# A Novel Approach of Unit Conversion in the Lattice Boltzmann Method
## 3 Results and Discussions
### One-Dimensional Examples
#### 3.1.3 1D Melting in a Half-Scale (Phase Change)  
A semi-infinite slab of pure aluminum initially is at the uniform melting temperature (660 \({}^{\circ}\)C). Suddenly, the left side of the slab (at \(x=0\)) is subjected to a constant source of a temperature of 750 \({}^{\circ}\)C at times \(t\geq 0\). As a result, a melting begins from the left side, where the liquid/solid interface moves in the positive x-direction as time progresses. Since the solid phase is at a constant melting temperature, the problem is considered a one-region problem. The temperature distribution in the liquid phase for a given time can be written as Refs. [24; 25]:  
\[T=T_{x=0}+(T_{x=0}-T_{x=L})\frac{erf\left(\frac{x}{2\sqrt{at}}\right)}{erf( \lambda)}, \tag{81}\]  
where \(\lambda\) are solutions to the transcendental equation \(\lambda e^{\lambda^{2}}erf(\lambda)=\frac{St}{\sqrt{\pi}}\). \(St=\frac{C_{p}(T_{x=0}-T_{x=L})}{h_{fg}}\) is the Stefan number, where \(h_{fg}\) is the latent heat of aluminum. Table 6 summarizes the thermo-physical properties of pure aluminum in the SI unit system [15]. Figure 7: Comparison of the LBM and finite difference method (FDM) for different mesh sizes. The LBM solution is achieved at the lattice-scale then converted to the physical-scale. \begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{3}{c}{**Reference Variables in Physical-Scale and Lattice-Scale**} \\ \hline
**Reference Variables** & **Physical-Scale** & **Lattice-Scale** \\ \hline \(m_{o}\) & 1.2 kg & 6.4 \(\times\) 10\({}^{4}\) mu \\ \hline \(x_{o}\) & 1 m & 40 lu \\ \hline \(t_{o}\) & \(1.0\times 10^{3}\) s & 206.97 ts \\ \hline \(\Theta_{o}\) & 373 K & 1 \({}^{*}\) tu \\ \hline \multicolumn{3}{c}{Converting properties between scales} \\ \hline Property & Physical-scale & Dimensionless-scale & Lattice-scale \\ \hline Length (\(L\)) & 1 m & 1 & 40 \({}^{*}\)lu \\ \hline Advection velocity (\(u\)) & 0.001 m/s & 1 \({}^{*}\) & 0.1933 lu.ts\({}^{-1}\) \\ \hline \(\phi_{x=L}\) & 298 K & 0.799 & 0.799 tu \\ \hline \(\phi_{x=0}\) & 373 K & 1 & 1 tu \\ \hline Density (\(\rho\)) & 1.2 kg/m\({}^{3}\) & 1 \({}^{*}\) & 1 \({}^{*}\)mu.lu\({}^{-3}\) \\ \hline Specific heat (\(C_{p}\)) & 1005 J/kg.K & 3.75 \(\times\) 10\({}^{11}\) & 1.4 \(\times\) 10\({}^{10}\)lu\({}^{2}\).ts\({}^{-2}\).tu\({}^{-1}\) \\ \hline Thermal conductivity (\(k\)) & 0.026 W/m.K & 8.1 \(\times\) 10\({}^{9}\) & 2.33 \(\times\) 10\({}^{9}\)mu.lu.ts\({}^{-3}\).tu\({}^{-1}\) \\ \hline Thermal diffusivity (\(\alpha\)) & 2.156 \(\times\) 10\({}^{-5}\) m\({}^{2}\)/s & 0.0216 & 0.1667 lu\({}^{2}\).ts\({}^{-1}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Converting data for the advection-diffusion problem. The values with \({}^{*}\) represent assumed values. For the LBM simulation, [PERSON] et al. 's [26] phase change model is used where the following data are used:  
\[\tau_{e}=1,\text{ and} \tag{82}\]  
\[S_{i}=-w_{i}\frac{h_{f_{S}}}{C_{p}}\frac{\partial f_{l}}{\partial t}\, \tag{83}\]  
where \(f_{l}\) is the volume-phase fraction of the liquid phase and is zero for the solid region and unity for the liquid region. In the present example, the non-dimensional LBM form is used. Therefore, the dimensionless-scale data listed in Table 6 are applied. Figure 8 displays the temperature distribution along the slab for dimensionless time \(t^{*}=1\) (i.e., for 4.6267 \(\times\) 10\({}^{2}\) s). The good agreements between the results can be shown in Figure 8.