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Isothermal Isobaric Reactive Flash Problem Gerardo Ruiz and Lakshmi N. Sridhar* Chemical Engineering Dept, UniVersity of Puerto Rico, Mayaguez, Puerto Rico 00681-9046
Raghunathan Rengaswamy Department of Chemical Engineering, Clarkson UniVersity, Potsdam, New York 13699-5705
In this article, we present some results for isothermal isobaric reactive separation process problems. We also demonstrate that, even in isothermal isobaric reactive separation processes, which are probably the least nonlinear of all reactive separation processes, we get nonlinear phenomenon such as Hopf bifurcations. While it has been shown that Hopf bifurcations are impossible in isothermal continuous stirred tank reactor (CSTR) problems involving the methyl tert-butyl ether (MTBE) and tert-amyl methyl ether (TAME) reactions, and also in nonreactive flash problems, we demonstrate in this paper that isothermal reactive flash processes involving both MTBE and TAME mixtures exhibit Hopf bifurcations. This shows that instabilities and oscillations can occur even in isothermal reactive separation systems and are not necessarily due to multiple stages. Additionally, we show that the Rachford-Rice procedure can be extended to reactive systems. Introduction During the past decade, there has been a tremendous interest in the field of reactive distillation. A review of the various models used in reactive distillation can be found in the paper by Taylor and Krishna.1 Of special interest is the existence of multiple steady states in these problems, since the combination of separation and reaction can, in principle, introduce the nonlinearity that can cause multiplicity. Multiple steady states in reactive distillations were demonstrated by several workers.2-16 The most commonly investigated situations include the methyl tert-butyl ether (MTBE) synthesis in the Jacobs-Krishna2 column configuration and the tert-amyl methyl ether (TAME) synthesis in the column of Mohl et al..11 The multiple steady states for these two columns were investigated by Chen et al.,16 who conclude that multiplicities are lost for high values of Da for TAME, while the opposite is found for MTBE. This conclusion, however, is specific to the column configurations descrbed in Jacobs-Krishna2 and Mohl et al.11 Rodriguez et al.17,18 discuss causes for the existence of multiple steady states in binary and ternary systems. The most important reactive separation process problems where multiplicity exists, such as MTBE and TAME processes, involve more than three components. To understand what causes multiplicity in these problems, one must look at the simplest reactive separation process problem involving the MTBE and TAME mixture, and hence, we are motivated to look at the isothermal reactive flash problem. Mohl et al.11 prove that isothermal continuous stirred tank reactor (CSTR) problems involving the MTBE and TAME reactions do not exhibit Hopf bifurcations, while on the other hand, for nonreactive isothermal flash processes involving homogeneous mixtures, Hopf bifurcations are impossible.19 However, we demonstrate that isothermal reactive flash processes involving both TAME and MTBE exhibit Hopf bifurcations, and that is one of the important contributions of this paper. This paper is organized in the following manner. First, a brief description of the isothermal reactive flash process is given along with the equations involved. We then demonstrate the existence of Hopf bifurcations in the isothermal reactive flash processes involving both the MTBE and TAME mixtures. Dynamic simulations are performed demonstrating the existence of limit
cycles that are a characteristic feature of problems with Hopf bifurcations, and the behaviors of these Hopf bifurcation points with temperature and pressure variations are presented. A modified Rachford-Rice procedure for solving the isothermal reactive separation process problem is then presented in the Appendix. Isothermal Reactive Separation Flash Problem For a single-stage reactive separation unit with a single reaction, we have c material-balance equations,
(Fzi - Lxi - Vyi) + Hνi ) 0
c eqs
(1)
where F is the external feed, L is the liquid flow, V is the vapor flow, H is the holdup, is the extent of reaction, and ν is the stoichiometric coefficient. We also have the phase-equilibrium equations,
yi ) Kixi
c eqs
(2)
where the K-value is
γiPsat i Ki ) P c
yi ) 1 ∑ i)1
(3)
c
xi ) 1 ∑ i)1
(4)
The total number of equations ) 2c + 2. The variables are xi, yi, L, and V. The total number of variables is (2c + 2). The specifications are T, P, and H. While this set of equations can be solved using a variety of techniques, a modified Rachford-Rice procedure for solving the isothermal isobaric reactive flash problem is presented in the Appendix. MTBE Process One of the most researched processes in reactive separation is the MTBE (methyl tert-butyl ether) process. The reaction
10.1021/ie060249a CCC: $33.50 © 2006 American Chemical Society Published on Web 08/17/2006
Ind. Eng. Chem. Res., Vol. 45, No. 19, 2006 6549
Figure 1. Continuation diagram.
Figure 2. Hopf point 1: Convergence to steady state.
Figure 3. Hopf point 1: periodic oscillation.
involved is
The Wilson binary interaction parameters and the Antoine coefficients were taken from Chen et al..20
i-butene + MeOH a MTBE
(5)
The inert compound present is n-butane. The rate model20 is
(
)
aMTBE r ) kf ai-buteneaMeOH Keq
TAME Process The TAME synthesis reaction can be written as16
(6)
2M1B + 2M2B + 2MeOH S 2(TAME)
(10)
The rate model also given by Chen et al.16 is
where
Keq ) 8.33 × 10-8(exp(6820/T))
(
(7)
) kf
and
kf ) 4464 exp(-3187/T)
(8)
The liquid-phase activity coefficients were obtained using the Wilson equation, and the Antoine equation has the form
ln Psat ) A + B/(T + C)
(9)
)
a2M1B 1 aTAME 2 aMeOH K1 a MeOH
(11)
where
( (
kf ) (1 + Λ)(1.9769 × 1010) exp -
10 764 T
K1 ) (1.057 × 10-4) e4273.5/T
))
(12) (13)
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Figure 4. Behavior of the Hopf bifurcation points at various temperatures.
and
Λ ) 0.648 e899.9/T
(14)
Solution Procedure Defining the Damkohler number as
Da )
H F(Kf,ref)
eq 1 can be rewritten as
(zi - θLxi - θVyi) + Da(kf,ref)νi ) 0
c eqs (15)
where θL ) L/F, θV ) V/F, and kf,ref is the forward rate constant evaluated at the boiling point of the lowest-boiling pure component in the system. This temperature value is 328.15 K for the MTBE process and 334.15 K for the TAME process. Using the Damkohler number as the continuation parameter, we solve eqs 15 and 2-4 using the program CL_MATCONT.21 Details of the algorithm and the strategy for obtaining the location of the bifurcation points are presented by Dhooge et al.21 Hopf Bifurcations in a MTBE TP Reactive Separation Flash In this section, we demonstrate the existence of Hopf bifurcations in an isothermal isobaric (TP) reactive separation flash problem. The condition for the existence of the Hopf bifurcations is given by Dhooge et al.21 Consider a reactive separation TP flash for the MTBE synthesis problem. In this problem, isobutene reacts with methanol to produce MTBE and the inert component is n-butane. The rate model and the activity
coefficient parameters are given by Chen et al.20 The components are ordered as [isobutene, methanol, MTBE, and n-butane]. For the feed composition of [0.163, 0.005, 0.081, 0.751], pressure of 11 atm, and temperature of 363.22 K,22 we used the program CL_MATCONT to draw the continuation curve by using the Damkohler number as the continuation parameter and obtained two Hopf bifurcation points at the Damkohler values of 1.495 and 5.128, as shown in Figure 1. We performed a dynamic simulation, and for two different starting points, for each of the Hopf points, we got a periodic oscillation and a convergence to steady state; this is a characteristic feature of Hopf bifurcation points. The convergence to steady state and the periodic oscillation at the first Hopf points are shown in Figures 2 and 3, respectively. Similar results are found for the second Hopf bifurcation point. Figure 4 shows the Hopf points at various temperatures. It can be observed that, at and beyond the temperature of 363.25 K, one of the Hopf points disappears. Figure 5 shows the behavior of the Hopf points at various pressures. It is seen that, as the pressure is lowered below 11 atm, one of the Hopf points disappears. Hopf Bifurcations in an Isothermal Isobaric TAME Process For the TAME problem, the Wilson model is used while the Antoine equation has the form
ln Psat ) A + (B/T) + C ln T + DTE
(16)
All the constants have been taken from Chen et al.16 If we order the components as MeOH, 2M1B, 2M2B, TAME, and n-pentane, for a feed composition of [0.2647, 0.0463, 0.2846, 0, 0.4044], a temperature of 335 K, and a pressure of 2.55 atm, we get a Hopf bifurcation point at a Damkohler
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Figure 5. Continuation diagram at different pressure values when the Hopf points appear.
Figure 6. Continuation diagram for TAME TP problem.
number of 0.461 999. This is shown in Figure 6. Figures 7 and 8 show that at this Damkohler value we get the existence of a limit cycle and a steady state. Figure 9 shows the existence of the Hopf bifurcation point at various temperatures, while Figure 10 shows the Hopf bifurcation points at various pressures.
Discussion of Results These results clearly demonstrate the existence of Hopf bifurcations, which cause the coexistence of a stable steady state and an unstable limit cycle in isothermal reactive flash processes.
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process problems involving MTBE and TAME mixtures do exhibit Hopf bifurcations. 2. In the neighborhood of these Hopf bifurcations, both limit cycles and steady states can be observed. 3. These Hopf bifurcations are, therefore, not necessarily due to multiple stages. 4. As shown in the Appendix, the Rachford-Rice procedure used to solve nonreactive flash isothermal isobaric flash processes can be extended to reactive systems. Appendix Modified Rachford-Rice Procedure. In this Appendix, we show how the Rachford-Rice procedure can be extended to isothermal isobaric reactive flash problems. Defining
Hνi ) Ri
Figure 7. Hopf point: periodic oscillation for TAME TP Problem.
and c
(
∑ Ri)/F ) Ξ i)1
we obtain after division of eq 1 by F,
Ri
(z - FLx - FVy ) + F ) 0 i
i
c eqs
i
(17)
Substituting yi ) Kixi and rearranging, we get
(zi) +
(
)
Ri L V + K x ) F Fi F i i
c eqs
(18)
or Figure 8. Hopf point: convergence to steady state for TAME problem.
It is possible that, under certain operating conditions, these Hopf bifurcations can exist in multistage columns too, and such columns may need special control mechanisms such as delayed feedback control. Mohl et al.11 showed that an isothermal CSTR problem involving the MTBE and TAME reactions cannot exhibit Hopf bifurcations. The work of Lucia19 clearly shows that an isothermal nonreactive flash problem cannot exhibit any multiple steady states. Furthermore, as can be seen in Figure 1, at large Damkohler numbers these bifurcation points do not exist. Therefore, we conclude that it is the combination of the phase equilibrium and the reaction that causes these Hopf bifurcations. Just as two nonsingular Jacobian matrixes can be added/ combined to give a singular Jacobian matrix, so also two processes that cannot by themselves produce limit cycles can be combined to produce highly nonlinear phenomenon like Hopf bifurcations. Additionally, this paper demonstrates that such instabilities and oscillations are not necessarily due to multiple stages and can occur in even in isothermal reactive separation process problems.
xi )
Ri F
(19)
L V + K F F i
(
)
and
y i ) Ki
(zi) +
Ri F
L V + K F F i
(
)
(20)
Since c
(yi - xi) ) 0 ∑ i)1 we have
(zi) +
c
∑ i)1 L
(
Ri F
(Ki - 1) ) 0
)
V + Ki F F
Conclusions The main conclusions of this paper are as follows. 1. While isothermal CSTR problems involving the MTBE and TAME reactions and isothermal nonreactive flash problems do not exhibit Hopf bifurcations, isothermal reactive flash
(zi) +
If
V L ) R, )1-Ξ-R F F
(21)
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Figure 9. Behavior of the Hopf bifurcation points at various temperatures for TAME system.
Figure 10. Behavior of the Hopf bifurcation points at various pressures for TAME system.
The derivative of this function with respect to R will be
we get
c
(zi) +
Ri F
∑ i)1 [(1 - Ξ - R) + (R)K ] i
c
(Ki - 1) ) Φ(R) ) 0 (22)
-
∑ i)1
(zi) +
Ri F
[(1 - Ξ - R) + (R)Ki]
2
(Ki - 1)2 ) Φ′(R)
(23)
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Using the method of Newton we can compute R in the inner loop and obtain both V/F ) R and L/F ) 1 - Ξ - R. The liquid and vapor compositions can be corrected in the outer loop using eqs 19 and 20. This procedure can be taught in undergraduate courses when reactive separation is introduced. Symbols A ) activity F ) feed V ) vapor L ) liquid H ) holdup T ) temperature P ) pressure Greek Symbols γ ) activity coefficient θ ) phase fraction Literature Cited (1) Taylor, R.; Krishna, R. Modeling reactive distillation. Chem. Eng. Sci. 2000, 55, 5183. (2) Jacobs, R.; Krishna, R. Multiple solutions in reactive distillation for methyl tert-butyl ether synthesis. Ind. Eng. Chem. Res. 1993, 32, 1706. (3) Nijhuis, S. A.; Kerkhof, F. P. J. M.; Mak, A. N. S. Multiple steady states during reactive distillation of methyl tert-butyl ether. Ind. Eng. Chem. Res. 1993, 32, 2767. (4) Bravo, J. L.; Pyhalahti, A.; Jarvelin, H. Investigations in a catalytic distillation pilot plant. Vapor/liquid equilibrium kinetics and mass-transfer issues. Ind. Eng. Chem. Res. 1993, 32, 2220. (5) Hauan, S.; Hertzberg, T.; Lien, K. M. Multiplicity of reactive distillation of MTBE. Comput. Chem. Eng. 1997, 21, 1117. (6) Hauan, S.; Hertzberg, T.; Lien, K. M. Why methyl tert-butyl ether production by reactive distillation may yield multiple solutions. Ind. Eng. Chem. Res. 1995, 34, 987. (7) Hauan, S.; Schrans, S. M.; Lien, K. M. Dynamic evidence of the multiplicity mechanism in methyl tert-butyl etherreactive distillation. Ind. Eng. Chem. Res. 1997, 36, 3995. (8) Sneesby, M. G.; Tade, M. O.; Smith, T. N. Steady-state transitions in the reactive distillation of MTBE. Comput. Chem. Eng. 1998, 42, 879.
(9) Eldarsi, H. S.; Douglas, P. L. Methyl-tert-butyl-ether catalytic distillation column. Part I: Multiple steady states. Trans. Inst. Chem. Eng. 1998, 76 (A4), 509. (10) Mohl, K.-D.; Kienle, A.; Gilles, E.-D. Multiple Steady States in a Reactive Distillation Column for the Production of the Fuel Ether TAME. I. Theoretical Analysis. Chem. Eng. Technol. 1998, 21 (2), 133. (11) Mohl, D. D.; Kienle, A.; Gilles, E. D.; Rapmund, P.; Sundmacher, K.; Hoffman, U. Steady-state multiplicities in reactive distillation columns for the production of fuel ethers MTBE and TAME: Theoretical analysis and experimental verification. Chem. Eng. Sci. 1999, 54, 1029. (12) Rapmund, P.; Sundmacher, K.; Hoffman, U. Multiple steady states in a reactive distillation column for the production of the fuel ether TAME. Part 2: Experimental validation. Chem. Eng. Technol. 1999, 21 (2), 136. (13) Higler, A.; Krishna, R.; Taylor, R. Nonequilibrium modeling of reactive distillation: Multiple steady states in MTBE synthesis. Chem. Eng. Sci. 1990, 54, 1389. (14) Guttinger, T. E.; Morari, M. Predicting Multiple Steady States in Equilibrium Reactive Distillation. 1. Analysis of Nonhybrid Systems. Ind. Eng. Chem. Res. 1999, 38, 1633. (15) Guttinger, T. E.; Morari, M. Predicting Multiple Steady States in Equilibrium Reactive Distillation. 2. Analysis of Hybrid Systems. Ind. Eng. Chem. Res. 1999, 38, 1649. (16) Chen, F.; Huss, R. S.; Doherty, M. F.; Malone, M. F. Multiple steady states in reactive distillation: Kinetic effects. Comput. Chem. Eng. 2002, 26, 81. (17) Rodriguez, I. E.; Zheng, A.; Malone, M. M. F. The stability of a reactive flash. Chem Eng. Sci. 2001, 56, 4737. (18) Rodriguez, I. E.; Zheng, A.; Malone, M. M. F. Parametric dependence of solution multiplicity in reactive flashes. Chem Eng. Sci. 2004, 59, 1589. (19) Lucia, A. Uniqueness of solutions to single-stage isobaric flash processes involving homogeneous mixtures. AIChE J. 1986, 32, 1761. (20) Chen, F.; Huss, R. S.; Doherty, M. F.; Malone, M. F. Simulation of kinetic effects in reactive distillation. Comput. Chem. Eng. 2000, 24, 2457. (21) Dhooge, A.; Govaerts, W.; Kuznetsov, Y. A.; Mestrom, W. Riet, A. M. CL_MATCONT; A continuation toolbox in Matlab. July 2004. (22) Cisneros, P.; Gani, R.; Michelsen, M. M. Reactive separation systems. II. Computation of Physical and Chemical Equilibrium. Chem. Eng. Sci. 1997, 52, 527.
ReceiVed for reView February 28, 2006 ReVised manuscript receiVed July 3, 2006 Accepted July 21, 2006 IE060249A
