## 2 Materials and Methods
### Solar System
The total heat loss rate from the receiver's aperture due to radiation can be determined as:  
\[h_{rad}=\frac{Nu_{rad}\times k_{a}}{D_{rec,\ outer}} \tag{16}\]  
\[\dot{Q}_{loss,\ rad}=h_{rad}A_{rec}(T_{s}-T_{\infty}) \tag{17}\]  
\(\bullet\) Convection heat loss  
The total heat transfer coefficient for convection heat loss from the cavity is determined by using forced and natural convection equations [28]:  
\[h_{forced}=4.22\left(V_{wind}{}^{0.805}\right) \tag{18}\]  
\[h_{natural}=1.45(T_{s}-T_{\infty})^{0.333} \tag{19}\]  
Figure 2:0  
For the hemispherical cavity receiver surface, the convection heat loss rate is presented by Equation (21):  
Figure 2:1  
Numerical methods are used to calculate the net heat transfer rate along the tube length according to Ref. [8]. Note that the fluid enters at the bottom of the investigated cavity receiver. The cavity receivers were divided into smaller elements along the tube length. A view of the location of the elements is presented in Figure 2:2  
Figure 2:3 A view of the location of elements for the hemispherical cavity receiver. The useful heat is calculated as:  
Figure 2:4  
The Nusselt number is determined with Equation (23):  
Figure 2:5  
The friction factor (\(f_{r}\)) is found using following equation:  
Figure 2:6  
Furthermore, the heat transfer coefficient is determined with Equation (25):  
Figure 2:7  
Therefore, based on Equation (1) and using SolTrace to get \(\dot{Q}^{*}_{n}\), the net heat gain can be found using Equations (22) and (26):  
Figure 2:8  
Figure 2:9  
The analysis is performed using real meteorological data from a typical day (20 October 2016) in Tehran, Iran at 12:00 p.m. (see Table 2). It should be emphasized that the receiver tube of the cavity receiver is divided into a number of shorter sections, where each round of cavity tube is assumed as one element. Furthermore, the thermal modeling was developed using Maple software [18]. Oxide nanoparticles including Al\({}_{2}\)O\({}_{3}\), CuO, and SiO\({}_{2}\) (30-50 nm), were examined in the solar system. Table 3 shows the thermal properties of the nanoparticles at ambient temperature of 25 \({}^{\circ}\)C. It should be mentioned that the nanofluids were investigated based on volume fraction of 3%. Note that these nanoparticles have a low specific heat, high thermal conductivity and high density. \begin{table}
\begin{tabular}{c c c} \hline \hline Wind speed (m/s) & 2.1 \\ Solar beam irradiance (W/m\({}^{2}\)) & 632.97 \\ Ambient temperature (\({}^{\circ}\)C) & 20.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Solar irradiance, ambient temperature, and wind velocity on 20 October (2016) at 12:00. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Property** & **Al\({}_{2}\)O\({}_{3}\)** & **SiO\({}_{2}\)** & **CuO** \\ \hline Specific heat (J/kg K) & 765 & 745 & 532 \\ Thermal conductivity (W/m K) & 40 & 1.4 & 77 \\ Density (kg/m\({}^{3}\)) & 3970 & 2220 & 6320 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Properties of nanoparticles [12]. The thermal properties of the nanofluids are determined using Equations (28)-(31), according to the properties of the nanofluid (\(\eta f\)), the nanoparticle (\(\eta p\)) and the base fluid (\(bf\)). The thermal conductivity is calculated as shown below [29], where \(\beta\) (assumed to be 0.1 [30]) is the ratio of the nanolayer thickness to the diameter of the nanoparticles. \[k_{nf}=k_{bf}\cdot\frac{k_{np}+2\cdot k_{bf}+2\cdot\left(k_{np}-k_{bf}\right) \cdot(1+\beta)^{3}\cdot\phi}{k_{np}+2\cdot k_{bf}-\left(k_{np}-k_{bf}\right) \cdot(1+\beta)^{3}\cdot\phi} \tag{28}\]  
The nanofluid density and specific heat capacity is calculated as: ([31; 32]),  
\[\rho_{nf}=\rho_{bf}\cdot(1-\phi)+\rho_{up}\cdot\phi \tag{29}\]  
\[c_{p,nf}=\frac{\rho_{bf}\cdot(1-\phi)}{\rho_{nf}}\cdot c_{p,nf}+\frac{\rho_{ np}\cdot\phi}{\rho_{nf}}\cdot c_{p,np} \tag{30}\]  
The dynamic viscosity is determined using the Batchelor model [33]:  
\[\mu_{nf}=\mu_{bf}\cdot\left(1+2.5\cdot\phi+6.5\cdot\phi^{2}\right) \tag{31}\]