Impact of hydraulic tortuosity on microporous and nanoporous media flow

Using two-dimensional porous structures made up of homogeneously arranged solid obstacles, we examine the effects of rarefaction on the hydraulic tortuosity in the slip and early transition flow regimes via extended lattice Boltzmann method. We observed that modification in either the obstacle's arrangement or the porosity led to a power-law relation between the porosity-tortuosity. Along with this, we also found that in the slip-flow regime, the exponent of this relation contains the effect of finite Knudsen number (Kn). In addition, we observed that on properly scaling Kn with porosity and hydraulic tortuosity, a generalized correlation can be obtained for apparent permeability.

Figure 1

Schematic of representative porous media designed as an array of circular obstacle in two dimensions. (a) ${\theta }_0,\varphi =0.75$, (b) ${\theta }_6,\varphi =0.75$, and (c) ${\theta }_6,\varphi =0.90$.

Figure 2

Tortuosity as a function of porosity for various arrangements in porous media. Solid lines are the best fit to $T-1=p_n{(1-\varphi )}^{{\gamma }_n}$, where $n(0,1,...,6)$ defines the alignment.

Figure 3

Steady-state velocity streamlines for porosity, $\varphi =0.75$ and $\varphi =0.90$, porous arrangement, ${\theta }_0$ and ${\theta }_6$, and at Kn = 0.01, Kn = 0.1, and Kn = 1.0.

Figure 4

Tortuosity as a function porosity at pore alignment ${\theta }_1$, ${\theta }_3$, and ${\theta }_6$ with varying Kn. Dashed lines are the best fit to the function $T-1=p_n{(1-\varphi )}^{({\gamma }_n+{\mathrm{f}}_n\left(\mathrm{Kn}\right))}$, where the value of $n(0,1,2,...,6)$ depends on the porous arrangement.

Figure 5

The Kn-dependent addition in the exponent of power-law behavior, $f_n$, for various Kn. Here $n(0,1,...,6)$ describes the alignment of porous arrangement. The inset magnifies the slip-flow regime, which shows that $f\left(\mathrm{Kn}\right)\sim {\mathrm{Kn}}^{0.6}$.

Figure 6

The permeability correction factor (PCF) as a function of Kn at fixed porosity and varying alignments. (a) $\varphi =0.75$, (b) $\varphi =0.83$, and (c) $\varphi =0.90$.

Figure 7

The permeability correction factor (PCF) as a function of Kn at fixed alignment and varying porosity. (a) ${\theta }_0$, (b) ${\theta }_3$, and (c) ${\theta }_6$.

Figure 8

The permeability correction factor (PCF) as a function of an effective ${\mathrm{Kn}}^{★}$ at randomly selected porous media configuration. The first-order correlation by Klinkenberg [Eq. (5)] [12], the second-order correlation given by Beskok and Karniadakis-Civan [Eq. (6)] [36, 37], and the correlation in the slip flow regime suggested by Civan [Eq. (8)] [37] are also presented as a reference.

Figure 9

Unit normals at the solid boundaries.