1-A Comparison of Steady-state Eq and Rate-based Models

8/2/2019 1-A Comparison of Steady-state Eq and Rate-based Models

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A C o m p a r i s o n o f S t e a d y - S t a t e E q u i l i b r i u m a n d R a t e - B a s e d M o d e l s

f o r P a c k e d R e a c t i v e D i s t i l la t i o n C o l u m n s

J i a nj un P e ng , Se ba s t i e n L e x t r a i t , T ho m a s F . E dg a r , * a nd R . B r uc e E l dr i dg e

Dep ar tm en t of Ch em ica l E n gin eeri n g, T h e S epa ra ti on s R esea rch Pr ogr am , T h e U n iv ers it y of T exa s at A u st in ,

A u st in , T X 78 71 2-1 06 2

A steady-state equilibrium model and a rate-based model were developed and compared forpacked r eactive distillation column s for th e pr oduction oftert-amyl methyl ether (TAME) andm e t h yl a ce t a t e . F or t h e m e t h yl a ce t a t e s ys t e m , b ot h m od e ls y ie ld g ood a g r ee m en t wit hexperimental data. The results predicted by the equilibrium and rate-based models are similarwith very few differences found under all simulation conditions. However, the rate-based modelis mu ch more complicated th an the equilibrium model and also more difficult to converge. Theinfluence of the reflux ra tio, th e operat ing pressure, th e cat alyst am ount, an d th e heat loss wasstu died. It was found t ha t r eactive distillation behaves very differen tly from ordinar y distillat ion.The existence of an optimal reflux ratio and an optimal pressure is predicted by both models.

1 . I n t r o d u c t i o n

T her e has been m uch i nt er est i n t he m odel i ng of reactive distillation since successful industrial applica-

tions were intr oduced in the 1980s. A reliable model isimportant for understanding the behavior of reactivedistillation. Two differen t t ypes of models ar e ava ilablein th e litera tur e for rea ctive distillation: the equ ilibriummodel a nd the rate-based model (or nonequilibriummodel). The equilibrium models1,2 assum e vapor-liquidequilibrium at each sta ge. The depa rtu re from equilib-rium is accounted for by tray efficiency (tray columns)or the height equivalent of a theoretical plate (HETP,packed column s). The ra te-based models3-8 assume thatthe vapor-liquid equilibrium occurs only at the inter-face and use the Maxwell-Stefan equa tion to describet he m ass t r ansf er bet w een t he vapor phase and t heliquid phase. Lee and Dudukovic9 compared an equi-l ibr ium m odel w it h a r at e-based m odel for a t r ay

reactive distillation column for the production of ethylacetate. They concluded that the rate-based model ispreferred because the Murphree tray efficiency is dif-ficult to predict a priori. However, no experimenta l dat awere available to support their conclusion that rate-based m odel should be pr efer r ed. B aur et al .8 alsocompar ed an equilibrium m odel with a r ate-based modelfor r eactive distillation. The meth yl tert-butyl ether(MTBE) and ethylene glycol systems were st udied forboth models. They found that there are multiplicitiesin both the equilibrium model and the rate-based modelbut that the window within which steady-state mul-tiplicity is observed is much narrower with the rate-based model.

The objective of this paper is to compare the equilib-r i um m odel w i t h t he r at e-based m odel f or packedreactive distillation columns, focusing on t emperat urean d composition pr ofiles. The cont ra st bet ween rea ctivedistillation and ordinary distillation is also discussed.Both the tert-amyl methyl ether (TAME) system a nd th emethyl acetate system were studied with the equilib-rium m odel and the rate-based model. For th e methylacetate system, the simulation results from the equi-

librium model and the rate-based model were comparedto the experimental data published by Popken et al .10

I t w as f ound t hat t he r esul t s pr edi ct ed by bot h t heequilibrium model and the rate-based model agree with

the experimental data reasonably well.

2 . E qu i l i b r i u m M o d e l a n d R a t e - B a s e d M o d e l

The equilibrium m odel used in t his pa per consists ofthe conventional MESH equations. The reactions areassumed to be pseudohomogeneous. The HETPs of boththe reactive packing a nd the nonreactive pa cking arechosen empirically, thu s determining the number of theoretical stages used in the simulation.

The rate-based model used in this paper follows theapproach of Taylor et al.11 for conventional distillationin combinat ion with t he a ssump tions listed below. Thepacked r eactive distillation column is vertically dividedinto a number of control volumes. Each control volume

contains l iquid pha se, vapor phase, a nd catalyst; ishomogeneous in temperature and composition; and isreferred to as a segment in t his paper . The configurationfor each segment is shown in Figure 1. The followingassu mpt ions were ma de to simplify th e model: (1) Ea chphase is perfectly mixed in each segment. (2) Vapor-liquid equilibrium is assu med only at the interface. (3)At the liquid-catalyst interface, a pseudohomogeneousreaction is assumed. Thus, reaction and diffusion insideth e catalyst ar e not considered. (4) The finite-flux ma ss-tra nsfer coefficients a re assu med to be the sa me as thelow-flux mass-transfer coefficients. This assumption hasbeen just ifi ed by sim ul at ion r esult s . (5) T he heat -tra nsfer coefficients a re a ssumed to be consta nt for a llsegments. This assumption has also been justified by

simulation results. (6) The condenser and the reboilerare treated a s equilibrium st ages.

The equations for the equilibrium model and the rate-based m odel are given in Tables 1 an d 2 for comparison.The equat ions u sed to calculate physical an d tra nsportproperties12-15 can be found in the Supporting Informa-tion.

The overall mass-transfer coefficient approach is asimplified rate-based approach in which the tempera-ture in the l iquid phase is assumed to be the same astha t in the vapor pha se. The mass -tra nsfer coefficientsfor the liquid phase and the vapor phase are combined

* Corresponding author. Phone: +1 (512) 471 3080. Fax:+1 (512) 471 7060. E-mail: edgar@ma il.utexas.edu.

2735In d . E n g. Ch em . R es. 2002, 41, 2735-2744

10.1021/ie010969b CCC: $22.00 2002 American Ch emical SocietyPublished on Web 04/24/2002


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to calculate the overall mass-transfer coefficients. Thetemperature and composition at the interface are notvariables in the overall ma ss-transfer coefficient ap-proach.

Both the equilibrium a nd rat e-based m odels (includ-ing th e overall mas s-tran sfer coefficient a pproach) wereimplemented in gPROMS, an equation-oriented simula-tor. Radfrac an d Rat efrac in AspenPlus were a lso usedto carry out simulations. The results from AspenPluswere compa red with t hose from th e models in th is work.The advan tage of implement ing models in an equat ion-oriented simulator is that the model is known exactlyand can be m odifi ed t o t est t he vali di t y of var iousassumptions. There is also more flexibility when con-vergence problems a re encoun tered. H owever, one dis-advant age i s t hat sol vi ng a m odel i n a n equat i on-oriented simulator is usually slower and convergenceis usually poorer than for a sequential simulator with

predefined modules, such as AspenPlus.

3 . S i m u l a t i o n R e s u l t s

Two packed reactive distil lation columns for theproduction of TAME and methyl acetate were used tocarry out simulations with the equilibrium and rate-based models developed in this work. The purpose ofthe simulations is to compare the equilibrium modelwith the rate-based model under various conditions andto stu dy th e beha vior of reactive distillation.

There a re five components in t he TAME system:meth an ol, 2-meth yl-1-but ene (2M1B), 2-meth yl-2-but ene(2M2B), tert-amyl meth yl ether (TAME), and n -pentane.There ar e thr ee reversible reactions in th is system, theetherification reactions of 2M1B and 2M2B and theisomerization reaction between 2M1B a nd 2M2B. Thekinet ic model of Rihko et al.16 was used t o calculate thereaction rates. The UNIQUAC equation was used tocalculate the liquid activity coefficients with binaryinteraction para meters from AspenPlus (version 10.2).

The r eactive distillation column configurat ion at theSeparations Research Program (SRP) facility in Austin,Texas (UT-SRP), was used as the basis for simulationsof the TAME system. The inner diameter of the columnis 0.1615 m. The reactive packing is KATAMAX, andth e nonr eactive packing is FLE XIPAC, both from Koch-Glitsch. The height of the packing for each section isshown in Figure 2.

The methyl acetate system was the first commercialapplication of reactive distillation. There are four com-ponents in th is system: meth an ol, acetic acid, methylacetate, and water. There is only one reversible reactionin this system, the esterification reaction of acetic acidwith met ha nol. The kinet ic model of Popken et al.17 wa sused to calculate the reaction rate, and the UNIQUACequat i on w as used t o cal cul at e t he l iquid act i vi t ycoefficients with binary intera ction pa ram eters fromPopken et al.17

The vapor-liquid equilibrium (VLE) of the methylacetate system is much more complicated than that ofthe TAME system because of the dimerization of aceticacid in the vapor phase. The approach of Marek18 wa sfollowed t o calculate the VLE. The VLE experiment aldata for this system published by Sawistowski et al.19

were used to check the accuracy of the VLE equationsu s ed in t h is p a pe r . T h e a v er a ge e r ror is 0 .0 14 incomposition and 1.54 K in temperature.

The one-feed setup and two-feed synthesis setup usedby Popken et al.10 for reactive distillation experimentswere used for the simulations of the methyl acetatesystem. The inner diameter of the column is 0.05 m.The reactive packing is Kata pak-S, and t he n onrea ctive

packing is Sulzer BX, both from Sulzer Chemt ech Ltd.3 .1 . Co m p a r i s o n w i t h R a d f r a c a n d R a t e f r a c . To

check the models developed in this paper, both Radfrac(equilibrium model) and Ratefrac (rate-based model) inAspenPlus were used for simulations of th e TAMEsystem and t he methyl acetate system. The temperatur eand composition profiles of the TAME system from theequilibrium model were compared with those fromRadfrac under the same conditions, with excellentagreement. Run number 1-12 of the one-feed setup inPopken et al .10 was used to compare the equilibriummodel with Radfrac for t he m ethyl acetate system. Theagreement was also excellent.

The temperature and composition profiles of th e

TAME system predicted by th e ra te-based m odel are inexcellent agreement with those from Ratefrac under thesame conditions, even though different methods wereused to calculate the diffusion coefficients. The detailsof t he com par isons can be found i n t he S uppor t i ngInforma tion (Figures 1s, 2s, and 3s).

3 .2 . I n f l u e n c e o f t h e N u m b e r o f S e g m e n t s . Whenthe number of segments for the rate-based model ischosen t o be t he sam e as t he num ber of t heor et i calstages for th e equilibrium model, th ere is a significan tdifference in the temperature and composition profilespredicted by the two models for both the TAME and themethyl acetate systems (see Figures 3 and 4, respec-tively). However, th e differen ce in th e pr ofiles betweenthe t wo models at t he t op and t he bottom of the columnis less tha n th at in the m iddle of the column. When th enumber of segments for the rate-based model is in-creased, both the tempera tur e an d composition profileschange significan tly (see Figures 5 an d 6). The num berof segm ent s can be consider ed as a par am et er t hataccounts for backmixing in the disti llation column.When t he num ber of segm ent s i s ext r em el y l ar ge,essentially no backmixing occurs in the disti llationcol um n, and t her efor e, t he separ at i on i s t he best .Because backmixing and other nonideal conditions doexist in real columns, an appropriate number of seg-ments should be used. However, that number cannotbe determined in advance. When experimental dat a a reavailable, the simulation results can be compared to

F i g u r e 1 . Schematic diagram of a ra te-based segment.

2736 Ind. Eng. Chem. Res., Vol. 41, No. 11, 2002


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experimental data , and the optimal num ber of segmentscan be determined. The influence of the n umber of segments for ordinary distillation without reaction wasalso studied an d was found to be much weaker.

3 . 3 . R e l a t i o n s h i p b e t w e e n t h e E q u i l i b r i u m a n dR a t e -B a s e d M o d e l s . There is a relationship betweent he equi li br i um and r at e-based m odels. When t henumber of segments in the rate-based model is chosen

to be the same as the number of theoretical stages inthe equilibrium model and th e vapor-liquid interfacialarea is increas ed, the pr ofiles from th e ra te-based modelbecome closer to those from the equilibrium model.When the vapor-l iquid interfacial area is about 100times as large as the actual area, the profiles from therat e-based m odel are a lmost ident ical to th ose from th eequilibrium model (see Figure 7). This is reasonable

T a bl e 1 . E q u a t io n s f o r t h e C o n d e n s e r a n d R e b o i l e r

total condenser (j ) 1) t ota l r eboiler (j ) N)

material balance Mi1 ) V2y i,2 - (1 + r1L

)L 1x i,1 ) 0 (1 ) Mi,N ) L 1x i,N-1 - VNy i,N - LNx i,N ) 0 (2 )

equilibrium Qi,N ) y i,N - Ki,Nx i,N ) 0 (3 )

summation S x1 )i)1

c

x i,1 - 1 ) 0 (4 ) S xN )i)1

c

x i,N - 1 ) 0 (5 )

S yN )i)1

c

y i,N - 1 ) 0 (6 )

energy balance E1 ) V2H2V - (1 + r1L)L 1H1L - Qc ) 0 (7 ) EN ) LN-1HN-1L - VNHNV - LNHNL + Qr ) 0 (8 )variables xi1, Qc, L 1, x i,N, y i,N, Qr, VN, TNequations Mi1, E1, S x1, Mi,N, Qi,N, EN, S xN, S yN

T a b l e 2 . E q u a t i o n s f o r th e jt h S t a g e s ( S e g m e n t s )

E qu ilibr iu m Model Ra t e-Ba sed Model

Material Balance

Mi,j ) Vj+1y i,j+1 + Lj-1xi,j-1 + Fjxi,jF-

Vjy i,j - Ljxi,j - WcjR i,j ) 0 (9 )

Mi,jV) Vj+1yi,j+1 - Vjy i,j -

4DT

2lsjaejNi,j

m) 0 (1 0)

Mi,jL) Lj-1x i,j-1 + Fjxi,j

F- Ljxi,j +

4DT

2lsjaejNi,j

m- WcjR i,j ) 0 (1 1)

E qu ilibr iu m E qu ilibr iu m a t t h e In t er fa ceQi,j ) Ki,jx i,j - yi,j ) 0 (1 2) Qi,j

I) yi,j

I- Ki,jxi,j

I) 0 (1 3)

Summation

S xj )i)1

c

x i,j - 1 ) 0 (1 4) S xj )i)1

c

xi,j - 1 ) 0 S yj )i)1

c

yi,j - 1 ) 0 (1 6)

S yj )i)1

c

y i,j - 1 ) 0 (1 5) S xjI)

i)1

c

xi,jI- 1 ) 0 S yj

I)

i)1

c

yi,jI- 1 ) 0 (1 7)

Energy Balance

Ej ) Vj+1Hj+1V+ Lj-1Hj-1

L+ FjHj

F-

VjHjV- LjHj

L- Qj ) 0 (1 8)

EjV) Vj+1Hj+1

V- VjHj

V-

4DT

2lsjaej[hj

V(Tj

V- Tj

I) +

i)1

c

Ni,jm

Hh i,jV

] ) 0 (1 9)

EjL) Lj-1Hj-1

L+ FjHj

LF- LjHj

L- Qj +

4DT

2lsjaej[hj

L(Tj

I- Tj

L) +

i)1

c

Ni,j

mHh

i,j

L] ) 0 (20)

EjI) hj

V(TjV- Tj

I) +i)1

c

Ni,jm

Hh i,jV- hj

L(TjI- Tj

L) -

i)1

c

Ni,jm

Hh i,jL) 0 (21)

Mass-Transfer Rates

RjV) Nj

m- Ctj

Vk

V(yj - yjI) - yj

i)1

c

Ni,jm) 0 (i ) 1, ..., c-1) (22)

RjL) Nj

m- Ctj

Lk

L(xj

I- xj) - xj

i)1

nc

Ni,jm) 0 (i ) 1, ..., c-1) (23)

The a pproach of Taylor et al.11 is followed to calculate

kL

a n d kV

.Variables

x i,j, yi,j , Lj, Vj, Tj xi,j, y i,j, xi,jI

, yi,jI

,TjL

, TjV

, TjI, Lj, Vj, Ni,j

m

EquationsMi,j, Qi,j, S xj, S yj, Ej Mi,j

L, Mi,j

V, Qi,j, R i,j

L, Ej

L, Ej

V, Ej

I, S xj, S yj, S xj

I, S yj

I, R i,j

V

Ind. Eng. Chem. Res., Vol. 41, No. 11, 2002 2737


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because, when the interfacial area is very large, thevapor and liquid phases sh ould contact each other very

well an d be in equilibrium. This relationship is impor-tant for improving the convergence of the rate-basedmodel. Solving t he rat e-based m odel was always m uchmore difficult than solving the equilibrium model forall of the simulations carried out in this work. Theequilibrium model can be solved first, and the resultscan be used a s th e initial guess for the r ate-based model.Then, a lar ge value can be used for t he inter facial area,and the solution of the rate-based model can be reachedeasily, because the initial guess is close to the solution.After t hat , the int erfacial ar ea can be decreased gradu -ally to the actua l value to yield the solution for t he r ealconfiguration.

3.4. C omparison w ith E xperime ntal D ata. For themethyl acetate system, experimental reactive distilla-t i on dat a ar e avail abl e.10 Therefore, a compar isonbetween the experimental data and model predictionsis sum mar ized in Ta ble 3 (one-feed setu p) and Table 4(t he t w o-feed set up). N o par am et er s i n t he m odelsdeveloped in this paper were fitted from experimentaldat a. F or t he one-feed set up, t he accur acies of t heequilibrium model and th e ra te-based m odel are a lmostthe same. Increasing the number of segments for therate-based model can slightly reduce the error in thepredicted compositions. For the two-feed setup, theaccuracies of the equilibrium and rate-based models are

F i g u r e 2 . Pilot-plant reactive distillation column at UT-SRP.

F i g u r e 3 . Comparison of temperature and composition profilesbetween the equilibrium model and the rate-based model for theTAME system.

F i g u r e 4 . Comparison of temperature and composition profilesbetween the equilibrium model and the rate-based model for themethyl acetate system.



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close in genera l. However, increasing t he n umber of seg-ments for the rate-based m odel increases th e error inthe predicted compositions for run number S-1 signifi-cantly. Therefore, a lar ger nu mber of segment s does notalways yield better results. It is interesting to note that,when a larger n umber of segments is used in t he ra te-based m odel, the r esults a re very close to those obtain edfrom the equilibrium model. For all of the runs, onlyminor differences could be observed between t he r esultsfrom the equilibrium model and the rate-based model.

For the TAME system, no experimental data havebeen published in th e open literat ure. Pilot-plan t r eac-tive distillation experiments will be carried out at theSeparations Research Program (SRP) facility in Austin,Texas (UT-SRP). The experimental data will be com-pared with the predictions from the models developedin this work.

3 . 5 . C o m p a r i s o n o f C o m p u t a t i o n a l E f f o r t . Th erate-based model is m uch m ore complicated than theequilibrium m odel. The num ber of equat ions in th e rat e-based model is 5-7 t imes th e nu mber of equations inthe equilibrium model if the nu mber of segment s in t herate-based model is th e sam e as the n umber of theoreti-cal s t ages i n t he equi li br i um m odel. H ow ever , t he

number of segments in the rate-based model requiredto obtain an accurate solution is usually several timesthe required n umber of theoretical sta ges in th e equi-

F i g u r e 5 . Influence of the number of segments on temperatureand composition profiles for the rate-based model for the TAMEsystem.

F i g u r e 6 . Influence of the number of segments on temperatureand composition profiles for the rate-based model for the methylacetate system.

F i g u r e 7 . Relationship between the equilibrium model and therate-based model for the TAME system.



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librium model. Therefore, th e total nu mber of equa tionsin the ra te-based model is at least an order of ma gnitudehigher t han that in the equilibrium model.

I t is n ot easy to compare the computat ional effortsrequired by the two models because convergence is not

guar an teed for either m odel. If an a rbitra ry initial guessis used for both models, both models will fail to convergein most cases. One comparison method is to use thepreviously converged solution a s t he initial guess an d,therefore, to obtain the CPU time needed to performone iterat ion. Anoth er compa rison is to use t he pr evioussolution as the initial guess an d increase th e feed flowrate by a certain percentage. The convergence of themethyl acetate system is poorer t han tha t of the TAMEsystem becaus e of th e complicat ed VLE equat ions. If th efeed flow rate is increased by more tha n 20%, both theequilibrium model and the rat e-based model frequentlyfail to converge.

A comparison of the computational efforts of the twomodels is given in Tables 5 and 6. It is obvious that therate-based model requires much more computation timethan the equilibrium model and t hat the convergenceof the rate-based model is also poorer than that of theequilibrium model. For the TAME system, when thefeed flow rate was doubled, the rate-based model failedto converge for any number of segments, whereas theequilibrium model converged quickly. It was observedthat, if the num ber of segments is increased, the CPUtime increases almost quadratically.

3 .6 . I n fl u e n c e o f H e a t -T ra n s f e r C o e ff i c i e n t s .There is no good way to calculate the heat-transfercoefficients in a multicomponent distillation column.The Chilton-Colburn ana logy is m ost commonly usedto calculate the heat -tra nsfer coefficients from th e ma ss-

tr an sfer coefficients. However, for mu lticomponent dis-tillation, th e j-factor for mass transfer is a matrix, butt h e j-factor for heat transfer is a scalar. An averagediffusion coefficient could be used to calculate the j-factor for mass transfer. However, the method used to

T a b l e 3 . C o m p a r i s o n o f E x p e r i m e n t a l D a t a w i t h M o d e l P r e d i c t i o n s f o r t h e M e t h y l Ac e t a t e S y s t e m ( On e - F e e d S e t u p )

E Qb RB c E Qb RB c

E XP a E XP ast a ges (segm en ts) 8 16 32 64 8 16 32 64

r u n n u m ber 1-11 1-12xD_MeOH 0.410 0.342 0.342 0.354 0.361 0.347 0.311 0.314 0.320 0.324xD_MeAc 0.584 0.638 0.628 0.625 0.623 0.647 0.681 0.674 0.674 0.673xD_H 2O 0.006 0.020 0.028 0.019 0.015 0.006 0.008 0.012 0.006 0.003xB_MeOH 0.127 0.163 0.157 0.154 0.151 0.003 0.029 0.031 0.026 0.024xB_H AC 0.322 0.294 0.288 0.292 0.293 0.244 0.256 0.261 0.262 0.263xB_MeAc 0.049 0.051 0.062 0.060 0.061 0.000 0.002 0.004 0.003 0.003

xB_H 2O 0.502 0.492 0.493 0.494 0.496 0.753 0.713 0.704 0.709 0.710MeOH con ver sion 0.523 0.524 0.532 0.527 0.526 0.645 0.653 0.648 0.647 0.645T_bot t om , C 83.1 84.2 82.7 82.0 82.0 95.1 98.7 99.1 98.6 98.9

x_error d 0.026 0.028 0.023 0.021 0.016 0.017 0.015 0.015

a Experimental data.10 b Equilibrium model predictions. c Rate-based model predictions. dx_error is the average error of the molefractions a t t he top and the bottom.

T a b l e 4 . C o m p a r i s o n o f E x p e r i m e n t a l D a t a w i t h M o d e l P r e d i c t i o n s f o r t h e Me t h y l A c e t a t e S y s t e m ( T w o - F e e d S e t u p )

E Qb RB c E Qb RB c

E XP a E XP ast a ges (segm en ts) 23 25 50 100 23 25 50 100

r u n n u m ber S-1 S-2xD_MeOH 0.107 0.041 0.086 0.059 0.044 0.028 0.010 0.031 0.017 0.011xD_H AC 0.000 0.000 0.003 0.001 0.001 0.000 0.000 0.005 0.003 0.002xD_MeAc 0.877 0.932 0.876 0.912 0.932 0.966 0.968 0.929 0.955 0.966xD_H 2O 0.016 0.027 0.035 0.028 0.023 0.006 0.022 0.034 0.025 0.020xB_MeOH 0.070 0.069 0.079 0.071 0.066 0.013 0.052 0.067 0.057 0.052xB_H AC 0.104 0.065 0.113 0.082 0.064 0.175 0.136 0.163 0.144 0.135xB_MeAc 0.002 0.000 0.000 0.000 0.000 0.013 0.000 0.000 0.000 0.000xB_H 2O 0.824 0.866 0.808 0.847 0.870 0.812 0.812 0.771 0.799 0.813MeOH con ver sion 0.876 0.889 0.836 0.909 0.889 0.924 0.929 0.892 0.917 0.928T_bot t om , C 91.7 92.4 92.8 92.7 92.7 99.6 95.6 94.9 95.3 95.5

x_error d 0.027 0.010 0.018 0.027 0.016 0.024 0.018 0.016

a Experimental data.10 b Equilibrium model predictions. c Rate-based model predictions. dx_error is the average error of the molefractions a t t he top and the bottom.

T a b l e 5 . C o m p a r i s o n o f t h e C o m p u t a t i o n a l E f f o r t sa o f

t h e T w o M o d e l s f o r t h e T A M E S y s t e m

m odel equ ilibr iu m r a t e-ba sed

stages(segments)

14 14 33 66 132

number ofequations

600 4430 10 225 20 650 41 212

CPU time, s(one it erat ion)

0.16 1.3 7.4 27 129

CPU time, s(F) 20%)

0.41 3.1 15.9 65 306

CPU time, s(F) 50%)

0.55 3.2 16.7 71 fa iled

CPU time, s(F) 100%)

2.51 fa iled fa iled fa iled fa iled

a PIII 550 PC with 512-M memory.

T a b l e 6 . C o m p a r i s o n o f t h e C o m p u t a t i o n a l E f f o r t sa o f

t h e T w o M o d e l s f o r O n e - F e e d S e t u p o f t h e M e t h y lA c e t a t e S y s t e m

m odel equ ilibr iu m r a t e-ba sed

stages(segments)

8 8 16 32 64

number ofequations

407 1899 3707 7323 14 555

CPU time, s(one it erat ion)

0.1 0.4 0.7 2.2 8.9

CPU time, s(F) 10%)

0.4 0.9 2.5 7.7 28.3

a PIII 550 PC with 512-M memory.



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average t he diffusion coefficients is ar bitrary, a nd thecalculated values of diffusion coefficients from differentmethods can differ widely. It was observed from thesimulation results that heat-transfer coefficients do notchange very much over the column length. Therefore,constant values based on the overall column averageswere assumed in the rate-based model. I t was foundthat the influence of the heat-transfer coefficients onthe temperature and composition profiles is negligible.Even if th e values of the heat -tra nsfer coefficients arechanged by an order of magn itude, the largest differencein composition profiles is only 0.0056. Therefore, as-suming consta nt heat -tra nsfer coefficients a ppears jus-tifiable.

3.7. Overall Mass-Transfer Coefficien t Approach .The overall mas s-tran sfer coefficient a pproach was usedfor simulating the TAME system. The configurationused for this simulation is slightly different from thatin Figure 2, as t he 1.2 m of Flexipac packing above th efeed is removed from th e column. It wa s found tha t t heresults are very different from t hose of the rigorous ra te-based m odel and the equilibrium model (Figure 8).Although no experimental data are available for thissystem, t he metha nol composition pr ofile pr edicted byth e overa ll mass-tra nsfer coefficient a pproach is clearlynot correct. Th e overall m ass-tran sfer coefficient ap-

proach is even worse for ordinary disti llation (seeSupporting In forma tion). Therefore, this a pproach sh ouldbe discarded. A similar conclusion was reported byAlopaeus and Aittamaa 20 for conventional tray distil-lation columns.

3 .8 . P a r a m e t r i c B e h a v i o r . The two reactive distil-lation models were used t o study t he influence of severalpara meters, including th e reflux ra tio, pressure, catalystamount, and heat loss. The configuration used for thesesimulations is the same as that in Figure 2 except thatth e 1.2 m of Flexipac packing a bove th e feed is removedf r om t he col um n. I t w as f ound t hat t he par am et r i cbehavior of reactive distillation is very different fromtha t of ordina ry distillation.

The influence of the reflux rat io on product pur ity isshown in Figures 9 a nd 10. Both the equ ilibrium modeland the rate-based model predict an optimal reflux ratioof around 2.2 for the TAME system an d a round 1.5 forthe methyl acetate system. This behavior was validatedexperimentally for the methyl acetate system.10,21 Th eproduct purity initially increased with the reflux ra tio,reached a maximum, and then decreased if the refluxratio was increased further. However, for ordinarydistillation, th e product pur ity should always increase

F i g u r e 8 . Profiles predicted by the overall mass-transfer coef-ficient approach for reactive distillation of the TAME system.

F i g u r e 9 . Influence of the reflux ratio on product purity for theTAME system.

F i g u r e 1 0 . Influence of the r eflux ra tio on pr oduct pu rity for t hemethyl acetate system.



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with increasing reflux ratio as long as the column isoperating normally. The counterintuitive behavior ofreactive distillat ion is caused by th e complex intera ctionbetween reaction and distillation. When the reflux ratiois small , the separation is not good and the productpurity is low. When the reflux ratio is too large, theresidence time in the reactive zone is not long enoughto a llow sufficient product to be generated. Therefore,the product purity decreases.

As shown in Figures 11 a nd 12, both the equ ilibriumm odel and t he r at e-based m odel pr edict an opt im alproduct purity near 3 atm for the TAME system andnear 0.7 at m for t he m et hyl a cet at e syst em . A l owpressure yields a reduced temperatur e in the reactivezone and a low reaction rate. Therefore, little productis generated by rea ction. When t he pr essure is t oo high,the temperature is high enough that the reaction rateis high. However, a higher temperature will favor thereverse reaction and result in lower conversion becausethe forward reaction is exother mic. The product pu ritydecr eases because l ess pr oduct i s gener at ed i n t hereactive zone.

The effect of the amount of catalyst on the reactantconver sion i s show n i n F i gur es 13 and 14. I f m or e

cat al yst i s put i nt o t he packi ng w hi l e t he hei ght of packing is kept t he same, the conversion is higher.However, th e increase in conversion is n ot significan teven if th e am ount of catalyst is doubled. This is causedby the interaction between reaction and disti l lation.When m or e cat al yst i s used, m or e pr oduct w il l begenerated. However, if the product generated by thereaction is too great to be separated from the system,chemical equilibrium will force the reaction in the wrongdirection. Therefore, it is not helpful to put too muchcat al yst i nt o t he packing. A bet t er w ay t o i ncr easeconversion is to increase t he height of the column.

It was observed tha t h eat loss along the column doesnot influence the product purity significantly for a heatloss of up to 60% of the reboiler duty (see SupportingInformation for details). However, the liquid and vaporflow rate profiles change significantly. When heat lossincreases, both t he liquid and vapor flow rat es increaseto offset the heat loss.

4 . Co n c l u s i o n s

A steady-state equilibrium model and a rate-basedmodel were developed for a packed reactive distillation

F i g u r e 1 1 . Influence of the operating pr essure on product pur ityfor the TAME system.

F i g u r e 1 2 . Influence of the operating pr essure on product pur ityfor the methyl acetate system.

F i g u r e 1 3 . Influence of the catalyst amount on conversion forthe TAME system.

F i g u r e 1 4 . Influence of the catalyst amount on conversion forthe methyl acetate system.



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column. The two models were compared for both theT AM E s ys t em a n d t h e m e th y l a c et a t e s ys t em . I ngeneral, the predictions from the equilibrium a nd r ate-based models are similar, with no major differencesbeing found for a range of simulation conditions. How-ever, the rate-based model is much more complicatedthan the equilibrium model and is also more difficultto converge.

It was found that there is a relationship between theequilibrium model and th e ra te-based m odel. When thenumber of segments in the rate-based model is chosento be the same as the number of theoretical stages inthe equilibrium model and th e vapor-liquid interfacialarea is increased, the pr ofiles from th e ra te-based modelapproach th ose from th e equilibrium model. When th evapor-liquid interfacial ar ea is about 100 times a s largeas t he rea l area, th e profiles from th e rat e-based modelar e al m ost i dent ical t o t hose fr om t he equi li br i ummodel.

The hea t-tra nsfer coefficient s in th e rat e-based m odelcan be assum ed t o be const ant for bot h t he T AM Esyst em a nd t he m et hyl acet at e syst em . E ven i f t hevalues of the h eat-tran sfer coefficients are changed byan or der of m agni t ude, t her e i s l i t t l e change i n t he

tempera tur e a nd composition profiles.The two models were used to study the influence of

several parameters, including the reflux ratio, pressure,catalyst amount, and heat loss. I t was found that thebehavior of reactive distillation is very different fromth at of ordinar y distillation. However, the t wo chem icalsystems behave quite similarly. An optimal reflux ratioand an optimal operat ing pressure were found for boththe TAME system and the methyl acetate system. Theoptimal values predicted by t he equilibrium model andthe rat e-based model were very close. The influence ofthe amount of catalyst in the reactive section is notstrong. Increasing the amount of catalyst will increasethe conversion only slightly. It was found that heat lossaffects only the liquid and vapor flow rates inside the

column. It does not affect t he t empera tur e an d composi-tion pr ofiles significan tly for a hea t loss of up t o 60% ofthe reboiler duty.

A c k n o w l e d g m e n t

This project was part ially supported by Aspen Tech-nology, Inc. The first author thanks Dr. Tim Popken forhelp on th e met hyl acetate system, Dr. Ross Ta ylor foranswering general questions on the rate-based model,and Dr. Thomas Badgwell for reviewing the paper.

S u p p o r t i n g I n f o rm a t i o n Av a i l a b l e : More figur esof the simulation results and a table describing the

equat i ons used t o cal cul at e physical and t r anspor tproperties. This material is available free of charge viathe Internet at http://pubs.acs.org.

N o m e n c l a t u r e

ae ) specific effective interfacial area between the vaporand liquid phases (m 2/m 3)

c ) num ber of componentsCt ) total molar concentration (mol/m 3)

DT ) diameter of the column (m)F) feed flow rate (mol/s)

HL ) liquid enthalpy (J/mol)HV ) vapor ent halpy (J/mol)

h L ) liquid h eat-tr an sfer coefficients [W/(m 2 K)]

h V ) vapor heat -tran sfer coefficients [W/(m 2 K)]i ) component index

j ) section indexk L ) liquid mass-transfer coefficient matrix (m/s)k V ) vapor mass-transfer coefficient matrix (m/s)K) vapor-liquid equilibrium constantlsj ) height of section j (m )

Lj ) liquid flow rate in section j (mol/s)N) number of stages (segments) in the column, including

condenser and reboiler

Nm ) mas s flux [mol/(m 2 s)]Qj ) heat loss in section j (J/s)

R i,j ) reaction rate of component i in section j [mol/(kg s)]

TjI) interface temperature of section j (K)

TjL) liquid temperature of section j (K)

TjV) vapor tempera tur e of section j (K)

Vj ) vapor flow rate in section j (mol/s)Wcj ) weight of dry catalyst in section j (kg)

xi,j ) liquid molar fraction of component i in section j

xi,jI) liquid molar fraction of component i at the interface

of section jyi,j ) vapor molar fraction of component i in section j

yi,jI) vapor molar fraction of component i at the interface

of section j

L i t e r a t u r e C i t e d

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transfer a nd m ixing on the performance of a tray column forreactive distillation. Chem . E n g. S ci. 1999, 54 , 2873.(7) Higler, A.; Krishna, R.; Taylor R. Nonequilibrium modeling

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(9) Lee, J.; Duduk ovic, M. A compa rison of th e equilibrium an dnonequilibrium models for a multicomponent reactive distillationcolumn. Comput. Chem . E ng . 1998, 23 , 159.

(10) Popken T.; Steinigeweg S.; Gmehling, J . Synthesis andhydrolysis of methyl acetate by reactive distillation using struc-tured catalytic packings: Experiments and simulation. In d. E n g.Chem . R es. 2001, 40 , 1566.

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(16) Rihko, L.; Kiviranta-Paakkonen, P.; Krause, A. Kineticmodel for the et herificat ion of isoamylene wit h met han ol. In d. E ng .Chem . R es. 1997, 36, 614.

(17) Popken T.; Gotze, L.; Gmehling, J. Reaction kinetics a ndchemical equilibrium of homogeneously and heterogeneously



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catalyzed acetic acid esterification with methanol a nd methylacetate hydrolysis. In d. E n g. Chem . R es. 2000, 39 , 2601.

(18) Marek, J . Vapor-liquid equilibrium in mixtures containingand associating substance. II . Binary mixtures of acetic acid atatmospheric pressure. Collect. Czech . Chem . C o mmu n . 1955, 20 ,1490.

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R eceiv ed for rev iew November 30, 2001

R evi sed m an u scr ip t recei ved February 19, 2002

A ccept ed March 18, 2002

IE010969B


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