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Reports: DNI852520-DNI8: Direct Simulation of Slip and Transitional Flows of Gas in Nanoporous Media using Lattice Boltzmann and DSMC

Xiaolong Yin, PhD, Colorado School of Mines

During Year 1, we derived and built a lattice-Boltzmann slip model that recovers the Maxwell’s first-order slip boundary condition (Maxwell, 1879). Then, by simulating fluid flow in one-dimensional and two-dimensional geometries, the accuracy and the robustness of the slip model were successfully validated. Preliminary results of flow through simple cubic arrays of spheres were also reported.
Our study in Year 2 focused on the parallelization of the lattice-Boltzmann slip model and simulation of fluid flow through three kinds of periodic arrays of spheres including simple cubic arrays (SC), body-centered cubic arrays (BCC) and face-centered cubic arrays of spheres (FCC).
The computational cost of lattice-Boltzmann simulation is relatively high. For example, a SC simulation cell with equal side lengths of 80 and sphere volume fraction of 0.1131 needs about 8 hours to reach the equilibrium state using one processor on MIO, a cluster at Colorado School of Mines. Hence, it is necessary to parallelize the code to speed up the simulations, especially in consideration of future application in large geometries. A parallel scheme dividing the geometry into equal sub-domains based on the number of processors called is implemented for the lattice-Boltzmann model. Fig. 1 depicts the speed up achieved as a function of the number of processors used. As can be seen, the simulation speed of slip flow through an 803 SC array of spheres with sphere volume fraction of 0.1131 keeps increasing nearly linearly with increasing number of processors called.
In the slip flow regime defined by Knudsen number Kn in the range of 0.001~0.1, series of simulations have been conducted for slip flow through three kinds of periodic arrays of spheres. Here, Kn is defined as the ratio of the mean free path λ to the smallest aperture between two adjacent spheres. Additionally, we introduce a slip factor S as a dimensionless form of the slip length (2 – σ) λ / σr where σ is the tangential momentum accommodation coefficient and r is the radius of the sphere making up the periodic arrays. While the Knudsen number gives a measure of the degree of rarefaction in the smallest gap, which provides a limit of the lattice Boltzmann model (the model should not be used when Kn exceeds 0.1), the slip factor S uses the particle size, which, together with the solid volume fraction, are more traditional measures of particle assemblies.
Fig. 1 summarizes the relations between the permeability ratio ks / kns and the slip factor for three different sphere solid volume fractions of 0.0335, 0.1131 and 0.2681 for SC, BCC and FCC arrays. Here, ks is the permeability with slippage, and kns is the one without. It can be seen that the permeability ratios for SC, BCC and FCC arrays of spheres exhibit similar trends. Besides, at any given solid volume fraction f, they are also quantitatively close. Take the BCC array of spheres with volume fraction of 0.1131, for an example: The permeability ratio increases with increasing slip factor and the trend may be fit as a cubic function of S. Given a slip factor of 0.21, the slip permeability is 1.49 times as large as the no-slip permeability or Darcy permeability, which is significant.
Widely used in describing the gas slip flow through porous media, the Klinkenberg’s law (Klinkenberg, 1941) correlates the slip permeability to the no-slip permeability by ks = kns (1 + b / P), where P is the average pressure of core inlet and outlet and b is the Klinkenberg coefficient. As the factor b / P is proportional to the mean free path, the Klinkenberg law implies that ks / kns should be a linear function of S.
As can be observed in Figures 2-4, the data points from lattice-Boltzmann simulations only demonstrated linearity at the highest solid fraction (0.2681). For the other two solid fractions, ks / kns showed nonlinearity that can be fit well by cubic functions ks = kns (aS3 + bS2 + cS + 1), where a, b and c are fitting parameters that are dependent on the solid volume fraction. Note that even for f = 0.2681, there is some curvature. The nonlinearity in S increases with decreasing solid fraction. <p">In summary, with the new lattice-Boltzmann slip model, we have obtained the exact expression of slip permeability in terms of no-slip permeability, slip factor and sphere volume fraction. The smooth curves connecting the simulation data points in Figures 2-4 are drawn based on the new cubic correlation. The data directly supported the hypothesis raised in the proposal that increasing the size ratio between pore bodies and pore throats increases the slip. Here, in the periodic arrays, the size ratio is increased by increasing the solid fraction. In addition, it brings in new data that suggest corrections to the Klinkenberg law, which needs to be further studied.
References
Maxwell, J. C., 1879. On stresses in rarefied gases arising from inequalities of temperature. Philosophical Transaction of the Royal Society of London, 170, 231-256.
Klinkenberg, L. J., 1941. The permeability of porous media to liquids and gases, Drilling and Production Practices, API, 200-213.
Figures
Figure 1: Speed up performance of the parallel lattice-Boltzmann slip model for flow through a SC array of spheres. The size of the cubic simulation cell is 803 with the sphere volume fraction of 0.1131.
Figure 2: Permeability ratio as a function of slip factor for SC arrays of spheres. The three sets of data correspond to three different sphere volume fractions.
Figure 3: Permeability ratio as a function of slip factor for BCC arrays of spheres. The three sets of data correspond to three different sphere volume fractions.
Figure 4: Permeability ratio as a function of slip factor for FCC arrays of spheres. The three sets of data correspond to three different sphere volume fractions.