Optimization of a stepped converging-diverging nozzle
Optimization of a stepped converging-diverging nozzle Tate Fanning and Matthew Searle April 12, 2016
Abstract We develop an analytical model for the head loss in a stepped converging-diverging (CD) nozzle through which liquid water flows. Under these conditions, the liquid water is modeled as steady and incompressible. The CD nozzle is composed of a series of finite steps. The head loss including major and minor losses is obtained for a variable number of steps as a function of the area of each step and the length of these stepped segments. The optimization problem considered here is to obtain the set of cross-sectional areas which minimize head loss. For the diffuser, we constrain each cross-sectional area in a segment to be greater than or equal to the area of the segment immediately upstream. For the nozzle, we constrain each cross-sectional area in a segment to be less than or equal to the area of the segment immediately upstream. We consider a constraint where we specify the velocity at a specific segments of the nozzle and of the diffuser supposing that measurements or process requirements mandate this constraint. Under these constraints, we obtain the optimal head loss and corresponding design variables for cases including or excluding friction and for cases with and without the velocity constraint. We find that minimizing minor losses dominates the selection of cross-sectional areas. The optimal solution adapts to the addition of major losses by increasing the areas of each segment slightly. We find that the concavity of the optimal cross-sectional area profile depends on the length of the diffuser with concave-up profiles encountered for short lengths and profiles which transition from concave-up to concave-down for longer lengths.
1
Introduction
Nozzles are widely used to accelerate a fluid for many applications. Nozzles typically increase fluid velocity while reducing the pressure and temperature of the fluid [1]; however, many designs of nozzles exist to perform different functions. Simple turbojet turboprop engines often use a fixed geometry convergent nozzle, while turbofan engines usually employ a more complicated co-annular nozzle [4]. One specific nozzle configuration, relevant to turbomachinery, is the convergent-divergent (CD) nozzle. The CD nozzle consists of a converging section followed by a diverging section as shown in Fig. 1.
Figure 1: Illustration of typical convergent-divergent nozzle. These nozzles can have a fixed or variable geometry. Afterburning turbojets and turbofans employ a variable- geometry CD nozzle, while rocket engines typically use a fixed-geometry CD nozzle. Exhaust gases 1
leave the combustion chamber and converge at the throat of the CD nozzle (see Fig. 1). The throat size is critical to the overall design and changes for various applications. Throat size is chosen to choke the flow to define the mass flow rate through the system [2]. For applications where supersonic speeds at CD nozzle output are desirable, the throat is sized to constrain the flow to be sonic, with a Mach number of 1. The flow is subsonic in the converging nozzle, reaches unity at the throat, and expands isentropically in the diverging nozzle, causing the fluid to accelerate to supersonic speeds. The analysis performed by Kanpur on simple nozzles and diffusers [3] produces a governing equation that explains this unintuitive phenomenon. dV dA =− [1 − M a2 ] (1) A V It is clear from this relationship that for subsonic flow (M a < 1), velocity and area changes are inversely related. However, for supersonic (M a > 1) flows, area and velocity changes are directly related. The throat provides a region by which a subsonic nozzle and supersonic diffuser are joined. Through this combination, the desired flow acceleration achieved. For this work, we consider a simplified CD nozzle to be used as a feed to an impulse turbine as seen in Fig. 2. To improve turbine efficiency, we seek to optimize a CD nozzle geometry that minimizes losses in the nozzle. Nozzles
Fluid Jets
Figure 2: Illustration of impulse turbine. The considered nozzles are CD nozzles. Adapted from [5].
2 2.1
Analysis and Methods Model
To design the CD nozzle, we first assume the nozzle is composed of multiple steps as shown in Figure 4. In this way we can perform a series of control volume analyses on each step using the control volume shown in Fig. 3. To make an analytical model possible, we consider water as the working fluid. Therefore, we assume the flow is also incompressible and steady. We assume the pipe roughness is uniform. The pipe cross section is assumed to be circular.
Figure 3: Control volume for a single step. Many steps were analyzed and evaluated in series to approximate a CD nozzle.
2
V1
Ai , Li , i
Lno
Ld
Figure 4: The stepped constriction is constructed using a stepped nozzle and a stepped diffuser. V1 is the inlet velocity. Lno and Ld are the lengths of the nozzle and diffuser, respectively. Each segment of the constriction has a cross-sectional area, length, and roughness indicated by Ai , Li , and , respectively. The objective is to minimize the diffuser head loss due to major and minor losses. The head loss for a single sudden expansion is derived in [5]. This derivation requires that a uniform velocity profile be present at the inlet and outlet of the step. For each step, the minor loss coefficient, KL,m , is KL,m
2 A1 = 1− A2
(2)
where A1 is the area prior to the step and A2 is the area following the step. The minor head loss associated with this step is V2 hL,m = KL,m 1 (3) 2g We can write this headloss for an arbitrary location in a diffuser with many steps. 2 V 2 A1 hL,m,j = KL,m,j 1 2g Aj
(4)
A1 and V1 are respectively the first cross-sectional area and velocity in the diffuser. Aj is the area prior to the jth step and Aj+1 is the area following. Substituting for the loss coefficient, 2 2 2 Aj V1 A1 hL,m,j = 1 − (5) Aj+1 2g Aj and summing over the n − 1 steps the total minor head loss may be obtained, hL,m =
n−1 X j=1
Aj 1− Aj+1
2
V12 2g
A1 Aj
2 (6)
p The major loss may also be determined. Assuming circular pipes, the diameter is Di = 2 Ai /π where i is the index used to traverse each segment of the stepped diffuser to sum major loss, i = 1...n. From continuity, Vi = V1 A1 /Ai . A vector of lengths, L, and a vector of roughnesses, , are initialized corresponding to each cross-sectional area. The friction factors may be calculated using the Haaland equation [5]. ( " #)−2 1.11 6.9 i Di + (7) fi = −1.8log 3.7 ReD The Reynolds number for each section is defined as Rei = Vi Di /ν, where ν is the kinematic viscosity. The major head loss for each section is li Vi2 hL,M,i = fi (8) Di 2g 3
The total major head loss may be obtained by summing hL,M,i from i = 1...n. hL,M =
n X i=1
fi
li Vi2 Di 2g
(9)
The objective function is the total head loss which is the sum of the major and minor head loss hL = hL,m + hL,M
(10)
2 2 X r (V1 ∗ A1 )2 X Aj 1 li π A21 ∗ V12 hL = ∗ 1− ∗ + fi ∗ ∗ ∗ 2g Aj+1 Aj 2 Ai A2i ∗ 2g
(11)
and is found to be
for the diverging section of the CD nozzle. Through a similar derivation, the total head loss in the converging section can be found, and is here defined as 2 r X (Vj ∗ A1 )2 Aj li π A21 ∗ V12 hL = ∗ 1− + (fi ∗ ∗ ∗ 2g Aj−1 2 Ai A2i ∗ 2g
(12)
for the converging section of the CD nozzle. We scale the objective for both the nozzle and diffuser to be O(1). The design variables are the cross-sectional areas, Aj j = 1...n. The lengths and surface roughnesses are fixed. We apply several types of constraints to the problem. The first is that the cross-sectional area downstream of each step is greater than or equal to the cross-sectional area upstream, Aj+1 ≥ Aj The second type of constraint is an equality constraint which requires the velocity at a certain point to be equal to some fraction of the inlet velocity. This constrains the throat area such that a set velocity is obtained at the throat, therefore constraining mass flow through the nozzle; Vloc = f (V1 ).
2.2
Optimization algorithm
To find an optimal nozzle geometry to minimize head loss, the MATLAB function fmincon was used. This function performs a gradient-based optimization using the interior-point algorithm. Prior to selecting this algorithm, we considered the CD nozzle geometry using gradient-based and gradient-free approaches. The gradient-free approach, fminsearch, implements the Nelder-Mead simplex. It produced unsatisfactory results as convergence was slow and the solution lacked the accuracy of the gradient-based solution. We also considered the active-set algorithm for use with fmincon. It completed the optimization in a comparable time and number of function calls. However, it achieved a higher first order optimality condition and constraint violation. We selected the interior-point algorithm. We used built-in finite-differencing to provide the gradients for the optimizer.
3 3.1
Results and Discussion Optimized Areas, Fixed Segment Lengths
We present optimized profiles for the stepped nozzle and stepped diffuser. Parameters for these solutions are given in Table 1. Figure 5 contains two panels. In each panel, there is a plot of cross-sectional area as a function of axial length. Panel (a) is the area length relationship for the diffuser and panel (b) is the area length relationship for the nozzle. Both panels plot the results obtained under four combinations of two settings: no friction with no velocity constraint, no friction with a velocity constraint, friction with no velocity constraint, and friction with a velocity constraint. The cross-sectional area decreases with increasing length for the nozzle and increases with increasing length for the diffuser. The rate of change in area for the converging nozzle decreases with increasing length while the rate of change of area for the diffuser increases. The velocity constraint requires that the velocity at the middle segment be 2.188 times the velocity at the inlet of the 4
Table 1: Parameters utilized for stepped diffuser and stepped nozzle optimization. Parameter L N Lseg A1 An V1 Tolerance ν
Diffuser 5, 10, 15 m 10, 20, 30 2m 1 m2 2.5 m2 1 m/s 1 × 10−6 1.12 × 10− 6 m2 /s 0.0045 mm
Nozzle same same same 2.5 m2 1 m2 0.4 m/s same same same
nozzle and 0.875 times the velocity at the inlet of the diffuser. The effect of this constraint is to reduce the cross-sectional area and match the outlet area of the of the nozzle and the inlet area of the diffuser. Consequently, the head loss increases significantly. We consider the diffuser. For the no friction case, the increase is 22%. For the friction case, the increase is 20%. For the diffuser under the velocity constraint, the velocity at the middle segment is higher than the velocity at that segment in the optimal case. To achieve this constraint, the optimizer increases the cross-sectional areas at each step by a smaller rate with length than in the optimal solution and then increases the cross-sectional areas at a greater rate with length following the constraint location. The cross-sectional area increases slightly when the objective function includes major losses. For incompressible flows, Vi Ai must be constant to satisfy continuity. Since hL,M,i ∝ Vi2 , the cross-sectional areas increase to reduce the major head loss (Figure 5). However, since this head loss is less dominant than the minor head loss, the change in cross-sectional areas is small. The influence of major losses may be considered by examining Table 2. We consider the diffuser. Without the velocity constraint, the head loss increases 126% with the addition of major losses. With the velocity constraint, the head loss increases by 155% with the addition of major losses. This larger increase in the case where velocity constrained is reasonable because the diffuser cross-sectional area is substantially reduced. This increasing the losses due to friction for a given flow rate. 2.5
2.5
Cross−sectional Area (m2)
no friction, no velocity constraint no friction, velocity constraint friction, no velocity constraint friction, velocity constraint
2
Cross−sectional Area (m )
no friction, no velocity constraint no friction, velocity constraint friction, no velocity constraint friction, velocity constraint
2
1.5
1
0
0.5
1
1.5
2 2.5 3 Axial Length (m)
3.5
4
4.5
2
1.5
1
5
(a) Converging nozzle
0
0.5
1
1.5
2 2.5 3 Axial Length (m)
3.5
4
4.5
5
(b) Diffuser
Figure 5: Four cases are plotted for the converging nozzle and diffuser. We explore four scenarios achievable with or without the velocity constraint and with or without friction. The cross-sectional area is plotted as a function of axial length.
5
Table 2: The head loss for the diffuser and nozzle are given for the cases considered in Figure 5. Friction No No Yes Yes
Velocity constraint No Yes No Yes
Nozzle head loss (m) 0.006 0.0073 0.0085 0.0102
Diffuser head loss (m) 0.00097 0.0012 0.0022 0.0028
If we fix the length of segments and vary the number of segments so that the total diffuser length changes, we notice that the concavity of the profile changes as the diffuser length increases (Fig. 6). Figure 6 consists of two panels, (a) and (b). Panel (a) shows the optimal nozzle profiles for nozzles. Similarly, panel (b) shows the optimal diffuser profiles for diffusers. For shorter diffusers, the losses are minimized by a concave-up profile. However, as the diffuser length decreases the losses are minimized by a profile which transitions between concave-up and concave-down. However, the profiles for the nozzles remain concave-up and the change in area with respect to position decreases as the length of the nozzle increases. To accompany Fig. 6, we give the head loss for the nozzle and for the diffuser as a function of length in Table 3. We note the surprising result that the head loss decreases with increasing length and attribute this to the fact that the reduction in the minor loss resulting from decreasing the size of each step is greater than the increase in the total major loss from increasing length.
2.5
2.5
Cross−sectional Area (m2)
2
Cross−sectional Area (m )
L=5m L = 10 m L = 15 m
2
1.5
2
1.5
L=5m L = 10 m L = 15 m 1
0
5
10
1
15
Axial Length (m)
0
5
10
15
Axial Length (m)
(a) Converging nozzle
(b) Diffuser
Figure 6: The optimal profiles expressed as cross-sectional area as a function of axial length are plotted for three nozzle lengths panel (a) and for three diffuser lengths panel (b). The length of all steps are the same requiring that the number of steps increase as the diffuser length increases.
Table 3: The head loss is given for nozzles and diffusers of different lengths. The segment length for each case is 0.5 m. Length (m) 5 10 15
Nozzle head loss (m) 0.014 0.0086 0.0076
6
Diffuser head loss (m) 0.0033 0.0032 0.0035
3.2
Optimized areas and segment lengths
Our model was insufficient to optimize segment lengths along with areas. This is because the step model assumes that the flow is fully developed before and after the change in area. For closely-spaced sudden expansions or contractions, this is not the case. Effectively, the minor loss model does not account for the influence of spatial location on the minor losses produced by the sudden steps. Thus, the result of the present diffuser optimization is to step through the changes in size over segments which have lengths constrained by the lower bound length until the largest area is reached for the final pipe segment which has sufficient length to extend to the end of the diffuser. This minimizes the objective function with respect to the constraints because it makes the changes in area small for each sudden step and reduces the friction losses by making the pipe diameter as large as possible for the greatest possible length. While valid for the objective function and constraints, this model neglects additional important flow physics such as the regions of recirculation which would exist under these conditions. If an additional penalty was added to the objective function in the form of a pressure drop obtained with a reduced order model, better designs could be realized.
4
Conclusions
From the presented results, we see head loss is higher in the converging part of the overall nozzle. As the CD nozzle area constricts to the throat, the velocity increases, increasing the pressure losses. This effect is only increased by further constraining velocity at another point partway between the inlet and throat or the throat and outlet for the converging and diverging sections, respectively. Therefore, it is desirable to only constrain the velocity at the throat, and allow for a smooth constriction and expansion. As overall CD nozzle length is increased, the optimized shape changes. Gradual constrictions expansions are characteristic of longer CD nozzles. It is interesting to note that there exists two diverging section shape regimes. At shorter lengths, the optimal nozzle shape is concave-up. At longer lengths, the nozzle geometry shifts to concave-down. This is likely a function of the limitations of the derived model. As step lengths decrease, the assumption of fully developed flow at the inlet and outlet of each step becomes less accurate. The model presented here provides a better solution for longer segment lengths. We also see that minor losses have a larger effect than major losses on total head loss, according to the model developed here. This is a reasonable result, as the friction losses will remain consistent for nozzles of similar length and material. Cross-sectional area has only a marginal effect on the major losses. In the considered CD nozzle geometry, cross-sectional area changes significantly, and the resulting pressure losses dominate the overall loss. Additionally, we do not address length as a design variable in our model, so the optimizer gives additional weight to minimize minor losses over major losses. We recognize this skews the results, and is a function of the simplifying assumptions. We also recognize the other limitations of the developed model, and recommend further work be done to resolve the inaccurate assumption of fullydeveloped flow at each step boundary. Furthermore, the incompressible assumption should be relaxed, and the model should be modified to allow for a gas as the working fluid and capture the compressibility effects present in a real supersonic CD nozzle.
References [1] Trung Duc Nguyen Bodgan-Alexandru Belega. Analysis of flow in convergent-divergent rocket engine nozzle using computational fluid dynamics. In International Conference of Scientific Paper, AFASES, 2015. [2] Nancy Hall. Nozzle design: Converging-diverging (cd) nozzle. Website. [3] IIT Kanpur. Compressible flow. Online, December 2009. [4] S. A. K. Jilani Kunal Pansari. Numerical investigation of the performance of convergent divergent nozzle. International Journal of Modern Engineering Research, 3(5):2662–2666, October 2013. [5] Bruce R. Munson, Donald F. Young, Theodore H. Okiishi, and Wade W. Huebsch. Fundamentals of Fluid Mechanics. Wiley, 6th edition, 2009.
7
Abstract We develop an analytical model for the head loss in a stepped converging-diverging (CD) nozzle through which liquid water flows. Under these conditions, the liquid water is modeled as steady and incompressible. The CD nozzle is composed of a series of finite steps. The head loss including major and minor losses is obtained for a variable number of steps as a function of the area of each step and the length of these stepped segments. The optimization problem considered here is to obtain the set of cross-sectional areas which minimize head loss. For the diffuser, we constrain each cross-sectional area in a segment to be greater than or equal to the area of the segment immediately upstream. For the nozzle, we constrain each cross-sectional area in a segment to be less than or equal to the area of the segment immediately upstream. We consider a constraint where we specify the velocity at a specific segments of the nozzle and of the diffuser supposing that measurements or process requirements mandate this constraint. Under these constraints, we obtain the optimal head loss and corresponding design variables for cases including or excluding friction and for cases with and without the velocity constraint. We find that minimizing minor losses dominates the selection of cross-sectional areas. The optimal solution adapts to the addition of major losses by increasing the areas of each segment slightly. We find that the concavity of the optimal cross-sectional area profile depends on the length of the diffuser with concave-up profiles encountered for short lengths and profiles which transition from concave-up to concave-down for longer lengths.
1
Introduction
Nozzles are widely used to accelerate a fluid for many applications. Nozzles typically increase fluid velocity while reducing the pressure and temperature of the fluid [1]; however, many designs of nozzles exist to perform different functions. Simple turbojet turboprop engines often use a fixed geometry convergent nozzle, while turbofan engines usually employ a more complicated co-annular nozzle [4]. One specific nozzle configuration, relevant to turbomachinery, is the convergent-divergent (CD) nozzle. The CD nozzle consists of a converging section followed by a diverging section as shown in Fig. 1.
Figure 1: Illustration of typical convergent-divergent nozzle. These nozzles can have a fixed or variable geometry. Afterburning turbojets and turbofans employ a variable- geometry CD nozzle, while rocket engines typically use a fixed-geometry CD nozzle. Exhaust gases 1
leave the combustion chamber and converge at the throat of the CD nozzle (see Fig. 1). The throat size is critical to the overall design and changes for various applications. Throat size is chosen to choke the flow to define the mass flow rate through the system [2]. For applications where supersonic speeds at CD nozzle output are desirable, the throat is sized to constrain the flow to be sonic, with a Mach number of 1. The flow is subsonic in the converging nozzle, reaches unity at the throat, and expands isentropically in the diverging nozzle, causing the fluid to accelerate to supersonic speeds. The analysis performed by Kanpur on simple nozzles and diffusers [3] produces a governing equation that explains this unintuitive phenomenon. dV dA =− [1 − M a2 ] (1) A V It is clear from this relationship that for subsonic flow (M a < 1), velocity and area changes are inversely related. However, for supersonic (M a > 1) flows, area and velocity changes are directly related. The throat provides a region by which a subsonic nozzle and supersonic diffuser are joined. Through this combination, the desired flow acceleration achieved. For this work, we consider a simplified CD nozzle to be used as a feed to an impulse turbine as seen in Fig. 2. To improve turbine efficiency, we seek to optimize a CD nozzle geometry that minimizes losses in the nozzle. Nozzles
Fluid Jets
Figure 2: Illustration of impulse turbine. The considered nozzles are CD nozzles. Adapted from [5].
2 2.1
Analysis and Methods Model
To design the CD nozzle, we first assume the nozzle is composed of multiple steps as shown in Figure 4. In this way we can perform a series of control volume analyses on each step using the control volume shown in Fig. 3. To make an analytical model possible, we consider water as the working fluid. Therefore, we assume the flow is also incompressible and steady. We assume the pipe roughness is uniform. The pipe cross section is assumed to be circular.
Figure 3: Control volume for a single step. Many steps were analyzed and evaluated in series to approximate a CD nozzle.
2
V1
Ai , Li , i
Lno
Ld
Figure 4: The stepped constriction is constructed using a stepped nozzle and a stepped diffuser. V1 is the inlet velocity. Lno and Ld are the lengths of the nozzle and diffuser, respectively. Each segment of the constriction has a cross-sectional area, length, and roughness indicated by Ai , Li , and , respectively. The objective is to minimize the diffuser head loss due to major and minor losses. The head loss for a single sudden expansion is derived in [5]. This derivation requires that a uniform velocity profile be present at the inlet and outlet of the step. For each step, the minor loss coefficient, KL,m , is KL,m
2 A1 = 1− A2
(2)
where A1 is the area prior to the step and A2 is the area following the step. The minor head loss associated with this step is V2 hL,m = KL,m 1 (3) 2g We can write this headloss for an arbitrary location in a diffuser with many steps. 2 V 2 A1 hL,m,j = KL,m,j 1 2g Aj
(4)
A1 and V1 are respectively the first cross-sectional area and velocity in the diffuser. Aj is the area prior to the jth step and Aj+1 is the area following. Substituting for the loss coefficient, 2 2 2 Aj V1 A1 hL,m,j = 1 − (5) Aj+1 2g Aj and summing over the n − 1 steps the total minor head loss may be obtained, hL,m =
n−1 X j=1
Aj 1− Aj+1
2
V12 2g
A1 Aj
2 (6)
p The major loss may also be determined. Assuming circular pipes, the diameter is Di = 2 Ai /π where i is the index used to traverse each segment of the stepped diffuser to sum major loss, i = 1...n. From continuity, Vi = V1 A1 /Ai . A vector of lengths, L, and a vector of roughnesses, , are initialized corresponding to each cross-sectional area. The friction factors may be calculated using the Haaland equation [5]. ( " #)−2 1.11 6.9 i Di + (7) fi = −1.8log 3.7 ReD The Reynolds number for each section is defined as Rei = Vi Di /ν, where ν is the kinematic viscosity. The major head loss for each section is li Vi2 hL,M,i = fi (8) Di 2g 3
The total major head loss may be obtained by summing hL,M,i from i = 1...n. hL,M =
n X i=1
fi
li Vi2 Di 2g
(9)
The objective function is the total head loss which is the sum of the major and minor head loss hL = hL,m + hL,M
(10)
2 2 X r (V1 ∗ A1 )2 X Aj 1 li π A21 ∗ V12 hL = ∗ 1− ∗ + fi ∗ ∗ ∗ 2g Aj+1 Aj 2 Ai A2i ∗ 2g
(11)
and is found to be
for the diverging section of the CD nozzle. Through a similar derivation, the total head loss in the converging section can be found, and is here defined as 2 r X (Vj ∗ A1 )2 Aj li π A21 ∗ V12 hL = ∗ 1− + (fi ∗ ∗ ∗ 2g Aj−1 2 Ai A2i ∗ 2g
(12)
for the converging section of the CD nozzle. We scale the objective for both the nozzle and diffuser to be O(1). The design variables are the cross-sectional areas, Aj j = 1...n. The lengths and surface roughnesses are fixed. We apply several types of constraints to the problem. The first is that the cross-sectional area downstream of each step is greater than or equal to the cross-sectional area upstream, Aj+1 ≥ Aj The second type of constraint is an equality constraint which requires the velocity at a certain point to be equal to some fraction of the inlet velocity. This constrains the throat area such that a set velocity is obtained at the throat, therefore constraining mass flow through the nozzle; Vloc = f (V1 ).
2.2
Optimization algorithm
To find an optimal nozzle geometry to minimize head loss, the MATLAB function fmincon was used. This function performs a gradient-based optimization using the interior-point algorithm. Prior to selecting this algorithm, we considered the CD nozzle geometry using gradient-based and gradient-free approaches. The gradient-free approach, fminsearch, implements the Nelder-Mead simplex. It produced unsatisfactory results as convergence was slow and the solution lacked the accuracy of the gradient-based solution. We also considered the active-set algorithm for use with fmincon. It completed the optimization in a comparable time and number of function calls. However, it achieved a higher first order optimality condition and constraint violation. We selected the interior-point algorithm. We used built-in finite-differencing to provide the gradients for the optimizer.
3 3.1
Results and Discussion Optimized Areas, Fixed Segment Lengths
We present optimized profiles for the stepped nozzle and stepped diffuser. Parameters for these solutions are given in Table 1. Figure 5 contains two panels. In each panel, there is a plot of cross-sectional area as a function of axial length. Panel (a) is the area length relationship for the diffuser and panel (b) is the area length relationship for the nozzle. Both panels plot the results obtained under four combinations of two settings: no friction with no velocity constraint, no friction with a velocity constraint, friction with no velocity constraint, and friction with a velocity constraint. The cross-sectional area decreases with increasing length for the nozzle and increases with increasing length for the diffuser. The rate of change in area for the converging nozzle decreases with increasing length while the rate of change of area for the diffuser increases. The velocity constraint requires that the velocity at the middle segment be 2.188 times the velocity at the inlet of the 4
Table 1: Parameters utilized for stepped diffuser and stepped nozzle optimization. Parameter L N Lseg A1 An V1 Tolerance ν
Diffuser 5, 10, 15 m 10, 20, 30 2m 1 m2 2.5 m2 1 m/s 1 × 10−6 1.12 × 10− 6 m2 /s 0.0045 mm
Nozzle same same same 2.5 m2 1 m2 0.4 m/s same same same
nozzle and 0.875 times the velocity at the inlet of the diffuser. The effect of this constraint is to reduce the cross-sectional area and match the outlet area of the of the nozzle and the inlet area of the diffuser. Consequently, the head loss increases significantly. We consider the diffuser. For the no friction case, the increase is 22%. For the friction case, the increase is 20%. For the diffuser under the velocity constraint, the velocity at the middle segment is higher than the velocity at that segment in the optimal case. To achieve this constraint, the optimizer increases the cross-sectional areas at each step by a smaller rate with length than in the optimal solution and then increases the cross-sectional areas at a greater rate with length following the constraint location. The cross-sectional area increases slightly when the objective function includes major losses. For incompressible flows, Vi Ai must be constant to satisfy continuity. Since hL,M,i ∝ Vi2 , the cross-sectional areas increase to reduce the major head loss (Figure 5). However, since this head loss is less dominant than the minor head loss, the change in cross-sectional areas is small. The influence of major losses may be considered by examining Table 2. We consider the diffuser. Without the velocity constraint, the head loss increases 126% with the addition of major losses. With the velocity constraint, the head loss increases by 155% with the addition of major losses. This larger increase in the case where velocity constrained is reasonable because the diffuser cross-sectional area is substantially reduced. This increasing the losses due to friction for a given flow rate. 2.5
2.5
Cross−sectional Area (m2)
no friction, no velocity constraint no friction, velocity constraint friction, no velocity constraint friction, velocity constraint
2
Cross−sectional Area (m )
no friction, no velocity constraint no friction, velocity constraint friction, no velocity constraint friction, velocity constraint
2
1.5
1
0
0.5
1
1.5
2 2.5 3 Axial Length (m)
3.5
4
4.5
2
1.5
1
5
(a) Converging nozzle
0
0.5
1
1.5
2 2.5 3 Axial Length (m)
3.5
4
4.5
5
(b) Diffuser
Figure 5: Four cases are plotted for the converging nozzle and diffuser. We explore four scenarios achievable with or without the velocity constraint and with or without friction. The cross-sectional area is plotted as a function of axial length.
5
Table 2: The head loss for the diffuser and nozzle are given for the cases considered in Figure 5. Friction No No Yes Yes
Velocity constraint No Yes No Yes
Nozzle head loss (m) 0.006 0.0073 0.0085 0.0102
Diffuser head loss (m) 0.00097 0.0012 0.0022 0.0028
If we fix the length of segments and vary the number of segments so that the total diffuser length changes, we notice that the concavity of the profile changes as the diffuser length increases (Fig. 6). Figure 6 consists of two panels, (a) and (b). Panel (a) shows the optimal nozzle profiles for nozzles. Similarly, panel (b) shows the optimal diffuser profiles for diffusers. For shorter diffusers, the losses are minimized by a concave-up profile. However, as the diffuser length decreases the losses are minimized by a profile which transitions between concave-up and concave-down. However, the profiles for the nozzles remain concave-up and the change in area with respect to position decreases as the length of the nozzle increases. To accompany Fig. 6, we give the head loss for the nozzle and for the diffuser as a function of length in Table 3. We note the surprising result that the head loss decreases with increasing length and attribute this to the fact that the reduction in the minor loss resulting from decreasing the size of each step is greater than the increase in the total major loss from increasing length.
2.5
2.5
Cross−sectional Area (m2)
2
Cross−sectional Area (m )
L=5m L = 10 m L = 15 m
2
1.5
2
1.5
L=5m L = 10 m L = 15 m 1
0
5
10
1
15
Axial Length (m)
0
5
10
15
Axial Length (m)
(a) Converging nozzle
(b) Diffuser
Figure 6: The optimal profiles expressed as cross-sectional area as a function of axial length are plotted for three nozzle lengths panel (a) and for three diffuser lengths panel (b). The length of all steps are the same requiring that the number of steps increase as the diffuser length increases.
Table 3: The head loss is given for nozzles and diffusers of different lengths. The segment length for each case is 0.5 m. Length (m) 5 10 15
Nozzle head loss (m) 0.014 0.0086 0.0076
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Diffuser head loss (m) 0.0033 0.0032 0.0035
3.2
Optimized areas and segment lengths
Our model was insufficient to optimize segment lengths along with areas. This is because the step model assumes that the flow is fully developed before and after the change in area. For closely-spaced sudden expansions or contractions, this is not the case. Effectively, the minor loss model does not account for the influence of spatial location on the minor losses produced by the sudden steps. Thus, the result of the present diffuser optimization is to step through the changes in size over segments which have lengths constrained by the lower bound length until the largest area is reached for the final pipe segment which has sufficient length to extend to the end of the diffuser. This minimizes the objective function with respect to the constraints because it makes the changes in area small for each sudden step and reduces the friction losses by making the pipe diameter as large as possible for the greatest possible length. While valid for the objective function and constraints, this model neglects additional important flow physics such as the regions of recirculation which would exist under these conditions. If an additional penalty was added to the objective function in the form of a pressure drop obtained with a reduced order model, better designs could be realized.
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Conclusions
From the presented results, we see head loss is higher in the converging part of the overall nozzle. As the CD nozzle area constricts to the throat, the velocity increases, increasing the pressure losses. This effect is only increased by further constraining velocity at another point partway between the inlet and throat or the throat and outlet for the converging and diverging sections, respectively. Therefore, it is desirable to only constrain the velocity at the throat, and allow for a smooth constriction and expansion. As overall CD nozzle length is increased, the optimized shape changes. Gradual constrictions expansions are characteristic of longer CD nozzles. It is interesting to note that there exists two diverging section shape regimes. At shorter lengths, the optimal nozzle shape is concave-up. At longer lengths, the nozzle geometry shifts to concave-down. This is likely a function of the limitations of the derived model. As step lengths decrease, the assumption of fully developed flow at the inlet and outlet of each step becomes less accurate. The model presented here provides a better solution for longer segment lengths. We also see that minor losses have a larger effect than major losses on total head loss, according to the model developed here. This is a reasonable result, as the friction losses will remain consistent for nozzles of similar length and material. Cross-sectional area has only a marginal effect on the major losses. In the considered CD nozzle geometry, cross-sectional area changes significantly, and the resulting pressure losses dominate the overall loss. Additionally, we do not address length as a design variable in our model, so the optimizer gives additional weight to minimize minor losses over major losses. We recognize this skews the results, and is a function of the simplifying assumptions. We also recognize the other limitations of the developed model, and recommend further work be done to resolve the inaccurate assumption of fullydeveloped flow at each step boundary. Furthermore, the incompressible assumption should be relaxed, and the model should be modified to allow for a gas as the working fluid and capture the compressibility effects present in a real supersonic CD nozzle.
References [1] Trung Duc Nguyen Bodgan-Alexandru Belega. Analysis of flow in convergent-divergent rocket engine nozzle using computational fluid dynamics. In International Conference of Scientific Paper, AFASES, 2015. [2] Nancy Hall. Nozzle design: Converging-diverging (cd) nozzle. Website. [3] IIT Kanpur. Compressible flow. Online, December 2009. [4] S. A. K. Jilani Kunal Pansari. Numerical investigation of the performance of convergent divergent nozzle. International Journal of Modern Engineering Research, 3(5):2662–2666, October 2013. [5] Bruce R. Munson, Donald F. Young, Theodore H. Okiishi, and Wade W. Huebsch. Fundamentals of Fluid Mechanics. Wiley, 6th edition, 2009.
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