# Thermal Performance Analysis of a Double-Pass Solar Air Collector: A CFD Approach
## 2 Materials and Methods
### Mathematical Model
#### 2.2.2 Boundary Conditions  
The momentum boundary conditions in the walls were considered no-slip stationary with a constant rugosity of 0.5. The thermal boundary conditions modeled the incident solar radiation on the cover and the absorber plate as a heat generation source, calculated using Equations (13) and (14).  
\[q_{c}(t)=\bigg{(}I(t)\left[\overline{\alpha}_{c}+\overline{\alpha}_{c} \overline{\tau}_{c}\overline{c}_{p}\varrho_{r}^{2(n)}\right]\frac{A_{c}}{A_{p} }\bigg{)}/w_{c} \tag{13}\]  
\[q_{p}(t)=\bigg{(}I(t)\ \overline{\tau}_{c}\varrho_{r}^{n}P_{g}\left[\overline{ \alpha}_{p}+\overline{\alpha}_{p}\overline{\varrho}_{c}\overline{\varrho}_{c }^{2(n)}\frac{A_{p}}{A_{c}}\right]\frac{A_{c}}{A_{p}}\bigg{)}/w_{p} \tag{14}\]  
where \(q(t)\) is the heat generation, \(I(t)\) is the solar irradiance, \(P_{g}\) is the gap loss factor (0.96), \(\alpha\) is the absortivity, \(\tau\) is the transmissivity, \(\rho\) is the reflectivity, \(A_{c}\) is the cover area, \(A_{p}\) is the absorber plate area, and \(w_{c}\) and \(w_{g}\), are the thickness of the cover and plate, respectively. The optical properties of the cover are \(\alpha_{c}=0.05\), \(\tau_{c}=0.89\) y \(\varrho_{c}=0.05\), the receiver are \(\alpha_{p}=0.95\) and \(\varrho_{p}=0.05\), and from the reflector \(\rho_{r}=0.91\).  
The heat transfer coefficient from the cover to the environment (HTC\({}_{\text{c-a}}\)) was obtained by applying the flow around finite flat-plates methodology reported in [30], while the convection losses of the external walls were calculated using the heat transfer coefficient (HTC\({}_{\text{b}}\)) correlation proposed by [31]. Additionally, in laminar flow, the heat transfer coefficient from the flat-plate receiver to the fluid in the cavity (HTC\({}_{\text{p-cav}}\)) was estimated using the discretized Fourier's law, considering the local temperature normal to the wall [26].  
Moreover, in turbulent flow, the HTC\({}_{\text{P-cav}}\) was determined using the law of the wall for estimating the local temperature of the fluid by applying the Reynolds analogy [26].  
The heat conduction in the exterior walls of the CPC was modeled as shell conduction, whereas the radiation losses in the cover were calculated with an emissivity value of \(\varepsilon_{c}=0.81\), and the sky temperature (\(T_{s}\)) was calculated with the correlation proposed by Swinbank (Equation (15)), reported in [32], where \(T_{a}\) refers to the ambient temperature.  
\[T_{s}=0.0552T_{a}^{1.5} \tag{15}\]  
Regarding the turbulence parameters, a turbulence intensity of 5% and a turbulent viscosity ratio of 10 were applied. Table 4 summarizes the parameters of the boundary conditions of the CFD modeling.  
The pressure-based solver with the Coupled scheme was selected. In addition, a Second-Order scheme for spatial discretization was implemented because of numerical simulation stability. The formal truncation errors of individual terms in the governing equations were calculated; the error for the HTC\({}_{\text{P-cav}}\) was 1.27%, 0.051% for the shear stress, and 0.05% for the Nusselt number (with a security factor of Fs = 3) [32]. In addition, values of 1 \(\times\) 10\({}^{-4}\) for the mass residual and a mass imbalance of 5.3 \(\times\) 10\({}^{-8}\)% and 2 \(\times\) 10\({}^{-7}\) energy residual were accomplished. The verification of the results was carried out by quantifying the uncertainty of the numerical calculations. For Nu, the spatial error of 0.51% was obtained, while for HTC\({}_{\text{P-cav}}\) and shear stress were 0.66% and 0.37%, respectively. The grid convergence index was also verified, finding out 0.05% for Nu, 0.19% for shear stress, and 0.09% for HTC\({}_{\text{P-cav}}\).