effectiveness factor

8/3/2019 Effectiveness Factor

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DIFFUSION AND REACTION INPOROUS CATALYSTS

N O R I A K I W A K A O A N D J . M . M I T HC ; l i r e r s i t j o/ Culi/ornin. n a r i s . CulJ

Using a previously developed concept of diffusion in bidisperse porous catalyst pellets, an expression forthe effective diffusivity i s der ived for diffusion under reaction conditions. This diffusivity i s a function of theeffectiveness factor, E , , of the microporous particles composing the catalyst pellet and reduces to the normaldiffusivity when E i becomes unity. The diffusion results are ap pli ed to the pro blem of evalu ating the effec-tiveness factor, E , , of the entire pellet. Charts are given for predicting E, for a first-order, isothermal reac-tion, in terms of a micropore diffusion parameter, a macropore diffusion parameter, and a reaction rateparcmeter. To evaluate E requires a knowledge of the macro- and micropore size distributions, the microeffectiveness factor, E t , and either the avera ge reaction rate per pel let or the reaction rate constant, k,.The method i s based upon a simple model of the pore structure. The model contains some unrealistic sim-plifications with respect to the actual mass transfer processes in porous materials. Hence the procedureshould not be re gard ed as a general one applic able t o all types of porous catalysts.

m m the effectiveness factor \ vas in t roduced by Thie leS:13) here h a s been copciderable ipterest-for exam pleil . -7. 1 5 )P in eva lua t ing the averaqe reac t ion ra te in a porouscatalys t. This reqilirrs a n effective diffusivity for the re-act ant fluid in the porei of the solid mate rial. Accordin qly,a t ten t ion has been focused recen t ly on measurement andprediction of diff(ision rates (1-6. . 7 7 . 72. 77). Thesestudies have al\va>-s been co nsidered fro m the s t and po int ofdiflusion in the abpence of chem ical reaction. Expe rime ntaln ieasuremer ts have been geperally made for diffusion thr0ug.hcatalyqt pellets . which in itself is not equivalent. in most in-s tance- . to diffur ion unde r reac t ion condi t ions . In the presen tLvork the first objective is the effect of chemical reaction on thedifTusion rates . Usi ng the results E O obtained. expressions arethen developed for predicting the effectiveness factor for anisothermal. firs t-order reaction in a spherical catalyst pelletof known pore s ize diqtribution.

I n this \vork the catalVtic material is supposed to be a pelletLvhich copta ins bo th macro- an d micropores . This type ofmateria l is ohtainrd Lvhen the catalyst is prepared by pelletingmicropo rous po\vder. T h e effectivenecs factor devel opm enti r rirnilar to that of Min g le a n d S mi th (7). in tha t ca ta lys tsLvith bidicperse pore systems ar e trea te d. Hokvever. sincethe ea r l ie r paper a more accur a te express ion has been der ivedf,?. (?. 7-71 for diffu5 ion in pores where bo th Knudsen and bu lkdiffusion are s ignificant. Th is result has been used by LVakaoa n d S mi th ( 7 I ) . along \vith a new concept of pore geometry .to deve lop a method of predicting diffusion rates as a functionof pore 4ze d is t r ibu t ion . Th e method gave resul ts in gooda g re e me n t w i th ru p e r ime n ta l d i f fu sion d a ta for a lum ina pe lle tscovering a wide denri ty range . T he ba i ic conc lus ion tha t thediffu.ion rar e is pro por tion al to the square of the poros i ty( E q u a t i o n 3 ) \vas a lso drve loped independent ly by \leisza n d S c h wa r tz (76 ) . This permits one to de te rmine the e ffec tof reaction upon diffusion-the firs t objective of this pa pe r.

The result can then be used to calculate effectiveness factorswithout making assumptions regarding. the controlling type ofdiffusion in the micro- avd macropores. ar \vas necessary inthe ea r l ie r paper ( 7 ) . For example. diffuriori in the macro-pores ( the space be t lveen powder par t ic le? ) may be pre -dom inant ly by a bu lk m echanism in lo \v-den


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Figure 1 i s the mode l p roposed by the au thors (14) for dif-fusion in a pe lle t . I t is c lea r tha t the a rran gem ent of poresand so l id mate r ia l in the mode l does no t represen t the rea lgeometry of an ac tua l porous mate r ia l . For example . theasqumption tha t a l l the m acropores a re para l le l to the d irec t ionof diffuFion is not true. How ever. the model can be analy zedquant i ta t ive ly for the problem of combined diffurion and re-action . Fo r these conditions the driving forces for reactionare ind ica ted a t the s ide of F igure 1 . It is supposed that thepellet is a s tatis tical assembly of microporous particles and thatvo lume void frac t ions a re the same as a rea vo id frac t ions .Ih e d o t t e d s q u a re s (F ig u re 1) represen t the microporousparticles . Denoting t,: t i . a n d td as the several volume frac-tions,r l

Ea + + 8 = 1 ( 2 )T he s tat ist ica l a rrang eme nt of macrop ores is such tha t the

effective area void fraction on a contact plane is e a 2 . Simi-la r ly . the a rea frac t ion of pa r t ic le -par t ic le contac t is (1 -T h e l e ng t h of the un i t ce l l . in F igure 1: is taken as the

distance Ax be tween the cen te r o f one par t ic le and tha t of th enext one . T he concentra t ion d iffe rence over the d is tanceis ( d y l d x ) Ax in the macropores and E,(dy d ~ )u in the micro-pores. Diffusion in a uni t cell is assumed to be the sum of thefollowing contribution5 :

1 .2 .

Diffus ion th rough the macropores with a n a rea of ea 2 a n dconcentra t ion d iffe rence (4) ; 'dx) Ax .Diffus ion fro m a par t ic le to the ad jacen t pa r t ic le th ro ughthe micropores with an a rea of (1 - e n ) Z and concentra t iondiffe rence E , (('d.u)k. he effective void fraction of micro-pores per unit area of this section is the square of the voidf ra c t io n : [ e Z ' e , + , ) ] * = [ e , (1-Diffus ion th r ough macro-m icropores in se r ies with a narea of 1 - a2 - (1 - , ) 2 = 2e,(1 - a ) , B e c a u s e o f th e lo wdiffusion rate in the micropor'es as compared ivith that in themacropores . i t is a s sumed tha t d iffus ion th rough the micropo resis the contro l l ing . s tep in the se r ies p ' th .3 .

Th e d iffusion from macropores in to the microporou s par tic lesin the la te ra l d i rec t ion is accounted for in the microporeeffectiveness factor in Equaiion 1 . Hence . the sum of th ethree lis ted contributions gives the total diffusion rate ofreac tan t A per unit cross-sectional area of the pellet. T heresult is :

(Mechanism 1 )

(Mechanism 2 ) (Me c h a n i s m 3 )

A s described by LVakao and Smith ( 7 4 ) . the fac tor o f 4 in thecontribution of mechanism 3 arises from a factor of 2 f ro m th earea and a second facror of 2 f rom the d iffus ion pa th leng thof At '2 . The composite diffusivities . D, n d D,.or macro-and micropores as deve loped in the d iffus ion pape r (74) are :

1

e 1 2 (1 - a y(I - ay) D , + 1 D,,, =

DRIVING FORCE FOR RE ACT I O N6: - % ;,MA C R O PORES/ Y - Ye_ _ _ _ _ _

'Y,)

L .:. *~' E , ' I-,Figure 1 . Model of diffusion and reaction in catalyst pellet

He re Db is the bulk (mo lecula r) diffusion coefficient in th esystem

A+R,a = 1 + (.VR '.YA) (6)a n d D,;,nd D,, re the Knudsen diffusivities for reactantspecies A in the mac ro- an d micropores. respectivel). . Th eyare eva lua ted from mean macro- and micropore rad i i , a,a n d ai: as follows:

Dka = 2 FAa , '3 : D,, 2 F A n r ' 3 ( 7 )Subs t i tu t ing Equa t ions 4 and 5 in to 3 , one ob ta ins

wh e re

Equa t ions 8 an d 9 are based upon cons tan t to ta l p ressure .T h i s is exactlv true in a reaction system only when there is nochang e in nu mb er of moles as a result of reactions. Hobv-ever . the equa t ions a re approximate ly t rue for pseudo-f i rs t -order reactions if the change in moles is not large, as pointedout by Scot t (70). Equat ion 9 shows that the effective dif-fusivity und er reactio n conditio ns io a function of the particle(microporous particle) effectiveness factor. E ,. A s the reac-tion rate decreases, E , approaches un i ty and Equa t ion 9becomes an expression for the diffusivity in a nonreac tivesystem. This result is identical to tha t ob ta ined in the d if-fusion paper (74) when the assumption regard ing- contr ibu t ion( 3 ) o the diffusion rate is taken into account. If the reac t ionrate is very fast. E , decreases and the micropore contr ibu t ion tothe diffusion rate decreases. This means th at the equil ibriumgas concentration prevails everywhere in the mic;oporousparticles . Diffusion then is limited to the macropores-thatis . only the firs t term of E quatio n 9 is retained.Effectiveness Factor fo r Catalyst Pellet

. i t steadv state . the reaction rate in a differential volume ofpellet is equal to the difference in the diffusion rate across thevolume For example . in a one-dimensional case :

Substitution of Equations 1 a n d 8 in to 10givesP dRT d r- - ( D 2) + k , . p B E L ( y v e ) = 0 i l l )

1 2 4 l & E C F U N D A M E N T A L S


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orA l l 2

E, =~;i3~~;J ztw (31)Equat ions 22. 23, and 31 \yere solved numerically using anIBM 1620 computer for various valuei of the microdiffusiori

p a ra me te r , B , macrodiffus ion paramete r , a , and reac t ion ra tep a ra me te r . A. T h e re su l ts ?shown as plots of the pellet effective-ness factor. E,, are given in Figure 2 .

For the case of a = 0 (corresponding to a = O ) , E q u a t i o n 1 7gives the solution for E, . In te rms of dimensionless pa-ramete rs Equa t ion l - may be w ri t ten

" * c o th ( ----)Iz x(B$-, B + 1'( 1 (B-+-> - 1rn-" F = 3 ~~~~~ . x

This analytical result was compared to the results obtained byextrapolating the numerical solution for finite values of a ton = 0. T h e agreem ent w as good in a11 cases, lend ing c on-fidence to the numerical rvork.

F igure 2 shows the expected decrease in E , with anincrease in reaction rate--i.e . , reaction rate para me ter X.\Vhen the average pore radii increase. D,, a n d D,i lsoincrease. F igure2 shows the magni tude of the increase in E , due to this re-duction in diffusional resistances.

Th e method of using F igure 2 ro determine the pelleteffectiveness factor may be briefly sum ma rize d.

Firs t it is assumed that the reaction rate is firs t-order-thatis . Equat io n 15 is app l icab le .The effectiveness factor for particle Ei ust be krio\vn. I tcan be de te rmined from ra te measurements on the par t ic lesas descr ibed by Ra o . Lt-akao . and Smit h (8)or the or tho-hydrogen co nversion. If the poxvder particles are small (lessth a n a few h u n d re d mic ro n s ) . E , is likel>- o be nea rly 1 O.From pore s ize measurements rhe quantities e, e l , a n d a,,a n d c a n b e o b ta in e d , a s d e sc r ibe d b y Wa k a o a n d S mi th( I 1) . 'The value of a w n be ascer ta ined fr om th e s to ich i-ometr y of the reac t ion. Th en param ete rs B a n d a a re e s t a b -l ished fro m Equa t io ns 26 a n d 2'.If rhe reac t ion ra te cons tan t . kw . i s known, X i s de te rminedf ro m E q u a t io n 28 a n d E , read from Figure 2 . If insteadthe reaction rate: i.. for the pellet has been established, a trialand error procedure is necessar>-. A va lue of k , or X is as-s u me d . a n d & is obtained from Figure 2 . T h e n t h eva lue of the produc t (k,E,) is checked by comparison withE q u a t io n 15 .

Evaluation of the method Jv i th da ta fo r the or tho-para -hydrogen reaction is considered in the following paper (8).Acknowledgment

'This results in hig her values of u a n d B .C 0

lo0

IO 0

Th e au thor s express the ir apprec ia tion to R. De Vogelaere ,Dep artm ent of Math ematics . Universi ty of California. Berkeley,for devising a numerical method for the solution of E q u a t io n13. Yu Chang ca rr ied ou t the necessa ry compute r rvork a n dhis assistance is gratefully acknowledged.

loo NomenclatureI D.,?- R = macropore d iffus ion paramete r de f ined by Equa t ion

270 ,fiiBI)D h

= mean rad ius for d iffusion in m acropores? cm .= mean radius for diffusion in micropores. cm.= micropore d iffus ion paramete r de f ined by E qua t ion 26= effective diffusivit?. in catalyst pellet under reaction= binary bulk diffusivity. sq. cm., sec.conditions. s q . cm. sec .

Figure 2. Pellet effectiveness factor

1 2 6 I & E C F U N D A M E N T A L S

(32)


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D , D = effective diffusivity for microporous particle . definedfi,, = mean Knudsen d iffus iv i ty o f gas A in macropores :Dk, = mean Knudsen d iffus iv i ty o f gas A in micropores ,E, = effectiveness fac tor of cataly st pelletE , = effectiveness factor defined for microporous powderpart ic leshI,, h , = Thie le modulus def ined by Equa t ions 16 and 17 ,respectively

= reac t ion ra te cons tan t de f ined per un i t t ime per un i tmass of catalI,st, g. mole/g. sec.= half thickness of s lab-shaped pellet. cm.

= diffusion rate per unit time per unit area of pellet,

by Equa t ion 18 : sq. c m. , s e c .sq . cm. , sec .sq. cm . Isec.

g . mo le, ./ sq . c m ~ec .= to ta l p ressure , a tm.= average reac t ion ra te pe r un i t t ime per un i t mass ofca ta lys t pe l le t , g . mole /g . sec .= reac t ion ra te ( ra te of convers ion of reac tan t ) pe r un i tt ime per un i t mass of powder par t ic le , g . mole /g. sec.= gas cons tan t , cc . a tm. /g . mole O K .= X 2 X/X,= t e m p e r a t u r e , K.= mean molecu la r ve loc i ty , cm. /sec .= dis tance var iab le , cm.= mean rad ius of powder par t ic les , cm.= me a n ra d iu s of spherica l pe l let , cm .= reactant gas mole fraction in macropores of pellet= equi l ib r ium mole frac t ion= reac tan t gas mole frac t ion a t ex te rna l surface of pellet= a iy - e ) / ! J o - J , )= 1 + (1VR/lVA)= macrovoid frac t ion in pe l le t

e te l pe,XP Bp p

= rnicrovoid fraction in peliet= microvoid fraction in particle= solid fraction in pellet= reac t ion paramete r de f ined by Equa t ion 28= density of pellet, g Icc.= densit! of particle , g. ;cc.

literature Cited(1) Beek, John, A.I .Ch.E. J . 7 , 337 (1961).(2) Carberry. J . J . . Ibid . , 7 , 351 (1961).(3 ) Evans, R. B ., LVatson, G. M., Mason, E. .4..Gaseous Diffu-sion in Porous Media at Uniform Pressure, IMP-AEC-15,Inst. for Molecular Phvsics. Lniv. of Marvland, June 1 . 1961.(4) Henry , J. P.. Chennakesanan. B.. Smith . J . M. . A.Z.Ch.E. J .7 . 10 (1961)(5 ) Ho&sch&en, J . , znd. Eng. Chem. 47 , 906 ( 1 955 ) .(6 ) Masamune , S.. Smith. J. M . , A.I .Ch.E. J . 8, 21 7 (1962).,( 7 ) Mingle, J. O. , Smith . J . M. , Ibid . , 7, 243 (1961).(8) Rao: M. R. , Wakao. Noriaki, Smith, J. M., I N D . N C .C H E M .FUKDAYESTALS, 12 7 (1964).(9 ) Rothfeld, I,. B ., LVatson, C . C., Gaseous Countrr Diffusion inCatalyst Pellets . 54th 4nnual Meeting: A.I .Ch .E. , hTew York,Dec. 3-7. 1961.(10) Scott, I). ., Can. J . Chem. Ene., to br published.(11) Scott, I). S.. Cox. K. E ., J . Chim. Phys. 57 , 1010 (1960).(12) Scott. L). S . . Dullirn, F. A . L .. A.I .Ch .E. J. . to be published.(13) Thiele, E . i V . , Znd. Ene. Chem. 31 , 916 ( 1939) .(14) klakao. Noriaki, Smith, J . M., Chem. Erie. Sci. 17 , 82 5 (1962) .(15) Whmler, Alborn. Catalysis , Vol. 11, Reinhold, New York,(16) Weisz, P. B., Schwartz, A . B., J . Catalyris 1, 39 9 (1962)(17) Wicke, E . , Kallenback, R., Kolloid Z. 97 , 135 (1941).

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R E C E I V E Dor review April 5, 1963ACCEPTED ecember 30 , 1963Project carried out w ith the financial assistance of the U. S. ArmyResearch Office, Grant No. DA-ARO(D)-31-124-G191.

DIFFUSION AND REACTION RATES IN THEORTHO-HYDROGEN CONVERSION

M . R A J A R A O , N O R l A K l W A K A O , A N D J . M . S M I T HCni~ersity f Californza, Davis . Calif.

Rate studies were carried out for the orth o-para -hydrogen conversion using single-pellet catalysts of N i Oon A1203. Measurements were also made with the powder particles of catalyst used to prepare the pellets,From these data the effectiveness factor, Ea, was evaluated for pellets of three different densities and, hence,different macropore properties. Pore size distributions, void volumes, and part icle sizes were also meas-ured. This information was sufficient to apply the theory in the preceding pape r to calculate theoreticaleffectiveness factors. The agreement between the experimental and predicted results indicated that thetheory was satisfactory for the specific, bidisperse catalysts used in this study. The effectiveness factor, E t ,for the microporous particles in the pellets was found to be unity. It appear s that E a will b e close to 1.0except for very rapid reactions using pellets prepared from unusually large catalyst particles.

HE effect of pole diffusion on rates of solid catalytic reac-T ions was firs t analyzed by Thiele (6). Since then a con-s iderab le vo lume of l i te ra ture has accumula ted-- fo r example(3 . 5. 7 , 72, 74. 75). Ho we v e r , no experimenta l work hasbeen reported for d iffus ion measurements under reac t ion con-d i t ions Part icu la r ly the re la t ion be tween d iffus iv i ty and poregeometrv . wi th simultaneous reac t ion , has no t been s tud ied

For the experimenta l p r oqr am , reac t ion ra tes were measuredfor s lab- type ca ta lys t pe l le ts under condi t ions ana logous tothose used to develo p the diffusion theory. In addit ion. rateda ta w ere measured for the ca ta lys t pa r t ic les used to p reparethe pel le ts . For these s tud ies the or tho-para -h ydrogen reac t ionwa s e mp lo y e d w in g a 25 % N i O o n A 1 ? 0 3 catalyst. Thisreaction was chosen for several reasons:I n a preced ing paper , a mode l deve loped for d iffus ion in b i-

disperse pore systems (70) is appl ied to the reac t ion case . T h eobject of the present s tudy is to compar e experimenta l resu l tswith the foregoing theory.

The diffusion process is e q u imo la l a n d c q u n te rc u r re n t .Tem per a tu re grad ien ts . even w ith la rge pe l le ts . a re negl ig ib leThis s implifies th e diffusivitv equ ations. because (Y = 0 .because of the low heat of r e a c t io n .

V O L . 3 NO . 2 M A Y 1 9 6 4 12 7

effectiveness factor

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