Diffusion in Microporous Membranes: Measurements and Modeling

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Ind. Eng. Chem. Res. 2008, 47, 5797–5811

5797

Diffusion in Microporous Membranes: Measurements and Modeling George R. Gavalas* DiVision of Chemistry and Chemical Engineering California Institute of Technology, Caltech 210-41, Pasadena, California 91125

Permeation of gases and vapors through microporous membranes, principally zeolite membranes, is reviewed, focusing on macroscopic diffusion models, measurement techniques, and applications of models to experimental data. Brief reference is made to the choice of adsorption isotherms for single components or mixtures. Concerning diffusion, the Maxwell-Stefan (M-S) model is reviewed in some detail, as it is the one universally adopted in recent literature. Emphasis is placed on the coverage dependence of the diffusion coefficients and on the proper handling of the cross terms. In the experimental technique section, the key distinctions are between steady state and transient measurements, and between the use or nonuse of a sweep gas. A few special techniques are also briefly reviewed. Interpretation of transient measurements using the time lag method is reviewed in some detail, especially for coverage-dependent diffusion coefficients. Several of the studies reviewed focus on the ability of the M-S model to match the measurements, and specifically on the suitability of various simplifying approximations. Two common approximations are (i) to treat the M-S diffusion coefficients as coverage-independent and (ii) to neglect the cross terms. 1. Introduction Microporous membrane materials include zeolites and the related ALPO and SAPO, amorphous silica, and microporous carbon. The bulk of the literature so far has focused on zeolite membranes, because of their superior selectivity properties and their thermal stability; therefore, this review will be mainly concerned with zeolite membranes. However, the methodology is general and applies to other types of microporous membranes, such as silica and carbon. In a further narrowing of scope, the review will be limited to macroscopic measurements and macroscopic models that are still the basic tools for membrane characterization and for the prediction and optimization of membrane operation. Molecular simulations, currently a very active and fruitful area, will be referenced only in conjunction with the evaluation of macroscopic models. A central principle underlying microporous membrane transport is that the fluxes are jointly controlled by the adsorption and diffusion properties. Transport through polymeric membranes also involves this joint control, but adsorption in the latter case is governed by somewhat different types of isotherms and is often accompanied by drastic changes of membrane structure. Thus, although several of the measurement techniques and data analyses addressed in this review were first developed in the context of polymeric membranes, the latter subject will not be included here. To describe and predict membrane permeation, the adsorption isotherms must be considered alongside the diffusion models. Measurement of adsorption isotherms by volumetric or gravimetric techniques is, in most cases, conducted on powders rather than films or membranes. In such measurements, the equilibrium state provides the isotherm while the transient part contains information about the diffusion parameters. However, interpreta* To whom correspondence should be addressed. Tel.: (626)3954152. Fax: (626)568-8743. E-mail address: [email protected].

tion of the transient is obscured by the nonuniformity of particle size and by film mass and heat transfer. Even when these parasitic effects are eliminated, the properties of a single crystal or a collection of crystallites are different than those of a membrane, with the latter having grain boundaries and other defects generated during the membrane casting, growing, and thermal processing. Hence, membrane properties can differ from those of individual crystals and must be obtained from permeation rather than uptake measurements. The review contains three sections: the first examines diffusion models, the second addresses measurement techniques, and the third involves the application of measurements and models to specific systems. 2. Diffusion Models for Microporous Materials In this first section, we review the basic macroscopic concepts and equations of pure-component and mixture diffusion through microporous membranes. The readers are referred to the wellknown book of Karger and Ruthven1 for a detailed acount of diffusion in zeolites, microporous glasses, and carbons, and to Karger et al.2 for a recent review of diffusion in zeolites. Mechanistic considerations of diffusion in microporous materials lead to a distinction between “surface diffusion” and “activated diffusion” (also known as “gas translation diffusion”). Surface diffusion is dominant for strongly adsorbing molecules at low or moderate temperatures and is the regime that has been commonly postulated in the studies referenced in sections 3 and 4. Activated gas diffusion becomes important for weakly adsorbing molecules at higher temperatures and has different temperature and concentration dependencies. Xiao and Wei3 provided a lucid treatment of the two regimes specifically for zeolites. Bakker et al.4 and Van de Graaf et al.5 discussed the two types of diffusion and estimated the respective parameters from measurements of single-component permeation through a silicalite-1 membrane. Most of the studies reviewed here deal

10.1021/ie800420z CCC: $40.75  2008 American Chemical Society Published on Web 07/19/2008

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with Silicalite-1 or ZSM-5 membranes. These two zeolite structures belong to the MFI structure family and differ only by their Al:Si ratio. 2.1. Single-Component Diffusion. In Fick’s Law (N ) -D dc/dz), the driving force is the concentration gradient, but the Fick’s diffusion coefficient (D) is generally concentrationdependent. To obtain a handle on the concentration dependence of D, it is useful to adopt the chemical potential gradient as the driving force, leading to what is known as the Maxwell-Stefan (M-S) model, which was originally developed for diffusion in gases and later generalized to diffusion in porous solids. For one diffusing component, the M-S model can be obtained heuristically, using the following force balance on a single molecule: -

dµ˜ V ) dz b˜

(2.1a)

where µ˜ is the molecular chemical potential; hence, -dµ˜ /dz is the force exerted on the molecule by the potential gradient and V is the mean velocity. The quantityb˜ is the mobility, and 1/b˜ is the force of resistance exerted on the diffusing molecule by the solid and by the other molecules. This resistance obviously is dependent on the concentration of the diffusing molecules, as well as the temperature. Dividing by the Avogadro number A, one obtains the balance in molar units (where µ ) Aµ˜ and b ) b˜/A): dµ V ) (2.1b) dz b Multiplying eq 2.1b by the molar concentration c, expressed on a per unit membrane volume basis, yields

is known as the thermodynamic factor. Both Γ and Dc are generally functions of concentration, but, in the so-called Darken approximation, Dc is assumed to be concentration-independent. At low coverage θ ) c/c* the activity is proportional to the concentration (Henry’s law) so that the thermodynamic factor is unity. Under these conditions, the mobility is also concentration-independent so that the diffusion coefficients D and Dc are equal and dependent only on the temperature. Transition-state analysis yields the corrected diffusion coefficient in the Arrhenius form, Dc ) A(T) e-E⁄(RT)

where the temperature dependence of A is weak and is usually neglected when matching eq 2.9 to the experimental data. The thermodynamic factor Γ is calculated by differentiation of the adsorption isotherm: c ) c/h(p, T)

dµ (2.2) dz When the chemical potential is expressed in terms of the activity a, µ ) µ° + RT ln a, one obtains dln a dln a dln c RT dln a dc dµ ) RT ) RT ) dz dz dln c dz c dln c dz and eq 2.2 becomes

(2.3)

(2.10)

where c* is the saturation concentration of the adsorbate and h is a dimensionless function. Among several functional forms used for h, two of the most popular are h) h)

c/Kp 1 + Kp

c/1K1p c/2K2p + 1 + K1p 1 + K2p

(Langmuir isotherm)

(2.11)

(dual - site Langmuir isotherm)

-

N ) -bc

(2.9)

(2.12) A three-site Langmuir isotherm defined similarly to the dualsite Langmuir isotherm has also been used. The parameters in these and other models have been estimated from experimental data on powders or from molecular simulations. In the case of light gases, the saturation concentrations cannot be measured experimentally, because of the high pressures required; hence, resorting to simulations is essential (according to Li et al.6). The major fundamental issue addressed in recent research is the dependence of the diffusion parameters on the coverage:

is the familiar Fick’s diffusion coefficient, which is also known as the transport or “chemical” diffusion coefficient. When the gas phase can be treated as ideal, the activity coefficient a in eq 2.5 can be replaced by the pressure; however, at higher pressures, the activity is related to the fugacity (a ) f/f°, where f° is the standard state fugacity, equal to 1 bar for gases). Based on eq 2.5, one defines the corrected diffusion coefficient (Dc) as follows:

c (2.13) c* Earlier experimental and modeling studies had been content to treat the M-S diffusivity as being independent of θ, i.e., to accept the Darken approximation. However, several measurements have shown that, for many systems, the coefficient Dc, which is estimated by ignoring the coverage dependence, varied with the feed pressure. To resolve this issue, the θ dependence of Dc was recently investigated by molecular simulation. Molecular dynamics simulations by Skoulidas and Scholl7,8 and Skoulidas et al.9 showed that the Darken approximation is accurate in some systems but has considerable error in others. Specifically, these authors found that, for diffusion through ZSM-5, the Dc value of CH4 is consistent with the Darken approximation, but the Dc value of CF4 is concentrationdependent, approximately obeying the relationship

Dc ) RTb

Dc(c) ) Dc(0)(1 - θ)

θ)

dc dz

(2.4)

dln a dln c

(2.5)

N ) -D where D ) RTb

(2.6)

This is also known also as the M-S diffusion coefficient. The Fick’s coefficient is then given by D ) ΓDc

(2.7)

dln a dln c

(2.8)

where Γ)

Figure 1 shows the coverage dependence of the Fick’s and M-S diffusion coefficients calculated from the simulations. The figure also shows the self-diffusion coefficient, which differs from the other two and will not be needed in the following. If the aforementioned linear relationship is combined with the thermodynamic factor derived from the Langmuir isotherm (Γ ) (1 - θ)-1) the resulting Fick’s diffusion coefficient, in eq 2.7, is coverage-independent, as observed experimentally for

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Figure 1. Maxwell-Stefan (M-S), Fick, and self-diffusion coefficients of (a) CH4 and (b) CF4 in MFI, versus loading at 298 K. The symbols are molecular dynamics calculations. The lines for the M-S coefficient are DCH4_c ) DCH4_c(0); DCF4_c ) DCF4_c(0). The lines for the Fick’s coefficients are given by eq 2.7. (Reproduced with permission from Skoulidas et al.9 Copyright 2003, American Chemical Society).

Figure 2. M-S diffusion coefficient of CH4 in (a) intersecting channel zeolites and (b) cagelike zeolites, as a function of loading (proportional to coverage) at 300 K for all structures in panel a, and at 300 K for CHA and FAU, 750 K for LTA, 600 K for ERI, and 373 K for DDR. (Reproduced with permission from Krishna et al.13 Copyright 2007, American Chemical Society).

some systems. This is not the case in Figure 1, evidently because the adsorption data are not described adequately by the Langmuir isotherm. The coverage dependence of the M-S diffusion coefficient has been determined to vary among different zeolite structures, as well as among different permeants. Krishna et al.10 examined the coverage dependence of several gases/vapors in different zeolite structures using kinetic Monte Carlo simulations on square and cubic lattices of different coordination numbers. The dependence of Dc on coverage was explained in terms of molecule-molecule interactions that vary among the different zeolite topologies. A previous correlation of Reed and Ehrlich11 gave a good fit of the coverage dependence in the simulation results, and also matched previous results8 generated using full molecular dynamics. In all cases, the M-S diffusivity vanishes as θ f 1. Moreover, increasing coverage reduces the diffusion activation energy, because of the repulsive forces that weaken the effective adsorption energy. Molecular dynamics simulations were performed by Krishna and van Batten12 to calculate the thermodynamic factor and the M-S diffusion coefficient of linear alkanes in different zeolite structures. Both properties had coverage functionality that was dependent on the zeolite involved. In AFI zeolite, for example, the two factors approximately canceled each other so that the Fick coefficient (eq 2.7) was coverage-independent. Krishna et al.13 conducted detailed simulations and modeling of CH4 and CO2 permeation in zeolite topologies involving intersecting channels (e.g., MFI such as ZSM-5, or silicalite), one-

dimensional channels (e.g., FER such as ferrierite), and cagelike structures (e.g., FAU such as zeolites X and Y). A three-site Langmuir isotherm fitted the adsorption data generated by Monte Carlo simulation well, and the Reed-Ehrlich correlation described the coverage dependence of the M-S diffusion calculated from the simulations well. The coverage dependence was pronounced and varied among the different zeolite structures, vanishing or reducing to a very small value as θ f 1. Figure 2 shows the coverage dependence in intersecting channels and in cagelike structures. Most of the aforementioned studies were conducted using molecular simulation on defect-free crystalline structures. Although the results were in good agreement with a limited sample of experimental results, such agreement cannot always be expected for membranes that have a high incidence of defects. The same caveat, of course, applies to multicomponent diffusion. 2.2. Multicomponent Diffusion. Multicomponent diffusion has been described by three formalisms: the Fickian formalism, the nonequilibrium thermodynamic formalism, and the generalized Maxwell-Stefan (M-S) formalism. The Fickian formalism is simply n

Ni ) -

∑D

ij

j)1

dcj dz

(i ) 1,...,n)

(2.14)

where all Dij are functions of concentrations. The second formalism is based on irreversible thermodynamics and expresses the fluxes in terms of the chemical potential gradients:

5800 Ind. Eng. Chem. Res., Vol. 47, No. 16, 2008 n

Ni ) -

∑L

ij

j)1

dµj dz

(i ) 1,...,n)

(2.15)

where the Onsager’s reciprocal relations dictate Lji ) Lij. The third and more widely used M-S formalism has been pioneered by Krishna and co-workers (for example, see refs 14 and 15). It can be derived heuristically by generalizing to several components the momentum argument employed earlier in the single-component case. For the ith species of an n-component mixture, the force balance reads as follows: -

n dµi Vi (Vi - Vj)θj ) + dz bi j ) 1 bij

∑

(2.16)

i*j

The first term on the righthand side represents the force on one molecule of species i due to the solid and the other molecules of the same or different adsorbed species. The jth term in the sum represents an additional resistance exerted on one molecule of species i by the species j that is present, where θj ) cj/cj/ is the (fractional) coverage of the jth species. In eq 2.16, we have already divided by the Avogadro number, to use molar units rather than molecular units. Equation 2.16 is usually written in terms of the corrected, or M-S, diffusivities Dic ) RTbi, Dijc ) RTbij

(2.17)

so that replacing the velocities with the molar fluxes, Vi ) Ni/ Fqi, eq 2.16 is rewritten as -

n Ni θi dµi cjNi - ciNj ) / + / / RT dz c D j)1 D c c i

ic

∑ j*i

(2.18)

ijc i j

Skoulidas et al.9 showed that the three formalisms are equivalent, in the sense that the coefficients of each formalism can be expressed in terms of the coefficients of any of the other two. All coefficients are generally concentration-dependent. When translated to the M-S parameters, the Onsager reciprocal relations Lij ) Lji imply an equality of the cross coefficients (Dijc ) Djic). In section 4, eq 2.18will be needed to be expressed in terms of concentration gradients rather than chemical potential gradients. For this purpose, one requires the multicomponent form of eq 2.3: n n ∂µi ∂ln ai ∂cj ∂ln pi ∂cj ) ) ∂z j)1 ∂cj ∂z j)1 cj ∂ ln cj ∂z

∑

∑

(2.19)

Following previous authors and assuming an ideal gas phase (ai ) pi/pref), we define a thermodynamic matrix Γ with elements Γij )

ci ∂ln pi cj ∂ln cj

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

(2.20)

and a mobility matrix B with elements Bii )

n cj 1 + , Dic j)1,j*i Dijc

∑

Bij ) -

ci Dijc

(i, j ) 1,...,n) (2.21)

The flux vector N is then given by dc (2.22) dz The thermodynamic matrix Γ is calculated from the isotherms of each species in the mixture, and it turns out that accurate isotherms are essential for good estimates of the M-S diffusion N ) -B-1Γ

parameters. Such isotherms were obtained by fitting adsorption models to measurements on powders or to molecular simulation results. One popular isotherm is the multicomponent Langmuir16–18 θi )

ci /

c

Kip

)

n

1+

∑Kp

(2.23)

i

i)1

where c* is the saturation concentration (expressed in terms of the number of moles per unit membrane volume), implicitly assumed, independent of composition. Another model is based on the “ideal adsorbed solution theory” (IAST) of Myers and Prausnitz19 and has been adopted by several authors.20–22 The IAST model does not provide a closed-form expression but, similar to the multicomponent Langmuir, its parameters can be calculated from pure component adsorption parameters. Another multicomponent isotherm based on statistical thermodynamics has been used in refs 6 and 23. The physical significance and the coverage dependence of Dic and Dijc has been central issues in the literature. In the heuristic derivation leading to eq 2.18, the coefficient Dic has a similar physical interpretation as in the one-component system, namely because it is the reverse of the resistance experienced by a molecule i by interacting with the solid and the other molecules of the same or different components. The coefficients Dic in a mixture are not identical to the corresponding singlecomponent coefficients, because the environment of adsorbed species differs between the two situations. At sufficiently low coverage, the diffusing molecules interact only with the solid and the two coefficients become identical. For higher coverage, one often-used approximation22 is Dic(θ1, ..., θn) ) Dpure ic (θ)

(2.24)

where θ ) θ1 + · · · + θn. The cross-coefficients Dijc, which are also known as “exchange coefficients”, are due to additional interactions between molecules i and j that are not included in the coefficients Dic and Djc. According to Skoulidas et al.,9 these interactions include vacancy correlations, momentum exchange, or concerted motions of clusters. One of their effects is to slow the fast species and speed up the slower species.22 As Dijc f ∞, the interactions disappear and the cross terms in the summation given in eq 2.18 vanish.5,16 The limit Dic f ∞ is often postulated for channel structures (e.g., ZSM-5), where diffusion proceeds in “single file” mode, i.e., where the molecules cannot overtake each other in a channel. Even when the cross terms are negligible, however, the diffusing components influence each other through the coefficients Dic, which are dependent on all the individual coverages. A central problem in multicomponent diffusion is determination of the cross coefficients. Being functions of all component coverages, their direct estimation from simulations or from measurements is not practical. Approximate estimation from single-component information is the recourse, and the M-S model is well-suited for this purpose. An empirical correlation originally proposed for bulk binary mixtures24 and introduced to surface diffusion by Krishna,25 Dijc ) Dicci⁄(ci+cj)Djccj⁄(ci+cj)

(2.25)

was used in several modeling studies (for example, ref 17, as will be discussed in section 5. Subsequently, Krishna and van Baten,15 and Yu et al.,22 noted that, when rewritten in terms of the Onsager coefficients (Lij), the cross coefficients obtained

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Figure 3. Steady-state ∆p setup for single-component permeation.

from eq 2.25 do not satisfy the reciprocal relations Lij ) Lji, and they proposed the modified relations c/j Dijc ) [c/j Diic]ci⁄(ci+cj)[c/i Djjc]cj⁄(ci+cj)

(2.26)

which are consistent with the reciprocal relations. The last equation introduces the additional parameters Diic, which are called “self-exchange” coefficients that were estimated from the simulation results, based on the single-component data. Clearly, eqs 2.25 and 2.26 are identical at equal saturation concentrations. 3. Measurement Techniques Microporous membranes commonly consist of a selective layer grown on a macroporous or mesoporous support. The support is necessary for mechanical strength but it adds a resistance in series with that of the selective layer and often must be taken into account in the data analysis. Because of the support resistance and the nonlinear relation of adsorption and diffusion to coverage, the orientation of the membrane (i.e., whether the feed is in contact with the support side or the selective layer side) can have a large effect on the fluxes.26 In most the examples included below, the feed was in contact with the selective layer side of the membrane. 3.1. Steady-State Measurements. 3.1.1. The ∆p Technique for Pure Components. Steady-state permeation of pure components can be performed by maintaining a steady pressure difference across the membrane and measuring the permeate flow rate. Figure 3 shows a schematic of a setup for steady-state permeation through a tubular membrane. The feed pressure is controlled using a back-pressure regulator, while the permeate pressure is usually atmospheric. For a detailed study of the pressure dependence of the permeance or the diffusion coefficient, a lower permeate pressure can be imposed by means of a vacuum pump (not shown), or a higher pressure can be maintained by means of a needle valve or a pressure regulator. The permeate flow rate can be measured using a bubble flow meter or a mass flow meter; in each case, the flow meter is located downstream of the regulating component. A batch setup can also be used, in which the feed compartment is pressurized and isolated while the permeate side is under vacuum.27,28 Measuring the pressure decline in the feed compartment provides the permeance, which yields the diffusion coefficient, if the adsorption isotherm is known. The batch setup is also suitable for transient measurements, as outlined further below. 3.1.2. The W-K Technique for a Single Component or for Mixtures. Figure 4 shows a schematic of a setup for mixture permeation through a tubular membrane. As in Figure 3, the feed and the permeate pass through the tube interior and the annulus, respectively. In this example, the feed is a threecomponent mixture that consists of the sweep gas and two components that are being tested. A sweep gas in the feed is used when it is desired to reduce the partial pressures of the other two components while maintaining an atmospheric or higher total pressure. The sweep gas at the permeate side is needed to prevent concentration polarization and to reduce the

Figure 4. Steady-state ∆p setup for mixture permeation.

partial pressure of the test gases while maintaining atmospheric or higher total pressure. This steady-state measurement is often referred to as the Wicke-Kallenbach (W-K) method (so named after the two workers who first used this setup to measure diffusion coefficients in macroporous or mesoporous catalysts). In the case of macroporous or mesoporous membranes, viscous flow often overwhelms diffusion so that to eliminate the viscous contribution feed and permeate sides are kept at equal pressures. In the case of microporous membranes, viscous flow can occur only through large membrane defects (pinholes), which are normally rare, so that the two pressures can be chosen independently. The W-K technique was used in refs 26 and 28 to study the effect of several operating conditions, including membrane orientation on the separation of n-butane and isobutane. The work of Funke et al.29 can be consulted for a design of the measurement cell and the flow layout. One concern with the W-K technique is the influence of the sweep gas on the fluxes of the components of interest.26,28 To extrapolate from W-K measurements to industrial operation, where there is no sweep gas, it would be necessary to have an accurate predictive model. Steady-state measurements have been reported in several additional publications.29–39 3.2. Transient Measurements. Unlike steady-state measurements, transient measurements contain information about adsorption as well as diffusion. In view of the measurement error, however, information about adsorption parameters can be extracted only in the case of thick membranes. Supported zeolite membranes, for example, must be more than 10 µm thick to provide adsorption information using present-day experimental instrumentation. Most of the transient measurements reviewed below involve single-component permeation. 3.2.1. Uptake Measurements. The simplest possible measurements are the volumetric or gravimetric uptake of powders of the material of interest. In principle, they provide adsorption and diffusion information.1 The equilibrium quantity of adsorbate on the solid is measured at different pressures or concentrations for the case of a liquid, to generate the adsorption isotherm. The transient approach to equilibrium can be used to estimate the diffusion coefficient, although mass-transfer resistance of the gas film and release of the heat of adsorption often make the interpretation of the transient data problematic. The zerolength-column technique has been devised to overcome these interferences40,41 but shape and size differences among the individual particles present an additional obstacle to data analysis. Extensive application of uptake experiments has been made to polymeric films42–45 not in membrane form. Such measurements are meaningful, provided the time for stabilizing the experimental boundary conditions such as the feed side pressure is much shorter than the characteristic diffusion time. Spatial nonuniformity in the film is obviously a complication so that uptake measurements are not suitable for thin skin asymmetric membranes. Inorganic films not in membrane form can also be examined in an adsorption setup; however, in the presence of micrometer-sized membrane defects, the parameters estimated

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Figure 5. Transient ∆p setup for single-component permeation. Figure 7. Transient setup with choked flow for one-component permeation.

Figure 6. Transient ∆p setup for single-component permeation.

from uptake transients may differ from the parameters obtained from the membrane permeation measurements discussed below. 3.2.2. Transient Measurements under ∆p (No Sweep Gas). As will be discussed in the next section, transient measurements allow determination of adsorption as well as diffusion parameters, whereas steady-state measurements only yield the diffusion coefficient, assuming that the adsorption isotherm is available. Similar to measurements at steady state, transient measurements can be conducted in the absence or presence of a sweep gas. In the setup of Figure 5, the feed compartment, permeate compartment, and membrane are initially evacuated and degassed. The measurements start by opening valve v1 and shutting valves v2 and v3 to admit the feed gas. A back-pressure regulator can be used for more accurate control of the feed pressure. The rising permeate pressure is measured by a capacitance gauge or some other sensitive gauge and provides the cumulatiVe molar flow at the permeate side of the membrane, as a function of time. For the purpose of data analysis, the transmembrane pressure difference can be considered to be constant to a high degree of approximation, because the permeate pressure remains very low throughout the measurement. Several variants of the basic setup have been reported. In early studies,46–48 a three-compartment setup (shown schematically in Figure 6) was used to measure the diffusion coefficient in a carbon compact. The additional compartment A was initially pressurized and isolated while the feed and permeate compartments were evacuated, as done with the other measurements previously. When valve v2 is opened, the pressure in the feed compartment increases instantaneously to a level determined by the initial pressure in compartment A and the volumes of compartment A and the feed compartment. The feed pressure slowly declines as the flux through the feed side of the membrane commences, but the low flux again allows the transmembrane pressure difference to be treated as being constant, for the purpose of data analysis. In addition to measuring the slowly rising pressure of the permeate compartment, the slowly declining feed compartment pressure can also be measured using a very sensitive differential pressure capacitance gauge.47 The instantaneous feed side and permeate side pressures provide the cumulative flow through the feed side (upstream) and permeate side (downstream) of the membrane, respectively. Accurate temperature control is necessary to perform these sensitive measurements. When testing a spatially uniform membrane, the two cumulative flows provide duplicate

information about the adsorption and diffusion coefficients, as will be discussed later in this work. Additional variations of the basic setup have been applied. In one variation,46 which is sometimes called a “desorption experiment”, the feed and permeate compartments, as well as the membrane, are initially saturated with the permeant gas at some specified pressure. The permeate compartment is then quickly evacuated and isolated by closing valves v1-v3. Measurement of the rising pressure at the permeate side once more yields the cumulative transfer through that side of the membrane. In this measurement, the feed supply can be shut off or maintained at the initial pressure. A combination of the adsorption and desorption measurements was used to investigate spatial inhomogeneities.46,49–51 Instead of measuring the pressures at the feed or permeate compartments, Watson and Baron52 measured the permeate side flux directly by utilizing choked flow, in a setup shown schematically in Figure 7 for a flat sheet membrane. Feed and permeate chambers, and membrane are initially evacuated and degassed. The measurements start with rapid filling of the feed compartment to the desired pressure, which remains constant thereafter. Governed by choked flow, the flux through the orifice is proportional to the pressure in volume V (between the orifice and the membrane). The proportionality constant (orifice constant) can be measured by suitable calibration. The flux through the permeate side of the membrane differs from the flux through the orifice by the rate of accumulation in the volume V; however, this difference can be minimized by reducing the volume V and, in any case, a simple differential equation can be used to make the necessary correction. Choked flow can be profitably utilized in adsorption measurements,53 as well as in permeation measurements. 3.2.3. Transient Measurements Using a Sweep Gas (W-K Setup). Measuring the cumulative flow by following the pressure at the feed side of the membrane demands accurate differential pressure measurements and temperature control. By contrast, measurement of the pressure at the permeate side is simple and sensitive when the compartment has been initially evacuated to a low residual pressure. Otherwise, differential pressure measurement and accurate temperature control is again required. An alternative experimental setup involves introducing a sweep gas at the permeate side to maintain a constant ∆pi value across the membrane. Figure 7 is still applicable, except that there is only one test component and the sweep gas. The transient concentration buildup at the permeate side recorded using mass spectrometry54 provides the instantaneous flux at the permeate side of the membrane. The measurements commence by switching the valve to conduct the test gas to the inner tube and the sweep gas to the vent while sweep gas continues passing through the annulus. Depending on the flow rate of the test gas and the volume of the permeate compartment, the concentration of the test gas reaches its final value within a

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Figure 8. Pseudo-steady-state setup for single-component permeation.

fraction of a second to a few seconds. The pressures at the two sides of the membrane can be controlled independently, and the feed can be diluted with sweep gas to obtain a lower partial pressure at that side. As in the case of steady-state measurements, the sweep gas introduces an additional component that modifies the fluxes of the test gases, to some extent. 3.2.4. Frequency Response Measurements. Frequency response measurements have been used by van den Begin et al.55,56 to determine the diffusion coefficients in zeolite powders. Their setup involved measuring the pressure variation resulting from a periodic volume variation implemented by a mechanical device. A similar setup can be used to measure adsorption and diffusion coefficients in membranes, as indicated in Figure 7. Feed and permeate chambers are initially filled with the test gas to equal pressures, followed by shutting off the supply. The feed compartment then undergoes a square wave volume variation with an amplitude of a few percent, during which the pressures are measured continuously at both sides. To obtain meaningful transient data, the two volumes should be minimized and the pressures must be accurately measured, preferably using a differential pressure gauge between the two chambers. Useful features of this technique include the following: (i) one has the ability to scan a wide range of frequencies; (ii) there are almost equal pressures at both sides, so that, at any given pressure, the diffusion and adsorption coefficients are essentially constant and the problem can be treated as a linear one; and (iii) repeating the measurements at a series of pressures provides the adsorption isotherm and the concentration-dependent diffusion coefficient. 3.2.5. Pseudo-Steady-State Measurements. In most of the setups previously discussed, the feed pressure was kept constant while measuring the permeate pressure or composition during the brief period before steady state ensued. If the feed pressure is allowed to decline, permeation will remain close to steady state, corresponding to the declining feed-side pressure, provided the experimental setup satisfies certain constraints. Figure 8, which is similar to Figure 6, shows a schematic of this arrangement. It shows a batch measurement with the feed and permeate compartments shut-off during the measurements. The volume of tank A is chosen to be large enough to slow the decline of the feed pressure and thus ensure pseudo-steadystate permeation. Measuring the feed-side and permeate-side pressures provides two independent measures of the transmembrane flux. For more-accurate pressure measurements, an auxiliary compartment (A) can be added to provide a reference pressure for differential measurements. That tank is shut off by valve v5 before the measurements commence. The pseudosteady-state measurements do not provide information about adsorption, but when the adsorption isotherm is available from other sources, the diffusion coefficient can be estimated, as a function of concentration from a single experimental transient (according to Villet and Gavalas57). The choked flow measurement described earlier52 can be conducted at pseudo-steady state by simply allowing the feed-side pressure to decline slowly during permeation. By spanning a pressure range rather than a single pressure, the concentration dependence of the diffusion

Figure 9. Permeance (Π) of propane versus temperature measured by different setups: (O) W-K setup (101 kPa helium sweep) with a zeolite layer at the feed side, (4) W-K setup (101 kPa helium sweep) with a zeolite layer at the permeate side, and (9) batch ∆p setup. (Reproduced with permission from van de Graaf et al.28 Copyright 1998, Elsevier).

coefficient can be accurately obtained without postulating a diffusion model. 3.2.6. Isotropic Tracer Measurements. Isotopic tracer techniques have been very useful in elucidating mechanisms in heterogeneous catalysis. Application of isotopic tracers to pervaporation of several compounds as pure and in mixtures through a Ge-ZSM-5 membrane was made by Tanaka et al.58 and Bowen et al.59 Diffusion in these experiments occurs at steady state, with respect to the total concentration; however, at the same time, diffusion is transient, with respect to the isotopic concentration. The measurements can also be performed so that the total concentration is spatially uniform but the isotope is diffusing under a concentration gradient, so that more-accurate estimates of single-component and mixture diffusion parameters can be obtained. 4. Data Analysis for Single-Component Systems 4.1. Steady State Measurements. 4.1.1. One-Component Systems. Most permeation experiments have been used to evaluate the membrane potential for specific separations and to test membrane preparation conditions, rather than to investigate fundamental transport issues. Therefore, it has been sufficient to measure the permeances Ji, Ji )

Ni pi1 - pi2

(4.1)

of gases as single components and in mixtures, as functions of temperature and feed pressure. Reduction of the data to estimate the diffusion coefficients has not been performed in most cases. Obtaining the diffusion coefficient, on the other hand, is essential to separate the contributions of adsorption and diffusion and to develop a membrane model. In mixture permeation, several diffusion parameters must be estimated for this purpose. A single-component, concentration-independent coefficient is immediately calculated from the flux: D)

N l[h(p1) - h(p2)]

(4.2)

where c ) h(p) (4.3) is the adsorption isotherm (presumed known) that relates the permeant concentration inside the membrane to the external

5804 Ind. Eng. Chem. Res., Vol. 47, No. 16, 2008

pressure. Unlike the permeance, the diffusion coefficient requires knowledge of the membrane thickness, which is not well-defined in some types of supported membranes. The relative diffusion coefficients of two or more permeant gases can obviously be obtained without knowing the thickness. Concentration-dependent diffusion coefficients can be estimated by measuring the transmembrane flux at a series of feedside pressures, while keeping the permeate side at the lowest pressure of interest. The basic flux relationship N ) -D(c)

dc dz

(4.4)

with z being the position inside the membrane, applies to tubular as well as to flat membranes. For a flat membrane (0 e z e l), N is constant, so that integrating eq 4.4 yields

∫

∫

1 h(p1) 1 co D(c) dc) h(p ) D(c) dc (4.5) 2 l cl l For an asymmetric tubular membrane, such as a supported zeolite membrane, the active layer can be treated as flat, because its thickness is much smaller than the tube radius. Equation 4.5 then still applies. When the membrane is an unsupported tube (r1 e r e r2), rN is now constant and the integration of eq 4.4 yields N)

N)

∫

h(p1)

h(p2)

D(c) dc (4.6)

r ln(r2 ⁄ r1)

Differentiating eq 4.5 with respect to the feed pressure gives dh ( dN dp ) ( dp )

D(c1) ) l

p1

(4.7)

p1

where the term dN/dp must be obtained by numerical differentiation, fitting the measurements to polynomials or splines. This calculation gives D(c) as a numerical table that can be used to compare different models. Alternatively, having specified a model for D, eq 4.5 can be used to directly estimate the model parameters, thus avoiding numerical differentiation. Van de Graaf et al.28 reported single-component steady-state measurements using the W-K setup with a helium sweep, and measurements using the batch ∆p setup (Figure 5) without a sweep gas. Figure 9 compares the permeances of propane, as a function of temperature, measured using the batch technique and the W-K technique. The permeance measured using the batch method is higher, but the difference decreases as the temperature increases. The maximum in the permeance being observed at ∼360 K is explained by the opposite trends of adsorption and diffusion with rising temperature. This figure also shows that passing the feed from the selective layer side gives higher permeance than passing it at the support side. Figure 10 shows the highly nonlinear effect of the permeate pressure, the difference in transmembrane pressure, and the measurement setup on the ethane permeance. To estimate the diffusion coefficient from the data of Figure 9, the authors used eq 2.7, along with eq 2.9 for the temperature dependence. The Langmuir isotherm, c ) f(p) ) c *

Kp 1 + Kp

(4.8)

was postulated to express the diffusion coefficient as D ) Dc

Figure 10. Permeance (Π) of ethane at (a) 303 and (b) 473 K versus the feed pressure measured by different setups: (0) batch ∆p, permeate side, vacuum; (9) batch ∆p, permeate side, 101 kPa; ([) W-K setup, helium sweep gas. (Reproduced with permission from van de Graaf et al.28 Copyright 1998, Elsevier).

Fc* Fc* E ) Do exp 1-θ 1-θ RT

( )

(4.9)

where θ ) c/c* is the (fractional) coverage and the terms Dc and Do are coverage-independent. Using their measurements and

adsorption data from the literature, the authors estimated the constant Do and the activation energy. Unlike the permeance, the diffusion coefficients estimated using the two different measurement methods are much closer to each other, which is consistent with the postulated adsorption-diffusion model. The batch ∆p setup shown schematically in Figure 3 was also used by Li et al.31 in single-gas and mixture permeation measurements of CH4 and CO2 in a supported SAPO-34 membrane. Typical results given in Figure 11 include singlegas permeances and ideal selectivity (the ratio of the singlegas selectivities) versus temperature. After a shallow maximum, the permeance of the more strongly adsorbed CO2 declines with temperature while the permeance of the less strongly adsorbed CH4 undergoes little change. The data was again analyzed using the M-S model combined with the Langmuir isotherm. Inserting eq 4.9 in eq 2.4 and integrating yields N)

(

1 + Kpf c * Dc ln l 1 + Kpp

)

(4.10)

and setting as before Dc ) Do exp(-E/RT), the flux measurements yielded Do and E. The authors examined the adequacy of eq 4.10 to describe the measured temperature and pressure dependence of N. Deviations from the Arrhenius temperature dependence were attributed to a shift in the diffusion mechanism from surface to activated gas diffusion, while deviations from the pressure dependence were attributed to Do not being strictly concentration-independent, as assumed in integrating eq 4.9. More-elaborate models for D can obviously be accommodated in the integration of eq 4.5. Pervaporation fluxes of methanol through a microporous methylated silica membrane was measured at 60-155 °C and up to a pressure of 16 bar.60 The measurements were described well by a simplified diffusion model, along with the dual-site Langmuir adsorption isotherm

Ind. Eng. Chem. Res., Vol. 47, No. 16, 2008 5805

[

Ql ) cf

(

∞

Dt l 2l (-1)n -Dn2π2 - - 2 exp 2 l 6 π n)1 n l2

∑

)]

(4.17)

As time increases, the cumulative fluxes assume the straightline asymptotes

( Dtl + 3l ) Dt l )c( - ) l 6

Q∞o ) cf

(4.18)

Q∞l

(4.19)

f

Fitting the large time measurements of Ql to 4.19 yields the slope cfD/l, and the intercept on the t-axis, using Figure 11. Single-gas permeances of CO2 and CH4 through a SAPO-34 membrane, and CO2:CH4 ideal gas selectivities versus temperature, ∆p setup at a feed pressure of 222 kPa and a transmembrane pressure difference of 138 kPa. (Reproduced, with permission, from Li et al.31 Copyright 2004, Elsevier).

(eq 2.12), where the fugacity replaced the pressure as appropriate for a liquid feed. 4.2. Transient Measurements and the Time Lag Method. The previous sections reviewed the estimation of D from steady-state measurements. Transient measurements contain information about the adsorption as well as the diffusion properties of the membrane. Much of the analysis of transient measurements has relied on the “time lag” method outlined in the next subsection. 4.2.1. Time Lag for a Single Component with a Constant Diffusion Coefficient (∆p Technique). The base case is a spatially uniform membrane with a constant diffusion coefficient, so that permeation is governed by the linear differential equation ∂c ∂2c )D 2 ∂t ∂x

(4.11)

In the most common specification, the membrane is initially evacuated and, at time zero, the feed side is quickly pressurized and maintained at constant pressure, while the permeate side is kept essentially at zero pressure: c(x, 0) ) 0;

(0, t) ) cf ;

c(l, t) ) 0

(4.12)

where cf ) f(pf).The solution is given in the work by Crank,45 ∞

(

c 1 nπx -Dn2π2t x 2 sin )1- exp cf l π n)1 n l l2

∑

( )

)

(4.13)

which also provides solutions for more general boundary conditions. The fluxes at the feed side (x ) 0) and the permeate side (x ) l) are given by c(x,0) ) 0; c(0,t) ) cf; c(l,t) ) 0 at x ) 0 and x ) l, respectively: No )

[

(

∞ Dcf –Dn2π2t exp 1+2 l l2 n)1

[

∑

(

)]

∞ Dcf –Dn2π2t Nl ) 1 + 2 cos(nπ) exp l l2 n)1

∑

(4.14)

)]

(4.15)

Integrating these two expressions, the cumulative fluxes up to time t are obtained as

[

Qo ) cf

∞

(

Dt l 2l 1 -Dn2π2t + - 2 exp 2 l 3 π n)1 n l2

∑

)]

l2 (4.20) 6D This intercept is the celebrated “time lag” that gives an estimate for the diffusion coefficient D. Having obtained D, the slope yields the concentration c(pf). If adsorption is in the Henry’s law regime, the constant H can be obtained from c(pf); however, to estimate the two parameters of the Langmuir isotherm, one needs an additional experiment at a different feed pressure pf. The asymptote of Qo yields the same slope as that obtained previously (cfD/l), but the intercept is tal )

tao ) -

l2 3D

(4.21)

This is also called a time lag. Measuring the cumulative flux at the permeate side by following the pressure increase from vacuum is simpler and more accurate, explaining the prominence of the time lag tla in the literature. The two aforementioned time lags are often called “adsorption” time lags, because of the rising permeant concentration in the membrane, using the superscript a. Desorption time lags tod,tld can be defined for a “desorption” run, where the initial concentration in the membrane is equal to the feed concentration, whereas the permeate-side pressure continues being close to zero. Derivations and application to membrane characterization can be found in refs 46–51 and 61, 62 In refs 46, 47, 49–51, the time lag technique was used to investigate spatial membrane inhomogeneities. Other more-general boundary conditions have also been considered, and analytical solutions for the time lag have been obtained in refs 63–65. 4.2.2. Time Lag for a Single Component with a Concentration-Dependent Diffusion Coefficient (D). When the time lag method is applied to a system with a concentrationdependent diffusion coefficient, one obtains some type of an average diffusion coefficient,66,67 which, however, differs from the concentration-averaged coefficient that enters in eq 4.6. A more-precise treatment of time lag for the concentrationdependent case starts with the diffusion equation that is associated with the same initial and boundary conditions as those used in eq 4.22. ∂ ∂c ∂c ) D(c) (4.22) ∂t ∂x ∂x Frisch68 showed that a time lag for this nonlinear equation can be obtained in terms of integrals of D(c) as follows. Integrating eq 4.22 from x to l and using the boundary conditions yields

(

)

∂c ∂c(l, t) ∂c dx ′ ) D(c ) - D(c) ) -N - D(c) ∫ ∂c ∂t ∂x ∂x ∂x l

(4.16)

x

l

l

(4.23)

5806 Ind. Eng. Chem. Res., Vol. 47, No. 16, 2008

Integrating further, between 0 and l, gives ∂c ) -lN - ∫ D(c) dx ) -lN + ∫ ∫ ∫ ∂c ∂t ∂x l

l

0

x

l

l

cf

l

0

0

D(c) dc

Langmuir isotherm and the M-S model with a constant M-S diffusivity, in which case the steady-state concentration satisfies the equation

(

This last equation is integrated once more, with respect to time, between 0 and t, to obtain

∫ ∫ c(x ′ , t) dx ′ ) -lQ + t∫ l

l

0

x

cf

l

0

D(c) dc

)

Dc dcss d )0 dx 1 - θss dx

(4.24)

The initial and boundary conditions are

(4.25)

The integral on the lefthand side can be integrated, in parts,

css(x, 0) ) 0

(4.30a)

css(0, t) ) cf

(4.30b)

css(l, t) ) cl

to

∫ ∫ c(x ′ , t) dx ′ )∫ xc(x, t) dx l

l

l

0

x

0

(4.26)

As the time increases, the integral on the righthand side becomes asymptotically ∫0l xcss (x) dx, where css(x) is the steady-state concentration profile. The cumulative flux Ql at the permeate side of the membrane has the large time asymptote

∫

∫

1 t xc dx + 0 D(c) dc (4.27) l 0 ss l Hence, fitting a straight line to the large-time measurements of Ql provides estimates for the two integrals in eq 4.27. The steady-state concentration css(x) is also dependent on D(c) and cf and, if desired, can be expressed fully in terms of integrals of D.68 Having postulated a model for the diffusion coefficient, the two integrals in eq 4.27 obtained by fitting long-time measurements can be used to estimate two diffusion parameters, or one diffusion parameter and one adsorption parameter. Multiple runs at different feed pressures are needed to evaluate diffusion and adsorption models that contain additional parameters. A similar integration of eq 4.23 gives Q∞l ) -

Qf∞ )

t l

cf

l

cf

l

∫ D(c) dc + ∫ (l - x)c

ss(x) dx

0

(4.28)

(4.29)

(4.30c)

Integrating once gives the steady-state flux as Dc dcss ) -Nss 1 - θss dx

(4.31)

Integrating twice yields the steady-state concentration profile,

(

css(x) ) c * -(c * -cf)

c / -cl c / -cf

)

(4.32)

Integrating this profile between 0 and l, the total quantity adsorbed at steady state is obtained as

( ( ))

c * + cf - cp c* Qss ) Ac * l c * -cf ln c * -cl

(4.33)

where A is the membrane area. Introducing the Langmuir isotherm, this expression can be rewritten as

[

K(pf - pl)

Qss ) lc * 1 -

(

(1 + Kpf)(1 + Kpl) ln

1 + Kpf 1 + Kpl

0

The slope is the same as noted previously, and the time lag provides a second relation for estimating an additional parameter. Measurement of the fluxes at both sides of the membrane, using differential pressure measurements, as discussed in section 3 can provide more complete information. By utilizing only long-time measurements, the time-lag method does not make full use of the available information and is useful mainly when the diffusion coefficient is represented by a one-parameter model. Several experiments at a series of feed pressures are needed to determine parameters in morecomplex models. Full nonlinear parameter estimation must be used for that purpose. In the report of Shah et al.,54 nonlinear least-squares estimation was applied to simulated measurements of transient pervaporation of organics through a polymeric membrane. In this application, time-lag analysis was used to obtain starting values for the sorption and diffusion parameters. 4.2.3. Transient Measurements for a One-Component Test Gas Using a Sweep Gas (W-K Setup). As mentioned in the Experimental Section, a sweep gas facilitates flux measurements and control of the test gas pressures at the two sides of the membrane. Gardner et al.67,69,70 developed a novel method to estimate membrane thickness, as well as adsorption and diffusion parameters, in a two-step procedure. First, thickness and adsorption parameters were estimated by matching the calculated and experimental values of the total permeant adsorbed in the membrane after steady state is reached. In the second step, the diffusion coefficient was estimated from the measured steady state flux. Their derivation postulates the

x⁄l

)

]

(4.34)

Using a mass balance, the quantity adsorbed at steady state can also be expressed in terms of the transient fluxes: Qss ) A

∫

tss

(Nx)0 - Nx)l) dt

0

(4.35)

where tss is large enough so that the flux has attained its steadystate value, within the measurement error. The flux at the permeate side of the membrane, x ) l, can be measured accurately using the W-K setup. However, being the difference of two almost equal numbers and being influenced by the uncertain flux of the sweep gas, the flux at the feed side x ) 0 cannot be measured accurately. To overcome this difficulty the authors67 introduced the approximation

∫ (N F) ∫ (N tss

0

x)0 - Nx)l)

tss

ss - Nx)l)

0

dt

)3

(4.35a)

dt

so that the experimental Qss value becomes tss

Qss ) 3A

∫ (N

ss - Nx)l)

dt

(4.36)

0

Using the least-squares method to match the experimental Qss with the model-calculated Qss value given by eq 4.34, one can estimate the three parameters l, K, and c*. With these three parameters determined, one can estimate Dc using eq 4.10, after css, given in eq 4.32, has been inserted as θss ) css/c* in eq 4.31. This method was applied67 to estimate adsorption and

Ind. Eng. Chem. Res., Vol. 47, No. 16, 2008 5807

to make progress. Krishna and Paschek71 studied permeation of n-hexane (n-C6) and 2,2-dimethylbutane (DMB) in ZSM-5 membranes. The dual-site Langmuir isotherm was adopted, where A sites (intersections between straight and zigzag channels) are those that can accommodate both isomers, whereas B sites (straight channel interior) are those that can adsorb only n-C6. In obvious notation, the isotherms are cA/ KA1p1 c/BK1Bp1 c1 ) + KA1p1 1 + K1Bp1 + K2Bp2

Figure 12. Adsorption isotherms of CO2 calculated from transient permeation measurements using (2) membrane M1 and ([) membrane M2. Dashed line is model fit, and solid lines are isotherms from the literature. (Reproduced from Gardner et al.67 Copyright 2000, with permission of John Wiley).

Figure 13. Single-component M-S diffusion coefficients versus feed partial pressure calculated using eq 4.33 for permeation of (9) N2,([) CH4, and (2) CO2 through membrane M1 (solid lines) and membrane M2 (dashed lines). (Reproduced from Gardner et al.67 Copyright 2000, with permission of John Wiley).

diffusion parameters of N2, CO2, and CH4, and the two butanes69 in thick ZSM-5 membranes. Figure 12 shows good agreement between the estimated adsorption isotherm of CO2 with previous calorimetric adsorption measurements. Similarly good agreement was obtained for N2. Figure 13 from the same paper plots the M-S diffusion coefficient for different feed pressures. This coefficient shows a modest increase with increasing feed pressure, which suggests some dependence on coverage. In a subsequent paper,70 Gardner et al. showed that eq 4.35 is exact when the Fick’s coefficient is independent of coverage. When the M-S diffusion coefficient is, instead, coverage-independent, F is still close to 3 at low coverage, but decreases to 2 as the coverage increases. Mass-transfer resistance in the support, and film resistance at the permeate side, also modify the factor F. In their application to butane mixture permeation in H-ZSM-5 and Na-ZSM-5 membranes,69 the authors took into account the resistance in the transport and used an iterative procedure to avoid making the assumption F ) 3. 5. Multicomponent Systems 5.1. Steady-State Measurements. In multicomponent permeation, an increased number of adsorption and diffusion parameters are encountered. For example, the M-S model includes three diffusion coefficients for a binary system and six for a ternary system. The simple Langmuir isotherm involves two parameters, but more parameters are involved in the dualsite and other more-complex isotherms. Full nonlinear estimation is not practical for such multiparameter models, because of the measurement error and the statistical correlations among the parameters. Various simplifying approaches have been adopted

c2 )

c/BK2B 1 + K1Bp1 + K2Bp2

The parameters were determined by fitting the results of Monte Carlo simulations. Neglecting binary diffusion interactions (D12 set to ∞), only the pure component parameters D11 and D12 of the M-S model had to be estimated, and single-component permeation measurements were sufficient for that purpose. Using the adsorption and diffusion parameters obtained from these assumptions, they obtained fair agreement with previous measurements.72 Van den Broeke et al.20 modeled permeation of CH4-N2, CO2-N2, and CO2-CH4 mixtures through a silicalite membrane. They made calculations for the “Extended Langmuir Isotherm” and the IAST isotherm, with the parameters of both isotherms estimated from the single-component isotherms. They estimated the single-component diffusion parameters from their previous measurements73 and, neglecting the cross terms, calculated the fluxes and separation factors in mixture permeation. The model results were in reasonable agreement with the measurements only when the IAST binary isotherms were used. The results calculated using the extended Langmuir isotherms were not consistent with the measurements. Permeation through a Silicalite-1 membrane of single components and binary mixtures of methane, ethane, and propane were measured using the W-K setup by van de Graaf et al.17 at different feed pressures, feed compositions, and temperatures. The results were modeled using the multicomponent Langmuir isotherm, and the M-S model with the coefficients Dijc computed using the Vignes correlation (eq 2.25). Calculations were also performed omitting the cross terms (setting Dijc ) ∞). In each case, the Dic coefficients were obtained from single-component data and were treated as being coverage-independent. In the permeation of methane-ethane mixtures, the measured ethane flux was matched well by the M-S model, regardless of whether or not the cross terms were included. However, neglecting the cross terms overestimated the methane flux and greatly underestimated the ethane-to-methane selectivity. Similar but stronger trends were observed for methane-propane mixtures. The flux of methane was grossly overestimated if the cross terms were neglected while the flux of propane was only slightly overestimated, as shown in Figure 14. At 300 K, the ideal propane: methane selectivity (single-component permeance ratio) at a feed pressure of 1 bar was ∼0.03, but in the 50:50 mixture, the propane:methane selectivity rose to ∼40. These trends are consistent with the general observation that the cross terms raise the fluxes of the slow components and suppress the fluxes of the fast component, where the effect on the flux of the faster component is more pronounced. The extended Langmuir isotherm as modified by assuming equal saturation coverages was satisfactory except for the propane-methane mixtures. Krishna and Paschek71 applied the M-S model to previous measurements72 of n-hexane-2,2-dimethylbutane mixture permeation through a ZSM-5 membrane. Using a dual-site Lang-

5808 Ind. Eng. Chem. Res., Vol. 47, No. 16, 2008

Figure 14. Fluxes of ([) propane and (b) methane, and (9) propane: methane selectivity versus propane feed partial pressure (total pressure ) 101 kPa) in permeation of a propane-methane mixture through a silicalite membrane measured using the W-K setup (with helium as the sweep gas). Solid lines were calculated using the M-S model, and dashed lines represent the M-S model neglecting the cross terms. (Reproduced from van de Graaf et al.17 Copyright 1999, with permission of John Wiley).

muir isotherm and neglecting the cross terms and the coverage dependence of the M-S diffusion coefficients, they obtained qualitative agreement with the measurements. Of great interest in the membrane literature has been the relationship between the flux of a pure gas and the flux of the same gas in a mixture, at the same partial pressures. Even when cross terms are neglected, the two fluxes generally differ, because of the thermodynamic matrix and the coverage dependence of the M-S diffusion coefficient. Obviously, the difference between the single-component and the mixture fluxes is responsible for the difference between ideal selectivities (the ratio of permeances in single-component measurements) and mixture selectivities. These differences were observed in numerous experiments, including the permeation (W-K setup) of pure gases/vapors and binary mixtures through ZSM-5 membranes by Keizer et al.73 for n-C6H14-2,2 dimethylbutane, and by van den Broeke et al.20,74 for CH4, N2, and CO2. Zhu et al.75 applied the M-S model to their measurements of CH4-CO2 mixture permeation through a Silicalite-1 membrane. The measurements were performed at 303-408 K, using the W-K setup with helium as the carrier gas. The one-component Langmuir isotherms for single components and the IAST model for binary mixtures were used, along with the M-S model. The cross terms were included in some of the calculations, using the Vignes correlation for D12c, but the coefficients D1c and D2c were treated as coverage independent. The model predictions were in fair agreement with the measurements, and some of the differences observed were attributed to the effect of intercrystalline barriers. Permeation fluxes of single-component and binary mixtures of CO2, CH4, and N2 across a SAPO-34 membrane were measured by Li et al.,6 using the ∆p method with both feed and permeate under flow. The feed pressure was as high as 7.3 MPa, whereas the permeate pressure was 84 kPa. Their data analysis used a statistical thermodynamics model for the pure component isotherms and the IAST model for the mixture isotherms. Two variants of the M-S model were investigated. In the first variant, the coverage dependence of the pure component coefficients was neglected but the cross terms were taken into account using the Vignes correlation. In the second variant the coverage dependence was taken into account using the Reed-Ehrlich model, but the cross terms were neglected (Dijc ) ∞). The first variant resulted in good agreement with the measurements, whereas the second variant’s results differed markedly from the measurements, as illustrated in Figure 15. It

Figure 15. Fluxes in mixture permeation through a SAPO-34 membrane. Equation 18 in the reference is the model, taking into account the coverage dependence of the one-component coefficients and ignoring the cross terms. (Reproduced with permission from ref 6. Copyright 2007, American Chemical Society).

may be concluded that, in the SAPO-34 structure (cages and narrow windows structure), the coverage dependence of the single-component M-S coefficients is pronounced but the cross terms can be neglected. In a subsequent paper, the same authors23 performed experiments, including the additional components helium, hydrogen, and argon. They confirmed the strong coverage dependence of Dic for all gases except helium and hydrogen, for which the coefficients were essentially constant. The measurements showed that the Dic values in the mixture were similar to the pure component Dic when the total coverage was substituted in the coverage dependence obtained with the pure component. Taking into account the coverage dependence and omitting the cross terms gave good agreement with the measurements for the CO2-CH4, CO2-N2, and CH4-N2 mixtures. For the mixtures that contain H2, the flux of the heavier gas was moderately higher than the pure component flux at the same coverage, whereas the hydrogen flux was slightly lower. The slowing down of hydrogen was explained as possibly being due to neglecting the cross terms, or due to the error in adopting the approximate equation 2.19. The model parameters determined from measurements as illustrated in the aforementioned studies refer to the particular membranes being tested. The same parameters may be applicable to mixtures that contain the same components, but over a broader range of composition and temperature, for the purposes of industrial design and optimization. The same model parameters would also be expected to apply to other membranes prepared using the same protocol. However, this extension would be frustrated by the often-observed lack of reproducibility caused by small uncontrolled changes in the support structure or in the preparation protocol. Such variations would be minimized when the membranes are prepared on an industrial scale using well-controlled equipment. Membranes prepared using different supports or protocols generally differ in their defect structure and cannot possibly be described by a common set of parameters. In the models discussed previously, the adsorption isotherms were, in most cases, obtained from data on powders or from molecular simulation. Adsorption parameters were, in a few cases,67,69,70 estimated directly from permeation measurements on thick MFI membranes. Although the results were in good agreement with powder measurements, more work is required to compare powder and membrane measurements more broadly. Detailed X-ray analysis by Jeong et al.76 has shown that supported MFI membrane layers are in compressive stress parallel to the membrane surface and in tensile stress in the perpendicular direction. As a result of these stresses, the crystallite unit cell dimensions in the directions parallel to

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the surface were smaller, compared to those of free crystals. This difference may be large enough to affect the adsorption properties of molecules such as xylenes that have a molecular diameter similar to that of the MFI channels. Additional research is needed to develop adsorption measurement techniques that are applicable to thin, micrometer-sized membranes. Another feature that complicates application of the models discussed previously is the effect of adsorption on the framework structure of certain zeolites. It has been known for some time (for example, in the work of Funke et al.29 and references therein) that the MFI structure exhibits some flexibility and can adsorb molecules whose size is slightly larger than the channel diameter (such as n-octane). Yu et al.77 studied the permeation of n-hexene, dimethylbutane (DMB), and trimethylbenzene as single components or in mixtures through MFI membranes that possess considerable nonzeolitic pores. They found that the adsorption of n-hexane expands the zeolitic channels, causing shrinkage of the nanosized nonzeolitic pores and resulting in sharp differences of mixture versus pure-component permeation. As an example, the ratio of DMB to n-hexane pure component permeances was 17, but it decreased to 0.1 for a 50:50 mixture (in both cases, at 303 K). Knowledge of the pure-component adsorption or diffusion parameters would be insufficient in this case for estimating mixture parameters using the correlations discussed previously. 6. Conclusions Macroscopic models of adsorption and diffusion are now capable of describing measurements of single-component and mixture permeation in microporous membranes satisfactorily. In single-component adsorption, the standard Langmuir isotherm is insufficient in many instances, and two- or even three-site Langmuir isotherms have been used to obtain satisfactory results. Mixture isotherms have been successfully modeled using the IAST theory. The so-called extended Langmuir isotherm has been found to be insufficient in several cases. Adsorption parameters obtained for powders may differ from those of membranes that have extensive defects or structural differences from the loose crystals. This matter has not been adequately investigated. With regard to diffusion, the M-S model has been universally and successfully applied. The two major problems encountered in applying this model are (i) the coverage dependence of the coefficients Dic in single-component and mixture permeation, and (ii) the estimation of the exchange coefficients Dijc from single-component measurements. Great progress has been made on both these problems, using molecular simulation. It was found that the coefficients Dic generally have a strong dependence on coverage, which varies among different crystalline structures of the membrane material. These coefficients vanish or decline to a small finite value as θ f 0. With regard to the cross terms, the original or updated Vignes correlation provides a practical way of estimating the exchange coefficients from the single-component coefficients. Significant progress has been made in applying the M-S model to permeation measurements. With respect to singlecomponent permeation, neglecting the coverage dependence results in estimates of Dc that vary with the upstream pressure, and when the coverage dependence is included, better agreement with the measurements is obtained. In mixture permeation, several studies have used coverage-independent Dic values. In some cases, the cross terms were also neglected. In view of measurement error and the membrane defects (grain boundaries and other), the data are often insufficient to test the necessity

of including the coverage dependence or the exchange terms. Nevertheless, as other uncertainties are eliminated, a predictive model would have to include an accurate isotherm, to take into account the coverage dependence of the M-S coefficients, and for many systems to also include the cross terms. Model extensions and modifications are necessary for membranes having two parallel pore systems, especially when these systems interact through structural changes. From the standpoint of measurement techniques, the W-K setup was used in most cases, because it allows good control of partial pressures at the membrane boundaries. As membranes become adopted industrially, the ∆p setups become more relevant. Precise differential pressure measurements and accurate composition measurements with online mass spectrometry or FTIR can enhance the accuracy of the data. Determination of adsorption isotherm parameters from transient measurements is useful, especially where traditional gravimetric or volumetric techniques are not feasible, either because the membrane is very thin or because it contains defects or extraneous matter. Literature Cited (1) Karger, J.; Ruthven, D. M. Diffusion in Zeolites; Wiley: New York, 1992. (2) Karger, J.; Vasenkov, S.; Aurbach, S. M. In Diffusion in Zeolites in Handbook of Zeolite Catalysts and Microporous Materials; Aurbach, S. M., Carrado, K. A., Dutta, P. K., Eds.; Marcel Dekker: New York, 2007; Chapter 12, pp 1-97. (3) (a) Xiao, J.; Wei, J. Diffusion mechanism of hydrocarbons in zeolites;I. Theory. Chem. Eng. Sci. 1992, 47, 1123–1141(b) Xiao, J.; Wei, J. Diffusion mechanism of hydrocarbons in zeolites;II. Analysis of experimental observations. Chem. Eng. Sci. 1992, 47, 1143–1159. (4) Bakker, W. J. W.; van den Broeke, L. J. P.; Kapteijn, F.; Moulijn, J. A. Temperature Dependence of One-Component Permeation through a Silicalite-1 Membrane. AIChE J. 1997, 43, 2203–2214. (5) Van de Graaf, J. M.; Kapteijn, F.; Moulijn, J. A. Permeation of Weakly Adsorbing Components through a Silicalite-1 Membrane. Chem. Eng. Sci. 1999, 54, 1081–1092. (6) Li, S.; Falconer, J. L.; Noble, R. D.; Krishna, R. Modeling Permeation of CO2/CH4, CO2/N2, and N2/CH4 Mixtures across SAPO-34 Membrane with the Maxwell-Stefan Equation. Ind. Eng. Chem. Res. 2007, 46, 3904– 3911. (7) Skoulidas, A. I.; Sholl, D. S. Direct Tests of the Darken Approximation for Molecular Diffusion in Zeolites Using Equilibrium Molecular Dynamics. J. Phys. Chem. B 2001, 105, 3151–3154. (8) Skoulidas, A. I.; Sholl, D. S. Transport Diffusivities of CH4, CF4, He, Ne, Ar, Xe, and SF6 in Silicalite from Atomistic Simulations. J. Phys. Chem. B 2002, 106, 5058–5067. (9) Skoulidas, A. I.; Sholl, D. S.; Krishna, R. Correlation Effects in Diffusion of CH4/CF4 Mixtures in MFI Zeolite. A Study Linking MD Simulations with the Maxwell-Stefan Formulation. Langmuir 2003, 19, 7977–7988. (10) Krishna, R.; Paschek, D.; Baur, R. Modeling the Occupancy Dependence of Diffusivities in Zeolites. Microporous Mesoporous Mater. 2004, 76, 233–246. (11) Reed, D. A.; Ehrlich, G. Surface Diffusivity and the Time Correlations of Concentration Fluctuations. Surf. Sci. 1981, 102, 588–609. (12) Krishna, R.; van Baten, J. M. Linking the Loading Dependence of the Maxwell-Stefan Diffusivity of Linear Alkanes in Zeolites with the Thermodynamic Correction Factor. Chem. Phys. Lett. 2006, 420, 545–549. (13) Krishna, R.; van Batten, J. M.; Garcia-Perez, E.; Calero, S. Incorporating the Loading Dependence of the Maxwell-Stefan Diffusivity in the Modeling of CH4 and CO2 Permeation across Zeolite Membranes. Ind. Eng. Chem. Res. 2007, 46, 2974–2986. (14) Krishna, R.; Wesselingh, J. A. The Maxwell-Stefan Approach to Mass Transfer. Chem. Eng. Sci. 1997, 52, 861–911. (15) Krishna, R.; van Baten, J. M. Diffusion of Alkane Mixtures in Zeolites: Validating the Maxwell-Stefan Formulation Using MD Simulations. J. Phys. Chem. B 2005, 109, 6386–6396. (16) van den Broeke, L. J. P. Simulation of Diffusion in Zeolitic Structures. AIChE J. 1995, 41, 2399–2413. (17) van de Graaf, J. M.; Kapteijn, F.; Moulijn, J. A. Modeling Permeation of Binary Mixtures through Zeolite Membranes. AIChE J. 1999, 45, 497–511.

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ReceiVed for reView March 14, 2008 ReVised manuscript receiVed May 5, 2008 Accepted May 16, 2008 IE800420Z