Mass conservative lattice Boltzmann scheme for a three-dimensional diffuse interface model with Peng-Robinson equation of state

Peng-Robinson (P-R) equation of state (EOS) has been widely used in the petroleum industry for hydrocarbon fluids. In this work, a three-dimensional diffuse interface model with P-R EOS for two-phase fluid system is solved by the lattice Boltzmann (LB) method. In this diffuse interface model, an Allen-Cahn (A-C) type phase equation with strong nonlinear source term is derived. Using the multiscale Chapman-Enskog analysis, the A-C type phase equation can be recovered from the proposed LB method. Besides, a Lagrange multiplier is introduced based on the mesoscopic character of the LB scheme so that total mass of the hydrocarbon system is preserved. Three-dimensional numerical simulations of realistic hydrocarbon components, such as isobutane and propane, are implemented to illustrate the effectiveness of the proposed mass conservative LB scheme. Numerical results reach a better agreement with laboratory data compared to previous results of two-dimensional numerical simulations.

Figure 1

Two-phase coexistence curve.

Figure 2

Numerical results of $n{C}_4$ at different time points: (a) $t=200$, (b) $t=500$, (c) $t=1000$.

Figure 3

Numerical results of $C_3$ at different time points: (a) $t=100$, (b) $t=500$, (c) $t=1500$, (d) $t=4000$.

Figure 4

3D profile along $Z=L/2$ after convergence ($n{C}_4$ at T=333.28 K): (a) surface tension contribution of Helmholtz free-energy density, (b) cross profile of (a); (c) homogeneous contribution of chemical potential, (d) cross profile of (c); (e) thermal pressure, (f) cross profile of (e).

Figure 5

Energy dissipation and mass conservation of 3D numerical simulation of $n{C}_4$: (a) energy dissipation, (b) total mass variation with time.

Figure 6

(a) velocity field of the $n{C}_4$ droplet at T=333.28 K, (b) time history of the average kinetic energy.

Figure 7

The width of the diffuse interface.

Figure 8

Comparison of surface tension between numerical predictions and laboratory data; (a) $n{C}_4$, (b) $C_3$.

Figure 9

Comparison with Laplace law: (a) $n{C}_4$, (b) $C3$.