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Ind. Eng. Chem. Res. 2008, 47, 2808-2816
Singularities in Reactive Separation Processes Gerardo Ruiz, Misael Diaz, and Lakshmi N. Sridhar* Chemical Engineering Department, UniVersity of Puerto Rico, Mayaguez, Puerto Rico 00681-9046
In this article, we demonstrate the existence of various types of singularities in equilibrium and nonequilibrium reactive separation process problems. First, the equilibrium reactive separation process problem is posed as a set of differential algebraic equations (DAEs), and it is shown that Hopf bifurcation points can exist, even for this formulation, for both the methyl tert-butyl ether (MTBE) and tert-amyl methyl ether (TAME) systems. Extension of this analysis to nonequilibrium models shows the existence of limit points, in the case of the TAME mixture, and isolas with intersecting branches, in the case of the MTBE mixture. The component mass balance equations are
( )
Introduction Multiple steady states in reactive distillations have been demonstrated by several workers.1-15 The most important cases of reactive distillation that have been known to have solution multiplicities (both experimental and theoretical) involve the methyl tert-butyl ether (MTBE) and tert-amyl methyl ether (TAME) systems, and the reactive separation processes that involve these two systems are considered by many to be the classic cases that exhibit multiple steady states. However, such multiplicities have been demonstrated for large columns, and to understand this phenomenon better, one must (a) examine the simplest reactive separation units that involve these two mixtures and (b) demonstrate the existence of the singularities that causes these multiplicities. In this paper, we will address these two issues. In a recent article, Ruiz et al.16 analyzed the isothermal isobaric reactive flash problem and showed that the MTBE and TAME reactive flash process, under isothermal, and isobaric conditions, exhibited Hopf bifurcations. In this article, we modify the approach used by Ruiz et al.,16 where a continuation procedure implemented in the CL_MATCONT program was used to solve all the equations for the reactive flash taken together. Here, we express the reactive flash problem as a set of differential algebraic equations (DAEs) and demonstrate Hopf bifurcations for the MTBE and TAME systems, even when such an approach is used. We then investigate the nature of the singularities in the nonequilibrium reactive isothermal flash problem described in Sridhar et al.17 and show that the effect of imposing mass-transfer rate equations on the equilibrium reactive flash problem causes the emergence of singularities such as limit points and bifurcation (branch) points. This paper is organized as follows. First, the DAE formulation for the equilibrium reactive flash process is described. Next, we demonstrate the solution procedure and the existence of Hopf bifurcations for the MTBE and TAME isothermal reactive flash problems. We then implement a similar procedure for the nonequilibrium isothermal and non-isothermal reactive flash problems and show the existence of limit points in the case of the TAME mixture and bifurcation points and isolas in the case of the MTBE system. DAE Formulation for the Isothermal Isobaric Equilibrium Reactive Flash Problem The differential algebraic formulation for the isothermal isobaric reactive flash problem can be derived as follows.
dxi Da ) zi - θLxi - θVyi νR dt kf,ref i
(1)
(where Da is the Damko¨hler number) while the overall mass balance equation is
( )
Da ν R)0 kf,ref T
1 - θL - θV -
(2)
n The summation expression for the vapor phase, ∑i)1 yi ) 1, and the phase equilibrium relation, yi ) Kixi, can be combined n Kixi ) 1. Differentiation with time will yield to yield ∑i)1
dxi
n
n
Ki + ∑ xi ∑ dt i)1 i)1
dKi dt
)0
(3)
sat Because Ki ) (Psat i /P)γi and dKi/dt ) (Pi /P) dγi/dt, and by applying the chain rule to dγi/dt, we get
dγi
∂γi dxj
n
)
dt
∑ j)1 ∂x
j
(4)
dt
This would yield the following equations:
dKi
)
Psat i
dt
P
n
∂γi dxj
∑ j)1 ∂x
(5)
dt
j
and
dxi
n
∑ i)1
Ki
+
dt
n
1
∑
P i)1
∂γi dxj
n
Psat i
∑ j)1
xi
∂xj dt
)0
(6)
Expressing ∂γi/∂xj in logarithmic form, we get ∂γi/∂xj ) γi ∂(ln γi)/∂xj, and substitution yields n
dxi
1
n
Ki + ∑ ∑ dt P i)1 i)1
[
Psat i
n
(
γi xi ∑ j)1
)]
∂(ln γi) dxj ∂xj
dt
)0
(7)
Substitution of the component mass-balance equation yields
10.1021/ie0716159 CCC: $40.75 © 2008 American Chemical Society Published on Web 03/14/2008
[
n
∑ i)1
( ) ] Da
Ki zi - θLxi - θVyi 1
n
∑
P i)1
Ind. Eng. Chem. Res., Vol. 47, No. 8, 2008 2809
kf,ref
{
n
(
γi xi ∑ j)1
Psat i
[
)
∂(ln γi) ∂xj
zj - θLxj - θVyj -
θV ) ×
( ) ]} Da
kf,ref
∑ i)1
νjR
kf,ref
) 0 (8)
kf,ref
νiR +
n
Psat i γi
Kiyi +
xi
P
i)1
∂(ln γi)
n
j)1
∂xj
yj
) 0 (9)
or
{ ( ) ∑ ( )[ ( ) ]} {∑ [ ∑ ( ) ]} {∑ [ ∑ ( ) ]}
n
Ki ∑ i)1
zi -
Da
∂(ln γi)
n
νiR +
Da
zj νj R ∂xj kf,ref n n ∂(ln γi) θL Ki xi + xi xj ∂xj i)1 j)1 n n ∂(ln γi) Ki yi + xi yj ) 0 (10) θV ∂xj i)1 j)1 kf,ref
xi
j)1
Furthermore, because θL ) 1 - θV - (Da/kf,ref)νTR, we would get n
{ ( )
Ki ∑ i)1
)[ ( ) ]} ( ) ){∑ [ ∑ ( ) ]} {∑ [ ∑ ( ) ]} Da
n
kf,ref
νiR +
∑ j)1
(
∂(ln γi)
xi
∂xj
zj -
Da
νj R kf,ref n n ∂(ln γi) Da 1 - θV νTR Ki xi + xi xj kf,ref ∂xj i)1 j)1 n n ∂(ln γi) θV Ki yi + xi yj ) 0 (11) ∂xj i)1 j)1
[
zi -
We can then solve for θV:
([ ( ) ]{∑ [ ∑ ( ) ]} ∑ { ( ) ∑( )[ ( ) ]})/ ∑ [ ∑ ( ) )]
θV )
1-
Da
kf,ref
n
i)1
n
Ki zi -
i)1
Da
kf,ref
Ki xi +
νTR
Da
kf,ref
xi
j)1
n
νiR +
n
j)1
n
i)1
which can be simplified to
∂xj ∂(ln γi)
xi
K i x i - yi +
νjR
∂(ln γi)
n
xi
j)1
∂xj ∂(ln γi) ∂xj
j)1
xj - z i + kf,ref ∂xj j)1 ∂(ln γi) Da xi zj νjR x i - yi + ∂xj kf,ref n ∂(ln γi) xi (xj - yj) (13) ∂xj j)1 xi
( )
Da ν R kf,ref T
yi ) Kixi
-
xj
∂xj
j)1
νi R -
∂(ln γi)
n
νTR xi +
(14)
Therefore, the DAE equation set for the reactive flash are eqs 13, 14, and 1, and the phase equilibrium relationship
Kixi +
∂(ln γi)
xi
n
i)1
∂xj
j)1
×
i)1
P
i)1
xi
- θL
νjR
kf,ref
n
Da
θL ) 1 - θV -
n
Da
n Psat i γi
n
∂(ln γi)
n
P
i)1
zj -
θV
Psat i γi
n
1-
Equation 2 can be rewritten as
[ ( ) ] ∑{ ∑( ) [ ( ) ]} {∑ ∑[ ∑ ( ) ]} {∑ ∑ [ ∑ ( ) ]}
Ki zi -
Da
∑ i)1
Da
Grouping similar terms for θL and θV yields n
({[ ( ) ][ ∑ ( ) ] ( ) ∑ ( )[ ( ) )]}/[ ∑( ) ] n
νiR +
xj
For a given temperature, pressure feed composition, and Da value, and an initial guess for the liquid-phase composition, we can calculate the vapor-phase composition and the activity coefficient, evaluate θL and θV, and solve the differential equation shown as eq 1 at steady state. The initial solution point is then fed into the CL_MATCONT program to obtain the solution curve. Non-Equilibrium Reactive Flash Problem The motivation for analyzing the nonequilibrium reactive TP problem is to investigate the effect of the mass-transfer equations on the single-stage reactive flash problem. Consequently, we examine a reactive analog of the single-stage nonequilibrium flash problem that has been described in the work of Sridhar et al.17 The equations in this problem include the material balance for the vapor and liquid phase, the phase equilibrium relationships, and the transfer-rate equations. Assuming that the reaction occurs in the liquid phase, the equations are very similar to those described in the work of Sridhar et al.,17 except for the reaction term in the liquid-phase material balance equation. The equations for the single-stage nonequilibrium reactive flash problem include the material balance equations for the vapor and liquid phases:
(16)
fLi - Lxi + NLi anet + LViR ) 0
(17)
and
The phase equilibrium relationships at the interface can be written as
(18)
The transfer-rate equations for the vapor and liquid phases are
zj -
(xj - yj
fVi - Vyi - NVi anet ) 0
yIi ) KixIi -
(15)
(NV) ) cVt [κV](y - yI) + NVt (y)
(19)
(NL) ) cLt [κL](xI - x) + NLt (x)
(20)
and
(12)
The mass-transfer coefficients are obtained using the AIChE method,18 with the modification that was reported by Bennet et
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Ind. Eng. Chem. Res., Vol. 47, No. 8, 2008
Figure 1. Algorithm flowchart for isothermal nonequilibrium problem.
al.,19 whereas the interfacial area was calculated using the procedures given in the work of Zuiderweg.20 These procedures are also described in the work by Taylor and Krishna.21 The interface material balance equations would be
NVi |I
)
N Li |I
Table 1. Equilibrium Problem for MTBE System (Feed Conditions and Specifications) stage specification temperature, T pressure, P zisobutene zMeOH zMTBE zn-butane
(21)
In addition, we have the summation equations for the interface compositions:
∑ xIi ) 1
(22)
∑ yIi ) 1
(23)
and
To obtain the singularities in the problem, we must divide this set of equations into differential and algebraic equations. The ordinary differential equations (ODEs) that we will incorporate in the CL_MATCONT program are the material balance equations for the vapor and liquid phases:
dVi ) fVi - Vyi - NVi anet dt and
( )
DaFLVi dLi R ) fLi - Lxi + NLi anet + dt kf,ref
(24)
(25)
These are the two equations that will be incorporated by the CL_MATCONT program to investigate the singularities. The other equations will be the algebraic equations that will be solved along with the ODEs. Using the definitions Li ) Lxi and Vi ) Vyi, we can rewrite eqs 24 and 25 as
dyi 1 V ) (f - Vyi - NVi anet) dt L i and
[
( )]
dxi 1 L DaFLVi ) L fi - Lxi + NLi anet + R dt kf,ref
value 364.25 K 11.25 bar 0.46241 0.08385 0.00128 0.45245
For a given temperature and pressure and fixed feed composition, the algebraic system of equations are solved first. This is done by first fixing an initial guess for xIi , yIi , NVi , NLi , V, and L. The physical properties FL, FV, σ, µV, DV, and DL are then estimated (see Appendix A) and the tray design procedure (given in Appendix B) is executed. The multicomponent masstransfer coefficients are then obtained, and the steady-state version of the ODE system is then solved. When the two systems of equations converge, the liquid and vapor holdups are calculated by the empirical co-relations in the tray design procedure. This gives us the initial value of Da, which is used as a continuation parameter. The design parameters that constitute the tray specifications are then fixed. With these design parameters maintained constant, the continuation procedure using the CL_MATCONT program is then executed using Da as the continuation parameter. In the non-isothermal nonequilibrium case, we also consider the bulk energy balance equations in the liquid and vapor phase, which are
dU L ) F LH LF - LHL + E Lanet + Q°L dt
(28)
dUV ) FVHVF - VHV - EVanet + Q°V dt
(29)
and
(26)
where UL ) LHL, UV ) VHV, and Q° is the heat-transfer rate from the surroundings. We also consider the interface energy balance equation, which is given as
(27)
E V | I - E L| I ) 0
(30)
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Figure 2. Algorithm flowchart for non-isothermal nonequilibrium problem. Table 2. Singularity in the Equilibrium MTBE Problem parameter
Table 5. Feed Conditions and Specifications for the Nonequilibrium TAME Problem
value
Value
singular point xisobutene xMeOH xMTBE xn-butane yisobutene yMeOH yMTBE yn-butane
0.33042 0.21338 0.06432 0.39189 0.45653 0.07385 0.01222 0.4574
stage specifications flow rate, F temperature, T pressure, P zMeOH z2M1B z2M2B zTAME zn-pentane
θL 1.65 × 10-7 θV 0.98919 Damko¨hler number, Da 0.07259 singularity nature Hopf point Table 3. Equilibrium Problem for the TAME System (Feed Conditions and Specifications) stage specifications
Value
335 K 2.46 bar 0.30556 0.03889 0.23889 0.00000 0.41666
Table 4. Singularity in the Equilibrium TAME Problem parameter
value
singular point xMeOH x2M1B x2M2B xTAME xn-pentane yMeOH y2M1B y2M2B yTAME yn-pentane
0.3603 0.01697 0.18645 0.14614 0.29013 0.28702 0.02658 0.23192 0.02669 0.42779
point 2
singular point xMeOH x2M1B x2M2B xTAME xn-pentane yMeOH y2M1B y2M2B yTAME yn-pentane liquid-phase temperature, TL vapor-phase temperature, TV Damko¨hler number, Da singularity nature
0.035660 0.291346 0.350054 0.156476 0.166464 0.197661 0.326563 0.302922 0.014808 0.158046 348.76 K 345.73 K 0.01337 limit point
0.003961 0.282852 0.340646 0.199069 0.173472 0.197661 0.326563 0.302922 0.014808 0.158046 357.10 K 345.73 K 0.007684 limit point
c
2.00 × 0.974 0.47492 Hopf point
NVi HVi ∑ i)1
(32)
The liquid and vapor heat-transfer coefficients are defined as hL ) κLFLt CLp Le1/2 and hV ) κVFVt CVp Le2/3 (from Kooijmann22), respectively, where κ is the mass-transfer coefficient, Ft the density, Cp the specific heat, and Le the Lewis number. The enthalpies are calculated using a procedure similar to that given by Chen et al.23 Figures 1 and 2 show the algorithm flowcharts for the isothermal and non-isothermal nonequilibrium problems. Results and Discussion
c
∑ i)1
point 1
EV ) hV(TV - T I) +
The energy fluxes, which include the convective and conductive contributions, are
E L ) hL(T I - T L) +
parameter
and
10-8
θL θV Damko¨hler number, Da singularity nature
vapor phase 252.00 kmol/h 346.31 K 2.00 bar 0.197793 0.326756 0.302763 0.014649 0.158039
Table 6. Non-Isothermal Nonequilibrium Singular Point for the TAME Problem
value
temperature, T pressure, P zMeOH z2M1B z2M2B zTAME zn-pentane
liquid phase 216.00 kmol/h 325.31 K 2.00 bar 0.090 0.300 0.350 0.100 0.160
N Li H Li
(31)
We found singularities for both the equilibrium and nonequilibrium models for both the MTBE and TAME systems. In the
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Table 7. Isothermal Nonequilibrium Singular Point for the TAME Problem Value parameter
point 1
point 2
singular point xMeOH x2M1B x2M2B xTAME xn-pentane yMeOH y2M1B y2M2B yTAME yn-pentane temperature, T Damko¨hler number, Da singularity nature
0.023890 0.291711 0.355515 0.162988 0.165896 0.197662 0.326561 0.302917 0.014812 0.158048 329 K 0.022223 limit point
0.002299 0.283698 0.343362 0.197949 0.172692 0.197662 0.326561 0.302917 0.014812 0.158048 329 K 0.012108 limit point
Table 8. Feed Conditions and Specifications for the Nonequilibrium MTBE Problem Value stage specifications flow rate, F temperature, T pressure, P zisobutene zMeOH zMTBE zn-butane
liquid phase
vapor phase
216.00 kmol/h 380.01 K 18.04 bar 0.3500 0.3500 0.1980 0.2770
252.00 kmol/h 401.01 K 18.04 bar 0.6069 0.1783 0.0475 0.1674
Figure 3. Hopf point for the isothermal equilibrium MTBE problem.
The inert compound that is present is n-butane. The rate model from Chen et al.23 is
(
r ) kf ai-buteneaMeOH -
point 1
kf ) 4464 exp
0.31290 0.37673 0.23400 0.07637 0.17857 0.04759 0.08675 0.68709 392.747 398.216 0.00421 closed curve with intersecting branch
Table 10. Isothermal Nonequilibrium Singular Point for the MTBE Problem
singular point xisobutene xMeOH xMTBE xn-butane yisobutene yMeOH yMTBE yn-butane temperature, T Damko¨hler number, Da singularity nature
(35)
ln Psat ) A +
point 2
0.09894 0.10549 0.35948 0.43609 0.60607 0.17871 0.04784 0.16739 381.13 K 0.01421 closed curve with intersecting branch
0.34818 0.17436 0.19934 0.27813 0.60607 0.17871 0.04784 0.16739 381.13 K 0.00023 closed curve with intersecting branch
(36)
B T+C
(37)
The Wilson binary interaction parameters and the Antoine coefficients were taken from Chen et al.23 The TAME synthesis reaction can be written as15
2M1B + 2M2B + 2.0MeOH S 2.0(TAME)
(38)
The rate model, which is also given in the work by Chen et al.,15 is
r ) kf
point 1
(-3187 T )
The liquid phase activity coefficients were obtained using the Wilson equation, and the Antoine equation has the form
Value parameter
(6820 T )
Keq ) 8.33 × 10-8 exp
point 2
0.34801 0.35144 0.19837 0.10218 0.17857 0.04759 0.08675 0.68709 381.849 398.216 0.00027 closed curve with intersecting branch
(34)
and
Value parameter
)
where
Table 9. Non-isothermal Nonequilibrium Singular Point for MTBE Problem
singular point xisobutene xMeOH xMTBE xn-butane yisobutene yMeOH yMTBE yn-butane liquid-phase temperature, TL vapor-phase temperature, TV Damko¨hler number, Da singularity nature
aMTBE Keq
[
( )]
a2M1B 1 aTAME 2 aMeOH K1 a MeOH
(39)
where
(
kf ) (1 + Λ)(1.9769 × 1010) exp -
10764 T
(4273.5 T )
K1 ) 1.057 × 10-4 exp
)
(40) (41)
and
Λ ) 0.648 exp
(899.9 T )
(42)
MTBE system, the reaction involved is
i-butene + MeOH a MTBE
(33)
The DAE system that was solved using the CL_MATCONT program produced the following results. The equilibrium
Ind. Eng. Chem. Res., Vol. 47, No. 8, 2008 2813
Figure 4. Hopf point for the isothermal equilibrium TAME problem.
Figure 7. Singular points (the two limit points) for isothermal nonequilibrium TAME problem.
Figure 5. Singular points (intersection points of isola with the branch) for isothermal nonequilibrium MTBE problem.
Figure 8. Singular points (the two limit points) for non-isothermal nonequilibrium TAME problem.
Figure 6. Singular points (intersection points of isola with the branch) for non-isothermal nonequilibrium MTBE problem.
reactive flash problem showed the existence of Hopf bifurcations, as observed in Ruiz et al.,16 whereas the nonequilibrium problem revealed the existence of limit points (turning points) in the case of the TAME mixture and isolas with an intersecting branch in the case of the MTBE problem. Tables 1-10 show the feed conditions and specifications and the location of the
singular points for the equilibrium and nonequilibrium problems. Figures 2-8 show the solution curves with the singular points for all the problems. Two important issues can be observed from these results. First, in regard to reactive separation processes in general, note that, in nonreactive problems, Sridhar et al.17 have shown that the imposition of the mass-transfer equations on the equilibrium reactive separation process problem does not cause any additional multiplicities. However, in the case of reactive separation process problems, the situation is slightly different. The imposition of the mass-transfer equations on the equilibrium reactive separation process problem causes the occurrence of limit points and branch points that are caused by the intersection of isolas with isolated branches. The additional nonlinearity that is produced by the reaction kinetics interacting with the masstransfer co-relations does indeed cause the birth of the limit points and branch points. The second important conclusion is the fact that the MTBE and TAME mixtures seem to react differently to the imposition of the mass-transfer equations. However, this result is not surprising to us. As far as multiple steady states are concerned, Chen et al.15 have demonstrated that, for certain specifications, both mixtures show significantly different behavior by demonstrating that, for TAME, multiplicities are lost for high Da values, whereas for MTBE, multiplicities are lost for low Da
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values. This is because of the difference in kinetics in both of these cases. This difference in the kinetics also seems to influence the nature of the singularities that are produced by the imposition of the mass-transfer co-relations on the reactive separation process problem. The MTBE process and the TAME process models do react differently when the additional masstransfer co-relations are incorporated. The incorporation of the mass-transfer co-relations causes the occurrence of limit (turning) points in the case of the TAME problem and branch (bifurcation) points and isolas in the case of the MTBE problem. This is due to the two different kinetic mechanism equations interacting with the mass-transfer co-relations. Conclusions
Kesler correlation.27 The surface tension is estimated using the Macleod-Sugden correlation.26 Appendix B The tray design procedure is briefly described in this appendix. We follow the procedure described by Locket28 and Kooijman.22 By fixing the tray spacing (Ts), the number of passes (Npasses), the weir height (hw), the fractional perforated tray area (φ), the downcomer clearance (hc), and the hole diameter (dh), the tray thickness (tt) will be given by the relation tt ) 0.43dh. The volumetric liquid flow (QL) is known, which enables us to calculate the downcomer area (Ad) and the liquid velocity in the downcomer (ud) on a vapor-free basis, using the relationship
This paper investigates the nature of the singularities in equilibrium and nonequilibrium reactive separation units. It is shown that the differential algebraic equation (DAE) formulation of the equilibrium reactive separation process problem produces Hopf bifurcations such as those observed in the work of Ruiz et al.16 It is also shown that the imposition of the mass-transfer equations on the reactive flash problem leads to the formation of limit points in the case of the TAME mixture and isolas with intersecting mixtures in the case of the MTBE problem.
The flow parameter (FP) is then calculated as
Appendix A
and if FP is determined to be 3.0bh*L, where b is the weir length per unit bubbling area:
(B13)
and if the weir load is more that the maximum value, the number of passes is increased incrementally. With the superficial vapor velocity being defined as us ) QV/Ab, the clear liquid height (hL) is computed using expressions from Bennett et al.:19
hL ) Re hw + C
ahf )
(B17)
where
QV uh ) Ah
(B18)
where g is the acceleration due to gravity and Fs is the superficial flow factor. The froth height (hf) is given by
We start with
hf ) φ)
Ah Ab
(B19)
and obtain Ah. This is used to find uh, which is used to obtain the Froude number Frh. If the Froude number Frh is not suitable, a new value of φ is chosen and the procedure is repeated. The parameter ahf (where a is the interfacial area per unit volume of froth and hf is the froth height) is calculated using the Zuiderweg method,20 which takes into account the nature of the flow regime (spray or mixed froth emulsion). Spray regime:
(
)
2 V 40 us Ft h* LFP ahf ) 0.3 σ φ
Mixed-froth emulsion flow regime:
0.37
hL R
(B31)
the liquid and vapor holdups and the net interfacial area are calculated using the equations
L ) AbhLcLt
(B32)
(1 -R R)
(B33)
anet ) ahfAb
(B34)
V )
L
and
Acknowledgment
(B20)
This work was supported by NSF (through Grant No. CTS 0341608). Part of this work was done in Clarkson University in the summer of 2006 by Mr. Gerardo Ruiz. L.N.S. thanks Dr.
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Raghunathan Rengaswamy, for his help with the DAE formulation and the numerical implementation, and Dr. Ross Taylor, for his help with the mass-transfer calculations. Literature Cited (1) Jacobs, R.; Krishna, R. Multiple Solutions in Reactive Distillation for Methyl tert-Butyl Ether Synthesis. Ind. Eng. Chem. Res. 1993, 32, 17061709. (2) 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, 2767-2774. (3) 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, 2220-2225. (4) Hauan, S.; Hertzberg, T.; Lien, M. Why Methyl tert-Butyl Ether Production by Reactive Distillation May Yield Multiple Solutions. Ind. Eng. Chem. Res. 1995, 34, 987-991. (5) Hauan, S.; Hertzberg, T.; Lien, M. Multiplicity in reactive distillation of MTBE. Comput. Chem. Eng. 1997, 21, 1117-1124. (6) Hauan, S.; Schrans, S.; Lien, K. Dynamic Evidence of the Multiplicity Mechanism in Methyl tert-Butyl Ether Reactive Distillation. Ind. Eng. Chem. Res. 1997, 36, 3995-3998. (7) Sneesby, M. G.; Tade, M. O.; Smith, T. N. Steady-state transitions in the reactive distillation of MTBE. Comput. Chem. Eng. 1998, 22, 879892. (8) Eldarsi, H. S.; Douglas, P. L. Methyl tert butyl ether catalytic distillation column, Part I. Multiple steady-states. Trans. Inst. Chem. Eng., Part A 1998, 76, 509-516. (9) Mohl, K. D.; Kienle, A.; Gilles, E. D. Multiple Steady States in a Reactive Distillation Column for the Production of the Fuel Ether TAME. 1. Theoretical Analysis. Chem. Eng. Technol. 1998, 21 (2), 133-136. (10) Mohl, K. D.; Kienle, A.; Gilles, E. D.; Rapmund, P.; Sundmacher, K.; Hoffmann 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, 1029-1043. (11) Rapmund, P.; Sundmacher, K.; Hoffman, U. Multiple Steady States in a Reactive Distillation Column for the Production of the Fuel Ether TAME. 2: Experimental Validation. Chem. Eng. Technol. 1998, 21 (2), 136-139. (12) Higler, A.; Taylor, R.; Krishna, R. Nonequilibrium Modeling of Reactive Distillation: Multiple Steady States in MTBE Synthesis. Chem. Eng. Sci. 1999, 54, 1389-1395. (13) Guttinger, T.; Morari, M. Predicting Multiple Steady States in Equilibrium Reactive Distillation. 1. Analysis of Nonhybrid Systems. Ind. Eng. Chem. Res. 1999, 38, 1633-1648. (14) Guttinger, T.; Morari, M. Predicting Multiple Steady States in Equilibrium Reactive Distillation. 2. Analysis of Hybrid Systems. Ind. Eng. Chem. Res. 1999, 38, 1649-1665.
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ReceiVed for reView November 27, 2007 ReVised manuscript receiVed February 1, 2008 Accepted February 11, 2008 IE0716159
