Nonequilibrium Reactive Mole Fraction Curve Maps

RESEARCH NOTES

Nonequilibrium Reactive Mole Fraction Curve Maps

Gerardo Ruiz and Lakshmi N. Sridhar*

Chemical Engineering Department, UniVersity of Puerto Rico, Mayaguez Puerto Rico 00681-9046

In this research and development (R&D) note, we derive new expressions to calculate the nonequilibriumcomposition curve maps for reactive separation processes incorporating mass transfer effects. The strategy tosolve the differential algebraic equation (DAE) system to find the nonequilibrium reactive mole fraction curvesis 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 studiedby several workers1-12 in the design and synthesis of reactiveand nonreactive separation processes. The RCM are used toestablish feasible splits by distillation of azeotropic mixturesdue to the presence of nonreactive azeotropes after the reaction,reactive azeotropes, and distillation boundaries for continuousdistillation at infinite reflux. In a simple distillation process theliquid composition changes dynamically because the vapors arericher in the more light components than the liquid from whichthey came. The path of liquid compositions starting from someinitial condition is called a residue curve, and the collection ofall such curves for a given mixture is called a residue curvemap.1 An RCM contains the same information as a phasediagram for a mixture. Barbosa and Doherty6 and Ung andDoherty2 have derived autonomous differential equations de-scribing the dynamics of simple homogeneous reactive distil-lation using a set of transformed composition variables. Howeverall these works involved the use of the equilibrium modelassuming that the liquid and vapor phase composition are inequilibrium and that there are no differences between theinterface and bulk composition profiles. The real reactiveseparation process operates distantly from the physical equi-librium resulting in mass transfer fluxes between phases(nonequilibrium phase) as a function of the mass transfergradient.

Castillo and Towler13 established a general relationshipbetween the vapor and liquid compositions that leave a tray attotal reflux condition to take into account the mass transfer effectin the nonreactive RCM. They assume that the behavior of astage column could be approximated to a packed columnbecause is has been demonstrated that residue curves representoperating liquid composition profiles of continuous columns ata total reflux condition.10 This approach is used by Taylor etal.14 for the nonreactive separation case to calculate equilibriumRCM and composition trajectory maps (CTM) considering masstransfer effects. Sridhar et al.,15,16 addressed departures fromequilibrium to draw nonequilibrium composition trajectories andlocate azeotropes. They conclude that the stationary points ofthese models are the same, but nonequilibrium modeling isnecessary to compute distillation boundaries.

This research and development (R&D) note is organized asfollows. First, a system of equations is established and discussed

to incorporate mass transfer effects and design aspects tocalculate composition curve maps for reactive separationprocesses. Next, a strategy is established to solve the differentialalgebraic equation (DAE) system for the nonequilibrium reactivecomposition curve maps, and the case when stationary reactivepoints calculated by equilibrium and nonequilibrium approachesdo 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 curvemaps in the limit of reaction equilibrium are reported.

Derivation of the Equations

Doing a component material balance (plug flow model) forthe vapor phase moving through the tray17 (Figure 1):

Where Vi is the molar flow rate of component i, Ni is the masstransfer flux of component i, a the interfacial area per unitvolume of froth, and Ab is the active bubbling area

* To whom correspondence should be addressed. E-mail:[email protected]. Figure 1. Diagram of the froth on a distillation tray.

dVi

dh) -NiaAb (1)

Ind. Eng. Chem. Res. 2009, 48, 3678–36843678

10.1021/ie8013082 CCC: $40.75 2009 American Chemical SocietyPublished on Web 03/02/2009

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

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

Substituting (2) into (3):

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

Now, defining (JV):

where ctV 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

ctL is the total molar concentration for liquid phase, [M] is the matrix

of equilibrium constant [M] ) [K][Γ], where [K] is a diagonal matrixof the vapor liquid equilibrium ratios Ki ) γiPi

S/P, and [Γ] is thethermodynamic factor matrix,

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

Substituting (6) into (5)

Integrating (9) over the dispersion height:

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

Rearrange (12)

Adding (yI) and subtracting (yE) on both sides of (14)

Defining [E] ) [I] - [Q] then,

Differentiating (17) with respect to z, where z ) z′/D is thedimensionless coordinate respect to total diameter of the stage.The coordinate system is shown in Figure 2

Assuming the matrix [E] is constant,

assuming that d(yE)/dz ) 0,20 where vapor is completely mixedbetween trays and the direction of liquid flow on successivetrays is immaterial. Then (18) is simplified to

The equilibrium vapor composition is related with the liquidbulk composition trough the expression

Figure 2. Coordinate system.

Table 1. Tray Specifications

system MTBE TAMEtray type sieve sieveWeir height (hw) 0.092 m 0.092 mdowncomer area (Ad) 0.047 m2 0.041 m2

bubbling area (Ab) 0.50 m2 0.93 m2

total tray area 0.60 m2 1.00 m2

Weir length (W) 0.59 m 0.62 mdowncomer width (Wd) 0.11 m 0.09 mliquid flow path length (Z) 0.65 m 0.95 mhole pitch (p) 0.015 m 0.015 mhole diameter (dh) 0.005 m 0.005 m

Table 2. Reactive Saddle Point Coordinates

NEQ model EQ model

X1 0.035091 0.008008X2 0.010644 0.002103X3 0.954265 0.989889T (K) 357.65 357.55

dVdh

) -NtaAb (2)

Vdyi

dh+ yi

dVdh

) -(JiV + yiNt)aAb (3)

Vdyi

dh) -Ji

VaAb (4)

Vd(y)dh

) -(JV)aAb (5)

(JV) ) ctV[KOV](y - y*) (6)

[KOV]-1 ) [κV]-1 +ct

V

ctL

[M][κL]-1 (7)

Γij ) δij + xi

∂ln γi

∂xj|T,P

δij ) { 1, i ) j0, i * j

(8)

d(y)dh

) 1V

ctV[KOV](y* - y)aAb (9)

∫(yE)

(yL) d(y)(y* - y)

) ∫0

hf 1V

ctV[KOV]aAb dh (10)

(y* - yL)

(y* - yE)) exp[-NOV] (11)

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

[NOV] ) ∫0

hf 1V

ctV[KOV]aAb dh (13)

(yL) - (y*) ) [Q](yE) - [Q](y*) (14)

(yL - yE) ) (y*) - [Q](y*) - (yE) + [Q](yE) (15)

(yL - yE) ) [[I] - [Q]](y* - yE) (16)

(yL - yE) ) [E](y* - yE) (17)

[E]d(y*)

dz)

d(yL)

dz+ [[E] - [I]]

d(yE)

dz(18)

[E]d(y*)

dz)

d(yL)

dz(19)

(y*) ) [M](x) + (b) (20)

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

Differentiating (20) with respect to z where [M] and (b) areindependent of z

Performing a material balance at steady state at any point inthe z′ direction using Figure 9.4 in the work of Lockett20 (alsosee Figures 1 and 2), and considering chemical reaction in liquidphase,

Where L′ ) L/W, V′ ) V/Ab, (ν) is the vector of stoichiometriccoefficients, and R is the rate of reaction. Substituting L′ andV′ into (22)

Assuming that V ) VL ) VE (nonheat effects)

Doing a total mass balance

νT is the net stoichiometric coefficient, νT ) ∑νi

Substituting (25) and changing z′ to z in (24)


Replacing (27) in (19)

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)

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 liquidat the tray exit (z ) 0)

Defining (YjL) as the average vapor composition above the liquid,(YjL) ) (yjL - yE)

Combining (30) with (31) and solving

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

d(y*)dz

) [M]d(x)dz

(21)

VE′(yE)dz′ + L′(x)|z′ - (ν)RhL dz′ ) VL

′(yL) dz′+L′(x)|z′+∆z′(22)

d(Lx)dz′ )

VL

AbW(y)L -

VE

AbW(y)E + WhL(ν)R (23)

Ld(x)dz′ + dL

dz′ (x) ) VWAb

(yL - yE) + WhL(ν)R (24)

dLdz′ ) WhLνTR (25)

d(x)dz

) WDAb

VL

(yL - yE) +WDhLR

L(ν - νTx) (26)

d(y*)dz

) [M](WDAb

VL

(yL - yE) +WDhLR

L(ν - νTx)) (27)

d(yL)

dz) WD

Ab[E][Λ](yL - yE) +

WDhLR

V[E][Λ](ν - νTx)

(28)

d(YL)

dz) A[E][Λ](YL) + B[E][Λ](ν - νTx) (29)

(YL) ) BA

[exp[A[E][Λ]z] - [I]](ν - νTx) +

exp[A[E][Λ]z](YL0) (30)

(YjL) ) ∫0

1(YL) dz (31)

(YjL) ) BA

[[exp[[E][Λ′]] - [I]][Λ′]-1[E]-1 - [I]](ν -

νTx) + [exp[[E][Λ′]] - [I]][Λ′]-1[E]-1(YL0) (32)

3680 Ind. Eng. Chem. Res., Vol. 48, No. 7, 2009

Where [Λ′] ) A[Λ]. Defining [EMV′] as a square matrix ((c -1) × (c - 1)) of multicomponent Murphree tray efficienciesrelative to [Λ′]

If A ) 1 that is for a rectangular tray of width W and length Z,eq 33 is the same expression of multicomponent Murphree trayefficiencies defined by Taylor and Krishna.17


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

Using (17) with y* ) y0* in (35)

Again, at total reflux condition (x) ) (yE) and (y*) ) [K](x), eq36 can be expressed as

The second term in the right of (37) is the (c -1)-dimensionalmatrix 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 equationsof the simple distillation residue curves with the solutions of

the differential equations that govern the continuous distillationcolumns at total reflux condition. We use the same approachfor the reactive case, taking the transformed composition spacethat model the simple reactive separation residue curves2,4 witha nonequilibrium relation between y and x bulk compositionsgiven by eq 37.

Now, using the transformed composition variables X and Y,and the representation of residue curve maps in the transformedcomposition variables2 that model the simple reactive separationprocess,

Where [νref]-1 is the inverse of the square matrix of stoichio-metric coefficients for the R reference components in the Rreactions,

[νref] ) [ν(c-R+1)1 · · · ν(c-R+1)R

l νir lνc1 · · · νcR

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

(xref) ) (xc-R+1

lxc

), (yref) ) (yc-R+1

lyc

)(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 vaporphase (dashed blue lines). The nonequilibrium reactive saddle point is indicated by an arrow.

[EMV′] ) [exp[[E][Λ′]] - [I]][Λ′]-1 (33)

(YjL) ) BA

[[EMV′][E]-1 - [I]](ν - νTx) + [EMV′][E]-1(YL0)

(34)

(yjL - yE) ) BA

[[EMV′][E]-1 - [I]](ν - νTx) +

[[EMV′][E]-1(yL0 - yE) (35)

(yjL - yE) ) BA

[[EMV′][E]-1 - [I]](ν - νTx) + [EMV′](y0* - yE)

(36)

(yjL) ) BA

[[EMV′][E]-1 - [I]](ν - νTx) + [[I] + [EMV′][K] -

[EMV′]](x) (37)

Xi )xi - (νi)[νref]

-1(xref)

1 - (νT)[νref]-1(xref)

i ) 1, ...,c - R (38)

Yi )yi - (νi)[νref]

-1(yref)

1 - (νT)[νref]-1(yref)

i ) 1, ...,c - R (39)

dXi

dτ) Xi - Yi i ) 1, ...,c - R - 1 (40)


The transformed molar fractions satisfied the summationequations,

The temperature of the system is given by the thermodynamicreaction equilibrium equation,

Where the reaction equilibrium constant KR is given by

Solution Strategy

To obtain the nonequilibrium (NEQ) composition maps, it isnecessary to solve a system of differential and algebraicequations (DAE). The algorithm by Ung and Doherty4 is usedbut is modified in the way that the relation between y and xdescribed by eq 37 is used. This increases the number ofalgebraic equations because now y, V, and L appear as implicitvariables. The thermodynamic factor matrix [Γ] is calculatedwith the Wilson model; the vapor and liquid mass transfercoefficients, and the interfacial area Anet are obtained as describedin the work of Ruiz et al.21 An important issue is theincorporation of design aspects into the NEQ reactive model.These design aspects are summarized in Table 1. They werecalculated for the MTBE and TAME nonequilibrium reactiveseparation processes.21

Case Study 1: MTBE

The MTBE ((CH3)3COCH3) is produced by liquid phaseesterification reaction from methanol (MeOH) and i-butene. The

reaction is highly selective for MTBE production only with thepresence of other olefins,22 which operates as an inert compo-nent, such as n-butane:

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

where T is the temperature in Kelvin, and a is the activity. Thenonideality of the liquid phase is represented by the Wilsonequation using the thermodynamic data taken from Table 3.3in the work of Barbosa5 and Table 3 in the work of Ung andDoherty.4

The MTBE synthesis is used as a case study to draw theequilibrium and nonequilibrium curve maps. The referencecomponent (zref) is MTBE, and the vectors of stoichiometriccoefficients are ν ) [ -1; -1; +1; 0] and νT ) -1.

Using eq 38, the transformed composition variables areobtained:

The equilibrium and nonequilibrium residue composition mapsfor 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 transformedcomposition variables for TAME synthesis.

∑i)1

c-R

Xi ) 1 (41)

∑i)1

c-R

Yi ) 1 (42)

KR ) ∏i)1

c

(γixi)Vi (43)

KR ) exp[-∆GR◦(T)

RT ] (44)

i-butene + MeOH + n-butane a MTBE + n-butane(c1) (c2) (c4) (c3)

(45)

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

aMTBE

8.33 × 10-8 exp(6820/T(K))) (46)

X1 )x1 + x3

1 + x3(47)

X2 )x2 + x3

1 + x3(48)

X4 )x4

1 + x3(49)


composition map is a triangle where each corner represents apure component of the system and each side of the trianglerepresents a binary mixture, two nonreactive (n-butane-methanol and n-butane-i-butene) and one reactive side (metha-nol-i-butene).

The two types of models converge to one nonreactiveazeotrope at T ) 354.27 K in the n-butane-methanol axis. Theconvergences of these two models does not take place for theother stationary point which is a saddle point as shown intransformed composition phase diagram (see Figure 4). Thesaddle point, in the nonequilibrium composition domain, isreferred to as the nonequilibrium saddle point. This azeotropeappears in the middle of the reactive side (methanol-i-butene);for all practical purposes, it is not possible to separate beyondthis point, and the nonequilibrium saddle point exhibits the sameproperty.4 Figure 4 shows the relationship between liquid phase(solid red lines), vapor phase (dashed blue lines), and thetemperature of the system. Here it is possible to visualize boththe nonreactive azeotrope and the stationary point.

When the differential part of the DAE system is set equal tozero (steady-state condition), the system is solved as a nonlinearalgebraic equation system. In this study, we found a reactivesaddle point in the vicinity of the n-butane vertex using the EQ(in this case this is the reactive azeotrope) and NEQ modelsbut unlike in nonreactive reactive case the two models do notconverge to the same point. Table 2 shows the reactive saddlepoint coordinates for both models. The reactive azeotrope pointhas been reported previously by Ung and Doherty2 and Tayloret al.11 using the EQ model approach. The effect of thenonequilibrium calculations is prevalent in the neighborhoodof the reactive azeotrope.

Case of Study 2: TAME

We now illustrate the equilibrium and nonequilibrium com-position curve maps in the limit of reaction equilibrium for thesynthesis of tertiary amyl methyl ether (TAME). TAME isgenerated from methanol and a mixture of isoamylenes 2-meth-yl-1-butene (2MB1) and 2-methyl-2-butene (2MB2) reacting inliquid phase using a sulfonic acid ion-exchange resin ascatalyst24-26 and n-pentane as inert. Three reactions take placesimultaneously,

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

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

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

The equilibrium constants28

and

In addition the thermodynamic equilibrium constants are takenfrom the work of Oost et al.25 For this study, we have to considertwo simultaneous reactions: the TAME synthesis from theisoamylenes (eq 53) and the isomerization (eq 52),

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 thisreactive system is two (c - R - 1 ) 5 - 2 - 1 ) 2), and thereactive composition curve map can be represented in a two-dimensional transformed composition coordinates. Two refer-ence components must be chosen; A2 and A3 are suitable choicessince [νref] is nonsingular, then

(xref) ) (x2

x3)

[νref] ) [-1 -1-1 1 ]

and

(νT) ) (-2, 0)

The transformed composition variables are

Only two transformed variables are independent due to eq 41and X1 and X5 are chosen as independent variables. For this setof coordinates, the composition space is contained by atrapezoid. The equilibrium and nonequilibrium compositionmaps for TAME synthesis at P ) 2.5 atm are shown in Figure5. The two types of composition maps localize one nonreactiveazeotrope at T ) 330.089 K in the methanol-n-pentane sidethat remains after the reaction. We deduce from this residuecomposition map that there are no reactive azeotropes.

Discussion of Results

For the MTBE system, we have both reactive and nonreactiveazeotropes. In this case, the equilibrium and nonequilibriummole fraction curves converge to the same nonreactive azeotropeas expected. The saddle points for the nonequilibrium molefraction curves do not coincide with the saddle point of the

Reaction 1:2MB1 + MeOH a TAME (50)

Reaction 2:2MB2 + MeOH a TAME (51)

Reaction 3:2MB1a 2MB2 (52)

Reaction 4:(2.0)MeOH + 2MB1 + 2MB2a(2.0)TAME (53)

R4 ) kf 4(a2M1B

aMeOH- 1

K1

aTAME

aMeOH2) (54)

kf 4 ) (1 + K3)(1.9769 × 1010)exp(-10764T ) (55)

K1 ) 1.057 × 10-4e4273.5/T (56)

K3 ) 0.648e899.9/T (57)

2A1+A2 + A3a 2A4 (58)

A2h A3 (59)

X1 )x1 - x3 - x2

1 - x3 - x2(60)

X4 )x4 + x2 + x3

1 - x3 - x2(61)

X5 )x5

1 - x3 - x2(62)


equilibrium curves which happens to be the reactive azeotrope.The nonlinearity imposed by the mass transfer is by itself notstrong enough to change the location of the singular point,which happens to be the nonreactiVe azeotrope. In the case ofthe reactiVe azeotrope, the combined nonlinearity of the masstransfer equations and the reaction is strong enough to changethe location of the stationary point away from the reactiVeazeotrope, and this is one of the important messages conVeyedby this paper. In the case, of the TAME mixture both theequilibrium and nonequilibrium composition curve maps con-verge at the nonreactive azeotrope as expected.

Conclusions

We have derived a new expression to relate liquid and vaporbulk composition for reactive separation processes at total refluxcondition in terms of the mass transfer coefficients andparameters related to column hardware to draw nonequilibriumcomposition maps. In this work it was also demonstrated thatthe saddle points in the MTBE synthesis for the equilibriumand nonequilibrium composition maps are not the same, contraryto nonreactive separation systems where all stationary pointsare similar for the two models. For TAME synthesis, thenonequilibrium and equilibrium reactive composition curvemaps in the limit of reaction equilibrium were reported and thetwo models converged to one nonreactive azeotrope.

Acknowledgment

This work was supported by NSF (through Grant No. CTS0341608) and the AGEP Puerto Rico program.

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

Accepted February 10, 2009

IE8013082


Nonequilibrium Reactive Mole Fraction Curve Maps

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