A Kinetic Model Framework for Combined Diffusion and Adsorption Processes

Transcription

1 BRINKMANN, E.A. and KING, R.P. A kinetic del fraewrk fr cbined diffuin and adrptin prcee. APCOM 87. Prceeding f the Twentieth Internatinal Sypiu n the Applicatin f Cputer and Matheatic in the Mineral Indutrie. Vlue 2: Metallurgy. Jhanneburg, SAIMM, pp A Kinetic Mdel Fraewrk fr Cbined Diffuin and Adrptin Prcee RA. BRINKMANN and R.P. KING Departent f Metallurgy and Material Engineering, Univerity f the Witwaterrand, Jhanneburg A kinetic del fraewrk fr ingle particle kinetic in cbined diffuin and adrptin prcee i preented. The fraewrk applie t the general cae in which bth diffuin and the adrptin reactin play iprtant rle in deterining the verall rate in prcee where the igrating pecie can bth adrb and accuulate. The technique, which huld be applicable t a wide range f reactin echani, cnit f an iterative integratin prcedure fr the cupled diffuin and reactin equatin. The cae f a reverible linear adrptin reactin rate i develped. In rder t verify the technique a a legitiate ethd f del develpent, an analytical lutin fr the identical cae i derived and cpared t the nuerical lutin. Excellent agreeent exit between the tw lutin. Intrductin Interactin between igrating pecie in a fluid and ru particulate phae, whereby e f the igrating pecie bece ibilized with repect t further fluid igratin a diffuin prceed, are fairly cn in adrptin-type prcee uch a carbn in pulp, rein in exchange and reval f gaeu pllutant. The tranprt-react in echani perative in prcee where the igrating pecie can bth adrb and accuulate are:(l) external a tranfer fr the bulk fluid t the uter urface f the pru particle, (2) intraparticle a tranfer thrugh the pre fluid and alng the wall f the pre, and (3) adrptin at a ite interir t the particle. A variety f del baed n the abve echani have been develped, frequently with the auptin that the adrptin reactin i intrinically rapid with repect t the 'a tranfer prcee uch that equilibriu i aintained at all tie between the free and adrbed cpnent f the diffuing pecie at the adrptin ite. When the apprach t yte equilibriu i uch lwer than can be accunted fr by diffuin cntrl alne r when the cntrlling echani in a prce are uncertain, the validity f thi auptin bece quetinable. In the preent paper, a del fraewrk i preented fr the general cae where bth a tranfer and the adrptin reactin ay play iprtant rle in deterining the verall rate. The ain bjective are: (1) t illutrate the lutin technique, which cau be ued fr a variety f reverible adrptin reactin type, and (2) t verify the nuerical technique reult by cparin with an analytical lutin fr the cae f a firt-rder reverible adrptin reactin. A KINETIC MODEL FRAMEWORK FOR DIFFUSION & ADSORPTION 189

2 Baic equatin The prble cnit f ru pherical particle f radiu R cpried f a lid that i capable f adrbing a fluiddiffuing reactant pecie at cncentratin C. The pecific cae where the pecie can bth adrb and accuulate i cnidered. In rder t iniize the cplexity f the prble, intraparticle diffuin ha been attributed entirely t pre diffuin. Thu, a diffuin prceed, the igrating pecie Cat cncentratin C) i adrbed and ibilized, prducing a nn-diffuing adrbed prduct pecie at cncentratin S. Under thee cnditin, the dynaic a balance fr the adrbate i (i D OC) _ i L S = i C r p r Ep t t with bundary cnditin CC r=r) = C b C -Cr=O) = r [1 ] [2 ] [3 ] Fr reverible adrptin, the reactin rate i gverned by S _ t - fcc,s) [4] with the initial cnditin SCt=O) = [5] The auptin f quai-tatinary behaviur and dilute lutin perit the refrulatin f Eq.[l] (l ) = l S r r Dp Ep t [ 6] Evidently, the external a tranfer reitance ha been neglected. In a wellixed idrber, thi reitance ay be acribed priarily t diffuin thrugh a hydrdynaic bundary layer at the uter urface f the particle. Therefre, the auptin f negligible external a tranfer reitance can be readily relaxed and the apprpriate bundary cnditin can be accdated within thi fraewrk. 19 Fr the ake f iplicity, the bulk cncentratin f the fluid phae wa aued t be cntant. Thee factr were nt included a the bjective i t cnider the repective rle f internal diffuinal effect and the adrptin reactin rate exprein in the verall rate prce. Obviuly, the excluin f external a tranfer effect and variable bulk fluid cncentratin caue eci liitatin under certain cnditin,,?,, a batch envirnent. Slutin technique The lutin technique cnit f an iterative integratin prcedure iilar t that decribed by Saltelli and Offecan 1 fr Eq. [6]. the cupled equatin Eq. [4] and In thi prcedure, the igrating pecie i firt allwed t diffue with n adrptin taking place. Then, after each tie tep f diffuin, adrptin i allwed t ccur befre taking the next tie tep f diffuin. In each adrptin calculatin, the lutin btained fr the previu integratin i ued a the initial cnditin. In rder t illutrate the lutin technique, prvide nuerical reult and verify the technique, the ca f a firtrder reverible adrptin reactin i cnidered. E =kc-ks t pad and Eq.[6] bece Peud-analytical lutin In thi cae, Eq.[4] bece 2 ka 2 kd r -C-r LD- D E P P P [7] [8] In rder t illutrate the lutin technique, eud-analytical lutin ver tw diffuin tie tep i preented fr the cae f reverible adrptin with a linear rate. METALLURGY: GOLD PROCESSING

3 Firt tie tep Accrding t Eq. [5] the value f S at tie zer i equal t zer uch that the ecnd ter n the right f Eq.[8] i eliinated. After integratin f the dified Eq.[8] and applicatin f the bundary cnditin in Eq.[2] and [3] the cncentratin f the diffuing pecie in the pre fluid wa btained R inh(r/ck ID )) p Cb r inh-rr/ck ID [9] A the bulk fluid cncentratin wa aued cntant, the ad r bed pecie cncentratin at tie t1 wa btained by the ubtitin f Eq.[9] int Eq.[7], integratin and applicatin f the initial cnditin given by Eq.[5] E p Cb p Secnd tie tep R k a r kd [1 - exp Si( r/dp)) inh(r/ck ID ) T C-k dt 1)] [1] The cncentratin f the diffuing pecie in the pre fluid at tie t2 wa deterined by ubtitin f Eq. [1] int Eq.[8] fllwed by integratin and applicatin f the bundary cnditin given by Eq. [2] and [3] R inh(r/ck ID )) CC r, t 2 ) = Cb r Sir;-h[R/(k/D T + R2 Cb 2-- I(k ID ) cth (RICk ID )) rap C B. ICk ID ) b 2 [11] The cncentratin f the adrbed pecie at tie t2 wa calculated by ubtitin f Eq.[ll) int Eq.[7], integratin and applicatin f the initial cnditin given by Eq. [1] E k R inh(r/ck ID )) ) p a SCr,t2 = Cb P k r inh(r/ck/d) [1 - exp C-k d t 2 )] E k R2 + C -- I(k ID ) cth(r/ck ID )) b P kd 2 rap inh(r/ck ID )) [F] inh(r/ck 7DJl" E k R - C - ICk ID ) b P kd 2 ch(r/cka/dp)) where inh(r/ck ID )) F = 1 - exp Ck d t 1 -k d S) - exp C-k d t 1 ) + exp C - k d t2 ) [ 12) [13] The prce culd be repeated, with the initial cnditin given by being ued t evaluate SCr,t.). 1 Nuerical lutin SC r, t. 1) 1- In rder t prvide nuerical reult which can be utilized in the verificatin f the lutin technique, the cae f reverible adrptin with a linear rate i preented. The nuber f independent paraeter t be cnidered fr nuerical analyi wa iniized by aking Eq.[7) and [8) dieninle. Fur dieninle variable and ne dieninle paraeter were chen: Ca) dieninle radial crdinate, A, r f_ = R [14] Cb) dieninle igrating pecie cncentratin f the pre fluid, a, [15] Cc) dieninle cncentratin f adrbed pecie,, S =--- Sequil Cd) dieninle tie, e, [16] A KINETIC MODEL FRAMEWORK FOR DIFFUSION & ADSORPTION 191

4 (e) and dieninle paraeter,, /(i k ID ) (17) (18) The nuerical lutin prcedure fr the cae f reverible linear kinetic iic the peud-analytical technique thrugh tie by applicatin f a finite difference technique fr the evaluatin f a with being evaluated 'analytically'. A dified tridiagnal Gauian eliinatin technique 2 wa ued t lve the et f iultaneu equatin. Nueru prgra run were executed in rder t evaluate the effect f the nuber f pacing interval alng the particle radiu (i.e. the nuber f ndal pint in the finite difference analg) and the increental ize f the dieninle tie tep. In all cae it wa fund that, fr articular value f 8, value aypttically decreaed a the nuber f ndal pint increaed and aypttically increaed a the increental ize f the tie tep decreaed. Althugh the lutin cnverged, it wa cncluded that apprxiately 1 ndal pint and dieninle tie increent a all a,1 were required. Analytical lutin In rder t evaluate the nuerical lutin technique, an analytical lutin wa develped fr the cae f a reverible linear adrptin reactin rate. Fr the purpe f cparin with the nuerical reult, the analytical lutin wa evlved in ter f the dieninle quantitie decribed by Eq.[14-18). Slutin fr the dieninle igrating pecie cncentratin in the pre fluid, a, and the dieninle adrbed pecie cncentratin,, were btained by applicatin f the Laplace tranfratin with repect t the dieninle tie 192 variable, 8. The ethd f lutin fllw Crank 4. that utlined by Wiln 3 and The dieninle equivalent f Eq. (7) and (8) are M = a - 8 -L (,? <!.) = 2,,..2 ( a- ) OA. OA. with crrepnding bundary cnditin a(a.=l) = 1 a(a.=o) = OA. and initial cnditin (8=)= (19) (2) (21) (22) [23 J After tranfratin int the Laplace dain, Eq.[19-22) bece = a +l a OA.2 A. OA. a(a.=l) = 1 - a -(A.=O) = OA. 2- a +l (24) (25) (26) (27) A differentiatin f ex with repect t d nt appear in Eq. (25), the equatin ay be lved a an rdinary differential equatin. After integratin and applicatin f the bundary cnditin, the lutin t Eq.(25) wa btained a(a.,) 1 inh A. inh (28) The invere tranfr f a wa deterined by uing the reidue at the ple 5 a(a.,8) = 2 2 (_-1) n 1 + TeA. n=l n ine ntea.) n Te exp 2 2 [-n Te 8 ] n Te and the invere tranfr f after there 6 (A.,8) applicatin f the 1-2 exp (-8) + -- z TeA. n=l (29) wa btained cnvlutin METALLURGY: GOLD PROCESSING

5 2 2 in(nna.) exp [-n J <jj n n 2 na. (_l)n exp (-8) L in( nna.) [3] n n=l Technique verificatin The nuerical and analytical lutin btained fr the cae f reverible linear kinetic prvide dieninle tie dependent prfile f dieninle adrbed pecie cncentratin veru dieninle radial crdinate f the pru particle. In Fiure 1-3, the nuerical reult btained by applicatin f the lutin technique are cpared t the btained fr the analytical lutin at different value f the dieninle paraeter, <jj. In the nuerical lutin, 1 ndal pint in the finite difference analg and cntant dieninle tie tep f,1 w'ere ued. The evaluatin f the infinite erie prtin f the analytical lutin wa terinated when the nth ter f the generated peud pattern 7 wa le than 1-5 Excellent agreeent exit between the tw lutin at all three value f the dieninle paraeter, <jj. Thu, it i cncluded that the lutin technique prvide a valid ean f del develpent which huld be applicable fr a wide range f reactin rate exprein in cbined diffuin and adrptin prcee. Additinal reark A del fraewrk, which huld be applicable t a wide range f adrptin reactin rate, ha been decribed and verified a a viable technique fr prcee in which the igrating pecie can bth adrb and accuulate by cparin with an analytical lutin fr the cae f a reverible linear adrptin reactin rate. A the del fraewrk i till under develpent, n attept ha yet been.2 en en G = en W =========.1 G 3. 1j; = 1. NUMERICAL SOLUTION. ANALYTICAL SOLUTION. ANALYTICAL-NUMERICAL 'TI 'TI rn ::D -. rn C b L L O DIMENSIONLESS RADIAL COORDINATE FIGURE 1. Dieninle adrbed pecie prfile fr reverible linear kinetic. t/; f nuerical and analytical lutin 1.; cparin A KINETIC MODEL FRAMEWORK FOR DIFFUSION & ADSORPTION 193

6 :LO.125 (\.9.1 I- er:.. I I- O.B \,,-----,-,_ U.5.7 \J, ---"'--. U O.S ID er:.5 <t J \ "".25..,.. -v'--=--_""'=" jJ NUMERICAL SOLUTION. \\-\ -.25 '\' O.S.7 O.B.9 :LO DIMENSIONLESS RADIAL COORDINATE " " JJ z n FIGURE 2. Dieninle adrbed pecie prfile fr reverible linear kinetic. y; = 1.; cparin f nuerical and analytical lutin :LO, S.9 I- i <t fe O.B i O'7 O.S! ID er: < -.J 1jJ NUMERICAL SOLUTION. ANALYTICAL SOLUTION. ANALYTICAL - NUMERICAL. -, , , , '---._ \\ -..4, I = O.O O.S.7 O.B.9 1. DIMENSIONLESS RADIAL COORDINATE " JJ rn n FIGURE 3. Dieninle adrbed pecie prfile fr reverible linear kinetic. y; = 1.; cparin f nuerical and analytical lutin 194 METALLURGY: GOLD PROCESSING

7 ade t cpare del reult btained by the technique with publihed experiental data n cbined diffuin and adrptin prcee. Iprveent in the fraewrk can be btained in everal way: (a) Incluin f external a tranfer effect. (b) Ipleentatin f an iprved nuerical prcedure (i. e. variable ize f nde pacing and dieninle tie tep). (c) Expanin t ther reactin echani. C Dp F r R S t Greek letter Ntatin igrating pecie cncentratin in the pre fluid, -3 g c f ; Cb igrating pecie cncentratin in the bulk fluid effective intraparticle diffuivity (baed n vid and lid area f the 2-1 particle), cf+ functin f tie defined by Eq.[13] -1 adrptin rate cntant, -1 derptin rate cntant, radial ditance fr the center f the pru particle, cf+ radiu f the pru particle, cf+ tranfr variable cncentratin f adrbed pecie, -1 gag tie, a dieninle igrating pecie a cncentratin f pre fluid, defined by Eq. [15] Lap1ace tranfr f a(k,e) e p <jj l. 2. dieninle cncentratin f adrbed pecie, defined by Eq.[16] Laplace tranfr f (k,e) internal prity f 3-3 particle, cfc f + dieninle tie, Eq. [17] dieninle defined by Eq.[14] radial the defined pru by crdinate, apparent deni ty f the pru -3 particle, gc + f dieninle paraeter, defined by Eq. [18] Reference SALTELLI, A. and OFFERMAN, P. Abut the delling f tranprt with cheical reactin. Cheical Engineering Science. vl. 41 n. l pp BE CKE TT, R. and URT, J. Nuerical Calculatin and Algrith. McGraw ill, New Yrk, WILSON, A.. A diffuin prble in which the aunt f diffuing ubtance i finite - I. The Philphical Magazine. vl.39 n pp CRANK, J. The Matheatic f Diffuin. 2nd editin, Oxfrd Univerity Pre, Lndn, KUFITTIG, P.K.F. Intrductin t the Laplace Tranfr. Yrk, Plenu Pre, New 6. CURCILL, R.V. Operatinal Matheatic. McGraw-ill, New Yrk, BRINKMANN, E.A. Ph.D. thei, Univerity f the Witwaterrand. ubitted. T be A KINETIC MODEL FRAMEWORK FOR DIFFUSION & ADSORPTION 195