## 2 Mathematical Model
### Heat Pipe Model  
The proper function of the heat pipe depends upon the pressure drop in the fluid flow embedded in the heat pipe that has to be compensated by the pumping pressure in the wick and the capillarity, as prescribed by [PERSON] [16; 31], [PERSON] [33], [PERSON] [19], and [PERSON] [32]:  
\[\Delta P_{p}=\Delta P_{i}+\Delta P_{v}+\Delta P_{g} \tag{18}\]  
where \(\Delta P_{p}\), \(\Delta P_{l}\), \(\Delta P_{v}\), and \(\Delta P_{g}\) are the total pumping pressure, pressure drop for liquid return from the condenser, pressure drop for vapor flow in the evaporator, and gravity head, respectively. \"The heat transfer limit for a heat pipe depends on the construction of the heat pipe and the operating environment. The design of the heat pipe and the wick properties are determined by the thermophysical properties of the working fluid used [32]. The heat pipe heat transfer capillary, sonic, entrainment, boiling, frozen start-up, continuum vapor, vapor pressure, and condenser effects determine the physical phenomena that establish the lowest limit of these phenomena and are considered as a design limit. The main design limitations were extensively discussed in [PERSON] [33], [PERSON] [32], [PERSON] [34], and [PERSON] [16]\". Readers interested in the capillary, sonic, entrainment, and boiling limits of the heat pipes are advised to consult the aforementioned references and others reported in the literature. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Properties** & **O** & **CuO** & **Fe\({}_{3}\)O\({}_{4}\)** & **SiO\({}_{2}\)** \\ \hline \(C_{pnf}\) & b = 0.1042a + 6226.5 & b = 0.2011a + 5730.8 & b = 0.8318a + 4269.8 & b = 0.6187a + 4293.2 \\ \(K_{nf}\) & b = 2 \(\times\) 10\({}^{-5}\)a + 1.4888 & b = 5 \(\times\) 10\({}^{-5}\)a + 1.3703 & b = 0.0002a + 1.0209 & b = 0.0001a + 1.0265 \\ \(h\) & b = 0.0031a + 73.092 & b = 0.0031a + 73.073 & b = 0.003a + 73.225 & b = 0.003a + 73.231 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Thermophysical properties as a function of magnetic field forces in Gauss. \"Several factors are considered in selecting the working fluid inside the heat pipe, in particular, the working fluid temperature range is an important criterion to be fulfilled. Refrigerants and refrigerant mixtures are widely used as working fluids for low-temperature heat pipe applications because of lower vapor pressure and boiling points\". \"The energy conversion and efficiency of heat transfer from the evaporator section of the heat pipe to the condenser section in the solar application is one of the most important selection criteria for the working fluid. Hence, the use of working fluid with higher latent heat is very beneficial to applications of heat pipes in solar energy conversion\". Water and different refrigerants, such as R-134a, R-123, R-32, R-125, R-152a, R-1234 ze, and R-1234z, as well as refrigerant mixtures [38], are considered in this investigation as working fluids inside the heat pipe. \"A heat pipe using two-phase thermosyphon principles has been reported in the literature by [PERSON] [19] in water heating applications. References [14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32]\" are also discussed. In the current study, the \"two phases were in forced convective condensers under either a crossflow or parallel flow pattern. In this study, the thermosyphon condition was under consideration, as illustrated in Figure 1. The condenser section of the heat pipe was placed in a thermal storage tank where natural convective heat transfer took place. It was assumed that the condensing water flow in the thermal tank was stagnant, and there was no temperature gradient along the axis of the heat pipe\".