# Thermal Performance Analysis of a Double-Pass Solar Air Collector: A CFD Approach
## 2 Materials and Methods
### Mathematical Model
#### 2.2.1 Turbulence Model  
A preliminary hydrodynamic analysis performed in the SolidWorks 2013 SP2.0 software determined that the flow separates due to the sudden expansion at the cavity inlet. Furthermore, the flow was found to be under development (L\({}_{\rm h,turbulent}<\) L\({}_{\rm Collector}\)), and the calculated average Reynolds numbers (Re) were very low (for flow 1 (0.01 kg/s) was Re\({}_{\rm cav-1}\) = 2972, and for flow 2 (0.02 kg/s) was Re\({}_{\rm cav-2}\) = 5961). The k-\(\omega\) turbulence models are better at predicting adverse pressure gradient boundary layer flows and separation, and they also have the ability to simulate the laminar-turbulent transition of wall boundary layers [26]. Additionally, the k-\(\omega\) models have low-Reynolds number terms (Re \(<\) 10\({}^{4}\)) that mimic laminar-turbulent transition processes. However, this function is not widely calibrated in the SST k-\(\omega\) model; therefore, it is recommended to use the Transition SST k-\(\omega\) model [27,28].  
The Transition SST model was selected based on the described above and considering the required accuracy to predict heat transfer from the absorber plate to the air as the flow. Furthermore, the buoyancy effects were adjusted to full, and the viscous heating was activated. The turbulence modeling consisted of implementing the Transition SST k-omega model described in Equations (5)-(12). Equation (5) corresponds to the transport equation for intermittency (\(\gamma\)), whereas Equations (6) and (7) represent the transition sources \(P_{\gamma 1}\) and \(E_{\gamma 1}\), respectively; and Equations (8) and (9), the destruction/re-laminarization sources \(P_{\gamma 2}\) and \(E_{\gamma 2}\)  
Figure 3: Solar collector mesh. (**a**) Transversal section; (**b1**) coarse mesh in longitudinal section; (**b2**) refined mesh in longitudinal section.  
Flows 1 and 2 were selected based on a preliminary analysis using the thermal model described in [29]; among those flows, the best balance between air outlet temperature and thermal efficiency was found. In the calculation of the Reynolds number, the cavity was approximated as a trapezoidal cross-section duct for calculating the hydraulic diameter (\(D_{h}\)).  
\[\frac{\partial(\rho\gamma)}{\partial t}+\frac{\partial\big{(}\rho U_{J}\gamma \big{)}}{\partial x_{j}}=P_{\gamma 1}-E_{\gamma 1}+P_{\gamma 2}-E_{\gamma 2}+\frac{ \partial}{\partial x_{j}}\Bigg{[}\Big{(}\mu+\frac{\mu_{t}}{\sigma_{\gamma}} \Big{)}\frac{\partial}{\partial x_{j}}\Bigg{]} \tag{5}\]  
\[P_{\gamma 1}=C_{a1}F_{length}\rho S[\gamma_{Fonset}]^{c_{\gamma 3}} \tag{6}\]  
\[E_{\gamma 1}=C_{c1}P_{\gamma 1}\gamma \tag{7}\]  
\[P_{\gamma 2}=C_{a2}\rho\Omega\gamma F_{turb} \tag{8}\]  
\[E_{\gamma 2}=C_{c2}P_{\gamma 2}\gamma \tag{9}\]  
On the other hand, Equation (10) refers to the interaction of the transition model with the SST turbulence model by modifying equation \(k\), where \(G_{k}^{*}\) and \(Y_{k}^{*}\) are the original production and destruction terms of the SST model [28].  
\[\frac{\partial(\rho k)}{\partial t}+\frac{\partial(\rho ku_{i})}{\partial x_ {j}}=\frac{\partial}{\partial x_{j}}\Bigg{(}\Gamma_{k}\frac{\partial k}{ \partial x_{j}}\Bigg{)}+G_{k}^{*}-Y_{k}^{*} \tag{10}\]  
\[G_{k}^{*}=\gamma_{ef}\,f\widetilde{G}_{k} \tag{11}\]  
\[Y_{k}^{*}=\min\Big{(}\max\Big{(}\gamma_{ef}\,f,0.1\Big{)},1.0\Big{)}\,Y_{k} \tag{12}\]  
The pressure-based solver with the Coupled scheme was selected, and a second-order spatial discretization scheme was implemented. The gradient evaluation method selected was based on Least Squares Cell-Based, and the high-order term relaxation option was used. The pressure factor was adjusted to 0.1, and the Flow Courant Number to 4.