## 2 Materials and Methods
### Heat Transfer Calculation  
The following assumptions were made for the heat transfer calculations:  
1. the ambient temperature and the air temperature inside the container distribute homogeneously; and,
2. the PCM's temperature keeps constant and distributes homogeneously during the phase transition process. Figure 4 shows the heat transfer schema of the container at a high ambient temperature as an example. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Water** & **OP5E** \\ \hline Specific heat capacity/kJ/(kg\textasciitif{}K) & 4.2 & 2.0 \\ Solid density (\(-15\) \({}^{\circ}\)C)/kg/L & 0.996 & 0.88 \\ Liquid density (20 \({}^{\circ}\)C)/kg/L & 0.998 & 0.76 \\ Thermal conductivity liquid/W/(m\textasciitif{}K) & 0.56 & 0.2 \\ Volume expansion/\% & 0.3 & 13 \\ Total mass used in the container (kg) & 10.87 & 8.28 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Technical parameters of water and OP5E. OP5E: an alkane mixture manufactured by Ruhr New Material Technology Co., Ltd. in Hangzhou in China. Figure 4: Heat transfer schema of the cold-chain insulated container at a high ambient temperature. VIP: vacuum insulation panel; PU: polyurethane. Figure 3: Partial enthalpy distribution of OP5E. OP5E: an alkane mixture manufactured by Ruhr New Material Technology Co.,Ltd. in Hangzhou in China. The average heat transfer area \(F\) of the container can be calculated according to Equation (1) as follows:  
\[F=\sqrt{F_{\mathrm{a}}\times F_{\mathrm{b}}} \tag{1}\]  
where \(F_{\mathrm{a}}\) and \(F_{\mathrm{b}}\) are the external and internal area of the container respectively. It was obtained that \(F\) = 0.856 m\({}^{2}\) by substituting data into Equation (1). The overall heat transfer coefficient \(K\) of the container is obtained as follows:  
\[K=\frac{1}{\frac{1}{\alpha_{1}}+\frac{\delta_{1}}{\lambda_{1}}+\frac{\delta_{2 }}{\lambda_{2}}+\frac{\delta_{3}}{\lambda_{3}}+\frac{1}{\alpha_{2}}} \tag{2}\]  
where \(\alpha_{1}\) and \(\alpha_{2}\) are the heat transfer coefficient of air outside and inside the container. The air flows outside and inside the container can be considered as natural convection since the containers are normally transported in a closed cabin of refrigerated trucks or vans. The coefficient is 7.0\(\cdot\)9.3 W/(m\({}^{2}\)K) for natural convection [8], here \(\alpha_{1}\) = \(\alpha_{2}\) = 8.0 W/(m\({}^{2}\)K). \(\lambda_{1}\), \(\lambda_{2}\) and \(\lambda_{3}\) are the thermal conductivity of PU, VIP, and PCM, respectively, here \(\lambda_{1}\) = 0.022 W/(m\(\cdot\)K), \(\lambda_{2}\) = 0.005 W/(m\(\cdot\)K), \(\lambda_{3}\) = 0.2 W/(m\(\cdot\)K). \(\delta_{1}\), \(\delta_{2}\) and \(\delta_{3}\) are the thickness of PU, VIP, and PCM, respectively. Heat is transferred from the ambient into the container or conversely by conduction and convection. The heat flow \(q_{1}\) can be estimated with the overall heat transfer coefficient \(K\), the average heat transfer area \(F\), the ambient temperature \(T_{\mathrm{a}}\) and the air temperature \(T_{\mathrm{b}}\) inside the container as follows:  
\[q_{1}=K\times F\times(T_{\mathrm{a}}-T_{\mathrm{b}}) \tag{3}\]  
\(T_{\mathrm{b}}\) is taken as the maximum temperature allowed in the container under extremely high temperature condition (\(T_{\mathrm{b,max}}\) = 8 \({}^{\circ}\)C) or the minimum temperature allowed under extremely low temperature condition (\(T_{\mathrm{b,min}}\) = 2 \({}^{\circ}\)C). The heat flow \(q_{2}\) caused by air leakage is obtained according to Equation (4):  
\[q_{2}=\beta\times q_{1} \tag{4}\]  
where \(\beta\) is the additional thermal load factor, here \(\beta=0.2\). The total heat flow \(q\) is the sum of the heat flow \(q_{1}\) and \(q_{2}\):  
\[q=q_{1}+q_{2} \tag{5}\]  
The heat capacity \(\Delta H\) of PCM can be determined as follows:  
\[\Delta H=m\times\Delta h \tag{6}\]  
where \(m\) is the total mass of PCM and \(\Delta h\) is the melting/freezing enthalpy of PCM in a given temperature interval.