## 2 Mathematical Modeling
### Mass and Energy Balances

Table 3: Equations of mass and energy balances for the HSR cycle [33; 34; 35]. In which, \(\dot{\mathrm{Q}}_{net}\) is of total heat rate absorbed by the solar receiver, \(\dot{\mathrm{Q}}_{S}\) and \(\dot{\mathrm{Q}}_{loss}\) are the solar energy input rate, and losses rate. The equation regarding heat loss rate can be expressed as [39]:  
\[\dot{\mathrm{Q}}_{loss}=\dot{\mathrm{Q}}_{loss,\ rad}+\dot{\mathrm{Q}}_{loss, conv} \tag{14}\]  
Here, \(\dot{\mathrm{Q}}_{loss,\ rad}\) and \(\dot{\mathrm{Q}}_{loss,conv}\) are thermal energy losses associated with radiation and convection. For the radiation form of thermal power losses, the following equation can be applied [39]:  
\[\dot{\mathrm{Q}}_{loss,rad}=\sigma\cdot\varepsilon_{re}\cdot A_{re}\big{(}T_{ re}^{4}-T_{am}^{4}\big{)} \tag{15}\]  
In Equation (15), \(T_{re}\) and \(T_{am}\) denotes the temperature of the receiver and the ambient environment. \(A_{re}\) and \(\varepsilon_{re}\) denote the area of the receiver and surface emissivity of the receiver. \(\sigma\) represent the [PERSON] constant. The subscript redefines the receiver. Additionally, the convection form of the thermal power losses is given as [39]:  
\[\dot{\mathrm{Q}}_{loss,conv}=h_{forced}\cdot A_{re}(T_{re}-T_{am}) \tag{16}\]  
Here, \(h_{forced}\) denotes the forced heat transfer coefficient of convection. Further information can be found in Reference [39]. For the solar energy radiation absorbed by an HSR in Equation (13), the following equation can be used [40]:  
\[\dot{\mathrm{Q}}_{S}=\eta_{opt}\alpha A_{ap}G_{b} \tag{17}\]  
Here, \(\alpha\) denotes the solar receiver absorption factor, and \(\eta_{opt}\) is the optic efficiency of the heliostat. \(A_{ap}\) defines the total area of the aperture. Additionally \(G_{b}\) represents the solar direct beam irradiation presented in Equation (12). The optic efficiency of heliostat (\(\eta_{opt}\)) can be formulated as follows [41]:  
\[\eta_{opt}=\eta_{ref}\times\eta_{s6b}\times\eta_{at}\times\eta_{spillage} \times\eta_{cos} \tag{18}\]  
Here, \(\eta_{ref}\) denotes the efficiency of the mirror reflectivity rate that is assumed to be 95% in this study [41]. \(\eta_{s6b}\) is the efficiency of shadowing and blocking, \(\eta_{at}\) represents the efficiency of atmospheric attenuation, \(\eta_{spillage}\) defines the efficiency of spillage, and \(\eta_{cos}\) is the cosine efficiency. The shadowing and blocking efficiency, the blocking defines as a heliostat blocks a heliostat in the backside. Additionally, the shadowing occurs as a heliostat casts the shadow on an adjacent heliostat. In this study, the shadowing and blocking efficiency was assumed to be 95% [42]. The atmospheric attenuation efficiency depends on two parameters, weather conditions and distance of heliostat and the receiver. For a distance more than 1000 m following equation can be expressed [41]:  
\[\eta_{at}=exp(-0.0001106d) \tag{19}\]  
Additionally, for a distance lower than 1000 m the atmospheric attenuation efficiency is given as [41]:  
\[\eta_{at}=0.99321-0.0001176d+1.97\times 10^{-8}d^{2} \tag{20}\]  
In which d is the distance mentioned. The cosine efficiency can be calculated using the following equation [43]:  
\[\eta_{cos}=\frac{\sqrt{2}}{2}\big{(}1+sin(\alpha_{s})\cdot cos(\lambda_{s})- cos\big{(}\varphi_{surf}-\varphi_{s}\big{)}cos(\alpha_{s})\cdot sin(\lambda_{s}) \big{)}^{0.5} \tag{21}\]In which, \(\alpha_{s}\) represents the solar altitude angle [44], \(\lambda_{s}\) denotes angle between reflected irradiations and vertical direction [43], \(\varphi_{s}\) and \(\varphi_{surf}\) are respectively the solar azimuth angle, and the angle of the surface [45]. For further information Reference [46; 47] can be used. Additionally, the spillage efficiency defines the failure of receiving reflected radiation of the heliostat field by the receiver is given as [48]:  
\[\eta_{spillage}=\int_{-r_{qp}}^{\alpha_{qp}}f_{1}\times d \tag{22}\]  
In which, \(r_{qp}\) denotes the aperture effective size of the receiver, \(f_{1}\) is a parameter adapted from Reference [48].