## 2 Materials and Methods
### Mathematical Description  
The heat transfer process inside a combined solar chimney is a complex dynamic process involving heat convection and radiation. To simplify the theoretical analysis and numerical calculations, assumptions were made as follows:  
1. The air inside the chimney was an incompressible Newtonian fluid;
2. The heat transfer was in a steady-state [16]. Both the solar irradiation and the outdoor ambient temperature were constant;
3. All relevant physical properties of the chimney material were independent of the temperature [20];
4. Air penetration was ignored. Heat storage on the storage wall was also ignored to simplify the heat transfer process;
5. This simulation only considered the ventilation performance of a solar chimney under hot-pressing [18]. Three-dimensional steady-state turbulence control equations [24; 25]: \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Model** & **H (m)** & **h (m)** & **W (m)** & **d (m)** & **\(\theta\) (\({}^{\circ}\))** & **Size of Air Vent (m\({}^{2}\))** \\ \hline CSC & 2.5 & 1 & 1.5 & 0.5 & 45\({}^{\circ}\) & 0.5 \(\times\) 0.3 \\ TW & 2.5 & – & 1.5 & 0.5 & – & 0.5 \(\times\) 0.3 \\ IRSC & – & 1 & 1.5 & 0.5 & 45\({}^{\circ}\) & 0.5 \(\times\) 0.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Geometrical parameters of the different models of a solar chimney. Figure 2: Models of a solar chimney with different structures: (**a**) CSC; (**b**) TW; and (**c**) IRSC. Continuity equation:  
\[\frac{\partial v_{i}}{\partial x_{i}}\;=\;0. \tag{1}\]  
Momentum equation:  
\[\frac{\partial\big{(}v_{i}v_{j}\big{)}}{\partial x_{j}}\;=\;-\frac{1}{\rho}\frac {\partial P}{\partial x_{i}}\;+\;\frac{\partial}{\partial x_{j}}\Bigg{[}(v\;+ \;v_{t})\frac{\partial v_{i}}{\partial x_{j}}\Bigg{]}\;-\;g_{i}\beta(T\;-\;T_{ \infty}). \tag{2}\]  
Energy equations:  
\[\frac{\partial\big{(}v_{j}T\big{)}}{\partial x_{j}}\;=\;\frac{\partial}{ \partial x_{j}}\Bigg{(}\Gamma\frac{\partial T}{\partial x_{j}}\Bigg{)} \tag{3}\]  
\[\Gamma\;=\;\frac{v}{\mathrm{Pr}}\;+\;\frac{v_{t}}{\sigma_{t}}. \tag{4}\]  
Turbulent kinetic energy:  
\[\frac{\partial(kv_{i})}{\partial x_{i}}\;=\;\frac{1}{\rho}\frac{\partial}{ \partial x_{j}}\Bigg{[}\bigg{(}v\;+\;\frac{v_{t}}{\sigma_{k}}\bigg{)}\frac{ \partial k}{\partial x_{j}}\Bigg{]}\;-\;\varepsilon+G_{k}. \tag{5}\]  
Dissipation rates:  
\[\frac{\partial(\varepsilon v_{i})}{\partial x_{i}}\;=\;\frac{1}{\rho}\frac{ \partial}{\partial x_{j}}\Bigg{[}\bigg{(}v\;+\;\frac{v_{t}}{\sigma_{k}}\bigg{)} \frac{\partial\varepsilon}{\partial x_{j}}\Bigg{]}\;+\;\frac{\varepsilon}{k} (c_{1}G_{k}\;-\;c_{2}\varepsilon) \tag{6}\]  
\[G_{k}\;=\;\frac{v_{t}}{\rho}\frac{\partial u_{i}}{\partial x_{j}}\Bigg{(} \frac{\partial u_{i}}{\partial x_{j}}\;+\;\frac{\partial u_{j}}{\partial x_{ i}}\Bigg{)} \tag{7}\]  
where \(v_{i}\) is the average velocity component in the \(x_{i}\) direction, m/s; \(x_{i}\) is the coordinate; \(\rho\) refers to the air density, kg/m\({}^{3}\); \(P\) is the pressure, Pa; \(v_{t}\) and \(v\) are the turbulent viscous coefficient and laminar viscous coefficient, respectively; \(g_{i}\) is the gravitational acceleration in the \(i\) direction, m/s\({}^{2}\); \(\beta\) is the air thermal expansion coefficient, L/K; \(T\) and \(T_{\infty}\) are the average temperature and reference point temperature, respectively, K; \(\Gamma\) is the generalized diffusion coefficient; \(k\) is the turbulent kinetic energy; \(\varepsilon\) refers to the dissipation of kinetic energy; \(G_{k}\) refers to the generation of turbulence kinetic energy; \(Pr\) is the Prandtl number; and \(\sigma_{k}\), \(\sigma_{\varepsilon}\), \(\sigma_{t}\), \(c_{1}\), and \(c_{2}\) are empirical values of 1.0, 1.2, 0.9, 1.44, and 1.9, respectively [26; 27]. The flowing fluid in the model was air, conforming to the Boussinesq hypothesis [28]. The physical parameter of the main materials and air are listed in Tables 2 and 3.