Nonequilibrium Reactive Mole Fraction Curve Maps - Industrial

Mar 2, 2009 - The strategy to solve the differential algebraic equation (DAE) system to find the nonequilibrium reactive mole fraction curves is discu...
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Ind. Eng. Chem. Res. 2009, 48, 3678–3684

RESEARCH NOTES Nonequilibrium Reactive Mole Fraction Curve Maps Gerardo Ruiz and Lakshmi N. Sridhar* Chemical Engineering Department, UniVersity of Puerto Rico, Mayagu¨ez Puerto Rico 00681-9046

In this research and development (R&D) note, we derive new expressions to calculate the nonequilibrium composition curve maps for reactive separation processes incorporating mass transfer effects. The strategy to solve the differential algebraic equation (DAE) system to find the nonequilibrium reactive mole fraction curves is discussed and illustrated for the methyl tert-butyl ether (MTBE) and tertiary amyl methyl ether (TAME) synthesis problems. Introduction The distillation residue curve maps (RCMs) have been studied by several workers1-12 in the design and synthesis of reactive and nonreactive separation processes. The RCM are used to establish feasible splits by distillation of azeotropic mixtures due to the presence of nonreactive azeotropes after the reaction, reactive azeotropes, and distillation boundaries for continuous distillation at infinite reflux. In a simple distillation process the liquid composition changes dynamically because the vapors are richer in the more light components than the liquid from which they came. The path of liquid compositions starting from some initial condition is called a residue curve, and the collection of all such curves for a given mixture is called a residue curve map.1 An RCM contains the same information as a phase diagram for a mixture. Barbosa and Doherty6 and Ung and Doherty2 have derived autonomous differential equations describing the dynamics of simple homogeneous reactive distillation using a set of transformed composition variables. However all these works involved the use of the equilibrium model assuming that the liquid and vapor phase composition are in equilibrium and that there are no differences between the interface and bulk composition profiles. The real reactive separation process operates distantly from the physical equilibrium resulting in mass transfer fluxes between phases (nonequilibrium phase) as a function of the mass transfer gradient. Castillo and Towler13 established a general relationship between the vapor and liquid compositions that leave a tray at total reflux condition to take into account the mass transfer effect in the nonreactive RCM. They assume that the behavior of a stage column could be approximated to a packed column because is has been demonstrated that residue curves represent operating liquid composition profiles of continuous columns at a total reflux condition.10 This approach is used by Taylor et al.14 for the nonreactive separation case to calculate equilibrium RCM and composition trajectory maps (CTM) considering mass transfer effects. Sridhar et al.,15,16 addressed departures from equilibrium to draw nonequilibrium composition trajectories and locate azeotropes. They conclude that the stationary points of these models are the same, but nonequilibrium modeling is necessary to compute distillation boundaries. This research and development (R&D) note is organized as follows. First, a system of equations is established and discussed * To whom correspondence should be addressed. E-mail: [email protected].

to incorporate mass transfer effects and design aspects to calculate composition curve maps for reactive separation processes. Next, a strategy is established to solve the differential algebraic equation (DAE) system for the nonequilibrium reactive composition curve maps, and the case when stationary reactive points calculated by equilibrium and nonequilibrium approaches do not match is shown for the methyl tert-butyl ether (MTBE) production. For tertiary amyl methyl ether (TAME) synthesis, the nonequilibrium and equilibrium reactive composition curve maps in the limit of reaction equilibrium are reported. Derivation of the Equations Doing a component material balance (plug flow model) for the vapor phase moving through the tray17 (Figure 1): dVi ) -NiaAb dh

(1)

Where Vi is the molar flow rate of component i, Ni is the mass transfer flux of component i, a the interfacial area per unit volume of froth, and Ab is the active bubbling area

Figure 1. Diagram of the froth on a distillation tray.

10.1021/ie8013082 CCC: $40.75  2009 American Chemical Society Published on Web 03/02/2009

Ind. Eng. Chem. Res., Vol. 48, No. 7, 2009 3679

Summing (1) and considering that ∑Vi ) V,

[KOV]-1 ) [κV]-1 +

dV ) -NtaAb dh

dyi dV + yi ) -(JVi + yiNt)aAb dh dh

(3)

Substituting (2) into (3): V

[M][κL]-1

(7)

dyi ) -JVi aAb dh

(4)

cLt is the total molar concentration for liquid phase, [M] is the matrix of equilibrium constant [M] ) [K][Γ], where [K] is a diagonal matrix of the vapor liquid equilibrium ratios Ki ) γiPSi /P, and [Γ] is the thermodynamic factor matrix, Γij ) δij + xi

∂ln γi ∂xj

|

δij ) T,P

{

1, i ) j 0, i * j

d(y) 1 ) cVt [KOV](y* - y)aAb dh V

d(y) ) -(JV)aAb dh

(5)

(yL)

(yE)

Now, defining (J ): (JV) ) cVt [KOV](y - y*)

(6)

where cVt is the total molar concentration for vapor phase, y* is the equilibrium vapor molar composition with a bulk liquid composition, and [KOV] is the overall mass transfer coefficient matrix defined as

(9)

Integrating (9) over the dispersion height:

∫

V

(8)

[κV] and [κL] are the mass transfer coefficients matrices (c - 1 × c 1) for vapor an liquid phase respectively; they are obtained using the AIChE method18 with the modification of Bennett et al.19 Substituting (6) into (5)

Combining (4) in (c -1)-dimensional matrix form V

cLt

(2)

and Ni ) JVi + yiNt, Vi ) yiV, substituting these two definitions in (1): V

cVt

d(y) ) (y* - y)

∫

hf

0

1 V c [K ]aA dh V t OV b

(10)

(y* - yL) ) exp[-NOV] (y* - yE)

(11)

(y* - yL) ) [Q](y* - yE)

(12)

Where [Q] ) exp[-NOV] and [NOV] is the overall number of transfer units for the vapor phase [NOV] )

∫

hf

0

1 V c [K ]aA dh V t OV b

(13)

Rearrange (12) (yL) - (y*) ) [Q](yE) - [Q](y*)

(14)

Adding (yI) and subtracting (yE) on both sides of (14) (yL - yE) ) (y*) - [Q](y*) - (yE) + [Q](yE)

(15)

(yL - yE) ) [[I] - [Q]](y* - yE)

(16)

Defining [E] ) [I] - [Q] then, (yL - yE) ) [E](y* - yE) Figure 2. Coordinate system. Table 1. Tray Specifications system tray type Weir height (hw) downcomer area (Ad) bubbling area (Ab) total tray area Weir length (W) downcomer width (Wd) liquid flow path length (Z) hole pitch (p) hole diameter (dh)

MTBE sieve 0.092 m 0.047 m2 0.50 m2 0.60 m2 0.59 m 0.11 m 0.65 m 0.015 m 0.005 m

TAME sieve 0.092 m 0.041 m2 0.93 m2 1.00 m2 0.62 m 0.09 m 0.95 m 0.015 m 0.005 m

Table 2. Reactive Saddle Point Coordinates

X1 X2 X3 T (K)

Differentiating (17) with respect to z, where z ) z′/D is the dimensionless coordinate respect to total diameter of the stage. The coordinate system is shown in Figure 2 Assuming the matrix [E] is constant, [E]

d(yL) d(yE) d(y*) ) + [[E] - [I]] dz dz dz

EQ model

0.035091 0.010644 0.954265 357.65

0.008008 0.002103 0.989889 357.55

(18)

assuming that d(yE)/dz ) 0,20 where vapor is completely mixed between trays and the direction of liquid flow on successive trays is immaterial. Then (18) is simplified to [E]

NEQ model

(17)

d(yL) d(y*) ) dz dz

(19)

The equilibrium vapor composition is related with the liquid bulk composition trough the expression (y*) ) [M](x) + (b)

(20)

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Figure 3. Nonequilibrium reactive composition curves (solid red lines) and equilibrium reactive composition curves (dashed blue lines) in transformed composition variables for MTBE synthesis.

Differentiating (20) with respect to z where [M] and (b) are independent of z d(x) d(y*) ) [M] dz dz

(21)

Performing a material balance at steady state at any point in the z′ direction using Figure 9.4 in the work of Lockett20 (also see Figures 1 and 2), and considering chemical reaction in liquid phase, VE(yE)dz′

+ L′(x)| z′ - (ν)RhL dz′ )

VL(yL)

dz′+L′(x)| z′+∆z′ (22)

Where L′ ) L/W, V′ ) V/Ab, (ν) is the vector of stoichiometric coefficients, and R is the rate of reaction. Substituting L′ and V′ into (22) VL VE d(Lx) ) W(y)L W(y)E + WhL(ν)R dz′ Ab Ab

dL W d(x) + (x) ) V (yL - yE) + WhL(ν)R dz′ dz′ Ab

Replacing (27) in (19) d(yL) WDhLR WD ) [E][Λ](ν - νTx) [E][Λ](yL - yE) + dz Ab V (28) Where [Λ] ) (V/L)[M]. At total reflux condition, yE ) x, and x is independent of yL and z. Defining (YL) ) (yL - yE), A ) (WD)/Ab, B ) (WDhLR)/V and substituting in (28) d(YL) ) A[E][Λ](YL) + B[E][Λ](ν - νTx) dz

(29)

Solving (29) as a first order matrix differential equation with (YL) ) (YL0) at z ) 0 as initial conditions, where (YL0) ) (yL0 - yE) and yL0 is the composition of the vapor above the liquid at the tray exit (z ) 0)

(24)

(YL) )

Doing a total mass balance dL ) WhLνTR dz′

)

(23)

Assuming that V ) VL ) VE (nonheat effects) L

(

WDhLR WD V d(y*) ) [M] (yL - yE) + (ν - νTx) (27) dz Ab L L

(25)

B [exp[A[E][Λ]z] - [I]](ν - νTx) + A exp[A[E][Λ]z](YL0) (30)

j L) as the average vapor composition above the liquid, Defining (Y j (YL) ) (yjL - yE) j L) ) (Y

νT is the net stoichiometric coefficient, νT ) ∑νi Substituting (25) and changing z′ to z in (24)

∫

1

0

(YL) dz

(31)

Combining (30) with (31) and solving WDhLR WD V d(x) ) (yL - yE) + (ν - νTx) dz Ab L L Substituting (26) into (21)

(26)

j L) ) B [[exp[[E][Λ′]] - [I]][Λ′]-1[E]-1 - [I]](ν (Y A νTx) + [exp[[E][Λ′]] - [I]][Λ′]-1[E]-1(YL0) (32)

Ind. Eng. Chem. Res., Vol. 48, No. 7, 2009 3681

Where [Λ′] ) A[Λ]. Defining [E ′] as a square matrix ((c 1) × (c - 1)) of multicomponent Murphree tray efficiencies relative to [Λ′] MV

[EMV′] ) [exp[[E][Λ′]] - [I]][Λ′]-1

(33)

If A ) 1 that is for a rectangular tray of width W and length Z, eq 33 is the same expression of multicomponent Murphree tray efficiencies defined by Taylor and Krishna.17 Substituting (33) into (32) j L) ) B [[EMV′][E]-1 - [I]](ν - νTx) + [EMV′][E]-1(YL0) (Y A (34)

the differential equations that govern the continuous distillation columns at total reflux condition. We use the same approach for the reactive case, taking the transformed composition space that model the simple reactive separation residue curves2,4 with a nonequilibrium relation between y and x bulk compositions given by eq 37. Now, using the transformed composition variables X and Y, and the representation of residue curve maps in the transformed composition variables2 that model the simple reactive separation process, Xi )

Rewriting (34) in terms of yL, yL0, and yE (yjL - yE) )

B MV′ [[E ][E]-1 - [I]](ν - νTx) + A [[EMV′][E]-1(yL0 - yE) (35)

Using (17) with y* ) y0* in (35) (yjL - yE) )

B MV′ [[E ][E]-1 - [I]](ν - νTx) + [EMV′](y* 0 - yE) A (36)

Again, at total reflux condition (x) ) (yE) and (y*) ) [K](x), eq 36 can be expressed as B (yjL) ) [[EMV′][E]-1 - [I]](ν - νTx) + [[I] + [EMV′][K] A [EMV′]](x) (37) The second term in the right of (37) is the (c -1)-dimensional matrix form of eq 14 in the work of Castillo and Towler.13 For the nonreactive case, Van Dongen and Doherty10 demonstrated the similarity between finite difference equations of the simple distillation residue curves with the solutions of

Yi )

xi - (νi)[νref]-1(xref) 1 - (νT)[νref]-1(xref) yi - (νi)[νref]-1(yref) 1 - (νT)[νref]-1(yref)

dXi ) Xi - Yi dτ

i ) 1, ...,c - R

(38)

i ) 1, ...,c - R

(39)

i ) 1, ...,c - R - 1

(40)

Where [νref]-1 is the inverse of the square matrix of stoichiometric coefficients for the R reference components in the R reactions,

[

ν(c-R+1)1 · · · ν(c-R+1)R νir l [νref] ) l νc1 · · · νcR

]

(xref) and (yref) are column vectors of dimension R, (xref) )

( )

xc-R+1 l , xc

(yref) )

( ) yc-R+1 l yc

(νi) and (νT) are row vectors of dimension R, (νi) ) (νi1, νi2, · · · ,νiR),

(νT) ) (νT1, νT2, · · · ,νTR)

Figure 4. Phase diagram in transformed composition variables with temperature for MTBE synthesis at P ) 11 atm: liquid phase (solid red lines) and vapor phase (dashed blue lines). The nonequilibrium reactive saddle point is indicated by an arrow.

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The transformed molar fractions satisfied the summation equations, c-R

∑X

i

)1

(41)

i)1

reaction is highly selective for MTBE production only with the presence of other olefins,22 which operates as an inert component, such as n-butane: i-butene + MeOH + n-butane a MTBE + n-butane (c1) (c2) (c4) (c3) (45)

c-R

∑Y

i

)1

(42)

i)1

The temperature of the system is given by the thermodynamic reaction equilibrium equation, c

KR )

∏ (γ x )

Vi

(43)

i i

i)1

[

∆G◦R(T) RT

]

(

R ) 4464 exp(-3187/T(K)) (ai-butene)(aMeOH) aMTBE -8

8.33 × 10

Where the reaction equilibrium constant KR is given by KR ) exp -

The thermodynamic equilibrium constant, the reaction rate constant, and the rate equation were taken from the work of Venimadhavan et al.23 The rate model that describes the kinetics of MTBE synthesis catalyzed by H2SO4 is

(44)

Solution Strategy To obtain the nonequilibrium (NEQ) composition maps, it is necessary to solve a system of differential and algebraic equations (DAE). The algorithm by Ung and Doherty4 is used but is modified in the way that the relation between y and x described by eq 37 is used. This increases the number of algebraic equations because now y, V, and L appear as implicit variables. The thermodynamic factor matrix [Γ] is calculated with the Wilson model; the vapor and liquid mass transfer coefficients, and the interfacial area Anet are obtained as described in the work of Ruiz et al.21 An important issue is the incorporation of design aspects into the NEQ reactive model. These design aspects are summarized in Table 1. They were calculated for the MTBE and TAME nonequilibrium reactive separation processes.21 Case Study 1: MTBE The MTBE ((CH3)3COCH3) is produced by liquid phase esterification reaction from methanol (MeOH) and i-butene. The

exp(6820/T(K))

)

(46)

where T is the temperature in Kelvin, and a is the activity. The nonideality of the liquid phase is represented by the Wilson equation using the thermodynamic data taken from Table 3.3 in the work of Barbosa5 and Table 3 in the work of Ung and Doherty.4 The MTBE synthesis is used as a case study to draw the equilibrium and nonequilibrium curve maps. The reference component (zref) is MTBE, and the vectors of stoichiometric coefficients are ν ) [ -1; -1; +1; 0] and νT ) -1. Using eq 38, the transformed composition variables are obtained: X1 )

x1 + x3 1 + x3

(47)

X2 )

x2 + x3 1 + x3

(48)

X4 )

x4 1 + x3

(49)

The equilibrium and nonequilibrium residue composition maps for MTBE synthesis at P ) 11 atm are shown in Figure 3. The

Figure 5. Nonequilibrium reactive composition curves (solid red lines) and equilibrium reactive composition curves (dashed blue lines) in transformed composition variables for TAME synthesis.

Ind. Eng. Chem. Res., Vol. 48, No. 7, 2009 3683

composition map is a triangle where each corner represents a pure component of the system and each side of the triangle represents a binary mixture, two nonreactive (n-butanemethanol and n-butane-i-butene) and one reactive side (methanol-i-butene). The two types of models converge to one nonreactive azeotrope at T ) 354.27 K in the n-butane-methanol axis. The convergences of these two models does not take place for the other stationary point which is a saddle point as shown in transformed composition phase diagram (see Figure 4). The saddle point, in the nonequilibrium composition domain, is referred to as the nonequilibrium saddle point. This azeotrope appears in the middle of the reactive side (methanol-i-butene); for all practical purposes, it is not possible to separate beyond this point, and the nonequilibrium saddle point exhibits the same property.4 Figure 4 shows the relationship between liquid phase (solid red lines), vapor phase (dashed blue lines), and the temperature of the system. Here it is possible to visualize both the nonreactive azeotrope and the stationary point. When the differential part of the DAE system is set equal to zero (steady-state condition), the system is solved as a nonlinear algebraic equation system. In this study, we found a reactive saddle point in the vicinity of the n-butane vertex using the EQ (in this case this is the reactive azeotrope) and NEQ models but unlike in nonreactive reactive case the two models do not converge to the same point. Table 2 shows the reactive saddle point coordinates for both models. The reactive azeotrope point has been reported previously by Ung and Doherty2 and Taylor et al.11 using the EQ model approach. The effect of the nonequilibrium calculations is prevalent in the neighborhood of the reactive azeotrope.

(

kf 4 ) (1 + K3)(1.9769 × 1010)exp -

Reaction 2: 2MB2 + MeOH a TAME Reaction 3: 2MB1 a 2MB2

Reaction 4: (2.0)MeOH + 2MB1 + 2MB2a(2.0)TAME

(53)

Since the isomerization (reaction 3) is very fast in comparison to the TAME reactions,24 the rate model for reaction 4 is

(

a2M1B 1 aTAME 2 aMeOH K1 a MeOH

K3 ) 0.648e899.9/T

(57)

and

In addition the thermodynamic equilibrium constants are taken from the work of Oost et al.25 For this study, we have to consider two simultaneous reactions: the TAME synthesis from the isoamylenes (eq 53) and the isomerization (eq 52), 2A1+A2 + A3 a 2A4

(58)

A2 h A3

(59)

where methanol is component A1, 2-methyl-1-butene (2MB1) is component A2, 2-methyl-2-butene (2MB2) is component A3, and TAME is component A4, with the presence of n-pentane (A5) as inert. The composition degrees of freedom for this reactive system is two (c - R - 1 ) 5 - 2 - 1 ) 2), and the reactive composition curve map can be represented in a twodimensional transformed composition coordinates. Two reference components must be chosen; A2 and A3 are suitable choices since [νref] is nonsingular, then (xref) )

)

(54)

And for a catalyst activity of 1.2 (equiv H+)/(kg catalyst):27

[

() x2 x3

-1 -1 -1 1

]

and (νT) ) (-2, 0) The transformed composition variables are X1 )

x1 - x3 - x2 1 - x3 - x2

(60)

X4 )

x4 + x2 + x3 1 - x3 - x2

(61)

X5 )

x5 1 - x3 - x2

(62)

(51)

Only two of the above three reactions are independent. Adding eqs 50 and 51,

R4 ) kf 4

(56)

(50)

(52)

(55)

K1 ) 1.057 × 10-4e4273.5/T

[νref] )

Reaction 1: 2MB1 + MeOH a TAME

)

The equilibrium constants28

Case of Study 2: TAME We now illustrate the equilibrium and nonequilibrium composition curve maps in the limit of reaction equilibrium for the synthesis of tertiary amyl methyl ether (TAME). TAME is generated from methanol and a mixture of isoamylenes 2-methyl-1-butene (2MB1) and 2-methyl-2-butene (2MB2) reacting in liquid phase using a sulfonic acid ion-exchange resin as catalyst24-26 and n-pentane as inert. Three reactions take place simultaneously,

10764 T

Only two transformed variables are independent due to eq 41 and X1 and X5 are chosen as independent variables. For this set of coordinates, the composition space is contained by a trapezoid. The equilibrium and nonequilibrium composition maps for TAME synthesis at P ) 2.5 atm are shown in Figure 5. The two types of composition maps localize one nonreactive azeotrope at T ) 330.089 K in the methanol-n-pentane side that remains after the reaction. We deduce from this residue composition map that there are no reactive azeotropes. Discussion of Results For the MTBE system, we have both reactive and nonreactive azeotropes. In this case, the equilibrium and nonequilibrium mole fraction curves converge to the same nonreactive azeotrope as expected. The saddle points for the nonequilibrium mole fraction curves do not coincide with the saddle point of the

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equilibrium curves which happens to be the reactive azeotrope. The nonlinearity imposed by the mass transfer is by itself not strong enough to change the location of the singular point, which happens to be the nonreactiVe azeotrope. In the case of the reactiVe azeotrope, the combined nonlinearity of the mass transfer equations and the reaction is strong enough to change the location of the stationary point away from the reactiVe azeotrope, and this is one of the important messages conVeyed by this paper. In the case, of the TAME mixture both the equilibrium and nonequilibrium composition curve maps converge at the nonreactive azeotrope as expected. Conclusions We have derived a new expression to relate liquid and vapor bulk composition for reactive separation processes at total reflux condition in terms of the mass transfer coefficients and parameters related to column hardware to draw nonequilibrium composition maps. In this work it was also demonstrated that the saddle points in the MTBE synthesis for the equilibrium and nonequilibrium composition maps are not the same, contrary to nonreactive separation systems where all stationary points are similar for the two models. For TAME synthesis, the nonequilibrium and equilibrium reactive composition curve maps in the limit of reaction equilibrium were reported and the two models converged to one nonreactive azeotrope. Acknowledgment This work was supported by NSF (through Grant No. CTS 0341608) and the AGEP Puerto Rico program. Literature Cited (1) Doherty, M. F.; Malone, M. F. Conceptual Design of Distillation Systems; McGraw-Hill: New York, 2001; p 568. (2) Ung, S.; Doherty, M. F. Calculation of residue curve maps for mixtures with multiple equilibrium chemical reactions. Ind. Eng. Chem. Res. 1995, 34 (10), 3195–3202. (3) Ung, S.; Doherty, M. F. Theory of phase equilibria in multireaction systems. Chem. Eng. Sci. 1995, 50 (20), 3201–3216. (4) Ung, S.; Doherty, M. F. Vapor-liquid phase equilibrium in systems with multiple chemical reactions. Chem. Eng. Sci. 1995, 50 (1), 23–48. (5) Barbosa, D. Distillation of ReactiVe Mixtures; University of Massachusetts: Amherst, MA, 1987. (6) Barbosa, D.; Doherty, M. F. The simple distillation of homogeneous reactive mixtures. Chem. Eng. Sci. 1988, 43 (3), 541–550. (7) Lucia, A.; Taylor, R. The geometry of separation boundaries: I. Basic theory and numerical support. AIChE J. 2006, 52 (2), 582–594. (8) Doherty, M. F.; Perkins, J. D. On the dynamics of distillation processes--III: The topological structure of ternary residue curve maps. Chem. Eng. Sci. 1979, 34 (12), 1401–1414. (9) Pham, H. N.; Doherty, M. F. Design and synthesis of heterogeneous azeotropic distillations. I. Heterogeneous phase diagrams. Chem. Eng. Sci. 1990, 45 (7), 1823–1836.

(10) Van Dongen, D. B.; Doherty, M. F. Design and synthesis of homogeneous azeotropic distillations. 1. Problem formulation for a single column. Ind. Eng. Chem. Fundam. 1985, 24 (4), 454–463. (11) Taylor, R.; Miller, A.; Lucia, A. Geometry of Separation Boundaries: Systems with Reaction. Ind. Eng. Chem. Res. 2006, 45 (8), 2777– 2786. (12) Mulopo, J. L.; Hildebrandt, D.; Glasser, D. Reactive column profile map topology: Continuous distillation column with non-reversible kinetics. Comput. Chem. Eng. 2008, 32 (3), 622–629. (13) Castillo, F. J. L.; Towler, G. P. Influence of multicomponent mass transfer on homogeneous azeotropic distillation. Chem. Eng. Sci. 1998, 53 (5), 963–976. (14) Taylor, R.; Baur, R.; Krishna, R. Influence of mass transfer in distillation: Residue curves and total reflux. AIChE J. 2004, 50 (12), 3134– 3148. (15) Sridhar, L. N.; Maldonado, C.; Garcia, A. M. Design and analysis of nonequilibrium separation processes. AIChE J. 2002, 48 (6), 1179–1191. (16) Sridhar, L. N.; Maldonado, C.; Garcia, A.; Irizzarry, J. Heterogeneous nonequilibrium mole fraction curve maps. Ind. Eng. Chem. Res. 2005, 44 (8), 2845–2847. (17) Taylor, R.; Krishna, R. Multicomponent Mass Transfer; WileyInterscience: New York, 1993; p 616. (18) Distillation Subcomittee. Bubble-tray Design Manual; prediction of fractionation efficiency; American Institute of Chemical Engineers: New York, 1958; p 94. (19) Bennett, D. L.; Agrawal, R.; Cook, P. J. New Pressure Drop Correlation for Sieve Tray Distillation Columns. AIChE J. 1983, 29 (3), 434–442. (20) Lockett, M. J. Distillation Tray Fundamentals; Cambridge University Press: Cambridge, 1986; p 256. (21) Ruiz, G.; Diaz, M.; Sridhar, L. N. Singularities in Reactive Separation Processes. Ind. Eng. Chem. Res. 2008, 47 (8), 2808–2816. (22) Rehfinger, A.; Hoffmann, U. Kinetics of methyl tertiary butyl ether liquid phase synthesis catalyzed by ion exchange resin--I. Intrinsic rate expression in liquid phase activities. Chem. Eng. Sci. 1990, 45 (6), 1605– 1617. (23) Venimadhavan, G.; Buzad, G.; Doherty, M. F.; Malone, M. F. Effect of kinetics on residue curve maps for reactive distillation. AIChE J. 1994, 40 (11), 1814–1824. (24) Oost, C.; Hoffmann, U. Synthesis of tertiary amyl methyl ether (TAME): microkinetics of the reactions. Chem. Eng. Sci. 1996, 51 (3), 329– 340. (25) Oost, C.; Sundmacher, K.; Hoffmann, U. Synthesis of tertiary amyl methyl ether (TAME): equilibrium of the multiple reactions. Chem. Eng. Technol. 1995, 18 (2), 110–117. (26) Rihko, L. K.; Krause, O. I. Kinetics of heterogeneously catalyzed tert-amyl methyl ether reactions in the liquid phase. Ind. Eng. Chem. Res. 1995, 34 (4), 1172–1180. (27) 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 (8), 1029–1043. (28) Rihko, L. K.; Linnekoski, J. A.; Krause, A. O. I. Reaction equilibria in the synthesis of 2-methoxy-2-methylbutane and 2-ethoxy-2-methylbutane in the liquid phase. J. Chem. Eng. Data 1994, 39 (4), 700–704.

ReceiVed for reView April 29, 2008 ReVised manuscript receiVed February 10, 2009 Accepted February 10, 2009 IE8013082