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 alsodemonstrate that, even in isothermal isobaric reactive separation processes, which are probably the leastnonlinear of all reactive separation processes, we get nonlinear phenomenon such as Hopf bifurcations. Whileit has been shown that Hopf bifurcations are impossible in isothermal continuous stirred tank reactor (CSTR)problems involving the methyltert-butyl ether (MTBE) andtert-amyl methyl ether (TAME) reactions, andalso in nonreactive flash problems, we demonstrate in this paper that isothermal reactive flash processesinvolving both MTBE and TAME mixtures exhibit Hopf bifurcations. This shows that instabilities andoscillations can occur even in isothermal reactive separation systems and are not necessarily due to multiplestages. Additionally, we show that the Rachford-Rice procedure can be extended to reactive systems.

Introduction

During the past decade, there has been a tremendous interestin the field of reactive distillation. A review of the variousmodels used in reactive distillation can be found in the paperby Taylor and Krishna.1 Of special interest is the existence ofmultiple steady states in these problems, since the combinationof separation and reaction can, in principle, introduce thenonlinearity that can cause multiplicity. Multiple steady statesin reactive distillations were demonstrated by several workers.2-16

The most commonly investigated situations include the methyltert-butyl ether (MTBE) synthesis in the Jacobs-Krishna2

column configuration and thetert-amyl methyl ether (TAME)synthesis in the column of Mohl et al..11 The multiple steadystates for these two columns were investigated by Chen et al.,16

who conclude that multiplicities are lost for high values ofDafor TAME, while the opposite is found for MTBE. Thisconclusion, however, is specific to the column configurationsdescrbed in Jacobs-Krishna2 and Mohl et al.11 Rodriguez etal.17,18discuss causes for the existence of multiple steady statesin binary and ternary systems. The most important reactiveseparation process problems where multiplicity exists, such asMTBE and TAME processes, involve more than three compo-nents. To understand what causes multiplicity in these problems,one must look at the simplest reactive separation processproblem involving the MTBE and TAME mixture, and hence,we are motivated to look at the isothermal reactive flashproblem. Mohl et al.11 prove that isothermal continuous stirredtank reactor (CSTR) problems involving the MTBE and TAMEreactions do not exhibit Hopf bifurcations, while on the otherhand, for nonreactive isothermal flash processes involvinghomogeneous mixtures, Hopf bifurcations are impossible.19

However, we demonstrate that isothermal reactive flash pro-cesses involving both TAME and MTBE exhibit Hopf bifurca-tions, and that is one of the important contributions of this paper.This paper is organized in the following manner. First, a briefdescription of the isothermal reactive flash process is given alongwith the equations involved. We then demonstrate the existenceof Hopf bifurcations in the isothermal reactive flash processesinvolving both the MTBE and TAME mixtures. Dynamicsimulations are performed demonstrating the existence of limit

cycles that are a characteristic feature of problems with Hopfbifurcations, and the behaviors of these Hopf bifurcation pointswith temperature and pressure variations are presented. Amodified Rachford-Rice procedure for solving the isothermalreactive separation process problem is then presented in theAppendix.

Isothermal Reactive Separation Flash Problem

For a single-stage reactive separation unit with a singlereaction, we havec material-balance equations,

whereF is the external feed,L is the liquid flow,V is the vaporflow, H is the holdup,ε is the extent of reaction, andν is thestoichiometric coefficient.

We also have the phase-equilibrium equations,

where theK-value is

The total number of equations) 2c + 2. The variables arexi,yi, L, andV. The total number of variables is (2c + 2). Thespecifications areT, P, andH.

While this set of equations can be solved using a variety oftechniques, a modified Rachford-Rice procedure for solvingthe isothermal isobaric reactive flash problem is presented inthe Appendix.

MTBE Process

One of the most researched processes in reactive separationis the MTBE (methyltert-butyl ether) process. The reaction

(Fzi - Lxi - Vyi) + Hνiε ) 0 c eqs (1)

yi ) Kixi c eqs (2)

Ki )γiPi

sat

P

∑i)1

c

yi ) 1 (3)

∑i)1

c

xi ) 1 (4)

6548 Ind. Eng. Chem. Res.2006,45, 6548-6554

10.1021/ie060249a CCC: $33.50 © 2006 American Chemical SocietyPublished on Web 08/17/2006

involved is

The inert compound present isn-butane. The rate model20 is

where

and

The liquid-phase activity coefficients were obtained using theWilson equation, and the Antoine equation has the form

The Wilson binary interaction parameters and the Antoinecoefficients were taken from Chen et al..20

TAME Process

The TAME synthesis reaction can be written as16

The rate model also given by Chen et al.16 is

where

Figure 1. Continuation diagram.

Figure 2. Hopf point 1: Convergence to steady state.

i-butene+ MeOH a MTBE (5)

r ) kf(ai-buteneaMeOH -aMTBE

Keq) (6)

Keq ) 8.33× 10-8(exp(6820/T)) (7)

kf ) 4464 exp(-3187/T) (8)

ln Psat) A + B/(T + C) (9)

Figure 3. Hopf point 1: periodic oscillation.

2M1B + 2M2B + 2MeOHS 2(TAME) (10)

ε ) kf(a2M1B

aMeOH- 1

K1

aTAME

aMeOH2) (11)

kf ) (1 + Λ)(1.9769× 1010)(exp(- 10 764T )) (12)

K1 ) (1.057× 10-4) e4273.5/T (13)

Ind. Eng. Chem. Res., Vol. 45, No. 19, 20066549

and

Solution Procedure

Defining the Damkohler number as

eq 1 can be rewritten as

whereθL ) L/F, θV ) V/F, andkf,ref is the forward rate constantevaluated at the boiling point of the lowest-boiling purecomponent in the system. This temperature value is 328.15 Kfor 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 thelocation of the bifurcation points are presented by Dhooge etal.21

Hopf Bifurcations in a MTBE TP Reactive SeparationFlash

In this section, we demonstrate the existence of Hopfbifurcations in an isothermal isobaric (TP) reactive separationflash problem. The condition for the existence of the Hopfbifurcations is given by Dhooge et al.21 Consider a reactiveseparation TP flash for the MTBE synthesis problem. In thisproblem, isobutene reacts with methanol to produce MTBE andthe inert component isn-butane. The rate model and the activity

coefficient parameters are given by Chen et al.20 The compo-nents are ordered as [isobutene, methanol, MTBE, andn-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 usedthe program CL_MATCONT to draw the continuation curveby using the Damkohler number as the continuation parameterand obtained two Hopf bifurcation points at the Damkohlervalues of 1.495 and 5.128, as shown in Figure 1. We performeda dynamic simulation, and for two different starting points, foreach of the Hopf points, we got a periodic oscillation and aconvergence to steady state; this is a characteristic feature ofHopf bifurcation points. The convergence to steady state andthe periodic oscillation at the first Hopf points are shown inFigures 2 and 3, respectively. Similar results are found for thesecond Hopf bifurcation point. Figure 4 shows the Hopf pointsat various temperatures. It can be observed that, at and beyondthe temperature of 363.25 K, one of the Hopf points disappears.Figure 5 shows the behavior of the Hopf points at variouspressures. It is seen that, as the pressure is lowered below 11atm, one of the Hopf points disappears.

Hopf Bifurcations in an Isothermal Isobaric TAMEProcess

For the TAME problem, the Wilson model is used while theAntoine equation has the form

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 of2.55 atm, we get a Hopf bifurcation point at a Damkohler

Figure 4. Behavior of the Hopf bifurcation points at various temperatures.

Λ ) 0.648 e899.9/T (14)

Da ) HF(Kf,ref)

(zi - θLxi - θVyi) + Da(kf,ref)νiε ) 0 c eqs (15)

ln Psat) A + (B/T) + C ln T + DTE (16)

6550 Ind. Eng. Chem. Res., Vol. 45, No. 19, 2006

number of 0.461 999. This is shown in Figure 6. Figures 7 and8 show that at this Damkohler value we get the existence of alimit cycle and a steady state. Figure 9 shows the existence ofthe Hopf bifurcation point at various temperatures, while Figure10 shows the Hopf bifurcation points at various pressures.

Discussion of Results

These results clearly demonstrate the existence of Hopfbifurcations, which cause the coexistence of a stable steady stateand an unstable limit cycle in isothermal reactive flash processes.

Figure 5. Continuation diagram at different pressure values when the Hopf points appear.

Figure 6. Continuation diagram for TAME TP problem.

Ind. Eng. Chem. Res., Vol. 45, No. 19, 20066551

It is possible that, under certain operating conditions, these Hopfbifurcations can exist in multistage columns too, and suchcolumns may need special control mechanisms such as delayedfeedback control.

Mohl et al.11 showed that an isothermal CSTR probleminvolving the MTBE and TAME reactions cannot exhibit Hopfbifurcations. The work of Lucia19 clearly shows that anisothermal nonreactive flash problem cannot exhibit any multiplesteady states. Furthermore, as can be seen in Figure 1, at largeDamkohler numbers these bifurcation points do not exist.Therefore, we conclude that it is the combination of the phaseequilibrium 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 twoprocesses that cannot by themselves produce limit cycles canbe combined to produce highly nonlinear phenomenon like Hopfbifurcations. Additionally, this paper demonstrates that suchinstabilities and oscillations are not necessarily due to multiplestages and can occur in even in isothermal reactive separationprocess problems.

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 problemsdo not exhibit Hopf bifurcations, isothermal reactive flash

process problems involving MTBE and TAME mixtures doexhibit Hopf bifurcations.

2. In the neighborhood of these Hopf bifurcations, both limitcycles and steady states can be observed.

3. These Hopf bifurcations are, therefore, not necessarily dueto multiple stages.

4. As shown in the Appendix, the Rachford-Rice procedureused to solve nonreactive flash isothermal isobaric flashprocesses can be extended to reactive systems.

Appendix

Modified Rachford-Rice Procedure.In this Appendix, weshow how the Rachford-Rice procedure can be extended toisothermal isobaric reactive flash problems.

Defining

and

we obtain after division of eq 1 byF,

Substitutingyi ) Kixi and rearranging, we get

or

and

Since

we have

If

Figure 7. Hopf point: periodic oscillation for TAME TP Problem.

Figure 8. Hopf point: convergence to steady state for TAME problem.

Hνiε ) Ri

(∑i ) 1

c

Ri)/F ) ¥

(zi - LF

xi - VF

yi) +Ri

F) 0 c eqs (17)

(zi) +Ri

F) (L

Fi+ V

FKi)xi c eqs (18)

xi )(zi) +

Ri

F

(LF + VF

Ki)(19)

yi ) Ki

(zi) +Ri

F

(LF + VF

Ki)(20)

∑i)1

c

(yi - xi) ) 0

∑i)1

c(zi) +

Ri

F

(L

F+

V

FKi)

(Ki - 1) ) 0 (21)

VF

) R,LF

) 1 - ¥ - R

6552 Ind. Eng. Chem. Res., Vol. 45, No. 19, 2006

we get The derivative of this function with respect toR will be

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.

∑i)1

c(zi) +

Ri

F

[(1 - ¥ - R) + (R)Ki](Ki - 1) ) Φ(R) ) 0 (22) -∑

i)1

c(zi) +

Ri

F

[(1 - ¥ - R) + (R)Ki]2(Ki - 1)2 ) Φ′(R) (23)

Ind. Eng. Chem. Res., Vol. 45, No. 19, 20066553

Using the method of Newton we can computeR in the innerloop and obtain bothV/F ) R and L/F ) 1 - ¥ - R. Theliquid and vapor compositions can be corrected in the outer loopusing eqs 19 and 20. This procedure can be taught inundergraduate courses when reactive separation is introduced.

Symbols

A ) activityF ) feedV ) vaporL ) liquidH ) holdupT ) temperatureP ) pressure

Greek Symbols

γ ) activity coefficientθ ) 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 formethyl 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 steadystates during reactive distillation of methyltert-butyl ether.Ind. Eng. Chem.Res. 1993, 32, 2767.

(4) Bravo, J. L.; Pyhalahti, A.; Jarvelin, H. Investigations in a catalyticdistillation pilot plant. Vapor/liquid equilibrium kinetics and mass-transferissues.Ind. Eng. Chem. Res. 1993, 32, 2220.

(5) Hauan, S.; Hertzberg, T.; Lien, K. M. Multiplicity of reactivedistillation of MTBE. Comput. Chem. Eng.1997, 21, 1117.

(6) Hauan, S.; Hertzberg, T.; Lien, K. M. Why methyltert-butyl etherproduction 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 themultiplicity mechanism in methyltert-butyl etherreactive distillation.Ind.Eng. Chem. Res.1997, 36, 3995.

(8) Sneesby, M. G.; Tade, M. O.; Smith, T. N. Steady-state transitionsin the reactive distillation of MTBE.Comput. Chem. Eng.1998, 42, 879.

(9) Eldarsi, H. S.; Douglas, P. L. Methyl-tert-butyl-ether catalyticdistillation 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 aReactive 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 columnsfor the production of fuel ethers MTBE and TAME: Theoretical analysisand experimental verification.Chem. Eng. Sci.1999, 54, 1029.

(12) Rapmund, P.; Sundmacher, K.; Hoffman, U. Multiple steady statesin 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 ofreactive distillation: Multiple steady states in MTBE synthesis.Chem. Eng.Sci.1990, 54, 1389.

(14) Guttinger, T. E.; Morari, M. Predicting Multiple Steady States inEquilibrium Reactive Distillation. 1. Analysis of Nonhybrid Systems.Ind.Eng. Chem. Res.1999, 38, 1633.

(15) Guttinger, T. E.; Morari, M. Predicting Multiple Steady States inEquilibrium 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. Multiplesteady 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 areactive flash.Chem Eng. Sci.2001, 56, 4737.

(18) Rodriguez, I. E.; Zheng, A.; Malone, M. M. F. Parametricdependence of solution multiplicity in reactive flashes.Chem Eng. Sci.2004,59, 1589.

(19) Lucia, A. Uniqueness of solutions to single-stage isobaric flashprocesses involving homogeneous mixtures.AIChE J.1986, 32, 1761.

(20) Chen, F.; Huss, R. S.; Doherty, M. F.; Malone, M. F. Simulationof 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 separationsystems. II. Computation of Physical and Chemical Equilibrium.Chem.Eng. Sci.1997, 52, 527.

ReceiVed for reView February 28, 2006ReVised manuscript receiVed July 3, 2006

AcceptedJuly 21, 2006

IE060249A

6554 Ind. Eng. Chem. Res., Vol. 45, No. 19, 2006