Maxwell-Stefan-theory-based lattice Boltzmann model for diffusion in multicomponent mixtures

The phenomena of diffusion in multicomponent (more than two components) mixtures are universal in both science and engineering, and from the mathematical point of view, they are usually described by the Maxwell-Stefan (MS)-theory-based diffusion equations where the molar average velocity is assumed to be zero. In this paper, we propose a multiple-relaxation-time lattice Boltzmann (LB) model for the mass diffusion in multicomponent mixtures and also perform a Chapman-Enskog analysis to show that the MS continuum equations can be correctly recovered from the developed LB model. In addition, considering the fact that the MS-theory-based diffusion equations are just a diffusion type of partial differential equations, we can also adopt much simpler lattice structures to reduce the computational cost of present LB model. We then conduct some simulations to test this model and find that the results are in good agreement with the previous work. Besides, the reverse diffusion, osmotic diffusion, and diffusion barrier phenomena are also captured. Finally, compared to the kinetic-theory-based LB models for multicomponent gas diffusion, the present model does not include any complicated interpolations, and its collision process can still be implemented locally. Therefore, the advantages of single-component LB method can also be preserved in present LB model.

Figure 1

The profiles of mole fraction ${\xi }_1$ at different time [Solid line: Eq. (68a), $◯$: $t=1.0,\square$: $t=5.0,⊳$: $t=20.0]$.

Figure 2

The profiles of diffusion flux ${\xi }_1$ at different time [Solid line: Eq. (68b), $◯$: $t=1.0,\square$: $t=5.0,⊳$: $t=20.0]$.

Figure 3

The errors of MRT-LB model for mole fraction ${\xi }_1$ at the time $t=5$, the slope of the inserted line is 2.0, which indicates that the MRT-LB model has a second-order convergence rate in space.

Figure 4

The errors of MRT-LB model for mole fraction ${\mathbf{J}}_1$ at the time t = 5, the slope of the inserted line is 2.0, which indicates that the MRT-LB model has a second-order convergence rate in space.

Figure 5

The mole fraction ${\xi }_i(i=1,2)$ at different time $\left(x=0.72\right)$.

Figure 6

The diffusion flux ${\mathbf{J}}_1$ and the negative of mole fraction gradient $\left(-{\partial }_x{\xi }_1\right)$ at different time $\left(x=0.72\right)$.

Figure 7

The diffusion flux ${\mathbf{J}}_2$ and the negative of mole fraction gradient $\left(-{\partial }_x{\xi }_2\right)$ at different time $\left(x=0.72\right)$.

Figure 8

The distributions of mole fraction ${\xi }_1$, diffusion flux ${\mathbf{J}}_1$ and negative of mole fraction gradient $\left(-{\partial }_x{\xi }_1\right)$ at different time.

Figure 9

The distributions of mole fraction ${\xi }_2$, diffusion flux ${\mathbf{J}}_2$ and negative of mole fraction gradient $\left(-{\partial }_x{\xi }_2\right)$ at different time.

Figure 10

Schematic of the three-component diffusion in the Loschmidt tube.

Figure 11

The average mole fraction ${\overline{\xi }}_1$ at different time (Solid and dashed lines: Present results; $•$ and $◯$: Experimental data [1]; $\blacksquare$ and $\square$: Linearized theory [1]; h: hour).

Figure 12

The average mole fraction ${\overline{\xi }}_2$ at different time (Solid and dashed lines: Present results; $•$ and $◯$: Experimental data [1]; $\blacksquare$ and $\square$: Linearized theory [1]).

Figure 13

The profiles of mole fraction $\overline{{\xi }_1}$ along $y$ direction.

Figure 14

The profiles of mole fraction $\overline{{\xi }_2}$ along $y$ direction.