## 2 Mathematical Model for Conjugate Heat Transfer with Phase Change  
According to the configuration standard of a building envelope in the hot summer and cold winter region of China, the current research focuses on the heat transfer capability and energy saving characteristics of the following three different roofs: (1) Ordinary roof, its configuration from the top to the bottom is: A 20 mm thick cement layer, a 100 mm thick reinforced concrete layer, and a 20 mm thick lime layer; (2) a PCM roof, its configuration from the top to the bottom is: A 20 mm thick cement layer, a 100 mm thick reinforced concrete layer, a 30 mm thick phase change material (PCM) layer, and a 20 mm thick lime layer; (3) a SIM roof, its configuration from the top to the bottom is: A 20 mm thick cement layer, a 100 mm thick reinforced concrete layer, a 30 mm thick sensible insulating material (SIM) layer, and a 20 mm thick lime layer. In order to simplify the mathematical complication, the following reasonable assumptions are made for the current modelling: (1) The thermophysical properties of PCM, SIM, and other materials are constant. (2) The volume expansion of PCM during solidification and melting is negligible, and its melting temperature is constant. (3) The contact thermal resistance between different material layers is neglected. (4) The effects of people, furniture, and other heat sources in the room are not considered. (5) The temperature distribution inside the room is uniform. Based on the above conditions, the transient conjugate heat transfer with phase change inside the PCM roof is governed by the following energy equation:  
\[\rho_{i}\frac{\partial H_{i}}{\partial t}=k_{i}\frac{\partial^{2}T_{i}}{ \partial y_{i}^{2}} \tag{1}\]  
where \(\rho_{i}\) is the density, \(k\) is the thermal conductivity, \(T\) is the temperature, \(t\) is the time, \(y\) is the Cartesian coordinate in the vertical direction, \(H\) is the enthalpy, and the index \(i=1\sim n\) represents the ith layer material of a building roof. For the PCM layer, the enthalpy could be defined as:  
\[H=c_{p}(T-T_{r})+f_{l}h_{s} \tag{2}\]  
where \(c_{p}\) is the specific heat, \(T_{r}\) is the reference temperature, \(f_{l}\) is the liquid fraction of the PCM, and \(h_{s}\) is the latent heat. For other sensible material layers, the enthalpy \(H\) is expressed as:  
\[H=c_{p}(T-T_{r}). \tag{3}\]  
At the external surfaces of a roof (\(i=1\) and \(y_{1}=0\) mm), the solar radiation and convective heat transfer between the environment and the cement layer are taken into account, which could be given by the third kind of thermal boundary condition as:  
\[-k_{1}A\frac{\partial T_{1}}{\partial y_{1}}=h_{out}A(T_{e}-T_{1})+Q_{rad} \tag{4}\]  
where \(h_{out}\) is the convective heat transfer coefficient of the external roof surface, which is used as \(h_{out}=19\) W/(m\(\cdot\)K) for summertime and \(h_{out}=23.3\) W/(m\(\cdot\)K) for winter time [2]; \(A\) is the roof surface area, \(T_{e}\) is the environmental temperature, and \(Q_{rad}\) is the solar radiation. The environment temperature, \(T_{e}\), and solar radiation, \(Q_{rad}\), could be simultaneously considered by introducing the equivalent temperature \(T_{eq}\)[2]:  
\[T_{eq}=T_{e}+\alpha I/h_{out}-I_{up}/h_{out} \tag{5}\]  
\(\alpha\) is the solar absorption coefficient of the roof surface, which was chosen to be \(\alpha=0.8\) for the cement material; \(I\) is the solar radiation intensity. For the building roof, the term \(I_{up}/h_{out}\) was 3.5-4 K [2], and \(I_{up}/h_{out}=3.5\) K was used in the current work. For the interior surface of the roof (\(i=n\) and \(y_{n}=L\) mm), where \(L\) is the roof thickness, the third kind of thermal boundary condition was applied:  
\[-k_{n}A\frac{\partial T_{n}}{\partial y_{n}}=h_{in}A(T_{n}-T_{in}) \tag{6}\]  
\(h_{in}\) is the convective heat transfer coefficient of the interior roof surface, which is \(h_{in}=8.7\) W/(m\(\cdot\)K) [2].