## 3 Design of the Flat Plate Solar Collectors with Elliptical and Circular Serpentine Tubes  
Figures 4 and 5 indicate the proposed elliptical and circular serpentine tubes in 3D and cross-sectional views. The technical manufacturing data and design parameters are summarized in Tables 2 and 3. Table 2 shows the materials' optical and thermal properties of the collector's components, whereas the collector's dimension and contributing parameters to determine \(F_{R}\) are listed in Table 3.  
An analytical investigation was conducted on each design of an elliptical and a circular serpentine solar collector. Figure 6 represents the preliminary design of an elliptical and a circular serpentine solar collector using water as the working fluid. It is worth mentioning that all numerical calculation was carried out under the standard test conditions (\(S=1000\) W m\({}^{-2}\), \(T_{a}=25\)\({}^{\circ}\)C). To investigate non-circular cross-sections such as elliptical,  
Figure 4: 3D view of the proposed flat plate serpentine designs.  
Figure 5: The cross-section view is (**a**) a circular serpentine tube and (**b**) an elliptical serpentine tube.  
the authors used the concept of hydraulic diameter (\(D_{h}\)). For an ellipse, the hydraulic diameter is defined by Equation (25) [57]:  
\[D_{h}=\frac{a\times b}{\sqrt{\frac{a^{2}+b^{2}}{2}}} \tag{25}\]  
where \(a\) and \(b\) are the minor and major axes of the ellipse, respectively. For a circle, \(D_{h}\) is equivalent to circle diameter. To investigate the role of working fluid patterns within the tubes with different cross-section geometry, the Reynolds number (\(Re\)) assumed laminar (\(Re<2300\)), transition (\(2300<Re<4000\)), and turbulent (\(Re>4000\)). The bond conductance (\(C_{b}\)) is an important parameter in evaluating solar thermal collector performance. The experiments have proved that good contact between the absorbing plate and the tube in the thermal collectors leads to a bond conductance greater than 30 W/m \({}^{\circ}\)C and prevents significant heat loss [53]. In this study, it was supposed that the absorbing plate was bound to the tubes effectively. Thus, the term (\(\frac{1}{C_{b}}\)), in Equation (17), is negligible.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Components** & **Material** & **Thickness (\%)** & **Optical and Thermal Properties** \\ \hline Metal box & Aluminum & 35 mm & \(K_{Aluminum}=205\left(\mathrm{W\,m^{-1}K^{-1}}\right)\) \\ Cover & Glass with anti-reflecting coating & 5 mm & \(\varepsilon_{g}=0.88\)\(N_{\mathrm{g}}=1\) \\ Metal absorber plate & Zinc oxide & 1 mm & \(\varepsilon_{p}=0.13\), \(K_{p}=60\)\(\left(\mathrm{W\,m^{-1}K^{-1}}\right)\) \\ Serpentine tubes & copper & 1 mm & \(K_{Copper}=385\)\(\left(\mathrm{W\,m^{-1}K^{-1}}\right)\) \\ Back insulation & Glass wool & 20 mm & \(K_{bin}=0.04\)\(\left(\mathrm{W\,m^{-1}K^{-1}}\right)\) \\ Side insulation & Glass wool & 20 mm & \(K_{cin}=0.04\)\(\left(\mathrm{W\,m^{-1}K^{-1}}\right)\) \\ \hline \end{tabular}
\end{table}
Table 2: Technical data on manufacturing serpentine tubes with elliptical and circular cross-sections.  
Figure 6: Schematic diagram of the proposed flat plate serpentine design with N = 10, W = 0.08 m, and Ls = 1.1 m.