Investigation of the Klinkenberg effect in a micro/nanoporous medium by direct simulation Monte Carlo method

The pressure-driven gas transport characteristics through a porous medium consisting of arrays of discrete elements is investigated by using the direct simulation Monte Carlo (DSMC) method. Different porous structures are considered, accounting for both two- and three-dimensional arrangements of basic microscale and nanoscale elements. The pore scale flow patterns in the porous medium are obtained, and the Knudsen diffusion in the pores is studied in detail for slip and transition flow regimes. A new effective pore size of the porous medium is defined, which is a function of the porosity, the tortuosity, the contraction factor, and the intrinsic permeability of the porous medium. It is found that the Klinkenberg effect in different porous structures can be fully described by the Knudsen number characterized by the effective pore size. The accuracies of some widely used Klinkenberg correlations are evaluated by the present DSMC results. It is also found that the available correlations for apparent permeability, most of which are derived from simple pipe or channel flows, can still be applicative for more complex porous media flows, by using the effective pore size defined in this study.

Figure 1

Schematics of flow past arrays of two-dimensional square elements.

Figure 2

Velocity distribution and streamlines for $D$ = (a) 1 μm, (b) 400 nm, (c) 100 nm, and (d) 20 nm, $\varepsilon =0.75$.

Figure 3

Distribution of $V_x$ along $Y$ for different (a) locations and (b) element sizes.

Figure 4

Velocity distribution and streamlines for $\varepsilon =$ (a) 0.84, (b) 0.75, (c) 0.556, and (d) 0.36, $D=400\mathrm{nm}$.

Figure 5

Dimensionless intrinsic permeability for various porosities [31, 32, 33, 34].

Figure 6

Variation of $K_a/K_{\infty }$ with $\mathrm{K}{\mathrm{n}}_D$ for various porosities.

Figure 7

Variation of $K_a/K_{\infty }$ with $\mathrm{K}{\mathrm{n}}_p$ for various porosities [8, 9, 10, 12, 23, 24, 35, 36, 37, 41].

Figure 8

Porous structures consisting of two-dimensional arranged (a) circular and (b) rectangular elements, and (c) three-dimensional arranged cubic elements.

Figure 9

Variation of $K_a/K_{\infty }$ with $\mathrm{K}{\mathrm{n}}_p$ for various pore structures [35]: first-order correlation [Eq. (20)], second-order correlation [Eq. (21)].

Figure 10

Parameter map of the present study.

Figure 11

Streamwise variation of the gas pressure for various Mach numbers.