a detailed model for nonisothermal sorption.pdf

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C h e m i c a l E n g i n e e r i n g Science, Vol . 42. No . 7. pp. 1585-1593. 1987. 0009~2509/87 S3.00+0.00Printed in Great Britain. Q 1987 Pergamon J ournals Ltd.

DET ILED MODEL FOR NONISOTHERM L SORPTION

IN POROUS DSORBENTS

L . M. SUN and F. ME UNI ERL aboratoire de T hermodynami que des F luides, U-A. C .N.R.S. 874, Bat 502 ter C ampus Un iversitaire, 91405

Orsay, France

(Receiv ed 16 Ma y 1986; a c c ep t e d f o r p u b l i c a t i o n 9 O c t o b er 1986)

Abstract-A more detailed model for nonisothermal sorption in a single micr oporous particle is proposed,based on the gradient of chemical potential as driving force and simultaneous gas and adsorbed phasediffusion. T he system of the simultaneous mass diffusion an d heat conduction equations, lineari zed tosimplify the analysis, is solved analytically, yielding the internal concentration. temperature profile and thekin etic uptake of the particle. Its shown that two parameters play an important role: Lewi s number L e andthermal B iot number Bi . Th e isothermal model would be valid if L eBi > 100 and the unifor m temperatureprofile model proposed by L ee and R uthven would be a good one if L .e z- 10. In other cases, the morecomplete model p resented herein should be used.

INTRODUCTION

The sorption in micr oporous adsorbents is alwaysassociated with large heat generation, and the influencemay be all the more impor tant since the diffusingprocess is rapid and the heat of sorption is high.However, the thermal effects were not taken intoaccount until conflicting results for diffusivities ob-tained by the traditi onal tr ansient uptake method andthe Nuclear Magnetic R esonance (NMR) methodswere observed. Fr om then on, different models havebeen tested to analyze nonisothermal sorption inmicr oporous adsorbents and i t turn ed out that theintrusion of thermal effects may be one of possibleexplanations for discrepancies in measurements ofdiffusivities.

Several models describing nonisothermal sorptionin microporous adsorbents can be found in theliterature. Chihara et al. presented a simplified modelneglecting the internal heat and mass transfer resist-ances (1976) whi le the model of L ee and R uthvenassumed an i nfinite thermal conductivi ty so that thetemperature was uni form throughout the adsorbentparti cle (1979). B ased on this model, they investigatedtwo different cases-intr acrystalline diffusion controland bed diffusion control (1980, 1981). T he samemodel was used by Kociri k for constant volume-vari able pressur e condi tions (1984). Br unovska et al.and I lavsky et al. proposed a simu ltaneous heat and

mass transfer model for a porous spherical particle offinite thermal conductivity and the comparison be-tween experimental and numerical results were pre-sented (1978, 198Oa, 1980b, 198 I ). A simi lar model wasproposed by Haul and Stremming and an analyticalsolution was obtained.

T he models above cited made the following import-ant assumptions:

(i) A simple Fickian equation for mass diffusion isused, although its known that the true driving force(when temperature gradients exist) is the gradient ofchemical potential.

(ii) Only the gas phase diffusion is considered.I n a more r ecent paper, we proposed (Sun et a l . ,

1986) an analyti cal model using the pr essur e gradientas driving force and taking into account two phasediffusions--vapor and adsorbed phase diffu sions.

We pr opose herein a more detailed model fornonisothermal sorption in micr oporous adsorbents.This model will use the gradient of chemical potentialas driving force and consider both vapor and adsorbedphase diffusion.

THEORY

The model to be presented is based on the followingconditions:

(i) T he domin ant mass transfer r esistance is intra-crystalline diffusion.

(ii) The mass diffusion i n the pore volume mayoccur by two different mechanisms:

-sur face diffusion of adsorbed molecules-pore diffu sion of gas molecul es.

(iii) The chemical potential plays a similar role ingoverning the mass diffusion as temperature does forthe heat diffusion so that the true driving force formass diffusion is the gradient of chemical potentialrather than that of concentration (Ru thven andL oughl in , 1971; R uthven, 1984).

The mass diffusion rate is related to the gradient of

chemical potential through the mobility B:J = - BmVp. (1)

This equation i s applied to both the vapor ph ase andthe adsorbed phase.

(iv) R ates of adsorption are generally controlled bytransport processes within the pore network ratherthan by the intrinsic kin etics of sorption at the surfaceso that we do suppose that the equation of statedescribing the equilibrium between the adsorbent andthe vapor is valid. As an approximation, the vapor isconsidered as an ideal gas. The chemical potential of

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Nonisothermal sorption in porou s adsorbents 1587

source term corr esponding to the phase transition:

p&g = V(niVT)+p,l,AH (22)

where AH is the heat of adsorption.The intr oduction of the mass source term yields:

A@$+ ,,,Cp+AH ~)q)~=(avT)+ AHV [D,(Vq + SVT)]. (23)

Al l transfer coefficients in this set of tr ansfer eqs (20)and (23) are, a p r i o r i , functions of adsorbed phaseconcentration q and temperature T .

L e t u s n o w present results obtained wi th that set ofequations in a particular case.

Numerical simulationWe consider a spheri cal particle of porous adsorbent

subjected to a small change in sorbate pressure atini tial time. T his case is of i nterest since it corr espondsto an experimental procedure used to determine themass diffusion coefficient. In order to simplify thetheoretical analysis the following approximations wil lbe used:

( i ) The stepwise changes of sorbate concentrationand temperature are sufficiently small, thus all coef-fici ents in transfer equations can be considered asindependent of concentration and temperature.

(ii) E quili brium of sorbate at the particle sur face isinstantaneous.

Based on these approximations, we obtain a simpli-fied linear set of differential equations for heat andmass transfer:

= a,+ ~D,AHlp,C, ar2

+ (25)

where a4 = I /p,C , is the therm al diffusivity .The appropri ate initial and boundary condi tions ar e

as follows.

Initial conditions:

t = 0 : q = q , , and T = To . ( 2 6 )

Boundary conditions:

r=O a4ar = J = 0 CTar r=O = 0 (27)r = r ,

p

po 4 >41r=ro-40)+T >T (w,=ro--o) = PI4(28)

where P I and Tr are pressure and temperature of theambient fluid, eq. (28) repr esents pressure instan-taneous equi libr ation condi tion and eq. (29) heatbalance at the particle surface.

These governing equations are made dimensionlessby introduci ng the following variables and coefficients:

Reduced variables:

q--40 e= D tw=--- - - q=l r = - .qm - -4o r o r ,

Dimensionless coefficients:

su = PI--PO

Cl = K ( F 1 - 1)T

G2= l+KF ,

q m is the final concentration of sorbate.The eqs (24k(29) are then wri tten:

r= 0 w=o e l

aw0

ae

atf I=o = zj qxo =0

WI,=, +cP et,=l -6,) = su

( 3 2 )

( 3 3 )

( 3 4 )

The important parameters i n this set of dimension-less differential transfer equations are the thermal Bi otnumber Bi = hr/d, representing the ratio of thermalinternal resistance and external surface resistance;/3 = D J D a ratio of vapor phase diffusion coefficientand total diffusion coefficient and the Lewis numberI_.e = as/D-rate of heat conduction/rate of massdiffusion.


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1588 L. M. SUN nd F. MEUNIER

E x a m p l e s o f si m u l a t i o n

The set of linear partial differential eqs (30)-(35) issolved analytically by the Laplace transformationmethod. The analytical solution is given in theAppendix.

In the simulations reported here, a reference casecorresponding to the sorption of water into a singlespherical particle of zeolite is chosen: r0 = 1 mm , E= 0.32 and pOC, = 2.0 x 106 . Estimated values for the

effective con ductivity and the overall heat transfercoefficient: ;1= 0.3 W/m K and h = 30 W/m K. Theinitial and final conditions: To = 80C PO = 30mband T, = 8 O C , P I = 50mb. The two coefficients inthe Dubinins equation: n = 2 and d = 1.8023 x 10e7.The heat of sorption AH can be determined fromadsorption isotherms (AH = 2.78 x 106). As to the diffu-sion coefficients D,, D, and D,the following values areused: D = 2 x lo-cm2/s ( D , = D , = 10-5cm2/s).Finally, the entropy at standard state of ideal watervapor is S, = 10465.

The above evaluated parameters yield: a = - 15,I = 0.5, K = 8.7 x lo-, Bi = 0.1, Le = 75, y

= 5.7 x 10 e4, 4 = 25.6. It has to be noted that the valueof the Lewis numb er is quite large du e to the low valueof the diffusivity.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0n

e

1.016

l.OlO_l /-

1.04/ 2s1.0061.004

l.oool-_~r0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

b) 9

Fig. 1. doncentr ation (a) and temperatu re (b) profi les.(Xx = 75, fl = 05, B i = 0.1) for 5: (1) 0.0001, (2) O.ooO5,(3) 0.001, (4) 0.0025, (5) 0.005, (6) 0.0075, (7) 0.01, (8) 0.033,

(9) 0.066, (10) 0.1.

A parametric study is now presented in which thevalues of some param eters are varied so as to de-termine their influence on the results of the numericalsimulation.

RESULTS AND DISCUSSIONS

In Fig. 1, we show concentration and temperatureprofiles within the particle for the reference case. Wemay note that gradients of concentration are relatively

important with regard to temperature gradients. Infact, the high value of Lew is numb er (IX = 75) in-dicates a rapid h eat transfer and a relatively slow massdiffusion process. We may observe that, after anincrease in a short time, temperatures decrease (almostlinearly) due to the heat transfer with the ambient fluid.The maximum dimensionless temperature rise in thiscase is approximately 0.016 (corresponding to 56C).

E f i c t of Bifor d i f f e r e n t L eFigures 2 and 3 present the effect of therm al Biot

numb er on volumic averaged concen tration and sur-face temperature for Le = 75 and 7 .5 respectively.

1 .oo ,_ __-. ;y~~;;T*Y -.w=~= ~-Y-.Aw.*

0.90rx)/..,/.,. ___----

,.., _,. ,A

0.80

0.70- / ,i ,/

~~~

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0Ia) 7

e , Bi = 0

1.018- --__--._

0.01

1.016- ,, --._

1.0?4- .\---___

1.012- \ .,

--__

(b)

Fi g. 2. E ffect of B iot num ber on (a) averaged concentrationand (b) surface temperatur e (Le = 75, 13 = 0.5).


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Nonisothermal sorption in porous adsorbents 1589

0 1 2 3 4 5(a) I

Bi = 0

0 1 2 3 4 5

b)f

Fi g. 3. E ffect of B iot n umber on (a) averaged concentrationand (b) sur face temperatu re (L e = 7.5, @ = 0.5).

From these figures, we can see:(i) For a small heat transfer coefficient h, the process

may be controlled by thermal di ffusion. I n this case, anisothermal (Bi = co) model would yield err oneousresults.

(ii) I n order that the isothermal condition be justi-fied for a given diffusion process, the value of h must besufficiently high. A cri teri on is proposed here: sorptionmay be assumed isothermal if L eBi > 100 (forexample Bi > 1.4 for L e = 75).

E f f e ct o f j 3

j ? , the ratio of gas phase diffusivity and totaldiffusivity, has not an important influence on ki neticsof sorption for moderate L e, as we can see from F ig. 4(the total diffusivity D is the same for different valuesof p). Thi s influence becomes negligible if L e Z= 10, thismeans that we can take a single effective diffusivity ifL e> 10.

E f l e c t o f c o n d u c t i o i t y

The influ ence of conductivity incr eases when theL ewis number decreases, an example i s presented in

w

0.30 -I I

0.20

0.10i

O.OOT_(a)

I

1.0824-I

0 i 2 4 57

lb)

Fig. 4. Influence of r9 (ratio of vapor phase diffusivity andtotal diffusivity) n (a) the uptake and (b) surface emperatur e(I X = 7.5, Bi = 0.1). -~=o,---~=o.5,---~B I .

F ig. 5 for L e = 7.5. Notable differences due to the0conductivity may be observed only for very smallL ewis numbers.

E f l e ct o f t h e d r i v i n g f o r c e f o r m a s s f i o w

T h e effect of the choice of the drivi ng force for massflow i s shown in F ig. 6. The curves presented areobtained with the present model and with anothermodel in which the driving force for the mass flow isthe pressure gradient i nstead of the chemical potentialgradient. The larger the difference, the smaller the

L ewis number. For large L ewis numbers (Lc > 10) thedifference becomes negligible.

E f f e c t o f t h e s i z e o f t h e p a r t i c l e

When the size of the parti cle is modified, the heattransfer between the particle and its surroundings ismodified. The curves pr esented on the F ig. 7 have beenobtained taking the same value (per square meter) forthe heat exchange coeffici ent. I t is seen that the smallerthe particle, the larger the effects due to temperatureinhomogeneity; for a particle size larger than 30mm


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L. M. SUN and F . MEUNIER

0 1 2 3 4 5I

a)

tl _

1 .ozti -

1.024 -

0 1 2 3 4 5T

bl

Fig. 5. E ffect of the effective conductivity. (a) averagedconcentration; (b) surface temperature (Le = 7.5, fi = 0.5)

- L = 0.03, --- a = 0.3, --- A = 3.

diameter in th e reference case, the result s are nearly th e T he r esults obtained with the present model ar esame as those obtained wi th an isothermal model. compared to those obtained with simplified models. It

So, criteria for different models, as illustrated in is shown that the kinetics of sorpti on can be describedFig. 8, can be defined as follows: by one out of three models depending on the values of

I X > l&--uni form temperatur e profile model the L ewis and Bi ot numbers (Fig. 8). The limitsL eBi > lO& -isothermal model between the domains where the different models areother-present model (or similar model). valid are defined as follows:

CONCLUSIONS

The model presented here provides a more detailedanalysis for non isothermal sorption in a singlemicr oporous particle taking into account the influence

of temperature gradients within the particle. Theimportant parameters of this model are: L ewis number(ratio between thermal diffusion rate and mass diffu-sion rate), and thermal B iot number (ratio betweeninternal and external heat transfer resistances).

Two conclusions can be inferred:0 when L e > 10, the heat transfer inside the par-

ticle is so rapid that the uni form temperature pr ofilemay be assumed.

0 when the size of the parti cle is increased, the Biotnumber incr eases at constant L ewis number, so that wego into the isotherm al domain (see F ig. 8).

o.oo* , , , , , , , , , , , , , , , , , , , , , , , 1 , , , ,0 1 2 3

4 I 5

e(a)

1.024 -

1. 0007 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,

0 1 2 3 4 5*

(b)

Fig. 6. Comparison of different driving forces (Le = 7.5, Bi= 0.1). (a) averaged conc entration; (b) surface temperature.

- gradient of chemical potential as driving force.--- gradient of pressure as driving force.

the isothermal model is valid i f LeBi > 100, inthat case, only one parameter D is necessary to describethe kinetics of the sorption.

m the uni form temperatur e profile model i s valid ifL .e > 10; two par ameters D and h are necessary to

describe the kinetics.m out of those lim its, the present model shou ld beused; in that case, the gradient of the chemical potentialhas to be used as the driving force. Two diffusioncoefficients corresponding to gas and adsorbed phasediffusion are necessary. Thus, four parameters D,, D,,h and A are necessary.

To observe directly experimentally the effects of anon uniform temperature profile, experiments have tobe performed at very short times after either a pressurestep or a temperature step in which the temperaturemust be measured i n several points w ithin the particle.


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Noni sothermal sorption in porous adsorhents 1591

0.00 , , / , , , , T0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

T

0.0 0.1 0.2 0.3 0 .4 0.5 0.6 0.7 0.8 0.9 1.0

b)

F ig. 7. In fluence of the parti cle size on the ki netics (Le = 75,jl = 0.5). (a) averaged concentr ation; (b) surface temperatur e- r0 = lmm, --- r0 = 2.5, --- r0 = 5. ---- Bi

= cc (isothermal case).

T he pair adsorbent-adsorbate has to be selected sothat it cor responds to a Lewis number less than 10: theeffect wil l be all the more impor tant since the diffusingprocess is rapid, the size of the particle is small and theheat of sorption is high.

This model can be applied only to monodisperseadsorbents but it is being extended in our laboratory tobidisperse adsorbents.

A c k n o w l e d g e m e n t s - T h i s research was supported byC.N.R.S.-P.I.R.S.E .M. (Programme lnterdisciphnai re deRecherch es sur l es Sciences pour 1E nergie et les Mat&es

Pr emieres) and A.F.M.E . (Agence F ranqaise pour la Maitrisede 1E nergie).

NOTATION

a4B

4Bi4

thermal diffusivity, m2/smobility, smobility of the adsorbed phase, sBiot numbermobility of vapor phase, svapor phase concentr ation, kg/meffective heat capacity, J /kgKconstant in the Dubinin s equation

Fi g. 8. Il lustration of limitin g conditions for differentmode&.

D

D a

D h

h *

1,

1

J,J m aJ m v

J,KL e

L e111n

P

P *

P O

P l

P ,

440

4r

rR

S

su

;

T O

T ,

total diffusivity, m2/sadsorbed phase diffusivity, m2/svapor phase diffusivity, m2/soverall heat transfer coefficient between ad-sorbent particl e and ambient flui d, w/m Kenthalpy of vapor at reference state, J /kgmass source of adsorbed phase, s-lmass source of vapor phase, s-ltotal mass flow density, kg/m2 s

mass flow density of adsorbed phase, kg/m2 smass flow density of vapor phase, kg/m2 sheat flow rate, w/m2dimensionless parameter defined in textLewis numberdimensionless parameter defined in textconcentration, kg/m3constant in Dubinin s equationsorbate pressure, Papressure of vapor at reference state, Painitial pressure, Papressure of ambient fluid, Pasaturated pressure, Paadsorbed phase concentration, kg/m3initial adsorbed phase concentr ation, kg/m3final adsorbed phase concentr ation, kg/m3radial coordinate, madsorbent particle radius, mgas constant, J /kg Kentropy of vapor at reference state, J /kgKdimensionless parameter defined in texttime, sabsolute temperature, Kinitial temperature, Ktemperature of ambient fluid, K


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1592 L. M. SUN and F. MEUNIER

G r eek s ym b o l s

;dimensionless parameterratio of vapor phase diffusivity and totaldiffusivity

Y dimensionless parameterAH heat of adsorption, J /kg

1porosity of adsorbent particleeffective thermal conductivity, W/m K

P chemical potential J /kg

rl normalized radial coordinate ( r / r , ) , functionof temperature defined by eq. (3)CO normalized adsorbed phase concentration

((4 -qo)/(qm --4o))w normalized adsorbed phase concentration

averaged over a particlePa density of adsorbed phase, kg/m3P density of vapor phase, kg/m3

PO density of solid, kg/m3z normalized time variable ( D t / r i )8 normalized temperature variable ( T / To )

S u b s c r i p t s

a adsorbed phaseV vapor phase0 initi al stateco final state

REFERENCES

Brunovska, A., H lavacek, V., I lavsky, J . and Valtyni , J ., (a)1978. An analysis of a nonisothermal one-componentsorption in a single adsorbent particle. C h e m . Engng Sci . 33,1385. (b) 1980, Non isotherm al one component sorption i na single adsorbent parti cle, effect of external heat transfer.Chem . En gn g Sci . 35 ,757.

Br unovska, A., Ilavsky, J . and H lavacek, V ., 1981, An analysisof a nonisothermal one component sorption i n a singleadsorbent particle-a simplified model. C h e m . E n g n g S c i .36, 123..

C hihara, K ., Suzuki , M. and K awazoe, K ., 1976, E ffect of heatgeneration on measurement of adsorption rate by gravi-metric method. Chem. E n g n g S c i . 3 1 , 5 0 5 .

H aul, R . and Stremming, H., 1984, Non isotherm al sorptionkin etics in porous adsorbents. .J . C o l l o i d I n t er f a c e S c i . 9 7 ,348.

I lavsky , J ., Br unovska, A. and H lavacek , V., 1980,Experimental observation of temperature gradients occur-ring in a single zeolite pellet. C h em . E n g n g Sci. 35, 2475.

K ociri k, M., Struve, P. and Bul ow, M., 1984, Analyticalsolution of simultaneous mass and heat transfer in zeolitecrystals under constant-volume/variable pressure con-ditions. J . Chem. Sac. F a r a d a y Tr a n s . I 8 0, 2 16 7.

k , L . K . and R uthven, D. M ., 1979, Analysi s of thermaleffects in adsorption rate measurements. J . Chem. Sot .F a r a d a y T r a n s . I 7 5, 2 4 06 .

R uthven, D. M. and L oughlin, K. F ., 1971, Corr elation andinterpretation of zeolitic diffusion coefficients. Tr a n s .F a r a d a y S o t . 67 , 1 6 6 1.

R uthven, D. M ., Lee, L . K . and Yucel, H ., 1980, K inetics ofnonisothermal sorption in molecular sieve crystals.A.1.Ch.E. J . 26 , 16 .

R uthven, D. M. and-Lee, L . K ., 1981, K inetics of non -isothermal sorption: systems with bed diffusion control.A.I .Ch.E. J . 27 , 654.

Ruthven, D. M., 1984, P r i n c i p l e s o f A d so r p t i o n a n d A d so r p t i o nPr ocesses. Wiley-Interscience.

Sun, L. M., M eunier, F . and Mjschler, B ., 1986, E tudeanalytique des distribu tions de temp&r ature et de concen-tration g lintkr ieur dun grain sphkri que dadsorbant solide

soumis P un 6chelon de pr ession de vapeur adsorbable. lnt.J . H e a t M a s s Tr a n s f er 2 3 , 1393.

APPENDIX

The li near differenti al system formulated in the text (eqs30-35) can be solved by use of L aplace tr ansformati onmethod.

After simple algebraic m anipulations, we may obtain fromeqs (30) and (31)

whereaI 1 = (BF z K - G&Q 012 = (Fz J X -a&)/Q

~21 = (GI -BF , K )IQ a22 = (G --FL m)/QQ = F 2 G , - F I G * .

The I -aplace form of these equation s i s:

(A.3)

(A.4)

The corresponding equations for boundary conditions are:

(A.3

The solution of t h i s set of differential equations i n Laplaceform is described as follows:

8= -- CP.WsWhsJ;)_- ~(s)sh(hv )l (A.91where

G.2 =al~+a22~C (a11+~n22)2-44(a11a22-a12a21)7Z

2hla22-a~2a2l)

E I = -l+(A,+ )y+Bi E z = -l+(A,++)u+Bi

F, = bl(Bi-E el) F Z = b2(Bi-EEl)

el = E L Bi ;u- ) (A, + 4) e2 = E2- Bi s: - ) (A2 + 4)

=I = Fz(A1+4) =z = FICA,+ )

G = (AI+~)-EI(Az+~)

P I @) = eL s h ( b l & ) + F,& c h ( b l & )

Pz(s) = ezsh(b&)+F,&ch(b,&)

Q(S) = Gsh&& )sh(b&)

+z&sh@&)ch(b,fi)

--&ch(b,&)sh(b&).


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Noni sothermal sorption in porous adsorbents 1593

T he expressions for sorbate concentration and temperature Qb.1 = C% -zdcos hp,)sin &PO)evolution in the real time domain are obtained by use ofresidue inversion method: + (Gbt + =I ) sin (b, P,) cos (b, P,,)

m 2Suexp(--pit) +P&I ~, -z,b,)cos(b~~,)cos(bzp,)o=l+E {AI PZ (p,)sin (bt ftpn)

n=, rlp,Q (P,)- ~,(z~b~ -zzbl)sin (b,p,)sin (b,p,).

- PI (p,)sin(bzqp,,)l (A.10)P, is given by the roots of:

m 2Suexp(-pir)E=e,+ c

VP~Q (P,)tPz (p,)sin (b, VP)

Gsin (blp,)sin (b,p,) + zlposin (bl~,)cos (bzp,,)

=I - z,p,cos (b,p,)sin (b2pn) = 0.(A.12)

- PI (P,) sin (bzqp,)>(A.ll) T he sorbate concentration averaged over the particle is

where m,=1 x

=I

pi= --s sin (b, P,)

+ (hip,)* >

P, (p,) = cl sin (blp,)+F 1~,cos (bt~,)

pz (p,) = ez sin (bzpn) + FZP~COS (bzp,)

- PI (P,) - (A.13)

a detailed model for nonisothermal sorption.pdf

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