## 2 Materials and Methods  
### Evaluation of Heat Removal Factor  
The performance of a flat plate solar collector can be defined by several different operational and design factors [33]. The instantaneous thermal energy per unit time or the useful energy gain (\(Q_{u}\)) generated by an FPSC can be expressed using the Hottel and Whillier-Bliss Equation [32; 49; 50], as in Equation (1):  
\[Q_{u}=A_{C}F_{R}[(G_{T}(\tau\alpha)-U_{L}(T_{i}-T_{a}))] \tag{1}\]  
According to the Hottel and Whillier-Bliss Equation, the contributing variables are the incident solar flux (\(G_{T}\)), the heat removal factor (\(F_{R}\)), the collector area (\(A_{C}\)), the total heat loss coefficient (\(U_{L}\)), and the effective transmittance-absorptance product of the cover system (\(\tau\alpha\)). \(T_{a}\) and \(T_{i}\) are the ambient and the fluid inlet temperature, respectively. \(S\) is equal \(G_{T}\) (\(\tau\alpha\)) and represents the absorbed solar energy per unit time per unit area of the collector and is equal to \(G_{T}\) (\(\tau\alpha\)). Thus, \(F_{R}\) can be expressed as in Equation (2) [32; 51]:  
\[F_{R}=\frac{\dot{m}C_{P}(T_{o}-T_{i})}{A_{C}[S-U_{L}(T_{i}-T_{a})]} \tag{2}\]  
In Equation (2), the working fluid's mass flow rate is expressed by \(\dot{m}\). \(C_{p}\) and \(T_{o}\) denote the fluid specific heat and the outlet temperature, respectively. If \(F_{R}\) and \(U_{L}\) values are computed appropriately, it is possible to calculate the useful energy gain of the FPSCs (Equation (1)). \(F_{R}\) determines the effectiveness of a solar thermal collector design, and all factors that contribute to FPSC efficiency are combined into it [51; 52]. It is explained as the ratio of actual \(Q_{u}\) to the maximum possible \(Q_{u}\) of a collector. The highest possible \(Q_{u}\) (heat transfer) is achieved when the collector temperature is the same as the working fluid's \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Authors** & **Year of Study** & **Cross-Section** & **Type of Study** & **Location** \\ \hline [PERSON] et al. [44] & 2020 & Circular & Experimental & China \\ [PERSON] and [PERSON] [30] & 2018 & Circular & theoretical & Greece \\ [PERSON] et al. [45] & 2018 & Circular & Experimental and simulation & India \\ [PERSON] et al. [46] & 2017 & Circular & Experimental & Malaysia \\ [PERSON] et al. [47] & 2016 & Circular & theoretical and experimental & Malaysia \\ [PERSON] et al. [23] & 2014 & Circular, rectangular, and square & Theoretical & Malaysia \\ [PERSON] et al. [48] & 2009 & Circular & Theoretical and simulation & Malaysia \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Literature reviews regarding serpentine designs with different cross-sections. inlet temperature and the collector's heat loss is minimum. Figure 3 shows the schematic diagram of a flat plate serpentine tube collector with inside and outside diameters of \(D_{i}\) and \(D_{o}\), respectively. The collector consists of an absorbing metal plate with a thickness of \(\delta_{P}\) and thermal conductivity of \(K_{P}\). The N segments of the serpentine tube with a spacing of \(W\) and length of \(L_{s}\) are attached to the plate. The first and last segments are placed at a distance of \(\frac{W}{2}\) from their respective ends of the plate. The area (\(A_{C}\)) of the collector was described as \(A_{C}=W\times N\times L_{s}\)[32, 49, 50, 51, 52, 53, 54, 55]. Due to a serpentine collector's complicated heat transfer model, the heat removal factor cannot be simply expressed using Equation (2). Several authors [51, 52, 53, 54, 55] have proposed various analytical solutions and modifications to obtain the heat removal factor of a serpentine design during the last four decades. [PERSON] and [PERSON] [32, 49] used matrix differential equations to modify the serpentine flow equations [44]. They pointed out that \(F_{R}\) can be determined through a set of non-dimensional parameters \(F_{1}\)-\(F_{6}\) within Equation (3) as defined below. Equations (3)-(9) are proper for any number of bends if \(F_{3}\) is greater than about 1 [32, 53].