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Steady State Behavior of Coupled Nonlinear Reactor-Separator Systems: Effect of Different Separators Anil Painuly, S. Pushpavanam,* and A. Kienle† Department of Chemical Engineering, Indian Institute of Technology Madras, Chennai 600036, India, and Max-Planck Institute for Dynamics of Complex Technical Systems, Sandtorstrasse 1, Magdeburg 39106, Germany
In a typical plant, downstream separators are coupled to upstream reactors through recycle streams. The bifurcation behavior of the coupled system is different from that of the individual units. The bifurcation diagram depicts the dependence of the steady state on an operating parameter. In this work, we analyze the behavior of three different coupled reactor-separator systems. Each system is characterized by a different separator. The three different separator units investigated are a flash unit, a nanofiltration unit, and an ultrafiltration unit. The reactor is modeled as a CSTR sustaining a cubic autocatalytic reaction. The stand-alone reactor can have a maximum of three solutions. The coupled system is investigated for the situation when the fresh feed is flow controlled and the reactor level is maintained constant using the reactor effluent flow stream. It is shown that the coupled reactor-flash system has a maximum of two steady solutions, while the two coupled reactor-membrane separation systems have a maximum of three solutions. Each coupled process is analyzed for two situations: (i) a binary system and (ii) ternary system. The physical cause for the difference in these behaviors is analyzed. I. Introduction A coupled reactor-separator network can be viewed as representing the essence of a chemical plant. In a typical chemical process, reactants are partially converted in the reactor. In a downstream unit, the unconverted reactants are separated from the products and these are recycled back to the reactor. When the conversion of the reactant is low and the reactant is very expensive, this recycle is necessary to render the process economically viable. The behavior of coupled reactor-separator systems has been extensively investigated in the past. In these studies the reactor was modeled as a CSTR and the separator as a flash. Luyben1 has studied the sensitivity of the ratio of the recycle flow rate to the fresh feed flow rate for an isothermal reactor coupled to a perfect separator. The recycle stream contained only the reactant, and the process effluent stream contained only the product. Pushpavanam and Kienle2 studied an exothermic reaction in the CSTR. The flash was operated at a constant temperature and pressure. The VLE was assumed to be ideal. They established that the coupled system can have a maximum of two steady states when the fresh feed to the system is flow controlled. For high values of this flow rate, the system had no steady state solution. Kiss et al.3 have analyzed the behavior of polymerization reactions in a coupled reactor-separator system. They found that steady state operation was feasible only when the reactor volume exceeded a certain critical value. Sagale and Pushpavanam4 have analyzed the behavior of an autocatalytic reaction in a coupled reactor-separator system. They analyzed the * To whom correspondence should be addressed. Phone: +91-44-22578218. Fax: +91-44-22570509. E-mail: spush@ iitm.ac.in. † Max-Planck Institute for Dynamics of Complex Technical Systems.
system for three different control strategies. They found that the strategy when the molar holdup is allowed to vary results in a unique steady state. Balasubramaniam et al.5 have considered the effect of delay or transport lag from the reactor to the separator. They studied the effect of this lag when different control strategies were used. They found that when fresh feed was flow controlled, delay did not destabilize the system and that when reactor effluent was flow controlled, delay could induce dynamic instability. The nonlinearity in these systems arises from the positive feedback effect present in the autocatalytic or exothermic reactions considered. Different modes of operation of the separator were studied, that is, isothermal isobaric flash or a constant heat duty flash, and so forth (Zeyer et al.6). The sensitivity of the system to different variables, the maximum number of solutions, and the existence of periodic solutions were analyzed in these investigations. In this work our objective is to compare the qualitative behavior of three different coupled reactor-separator systems. The reactor is modeled as an ideal CSTR. We consider the effect of different separators on the performance of the coupled systems. We study a flash unit, a nanofiltration unit, and an ultrafiltration unit as the separator. Earlier studies have investigated and compared the behaviors of different control strategies for the coupled reactor flash system.2 In this work we restrict ourselves to analyzing only the strategy when the fresh feed is flow controlled. The holdup in the reactor is controlled using the reactor effluent flow rate. The reaction is assumed to occur in the liquid phase. The effect of using different separators is investigated here. The aim is to find whether a choice of the different separators gives rise to different qualitative behavior of the coupled system. We want to physically understand the cause of any differences which may arise when we use different separators.
10.1021/ie0496840 CCC: $30.25 © 2005 American Chemical Society Published on Web 02/22/2005
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Figure 1. Schematic of a coupled reactor-flash system.
A fundamental difference exists in the way membrane separators and distillation columns are operated. Membrane separation units are best described using rate based models.7 The composition of the streams leaving the separator is hence determined by the operating conditions such as pressure drop across the unit, influent concentration, and so forth. Consequently, there are no feasibility constraints imposed on the system when membrane processes are used for separation. In contrast to this, the flash or distillation column is analyzed classically using an equilibrium model.2 More recently, rate based models have been used. In these units, composition controllers are usually placed on the effluent streams. This fixes the composition of the distillate and bottom streams leaving the column and constrains the performance of the system for a binary feed. Consequently, here a minimum reactor volume is required to ensure that there is sufficient conversion in the reactor, which ensures that the coupled system can have a steady state. We would like to understand if this difference in operation of separation systems where on one hand the compositions of the effluent streams are fixed and on the other hand they are not gives rise to any fundamental qualitative difference in the behavior of the coupled reactor-flash separation systems and the reactor-membrane separation systems. In this work we analyze the behavior of an autocatalytic reaction of the form
A + 2B f 3B The autocatalytic effect is representative of many biochemical reactions where the objective is the production of biomass. Here A can be viewed as a substrate on which the biomass B is growing. These biochemical systems are also autocatalytic systems possessing a positive feedback effect, and the choice of the above reaction captures this autocatalytic effect. This choice makes the analysis tractable. The steady state behavior of the stand-alone reactor sustaining this reaction has been studied earlier.8 When the mole fraction of A in the feed stream is xA0 and the reactor effluent stream mole fraction is zA, the steady state is governed by
Da ) (xA0 - zA)/{zA(1 - zA)2}
(1)
where Da is a dimensionless Damkohler number defined as VRkF3/F0, Vr is the reactor volume [m3], k is the
reaction rate constant [mol-2 m6 s-1], F is the molar density [mol/m3], and F0 is the molar flow rate of the fresh feed [mol/s]. For xA0 > 8/9, the system governed by (1) has three solutions for some Da, while, for xA0 < 8/9, the system possesses a unique steady state for all Da. Hence, multiplicity occurs for a range of Da when the concentration of A in the feed stream is sufficiently high. We now study how this behavior of the stand-alone reactor is modified when it is coupled to different downstream separators in a process plant. The steady state multiplicity has been caused by the positive feedback induced by the autocatalytic reaction. II. Reactor-Flash System A typical coupled reactor-flash separator system is depicted schematically in Figure 1. The autocatalytic reaction involves only two components: A and B. This system is hence a binary system. The presence of inerts or solvents can render this a ternary system. We now discuss the steady state bifurcation characteristics of these two systems. For the case of the binary system, we treat A as a solvent, and for the analysis of the ternary system, we treat A and B as solutes. (A) Binary System. Sagale and Pushpavanam4 have analyzed the steady state behavior of a coupled reactorflash system. They considered the cubic autocatalytic reaction in a CSTR. They considered the flash to be operated under isothermal, isobaric conditions. For an ideal binary system, operating under equilibrium, this uniquely fixes the mole fraction of the streams leaving the flash, that is, xA and yA. Here, the steady state is governed by
Da ) (xA0 - yA)/{zA(1 - zA)2}
(2)
where Da, the dimensionless Damkohler number, is defined as VRkF3/F0. The coupled system governed by (2) has no steady states below a critical value of Da. Above this critical value, the system has two steady states. The behavior of the coupled reactor-flash system is hence qualitatively significantly different from that of the stand-alone reactor. A typical bifurcation diagram is depicted in Figure 2. Here we have plotted the dependence of the mole fraction in the reactor on the Damkohler number. The parameter values for which the diagram is drawn
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F0xA0 + LxA - FzA - VRkF3zAzB2 ) 0
(3a)
F0xB0 + LxB - FzB + VRkF3zAzB2 ) 0
(3b)
(3) Overall Balance for the System. Since the molar holdup in the reactor and flash is a constant, we obtain the overall balance for the system as
F0 ) V
(4)
This is valid at every time instant. (4) Overall Balance across the Flash. Since the molar holdup in the flash is a constant, we obtain the overall balance across the flash as
F)V+L Figure 2. Bifurcation diagram for a coupled reactor-flash binary system: solid line, stable state; dashed line, unstable state.
are chosen as xA0 ) 0.99, xB0 ) 0.01, xA ) 0.9, and yA ) 0.01. We can see that above a critical value of Da ) 6.6 the system has two solutions, and below this value there is no solution. In this diagram the stable solution is depicted as a solid line while the unstable solution is shown as a dashed line. We now interpret this result in terms of the physical processes occurring in the system. The flash has been represented using an equilibrium model. For an isothermal isobaric flash and a binary mixture, this uniquely fixes the composition of the distillate and bottom streams. This constrains the feed composition of the flashsa binary mixturesto lie between the composition values of the distillate and bottom streams. Since the distillate stream composition which leaves the system is fixed uniquely, the conversion in the system is fixed when the reactor and flash holdups are constant. For high flow rates or low Da, the coupled system cannot satisfy this constraint and may not have a feasible steady state. For low Da, the residence time in the reactor is very low. Consequently, the conversion in the reactor is very low. The recycle flow rate only serves to reduce the residence time and, hence, to reduce the conversion further; consequently, the constraint of a fixed conversion cannot be satisfied. When the fresh feed flow rate is low (or Da is high), the recycle flow rate can adjust itself to a large value and ensure that the constraint on the conversion is satisfied. (B) Ternary System. To determine if the bifurcation behavior in Figure 2 is caused by the VLE which determines the compositions uniquely for a binary system, we now analyze a ternary system. Here, we again analyze the same cubic autocatalytic reaction system. However, we consider an additional component C to be present which acts as an inert. This component can be viewed as a solvent. All three components, A, B, and C, are assumed to be volatile with equilibrium ratios KA, KB, and KC, respectively. For this case, the governing equations are as follows: (1) Component balances across the reactor are based on the general relationship
input - output + generation ) accumulation (2) Accumulation Term Set to Zero. Since we are interested in the steady state behavior, we set the accumulation term to zero. This yields for species A and B
(5a)
This is also valid at every time instant. (5) Component balances across the flash for species A and B yield
FzA ) VyA + LxA
(5b)
FzB ) VyB + LxB
(5c)
(6) Assuming the streams leaving the flash to be in equilibrium yields
yi ) Kixi
i ) A, B, C
(6)
(7) Summation Conditions. We also need the summation conditions:
∑yi ) 1
∑zi ) 1
∑xi ) 1
(7)
This stems from the fact that the sum of all mole fractions must add to unity. This system of equations is used to determine the flow rates of streams F, V, and L (see Figure 1) and their corresponding compositions. Hence, we have 12 equations in as many variables. For an ideal system, the equilibrium ratios Ki are a function of temperature and pressure. For an isothermal isobaric flash operation, they are, however, a constant. By manipulating the above system of equations, it can be shown that the composition of A in the stream leaving the system is given by
yA ) [KAKB(1 - xC0/Kc) - {KA(xA0 + xB0)}]/(KB - KA) (8) For a given inlet composition x0, this is hence a constant. We obtain similar expressions for other compositions yi as well. We conclude that the mole fractions of the stream exiting the system are uniquely determined by the mole fractions of the inlet stream. Here too, the steady state is governed by
Da ) (xA0 - yA)/(zAzB2)
(9)
Here again only two steady states can be obtained above a critical Da, while there is no solution below the critical Da. The bifurcation diagram is depicted in Figure 3. The chosen parameter values are KA ) 0.02, KB ) 2.5, KC ) 1.2, xA0 ) 0.1, xB0 ) 0.5, and xC0 ) 0.4. Here, the critical Da is 2.42. The system has two steady states for high Da and no steady state for low Da. This is a generic behavior which we have observed for other initial
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(iv) The pressure drop across the membrane is maintained constant. (v) Osmotic pressure varies linearly with concentration. (vi) We neglect the effect of concentration polarization and assume the molar density is a constant throughout the system. (vii) The reaction occurs only in the reactor, not in the pipelines or in the separator unit. (viii) No fouling is assumed to occur on the membrane. The equations governing the coupled reactor-nanofiltration system are as follows:7,10 (1) Across the Reactor. (a) Overall Volumetric Balance. Since the reactor is operated at a constant holdup and the liquid is assumed to be incompressible, we obtain
q0 + qp ) qf Figure 3. Bifurcation diagram for a coupled reactor-flash system, ternary case: solid line, stable state; dashed line, unstable state.
compositions and parameters. We conclude that even for multicomponent systems the coupled reactor flash system has a maximum of two steady states when the fresh feed is flow controlled and the flash is operated at a constant temperature and pressure.
(10a)
(b) The mass balance for species A yields
q0CA0 + qpCAp - qfCAf - VRkCAfCBf 2 ) 0
(10b)
Here we have written the expression in terms of volumetric flow rates and concentrations. (c) Stoichiometric Relationship
CAf + CBf ) C0
(10c)
III. Reactor-Nanofiltration System The flash unit above was modeled assuming that its effluent streams are in equilibrium. In contrast to this, the membrane separators are modeled as rate controlled processes.6 We now discuss a coupled reactor membrane separation system. The membrane separator is assumed to work as a nanofiltration unit. A typical coupled reactor-membrane separator system is depicted schematically in Figure 4. Here q0, qf, qp, and qr represent the volumetric flow rate of fresh feed, reactor effluent, permeate, and retentate streams, respectively. Component A, the reactant, has been treated as a solvent. We assume the permeate stream is rich in the reactant A and is recycled. Here Ci0, Cif, Cip, and Cir represent the concentration of species “i” in the fresh feed, reactor, permeate, and retentate streams, respectively. C0 is the total concentration of A and B in the fresh feed (i.e. C0 ) CA0 + CB0). In the case of membrane separators, the modeling equations are in terms of fluxes. These are determined by the concentration of solutes. Thus, it becomes more useful to work in terms of concentrations here. This is in contrast to the flash separation unit, where the equilibrium relationships are in terms of mole fractions, which is the reason we have used mole fractions to describe compositions there. (A) Binary System. In this section, we restrict the analysis to the case where all streams are, at best, binary mixtures of components A and B. We treat the reactant A as a solvent and the product B as a solute. The permeate is rich in A and is recycled. For the coupled reactor-nanofiltration unit, we propose a model based on the following assumptions:7,9 (i) Nanofiltration occurs by the solution-diffusion mechanism. (ii) We assume that the membrane separator is a flat-sheet module. (iii) The solute and solvent permeability are constant.
This follows from the fact that the overall reaction can be viewed as one mole of A gives one mole of B. (2) Across the Nanofiltration Unit. (a) The mass balance for species A yields
qfCAf ) qrCAr + qpCAp
(11a)
(b) The stoichiometric relationship is obtained by writing (10b) and (11a) for species B and adding the balances for the respective components in the respective units:
CAr + CBr ) CAp + CBp ) C0
(11b)
(c) The solvent flux Jw in terms of the pressure drop ∆P and the permeability constant A1:
Jw ) A1[∆P - R{CBr - CBp}]
(11c)
(d) The solute flux JSB in terms of the permeability constant BB of the solute is
JSB ) BB(CBr - CBp) ) Js
(11d)
(e) The concentration of the solute B in the permeate is obtained using
CBP ) {JsB/(Jw + Js)}Fpermeate
(11e)
(f) and the flow rate of the permeate is obtained by
qp ) {(Jw + Js)/Fpermeate}sm
(11f)
where sm is the membrane area. (3) Overall Balance. Assuming that the holdups for the two units are constant, we obtain the relationship
qr ) q0
(12)
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Figure 4. Schematic of a coupled reactor-membrane separation system.
Figure 5. Dependence of inflow of A and removal of A on CAf*.
The above set of equations determines the steady state of a coupled reactor-nanofiltration unit. The steady state of the coupled reactor-nanofiltration unit, in terms of the dimensionless concentrations, is given by
Da ) (CA0* - CAr*)/(CAf*(C0* - CAf*)2)
(13)
where Da is the dimensionless Damkohler number defined here as VRkC02/q0. Here, an asterisk is used to denote dimensionless variables. Concentrations and flow rates have been made nondimensional by using C0 and q0 as scale factors. For this binary reactor-nanofiltration system, we found that CAr* is a function of CAf* and qf, and is given by
CAr* ) (q0 - qf)/(RqfC0){∆P - (qf - q0)/(A1Sm)} + CAf* (14) In Figure 5 we have plotted (CA0* - CAr*) (i.e. the net inflow of A for the coupled system) against CAf* by a solid line. The removal term of A due to the reaction, that is, Da(CAf*(C0* - CAf*)2) is the same as that for the stand-alone reactor system, and is depicted in
Figure 6. Bifurcation diagrams for a coupled reactor-nanofiltration system, binary case: solid line, stable state; dashed line, unstable state.
Figure 5 by a dotted line. The two curves intersect at three points for Da ) 6.5. This guarantees the existence of three steady states for the system. Hence, this coupled system exhibits qualitatively the same behavior as the stand-alone reactor. Figure 6 shows simulation results for different values of CA0*. The bifurcation diagrams show three steady states for a range of Da for high CA0* ()0.82) (Figure 6a), and a unique steady state for all Da for low CA0*
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()0.485) (Figure 6b). Here, C0 is taken as 330 [mol/m3], and the other parameters chosen for the simulation are A1 ) 1 × 10-8 [mol solvent/(m2‚Pa‚s)], BB ) 1 × 10-8 [m/s], R ) 8.0 × 103 [Pa‚m3/mol], q0 ) 5 × 10-3 [m3/s], ∆P ) (27 ( 1.013) × 105 [Pa], and sm ) 180 [m2]. From this we conclude that if we want to avoid static instabilities due to multiple solutions, it is necessary to decrease the concentration of A in the feed stream. (B) Ternary Case. We now consider the situation where the reaction takes place in a solvent C which is considered inert. The components A and B are now both treated as solutes. Here, in addition to the earlier equations (eqs 10ab, 11a, 11c-f, 12), we include the species balance of B across the reactor:
q0CB0 + qpCBp - qfCBf + VRkCAfCBf 2 ) 0
(15)
Across the nanofiltration unit, we include the species B balance:
qfCBf ) qrCBr + qpCBp
(16a)
The solvent and solute flux equations are modified as
Jw ) A1[∆P - R{(CAr + CBr) - (CAp + CBp)}]
(16b)
JsA ) BA(CAr - CAp)
(16c)
Js ) JsA + JsB
(16d)
CAp ) {JsA/(Jw + Js)}Fpermeate
(16e)
The steady state of the system is obtained by solving the above set of equations. Simulating these, we determine the dimensionless concentration of A in the reactor CAf* for different values of Da. This is plotted in Figure 7 as bifurcation diagrams. Here too, the bifurcation diagrams show three steady states for a range of Da for a high value of CA0* ()0.97) (Figure 7a), while a unique steady state exists for all Da for low CA0* ()0.8) (Figure 7b). Here, C0 is taken as 30 [mol/m3], and the other parameter values used for simulating the system are BA ) 1.82 × 10-8 [m/s], BB ) 1.82 × 10-9 [m/s], R ) 8.0 × 103 [Pa‚m3/mol], q0 ) 1 × 10-4 [m3/s], ∆P ) (38 ( 1.013) × 105 [Pa], sm ) 180 [m2], and A1 ) 3 × 10-8 [mol solvent/(m2‚Pa‚s)]. As can be observed from the equations, here also, the output compositions CAr and CBr are not fixed, and hence, results similar to those for the binary system are expected. We observe that, for the reactor-flash system, we always get two steady states for high Da. There is a minimum value of Da below which the system has no steady states. In contrast to this, in a reactor-nanofiltration system, we see that the constraint of the minimum Damkohler number needed to obtain a steady state vanishes. Further, we obtain three steady states for high values of CA0* for a range of Da, while we have a unique steady state for all Da for low values of CA0*. This behavior is analogous to that of the stand-alone reactor. In the reactor-flash system, as explained earlier, the composition leaving the system is uniquely fixed for a given feed composition. This implies that the conversion of the system as a whole is fixed. For sufficiently low flow rates, high Da, the recycle flow rate can adjust itself
Figure 7. Bifurcation diagrams for a coupled reactor-nanofiltration system, ternary case: solid line, stable state; dashed line, unstable state.
to satisfy this constraint. If the flow rate is sufficiently high, this constraint of constant conversion cannot be satisfied. In contrast, in the reactor-nanofiltration system, the composition leaving the system is not fixed. This is dependent on the feed composition to the nanofiltration unit and the flow rate qf. Here, the composition is defined in terms of the component flux. In a reactor-nanofiltration unit, the output from the nanofiltration adjusts itself to satisfy the material balance for different compositions and flow rates of reactor effluent even when the reactor volumes are low. Here, there is no critical value of Da below which no solutions exist. IV. Reactor-Ultrafiltration System We now discuss the performance of one more membrane separation technique: ultrafiltration. In the analysis of this case, we have assumed the formation of gel on the retentate side. This is commonly encountered in ultrafiltration.6 (A) Binary Case. Here, we assume A to be the solvent and the product B to be a molecule causing gelation. Also, the rejection ratio of solute B is assumed to be a constant, RB. With retentate going out of the system and permeate being recycled, the equations governing the system are similar to those for the binary reactor-nanofiltration system, except that now eq 11d is not needed. The
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Figure 8. Bifurcation diagrams for a coupled reactor-ultrafiltration system, binary case: solid line, stable state; dashed line, unstable state.
equations of solvent flux and solute concentration, that is, eq 11c and 11e, are now replaced by6
Jw ) Kw ln(Cg/CBr)
(17a)
CBp ) Cg(1 - RB)
(17b)
where Cg ) gel concentration of B [mol/m3] and RB ) rejection ratio of B. From these equations, it can be easily shown that CBr is a nonlinear function of CBf:
CBf ) [q0CBr + KwSmCg(1 - RB) ln(Cg/CBr)]/[q0 + KwSm ln(Cg/CBr)] (17c) In this case, the dependence of CBr on CBf is highly nonlinear. The steady state dependence of CAf* on Da for this case is defined as for the nanofiltration, that is, eq 13. Here again, the output composition leaving the system is not constant. Hence, the bifurcation diagrams of CAf* versus Da are qualitatively similar to that of the coupled reactor-nanofiltration system. Figure 8 shows the bifurcation diagrams for this case. In Figure 8a, the following parameter values were used: RB ) 0.97, kw ) 1 × 10-6 [m3/m2‚s], q0 ) 1 × 10-4 [m3/s], cg ) 31 [mol/m3], sm ) 80 [m2], CA0 ) 28, and CB0 ) 2 [mol/m3]. For Figure 8b, the inlet feed composition was changed to CA0 ) 20 and CB0 ) 10 [mol/m3].
Figure 9. Bifurcation diagrams for a coupled reactor-ultrafiltration system, ternary case: solid line, stable state; dashed line, unstable state.
We see that for low CA0 the system has a unique steady state for all Da while for high CA0 there is a range of Da where the system has three steady states. (B) Ternary Case. We now extend the above analysis, when the separator is an ultrafiltration unit, to a ternary system. Here, we assume the third component C to be a solvent. Components A and B are taken as solutes. Gelation is assumed to be caused only due to B. The equations governing this case are similar to those that describe the binary system. We need only one more equation which redefines the concentration of solute A in terms of RA, the rejection ratio of solute A:6
CAp ) MACAr(1 - RA)
(18a)
Figure 9 shows the bifurcation diagrams for this case. This again shows a region of three steady states for some Da for high CA0. For Figure 9a, the parameters chosen were RB ) 0.97, RA ) 0.9, MA ) 25, kw ) 1 × 10-5 [m3/m2‚s], q0 ) 1 × 10-4 [m3/s], cg ) 31 [mol/m3], sm ) 80 [m2], CA0 ) 25, and CB0 ) 5. For Figure 9b, the inlet feed concentrations were changed to CA0 ) 17 and CB0 ) 13. Since the concentration of A is low, the system has a unique solution for all Da. V. Conclusions In this work, we have analyzed three different reactor separator systems. All systems were analyzed for the case when the fresh feed flow rate F0 is flow controlled.
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The reaction was considered to be autocatalytic. However, the approach used in this work is applicable to all reaction systems. When we analyze membrane separation systems and the feed is a binary mixture, we treat A as a solvent and B as a solute. For ternary feed mixtures, we treat A and B as solutes and C as a solvent. This has been done only to give some physical understanding of the process under investigation. In our analysis, we do not make any assumption of perfect separation and, hence, all components are present in all streams. Their concentrations in the different streams are, however, different. In a typical distillation column, compositions of streams leaving the separator are controlled using local composition controllers. This mode of operation constrains the conversion of the coupled system. The residence time in the reactor must be sufficiently high to satisfy this constraint. Here, when the fresh feed rate is sufficiently low, the recycle flow rate attains a value such that we get the desired conversion. When the fresh feed flow rate is sufficiently high, the recycle flow rate serves to reduce the residence time even further. Since recycle flow rates cannot become negative, the system cannot satisfy the constraint of fixed conversion for low Da. Hence, we have no steady states for low Da. When a membrane unit is used as a separator, the compositions leaving the separator are not controlled. The separation is defined in terms of fluxes. These are in turn determined by compositions. Here, the compositions and the flow rates adjust themselves so that a feasible steady state is obtained. Consequently, the behavior of the stand-alone reactor is imprinted on the behavior of the coupled system. The coupled reactor-flash ternary system exhibits a maximum of two solutions, when the third component is an inert solvent. We have also investigated the autocatalytic reaction system
2A + 2B f 3B + C This is a ternary system where component C is formed as a product due to the above reaction. The kinetics is assumed to follow rB ) kCACB2, as before. Here again, the qualitative behavior of the coupled system is such that it has two steady states for high Da and no steady state for low Da. This is the same as the case when C is an inert solvent. The qualitative bifurcation behavior of the ternary coupled reactor system appears to be similar irrespective of whether C is an inert solvent or it takes part in the reaction. In general, we conclude that when the separator operates at equilibrium or when there are composition controllers in the separator, the behavior of the coupled system is markedly different from that of the standalone reactor. We hypothesize that when the separator is such that its performance is determined by fluxes, the behaviors of the coupled system and the stand-alone reactor are qualitatively similar. VI. Notations A1 ) solvent permeability constant (mol solvent/(m2‚Pa‚ s)) BA, BB ) permeability constants of solutes A, B, respectively (m/s) CA0, CB0 ) concentrations of A and B in the fresh feed, respectively (mol/m3)
CAp, CBp ) concentrations of A, B in the permeate (mol/ m3) CAf, CBf ) concentrations of A, B in the reactor (mol/m3) CAr, CBr ) concentrations of A, B in the retentate (mol/m3) C0 ) constant molar density (mol/m3) Cg ) gel concentration of solute B (mol/m3) F0 ) molar flow rate of feed (mol/s) F ) molar flow rate of reactor effluent (mol/s) Jw ) molar flux of solvent (mol/m2‚s) JsA, JsB ) molar flux of solutes A and B, respectively (mol/ m2‚s) Js ) molar flux of all solutes (mol/m2‚s) KA, KB, KC ) equilibrium ratios for A, B, C k ) reaction rate constant (mol-2 m6 s-1) Kw ) solvent permeability constant for ultrafiltration (mol/ m2‚s) L ) molar flow rate of liquid (residue) from flash (mol/s) MA ) constant for solute A to account for concentration polarization ∆P ) pressure difference across membrane (Pa) q0 ) volumetric flow rate of feed (m3/s) qp ) volumetric flow rate of permeate (m3/s) qf ) volumetric flow rate of reactor effluent (m3/s) qr ) volumetric flow rate of retentate (output of coupled system) (m3/s) sm ) membrane area (m2) V ) molar flow rate of vapor (distillate) from flash (mol/s) VR ) reactor volume (m3) xA0, xB0, xC0 ) mole fractions of A, B, C in fresh feed for reactor-flash system xA, xB, xC ) mole fractions of A, B, C in residue from flash for reactor-flash system yA, yB, yC ) mole fractions of A, B, C in distillate from flash for reactor-flash system zA, zB, zC ) mole fractions of A, B, C in reactor for reactorflash system R ) proportionality constant for osmotic pressure (Pa‚m3/ mol) Fpermeate ) molar density of permeate {) (CA0 + CB0) for binary case and ) [CA0 + CB0 + (1 × 106)]/18 for ternary case} Subscripts A, B, C, i ) components f ) feed to separator m ) membrane p ) permeate r ) retentate R ) reactor w ) solvent 0 ) fresh feed Superscript * ) nondimensionalized quantities
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Received for review April 17, 2004 Revised manuscript received January 5, 2005 Accepted January 19, 2005 IE0496840
