simulación de la sintesis de mtbe

8/12/2019 Simulacin de La Sintesis de MTBE

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KINE TIC S, CATALYSIS , AND RE ACTION E NG INE ERI NG

Heterogeneous Models of Tubular Reactors Packed with

Ion-Exchange Resins: Simulation of the MTBE Synthesis

Rosa M. Quinta Ferreira* and Cri stina A. Almeida-CostaDepart ment of Chemi cal E ngi neeri ng, Uni versi t y of Coi mbra, Largo Marques de P ombal ,

3000 Coi mbra, P ort ugal

Alrio E. Rodrigues

Depart m ent of Chemi cal E ngi neeri ng, U ni versi t y of P ort o, Rua dos B ragas, 4099 P ort o Codex, P ort u gal

The study of the beha vior of fixed-bed reactors using ion-excha nge resins a s cat a lysts w a s car riedout by making use of a complete bidimensional heterogeneous model for the reactor, whichincluded the r esista nces inside th e ion-excha nge resin par ticles, considered wit h a ma croreticularstru ctu re. Th e a ctive si te s w e re locate d in sid e th e g e l p h ase of t h e re sin , re pre sen te d b ymicrospheres, and on the ma cropores w a lls. The overa ll efficiency of such heterogeneous cat a lystpart icles w a s defined by the ma croeffectiveness a nd microeffectiveness fa ctors a ccounting forthe process behavior on t he ma cropores a nd inside the m icrospheres. The synt hesis of methy l

tert-butyl ether, MTBE, a liquid-phase reversible exothermic reaction between methanol andisob u te n e, w as con sid ere d a s a re fe ren ce case . Th is syste m w as stu d ied in th e t e m pe ratu rera nge of 313-338 K, a nd t he effect of the t hermodynamic equilibrium conditions wa s examined.The results predicted by the complete heterogeneous model were compared with those obtainedwit h t he simple pseudohomogeneous m odel, wh ich r evea led higher hot spots. Moreover, acom p arison b etw e en b id im e n sion al an d u n id im e n sion al m od e ls wa s also p erform e d . Th eorthogonal collocation method was used for the discretization of the differential equations insidethe cat a lyst pa rticles, wh ich w ere reduced from t hree (corresponding to t he thr ee mass ba lancesfor the thr ee compounds, isobutene, metha nol, a nd MTB E) to only one differentia l equa tion, byusing t he concept of t he generalized va ria ble.

Introduction

Ion-exchange synthetic resins are produced through

a copolymerization procedure with styrene and divinyl-ben ze ne u s ed a s a cros s -l in k ing a g e n t . Th e a c t ivegroups, containing hydrogen ions in the form of sulfonica c id g r ou ps , a r e a t t a ch ed t o t h e p ol y me ri c m a t r i xdeveloped in t he gel pha se by long polystyrene chainsfixed by bridges of divinylbenzene, leading to a sta blean d rigid structure a lso called a solid acid. This paperdeals w ith th e ana lysis of the performa nce of a reactorpacked with such a st rongly a cidic resin, Amberlyst 15.The behavior of the catalyst part icles was studied byu s in g a mode l p res e nt e d by I h m et al . (1982), whichaccounts for th e ma croreticular structure of the resin.The gel phase of the polymeric matr ix wa s a ssumed tobe represented by microspheres of uniform size a nd t hefree space betw een them a ccounted for the ma cropores,

Figure 1. Therefore, tw o types of react ive sites w ereconsidered: a fra ct ion, , lo c a t e d o n t h e wa l ls o f t h ema cropores a nd t he rema ining fract ion, 1- , locatedin s ide t h e micros ph e res . We h a v e a lre a dy u s ed t h ismodel in a previous work (Quinta Ferreira and Rod-rigues, 1993) where the behavior of such a macroreticu-lar resin with a zero-order rea ct ion w as an alyzed, onlyat the pa rt icle level.

It is our objective to illustrate the strategy proposedby Aris (1975) at the ca ta lyst level, for r eactive syst emswit h L a n g mu ir-Hinshelwood kinetics. This method

points out t he use of only one ma ss ba lan ce involving ageneralized variable instead of the various partial massbala nce equations corresponding to t he different reac-t ive compounds usually required for calculat ing their

concentrat ion in each spatial posit ion of the catalyst .The individual composit ion of ea ch species will bere la t e d t o t h a t g en e ra l ize d v a ria ble t h rou g h s implea lgebraic expressions. U sing this methodology in bothtypes of act ive sites of the macroreticular resin, i t ispossible to significantly reduce the number of second-order ODE s to be solved simulta neously. In fa ct , oneneeds to deal with only one differential equation on thema cropore w alls an d a nother inside the microspheres,no matter the number of react ive compounds presentin th e fluid mixture.

The other issue we would like to strengthen in thisw o rk i s t h e e ff ect of t h e r ea c t or op er a t i on u n d ertherm odyna mic equilibrium conditions on th e behaviorof such heterogeneous systems running with reversible

* To w hom correspondence should be a ddressed. Tele-phone: 351-39-28392. Fa x: 351-39-27425.

Figure 1. S ketch of a resin part icle.

3827Ind. Eng. Chem. Res. 1996, 35 , 3827

S0888-5885(96)00242-4 CCC : $12.00 1996 American C hemica l Society


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react ions. In order to fulfill these objectives, we ha vechosen the MTBE synthesis as a reference case due toits industrial importa nce. This ana lysis wa s car ried outconsidering that the heterogeneous model of the catalystparticles was included in a fixed-bed reactor which wassimulated through one- and two-dimensional heteroge-n e ou s mode ls (Q u in t a F e rreira et a l . , 1994). Theimporta nce of ta king into a ccount the cat alyst gra dients

wa s esta blished by compar ing those predictions with theresults obta ined w ith the pseudohomogeneous model,w h e r e a l l t h e i n t e r n a l a n d e x t e r n a l r e s i s t a n c e s a r eneglected. The effect of the therm odyna mic equilibriumcondit ions w as emphasized in the feed range t empera-ture of 313-338 K.

The restrict ive la ws about environmental pollutionrequire the use of unleaded gasolines in automobiles.Amon g t h e a v a i la ble p roce ss es t o re duce t h e lea dcontent , the ad dition of methyl tert-butyl ether (MTBE)wa s re v e a le d t o be o n e o f t h e mo s t in t e re s t in g t e c h -niques; because of its high octane index, only a loweram ount of the dangerous octane enhan cer t etraethylleadmust be used, without changing the performance of theengine. Therefore, higher interest ha s been put in the

production of that compound, which is obtained throughthe reversible reaction between metha nol and isobutenein the liquid pha se:

This moderate exothermic reaction can be catalyzed bymin e ra l a c ids in t h e h o mo g e n e o u s p h a s e , wh ic h h a sbeen associated, however, with quite severe corrosionproblems and difficulties in the separation of the acids.Th u s , a h e t erog e ne ou s p roce s s is u s u a l ly u s ed inindustry, the reaction being performed in multitubularfixed-bed rea ctors wh ere a cidic ion-excha nge resins a reused as solid cat alysts. Isobutene conversions higher

t h a n 98% a re u s u a l ly obt a in e d in in dus t r ia l p la n t s ,where the rea ction temperatur es range betw een 303 and373 K a n d t h e p res su r e b et w een 7 a n d 14 a t m . Acomprehensive review of the studies concerning thekinetics of MTBE synthesis was presented by Hutchingset al. (1992).

The experimental studies concerning the kinetics ofthe MTBE synthesis w ith Amberlyst 15 performed byGicquel and Torck (1983) in the temperature range of323-368 K r ev ea l ed t h a t t h e r ea c t ion t a k es p la c ebetween the isobutene in the solution and the adsorbedmethanol at the catalyst surface; the MTBE producedis then desorbed to th e solution. According t o thesea u t h o rs a n d f o l lo win g t h e me c h a n is m o f L a n g mu ir -Hinshelwood, the expression for t he rea ction r at e refer-

Table 1. Operating Conditions and Reaction, Reactor,and Catalyst Characteristics

ope ra t i ng con d it ion s r ea ct i on

P) 9 a t m -H)37.3 kJ /molu0) 4. 4 10 -3 m /s EB ) 82 kJ /m olCC 4

)3840 m ol/m 3 E*) 122 kJ /molCMeOH /CC 4

)1.10 E) -22 kJ /m olCinerts/CC 4

)1.348 KB0) 2.74 10 7 m 3/(K g ca t s )T0) 31 3-338 K KA0R 0) 9.99 10 15 m ol/(K g ca t s )

R 0) 7.41 10 -5

r ea ct or ca t a lyst

L )7.5 m da) 8 10 -4 mDt) 0.027 m di) 3 10 -8 mFb) 760 kg /m 3 p) 0.36) 0. 4 c) 1.69

Fc) 1980 kg /m sol 3)0.05

C H 3OH +(C H 3)2CdC H 2 h (CH 3)3C H 3

Table 2. Dimensionless Equations for theBidimensional, Heterogeneous Model (HT)

F lu id P h a s e

m a s s b a la n c e

Yk,b

z* ) RkjovD aRb+

L2

R02

1

Perm[

2Yk,b

r*2 + 1

r*

Yk,b

r*] (k)C 4, M eO H , M TB E ) (1)

thermal balance

b

z*) jovD aBRb+

L2

R02

1

Perh[2b

(r*)2+ 1

r*

b

r*] (2)boundary conditions

z*) 0; r* g 0: YC 4 )1; YMeOH)CMeOH

CC 4; YMTBE) 0

b) 1 (3)

z* g 0; r*) 0:Yk,b

r*|r*)0 )b

r*|r*)0 )0

r*) 1:Yk,b

r*|r*)1 )0 (k) C 4, MeOH, MTBE )

-b

r*|r*)1 ) B i(b- ) (4)

Fluid/P ar ticle Interfa ce

thermal balance

s) b+ BNf,C 4

Nfh(YC 4,b

- YC 4,s) (5)

Ca t a ly s t P a r t ic le

macropores

2

Yk,a

xa2 +

2

xa

Yk,a

xa ) -Rkk,a

2

Ra+ 3k,a

2

k,i2

Yk,i

xi|xa ;xi)1(k)

C 4, M eO H , M TB E ) (6)

boundary conditions

z* g 0, r* g 0: xa) 0Yk,a

xa)0

xa) 1 YC 4,s ) YC 4,b - D a

Nf,C 4

ovRs

YMeOH,s) YMeOH,b-Kf,C 4

Kf,MeOH(YC 4,b

- YC 4,s)

YMTBE,s) YMTBE,b- Kf,C 4

Kf,MTBE(YC 4,b- YC 4,s) (7)

microspheres

2

Yk,1

xi2

+ 2xi

Yk,i

xi) -Rk(1- )k,i

2Ri (k)

C 4, M eO H , M TB E ) (8)

boundary conditions

z* g 0, r* g 0, xa g 0: xi) 0Yk,i

xi)0 (9)

xi) 1 Yk,i ) Yk,a (xa )

3828 Ind. E ng. C hem. Res. , Vol. 35, No. 11, 1996


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ring to th e ma ss of cat a lyst (mol/(K g ca t s)) is given by

wh e re KB (m f3/(K gca t s)) and KA (mol/(K g ca t s)) are thef or w a r d a n d b a ck w a r d r a t e con s t a n t s . Th e s u r fa c efractions covered by methanol and MTBE are given by

wh e re bMeOH a nd bMTBE are the adsorption coefficientsfor metha nol a nd MTB E, respectively. The reaction ra tewill then become

with the rate constants defined by

wh e re EB a n d EA a r e t h e a c t iv a t i on e ne rg ie s of t h esynthesis and the decomposition of MTBE, respectively;E) MTBE - MeOH represents the difference betweenthe heats of adsorption of MTBE and methanol, and thetherma l increment E* is given by E*) EA - E.

In t he present st udy, we ha ve used this kinetic model,which wa s a lso pointed out t o be the most a dequat e forthe MTB E synthesis with Amberlyst 15 by the w ork ofAli and B hat ia (1990). Ca etano et al. (1994) presented

a kinetic model very similar t o the one of G icquel an dTorck (1983) for the production of MTBE catalyzed byAmberlyst 18. More recently, ra te expressions of t heL a n g mu ir-Hinshelwood t ype have been reported a s afunction of U NIFAC liquid-phase act ivit ies instead ofconcentra t ions (Rehfinger and Hoffman, 1990a; P ar raet al., 1994), due to the high nonideality of the alcohol-ether-hydrocar bons mixture. For processes in w hichis obu t en e is p res en t in la rg e e xce ss , R e h fin g er a n dH o ff ma n (1990b) a n a ly zed t h e e ff ect of ma c rop orediffusion of metha nol, r epresented by a pseudobinarydiffusion model, on the act ivity and select ivity of thema c rop orou s re s in ca t a ly s t a n d con clu ded t h a t t h eobserved reaction could be significantly affected by thediff u sion a l re sis t a n ce s in t h e ma crop ore s. B e rg a n dHarris (1993) point out that for commercial processes,where the rat io of methanol to isobutene is stoichio-metric, it is necessary to include multicomponent dif-fusion effects.

The opera t ing condit ions, r eact ion para meters, a ndreactor and catalyst characterist ics used in this workar e shown in Table 1.

Model Equations for Steady-State Operation ofFixed-Bed Reactors with Ion-Exchange Resins

The model equations of the system representing thes t e a dy - s t a t e re g ime we re u s e d in t h e dime n s io n le s sform. The concentra tions of isobutene (C4), methanol(M e OH ), a n d me t h y l tert-bu t y l e t h er (M TB E ) we re

Table 3. Model Parameters for the Heterogeneous Model(HT)

Arrhenius no. b ) EBR T0

29

*) E*

R T0

43

) E

RT0

8

Da mkh oler no.D a) FbKB(T0)

5.6

dimensionless adiaba t ictemp rise B)

(-H)CC 4T0Ffcpf

0.3

no. of film mass-transferu n it s Nf,C 4

)1-

Kf,C 4ap

2.26 10 5

no. of film heat-transferu n it s Nfh )

1 -

h ap

Ffcpf3.98 10 3

radial mass Peclet no. Perm )L u0

Der

85812*

radial heat Peclet no.Perh )

L u0Ffcpfer

49122*

wall heat Biot no. B iw)hwR0

er

11.2

pore Thiele modulusk,a ) Ra

Fc(1- p)KB(T0)Dk,a

0.5

microspheres Thielemodulus k,i ) Ri

FcKB(T0)Dk,i

0.1

radical P eclet nos. basedon the part icle diam Perm (dp))

dpu0

Der

9.2

Perh (dp))dpu0Ffcpf

er

7.5

Table 4. Generalized Equations for the CatalystParticle

Ca t a ly s t P a r t ic lemacropores

2

Ua

xa2+ 2

xa

Ua

xa)a

2Ra (Ua )+3

a2

i2

Ri(Ui) 1)

Ra (Ua) 1)

Ui

xi|xa ,xi)1 (10)boundary conditions

z* g 0, r* g 0: xa) 0Ua

xa)0

xa) 1 Ua) 1 (11)

microspheres

2Ui

xi2+ 2

xi

Ui

xi)(1- )i

2Ri(Ui) (12)

Bounda ry condit ions

z* g 0, r* g 0, xa g 0: xi) 0Ui

xi)0

xi) 1 Ui) 1 (13)

w h e r e

a) RaFc(1- p)KB(T0)

C aRa (Ua) 1)

i) RiFcKB(T0)

C iRi(Ui) 1)

R)KBCC 4MeOH- KAMTB E (14)

MeOH) bMeOH CMeOH

bMeOH CMeOH+ bMTB E CMTBE

MTBE) bMTBE CMTB EbMeOH CMeOH+ bMTB E CMTBE(15)

R)KBCC 4CMeOH- KAR CMTBE

CMeOH+ R CMTBE(16)

KB )KB0e-EB/ (R T )

KA )KA0e-EA/ (R T )

R )bMTB E

bMeOH)R 0e

E/ (R T )

KAR ) KA0R 0e-E * / ( R T ) (17)

Ind. Eng . C hem. R es. , Vol. 35, No. 11, 1996 3829


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normalized by the isobutene concentration at the reactorinlet, CC 4: Yk) Ck/CC 4, with k) C 4, MeOH, MTBE; thereduced temperatur es inside the r eactor were r eferredto as the feed temperatur e,T0: ) T/T0. We have alsoused dimensionless ra te equa tions,R, by dividing th ea c t u a l v a lu es , R, by t h e ra t e v a lu e s c a lc u la t e d a t t h econdit ions of the reactor entra nce, R0 (R ) R/R0,where, R0 ) KB(T0)CC 4). U s i n g t h e n o r ma l i ze d v a r i -ables for the concentra t ions and t emperat ures, the rateequation defined by eq 16 will then become

I n o rde r t o re f e r t o t h e v a lu e s o f t h e re a c t io n ra t e s ,concentra t ions, and tempera tures occurring in t he dif-f e re n t p h a s e s o f t h e s y s t e m, we h a v e u s e d di f f e re n tsubscripts: b for the bulk phase, s for the resin surfa ce,a for t he ma cropores, and i for the microspheres.

The dimensionless equations for the bidimensionalheterogeneous model (HT) w hich accounted for themacroreticular structure of the solid particles are writ-ten in Ta ble 2. For the fluid pha se, eq 1 represents th e

three ma ss bala nces for th e three compounds: isobutene,k ) C 4 a nd Rk ) -1; methanol, k ) M e O H a n d Rk )-1; an d methyl tert-butyl ether, k ) MTBE and Rk) 1.The therm al bala nce is represented by eq 2, the bound-ary condit ions of t hese equations being defined by eqs3 and 4.

The catalyst part icles were considered with an iso-thermal behavior, and the solid temperature,s, in eachradial and axial posit ion of the reactor was calculatedthrough a thermal balance in the fluid -part icle inter-f a c e, e q 5 in Ta ble 2. I n s ide t h e s o lid p a rt ic le s, t h eprocess wh ich ta kes place on th e pore wa lls depends onthe diffusional f lux and chemical react ion occurringthere and also on the diffusional f lux going on insidethe microspheres, eqs 6 and 7. The ma ss bala nces forthe microspheres consider t heir internal diffusiona lfluxes an d chemica l reaction, eqs 8 and 9. The reactionra t e s a t t h e re s in s u rf a c e , Rs, a t t h e p o re wa l ls , Ra ,a n d in s ide t h e micros ph e res , Ri, w e r e ca l cu la t e dt h rou g h e q 18 by u s in g t h e re s pe ct iv e n o rma lize dconcentrations Yk, s, Yk,a , a n d Yk,i for each of the threecompounds (k) C 4, MeOH, MTB E) an d considering th enormalized solid temperature (s) in all the cases. Themo de l p a ra me t e rs c a lc u la t e d a t 329 K a re s h o wn inTable 3.

For the macropores and for the microspheres, thethr ee second-order differentia l equa tions, correspondingto the thr ee mass ba lan ces of the th ree compounds, canb e r e d uce d t o on l y on e, b y t a k i n g i n t o a c cou n t a n

adequa te va riable change (Aris, 1975). A generalizedconcentra t ion in the pores, Ua , a n d in s ide t h e micro-spheres, U i, can now replace the d imensionless concen-trat ions, Yk,a a n d Yk, i, through the following equations:

Subst ituting these expressions in eq 18, th e dimension-less react ion rates for the catalyst part icles were alsodefined in terms of the generalized concentrations, Ua

a nd Ui, a s follows:

Moreover, normalized reaction rates for the macropores,Ra , and for the microspheres, Ri, we re u s e d wh ic hwe re o bt a in e d by div idin g t h e a c t u a l v a lu e s , Ra a n dRi, by t h e ra t e s wh ic h wo u ld t a k e p la c e a t t h e c on di-t ions of the resin surface (Yk, a ) Yk, s) for the react ionon the m acropores a nd at the condit ions of the micro-spheres surfa ce (Yk,i ) Yk,a ) for the reaction ra tes insidethe microspheres. In these condit ions, Ua a n d Ui a reequal to one (eq 19), and therefore,

Subst ituting eq 20 in t hese expressions, one finally gets

The variables involved in the above expressions are

defined as follows:

E1) e-(b/s)-1

E2) KA0R 0

K(TB0)CC 4e-

*/s (22)

E3) R 0e/s

C a) ( 1DMeOH,a - E3

DMTBE,a)

-1

C i) ( 1

DMeOH,i -

E3

DMTBE,i)-1

(23)

a) YMeOH,s+ E3YMTBE,s - 1

i) YMeOH,a+ E3YMTBE,a- 1 (24)

a) E1YC 4,s YMeOH,s+

E1(-YC 4,s C aDMeOH,a -YMeOH,sC a

DC 4,a+

C a2

DC 4,a2

DMeOH,a)+E2(-YMTBE,s- C aDMTBE,a )

R)

e-b((1/)-1)

YC 4YMeOH-

KA0R 0KB(T0)CC 4

e(-*/)

YMTBE

YMeOH+ R 0e/YMTBE

(18)

macropores Yk, a) Yk, s+ RkDk, a

(1- Ua )C a

microspheres Yk, i) Yk, a+ RkDk, i

(1- Ui)C i (19)

macropores Ra (Ua ))a+ a Ua+ a Ua

2

a+ Ua

microspheres Ri(Ui))i+ iUi+ iUi

2

i+ U i(20)

Ra (Ua )) Ra (Ua )

Ra (Ua)1) Ri(Ui))

Ri(Ui)

Ri(Ui) 1)

macropores Ra (Ua ))a+ a Ua+ a Ua

2

a+ a+ a

a+ 1a+ Ua

microspheres Ri(Ui))i+ iUi+ iUi

2

i+ i+ i

i+ 1i+ Ui

(21)



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i) E1YC 4,a YMeOH,a+

E1(-YC 4,a C iDMeOH,i - YMeOH,aC a

DC 4,i+

C i2

DC 4,iDMeOH,i)+

E2(-YMTBE,a - C iDMTBE,i) (25)

a) E1

(YC 4,s

C a

DMeOH,a

+YMeOH,sC a

DC 4,a

-

2C a

2

DC 4,aDMeOH,a)+ E2

C aDMTBE,a

i) E1(YC 4,a C iDMeOH,i +YMeOH,aC i

DC 4,i-2

C i2

DC 4,iDMeOH,i)+

E2

C iDMTBE,i

(26)

a) E1C a

2

DC 4,a DMeOH,a

i) E1C i

2

DC 4,iDMeOH,i

(27)

Taking into a ccount t he generalized va riables defineda b ov e, t h e m a s s b a l a n ce s f or t h e m a c rop or es a n dmicrospheres represented by eqs 6-9 in Table 2 werereduced to generalized expressions presented in Table4, eqs 10-13. In this wa y, the three intra part icle ma ssba la n c e e q u a t ion s f or t h e t h re e c omp ou n ds we re re -placed by only one generalized balance for the macroporesand one for the microspheres.

EffectivenessFactors. The overa ll efficiency of the

solid resins has to account for the effectiveness factorsassociat ed wit h t he microspheres,ji, an d th e ma cropores,a , as defined by Ihm et al. (1982) a nd a lso used in ourprevious pa per (Quinta Ferreira a nd R odrigues, 1993).For the microspheres, ji represents the average of theindividual effectiveness factors, i, also called microef-fectiveness factors, which are given by

or

Introducing the dimensionless variables, one gets

Thus, the average microeffectiveness factor, ji, will be

or

Replacing eq 28, it will become Using the dimensionless

varia bles, one gets

For the process taking place in the macropores (reactionand mass transfer), the macroeffect iveness factor, a ,is defined a s

Then,

or

i) observed reaction ra te in t hemicrosphere/int rinsic rea ction ra te in

the m icrosphere a t the sur face conditions

i) Vi

Rid V

i

Ra Vi(28)

i) 301Ri(Ui)xi

2d xi (29)

ji) observed rea ction ra te in a ll themicrospheres/int rinsic rea ction ra te in

a ll the microspheres a t their surfa ce conditions

Figure 2. H T model. Radial mean t emperatur es (a) and isobuteneconcentrat ion profiles (b) in t he bulk phase for different reactorfeed temperatures.

ji)Va [ViRi d Vi] d Va

VaRa Vid Va(30)

ji) Vai

Ra dVa

VaRa dVa(31)

ji)0

1iRa (Ua )xa

2dxa

01Ra (Ua )xa

2dxa

(32)

a) (observed reaction rate in the pores+observed reaction rate in the microspheres)/

(intrinsic reaction rat e in the pores at

concentration Cs+ re action ra te in al l th emicrospheres if t heir surfa ce concentra tion w a s Cs)

a)VaRa dVa+ (1- )jiVaRa dVa

RsVa+(1- )j iRsVa(33)



6/15

Figure 3. H T model. Isobutene conversion in the bulk phase a t t he operat ing condit ions, X, a nd isobutene conversion a t equilibriumconditions, Xeq, a long the reactor axis for four inlet temperatur es: (a) T0) 313 K; (b) T0) 323 K; (c) T0) 329 K; (d) T0) 338 K.

Figure 4. H T m od el . R e a ct or r a t e s in t h e b u lk p h a s e a lon g t h e r e a c t or a x is : g lob a l r e a c t ion r a t e , Rb, direct react ion rate, RB , a n dinverse react ion ra te, RA , for four inlet temperatur es: (a) T0) 313 K; (b) T0) 323 K; (c) T0) 329 K; (d) T0) 338 K.



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The use of the genera lized varia bles, Ua , a l lo ws u s t owri t ea in the following form: The overall effectiveness

factor, ov, is defined as

or

Comparing this expression with eq 34, one can see that

ov c a n a l s o b e ca l cu la t e d a f t e r k n ow i n g a a n d jithrough the following expression:

This overall effectiveness factor referring to the condi-tions at the surfa ce part icles,ov, is related to the overalleffectiveness fa ctor referring to th e bulk conditions, jov,by the reaction rates calculated with the concentrationsand temperatures occurring in the fluid phase, Rb, a n dat the surface of the resin, Rs:

Figure 5. H T model. Radia l mean va lues of the three effect ive-ness factors : (a) overall effect iveness factor, ov; (b) macroeffec-t iveness factor, a ; (c) average microeffectiveness factor, j i.

Figure 6. H T model. Radia l profiles of (a) temperatures, (b)isobutene concentration, and (c) isobutene concentration at localand equilibrium conditions, in the hot spots of the bulk phase forfour different reactor feed temperatures: T0) 313 K (z*) 0.17);T0) 323 K (z*) 0.14); T0) 329 K (z*) 0.08); T0) 338 K (z*)0.04).

a) 1

RsVaVaRa dVa (34)

a) 301Ra (Ua )xa

2 dxa (35)

ov) (observed reaction rate in the pores+observed reaction rate in the microspheres)/

intrinsic reaction rate in all the resin particle(pores+ microspheres) if the concentr a tion

wa s th e on e at t h e surface

ov)VaRa dVa+ (1-)jiVaRa dVa

RsVa(36)

ov)+(1-)ji

RsVaVaRa dVa (37)

ov) a [+(1- )j i] (38)

jov)Rs

Rbov (39)



8/15

The cat a lyst m odel defined here to describe the behaviorof t he r esin part icles with heterogeneous structure (0< < 1) can a lso be used in th e case of uniformly porouscata lysts, by considering ) 1, which will lead to thek n own re la t io n ov ) a . Th a t m od el ca n s t il l b eextended to bidisperse porous catalysts by making )0, which meansov) aji. In the first case, the chemicalreact ion takes place only on the pore walls where theactive sites are concentrated, and in the second case,a l l t h e c a t a ly t ic s i t e s a re lo c a t e d in s ide t h e mic ro -spheres.

Wh e n in t e rn a l a n d e xt e rn a l s ol id re s is t a n c es a reneglected, i.e. , when the catalyst efficiency is assumedto be equal to one, one gets the pseudohomogeneousmodel (P H m odel), wh ich is only described by th e ma ssand t hermal ba lances to the fluid phase. For this model,eqs 1-4 from Table 2 can be used by putt ing jov ) 1.

The numerical solution of the system of the part ialdifferential equations of those bidimensional models was

achieved through the method of lines. The ma ss a ndt h e rma l ba la n c es f or t h e f lu id p h a s e , e q s 1-4 (Table2), were solved by using the pa ckage PD EC OL (Mad sena nd S incovec, 1975), which discretized a utoma tically t hevariables on the reactor radial position, r*, through theorthogonal colloca tion meth od on finite elements usin gcubic B-Splines a s t rial functions. We ha ve used onlyone finite element, w ith t wo int erior colloca tion points,s in ce t h e u s e of t w o s u bi n t er v a ls d id n ot l ea d t osignificative changes on the simulated profiles, requir-ing, however, higher computing t imes. The result ingODEs were solved by t he implicit integrator G EARIB(H i n dm a r s h , 1976) i n cl ud ed i n P D E C O L . F or t h eunidimensional models, the init ial value problem as-sociat ed with the ordinary differential equat ions of the

bulk phase w as solved by using the G EAR code (Hind-ma rsh, 1974). For t he heterogeneous m odels, the in-tra part icle calculat ions were performed, in each ra diala n d /or a x i a l p os it i on of t h e r ea c t or , b y u s in g t h egeneralized equations for the macropores, eqs 10 and11, and for the microspheres, eqs 12 and 13, Table 4(instead of the dimensionless equations 6-9, Table 2).Their numerical solution w as carried out by using theorthogona l colloca tion m ethod described in Appendix A.The effect iveness factors of the resin part icles werecalculated t hrough numerical q uadr at ure following t hemethodology a lso referred t o in Appendix A.

Computer Results

For various feed temperatures, Figure 2 shows thera dia l me a n v a lu e s o f t h e bu lk t e mp e ra t u re in a a n disobutene concentra tion in b along the rea ctor when th ecomplete bidimensiona l het erogeneous model wa s used.Increasing T0, the hot-spot magnitude also increasesan d its locat ion a pproaches the reactor entra nce. Theser es u lt s a l s o s h ow t h a t t h e op t im a l op er a t i n g i n le ttemperature, for which a maximum outlet isobuteneconversion is obtained, is approximately T0 ) 329 K.Co mp a rin g t h e ra dia l me a n v a lu e s o f t h e is o bu t e n e

conversion obtained along the reactor bed,X, with thosewhich would be achieved if the chemical react ion wasat equilibrium in each reactor posit ion, Xeq, F ig u re 3shows tha t t he increase of the feed temperat ure allowsthe reactor operat ion to be performed closer to thoseequilibrium conditions. While for T0) 329 K (Figure3c) the equilibrium condit ions are reached just at thereactor exit , for higher feed t empera tures, a s for T0)388 K (Figure 3d), th ose conditions a re a chieved some-where inside the reactor; after that point, the isobuteneconversion was limited by the equilibrium value calcu-lated a t the sam e temperat ure. Since, for exothermicprocesses, the equilibrium conversion decreases withtempera ture, the operat ing tempera tures a bove 329 Kwill lead t o a decrease of the f inal r eactant conversion.

Th is t h e rmody n a mic e ff ect ca n a ls o be obs erv ed inFigure 4, where the global react ion rates (mean radialv a lu es ) c a lcu la t e d a t t h e bu lk con dit ion s , Rb, a r ecompared with the values of the direct, RB , and inverse,RA , react ion rates (Rb) RB - RA ). The overa ll reactionra t e , Rb, tends to zero for the higher operating temper-at ures when t he equilibrium conditions are a pproa ched,as shown in Figure 4d for T0) 338 K.

Th e e ff icien cy of t h e ca t a ly st ca n be obs erv ed inF ig u re 5, wh e re t h e me a n ra dia l v a lu e s o f t h e t h re eeffectiveness factors a ssociat ed with the ma croreticularresins (ov,a , andji) ar e represented along the cata lyticbed. The globa l efficiency of the ca ta lyst is ma inly dueto the process occurring on the macropores, since theefficiency in the microspheres, j

i, is unity, as shown in

Figure 5c. Therefore, ov follows a very closely, asdepicted in F igure 5a an d b.

For the inlet tempera tures a na lyzed before (T0) 313,323, 329, a n d 338 K ) , F ig u re 6 p res en t s , in a , t h ethermal radial profiles observed in the hot spots and,in b , t h e c o rre s p o n din g ma s s ra dia l p ro f i le s f o r t h eisobutene. One can see that for high temperatures, thet h e rma l ra dia l g ra die n t s ca n be import a n t (40 K forT0)338 K). The mass r adia l gradients for the highestt e mpe ra t u re (cu rv e D ) a re les s p ron ou n ce d t h a n t h eones for lower temperatures (curves B and C), due totherm odynam ic equilibrium effects. Compar ing in sepa-ra te plots (Figure 6c) these ra dia l isobutene concentra -tion profiles, YC 4,b , t o t h e corre sp on din g p rof iles i f

Figure 7. HT model. Concentra tion profiles for isobutene insidethe microspheres located at the part icle surface (xa ) 1 ) in t h ehot spot , (a) a t the rea ctor center, r*) 0; and (b) a t the rea ctorw a l l , r*) 1, for four inlet temperatures: T0) 313 K (z*) 0.17);T0) 323 K (z*) 0.14); T0) 329 K (z*) 0.08); T0) 338 K (z*)0.04).



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equilibrium condit ions were reached, Yeq, i t c a n b eobserved tha t such conditions a re approached when t hefeed t empera tures increase. The reverse react ion isthen favored, leading to higher isobutene concentra-t ion s , w h ich h a p p en s p redomin a n t ly a t t h e re a ct o rcenter, where the local t empera tures a re higher. Thisis the case reported for T0) 338 K (curve D in Figure6b), where the isobutene concentrat ion in the centerregion of the reactor is higher than the one observedfor T0) 329 K (curve C).

Our results point that mass and heat external resis-tances are not very significant in the react ion system;therefore, the cat alyst tempera ture is very close to theone of the bulk pha se (isotherma l behavior for the resinwa s considered). Also the ma ss gradients in t he filmare not very importa nt . The isobutene concentra t ioninside the microspheres (YC 4,i) a n d o n t h e p o re wa l ls(YC 4,a ) of the r esin part icles locat ed a t the hot spot canbe observed in Figures 7 and 8, respectively, for tworeactor radia l positions: (a ) r*) 0 and (b) r*) 1. Insidethe microspheres, the mass gradients are almost absentin all cases as observed before, through the unit values

of i (Figure 5c), also reported by Ihm et al. (1988).I n w h a t con ce r ns t h e p r oce ss occu r r in g on t h e

m a c rop or e w a l ls , t h e s e r e s ul t s s h ow t h a t t h e m a s sgra dients on the ma cropores can be low er at the rea ctorcenter (r* ) 0 ) t h a n a t t h e r e a c t o r w a l l (r* ) 1), asdepicted in Figur e 8a a nd b. The concentra tion profileis practically flat in the reactor center, in particular for

T0) 338 K (curve D in Figure 8a), where equilibriumconditions w ere closely a pproa ched. This is shown inFigure 8c, where the actual profiles for the same T0a recompared to the corresponding equilibrium concentra-t ions in both radial locat ions (r* ) 0 a n d r* ) 1). Atthe rea ctor wa ll, where the loca l solid tempera ture (347K ) is lo we r t h a n a t t h e re a c t o r c e n t e r ( 387 K ) , t h esystem is far from equilibrium, so the mass gradientsa re h ig h e r . I n s p i t e of t h a t , t h e c a t a ly s t e f ficien cy a tthe center (ov ) 0.39) is lower than at the wall (ov )0.92). In order to clar ify these results, we representedin Figure 8d th e profiles of the overa ll reaction ra te (Ra )on the macropore walls of the resin particles located atr*) 0 a n d r*) 1. I t can be observed tha t for r*) 0,t h e e q u il ibr iu m is re a c h e d t o wa rd t h e c e n t e r o f t h ecatalyst part icle (Figure 8c), so the react ion rate willapproach zero in the inner posit ions of the ma croporewa lls (Figure 8d). However, it ha s a compar at ively highvalue at the surface, which results from the effect ofthe high resin temperature (387 K) on the rat e constantsand the slight deviation of concentration from equilib-r iu m con dit ion s . I n f a ct , w h e n t h e re a c t ion ra t e s a recalculat ed, those differences ar e augment ed due to thehigh va lues of the kinetic constant s. The steep gradientof the overall rea ction ra te so obta ined explains the lowvalue ofov, since this para meter is defined as the ra t iobet we e n t h e a v e ra g e re a c t ion ra t e in s ide t h e ca t a ly s ta n d i t s s u rf a c e v a lu e. F o r r*) 1, the overall reactionrate profile inside the catalyst particle is more uniform

Figure 8. H T model. Concentra t ion profiles for isobutene on t he pore walls of the resin par t icles located a t t he hotspot , (a) a t t hereactor center, r*) 0; and (b) a t the rea ctor w all, r*) 1, for four inlet temperatures: T0) 313 K (z*) 0.17); T0) 323 K (z*) 0.14); T0)329 K (z*) 0.08); T0) 338 K (z*) 0.04). (c) Isobutene concentr at ion and corresponding equ ilibrium va lues for T0) 338 K; (d) globalreact ion rat es for T0) 338 K.



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(Figure 8d), in spite of the less uniform concentrationprofile (which is still very far from equilibrium, Figure8c), leading then to a higher efficiency.

T h e ra dia l a v e ra g e v a lu e o f ov a t t h i s p a r t i c u l a r

point , w here the hot spot occurs, is 0.63, a s shown inFigure 5a for T0 ) 338 K. This corresponds to t hemin imu m v a lu e o f t h e ca t a ly st e ff icien cy a lo n g t h ereactor, since, in this region of higher t empera tures, th eg ra die n t s o f t h e re a c t io n ra t e o n t h e ma c ro p o re s a remore pronounced. The sam e expla na tion stan ds for thes i t u a t ion s wh e re t h e p roce ss ru n s n e a r e q u il ibr iumcondit ions (curves C a nd D in Figure 5a). In t he caseswhere equilibrium is far from being reached (curves Aand B ), the effect iveness factors a re determined by th ecompetit ion between react ion rate and diffusional re-s ist a n c e s in s ide t h e re sin p a rt icles . I n t h e h ot -s potregion, the highest temperature of the cata lyt ic bed isat ta ined. Therefore, the reaction ra te suffers the maxi-

mu m in cre a s e in re la t io n t o t h e ra t e of di ff u sion a ltra nsport, w hich imposes t he most pronounced concen-t ra t ion g ra dien t s . Co n se q u en t ly , t h e h ig h es t ov era l lrat e profiles are obtained a t t his point , leading then t othe minimum efficiencies.

Th e comp a ris on of t h e re su lt s p redict e d by t h eheterogeneous and pseudohomogeneous models is shownin Figure 9. For low feed temperatures, T0 ) 313 K,the temperatur e rise in the cat alyt ic bed is not signifi-cant and the catalyst efficiency is near one (Figure 5a),the solid gradients than being negligible; therefore, thesystem performa nce can be adequa tely represented bythe simple P H model a s shown in Figure 9a. Increasingthe feed temperat ure to T0) 323 K, there is an increaseof the concentra tion gra dients inside the solid part icles.

Thus, the efficiency of the resin particles will decrease(Figure 5b) a nd the predictions of th e complete hetero-geneous model will be different from th e ones obta inedwhen t hose internal gra dients ar e not accounted for, as

depicted in Figure 9b. These two cases r efer t o non-e q u il ibr ium s i t u a t ion s , a s p rev iou s ly s ee n . I f t o t a lequilibrium could be achieved all over the reactor, theconcentration profiles inside the resin would certainlybe flat (the equilibrium concentration is only tempera-ture dependent and the part icles are taken as isother-ma l). In this limit ing case, t he pseudohomogeneousmodel would totally match the heterogeneous modelpredictions (efficiencies would be unity). As a na lyzedbefore, increa sing th e feed tempera tur e aboveT0) 329K, the process runs closer to this limiting case; i.e. , wewo u ld e xp ect a bet t e r re pre se n t a t ion of t h e s y s t embehavior through the simpler pseudohomogeneous model.I n de e d, t h is is wh a t h a p p e n s in mo s t o f t h e c a t a ly t ic

bed (Figure 9c and d). How ever, in the hot-spot zone,the predictions of both models still present some differ-ences, which is due to the low efficiencies observed inthis region, as explained above.

Representing the final isobutene conversion, at there a ct o r e xi t , wh e n p redict e d by t h e h e t erog en e ou smodel,XHT, and by the pseudohomogeneous model, XP H ,for different inlet temperatures, Figure 10 shows oncemo re t h a t a f t e r 329 K , t h e f in a l re a c t a n t c o n v e rs io napproaches a ssymptotically t he equilibrium conversion,Xeq. In t hese conditions, the pseudohomogeneous modelcan predict quite well the final isobutene conversion atthe reactor exit. However, and a s ca n be seen in Figure9, the maximum temperatures inside the reactor cal-culated by this model are higher than for the hetero-

Figure 9. Radial mean temperature profiles in the bulk phase for the heterogeneous model, HT, and pseudohomogeneous model, PH, forfour inlet temperatur es: (a) T0) 313 K; (b) T0) 323 K; (c) T0) 329 K; (d) T0) 338 K.



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geneous model. Thus, a deta iled a na lysis of the systemwill require the more realist ic heterogeneous model.Finally, we present in Figure 11 a comparison betweenthe results obtained with the heterogeneous model whenthe radial gradients are taken into account, H T2D , a n dwh e n t h e y a re n e g le c t e d, H T1D . The predict ions ob-s erv ed in bot h ca s e s re f lect t h a t t h e t e mpe ra t u re s

obtained with the unidimensional model are not veryf a r f rom t h e ra dia l a v era g e t e mpe ra t u re p re dict e d bythe bidimensional model. How ever, the use of the firsts impler mode l wil l n ot a l low t h e k n owle dge of t h et e mpe ra t u re p rof iles a lo ng t h e ra dia l coordin a t e . I nsome cases, these gra dients can be significan t , leadingto quite higher temperatures in the center of the reactor,as we ha ve observed before, in Figure 6a. This can bei mpor t a n t w h en t h e t h er m a l s t a b il it y of t h e i on -exchange resins ha s t o be a ccounted for.

Conclusions

An a n a ly sis o f t h e beh a v ior of a ca t a ly t ic t u bu la rreactor packed with ion-exchange resins for a reversibleprocess wa s performed by using a ma themat ical modela c cou n t in g f o r t h e g ra dien t s in s ide t h e s o lid . Th ema croreticular cat alyst part icles w ere envisaged as a nagglomeration of microspheres, representing the gelphase of the resins, wh ich formed betw een them th e so-called macropores. The active sites were loca ted insideof these microspheres and also at their surface, i.e. , on

the ma cropores wa lls. An isothermal behavior for th esolid par t icles wa s assumed. However, separ at e con-centration gradients were established on the macroporesand inside the microspheres, leading also to differenteffectiveness factors, the macroeffectiveness factors, andmicroeffect iveness factors, further used on the deter-minat ion of the overall effect iveness factor of t hesecata lysts with heterogeneous structure. The synthesisof MTBE was used as a reference process in this paperto illustrate the behavior of such reversible systemscata lyzed by ma croreticular ion-exchange resins. Theuse of generalized variables for the mass balances onthe ma cropores wa lls and inside the microspheres of thecatalyst part icles allowed, in each case, the reduction

o f t h e t h re e di f f e re n t ia l e q u a t io n s o f t h e t h re e c o m-pounds, isobutene, methanol, and MTBE, to only onedifferential equa tion for th e ma cropores an d one for th emicrospheres. The discretiza tion of th ese second-orderdifferential equations was achieved by making use ofthe orthogonal collocation method.

The performa nce of the syst em wa s stud ied in the feedtempera ture ra nge 313-338 K, and our result s showedthat the process approaches thermodynamic equilibriumconditions aft er 329 K. In t hese situat ions, the concen-

Figure 10. I n f lu en c e of t h e in let t e m p er a t u r e on t h e f in elisobutene conversion for the heterogeneous model, XHT, a n dpseudohomogeneous model, XP H , and on t he isobutene equilibriumconversion, Xeq.

Figure 11. R a d i a l m e a n t e m pe r a t u r e p r of i le s in t h e b u lk p h a s e f or t h e h e t er oge n eous b idim en s ion a l m od el , H T2D , a n d f o r t h eheterogeneous unidimensional model, H T1D , for four inlet temperat ures: (a) T0) 313 K; (b) T0) 323 K; (c) T0) 329 K; (d) T0) 338 K.



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tra t ion gradients inside the solid part icles a re reducedand the results predicted by the heterogeneous modelsbecome closer to those obtained with the pseudohomo-geneous models. However, and even if the fina l reactantconversion can be well calculated through the PH model,t h e ma x imu m t e mpe ra t u re s in s ide t h e re a ct o r w illdeviate from the ones predicted by the more realist icheterogeneous model. A comparison of t he bidimen-sional models an d the unidimensiona l models wa s a lsoa c h ie ve d. O u r res u lt s s h owe d t h a t t h e ra dia l t h e rma lp ro f i le s c a n be imp o rt a n t , de s p it e t h e s imila r me a nradia l tempera ture values to those calculated when t hera dia l grad ients a re neglected. Therefore, it is our belieft h a t a mo re c a re f u l s t u dy o f t h e p ro c e s s in o rde r t oi de nt i fy t h e l oca l m a x im u m t e mp er a t u r es t h a t a r epossible to occur inside the system will require the useof the complete t wo-dimensional model.

This st udy points out the effect of thermodynamicson the control of the beha vior of het erogeneous revers-ible systems. When the local condit ions a re far fromequilibrium , the process is controlled by th e competitionbetween chemical react ion and diffusional transport .When equilibrium is approached, the thermodynamiceffects are predominant over those phenomena.

Nomenclatureap) specific particle ar ea, m -1B ) d im e n si on l es s a d i a b a t i c t e m pe r a t u r e r i s e (B )

(-H)CC 4/T0Ffcpf)B iw) w a l l h e a t B i ot n u m b e r (B iw) hwR0/er )bMeOH ) ad sorption coefficients for metha nolbMTBE ) adsorption coefficients for MTBECk) concentr a tion of compound k, mol/m 3CC 4

) inlet isobut ene concentra tion, m ol/m 3CMeOH) inlet met ha nol concentr a tion, mol/m 3cpf) heat capacity of the fluid, cal/(kg K )Ca , C i) pa ra met ers defined by eq 10D a) Da mkh oler num ber (D a) FbKB(T0))Der ) effective diffusivity in t he reactor ra dial coordina te,

m 2/sDk) effective d iffusivity of compound k, m 2/sDt ) rea ct or dia met er , mda ) diameter of the resin particle, mdi) diameter of the microspheres, mEB, EA ) a ct iva t ion energies of t he forw a rd a nd ba ckw a rd

rea ctions, cal/molE*) t herma l increment (E*) EA - E)E ) difference between the adsorption heats of MTBE and

met ha nol (E) MTBE - MeOH )E1, E2, E3) pa ra met ers defined by eq 9-H ) heat of rea ction, cal/molh) film hea t t ra nsfer coefficient, ca l/(m2 s K )hw) wa ll hea t tra nsfer coefficient, cal/(m 2 s K )KB ) forw a rd rea ct ion kinet ic cons t a nt , m f3/(K g ca t s)KA ) backwa rd reaction kinetic constan t, m ol/(K g ca t s)Kf) film ma ss-tr a nsfer coefficient of th e compound k , m/sk a , k i) pa ra met ers defined by eq 11L ) rea ct or lengt h, mNf,k ) number of film ma ss-tra nsfer units for compound k

(Nf,k ) [(1 - )/]Kf,kap)Nfh ) number of f i lm hea t -t ra ns fer unit s (Nfh ) (1 - )-

h ap/Ffcpf)P) t ot a l pres s ure, a t mPerh ) ra dia l hea t P eclet number (Perh ) L u0Ffcpf/er )Perh (dp) ) ra dia l hea t P eclet number ba s ed on pa rt icle

dia met er (Perh (dp)) dpu0Ffcpf/er )Perm ) ra dia l ma s s P eclet number (Perm ) L u0/Der )Perm (dp) ) rad ial ma ss Peclet number based on the particle

dia met er (Perm (dp)) dpu0/Der )R) idea l ga s cons t a nt

R0) rea ct or ra dius , mRa ) radius of the resin particle, mRi) radius of a microsphere, mR ) ra t io of t he a ds orpt ion coefficient s of met ha nol a nd

MTBE (R ) bMTBE/bMeOH )R ) chemica l rea ction ra te, m ol/(K gca t s)Ro) reaction ra te a t t he reactor ent ra nce, mol/(K g ca t s)R) dimensionless reaction rate (R) R/Ro)Ra ) normalized reaction rate for the macropores (Ra )

Ra (Ua )/Ra (Ua ) 1))Ri) normalized rea ction ra te for the microspheres (Ri)

Ri(Ui)/Ri(Ui) 1))r) rea ct or ra dia l coordina t e, mr*) dimensionless reactor radial coordinate (r*) r/R0)ra) spat ial position on th e ma cropores of the resin pa rticle,

mri ) s p a t i a l p os i t ion i n t h e m i cr os ph e re s of t h e r e si n

particle, mT) a bs olut e t empera t ure, KT0) feed temperature, KUa , U i) generalized concentrat ions on t he ma cropores a nd

in the microspheres defined by eq 6U a , U i) approximated generalized concentration of Ua a nd

Uiu0) superficia l velocity , m/sX) isobutene conversionX

eq) isobutene conversion at equilibrium conditions

XH T) f inal isobutene conversion predicted by t he hetero-geneous model

XP H) f inal isobutene conversion predicted by the pseudo-homogeneous model

xa ) dimensionless spatial position on the macropores oft he res in pa rt icle (xa ) ra /Ra )

xi) dimensionless spat ial position in t he microspheres (xi) r i/Ri)

Yk ) dimensionless concentration for compound k (Yk )Ck/CC 4

)z) reactor axial coordinate, mz*) dimensionless reactor axial coordinate, (z*) z/L )

Greek Symbols

Rk) stoichiometric coefficient of compound k) fraction of the active sites on the macropores walls of

the resin particleb, *, ) Arrhenius numbers (b ) EB/R T0) (*) E*/R T0)

() E/R T0)a , i) pa ra met ers defined by eq 12a , i) pa ra met ers defined by eq 13a , i) pa ra met ers defined by eq 14) bulk porosityp) catalyst porositya ) effectiveness fa ctor of th e ma croporesi) effectiveness factor of a microsphereji) average effectiveness factor of the microspheres of the

resin particleov ) overa ll ef fect ivenes s fa ct or of t he res in pa rt icle

referring to the resin surface conditions

jov ) overa ll ef fect ivenes s fa ct or of t he res in pa rt iclereferring to the bulk conditions) dimensionless temperature () T/T0)er ) rad ial effective therma l conductivity, cal/(m s K )MeOH ) a ds orpt ion h ea t s of met ha nolMTBE ) adsorption heats of MTBEFb) bulk densit y, kg/m 3Ff) fluid densit y, kg/m 3Fc) cata lyst density, kg/m 3MeOH ) catalytic surface fractions covered by methanolMTBE ) catalytic surface fractions covered by MTBE) space time for the fluid () L /u0)c) ca t a lys t t ort uos it yk,a ) Thiele modulus of compound k for the pores (k, a )

RaFc(1- p)KB(T0)/Dk,a )

3838 Ind. Eng. Chem. R es. , Vol. 35, No. 11, 1996


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k,i ) Thiele modulus of compound k for the microspheres(k,i ) RaFcKB(T0)/Dk,i )

a ) generalized Thiele modulus for the ma cropores (a) RaFc(1- p)KB(T0)Ra (Ua) 1)/C a )

i) generalized Thiele modulus for t he m icrospheres (i) RiFcKB(T0)Ri(Ui) 1)/C i)

Subscripts

a ) macroporesb) bulk conditionsi) microspheresk) compound: isobutene, k ) C 4; met ha n ol, k ) MeOH;

MTBE, k ) MTBEs ) resin surfacew) rea ct or w a ll

Appendix A

I n t h e p re se nt s t u dy , t h e m et h od of or t h og on a lcollocation (Finla yson, 1972; Villad sen a nd Michelsen,1978) was used to solve the generalized differentialequations of the catalyst part icle, eqs 10-13, Table 4.Ta king into a ccount th e symmetr y of the concentra tionprofiles, a cha nge of the spa ce coordina tes xa a nd xiw a sintroduced:

Those mass ba lances may now be rewrit t en as follows:

F or e a ch ca t a ly s t p a rt icle, t h e e xa c t p rof ile o f t h eg en e ra l ize d v a ria ble o n t h e p ore wa l ls , Ua (h) , w a sapproximated by a trial function, U a (h), through

wh e re aj are unknown coefficients to be calculated andlj(h) a re s p ecified f u n ct ion s wh ich mu s t s a t is f y t h eboundary conditions.

The gra dients in side the m icrospheres, Ui(v), locat edo n e a c h s p a t ia l p o s i t io n o f t h e ma c ro re t ic u la r re s in(including the ones located at the surface) were alsoapproximat ed by the functions U i(v):

Selecting for the trial functions lj(h) (j) 1, 2, ..., N+1)a n d lj(v) (j) 1, 2, ..., M+1), the Lagrange polynomials

which sat isfy the relat ions

a nd

the unknown par a meters for the ma cropores and for themicrospheres a re identified w ith the correspondingapproximat ed solutions calculated a t the sa me points;i.e., U a (hj)) aj (j) 1, 2, ..., N + 1) and U i(vj)) bj (j)1, 2, ..., M+1). The La gra nge interpollat ion polynomi-a ls lj(h) a n d lj(v) are defined by

wh e re PN+1(R,)(h)) (h- h1)...(h- hN+1) a n d PM+1

(R,) (v)) (v- v1)...(v- vM+1) are the n odal polynomials with degreesN + 1 a n d M + 1, respectively; PN+1

(R,)(hj)(1) a nd PM+1

(R,)

(vj)(1) ar e the first derivat ives of t hose polynomials, in

order of the spa tial posit ions h a n d v when calculated

at the posit ions h) hj a n d v) vj, respectively.The approximated solutions (A.5) and (A.6) are thencalculated by

The profiles on the pore walls were obtained with Ninterior collocation points with N unknown variables U a) hj(j) 1, 2, ..., N) and one boundary point at the resin

surface where the solution is known: h

N+1) 1,U

a) 1,eq A.2. Inside t he microspheres, located on ea ch posi-tion hjof the ma cropores (including th e resin sur face, j) 1, 2, . . . , N + 1), M interior collocation points wereused also with M unknown variables; as seen before, atthe surface of the microspheres, the generalized va ri-ables are known: vM+1) 1, Ui(1)) 1, eq A.4. Thus, itis necessary to ca lcula te Nva ria bles for th e macroporesa nd Munknowns for the microspheres located on eachone of th e N+1 collocation points of the macropores inorder t o get t he solution of the system. Following themeth odology of th e ort hogonal collocat ion method, th eseunknowns are determined by set t ing the residuals ofthe approximated differential equations equal to zeroon each interior collocation point. Substit uting the tria l

Ui(v) = U i(v)) j)1

M+1

bjlj(v) (A.6)

lj(hn)){0 j*n1 j)n (A.7)

lj(vn)){0 j* n1 j) n (A.8)

lj(h)) PN+1

(R,)(h)

(h-hj)PN+1(R,)(hj)

(1) (A.9)

lj(v)) PM+1

(R,) (v)

(v- vj)PM+1(R,) (vj)

(1) (A.10)

U a (h))j)1

N+1

U a (hj)lj(h) (A.11)

U i(v))j)1

N+1

U i(vj)lj(v) (A.12)

h) xa 2; v) xi2

macropores

4h

2Ua

h2

+6Ua

h )a

2Ra (Ua )+

6a

2

i2

Ri(Ui) 1)

Ra (Ua) 1)Ui

xi|

h,v)1(A.1)

boundary conditions

z* g 0, r* g 0: h) 0Ua

h f in it e (A. 2)

h) 1 Ua) 1

microspheres

4v

2Ui

v2

+6Ui

v )(1-)i

2Ri(Ui) (A.3)

boundary conditions

z* g 0, r* g 0, hg 0: v) 0Ui

v fin it e (A.4)

v) 1 Ui) 1

Ua (h) = U a (h))j)1

N+1

ajlj(h) (A.5)



14/15

function (A.11) int o eq s (A.1) an d (A.2), one g ets Ndifferential equa tions defined for the pore w alls:

F o r t h e mic ro s p h e re s , lo c a t e d a t e a c h o f t h e N + 1collocation points, the substitut ion of t he tr ial functionsdefined by eq A.12 into eqs A.3 and A.4 generates Mdiff ere n t ia l e q u a t ion s w it h n u ll re s idua ls a t t h e Mcollocation points:

The first a nd second deriva tives of the tr ial fun ction oneach collocation point are given by

wh e re

These weights of the approximated derivat ives wereca l cu la t e d b y m a k in g u s e o f t h e s u br ou t e D F O P R(Villadsen and Michelsen, 1978).

Subst itutin g relat ions A.15 an d A.16 into differentialequa tions (A.13) a nd (A.14), one get s the collocatedequations for the macropores and microspheres:

microspheres (located on th e N+1 collocation points of

the macropores)

This system of N + (N + 1)M nonlinear algebraicequations (Nequations on the macropores and (N+1)-M in the microspheres) was solved by the Newton-

Ra phson method. After the calculation of the a pproxi-ma ted solutions of the generalized var iables on the porewalls and in the microspheres, the concentration profilesof the different compounds were achieved t hrough eq19. The react ion ra tes were then calculat ed by makinguse of eq 18.

EffectivenessFactors. The effectiveness fa ctors inthe microspheres, i, ar e given by eq 29:

I n t ro ducin g t h e ch a n g e of t h e s p a t ia l v a r ia ble w h ichtakes into account the symmetry condition, v) xi2, onegets

wh e re (v)) v(1- v)R ) v1/2 ( ) 1/2; R ) 0) is theweight function. Using the M +1 collocation points int h e mic ro s p h e re s , t h e R a da u q u a dra t u re a l lo ws t h eca lc ula t ion of t h e a p p rox ima t e d v a lu e of i on eachcollocation point of the macropores (m ) 1, 2, ..., N +1):

wh e reja re t h e q u a dra t u re we igh t s c a lcu la t e d by t h e

subroutine Rada u (Villadsen and Michelsen, 1978).Since the boundary point init ia lly f ixed vM+1 w a s a l sou s ed , w e w i ll h a v e a n e xa c t R a d a u q u a d r a t u r e f or apolynomial with a degree e 2M, i f t h e M q u a d r a t u r ep o in t s a re t h e ze ro s o f a p o ly n o mia l , wh ic h mu s t beorthogonal with respect to the weight function (v)(1- v)) v1/2(1- v) for any polynomial Pj(v), j e M - 1.Thus, t he concentrat ions of the compounds must becalculated at those points, so the interior collocat ionp oin t s u s ed in t h e micros ph e re f o r t h e ort h o gon a lcollocation method w ere th e zeros of t he J a cobi polyno-

mial of degree M, Pm(1, 1/2) (v), sa t isfying the orthogonal-

ity relat ion

Villadsen and Michelsen (1978) developed the J COBIsubroutine, which w as used to calculate t he roots of tha tpolynomial.

The average microeffectiveness factor, ji, is definedthrough eq 32:

and can be replaced by

4h

2U a

h2|

h)hm

+6U ah

|h)hm

-a2Ra (U a )-

6a

2

i2

Ri(Ui) 1)

Ra (Ua) 1)U iv

|hm,vM+1)1

)0 (m)

1, 2, ..., N) (A.13)

4v

2U i

v2|

vk

+6U iv

|vk

-(1- )i2Ri(U i)|vk )0

(k) 1, 2, ..., M) (A.14)

d Ua

d h |hm )j)1N+1

A m jU a (hj)

d 2U a

dh2|

hm

)j)1

N+1

Bm jU a (hj) (m)1, 2, ..., N) (A.15)

dU i

dv|

vk

) j)1

M+1

A IkjU i(vj)

d 2U i

dv2|

vk

) j)1

M+1

B Ik jU i(vj) (k) 1, 2, ..., M) (A.16)

A m j)d lj(h)

dh |hm Bm j)d

2lj(h)

dh2|

hm

A Ikj)d lj(v)

dv |vk B Ikj)d

2lj(v)

d v2|

vk

macropores

4hmj)1

N+1

Bm jU a (hj)+6 j)1

N+1

A m jU a (hj)-a2Ra (U a )|hm -

6a

2

i2

Ri(Ui) 1)

Ra (Ua) 1)j)1

M+1

A IM+1,jU i(vj)|hm )0 (m)

1, 2, ..., N) (A.17)

4vkj)1

M+1

B IkjU i(vj)+6j)1

M+1

A IkjU i(vj)-

(1- )i2Ri(U i)|vk )0 (k)1, 2, ..., M) (A.18)

i) 301Ri(Ui)xi

2dxi (A.19)

i)320

1v

1/2Ri(Ui) d v (A.20)

i =3

2j)1

M+1

jRi(U i)|vj (m)1, 2, ..., N+1) (A.21)

01

v1/2 (1- v)Pj(v)PM(1, 1/2)(v) d v) 0 (j)1, 2, ..., M- 1) (A.22)

ji)0

1iRa (Ua )xa2

dxa

01Ra (Ua )xa

2 dxa

(A.23)



15/15

after substituting the resin radial coordinate by h) xa 2.The Radau quadrature was used again for the calcula-tion of both integra ls wh ich include the weight function(h)) h(1- h)R ) h1/2() 1/2;R ) 0):

In order to sat isfy the precision requirements of theR a da u q u a dra t u re wh e n a f ix e d n o de is u s e d (uN+1 )1), the N interior quadrature points must be the roots

of the Nth degree J a cobi polynomial, PN(1, 1/2)(h), defined

by t he orthogona l relat ion

Th e s a m e s pa t i a l p os it i on s w e r e a l s o u s ed f or t h einter ior colloca tion points on t he ma cropores of the solidre sin s . Th e q u a dra t u re we ig h t s, j, a n d t h e ze ro s o fthe polynomial were also calculated through the sub-routines RADAU and J COB I, respectively (Villadsenan d Michelsen, 1978). The ma croeffectiveness factor,a , is given by eq 35

or, introducing t he va riable change h) xa 2,

U s in g t h e R a da u q u a dra t u re f o r t h e s a me re a s o n s a s

referred to above, the numerical calculations of a willbe performed by

The quadrature points and the weighting functions jare the ones referred to before for the determination ofj i.

Literature Cited

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Aris, R. The M athemat ical Theory of Diffu sion Th eory of Di ffusionand Reaction i n Permeable Catalysts; Clarendon: Oxford, 1975.

B e r g , D . A. ; H a r r is , T. Ch a r a c t er iz a t ion of M u lt icom pon en tDiffusion Effects in MTBE Synthesis . I n d. E n g. Chem. Res.1993, 32, 2147.

Caetano, N. S.; Loureiro, J . M.; Rodrigues, A. E. MTBE synthesiscatalyzed by ion exchange resins: kinetic studies and modelingof mult iphase batch rea ctors. Chem. E n g. S ci. 1994, 4 9, 4589.

Finlayson, B. A. T h e M et h o d o f Wei g h t ed Resi d u a l s a n d V a r i a - t i o n a l P r i n c i p l es ; Academ ic: New York, 1972.

Gicquel, A.; Torck, B. Synthesis of Methyl Tert iary Butyl EtherCa t a ly z ed b y I on -Ex c h a n g e R e s in . I n f lue n ce of M e t h a n ol

Concentra t ion a nd Temperature. J. Catal. 1983, 83, 9.H in d m a r s h , A. C. R e por t U CI D -3 00 01 ; L a w r e n ce L iv er m or e

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H in d m a r s h , A . P r e lim in a r y d oc um e n t a t ion of G E AR I BsReportUC ID-30130; Law rence Livermore La boratory, 1976.

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Ihm, S . K. ; C hung, M. J . ; Pa rk, K. Y. Activity difference betweenthe internal and external sulphonic groups of macroret icularion-exchan ge resin cata lysts in isobutylene hydra tion. I nd. E ng.Chem. Res. 1988, 27, 41.

Pa rra , D . ; Tejero, X.; Cun ill , F. ; Iborra , M.; Izquierdo, J . Kinetic

study of MTBE liquid-phase synthesis using C4 olefinic cut.Chem. E n g. Sci. 1994, 49, 4563.

Quinta Ferreira , R. M.; Rodrigues, A. E. Diffusion and Catalyt icZero-Order React ion in a Macroret icular Ion E xchange Resin.Chem. E n g. Sci. 1993, 48, 2927.

Q u in t a -F e r r eir a , R . M . ; Alm eid a -Cos t a , C. ; S p n ola , A . D . C. ;Rodrigues, A. E. Modeling of MTBE Synthesis Reactor WithMacroret icular Ion Exchange Resins. Proc. 4 4 t h Ca n . Chem.E n g. Conf . , Ca lg a r y , Ca n a d a , 1 9 94 .

Madsen, N.; Sincovec, R., PDECOL: General Collocation on FiniteElements. Chem. E n g. Sci. 1975, 30, 587.

Rehfinger, A.; Hoffman, U . Kinetics of methyl t ert iary butyl etherl iq u id p h a s e s y n t h e s is c a t a ly z ed b y ion e xc h a n g e r e s insI .Intrinsic rate expression in liquid phase activities. Chem. E n g.Sci. 1990a, 45, 1605.

Rehfinger, A.; Hoffman, U . Kinetics of methyl t ert iary butyl ether

l iq u id p h a s e s y n t h e s is c a t a ly z e d b y ion e xc h a n g e r e s ins

I I .Methanol as rate-controlling step. Chem. E n g. S ci. 1990b, 4 5,1619.

Villadsen, J . ; Michelsen, M. L. S o l u t i o n o f D i f f er e n t i a l E q u a t i o n M od el s b y P o l y n omi a l A p p r o xi m a t i o n ; P r e n t ice -H a l l : En g le -wood C liffs, NJ , 1978.

Received for review April 22, 1996Accepted May 28, 1996X

IE9602421

X Abstract published in A d vanceA CS A bst ract s,September15, 1996.

ji)0

1h

1/2iRa (Ua ) d h

01

h1/2

Ra (Ua ) d h(A.24)

ji =

j)1

N+1

jRa (U a )i|hj

j)1

N+1

jRa (U a )|hj

(A.25)

01

h1/2

(1- h)Pj(h)PN(1, 1/2)

(j)1, 2, ..., N-1)

(A.26)

a) 301

Ra (Ua )xa2 dxa (A.27)

a)3

20

1h

1/2Ra (Ua ) d h (A.28)

a =3

2j)1

N+1

jRa (Ua )|hj (A.29)


simulación de la sintesis de mtbe

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