## 2 A New Transient Numerical Model
### Governing Equations
#### 2.2.3 Liquid Control Volume  
The energy equation of the liquid slug is as follows:  
\[(\rho_{I}A_{c}dx)c_{pl}\frac{\partial T_{I}}{\partial t}=\lambda_{I}A_{c}\frac {\partial T_{I}}{\partial x}+h_{I}A_{i}(T_{w}-T_{I})+h_{nb}(T_{w}-T_{sat}) \tag{17}\]  
The heat transfer between the pipe wall and the liquid slug may include the convective heat transfer, the subcooled boiling, and the saturated nucleate boiling, and thus \(q_{\text{wf}}\) in Equation (1) can be calculated as follows:  
\[q_{\text{wf}}=\left\{ \begin{array}{ll}h_{I}A_{i}(T_{w}-T_{I})&\text{single phaseforcedconvection}\\ h_{I}A_{i}(T_{w}-T_{I})+h_{nb,sub}A_{i}(T_{w}-T_{sat})&\text{subcooled boiling}/ \text{saturated nucleate boiling}\end{array} \right. \tag{18}\]  
The convective heat transfer coefficient of the liquid slug (\(h_{\text{I}}\)) is calculated by [27]:  
\[h_{I}=\left\{ \begin{array}{ll}\frac{\lambda_{I}}{d_{i}}\left(4.364+\frac{0.72 2\text{RePr}d_{i}}{L_{ex}}\right)&\text{Re}_{l}<2000,\text{ RePr}/d_{i}\leq 33.3\\ 1.953\frac{\lambda_{I}}{d_{l}}\left(\frac{\text{RePr}d_{i}}{L_{ex}}\right)^{0.333}&\text{Re}_{l}<2000,\text{ RePr}/d_{i}\geq 33.3\\ \frac{\lambda_{I}}{d_{i}}\frac{(f_{v}/8)(\text{Re}_{l}-1000)\text{Pr}_{l}}{1+ 12.7(f_{v}/8)^{1/2}(\text{Pr}_{l}^{2/3}-1)}&2000\leq\text{Re}_{l}<10,000\\ \left(\frac{\lambda_{I}}{d_{i}}\right)\times 0.23\left(\text{Re}_{l}^{0.8}\text{Pr}^{0.4}\right)&\text{Re}_{l}\geq 10,000,\text{ evaporation section}\\ \left(\frac{\lambda_{I}}{d_{i}}\right)\times 0.23\left(\text{Re}_{l}^{0.8}\text{Pr}^{0.3}\right)&\text{Re}_{l}\geq 10,000,\text{ condensation section}\end{array} \right. \tag{19}\]  
When the temperature difference of the tube wall and the saturated temperature of the liquid slug is higher than the superheat required for the nucleate boiling, and the temperature of the liquid slug is lower than the saturation temperature, subcooled boiling can occur in the PHP. The superheat needed for the nucleate boiling is calculated by [23; 24]:  
\[\Delta T_{\text{sat}}=\left(\frac{8\sigma T_{sat}q}{h_{f}g_{sat}\lambda_{I}} \right)^{1/2} \tag{20}\]  
In the subcooled boiling, bubbles are generated on the tube wall, but can easily collapse in the liquid slug, because the bulk temperature of the liquid is lower than the saturation temperature. Thus, under this circumstance, no new vapor plug will be formed. When the temperature of the liquid slug increases and reaches the saturation temperature, the saturated nucleate boiling can occur in the PHP, and new bubbles can be formed in the liquid slug. The nucleate boiling heat transfer coefficient can be calculated by [24]:  
\[h_{nb}=\frac{0.0012\Delta T_{sat}{}^{0.24}\Delta P_{sat}{}^{0.75}c_{pl}{}^{0. 45}\rho_{l}{}^{0.49}\lambda_{l}{}^{0.79}}{\sigma^{0.5}\lambda^{0.24}\mu^{0.29} \rho_{v}{}^{0.24}} \tag{21}\]  
When the saturated nucleate boiling occurs, the mass change in a liquid slug is calculated as follows:  
\[dm_{l}=\sum_{i}h_{nb}(\pi d_{i}Le_{l})(T_{w}-T_{l})/h_{fg} \tag{22}\]  
When \(dm_{l}\geq\;2\rho_{v,sat}A_{c}dx\), a new vapor generates in the liquid slug. 2.2.4. Control Volumes in the Interface between Liquid Slug and Vapor Plug  
For the vapor-liquid control volume as shown in Figure 2, the mass conservation of the liquid phase in the vapor-liquid control volume is as follows:  
\[\frac{dm}{dt}=-\frac{dm_{v}}{dt}-\frac{dm_{l}}{dt} \tag{23}\]  
where \(x_{1}\) represents the liquid length in the vapor-liquid control volume. The mass change in the liquid phase of the vapor-liquid control volume includes the mass change caused by the phase change in the vapor plug (\(dm_{v}\)) and the phase change in the liquid slug (\(dm_{l}\)).