## 2 Mathematical Modeling and Simulating
### Modeling
#### 2.1.2 Heat Pipe Evacuated Tube Solar Collectors  
A diagram of an evacuated heat pipe solar collector and a diagram of a wickless heat pipe using the thermosiphon effect are shown in Figure 3a,b, respectively. The modeling of this collector was developed, as suggested by [46].  
The useful heat of the evacuated tube solar collector with heat pipes is determined as follows:  
\[Q_{u}=F_{R}A_{c}[I\tau\alpha-U_{L}(T_{i}-T_{a})], \tag{21}\]  
The efficiency of the evacuated tube solar collector with heat pipes is calculated as follows:  
\[\eta_{col}=F_{R}(\tau\alpha)-F_{R}U_{L}\bigg{(}\frac{T_{i}-T_{a}}{l_{t}} \bigg{)}, \tag{22}\]  
The collector heat removal factor can be calculated using Equation (3), where:  
\[F^{\prime}=\frac{1}{1+R_{llr}+R_{cr}}, \tag{23}\]  
The thermal resistant, \(R_{llr}\) and \(R_{cr}\), are considered by the following:  
\[R_{llr}=\frac{R_{hp}}{R_{rad}}, \tag{24}\]  
\[R_{cr}=\frac{R_{cond}}{R_{rad}}, \tag{25}\]  
Figure 3: (**a**) Diagram of the evacuated heat pipe collector. (**b**) Diagram of the heat pipe.  
The thermal resistance of the evaporator is represented by \(R_{hp}\) and is a combination of three resistances: the resistance of the circular fin, the resistance of the copper tube, and the boiling film condensation resistance inside the evaporator tube. Therefore, \(R_{hp}\) can be written as follows:  
\[R_{hp}=\frac{\ln\left(\frac{D_{g,fin}}{D_{j,fin}}\right)}{2\pi k_{fin}L_{e}}+ \frac{\ln\left(\frac{D_{g,fin}}{D_{j,fin}}\right)}{2\pi k_{c}L_{e}}+\frac{1}{h_{e }A_{e}}, \tag{26}\]  
where, \(D_{g,fin}\), \(D_{i,fin}\), represent the outer and inner diameter of the evaporator, respectively. The values for \(L_{e}\), \(k_{e}\), and \(A_{e}\) indicate the length, conductivity, and area of the evaporator, respectively.  
The thermal resistance of the condenser section, \(R_{cond}\), which is the sum of the film condensation resistance in the condenser, the copper pipe section of the condenser, and the dry manifold condenser resistance. The thermal contact resistance in the dry condenser is assumed to be insignificant.  
\[R_{cond}=\frac{\ln\left(\frac{D_{g,cm}}{D_{i,c}}\right)}{2\pi k_{c}L_{c}}+ \frac{\ln\left(\frac{D_{g,cm}}{D_{j,cm}}\right)}{2\pi k_{mant}L_{mm}}+\frac{1}{ h_{e}A_{cond}}, \tag{27}\]  
where \(D_{g,c}\) and \(D_{i,c}\), are the outside and inside diameter of the condenser, and \(D_{o,mm}\), \(D_{i,man}\), \(k_{mm}\), and \(L_{mm}\) are the outer and inner diameter, conductivity, and length of the manifold, respectively.  
The thermal resistance due to radiation is \(R_{rad}\), which can be calculated and represented as follows:  
\[R_{rad}=\frac{\left(T_{p}-T_{a}\right)}{e\sigma\left(T_{p}^{4}-T_{a}^{4}\right) A_{r}}, \tag{28}\]  
Evaporator Heat Transfer Coefficient  
The convection heat transfer coefficient in the evaporator, he, is determined as follows:  
\[h_{e}=0.555\Bigg{[}\frac{g\rho_{l}(\rho_{l}-\rho_{v})h_{fg}k_{l}^{3}}{\mu_{l} \left(T_{p}-T_{k}\right)D_{e}}\Bigg{]}^{1/4}, \tag{29}\]  
where \(g\), is the gravity and \(T_{p}\) and \(T_{k}\) are temperature of the plate and sky, respectively; \(\rho_{l}\), \(k_{l}\), and \(\mu_{l}\) are the density, conductivity, and viscosity of the liquid, respectively; \(\rho_{v}\), \(h_{fg}\), are the vapor density and phase change enthalpy, respectively.  
Condenser Heat Transfer Coefficient  
The convection heat transfer coefficient in the condenser, \(h_{c}\), is determined as follows:  
\[h_{c}=0.555\Bigg{[}\frac{g\rho_{l}(\rho_{l}-\rho_{v})h_{fg}k_{l}^{3}}{\mu_{l} \left(T_{p}-T_{k}\right)D_{c}}\Bigg{]}^{1/4}, \tag{30}\]