Chapter 9. Multicomponent diffusion in zeolites and multicomponent surface diffusion

W. Rudzifiski, W.A. Steele and G. Zgrablich (Eds.) Equilibria and Dynamics of Gas Adsorption on Heterogeneous Solid Surfaces Studies in Surface Science and Catalysis, Vol. 104 9 1997 Elsevier Science B.V. All rights reserved.

487

Multicomponent diffusion in zeolites and multicomponent surface diffusion YuDong Chen a and Ralph T. "fang b aThe BOC Group, Inc., 100 Mountain Avenue, Murray Hill, NJ 07974 bDepartment of Chemical Engineering, University of Michigan, Ann Arbor, MI 48109

Recent development of theoretical models and experimental results on multicomponent surface diffusion and diffusion in molecular sieves have been briefly summarized. The model development of multicomponent diffusion have been focused on four most attractive approaches which are: single-file approach, Maxwell-Stefan approach, irreversible thermodynamics approach and kinetic approach and with a more detailed model derivations on the last one. A step by step procedures for predicting multicomponent diffusion from single component information have been demonstrated in the examples. Besides, a direct comparison between these four models for co- and counter-diffusion cases are also presented. The single-file and irreversible thermodynamic models have the advantage of being easy to use and having no adjustable parameters. The only information required for this prediction is single component diffusivities. They can be used to predict multicomponent diffusion, especially, at low surface loadings. The kinetic model is capable of predicting multicomponent diffusion between Aij from 0 to 1. The Aij is indicative of the level of molecular interactions which can be directly obtained from single component information. Based on these comparisons, it is clear that the kinetic model is superior to the others.

1. I N T R O D U C T I O N Despite the fact that nearly all adsorption processes of industrial interest involve multiple adsorbates, studies on surface diffusion and diffusion in molecular sieves have been largely limited to single component systems. The literature on multicomponent diffusion in these systems is scanty. Habgood[1] studied the diffusion of mixtures of nitrogen and methane in 4A molecular sieves in 1958. He observed that nitrogen diffuses faster than methane and is preferentially adsorbed at the beginning. The preferentially adsorbed nitrogen is later displaced by methane, resulting in an overshoot phenomenon with a maximum in the amount of nitrogen adsorbed over time. Using chemical potential as the driving force for diffusion, Round et al.[2] presented a numerical solution to the equations describing the sorption of a binary mixture. Kokoszka[3] studied the rate of sorption of propane and butane mixtures from helium in

488 5A molecular sieves. He reported that the rates of sorption of a ternary system of propane and butane in helium are lower than the respective rates of sorption from pure components. Riekert[4] studied the rates of exchange between CO2 and C2H6 in hydrogen and sodium mordenites. He found that a countercurrent migration is possible in both zeolites, and observed a reduction in diffusivity of about an order of magnitude as compared with the single component sorption. Ma and Roux[5] studied rates of sorption of SOs and COs in sodium mordenite by using a volumetric method. They observed a large overshoot for the fast diffusing component COs. Later Ma and Lee[6] reported binary diffusion data on isobutane/n-butane and isobutane/1-butene in X zeolite and found that the binary diffusivities are smaller than those for pure components. The overshoot phenomenon which was caused by displacement was also observed. Ks and Billow[7] investigated the binary diffusion system of benzene/n-heptane in NaX zeolite and used a irreversible thermodynamic approach to derive a set of binary diffusion equations which has the same form as that given earlier by Habgood[1]. Those equations provided a satisfactory description of their binary experimental data. Ruthven and Kumar[8] reported binary diffusion data for methane and nitrogen mixture in 4A molecular sieve by using a chromatographic method. They found that their results are consistent with the hypothesis that each species in the mixture diffuses independently with the same intrinsic mobility as that for single component diffusion at the same temperature. This notion was confirmed by Kumer et al.[9] by using the same experimental technique. Palekar and Rajadhyaksha[10] studied binary sorption in channel type zeolites by a Monte Carlo method. From their uptake simulations, they concluded that the apparent diffusivity of the fast moving component decreases with an increase in the occupancy of the slower diffusing component and increases with the fast diffusing component. Yasuda et al.[ll] measured binary diffusion of CH4/He and CH4/Kr over 5A zeolite. They concluded that each component never diffuses independently in spite of the low equilibrium partial pressures. Both of the binary systems involve two intracrystalline diffusion processes with diffusivities different from those of the pure components. Also, a negative cross-term diffusivity was shown. Ks and Pfeiffer[12] extended a nuclear magnetic resonance (NMR) spectroscopy technique to investigate multicomponent diffusion in zeolites. Marutorsky and B/ilow[13] were able to evaluate the Fickian diffusion coefficient matrix in a microporous sorbent under constant pressure conditions. The method involved the solution of the reverse problem of calculating the matrix of diffusion coefficients based on experimental uptake profiles. In their study, a binary uptake experiment of the n-hexane/ammonia mixture in NaCaA zeolite at 200~ was conducted. The calculated results showed a negative cross-term diffusion coefficient for the co-diffusion case. Carlson and Dranoff[14] measured rates of adsorption and desorption of methane and ethane in 4A zeolites by a differential adsorption bed (DAB) technique. A large overshoot of the fast diffusion component methane was observed for the co-diffusion case. Also, they solved the binary Fickian diffusion equations analytically by assuming constant (i.e., concentration independent) main- and cross-term diffusivities. The diffusivities obtained from regression of their binary experiments showed that the cross diffusion coefficients are quite small in comparison with the main-term diffusivities, but they were significant

489

enough to improve the fit of the experimental data. Yasuda and Matsumoto[15] used the frequency response method to determine diffusivities of a binary N2/O2 mixture in 4A zeolite. They concluded that even at low occupancies, the binary interactions are important. Micke and Biilow[16,17] used Valterra integral equations to model the sorption kinetic processes of multicomponent mixtures under constant and variable concentration conditions. This sorbent was surrounded by the fluid bulk phase. The model was used to calculate the sorption kinetics in single-step arrangements. Krishna[18,19] used the Maxwell-Stefan equation to describe the multicomponent diffusion in zeolites. The vacant sites were treated as the (n+l)th component in the diffusing mixtures. The coefficients, Dij, which describe the facility for counter-exchange between the adsorbed species i and j were related to the coefficients 7)iv and Djv by using an empirical Vignes[20] formula which was correlated from liquid mixtures. Here :Div is the facility for diffusive exchange between species i and the vacant sites. This model can be used to calculate multicomponent diffusivities but it should not be considered as a predictive model. It is appropriate to quote below the discussion of Ks and Ruthven[21] on Krishna's model: "In principle the Krishna model does not imply any assumptions concerning the concentration dependence of the Stefan-Maxwell diffusivities. In a formal sense it may therefore be regarded as simply a transformation of one set of parameters (the phenomenological coefficients or the Fickian diffusivities) into another set of equivalent parameters (the Stefan-Maxwell diffusivities). The main advantage is the Stefan-Maxwell diffusivities are somewhat more amenable to a microdynamic interpretation than either the Fickian diffusivities or the phenomenological coefficients. However, the physical basis of the model can be questioned. The Stefan-Maxwell formulation is derived from momentum transfer arguments and it is not immediately obvious that the extension to an adsorbed phase in which, instead of considering the transfer of momentum to the adsorbent, the vacancies are treated as an additional component, is physically justified." Kraaijeveld and Wesselingh[22] reported that negative Maxwell-Stefan diffusion coefficients were possible as did Krishna[18,19]. Tsikoyiannis and Wei[23,24] used the stochastic theory of Markov processes to model diffusion and reaction in zeolites. The Markov theory was applied to a system where each site can be only occupied by one molecule. Those molecules migrate on the surface as a single file. This model can be also used to describe different site-site and transition-site interactions for different types of dependency of single and binary Fickian diffusivities on occupancy. A simple formula of binary diffusivities based on these processes was derived for predicting binary diffusion phenomenon. This formula has the same form as derived from the lattice-gas model of Sundaresan and Hall[25]. Qureshi and Wei[26,27] further extended the single-file diffusion model to describe multicomponent apparent diffusivities in ZSM-5 which is a channel-type zeolite. They compared this model with the benzene/toluene binary experimental results measured by using the Wicke-Kallenbach method. Satisfactory results were obtained. More recently, Nelson and Wei[28] developed

490 another single-file diffusion model by using the lattice-gas model[29] to simulate co- and counter-diffusion cases in self-diffusion. From the simulation results, they reached a simple binary diffusion formula which embodied a correlation factor. From their model, binary diffusivities can be predicted from single component information. However, the theory is thus far developed only for self-diffusion. Yang et a1.[32] derived a binary diffusion model based on irreversible thermodynamic approach which included the cross-coefficient L12. In their model, they introduced an interaction parameter w which takes into account the molecular interaction between two diffusing species. The value of w can be obtained from the interaction energy as will be discussed later. Thirteen sets of co- and counter-diffusion data for CO2/C2H6 in 4A zeolite which were measured by the DAB method were also reported and compared with the model. Recently, Chen and Yang[33] and Chen et a1.[34] derived a fully predictive multicomponent diffusion model based on kinetic and irreversible thermodynamic approaches. This model has considered interactions between the same and different molecules and use only the information from single component experimental data to predict multicomponent diffusivities. The predicted data by this model have been compared with binary experimental data from literature and their own experimental data[35,36] with fair agreements. This is the only simple explicit model available for predicting multicomponent diffusion that considers adsorbate interactions. The detail derivation of this model will be given in the following sections. Dahlke and Emig[37] reported results of Monte Carlo simulations of binary co-diffusion in cage type zeolites. In their simulation, the number of diffusing molecules in each cage were variable and the maximum loading of a cage can be two molecules. For the two apparent diffusivities, higher values were found when small amounts of the other component were present. The cross-term diffusivity, Dij, of the Fickian diffusion matrix reached the same order of magnitude as the main-term diffusivity, Dii, if the surface occupancy was high. Karge and Niessen[38] developed a technique, using Fourier transform infrared spectroscopy (FTIR), to measure single and multicomponent transient uptake. The experimental results of co- and counter-diffusion for benzene/ethylbenzene in ZSM-5 have been reported and were compared with two different models for the Fickian diffusivities matrix. Van den Broeke et a1.[39] used Monte Carlo simulations to study single and multicomponent diffusion in channel type zeolites and obtained a similar results as the single-file model. Andersson and Agren[40] developed a formalism based on previous work in the literature for implementation on the computer. Hu and Do[41,42] derived a multicomponent kinetic model based on different equilibrium isotherms. They used the ideal adsorption solution model[43] instead of the extended Langmuir model in the phenomenological equation to predict binary diffusion coefficient. Meanwhile, three combination of binary diffusion data of ethane, n-butane and n-pentane in activated carbon were reported by using the DAB experimental method[44]. Later, Do and Hu[45] used a heterogeneous extended Langmuir model proposed by Kapoor et a1.[46] to describe the multicomponent equilibrium of ethane and propane in activated carbon. This model computes the gas mixture equilibrium by using an extended Langmuir isotherm on a patch of surface and then integrates it over an uniform energy distribution. The predicted results are satisfactory.

491 2. L A T T I C E - G A S

SINGLE FILE APPROACH

The single-file theories are developed by Sundaresan and Hall[25] with quasichemical approach and by Wei and coworkers[23,26], and have been used successfully to interpret diffusion data in channel-type zeolites, specifically ZSM-5127]. The assumptions of this theory are as follows. The adsorbed molecules are found only on the sites. Each site can be occupied by one sorbate molecule. The diffusion of the adsorbed molecules may be viewed as a succession of discrete jumps from site to site; jumps to occupied sites are forbidden. Molecules move through the pores in single file, thus two molecules cannot cross each other moving in opposite directions in a pore. The migration of a molecule from one site to an adjacent unoccupied site through the pore connecting these sites is an activated process. Consider a lattice containing M sites. Let NA and NB be the respective numbers of molecules of A and B in the lattice. The number of unoccupied sites, No, is equal to M - NA -- Ns. Therefore, the lattice coverage 0 has the relation:

(i)

OA + OB + Oo = I

where Oi - Ni/M is the lattice sites covered by molecule i. In zeolite crystals, the flux of species i under isothermal conditions, Ji, can be expressed as

Ji = - L i V ( # i / R T )

(2)

where L{ denotes the phenomenological coefficient for species i and # is the chemical potential. The phenomenological coefficient, in general, depends on the sorbate composition as well as the lateral interaction energies between the sorbate molecules. In the absence of lateral interactions between adsorbates, it can be shown that, for a random walk in an unblocked lattice[47]

L, = ~-~-120,0og(6)

(3)

where kmi is the rate constant for the migration of species i in the lattice, I is the distance between adjacent sites, Z is the number of nearest neighboring sites for every site in the interior of the crystal and g(6) is the permeability of the lattice. The chemical potential #~ in Eq.2 is determined from

#i(OA, OB, T) = #~

+ RTln (O~o)

i = A,B

(4)

where p~ is the standard chemical potential of adsorbed species i molecules. It should he noted that, in the above derivation, it is assumed that there are no interactions between the adsorbed molecules and the adsorbed molecules are presented as a single phase. Based on these assumptions, the binary diffusivities are Dij = L~

O(,,/RT) 00j

i, j = A, B

(5)

492 Substitute Eqs.l,3 and 4 into Eq.5, one gets

o=(o

0

D~

OB

0A)

1 - OA

(6)

where

Di = ~ - 12g(5)

(7)

The Di is the single component diffusivity of species i and it is concentration independent. Equation 6 has been used extensively by Tsikoyiannis and Wei[23,24] and Qureshi and Wei[26,27] to predict the binary diffusivities from the constant single component diffusivities in order to compare the predicted data with their Monte Carlo simulation results. More recently, Eq.6 has been modified by Nelson and Wei[28]. They use molecular simulation employing Poisson-distributed event times described by Nelson et a1129] with two diffusing species A and B. In their derivation, components A and B are assumed to have identical diffusive and adsorptive properties. The apparent diffusivity of component i, D +, is defined as

Ji = -D+VOi

(8)

where Ji is the flux of component i between adjacent rows and VOi is the concentration gradient of component between the same two adjacent rows. From the simulation results at steady state, the following conclusions are reached For co-diffusion: DA+ = Ds+ = Do

(9)

OB

V0s = ~ VOA

(10)

For counter-diffusion: D + = D~ = Do (1 - OT) f (Or)

(11)

VOA = --VOB

(12)

where f(OT) is the correlation factor[30,31] for self-diffusion, it has the form

f (OT) =

1 + (cos r 2-1 + --OT (cos r

(13)

For a square lattice (cos C) = -0.36338023. In Eqs.9 and 11, Do is the single component diffusivity and OT = OA + OB. The binary diffusion equation can be written as JB

=-

D21 D22

V0s

493 Equating the fluxes in Eq.14 with the fluxes obtained from the co- and counter-diffusion simulation results, Eqs.9-12, and solving for the diffusivities Dij gives

D = D~ ( 1 - OBgOBg 1 -- oAgOAg )

(15)

where

9 (OT) = f (OT) +

1 - f(OT) OT

(16)

the factor 9(OT) can be calculated from Eq.13 and is approximately equal to 1.571 0.5710T. Equations 6 and 15 can be used to predict binary diffusion. These two equations have the advantage of easy to use and have no adjustable parameters. The only information needed for this binary prediction is the single component diffusivity data which can be obtained from regression of the single component uptake results.

3. MAXWELL-STEFAN (M-S) APPROACH This approach[18] treats the vacant sites as the (n + 1)th component in the diffusing mixture. Therefore, for surface diffusion in the (n + 1) component system which consists of n adsorbed species, the bulk fluid phase Maxwell-Stefan diffusion equation[48,49] is used for this system: 0k RT VP ~ =

/=1

OiNi - OiUj ~ OvNi ntDij ntDiv

i# j

(17)

where ~)ij denotes the M-S diffusivity describing the facility for counter-exchange between the adsorbed molecules i and j, •iv for exchange between molecule i and vacant sites, nt is the total surface concentration, and 04 is the fractional surface coverage of molecule i that follows n

O~ + ~ Oi = 1

(18)

i=1

In Eq.17, #i represents chemical potential of the adsorbed species i. For thermodynamic consistency, the chemical potential #i should obey the Gibbs-Duhem equation[43]" n+l

}2 o,v , = o

(19)

i=l

Therefore, only n of the equations in Eq.17 are independent. The chemical potential of species i can be expressed as

#i = #o + RTln (fi)

(20)

where fi is the fugacity of component i in the bulk fluid phase in equilibrium with the adsorbed mixture. Substituting Eq.20 into Eq.17, the driving force on the LHS of Eq.17 becomes Oi n dOj RT V # / = j=l~FiJ dz

(i = 1,2, . . . , n )

(21)

494 where

Fij = Oi 00---7-

(i,j = 1 , 2 , . . . , n )

(22)

Here [F] have been called thermodynamic factors. They can be calculated from equilibrium isotherms. For example, if one assumes the gas phase in equilibrium with the adsorbed species is an ideal gas mixture (fi = Pi), which implies no interactions between molecules, then the extended Langmuir isotherm can be applied and one gets

Oi r~j = ~s + O~

(i,j

= 1,2, "'" ,n)

(23)

where 8is is Kronecker delta (=1 if i = j, =0 if i # j). In order to make the M-S model predictive, Krishna used the empirical Vignes relation [20] which was obtained for diffusion in liquid mixtures:

Oj

O~

z~,s = [z~s~]~ + os [z~,~]o, + os

(24)

Furthermore, let N = J and from the Onsager reciprocal relation it follows that those counter-sorption M-S diffusion coefficients Dis and :Dsi are symmetric that is

~,s = z~j,

(25)

Therefore, from single component M-S diffusivities :Div and :Dj~, which can be obtained from regression of single component experimental data, one can calculate Dis and Dji values based on Eq.24. The multicomponent fluxes can then be obtained by [J] =

-n,[B]-'[r]VO

(26)

where the elements for the n-dimensional matrix of M-S diffusivities [B] are Oi

~

0j

Bq = ~ + j=x ~s Bij = -Oi ( ~ j

Divl )

4. I R R E V E R S I B L E

[i(j

# i) = 1, 2,... , n]

[i,j(i#j)=l,2,...,n] THERMODYNAMICS

(27)

(28)

APPROACH

For the theoretical description of multicomponent surface diffusion and diffusion in zeolites, a number of models based on irreversible thermodynamics with a Fickian diffusion coefficient matrix have been derived by using the Onsager formalism with a vanishing cross-coefficient L12[1,2,7,33]. "fang et a1.[32] included the cross-coefficient Lx2 in the description of binary diffusion in zeolites based on the Onsager formalism. The main concept of the derivations by using this approach is discussed below. The phenomenological expression for flux J is given by: n

Ji = - ~ LijV#j j=l

(29)

495 where the chemical potential # has been assumed as the driving force. If the gas phase behaves like an ideal gas mixture, and the chemical potentials of the adsorbed phase and the gas phase are equal under the equilibrium condition, then the chemical potential can be expressed by #, = #o + R T In Pi

(30)

where P~ is the partial pressure which is a function of the adsorbed amounts qi(i = 1 , 2 , . . . , n). Substitute Eq.30 into Eq.29, one gets n

Ji = - ~_~ Dij Vqi

(31)

j=l

where 0 In Pi N 0 In Pk Dij = R T Li, Oqj + y~ Lik k=l Oqj

(k r i)

(32)

Next, one may assume all cross-term phenomenological coefficients follow the Onsager relation and are equal to zero L# = Lji = 0

ir j

(33)

This assumption implies that all drag effects by other components are negligible. Therefore, by introducing the following relation[54] Di0 = R T L , qi

(34)

one can get multicomponent diffusivities from Eq.32 O ln Pi Dij = Dio qi Oqj

(35)

For the Langmuirian case, i.e., the multicomponent adsorption isotherms are expressed by the extended Langmuir equation: q, biPi

qi -"

(36)

n

1 i--1

By substituting Eq.36 into Eq.35, one can obtain the diffusion coefficients for multicomponent diffusion with no interactions between adsorbate molecules:

496 where qv = qs - ~2i~1 qi. The multicomponent diffusivities in Eqs.37 and 38 are predictable from single component diffusivities Dio; and are concentration dependent. It should be noted that Eq.36 is the same as the IAS theory when qsbi for all components are equal. This irreversible thermodynamic approach for multicomponent diffusion has been extended by Yang et a1.[32] by considering the non-zero cross-term phenomenological coefficients where they introduced an interaction parameter w which indicates the extent of interactions between the two diffusing species. For a binary diffusion system, they combined with the Onsager reciprocal relation and noting that Ll1 > 0 and L22 > 0,

Lx2 = L21 = w(Lx,L22) 1/2

(39)

where

1

(40)

Following the derivation procedure discussed above, for w = 0, the diffusivity formula Eqs.37 and 38 are obtained for the binary diffusion case when a binary Langmuir isotherm equation is applied to express the gas phase partial pressures. However, when the absolute value of w is between 0-1, one obtains

Dij = Dioqi 0 Oqj In Pi +w(qiqJ)~(Di~176 ,

Oqj

i,j = 1,2

(41)

where Pi is a function of ql and q2, and can be correlated by using binary equilibrium isotherm data. w is the interaction parameter for diffusing molecules. It can be positive or negative depending on whether attractive or repulsive molecular interactions are operative. The value of w must be pre-determined based on information on single component diffusion, and it should satisfy the following conditions 0~(0j) .... >0

Eij

>0

~

>0

w

)0

Following the constraints above, one possible relationship between w and Eij is: w = (1

-

e -Ei'/'T) OiOj

(42)

where Ei i is the interaction energy between two different diffusing species. It can be calculated from E/i and Ejj where the energies E/i and Ejj can, in principle, be obtained from single component diffusion data. Here the geometric-mean rule or mixing rule is needed to calculate the cross-term energy

Eij= ~/Eii Ejj

(43)

Equation 42 is an empirical correlation that has the correct limits. More recently, Chen, et a1.[34] have taken an entirely different thermodynamic approach by considering the rate processes involved. For the diffusion of a multicomponent system, the phenomenological flux equation for component i is

Ji = - L i V ~ i .

(44)

497 where i* denotes the activated molecule i. The underlying assumption for this equation is that all migration steps must involve the activated species. The interactions between the unlike molecules will be accounted for in the calculation of V#i., so no cross-term is needed in Eq.44. Assuming ideal behavior where the activity coefficient is equal to unity (it needs to be stressed here, however, that this model is not limited to this condition; an activity coefficient may be added to take into account the lateral interactions between molecules), the chemical potential gradient is expressed as V # i = R T V In Oi

(45)

where 0 is the surface coverage. The relationship between the phenomenological coefficient Li and the Fickian diffusion coefficient at zero surface concentration Di0 is already expressed by Eq.34. Therefore, the Hux equation can be recast to Ji = - D i o qiV in 0i.

(46)

where qi = qisOi, and qis is the saturated amount adsorbed. In order to relate 0i and 0i., one needs to understand the rate process. For diffusion in zeolite, the rate processes are[33]" Activation:

Mi k,)

(47)

Mi. + V

Deactivation (to a vacant site)" (48)

mi. + V k'r ~ Mi

Deactivation (to an occupied site)" (49)

Mi. + Mj k,,> Mj. + Mi

In the step indicated by Eq.49, the exchange of the activated molecule i* with the adsorbed molecule j causes the adsorbed molecule to become activated due to the exchange of energy while i becomes adsorbed. The rate of formation for the activated i* is given by OOi. '~ Ot = k~O~ - k~O~.O~ - y ~ k~jO~.Oj

(50)

j----1

where the first term on the RHS in Eq.50 is the rate of activation, the other terms are the rate of deactivation on a vacant site (ki~Oi.O~) and sites already occupied by j; here j could be equal to i. The deactivation terms include both forward and backward movements, Furthermore, the steady-state condition stipulates that 00i. =0 Ot

498 It follows then kiOi

kiv 1 - ~ (1 - Aij) Oj j=l where O~ = 1 - ~in__l Oi, and ki...,i _ sticking probability of molecule i on adsorbed molecule j -ei,,-eij)/RT (53) Aij = kiv sticking probability of molecule i on vacant site = e

In Eq.53. kij and ki,, are, respectively, the rate constants for an activated i" to land on and stick to an adsorbed molecule j or a vacant site. A further underlying assumption for this equation is that the transit time between sites is negligible relative to the residence time at either site (whether vacant or occupied). Substituting Eq.52 into Eq.46, one gets J = -[D]V~

(54)

where J and V~ are two vectors of n components for, respectively, flux and concentration gradient. The diffusivity matrix [D] is given by

Dii = D,o 1 +

(ln-A,,)O, 1 - ~(1

-

(55)

A,j)Oj

j=l

(i - Aij )Si

Dij = Dio

n D

~(1 j:l

j #{

(56)

- )~,j )oj

Procedures for calculating the values of A and application of the above equations will be shown in the next section. Based on Eq.35, which was derived from the phenomenological flux equation, Hu and Do[41,42] have modified it by using different equilibrium isotherm formula. In their models, they used single component Langmuir or Toth isotherms to formulate the single component diffusion equation and applied IAS (ideal adsorption solution) model to compute the multicomponent adsorption equilibrium. Since the IAS model can be used to predict multicomponent equilibrium adsorption data from single Langmuir or Toth models without requiring any additional information, this model can also be used to predict multicomponent diffusion behaviors of the adsorbed species from the information of single component diffusion. More recently, Hu and Do[45] further modified Eq.35 by using the heterogeneous extended Langmuir model proposed by Kapoor et a1.[46]. This model computes the gas mixture equilibrium by using an extended Langmuir isotherm on a patch of surface and then integrates it over a uniform energy distribution, From the information obtained by fitting

499 single component dynamic data, they were able to predict the multicomponent diffusivities.

5. K I N E T I C ) k P P R O A C H 5.1. Model

formulation

Diffusion in zeolite and on surface are activated processes in which the adsorbed molecule must be activated in order to migrate to an adjacent site. Migration is then the result of a hopping process of the activated molecule. Before the derivation for binary diffusion, some assumptions must be made. First, the pore spaces are small but an adsorbed molecule will not cause pore blocking. Second, unlimited multilayer adsorption is not allowed but the adsorbed molecules may grow as a cluster[50]. The second assumption is equivalent to stating the desorption of the adsorbed molecules A or B requires the same activation energy regardless whether they are activated directly from the sorbent surface or from the occupied site. Following the above assumptions, a binary mixture of A and B may undergo the following rate processes for molecule A located at lattice site at (x - ~), where x is the distance coordinate and 5 is the inter-site distance (each hop covers a distance of 5). A potential energy diagram for the diffusion process is given in Figure 1.

A*

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

x'--~2

x

x+~

Figure 1: Potential energy diagram for activated diffusion along distance coordinate x. 1. Activation (A)x_ ~

'

ka

) (A')x_~6

(57)

rl = ka~A,~_~

2. Deactivation (to vacant site) (A*)=_~ + (Y)=_~ r2 --

km)

(A)=_~

km~A',x-~ ,x-~

(58)

500 3. Forward Migration (to vacant site) (a*) x - 6 ~ + (V)x+~ ,k,%(a)~+,~r3 -- k m O A , , x _ _ ~o~, ~+~

(59)

4. Forward Migration (to site occupied by A) (A*)~_~ + (A)~+~ koo>(A 9A)~+~ r4 -- kaaOA.,x_~OA, x+ ~

(60)

5. Backward Migration (to site occupied by A) (a*)~_~ + (a)~_~ % ( a . Ak_, r5 -- kaaOA.,x_ ~ 0 A

(61)

6. Forward Migration (to site occupied by B)

(a*)~_~ + (B)~+~ k~ (B. A)~+~ r6 = kobOA.,~_~Os,~+ ~

(62)

7. Backward Migration (to site occupied by B)

(a*)x_~ + (B)~_~ k.b>(B. A)~_~ r7 = kobOa.,~_~Os,~_ ~

(63)

Similarly, for molecule B at lattice (x - ~) one can have the following steps: 1. Activation (B)~_~ kb> (B*)~_~

r~ = kbOs,~_~

(64)

2. Deactivation (to vacant site) (B*)~_~ + (V)~_~ --~ (B)~_~ r2 = k~OB.,~_~O~,~_ ~

(65)

3. Forward Migration (to vacant site)

(B')~_~ + (V)~+~ kn >(B)~+~ r3 = knOB.,~_~O,,,~+~

(66)

4. Forward Migration (to site occupied by B) (B')x_~ + (B)~_~ kbb,, (B 9B)~+~ r4 -- kbbOB,,x_~OB,x+ ~

(67)

501 5. Backward Migration (to site occupied by B)

(B'),~_~ + (B)x_~ % (B. B),~_~ rs = kbbOs.,~_~Os,~

~_~

(68)

6. Forward Migration (to site occupied by A) (B')~_~

+ (A).+,

~ kbo, , (A 9 B)~+,

r6

kba 0 B,, x- ~,0A z+ ,

=

(69)

7. Backward Migration (to site occupied by A) (B')~_~ + ( A ) . _ ~ ~

=

k~o,, (A. B)~_~

kbo0s.,~_~0A,~_~

(70)

One advantage of using these rate equations is that one does not need to have the knowledge of the actual events occurring during the molecule migration on the surface or pores. In the RHS of Eqs.60-63, for example, molecule A can either stick on sites occupied by molecule A or B, or switch between A or B (exchange energies); the rate equations are the same. As mentioned earlier, for diffusion in narrow channels and pores, blockage or partial blockage (by adsorbed A or B) may occur, then two additional rate processes (blocking by A and B) should be included. Such a blockage process has been attributed to be the cause for the decrease of diffusivity with increasing concentration, and has been treated previously[51] for the case of single-component diffusion. For simplicity, the blockage processes are not included in this treatment. In steps 60-63, the mobile molecule sticks to the adsorbed A or B by forming a bond. There is a possibility, however, that it does not stick. In that event, two possibilities arise (for one-dimensional diffusion): it continues the movement either forward or backward (to make multiple jumps). For simplicity, we assume that these two probabilities are equal and therefore the subsequent events do not contribute to the net flux. 5 The net rates of forward migration for molecules A and B located at lattice site ( x - 5) are. respectively: MA, x-~ -- kmOA" ,z-~'Ov, x+~' + kaaOA',z-~OA, x+~, + kabOa. ,x- ~OB,x+~'

(71)

' z - ~Os,~+ ~, + kb~Os ., ~_ ~0a,~+~ B , ~_ 5~ = k,O s , ,z-~~O~+~+kbbOs. ,

(72)

Following the same procedure, one may write for the net rates of backward migration for molecules A and B located at lattice site (z + MA ~+~) MA, z+ ~ = kmSa.,x+~Ov, z_ ~ + ka~SA.,z+~OA, z_ ~ + kabSA.,z+~SB,z_ ~

(73)

Ms~+ ~ = k , Os.,~+~O~ ~_~ + kbbOs . ~+~0s ~_~ + kb~Os. ~+~OA~_ ~

(74)

502 The net rates for formation of activated molecules, A* and B* at two lattice sites are:

Ot

'

~-

'

~-~

-~o~o~.~ (o~,~_~ + o~,~+~) (rs) oo~. ~

=

Ot

~o~ ~

-

'

~oo~. ~ (<,

, +

o~ ~+~) - ~o~. ~ (o~,~_~ + o~,~+~) '

~-~

~

-k~oe~..•

(o.~_~ + o.=+~) (76)

The steady-state theory stipulates that the net rates of formation of the activated species are zero. Hence, the concentrations for the activated species are k=OA'~'+~

(77)

o~. ~ = ~ (<.~_~ + o~.~+~) + ~oo (o.~_~ + o~.~+~) + ~o~ (o~.~_~ + o~.~+~)

o~. ~ = ~o (<,~_~ + o~,~+~)+ ~ (o~ ~_~ + o~,~+~) + ~o (o.~_~ + o~,~+~)

(78)

The net rates of migration of molecules A and B are

~A = MA,~_~ =

~

-

M~,~+~

(o~.~_~o~+~-o~..~<~_~)

+ ~oo ( o ~ . ~ _ ~ o . ~ + ~ - o ~ . ~ + ~ o . ~ _ ~ )

+ko~ (oA.. ~_~e,,,.+~ - Oa...+~eB, =_ ~)

~s

=

Ms,._~ - Us,.+~

-

k~ (o~,.~_~< . ~ + ~ - o~ .. ~+~o~ _ .~_~ ) + k~ (oB.~_ ~ oB .~+~ - oB. ~+~os, ~_ ~)

+k~o (o~.._~o.~+~ - o~. ~+~o.~_~)

(79)

(8o)

If the lattice parameter ~ is sufficiently small, one may approximate: OOA,.

OA,.+ ~ = OA,. + 2

0,~,~+~ = 0,,,~ +

2

Ox

Ox

(81)

(82)

At any lattice site OA,. + OB,~ + 0~,~ = 1

(83)

503 Moreover, the mass fluxes can be related to the rates of migration:

JA (A,)

=

JB (As)

= rB (V~)

JA

"A

= 5rA = 5rs

J8

(84)

(85)

where V~ is the volume of the lattice sites, As is the cross section area of the sites and is the distance between lattice sites. Substitute Eqs.79-83 into Eqs.84 and 85, one gets: JA -- k~km 52

2

OA, xOOv'x __ Ov aOA,x

o~

,x a~

kmO,,,~+ kaaOA, z + kabOB,z

kbkn 52

Js = --~

OB, X ~ az

-- Ov,z OOB, Oz x

k,~O,,,~:+ kbbOs,~ + kb,~OA,x +

kakab -2 + .... 2 b

kbkb~ 5~ 2

OA

OOB,x __ OB,x

,~ 0~

OOA x ,

,

a~

kmOv, z + kaaOA, z + kabOB,z

0.. oe~.~ _ OA. "

knOv,~+ kbbOB,z + kbaOA, x

(86)

(87)

Let ~_552 = -~--5 l,/a 2e e,,/RT DAO = 2

(88)

kb 52 = 2 52eCb/R T DBO = "~

(89)

where u is vibration frequency of the bond holding the molecule to the site and e is the effective energy of that bond, i.e. the difference in energy between the states corresponding respectively to adsorption at the ground vibrational level of the bond and to free mobility on the surface. We further define A as that in Eq.53 k~ )~AA --" k m

--

sticking probability on adsorbed A sticking probability on vacant site

(90)

The second equality in the above equation arises because ka~ and km are, respectively, the rate constants for an activated A* to land on and stick to an adsorbed A or to stick to a vacant site. A further reason for the equality is that the transit time between sites is negligible relative to the residence time at either site (vacant or occupied). Following the above derivation, one can have

AAA = e -(~~176

(91)

Similarly, kbb

(eb~--,bb)/nT .

(92)

kab = e_(e,.,_e.b)lR T

(93)

~ B B "- ~

AAB = -~

-" e -

504

ABA = ~kba = e- (ebv--eba)/RT

(94)

where e=~ and eb~ are the effective bond energies between molecules A and B, respectively, with the vacant site; e== is the effective bond energy between molecules A and A; e=b is the effective bond energy between molecules A and B. Since diffusion is concerned, these effective bond energies are the activation energies for diffusion. Substituting Eqs.88-94 into Eqs.86 and 87, one has

OA,= + 0,,,~ + AABOB,, ] OOA,,: JA = -- DAO Ov, x dr. AAAOA, z _~_ AABOB, xj OX

(1 - :~A,~) O~,~

1 oe~,~

-- DAO Ov,z q_ AAAOA, z + AABOB, xj

OX

(95)

(1 -- ABA) OB,~ ] OOA,z JB = -- DBO Ov,x + ABBOA,z + ABAOA,z COX [ OB'z + Ov'z + /~BAOA'z I OOB'x

(96)

In comparison with Fickian equations and dropping the subscript x, one has

JA =--DAA~

O0.._~B DAB Oqx

JB = --DBA~--:

DBB OZ

(97)

00B

(98)

The concentration-dependent Fickian diffusivities are

DAA = DAO

1 -

1 - (1

(1 -

-/~AA)OA 1 -

(1

AAB)OB

(99)

-- (1 - AAB)OB

- AAB)OA

]

DAB = DAO 1 -- (1 - ,~AA ) OA -- (1 - AAB) OB 1 - (1 - AAS)OB

]

DBA = DBo 1--(1--ABB)OB--(1--ABA)OA D B B = DBo

1 - (1 - ABA) OA ] 1 -- (1 -- ~BB) OB -- (1 -- ~SA) OA

(loo) (~ol) (~o2)

From pure-component diffusion, by following the same procedure, the concentration- dependent Fickian diffusivity is[52]:

DA 1 = DAO 1 -- (1 - )~AA)OA

(103)

505 which is the same result as that of Yang et a1.[53]. From pure-component diffusivity data the value of "~AA may be obtained by regression. Using Eq.91, the value for e~a can be calculated e~a = ear + R T

In AAA

(104)

where ear is the activation energy for diffusion of molecule A from a bare surface. For some systems, it can be estimated from the heat of adsorption at zero surface coverage. The same procedure is applied to pure-component B and the value of ebb may be obtained from the pure-component diffusion data. For the interaction energies between unlike molecules, A on B or B on A, that is, e~b or cb~, we use the following geometric-mean rule or "mixing rule"[55] (105)

Cab = ~ba = (5aaC.bb) 1/2

The geometric-mean rule, which has been derived by London for nonpolar molecules, has been used frequently and fruitfully in equations of state for gas mixtures and in theories of liquid solutions[55]. It has also been used in Monte Carlo simulations of mixed-gas adsorption[56,57]. Using this rule and the values of cab and cba from the rule, one may calculate the values for AAB and s directly from Eqs.93 and 94. By using the geometric-mean rule for interactions between unlike adsorbate molecules, one is able to obtain a first approximation for binary diffusion on surfaces and in zeolites. The geometric-mean rule is a good approximation for nonpolar molecules with comparable sizes, shapes and adsorption potentials. These restrictions may be relaxed by introducing an additional parameter, Z~b[55,ss,57] ~o~ = ~ o

)1/2

= (~oo~bb

(1 --~o~)

(106)

where ~b is a constant and is characteristic of the A - B interaction. As discussed by Prausnitz et a1.[55], ~b can be expressed in terms of molecular parameters from London's theory of dispersion forces. The values of ~ab are small compared to unity, usually within the range -0.1 to 0.1 for hydrocarbon mixtures. For binary diffusion, f~b may be considered as an empirical interaction parameters. The power of Eq.106 is that once the value of ~b is determined from one data point, this equation can be employed for all conditions for the given mixture A and B, e.g. all compositions and amounts adsorbed. To summarize the result of the theory, Eq.104 is first used to obtain the like-molecule interaction parameters "~AA (and ABs) from pure-component Fickian diffusivities and activation energy or heats of adsorption. The interaction energy for the unlike molecules, e~b (= cb:) is calculated by using the geometric-mean rule, Eq.105 or 106, which yields values for parameters '~AB and "~BA directly from Eqs.93 and 94. The binary Fickian diffusivities are then calculated by using Eqs.99-102. This is indeed a very simple calculation procedure. This theory has been extended to multicomponent mixtures without requiring the introduction of new principles or assumptions[34]. 5.2. L i m i t i n g cases For surface diffusion and diffusion in zeolites, the interaction parameter )~ takes a positive value, ranging from 0 to 1. When "~AA --" '~BB --" O, the binary values of )~AB and

506 ~, 2.5 o

-.~ e~

2.0 1.5 1.0 0.5 0.0 0.0

0.2

0.4

0.6

0.8

1.0

XA

Figure 2: Plot of binary Fickian diffusivities with surface concentration at fixed total surface loading OT ~- 0.7 (• OA + OB) and DAo/Dso = 10. The dashed lines are for Aij = 0 and solid lines for Aij = 0.2. The x axis XA = OA/(OA + On). )~BA a r e also zero. Consequently the binary Fickian diffusivities, expressed by Eqs.99-102, reduce to Eqs.37 and 38 which are the forms for ideal binary diffusion. In deriving Eqs.37 and 38, however, the chemical potential in the gas phase is assumed to be equal to the partial pressure and the Langmuir isotherm is also assumed. The result derived here by using kinetic approach is not subjected to these restrictions. In the other extreme case, when '~AA "- '~BB : 1 , and the activation energy for component A and B on bare surface are equal, i.e. "~AB -" /~BA " - 1, the main-term diffusivities (Dii) in binary diffusion are independent of concentration, and the cross-term diffusivities (Di.i) are zero. Consequently the system is well-represented by single-component Fickian diffusion equations with constant (but concentration independent) diffusivities. It is seen that the origin of the concentration dependence lies in the difference between activation energy (e~v) and bond energy between adsorbate (e~). When e~v = e~, there is no difference between landing on a vacant site or occupied site; hence, there is no concentration dependence. 5.3. Parametric

behavior

The binary diffusivities Dij v s OA from Eqs.99-102 have been plotted in Figure 2 for a fixed total surface concentration 6T = 0.7(0T = OA + OB) for two special cases. In case 1, one assumes all interaction coefficients Aij are equal to zero, which results in ideal binary diffusivities (Eqs.37 and 38). In the other case, /~AA -- )~BB "-" 0.5 and e~. = ebb. By comparing these two cases, one finds that a small change of the A values can have significant effects on the diffusivities. If A equals to zero, the diffusivities are increased (component A) or decreased (component B) dramatically at high surface loadings, and eventually reach infinity after the surface is saturated. At high surface loadings, the molecular interactions are important. Therefore. it can cause errors by using Eqs.37 and 38 to calculate diffusivities under this conditions. Experimental data are generally available in the form of uptake rates, i.e. the cumulated amount of diffusion vs time. In order to calculate the uptake rates, solutions to the

507

0.8

I

I

r

Z,B

Z,A

I

I

B

0.6

0.8

I

~

~.~.

~

~

0.0 0.5 1.0

I

r

I

0.6

., 0.o B

,~,

-

,,"

t~

,."

s

s

,...

s

~. -

,. -- " 0.5 - . - " ...- 1.0

~- 0.4

S

04-

'1

II

o

o f:a.,

I

J

"

Single

J

Single

l J

.-

0.2 -, 9 / ." 0.0 0.0

0.2

A

0.2

. . . . ~. . . . . i . . . . . 0.4 0.6 0.8 1.0

0.0 0.5 1.0

Time z

-,o

m

I",,,,~ 0.0 0.0

-

" 0.2

........ i i " 0.4 0.6 0.8

0.5~176 1.0 1.0

Time z

Figure 3" Uptake behavior for binary co-diffusion. Where DAo/DBo = 20 and dimensionless time r = D A o t / R 2. Cases I and II with initial clean particles, 8A0 = 0B0 = 0, subjected to a step increase at the surface to 8A = 0.05 and 0B = 0.8. AB = 0 in case I and /~A = 0 in case II. diffusion equations are needed. Assuming spherical particles, the binary Fickian diffusion equations are

OqA 1 0 Ot = r2 Or [r2 (DAA ~ r

Ot - r 2 0r

r2

DBA--

+DAB

_~rB)]

(107)

+ DBB'-~" r

where the binary Fickian diffusivities Dij are given by Eqs.99-102. These equations can be solved numerically with proper boundary and initial conditions, i.e.,

I.C.

t = 0

qA = qao, qB = qBO

B.C.

r-O

--=0 Or Or qA = qA~, qB = qBcr

r = R

~qA

CgqB

(109)

The calculated parametric results in terms of uptake rates are shown for two cases of codiffusion (fluxes of A and B in the same direction) and two cases of counterdiffusion (fluxes of A and B in opposite directions). In all cases, predictions based on assuming pure-component diffusion are also included for comparison. This comparison will show the importance of inclusion of the cross-term diffusivities. In the case of single component prediction, the cross-term diffusivities are assumed to be zero and Eq.103 is used for main-term diffusivities. The results for codiffusion are shown in Figure 3 and that for counterdiffusion Figure 4. One notable conclusion from these results is that for codiffusion, the diffusion of

508

0.51 o

~.o

0.4

~

.

.

9

.

.

.

.

.

.

.

.

.

.

0.5

.

00

~

~

0.4

1

~ II

l

i

i

Single . 9. . . . . . . . . . . . . . . . .

1.0 0.5

t-q

0.3

"

.'

0.3 " "

"-., ~... A

0.2 '

-

~'~

"t3"~"~ : ' :

D~ :.- :'.

1.0 0.5 0.0

-

o.o

A

~' B~ ~ ~~"-~---'"- 1.0 ' ~ 0.5

0.2

0.0

0.1 0.0 0.0

0.2

0.4

0.6

0.8

1.0

0.1 0.0 0.0

0.2

Time x

I

I

I

0.4

0.6

0.8

1.0

Time x

Figure 4" Uptake behavior for binary counter-diffusion. Where DAo/DBo = 20 and dimensionless time r = D A o t / R 2. Cases I and II with initial 8AO = 0 and 8so = 0.4, subjected to a step change at the surface to 0 A = 0.4 and 8B = 0.05. AB = 0 in case I and AA = 0 in case II. both components are increased (or facilitated) by the presence of the flux of the other component (Figure 3). While for counterdiffusion, the diffusion of both components are decreased (or hindered) by the other component (Figure 4). The other notable conclusion is that, for both codiffusion and counterdiffusion, the influence exerted by the other component is much stronger on the fast diffusing species; the slower component is much less affected by the fast component. From Figure 3, it is seen that varying A for the fast component has little effect on the diffusion of the slow component, while varying A for the slow component has a large effect on the fast component. Also shown in these figures are the "overshoot" phenomenon for the fast component, which has been discussed extensively in the literature and will not be further discussed here. It is also clearly shown in these results that the differences between single-component diffusion and binary diffusion formulations are large and large errors can result if the single-component equations are used for binary diffusion. However, for a dilute binary diffusion system, it is possible to predict the uptake rate of slow diffusion component by using single-component diffusion model. 5.4. E x a m p l e s

for p r e d i c t i n g m u l t i c o m p o n e n t

diffusion

In order to predict multicomponent diffusivities with this model, the concentration dependent single-component diffusivities, activation energies (or heats of adsorption) and the saturated amount adsorbed for each component are required. Example

1

Binary diffusion of 02N2 in Bergbau-Forschung carbon molecular sieve at 27~ were measured by using the DAB technique[35]. By fitting the 02 and N2 isotherm data with

509 the Langmuir equation q~bP q = l + bP

(110)

one can obtain the saturated amounts of adsorption q~ for each component. Therefore, the surface coverage 8(8 = q/qs) can be calculated. The heat of adsorption for each component can be calculated by using at least two sets of isotherm data at different temperatures

OP)

H~t

q= RT 2

(111)

where H st is the isosteric heat of adsorption. The single component diffusivities Do and the interaction parameters A can be obtained from concentration dependence of single component diffusion experiments. By least-square fit of the diffusivities D with surface coverage ~ from Eq.103, one can get Do and A values for each component. As a first approximation, one may assume that the diffusional activation energies for each component from the bare surface is equal to the heat of adsorption at zero surface coverage. Here, it is worth noting that the activation energy, for many cases, can be related to the heat of adsorption by an empirical correlation: H st

ev =

(112) m

where the empirical constant m is found to be integers varying between 1 to 3[59] and it can be less than 1 for zeolite[35]. In this example, the activation energy is assumed to be equal to the heat of adsorption. Therefore, from Eq.104, one can calculate ea~ which is the interaction energy between two like molecules. The interaction energy between two unlike molecules e~b can be calculated from Eq.105. All calculated results are listed in Table 1. These results will be used to predict binary O2N2 co- and counter-diffusion. In order to predict the binary uptake rates, one needs to solve diffusion equations, Eqs.107 and 108, with the binary diffusivities, Dij ( i , j = A,B), given by Eqs.99-102. In these diffusion equations, the value of qs for the binary mixture is calculated from[54] 1 --

XA =

qs

XB t

qsA

(113)

qsB

where x is the adsorbed phase mole fraction at equilibrium which may be determined from the extended Langmuir equation for mixtures: qsabA P] a qA = 1 + bAP] a + bBP~ s

(114)

with a similar expression for qB. za =

qA

(115)

qA + qB

Equations 107 and 108 are solved numerically with proper experimental boundary and initial conditions. The predicted co- and counter-diffusion results are shown in Figure 5

510 0.20

I

I

I

I

CD

I

N2 o

,~ 015

0"35t ~ O.3O II

'

*

*

~

,-. 025 o

020

0, 0.10

. . . . . . . . . . . . . . . . . . .

,,,,F

015

02

0.10

005

.9 ....

0.05 0.0;

0

t

I

t

I

10

20

30

40

50

Time (min)

0.0 ~ v 0

o:1

. . . . . . . . . . . . . . . . . . .

~

i

I

10

20

30

I

/

40

50

Time (min)

Figure 5: Co- and counter-diffusion of 02N2 in Bergbau-Forschung carbon molecular sieve at 27~ Symbols are experimental data. Curves are predictions using single- component diffusivities (dashed line) and theoretical binary diffusivities (solid line). Case I with initial ~O2 - - ~N2 - - 0, subjected to a step change at the surface to 8o2 = 0.105 and ~N2 = 0.333. Case II with initial 0o2 = 0.376 and ON2 = 0, subjected to a step change at the surface to 002 = 0.105 and ~S2 -- 0.328 (Data are listed in table 1). (cases I and II), with a comparison of the predicted binary diffusion results which assuming single component diffusion. The single component diffusion results are calculated by assuming that the cross-term diffusivities are zero and the main-term diffusivities follow Eq.103. Points on the figures are the experimental data. By comparing the predicted results from binary and single component model with the experimental results, the superiority of the binary model is clearly demonstrated, especially, (case I) for co-diffusion case.

Example

2

Binary diffusion of benzene/toluene in ZSM-5 zeolite at 65~ were measured by using the Wicke-Kallenbach method[27]. The heats of adsorption which used in this prediction were from Tsikoyiannis and Well24]. The calculated single component parameters are listed in Table 1. The predicted binary uptake rates for benzene/toluene are shown in Figure 6. By comparing the predicted results with their experimental data, the binary model is indeed satisfactory. Example

3

Binary diffusion of CHa/C2H6 in a carbon molecular sieve membrane at three different temperatures (24, 50 and 80~ were measured by using the Wicke-Kallenbach method[36]. This carbon molecular sieve membrane was prepared by pyrolysis of polyfurfuryl alcohol supported on a macroporous graphite substrate. The single component

511 0.5[

,

,

,

,

,

i

0

0.2

F

.-"

-

0.1 0.

-

0

5

10

15 20

25

30

Time (min) Figure 6: Counter-diffusion of benzene/toluene in ZSM-5 zeolite at 65~ Symbols are experimental data. Curves are predictions using single-component diffusivities (dashed line) and theoretical binary diffusivities (solid line). isotherm data were fitted by the modified Langmuir equation q~bP n q = 1 + bP '~

(116)

The Do and A values were obtained by an integral analysis of the data on flux vs partial pressure. The flux J for gas diffusion through a membrane can be expressed in the Fickian form (117)

dq J = - D d--~

Substitute Eq.103 into Eq.ll7, and integrating from qH to qn, one gets j __. Do qs Ax(1 - A )

In

(118)

where subscripts H and L denote high and low concentrations on the two sides of the diffusion cell. For diffusion through the carbon molecular sieve layer, qL << qH. Therefore, Eq.ll8 can be further simplified by neglecting the amount adsorbed on the low concentration side and substituting Eq.ll6 into Eq.ll8, one gets Do J =

Az

qs (1 -

ln[1-(1-A) A)

bPn 1 --!-

]

bP'~J

(119)

where P is the partial pressure of the diffusing gas on the high concentration side. By fitting single component J vs P data with Eq.ll9, the values for Do and A can be obtained. The activation energies can be calculated from the temperature dependence of diffusivities by using the Arrhenius equation Do = ,Uoe -,. -ev/RT

(120)

BergauForschung CMS

ZSM-5

CMS

27

65

24

1.15

2.07

CH4

C2H6

15.2

0.37

1.24

0.84

1.46.10 -9

1.70.10 -s

1.15.10 -l~

1.77-10 -2

CrHr

0.16

6.46.10 TM

6.65. I0-"

1.47

N2

0.12 \

C6H6

1.87

Isotherm parameters Diffusivity q~(mmol/g)_ b(1/atm)' ' n .... Do(cm2/s)

02

Sorbate

C211s

0.021 0.580 0.430 0.317 0.625 0.125

CH4

0.611 0.138 0.093 0.465 0.349 0.218

Mol fraction in

C2H6 0 1.68.10 -3 1.39.10 -3 7.22.10 -4 2.37.10 -a 3.23.10 -4

CH4

2 . 7 0 . 1 0 -3 4 . 6 0 . 1 0 -4 2 . 8 6 . 1 0 -4 1.93.10 -3 1 . 8 2 . 1 0 -3 7 . 8 1 . 1 0 -4

mol fraction out

1.88.10 -s 3 . 1 8 . 1 0 -9 1.99.10 -9 1.36.10 -s 8 . 0 4 . 1 0 -9 5 . 4 9 . 1 0 -9

CH4

0 1.16.10 -s 9 . 6 0 . 1 0 -9 5 . 0 9 . 1 0 -9 1.04.10 -s 2 . 2 6 . 1 0 -9

C2H6

Expt. flux (mol/cm2s)

0

5 . 6 0 . 1 0 -2

1.24.10 -1

8.73.I0-'

1.29.10 -2

5 . 2 8 . 1 0 -2

A

C2H6 5 . 6 3 . 1 0 -1~ 6 . 0 7 . 1 0 -9 5 . 3 6 . 1 0 -9 3 . 9 7 . 1 0 -9 5 . 8 5 . 1 0 -9 2 . 5 6 . 1 0 -9

CH4 1.75.10 -s 1.67.10 -9 1.19.10 -9 8 . 5 3 . 1 0 -9 4 . 7 8 . 1 0 -9 4 . 8 7 . 1 0 -9

Predicted flux from binary theory (mol/cm2s)

Table 2: Binary diffusion of CH4/C2H6 in carbon molecular sieve at 24~

Sorbent

T(~

Table 1" Parameters from single component isotherm and diffusivity data.

El/

Chen and Yang 1994 [36]

Qureshi and Wei 1990127]

Chen et al. 19941351

Reference

C2H~ 4 . 3 9 . 1 0 -1~ 5 . 9 4 . 1 0 -9 5 . 2 7 . 1 0 -9 3.58.10 -9 5 . 4 7 . 1 0 -9 2.42.10 -9

CH4 1.59.10 -s 9.89.10-1~ 7 . 3 7 . 1 0 -l~ 5 . 6 6 . 1 0 -9 2 . 8 2 . 1 0 -9 3.67 9 10 -9

Predicted flux assuming single-comp. diff. (mol/cm2s)

2.97

2.51

12.80

13.20

6.85

5.59

(kcal/mol)

t,o

513 where ev is the activation energy and D; is the pre-exponential factor. The calculated single component data are listed in Table 1. The binary fluxes can be predicted by substituting diffusivities, Eqs.99-102, into the flux equation

Ji = -

Dij--~x

i = A, B

(121)

j"-I

Integrating the flux equation over qA by keeping the other component at a constant average qB over the membrane, one gets

Ji =

Axl ~

[---~,j (qjo~t - qj~)]

i = A, B

(122)

j--1

where qA

D---AA =

out

1 f DAA (qA, "qB) dqA qA out -- qA in q A i n

(123)

q B o~t

_

DAB -_

1

qB out -- qB in

f

DAB (-qA, qB) dqB

(124)

qB in

Similarly, one gets -DBA and -'DBB. The subscripts in and out stand for conditions on the two sites of the membrane. The examples of predicted binary flux results at 24~ are listed in Table 2. Comparing the experimental fluxes with the predictions (Table 2), it is clear that the binary theory predictions were consistently superior than predictions ignoring the cross-term effects, i.e., assuming single-component diffusion.

Example 4 A comparison of the Chen-Yang model with other models is in order. The three other models are SFM (single-file model)J23,25,26], MSM (Maxwell-Stefan model)[18] and ITM (irreversible thermodynamics model)[1,2,7,32]. A direct comparison between these models for co- and counter-diffusion cases are shown in Figure 7 (cases I and II). For co-diffusion, the four models show only little differences in their initial uptake slopes at very short time. After this period, the Maxwell-Stefan model shows a more enhanced overshoot for the fast diffusing component. This additional overshoot is attributed to the drag effect by considering the counter-exchange coefficient 7912 in the M-S model. Due to the vanishing 7912 (which is equivalent to saying L12 = 0 in the phenomenological equation), the M-S model is eventually reduced to the irreversible thermodynamic model (Eqs.37 and 38) which is indicated by long dashed lines in Figure 7. After the maximum peaks, the single-file model gives a different behavior than the others; it takes much longer to reach equilibrium. This behavior can be understood from the basic assumption made in this model: At high surface concentrations, an activated molecule will have little chance to migrate to adjacent sites because of the limited vacant sites available. In the basic assumption, molecules moving to occupied sites is forbidden. Therefore, a molecule

514 0.6 r

I

i

I

I

I

~

0.5

t.,

o

0.4

=.

0.3

...... ......

~

CYM SFM ITM MSM

'

'

0.4 I-.....

=

0.3

'-

'cY

.... ITM 9. . . . . M S M l .................. .~..,

D

0.2

I

..,...,

....

~

,

~ " " 2"- " " " " " " " - "-

0.2 0.1 0.0 0

1

2

3

4

5

Time x

6

0.1-

""

0.0 0

1

................

2

3

4

5

6

Time 'z

Figure 7: Comparison of theoretical predictions of co- and counter-diffusion between SFM (Single-file Model), MSM (Maxwell-Stefan Model). ITM (Irreversible Thermodynamics Model) and CYM (Chen-Yang Model) (with ,~ij = 0.2). The uptake rate of mixture in a particle is specified by the diffusivity ratio DAo/DBo = 35 and dimensionless time r = D A o t / R 2. Case I with initial 0A0 = 0 and 0S0 = 0, subjected to a step change at the surface to 0A = 0.5 and 0B = 0.1. Case II with initial 0 A - - 0.5 and 0B = 0, subjected to a step change at the surface to 0 A --" 0 and 0B = 0.5. will take longer to migrate on the surface at high loadings. The Chen-Yang kinetic model has a wide range of predictability. As discussed in the previous section, when Aij = 0 (no interactions between molecules), the Chen-Yang model is essentially reduced to the irreversible thermodynamic model. However, when )~ = 1, the Chen-Yang model will predict two independent single component diffusion in a binary system. Here there are no cross-term diffusivities (Dij = O, i 5~ j) and the main-term diffusivities are concentration independent. In other words, the Chen-Yang model is capable of predicting binary diffusion between ~ij from 0 to 1. The Aij is indicative of the level of molecular interactions which can be obtained from single component information. For counter-diffusion, the Maxwell-Stefan model shows a large deviation from the other models. The drag force exerted between two counter diffusing molecules tends to reduce both migration rates. Therefore, the results of considering the drag effect in the system will cause a retardation of molecular movement. This explanation is consistent with the model calculation results. On the other hand, the single-fiIe model shows a slightly slower rate of uptake to reach equilibrium. This phenomenon can also be explained by the reason discussed above for the co-diffusion case. The experimental results for counter-diffusion of 02/N2 in Bergbau-Forschung CMS and CsHs/CrH8 in ZSM-5 zeolite with the predicted results from the Chen-Yang model have been shown in Figures 5 (case II) and 6 with fair agreements. Therefore. it is clear that the Chen-Yang model is superior to the others based on these comparisons.

515 0.6 / (:D ~D

,

,

,

i

0.5

-

O.4 0.3 0.2

0.1 00 0

._ 2

4

B 6

8

Time "c Figure 8: Example for ternary diffusion prediction by the Chen-Yang kinetic model. The parameters used in prediction are D A o / D B o = 2, D A o / D c o = 50, ~AA = 5.3-10 -2, )~BB = ,kCC = 1.3 910 -2, E~v = 5.6, Ebv = 6.5 and Ecv = 6.9 kcal/mol. Dimensionless time r = D A o t / R 2.

Example 5 An example for ternary diffusion predictions are shown in Figure 8. From this figure, both fast diffusing components A and B exhibit the overshoot phenomenon, with component A showing a sharper maximum since component A is the fastest. Component A competes better for the vacant sites at the initial stage and will then be quickly displaced by components B and C since Ebv and Ecv are larger than E~v. This figure demonstrates the capability of the Chen-Yang model for predicting multi-component diffusion with only single component information. Experimental data from the literature have been compared with theoretical predictions by the Chen-Yang model for co- and counter-diffusion shown in Figures 5 and 6. Considering the simplicity of the theory, the agreements between the theory and the experiments are indeed excellent. As mentioned, large errors can arise by using single-component diffusivities for binary diffusion systems. The results using single-component diffusivities are also shown in these figures and large errors are clearly seen. 5.5. C o n c l u d i n g remarks A general and simple binary diffusion model based on kinetic theory is developed for surface diffusion and diffusion in zeolites. Predictions by the theory compare well with the experimental data. To use the model, single component isotherms and concentrationdependent single-component diffusivities are needed. Moreover, the main-term diffusivities are always positive, and the cross-term diffusivities can be either positive or negative. Comparing the different models (SFM, MSM and ITM), the single-file and irreversible thermodynamic models have the advantage of being easy to use and having no adjustable parameters. It can be used to predict multicomponent diffusion, especially at low surface loadings with fair agreement. The Chen-Yang model, however, is superior to the others.

516 This model can be used to predict a wide range of multicomponent diffusion systems by taking into account of the interactions between the diffusing molecules. Furthermore, it can be easily extended to heterogeneous surfaces by considering surface energy distributions of e~v. The lateral interactions may be accounted for by using a non-unity activity coefficient in Eq.45.

6. L I S T O F S Y M B O L S

As b B D T~ Do

f~ f(er)

g(a) HSt J k l L M n

N P q q, r

R t T V

y~ Z

Z

cross section area of the lattice sites Langmuir constant Maxwell-Stefan diffusivity Fickian diffusivity Maxwell-Stefan diffusivity Fickian diffusivity at zero surface coverage fugacity of component i correlation factor given by Eq.13 permeability of the lattice isosteric heat of adsorption flux rate constant distance between adjacent sites phenomenological coefficient rate of migration or lattice sites Langmuir constant or number of components number of molecules pressure or partial pressure amount adsorbed saturated amount adsorbed rate or radial distance gas constant or particle radius time absolute temperature vacant site volume of the lattice sites coordinator number of nearest neighboring sites

517 Greek letters correction constant for mixing rule thermodynamic factor, Eq.22 F distance between two adjacent adsorption sites g effective bond energy activation energy for diffusion of A on site covered by B gab gay activation energy for diffusion of A on bare surface surface coverage or dimensionless surface concentration, (= q/qs) ratio of rate constants defined by Eq.90 or molecular interaction parameter chemical potential # bond vibration frequency I/ dimensionless time (= DAot/R 2) T interaction parameter defined by Eq.39 Superscript * a, b A, B i, j x 0

or Subscripts activated species species species species distance coordinate at q=O

REFERENCES

~

2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

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