Upscaled Model for Multicomponent Gas Transport in Porous Media Incorporating Slip Effect

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.

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)

$$\begin{aligned} \displaystyle \frac{1}{4} \sum _{i} \left. c_{i} \overline{C}_{i} M_{i} v_{i}^{t}\right| _{y = \alpha _{i} \ell _{i}} = \displaystyle \frac{1}{4} \sum _{i} \left. \rho _{i} \overline{C}_{i} v_{i}^{t}\right| _{y = \alpha _{i} \ell _{i}} \end{aligned}$$

(68)

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

$$\begin{aligned} \displaystyle \frac{1}{4} \sum _{i} \left. \rho _{i} \overline{C}_{i} v_{i}^{t}\right| _{y_{i}=\alpha _{i} \ell _{i}} = \mu \left. \displaystyle \frac{\mathrm{d}v^{t}}{\mathrm{d}y} \right| _{y=0} \end{aligned}$$

(69)

where $\mu$ is the dynamic viscosity of the fluid defined as (see Bird et al. 2002)

$$\begin{aligned} \mu = \displaystyle \frac{1}{2} \sum _{i} \rho _{i} \overline{C}_{i} \alpha _{i} \lambda _{i} \end{aligned}$$

(70)

Hence,

$$\begin{aligned} \displaystyle \frac{1}{4} \sum _{i} \left. \rho _{i} \overline{C}_{i} v_{i}^{t}\right| _{y_{i} = \alpha _{i}\ell _{i}} = \displaystyle \frac{1}{2}\sum _{i} \rho _{i} \overline{C}_{i} \alpha _{i} \lambda _{i} \left. \displaystyle \frac{\mathrm{d}v^{t}}{\mathrm{d}y} \right| _{y=0} \end{aligned}$$

(71)

A Taylor development of the left-hand side of the previous equation leads to

$$\begin{aligned} \displaystyle \frac{1}{4} \left[ \sum _{i} \rho _{i} \overline{C}_{i} \left( \left. v_{i}^{t}\right| _{y =0} + \alpha _{i} \lambda _{i} \left. \displaystyle \frac{\mathrm{d}v_{i}^{t}}{\mathrm{d}y}\right| _{y=0} \right) \right] = \displaystyle \frac{1}{2}\sum _{i} \rho _{i} \overline{C}_{i} \alpha _{i} \lambda _{i} \left. \displaystyle \frac{\mathrm{d}v^{t}}{\ d y}\right| _{y=0} \end{aligned}$$

(72)

or

$$\begin{aligned} \displaystyle \frac{1}{4} \sum _{i} \rho _{i} \overline{C}_{i} \left. v_{i}^{t}\right| _{y =0} = \displaystyle \frac{1}{4} \left[ \sum _{i} \rho _{i} \overline{C}_{i} \alpha _{i} \lambda _{i} \left( 2\left. \displaystyle \frac{\mathrm{d}v^{t}}{\ d y}\right| _{y=0} - \left. \displaystyle \frac{\mathrm{d}v_{i}^{t}}{\mathrm{d}y}\right| _{y=0} \right) \right] \end{aligned}$$

(73)

An evaluation of the different orders of magnitude leads to

$$\begin{aligned} \sum _{i} \left. \rho _{i} \overline{C}_{i} v_{i}^{t}\right| _{y=0} \simeq \left( \sum _{i} \rho _{i} \overline{C}_{i} \alpha _{i} \lambda _{i} \right) \left. \displaystyle \frac{\mathrm{d}v^{t}}{\mathrm{d}y}\right| _{y=0} = \mathscr {O}\left[ \left( \sum _{i} \rho _{i} \overline{C}_{i} \alpha _{i} \right) v_{m} \displaystyle \frac{\lambda }{L} \right] \end{aligned}$$

(74)

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

$$\begin{aligned} \sum _{i} \left. \rho _{i} \overline{C}_{i} v_{i}^{t}\right| _{y=0} = 0 \end{aligned}$$

(75)

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

$$\begin{aligned} \sum _{i} \displaystyle \frac{\omega _{i}}{\sqrt{M_{i}}} \left. \left( v + u_{i}\right) \right| _{y=0} = 0 \end{aligned}$$

(76)

leading to

$$\begin{aligned} \left. v \right| _{y=0} = u_{s} = - \displaystyle \frac{\displaystyle \sum \nolimits _{i} \omega _{i} \left. M_{i}^{-1/2} u_{i} \right| _{y=0}}{\displaystyle \sum \nolimits _{i} \omega _{i} M_{i}^{-1/2}} = - \displaystyle \frac{\displaystyle \sum \nolimits _{i} \omega _{i} \left. M_{i}^{-1/2} (v_{i}^t - v) \right| _{y=0}}{\displaystyle \sum \nolimits _{i} \omega _{i} M_{i}^{-1/2} } \end{aligned}$$

(77)

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

$$\begin{aligned} {{\varvec{\nabla }}}x_{A} + \left( x_{A} - \omega _{A} \right) \displaystyle \frac{{{\varvec{\nabla }}}p}{p} = x_{A} x_{B} \mathbf {D}_{AB}^{-1} \left( \displaystyle \frac{\langle \mathbf {j}_{B} \rangle }{\rho _{B}} - \displaystyle \frac{\langle \mathbf {j}_{A} \rangle }{\rho _{A}} \right) \end{aligned}$$

(78)

Inverting this relation with the condition $\langle \mathbf {j}_{A} \rangle + \langle \mathbf {j}_{B} \rangle = 0$ yields:

$$\begin{aligned} \left\langle \mathbf {j}_{A} \right\rangle = - \left\langle \mathbf {j}_{B} \right\rangle = - \rho \mathbf {D}_{AB} \left[ {{\varvec{\nabla }}}\omega _{A} + \displaystyle \frac{M_{A} M_{B}}{M^{2}} \left( x_{A} - \omega _{A} \right) \displaystyle \frac{{{\varvec{\nabla }}}p}{p} \right] \end{aligned}$$

(79)

The Darcy velocity is given by

$$\begin{aligned} \left\langle \mathbf {v}\right\rangle = - \displaystyle \frac{\mathbf {K}}{\mu } \cdot {{\varvec{\nabla }}}p - \displaystyle \frac{1}{\rho } \displaystyle \frac{M_{A}^{-1/2} - M_{B}^{-1/2}}{\omega _{A} M_{A}^{-1/2} + \omega _{B} M_{B}^{-1/2}} \left\langle \mathbf {j}_{A}\right\rangle \end{aligned}$$

(80)

The transport equation can be put into the form

$$\begin{aligned} n_{f} \rho \displaystyle \frac{\partial \omega _{A}}{\partial t} = - {{\varvec{\nabla }}}\cdot \langle \mathbf {n}_{A} \rangle \end{aligned}$$

(81)

where the mass flux $\langle \mathbf {n}_{A} \rangle$ in the laboratory frame is given by

$$\begin{aligned} \left\langle \mathbf {n}_{A} \right\rangle= & {} \omega _{A} \rho \langle \mathbf {v}\rangle + \left\langle \mathbf {j}_{A}\right\rangle \nonumber \\= & {} - \rho \omega _{A} \left( \displaystyle \frac{\mathbf {K}}{\mu } + \displaystyle \frac{\sqrt{M_{A}}}{\omega _{A} \sqrt{M_{B}} + \omega _{B} \sqrt{M_{A}}} \displaystyle \frac{M_{A} - M_{B}}{M} \displaystyle \frac{\mathbf {D}_{AB}}{p} \right) \cdot {{\varvec{\nabla }}}p \nonumber \\&- \displaystyle \frac{\sqrt{M_{A}}}{\omega _{A} \sqrt{M_{B}} + \omega _{B} \sqrt{M_{A}}} \rho \mathbf {D}_{AB} \cdot {{\varvec{\nabla }}}\omega _{A} \end{aligned}$$

(82)

An equivalent relation can be written in terms of molar flux $\langle \mathbf {N}_{A} \rangle$ and molar fraction $x_{A}$

$$\begin{aligned} \left\langle \mathbf {N}_{A} \right\rangle = \displaystyle \frac{\left\langle \mathbf {n}_{A} \right\rangle }{M_{A}}= & {} - c \, x_{A} \left( \displaystyle \frac{\mathbf {K}}{\mu } + \displaystyle \frac{M_{A} - M_{B}}{\sqrt{M_{B}}} \displaystyle \frac{1}{x_{A} \sqrt{M_{A}} + x_{B} \sqrt{M_{B}}} \displaystyle \frac{\mathbf {D}_{AB}}{p} \right) \cdot {{\varvec{\nabla }}}p \nonumber \\&- \displaystyle \frac{\sqrt{M_{B}}}{x_{A} \sqrt{M_{A}} + x_{B} \sqrt{M_{B}}} c \mathbf {D}_{AB} \cdot {{\varvec{\nabla }}}x_{A} \end{aligned}$$

(83)

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:

$$\begin{aligned} \sqrt{M_{A}} \, \left\langle \mathbf {N}_{A} \right\rangle + \sqrt{M_{B}} \, \left\langle \mathbf {N}_{B} \right\rangle = 0 \end{aligned}$$

(84)

In order to compare the preceding results with the case without slip effects, the standard Fick relations are recalled. Without slip effect, it yields

$$\begin{aligned} \left\langle \mathbf {n}_{A} \right\rangle= & {} - \omega _{A} \rho \displaystyle \frac{\mathbf {K}}{\mu } \cdot {{\varvec{\nabla }}}p - \rho \mathbf {D}_{AB} \cdot \left( {{\varvec{\nabla }}}\omega _{A} + \omega _{A} \omega _{B} \displaystyle \frac{M_{B} - M_{A}}{M} \displaystyle \frac{{{\varvec{\nabla }}}p}{p}\right) \end{aligned}$$

(85)

$$\begin{aligned} \left\langle \mathbf {N}_{A} \right\rangle= & {} - x_{A} c \left( \displaystyle \frac{\mathbf {K}}{\mu } + x_{B} \displaystyle \frac{M_{B} (M_{B} - M_{A})}{M^{2}} \displaystyle \frac{\mathbf {D}_{AB}}{p} \right) \cdot {{\varvec{\nabla }}}p - c \displaystyle \frac{M_{B}}{M} \mathbf {D}_{AB} \cdot {{\varvec{\nabla }}}x_{A} \end{aligned}$$

(86)

and in the absence of pressure gradient, the Fick’s law is recovered

$$\begin{aligned} M_{A} \left\langle \mathbf {N}_{A} \right\rangle + M_{B} \left\langle \mathbf {N}_{B} \right\rangle = 0 \end{aligned}$$

(87)