## 2 Materials and Methods
### Evaluation of Heat Removal Factor
\[F_{R}=F_{1}\times F_{3}\times F_{5}\times\left[\frac{2F_{4}}{F_{6}\times exp \left(\frac{-\sqrt{1-F_{2}^{2}}}{F_{3}}\right)+F_{5}}-1\right] \tag{3}\]  
\[F_{1}=\frac{NkL_{S}}{U_{L}A_{C}}\frac{kR(1+\gamma)^{2}-1-\gamma-kR}{\left[kR (1+\gamma)-1\right]^{2}-\left(kR\right)^{2}} \tag{4}\]  
\[F_{2}=\frac{1}{kR(1+\gamma)^{2}-1-\gamma-kR} \tag{5}\]  
\[F_{3}=\frac{\dot{m}C_{P}}{F_{1}U_{L}A_{C}} \tag{6}\]  
\[F_{4}=\left[\frac{1-F_{2}^{2}}{F_{2}^{2}}\right]^{\frac{1}{2}} \tag{7}\]  
\[F_{5}=\frac{1}{F_{2}}+F_{4}-1 \tag{8}\]  
Figure 3: Schematic diagram of a flat plate thermal collector using serpentine design [32, 49, 50, 51, 52, 53, 54, 55]. \[F_{6}=1-\frac{1}{F_{2}}+F_{4} \tag{9}\]  
where the parameters \(n,\ k\), and \(\gamma\) are non-dimensional quantities and can be calculated as follows [39; 40; 41; 42; 43; 44]:  
\[n =\sqrt{\frac{U_{L}(W-D_{o})^{2}}{K_{p}\delta_{p}}} \tag{10}\] \[F_{R}=F_{1}\times F_{3}\times F_{5}\times\left[\frac{2F_{4}}{F_{6}\times exp \left(\frac{-\sqrt{1-F_{2}^{2}}}{F_{3}}\right)+F_{5}}-1\right] \tag{3}\]0 (11) \[F_{R}=F_{1}\times F_{3}\times F_{5}\times\left[\frac{2F_{4}}{F_{6}\times exp \left(\frac{-\sqrt{1-F_{2}^{2}}}{F_{3}}\right)+F_{5}}-1\right] \tag{3}\]1  
The value of the total heat loss coefficient (\(U_{L}\)) is obtained through subsequent equation [15; 42]:  
\[U_{L}=\left( \begin{array}{c}\frac{N_{g}}{\frac{\varepsilon}{m}\left[\frac{T_ {g}-T_{g}}{K_{p}+f}\right]^{\alpha}}+\frac{1}{h_{w}}\end{array} \right)^{-1}+ \left( \begin{array}{c}\frac{\sigma(T_{m}+T_{o})\left(T_{m}^{2}+T_{g}^{2} \right)}{\frac{1}{F_{p}^{2}\gamma^{2}\cosh{n}}+\frac{2N_{g}+f-1+0.13k_{p}}{K_ {g}}-N_{g}}\end{array} \right) \tag{13}\]  
\[ \begin{array}{c}c=520\Big{(}1-0.000051\theta^{2}\Big{)}\ for\ 0<\theta<70^{\circ} \end{array}  \tag{14}\]  
\[e=0.430\Big{(}1-\frac{100}{T_{m}}\Big{)} \tag{15}\]  
\[f=\left(1+0.089h_{w}-0.1166h_{w}e_{p}\right)\left(1+0.07866N_{g}\right) \tag{16}\]  
The parameter \(R\) is the thermal resistance between the absorbing plate and working fluid, and is written as [29; 39; 40; 41; 42; 43; 44]:  
\[R=\frac{1}{\pi h_{fi}D_{i}}+\frac{1}{C_{b}} \tag{17}\]  
where, \(h_{fi}\) is the heat transfer coefficient:  
\[h_{fi}=\frac{(Nu)K_{w}}{D_{i}} \tag{18}\]  
In Equation (18), the Nusselt number \((Nu)\) is gained by Equations (19)-(21) [55; 56]:  
\[Nu =1.86+\left(\frac{RePrD_{i}}{L}\right)^{\frac{1}{3}}\bigg{(}\frac{h_{b}}{ \mu_{w}}\bigg{)}^{0.14}\ for\ laminar\ flow \tag{19}\]  
\[Nu =0.023(Re)^{\frac{2}{3}}(\text{Pr})^{0.4}\bigg{(}\frac{\mu_{b}}{\mu_{w}} \bigg{)}^{0.14}\ for\ transition\ flow \tag{20}\]  
\[Nu =0.023(Re)^{0.8}(\text{Pr})^{\frac{1}{3}}\bigg{(}\frac{\mu_{b}}{\mu_{w}} \bigg{)}^{0.14}\ for\ turbulent\ flow \tag{21}\]  
while the Reynolds (\(Re\)) and the Prandtl (\(Pr\)) numbers are defined by Equations (22) and (23) [55]:  
\[Re =\frac{D_{i}V\rho}{\mu_{d}} \tag{22}\] \[Pr =\frac{\mu_{d}C_{p}}{K_{w}} \tag{23}\]The thermal efficiency (\(\mu_{Th}\)) of the collector is calculated by Equation (24) [30]:  
\[\eta_{Th}=\frac{Q_{u}}{A_{C}S} \tag{24}\]  
where, \(Q_{u}\), \(A_{C}\), and \(S\) denotes the useful energy gain, collector area, and absorbed solar energy, respectively.