## 2 Mathematical Modeling and Simulating
### Model Validation  
The mathematical model presented in Section 2.1 was programmed in the specialized software Engineering Equation Solver (EES), which solves and identifies a set of equations  
Figure 4: (**a**) Diagram of the heat pipe. (**b**) Diagram of the evacuated heat pipe collector.  
and/or simultaneous operations [47]. The mathematical models were validated by comparing the results obtained in this work with those published by [30; 46; 48] for flat plate, evacuated tube, and flat plate collectors with heat pipes, respectively, using the data in Table 1. The values of radiation, as well as ambient and entrance temperature to collector were 750 W/m\({}^{2}\), 23 \({}^{\circ}\)C and 25 \({}^{\circ}\)C, respectively.  
Figure 5 shows the validation results, with the relative error for each collector evaluated. From Figure 5, the maximum error was less than 6.2%, \(-\)4.5%, and 0.8% for the flat plate, evacuated tube, and flat plate collectors with vacuum tubes, respectively. Figure 5a shows the efficiency behavior with respect to mass flow rate, where the same trend as the efficiency results can be seen compared to those shown by [47]. Figure 5b presents the relative error of the results compared to [47]. From the figure, a maximum error of 6.2% is observed. This may be due to the uncertainty in the thermophysical properties of the fluids. Figure 5c shows the collector efficiency with respect to the working fluid inlet temperature. The results were consistent with those of [46]. Figure 5d shows a maximum error of \(-\)4.5%. Furthermore, this figure shows that the error is greater at a low working fluid inlet temperature. As this temperature increases, the error approaches 0%. Figure 5e shows the collector efficiency over a period of the day. The results are very similar to those shown by [30]. Therefore, the error shown in Figure 5f is as low as 0.8%. According to the error shown in the validation and in [49; 50], which state that, if the error is at most 10%, the results present a good validation. Therefore, the mathematical models of Section 2.1, with different operating conditions, were used for the simulation of the three types of collectors evaluated here. Through a sensitivity analysis, it was found that the condensation temperature had the greatest effect on the accuracy of the model.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **FPC**[48] & **HPETC**[46] & **HPFFC**[30] \\ \hline Collector area, \(Ac_{c}\) [m\({}^{2}\)] & 1.13 & 1.78 & 2.83 \\ Number of ris pipes, N & 6 & & \\ Length of collector, \(L_{col}\) [m] & 1.13 & 1.7 & 1.843 \\ Width of collector, \(W_{col}\) [m] & 1 & & 1.54 \\ Internal pipe diameter, \(D_{int}\), [m] & 0.013 & & \\ External pipe diameter, \(D_{out}\), [m] & 0.016 & & \\ Width between pipes, \(w\) [m] & 0.166 & 0.078 & 0.077 \\ Plate thickness, \(\delta\) [m] & 0.0002 & & 0.0015 \\ Thermal conductivity of the plate, \(K_{p}\left[W/mK\right]\) & 384 & & \\ Glass emittance, \(\varepsilon_{g}\) & 0.84 & 0.07 & \\ Plate emittance, \(\varepsilon_{p}\) & 0.05 & & \\ Thermal conductivity of the insulation, \(K_{i}\left[W/mK\right]\) & 0.038 & & \\ Thickness of the insulation, \(L_{ins}\) [m] & 0.0254 & & \\ Transmittance/Absorbance, \(\tau\alpha\), taken from [26] & 0.84 & 0.93 & \\ Condenser outside diameter. (m) & & 0.024 & 0.00635 \\ Evaporator outside diameter & & & 0.008 \\ Condenser length (m) & & 0.008 & 0.078 \\ Evaporator length, \(L_{E}\) & & 1.745 & 1.745 \\ Glass cover, \(e\), [m] & 0.01 & & 0.01 \\ Glass cover transmittance & & 0.9 & \\ Distance glass cover and absorber surface, \(a\) & 0.06 & & 0.06 \\ Manifold diameter, \(d_{man}\) & & 0.035 & 0.035 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Basic characteristics and specifications of the different solar collectors.