A Comparison of Control Strategies for a Nonlinear Reactor

Ind. Eng. Chem. Res. 2002, 41, 2005-2012

2005

PROCESS DESIGN AND CONTROL A Comparison of Control Strategies for a Nonlinear Reactor-Separator Network Sustaining an Autocatalytic Isothermal Reaction Ajit Arjun Sagale and S. Pushpavanam* Department of Chemical Engineering, Indian Institute of Technology, Madras, 600036 India

In this work, we compare the behavior of a stand-alone reactor with that of a coupled nonlinear reactor-separator system. The coupling between the two units arises because of the recycle of the reactant-rich stream from the downstream separator. The reaction considered is an elementary autocatalytic reaction of the form A + 2B f 3B. The reactor is assumed to be isothermal. Three different modes of operation of the coupled system corresponding to three different control strategies are investigated. The nonlinear system behavior is analyzed for these cases using singularity theory and the D-partition method. We obtain the preferred control strategy as that in which the reactor effluent flow rate and the fresh feed flow rate are flowcontrolled and the reactor holdup is allowed to vary. Introduction The steady-state and dynamic behaviors of nonlinear processes occurring in individual units have been studied extensively. For example, the steady-state and dynamic behaviors of a CSTR sustaining an exothermic reaction were classified using bifurcation theory by Uppal et al.1 They determined the conditions under which the system exhibits multiple steady states and sustained periodic solutions. Doedal et al.,2 Jorgenson and Aris,3 and Jorgenson et al.4 have studied periodic solution branches and oscillatory dynamics of a stirredtank reactor sustaining the series reaction A f B f C. Separation units can also exhibit similar characteristics. Jacobsen and Skogestad5 reported two different types of multiplicity in binary distillation columns with ideal vapor-liquid equilibrium. (a) For constant molar overflow (CMO), multiplicities can occur as a result of the nonlinear relationship between the mass (or volumetric) and molar flow rates. (b) Multiplicities can also be caused by the energy balance when the separator is nonisothermal. Guttinger et al.6 experimentally found multiple steady states in the homogeneous azeotropic distillation of the methanol-methyl butyrate-toluene system, and Muller and Marquardt7 experimentally verified multiple steady states in the heterogeneous azeotropic distillation of the ethanol-water-cyclohexane system. Lee et al.8 found limit cycles in the homogeneous azeotropic distillation of the methanol-methyl butyrate-toluene system using a CMO model. The sustained oscillations observed in that case result in a periodic oscillation of the profiles inside the column. A chemical process plant typically consists of reactors and separators. The fresh feed stream transformed * To whom correspondence should be addressed. E-mail: [email protected]. Fax: +91-44-2350509. Tel.: +91-444458218.

partially in the reactor is fed to the separator, where it is split into two streams of different compositions. The reactant-rich stream is recycled back to the reactor to render the process economical. The recycle stream couples the two units such that the downstream separator affects the performance of the upstream reactor. In this paper, we discuss how this coupling renders the behavior of the coupled system completely different from that of the individual units. In this connection, Morud and Skogestad9 investigated how the presence of a material or energy recycle can be viewed as a positive feedback effect. This feedback increases the system time constant. It increases the system sensitivity to slow disturbances and can sometimes give rise to instabilities. Jacobsen10 has studied the effect of recycle on dynamic behavior and described how recycling induces nonminimum phase behavior. Luyben and Luyben11 and Luyben12 have shown that, for some control configurations, the recycle system can exhibit a snowball effect. They analyzed the sensitivity of the steady-state recycle flow rate to changes in the fresh feed composition and the fresh feed flow rate for two different control strategies. In the first, the fresh feed flow rate was flow-controlled, and the reactor holdup was maintained constant by manipulating the reactor effluent rate. In the second control strategy, the fresh feed flow rate and the effluent flow rate were flowcontrolled, and the reactor holdup was allowed to vary. They conclude that, with the second control strategy, the state variables at steady state are less sensitive to changes in parameters such as the fresh feed flow rate and composition. However, they restricted their analysis to an isothermal reactor sustaining a first-order reaction coupled to an isothermal, isobaric separator, which is a linear system. Wu and Yu13 proposed a balanced scheme to overcome this snowball effect. Their approach uses

10.1021/ie010143+ CCC: $22.00 © 2002 American Chemical Society Published on Web 03/23/2002

2006

Ind. Eng. Chem. Res., Vol. 41, No. 8, 2002

the principle of absorbing the load changes evenly over all process units. Pushpavanam and Kienle14 have analyzed the steadystate and dynamic behavior of a coupled nonlinear reactor-separator system. The reactor was assumed to sustain a first-order exothermic reaction. They studied the case in which the reactor holdup is maintained at a constant value using an ideal controller and considered two different choices of the flow control variables: (a) the fresh feed rate and (b) the liquid recycle rate. The different steady-state bifurcation diagrams for the two cases were obtained using singularity theory, and the stabilities of the states were obtained from bifurcation theory. When the fresh feed rate was chosen as the independent control parameter, they found that the system can exhibit a maximum of two steady states. In particular, there were operating conditions for which no steady states existed. The operating conditions over which a feasible stable steady state with separation occurred was very narrow. In contrast, the system was more stable when the liquid recycle rate was flowcontrolled. In that case, the system can have a maximum of three steady states. They concluded that it is preferable to operate the coupled system with flow control of the liquid recycle flow rate. This approach cannot reveal any information about the stability of an operating point on the bifurcation diagram. In this work, we make a comparative study of three different control strategies for operating a coupled nonlinear reactor-separator. It will be assumed that the reactor sustains an elementary cubic isothermal autocatalytic reaction of the form A + 2B f 3B. The separator, modeled as a flash, is assumed to be isothermal and isobaric. The coupled reactor-separator system is analyzed using classical bifurcation theory. The different shapes of the bifurcation diagrams are classified elegantly as they are governed by polynomial nonlinearities. The D-partition method (Porter15) provides information regarding the stability of a particular operating point on the bifurcation diagrams in the auxiliary parameter space. This is independent of the nature of the bifurcation diagram determined by bifurcation theory. These two methods are superposed, and the information generated by them is discussed. The following three modes of operation are studied: (i) Fixed F0 and MR. In this mode of operation, the fresh feed flow rate to the system F0 is flow-controlled, and the reactor effluent flow rate F is used as a manipulated variable to control the reactor holdup MR (Figure 1a). (ii) Fixed F and MR. In this mode of operation, the reactor effluent flow rate F is flow-controlled using a flow controller, and the reactor holdup or level is controlled using F0 as the manipulated variable (Figure 1b). (iii) Fixed F and F0. Here, the fresh feed flow rate F0 and the reactor effluent flow rate F are both flowcontrolled (Figure 1c). This fixes the recycle flow rate L at steady state. The molar holdup of the reactor in this case is determined by the interaction between the reactor and the separator. In this paper, we discuss the nonlinear behavior of the coupled reactor-separator system for the three control strategies mentioned above. The rate of generation of A is assumed to be of the form rA ) -kcAcB2. The flash is assumed to operate at a constant temperature

Figure 1. Reactor-separator network and the three different modes of operation.

and pressure. Because the feed to the flash is a binary mixture, this fixes the mole fraction of the bottoms (xe) and the distillate (ye) leaving the flash, uniquely. It is well-known that a stand-alone CSTR sustaining this reaction can exhibit multiple steady states.16 It cannot sustain any limit-cycle oscillations. Here, we discuss how the behavior of the stand-alone reactor becomes modified when it is coupled to a downstream separator. In particular, we determine which of the modes of operation of the coupled system increase the region of instabilities and which make the system stable. Modeling Equations We now discuss the equations that govern the behavior of the reactor-separator network. The evolution of the molar holdup MR and of z, the mole fraction of A in the reactor, is governed by the following ordinary differential equations

overall mole balance dMR ) F0 + L - F dt

(1)

component balance of reactant A d(MRz) ) F0xaf + Lxe - Fz - (-rA)VR dt

(2)

Here, MR and VR represent the molar holdup and volume of the reactor, respectively. F0, L, F, and V are the fresh feed flow rate, the recycle rate from flash to the reactor, the effluent flow rate from the reactor, and the vapor flow rate from the flash, respectively, all in

Ind. Eng. Chem. Res., Vol. 41, No. 8, 2002 2007

moles per hour. xaf, xe, ye, and z represent the mole fractions of component A in the fresh feed, in the recycle stream (bottom from the flash), in the distillate stream from the flash, and in the reactor, respectively (Figure 1). rA is the rate of generation of component A. We neglect the holdup of the vapor and liquid phases in the flash. We assume that the flash is isothermal and isobaric. This yields, for the overall mass balance and the component balance, the algebraic equations

F)V+L

(3a)

Fz ) Vye + Lxe

(3b)

Because the flash is at a constant temperature and pressure and because its feed is a binary mixture, the composition xe, ye is fixed uniquely. The heat exchanger HX1 is used to change the temperature of the reactor effluent stream from Tf to Tflash. Similarly, HX2 is used to change the recycle stream temperature from Tflash to Tf. These heat exchangers decouple the two units energetically. Hence, the two units are coupled only through mass. We first discuss the behavior of the stand-alone reactor. Stand-Alone Reactor The mass balance for the reactor is

d(MRz) ) F0xaf - Fz - VRFm3kz(1 - z)2 dt

(4)

The molar flow rate into the reactor F0 must equal the molar flow rate out of the reactor F if the molar holdup of the reactor is constant. This is necessary to ensure steady-state operation. We assume that the molar density is constant and independent of composition. Now, the inlet and effluent volumetric flow rates across the reactor are also equal, which results in the reactor having a constant volume. Noting that MR ) VRFm, we make the above set of equations dimensionless to yield

dz ) xaf - z - Daz(1 - z)2 dτ

(5)

Here, the dimensionless variables are defined as

tF0 VRFm3k τ) Da ) F0 MR The steady-state mole fraction of A in the reactor is governed by

F(z,Da,xaf) ) xaf - z - Daz(1 - z)2

(6)

Steady-State Behavior. Case of Pure Feed. We discuss the case where the feed has pure A, i.e., no autocatalytic component B or inert component. This means that the mole fraction of A in the feed is xaf ) 1. The system admits z ) 1 as a solution for all values of the bifurcation parameter Da. This trivial state is a wash-out state because the B present initially in the reactor is completely washed out. The system also has two feasible nontrivial solutions for Da > 4. The dependency of the steady-state mole fraction of A in the reactor on Da is depicted in Figure 2. The unstable branch shown as a dashed line meets the trivial solution z ) 1 at Da ) ∞.

Figure 2. Bifurcation diagram depicting dependency of z on xaf for stand-alone reactor with pure feed. xaf ) 1.

Feed Containing the Autocatalytic Component B. For this case, xaf < 1, so the trivial branch z ) 1 is no longer an admissible solution. The equation determining the steady states of the system (eq 6) is a cubic equation. It can admit up to a maximum of three solutions for some values of bifurcation parameter. The system can hence exhibit two kinds of bifurcation diagrams, one when the system has only a unique solution for all values of the bifurcation parameter and the other when there is an interval of bifurcation parameter values where the system has up to three steady states. The parameter Da is identified as the bifurcation parameter. This choice is motivated by the fact that Da can be experimentally varied by controlling the flow rate into the reactor. It is also the classical choice in the literature. This allows us to compare our results with those of earlier workers. The bifurcation diagram represents the dependency of the state variable z on the bifurcation parameter Da. The nature of the bifurcation diagram is determined by the auxiliary parameter xaf. The critical value of xaf beyond which the system behavior becomes different is given by the hysteresis variety. This is defined as the solution to the equation F ) ∂F/∂z ) ∂2F/∂z2 ) 0. This yields as its solution z ) 2/3, Da ) 3, and xaf ) 8/9. For xaf < 8/9, the bifurcation diagram admits a unique solution for all Da. For xaf > 8/9, the bifurcation diagram of z vs Da has two turning points. The system has three solutions for Da values between these turning points and a unique solution outside this interval. This is depicted in Figure 3a,b. The two different bifurcation diagrams illustrate the qualitative features of the stand-alone reactor. However, they do not reveal any information about the stability of a particular operating point on the bifurcation diagram, unless it is explicitly estimated. We now use the D-partition method to determine the conditions for which a particular operating point on the bifurcation diagram is stable. When the operating point characterized by z is chosen such that z < 1/3, we have ∂F/∂z < 0. Hence, this operating point is stable for all xaf and Da. Again, for z > 1/3, ∂F/∂z < 0 when xaf < 2z2/(3z - 1). Hence, for a suitably chosen value of xaf satisfying this condition, Da can be chosen to satisfy the steady-state equation. This will ensure that the chosen operating point is stable. The stand-alone reactor system cannot exhibit any dynamic instability, i.e., sustained oscillations, because it is one-dimensional. We mention here that similar results can be obtained if xaf is chosen as the bifurcation parameter and Da as the auxiliary parameter. Reactor-Separator Network We discuss the behavior of the coupled system for the three different control strategies shown in Figure 1.

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Figure 4. Dependence of z(1 - z)2 on z.

Figure 3. Bifurcation diagram depicting dependency of z on Da in stand-alone reactor. (a) xaf ) 0.8, (b) xaf ) 0.9.

Fixed F0, MR Case. In this case, the fresh feed flow rate F0 and the molar holdup of the reactor MR are held constant. For the case of constant F0 and MR, the equations that govern the reactor are

F ) F0 + L

(8)

d(MRz) ) F0xaf + Lxe - Fz - VRFm3kz(1 - z)2 (9) dt

(Figure 4). The system has no solutions that are feasible and steady for R > 4/27. The bifurcation diagram showing the bifurcation behavior, i.e., dependence of z on Da, is presented in Figure 5. Because only the single parameter R characterizes the bifurcation diagram, the operating point is stable if z < 1/3. Hence, the operating points for z > 1/3 will always be unstable for the coupled system in this mode of operation. Fixed F, MR Case. In this mode of operation, the dimensionless differential equation governing the system is

dz (xaf - ye)(xe - z) - Daz(1 - z)2 ) dτ (xe - ye)

This equation arises when we eliminate the state variables F0, L, and V from eqs 8-11 by substituting for them in terms of xe, ye, z, and F. Here, Da is defined using F as the characteristic flow rate

The flash balance equations yield

z - ye L ) F0 xe - z

(10)

z - ye L ) F xe - y e

(11) F)

(12)

where the parameter Da is the same as was defined for the case of the stand-alone reactor. The steady-state reactor conversion in this case is governed by

F(z,xaf,xe,ye,Da) ) xaf - ye - Daz(1 - z)2 ) 0

VRFm3k tF and τ ) Da ) F MR The steady-state equation governing the variation of z is

The system of differential and algebraic equations (eq 9 and eqs 8, 10, and 11, respectively) is converted into a single ordinary differential equation in z. This is done by eliminating the state variables L and F in eq 9 using xe and ye. The result is the single equation

dz ) xaf - ye - Daz(1 - z)2 dτ

(14)

(13)

We now define (xaf - ye)/Da ) R for algebraic convenience. The maximum value of z(1 - z)2 is 4/27. For R < 4/ , the system admits two solutions in the feasible 27 range (0 < z 1

(xaf - ye)(xe - z) (xe - ye)

- Daz(1 - z)2 ) 0

(15)

The bifurcation features of this system are the same as those of the stand-alone reactor. We can see this when we observe that the linear terms in the two eqs 6 and 15 differ by only a multiplicative constant and are identical when xe ) xaf. Hence, the behavior of this coupled system is similar to that of the stand-alone reactor when we replace xe by xaf. The hysteresis variety (HV) for this operation is given by xe ) 8/9. This is the solution to F ) ∂F/∂z ) ∂2F/∂z2 ) 0. For xe > 8/9, there is a range of bifurcation parameter (Da) values where the system has three solutions. For xe < 8/9, the system has a unique solution for all Da. This variety is independent of ye and is shown as a dashed-dotted line in Figure 6. New features in the bifurcation diagram arise from the constraints imposed by the flash on z, namely, ye < z < xe. Because the limit point, i.e., the turning point, can cross the z ) ye boundary, this gives rise to the boundary limit set


Figure 5. Bifurcation diagram depicting dependency of z on Da for fixed F0 and MR mode of operation in reactor-separator. xaf ) 1, ye ) 0.3.

Figure 7. Different bifurcation diagrams for fixed F and MR mode of operation.

Figure 6. HV, BLS, and SL for autocatalytic reaction system in fixed F and MR mode of operation.

(BLS), which is defined as

F)

∂F ) 0 at z ) ye ∂z

This BLS is given by

xe )

2ye2 (3ye - 1)

(16)

For values ye < 0.5, xe is either negative (for ye < 1/3) or greater than unity (for 1/3 < ye < 0.5). Hence, the BLS is infeasible for 0 < ye < 0.5. It is valid only for 0.5 < ye < 1 and is shown in Figure 6. This is tangential to the hysteresis variety, as shown in Figure 6, as the minimum value of xe given by eq 16 is 8/9. The D-partition method can be applied for this case to determine the conditions under which the operating point is stable. Because our system is one-dimensional, the condition for stability is ∂F/∂z < 0 for fixed z. This yields a critical value for xe

xcr e )

2z2 (3z - 1)

(17)

This is shown in xe-ye plane in Figure 6 by the dark dashed line for z ) 0.8. It is called the singular locus (SL).

These critical surfaces (HV and BLS) divide the xeye plane into four regions (Figure 6), where the reactor effluent stream is separated into two streams by the separator, i.e., ye < z < xe. The SL is depicted for z ) 0.8 (Figure 6). Because the BLS and SL are given by similar expressions (eqs 16 and 17, respectively), they intersect. The SL extends only up to ye ) z. The operating point is in the infeasible region to the right of the vertical line through ye ) 0.8, because, here, ye > z. The SL divides regions 2 and 3 into subregions, where the location of the operating point relative to the bifurcation point is different. In region 1, the system has a unique stable solution in the feasible region ye < z < xe (Figure 7). In region 2, the system has three solutions in the feasible range of (ye, xe) for some Da. In region 2a (2b), the operating point is stable (unstable). In region 3, the lower turning point crosses the z ) ye boundary. Hence, in the feasible region, there can be a maximum of two solutions, one statically stable and the other unstable. In region 3a (3b), the operating point is in the stable (unstable) region. In region 4, both turning points occur below the feasibility boundary. Hence, the system has only one solution in the feasible region (Figure 7). To the right of the vertical line ye ) 0.8, the bifurcation diagrams exist. No qualitative change in the bifurcation diagram occurs as we cross this line. Here, however, the operating point z ) 0.8 is infeasible, as ye > 0.8. Fixed F0, F Case with Variable Reactor Holdup. We now discuss the behavior of the autocatalytic reaction in the coupled system when the reactor holdup is not constrained to be constant. The overall balance and the component balance for reactant species A are

dMR ) F0 + L - F dt

(18)

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d(MRz) ) F0xaf + Lxe - Fz - MRk0z(1 - z)2Fm2 dt

(19)

respectively. The equations describing the flash (eqs 10 and 11) are valid here as well. The differential algebraic equation model can be simplified in dimensionless form to give the following system of two coupled ordinary differential equations

(z - xe) dM* r )1+ dτ (xe - ye)

(20)

dz xaf - z (xe - z)(z - ye) ) + r - Daz(1 - z)2 dτ M* M*(xe - ye)

(21)

with

τ)

tFo MchFm2k0 and Da ) Mch F0

where Mch is a characteristic holdup used to scale the reactor holdup. Here, the parameter r is F/F0, and the dimensionless holdup M* is M/Mch. These equations can be solved explicitly for the dimensionless state variables M* and z at steady state to give

z ) xe M* )

[

Da xe -

xe - ye r

(xaf - ye)

]{ [

(22)

]}

(xe - ye) (xe - ye) 1 - xe r r

2

(23)

For this case, we choose r to be the bifurcation parameter because it can be experimentally controlled. The parameter Da does not have the significance of a dimensionless residence time because the holdup of the reactor varies. The stability of the system to infinitesimal disturbances is governed by the eigenvalues of the Jacobian matrix. We can establish that the determinant of this two-dimensional Jacobian matrix is always positive. This implies that this two-dimensional system cannot exhibit any static instability, i.e., the steady states of the system cannot be saddle points. This conclusion is consistent with the fact that the system has a unique solution for all values of the bifurcation parameter. This two-dimensional system can be dynamically unstable, i.e., the state can be an unstable focus, when the trace of the Jacobian matrix J becomes positive and the determinant is positive. Hence, the bifurcation diagrams can be qualitatively different because of dynamic instabilities. Double Hopf (DH) Locus The steady-state behavior of the system is such there is always a unique solution for all values of the bifurcation parameter. The bifurcation diagrams, however, can be dynamically different because the system is now a two-dimensional system. This admits the possibility of dynamic instabilities (when the system has a complex conjugate pair of eigenvalues in the right half plane). Hence, the bifurcation diagram can be such that it has a region of dynamic instability or it has no region of

Figure 8. (a) DH locus for autocatalytic reaction system in fixed F0 and F mode of operation in the xe-ye plane for Da ) 0.2. (b-d) Root loci and bifurcation diagrams in fixed F0 and F mode of operation across the DH locus shown in Figure 8a.

dynamic instability (eigenvalues are always in the left half plane). This is determined by the values of the parameters xe and ye for a fixed value of xaf. There is a locus in this parameter plane across which the bifurcation diagram becomes qualitatively different. For xeye parameter values along this locus, two Hopf bifurcation points on the bifurcation diagram coalesce for a particular value of r (the bifurcation parameter). Because the eigenvalues are purely imaginary at the Hopf bifurcation point, the condition for the double Hopf locus for a two-dimensional system is given by

F ) Re(λi) ) 0,

d [Re(λi)] ) 0 dr

(24)

such that Im(λi) * 0. For our two-dimensional system, this yields

F ) trace(J) ) 0,

d [trace(J)] ) 0 and dr det(J) * 0 (25)

where J is the Jacobian matrix governing the evolution of the linearized system. This locus is depicted in Figure 8a. The variation of the eigenvalues as we move along the bifurcation diagram for xe-ye values on the DH locus is schematically shown in Figure 8c. The corresponding bifurcation diagram is also depicted on the right side. Across this locus, the variation of eigenvalues and the corresponding bifurcation diagrams are shown schematically in Figure 8b and d. In Figure 8d, there is a


Figure 9. DH (dashed line) and NSL (solidline ) for fixed F0 and F mode of operation.

region of dynamic instability for a range of r. This does not exist in Figure 8b. D-Partition Method The DH surface obtained above divides the xe-ye space into different regions. In each region, the dynamic behavior along the bifurcation diagram is qualitatively different. However, this fact does not provide any information on the stability of a desired operating point on the bifurcation diagram. The engineer is interested in operating the reactor at a desired operating point, which is determined, for example, by specifying a state variable such as the reactor concentration or conversion. Typically, the engineer might want the concentration in the reactor to be maintained at a fixed level, e.g., at z ) 0.5. We now discuss how the D-partition method can be used to determine the stability of a given operating point of the system for this case. It can be used to isolate regions in the auxiliary parameter space in which an operating point is stable. The stability of the system is governed by the eigenvalues of the Jacobian matrix J, which are the roots of the quadratic equation

s2 - trace(J)s + det(J) ) 0

(26)

The coefficients of the characteristic equation depend on the operating point. The stability boundary of the system is the imaginary axis in the eigenvalue plane. This boundary is mapped onto the xe-ye parameter plane at the desired operating point. This mapping is performed by setting s ) iω in the equation, yielding

-ω2 - trace(J)iω + det(J) ) 0

(27)

Equating the real and imaginary parts of this equation to zero, we obtain

trace(J) ) 0 and det(J) ) ω2 We allow ω to vary from -∞ to ∞ and solve this system of equation for xe-ye for fixed z. This yields the nonsingular locus (NSL). The operating point becomes dynamically unstable as we cross this locus.15 The steady-state equations (eqs 22 and 23) are used to obtain the bifurcation parameter value, which allows us to operate the reactor at the desired operating point. Figure 9 depicts the NSL (solid line) and the DH locus in the xe-ye plane for the autocatalytic reaction system

Figure 10. Bifurcation diagrams for fixed F0 and F mode of operation in the three regions of Figure 9: (a) region 1, (b) region 2, (c) region 3.

when the reactor holdup is a state variable. The NSL is drawn for z ) 0.5. As we move across the NSL, the stability of the operating point on the bifurcation diagram changes. When we move across the DH locus, the nature of the bifurcation diagram changes. On one side of the DH locus, the bifurcation diagram has a solution that is dynamically stable for all values of the bifurcation parameter, and on the other side, there is a range of the bifurcation parameter values for which the bifurcation diagram has a solution that is dynamically unstable. These two loci divide the parameter space into three different regions. In region 1, the bifurcation diagram has a unique stable steady state for all values of the bifurcation parameter. In region 2, the bifurcation diagram has a region of dynamic instability, but the operating point is in the dynamically stable region. In region 3, the operating point is in the dynamically unstable region of the bifurcation diagram. The Hopf bifurcation points in the bifurcation diagram are indicated as ×’s and the operating point as an open circle in Figure 10. The region where dynamic instability occurs is shown as a dashed line. Conclusions In this work, we have studied the behavior of a coupled nonlinear reactor-separator system sustaining an elementary cubic autocatalytic reaction. It is seen

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that, when the fresh feed flow rate is flow-controlled and the molar holdup of the reactor is held constant, the system has a maximum of two solutions. For sufficiently low values of Da, the system has no steady-state solutions. When the reactor effluent flow rate is flow-controlled, the behavior of the reactor-separator network is the same as that of the stand-alone reactor. The constraints imposed by the separator modify the behavior of the coupled system from that of the stand-alone reactor by rendering some solution branches infeasible. When the reactor holdup is allowed to vary and the flow rates F and F0 are flow-controlled, the reactor exhibits a unique steady state. Consider the situation where a disturbance causes a decrease in z. This results in a decrease in L, the recycle stream flow rate, which causes the reactor holdup to decrease. Consequently, the residence time decreases, which results in a poorer conversion and an increase in z. Thus, in this mode, the variable holdup offsets the self-propagating effect of the autocatalytic reaction. This system can, however, be dynamically unstable. The region of dynamic instability can be avoided by choosing the parameter values xe and ye appropriately. Hence, this is the preferred mode of operation. Literature Cited (1) Uppal, A.; Ray, W. H.; Poore, A. B. On the dynamic behavior of continuous stirred tank reactors. Chem. Eng. Sci. 1974, 29, 967. (2) Doedal, E. J.; Heinemann, R. F. Numerical computation of periodic solution branches and oscillatory dynamics of stirred tank reactor with A f B f C reactions. Chem. Eng. Sci. 1983, 41, 1384. (3) Jorgenson, D. V.; Aris, R. On the dynamics of a stirred tank with consecutive reactions. Chem. Eng. Sci. 1983, 38, 45.

(4) Jorgenson, D. V.; Farr, W. W.; Aris, R. More on the dynamics of the stirred tank with consecutive reactions. Chem. Eng. Sci. 1984, 39, 1741. (5) Jacobsen, E.; Skogestad, S. Multiple steady state in ideal two-product distillation. AIChE J. 1991, 37, 499. (6) Guttinger, T. E.; Cornelius, D.; Morari, M. Experimental study of steady state in homogeneous azeotropic distillation. Ind. Eng. Chem. Res. 1997, 36, 794. (7) Muller D.; Marquardt, W. Experimental study of steady state in heterogeneous azeotropic distillation. Ind. Eng. Chem. Res. 1997, 36, 5410. (8) Lee, M.; Dorn, C.; Meski, G.; Morari, M. Limit cycles in homogeneous azeotropic distillation. Ind. Eng. Chem. Res. 1999, 38, 2021. (9) Morud, J.; Skogestad, S. Dynamic behavior of integrated plants. J. Process Control 1996, 6, 145. (10) Jacobsen, E. On the dynamics of integrated plantss Nonminimum phase behavior. J. Process Control 1999, 9, 439. (11) Luyben, M.; Luyben, W. Essentials of Process Control; McGraw-Hill: New York, 1997. (12) Luyben, W. L. Snowball effects in reactor-separator process with recycle. Ind. Eng. Chem. Res. 1994, 33, 299. (13) Wu, K.-L.; Yu, C.-C. Reactor/separator process with recycle. Candidate control structure for operability. Comput. Chem. Eng. 1996, 11, 1291. (14) Pushpavanam, S.; Kienle, A. Nonlinear behavior of an ideal reactor-separator network with mass recycle. Chem. Eng. Sci. 2001, 56, 2837-2849. (15) Porter, B. Stability Criteria for Linear Dynamical Systems; Oliver and Boyd: Edinburg, 1967. (16) Scott, S. K. Chemical Chaos; Clarendon Press: Oxford, U.K., 1991.

Received for review February 12, 2001 Revised manuscript received October 4, 2001 Accepted January 24, 2002 IE010143+

A Comparison of Control Strategies for a Nonlinear Reactor

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