Determination of the effective gas diffusivity of a porous composite medium from the three-dimensional reconstruction of its microstructure

The present work describes a procedure to calculate the effective diffusivity of a porous composite medium from the three-dimensional reconstruction of its microstructure. We perform Monte Carlo simulations based on the mean-square displacement method on numerical models of composite materials microstructures. First, computations of the effective diffusivity in the bulk diffusion regime account for the effect of the tortuosity of the geometry on gas diffusion. The Bruggeman equation, which is often used in the literature to relate the effective diffusivity to the porosity of the structure, appears to be inaccurate for porosities $\epsilon <0.40$. A more accurate correlation for this range of porosities is provided based on the results of our simulations. Second, the Bosanquet equation, which accounts for the effect of pore confinement on gas diffusion, is validated provided that the definition of the Knudsen number is based on the appropriate characteristic length. The procedure to calculate this characteristic length is demonstrated for analytical geometries. However, in practice, geometries obtained from experimental measurements are discrete. For discrete geometries, we show the effect of the resolution of the geometry on the accuracy of the calculation of the effective diffusivity and other properties of the porous material. In addition, the tesselation of solid surfaces affects the calculation of the chord-length distribution regardless of the resolution. This hinders the accurate estimation of the characteristic length necessary to compute the Knudsen number and the effective diffusivity.

Figure 1

Three-dimensional representation of a model geometry for the monodisperse case. Particles in light gray are of type A, and particles in dark gray are of type B.

Figure 2

Relative error ($3{\sigma }_{\mathit{MC}}$) of simulation results as a function of the number of test particles. Calculations are performed for 20 structures with monosized particle distributions and porosity $\epsilon =0.3$. The error decreases as $1/\sqrt{N}$ (full line).

Figure 3

Effective diffusivity as a function of the porosity for the different cases described in Table . Simulation results are compared to correlations provided by Bruggeman [1] [Eq. (2)] and Maxwell [38] [Eq. (14)]. A least-square fit of the simulation results is also presented: $D_e/D_b=1.6{\epsilon }^2$. Each data point is the average of the results obtained for 20 structures. Error bars are smaller than the size of the markers, and therefore they are not depicted on the figure.

Figure 4

Mean chord length as a function of the porosity for the different cases described in Table . The dashed line is a least-square fit of the calculations based on Refs. [43] and [44]. Each data point is the average of the results obtained for five structures.

Figure 5

Ratio $\langle l^2\rangle /2\langle l\rangle {}^2$ as a function of the porosity for the different cases described in Table . The dashed line corresponds to the expected value for a structure with an exponential chord-length distribution. Each data point is the average of the results obtained for five structures.

Figure 6

Coefficient $\beta$ as a function of the porosity for the different cases described in Table . The dashed line corresponds to the value $\beta =4/13$ predicted by Derjaguin [33]. Each data point is the average of the results obtained for five structures.

Figure 7

Ratio of effective diffusivity and bulk diffusivity as a function of the Knudsen number. Calculations were performed for a geometry with monosized particles and a porosity $\epsilon =0.31$. $ˆ$: $\mathrm{Kn}$ is calculated based on the mean pore size $\langle r\rangle$ given by Cannarozzo et al. [3]. $◊$: $\mathrm{Kn}$ is calculated based on the mean chord length $\langle l\rangle$. $\square$: $\mathrm{Kn}$ is calculated based on the correct characteristic length $d$. The dashed line gives the diffusivity predicted by the Bosanquet equation [Eq. (8)].

Figure 8

Porosity as a function of the resolution. Porosity is normalized by its theoretical value ${\epsilon }_{\text{th}}$.

Figure 9

Surface area as a function of the resolution. Surface area is normalized by its theoretical value $S_{\text{th}}$.

Figure 10

Ratio of effective diffusivity and bulk diffusivity as a function of the resolution. Calculations were performed in the bulk diffusion regime, i.e., $\mathrm{Kn}\to 0$. $D_e/D_b$ is normalized by its theoretical value $(D_e/D_b){}_{\text{th}}=1.6{\epsilon }_{\text{th}}^2$ [Eq. (15)].

Figure 11

Schematic drawing of the effect of discretization on collisions between a gas particle and a solid.

Figure 12

Chord-length distributions measured for an analytical geometry and the corresponding discrete geometry. The geometry is of type polyB1, with a theoretical porosity ${\epsilon }_{\text{th}}=0.303$ and an internal surface area $S_{\text{th}}=13.591$. The discrete grid has $500\times 500\times 500$ elements.