Multipseudopotential interaction models for thermal lattice Boltzmann method simulations

In this work, in the first instance, the multipseudopotential interaction (MPI) model's capabilities are extended for hydrodynamic simulations. This is achieved by combining MPI with the multiple-relaxation-time collision operator and with surface tension modification methods. A method of approaching thermodynamic consistency is also proposed, which consists of splitting the ${\varepsilon }_j$ term into separate terms. One of these terms is used in the calculation of the interparticle force, and the second one is used in the forcing scheme. Secondly, MPI is combined with thermal models in order to simulate droplet evaporation and bubble nucleation in pool boiling. Thermal coupling is implemented using a double distribution function thermal model and a hybrid thermal model. It is found that MPI thermal models obey the ${\mathrm{D}}^2$-law closely for droplet evaporation. MPI is also found to correctly simulate bubble nucleation and departure from the heating element during nucleate pool boiling. It can be suggested that MPI thermal models are comparatively better suited to thermal simulations at low reduced temperatures than single pseudopotential interaction models, although such cases remain very challenging. Droplet evaporation simulations are carried out at a reduced temperature $\left(T_r\right)$ of 0.6 by setting the parameters in the Peng-Robinson equation of state to $a=1/6272$ and $b=1/168$.

Figure 1

Deviation of MPI from the thermodynamic coexistence densities when the SRT and MRT collision operators are used and the performance of the modification for a broad temperature range.

Figure 2

Interparticle force generated by MPI and YS methods at the same density ratio. Carnahan-Starling $a$ parameter was set to 0.01, $b$ was set to 0.2, and $R$ was set to 1. The value of ${\varepsilon }_{\mathrm{Forcing},j}$ in MPI was set to $0.89{\varepsilon }_{\mathrm{EOS},j}$ and the values of $k_1$ and $k_2$ in the Huang-Wu forcing scheme for the SPI model were set to 0.0 and −0.2188, respectively.

Figure 3

Spurious velocity magnitude generated by the MPI and YS methods at $T_r=0.42$ (${\tau }_v=1$ and $\mathrm{\Lambda }=1/12$).

Figure 4

Spatial distribution of the Laplacian operator of temperature calculated using the second-order isotropic difference scheme and the isotropic discrete scheme for the case of droplet evaporation at time steps equal to 13 000, 25 000, and 38 000.

Figure 5

Density field on the left-hand side, temperature field in the middle, and pressure field on the right-hand side to illustrate the results of unmodified thermal MPI in bubble nucleation simulations. The black tube in the liquid phase is used as the heating element.

Figure 6

Verification of the ${\mathrm{D}}^2$-law during droplet evaporation for the unmodified and corrected thermal MPI. The initial droplet diameter $\left(D_0\right)$ was 60 lattice units. Reduced temperature was set to 0.86 and the superheat with respect to the saturation temperature was set to 0.14 times the critical temperature. Thermal conductivity was set to 2/3 and heat capacity at constant volume was set to 5. Acentric factor was set to 0.344, PR EOS $a$ to 3/49, $b$ to 2/21. Comparison to data in Ref. [33].

Figure 7

Comparison of droplet evaporation rates of YS and MPI thermal models. Reference [33] data plotted for validation purposes. The initial droplet diameter $\left(D_0\right)$ was 60 lattice units. $T_r=0.86$, initial droplet diameter $=60$, superheat $=0.14T_c, c_v=5$, thermal conductivity $=2/3$, acentric factor $=0.344$, PR $a=3/49$, and PR $b=2/21$.

Figure 8

Comparison of bubble nucleation and departure from a heating element. The image on the left-hand side was obtained using the YS DDF model, and the image on the right-hand side was obtained using the MPI DDF model.

Figure 9

Effect of reducing surface tension on the departure of a nucleated bubble from the heating element. Surface tension decreases from the right-hand side coefficient value of 1.0, through 0.6 and 0.2, down to 0.1 on the left-hand side.

Figure 10

Effect of increasing surface tension on the departure of a nucleated bubble. Surface tension coefficient increases from 1.0 on the left-hand side, though 1.4 and 1.8, up to 1.9 on the right-hand side.

Figure 11

Amount of time steps taken for the nucleated bubble to break free from the heating element at different values of surface tension.

Figure 12

Semi-log trend of the Weber number of the departing bubbles at different values of the surface tension coefficient.

Figure 13

Comparison of the evaporation rates of different models at a reduced temperature of 0.86. Reference [33] data included for comparison. $D_0$ was set to 60 lattice units.

Figure 14

Comparison of the evaporation rates of different models. The reduced temperature was equal to 0.8, and $D_0$ was equal to 60 lattice units.

Figure 15

Results of droplet evaporation simulations at a range of reduced temperatures. Simulations at lower reduced temperatures were made possible by lowering of the $a$ and $b$ parameters in the Peng-Robinson EOS. As in the previous cases, $D_0=60$.

Figure 16

Details of the effect of adjusting the Peng-Robinson EOS parameters on droplet evaporation at $T_r=0.6$ with $D_0=60$.

Figure 17

The effect of increasing bulk viscosity as a multiple of kinematic viscosity on the rate of droplet evaporation. Comparison to data in Ref. [33]. The initial droplet diameter $\left(D_0\right)$ was 60 lattice units.

Figure 18

The influence of modifying ε in the Huang-Wu scheme [36] for the adjustment of thermodynamic consistency. $k_1$ was set to 0 in order to avoid modifying surface tension, and $k_2$ was varied in order to change the value of ε. Evaporation rates are compared to data in Ref. [33] using the same initial droplet diameter of 60 lattice units.