Theoretical analysis of non-catalytic gas—solid in a single pellet diffusion reactor

Shorter Communtcattons

Theoretical

andysis

of non-astalytic gasadid diifusion reactor

mzacths

in a

single pellet

(Recerved 18 hdy 1977, accepted 1 Muy 1978) Non-catalwc gas-sobd reactions are unportant m many matenal pmcessmg and chemtcsl mdustrtes Such reachons call for specud attentton today regardmg the coal gasdication processes hvlng the past fifteen years, a number of papers on gas-sohd reachons has appeared m the hterature Recently Szekely et al [l] pubbshed a book on the subject The bulk of the literature descnbe mathematical models that can be used to predict the umverston vs tune relattons durmg gas-solid reactions However, only few papers have been pubhshed contammg experunental data or expenmental techmques to test cnttcally the vabdtty of the ex#stmg models In the general case of a gas-sobd reaction, as the solid IS consumed, the pore structure opens up, mcreasmg the rate of dtffuston of reactant gas wttb tune Stmultaneously, the reactive area decreases and the reactlon rate decreases Consequently, to descnbe the relatton between area and solid volume, a model of the pore structure 1s reqmred Yag~ and Kunuf2,3] presented a sunple model of gas-sold reactions called shrmkmg-core model or shell progresswe model However, tlus model offers only a valid descnphon of reactions m which the sobd reactant IS unpervrous to the gaseous reactant or when the reamon 15 strongly dlffuston tontrolIed More general models assume the solid to be a pseudo-homogeneous matertal Some authors assume that the reactton rate IS nth order with respect to sobd concentration[4]. or propotional to the surface area of the sohd[5] In the latter case, tt IS necessary to spectiy a model of the pore structure Several models of pore structure have been pubhshed[fLlO] for lsothermai and nonisothermal gas-solid reacttons[l l-131 The models of pore structure used tn ttus work are (a) Petersen’s [6] Random Pore Model, and (b) the umform packed spheres model or Gram model [7,8] The probkm of gas-solid reacttons can be conceptually reduced to a problem of stmukaneous iitffuston and reaction m a porous sohd with a tune dependent rate, which 1s mathemattcally analogous to the problem of dtffuslon m a solid catalyst under deacttvatton condtttons Recently, Petersen et al [14] and Petersen and Wolf[l5,16] have shown that the single pellet dtffuston reactor (SPDR) IS an expertmental techmque that allows one to dtscrmuna& among dtfferent potsonmg mechamsms The analogy, gas-s&d and catalyst deacttvatton. prompted the authors to utqture whether the SPDR could be used to test cntlcally the theorettcal models used m gas-solid reacbons The operatton and theory of the SPDR has been recently revtewed[17.18], consequently only a bnef descnptton unll be gwen here A schemata dtagram of a reacbve sobd pellet (catal* or n~talyttc) of slab geometry II shown m Fw l(a) Under d~Bus~on mfluencad condttums, the concentratton profile and thus the gaseous reactant centerplane concentration are umquely detemuned by the value of the T&le modulus Consequently. measurmg the centerplane fzoncenmon and the reactton rate allows one to calculate the Tlutle parameter and the rate constant Such measurement can be done by the sunple artifice of cutttng the pellet by half and seahng one of the pellet faces mto a closed chamber, as shown in Fw l(b) The co&ttions existmp at the peUet centerplane of Fu l(a) are reproduced m Fu 1t.b) The arrangement of Fqg l(b) constttute the basw destgn of tbe SPDR. Techmques as IR spectrocopy and gaschromatography[l5,161 can be used to secure the value of the CES Vd

33 No 114

CENTER PLANE CHAMBER

Fig

1 (a), Reactive

solid

pellet, reactor

(b), Smgle

pellet

Qffuston

centerplane concentration Expenments performed at temperatures in the range 38%5OV~[l6] proved that the SPDR can be successftilly applied to study reacttons under a wtde range of coudtttons An analysts sitmlar to the one used to dlscruntnate among different potsoning mechantsms IS apphed m thts work to assess the potenttal of the SPDR to dtscrumnate among the d&rent pore models used in gas-sohd reacttons TElXXtY

Cottstdcr the general gas-soltd reactton A. + 6B. + a0, + eE, where the porous sobd B IS emmonad as a pseudo homogeneous matenal havmg slab geometry The followmg atsumpttons are made 18 the tnmlel development (1). Equihbmun of reactants mstde the catalyst ts fast enough so that a pseudo-steady state condttion can be mvoked (2). N&&k extermd mass transfer resistance exist between gas and sobd (3). Transport of reactant msuie the soltd occurs by Muston only Assumption (1) IS generally accept&k for most gas-sohd reactions[8,22] Experunental condtttons can be ad~uatcd in order to sattsfy asmu (21 The thud assumptton m certamly w&d under con&ions of eqtumolar-cotmterdlfFumon, low reactant concentratton or small reactant fluxes Under these assumptions, the contmutty equatmn for reactant A is

(1)

1558 The conservation

Shorter CommuNca&ons equauon

for the sold

IS even

by (14)

The uutml and boundary

condmons

t=o

x20

t>0

x=0

for eqn (1) and (2) are

c,=c,

(3)

CA = CA(X &)

(4)

CA=&

(5)

where f = dr,, IS the pore radius relatwe to the mmal pore radms G IS a constant related to the unhal sohd porosity by

G’-rr,_+ 4e,

(15)

0

To obtam an exphcd relatton between a and Co, It IS more convement to use 6 as a parameter rather than t9e Then Co can be related to 6 Instead than @a by the followmg relauon Equation (4). whxh follows from assmnptlon (1). is the solution of eqn (1) for a constant mltial actwuy and dtiuslvlty Smce gas-sobd reacuoons take place at the sohd surface, the appropnate rate expression for a fhst order reactum IS rA = c(kC.4)

(7)

de,

dt=

For a first order gas-sohd reachon, the above Wruten m the followmg dunensionless form

As the sohd IS consumed, the surface area decreases, decreasmg the reaction rate Sunultaneously. the porosity increases. mcreasmg the effectwe ddfuslvlty For a porous solid. the effecuve dtiustvuy 1s assumed to change accordmg to 9n

= s,,lM

The tortuosity factor 0 IS expected Increases A relation between n and geneous catalysis[l91 and in W&r’s combustlon[20], IS fl = l/e Thus one general relation between f& and 0

Z!g= a7

e=l+a

(10) In order to account for the formauon of a sobd product (a&es). the folloWmg relation between ~9, and B has been developed The total void IS consIdered to be equal to the irutaat void pIus the void of the sobd product, the latter bemg proporuonal to the porosdy of the sohd product and the difference bctwecn (0, &) Thus the total porosdy 18 gwen by

e=i+a

(2-i

)

(11)

The relation between ge and the sobd surface area m provnied by the model of pore structure A3 noted earher, the packed sphere or gt’aN Node1 and random pore Nod&¶ are USed N thK+Work. Omen modef Accord& to the graN model (GM). I the reacuon proceeds m a well-delined coaceatnc front 111tJm 8raul, the reactive area pr umt volume of the sohd IS gwen by

02) Random pore model. In the random pore model, the surface area per total pellet volume IS related to e,, by the foUowlng equations c -= a0

(13)

can be

(17)

o 0 a0 (

#A

ea --I @o

>

. where the functton f(g$&,) u determmed by the pore model mmal and boundary condmons m dunenszonless form are 750.

where f&a,, = %Le &/& and &,, f&, are the IN&~ porosity and tortuoslty Thus, dependmg on the value of M, eqn (9) allows one to account for the dtierent functtonahues between B,e and B (a) M = 0, constant effectwe duTumvuy, (b) M = 1, constant tortuoslty factor and (c) M = 2. fl= l/g The sohd reactant concentratum can be related dnectly to 0, by

equations

~((-$“~)-hot(~)d*=o

(8) to decrease as the porosity B often lnvokcd in heteroexperunental work N coal can propose the followmg

roa.

#Ah 0) =

-t-1, 7>0.

r)

9 =

1,

The

cash Cbtl - ~111 cash h,, $1

=o.

&A(O.7) = 1

(23)

!$l,+=O

Equation (21) IS the soluuon correspondmg to a first order reaction The system of eqns (17)-(20) IS nonlmear and an analytmal solution IS not feasible A numeruzal soIuuon was obtruned usmg the qua.+hnearzauon tecluuque descriid by L&21] A four-pomt subsututum of the denvtives for fimte Merences was used after unstable solutions were found with the more commonly used Crank-Nlcolson substitution RBuLTg F-s 2 and 3 dmplay the solututn of eqns (17)-Q@ for the Gram and Randon Pore models m terms of the centerplane concentratum #(I. r) vs duuenstouless tune for dtierent values of &, M. P and &, As the reachon proceeds, #(l. 7) Increases vnth tune untd It reaches unity at whch poNt the sobd has been completely consumed As mentxmed earher, the uutml value of @(l. r) can be duectly measured m the SPDR. ~akmg the estunafion of the ThKle modutus posstble through the use of eqn (21) Smce the vahnz. of & and Q can be secured experunentally, the only undertermmed parameters are (a) the value of M and (b)theporeNodel FmZand3showtheeffectof hoandM on the cente@ane concenunuon For a gwen value of ko and merent values of M, the uuhal centerplane concentration Is the same, except for the RPM vntb M =0 In this later case, to obta~ consistent rates, rt was necessary to remove the steady state assumptmn Roth pore models show the same trend regard~g the effects of M and lb on the centerplane concentrahons Such effects can be exphuned m terms of the relive unportance of the dBusmn remstance dependmg on the matenaI and reaction

1559

06 Jl(

I,T)

,

04

z

5

Fig 2 Centerplane

concentratmn

vs dunenslonless

nrne Gram model

ho=25 ho= 50 __----

----_ /’ ,-*I &&EL--‘-_

__---

I

M-0 __-__---3

2

4

T Fa

3 Centerplane

concentraaon

vs dunenslonless

parameters (te iIf, &, a, &) As shown m Figs 2 and 3 for a @van pore model and &,. mcreasmg M Increases the effective tiuslvtty, thus decreasmg the ddfuslon reststance and therefore tbe tune reqmred to attam a Ipvcn converslon In contrast, mcreasmg & Increases the dlfiuslon re-ststance relatlvc to tbe reacuon, and thus Increases the tune requtred to attain a given converslon The effect of the sold ~mhal porosity and ashes porosity can also be related to theu relatwe effect on the d&uslon reastance Mater& havmg low amhal porostttes (80) and ash porostty (Q) present a h&$er dtiuslon resfstance than those wrth h& poroslties, thus mcreasmg the tune t+eqtured for complete conbersxon The relative effects of the reacm aixl matend parameters m the relahve reaction rates are &played m Fw 4 (M- 2) The reactton rates R are presented m duneaslodess form, usmg Z&, the uutml rate as a reference For M = 2. the relphve rates vs tune curves e&bat the daatmct feature of mcreasmg untd reachmg a maxunum and then decreasmg The maxuua 1s the combmed effect of an mcreasmg effective Mustvlty due to the openmg of the pore structure, therefore decreasmg the dlffusmn resistance, and a decrease m the reaction rate due to sobd

tune

Randon pore model

I

I

I

,‘--. zo-

, ’ I,I/’

0

0

/

‘8 ‘\ --x

I

RPM

-----

GM

-

M=2

\

1

r

I

2

\

I

3

4

T Fii

4 Dunenslonless

reaction

rate vs henslonless

tune

1560

shorter

commuNcatlons

NOTATION

a

b C DM B Gii ho k L MB M RPz SPDR t x

@= +(lr)-JI(I,o, reactlon rate vs normahxed concentration

CONcLuslONS A theorebcal analysis of a gas-solid reaction m a smgle pellet dIffuaon reactor indicates that this reactor has potentral to dlscnmmate among different pore models and effecuve dtffuslvuy functlonahties The model selection can he made Independently of adjustable parameters, except for the assumed rate expresof the centerplane slon, and IS based on the measurement concentrabon Wolf and Petersen1161 showed that such measurement can be carried out at Hughtemperature and pressure m the case of a catalyuc reaction It IS ltkely that the centerplane concentratton can also be measured m the case of gas-sold reaction, specmlly at the begmnmg of the experunent The fact that the solid may collapse near comptetion of the reacuon can mtroduce dltiicultles m the later part of the experunents Fortunately Frg 5 proves that the rnrhal results are the most s&cant to dtierentiate among pore models The determmatlon of the pore model that best describes the solid and the appropnate dtiustvlty dependence allows one to predtct accurately the conversion-tune relations GUILLERMO L GUZMAN EDUARDG E WOLF &putiment of Chemzcal Ea~meerut~

_

_

concentratxm

or Knudsen drffuslvrty of A effecuve ddfuslvuy of A gram model Ttuele modulus = Lt/( kad0& reaction rate constant per unit area of sohd width of pellet molecular weight of Z3 parameters definmg funcuonabty between 6 and 9.e reaction rate per total volume of pellet random pore model single pellet diffusion reactor tune distance inside the pellet from the external surface molecular

‘dunenstonless

centerplane

consumption For the same value of the material parameters, the maxuna predicted by the RPM IS larger than the one predicted by the Gram Model Fmre 5 presents the results m terms of relative rate and a normahxed centerplane concentration @ Ths variable replace dtmenslouless tune as Independent vanable and rt can be calculated from the measured values of #(i, 7) Smce b IS determmed by $(l, 0). the R/R0 vs @ curves depend only on the pore model selected since Q and 8, can be measured The untqueness of the curves shown in F~J 5 can then be used to select the appropriate value of M and pore model that best descnbe the data As shown m Fw 5, an approprmte value of M and pore model can be dtscnmmated when @ CO 6 As @ Increases towards umty, It IS more dficult to select the appropriate pore model For the case ho= 5 0. the effect of the pore model can only be distrngurshed for @ < 0 3, however, the effect of M can be clearly distinguished The foregomg dtscusslon indicates that the smgle pellet diffusion reactor IS an expenmental technique that has potential to dlscrmunate among different pore models, and to ascertam the value of M descnbmg the diffusivcty-porosity functronalrty m a gassolid reactton If experimental data obtained m a SPDR does not fit any of the theoretlcal results presented here, rt would suggest an Inaccurate pore model or effective ddfustvlty functlonabty

Unrversrty of-Notre Dame Notre Drrme IN 46556. USA

molar

of solid

Greek symbols

lo-g~l,ol

F~J 5 lZhmenslonless

sobd surface area per total volume stoicluometnc factor

distance x/L

solzd porosrty sohd true densuy dImensionless tune = bWBa,C,&pB duuensionless gas concentrauon C,+/C, tune7 void fraction of sohd product layer duuenslonless ratio T/r0 tortuositv factor normahxed

centerplane

concentration

=

at distance

u.

~r(l,7)-til,O] l,O- cMl.0)

Subscripts A gas reactant B 0

sohd reactant initial conditions

[l] Szekely J , Evans J W and Sohn H , Gas-Solid Reacrrons Academic Press, New York 1976 121 Yap S and Kunu D, 5th Symposrum (Zntemaftonal) on Cornbutton, p 231 Remhold, New York 1955 [3] Yg S and Kunu D , Chem Engng SCI 1%1 16 364 [4] Wen C Y , Ind Engng Chem 196&60(9) 34 I53 Calvelo A and Cunnmgbam R , Ind Engng Chem Funds. 1970, 9(3) 505 161 Petersen E , A I Ch E J 1957 3 443 (71 Calvelo A and Cunnmgham R E , J Cat 1970 17, 1 [8] Szekely J and Evans J W , Chem Engng SCI 1970, 25. 1091 I91 Sxekely I and Sohn H Y . Chem Engng Scr 1972 27 763 DO] Chn C , Chem Engng SC!. 1972,27 367 r113 lshida M , Wen C Y and Shua~ T , Chem Engng Scr 1971, 26 1043 1121 Calvelo A and Smith J M , Proc Chemeca 1970 r131 Shen 3 and Smith J M , Ind Eng Chem Fundls 1%5,4(3) 293 L L and Petersen E E , Chem Engng So 1972, [14] Fi$z I151 W&f E E and Petersen E E , Chem Engng Scr 1974 29 1500 I161 Wolf E E and Petersen E E , J Cat 1977,87 85 E E, “Expenmental Methods III Catalyllc [I71 Petersen Research” (Edited by Anderson R B 1, Vol II Academtc Press, New York 1975 [I4 Hegedus L and Petersen E E , Cat Rev 1974,9 X5 Reactnon Engmeer[I91 Carberry 1 3 , Chemrcaf and Catalpc mg, p 491 McGraw-H& New York 1976 WI Walker P L , Rusmko F and Austm L G , Adv Cal XI

1959. 178

[21] Lee E S, Quasrlmeanzatron and Inunnant Academic Press, New York 1968 I221 Blschoff K , Chem Engng Scl 1%3,18 711

ImbeaWmg