## 2 Establishment of the Model
### Mathematical Model
Therefore, the change of liquid volume fraction, \(f_{1}\), can describe the phase-change process: when \(f_{1}=0\), the phase-change material is a solid phase, and the temperature of the phase-change material is lower than the phase change temperature; when \(0<f_{1}<1\), the two phases coexist, and the temperature of the phase-change material is equal to the phase change temperature; and when \(f_{1}=1\), the phase change material is liquid phase, and the temperature of the phase-change material is higher than the phase change temperature. The expression of sensible heat \(h=\int_{T_{\text{m}}}^{T}\rho c\text{d}T\) and \(H\) is unified as \(H=h+\rho_{1}f_{1}h_{\text{m}}\), so the energy equation can be unified as the equation with the variable of sensible heat, \(h\), as follows:  
\[\frac{\partial h}{\partial t}+\overset{\rightarrow}{v}\cdot\
abla(h+f_{1}h_{ m})=\
abla\cdot(\frac{k}{\rho c}\
abla h)+\dot{q}-\rho_{1}h_{m}\frac{\partial f _{1}}{\partial t} \tag{6}\]  
Treatment of Liquid-Phase Velocity  
For the underwater monitoring device driven by temperature-difference energy studied in this paper, the volume change law of phase-change material and the total time required for phase change are the most important parameters, while the velocity field inside the liquid working medium is meaningless for the study of the whole system, and it is not necessary to spend a lot of time on solving the velocity field. Therefore, in order to simplify the calculation, this paper adopts a method of effective thermal conductivity [36]. The influence of liquid velocity on the melting and solidification process of phase change refrigerant is compensated by modifying the thermal conductivity. In this way, the term of velocity contained in the energy equation can be discarded, and the effect of convection can be reflected on the effective thermal conductivity. At the same time, because there is no velocity term, the phase transition process can be described by only one equation of the energy equation; that is, Equations (1)-(3) can be simplified to a control equation of the energy equation. The simplified governing equation is as follows:  
\[\frac{\partial h}{\partial t}=\
abla\cdot(\frac{k_{\text{eff}}}{\rho c}\
abla h )+\dot{q}-\rho_{1}h_{\text{m}}\frac{\partial f_{1}}{\partial t} \tag{7}\]  
In the formula, \(k_{\text{eff}}\) is the phase-change material's equivalent thermal conductivity, and \(\dot{q}\) is the unit mass solid (liquid)-phase volume heat source (sink). Because there is no internal heat source in the PCM system of the ocean temperature-difference drive device, so \(\dot{q}=0\). The Final Form of the Governing Equation  
After the simplification of the above three steps, the phase change heat transfer model is simplified as follows:  
\[\frac{\partial h}{\partial t}=\
abla\cdot(\frac{k_{\text{eff}}}{\rho c}\
abla h )-\rho_{1}h_{\text{m}}\frac{\partial f_{1}}{\partial t} \tag{8}\]  
For the one-dimensional cylindrical phase change heat exchanger in this paper, the energy equation is more convenient to be expressed in cylindrical coordinates, and its form is as follows:  
\[\frac{\partial h}{\partial t}=\frac{1}{r}\frac{\partial}{\partial r}[\frac{k_ {\text{eff}}}{\rho c}r\frac{\partial h}{\partial r}]-\rho_{1}h_{\text{m}} \frac{\partial f_{1}}{\partial t} \tag{9}\]The determination method of physical parameters is as follows:  
\[f_{1}=0\,,\ \ \ \left\{ \begin{array}{l}\rho=\rho_{\rm s}\\ c=c_{\rm ps}\\ k_{\rm eff}=k_{\rm s}\end{array} \right. \tag{10}\]  
\[f_{1}=1\,,\ \ \ \left\{ \begin{array}{l}\rho=\rho_{1}\\ c=c_{\rm pl}\\ k_{\rm eff}=k_{\rm Leff}\end{array} \right. \tag{11}\]  
\[0<f_{1}<1\,,\ \ \ \left\{ \begin{array}{l}\rho=(1-f_{1})\rho_{\rm s}+f_{1}\cdot \rho_{1}\\ c=(1-f_{1})c_{\rm ps}+f_{1}\cdot c_{\rm pl}\\ k_{\rm eff}=(1-f_{1})k_{\rm s}+f_{1}\cdot k_{\rm Leff}\end{array} \right.