ANALYTICAL EXPRESSION FOR THE NON-ISOTHERMAL EFFECTIVENESS FACTOR: The n th -order reaction in a slab geometry

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1 ANALYTICAL XPRSSION FOR T NON-ISOTRMAL FFCTIVNSS FACTOR: Th h -ordr racio i a slab gomry riqu Muñoz Tavra Dparm o Biogirig, Ric Uivrsiy, ouso, TX USA muoz@ric.du Absrac Th problm o calculaig h civss acor, or a porous slab o caalys pll udr o-isohrmal codiios, was rvisid. A ac ormal aalyical soluio was obaid or a h -ordr, ohrmic ad irrvrsibl chmical racio. A closd aalyical ormula was obaid Muñoz Tavra, 5 or h vry as racio limi, ad is applicabiliy was umrically sd or a sigiica rag o valus o h paramrs ivolvd i h modl, i. h hrmiciy group ß ad h Arrhius group.. Iroducio A wid variy o chmical procsss ivolv h us o hrogous caalyic racors as a cor ui opraio. Tradiioal chmical girig applicaios L, 985; Sarild, 97, li hydrogaio o orgaic compouds, oidaio, dhydrogaio, polymrizaio ad caalyic cracig, amog may ohrs, ar prormd by coacig a luid gas or liquid racig phas wih solid porous paricls, which possss a high surac-volum raio o hac coac bw h luid phas ad h acual caalyic ag, ypically a mal Ni, Pd, F, Cu, P or mal oid. Aalogous siuaio ariss i som biochmical girig procsss Bucholz, 98; gassr ad orvah, 976, i paricular i immobilizd zym racors, whr zyms ar id o a porous mari, such as agaros. From a chmical girig prspciv, oly macroscopic variabls i h racor ar accssibl or boh masurm ad corol. owvr, i h procsss dscribd abov, chmical or biochmical caalyic racios a plac a h surac o solid paricls, usually a h por suracs i h paricl irior, ad cosquly hy dpd o microscopic disribud paramrs, such as cocraio ad mpraur, which ar rlad o h macroscopic valus hrough a combiaio o mass ad ha rasr rsisacs, i addiio o h chmical racio isl. A simpliid aalysis o h ral siuaio is o cosidr wo rsisacs i sris L, 985; Sarild, 97, h irs o or raspor hrough h ral boudary layr which surrouds h paricl, ad h scod o du o h combiaio o diusio ad racio a h irior o h pors. A sady sa, h ovr all racio ra will b corolld by h highr rsisac, i.. h slows sp. For h cas i which racio ad diusio a h irior o h pors corols, h hory o civss acors ca b applid. Origially dvloppd idpdly by Damöhlr 935, Thil

2 939 ad Zldovich 939, his hory has bcom a impora ad usul cocp i hrogous racor aalysis, dsig ad corol or may yars. siv horical sudis hav b prormd or h calculaio o civss acors by assumig isohrmal codiios ad dir racio ras Bischo, 965, ad aalyical prssios ca b obaid or ypical cass: simpl h -ordr, Lagmuir isohrm Chu ad oug, 96 assumig simpl slab o-dimsioal gomris. Ohr shaps sphrs, cylidrs hav b sudid Amudso ad Luss, 967 i h co o h irsordr racio. Udr codiios wh ira-paricl hrmal rsisac bcoms impora, i.. oisohrmal pll, h aalysis bcoms mor complicad du o h uavoidabl couplig bw ha ad mass rasr quaios. Much horical wor has b prormd o sudy qusios rlad o uiquss ad sabiliy o soluios Aris, 969; Dro ad Aris, 969; laváč al., 969a,b; Luss, 968 Dspi h ihr mahmaical compliy o gral soluios dscribd abov, i may pracical cass h iral mpraur gradis ar comparaivly small laváč al., 969a bu o cssarily gligibl, ad or vry as irrvrsibl racios h cocraio o raca a h o h pll will b clos o zro, so uiquss o soluios i hos cass is guarad v o physical grouds. I his co, simpl mpirical aalyical prssios or h mpraur dpdc o h civss acor i h irs-ordr racio hav b rpord Liu, 969. A ac ormal aalyical soluio was dvlopd Muñoz Tavra, 5 or h oisohrmal civss acor i a slab gomry, or h cas o irrvrsibl ad ohrmic h -ordr racio ra. This gral ormal prssio is spcializd or h limi o a vry as chmical racio, whr h cocraio a h o h slab ca b assumd o b clos o zro, obaiig a aalyical ormula suiabl or dirc calculaios.. Thory.. Diiio o h civss acor For a porous caalys pll, whr h corollig rsisac is assumd o b diusio ad chmical racio occurrig a h irior pors, h civss acor is did by h prssio: RdV V Ravg pll Vpll η R R sur sur

3 whr R rprss h volumric racio ra isid h pll pors, ad V pll is h oal volum o h caalys pll. I wha ollows, w shall assum ha C is h raca cocraio a h irior o h pll pors. W shall also cosidr civ raspor propris isid h pors: D ad will b h civ diusio coici ad hrmal coduciviy. Th racio halpy will b Δ R, ad h soichiomric coici gaiv or h raca will b ν c. Udr sady-sa codiios, h mass ad rgy balac quaios ar: Mass Balac: D C ν R rgy Balac: T Δ R c R 3 Subsiuig h racio ra rom quaio io h civss acor diiio, ad ar applyig Gauss Thorm: η D C ds ˆ ν V R 4 c pll sur S pll prssio 4 is i gral valid or ay gomry. owvr, i may applicaios h irsig cass ar simpl gomris slabs, cylidrs, sphrs whos symmry allows us o assum ha h cocraio gradi is idpd o h posiio ovr h surac. Udr his las assumpio, quaio 4 ca b rducd o: η D S C 5 pll ˆ ν c V pll Rsur sur.. Pll wih h shap o a slab Cosidr a caalys pll wih h shap o a slab, o hicss L, wih L << S pll. For his gomry, h mass ad rgy balac quaios,3 rduc o h ollowig prssios: Mass Balac d C D ν c R d 6 rgy Balac d T Δ R R d 7 Th boudary codiios or his sysm o dirial quaios ar:

4 T ± L T, C ± L C I s s dt dc II, d d From quaios 6 ad 7, i is possibl o limia h racio ra. Ar igraio subjcd o boudary codiios I ad II, i dimsiolss orm o obais: θ 8 C T ζ,, θ, L C T s s Δ R D ν c Ts C s No ha or a ohrmic racio >. Th o-dimsioal group has b rrrd i h liraur Sarild, 97 as hrmiciy or ha graio ucio. I rprss h raio bw h ra o ha graio du o h chmical racio, ad h ra a which ha is raspord by hrmal coducio mchaisms. Th hrmiciy is h a dirc masur o o-isohrmal cs, ad i ollows rom 8 ha, or a vry as racio : T T ΔT s ma θ 9 Ts Ts I ca b cocludd rom 9 ha h limi rprss h isohrmal pll cas, whil o-isohrmal cs bcoms mor impora as h hrmiciy valu icrass. owvr, v or highly ohrmic racios laváč, 969a ; Sarild, 97, h hrmiciy rarly cds.. Tabl : primal valus laváč al., 969a or h paramrs ivolvd i h modl i som idusrial chmical racios Racio φ N 3 syhsis Oidaio o C 3 O o C O Syhsis o viylchlorid ydrogaio o hyl Oidaio o hyl Dissociaio o N O ydrogaio o bz Oidaio o SO

5 .3. Racio ra o arbirary igr ordr L s rsric h aalysis o a h -ordr racio iics, whr is a igr R R g T T s κ s C ad κ s is h valu or h iic cosa a surac mpraur T s. I ca b show Muñoz Tavra, 5 ha or his racio ra, a ac prssio or h civss acor is / η d φ By diig h ucioal [, ] as: [, ] w w w dw i ca b show Muñoz Tavra, 5 ha i saisis h idiy d φ [, ] 3 No ha by mas o q.3, h o-dimsioal cocraio a h o h slab, is did as a ucio o h gralizd Thil modulus a isohrmal codiios φ. Usig h sam oaio, h civss acor i 3 ca b prssd as: η [, ] 4 φ.4. ac Aalyical prssio As is dmosrad i dail i h Appdi, a ovl closd aalyical prssio or h igral has b dvlopd Muñoz Tavra, 5, i rms o wll-ow spcial ucios, h poial igrals Abramowiz ad Sgu, 97a.

6 [, ]!!! [ ] [ ] 5 I a vry rigorous calculaio is dsird, prssio 5 ca b subsiud io 3, ad prormig a umrical igraio, h ac valu o h o-dimsioal cocraio a h o h caalys pll ca b obaid, or ay igr racio ordr ad gralizd Thil modulus φ. Th calculaio rquirs a iraiv procdur, du o h o-liar dpdc bw boh paramrs. Oc is obaid, i is subsiud i 4 o obai h ac valu o h o-isohrmal civss acor. I h prs wor, a mor pracical approach is proposd, by assumig h cas o a vry as racio, whr h approima limiig codiio ca b applid. Udr his assumpio, rom h ac prssio 4, h corrspodig approimaio or h civss acor bcoms η [, ] 6 φ Taig h corrspodig limi i h ac aalyical prssio 5 η φ! /!! 7 To s h accuracy ad validiy o h limi 7, umrical calculaios whr prormd o obai h ac soluios φ rom quaio 3, or small valus o. Tabl displays h rsuls or h cass,. Also show is h rlaiv rror ivolvd i h calculaio o h civss acor by usig 7, compard wih h ac prssio 4, accordig o h ormula: η η η [,] [, ] 8 As pcd o physical grouds, Tabl rlcs h ac ha as φ. owvr, v or ii valus o h gralizd Thil modulus ad, h rlaiv

7 rror ivolvd i calculaig h civss acor by usig h aalyical approimaio 7 dos o cd 5% or mos cass show i Tabl. As h produc icrass, h miimum hrshold φ, mi i h Thil modulus o achiv a accpabl prcisio dcrass. Rcommdd hrshold valus or h applicabiliy o quaio 7 ar show i Tabl 3. Comparig h valus o h calculad Thil modulus or h dir cass prsd i Tabl, wih h primal valus or commo idusrial racios prsd i Tabl, i is cocludd ha h approimaio 7 is applicabl or may ral cass. Tabl : Numrical soluio o q. or φ. Also show is h rlaiv rror, as did i q.7, ivolvd i h calculaio o h civss acor by usig 6, compard wih h ac prssio. φ η/η φ η/η , , , , Th ohr irsig aur ha ca b obsrvd rom h rsuls prsd i Tabl 3, is ha o-isohrmal cs ar srogly dpd o h valus o h Arrhius group. As was poid ou prviously, i h isohrmal limi i is pcd ha ηφ or largφ.

8 owvr, du o iral mpraur gradis, ηφ raiss up o or zro-ordr, up o 5 or irs-ordr, ad up o 3 or scod-ordr iics, wh larg valus o ad ar aaid. Thos umbrs rprs corrcio acors o h isohrmal limi η / φ. This ac has b rpord i prvious umrical sudis Aris, 969; Dro ad Aris, 969; laváč al. 969a,b; Liu, 969; Luss, 968, ad a dpdc rlad o h produc as b proposd laváč al. 969a,b; Liu, 969. Such a dpdc also bcoms plici i a low ba approimaio o q. 7 Muñoz Tavra, 5, wih a proporioaliy o h / civss acor o h prssio. I agrm wih his asympoic rsul, a compsaio dcy or o-isohrmal cs is obsrvd, accordig o h rsuls i Tabl 3, or high valus o h racio ordr. φ,mi φ,mi Tabl 3: Rcommdd rshold valus or h gralizd Thil modulus, φ φ, mi, o apply h ormula 6 wih a rlaiv rror o cdig %, or irs ad scod ordr iics. 4. Coclusios Accordig o h sadard hory or civss acors, a ovl rigorous mahmaical rsul was drivd or h cas o a o-isohrmal caalys pll, wih h shap o a slab, or a ohrmic irrvrsibl racio o h -ordr. Ar his rsul is spcializd or h cas i which h racio is vry as, ad h cocraio i h o h slab is clos o zro, a aalyical prssio was drivd or his limi. This approimaio was compard wih umrical igraio rsuls, or dir ralisic valus o h paramrs ivolvd i h modl, showig good agrm or ii valus o h gralizd Thil modulus. This aalyical ormula is rlaivly simpl, ivolvig wll ow spcial ucios, ad hror is suiabl or compuaioal applicaios, li simulaio ad dsig. Noaio C Raca cocraio, mol m -3 C s Raca cocraio a pll surac, mol m -3 D civ diusio coici isid h pll pors, m s -

9 Acivaio rgy o h chmical racio, J mol - Dimsiolss cocraio Dimsiolss cocraio a h o h slab Δ R Racio halpy, J mol - civ hrmal coduciviy isid h pll pors, W m - K - R Volumric racio ra, mol m -3 s - R avg Avrag racio ra, mol m -3 s - R g Uivrsal cosa o gass, J mol - K - R sur Racio ra a ral surac codiios, mol m -3 s - S pll ral surac o h pll, m T Tmpraur, K T Tmpraur a h o h slab, K ΔT ma Maimum mpraur dirc, K T s Tmpraur a h pll surac, K V pll Toal volum o h caalys pll, m 3 L Characrisic hal hicss o h slab, m Thrmiciy group, dimsiolss φ Gralizd Thil modulus, dimsiolss Arrhius group, dimsiolss η civss acor, dimsiolss κ s Kiic cosa or h -ordr racio a surac codiios, mol - m -3- s - ν c Soichiomric coici, gaiv or raca, dimsiolss θ Dimsiolss mpraur Dimsiolss mpraur a h o h slab θ Aowldgms I is a plasur o ha Dr. duardo Myr or lighig discussio. Appdi ollowig sps Th ac aalyical igraio or h ucio [ ] w [, ] w dw w. Rarrag h rms i h umraor: / w w, was prormd hrough h A

10 [ ] dw w w /, A. Chag o variabl: w z [ ] dz z z, / A3 3. Chag o variabl: z / [ ], c d A4 4. Biomial pasio:!!! A5 5. Subsiu A5 io A4: [ ] d!!!, A6 6. Dcompos h igrals i A6 io wo rms: d d d A7 7. Apply h ollowig chag o variabls o h irs ad scod igrals i A7, rspcivly: / /

11 d d d A8 From h diiio o h poial Igral ucio Abramowiz ad Sgu, 97a: d A9 d A 8. Subsiuig h rsul A io A5, h ial prssio is obaid: [ ] [ ] [ ]!!!, A Rrcs Abramowiz, M., Sgu, I. A., 97a. adboo o Mahmaical Fucios. Dovr, 9 h d., pp Abramowiz, M., Sgu, I. A., 97b. adboo o Mahmaical Fucios. Dovr, 9 h d., pp Amudso, N.R., Luss, D., 967. O a Cojcur o Aris: Proo ad Rmars. A.I.Ch.. Joural 3, Aris, R., 969. O sabiliy criria o chmical racio girig. Chmical girig Scic 4, Bischo, K. B., 965. civss Facors or Gral Racio Ra Forms. A.I.Ch.. Joural, Buchholz, K., 98. Racio girig Paramrs or Immobilizd Biocaalyss. Advacs i Biochmical girig 4. Sprigr-Vrlag, Brli, pp Chu, C., oug, O. A., 96. Th c o adsorpio o h civss acor o caalys plls. Chmical girig Scic 7,

12 Damöhlr, G., 935. Th adsorpio vlociy o gass o porous adsorbs. Zischri ür physialisch A74, Dro, D.W., Aris, R., 969. Commuicaios o h hory o diusio ad racio I A compl paramric sudy o h irs-ordr, irrvrsibl ohrmic racio o a slab o caalys. Chmical girig Scic 4, gassr, J.M., orvah, C., 976. Diusio ad Kiics wih Immobilizd zyms. Biochmisry ad Biogirig. Immobilizd zym Pricipls. Acadmic Prss, Nw Yor, N.Y. pp. 7-. Gradshy, I.S., Ryzhi, I.M.,. Tabl o Igrals, Sris ad Producs. Acadmic Prss, 6 h d., p. 89. laváč,., Kubíč, M., Mar, M., 969a. Aalysis o Nosaioary a ad Mass Trasr i a Porous Caalys Paricl I. Joural o Caalysis 5, 7-3. laváč,., Kubíč, M., Mar, M., 969b. Aalysis o Nosaioary a ad Mass Trasr i a Porous Caalys Paricl II. Joural o Caalysis 5, 3-4. L,.., 985. rogous Racor Dsig. Burworh Pub., s d. Liu, S-L., 969. Sabl plici Dirc Approimaios o Parabolic Parial Dirial quaios. A.I.Ch.. Joural 5, Luss, D., 968. Suici codiios or uiquss o h sady sa soluios i disribud paramr sysms. Chmical girig Scic 3, Muñoz Tavra,., 5. Aalyical prssio or h o-isohrmal civss acor: h h -ordr racio i a slab gomry. Chmical girig Scic 6, Sarild, C.N., 97. Mass Trasr i rogous Caalysis. M.I.T. Prss, s d., pp Thil,.W., 939. Rlaio bw Caalyic Aciviy ad Siz o Paricl. Idusrial ad girig Chmisry 3, Zldovich, Ya.B., 939. Th hory o racios o powdrs ad porous subsacs. Aca Physicochimica URSS,