ARTICLE pubs.acs.org/Langmuir
Multicomponent Effective MediumCorrelated Random Walk Theory for the Diffusion of Fluid Mixtures through Porous Media Mauricio R. Bonilla and Suresh K. Bhatia* School of Chemical Engineering, The University of Queensland, Brisbane, QLD 4072 Australia ABSTRACT:
Molecular transport in nanoconfined spaces plays a key role in many emerging technologies for gas separation and storage, as well as in nanofluidics. The infiltration of fluid mixtures into the voids of porous frameworks having complex topologies is common place to these technologies, and optimizing their performance entails developing a deeper understanding of how the flow of these mixtures is affected by the morphology of the pore space, particularly its pore size distribution and pore connectivity. Although several techniques have been developed for the estimation of the effective diffusivity characterizing the transport of single fluids through porous materials, this is not the case for fluid mixtures, where the only alternatives rely on a time-consuming solution of the pore network equations or adaptations of the single fluid theories which are useful for a limited type of systems. In this paper, a hybrid multicomponent effective mediumcorrelated random walk theory for the calculation of the effective transport coefficients matrix of fluid mixtures diffusing through porous materials is developed. The theory is suitable for those systems in which component fluxes at the single pore level can be related to the potential gradients of the different species through linear flux laws and corresponds to a generalization of the classical single fluid effective medium theory for the analysis of random resistor networks. Comparison with simulation of the diffusion of binary CO2/H2S and ternary CO2/H2S/C3H8 gas mixtures in membranes modeled as large networks of randomly oriented pores with both continuous and discrete pore size distributions demonstrates the power of the theory, which was tested using the well-known generalized MaxwellStefan model for surface diffusion at the single pore level.
describe the coupling of diffusion fluxes and potential gradients in each pore segment and (ii) averaging of the single pore diffusion fluxes over all possible pore sizes, lengths and orientations. For the first task, a considerable number of approaches have been developed, most of them embedding the early Knudsen formulation11,12 to account for the effect of fluidwall interactions. Among these approaches, the well-known dusty gas model (DGM)13 is probably the most recurrent in the literature, although a number of outstanding alternatives, such as the socalled “Lightfoot model”,14 the mean transport pore model,15 or the velocity profile model,16 are also available. These approaches breakdown within very narrow pores, where fluid-wall interactions are not appropriately described by the Knudsen hardsphere hypothesis. Fortunately, novel theoretical models have recently been developed to tackle the transport of soft-spheres in
1. INTRODUCTION The transport of fluids in narrow pores and confined spaces has been of great interest due to its importance in catalytic and noncatalytic fluidsolid reactions, adsorptive separations, and electrochemical processes.1,2 Such interest has significantly increased in the last decades as a result of emerging applications in the fields of nanofluidics,3 lab-on-a-chip technology,4 and adsorptive energy storage5,6 and as a consequence of the exponential growth of novel nanoporous materials. Examples of such materials include carbon nanotubes,7 MCM-41 silicas and their analogues,8 metalorganic frameworks,9 and zeolite-based membranes,10 all of them considered to be promising for the aforementioned applications. In every case, fluids transport within the narrow pores of such materials plays an important role, and this has triggered much activity on the understanding of fluid behavior in confined spaces, particularly with regard to structural characterization and adsorption equilibrium. The development of transport models in porous media usually involves two tasks: (i) the use of appropriate flux equations to r 2011 American Chemical Society
Received: October 19, 2011 Revised: November 23, 2011 Published: November 29, 2011 517
dx.doi.org/10.1021/la2040888 | Langmuir 2012, 28, 517–533
Langmuir
ARTICLE
can be estimated as25
pores of a few molecular diameters width, such as the oscillator model17,18 and the distributed friction model19,20 from this laboratory, as well as the MaxwellStefan equation-based approach from Krishna and co-workers.21,22 These models allow for rigorous analysis of fluid transport and are easily incorporated within averaging schemes, offering considerable advantage over simulation methods such as molecular dynamics. The task of averaging the fluxes over a pore size distribution (PSD) was undertaken in early work by Stewart and coworkers,23,24 who employed the DGM to model the transport of a gas mixture at the single pore level and represented the pore structure as a collection of straight and infinitely long, crosslinked cylindrical capillaries. For axial diffusion of an n components gas within a single, infinite pore of radius r, the DGM equations can easily be cast in a Fickian form22 j ¼ Dðr, GÞ
dG dz
De ¼
ð1Þ
γ¼
where Z is the coordinate in the direction of the macroscopic potential field, γ is the tortuosity factor, taking a value of 3 for cross-linked pore networks and regular lattices, and ε(r) is the void fraction of pores with radius between r and r + dr. Equation 2, commonly known as the smooth-field approximation (SFA), is intrinsically inaccurate for the majority of pore network topologies, as it does not generally satisfy the mass balance equations at the “nodes”, i.e., at the pores intersections.1 Burganos and Sotirchos25 later combined the SFA with the effective medium theory (EMT) developed by Kirkpatrick26,27 for estimation of the effective conductance of random resistors networks, leading to what is known as the EMT-SFA. In summary, EMT replaces the actual resistors network by an equivalent network of uniform conductances, for which the SFA is found to be valid. For single component diffusion in a cylindrical pore of radius r, length l, and diffusion coefficient D, the transport conductance is defined as25
α
0
0
ð7Þ
dG ¼ ðB þ f ðrÞIÞj dz
ð8Þ
where matrices α and B are independent of the pore radius, α is a diagonal matrix, and f is a scalar function of the pore radius. A different approach to find the effective diffusivity matrix is based on solving the mass conservation balances in an arbitrary pore network with a given PSD and coordination number. This alternative has been used by numerous authors, either by solving the mass conservation equations in the nodes according to Kirchhoff’s law3234 or by solving the continuity equation in every pore along with the node balances.3538 Nevertheless, pore network models require time-consuming implementation and, for binary and ternary mixtures, require advanced numerical schemes to ensure global convergence, especially when chemical reactions are to be considered.35,39 In this work, an extension of the multicomponent EMT for general transport equations of the form (1) is introduced, which contains as a special case the expression of Sotirchos and Burganos.31 The development follows the lines of the original derivation by Kirkpatrick and, along with the CRWT, yields values for the Onsager coefficient that closely resemble those obtained by simulation of the transport of binary and ternary gas mixtures in randomly oriented networks, following either a discrete or a continuous PSD. The formulation requires the solution of an nxn nonlinear equations
ð4Þ
If the pore radii and lengths are randomly distributed according to a number distribution f (r,l), the effective conductance gm of the equivalent uniform network follows, according to EMT gm gðr, lÞ f ðr, lÞ dr dl ¼ 0 ½gðr, lÞ þ ðN=2 1Þgm
3ðN þ 1Þ N 1
Although the CRWT is less accurate than the EMT near the percolation threshold, a hybrid EMTCRWT approach combining eqs 57 has been found to yield better agreement with simulation.29 More recently, it has been used with the oscillator model from this laboratory17,18 at the single pore level to interpret the experimental variation of apparent tortuosity with temperature in mesoporous glass membranes for several light gases.30 Extension of the EMTCRWT is, thus, desirable for the analysis of transport of multicomponent mixtures in porous solids. While eq 7 is independent of the fluid properties and is therefore expected to remain unaltered with the number of diffusing species, this is obviously not the case for the EMT averaging expression. A multicomponent version of the EMT was reported by Sotirchos and Burganos,31 exploiting the special structure of the isobaric DGM equations. However, this treatment, elegantly based on the spectral theorem, is only applicable when eq 1 can be rewritten as
and it is related to the macroscopic fluxes jm and density gradients (dG/dZ)m by m dG ð3Þ jm ¼ De ðGÞ dZ
Z ∞Z ∞
ð6Þ
where Æ...æ represents an average over the distribution of pore lengths and radii. While eq 6 is accurate for symmetric lattices for which the value of γ is well established, it is not complete on its own when applied to randomly oriented networks, for which γ must be dealt with as a fitting parameter. In an alternative development, Bhatia28 analyzed the meandering of molecules between pore intersections in a network of randomly oriented pores, while considering the correlation between successive pores traversed because of the finite probability that a molecule retraces its path. This approach is known as the correlated random walk theory (CRWT), and provides a tool to predict the tortuosity in such networks. For the case of a uniform network of randomly oriented conductors, the CRWT provides
where z is the microscopic axial coordinate in the pore, j = [j1,...,jn]T, G = [F1,...,Fn]T, and ji and Fi are the flux and molar concentration of species i, respectively. Furthermore, D(r,G) is the diffusivity matrix, which is in turn a function of the binary diffusion coefficients {Dij(G)}i,j=1,...,n and the species Knudsen diffusivities {Dki(r)} i=1,...,n. Following Stewart and co-workers, the effective diffusivity matrix De(G) is given by Z 1 Dðr, GÞεðrÞ dr ð2Þ De ðGÞ ¼ γ
gðr, lÞ ¼ Dπr 2 =l
εgm Æl2 æ γÆr 2 læ
ð5Þ
where N is the coordination number, that is, the number of pores meeting at a node. Upon solving for gm, the effective diffusivity 518
dx.doi.org/10.1021/la2040888 |Langmuir 2012, 28, 517–533
Langmuir
ARTICLE
system, i.e., 4 for binary mixtures and 9 for ternary mixtures, making it very attractive and easy to implement to obtain reliable estimates of the Onsager coefficients matrix when the PSD and the (average) coordination number are known. For testing the multicomponent EMT, the generalized MaxwellStefan (GMS) model from Krishna and co-workers21,22,40 was chosen, as it is not possible to reduce it in the form of eq 8. The theory provides excellent agreement with simulation results, although some deviation is observed in the vicinity of the percolation threshold when dealing with discrete PSD’s. The interaction of the species comprising the fluid reveals a rich variety of behavior in the effective coefficients, as described in the following sections.
effect of the random network was represented by a homogeneous effective medium preserving the external field, with all of its bonds having the same conductance. This effective conductance was fixed by forcing the average of the potential fluctuations to be zero when the individual conductances replace the effective conductance in the effective medium. The same procedure can be applied to analyze the transport of multicomponent mixtures through pore networks when the flux vector is linearly dependent on the transport potential. While so far we have considered the molar concentration gradient as the transport potential, an analogous form of eq 1 can be written in terms of the chemical potential gradient through the Onsager formalism
2. MULTICOMPONENT EMT In this section, a multicomponent EMT formulation is derived, in the same spirit as the original derivation outlined by Kirkpatrick,27 for the transport of a single entity. For the diffusion law in eq 1, the corresponding flow rate vector inside pore k is given by wk ¼
πrk 2 Dðrk , GÞΔGk lk
j ¼ LðrÞ
πrk 2 Dðrk , GÞ lk
ð9Þ
ð10Þ
Here we have assumed that, within a single pore, the concentration drop is small enough to assume a constant diffusivity matrix, which can be suitably evaluated at a mean molar concentration F inside all the pores in the network. As a consequence, the following treatment is restricted to those cases where this condition is met. However, the pore length is generally far below the normal length scale in which significant concentration changes occur, making this assumption sufficiently accurate when the concentration drop along the membrane is small. For the analysis of membranes through which large concentration drops occur, but the concentration varies insignificantly inside individual pores, the current approach can be used to calculate De(G) at several values of G and then solve the continuity equation, which, for one-dimensional transport in the Z direction reads
dj d2 G dDe ðGÞ dG ¼ De ðGÞ 2 þ ¼0 dZ dZ dZ dZ
w 0 ¼ ðgm gab ÞΔq
εðN 1ÞÆl æ g De ¼ 3ðN þ 1ÞÆr 2 læ m
ð14Þ
Following Figure 1b, the extra potential drop δq necessary to generate w0 can be estimated if the network conductance between nodes a and b, G0ab, in the absence of gab is known, since 0
w 0 ¼ ðGab þ gab Þδq
ð15Þ
To obtain G0ab we can first calculate the conductance of the uniform network, Gab, and use the fact that G0ab = Gab gm. In turn, Gab can be extracted from a symmetry argument, analogous the one employed by Kirkpatrick in his original derivation for the analysis of electric circuits. Thus, w0 can be re-expressed as the sum of two contributions: one introduced at node a and extracted far away, in all directions, and the other introduced at infinity and extracted at b. Therefore, the flow through both the N equivalent pores connected to the node at which w0 is introduced, and the N equivalent pores connected to the node at which w0 is extracted is w0/N, leading to a total flow 2w0/N through pore ab. Thus, for the uniform network
ð11Þ
for given values of the boundary concentrations G(Z = 0) and G(Z=L), where L is the membrane thickness. After consideration of eq 7, the multicomponent equivalent eq 6 is given by 2
ð13Þ
The matrix of Onsager coefficients, L, satisfies LT = L. To maintain the generality, in the following derivation the transport potential will be denoted as rq, representing the gradient of either chemical potential, concentration or partial pressure, depending on the nature of the diffusion coefficient matrix D. Consider a pore of conductance gab between nodes a and b, oriented in the direction of the macroscopic potential field and surrounded by the effective medium, as depicted in Figure 1a. To the uniform field solution of the network equations, in which the potential drops by a constant amount from row to row, say Δq, an additional flow w0 must be introduced at a and extracted at b in order to satisfy the mass conservation balance. Such flow must clearly be given by
where, rk and lk represent the radius and length of pore k, respectively. Furthermore, wk = [w1,k,...,wn,k]T and wi,k is flow rate of species i inside pore k. From eq 9, the conductance matrix of pore k should be defined as gk ðrk , lk , GÞ ¼
dμ dz
2 w 0 ¼ gm Δqab N
ð12Þ
ð16Þ
which provides Gab = (N/2)gm. Consequently N 0 1 gm Gab ¼ 2
where gm is the multicomponent effective conductance matrix of the pores in an equivalent, uniform medium. In his original work, Kirkpatrick26,27 modeled the distribution of potentials in a regular network of resistors of random conductances as the superposition of two effects: “an external field” increasing the voltage by a constant amount per row of nodes and a fluctuating “local field”, whose average over a large region vanishes. The average
Substituting eqs 17 and 16 into eq 15 yields 1 N 1 gm þ gab ðgm gab ÞΔq δq ¼ 2 519
ð17Þ
ð18Þ
dx.doi.org/10.1021/la2040888 |Langmuir 2012, 28, 517–533
Langmuir
ARTICLE
where u(r,l) = ((N/2 1)g1(r,l) + gm1)1gm1(g(r,l) gm), In is the n n identity matrix and g represents the number average conductance g̅ ¼
Z ∞Z ∞ 0
gðr, lÞf ðr, lÞ dr dl
ð23Þ
0
Equation 22 provides a useful recursive formula for the solution of 20 within a successive approximations scheme, taking gm < g as the initial guess. Notice that, if g(r,l) is symmetric for all r and l, the resulting g will also be symmetric, independently of the particular pore size and length distribution. Therefore, the matrix Z ∞Z ∞ 0
ðgm þ uðr, lÞÞf ðr, lÞ dr dl
ð24Þ
0
is symmetric for every possible choice of f(r,l), which implies Figure 1. Schematic of network of resistors used in the multicomponent EMT formulation.
gm þ uðr, lÞ ¼ gm T þ uT ðr, lÞ
for all values of r and l. In particular, at r f 0 we must have g(r,l) f 0, and from here it is straightforward to see that u(r f 0,l) = uT(r f 0,l). As a consequence, gm = gmT whenever g(r,l) = gT(r,l), which, together with eq 12, means that the EMT formulation preserves the reciprocity of the effective Onsager coefficients for all possible pore size and length distributions. When the transport potential is based on concentration or partial pressure instead of chemical potential it can be proven, through the Maxwell-Stefan equations, that g(r,l) = (πr2/l)UL(r,l)V, where L is the Onsager coefficients matrix and matrices U and V are independent of pore radius. In that case, U1gmV1 will clearly be symmetric, leading again to a symmetric effective Onsager coefficients matrix, as we will show below. 2.2. Percolation Threshold. Consider a network following a bimodal distribution of pore radii
Finally, requiring the average of δq to vanish leads to the following condition determining gm 1 N 1 gm þ g ðgm gÞ ¼ 0 ð19Þ 2 If the pore radii and lengths are distributed according to a (continuous or discrete) number distribution function f (r,l), eq 19 can be written as Z ∞Z ∞ 0
0
1 N ðgm gðr, lÞÞf ðr, lÞ dr dl ¼ 0 1 gm þ gðr, lÞ 2
ð20Þ
f ðr, lÞ ¼ pðrÞ ¼ ð1 mÞδðr r0 Þ þ mδðr r1 Þ
which is a natural extension of the single component EMT, eq 5, and is valid for both 2D and 3D networks. Equation 20 provides a system of coupled nonlinear equations for the elements of the effective conductance matrix gm. Some interesting properties arising from the form of eq 20 are derived in the following subsection. 2.1. Reciprocity of the Effective Onsager Coefficients. A fundamental consequence from the principle of microscopic reversibility is the symmetry of the Onsager coefficients matrix, L. When the transport potential corresponds to the chemical potential, g(r,l) is proportional to L, and it is therefore symmetric. Hence, it is interesting to see whether the effective medium conductance obtained through EMT is symmetric and leads, through eq 12, to a symmetric effective Onsager coefficients matrix. To investigate this issue, consider Woodbury’s identity, valid for N > 2 1 1 N N 1 gm þ gðr, lÞ 1 ¼ gm 1 2 2
6 1m 7 6 7 7gðr1 Þ gm ¼ 6m 4 5 N 1 2
ð21Þ Combining 21 and 20 yields, after some manipulation " #1 0
uðr, lÞf ðr, lÞ dr dl
ð27Þ
= 2/N as the single leading to the same percolation threshold mEMT p component EMT. In deriving eq 27, it is assumed that the pores with radius r1 are accessible to all species. On the other hand, it is also possible that the pores of radius r0 are accessible to all species but one, say species i. In this case a percolation threshold mp,i for species i such that {gm,ij, gm,ji} f 0 for all j at m f mp,i is naturally expected. In Appendix I it is demonstrated that, as long as g(r,l) is symmetric (or, alternatively, as long as it can be expressed as = 2/N. Although the previous symmetry conUL(r)V), mEMT p,i straints are always satisfied in diffusion problems, it may be of interest for other fields in which flow of other entities through random networks occur (i.e., social or economic sciences) to investigate the effect of asymmetry in percolation effects. However, that is out of the scope of the present work. It is well-known that direct application of EMT leads to overestimation of the percolation threshold, mp. Sahimi et al.41,42 have applied renormalization methods to substitute the actual network, in the vicinity of mp, by a rescaled version which is
!1 2 1 N N 1 1 1 gm g ðr, lÞ þ gm gm 1 1 1 2 2
Z ∞Z ∞
ð26Þ
and with all of its pores having the same length. In 26, δ represents the Dirac delta function. If the pores with radius r0 are inaccessible to all species, i.e., g(r0) = 0, eq 20 yields, after some manipulation 2 3
gm ¼ g̅ In þ gm 1
ð25Þ
ð22Þ
0
520
dx.doi.org/10.1021/la2040888 |Langmuir 2012, 28, 517–533
Langmuir
ARTICLE
topologically equivalent but lies farther from criticality. In the rescaled network, EMT is more precise and yields values of mp that match the simulation results for square (mp ≈ 0.5) and cubic (mp ≈ 0.25) lattices. The latter value is very similar to our result in disordered lattices with N = 6, as we shall see below. 2.3. Relation with Prior Formulations. It can easily be shown that the multicomponent EMT from Burganos and Sotirchos31 is a special case of the present formulation for transport equations of the form in eq 8. As a matter of fact, it is easily verified that diagonalization of the diffusivity matrix in eq 8 leads to αΔG ¼ Z1 Λðr, lÞZw
ð28Þ
where the matrix Z is independent of r and l and Λ(r,l) is diagonal. Sotirchos and Burganos further defined Δ~ F = ZαΔG and Δ~ w = Zw, corresponding to the transport potential and flow vector, respectively, of a set of n dummy species with conductance matrix Λ(r,l). Since Λ(r,l) is diagonal, the n transport equations are fully independent and the single component EMT (eq 5) can be applied separately to each one of them. This leads to a diagonal effective conductance matrix for the dummy species Λm from which the effective conductance for the real species is recovered as gm = Z1Λm1(r,l)Zα. Using eq 20 with g(r,l) = Z1Λ1(r,l)Zα yields Z ∞Z ∞ 0
0
Figure 2. Scheme of one-dimensional transport through a porous membrane by a mixture of n gases. The curves represent the steadystate pseudobulk pressures profile. Notice that the diffusion resistance between the bulk fluid and the membrane surface has been considered negligible. Z represents the macroscopic diffusion direction, whereas z is the diffusion direction inside the pore.
1 N 1 gm þ Λðr, lÞ ðgm Λðr, lÞÞf ðr, lÞ dr dl ¼ 0 2
ð29Þ g/m
1 1
where = Zgmα Z . A procedure analogous to that described in section 2.1 easily demonstrates that g/m must diagonal, hence reducing eq 29 to a set on n independent equations. Consequently, Λm1 = g/m, which shows that Sotirchos and Burganos method is a particular case of the more general eq 20.
describing the drag exerted by the molecules of j on the molecules of i, and jis is the flux of i. Superscript “s” simply emphasizes the fact that diffusion is, in this case, an activated process. When considering transport through a porous membrane, as schematically depicted in Figure 2, it is usually the partial pressures in the bulk phases at Z < 0 and Z > L that are known. Therefore, it is convenient to express the chemical potentials of the adsorbed gases in terms of the corresponding pseudobulk pressures of a bulk gas mixture at the same chemical potentials. If we consider low-pressure diffusion such that the fugacity fi ≈ pi, the chemical potential gradient can rewritten as dμi/dz = (RT/pi) dpi/dz. It will be assumed that the (pseudobulk) pressure and the fractional coverage are related through an extended Langmuir isotherm
3. VALIDATION OF THE MULTICOMPONENT EMT 3.1. Generalized MaxwellStefan Model (GMS). The GMS model for diffusion in nanopores was developed by Krishna and co-workers2,22,40 in the same spirit as the DGM, by introducing a model for surface diffusion. An important aim of the approach is to be able to determine the individual species fluxes in multicomponent systems, based only on information on pure component diffusivities. The GMS is very convenient to test the present averaging approach since, unlike the DGM, it cannot generally be written in the form of eq 8, and as a result, it cannot be tackled through the single component EMT. The GMS equations for one-dimensional diffusion in a single pore oriented in the z direction are
θi dμi ¼ εFs RT dz
∑
j ¼ 1, j 6¼ i jsi Ci, sat Dsi
Ci, sat Cj, sat Dsij
i ¼ 1, :::, n
bi pi 1 þ
ð31Þ
n
∑
j¼1
bj pj
where the coefficients bi vary with temperature and pore radii following an Arrhenius-type formula
θj Cj, sat jsi θi Ci, sat jsj
n
þ
θi ¼
bi ¼ b0i expð Ei ðrÞ=RTÞ
ð32Þ
According to Hu,43 the activation energy Ei(r) can be taken as the minimum of the interaction potential between species i and the solid in a cylindrical pore of radius r. For consistency with this definition, we will follow Hu and take the interaction potential Φ(x) to be described by44
ð30Þ
Here T is temperature, θi = ci/ci,sat represents the fractional coverage of species i, ci is its adsorbed amount in mmol/g of adsorbent, and Ci,sat is the corresponding concentration at saturation. Further, Fs and ε are the solid density and porosity, respectively, Dis is the surface diffusion coefficient accounting for the drag exerted by the pore wall on i, Dijs is the binary diffusivity
( 2k 4 ∞ 2k ) 5 21 σi 10 ∞ x σi x ΦðxÞ ¼ εi π αk βk 2 32 r r r r k¼0 k¼0
∑
∑
ð33Þ 521
dx.doi.org/10.1021/la2040888 |Langmuir 2012, 28, 517–533
Langmuir
ARTICLE
where x is the radial coordinate, σi is the solidfluid collision diameter, and εi is the depth of the Lennard-Jones potential minimum for a single lattice plane. Further αk 1=2 ¼
Table 1. FluidSolid Parameters in the GMS Model Taken from Hu40 i
Γð 4:5Þ Γð 1:5Þ ; βk 1=2 ¼ Γð 4:5 kÞΓðk þ 1Þ Γð 1:5 kÞΓðk þ 1Þ
ð34Þ where Γ represents the gamma function. For estimation of the binary diffusion coefficients, Skoulidas et al.45 proposed a corrected Vignes-type correlation
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8RT=πMi
ð35Þ
dp dz
ð36Þ
ð37Þ
where p = [p1, ..., pn]T
Bij ðr, p̅ 1 , :::, p̅ n Þ ¼
8 > > > > > > >
> > j ¼ 1, > > > > : j 6¼ i
Ci, sat θi ðr, p̅ 1 , :::, p̅ n Þ i 6¼ j Cj, sat Dsij ðr, p̅ 1 , :::, p̅ n Þ θj ðr, p̅ 1 , :::, p̅ n Þ 1 þ s i¼j Di ðrÞ Dsij ðr, p̅ 1 , :::, p̅ n Þ
ð38Þ and Γðp̅ i Þ ¼ εFs diagðθi Ci, sat =p̅ i Þ
ð39Þ
In eqs 38 and 39 it is emphasized that, at the single pore level, the transport properties and fractional coverage are assumed constant and will be evaluated at the average partial pressures pi = (p0i + pLi )/L. Since p = RTG we have, after comparing 1 and 37 Ds ¼ RTB1 Γ
ð40Þ
In addition, the Onsager coefficient matrix is given by L¼
B1 Γ diagðp̅ i Þ RT
1.07 10
4.956 106 1.879 108
7.666 15.36 14.54
σi (nm)
ci,sat
0.33505
4.184
0.31465 0.3939
0.31465 0.3939
RT 1 εFs Cj, sat Dsij
ð42Þ
This result, along with the corrected Vignes formula in eq 35 yields a symmetric Onsager coefficients matrix. In order to validate the multicomponent EMT presented here, a binary CO2/ H2S mixture and a CO2/H2S/C3H8 ternary mixture diffusing through a disordered membrane having a random pore network has been selected as a test example. The adsorption parameters have been taken to be the same as those for H-mordenite zeolite, given by Hu43 and summarized in Table 1. The saturation concentration was assumed to be the same for the three species and to vary linearly with temperature as Ci,sat(mmol/g) = 0.0156T(K) + 8.5988, in agreement with the values provided by Hu. The solid density and porosity were taken as 1.8 g/cm3 and 0.5, respectively. It is stressed here that, although our membrane is a disordered one, the parameters of the crystalline zeolite material are used only for purposes of model testing and demonstration. Further, the use of the Vignes relation to model binary diffusivities, while using Knudsen values for the pure component diffusivities, is perhaps questionable, but has been done here purely for illustrative purposes. 3.2. Simulation of Diffusion in a Random Network. In order to validate the multicomponent EMTCRWT scheme, we have conducted numerical simulations of the diffusion in synthetic random networks. For this, several realizations of random pore networks consisting of an array of 40 40 40 cubic cells were built. Inside each cell, a node was randomly located and joined to its N nearest neighbors. To do this, every node was first assigned a bond with its first nearest neighbor following a random sequence (except for those nodes that had previously been connected to another node). This procedure was repeated N 1 times (forming bonds with the second, third, ..., Nth nearest neighbors) following a different sequence every time, until the coordination number for each node was N or less. The nodes with coordination number lower than N (that usually made up
