Fugacity-based lattice Boltzmann method for multicomponent multiphase systems

The free-energy model can extend the lattice Boltzmann method to multiphase systems. However, there is a lack of models capable of simulating multicomponent multiphase fluids with partial miscibility. In addition, existing models cannot be generalized to honor thermodynamic information provided by any multicomponent equation of state of choice. In this paper, we introduce a free-energy lattice Boltzmann model where the forcing term is determined by the fugacity of the species, the thermodynamic property that connects species partial pressure to chemical potential calculations. By doing so, we are able to carry out multicomponent multiphase simulations of partially miscible fluids and generalize the methodology for use with any multicomponent equation of state of interest. We test this fugacity-based lattice Boltzmann method for the cases of vapor-liquid equilibrium for two- and three-component mixtures in various temperature and pressure conditions. We demonstrate that the model is able to reliably reproduce phase densities and compositions as predicted by multicomponent thermodynamics and can reproduce different characteristic pressure-composition and temperature-composition envelopes with a high degree of accuracy. We also demonstrate that the model can offer accurate predictions under dynamic conditions.

Figure 1

Left: An immiscible system of a hydrocarbon phase composed of the components ${\mathrm{CH}}_4$, ${\mathrm{C}}_2{\mathrm{H}}_6$, ${\mathrm{C}}_3{\mathrm{H}}_8$ among others and an aqueous phase composed of the component ${\mathrm{H}}_2\mathrm{O}$. The components are restricted to their phases and there is no interfacial mass transfer. Right: A partially miscible system of hydrocarbons with a light vapor phase and dense liquid phase. Both phases are composed of the same components (${\mathrm{CH}}_4$, ${\mathrm{C}}_2{\mathrm{H}}_6$, ${\mathrm{C}}_3{\mathrm{H}}_8$ among others) and there is interfacial mass transfer.

Figure 2

The equilibrium profiles for (a) density, (b) composition of C3, and (c) composition of C5 using the PR EOS. The solid black line shows the results from LBM and the dashed blue and red lines show the theoretical values from a flash calculation for the liquid and vapor phase, respectively.

Figure 3

The equilibrium profiles for (a) density, (b) composition of C3, and (c) composition of C5 using the SRK EOS. The solid black line shows the results from LBM and the dashed blue and red lines show the theoretical values from a flash calculation for the liquid and vapor phase, respectively.

Figure 4

(a) The pressure-temperature envelopes for the C2-C5 mixture at different compositions of C2. The solid black lines represent the pure species with the line on the right representing pure C5 and the line on the left representing pure C2. (b) The black lines are the pure component curves and critical locus from (a), the blue lines represent the relevant isobars and the red lines represent the relevant isotherms.

Figure 5

Left: The $p\text{−}x$ envelopes at (a) $T=274.96$ K and (c) $T=387.70$ K, where the solid lines represent the theoretical results and the dots represent the results predicted by LBM. Right: The relative error (%) in the predicted values at (b) $T=274.96$ K and (d) $T=387.70$ K.

Figure 6

Left: The $T\text{−}x$ envelopes at (a) $p=30$ bar, (c) $p=45$ bar, and (e) $p=58$ bar, where the solid lines represent the theoretical results and the dots represent the results predicted by LBM. Right: The relative error (%) in the predicted values at (b) $p=30$ bar, (d) $p=45$ bar, and (f) $p=58$ bar.

Figure 7

The equilibrium profiles for the (a) density, (b) composition of C1, (c) composition of C2, and (d) composition of C3. The solid black line shows the results from LBM and the dashed blue and red lines show the theoretical values from a flash calculation for the liquid and vapor phase, respectively.

Figure 8

A uniform system separating into different phases shown at time (in lattice units) (a) $10000$, (b) $20000$, (c) $30000$, (d) $50000$, (e) $100000$, and (f) $500000$. The points marked A and B, in (f), represent the points in the liquid and vapor region, respectively, where relevant properties are measured.

Figure 9

The ratio of the droplet radius to the equilibrium radius versus time (in lattice units). The dots represent the raw data obtained from the simulation and the solid lines are the smooth curves fit to this data.

Figure 10

The liquid density vs pressure (blue) and vapor density vs pressure (red) at (a) $T=274.96$ K and (b) $T=387.70$ K. The solid lines represent the theoretical results, whereas the dots represent the results predicted by LBM.

Figure 11

The liquid density vs temperature (blue) and vapor density vs temperature (red) at (a) $p=30$ bar, (b) $p=45$ bar, and (c) $p=58$ bar. The solid lines represent the theoretical results whereas the dots represent the results predicted by LBM.