## 2 Model Setup and Validation
### Governing Equations  
The airflow and temperature distribution in the room are set to maintain conservation laws of mass, momentum and energy. The airflow is simulated as unsteady state, incompressible, and turbulent. Based on the above assumptions, the governing time-averaged equations are given by [14]:  
\[\frac{\partial u_{\rm i}}{\partial x_{\rm i}}=0 \tag{1}\]  
\[\frac{\partial u_{\rm i}}{\partial t}+\frac{\partial\big{(}u_{\rm i}u_{\rm j} \big{)}}{\partial x_{\rm j}}=-\frac{1}{\rho}\frac{\partial p}{\partial x_{\rm i }}+\frac{\partial}{\partial x_{\rm j}}\Bigg{(}\
u\frac{\partial u_{\rm i}}{ \partial x_{\rm j}}-\overline{u^{\prime}{}_{\rm i}u^{\prime}{}_{\rm j}}\Bigg{)} \tag{2}\]  
\[\frac{\partial\big{(}C_{\rm p}T\big{)}}{\partial t}+\frac{\partial\big{(}C_{\rm p }Tu_{\rm i}\big{)}}{\partial x_{\rm i}}=\frac{\partial}{\partial x_{\rm j}} \Bigg{[}\Bigg{(}\frac{\lambda}{\rho}+\frac{C_{\rm p}v_{\rm t}}{Pr_{\rm t}} \Bigg{)}\frac{\partial T}{\partial x_{\rm j}}\Bigg{]} \tag{3}\]  
\[-\overline{u^{\prime}{}_{\rm i}u^{\prime}{}_{\rm j}}=v_{\rm t}\Bigg{(}\frac{ \partial u_{\rm i}}{\partial x_{\rm j}}+\frac{\partial u_{\rm j}}{\partial x_ {\rm i}}\Bigg{)}-\frac{2}{3}k\delta_{\rm ij} \tag{4}\]  
\[v_{\rm t}=C_{\rm\mu}\frac{k^{2}}{\varepsilon} \tag{5}\]  
where \(x_{\rm i}\) represents the Cartesian coordinates while the subscripts i and j range between 1 and 3, separately, refer to the (\(x\), \(y\), \(z\)) directions in space. The term \(u_{\rm i}\) is the velocity of component i. \(p\) and \(T\) are the pressure and temperature, respectively. \(\rho\) and \(\lambda\) are the air density and thermal conductivity, respectively. \(\
u\) is the kinematic viscosity, \(v_{\rm t}\) is the turbulent eddy viscosity, \(k\) is turbulent kinetic energy, and \(\delta_{\rm ij}\) is the Kronecker delta (i = j, \(\delta_{\rm ij}\) = 1; i \(\
eq\) j, \(\delta_{\rm ij}\) = 0). The renormalization group (RNG) \(k\)-\(\epsilon\) turbulence model is used to simulate the three-dimensional turbulent airflow, as it has been shown to accurately describe the flow field of the near wall region and relatively better predict of the indoor environment than the standard \(k\)-\(\varepsilon\) model by introducing an additional term in the \(\varepsilon\)-equation [15; 16; 17]. The transfer equations of turbulent kinetic energy \(k\) and turbulence dissipation rate \(\varepsilon\) are given by:  
\[\frac{\partial k}{\partial t}+\frac{\partial}{\partial x_{\rm i}}(ku_{\rm i })=\frac{\partial}{\partial x_{\rm j}}\Bigg{[}\Bigg{(}\
u+\frac{v_{\rm t}}{ \sigma_{\rm k}}\Bigg{)}\frac{\partial k}{\partial x_{\rm j}}\Bigg{]}- \overline{u^{\prime}{}_{\rm i}u^{\prime}{}_{\rm j}}\frac{\partial u_{\rm j}}{ \partial x_{\rm i}}-\varepsilon \tag{6}\]  
\[\frac{\partial\varepsilon}{\partial t}+\frac{\partial}{\partial x_{\rm i}}( \varepsilon u_{\rm i})=\frac{\partial}{\partial x_{\rm i}}\Bigg{[}\Bigg{(} \
u+\frac{v_{\rm t}}{\sigma_{\rm z}}\Bigg{)}\frac{\partial\varepsilon}{ \partial x_{\rm j}}\Bigg{]}-C_{1\varepsilon}\frac{\varepsilon}{k}\overline{u^ {\prime}{}_{\rm i}u^{\prime}{}_{\rm j}}\frac{\partial u_{\rm i}}{\partial x_ {\rm i}}-C_{2\varepsilon}^{*}\frac{\varepsilon^{2}}{k} \tag{7}\]  
\[C_{2\varepsilon}^{*}=C_{2\varepsilon}+\frac{C_{\rm\mu}\rho\eta^{3}(1-\eta/\eta_{ 0})}{1+\zeta\eta^{3}} \tag{8}\]  
where \(\eta=S\) (\(k/\varepsilon\)) is the ratio between the time scales of the turbulence and the mean flow, \(S\) is the coefficient of surface tension and is defined by \(S=\sqrt{2S_{\rm ij}S_{\rm ij}}\), \(S_{\rm ij}=\big{(}\partial u_{\rm i}/\partial x_{\rm j}+\partial u_{\rm j}/ \partial x_{\rm i}\big{)}/2\)The constants appearing in the RNG \(k\)-\(\varepsilon\) model are: \(C_{\rm{\mu}}=0.0845\), \(C_{1\varepsilon}=1.42\), \(C_{2\varepsilon}=1.68\), \(\sigma_{\rm{k}}=\sigma_{\varepsilon}=0.7178\), \(\zeta=0.012\), \(\eta_{0}=4.38\). To observe the dynamic heat transfer process in envelope, a three-dimensional system of wall with thickness direction \(x\), width direction \(y\) and height direction \(z\) is established (see Figure 3a).