Diffusion in Randomly Overlapping Parallel Pore and Fiber Networks

Article pubs.acs.org/IECR

Diffusion in Randomly Overlapping Parallel Pore and Fiber Networks: How Pore Geometry and Surface Mobility Impact Membrane Selectivity Marziye Mirbagheri and Reghan J. Hill* Department of Chemical Engineering, McGill University, Montreal, Quebec H3A 0C5, Canada ABSTRACT: Pore-resolved computations are undertaken, within a continuum model framework, to explore surface diffusion as a selective mechanism for gas separations using membranes of randomly overlapping parallel cylindrical pores or fibers. Orders of magnitude of the model parameters are established using an intrinsic gas diffusivity that is selfconsistent with the Knudsen diffusivity obtained from Monte Carlo simulations reported in the literature. The relative contributions of surface and gas diffusion to the overall permeation show that the surface-diffusion flux increases with the specific surface area, whereas the gas flux increases with porosity. Thus, gas diffusion that is perpendicular to pores and fibers can be hindered by the increasing tortuosity while simultaneously promoting permeation via surface diffusion. The selectivity of pore structures with fibrous networks is examined for the dehumidification of air, natural gas, and carbon dioxide. Selectivities, defined as the ratio of the effective diffusion coefficients for the adsorbed phases (moisture) and the void-diffusing gas (air, natural gas, or carbon dioxide), reveal that selectivity is higher for (i) heavier gases, which have lower gas-diffusion fluxes, and (ii) capillary pores, which have higher surface connectivity than fibers.

1. INTRODUCTION The high cost and energy intensive characteristics of gas separations using distillation and pressure-swing adsorption have directed considerable attention to microporous membranes.1−7 Membrane separation efficiency depends on the diffusion resistance and gas-separation mechanism. Many studies have shown that structural properties of porous materials, such as porosity, surface connectivity, and the orientation of anisotropic particles/pores with respect to the bulk diffusion direction, can control the diffusion resistance.8−22 The gas-separation mechanism is influenced by the pore size and physicochemical interaction of the permeating molecules and pore surfaces. Thus, operating conditions, such as temperature, pressure, and feed gas composition are influential.23 For a nonadsorbing gas mixture, viscous flow (continuum diffusion) controls separation in macropores, whereas molecule−wall collisions in smaller pores give rise to Knudsen diffusion. Moreover, gas molecules may adsorb on the pore surfaces and migrate by surface diffusion. Note that physically adsorbed species are often highly mobile and may diffuse along the pore surface faster than in the voids.24 Thus, a large specific surface area enables surface diffusion to have a significant influence on the overall permeability and selectivity.3,7,23,25 Recently, Mirbagheri and Hill22 applied the theory of Albaalbaki and Hill20 in three-dimensional simple-cubic arrays of solid spheres and spherical cavities. They solved the tracer conservation equations in gas, solid, and surface phases, with interfacial boundary conditions to couple diffusion in three domains with interfacial exchange fluxes. The computations © XXXX American Chemical Society

quantify how the interaction between the gas and surface phase influence the permeability of disconnected solids, limiting the surface flux by hindered gas diffusion in the Knudsen regime. Moreover, the coupling of surface and gas diffusion is particularly significant for pore structures that have discontinuous surfaces.26However, Mirbagheri and Hill22 considered the diffusion of only one species, and therefore did not address the selectivity that arises from molecules that have different pore affinities. The surface flux promoted by nanosized pores, which hinder gas diffusion by limiting the mean-free-path, highlights the potential of surface diffusion to increase membrane selectivity by promoting the flux of adsorbed species and blocking voiddiffusing (low solubility) species. For example, Bakker et al.27 showed that the permeation of weakly adsorbing molecules in metal−supported silicalite-1 zeolites can decrease by 2 orders of magnitude in the presence of strongly adsorbing molecules. Similarly, selective surface-flow carbon membranes have been used to increase selectivity.28,29 A robust study of permeability and selectivity therefore hinges on a prescription of the gas and surface mobilities, and of surface affinity via an adsorption isotherm. To explore surface diffusion as a selective mechanism for gas separations (and its dependence on various structural and Received: Revised: Accepted: Published: A

February 10, 2017 March 23, 2017 March 27, 2017 March 27, 2017 DOI: 10.1021/acs.iecr.7b00573 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research

phase. For example, the parallel D∥e and perpendicular D⊥e scalar diffusion coefficients are obtained with B = [1,0,0] and [0,1,0], respectively. The first two terms (ϵDg + (1 − ϵ) HDs) are the volume weighted gas and solid diffusion coefficients. The analogue for surface diffusion is K Diτ,22 for which

physical properties) the continuum model of Albaalbaki and Hill is applied to three-dimensional arrays of randomly overlapping parallel cylindrical-pore and fiber networks. With an impenetrable solid, these structures are permeable when the void space is continuous. In the transverse direction, the surfaces are (i) connected, enabling capillaries to produce the highest possible surface flux,22 and (ii) disconnected, so that fibers maximize the coupling between the gas and surface phases.26 The equation-based modeling and physics-building capabilities of Comsol Multiphysics (Version 4.4) are used to solve the diffusion equations, and the LiveLink Matlab tool within Comsol Multiphysics is used to construct the randomly overlapping parallel structures. Appropriate orders of magnitude for the various model parameters are established according to the diffusion regime, temperature, pressure, surface affinity, relative humidity, porosity, pore size, and diffusion direction. A detailed analysis of the moisture effective diffusivity and surface-diffusion contribution in random structures is presented. Moreover, comparison of capillary and fiber packings is undertaken to understand the influence of pore morphology. Finally, the role of surface diffusion in membrane selectivity for dehumidification of natural gas, air, and carbon dioxide is discussed.

τ = Ω−1

only depends on the surface geometry. For example, for sphere arrays, it can be shown that τ = τδ, where τ = 2As/3 with As = Γi/Ω, the specific surface area.22 Similarly, for diffusion in parallel cylinder arrays, τ = τ⊥ δ + (τ∥ − τ⊥) e∥ e∥, where e∥ is the cylinder director, τ⊥ = As/2 and τ∥ = As. With an impenetrable solid, De can be decomposed into effective gas and surface diffusion coefficients, Deg and Dei, respectively. Therefore, the total effective diffusivity can be written in terms of the diffusivity and tortuosity of each phase as17 De = Deg + Dei =

1 ηg

ηi

i

(4)

−

∫Ω (∇gg·e /⊥) dΩg

and

g

1 τ /⊥ = As Γi

∫Γ (∇igi·e /⊥) dΓi i

(5)

∥

where e is perpendicular to e .

3. DIFFUSION COEFFICIENTS AND ADSORPTION ISOTHERM The gas-phase diffusivity Dg depends on the gas-phase diffusion mechanism. The continuum gas-phase diffusion coefficient Dgo is from the kinetic theory of gases.30 The Knudsen regime is considered in an approximate manner by eliminating surfaceand solid-phase diffusion, and calculating a direction-dependent tortuosity factor22 f (ϵ) ≡ ϵ/η = De /Dgk

(6)

where Dgk is the intrinsic Knudsen gas diffusivity, and η is the gas tortuosity in the absence of surface and solid diffusion. Next, the appropriate gas-phase Knudsen diffusion coefficient Dgk is calculated according to the pore or fiber radius a0, porosity, and diffusion direction as Dgk =

DK′ (ϵ, a0 , ...) f (ϵ)

(7)

where DK′ (ϵ, a0, ...) is the effective Knudsen diffusivity for either randomly positioned parallel capillaries11 and parallel fibers12 without surface or solid diffusion. For capillaries, DK′ is also available from Tomadakis and Sotirchos13 with Kn = λ/(2a0) = 100, where λ is the mean-free path. In this work, λ is from the formula for air: λ = π /8 (μ/u)/ ρp , where μ and ρ are the viscosity and density, p is the pressure, and u = 0.498744, derived from kinetic theory.31 Burganos and Sotirchos11 reported the effective Knudsen diffusivity, made independent of the pore size by scaling

∫Ω ∇gg dΩg + HDs ∫Ω ∇gs dΩs

∫Γ ∇igi dΓi)

/⊥

1 Ωg

⊥

J = −[ϵDg δ + (1 − ϵ)HDsδ + KDiτ ]·B

+ KDi

−1=

/⊥

1

is obtained from the total flux

g

A ϵ Dg + s KDi ηg ηi

where ηg and ηi are the gas and surface tortuosities, respectively. In turn, these tortuosities are related to the gas- and surfaceconcentration disturbances in eq 2 by the following reduced tortuosities:22

(1)

− Ω−1(Dg

(3)

i

2. MICROSCALE THEORY Details of the microscale theory adopted in this paper are provided elsewhere. 21,22 Briefly, there are three mass conservation equations for the gas, solid, and surface phases, and two boundary conditions to model interfacial exchange kinetics between the three domains. The dimensional parameters include diffusion coefficients Dg, Ds and Di for the gas-, solid-, and surface-phases, respectively, equilibrium solid solubility H, adsorption isotherm partition coefficient K, and the microstructural length scale l. These form dimensionless parameters (Πg, Πs, Πi, Πsg, and Πsi) appearing in the dimensionless equations. We adopt an impenetrable solid phase (Ds → 0) with low solubility (H = 0.01), which is appropriate for many porous solids. To compute the effective diffusion coefficient of a tracer in a porous structure using periodic boundary conditions, the tracer conservation equations are written in terms of periodic concentration disturbances gg, gs, and gi for the gas-, solid-, and surface-phases, respectively. These are computed when a periodic unit cell is subjected to a mean gas-phase concentration gradient B in a direction that is parallel or perpendicular to the primary axis of symmetry. The overall effective diffusivity tensor De, defined by J = −De ·B

∫Γ (δ − nn) dΓi

s

(2)

where the integrations are performed over the gas and solid volumes (denoted Ωg and Ωs) and the interfacial surface area (denoted Γi). Here, ϵ is the porosity, δ is the second-rank identity tensor, and ∇i = ∇ − n(n·∇) is the surface-gradient operator with n being the unit normal from the solid to the gas B

DOI: 10.1021/acs.iecr.7b00573 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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where n*g = p*/(kBT) with p* being the saturation pressure at absolute temperature T. In addition, the kinetic exchange coefficient kgi from the gas to the gas−solid interface, which is required to prescribe the nondimensional model parameters, is assumed to have the order of thermal velocity, taken to be kgi = 100 m s−1. For simple-cubic arrays of spherical solids and spherical cavities, Mirbagheri26 showed that the magnitude of kgi is only influential for porous structures with discontinuous surfaces, for which the interfacial exchange fluxes are especially important.

with the Knudsen diffusivity for a cylindrical pore with radius a0,32 D K (a 0 ) =

2 3

8kBT a0 πM

(8)

where kBT is the thermal energy and M is the molecular weight. However, Tomadakis and Sotirchos13 reported an effective Knudsen tortuosity that can be converted to D′K with the use of an intrinsic diffusivity from Bonsanquet’s weighting32 of Dgo and the Knudsen diffusivity for a cylindrical pore with radius equal to the averaged pore radius of the overlapping capillary structure, −ϵ DK (a0̅ ) = D K (a 0 ) (1 − ϵ) ln(1 − ϵ) (9)

4. RANDOM OVERLAPPING PARALLEL PORE AND FIBER STRUCTURES To construct quasi-three-dimensional randomly overlapping parallel structures, a rectangular unit cell with dimensions Lx/l, Ly/l and Lz/l in a Cartesian coordinate system (x, y, z) is considered. Here, Lx, Ly, and Lz are determined by the cylinder radius a0, cylinder aspect ratio γ = L/(2a0) (L is the cylinder length), porosity, and the number of cylinders N. The porosity of an infinite capillary array with uniform pore size is11

Figure 1 compares the scaled effective Knudsen diffusivity D′K/DK(a0) with a0 = λ/200 from the two studies.

2 ϵ = 1 − exp( −πLa ̅ 0)

(13)

where L̅ is the capillary length (with radius a0) per unit volume. Note that the porosity of a fiber structure is equivalent to the solid volume fraction of its inverse capillary structure, so Lx = L ,

Figure 1. Parallel (solid) and perpendicular (open) effective Knudsen gas diffusivity scaled with DK(a0) for randomly positioned parallel capillarypores with a0 = λ/200 from the Monte Carlo studies of Burganos and Sotirchos11 (red) and Tomadakis and Sotirchos13 (blue).

The multilayer adsorption theory of Chen and Yang33 provides the surface diffusion coefficient Di as a function of fractional surface coverage θ, relative humidity 0 < x < 1, temperature T, the specific adsorption-site area a2m, and the ratio of the rate constants between the first adsorbed layer and the higher adsorbed layers α. The Guggenheim−Anderson−de Boer (GAB) isotherm34 gives the fractional surface coverage as ckx (1 − kx)(1 − kx + ckx)

θ am2 xng*

− πN ln(1 − ϵ)

(14)

5. RESULTS AND DISCUSSION 5.1. Gas-Phase Knudsen Diffusion. The tortuosity factors f(ϵ) for randomly overlapping parallel pores and fibers are shown in Figure 3. Recall, that these are used to convert literature correlations of the effective Knudsen diffusivities (obtained without surface or solid diffusion) to an intrinsic Knudsen gas diffusivity that can be used in the continuum computations with surface and solid diffusion. The finiteelement computations (undertaken without surface or solid diffusion) have been used to fit the following empirical formulas. Capillary pores and fibers:

(11)

where the nondimensional parameter k allows the Brunauer− Emmett−Teller (BET) theory35 to better fit experimental data, and c is a dimensionless parameter comparing water−site and water−water affinities. Note that α can be approximated as 1/c when the heat of adsorption and activation energy on the first layer are equal and considerably larger than on higher layers.33 This assumption breaks down as k → 1, because k is a measure of the difference between the heat of adsorption on the first and higher layers.34 Moreover, the adsorption isotherm partition coefficient K in Allbaalbaki and Hill’s theory is prescribed as

K=

Ly = Lz = a0

For the computations reported below, γ = 5 and N = 12 were chosen for computational economy and to expedite the construction of geometries using the LiveLink Matlab tool. Additionally, the microstructural length scale l was chosen, without loss of generality, to be the cylinder radius, that is, l = a0. Next, a line in the x-direction with a random starting point (0, y, z) is prescribed, around which a cylinder of length L/l and radius a0/l is placed. If the starting point is positioned such that a portion of the cylinder lies outside the unit cell, a similar cylinder is copied symmetrically to preserve the unit-cell periodicity. This procedure is repeated until all 12 cylinders are present. By rejecting parts of cylinders that lie outside the unit cell, and combining overlapping objects, a periodic structure is generated with a porosity that is calculated using Comsol. Cross sections of representative geometries comprising monodisperse aligned cylinders are shown in Figure 2.

The gas diffusivity Dg → Dgb at any pore size is furnished by Bonsanquet’s weighting of the continuum and Knudsen diffusion coefficients:32 1 1 1 = + Dgb Dgk Dgo (10)

θ=

and

f (ϵ) ≈ ϵ

(12) C

(15) DOI: 10.1021/acs.iecr.7b00573 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Note that the parallel tortuosity factor for pore and fiber structures [eq 15] is exactly the same as Raleigh’s formula for square arrays of long impenetrable cylinders when aligned parallel to the diffusion direction.36 The transverse void-percolation thresholds, below which the gasphase is discontinuous and f(ϵ) = 0, are obtained as fitting parameters to be ϵ ≈ 0.6226 and 0.2298 for pores and fibers, respectively. Figure 4 shows the scaled intrinsic Knudsen gas diffusivity Dgk/DK(a0) obtained using f(ϵ), and the scaled effective

Figure 2. Cross sections of representative unit cells for unidirectional capillary (top) and fibrous (bottom) structures with ϵ ≈ 0.468 (top) and 0.498 (bottom), and N = 12.

Figure 4. Intrinsic (open) and effective (solid) Knudsen gas diffusivity scaled with DK(a0) parallel (circles) and perpendicular (squares) to pores (top) and fibers (bottom). The solid lines are empirical fits from eq 18−21 for DK′ /DK(a0), and the vertical lines identify the transverse void-percolation thresholds. The solid lines for Dgk/DK(a0) are obtained by dividing the corresponding formulas for DK′ /DK(a0) by those for f(ϵ).

Knudsen diffusion coefficient DK′ /DK(a0) taken from the literature.11,12 The following empirical formulas are fitted to DK′ /DK(a0) using four and five fitting parameters for the longitudinal and transverse diffusivities, respectively:

Figure 3. Tortuosity factor f(ϵ) = De/Dg for capillary-pore (top) and fiber (bottom) arrays without surface or solid diffusion, in the parallel (circles) and perpendicular (squares) directions. Solid lines are empirical fits to the finite-element computations (see the main text for details), and the vertical lines identify (apparent) transverse voidpercolation thresholds.

Capillary pores: DK′ /DK (a0) ≈

Capillary pores, four fitting parameters: ⊥

f (ϵ) ≈

1. 56711.7229 y 0.2726

, (2.5671 − y)1.7229

ϵ − 0.6226 y= 1 − 0.6226

DK′ ⊥ /DK (a0) ≈ (16)

(1.0096 − y)0.9883

,

y=ϵ (18)

0.1809y 0.3765

,

2.3296

(1.1328 − y)

y=

ϵ − 0.6704 1 − 0.6704 (19)

Fibers, two fitting parameters: f ⊥ (ϵ) ≈ y − 0.5562y(1 − y),

0.6879y 0.6549

Fibers: y=

ϵ − 0.2298 1 − 0.2298

DK′ /DK (a0) ≈

(17) D

1.6514y1.3497 (1.0174 − y)1.0509

,

y=ϵ (20)


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Industrial & Engineering Chemistry Research DK′ ⊥ /DK (a0) ≈

1.0776y 2.3463 0.7209

(y − 0.9947)

,

y=

ϵ − 0.2794 1 − 0.2794 (21)

The empirical lines for Dgk/DK(a0) are obtained by dividing the corresponding formulas for D′K/DK(a0) by those for f(ϵ). Here, the transverse void-percolation thresholds are estimated to be ϵ ≈ 0.6704 and 0.2794 for pores and fibers, respectively. Note that these percolation thresholds are obtained from simulations with larger N (e.g., 400 cylinders). The percolation thresholds obtained from Figure 3 are for smaller computational domains with N = 12 cylinders. Therefore, it is possible to realize percolating structures with void fractions smaller than but close to 0.6704 (0.2794). Similarly, the geometries can be nonpercolating with porosities larger than, but close to, 0.6704 (0.2794), as demonstrated in Figure 4 (bottom panel) with ϵ ≈ 0.34. Several theoretical studies have attempted to measure the continuum percolation threshold for various 2D and 3D geometries, such as disks, cylinders, spheres, and cubes, with high accuracy. The transverse void-percolation threshold for our randomly aligned cylinders when L = Lx corresponds to the percolating area-fraction threshold for random circle arrays. Quintanilla and Ziff37 reported the critical area fraction ϕc for percolating circles to be 0.6763475(6), and Mertens and Moore38 improved the accuracy, reporting ϕc = 0.67634831(2). Moreover, the critical area fraction for the inverse structure of circle arrays, representing fibrous structures, is 1− 0.67634831(2) = 0.3236516(9). Therefore, the vertical lines in Figures 6−8, and in Figure 10, adopt these more accurate values to identify the percolation thresholds. 5.2. Gas and Surface Diffusion. For these calculations, Dg according to eq 10 is adopted, so the gas diffusivity varies with ϵ. Additionally, we use c = 9.8 and k = 0.87, which are values for water vapor diffusion in Kraft paper at room temperature.39 Nevertheless, the influence of varying these parameters and the relative humidity x are considered; these and other model parameters for Figures 6−10 are summarized in Table 1. Note that each data point in the figures is from a

Figure 5. Dimensionless specific surface area for capillary-pore (blue) and fiber (red) structures. Solid lines are theoretical relations for infinite multidirectional, parallel-pore and parallel-fiber structures (eqs 22 and 23).

theoretical relations for infinite samples of multidirectional, uniformly sized parallel-pore and parallel-fiber structures obtained using geometrical probability.11,12,40 For fibers, Asa0 is related to the porosity by A sa0 = −2ϵ ln ϵ

(22)

and for pores by A sa0 = −2(1 − ϵ) ln(1 − ϵ)

(23)

According to eq 4, surface diffusion depends on As and ηi. For diffusion parallel to straight pores and fibers, the numerical calculations confirm that 1/ηi = τ/As = 1 (Figure 6, top panel)

Table 1. Model Parameters for Figures 6−10 parameter

value

T p am a0 Πi Πs Πg Πsi Πsg γ N x H c k

25 °C 1 atm 10−10 m 10−7 m ≈10 ≈10−10 ≈105 × [Dg (m2 s−1)] 0.01 0.01 5 12 0.2 0.01 9.8 0.87

Figure 6. Reduced surface 1/ηi − τ/As (top) and gas 1/ηg − 1 (bottom) tortuosity (red), and τ/As (blue) for diffusion parallel (circles) and perpendicular (squares) to pores (solid) and fibers (open). Solid lines are the theory of Albaalbaki and Hill20 (dilute arrays of infinitely long impenetrable solid cylinders), and vertical lines identify the theoretical transverse void-percolation thresholds.

single random realization of a capillary-pore or fibrous structure. Figure 5 shows that porosity affects the scaled specific surface area Asa0. Here, the computations of Asa0 are compared with

and, therefore, there are no concentration disturbances. Thus, the effective longitudinal surface diffusion coefficients D∥ei vary only according to the surface area, as shown in Figure 7 (top, left panel). For transverse diffusion, τ/As = 1/2, and the reduced E


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Figure 7. Scaled effective surface (left) and gas (right) diffusivities, parallel (circles) and perpendicular (squares) to cylindrical pores (solid) and fibers (open). Solid lines are the theory of Albaalbaki and Hill20 for dilute arrays of infinitely long impenetrable solid cylinders, and vertical lines identify the theoretical transverse void-percolation thresholds.

Figure 8. Parallel (solid) and perpendicular (open) surface diffusion fractions in capillary (top) and fibrous (bottom) structures. The vertical lines identify the transverse void-percolation thresholds.

transverse surface tortuosities are negative with magnitudes that are slightly less than 1/2. Similarly, the gas tortuosities are one for diffusion parallel to capillary-pores and fibers resulting in D∥eg/Dgb = ϵ as shown in Figure 7 (top, right panel). The perpendicular reduced gas tortuosities 1/ηg − 1 vary between −0.5 and 0.5 with large fluctuations near the percolation thresholds. The theory of Albaalbaki and Hill20 for a dilute array of infinitely long impenetrable solid cylinders predicts increasing 1/ηg − 1 with decreasing porosity, producing gas tortuosities less than one near the void-percolation threshold, and a minimum for D⊥eg/ Dgb at ϵ ≈ 0.5 (Figure 7, bottom, right panel). This may be because a larger interfacial uptake near the void-percolation threshold enhances the gas flux by increasing the local gasphase concentration gradient. From Figure 8, the surface-diffusion fraction is inversely proportional to ϵ,41 attaining its highest value at the voidpercolation threshold, where gas diffusion attains its minimum. At this critical point, below which both interfacial and void diffusion vanish, the surface flux is higher because Πi/Πg > 1. The surface-diffusion fraction is proportional to the ratio of the surface area to the gas-phase volume Ag = As/ϵ and the ratio Πi/Πg = KDi/(Dgl). Therefore, larger Ag and Πi/Πg at lower void fractions greatly enhance the surface flux. The ratio Πi/Πg increases at lower ϵ according to the manner in which Dg depends on ϵ in the Knudsen regime. The effects of the surface-interaction parameters, c and k, and of the relative humidity x on the scaled effective surface diffusivity for fibers and pores with ϵ ≈ 0.77 are shown in Figure 9. In each panel, the parameters are according to Table 1. As expected, the surface flux increases with surface affinity;3 however, the perpendicular diffusivity is almost insensitive to c at ϵ ≈ 0.77, perhaps because the transverse surface flux is negligible at this porosity according to Figure 7 (bottom, left panel). Similarly, D⊥ei/Dgb is independent of k and x. Recall, the GAB-isotherm constant k accounts for a smaller heat of adsorption on second and higher adsorbed layers.34 Thus,

Figure 9. Parallel (circles) and perpendicular (squares) scaled surface diffusivity versus surface affinity parameter c (left), GAB-isotherm constant k (middle), and relative humidity x for capillary pores with ϵ ≈ 0.768 (solid) and fibers with ϵ ≈ 0.765 (open).

increasing k increases surface adsorption and diffusion. Similarly, with larger moisture contents, adsorption on the higher layers is more significant. The intrinsic gas diffusivity Dgb for fibers is larger than for capillary pores in the parallel and perpendicular directions [Figure 4 and eq 10], so fiber arrays with the same ϵ are more permeable to gas. To compare the selectivities of these structures, we now consider the dehumidification of air, natural gas, and carbon dioxide using the capillary-pore and fiber structures with c = 9.8 and k = 0.87 at various porosities. The difference between the secondary gases (air, natural gas, and carbon dioxide) is their molecular weight and, thus, their intrinsic gas diffusivity. Water vapor diffuses with De = Deg + Dei from Figure 7, whereas the secondary gas can only diffuse though the void space. The effective diffusion coefficient of the nonadsorbing gas is f Dg with Dg prescribed by eq 10 and intrinsic gas-phase Knudsen diffusivity Dgk =

Dgk

2 D K (a 0 ) 3

8kBT a0 πM

(24) −3

Note that Dgk/DK(a0) is from Figure 4, and M = 19 × 10 /NA, 29 × 10−3/NA, and 44 × 10−3/NA kg for natural gas, air, and carbon dioxide, respectively, with NA being the Avogadro number. F


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Our model neglects interactions between the adsorbed phase (moisture) and the void-diffusing species (air, natural gas, and carbon dioxide). For example, Bell and Brown42 showed that a mobile adsorbed-phase can inhibit diffusion in the void space. This is because collisions of the gas-phase molecules with the surface-adsorbed molecules give rise to different transport behavior in the Knudsen regime than if the gas-phase molecules collided with a bare pore wall. Later, they attributed this to a different momentum exchange for the two types of collision.43 Despite neglecting these interactions, the present analysis still provides valuable insight into the relationship between S and ϵ, also distinguishing fibrous networks from capillary structures. 5.3. Comparison with Experimental Data. A comparison of our computations with experimental data from the literature is presented in Table 2. Calculations of specific surface areas are reasonable, considering As ∼ 1/a0, because As for our structures with pore sizes equal to a0 is ∼100 times smaller than for experimental media with pore sizes ∼0.01a0. The substantial influence of As on surface diffusion is evident from the numerical and experimental studies. For example, for methane diffusion in silica−alumina cracking catalysts with different specific surface areas (no. 16 and 17) at the same temperature and porosity, the medium with a larger As has almost twice the surface-diffusion flux. However, the value of As does not solely determine the surface-diffusion contribution to the overall permeability, because the porosity also plays an important role. For example, the surface-diffusion fraction for moisture diffusion in Kraft paper with porosity ϵ ≈ 0.7 (no. 3) is smaller than in a medium with ϵ ≈ 0.4 (no. 1), despite having a larger As. Therefore, a parameter that groups As and ϵ is expedient. One candidate is the ratio of the surface area to the gas-phase volume Ag = As/ϵ, so any medium with a larger Ag produces a higher surface-diffusion contribution at the same temperature. The surface-diffusion fraction is inversely proportional to the temperature, as also reported by Barrer and Gabor.41 For example, CH4 diffusion in glass membranes at T = 19 °C has almost the same surface-diffusion contribution as in silica− alumina cracking catalysts at T = 30 °C, despite the latter having a larger Ag. However, the effect of temperature is only apparent when there is a significant temperature difference. When comparing some of the other pairs (e.g., no. 9 and 10), a smaller difference in T does not compensate for a smaller Ag.

Figure 10. Capillary-pore (open) and fibrous (solid) membrane selectivity for dehumidification of natural gas (blue), air (yellow), and carbon dioxide (red) in the parallel (top) and perpendicular (bottom) directions. Gases are distinguished in the model only by their molecular weights. Vertical lines identify the transverse void-percolation thresholds.

As shown in Figure 10, the selectivity, defined by D S= e fDg

(25)

increases with decreasing porosity, because the effective gasphase diffusivity is hindered, but the surface flux increases at lower void fractions. Similarly, S is larger in the perpendicular direction near the percolation threshold with slower gas diffusion than in the parallel orientation. The membranes are more selective for dehumidification of a heavier gas, because the gas diffusivity decreases with M [eq 24]. Interestingly, capillaries are generally more selective than fibers, perhaps because the surfaces of capillary structures are always connected, giving the largest possible surface flux. Note also that fibers have larger void permeability than capillaries.

Table 2. Surface-Diffusion Fraction from This Study (Numerical) and Experimental (Literature) Studies no.

medium

gas

T (°C)

ϵ

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Kraft paper Kraft paper Kraft paper Kraft paper Kraft paper Kraft paper Kraft paper glass membrane glass membrane carbon membrane glass membrane carbon membrane glass membrane carbon membrane glass membrane silica−alumina cracking catalysts silica−alumina cracking catalysts

H2 O H2 O H2 O H2 O H2 O H2 O H2 O O2 N2 N2 Ar Ar Kr Kr CH4 CH4 CH4

25 25 25 25 25 25 25 19 19 25 19 25 19 25 19 30 30

0.40 0.50 0.70 0.3 0.39 0.5 0.7 0.3 0.3 0.5 0.3 0.5 0.3 0.5 0.3 0.4 0.5

As ×10−6 (m−1) Ag ×10−6 (m−1) pore size (nm) surface-diffusion fraction (%) 5.9 6.5 8 8 7.8 6.5 5.5 210 210 800 210 800 210 800 210 480 400

14.7 13 11.4 26.7 20 13 7.6 700 700 1600 700 1600 700 1600 700 1200 800 G

100 100 100 100 100 100 100 3 3 1.3 3 1.3 3 1.3 3 1.7 2.5

79 78 69 87 82 72 54

(∥) (∥) (∥) (∥) (∥) (∥) (∥) 34 36 59 35 60 40 70 35 36 19

ref this study (pores) this study (pores) this study (pores) this study (fibers) this study (fibers) this study (fibers) this study (fibers) Barrer and Barrie44 Barrer and Barrie44 Aylmore and Barrer45 Barrer and Barrie44 Aylmore and Barrer45 Barrer and Barrie44 Aylmore and Barrer45 Barrer and Barrie44 Barrer and Gabor41 Barrer and Gabor41


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A fair comparison can only be made when considering the influence of the ratio Πi/Πg = KDi/(Dgl) on surface diffusion. For example, the surface-diffusion fraction for moisture transport in a medium with Ag ≈ 13 × 106 (m−1) (no. 2 and 6) is greater than for nitrogen diffusion in a medium with 100 times larger Ag (no. 10) at the same temperature. Surface diffusion equally contributes to the transport of H2O (no. 6) and Kr (no. 14) in media with different Ag at the same temperature. Moreover, from the studies of Barrer and Barrie44 and Aylmore and Barrer,45 N2 and Ar have similar surface-diffusion fractions when diffusing in the same medium. However, Ar and Kr have completely different surface fluxes when diffusing under the same conditions (no. 11 and 13 or no. 12 and 14). Obviously, Πi/Πg for these studies are not reported in the table, because the dimensional parameters within are unknown.

Reghan J. Hill: 0000-0001-9735-0389 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Support from NSERC (Innovative Green Wood Fiber Network and Discovery Grant 262785-13) and McGill University (McGill Engineering Doctoral Award to M.M.) is gratefully acknowledged.

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NOMENCLATURE a0 cylinder radius, m a2m specific adsorption-site area, m2 As specific surface area, 1/m Ag ratio of surface area to gas volume, 1/m B dimensionless average gas-concentration gradient c surface affinity Dj intrinsic diffusivity, m2/s Dgo continuum gas diffusivity, m2/s Dgk Knudsen gas diffusivity, m2/s Dgb gas diffusivity at any pore size, m2/s Deg gas effective diffusivity, m2/s Dei surface effective diffusivity, m2/s De total effective diffusivity, m2/s De total effective diffusivity tensor, m2/s DK′ Knudsen effective diffusivity, m2/s DK(a0) Knudsen diffusivity for a cylindrical pore with radius a0, m2/s DK(a0̅ ) Knudsen diffusivity for a cylindrical pore with radius equal to the averaged pore radius of the overlapping cylinders, m2/s e director/unit vector f ≡ ϵ/η tortuosity factor gj dimensionless periodic concentration disturbance H equilibrium solid solubility J total flux, mol/(m2 s) k GAB-isotherm parameter kB Boltzmann contant, kg m2/(s2 K) kjk equilibrium kinetic exchange coefficient (phase j to k), m/s K adsorption partition coefficient, m Kn Knudsen number l microstructural length scale, m L cylinder length, m L̅ capillary length per unit volume, 1/m2 Lx/y/z unit cell dimensions, m M molecular mass, kg n*g saturation number density, 1/m3 n unit normal vector N number of cylinders NA Avogadro number p pressure, Pa p* saturation pressure, Pa S selectivity T temperature, K x relative humidity

6. CONCLUSIONS Surface diffusion was explored as a selective mechanism for membrane gas separations. Its dependence on various geometrical and physical properties was established by studying the relative gas- and surface-diffusion contributions to the overall permeability of quasi-three-dimensional arrays of randomly overlapping parallel cylindrical pores and fibers. Whereas gas and surface diffusion are coupled, this coupling is negligible for media with continuous surfaces. The gas and surface mobilities, and the adsorption isotherm partition coefficient were prescribed to study the role of surface diffusion in microporous media. The effect of other physical quantities, such as surface affinity, relative humidity, and the GAB-isotherm parameter, were explored. Surfaces with higher affinities, and closer heats of adsorption for the first and higher adsorbed layers, have higher surface-diffusion contributions. Similarly, the higher is the moisture content, the higher is the surface flux. Surface diffusion is mainly controlled by the specific surface area, especially in the parallel direction. The effect of concentration disturbances, even though more considerable in the transverse direction, are very weak compared to the role of surface area. This is in contrast to arrays of solid spheres and spherical cavities22 for which both surface area and concentration disturbances are important. The longitudinal scaled effective gas-phase diffusivity is proportional to porosity with a gas-phase tortuosity equal to 1. The transverse scaled gas diffusivity, having large fluctuations near the void-percolation threshold, generally increases with porosity. The fiber networks with larger intrinsic gas diffusivities are more permeable in the void space than capillary-pore arrays. Pore structures are more selective than fibers for dehumidification of air, natural gas, and carbon dioxide. In general, parallel diffusion is more selective at high porosities, whereas the perpendicular orientation furnishes the highest selectivities near the transverse void-percolation threshold. Selectivity, defined here as the ratio of the effective diffusion coefficient for an adsorbing species to that of a tracer that diffuses only in the void space, is inversely proportional to porosity. A comparison of our computations with experimental data highlights the importance of the ratio of the surface area to the gas-phase volume, temperature, and the ratio Πi/Πg on the relative magnitudes of surface and gas diffusion.

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AUTHOR INFORMATION

α γ Γi

Corresponding Author

*E-mail: [email protected]. Tel.: 514 398 6897. H

GREEK SYMBOLS rate constant ratio cylinder aspect ratio interfacial surface area, m2 DOI: 10.1021/acs.iecr.7b00573 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Article

Industrial & Engineering Chemistry Research δ ϵ ηj η θ λ μ Πj Πjk ρ τ τ ϕc Ω Ωj

■ g i j s

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unit tensor porosity tortuosity gas tortuosity in the absence of surface and solid diffusion fractional surface coverage mean-free path, m viscosity, Pa s dimensionless parameter dimensionless parameter relating kjk to kgi density, kg/m3 interfacial geometry tensor, 1/m scalar interfacial geometry, 1/m critical area fraction total volume, m3 phase volume, m3

SUBSCRIPTS gas surface/interface phase solid

SUPERSCRIPTS ∥ parallel direction ⊥ perpendicular direction REFERENCES

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Diffusion in Randomly Overlapping Parallel Pore and Fiber Networks

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