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687

NUMERICAL SIMULATION OF THE GENERALIZEDMAXWELESTEFAN MODEL FOR MULTICOMPONENT DIFFUSIONIN MICROPOROUS SORBENTS*

Jan-Baptist W. P. Loos, Peter J . T. VERHEIJENnd Jacob A . MOULIJNDepartment of Che mica l Process Technology, D el ft Unicersit.v of Technology,Jirlianalaan 136, 2628 B L De lf t , The Netherlands

Received Septem ber 13, 1991Accepted October 22, 1991

The resu l ts of s imulat ions of transient mult icomponent surface diffusion in a microporous ad-sorben t are given for various condit ions. T h e general ized Maxwell-Stefan theory was appliedto describe the diffusion phenomena. The diffusion equations were solved numerical ly. Thet ranq ien t up take of a b inary mix ture and the counterd i ffus ionof two sorb ate species in a slabwere s imulated . Som e s t r ik ing phen om ena of mul t icomponen t d i ffusion are p resen ted . Th erelat ion to the appro x im ate so lu t ion ob ta ined by the l inear ized theoryof mult icomponent masst ransfer is d iscussed . T he s imulat ions are co mp ared wi th var ious resu l t s foun d in the l i tera tu re .

Separation and conversion processes taking place on solid sorbents, e.g. zeolitesoften proceed under non-equilibrium conditions. The understanding and modellingof the diffusion behaviour of mixtures i n these materials has practical importancebecause of its influence on the selectivity of these se pa ratio ns an d reactions. Surfacediffusion is also of fundamental interest in physical chemistry.In principle, a mole-cule moving over a surface can be seen as a pro be of the interac tions with the surfaceand with the other molecules i n its vicinity.

Diffusion i n micropores (size smaller than2 nni) is strongly temperature and con-centration de pe nd en t. Th e diffusing molecules neker really leave the force fieldofthe surface. In zeolites the intracrystalline poreso r openings are of m olecular dimen-

sions and the diffusing molecules are likely to hinder each other at high surfaceoccupancies. Mu lticompon ent diffusion behav iour is of special interest. The transientuptake of a m ixture of a fast diffusing-weakly adsorbing species an d a slow diffusing--strongly adsorbing species i n zeolites is typical: the uptake of the fast diffusingspecies exceeds the equilibrium valuein the approach to the equilibrium situation(see e.g. refs *-4 ).

The transient uptake of mixture componentsi n microporous sorbents has been

. _ _ _ _ _ _ _ _ _

Prague. September 8- 1 3 1991.

Presen ted as a po ster a t theInternational SympoJium Z eo lite Ch emistry and CataIysis,

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688 Loos, Verheijen, Moulijn :

simulated numerically by Ro un d et al. (using a simplified Onsa ger ap pro ach ,neglecting cross term s). M aru to vsk y an d B u l ~ w ~ ~escribed multicomponentdiffusion in zeolites by the generalized Fick approach and the linear driving force

model. Krishna applied the generalized Maxwell-Stefan(GMS) approach tosurface or temperature activated diffusion. The linearized theory of multicomponentmass transfer839was used to obtain solutions of the last twomodels mentioned.

Here, the objective is to numerically simulate the transient uptake of mixturesin microporous adsorbent particles, with the most rigorous model possible, therebyavoiding the earlier used approximations.

THEORETICAL

The transient diffusion behaviour of a binary mixture in an infinite plane slab issimulated. Th e eq uation o f continuityor conservation of mass conn ects concentra-tion an d diffusive flux in the a ds orb en t:

n, v,ei(z , ) + V = J ~ ( Z ,) = o ( i = 1, 2 ... n ) .We may combine this equation with an equation describing the phenomenon ofthe diffusive surface flux to obtain the diffusion e quation. T o this en d the generalizedMaxwell-Stefan equations were applied. This results in a set of nonlinear coupledpartial differential equations PDE). With the proper initial and boundary conditions

this is sufficient to determine uniquely the functionsOi(z,t ) .

Genera l ized Maxwel l -S t e fan Theo ry

Th e generalized Maxwell-Stefan equations7

8 . J . - eiJj( i = 1 , 2 , ... n )( ,+ = c u . u . ) = c Jn + 1eiej+ 1 (2)@ i 1 JR T j = 1 g J . . j = t n , B i j

employ the gradients of the chemical potentials as driving forces, in agreement

with the theory of irreversible thermodynamics. The influence of the substrate isincorporated in the model by considering the vacant sitesV to be the ( n + 1)thconstituent of the system7. It is pseudo-species to which thermodyn amic quantitiesca n be assigned. Onlyn GM S-equations are independent because the G ibbs-Duhemequation gives an extra relation. Surface diffusion is a special case. There can beno drift velocity of the( n + 1) component mixture. Since the total number of sitesis fixed, the vacancy flux balances the fluxes of adsorbed species. For sorption oftwo species the physical interpretation ofEq. ( 2 ) is as followso. The driving forceof component 1 is balanced by the frictional dra g on molecules1 moving past those

of component 2 with a relative velocity ( u l u 2 ) , weight factor 0102 and inter-

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Generalized Maxwell-Stefan Model 689

molecular drag coefficientl/ka,, and of molecules 1 moving past the surface sites(component V )with a relative velocity(ul - uv), weigh t f ac to r 1 9 ~ 8 ~nd dra g coeffi-cient l/dIv he Onsager reciprocal relations show that the GMS diffusivities are

symmetrical:

gij Bji i , j = 1 , 2 , ... n , v . (3)

The G M S diffusivitiesgiV an be related to single sorbate diffusivities. Th e G M SdiffusivityB I Z , escribing the interchang e of species1 and 2 adsorbed o n the surface,is interpolated between two limiting values

by the mu lticompon ent Vignes equation

Th e gradient of the chemical potential can be expressed in the gradients of fractionalcoverages:

( 6 )e. d p doj doj

RT j = 1 doj d z j = 1 dzZ(pJT,+= eiC -I C ri j i = 1 , 2 , ... n) .

The result is that thermodynamic factorsTij are introduced in the equations. Thesefactors increase with total occupancy in th e case of convex isotherms. F or a Lang muirtype adsorption isotherm

biPiei =n

( i = 1 2, . . . 4 (7)1 + 2 bjpj

j = 1

describing the equilibrium between an ideal binary gas mixture and a solid ad-sorbent, the factors are given in the Appendix (Eq.(15) ) .

The GMS-equations(2) in combination with Eq. 6) may be written in n-dimen-sional matrix fo rm to give explicit relations for the fluxes

where [ B ] - contains the G M S diffusivities. Substitutionof Eq. (8) in the continuityequations (1) leads to the diffusion equations. From this notation the relation withthe generalized Fick approach becomes apparent:

[D] [ B ] - [ r ] (9)Collect. Czech. Chem. Commun. (Vol. 57) (1992)


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I n the Appendix the elements of [ B ] - and [D] re given. The GMS approach ,starting fro m physically meaningful dr ag coefficients, results in a co nc en tra tiondependence of the Fick diffusivities[D], composed of transport an d thermodynamic

factors.In general, different satu ration capacitiesn will exist for different molecules. The

surface occupancies may be expressed in the numberof sites covered per unit areainstead of the number of molecules adsorbed per unit a rea. The total num ber ofsites may be defined by e.g.T I , , , the sa turatio n capacity of species1. By this definitionthe number of sitesis fixed and the vacancy flux balances the fluxes of sites coveredby species 1 and 2. We only have to multiply the flux of sites covered by species2by ns,2/ns,lto obtain the mole flux of species2. Th e values of gi j o not dependon the definition.

Numerica l Solu t ion Method

The differential equations describing the model are non-linear and must be solvednumerically. The continuum equations were discretized in the x-coordinate by anexplicit finite difference app rox im atio n . This leads t o a set of coup led first ord erdifferential equations for each slice. The discretized versions of thePDE's wereintegrated in time using a fourth-o rder Runge-Kutta procedure with variable step-size. The procedure solves stiff equations by a semi-implicit method and non-stiffby an explicit me thod . Th e R unge-Kutta procedure estimates the stepsize withinthe maximum stepsizeAt allowed. Th e finite difference a pp roxim ation s for the fluxesare:

JNf1) = 0

The init ial conditions (IC) and the boundary conditions(BC) fora re

e i (x ,o ) = 0 ; - 1 < x < 1 ; t = 0 ,

e i ( + i , t ) = e i , ; x = + I ; t < o ,

a- e i ( o , t ) = o ; x = o ; t r o .ax

ransient uptake

A discontinuity exists at (x,t ) = k 1, 0) in the concentration functions Oi(x,t ) .Since [D] depends on the concentrations Eq. 16)) there is also a discontinuity

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____ _ _Generalized Maxwell-Stefan Model 691

in [D] t the start. An alternative scheme was also attemptedto avoid the disconti-nuity in [D]:

The local truncation erro rs introduced by these bou nda ry layer effects d o not pene-trate the solution a nd decay provided the m ethod is stable. Th e stability, th e growthof the deviation from the true solution, ca nn ot be derived for a non-linear problem .In that case the stability does not only depend on the finite difference scheme butalso on the solution being ob taine d.If the solution becomes unbound ed for bound edinput values and oscillates with increasing amplitudei t is termed unstable.

RESULTS AND DISCUSSION

Simulations were performed at b oun dary conditions n ear satu ration . In this situationthe molecules will hinder each other most and multicomponent effects are likely tobe more pronounced.

EfSect of Grid S i z e

The behaviour in the boundary layer is the main concern. The finite difference ap-proximation is a poor one at the boundary where the gradients at the start will

be infinitely large. Use ofEq. (10) leads to stable solutions depending on the inpu tvalues. Th e local truncation errors decay an d a n accurate solution is obta ined . Th isis shown in Fig. 1 (solid curves 2 - 4 coincide). The errors may oscillate with de-caying amplitude when the discontinuities in(0) and [D] are large. Decreasing Atan d increasing Ax may eliminate the oscillations.

Use of another difference scheme,Eq. (12 ) , does n ot lead to oscillations for theseinput values used, but th e solution is inaccu rate. Refining the grid prevents to a certain

1 0species 2

FIG.1Convergence of the numerical solutions. Eq.12): broken lines, Eq. (10) : solid lines.

Curves. 1 A x = 0.2; 2 A x = 0.10; 3 A x == 0.05: 4 A x = 0,025, AFo = 1 . IC:= 0.10; B z ( 1 , 1 ) = 0.85, B ~ v / ~ z v100e l ( x , o )= B , 0 = 0. BC: el(i-i,I =

.

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692 Loos, Verheijen, Moulijn:

extent the solution from being swamped by the boundary layer effects and thissolution converges to the solution using Eq.(10) (Fig. I) .

All simulations in Figs2 - 6 were carried out fo rAx = 0.025 and AFo = 1 . lo-'using Eq. (10).

Rou nd et a].' obtained solutionsof a simplified Onsager model numerically.These model equations can also be solved by the numerical procedure used to solvethe GMS model. The following equations for the fluxes were used:

The Onsager coefficientsL , , and L Z were assumed constant and the cross coeffi-cients L,, nd LZl were assumed zero. This means that the matrix[ B ] - may be

replaced in the numerical procedure by the diagonal matrix[ L ] . The results ofRound et 31.' were reproduced.

Eflect of the Ratio of GMS Digusivities and the Boundary Conditions

The transient uptake of a binary mixture of sorbate molecules was simulated forseveral diffusivity ratios 91v/5B2v. This is shown in Figs 2 and 3 for differentboundary conditions. In all cases a maximum in the uptake of the fastest diffusingspecies 1 is observed. T he heightof the maximum relative to the equilibrium value

1.01 I I

02

00 01 02 0 3 0 4 ,co 06

10 I , I I

8 , soecies 2

06

0.1

0.2

I I I0 01 0 2 03 0 4 F~ 0 6

FIG. FIG.Transient uptake of a binary mixture in Transient uptake of a binary mixture ina plane slab as function of the Fourier a plane slab as function of the Fouriernumber for various ratios of diffunivitiei number for variours ratioo of difhuivitiesBlv/ 32v, as indicated. IC: Bl(x, 0) = BlV/5BZv, s indicated. IC: Bl(x, 0) =

= 32(x, 0) = 0. BC: el(& 1 ) = 0.10; e = e, X, 0 = 0, BC: el(+ I, ) = 0 m e,( A l , ) = 0-85 ( 1, t ) = 0.75

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is approximately the same in Figs2 and 3. Decreasing l v esults in a lower andbroader maximum. The uptake of species2 is not much affected by variationof9, . he ratio of the uptake of both species is nearly c ons tant up to the maximumin the curve. From thenon molecules 1 are replaced by molecules2 and the curvesare sym metrical with respect t o a h orizontal line in Figs2 and 3.

The counterdiffusion behaviour, the (partial) replacement of species1 by species 2an d vice versa was also simulated. Both cases are shown in Fig.4. The exchangeof the slow diffusing component by the fast diffusing component is faster than theexchange of the fast diffusing component by a slow diffusing component. Thenumerical solution leads to th e same result as the appro xima te solution by Krishna'.The uptake of species 1 and desorption of species2 is accelerated because the de-creasing frictional dra g 1/Bl2) etween these comp onen ts with increasing conce ntra-tion of species 1 Eq. 5 ) ) facilitates the interchang e of molecules1 and 2.

Concentration Profiles

The concentration profiles in a half-slab are shown in Fig.5 for various Fouriernumbers. The solutions are symmetrical with respect to the center of the slab. Atthe start b oth species diffuse in the direction of decreasing conc entration. Fo r0.01 0.12) theprofiles have become symmetrical with respect to a horizontal line in Fig.5 andcounterdiffusion setsi n .

Comparison with the ApproximateSolut ion

Th e GM S model for transient uptake in microporous sorbents was solved by Kris hna7using the linearized theory of multicomponent mass t r a n ~ f e r ~ .his theory relies onthe assumption of constancy of the Fick diffusivity matrix[D] along the diffusionpath. However, [D] s a function of the concentrations an d will have to be calculatedat average concentrations when using this method. With this assumption, the uptakeof n species is the n-dimensional matrix ana logof the single sorb ate uptake solutionF . The square matrix function [F] of the averaged Fickian matrix can be evaluatedafter a small time interval AFo using Sylvesters theorem (see e.g. ref.7 ). T he newconcentrations so obtained are used to calculate [D] or the next time interval andso on . The procedure starts with th e Fickian matrix calculated for a n empty particle.Th e numerical proced ure star ts with very different values of[ D ] , especially when theboundary concentrations are high. The maximum calculated by the approximatemethod for the same conditions is much higher (Fig.6 ) . The better agreement withthe solutions obtained by use of Eq.(12) (Fig. 1, curve 1) can be understood from thefact that this scheme also starts with [D] calculated at zero concentrations. Forsmaller grid sizes this solution converges to the numerical solution obtained viaEq. (10). In general the approximation will be less accurate in the first part of thesolution and when the discontinuities at

1 0I 1

I 1

s p e c i e s 2

0 01 0.2 0.3 0.L Po 06

the bo unda ry are large.

FIG.Comparison with the approximate solution(AFo = 0.0018) obtained by the linearizedtheory. Ic ( x , 0) = e 2 ( x , 0) = 0 . BC:e l ( h i , ) = 0.10; e2(k , t ) = om. g lv/ 9 z v = 100

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Generalized M ax we llSte fan Model 695

Compar ison w i th Exper iment

The simulations in Figs 2 and 3 are very similar in shape to measurements of thetransient uptake of nitrogen and methane in zeoliteA by Habgood'. Th e loading

of nitrogen overshoots its equilibrium value asi n Fig. 2. For a higher boundaryconcentration of nitrogen it overshoots the loading of methane as well, as in Fig.3.The same was observed4 for the mixture upta ke of benzene an d hep tane in zeolite X.The uptake of mixtures of carb on dioxide an d sulfur dioxide in m orde nite was studiedby Ma a nd Roux'. These uptake curves ar e very similarin shape to the simulationsin Fig. 2.

CONCLUSIONS

Solutions of the GMS model for surface diffusion were obtained by this numericalmethod. T he solutions for transient upta ke com pare well with experimental transientuptake curves of binary mixtures in zeolites foundin the literature. The simulatedconcentration profiles in a slab clarify the multicomponent diffusion effects foundin zeolites.

To apply the Maxwell-Stefan app roac h in the descriptionof transient multi-component surface diffusion a numerical solution method is necessary. Applicationof the linearized theory may lead to a significant deviation. The reason for thedifferent results is the strong dependence of the Fick diffusivities on the concentra-

tions in a Langmuirian sorbed phase, especially when the total occupancy of thesites is high. This dependence is mainly due to the influence of the thermodynamiccorrection factors in the Fick diffusivities. Adsorption isotherms for microporoussorbents are usually strongly non-linear and near saturation this leadsto a highsensitivity of these factors on con cen tration. Th e linearized theory relies on theassumption of constant Fick diffusivities along the diffusion path. In many casesof surface diffusion this assumption is nota good one at the s tar t of the uptake anderrors will occur i n this region of the solution.

-Kurta procedure.

The authors would like to thank dr. i r. A . W erritsen f o r providing the code of the Runge-

SYMBOLS

(A1

bi[BIg i j GMS diffusivity, m/sZ[DlF

Fo = 22vt /12 Fourier number

= [ B ] - with elements A i jcoefflcient in Langmuir isotherm, Eq. 7), m 2 / Nmatrix of inverted GMS-diffusivities deflned byEq. ( 8 ) ,s /m2

matrix of Fick diffusivities with elementsDij , m 2 / ssingle sorbate function giving the approachto equilibrium

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column vector of diffusive fluxes,mol/msmatrix of Onsager coefficients with elementsL i j , m2/shalf-thickness of the slab,mnumber of sorbates

total surface concentration a t saturation, m ol/m2number of slices in the half-slabpartial pressure of i in the bulk phase, N/mZgas constant, J/mol/Ktime, stemperature, Kvelocity of diffusion, m / s= r l dimensionless distance from the center of the slabdistance from the center of the slab, mmatrix of therm odyna mic correction factors defined by Eq.chemical potential of i J /mol

spreading pressure, N /mcolum n vector of fractional surface occupancies

( 6 ) with elements r,,

Subscripts

refers resp. to species i, , I 2refers to the number of the slice in the slabrefers to surface propertyrefers to timerefers to gradient obtained under co nditionsof constant temperature and spreadin8pressurerefers to vacanciesrefers to coordinate along diffusion p ath

Mathematical Symbols

[ I

1.2.

3.

4.5.6.7.8.9.

10.

11.

squa re matrix of dimension nvector of dimension n

REFERENCES

Habgood H. W.: Can. J. Chem. 36, 1384 (1958).M a Y. H., Roux A . J.: AIChE J. 19, 1055 (1973).Carlson N. W., Dranoff J. S. in: Proc. Znd ng. Found. Conf. on Fund. of Adsorption 1987,(A. I. Liapis, Ed.), p. 129.Marutovsky R . M., Bulow M.: Z. Phys. Chern. (Lepizig)263, 849 (1982).Round G. F., Habgood H . W. , Newton R.: Sep. Sci. I, 219 (1966).Marutovsky R. M. , Biilow M.: Chem . Eng. Sci.42, 2745 (1987).Krishna R.: Chem . Eng. Sci.45, 1779 (1990).Toor H. .: AIChE J. 10, 448; 10, 460 (1964).Stewart W. E. , Prober R.: Ind. Eng. Chem., Fundam.3 , 224 (1964).Krishna R., Taylor R. in: Handbook of Heat and Mass Transfer Operations (N. P. Chere-misinoff, Ed.), Vol. 2, Chap. 7. Gulf, 986.Ames F. A.: Numerical M eth od sfo r Partial Differential Equations. 2 d ed., Chop.2. Academic

Press, New York 1977.

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~


APPENDIX'

Th e elements of [A] = [ B ] - for diffusion of a mixture of two sorbates in a sorbentare:

A , , = 9 1 v ( e 1 9 2 v 1 6 1 ) 9 1 2 ) / ( 0 1 9 2 Y 0 2 9 l Y e v 9 1 2 )

A 1 2 = ~ 1 9 2 ( 9 * V - 9 1 2 ) / ( e 1 9 2 v + f 3 2 9 1 v 0 v 9 1 2 )A 2 1 = e 2 9 1 v ( 9 2 v 9 1 2 ) / ( & 9 2 v + 0 2 9 1 v + & 9 , 2 )A 2 2 = 9 2 v ( e 2 9 1 v + 1 - 0 2 ) 9 1 2 ) / ( 6 1 9 2 v 0 2 9 1 v + &%2) (1 4 )

The elements of the matrix of thermodynamic factors [f] n the case of a binaryLangmuir adsorption isotherm are:

, r l l = 1 + r 1 2 ,1r12 _ _ _1 e l e2

, r t2 1 + r Z i . (15)rZl 6 21 el e2

Th e element of the Fickian matrix [D] re obtai ned by matrix multiplication(Eq. ( 9 ) ) :

Collect. Czech. Chem. Commun . Vol. 57) 1992)

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