# A New Ice Accretion Model for Aircraft Icing Based on Phase-Field Method
## 2 Materials and Methods
### Two-Dimensional Form of Governing Equation  
Aircraft icing is a special fluid-solid phase change phenomenon. However, both the Messinger model and the shallow water film model only study the continuous water film on the surface. In the phase-field icing model, the mixture of water and ice is taken as the research object. Therefore, the phase equation, the mass equation and the energy conservation equation of the mixture need to be established. By solving these governing equations, the mixture thickness and the mass fraction of ice in each grid are predicted. On this basis, the ice thickness and the ice shape can be obtained. In the control volume \(V(t)\), the mass of the mixture, water phase and ice phase are denoted by \(M\), \(M_{w}\) and \(M_{i}\), respectively. Then, we obtain [25]:  
\[M=M_{w}+M_{i},\rho V=\rho_{w}V_{w}+\rho_{i}V_{i}., \tag{1}\]  
Among them, \(V\), \(V_{w}\) and \(V_{i}\) are the total volume of mixture, the volume of water phase and the volume of ice phase in the control volume, respectively. Their expressions are as follows:  
\[V=s\cdot h,V_{w}=s\cdot h_{w},V_{i}=s\cdot h_{i}, \tag{2}\]  
where \(h\) is the average normal height of the mixture over the control volume, \(h_{w}\) is the average normal height of water phase and \(h_{i}\) is the average normal height of ice phase, \(h=h_{w}+h_{i}\). By combining Equations (1) and (2), we can obtain:  
\[\rho h=\rho_{w}h_{w}+\rho_{i}h_{i}, \tag{3}\]  
Figure 1: Diagram of ice model simplification: (**a**) the grid required for the calculation of airflow field and droplet trajectory; (**b**) the control volume of ice layer and water film; (**c**) the control volume of mixture. \[\rho h\mathbf{U}=\rho_{w}h_{w}\mathbf{U}_{w}+\rho_{i}h_{i}\mathbf{U}_{i}, \tag{4}\]  
where \(\mathbf{U}\) is the velocity of the mixture, \(\mathbf{U}_{w}\) is the velocity of the water phase and \(\mathbf{U}_{i}\) is the velocity of the ice phase. In this study, the ratio of ice mass to mixture mass is chosen to be the phase parameter \(C\), which means \(C=(\rho_{i}h_{i})/(\rho h)\), \(\rho_{w}h_{w}=1-C\). We obtain:  
\[\mathbf{U}=(1-C)\mathbf{U}_{w}+C\mathbf{U}_{i}. \tag{5}\]  
Assuming that the ice produced by freezing adheres to the surface without displacement, the average velocity is as follows:  
\[\mathbf{U}_{i}=0. \tag{6}\]  
Then, the velocity of the mixture can be described as:  
\[\mathbf{U}=(1-C)\mathbf{U}_{w}. \tag{7}\]  
The conservative equation of phase parameter \(C\) can be written as:  
\[\frac{\partial\rho hC}{\partial t}+\
abla\cdot(\rho hC\mathbf{U})=\dot{m}_{ ice}. \tag{8}\]  
In the phase equation, \(m_{ice}\) is the amount of liquid water frozen into ice, which can be described as:  
\[m_{ice}=f_{ice}\cdot(1-C)\rho h, \tag{9}\]  
where \(f_{ice}\) is the freezing rate, and can be written as [31]:  
\[V=s\cdot h,V_{w}=s\cdot h_{w},V_{i}=s\cdot h_{i}, \tag{2}\]0  
\[V=s\cdot h,V_{w}=s\cdot h_{w},V_{i}=s\cdot h_{i}, \tag{2}\]1  
\[V=s\cdot h,V_{w}=s\cdot h_{w},V_{i}=s\cdot h_{i}, \tag{2}\]2  
in which \(htc\) is the convective heat transfer coefficient. \(T_{w}\) is the temperature of the water film. \(Cp_{w}\) is the specific heat capacity and water. \(T_{m}\) is the the phase transition equilibrium temperature of water, \(T_{m}=273.15\) K. \(Le\) and \(Lf\) are the latent heat of water gasification and ice condensation. \(u_{d}\) is the velocity of a droplet impacting on the surface. \(\lambda\) is thermal conductivity of air. \(T_{\infty}\) and \(P_{\infty}\) are the temperature and pressure of the incoming stream. \(T_{total}\) is the total temperature of the incoming stream and \(P_{total}\) is the total pressure of the incoming stream, which can be specified as follows [18]:  
\[V=s\cdot h,V_{w}=s\cdot h_{w},V_{i}=s\cdot h_{i}, \tag{2}\]3  
\[V=s\cdot h,V_{w}=s\cdot h_{w},V_{i}=s\cdot h_{i}, \tag{2}\]4  
As shown in Figure 2, the conservation equation of mass for the mixture can be described as:  
\[V=s\cdot h,V_{w}=s\cdot h_{w},V_{i}=s\cdot h_{i}, \tag{2}\]5  
in which \(\dot{m}_{imp}\) is the water droplet collection rate.