## 2 Mathematical Modeling and Simulating
### Simulation
#### 2.3.2 Assumptions  
The assumptions made for each of the collectors are presented below for each of the solar collector systems. Flat plate solar collectors:  
* The absorber area of the collector is always at a uniform temperature. * The temperature gradient along the thickness of the absorber and around its perimeter is neglected. * Since the evaporator section is subject to a constant heat flux, and phase change occurs at a constant temperature, the temperature gradient along the longitudinal direction is negligible. \begin{table}
\begin{tabular}{c c c c c} \hline
**Parameter** & **FPC** & **HPETC** & **HPFFC** \\ \hline Mass flow rate, \(\bar{m}\), kg/s & 0.03–0.06 & 0.03–0.06 & 0.03–0.06 \\ Radiation, \(I\), W/m2 & 362–800 & 362–800 & 362–800 \\ Ambient temperature, \(T_{a}\), \({}^{\circ}\)C & 25–34.4 & 25–34.4 & 25–34.4 \\ Fluid temperature, \(T_{i}\), \({}^{\circ}\)C & 26–80 & 26–80 & 26–80 \\ \hline \end{tabular}
\end{table} 
Table 4: Range of evaluated parameters. \begin{table}
\begin{tabular}{c c c c} \hline
**Parameter** & **FPC** & **HPETC** & **HPFFC** \\ \hline Collector area, \(A_{c}\), \([\text{m}^{2}]\) & 1.13 & 1.13 & 1.13 \\ Number of risper pipes, N & 6 & 6 & 6 \\ Length of collector, \(L_{col}\) \([\text{m}]\) & 1.13 & 1.13 & 1.13 \\ Width of collector, \(W_{col}\) \([\text{m}]\) & 1 & 1 & 1 \\ Internal pipe diameter, \(D_{inv}\), \([\text{m}]\) & 0.013 & & \\ External pipe diameter, \(D_{out}\), \([\text{m}]\) & 0.016 & & \\ Width between pipes, \(w\) \([\text{m}]\) & 0.166 & 0.166 & 0.166 \\ Plate thickness, \(\delta\) \([\text{m}]\) & 0.0002 & & 0.0002 \\ Thermal conductivity of the plate, \(K_{p}\)\([W/mK]\) & 384 & & 384 \\ Thermal conductivity of the insulation, \(K_{i}\)\([W/mK]\) & 0.038 & & 0.038 \\ Thickness of the insulation, \(L_{inv}\)\([\text{m}]\) & 0.0254 & & 0.0254 \\ Transmittance/Absorbance, \(\alpha\)\(\alpha\), taken from [27] & 0.84 & 0.84 & 0.84 \\ Condenser outside diameter \([\text{m}]\) & & 0.0155 & 0.0155 \\ Evaporator outside diameter, \([\text{m}]\) & & 0.008 & 0.008 \\ Condenser length, \([\text{m}]\) & & 0.1 & 0.1 \\ Evaporator length, \([\text{L}_{c},[\text{m}]\) & & 1.13 & 1.13 \\ Glass cover, \(e\), \([\text{m}]\) & & 0.005 & 0.005 \\ Manifold inlet diameter, \(D_{inv}\), \([\text{m}]\) & & 0.0127 & 0.0127 \\ \hline \end{tabular}
\end{table} 
Table 2: Basic characteristics and specifications of the different evaluated solar collectors. \begin{table}
\begin{tabular}{c c c c c} \hline
**Parameter** & **Water [47]** & **EG 30\% [51]** & **PG 30\% [52]** & **EG50\% [51]** & **PG50\% [52]** \\ \hline Freezing temperature, \({}^{\circ}\)C & 0 & \(-14.44\) & \(-12.77\) & \(-36.11\) & \(-31.11\) \\ Boiling temperature, \({}^{\circ}\)C & 100 & 104.44 & 102.22 & 107.22 & 104.44 \\ Density, \([\text{kg}/\text{m}^{3}]\) & 997.1 & 1044.1 & 1026.6 & 1071.8 & 1040.1 \\ Specific heat, \(D_{inv}\), \([\text{m}]\) & 4.183 & 3.6634 & 3.8686 & 3.3452 & 3.5587 \\ Conductivity, \(L_{inv}\)\([\text{m}]\) & 0.5948 & 0.471 & 0.3243 & 0.3842 & 0.3478 \\ Viscosity, \(\mu\)\([\text{Pa}\text{ s}]\) & 0.0008905 & 0.00184 & 0.00243 & 0.00325 & 0.00505 \\ \hline \end{tabular}
\end{table} 
Table 3: Thermophysical properties of evaluated working fluids. The thermal contact resistance between the absorber area and the evaporator tube, and between the condenser section and the manifold, is neglected. The connection between the evaporator and condenser of the heat pipe is assumed to be adiabatic. Thermosiphon evacuated solar collectors:  
The mathematical model is one-dimensional. The thermal resistance caused by the welding between the absorber and the heat pipes is neglected. Inside the heat pipe, the vapor temperature is the same as the saturation value at the given pressure. The thermophysical properties of all solid substances are constant. The properties of the working fluid depend on the temperature. Thermosiphon flat plate solar collectors:  
Fully developed flow and stable conditions. The ambient temperature at the top and bottom of the collector are the same.