# Energy-Based Thermo-Hydraulic Performance of Roughened Absorber in Solar Air Heater Duct
## 2 Details of SAH Equipped with Conical Protrusion Ribs
### Thermal Analysis of SAH  
The heat transfer in various components of the SAH is important for the analysis of the useful heat gain and thermal losses. The system thermal efficiency is evaluated by accounting for thermal losses on the side cover, back cover and top glass cover. The thermal analysis takes into consideration the following assumptions: (i) the properties of the glass cover and absorber plate materials are temperature independent, (ii) the effect of sun location (longitude and latitude) is neglected, (iii) the effect of humidity is negligible, (iv) spatial temperature variation in absorber plate is negligible, (iv) optical losses are neglected and (v) air thermo-physical properties are calculated at the average temperature. The combined convection-radiation thermal resistance between the glass cover and the sky, as well as between the absorber plate and the glass cover, are calculated using the thermal network approach, shown in Figure 4. The thermal losses of the side insulation and back cover insulation are neglected in the thermal network of the SAH.  
Figure 3: Flow around protrusion rib.  
Figure 2: Conical protrusion rib.  
Absorber-to-glass-cover heat transfer:  
\[q_{\text{loss,}p-\text{c}}=h_{\text{c,}\text{c}-\text{p}}\big{(}T_{\text{p}}-T_{ \text{c1}}\big{)}+\frac{\sigma\big{(}T_{\text{d}}^{4}-T_{\text{c}}^{4}\big{)}}{ \frac{1}{i_{\text{p}}}+\frac{1}{i_{\text{c}}}-1}=\big{(}h_{\text{c,}\text{p}- \text{c}}+h_{\text{r,}\text{p}-\text{c}}\big{)}\big{(}T_{\text{p}}-T_{\text{c}} \big{)} \tag{1}\]  
where  
\[h_{\text{r,}\text{p}-\text{c}}=\frac{\sigma\big{(}T_{\text{p}}^{2}+T_{\text{c} }^{2}\big{)}\Big{(}T_{\text{p}}+T\Big{)}}{\frac{1}{i_{\text{p}}}+\frac{1}{i_{ \text{c}}}-1} \tag{2}\]  
\[R_{2}=\frac{1}{\big{(}h_{\text{c,}\text{c}-\text{p}}+h_{\text{r,}\text{c}- \text{p}}\big{)}} \tag{3}\]  
The convective heat transfer coefficient between the absorber and the glass cover is:  
\[h_{\text{c,}\text{p}-\text{c}}=\frac{Nu\cdot k}{L} \tag{4}\]  
The Nusselt number between the glass cover and the absorber plate due to natural convection is calculated as:  
\[\begin{array}{ll}Nu&=1+1.44[1-1708/\text{Ra}\text{Cos}\beta]^{+}\Big{\{}1- 1708(\sin 1.8\cdot\beta)^{1.6}/\text{Racos}\beta\Big{\}}\\ &+\Big{[}(\text{Ra}\text{Cos}\beta/5830)^{0.33}-1\Big{]}\end{array} \tag{5}\]  
\[Ra=\frac{g\cdot\beta^{\prime}\cdot(T_{\text{p}}-T_{\text{c}})\cdot L^{3}}{ \
u\cdot\alpha} \tag{6}\]  
The radiation resistance between the top cover and the sky accounts for radiative heat exchange with the sky. The radiative heat transfer coefficient is:  
\[h_{\text{r,}\text{c}-\text{a}}=\frac{\sigma\cdot\varepsilon_{\text{c}}(T_{ \text{c}}+T_{\text{s}})\big{(}T_{\text{c}}^{2}+T_{\text{s}}^{2}\big{)}(T_{ \text{c}}-T_{\text{s}})}{(T_{\text{c}}-T_{\text{a}})} \tag{7}\]  
The thermal resistance to the ambient is:  
\[R_{1}=\frac{1}{(h_{\text{r,}\text{c}-\text{a}}+h_{\text{w}})} \tag{8}\]  
Figure 4: Thermal resistance of SAH.  
The SAH-to-ambient heat transfer coefficient is calculated as:  
\[U_{\rm t}=\frac{1}{(R_{1}+R_{2})} \tag{9}\]  
For the edge and back sides, the heat transfer coefficients are calculated as follows:  
\[U_{\rm e}=\frac{1}{(R_{4})}=\frac{(UA)_{\rm edge}}{A_{\rm p}}\ \rm{and}\ U_{\rm b}= \frac{1}{(R_{3})}=\frac{k_{\rm ins}}{L^{\prime}} \tag{10}\]  
The overall heat transfer coefficient for thermal losses is:  
\[U_{\rm o}=U_{\rm t}+U_{\rm b}+U_{\rm e} \tag{11}\]  
Heat gain to the air stream is:  
\[Q_{\rm u}=\big{[}I(\alpha\cdot\tau)-U_{\rm o}\big{(}T_{\rm p}-T_{\rm a}\big{)} \big{]}A_{\rm p} \tag{12}\]  
The thermal efficiency is:  
\[\eta_{\rm th}=\frac{Q_{\rm u}}{A_{\rm p}\cdot I} \tag{13}\]