## 4 Results and Discussion  
Figure 7 indicates that \(Re\), \(Nu\), and \(F_{R}\) show an upward trend by increasing \(\dot{m}\) from 0.01 to 0.05 (kg s\({}^{-1}\)) for both cross-sections. The Nusselt number values for laminar flow (\(Re<2300\)) are less than 10, while they increase approximately two times for the Reynolds number greater than 2300 (non-laminar flow). It means that the heat transfer characteristics of the working fluid (water) will improve under non-laminar flow rates. The Nusselt number is slightly higher for the elliptical cross-section. Thus, the heat transfer will be higher compared to the circular one. The highest values of the heat removal factor correspond to the elliptical and circular cross-section under turbulent flow (\(\dot{m}=0.05\) (kg s\({}^{-1}\))) with 0.539 and 0.538, respectively. \begin{table}
\begin{tabular}{c c c} \hline
**Symbols** & **Parameters** & **Values** \\ \hline L\({}_{\rm S}\) & Length of the serpentine segments & 1.1 (m) \\ W & Serpentine spacing & 0.08 (m) \\ N & Number of segments in the serpentine tube & 10 \\ A\({}_{\rm c}\) & Collector area & 0.88 (m\({}^{2}\)) \\ C\({}_{\rm P}\) & Specific heat capacity of water at 25 \({}^{\circ}\)C & 0.6 (J kg\({}^{-1}\) K\({}^{-1}\)) \\ K\({}_{\rm w}\) & Thermal conductivity of water at 25 \({}^{\circ}\)C & 0.6 (W m\({}^{-1}\) K\({}^{-1}\)) \\ T\({}_{\rm a}\) & Ambient temperature & 298 (\({}^{\circ}\)K) \\ T\({}_{\rm m}\) & Mean plate Temperature & 297.5 (\({}^{\circ}\)K) \\ T\({}_{\rm i}\) & Inlet fluid temperature & 295 (\({}^{\circ}\)K) \\ \(\theta\) & Collector tilt & 15\({}^{\circ}\) \\ L\({}_{\rm bl}\) & Collector back length & 1.3 (m) \\ L\({}_{\rm bw}\) & Collector back width & 1 (m) \\ P\({}_{\rm c}\) & Collector perimeter & 4.8 (m) \\ h\({}_{\rm w}\) & Wind heat transfer coefficient & 10 (W m\({}^{-2}\) K\({}^{-1}\)) \\ H\({}_{\rm d}\) & Dynamic viscosity at 25 \({}^{\circ}\)C & 8.9 \(\times\) 10\({}^{-4}\) (Pa-s) \\ H\({}_{\rm b}\) & The bulk viscosity at 25 \({}^{\circ}\)C & 2.47 \(\times\) 10\({}^{-3}\) (Pa-s) \\ H\({}_{\rm w}\) & The wall viscosity at 25 \({}^{\circ}\)C & 1.00 \(\times\) 10\({}^{-3}\) (Pa-s) \\ \(\sigma\) & [PERSON] constant & 5.67 \(\times\) 10\({}^{-8}\) (W m\({}^{-2}\) K\({}^{-4}\)) \\ S & Absorbed solar energy & 1000 (W m\({}^{-2}\)) \\ \(\rho\) & Density of water at 25 \({}^{\circ}\)C & 997 (kg m\({}^{-3}\)) \\ \hline \end{tabular}
\end{table} 
Table 3: Values of different parameters to determine FR. As shown in Figure 8, raising \(\dot{m}\) from 0.01 to 0.05 (kg s\({}^{-1}\)) will lead to a higher \(h_{fi}\), \(F_{R}\), \(Q_{u}\), and \(\mu_{Th}\) for both cross-sections. It means that the cooling effect of water will increase by mass flow rate directly. The highest values for heat removal factor, useful energy gain, and thermal efficiency are possible when the heat transfer coefficient is maximum and vice versa for both cross-sections. All heat transfer parameters increase significantly by changing \(\dot{m}\) from 0.02 to 0.03 (kg s\({}^{-1}\)). It is due that water flow moves from laminar to transition states at these flow rates. The largest amounts of useful energy gain (493.8 W and 493 W) and thermal efficiency (56% and 55.8%) correspond to the elliptical and circular cross-sections, respectively, with \(\dot{m}=\)0.05 (kg s\({}^{-1}\)) (turbulent flow). Figure 7: Effects of \(\dot{m}\) on the \(Re\), \(Nu\), and \(F_{R}\). As it can be seen from Figure 9, \(Re\) values directly impact the heat transfer governing factors of the system, namely \(h_{fi}\), \(F_{R}\), \(Q_{u}\), and \(\mu_{Th}\). \(h_{fi}\) rises approximately three times by increasing \(Re\) from laminar to turbulent flow for both cross-sections. All governing parameters increase sharply by changing laminar to transition flow (\(Re\) > 2300). It is due that the Nusselt number (Figure 7) will increase significantly by altering laminar to transition flow; however, they raise smoothly under laminar flow (\(Re<2300\)) and from transition (\(2300<Re<4000\)) to turbulent (\(Re>4000\)) flow rates.