# Layout Design and Die Casting Using CAE Simulation for Household Appliances
## 2 CAE Simulation of Die-Casting Process
### Numerical Model of Die-Casting Process  
In the die-casting process, the rapid motion of the chamber plunge generates rapid movement and flow of molten metal and charges the molten metal at high pressure in the mold cavity. AnyCasting is designed to analyze the flow of molten metal as a 3D fluid flow with free surfaces and boundaries using a hybrid method. There are three main phenomena, mold filling, solidification and cooling, and stress and strain distribution, to be considered in casting process modeling [17].  
Fluid flow and heat transfer are described by mass balance, momentum balance, and energy conservation laws. According to the geometry of the melt flow path and the material properties, the following governing equations are non-linear. A set of simultaneous and algebraic equations are obtained by linearizing and discretizing them with numerical methods [7; 9].  
Continuity equation (when \(T>T_{S}\)):  
\[\frac{\partial\rho}{\partial t}+\frac{\partial}{\partial x_{j}}\big{(}\rho U_{ j}\big{)}=0 \tag{1}\]  
where \(t\)--time (s), \(x\)--space (m), \(\rho\)--density (Kg/m\({}^{3}\)), \(U\)--velocity (m/s), \(T\)--temperature (C, K), and \(T_{s}\)--solid temperature (C, K).  
Momentum equation (Navier-Stokes, when \(T>T_{S}\)):  
\[\frac{\partial}{\partial t}(\rho U_{i})+\frac{\partial}{\partial x_{j}}\big{(} \rho U_{j}U_{i}\big{)}=\frac{\partial\rho}{\partial x_{i}}+\frac{\partial}{ \partial x_{j}}\Bigg{(}\mu\frac{\partial U_{i}}{\partial x_{j}}\Bigg{)}+\rho g _{i} \tag{2}\]  
where \(t\)--time (s), \(x\)--space (m), \(\rho\)--density (Kg/m\({}^{3}\)), \(u\)--kinematic viscosity (m\({}^{2}\)/s), \(g\)--gravity (Kgf), \(U\)--velocity (m/s), \(T\)--temperature (C, K), and \(T_{s}\)--solid temperature (C, K).  
Volume of Fluid (VOF) for the open surface:  
\[\frac{\partial\text{F}}{\partial t}+U_{j}\frac{\partial\text{F}}{\partial x_ {j}}=0,\ 0\ \leq F\ \leq 1 \tag{3}\]  
where \(t\)--time (s), \(x\)--space (m), \(F\)--volume (m\({}^{3}\)), and \(U\)--velocity (m/s).  
The Fourier heat conduct equation is used for solidification modeling. Since heat flows from high temperature to low temperature, the movement of heat via heat conduction follows [PERSON]'s law [7]. Solidification is a complex problem of phase transformation involving the absorption and release of heat. This feature has nonlinear boundary conditions between different phases, making it impossible to interpret the temperature distribution without including many assumptions (uniform physical properties, contact thermal resistance of phase change, thermal resistance of mold boundary) accurately [18]. The following conservation equations can describe the solidification modeling of molten metal on the condition that the heat balance and distribution of castings are laminar fluid flow and heat transfer processes [19; 20].  
Energy equation:  
\[\frac{\partial}{\partial t}\big{(}\rho C_{p}T\big{)}+\frac{\partial}{\partial x _{j}}\big{(}\rho C_{p}U_{j}T\big{)}=\frac{\partial}{\partial x_{j}}\Bigg{(} \lambda\frac{\partial T}{\partial x_{j}}\Bigg{)}+Q \tag{4}\]  
where \(t\)--time (s), \(x\)--space (m), \(\rho\)--density (Kg/m\({}^{3}\)), \(C_{p}\)--heat capacity (J/K), \(\lambda\)--conductivity (W/m\({}^{2}\)K), \(U\)--velocity (m/s), \(T\)--temperature (C, K), and \(Q\)--heat source (C, K).  
Governing Differential Equation (FDE):  
\[\frac{\partial(\rho\varphi)}{\partial t}+\frac{\partial(\rho U_{j}\varphi)}{ \partial x_{j}}=\frac{\partial}{\partial x_{j}}\Bigg{(}\Gamma_{\varphi}\frac{ \partial\varphi}{\partial x_{j}}\Bigg{)}+S_{\varphi} \tag{5}\]where \(\varphi\)--generated variables which denotes mass, momentum, volume fraction of fluid and energy conservation, \(t\)--time (s), \(x\)--space (m), \(\rho\)--density (Kg/m\({}^{3}\)), \(U\)--velocity (m/s), \(\Gamma_{\varphi}\)--generated diffusion coefficient, and \(S_{\varphi}\)--source term.