Abstract
A two-scale model for multicomponent gas transport in porous media is developed. At the pore-scale, Stefan–Maxwell formulation is used to describe the multi-gas transport together with the mass and momentum conservation equations, whereas on the solid/fluid interface, a slip velocity according to the Kramers–Kistemaker condition is taken into account. The pore-scale equations are then upscaled using a formal homogenization procedure. The macroscopic model shows that the total average velocity is modified by a slip velocity which depends mostly on the diffusive flux of the light gas in the mixture. Application to hydrogen transport in electrochemical devices such as fuel cell, electrochemical hydrogen purifier/compressor, in which considerable contrast of molar mass between the gases occurs, is carried out. As a result, the gas slip effect can modify considerably the gas transport behavior within the porous medium of the devices. This is an important result because the gas transport mechanisms play a crucial role in their efficiency.
Similar content being viewed by others
References
Auriault, J.-L.: Heterogeneous media: Is an equivalent homogeneous description always possible? Int. J. Eng. Sci. 29, 785–795 (1991)
Auriault, J.-L., Lewandowska, J.: Non-linear diffusion in porous media. C. R. Acad. Sci. Paris 324, 293–298 (1997). Série IIb
Auriault, J.-L., Boutin, C., Geindreau, C.: Homogenization of Coupled Phenomena in Heterogenous Media. Wiley, New York (2009)
Bird, R.B., Stewart, W.E., Lightfoot, E.N.: Transport Phenomena, Second Edition, Chapitre 24 (2002)
Brandon, N.P., Brett, D.J.: Engineering porous materials for fuel cell applications. Philos. Trans. R. Soc. A 364, 147–159 (2006)
Cercignani, C.: The Boltzmann Equation and Its Applications, Applied Mathematical Sciences, vol. 67. Springer, Berlin (1988)
Chapman, S., Cooling, T.G.: The Mathematical Theory of Non-uniform Gases. University Press, Cambridge (1970)
Curtiss, C.F., Bird, R.B.: Multicomponent diffusion. Ind. Eng. Chem. Res. 38, 2515–2522 (1999)
Curtiss, C.F., Bird, R.B.: Additions and corrections. Ind. Eng. Chem. Res. 40, 1791 (2001)
Fu, Y., Jiang, Y., Poizeau, S., Dutta, A., Mohanram, A., Pietras, J.D., Bazant, M.Z.: Multicomponent gas diffusion in porous electrodes. J. Electrochem. Soc. 162, 613–621 (2015)
Graham, T.: On the law of the diffusion of gases. Trans. R. Soc. Edinb. 12(1), 222–258 (1834)
Hirschfelder, J.O., Curtiss, C.F., Bird, R.B.: Molecular Theory of Gases and Liquids, Wiley, Chapitre 11, 1954; corrected printing with notes added (1964)
Jackson, R.: Transport on Porous Catalysts, Chemical Engineering Monographs, vol. 4. Elsevier, Amsterdam (1977)
Kerkhof, P.J.A.M.: A modified Maxwell–Stefan model for transport through inert membranes: the binary friction model. Chem. Eng. J. 64(3), 319–343 (1996)
Kerkhof, P.J.A.M., Geboers, M.A.M., Ptasinski, K.J.: On the isothermal binary mass transport in a single pore. Chem. Eng. J. 83, 107–121 (2001)
Kerkhof, P.J.A.M., Geboers, M.A.M.: Toward a unified theory of isotropic molecular transport phenomena. AIChE J. 51, 79–121 (2005)
Kramers, H.A., Kistemaker, J.: On the slip of a diffusing gas mixture along a wall. Physica 10(8), 699–713 (1943)
Lasseux, D., Valdes-Parada, F.J., Ochoa Tapia, J.A., Goyeau, B.: A macroscopic model for slightly compressible gas slip-flow in homogeneous porous media. Phys. Fluids 26, 053102 (2014)
Mason, E.A., Malinauskas, A.P., Evans, R.B.: Flow and diffusion of gases in porous media. J. Chem. Phys. 46(8), 3199–3216 (1967)
Mason, E.A., Malinauskas, A.P.: Gas transport in porous media: the dusty-gas model. Chem. Eng. Monogr. (1983)
Maxwell, J.C.: On the stresses in rarefied gases arising from inequalities of temperature. Philos. Trans. R. Soc. Lond. 170, 231–256 (1879)
Pisani, L.: Multi-component gas mixture diffusion through porous media: a 1D analytical solution. Int. J. Heat Mass Transf. 51, 650–660 (2008)
Quintard, M., Bletzacker, L., Chenu, D., Whitaker, S.: Nonlinear, multicomponent, mass transport in porous media. Chem. Eng. Sci. 61, 2643–2669 (2006)
Rohland, B., Eberle, K., Ströbel, R., Scholta, J., Garche, J.: Electrochemical hydrogen compressor. Electrochim. Acta 43, 3841–3846 (1998)
Sanchez-Palencia, E.: Non-homogeneous Media and Vibration Theory, Lecture Notes in Physics. Springer, Berlin (1980)
Sdanghi, G., Dillet, J., Didierjean, S., Fierro, V., Maranzana, G.: Feasibility of hydrogen compression in an electrochemical system: focus on water transport mechanisms. Fuel Cells 20, 370–380 (2019)
Skjetne, E., Auriault, J.-L.: Homogeneization of Wall-slip gas flow through porous media. Transp. Porous Med. 36, 293–306 (1999)
Young, J.B., Todd, B.: Modelling of multi-component gas flows in capillaries and porous solids. Int. J. Heat Mass Transf. 48, 5338–5353 (2005)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendices
A Kramers and Kistemaker Relation
Consider a multicomponent gas mixture of N species. The velocity of a molecule of the component i is sum of a mass weighted velocity (\(v_{i}\)) and of a random velocity (\(C_{i}\)) obeying a Maxwell–Boltzmann distribution (\(1 \le i \le N\)) (Chapman 1970; Cercignani 1988). Each mole of species i brings a momentum parallel to the wall \(M_{i} v_{i}^{t}\) where \( v_{i}^{t}\) is the tangential velocity component (parallel to the wall). The molar flux of i molecules impacting the wall per surface and unit time is given for a Maxwell–Boltzmann velocity distribution by \(c_{i} \overline{C}_{i}/4\) where \(c_{i}\) is the molar concentration and \(\overline{C}_{i}\) the average velocity magnitude of the Maxwell–Boltzmann distribution for species i. Therefore, the tangential momentum transferred to the wall per unit area and per unit time is given by (Maxwell 1879; Jackson 1977)
The subscript \(y = \alpha _{i} \ell _{i}\), where y is the distance from the wall, indicates that the quantities must be evaluated at a distance \(\alpha _{i} \ell _{i}\) of the wall (where \(\ell _{i}\) is the mean free path for species i and \(\alpha _{i} \simeq 1\); in Bird et al. (2002), \(\alpha _{i} = 2/3\)) corresponding to the mean position of the last intermolecular collision of the molecules i before their contact with the wall.
Let us consider a diffuse reflection of the molecules at the wall. The previous quantity in (68) represents the net rate of momentum transfer to the wall that can be identified by the shear stress
where \(\mu \) is the dynamic viscosity of the fluid defined as (see Bird et al. 2002)
Hence,
A Taylor development of the left-hand side of the previous equation leads to
or
An evaluation of the different orders of magnitude leads to
where \(v_{m}\) is a macroscopic reference velocity, L a macroscopic length scale corresponding to the pore size and \(\lambda = \max {(\lambda _{i})}\). When \(\lambda \ll L\), a good approximation is
As \(\overline{C}_{i} = \displaystyle \frac{A}{\sqrt{M_{i}}}\) where A is a constant, \(\rho _{i} = \omega _{i} \rho \) and using velocity decomposition \(v_{i}^{t} = v + u_{i}\), it comes
leading to
which is known as the Kramers–Kistemaker slip velocity (Kramers and Kistemaker 1943).
B Graham’s Law for a Binary Gas Mixture
In the simple case of a binary ideal gas mixture with two components A and B, the Stefan–Maxwell relation is written as
Inverting this relation with the condition \(\langle \mathbf {j}_{A} \rangle + \langle \mathbf {j}_{B} \rangle = 0\) yields:
The Darcy velocity is given by
The transport equation can be put into the form
where the mass flux \(\langle \mathbf {n}_{A} \rangle \) in the laboratory frame is given by
An equivalent relation can be written in terms of molar flux \(\langle \mathbf {N}_{A} \rangle \) and molar fraction \(x_{A}\)
where c is the total molar concentration. Permuting A and B with \(\mathbf {D}_{AB}=\mathbf {D}_{BA}\), a similar relation can be obtained for B. Without pressure gradient, the Graham law is recovered:
In order to compare the preceding results with the case without slip effects, the standard Fick relations are recalled. Without slip effect, it yields
and in the absence of pressure gradient, the Fick’s law is recovered
Rights and permissions
About this article
Cite this article
Moyne, C., Le, T.D. & Maranzana, G. Upscaled Model for Multicomponent Gas Transport in Porous Media Incorporating Slip Effect. Transp Porous Med 135, 309–330 (2020). https://doi.org/10.1007/s11242-020-01478-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11242-020-01478-x