# Lattice Structure for Improving Cooling Uniformity in HPDC Mould Corners
## 3 CFD and Thermal Model  
The steady-state flow model was developed to simulate the coolant flow within the pipe for the aluminium die-casting process in this study. The cast part was modelled as a heat source, with volumetric heat generation evaluated based on the solidification energy throughout the entire casting cycle. The Reynolds-averaged Navier-Stokes (RANS) was employed to solve for the coolant fluid dynamics in the channels. The governing equations for fluid flow--mass, momentum, and energy--are expressed as follows:  
\[\
abla\cdot(\rho\
abla)=0 \tag{1}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Materials**} & **Density, \(\rho\)** & **Specific Heat** & **Thermal** \\  & **(kg/m3)** & **Capacity, \(C_{p}\) (J/kg K)** & **Conductivity,** \\  & & **k (W/mK)** \\ \hline AlSi9 Cu3(Fe)-Cast & 2760 & 963 & 130 \\ \hline H13 Steel (Mould) & 7750 & 460 & 24.7 \\ \hline Water & 997.561 & 4182 & 0.62 \\ \hline BCC Lattice & 7750 & 460 & 1.54 \\ \hline FCCZ Lattice & 7750 & 460 & 5.22 \\ \hline FBCCZ Lattice & 7750 & 460 & 6.29 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The physical properties of the three regions and the lattice structures [15; 20]. Figure 2: Polyhedral mesh distribution in the three regions. \[\
abla\cdot(\rho\
abla\times\
abla)=-\
abla\cdot\overline{p}\Pi \tag{2}\]  
\[\
abla\cdot(\rho\overline{E}\
abla)=-\
abla\cdot\overline{p}\,\
abla-\
abla\cdot \overline{q} \tag{3}\]  
In these equations, \(\rho\) represents the fluid density, \(\overline{v}\) denotes the mean velocity, \(\overline{p}\) is the pressure, \(\overline{E}\) signifies the total energy per unit mass, and \(\overline{q}\) represents the heat flux. \(I\) is the identity tensor. A shear-stress transport (SST) k\(-\omega\) turbulence model [21] was employed to model the additional turbulent shear stress. This model offers improved predictions for flow separation and positive pressure gradients and mitigates sensitivity to free-stream and inlet conditions [21]. The initial temperature of the cast was set to 750 \({}^{\circ}\)C. The coolant water was admitted at a constant volume flow rate of 5 L/min and a temperature of 70 \({}^{\circ}\)C, with a boundary pressure outlet at the pipe exit. The interface between the cast part and the mould was assumed as a thermal resistance corresponding to an interface heat transfer coefficient of 500 W/m\({}^{2}\)-K. These conditions were selected to align with previously reported experimental results [18]. The front and back surfaces of the mould and cast are defined as periodic boundaries, while the remaining boundary surfaces are defined as adiabatic. The cooling process involves extracting heat from the cast until the aluminium alloy material transfers from liquid to solid. This phase change alters the material properties, complicating the analysis. To simplify, the aluminium alloy cast is modelled as a solid with a constant density \(\rho\) with a constant volumetric heat generation \(q_{v}\) equivalent to the heat required to cool the aluminium from the injection temperature \(T_{melt}\) to the assumed demoulding or ejection temperature. \(T_{demould}\) is within the moulding time \(t\). Therefore, the volumetric heat generated by the cast can be evaluated by the following equation:  
\[\dot{q_{v}}=\frac{\rho\left[h_{f_{s}}+Cp_{Al,s}(T_{s}-T_{demould})+Cp_{Al,l}(T_ {melt}-T_{L})\right]}{t} \tag{4}\]  
The temperatures \(T_{S}\) and \(T_{L}\) represent the solidus and liquidus temperatures of the aluminium alloy, respectively. \(Cp_{Al,s}\) and \(Cp_{Al,l}\) denote the heat capacities of the solid and liquid phases of the aluminium alloy, respectively. Table 2 provides the values used in the model. The assumption of steady-state heat transfer does not affect the validity of the results or objectives of this study, as it effectively represents a single time step, with all subsequent time steps exhibiting similar behaviour but with different magnitudes. Additionally, the focus of this study is on comparative analysis.