analysis of axially dispersed systems with general ... - Semantic Scholar
Chemical EnginerringScieme Vol. 39, NO. 11, pp. 1571-1579.1984 Rintcdinthe U.S.A.
000!%2509/84 53.m+ .OO Pa~nman RCES Ltd.
ANALYSIS OF AXIALLY DISPERSED SYSTEMS GENERAL BOUNDARY CONDITIONS-I
WITH
FORMULATION SATISH J. PARULEKAR and DORAISWAMI RAMKRISHNA* School of Chemical Engineering, Purdue University, West Lafayette, IN 47907, U.S.A.
(Receded 6 May 1983; in revised form 19 October 1983; accepted 17 January 1984) Abstract-Transient analysis in the past of finite axially dispersed systems with or without a rate process has been subject to the use of the Danckwerts boundary conditions. We present here a formulation which suggests a general set of boundary conditions from which the Danckwerts conditions arise.as a special case.. The resulting boundary value problems neatly fit the mold of self-adjoiat operator theory. An integral transform approach is outlined for those not familiar with operator theory. Discussion of actual solutions to different casts is deferred to two other papers which appear as Parts II and III.
1. INTRODUcIlON The axial dispersion model has been a popular feature of chemical engineering analysis for many years. It has been used in a wide variety of dispersion problems, e.g., in the analysis of tubular chemical reactors, heat exchange processes, and in various separation processes[l]. The main attractiveness of the axial dispersion model is its power to fruitfully amend the predictions of the plug flow model while still retaining the latter’s unidimensional simplicity. Further, the early work of Taylor[2] that established the reducibility of important two dimensional situations of convective-diffusion of a solute in a solvent stream to one dimensional convective-dispersion problems generated greater interest in axial dispersion models. The formulation of axial dispersion models in tubes of finite length has been complicated by difficulties associated with the specification of appropriate boundary conditions. The ditficulty originates from the dependence of events at either boundary on those that occur outside of the tube, which are frequently unknown or difficult to model without substantial loss of simplicity. The boundary conditions, generally attributed to Danckwerts[3],t have been extensively used in the analysis of axial dispersion models. Since the publication of Danckwerts however, there has been considerable discussion of his boundary conditions in the literature[&lS], mostly with a view to assess the conditions of their validity. Some of these auempts[7, 10, 12, 15, 181 consist in appending a semi-infinite upstream section at the inlet to the tube and a semi-iqfinite downstream section at the outlet. Dispersion is presumed to persist in the appended sections with dispersion coefficients that may or may not he identical to that in the tube proper. The “practical” lengths of the
‘*Author to whom correspondence should be addressed. tMore recently, A&[41 has traced them further back to a paper by Langm*r[fl.
appended sections are however, dependent on the magnitudes of the dispersion coefficients in these sections, virtually vanishing when negligible dispersion occurs there. Much of the discussion conceming the validity of the Danckwerts boundary conditions has been co-ed to steady state situations for which, paradoxically enough, the Danckwerts houndary conditions suffice to describe the events in the tube proper. There is little that has heen said of the transient situation for which the foregoing arguments in essence imply the inapplicability of the Danckwerts boundary conditions, except of course when there is no dispersion outside of the tube. Transient analyses of axially dispersed systems have generally employed the Danckwerts boundary conditions[3,6,8,11,19]. Arguments in support have depended on, for example, not permitting the solute to disperse out of the medium (such as a porous bed) at the entrance where it is introduced, while at the other end a zero axial gradient prevents the occurrence of an interior maximum or minimum in solute concentration. We will address these issues granting that axial dispersion is appropriately represented by a Fickian flux expression resembling that of molecular diffusion. Thus it is not our purpose here $0 investigate substitutes leading to hyperbolic type models that have interested some workers. If the dispersing medium such as a porous bed is preceded by an inlet section, there is perhaps.not much reason to fqrbid a dispersive escape out at the entrance. Likewise, the zero gradient boundary condition at the exit is not necessary to prevent interior concentration or temperature extrema. In spite of the foregoing objection% the Dauckwerts boundary conditions have been valuable for transient. analysis in that their implications, although not, essential, are in some situations. nearly satisfied. The main advantage of these boundary conditions is that the “linite domain” nature of the.houndary value problem is retained. In addressing the transient problem with Danckwerts boundary conditions, one need only specify the initial
1571
1572
S. T.
P~nvm
state in the tube since the situation “outside” of the tube is irrelevant to the analysis. One would therefore expect that any departures of predictions on the foregoing basis from actual behavior should depend on the initial condition in the fore section (upstream appended section) and aft section (downstream appended section). Amundson[l9] points out that the major difficulty in appending semi-infinite fore and aft sections for transient analysis lies in the numerical solution to which one is constrained in dealing with nonlinear problems. While this is certainly true in dealing with nonlinearities, linear problems present some interesting possibilities that have not been explored in the past. The methods of this paper are relevant to axially dispersed systems featuring a linear rate process such as a Erst order chemical reaction, heat exchange or separation process occurring in absorption, ion exchange, membrane transfer, and chromatographic columns. With the foregoing in mind, there is no generality lost by considering the specific example of a Erst arder chemical reaction occurring in a tubular reactor of finite length. The actual reactor section will be viewed as extending throughout the infinite tube with chemical reaction occurring with a position which has a constant dependent rate “constant” value in the reactor section and vanishes outside it. This viewpoint is motivated by the fact that the resulting differential operator fits the mold of the established spectral theory of second order differential operators on unbounded domains. Although real reactors are not blessed with infinite appendages, the latter must be viewed as convenient mathematical abstractions whose validity will depend on the actual lengths outside the reactor which are influenced by events in the reaction section. Such abstractions are not new to engineering analysis; one such example is the use of semi-infinite domains connected with the concept of penetration thickness in dealing with mass transfer problems. There is some caution, however, that must be observed, the nature of which is best left to be expounded at a later stage. For the present, we let the appended sections be infinitely long if there is tinite dispersion there. On the other hand, it is possible to conceive of pre- or post-reactor sections in which the dispersion coefficient may be regarded as infinite because of the intensity of mixing there. Such sections, which cannot be assigned an infinite length (or volume), can arise, for example, when the reactor is either preceded or succeeded by a well-mixed vessel (of say, constant volume) serving as an interrnedlary open system. Finally, we must observe that the case of zero dispersion dispenses with the need for ascribing a definite length to the appended sections for then the Danckwerts boundary conditions will have been resuscitated. In summary, the formulation of appropriate boundary conditions is governed by the configurational details of the reactor assembly. Our objective in this paper is to expose a family of suitable
and D. RAMKRlsHN& “boundary conditions” to which the ones due to Danckwerts belong as a special case. Besides, the family is harmoniously within the scope of a unified treatment of a second order differential operator of the singular Sturn-Liouville type. It must be understood that, from a mathematical viewpoint, the boundary conditions are most simply stated (the concentration at - cc being the feed concentration while at + co it must remain Enite); of course the continuity of concentrations and mass fluxes at the reactor boundaries (the so-called Wehner-Wilhelm conditions) need be imposed. 2. SYSTEM
EQUATIONS
In,the ordinary course, one begins with identifying the appropriate differential equations followed by a discussion of boundary conditions. Since the focus here is on reactor boundary conditions, we will address these first. The reactor stretches between z = -L and z = L so that the reactor length is 2L. The dispersion coefficients in the reactor and its fore and aft sections are assumed to be uniform given by D, D-, D, respectively. The concentration field C(z, t) must satisfy the continuity conditions lim C = z_l~~+p C; z-.-L--o
C2) Equations (1) hold when the fore section is infinitely long with a Enite non-zero dispersion coefficient. Equations (2) hold when the aft section is infinitely long with a -finite non-zero dispersion coefficient. If one has a well-mixed pre-reactor section with volume V_, the conditions (1) must be replaced by lim
z--L+0
C=C_;
c--
z_~+opg--uq+G,
V_ dC_ A dr
where C, is the concentration of the reactant in the feed, C_ is the concentration of the reactant in the well-mixed fore section (continuous stirred tank), v is tbe uniform velocity of the fluid, and A is the uniform tube cross-section. Similarly, a well-mixed aft section of volume V+ yields in place of (2) the conditions Em C=C+, f-L-0
8C -Daz
lim
r-L-0
[
1
where C, is the concentration of the well-mixed aft section. It is interesting Danckwerts boundary condition may either of the reactor ends by allowing
reactant in the to note that the be obtained at either volume
Analysis of axially dispersed
systemswith general boundary conditions-l
V+ or V_ to vanish. Although less apparent, eqns (1) can also be used to establish the Dauckwerts houndary condition at the reactor inlet in the limit of vanishing D_ . Similar statements hold for the reactor outlet also. In addition to the foregoing boundary conditions, we must have
lim C=C, I-r-co
(5)
when au infinite fore section is used. When an infinite aft section is employed we require that litllC
000!%2509/84 53.m+ .OO Pa~nman RCES Ltd.
ANALYSIS OF AXIALLY DISPERSED SYSTEMS GENERAL BOUNDARY CONDITIONS-I
WITH
FORMULATION SATISH J. PARULEKAR and DORAISWAMI RAMKRISHNA* School of Chemical Engineering, Purdue University, West Lafayette, IN 47907, U.S.A.
(Receded 6 May 1983; in revised form 19 October 1983; accepted 17 January 1984) Abstract-Transient analysis in the past of finite axially dispersed systems with or without a rate process has been subject to the use of the Danckwerts boundary conditions. We present here a formulation which suggests a general set of boundary conditions from which the Danckwerts conditions arise.as a special case.. The resulting boundary value problems neatly fit the mold of self-adjoiat operator theory. An integral transform approach is outlined for those not familiar with operator theory. Discussion of actual solutions to different casts is deferred to two other papers which appear as Parts II and III.
1. INTRODUcIlON The axial dispersion model has been a popular feature of chemical engineering analysis for many years. It has been used in a wide variety of dispersion problems, e.g., in the analysis of tubular chemical reactors, heat exchange processes, and in various separation processes[l]. The main attractiveness of the axial dispersion model is its power to fruitfully amend the predictions of the plug flow model while still retaining the latter’s unidimensional simplicity. Further, the early work of Taylor[2] that established the reducibility of important two dimensional situations of convective-diffusion of a solute in a solvent stream to one dimensional convective-dispersion problems generated greater interest in axial dispersion models. The formulation of axial dispersion models in tubes of finite length has been complicated by difficulties associated with the specification of appropriate boundary conditions. The ditficulty originates from the dependence of events at either boundary on those that occur outside of the tube, which are frequently unknown or difficult to model without substantial loss of simplicity. The boundary conditions, generally attributed to Danckwerts[3],t have been extensively used in the analysis of axial dispersion models. Since the publication of Danckwerts however, there has been considerable discussion of his boundary conditions in the literature[&lS], mostly with a view to assess the conditions of their validity. Some of these auempts[7, 10, 12, 15, 181 consist in appending a semi-infinite upstream section at the inlet to the tube and a semi-iqfinite downstream section at the outlet. Dispersion is presumed to persist in the appended sections with dispersion coefficients that may or may not he identical to that in the tube proper. The “practical” lengths of the
‘*Author to whom correspondence should be addressed. tMore recently, A&[41 has traced them further back to a paper by Langm*r[fl.
appended sections are however, dependent on the magnitudes of the dispersion coefficients in these sections, virtually vanishing when negligible dispersion occurs there. Much of the discussion conceming the validity of the Danckwerts boundary conditions has been co-ed to steady state situations for which, paradoxically enough, the Danckwerts houndary conditions suffice to describe the events in the tube proper. There is little that has heen said of the transient situation for which the foregoing arguments in essence imply the inapplicability of the Danckwerts boundary conditions, except of course when there is no dispersion outside of the tube. Transient analyses of axially dispersed systems have generally employed the Danckwerts boundary conditions[3,6,8,11,19]. Arguments in support have depended on, for example, not permitting the solute to disperse out of the medium (such as a porous bed) at the entrance where it is introduced, while at the other end a zero axial gradient prevents the occurrence of an interior maximum or minimum in solute concentration. We will address these issues granting that axial dispersion is appropriately represented by a Fickian flux expression resembling that of molecular diffusion. Thus it is not our purpose here $0 investigate substitutes leading to hyperbolic type models that have interested some workers. If the dispersing medium such as a porous bed is preceded by an inlet section, there is perhaps.not much reason to fqrbid a dispersive escape out at the entrance. Likewise, the zero gradient boundary condition at the exit is not necessary to prevent interior concentration or temperature extrema. In spite of the foregoing objection% the Dauckwerts boundary conditions have been valuable for transient. analysis in that their implications, although not, essential, are in some situations. nearly satisfied. The main advantage of these boundary conditions is that the “linite domain” nature of the.houndary value problem is retained. In addressing the transient problem with Danckwerts boundary conditions, one need only specify the initial
1571
1572
S. T.
P~nvm
state in the tube since the situation “outside” of the tube is irrelevant to the analysis. One would therefore expect that any departures of predictions on the foregoing basis from actual behavior should depend on the initial condition in the fore section (upstream appended section) and aft section (downstream appended section). Amundson[l9] points out that the major difficulty in appending semi-infinite fore and aft sections for transient analysis lies in the numerical solution to which one is constrained in dealing with nonlinear problems. While this is certainly true in dealing with nonlinearities, linear problems present some interesting possibilities that have not been explored in the past. The methods of this paper are relevant to axially dispersed systems featuring a linear rate process such as a Erst order chemical reaction, heat exchange or separation process occurring in absorption, ion exchange, membrane transfer, and chromatographic columns. With the foregoing in mind, there is no generality lost by considering the specific example of a Erst arder chemical reaction occurring in a tubular reactor of finite length. The actual reactor section will be viewed as extending throughout the infinite tube with chemical reaction occurring with a position which has a constant dependent rate “constant” value in the reactor section and vanishes outside it. This viewpoint is motivated by the fact that the resulting differential operator fits the mold of the established spectral theory of second order differential operators on unbounded domains. Although real reactors are not blessed with infinite appendages, the latter must be viewed as convenient mathematical abstractions whose validity will depend on the actual lengths outside the reactor which are influenced by events in the reaction section. Such abstractions are not new to engineering analysis; one such example is the use of semi-infinite domains connected with the concept of penetration thickness in dealing with mass transfer problems. There is some caution, however, that must be observed, the nature of which is best left to be expounded at a later stage. For the present, we let the appended sections be infinitely long if there is tinite dispersion there. On the other hand, it is possible to conceive of pre- or post-reactor sections in which the dispersion coefficient may be regarded as infinite because of the intensity of mixing there. Such sections, which cannot be assigned an infinite length (or volume), can arise, for example, when the reactor is either preceded or succeeded by a well-mixed vessel (of say, constant volume) serving as an interrnedlary open system. Finally, we must observe that the case of zero dispersion dispenses with the need for ascribing a definite length to the appended sections for then the Danckwerts boundary conditions will have been resuscitated. In summary, the formulation of appropriate boundary conditions is governed by the configurational details of the reactor assembly. Our objective in this paper is to expose a family of suitable
and D. RAMKRlsHN& “boundary conditions” to which the ones due to Danckwerts belong as a special case. Besides, the family is harmoniously within the scope of a unified treatment of a second order differential operator of the singular Sturn-Liouville type. It must be understood that, from a mathematical viewpoint, the boundary conditions are most simply stated (the concentration at - cc being the feed concentration while at + co it must remain Enite); of course the continuity of concentrations and mass fluxes at the reactor boundaries (the so-called Wehner-Wilhelm conditions) need be imposed. 2. SYSTEM
EQUATIONS
In,the ordinary course, one begins with identifying the appropriate differential equations followed by a discussion of boundary conditions. Since the focus here is on reactor boundary conditions, we will address these first. The reactor stretches between z = -L and z = L so that the reactor length is 2L. The dispersion coefficients in the reactor and its fore and aft sections are assumed to be uniform given by D, D-, D, respectively. The concentration field C(z, t) must satisfy the continuity conditions lim C = z_l~~+p C; z-.-L--o
C2) Equations (1) hold when the fore section is infinitely long with a Enite non-zero dispersion coefficient. Equations (2) hold when the aft section is infinitely long with a -finite non-zero dispersion coefficient. If one has a well-mixed pre-reactor section with volume V_, the conditions (1) must be replaced by lim
z--L+0
C=C_;
c--
z_~+opg--uq+G,
V_ dC_ A dr
where C, is the concentration of the reactant in the feed, C_ is the concentration of the reactant in the well-mixed fore section (continuous stirred tank), v is tbe uniform velocity of the fluid, and A is the uniform tube cross-section. Similarly, a well-mixed aft section of volume V+ yields in place of (2) the conditions Em C=C+, f-L-0
8C -Daz
lim
r-L-0
[
1
where C, is the concentration of the well-mixed aft section. It is interesting Danckwerts boundary condition may either of the reactor ends by allowing
reactant in the to note that the be obtained at either volume
Analysis of axially dispersed
systemswith general boundary conditions-l
V+ or V_ to vanish. Although less apparent, eqns (1) can also be used to establish the Dauckwerts houndary condition at the reactor inlet in the limit of vanishing D_ . Similar statements hold for the reactor outlet also. In addition to the foregoing boundary conditions, we must have
lim C=C, I-r-co
(5)
when au infinite fore section is used. When an infinite aft section is employed we require that litllC
