## 2 Mathematical Model
### Heat Pipe Model
The energy balance under a natural convection heat transfer condition in the thermal storage tank, as per Figure 1, using a single control volume of heat pipe submerged in the thermal tank, is presented in Equation (18) [13; 16]:  
\[V_{w}P_{w}C_{w}\frac{dT_{w}}{dt}=\pi d_{o}I_{cond}h_{eff}\Big{(}T_{hp}-T_{w} \Big{)}-U_{\text{tan}}A_{\text{tan}}\Big{(}T_{hp}-T_{a}\Big{)} \tag{19}\]  
where \(V_{w}\) represents the water volume in the thermal tank and \(U_{\text{tan}}\) and \(A_{\text{tan}}\) are the overall heat transfer coefficient in the thermal tank and the equivalent heat transfer area in the tank, respectively. In addition, \(T_{hp}\), \(T_{w}\), and \(T_{a}\) are the temperatures of the heat pipe, water, and ambient air. On the other hand, the effective heat transfer coefficient \(h_{eff}\) of the condenser section of the heat pipe can be given by the following [13; 16]:  
\[h_{eff}=1/\left(\frac{1}{h_{cond}}+\frac{1}{h_{f}}+\frac{\delta_{hp}}{k_{hp}} \right) \tag{20}\]  
where \(h_{f}\) is the natural heat transfer coefficient \(h_{f}\) between the water and the condenser in the thermal tank, \(\delta_{hp}\) represents the thickness of the heat pipe material, and \(k_{hp}\) is the thermal conductivity of the working fluid in the heat pipe. Additionally, \(h_{cond}\) represents the film condensation heat transfer coefficient and can be obtained from the following correlation suggested by references [13; 25; 26; 27; 28] and using the Nusselt equation:  
\[h_{cond}=0.729\left(\frac{K_{l}^{3}\lambda(\rho_{l}-\rho_{r})g\sin\beta}{\mu_{ l}d_{i}\Big{(}T_{hp}-T_{w}\Big{)}}\right)^{\frac{1}{4}} \tag{21}\]  
where \(T_{hp}\), \(T_{w}\), represent the temperatures of the heat pipe and the water in the tank, respectively, and \(\rho_{l}\), \(\rho_{r}\) are the liquid and vapor densities of the working fluid inside the heat pipe, respectively. Additionally, \(\mu_{l}\), \(d_{i}\) are the viscosity and internal diameter of the heat pipe, respectively. The natural heat transfer coefficient, \(\eta_{t}\), between the water in the thermal tank and the condenser section of the heat pipe, can be developed using the Nusselt equation [13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32] as follows:  
\[\frac{h_{f}d_{o}}{k_{l}}=1.09\left(\frac{d_{o}^{3}\rho_{l}^{2}g \gamma\Big{(}T_{hp}-T_{w}\Big{)}C_{p}}{\mu_{l}k_{l}}\right)^{\frac{1}{5}} \tag{22}\]  
where \(d_{o}\) represent the outside diameter of the condenser section of the heat pipe and \(C_{p}\) is the specific heat of the working fluid inside the heat pipe. Additionally, \(\gamma\) is the specific heat ratio. \(\mu\) represents the liquid viscosity of the working fluid. To determine the heat capacity of the condenser section of the heat pipes, the following energy and mass balance equations are considered:  
\[Q_{cond_{hp}}=H_{eff}\times\Big{(}T_{hp}-T_{w}\Big{)}\times d_{o }\times I_{o}\times\pi \tag{23}\]  
where \(I_{o}\) is the length of the condenser section of the heat pipe. The efficiency \(\eta\) of the heat exchanger section of the heat pipes can be calculated as:  
\[Q_{evap_{hp}}=Q_{th}\times\eta \tag{24}\]  
On the other hand, the thermal energy dissipated into the thermal tank from the heat pipes and delivered to the domestic or industrial end user is:  
\[Q_{tt}=\eta_{hx}\times m_{w_{Qtt}}\times C_{p_{w}}\times(T_{12}-T_{13}) \tag{25}\]  
where the \(m_{w_{Qtt}}\) represents the water mass flow rate circulating between the thermal tank and the user application in question. \(T_{12}\) and \(T_{13}\) are the supply and return temperatures for the end user application, respectively. \(\eta_{hx}\) is the thermal tank efficiency. The water mass flow rate for supplying thermal energy to the end user application is:  
\[m_{w_{Qtt}}=\frac{Q_{tt}}{\eta_{hx}\times c_{p_{w}}\times(T_{12}-T_{13})} \tag{26}\]  
The efficiency of the solar PV panels can be expressed as follows:  
Figure 1.0  
where \(Q_{elec}\) is calculated by Equation (2) and \(Q_{collector}\) is obtained by Equation (3).