Refined energy-conserving dissipative particle dynamics model with temperature-dependent properties and its application in solidification problem

It has been observed previously that the physical behaviors of Schmidt number (Sc) and Prandtl number (Pr) of an energy-conserving dissipative particle dynamics (eDPD) fluid can be reproduced by the temperature-dependent weight function appearing in the dissipative force term. In this paper, we proposed a simple and systematic method to develop the temperature-dependent weight function in order to better reproduce the physical fluid properties. The method was then used to study a variety of phase-change problems involving solidification. The concept of the “mushy” eDPD particle was introduced in order to better capture the temperature profile in the vicinity of the solid-liquid interface, particularly for the case involving high thermal conductivity ratio. Meanwhile, a way to implement the constant temperature boundary condition at the wall was presented. The numerical solutions of one- and two-dimensional solidification problems were then compared with the analytical solutions and/or experimental results and the agreements were promising.

Figure 1

Comparison of Schmidt numbers of water for $0.91<T<1.24$. For the present eDPD simulations, the error bars are obtained by repeating the simulations five times. $r_c=1.96,\gamma =8.0,\rho =4$.

Figure 2

The relationship between $r_c$ and $s$ at different temperatures. Solid polygons: $\gamma =4.5$ and $\rho =4(r_{c,\mathrm{crit}}\sim 2.365)$. Empty polygons: $\gamma =8.0$ and $\rho =4\left(r_{c,\mathrm{crit}}\sim 1.952\right)$. $r_c$ must be chosen in such a way that $r_c>r_{c,\mathrm{crit}}$.

Figure 3

Comparison of Schmidt and Prandtl numbers of water for $0.84<T<1.24$. The error bars of the present eDPD simulations are obtained by repeating the simulation five times. $r_c=1.96,\gamma =8.0,\rho =4$.

Figure 4

Time evolutions of solid-liquid interface Ψ for cases (a) without mushy particle and (b) with mushy particles at different thermal diffusivity ratios (${\alpha }_{SL}$), i.e., 1.0, 2.0, 3.0, and 4.08. As ${\alpha }_{SL}$ increases, the speed $d\mathrm{\Psi }/dt$ increases.

Figure 5

The displacement measurement of solid-liquid interface Ψ($t$). $S$, $L$, and $f$ denote the solid phase, liquid phase, and transition phase (mushy zone), respectively. Each cell (bounded by thick solid line) is divided into five bins in the $y$ direction. The temperature in the transition zone is $T_f$.

Figure 6

The growth of mushy layer thickness as time elapses. ${\alpha }_{SL}=1.0$.

Figure 7

Instantaneous positions of the eDPD particles at (a) $t=70$, (b) $t=140$, and (c) $t=280$ colored by solid fraction ${\phi }_i$. Red: solid phase (${\phi }_i=1.0$). Blue: liquid phase (${\phi }_i=0.0$). eDPD solutions with and without mushy particles are displayed on the left and right columns, respectively. ${\alpha }_{SL}=4.08$.

Figure 8

Temperature distributions along the $y$ direction for ${\alpha }_{SL}=1.0,2.0,3.0$, and 4.08. eDPD method with mushy particle is used. Results at $t=140$ are shown.

Figure 9

Wall boundary condition. Red: wall particles. Green: Imaginary particles (reflection from wall particles about the wall boundary). Empty circle: interior fluid particle. Temperatures of imaginary particles are interpolated from interior fluid particles.

Figure 10

Temperature distributions along the $y$ direction for ${\alpha }_{SL}=4.08$ at different time levels. eDPD methods with and without mushy particles are used. The temperatures are averaged by 15 sample data.

Figure 11

The solidification process in a 2D square domain. The positions of solid-liquid interface at time (a) $t=7$, (b) $t=56$, (c) $t=105$, and (d) $t=210$ are shown. Solid and liquid particles are marked in red and blue colors, respectively.

Figure 12

Comparison of the instantaneous dimensionless positions of mushy particles ($x^{*},y^{*}$) with the analytical solution at different time levels.

Figure 13

Comparison of instantaneous dimensionless positions of mushy particles ($x^{*},y^{*}$) with the analytical solution for (a) $t=28$, (b) $t=56$, (c) $t=105$, (d) $t=126$, (e) $t=210$, and (f) $t=280$. The linear dashed line indicates $y^{*}=x^{*}$. The red solid circle represents the dimensionless position of the square center. For these particular instants, the eDPD solutions do not agree with the analytical solutions due to the semi-infinite assumption made in the latter.

Figure 14

Total number of mushy particles during the 2D solidification process.

Figure 15

Comparison of ${x_o}^{*}$ values obtained from experiments (Jiji et al. [67]) and eDPD for three conditions. Experiment A: $\mathrm{St}=1/33.9$ and $T_o^{*}=1.35(0.104<x_o^{*}<0.125)$; experiment B: $\mathrm{St}=1/22.4$ and $T_o^{*}=0.553(0.134<x_o^{*}<0.146)$; experiment C: $\mathrm{St}=1/19.6$ and $T_o^{*}=0.25(0.149<x_o^{*}<0.154)$.