## 2 Establishment of the Model
### Mathematical Model  
2.2.1. The General Form of Control Equation  
Continuity equation:  
\[\
abla\cdot\overrightarrow{v}=0 \tag{1}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Material** & **Melting Point/** & **Latent Heat of Phase Change/** & **Specific Heat Capacity \(c_{ps}\)/\(c_{pl}\)** & **Density \(\rho_{s}\)/\(\rho_{1}\)** & **Thermal Conductivity** \\  & \({}^{\circ}\)**C** & **kJ\(\cdot\)kg\({}^{-1}\)** & **kJ\(\cdot\)(kg\(\cdot\)K)\({}^{-1}\)** & **kg\(\cdot\)m\({}^{-3}\)** & **k\({}_{s}\)/k\({}_{1\)eff**} \\  & & & & **W\(\cdot\)(m\(\cdot\)K)\({}^{-1}\)** \\ \hline n-hexadecane & 18.15 & 236 & 1.64/2.09 & 864/773 & 0.14/1.48 \\ graphite & 3500 & – & 0.710/\(-\) & 2000/\(-\) & 129/\(-\) \\ \hline Mixture of & & & & & \\ n-hexadecane and & 18.15 & 212 & 1.547/1.952 & 978/896 & 0.277/2.312 \\ graphite (9:1) & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Physical parameters of phase change materials [38]. Figure 1: Element model example of cylindrical phase change heat exchanger. Momentum equation:  
\[\frac{\partial\overrightarrow{v}}{\partial t}+\overrightarrow{v}\cdot\
abla \overrightarrow{v}=f+\
abla\cdot(v\
abla\overrightarrow{v})-\frac{1}{\rho}\
abla P \tag{2}\]  
Energy equation:  
\[\frac{\partial H}{\partial t}+\overrightarrow{v}\cdot\
abla H=\
abla\cdot( \frac{k}{\rho}\
abla T)+\dot{Q} \tag{3}\]  
In the formula, \(H\) is the total enthalpy of the unit volume phase change material; \(t\) is the time; \(\overrightarrow{v}\) is the liquid-phase velocity vector; \(k\) is the thermal conductivity; \(\rho\) is the density; \(T\) is the temperature; \(\dot{Q}\) is the solid-phase (or liquid-phase) phase volume heat source (sink); \(f\) is the unit mass force; \(v\) is the kinetic viscosity; and \(P\) is the pressure. Hypothesis  
In order to simplify the calculation model, the following assumptions are made:  
(1) The axial and circumferential heat transfer of the cylindrical vessel is neglected, and only the radial heat transfer is considered;  
(2) There is no supercooling and performance degradation of phase change materials;  
(3) The solid-phase and liquid-phase thermophysical properties of phase change materials are different, but they have nothing to do with temperature and pressure. (4) Isotropy of phase change materials;  
(5) Ignoring the thermal resistance of the tube wall;  
(6) In the melting process of phase change materials, the influence of natural convection caused by gravity is reflected by effective thermal conductivity, and the phase change process is regarded as a heat transfer process dominated by heat conduction. The Unity of Enthalpy Field and Temperature Field  
Although the differential equation of the enthalpy model unifies the solid phase, liquid phase, or two-phase region in the form of the equation, the form of the enthalpy field, \(H\), and the temperature field, \(T\), is not unified. When the specific heat capacity of the solid and liquid phases is constant, respectively, there is the following relationship between them:  
\[H= \begin{cases}&\int_{T_{\rm m}}^{T}\rho_{\rm s}c_{\rm s}{\rm d}T\,,\ T<T_{ \rm m}\\ &\rho_{\rm f}{\rm f}_{\rm l}h_{\rm m}\,,\ T=T_{\rm m}\\ &\int_{T_{\rm m}}^{T}\rho_{\rm l}c_{\rm l}{\rm d}T+\rho_{\rm l}h_{\rm m}\,,\ T>T_{ \rm m}\end{cases}  \tag{4}\]  
In the formula, \(\rho_{\rm s}\) is the density of the solid-phase-change material; \(c_{\rm s}\) is the specific heat capacity of solid-phase-change materials; \(T_{\rm m}\) is the phase transition temperature; \(\rho_{1}\) is the density of liquid-phase-change materials; \(f_{1}\) is the volume fraction of liquid phase; \(h_{\rm m}\) is the latent heat of melting of phase-change materials; and \(c_{1}\) is the specific heat capacity of liquid-phase-change materials. For the isothermal phase-transition process, the liquid volume fraction, \(f_{1}\), is determined by Equation (5). \[f_{1}= \begin{cases}0,\ T<T_{\text{m}}\\ 1,\ T>T_{\text{m}}\end{cases}  \tag{5}\]  
The region for \(0<f_{1}<1\) is the mushy zone.