t )
(7.44)
The changes of flow quality and velocities across a boundary can be derived through conservation of flow enthalpy (equation 7.43) and conservation of mass flow rate. Combining the equation (7.10) and equation (7.43), we will find x 2 (t ) =
hx ,1 (t ) − hx , 2 (t ) hg − hx , 2 (t )
+
hg − hx ,1 (t )
hg − hx , 2 (t )
131
x1 (t )
(7.45)
Na,2
hf,2, hm,2, x2, vm,2
Sur2 K
hf,1, hm,1, x1, vm,1
Sur1 Na,1
Figure 7.4: Conditions across a boundary. Sur1 is a surface below but infinitely close to the boundary. Sur2 is the surface above but infinitely close to the boundary. Conditions on Sur1 include hf,1, hm,1, x1, and vm,1. Area number of the Sur1 is Na,1. Conditions on Sur2 include hf,2, hm,2, x2, vm,2. Area number of the Sur2 is Na,2. There is a pressure loss coefficient K on the boundary.
132
where, hx ,1 (t ) and hx , 2 (t ) denote enthalpies of water phase in the lower and upper nodes. If a node is single phase, hx ,i (t ) = hm ,i (t ) ; other wise, hx ,i (t ) = h f ,i (t ) . Non-dimensional density of flow is given by
ρ m (t ) =
Nρ Nr
(7.46)
N ρ N r + x(t )
Through conservation of mass flow rate across the boundary
ρ m,1 (t )v m,1 (t ) N a ,1 = ρ m, 2 (t )v m, 2 (t ) N a , 2
(7.47)
find
v m , 2 (t ) =
N a , 2 N ρ N r + x 2 (t ) N a ,1 N ρ N r + x1 (t )
v m ,1 (t )
(7.48)
Time derivatives of equations (7.43), (7.44), (7.45), and (7.48) show ODE changes across the boundary.
dhm , 2 (t ) dt
=
dhm ,1 (t ) dt
Nρ Nr 2 dx1 (t ) − v (t ) 2 m ,1 dt dh f , 2 (t ) dh f ,1 (t ) K (N ρ N r + x1 (t )) = − ∆Pdrv dt dt 2N ρ N r dv m ,1 (t ) v m ,1 (t ) dt N ρ N r + x1 (t ) 1 − x (t ) dhx ,1 (t ) 1 − hg − hx , 2 (t ) dt (h − h (t ) ) dhx , 2 (t ) dx 2 (t ) g x ,1 (1 − x1 (t ) ) = + (hg − hx , 2 (t ) )2 dt dt hg − hx ,1 (t ) dx1 (t ) hg − hx , 2 (t ) dt
133
(7.49)
(7.50)
(7.51)
dv m , 2 (t ) dt
=−
N a,2 N a ,1
N ρ N r + x 2 (t ) dx1 (t ) − N N x t dt + ( ) ρ 1 r dx 2 (t ) v m ,1 (t ) − N ρ N r + x1 (t ) dt N ρ N r + x 2 (t ) dv m ,1 (t ) v m ,1 (t ) dt
(7.52)
7.3. Closure of Boiling Boundary and Its Time-Dependent Derivative For a set of the expansion parameters, the ODEs given by (7.30), (7.32), (7.33) and (7.34) depend on evaluations of intermediate unknowns. These include the boiling boundary µ (t ) , the saturation enthalpy at the boiling boundary h f , µ (t ) , the inlet saturation enthalpy h f ,inlet (t ) , time derivative of inlet flow velocity
dvinlet (t ) dµ (t ) , time derivatives of the boiling boundary and dt dt
time derivative of the inlet saturation enthalpy
dh f ,inlet (t ) dt
. If these unknowns are determined as
functions of the expansion parameters, ODEs of the expansion parameters, thus, can be evaluated sequentially from the lowest node at the channel inlet to the highest one at the riser outlet. However, the unknowns do not depend on local expansion parameters, but all the expansion parameters. Analytical forms of the functions therefore are fairly complicated and derivations can be tedious processes. Alternative numerical evaluations of the unknowns require iterative processes. In this section, the numerical evaluation method is discussed. The pressure at the riser outlet is fixed as system pressure. Non-dimensional process and dependence of saturation enthalpy and local pressure lead to the saturation enthalpy at riser outlet fixed
h f ,r ,out (t ) = 1
134
(7.53)
For a set of expansion parameters a f ,i , a m ,i , si and vi , values of the unknowns have to get the boundary condition (7.53) satisfied.
µ (t ) , h f , µ (t ) , h f ,inlet (t ) and
dvinlet (t ) are inter-related. If any one of them is chosen as the dt
iterative variable, the rest three can be determined as functions of the chosen one by pressure balance along the single phase regions. Figure 7.5 shows an example of the case “boiling in the channel”. The number of nodes for the three regions are N 1 , N 2 and N 3 respectively, and the total number of nodes N = N 1 + N 2 + N 3 . Node lengths in the three regions from the channel inlet to the riser outlet are given by
∆z1 (t ) = µ (t ) N 1 ,
(7.54)
∆z 2 (t ) = (Lc − µ (t )) N 2
(7.55)
∆z 3 = Lr N 3
(7.56)
From the assumption 1 in the section 7.2.1, flow is at equilibrium state at the boiling boundary, thus flow enthalpy hm , N1 +1 is equal to the saturation enthalpy h f , µ (t ) . N1
h f , µ = hm , N1 +1 = hm ,inlet + ∑ a m ,i (t )∆z1 (t )
(7.57)
i =1
where hm ,inlet =
N sub dvinlet (t ) is the flow enthalpy at the channel inlet. The value of can be N flash dt
determined through integration of the single phase momentum equation (7.15) along the dashed line in the Figure 7.5 from the riser outlet to the boiling boundary.
dvinlet (t ) 1 = ∆Pext − ∆Pgrav − ∆Pfric − ∆Ploc dt Mass
[
where
135
]
(7.58)
Kr,out
h,f, r,out=1
Kdc,in
N3
Kc,out
Ldc hf, µ
N2 N1 µ
Kc,in
vinlet, dvinlet/dt
hf,inlet
Figure 7.5: Intermediate unknowns for ODE evaluations. Boiling boundary is in the channel in this example.
136
Mass = N a ,dc (Lc + Lr ) + N a ,c µ (t )
(7.59)
∆Pext = (h f , µ (t ) − 1)∆Pdrv
(7.60)
∆Pgrav
N1 (1 + N exp,h ,c hm ,i −1 (t ) + N dev ,h ,c ) ∑ ∆z1 (t ) + Fr i =1 N1 2 N exp,h ,c a m ,i (t ) ∆z1 (t ) = ∑ − i =1 Fr 2 (Lc + Lr ) F r
∆Pfric = N f ,1 (N a ,c vinlet (t )) µ (t ) 2
(
(7.62)
)
∆Ploc = k c ,in N a ,c + k dc ,in N a ,dc vinlet (t ) 2
2
(7.61)
2
(7.63)
Mass is non-dimensional mass along the dashed line, ∆Pext is the pressure between the boiling
boundary and riser outlet, ∆Pgrav is the pressure loss (and gain) due to gravity, ∆Pfric is the pressure loss due to friction, ∆Ploc is the pressure loss due to restrictions along the integration line. The value of inlet saturation enthalpy h f ,inlet (t ) is given by integration along the dashed line from riser outlet to the channel inlet
h f ,inlet (t ) =
1 ∆Pdrv
dvinlet (t ) Mass dc dt + ∆Pgrav ,dc + ∆Pfric ,dc + ∆Ploc ,dc + 1
(7.64)
where, Mass dc = N a ,dc (Lc + Lr )
(7.65)
∆Pext = (h f ,inlet (t ) − 1)∆Pdrv
(7.66)
∆Pgrav = −
(Lc + Lr ) Fr
∆Pfric = 0
(7.67) (7.68)
137
∆Ploc = k dc ,in (N a ,dc vinlet (t ))
2
(7.69)
The subscript “dc” denote the values in the downcomer region. For the moving boundary scheme, different cases of boiling are modeled differently, thus the approximate position of the boiling boundary is known at least roughly (out of the riser, in the riser, at the connection of the channel and riser, or in the channel). It is convenient to choose the boiling boundary as the iterative variable and other three unknowns are determined through equations (7.54 – 7.69) as functions of the boiling boundary. For the fixed boundary scheme, however, there is no previous knowledge about the position of boiling boundary µ (t ) . It can be in the channel, in the riser, at the connection between the channel and riser, or more specifically, out of the riser. A bad initial guess of the boiling boundary may lead to non-convergence of the iterative process. To avoid this difficulty, the saturation enthalpy at the channel inlet h f ,inlet (t ) , instead of the boiling boundary µ (t ) , is chosen as the a better candidate of the iteration variable. The
dvinlet (t ) , thus, is represented as a function dt
of h f ,inlet (t ) (equation (7.64)). Initial guess of the boiling boundary µ (t ) is at the riser outlet ( µ (t ) = Lc + Lr ). The flow enthalpy hm (t ) and saturation enthalpy h f (t ) are evaluated sequentially from the lowest node. Since the boiling boundary is initially assumed at the exit of the highest node, virtually all the nodes evaluated are single-phase. If in a specific node i, the flow enthalpy hm,i (t ) are larger than the saturation enthalpy h f ,i (t ) at the node exit, it indicates the boiling boundary is actually located at somewhere in the middle of the node i ( z i −1 < µ (t ) < z i ). The value of µ (t ) has to get h f , µ (t ) = hm , µ (t ) satisfied. From equation (7.19), (7.20) and (7.27), the value of µ (t ) and h f , µ (t ) is given by
138
µ (t ) = z i −1 −
2 E3
(7.70)
2
E 2 + E 2 − 4 E1 E3
h f , µ (t ) = hm , µ (t ) = hm ,i −1 (t ) + a m ,i (µ (t ) − z i −1 )
(7.71)
where, E1 =
E2 =
N exp,h
(7.72)
2 Fr ∆Pdrv 1 ∆Pdrv
dv m ,i (t ) (1 + N exp,h hm ,i −1 + N dev ,h ) 2 + + N f ,1v m ,i − a m ,i (7.73) Fr dt
E3 = h f ,i −1 − hm ,i −1
(7.74)
Once the boiling boundary is determined (either as iterative variable for the moving boundary scheme, or as function of the h f ,inlet (t ) for the fixed boundary scheme), saturation enthalpies above the boiling boundaries h f ,i , i = N µ ,
N + 1 can be determined sequentially
along the nodes from the boiling boundary to the riser outlet from equations (7.19-7.22) and (7.44). The saturation enthalpy at the riser outlet h f ,r ,out = h f , N +1 , therefore, is a single value function of the iterative variable y
h f ,r ,out (t ) = h f , N (t ) = g ( y (t ))
(7.75)
for the moving boundary scheme y (t ) = µ (t ) ; for the fixed boundary scheme y (t ) = h f ,inlet (t ) . A simple Newton’s method can be used to find the iterative variable that satisfies the equation ()
(
(
µ (t )(n +1) = µ (t )(n ) + 1 − g y (t )(n )
)) DG
(7.76)
where DG = dg ( y (n ) ) dy (n ) . It is found that the iteration of Newton’s method converges rather quickly. In most cases, less than 4 iterations are enough.
139
The afro-mentioned process can be applied similarly to evaluate the time derivatives of the boiling boundary
dh f ,inlet (t ) dµ (t ) and inlet saturation enthalpy . If there is no considerations of dt dt
the zero-length node along the channel and riser, it is shown that the time derivative of the saturation enthalpy at the riser outlet
dh f ,r ,out (t ) dt
is a linear function of either
dh f ,inlet (t ) dµ (t ) or . dt dt
Therefore only one Newton’s iteration is enough for the solution of time derivative of the boundary condition (7.53). If there is node with almost zero length, the solutions of (7.39-7.42) cause a nonlinear function of the
dh f ,r ,out (t ) dt
to the
method are required.
140
dh f ,inlet (t ) dt
. Multiple iterations of Newton’s
Chapter 8 Stability and Bifurcation Analyses of Natural-Circulation BWR Model
In this chapter, the stability and bifurcation analyses of the thermal-hydraulics and nuclear coupled thermal hydraulics models of the natural circulation BWR are carried out. The methodology is the same as that in section 2.2. For the thermal-hydraulics model, two sets of operating parameters listed in Appendix H and I are used for these studies. One corresponds to the natural circulation loop SIRIUS [44] [61] and the other corresponds to the natural circulation BWR Dodewaard reactor (the Netherlands) [45] [59] [62] [63]. Results obtained using BIFDD are first validated and compared with experiments carried out on the SIRIUS facility. Stability, bifurcation and sensitivity analyses are then carried out using the two sets of data. For the nuclear coupled model, only the set of operating parameters corresponding to the Dodewaard reactor are used. Characteristic of bifurcations and regions of in-phase/out-of-phase oscillations are determined and analyzed in different operating parameter spaces.
8.1. Validations and Comparisons
8.1.1 Convergence The set of parameters for SIRIUS facilities are used for testing convergence of different nodalization schemes. Figure 8.1 shows steady state non-dimensional boiling boundary ( µ ) and *
inlet flow ( vinlet ) as a function of inlet subcooling temperature ( ∆Tinlet ) calculated using the fixed
141
boundary scheme. The system pressure is Psys = 0.2 MPa, and heat flux in the channel is q ' = 6.0 (kW/m). Total numbers of nodes are N = 3, 5, and 9 respectively (the nodes with fixed boundaries in the channel and riser are equal, and N 1 = N 2 = 1, 2, and 4). The boiling boundary is affected little by the number of nodes. For the flow velocity, the results appear to converge for N = 9. As the subcooling increases, Figure 8.1(a) shows that the steady-state boiling boundary *
moves upward. For ∆Tinlet > 10.6 (K), the boiling boundary is in the riser and flashing induced *
boiling due to pressure drop, instead of heating, begins. For ∆Tinlet > 34.8 (K), the boiling boundary reaches the riser outlet, and no boiling at steady state occurs in the channel or the riser. It is noted that the value of subcooling for the no-boiling case strongly depends on the Boussinesq [64] expansion parameters N exp h and constant N devh (Appendix E). In this calculation, the parameters N exp h and N devh are assumed constant for all single phase nodes in the channel and *
riser, and they are evaluated using thermodynamic properties of water at pressure Phigh (section 7.2.1). The parameters N exp h and N devh in the downcomer are assumed to be zero. However, thermodynamic properties along the channel and riser may be quite different due to changes in the local pressure. Consequently, the values of these two parameters, N exp h and N devh , will be parametrically changed later in this chapter. Figure 8.1(b) shows the steady state inlet flow velocity. It is noted that the inlet flow velocity is increasing with inlet subcooling at low values of inlet subcooling due to lower friction as the boiling boundary moves upward. Inlet flow velocity starts to decrease with inlet subcooling as the boiling boundary reaches into the riser—because of lower density difference between the channel-riser and the downcomer, and thus a drop in the natural circulation driving force.
142
5
P2 P1
4
3
µ
P3
N=3 N=5 N=9 N=17
2
P4
1
P5 0
0
10
20
30
40
*
∆Tinlet (K)
(a)
7
P4
N=3 N=5 N=9 N=17
6
P3
P5
vinlet
5
4
3
P2 P1 2
1 0
10
20
30
40
*
∆Tinlet (K)
(b) Figure 8.1: Steady state boiling boundary and inlet flow velocity calculated with fixed boundary schemes. Total number of nodes are N = 3, 5, and 9. (a) boiling boundary µ ; (b) inlet flow velocity vinlet .
143
Figure 8.2 shows the stability results in the form of stability boundaries (SBs), determined using BIFDD, in the inlet subcooling temperature—heat flux parameter space. The system pressure is Psys = 0.2 MPa. Figure 8.2 (a) shows the SBs calculated with the moving boundary schemes. Total numbers of nodes are N = 3, 6, 12, and 24 (number of nodes in the three regions are equal). For small values of N , the stable region is larger, indicating insufficient nodalization may lead to non-conservative results. As N is increased, the stable area is decreased. The stable area for N = 12 and 24 are very close, and the SBs appear to have converged. Converged boundary is discontinuous at one point, and has a discontinuity in slope at another point. The discontinuity in slope is due to the fact that there are two different complex conjugate pairs, and as shown in chapter 3 for forced circulation systems, a different pair is responsible for the instability on the two sides of the discontinuity. This is explained in much more detail later in this chapter. Reason for the discontinuity in the SB along the separation boundary is discussed below. In addition to the SB, another boundary represented by the short dash-dot line is shown in Fig. 8.2, which separates the parameter space into two regions. The steady state boiling boundary in region in parameter space above this separation boundary is in the riser; while the boiling boundary in the region below the separation boundary is in the channel. The SB is thus divided (by this dash-dot separation boundary) into a segment along which the boiling boundary is in the riser and a segment along which the boiling boundary is in the channel. The SB is discontinuous where this transition occurs. (The discontinuity is exacerbated by an insufficient number of nodes.) The discontinuity can be explained by the fact that there is a qualitative change in the system when the boiling boundary moves from the channel into the riser. [These qualitative changes include a concentrated pressure drop at the channel exit; change in area; and no heat flux in the riser. Moreover, boiling (or flashing) in the riser is due to drop in pressure, unlike in the
144
15
15
N=3 N=6 N=12 N=24
14
Unstable
13
Unstable
12
*
11 10 9 8
11 10 9 8
Riser
7
N=3 N=5 N=9 N=17 N=33
13
∆Tinlet (K)
*
∆Tinlet (K)
12
14
Riser
Stable
7
Stable
6
6
channel 5
channel
5 0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.001
q' (kW/m)
0.002
0.003
0.004
0.005
0.006
0.007
q' (kW/m)
(a)
(b)
15 14
N=24, Moving Bound. N=33, Fixed Bound.
13
*
∆Tinlet (K)
12
Unstable
11 10 9 8
Riser
7
Stable
6
Channel
5 0.001
0.002
0.003
0.004
0.005
0.006
0.007
q' (kW/m)
(c)
Figure 8.2: Stability boundaries in inlet subcooling—heat flux parameter space for two different nodalization schemes. (a) moving boundary scheme; (b) fixed boundary scheme; (c) comparison of moving and fixed boundary schemes.
145
channel where heat addition leads to boiling.] The SB only indicates stability in response to an infinitesimal perturbation about a steady state. For this infinitesimal perturbation, it is possible that the system is stable for a steady state with a boiling boundary in the channel but infinitesimally close to the channel exit, but unstable for another steady state with the boiling boundary in the riser but infinitesimally close to the channel exit. Part of the reason for this discontinuity is the failure of the homogeneous equilibrium model used here to represent the two phase flow. With the inclusion of a subcooled boiling model and thermodynamics nonequilibrium, the discontinuity is likely to disappear. Figure 8.2(b) shows results for the fixed boundary scheme. Total numbers of nodes are N = 3, 5, 9, 17, and 33. Here the converged SB is discontinuous not only on the boundary that
separates the region where the boiling boundary is in the channel from the region where it is in the riser, but also at locations where the boiling boundary is in the channel or in the riser. These jumps in the SBs are more pronounced for smaller numbers of nodes. As the number of nodes is increased, the discontinuities are smoothed out, and the SBs converge to a limiting boundary. These discontinuities result due to the movement of the boiling boundary across fixed nodes. Every time when the boiling boundary moves across a fixed boundary, a node will change from single phase to a two phase or vice versa. This leads to changes in the dynamical system and dynamics of the system accordingly, thus, creating a jump in the SB. For the fixed boundary scheme, the SB is always assumed to be a single valued function of heat flux. This condition is valid when the heat flux is small and large. In these cases, SB segment exists either in region above the separation boundary (boiling in the riser) or below (boiling in the channel). However, for the intermediate value of heat flux, segments of SB can exist in both the two regions. To maintain the single valued condition, segments of the SB above the separation
146
boundary in Figure 8.2(b) are tailed off. The part of the SB segment where the boiling boundary is in the riser in the Figure 8.2 (b) does not smoothly approach the separation boundary. It instead suddenly jumps from the riser region to the channel region. Figure 8.2(c) compares the converged SBs for the moving and fixed boundary schemes. They agree with each other satisfactorily (except the part that is tailed off by the fixed boundary scheme). However, it takes more nodes in the fixed boundary scheme than the moving boundary scheme to reach a smooth and converged boundary. The results suggest that the moving boundary scheme is superior to the fixed boundary scheme for the purpose of stability analysis.
8.1.2. Model Validations and Comparisons with Experiments Preliminary evaluation of the model carried out by comparing steady state and stability results predicted by the model with experimental data is presented in this section. More detailed stability and bifurcation analyses are presented in the next section. Figure 8.3 compares converged steady state flow velocities with experimental data by Inada et al [61]. The system pressure for the experiments is Psys = 0.2 MPa. Heat flux in the channel is q ' = 1.7647, 2.9412, 4.1177, and 5.2941 (kW/m), respectively. Good agreement between the model based on HEM and experimental data is found when heat flux and subcooling are high (Figure 8.3 (c) and (d)). The model, however, over-estimates flow rates for low heat flux and inlet subcooling (Figure 8.3(a) and (b)). This over-estimation is (most likely) due to the lack of subcooled boiling model as well as due to the use of HEM. Figure 8.4 compares stability boundaries calculated using the model developed here with experimental results by Inada et al [61], in the inlet subcooling—heat flux parameter space. The system pressure is Psys = 0.1 MPa. The open squires represent the operating points with stable
147
1.5
1.5
Experiment Analysis
Experiment Analysis
*
vinlet (m/s)
1.0
*
vinlet (m/s)
1.0
0.5
0.5
0.0
0.0 5
10
15
20
5
10
15
*
20
25
30
*
∆Tinlet (K)
∆Tinlet (K)
(a)
(b) 1.5
1.5
Experiment Analysis
Experiment Analysis 1.0
*
*
vinlet (m/s)
vinlet (m/s)
1.0
0.5
0.5
0.0
0.0 10
15
20
25
30
10
15
20
25
30
35
*
*
∆Tinlet (K)
∆Tinlet (K)
(d)
(c)
Figure 8.3: Comparison of steady state flow rates between experiments and analyses. System pressure is Psys = 0.2 MPa. (a) heat flux q ' = 1.7647 (kW/m); (b) q ' = 2.9412 (kW/m); (c) q ' = 4.1177 (kW/m); (d) q ' = 5.2941 (kW/m)
148
40
flow rates found in the experiments. The solid circles represent those with unstable flow rates in the experiments. As shown in Figure 8.4, when the inlet subcooling is large, no boiling occurs in the natural circulation system. The flow is single phase and stable. As the subcooling is decreased, flashing starts in the riser and unstable intermittent waves occur. As the subcooling is decreased further, the flow is still unstable, but it evolves from the intermittent waves to the sinusoidal waves. For small inlet subcooling, amplitude of the sinusoidal wave is very small, and the flow is again stable. Figure 8.4(a) shows comparison of the stability boundary—which separates the stable twophase flow from the unstable sinusoidal wave flow—calculated using BIFDD with the experimental data,. Only the segment in the riser is plotted. Good agreement is found in the range 2.0 (kW/m) < q ' < 4.66 (kW/m). For the heat flux less than 2.0 (kW/m), the analytical stability results slightly over-estimate the stable region. This over estimation is evidently because of the lack of a subcooled boiling model. At point T (heat flux is 4.66 (kW/m)), the stability boundary (solid line) turns sharply back toward lower heat flux region. This result indicates that operating points with heat fluxes greater than 4.66 (kW/m) are unstable. The experimental data however shows at least one more case of stable flow for heat flux greater than 4.66 kW/m. The discrepancy between the analytical result and the experimental data beyond the point T might be explained by a mechanism similar to the crossing of eigenvalues in section 3.1 of Chapter 3. As it will be shown in the next section, the SB turns back because the pair of eigenvalues with the second largest real part (along the stability boundary shown) crosses the eigenvalues with the largest real part. Figure 8.4(a) also shows the continuation of the SB (dotted line) on which the real part of the second rightmost eqignvalue pair is equal to zero. The combined segments of the SB, A-T and the dotted line T-B capture the division between the stable and unstable experimental data points
149
50
50
Stable Unstable
*
∆Tinlet (K)
40
30
*
∆Tinlet (K)
40
Stable Unstable
E=1.0 E=1.5 E=2.0
30
20
20
B
A T 10 0.000
0.002
0.004
10 0.000
0.006
q' (MW/m)
0.002
0.004
0.006
q' (MW/m)
(a)
(b)
50
Stable Unstable
*
∆Tinlet (K)
40
E=1.5
30
20
B
A T 10 0.000
0.002
0.004
0.006
q' (MW/m)
(c) Figure 8.4: Comparisons of experimental data with the stability boundaries evaluated using the model developed in chapter 7. System pressure is Psys = 0.1 MPa. (a) Comparison of stability boundary; (b) Comparison of no-boiling boundary for different Boussinesq expansion parameter values; (c) Comparison of both stability boundary and no-boiling boundary.
150
satisfactorily. More detailed discussion of these issues will be presented in the next section. Figure 8.4(b) shows comparison of analytical no-boiling boundary (boundary in parameter space above which no boiling occurs in the system; flow is all single phase and stable) with the experimental data, which separates the stable single phase flow from the unstable intermittent wave flows. The analytical boundary is calculated by tracking the operating points on which boiling occurs exactly at the riser outlet. Above the boundary, the model predicts no-boiling in the natural circulation system. As described at the beginning of this section, the no-boiling condition strongly depends on the Boussinesq expansion parameter N exp h and N devh . Values of the *
parameters are evaluated at the system pressure Phigh , which is roughly the pressure at the channel inlet, and assumed constant in the entire single phase region. However, due to pressure drop, rate of the Boussinesq expansion is different along nodes in the channel and riser. To account for this effect, the expansion parameter is parametrically modified by multiplying N exp h and N devh by a factor E , varying E between 1 and 2. Figure 8.4 (b) shows the no-boiling boundaries for E = 1.0, 1.5 and 2.0. The boundary for E = 1.5 case is found to best match the experimental boundary separating the stable experimental data points from the unstable ones. Figure 8.4 (c) combines the results of Figures 8.4 (a) and (b). The segments A-T and T-B in Figure 8.4(a) and the no-boiling boundary in Figure 8.4(b) are plotted on the same figure. Comparison with the experimental data show good agreement, especially for large heat flux values. *
Similar comparisons are made for the system pressure Psys = 0.2 MPa (Figure 8.5). In Figure 8.5(a), the SB calculated by BIFDD is compared with the experimental data. When heat flux q ' is less (greater) than 4.0 (kW/m), boiling along the operating points on the SB occurs in
151
50
50
Stable Unstable
40
40
E=1.0 E=1.5 E=2.0
30
*
∆Tinlet (K)
*
∆Tinlet (K)
30
Stable Unstable
20
20
10
10
In Riser 0 0.000
0.002
In Channel
0.004
0 0.000
0.006
q' (MW/m)
0.002
0.004
0.006
q' (MW/m)
(a)
(b)
50
40
Stable Unstable
*
∆Tinlet (K)
E=1.5 30
20
10
In Riser 0 0.000
0.002
In Channel
0.004
0.006
q' (MW/m)
(c) Figure 8.5: Comparison of experimental data points with stability boundaries evaluated using the model developed in Chapter 7. System pressure is Psys = 0.2 MPa. (a) Comparison of stability boundary; (b) Comparison of no-boiling boundary for different values of the Boussinesq expansion parameter; (c) Comparison of both, stability boundary and no-boiling boundary with the experimental data.
152
the riser (channel). The SB separates the stable experimental data points from the unstable ones satisfactorily. Figure 8.5(b) plots the no-boiling boundaries for E = 1.0, 1.5 and 2.0. The boundary for E = 1.5 is again found to best match with the experimental data. The combined results are presented in the Figure 8.5(c).
8.2. Stability and Bifurcation Analyses of the Thermal-Hydraulics Model In this section, natural circulation thermal-hydraulics is studied as a dynamical system. Stability boundaries and oscillation curves are calculated and analyzed for the SIRIUS facility *
[44] under low pressure conditions; Psys = 0.1 and 0.2 MPa. Behavior of natural circulation *
system under higher pressure condition ( Psys = 1.07 MPa) is studied for the Dodewaard reactor system (without neutronics). Description and dimensions and parameters of the SIRIUS facility are given in reference [44] [61] and Appendix H. Figure 8.6 shows, in the inlet subcooling—heat flux parameter space, the stability boundary (solid line) and the boundary on which the second rightmost eigenvalue pair has zero real part (the dotted line). This second boundary is given only for the case when the boiling boundary is in the riser. [The second boundary has no significance for the case when the boiling boundary is in the channel. That is, when the boiling boundary is in the channel (and no neutronics), the other pairs of eigenvalues of Jacobian matrix are all far to the left of the pair with *
the largest real part.] System pressure is Psys = 0.1 MPa. Unlike the SB shown in Figure 8.4, here
both branches of the SB—that is, segment on which boiling starts in the riser and segment on which boiling starts in the channel—are plotted. They are separated by the separation boundary (short dash-dot line). The two branches do not exactly meet at the same point on the separation
153
boundary, D and E.. The discrepancy is again due to the sudden jump in system characteristics as the boiling boundary, constrained by a HEM, jumps from the channel into the riser. On the branch with boiling in the riser, the two pairs of complex conjugate eigenvalues *
cross at the point T ( q ' = 4.66 (kW/m), ∆Tinlet = 15.33 (K)). Along the segment of the SB from point A to T, real part of the right most pair of eigenvalues e1 is zero. As one moves from A toward
T, the second rightmost pair of eigenvalues e2 approaches the imaginary axis. The pair e2 crosses the pair e1 at the point T. Beyond the point T, the real part of e1 on the dotted line segment T-B is still zero, but the pair e1 is now the second rightmost pair. Real part of the rightmost pair of eigenvalue e2 is greater than zero and the system is unstable. The SB segment C-T is determined by the pair e2. Along the SB segment C-T and the dotted line segment T-C’, the real part of e2 is zero, but the segment T-C’ is in the unstable region. Similar to the crossing of eigenvalues described in the section 3.1, this change of eigenvalue pair responsible for the instability represents characteristic change in the type of oscillations that result as the SB is crossed. Moreover, this also explains the jump in slope in the SB in Fig. 8.2. The most significant difference is in the period of oscillations produced by the pair e1 and those due to e2. The imaginary part of the rightmost pair of eigenvalue determines the frequency of oscillations for the operating points close to the SB. In this case, the imaginary part for the pair
e1 is 16.01; while for the pair e2, the imaginary part is 3.32. Therefore, the frequency that results due to the pair e1 is about 5 times larger than that produced by the pair e2. This fact may help explain the slight discrepancy between the SB calculated and the experimental data (for one experimental data point) in section 8.1.2. In the region of the parameter space below the segment T-B and to the right of the segment C-T, the system is unstable, because the real part of the rightmost pair e2 is greater than zero. However, the real part of the second rightmost pair e1 is less
154
than zero. Low frequency oscillation produced by the pair e2 will hence grow, but high frequency oscillations produced by the pair e1 will initially decay. Since the high frequency oscillations are the easiest to observe and record, when operating point crosses the segment T-B, the experimental observation of high frequency (initially) decaying oscillations might have suggested a stable system, and the experiment might not have been continued long enough to observe the unstable low frequency oscillations. This might have led to the characterization of the data point for the rightmost value of heat flux studied as “stable.” Figure 8.7 shows 7.5% oscillation curves (OCs) calculated using BIFDD for different *
*
segments along the SB. The y-axis is ∆Tinlet − ∆Tinlet ,critical , where subscript “critical” denotes the value on the SB. For the SB branch on which boiling is in the riser, the oscillation curve for the segment A-T is in the unstable region, and the bifurcation is supercritical. On the other hand, the OC for the segment T-C is in the stable region, and the bifurcation is subcritical. For the SB branch on which boiling is in the channel, the bifurcation is subcritical along the segment D-E and along a part of the segment E-F. Bifurcation turns supercritical below a certain value of subcooling near the middle of the segment E-F, and stays supercritical for the entire segment F-G. *
Figure 8.8 shows results of the stability analysis for the system pressure Psys = 0.2 MPa. *
Similar to the case of Psys = 0.1 MPa, two pairs of complex conjugate eigenvalues cross in the complex plane along the branch of SB when the boiling boundary is in the riser. At the point T ( q ' *
= 3.62 (kW/m), ∆Tinlet = 11.04 (K)), both pairs of complex conjugate eigenvalues have zero real part. The imaginary parts (frequencies) of e1 and e2 at the point T are 14.60 and 3.68, respectively. Along the segment A-T, e1 is the dominant pair, while e2 is the dominant pair along T-D . Figure 8.9 shows 7.5% oscillation curves for different segments along the SB in Figure 8.8. The
155
25
Stability Boundary Boundary of the 2nd eigenvalue 20
C' B
T
15
*
∆Tinlet (K)
A
Stable 10
5
Riser
C
E
D Unstable
Channel
F G
0 0.000
0.002
0.004
0.006
q' (MW/m)
Figure 8.6: Stability boundary and the boundary on which the second rightmost pair of *
eigenvalues has zero real part. System pressure Psys = 0.1 MPa. The SB is divided into segments along which boiling is in the riser, and the other one along which boiling is in the channel.
156
0.010
T ∆Tinlet -∆Tinlet,critical (K)
0.008
T
*
0.008
0.006
0.004
0.002
0.000 0.000
0.006
*
*
*
∆Tinlet -∆Tinlet,critical (K)
0.010
Supercritical
0.004
C 0.002
Subcritical
A 0.002
0.004
0.000 0.000
0.006
0.002
0.004
0.006
q' (MW/m)
q' (MW/m)
(b)
(a)
E Subcritical
-0.05
*
*
∆Tinlet -∆Tinlet,critical (K)
0.00
-0.10
-0.15
-0.20
D
-0.25 0.000
0.002
0.004
0.006
q' (MW/m)
(c)
Figure 8.7: 7.5% oscillation curves for different segments of the SB shown in Figure 8.6. (a) segment A-T; (b) segment T-C; (c) segment D-E; (d) segment E-F; (e) segment F-G
157
0.012
0.008
*
∆Tinlet -∆Tinlet,critical (K)
E
0.004
*
Subcritical 0.000
Supercitical -0.004
F 0.000
0.002
0.004
0.006
q' (MW/m)
(d) 0.007
F
*
*
∆Tinlet -∆Tinlet,critical (K)
0.006
0.005
0.004
0.003
0.002
G
0.001
0.000 0.000
0.002
0.004
Supercritical
0.006
q' (MW/m)
(e)
Figure 8.7 (continue): 7.5% oscillation curves for different segments of the SB shown in Figure 8.6. (d) segment E-F; (e) segement F-G
158
16
Stability Boundary Boundary of 2nd Eigenvalue
14
Unstable
12
T D'
*
∆Tinlet (K)
B
A
10
8
Stable Riser
6
C D E
F
Channel
0.000
0.002
0.004
0.006
q' (MW/m)
Figure 8.8: Stability boundary and the boundary of the eigenvalues with the second largest real *
part. System pressure Psys = 0.2 MPa. The SB is divided by the separation boundary that divides the parameter space into a region where the boiling boundary is in the riser (above) and a region where the boiling boundary is in the channel (below).
159
0.006
0.000
*
∆Tinlet -∆Tinlet,critical (K)
T
*
∆Tinlet -∆Tinlet,critical (K)
T 0.005
0.004
Subcritical
-0.010 -0.015 -0.020
*
*
0.003
-0.005
0.002
A
-0.025 -0.030
0.001
Supercritical
0.000 0.000
0.002
0.004
-0.035 -0.040 0.000
0.006
C 0.002
C ∆Tinlet -∆Tinlet,critical (K)
0.0001
Supercritical
*
*
∆Tinlet -∆Tinlet,critical (K)
0.0002
D
0.12 0.10
*
0.08
0.0000
-0.0001
Subcritical
*
0.06 0.04
Subcritical
0.02 0.00 0.00407
0.006
(b)
(a)
0.14
0.004
q' (MW/m)
q' (MW/m)
0.00408
0.00409
0.00410
0.00411
-0.0002
-0.0003
-0.0004 0.000
0.00412
q' (MW/m)
E 0.002
0.004
F 0.006
q' (MW/m)
(c)
(d)
Figure 8.9: 7.5% oscillation curves for different segments of the SB shown in Figure 8.8. (a) segment A-T; (b) segment T-C; (c) segment C-D; (d) segment E-F.
160
bifurcation is supercritical for the segment A-T, but subcritical for the segment T-C and C-D. The bifurcation along the SB branch along which the boiling boundary is in the channel (segment E-F) moves back and forth from the subcritical to the supercritical case. Dynamics of a natural circulation loop under high pressure is studied using system parameters corresponding to the Dodewaard natural circulation reactor (the Netherlands). Note that neutronics is not included at this stage. Coupled system is studied in the next section. System *
pressure in Figure 8.10 is Psys = 1.07 MPa. Pressure loss coefficient K c ,out at the connection of the riser and channel in the data of Dodewaard reactor is not zero (1.095). Therefore, the branches of the SB corresponding to the cases of boiling boundary in the riser and boiling boundary in the channel are separated by not just one dashed line but two lines. The branch to the left of the dashed lines corresponds to boiling in riser, and the branch to the right of the dashed lines corresponds to boiling in channel. The region in the middle of the two dashed lines corresponds to the special case where the boiling boundary stays at the connection. Since the system pressure is relatively high, flashing in the riser is relatively less intense. Region in the parameter space where boiling (flashing) may start in the riser (to the left of the dashed lines) is much smaller than that of the region where boiling starts in the channel (to the right of the dashed lines). Most of the SB (segments C-D-E) is in the second region. Figure 8.11 shows 7.5% oscillation curves for different *
segments of the SB in ∆Tinlet —( q' - q' critical ) parameter space. Along the segment A-B, which is the SB branch along which boiling starts in the riser, the oscillation curve is in the stable region *
*
for ∆Tinlet > 3.04 (K) and ∆Tinlet < 1.88 (K), and the bifurcation is subcritical. It is in the unstable *
region for 1.88 (K) < ∆Tinlet < 3.04 (K), and the bifurcation is supercritical. For the segment C-D, characteristics of bifurcation rapidly changes from supercritical to subcritical as subcooling is
161
12
At Connection D
10
In Riser
In Channel
* P3
*
∆Tinlet (K)
8
6
P1
4
B 2
Unstable
Stable
A
P2
**
C
E
0 0.0
0.1
0.2
0.3
0.4
0.5
q' (MW/m)
Figure 8.10: Stability boundary calculated for system parameters corresponding to the Dodewaard *
reactor. System pressure Psys = 1.07 MPa. The SB is separated into a branch corresponding to boiling in the riser, and the other one corresponding to boiling in the channel.
162
12
Supercritical Subcritical
Supercritical Subcritical
10
10
8
8
∆Tinlet (K)
∆Tinlet (K)
12
6
4
C 2
A
0 -0.002
0 -0.0010
-0.0008
-0.0006
-0.0004
-0.0002
0.0000
0.0002
0.000
0.001
0.002
(b)
(a)
10
-0.001
q'-q'critical (MW/m)
q'-q'critical (MW/m)
12
6
4
B
2
D
Subcritical
Supercritical
D
∆Tinlet (K)
8
6
4
E
2
0 -0.0003
-0.0002
-0.0001
0.0000
0.0001
0.0002
q'-q'critical (MW/m)
(c)
Figure 8.11: 7.5% oscillation curves for the segments of the SB shown in Figure 8.10. (a) segment A-B; (b) segment C-D; (c) segment D-E.
163
increased. It turns back to supercritical again along the segment D-E and stays supercritical even for very low inlet subcooling. The analyses of stability and bifurcation properties under low and high pressure conditions reveal complexities of the dynamics of the natural circulation boiling/flashing systems.
8.3. Stability and Bifurcation Analyses of the Nuclear-Coupled Model In this section the nuclear coupled model is studied using the moving boundary scheme.
8.3.1. Effect of Nodalization Schemes Coupling among thermal-hydraulics, fuel dynamics and neutronics introduces extra feedbacks, which are crucial to the dynamics of the system. However, accurate evaluations of the feedback signals, the void fraction in the channel, and fuel temperature rely on nodalization scheme. The higher the number of nodes in each single-phase or two-phase region, the more accurate is the estimate of the void fraction and fuel temperature. Large number of nodes on the other hand dramatically increase the number of phase variables, leading to computational inefficiency. Therefore, an analysis of the impact of the number of nodes on the results of the stability analysis is carried out first. Stability boundaries (SBs) are calculated, for two different system pressures, using BIFDD for different nodalization schemes in the so-called inletsubcooling—reactivity parameter space (Figures 8.12) for parameter values that roughly correspond to the Dodewaard natural circulation *
*
BWR [45] [62]. System pressure for these two SBs are Psys = 7.0 MPa and Psys = 0.4 MPa. Here, the x axis, ρ ext , is the external reactivity introduced by control rod movement. The y axis, *
∆Tinlet , is the inlet subcooling (saturation temperature minus inlet temperature). Different regions
164
40
N=1 N=2 N=4 N=8 N = 16
4.5 4.0 3.5
30
∆Tinlet (K)
∆Tinlet (K)
5.0
N=1 N=2 N=4 N=8 N = 16
50
Unstable
20
Stable
10
3.0
Unstable
2.5 2.0 1.5
Stable
1.0 0.5
0
0.0
-0.04
-0.03
-0.02
-0.01
0.00
0.01
0.02
-0.04 -0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 0.04
0.03
ρext
ρext
(b)
(a)
Figure 8.12: SBs for different size nodes in ∆Tinlet — ρ ext parameter space for (a) Psys = 7.0 MPa, *
*
*
(b) Psys = 0.4 MPa. The boiling boundary is in the channel, and different regions are discretized to the same number of nodes characterized by N.
165
along the channel inlet to the riser outlet are discretized using identical number of nodes. Different nodalization schemes are characterized by N, which is the number of nodes in each (single phase and two-phase) region. Figure 8.12(a) for high system pressure case, shows that as the number of nodes increases, the SBs converge to a limiting SB. The results point out that insufficient nodalization may cause large deviation in both the area of stable regions and in the shapes of SBs from those obtained using sufficiently large number of nodes. Compared to the SB of N = 16 case, the one with N = 1 significantly over-estimates the area of the stable region. As N is increased, the SBs tend to converge. Good convergence of SB is achieved in regions of low and high ρ ext for the N = 8 case, but the SB may be over-estimated by as much as 10% in the middle range of ρ ext . For the low system pressure case (0.4 MPa), Figure 8.12(b) shows a much better convergence for relatively smaller number of nodes compared to the high system pressure case. Even the SB for the N = 4 case is very close to that of the converged SB. SBs for the N = 8 and N = 16 cases are too close for most values of ρ ext to be distinguished from each other. Results presented in the rest of the paper were obtained using N = 8.
8.3.2. In-phase and Out-of-phase Oscillations *
Figures 8.13 and 8.14 show SBs (solid lines) in ∆Tinlet — ρ ext space for system pressure of 7.0 MPa and 0.4 MPa, respectively. Nature of instability that results—in-phase or out of phase— as these SBs are crossed, may however vary significantly along different parts of the SB. As shown in part I of this dissertation for forced circulation BWR system, the two rightmost pairs of eigenvalues can be associated with in-phase and out-of-phase oscillations (section 3.1). The relative magnitude of the elements in the corresponding eigenvectors indicate as to which mode—
166
fundamental or first azimuthal—will be dominant in the unstable system as the corresponding eigenvalue crosses the imaginary axis. It was shown that the eigenvalue pair that corresponds to the first azimuthal mode may cross the imaginary axis before the eigenvalue pair that corresponds to the fundamental mode leading to predominantly out-of-phase oscillations. Hence, even for the natural circulation system it is important that boundaries associated with the eigenvalues with the largest and second largest real part be taken into stability consideration. Relative magnitude of the elements in the eigenvector can then be used to identify whether it is the in-phase or the out-ofphase mode that will be dominant in the oscillations when the system becomes unstable as the SB is crossed. Figures 8.13 and 8.14 show two boundaries associated with the two pairs of complex conjugate eigenvalues with largest real parts. One of these pairs has zero real part along the boundary with squares, while the second pair has zero real part along the boundary with triangles. For low system pressure (Figure 8.14) the two boundaries do not intersect, and hence the entire SB is composed of the boundary associated with one pair of eigenvalue—in this case the one with *
squares. However, Figure 8.13, for Psys = 7.0 MPa case, shows that the two boundaries intersect *
at point X ( ρ ext = -0.0335, ∆Tinlet = 8.65 K). Hence, the SB (the solid line) is composed of a segment of the boundary with squares and another segment with triangles. Characteristic difference of oscillations associated with the two rightmost pairs of complex conjugate eigenvalues is further illustrated by analyzing the eigenvalues and the corresponding *
eigenvectors at four points on the two boundaries. Point 1 ( ρ ext = -0.0278, ∆Tinlet = 20.0 K) and *
*
point 2 ( ρ ext = -0.0306, ∆Tinlet = 20.0 K) are on the stability map of the system with Psys = 7.0 *
*
MPa; point 3 ( ρ ext = 0.0124, ∆Tinlet = 2.0 K) and point 4 ( ρ ext = 0.0179, ∆Tinlet = 2.0 K) are on
167
55
Stability Boundary Boundary of 2nd largest real part
50
14
45
35
Unstable
B
∆Tinlet (K)
∆Tinlet (K)
40 IV
30 25 2 1
20
II
Stable
* *
I
10
See inset
X
10 8
C -0.02
I
6
III
2
A -0.03
Stable
IV
A
III
0 -0.04
X
Unstable
4
15
5
II
12
-0.01
0.00
0.01
0 -0.040
0.02
-0.038
-0.036
-0.034
-0.032
-0.030
ρext
ρext
(b)
(a)
*
*
Figure 8.13: Boundaries in ∆Tinlet — ρ ext space for Psys = 7.0 MPa. The solid line is the stability boundary, and the dotted line is the boundary on which the second largest real part is zero. The line with open squares is the boundary on which the real part of the in-phase eigenvalue is zero, and the line with open triangles is the boundary on which the real part of the out-of-phase eigenvalue is zero.
168
7
Stability boundary Boundary of the 2nd largest real part
6 II
∆Tinlet (K)
5
E
4
Unstable III
D 3
Stable
3
I
2
*
*
4
1 F 0 -0.04
-0.03
-0.02
-0.01
0.00
0.01
0.02
0.03
ρext *
*
Figure 8.14: Boundaries in ∆Tinlet — ρ ext space for Psys = 0.4 MPa. The solid line is the stability boundary (SB), and the dotted line is the boundary on which the second largest real part is zero. The two boundaries do not cross. The SB is the boundary on which the real part of the in-phase eigenvalue is zero, and the dotted line is the boundary on which the real part of the out-of-phase eigenvalue is zero.
169
*
the stability map of the system with Psys = 0.4 MPa. Table 8.1 lists the pure imaginary eigenvalues of the Jacobian E imag and magnitudes of the elements corresponding to n 0 (t ) and
n1 (t ) , EV n0 and EV n1 , in the corresponding eigenvectors at these points. Points 1 and 3 (2 and 4) are chosen to analyze the nature of oscillations associated with the eigenvalues that have a zero real part on the boundary with squares (triangles). For points 1 and 3, the magnitude of the element of the eigenvector corresponding to
n 0 (t ) is much smaller than that for n1 (t ) , suggesting large amplitude oscillations in n 0 (t ) , but much smaller amplitude oscillations for n1 (t ) at these two points. The pure imaginary eigenvalues at these two points and along the boundary with squares, thus, are called in-phase or fundamental mode eigenvlaues. For points 2 and 4, the magnitude of the element of the eigenvector corresponding to n1 (t ) is large, while that of n0 (t ) is very small. These eigenvalues, and those corresponding to the boundaries with triangles, are hence called out-of-phase or azimuthal mode eigenvlaues. Therefore, a characteristic change in the nature of oscillation along the SB (solid line) must be expected due to eigenvalue crossing at point X (Figure 8.13). To the left of point X, segment A-X is the SB due to out-of-phase oscillations. To the right of point X, segment X-B-C is the SB due to in-phase oscillations. Parameter space in Figure 8.13 is divided by these boundaries of fundamental and first azimuthal mode eigenvalues into four regions. In region I, all eigenvalues have negative real parts and hence the system is stable. Small perturbation will cause decreasing amplitude oscillations. In region II, the fundamental mode eigenvalue has positive real part and the first azimuthal mode eigenvalue has negative real part. The system is hence unstable, and small perturbations will lead to oscillations dominated by in-phase component. In region III, the first azimuthal mode
170
Table 8.1: Critical eigenvalues E imag and elements EV n0 and EV n1 (corresponding to n0 (t ) and
n1 (t ) ) respectively in the eigenvector at four operating points shown in Figures 8.13 and 8.14.
Point 1
Point 2
Point 3
Point 4
Eimag
±2.306 i
±1.709 i
±3.030 i
±4.079 i
EVn0
0.156
0.170×10-11
0.00972
0.110×10-13
EVn1
0.943×10-12
0.0595
0.324×10-13
0.00530
171
eigenvalue has positive real part and the fundamental mode eigenvalue has negative real part. The system is hence unstable, and small perturbations will lead to oscillations dominated by iout-ofphase component. In the last region (IV), both eigenvalues have positive real part, and hence the system is unstable and the oscillations will therefore be a combination of in-phase and out-ofphase modes. For low system pressure, instead of four regions, three regions of the parameter space are identified. The system is stable in region I. Growing oscillations in region II will be dominated by the fundamental mode; and in unstable region III both modes will contribute about equally to the growing oscillations (Figure 8.14).
8.3.3 Bifurcation Analyses In addition to in-phase and out-of-phase, the oscillations in a natural circulation BWR can also be characterized by whether they evolve to stable periodic oscillations or to unstable growing amplitude oscillations. As discussed in section 2.3, PAH bifurcation theorem provides us with conditions that lead to one or the other. According to PAH-B theorem, if the periodic solution (oscillations) exists in the unstable region, it is stable and the bifurcation is supercritical; if it exists in the stable region, the periodic solution is unstable and the bifurcation is subcritical. Nature of PAH-B (supercritical or subcritical) along the SB will again be represented by fixed amplitude oscillation curves. Figures 8.15 and 8.16 show 7.5% oscillation curves for different *
segments of the SBs shown in Figures 8.13 and 8.14 in ∆Tinlet —( ρ ext – ρext,critical ) parameter space. Here
ρext,critical
is the reactivity on the SB. The amplitude of the limit cycle along these curves is
about 7.5% of the magnitude of the corresponding elements in the eigenvector. In Figure 8.15(a), for the out-of-phase segment A-X, the oscillation curve is in the unstable region, indicating a
172
Subcritical
Supercritical
30
30
∆Tinlet (K)
∆Tinlet (K)
B
25
25 20 15 10
Subcritical
Supercritical
35
35
X
20 15 10
X
5
5 A
0 -0.0012
-0.0008
0 -0.0004
-0.03
0.0000
-0.02
-0.01
0.00
0.01
ρext - ρext,critical
ρext - ρext,critical
(b)
(a) Subcritical Supercritical 35 B
30
∆Tinlet (K)
25 20 15 10 5 C 0 0.000
0.005
0.010
0.015
0.020
0.025
0.030
ρext - ρext,critical
(c)
Figure 8.15: 7.5% oscillation curve along the SB shown in Figure 8.13. (a) segment A-X. (b) segment X-B. (c) segment B-C.
173
0.02
Supercrtical 4.0
Subcritical
Subcritical
3.0
∆Tinlet (K)
3.0
∆Tinlet (K)
E
3.5
D
3.5
2.5 2.0 1.5
2.5 2.0 1.5
1.0
1.0
0.5
0.5
0.0 -0.020
Supercritical
4.0
E
-0.015
-0.010
-0.005
0.000
0.005
0.0 -0.01
F
0.00
0.01
ρext - ρext,critical
ρext - ρext,critical
(a)
(b)
0.02
Figure 8.16: 7.5% oscillation curve along the SB shown in Figure 8.14. (a) segment D-E. (b) segment E-F.
174
0.03
supercritical PAH-B along the SB. From the out-of-phase segment (A-X) of the SB to in-phase *
segment X-B, oscillation curve has a jump from the unstable side to the stable side. For ∆Tinlet < 28.91 K, the oscillation curve of segment X-B is in the stable region, and type of PAH-B is *
subcritical. For 28.91 < ∆Tinlet < 31.42 K (point B), the oscillation curve returns to the unstable region, and consequently type of PAH-B along this segment of the SB is supercritical. For segment B-C, Figure 8.15(c) shows that oscillation curve along the B-C branch is always in the unstable region. Type of PAH-B along this segment is hence supercritical. Similar back-and-forth transitions from sub- to supercritical bifurcations are observed for the low system pressure case. In Figure 8.16(a), the type of bifurcation is supercritical along the D-E branch of the SB. Along the *
E-F branch of the SB (Figure 8.16(b)), when ∆Tinlet > 2.762 K, the bifurcation is supercritical; *
while for ∆Tinlet < 2.762 K, it is subcritical. *
For the out-of-phase segment A-X (Figure 8.15(a)) and low ∆Tinlet region of the in-phase segment B-C (Figure 8.15(c)), deviations of ρ ext from its critical value required for the 7.5% *
amplitude, are much smaller than those for the segment X-B and high ∆Tinlet region of the segment B-C, indicating that much larger amplitude oscillations (in-phase or out-of-phase) may result for the same deviation along the SB in the low subcooling region than those along the SB in the high subcooling region. Large amplitude oscillations at lower inlet subcooling are due to longer two-phase regions in the channel and riser. Similar trend can be found in the low system pressure case (Figure 8.16). The change in characteristic of bifurcation (supercritical or subcritical) and change in characteristic of oscillations (in-phase or out-of-phase) along the SB, thus, reveal complexity of the dynamic behavior of the coupled natural circulation BWR system.
175
8.3.4 Sensitivity Analyses Effect of operating parameters other than ∆Tinlet and ρ ext on the SBs is investigated. Since *
behavior of two-phase natural circulation systems under high system pressure conditions (close to 7.0 MPa) is relatively better understood, sensitivity analysis of operating parameters is focused on relatively low system pressure. To study the effect of system pressure, SBs are plotted in ∆Tinlet — ρ ext parameter space *
*
*
for five different values of Psys (Figure 8.17(a)). Impact of a change in Psys on systems with *
relatively large pressure ( Psys = 1.5, 1.0 MPa) is different from that on systems at low pressure (
Psys
*
= 0.4, 0.3, 0.2 MPa). Stable region decreases as
Psys
*
decreases if the system pressure is *
P relatively high, especially for small values of ρ ext . Thus, a drop in sys has a destabilizing effect under these conditions. If the system pressure is relatively low, however, a drop in
Psys
*
causes the
stable region to increase. Therefore, instead of destabilizing, it has a stabilizing effect on the dynamics of the system. Comparing these results with those of pure thermal-hydraulics model, where a drop in system pressure is a destabilizing factor, this new stabilizing effect can be attributed to the coupling between system thermal-hydraulics and neutron kinetics. At low system pressure, without neutronics coupling, small perturbation can quickly grow due to two-phase flow instability. In the coupled system, however, void fraction feedback acts as a stabilizing force. Perturbations in void fraction lead to negative feedback of reactivity, which suppresses thermalhydraulic perturbations. *
Figure 8.17(b) shows the effect of riser length Lr , for Psys = 0.4 MPa case. In natural
176
∆Tinlet (K)
OP
Lr = 0.30 m Lr = 3.06 m Lr = 5.00 m
OP
6 5
∆Tinlet (K)
7 6
7
Pexit = 0.2 MPa Pexit = 0.3 MPa Pexit = 0.4 MPa Pexit = 1.0 MPa Pexit = 1.5 MPa
8
5 4 3
3 2
2
1
1
0
0 -0.05 -0.04 -0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 0.04
Unstable
4
Stable
-0.04
-0.03
-0.02
-0.01
0.02
0.03
(b)
(a)
kr,out = 0.00 kr,out = 0.50 kr,out = 10.0
5.0 OP
4.0 3.5
∆Tinlet (K)
0.01
ρext
ρext
4.5
0.00
Unstable
3.0 2.5
Stable
2.0 1.5 1.0 0.5 -0.04
-0.03
-0.02
-0.01
0.00
0.01
0.02
0.03
ρext
(c) Figure 8.17: Sensitivity analysis results for different operating parameters. Out-of-phase oscillation SBs are indicated by OP. A change in slope of the SB indicates transition to in-phase. (a) System pressure; (b) Riser length; (c) Pressure loss coefficient at riser exit
177
kc,out = 0.00 kc,out = 1.10 kc,out = 10.0
5.0 4.5 4.0
∆Tinlet (K)
3.5 3.0
Unstable
2.5 2.0
Stable
1.5 1.0 0.5 0.0
-0.04 -0.03 -0.02 -0.01
0.00
0.01
0.02
0.03
ρext
(d)
kc,in = 0.00 kc,in = 1.73 kc,in = 10.0
6.0 5.5 5.0
OP
∆Tinlet (K)
4.5 4.0
Unstable
3.5 3.0 2.5 2.0 1.5
Stable
1.0 0.5 0.0 -0.04 -0.03 -0.02 -0.01 0.00
0.01
0.02
0.03
0.04
ρext
(e)
Figure 8.17 (continue): Sensitivity analysis results for different operating parameters. Out-ofphase oscillation SBs are indicated by OP. A change in slope of the SB indicates transition to inphase. (d) Pressure loss coefficient at channel exit; (e) Pressure loss coefficient at channel inlet
178
circulation system, increasing Lr increases the driving force leading to higher flow rate. Therefore, it has a strong stabilizing effect. However, Figure 8.17(b) also shows that out-of-phase oscillations are possible at low reactivity for long risers. Effect of pressure loss coefficient at riser outlet K r ,out is not as strong as those for Psys
*
and Lr (Figure 8.17(c)), but SBs indicate the possibility of out-of-phase oscillations for low reactivity if K r ,out is high. Figure 8.17(d) shows the effect of an increase in pressure loss at the outlet of the core. Increasing K c ,out generally tends to cause a loss of stability. The last operating parameter studied is the pressure loss coefficient at the core inlet, K c ,in . As shown in Figure 8.17(e), an increase in
K c ,in has a strong stabilizing effect.
179
Chapter 9 Numerical Simulations of Natural-Circulation Model
In this chapter, results of numerical simulations for a number of representative operating points in the parameter space are presented. These results are compared with the results of stability and bifurcation analyses presented in Chapter 8. Results for only the thermal hydraulic model are presented first in Section 9.1, followed by those for the nuclear-coupled thermal hydraulic system in Section 9.2.
9.1. Thermal-Hydraulic Model Simulation (numerical integration) of the complete nonlinear thermal hydraulic system of a natural circulation two-phase flow loop under low pressure is carried out first. The parameters correspond to the operating points P1 through P5 in Figure 8.1 (a). System pressure and heat *
fluxes for the operating points are fixed at Psys = 0.2 (MPa) and q ' = 6.0 (kW/m). A perturbation ( vinlet : vinlet , ss + 0.001) is introduced in the system at t = 0, where vinlet , ss is the steady-state value of the inlet velocity. Results are presented in Figures 9.1 through 9.5. [Realizing that the perturbations introduced are very small, the effect of larger amplitude perturbations are also studied, and those results are presented after the results for small amplitude perturbations.] *
For the point P1, the inlet subcooling is ∆Tinlet = 35.0 (K), which is high enough to prevent boiling in the channel and the riser. Figure 9.1(a) shows the straight line at the riser exit location, indicating that the boiling boundary is not in the heated channel or the riser, and hence that there is no boiling in this case. Figure 9.1(b) shows the time evolution of the inlet velocity. The single
180
phase flow corresponding to this operating point is seen to be quite stable, and the initial *
perturbation decays rapidly. Figure 9.2, which is for inlet subcooling ∆Tinlet = 31.496 (K) case (operating point P2 in Figure 8.1(a)), shows an example of the intermittent wave. The corresponding steady state boiling boundary and inlet velocity are µ ss = 4.367 and vinlet = 2.157. The boiling boundary is very close to the riser outlet (4.38). Figure 9.2(a) showing the evolution of the boiling boundary shows that the system at this operating point is unstable, and the boiling boundary oscillation amplitude increases rapidly in the beginning. At time t = 8.522, it reaches the riser outlet (4.38), and the flow in the channel and riser stays single-phase for a while before boiling starts again, starting at the top of the riser. The boiling boundary then moves downward. This leads to a series of intermittent waves in which boiling starts at the riser exit, followed by a downward movement of the boiling boundary. After reaching a minimum, the boiling boundary again travels upward and reaches the riser exit. For a brief interval, the flow is single phase in both the channel and the riser. The non-dimensional period of the intermittent oscillations is about 2.2, which is close to the time it takes for the fluid to travel from the channel inlet to the riser exit. Figure 9.2(b) shows the corresponding intermittent oscillations in inlet velocity. The inlet velocity and the boiling boundary oscillate in-phase. As the inlet subcooling is further decreased, the intermittent waves turn into sinusoidal *
waves. Figure 9.3 shows growing amplitude sinusoidal oscillations for ∆Tinlet = 15.0 K case (corresponding to point P3 in Figure 8.1(a)). The amplitudes of the boiling boundary and inlet velocity increase, clearly indicating that the operating point is unstable. However, before a limit cycle is reached—if there is one—the numerical algorithm fails to converge at about t = 5.0. These are high frequency oscillations where the period of the oscillations is about 0.5. Clearly, the compressibility effect is playing a role in these oscillations.
181
Boiling Boundary
5.5
5.0
4.5
4.0
3.5
3.0 0
2
4
6
8
10
12
14
16
18
20
Time
(a) 2.160
2.158
vinlet
2.156
2.154
2.152
2.150 0
2
4
6
8
10
12
14
16
18
20
Time
(b) Figure 9.1: Results of numerical simulation for operating parameters corresponding to point P1 in *
*
Figure 8.1(a). For this point, Psys = 0.2 MPa, q ' = 6.0 kW/m, and ∆Tinlet = 35.0 (K). (a) Nondimensional boiling boundary; (b) Non-dimensional inlet velocity.
182
Boiling Boundary
4.38
4.37
4.36
4.35
4.34
4.33
4.32 0
10
20
30
40
50
Time
(a) 2.170
2.165
vinlet
2.160
2.155
2.150
2.145 0
10
20
30
40
50
Time
(b) Figure 9.2: Results of numerical simulation for operating parameters corresponding to point P2 in *
*
Figure 8.1(a). For this point, Psys = 0.2 MPa, q ' = 6.0 kW/m, and ∆Tinlet = 31.496 (K). (a) Nondimensional boiling boundary; (b) Non-dimensional inlet velocity
183
Boiling Boundary
2.485
2.480
2.475
2.470
2.465 0
1
2
3
4
5
Time
(a) 5.52
5.50
vinlet
5.48
5.46
5.44
5.42
5.40 0
1
2
3
4
5
Time
(b) Figure 9.3: Results of numerical simulation for operating parameters corresponding to point P3 in *
*
Figure 8.1(a). For this point, Psys = 0.2 MPa, q ' = 6.0 kW/m, and ∆Tinlet = 15.0 (K). (a) Nondimensional boiling boundary; (b) Non-dimensional inlet velocity. 184
Boiling Boundary
0.80162
0.80160
0.80158
0.80156
0.80154
0.80152
0.80150 0
10
20
30
40
50
40
50
Time
(a) 6.5266 6.5264 6.5262 6.5260
vinlet
6.5258 6.5256 6.5254 6.5252 6.5250 6.5248 6.5246 0
10
20
30
Time
(b) Figure 9.4: Results of numerical simulation for operating parameters corresponding to point P4 in *
*
Figure 8.1(a). For this point, Psys = 0.2 MPa, q ' = 6.0 kW/m, and ∆Tinlet = 8.8 (K). (a) Nondimensional boiling boundary; (b) Non-dimensional inlet velocity.
185
Boiling Boundary
0.38036
0.38034
0.38032
0.38030
0.38028 0
10
20
30
40
50
40
50
Time
(a) 5.2568 5.2566 5.2564
vinlet
5.2562 5.2560 5.2558 5.2556 5.2554 5.2552 0
10
20
30
Time
(b) Figure 9.5: Results of numerical simulation for operating parameters corresponding to point P5 in *
*
Figure 8.1(a). For this point, Psys = 0.2 MPa, q ' = 6.0 kW/m, and ∆Tinlet = 5.0 (K). (a) Nondimensional boiling boundary; (b) Non-dimensional inlet velocity. 186
Results for numerical simulation for operating parameters corresponding to Point P4 (inlet *
subcooling, ∆Tinlet = 8.8(K)) are shown in Figure 9.4. Inlet subcooling is now so small that the boiling boundary at steady state ( µ ss = 0.802) is in the channel for this case. Operating point P4 is clearly unstable, but the rise in amplitude per period is much smaller than that for point P3. Figure 9.5 shows results of simulations for operating parameters corresponding to point P5 (inlet *
subcooling, ∆Tinlet = 5.0 (K)). The steady state boiling boundary is for this case even lower in the channel ( µ ss = 0.380), and the simulations show the operating point to be stable with single and two-phase natural circulation flows. The evolutions of the boiling boundary and inlet velocity oscillations as the inlet subcooling is decreased qualitatively agree with the experimental data obtained by Inada et al. [42]. Results of numerical simulations reported above are obtained following a rather small perturbation in the inlet velocity (about the steady state value). Additional results obtained by numerically integrating the set of governing ODEs for different set of parameter values and using different perturbation amplitudes are reported below. They are also compared with the results of stability analysis. Figures 9.6 through 9.8 show evolution of the boiling boundary and inlet velocity for three *
operating points—P1, P2, and P3—in the ∆Tinlet — q ' parameter space shown in Figure 8.10. The set of operating parameters correspond to those of the Dodewaard reactor, and system pressure is *
Psys = 1.07 (MPa). [Note that neutronics is not included at this stage.] Inlet subcooling and heat *
flux ( ∆Tinlet , q ' ) for points P1, P2 and P3 are given by (4.0, 379.0), (4.0, 379.9), and (8.5, 256.95), respectively. Points P1 and P2 are respectively on the stable and unstable side of the SB in a region where the bifurcation is supercritical. For point P1, a perturbation is introduced at t = 0
187
( vinlet : vinlet , ss → vinlet , ss + 0.01). Figures 9.6(a) and (b) show time evolutions of the boiling boundary and inlet velocity. As expected, the system is stable at the point P1. For point P2, the same initial perturbation is introduced. Figure 9.7 shows that the system evolves to a stable limit cycle with saturated oscillation amplitude. Point P3 is on the stable side of the SB in a region where bifurcation is subcritical. A quite large perturbation ( vinlet : vinlet , ss → vinlet , ss + 0.2, a m ,i :
a m ,i , ss + 0.2, i = 1,…,8) is introduced. Figure 9.8 shows the time evolutions for parameter values corresponding to point P3. Even though point P3 is in the stable region, since the perturbation given is large and the bifurcation at point 3 is subcritical, the perturbations grow, being repelled by the unstable limit cycle around the stable fixed point.
9.2. Nuclear-Coupled Thermal-Hydraulics Model of A Natural Circulation BWR Numerical simulations are carried out to gain further insight into the dynamics of the natural circulation, nuclear-coupled thermal hydraulic system as well as to compare the results obtained using numerical simulations with those obtained earlier—for stability and bifurcation analysis—using BIFDD. Results of numerical simulations are given here for eight points on the stability maps of high and low pressure systems. These points—labeled a through h—are shown in Figures 9.9(a) and (b). Table 2 lists operating parameters as well as characteristics of bifurcation (sub- or supercritical) and the nature of oscillations (in-phase or out-of-phase) predicted by BIFDD for these operating points. Points a and b are close to the out-of-phase segment of the stability boundary (Figure 10a).
188
Boiling Boundary
0.2070 0.2068 0.2066 0.2064 0.2062 0.2060 0.2058 0.2056 0
50
100
150
200
150
200
Time
(a) 1.920
vinlet
1.915
1.910
1.905
1.900 0
50
100
Time
(b) Figure 9.6: Results of numerical simulation for operating parameters corresponding to point P1 in *
*
Figure 8.10. For this point, Psys = 1.07 MPa, q ' = 379.0 kW/m, and ∆Tinlet = 4.0 (K). (a) Nondimensional boiling boundary; (b) Non-dimensional inlet velocity.
189
0.2070
Boiling Boundary
0.2065
0.2060
0.2055
0.2050
0.2045
0.2040 0
20
40
60
80
100
80
100
Time
(a) 1.920
1.915
vinlet
1.910
1.905
1.900
1.895
1.890 0
20
40
60
Time
(b) Figure 9.7: Results of numerical simulation for operating parameters corresponding to point P2 in *
*
Figure 8.10. For this point, Psys = 1.07 MPa, q ' = 379.9 kW/m, and ∆Tinlet = 4.0 (K). (a) Nondimensional boiling boundary; (b) Non-dimensional inlet velocity.
190
Boiling Boundary
0.80
0.78
0.76
0.74
0.72
0.70 0
20
40
60
80
100
120
140
160
180
200
220
140
160
180
200
220
Time
(a) 2.50 2.45 2.40
vinlet
2.35 2.30 2.25 2.20 2.15 2.10 0
20
40
60
80
100
120
Time
(b) Figure 9.8: Results of numerical simulation for operating parameters corresponding to point P3 in *
*
Figure 8.10. For this point, Psys = 1.07 MPa, q ' = 256.95 kW/m, and ∆Tinlet = 8.5 (K). (a) Nondimensional boiling boundary; (b) Non-dimensional inlet velocity. 191
Point a is in the unstable region, while point b is in the stable region. Bifurcation along this segment of the SB is supercritical. For point a, initial perturbation is given by ( n1 : n1, ss → n1, ss + 0.0001). Figure 9.10 shows results of numerical simulations for the expansion parameters of the fundamental and first azimuthal modes n0 (t ) and n1 (t ) . As expected, stable amplitude limit cycle of n1 (t ) and very small amplitude of n 0 (t ) result. For point b, the perturbations of n0 (t ) and n1 (t ) are given by ( n0 : n0, ss → n0, ss + 0.0001, n1 : n1, ss → n1, ss + 0.0001). Figure 9.11 shows decreasing amplitude oscillations for both n 0 (t ) and n1 (t ) . Points c, d and e are close to the in-phase segment of the SB. Bifurcation at point c is subcritical; while those for points d and e are supercritical. Perturbations for the point c is ( n0 : n0, ss → n0, ss + 0.003, n1 : n1, ss → n1, ss + 0.001). Figure 9.12, corresponding to point c, shows increasing amplitude of n 0 (t ) (and initially decaying n1 (t ) ) for large perturbation even though point c is in the stable region, consistent with the subcritical nature of the bifurcation. For point d (in the stable region) and e (in the unstable region), perturbations lead to decreasing amplitude oscillations (point d) and stable limit cycle oscillations (point e) for
n 0 (t ) while initially decreasing amplitude oscillations result for n1 (t ) in both cases (Figures 9.13 and 9.14). For the case of low system pressure, Figures 9.15, 9.16, and 9.17 show that oscillations at points f, g, and h are in-phase as suggested by the stability analysis. Large perturbation at point f, which is in the stable region, causes increasing amplitude oscillation consistent with the subcritical bifurcation there. Perturbations at point g (unstable region), and h (stable region) cause limit cycle and decreasing amplitude oscillations, respectively, for n 0 (t ) . Numerical simulations of systems with high and low system pressures agree with predictions of BIFDD satisfactorily.
192
The numerical simulations of nuclear coupled natural circulation BWR model show characteristic complexities of oscillations similar to the forced circulation BWR model. Depending on operating conditions, they both can generate in-phase or out-of-phase oscillations as well as supercritical or subcritical bifurcations. However, two differences between the forced and natural circulation systems can be pointed out. First; in the forced circulation case, the out-ofphase oscillations were observed (for the range of parameter values studied) only when the inhomogeneity of core loading is considered. If the core loading is homogeneous, the oscillations are in-phase for typical BWR operating parameters. However, for the natural circulation model, even if the core loading is homogeneous, for the set of operating parameters corresponding to the Dodewaard reactor, out-of-phase oscillations can still be generated in certain areas of the operating parameter space. This may suggest higher tendency towards out-of-phase instability for the natural circulation BWRs. Second; system pressure for forced circulation BWR is generally very high. Therefore it shows only trivial effects on the dynamics of the system. However, pressure is found to play a very important role in determining the characteristics of oscillations for natural circulation BWRs. The system pressure impacts decay ratios, amplitudes, frequencies, and phase and bifurcation properties. The differences are of importance in safety considerations for the natural circulation BWR designs.
193
35
Pexit = 7.0 MPa
30
∆Tinlet (K)
25
c
d e
**
*
20
Unstable
15
Stable
10
a b
**
5
OP
0 -0.04
-0.03
-0.02
-0.01
0.00
0.01
0.02
ρext (a)
Pexit = 0.4 MPa
g h
4.0
**
3.5
∆Tinlet (K)
3.0
Unstable
2.5
f
Stable
2.0
*
1.5 1.0 0.5 -0.04
-0.03
-0.02
-0.01
0.00
0.01
0.02
0.03
ρext (b) Figure 9.9: Operating points in the inlet subcooling—reactivity parameter space for numerical *
*
simulations of the nuclear-coupled model. (a) Psys = 7.0 (MPa); (b) Psys = 0.4 (MPa).
194
Table 9.1: Operating points and characteristics of oscillations and PAH-B predicted by BIFDD *
Psys (MPa)
ρ ext
*
∆Tinlet (K)
Characteristics Characteristics of oscillations
of PAH-B
Point a
7.0
-0.03413
6.0
Out-of-phase
Supercritical
Point b
7.0
-0.03400
6.0
Out-of-phase
Supercritical
Point c
7.0
-0.02781
20.0
In-phase
Subcritical
Point d
7.0
-0.00840
20.0
In-phase
Supercritical
Point e
7.0
-0.00823
20.0
In-phase
Supercritical
Point f
0.4
0.01237
2.0
In-phase
Subcritical
Point g
0.4
-0.02570
3.8
In-phase
Supercritical
Point h
0.4
-0.02540
3.8
In-phase
Supercritical
195
n0-n0,ss
3.0x10
-6
2.0x10
-6
1.0x10
-6
0.0 -1.0x10
-6
-2.0x10
-6
-3.0x10
-6
0
20
40
60
80
100
120
140
160
100
120
140
160
Time
n1-n1,ss
(a)
1.0x10
-4
8.0x10
-5
6.0x10
-5
4.0x10
-5
2.0x10
-5
0.0 -2.0x10
-5
0
20
40
60
80
Time
(b)
Figure 9.10: Results of numerical integration for point a in Figure 9.9(a). Perturbation at t = 0 is given by ( n1 : n1, ss → n1, ss + 0.0001). (a) time evolution of n 0 (t ) ; (b) time evolution of n1 (t ) .
196
n0-n0,ss
1.0x10
-4
8.0x10
-5
6.0x10
-5
4.0x10
-5
2.0x10
-5
0.0 0
50
100
150
200
250
300
200
250
300
Time
n1-n1,ss
(a)
1.0x10
-4
8.0x10
-5
6.0x10
-5
4.0x10
-5
2.0x10
-5
0.0 -2.0x10
-5
0
50
100
150
Time
(b)
Figure 9.11: Results of numerical integration for point b in Figure 9.9(a). Perturbations at t = 0 are given by ( n0 : n0, ss → n0, ss + 0.0001, n1 : n1, ss → n1, ss + 0.0001). (a) time evolution of n 0 (t ) ; (b) time evolution of n1 (t ) .
197
n0-n0,ss
4.0x10
-3
3.0x10
-3
2.0x10
-3
1.0x10
-3
0.0 -1.0x10
-3
-2.0x10
-3
-3.0x10
-3
0
100
200
300
400
300
400
Time
n1-n1,ss
(a)
1.0x10
-3
8.0x10
-4
6.0x10
-4
4.0x10
-4
2.0x10
-4
0.0 0
100
200
Time
(b)
Figure 9.12: Results of numerical integration for point c in Figure 9.9(a). Perturbations at t = 0 are given by ( n0 : n0, ss → n0, ss + 0.003, n1 : n1, ss → n1, ss + 0.001). (a) time evolution of n 0 (t ) ; (b) time evolution of n1 (t ) .
198
n0-n0,ss
1.2x10
-3
1.0x10
-3
8.0x10
-4
6.0x10
-4
4.0x10
-4
2.0x10
-4
0.0 -2.0x10
-4
-4.0x10
-4
-6.0x10
-4
0
100
200
300
400
300
400
Time
n1-n1,ss
(a)
1.0x10
-3
8.0x10
-4
6.0x10
-4
4.0x10
-4
2.0x10
-4
0.0 0
100
200
Time
(b)
Figure 9.13: Results of numerical integration for point d in Figure 9.9(a). Perturbations at t = 0 are given by ( n0 : n0, ss → n0, ss + 0.001, n1 : n1, ss → n1, ss + 0.001). (a) time evolution of n 0 (t ) ; (b) time evolution of n1 (t ) .
199
n0-n0,ss
1.2x10
-4
1.0x10
-4
8.0x10
-5
6.0x10
-5
4.0x10
-5
2.0x10
-5
0.0 -2.0x10
-5
-4.0x10
-5
-6.0x10
-5
-8.0x10
-5
0
100
200
300
400
300
400
Time
(a) 1.0x10
-9
-10
8.0x10
-10
6.0x10
-10
4.0x10
n1-n1,ss
-10
2.0x10
0.0 -10
-2.0x10
-10
-4.0x10
-10
-6.0x10
-10
-8.0x10
-1.0x10
-9
0
100
200
Time
(b) Figure 9.14: Results of numerical integration for point e in Figure 9.9(a). Perturbation at t = 0 is given by ( n0 : n0, ss → n0, ss + 0.0001). (a) time evolution of n 0 (t ) ; (b) time evolution of n1 (t ) .
200
n0-n0,ss
3.0x10
-2
2.5x10
-2
2.0x10
-2
1.5x10
-2
1.0x10
-2
5.0x10
-3
0.0 -5.0x10
-3
-1.0x10
-2
0
20
40
60
80
100
60
80
100
Time
n1-n1,ss
(a) 1.0x10
-10
8.0x10
-11
6.0x10
-11
4.0x10
-11
2.0x10
-11
0.0 -2.0x10
-11
-4.0x10
-11
-6.0x10
-11
-8.0x10
-11
-1.0x10
-10
0
20
40
Time
(b) Figure 9.15: Results of numerical integration for point f in Figure 9.9(b). Perturbation at t = 0 is given by ( n0 : n0, ss → n0, ss + 0.002). (a) time evolution of n 0 (t ) ; (b) time evolution of n1 (t ) .
201
n0-n0,ss
1.2x10
-4
1.0x10
-4
8.0x10
-5
6.0x10
-5
4.0x10
-5
2.0x10
-5
0.0 -2.0x10
-5
-4.0x10
-5
-6.0x10
-5
0
50
100
150
200
150
200
Time
n1-n1,ss
(a)
1.0x10
-4
8.0x10
-5
6.0x10
-5
4.0x10
-5
2.0x10
-5
0.0 0
50
100
Time
(b) Figure 9.16: Results of numerical integration for point g in Figure 9.9(b). Perturbations at t = 0 are given by ( n0 : n0, ss → n0, ss + 0.0001, n1 : n1, ss → n1, ss + 0.0001). (a) time evolution of n 0 (t ) ; (b) time evolution of n1 (t ) .
202
n0-n0,ss
1.2x10
-3
1.0x10
-3
8.0x10
-4
6.0x10
-4
4.0x10
-4
2.0x10
-4
0.0 -2.0x10
-4
-4.0x10
-4
0
100
200
300
400
500
400
500
Time
(a)
-3
1.0x10
-4
n1-n1,ss
8.0x10
-4
6.0x10
-4
4.0x10
-4
2.0x10
0.0 0
100
200
300
Time
(b) Figure 9.17: Results of numerical integration for point h in Figure 9.9(b). Perturbations at t = 0 are given by ( n0 : n0, ss → n0, ss + 0.001, n1 : n1, ss → n1, ss + 0.001). (a) time evolution of n 0 (t ) ; (b) time evolution of n1 (t ) .
203
Chapter 10 Summary, Conclusions and Future Work of Part II
This chapter summarizes development of the model for natural circulation BWR and discusses results of stability and bifurcation analyses and numerical simulations. Potential areas for future work are also discussed.
10.1. Summary of Natural Circulation BWR Model A new nuclear-coupled reduced order model is developed to simulate instabilities in natural circulation BWRs under either low or high system pressures. Thermal-hydraulic system of natural circulation BWR is simplified to two, parallel, heated channels with common inlet (lower plenum), two risers above the channels with common outlet (upper plenum), and a downcomer. Fuel temperature dynamics and neutronics of the model do not differ from those of the forcedcirculation model [30]. A key to catch the essence of flashing-induced instabilities when system pressure is low is to appropriately model dependence of water saturation enthalpy and local pressure. The model assumes a linear relation between the two. Homogeneous Equilibrium Model (HEM) is chosen to model two-phase flow dynamics due to its simplicity and computational efficiency. Two different nodalization schemes are applied. In the first, “moving boundaries,” that separate nodes,, move in proportion to the movement of the boiling boundary. This nodalization scheme has the advantage of simplicity for programming purposes, and is shown to be necessary for applications such as stability and bifurcation analyses. However, this nodalization scheme fails to simulate the cases in which the boiling boundary may cross from the channel to the riser, or 204
move out of the riser. In these special cases, the difficulty arises because of node(s) with near zero length. The second nodalization scheme uses “fixed boundaries” for each node and the only moving boundary in this case is the boiling boundary. The only zero length node in this case occurs when the boiling boundary approaches a fixed node boundary. This causes numerical difficulties during simulations, and it is addressed properly in the model development. Weighted residual approach is applied to mass, energy and momentum conservation PDEs for each node to reduce them into a set of ODEs. Time-dependent coefficients of step-wise linear spatial profiles of flow enthalpy, water saturation enthalpy (linearly varying with pressure), steam quality and flow velocity are chosen as phase variables of the set of ODEs. Boiling boundary is determined iteratively using Newton’s method. For the special case when node length is close to zero, the set of ODEs are very stiff. Two approaches are used to avoid numerical difficulty to integrate a set of stiff ODEs. First, a numerical integration code with better numerical stability is used. Second, evaluation of the specific ODEs for the node is stopped if the length is less than a given small number. Phase variables in that tiny node are assumed to be given by the values (flow enthalpy, saturation enthalpy, quality and velocity) at the inlet of the node.
10.2. Summary of Results The thermal-hydraulic model is validated for its convergence and accuracy first. The set of parameters for the validation exercise correspond to those of the SIRIUS facility [44]. As number of nodes increases, steady state velocities and boiling boundaries converge rapidly. Good agreement is found between model results and experimental data for steady state flow rate for large heat flux. The modle over-estimates the flow rate for lower values of heat flux. This is likely due to the fact that model ignores subcooled boiling effect. Convergence of the stability
205
boundaries for the “moving boundary scheme,” however, is found to be much faster than that for the “fixed boundary scheme.” Stability boundaries are found to be consistent with the experiments for lower heat flux values. For larger values of the linear heat flux, the stability boundary for the system pressure Psys = 0.1 MPa under-estimates the stable area. It is presumed that the disagreement is caused by the exchange of two pairs of complex conjugate eigenvalues with the largest real parts. No-boiling boundaries, which separate areas in the operating parameter space where boiling appears in the riser or channel from those where no boiling exists at the steadystate, are studied for different Boussinesq expansion parameters. Good agreement with experimental data is found for adjusted expansion parameters (E = 1.5). Stability and bifurcation for the thermal hydraulic model are analyzed for the set of parameters corresponding to the SIRIUS facility under low pressure conditions (0.1 and 0.2 MPa), and the Dodewaard reactor under higher pressure condition (1.07 MPa). Results of the SIRIUS facility indicate that the stability boundary is separated into two segments; one in which the boiling boundary is in the riser, and second in which the boiling boundary is in the channel. For the segment corresponding to boiling in the riser, similar to the case of forced-circulation BWR model with large in-homogeneity, two conjugate pairs of eigenvalues with the largest and second largest real parts play an important role in defining the stability boundary. Imaginary parts suggest very different frequencies of oscillations for the two pairs. For lower values of the heating power, the pair corresponding to the higher frequency is the one with the largest real part. When the heating power is increased, the second pair, corresponding to the lower frequency, approaches the imaginary axis. As the heating power crosses a threshold value, the second eigenvalue pair crosses the first and becomes the one with the largest real part. It also represents a transition of oscillations from high frequency to low frequency. Similar to the approach followed for studying
206
the forced-circulation BWR model, the stability boundary and the boundary corresponding to the second pair of eigenvalues when plotted together,help explain the slight discrepancy between the analytical stability boundary and the experimental data. For the stability boundary segment corresponding to the “boiling boundary in the channel” case, the pair with the second largest real part is always far to the left of the pair with the largest real part. Results of thermal hydraulics simulations for the Dodewaard reactor (without neutronics) indicate less intense flashing in the riser under higher system pressure conditions. Moreover, stability boundary segment corresponding to the “boiling boundary in the riser” case is much shorter than that corresponding to the “boiling boundary in the channel” case. Bifurcation analyses of the thermal hydraulic model show both supercritical and subcritical bifurcation along different segment of the stability bondary. The natural circulation system generally follows the same trend as the forced-circulation system. The supercritical bifurcation occurs in regions of parameter space with intermediate inlet subcooling values. The subcritical bifurcation, however, occurs in regions with large and small inlet subcooling. Finite amplitude oscillations curves also indicate that much less deviation from the stability boundary is required for the same amplitude of oscillation if the boiling boundary is in the channel. This implies much more intense oscillations are expected to occur as the stability boundary is crossed if the boiling boundary is in the channel. Dynamics of the nuclear coupled natural circulation BWR model are studied for the set of operating parameters corresponding to the Dodewaard reactor. Nominal system pressure Psys = 7.0 MPa and a much lower pressure Psys = 0.4 MPa are chosen to represent typical operating conditions a natural circulation BWR may encounter. Core loading is assumed to be homogeneous. For the high system pressure case, characteristic of oscillation is in-phase along
207
most of the stability boundary. However, in a small region of low reactivity, the azimuthal mode eigenvalue pair crosses the fundamental mode eigenvalue pair, and the oscillations in this region is hence expected to be out-of-phase. For the low system pressure, characteristics of oscillations are found to be always in-phase along the entire stability boundary. Bifurcation results show both supercritical and subcritical PAH-B along the stability boundaries for low as well as high system pressure operations. Sensitivity analysis of various operating parameters is also carried out. It is shown that increasing system pressure may lead to a more stable or a less stable system depending upon the value of the absolute system pressure. Increasing pressure loss coefficient at the outlet of core or riser has a destabilizing effect, while increase of pressure loss coefficient at the core inlet has a stabilizing effect. Numerical simulations of various operating points for thermal hydraulics system and nuclear coupled thermal hydraulic system are presented in Chapter 9. Results of numerical simulations of the thermal hydraulic model (without nuclear coupling) show characteristic similarity with experimental data. As inlet subcooling is decreased, inlet velocity and boiling boundary of the system experience, sequentially, stable single phase flow, intermittent wave, unstable sinuous wave, and stable two-phase flow. Non-dimensional period of the intermittent wave is about 2.2, which is close to the time it takes for the fluid to travel from the channel inlet to the riser exit. Results of numerical simulations of the nuclear coupled system show increasing or saturated amplitudes of oscillations, consistent with the bifurcation properties of the operating points (subcritical or supercritical, respectively). Numerical simulations also show large amplitude fundamental or azimuthal mode oscillations consistent with the predictions of the stability analysis results (in-phase or out-of-phase, respectively). In general, numerical simulation results
208
confirmed the characteristics predicted by the stability and bifurcation analyses. In addition, they also showed mixed oscillation characteristics with larger amplitude, that were beyond the scope of stability and bifurcation analyses.
10.3. Future Work In the thermal hydraulics model of the natural circulation system, the two phase flow is simulated using the Homogeneous Equilibrium Model (HEM). The HEM does not consider different velocities between the steam phase and water phase and assumes thermal equilibrium between the two phases. The assumptions of HEM are not valid for certain applications, such as the low flow rate and void fraction, and when there is a large subcooled boiling region. In future work, the HEM can be replaced by the more accurate drift flux model which takes into account the relative velocity between the two phases and the radially non-uniform distribution of the vapor phase. In addition, a subcooled boiling model would also help in achieving better agreement with the experimental data for lower values of the heating power. Validation of the model is currently limited due to limited access to experimental data. Only two sets of data (SIRIUS and Dodewaard) are used in the validation exercise. If more sets of experimental data and details of the facilities, such as those of PANDA and PUMA, become available in the future, additional validation exercises can be carried out.
209
Appendix A Non-dimensional Variables and Parameters in the Forced Circulation BWR Model
The non-dimensional variables and parameters were defined in the forced circulation BWR model by Karve et al. [30]. The following definitions are reproduced from Appendix A in the reference [30].
A.1. Thermal-Hydraulic Variables and Parameters h = h * ∆h fg v = v * vo
t = t * vo
N sub
*
z = z * L*
L*
P = P * ρ f vo*
*
Nρ = ρg
(
ρf*
*
(h =
ρ = ρ* ρ f *
*
* f , sat
Nr = ρ f
)
− hinlet ∆ρ fg *
∆h fg ρ g *
(
N f ,1 = f 1φ L* 2 Dh
*
)
*
*
*
N pch =
Fr = vo*
*
∆ρ fg
) *
∆ρ fg
*
A* vo ∆h fg , sat ρ f ρ g *
2
r = r * L*
210
2
L* q '
A.2. Fuel Dynamic Variables and Parameters T = T * To
*
(gL ) *
*
*
*
v = v * vo
z = z * L*
*
(
α p ,c = α p ,c * L* vo * *
Bic , g , p =
h∞ rc , g , p k c,g , p
*
)
*
*
c q = c q no L*
*
2
(k
N pch
Λ = Λ* vo *
Σ f ,a = Σ f ,a L*
*
) *
*
R = R * L*
To
∆ρ fg L* q ' = * * * * * A vo ∆h fg , sat ρ f ρ g
A.3. Neutronic Variables and Parameters n = n * no
* p
*
L*
ω ik = ω ik * L* vo *
211
Appendix B Impact of the Error of Intermediate Variable I6(t)
In the reduced order model by Karve et al. [30], the profile of single phase enthalpy and two phase steam quality are represented by quadratic profiles. Weighted residual approach is applied to reduce the conservation PDEs to a set of ODEs of the expansion parameters of the profiles. As the weighted residual procedure is applied to the two phase continuity and energy equations, a number of intermediate variables which are used in the final forms of ODEs are defined. These are included δ 1 (t ) , δ 3 (t ) , δ e (t ) , ( I i (t ) , i = 1, …6), and ( M i (t ) , i = 1, ..3). The original form of I6(i) is given by
δ 13 (t ) 1 + 2 + δ 1 (t ) − δ e (t ) δ 3 (t )[1 − δ e (t )δ 3 (t )] 2( ) δ 1 t [3 − δ 1 (t )δ 3 (t )] log 1 − δ e (t ) − 1 − δ 1 (t )δ 3 (t ) 1 δ 1 (t ) I 6 (t ) = 2 2 2 s1 (t ) − 4s 2 (t ) 1 − δ 1 (t )δ 3 (t ) + δ 1 (t )δ 3 (t ) − δ 13 (t )δ 3 3 (t ) δ 3 2 (t )[1 − δ 1 (t )δ 3 (t )] [1 − 3δ 1 (t )δ 3 (t )]log[1 − δ e (t )δ 3 (t )] δ 3 2 (t )[1 − δ 1 (t )δ 3 (t )]
[
]
+
(B.1)
~ Because steady-state δ 3 = 0 , it will cause both numerators and denominators are zero for terms
on the right-hand-side of (B.1). Taylor’s expansions are used for the certain trouble terms to avoid numerical difficulties. However, it is found the expanded expression given in the reference [30] (equation (E.28) in Appendix E) has a mistake. The wrong form of I6(t) in the reference [30] is given by
212
δ 13 (t ) δ 1 2 (t )[3 − δ 1 (t )δ 3 (t )] δ e (t ) log 1 − + + 1 − δ 1 (t )δ 3 (t ) δ 1 (t ) δ 1 (t ) − δ e (t ) 2 δ e (t ) − δ 2 (t ) + 2δ (t )δ (t ) + 1 1 e 2 2δ e 3 (t ) 1 3 2 2 + δ 1 (t ) − δ e (t )δ 1 (t ) + 2δ e (t )δ 1 (t )δ 3 (t ) + I 6 (t ) = 2 (B.2) s1 (t ) − 4 s 2 (t ) 3 4 2 3δ e (t ) − δ 4 (t ) + 2δ e (t )δ 1 (t ) + δ 2 (t )δ 2 (t ) + 2δ (t )δ 3 (t )δ 2 (t ) + 3 1 e 1 e 1 4 3 O δ 3 3 (t ) The correct form is given by
[
]
δ 13 (t ) δ 1 2 (t )[3 − δ 1 (t )δ 3 (t )] δ e (t ) log 1 − + + 1 − δ 1 (t )δ 3 (t ) δ 1 (t ) δ 1 (t ) − δ e (t ) 2 δ e (t ) − δ 2 (t ) + 2δ (t )δ (t ) + 1 1 e 2 3 1 2δ e (t ) 2 2 + δ e (t )δ 1 (t ) + 2δ e (t )δ 1 (t )δ 3 (t ) + I 6 (t ) = 2 s1 (t ) − 4 s 2 (t ) 3 4 3 3δ e (t ) + 2δ e (t )δ 1 (t ) + δ 2 (t )δ 2 (t ) + 2δ (t )δ 3 (t )δ 2 (t ) + 3 e 1 e 1 4 3 O δ 3 3 (t )
[
(B.3)
]
Figure B.1 compared the expressions (B.1), (B.2) and (B.3) when δ 3 (t ) is changed. Values for (s1, s2, δ1, δe) are fixed at (13.23, 0.0, -0.076, 0.50). The expression (B.3) has much closer agreement with the un-expanded expression (B.1) then the expression (B.2). It is noted that the mistake does not affect stability analysis results in the reference [30]. The reason is Jacobian matrices are always evaluated at steady states for the stability analysis, and they are the same value at the steady-states evaluated with the expression (B.2) or (B.3). Therefore, there is no difference of the eigenvalues of the Jacobian Matrices for the two
213
expressions, which determine linear stabilities of the system. However, higher order derivatives are required for bifurcation analyses and they are different for the two expressions. The mistake thus has a significant impact to characteristics of bifurcation of the system. For example, Figure B.2 shows comparisons of numerical simulations for the operating point 6 on the page 118 of reference [30]. The oscillation calculated using expression (B.2) indicates an increasing amplitude for large perturbation. Therefore, the system has subcritical PAH bifurcation close to this operating point. The one calculated using expression (B.3), however, indicates a decreasing amplitude for the same perturbation. The system should have supercritical PAH bifurcation. It is crucial to correct this mistake for appropriate results of bifurcation analyses.
214
0.0008
(B.1) (B.2) (B.3)
I6(t)
0.0007
0.0006
0.0005
0.0004 0.0
0.1
0.2
0.3
0.4
δ3(t)
Figure B.1: Comparisons of I6(t) evaluated with expressions (B.1), (B.2), and (B.3)
215
0.70
0.65
n0
0.60
0.55
0.50
0.45 0
20
40
60
80
100
120
140
100
120
140
time (s)
(a) 0.70
0.65
n0
0.60
0.55
0.50
0.45 0
20
40
60
80
time (s)
(b) Figure B.2: Comparisons of numerical simulations for operating point 6 on page 118 of reference [30]. Operating parameters ( N sub , ρ ext , ∆Pext ) are (1.123, 0.0, 8.2). Phase variables n0, a1,0, s1,0 and vinlet,0 are perturbed as follows—n0: 0.5800 → 0.5916; a1,0: 0.26608 → 0.25610; s1,0: 4.9306 → 4.7367; vinlet,0: 0.39400 → 0.39383). Oscillations are evaluated with (a) expression (B.2); (b) expression (B.3) 216
Appendix C Typical Dimensional Parameters of Forced-Circulation BWR
Typical dimensional parameters of forced-circulation BWR are listed in this appendix. They are reproduced from the Appendix H of reference [30]. A* = 1.442 × 10 −4 m 2
c p = 325.0 J kg −1 K −1
D * = 1.424 cm
c q = 4.6901 × 10 −6 W cm −3
L* = 3.81 m
f 1φ = 0.01467
R * = 2.32 m
f 2φ = 0.016357
*
Tinlet ,o = 551 K *
To = 561 K *
*
*
g = 9.81 m s −2 *
hg = 5678.2 W m −2 K −1 *
Tsat = 561 K
k c = 17.0 W m −1 K −1
c1,0 = −0.101
k exit = 5.74 × 10 −1 W m −1 K −1
c1,1 = −0.100
k inlet = 19
*
k p = 2.7 W m −1 K −1
*
n0 = 3.6245 × 10 7 cm −3
c 2, 0 = −4.3 × 10 −5 K −1 c 2,1 = −4.3 × 10 −5 K −1 *
c c = 330.0 J kg −1 K −1 *
c f = 5.307 × 10 3 J kg −1 K −1
*
*
P * = 7.2 × 10 6 N m −2 *
pc = 16.2 × 10 −3 m
217
*
ρ c * = 6.5 × 10 3 kg m −3
*
ρ f * = 736.49 kg m −3
*
ρ g * = 37.71 kg m −3
rc = 6.135 × 10 −3 m
rg = 5.322 × 10 −3 m rp = 5.2 × 10 −3 m
(
)
v * = 1 / νΣ f Λ* = 1.48987 × 10 4 m s −1 *
*
v o = 2.67 m s −1
λ* = 0.08 s −1 µ f * = 9.693 × 10 −5 N m −2 s
∆h fg = 1494.2 × 10 3 J kg −1
νΣ f * = 0.01678 cm −1
Λ* = 4.0 × 10 −5 s
ω 00 * = 0 s −1
*
Σ a = 0.016527585 cm −1
ω10 * = −140.08 s −1
α c * = 7.925 × 10 −6 m 2 s −1
ω 01* = −0.0571756 s −1
α p * = 0.797 × 10 −6 m 2 s −1
ω11* = −490.78 s −1
β = 0.0056
ξ h * = 3.855 × 10 −2 m
*
218
Appendix D Different Definitions of Subcooling and Phase Change Numbers for Forced and Natural Circulation Models
Multiple definitions of subcooling and phase change numbers ( N sub and N pch ) are present in this dissertation. In the forced-circulation BWR model, pressure along the flow channel is assumed constant. Definitions of the two numbers are given by
(h
N sub =
* f , sat
*
∆h fg ρ g *
(D.1)
*
∆ρ fg
L* q '
N pch =
)
− hinlet ∆ρ
A vo ∆h fg , sat ρ f ρ g *
*
*
(D.2)
where, thermo properties ρ f , ρ g , h f ,sat , hg ,sat and hinlet are evaluated at the constant system *
*
*
pressure. In the natural circulation model, however, pressure is varying from the channel inlet to the riser outlet. It is necessary to specify what pressure the thermo properties in the expressions of the dimensionless numbers are evaluated at. The definitions of subcooling and phase change numbers are given by h f , sat ,high − hinlet ∆ρ fg *
N sub =
N pch =
∆h fg , sat ,high L* q '
*
*
ρg ∆ρ fg
* * A* v o ∆h fg , sat ,high ρ f ρ g
where the subscript “high” and “low” specify the high and low reference pressures. 219
(D.3)
(D.4)
In the section 6.1 of chapter 6, results of natural circulation experiment performed by Inada et al. [42] are discussed. Another definition of the subcooling number is used in the reference [42], which is given by h f , sat ,low − hinlet ∆ρ fg *
N sub =
∆h fg , sat ,low
*
*
(D.5)
ρg
Because the water saturation enthalpy in the nominator is evaluated at the low reference pressure *
(system pressure), it can be less than the inlet flow enthalpy hinlet . The subcooling number can be negative for the cases of low inlet subcooling.
220
Appendix E Non-dimensional Variables and Parameters in the Natural Circulation BWR Model
The non-dimensional thermal hydraulic variables and parameters defined in the natural circulation BWR model are listed below. The asterisks indicate the original dimensional quantities. Non-dimensional fuel dynamic and neutronic variables and parameters do not differ from those defined for the forced-circulation BWR model. They can be found in Appendix A.
*
t = t * vo / Lc v = v * / vo
*
z = z * / Lc
*2
*
P = P * /(ρ f vo )
*
*
ρm = ρm / ρ f
Nρ =
*
*
h = (h * − h f , sat ,high ) /( h f , sat ,low − h f , sat ,high )
ρg*
Nr =
ρf*
ρf* ∆ρ fg
*
N f ,1 = f 1φ L* /(2 Dh )
*2
Fr = vo /( gL* ) N f , 2 = f 2φ L* /(2 Dh )
N pch =
h f , sat ,high − h f ,sat ,low ∆ρ fg *
N flash =
*
∆h fg , sat ,high
*
*
ρg*
L* q '
∆ρ fg
* * A* v o ∆h fg , sat ,high ρ f ρ g
*
(v )
* 2
N exp =
221
(h
o
* f , sat , high
− h f , sat ,low
*
)
h f , sat ,high − hinlet ∆ρ fg *
N sub =
N exp,h
*
∆h fg , sat ,high
(h =
*
f , sat ,low
*
*
*
N a = Ainlet / Alocal
ρg*
)
− h * f , sat ,high ∂ρ l
ρf*
*
∂hl
*
N dev ,h = −
* hinlet *
222
N exp,h N sub N flash
Appendix F ODEs of Thermal Hydraulics of Natural Circulation Loop In chapter 7, profiles of water saturation enthalpy, flow enthalpy, steam quality and flow velocity are assumed linear equations in a node. Their forms are given by h f ( z , t ) = h f ,i −1 (t ) + a f ,i (t )( z − z i −1 (t ))
(F.1)
hm ( z , t ) = hm ,i −1 (t ) + a m ,i (t )( z − z i −1 (t ))
(F.2)
x( z , t ) = xi −1 (t ) + N ρ N r si (t )(z − z i −1 (t ) )
(F.3)
v m ( z , t ) = v m , i −1 (t ) + vi (t )( z − z i −1 (t ))
(F.4)
ODEs of the expansion parameters can be derived through weighted residual approach.
Details of arriving at ODEs for a m,i (t )
ODE of flow enthalpy expansion parameter a m,i (t ) can be derived from the single phase energy conservation equation (7.14). Plugging the equation (F.2) into the (7.14) and integrating the governing PDE along the node, we will arrive at da m ,i (t ) dt
=
Am ,1 (a m , a f , s, v, ∆z (t )
)
+ Am , 2 (a m , a f , s, v,
)
(F.5)
where
Am ,1
N pch (t ) + N a ,local vinlet (t )ai (t ) + N = −2 flash dhi −1 ( z i −1 (t ), t ) − a (t ) dz i −1 (t ) i dt dt
223
(F.6)
Am , 2 = 0
(F.7)
Details of arriving at ODEs for a f ,i (t ) Two phase energy conservation PDE (7.17) can be used to derive ODE of water saturation enthalpy expansion parameter a f ,i (t ) . Form of the ODE is given by da f (t ) dt
=
A f ,1 (a m , a f , v, s,...) ∆z
+ A f , 2 (a m , a f , v, s,...)
(F.8)
where A f ,1 and A f , 2 are nonlinear functions defined by the weighted residual procedure. They are given by
A f ,1
v m ,i −1 (t )a f ,i (t ) + x t ( ) 1 − i vi (t ) N ρ N r (hg − h f ,i −1 (t ) )1 + N N (1 + xi −1 (t )) + r ρ = −2 N pch (t ) ( ) x t i 1 − (1 + xi −1 (t )) + 1 + N ρ N r N flash dh (t ) f ,i −1 − a (t ) dz i −1 (t ) f ,i dt dt
A f ,2
vi (t )a f ,i (t ) + x ( t ) i −1 vi (t ) N ρ N r (hg − h f ,i −1 (t ) )1 + N N I 1 (t ) + r ρ (hg − h f ,i −1 (t ) )si (t ) − vi (t ) N ρ N r I (t ) − 1 + xi −1 (t ) a f ,i (t ) 2 = − N N r ρ vi (t ) N ρ N r a f ,i (t ) si (t ) I 3 (t ) + N pch (t ) 1 + xi −1 (t ) I (t ) + 1 N flash N N ρ r N pch (t ) si (t ) I 2 (t ) N flash 224
(F.9)
(F.10)
where intermediate variables I 1 , I 2 and I 3 are defined by integration I 1 (t ) =
z 2 1 i 1 1 − dz (F.11) ∆z (t ) ∆z (t ) z∫i −1 1 − xi −1 (t ) − N ρ N r s i (t )( z − z i −1 (t )) 1 − xi −1 (t )
I 2 (t ) =
I 3 (t ) =
2 ∆z (t ) 2 2 ∆z (t ) 2
zi
( z − z i −1 (t )) dz i −1 (t ) − N ρ N r s i (t )( z − z i −1 (t ))
(F.12)
( z − z i −1 (t )) 2 ∫ 1 − xi −1 (t ) − N ρ N r si (t )( z − z i−1 (t )) dz zi −1
(F.13)
∫ 1− x
zi −1 zi
They can be expanded by Taylor’s expansion formula ∞
I 1 (t ) = ∑ TI 1,n (t )
(F.14)
n =1 ∞
I 2 (t ) = ∑ TI 2,n (t )
(F.15)
n =1 ∞
I 3 (t ) = ∑ TI 3, n (t )
(F.16)
n =1
where, 2(N ρ N r si (t ) ) ∆z (t ) n −1 n
TI 1, n (t ) =
(n + 1) ⋅ (1 − xi −1 (t ) )n +1
2(N ρ N r si (t ) ) ∆z (t ) n −1
(F.17)
n −1
TI 2,n (t ) =
(n + 1) ⋅ (1 − xi −1 (t ) )n
2(N ρ N r si (t ) ) ∆z (t ) n
(F.18)
n −1
TI 3,n (t ) =
(n + 2) ⋅ (1 − xi −1 (t ) )n
Details of arriving at ODEs for si (t ) 225
(F.19)
Mass conservation equations (7.16) is integrated along the node to obtain ODE of the quality expansion parameter si (t ) . It is given by ds (t ) S1 (a m, a f , v, s,...) = + S 2 (a m, a f , v, s,...) dt ∆z
(F.20)
where nonlinear functions x (z (t ), t ) 1 + i −1 i −1 vi (t ) − Nρ Nr S1 = 2 v m ,i −1 ( z i −1 (t ), t )si (t ) − 1 dxi −1 ( z i −1 (t ), t ) − s (t ) dz i −1 (t ) i N ρ N r dt dt
(F.21)
S2 = 0
(F.22)
Details of arriving at ODEs for vi (t ) ODE of vi (t ) comes from governing equation momentum conservation (7.18). Similarly, it has expression dv(t ) V1 (a m , a f , v, s,...) = + V2 (a m , a f , v, s,...) dt ∆z
(F.23)
where
x (t ) 1 2 − ∆Pdrv 1 + i −1 a f ,i (t ) − vi (t )v m ,i −1 (t ) − N f , 2 v m ,i −1 (t ) − Nρ Nr Fr V1 = 2 dz i −1 (t ) dv m ,i −1 ( z i −1 (t ), t ) − vi (t ) dt dt
(F.24)
2 V2 = ∆Pdrv si (t )a f ,i (t ) − vi (t ) 2 − N f , 2 2v m ,i −1 (t )vi (t ) + vi (t ) 2 ∆z (t ) 3
(F.25)
226
Appendix G Expansion Parameters When Node Length Is Close to Zero
Numerical difficulties will arise, if ODEs of the expansion parameters are evaluated for the node with very small length. In this case, the ODE system will be too stiff to integrate because of the first terms on the right hand sides of equations (7.30), (7.32), (7.33) and (7.34) (the terms with ∆z (t ) as denominators). To avoid this difficulty, a smallest node length is set up in the code. If the node length is smaller than this value, ODEs are not evaluated in the specific node. Instead, explicit expressions for the expansion parameters are used assuming the nominators of the first terms equal to zero, i.e. A f ,1 (a m , a f , v, s,...) = 0
(G.1)
Am ,1 (a m , a f , v, s,...) = 0
(G.2)
S1 (a m , a f , v, s,...) = 0
(G.3)
V1 (a m , a f , v, s,...) = 0
(G.4)
Replace the left hand side of (G.2) with the the equation (F.6). Explicit form of the expansion parameter ai (t ) is given by N pch dhi −1 (t ) N a ,local vinlet (t ) − dz i −1 (t ) ai (t ) = − + N dt dt flash The expansion parameters a f ,i (t ) , si (t ) and vi (t ) can be obtained by solving the equations (G.1), (G.3) and (G.4) From equation (G.1) and (F.9), we have 227
(G.5)
m1 (t )a f ,i (t ) + n1 (t )vi (t ) = e1 (t )
(G.6)
where, m1 (t ) = v m ,i −1 (t ) − n1 (t ) = N ρ N r
(h
dz i −1 (t ) dt
− h f ,i −1 (t ) ) 1 + xi −1 (t ) N ρ N r 1 − xi −1 (t )
g
N pch (t ) 1 1 + xi −1 (t ) + N N ρ N r 1 − xi −1 (t ) e1 (t ) = − flash dh f ,i −1 (t ) dt
(G.7)
(G.8)
(G.9)
From the equations (G.4) and (F.24), we have m2 (t )a f ,i (t ) + n2 (t )vi (t ) = e2 (t )
(G.10)
where, x (t ) m2 (t ) = ∆Pdrv 1 + i −1 N ρ N r
(G.11)
dz i −1 (t ) dt
(G.12)
n2 (t ) = −v m ,i −1 (t ) +
2 + N v t ( ) f , 2 m ,i −1 1 e2 (t ) = − (1 + N exp,h h f ,i −1 (t ) + N dev ,h ) + Fr dv (t ) m ,i −1 dt
(G.13)
Therefore, a f ,i (t ) and vi (t ) are given by a f ,i (t ) =
e1 (t )n2 (t ) − e2 (t )n1 (t ) m1 (t )n2 (t ) − m2 (t )n1 (t )
228
(G.14)
e2 (t )m1 (t ) − e1 (t )m2 (t ) m1 (t )n2 (t ) − m2 (t )n1 (t )
(G.15)
m3 (t )vi (t ) + n3 (t )si (t ) = e3 (t )
(G.16)
vi (t ) = Similarly, from (G.3) and (F.21), we have
where, m3 (t ) = 1 +
xi −1 (t ) Nρ Nr
n3 (t ) = −v m ,i −1 (t ) +
dz i −1 (t ) dt
(G.17)
(G.18)
e3 (t ) =
1 dxi −1 (t ) N ρ N r dt
(G.19)
si (t ) =
e3 (t ) − m3 (t )vi (t ) n3 (t )
(G.20)
si (t ) is given by
229
Appendix H Typical Dimensional Parameters of SIRIUS Facility The following dimensional parameters of SIRIUS facility [44] [61] are used in Chapter 8 and 9 in this dissertation. Note: water and steam densities ρ f and ρ g are evaluated at the *
*
(
given pressure at the riser outlet Plow = Psys . The pressure Phigh = ρ f g Lc + Lr *
*
*
*
*
*
) is the static
pressure at the inlet of channel. Thermo properties of water and steam related to the pressures *
*
Plow and Phigh are evaluated using ASME steam table routines.
*
*
Lr = 5.746 m
*
Ar = 5.925 × 10 −4 m 2
*
Dr = 2.747 × 10 −2 m
Lc = 1.7 m Ac = 4.798 × 10 −4 m 2
Dc = 9.2 × 10 −3 m *
*
*
*
Ad ,i = 2.663 × 10 −4 m 2
Ad ,o = 2.663 × 10 −4 m 2
K c ,in = 15.5
K c ,out = 0.0
K d ,in = 0.0
K r ,out = 10.0
f 1,c = 0.0309
f 1,r = 0.0309
g * = 9.81 m / s 2
Pr ,out = 0.2 MPa
q ' = 5.294 × 10 −3 MW / m
v o = 1.126 × 10 −1 m / s
*
*
*
Tsub = 18.739 K 230
Appendix I Typical Dimensional Parameters of Dodewaard Reactor I.1. Thermal-Hydraulic Parameters The following dimensional parameters of Dodewaard reactor [62] [63] are used in Chapter 8 and 9 in this dissertation. Note: water and steam densities ρ f and ρ g are evaluated *
*
(
at the given pressure at the riser outlet Plow = Psys . The pressure Phigh = ρ f g Lc + Lr *
*
*
*
*
*
) is the
static pressure at the inlet of channel. Thermo properties of water and steam related to the *
*
pressures Plow and Phigh are evaluated using ASME steam table routines.
*
*
Lr = 3.06 m
*
Ar = 1.573 × 10 −2 m 2
*
Dr = 1.415 × 10 −1 m
Lc = 1.794 m
Ac = 6.981 × 10 −3 m 2 Dc = 1.419 × 10 −2 m *
*
*
*
Ad ,i = 2.146 × 10 −2 m 2
Ad ,o = 1.598 × 10 −2 m 2
K c ,in = 1.73
K c ,out = 1.095
K d ,in = 0.5
K r ,out = 0.5
f1,c = 0.0164
f1,r = 0.0164
g * = 9.81 m / s 2
Pr ,out = 1.07 MPa
q ' = 8.497 × 10 −3 MW / m
vo = 7.233 × 10 −1 m / s
*
*
*
Tsub = 1.8 K 231
I.2. Fuel Dynamics and Neutronics Parameters The following fuel dynamics and neutronics parameters of Dodewaard reactorare used in Chapter 8 and 9 in this dissertation *
α o = 0.3849
*
Tavg ,o = 832.5 K
*
β = 0.0061
*
λ* = 0.084 s −1
*
Λ* = 5.0 × 10 −5 s
hg = 5678.2 W m −2 K −1
*
k ∞ = 1.01
α p * = 0.797 × 10 −6 m 2 s −1
L2 = 8.43 × 10 −4 m 2
α c * = 7.925 × 10 −6 m 2 s −1
νΣ f * = 0.01678 cm −1
rc = 6.135 × 10 −3 m
rg = 5.322 × 10 −3 m rp = 5.2 × 10 −3 m k c = 17.0 W m −1 K −1 k p = 2.7 W m −1 K −1
*
cc = 330.0 J kg −1 K −1
cα = −0.078
R * = 0.796 m
c D = −2.0 × 10 −5 K −1
*
*
To = 564.5 K
*
g = 9.81 m s −2
*
c q = 2.034 × 10 9 W cm −3
232
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