# Dynamic Investigation of a Solar-Driven Brayton Cycle with Supercritical CO2
## Material and Methods
### Modeling of the Solar Tower  
The modeling of the solar field is based on the basic energy balances of a CSP. The solar field consists of the heliostats and the solar tower, which is responsible for raising the temperature of the HTF. The optical efficiency (\(\eta_{\rm o}\)) of the solar field can be calculated using the equation below [27]:  
\[\eta_{\rm o}=\eta_{\rm mir}\times\eta_{\rm atm}\times\eta_{\rm spill}\times \eta_{\rm cos}\times\eta_{\rm S\&B} \tag{1}\]  
The mirror efficiency (\(\eta_{\rm mir}\)) represents the heliostat's efficiency of reflectivity which strongly depends on the material and the quality of the surface of a heliostat [27]. The atmospheric efficiency (\(\eta_{\rm atm}\)) represents the losses of irradiance which, although reflected by the heliostats, does not reach the central receiver of the solar tower but is either absorbed or scattered into the atmosphere [27]. The spillage efficiency (\(\eta_{\rm spill}\)) represents the losses  
Figure 1: System configuration of a solar-driven Brayton cycle. incurred due to various system errors, such as the orientation of the heliostats to monitor the sun's orbit. The last two efficiencies depend on the distance between the heliostat and the central receiver. As this distance increases, these efficiencies decrease. Moreover, as the total area of the heliostat increases, the spillage efficiency decreases [27]. The cosine efficiency (\(\eta_{\text{cos}}\)) depends on the relative position between the sun, the heliostat and the central receiver. For the north hemisphere, heliostats placed on the north side of the solar tower have a higher cosine efficiency [27]. Finally, the shading and blocking efficiency (\(\eta_{\text{IS\&B}}\)) represents the losses due to the arrangement of the heliostat field [27]. The total available solar irradiance (Q\({}_{\text{sol}}\)) is the total available heat input for the solar field and can be calculated using the equation below, where A\({}_{\text{mir}}\) represents the total area of the heliostats and DNI is the direct normal irradiance [14; 15]. \[\text{Q}_{\text{sol}}=\text{A}_{\text{mir}}\times\text{DNI} \tag{2}\]  
The total heat absorbed by the central receiver (Q\({}_{\text{abs}}\)) can be calculated with the below equation using the total optical efficiency of the solar field [13]. \[\text{Q}_{\text{abs}}=\eta_{\text{o}}\times\text{Q}_{\text{sol}} \tag{3}\]  
Additionally, the total heat absorbed can be calculated as the sum of the useful heat (Q\({}_{\text{u}}\)) and thermal losses (Q\({}_{\text{loss}}\)), as shown in the equation below [13]. \[\text{Q}_{\text{abs}}=\text{Q}_{\text{u}}+\text{Q}_{\text{loss}} \tag{4}\]  
The useful heat (Q\({}_{\text{u}}\)) can be calculated using the heat transfer from the central receiver to the HTF using the equation below. The overall heat transfer coefficient of the heat exchanger (U\({}_{\text{ex}}\)) represents the total heat transfer between the surface of the central receiver and its internal heat exchanger where the HTF flows. Additionally, the total area of this heat exchanger (A\({}_{\text{ex}}\)) needs to be known. The useful heat gained by the HTF is calculated based on the temperature difference between the central receiver's surface temperature (T\({}_{\text{r}}\)) and the mean fluid temperature (T\({}_{\text{fm}}\)) [30]. \[\text{Q}_{\text{u}}=\text{U}_{\text{ex}}\times\text{A}_{\text{ex}}\times( \text{T}_{\text{r}}-\text{T}_{\text{fm}}) \tag{5}\]  
The mean fluid temperature (T\({}_{\text{fm}}\)) can be calculated as the mean of the inlet (T\({}_{\text{in}}\)) and the outlet (T\({}_{\text{out}}\)) temperature of the HTF flowing through the internal heat exchanger of the central receiver, as shown in the equation below [23]. \[\text{T}_{\text{fm}}=\frac{\text{T}_{\text{s,out}}+\text{T}_{\text{s,in}}}{2} \tag{6}\]  
Additionally, the useful heat can be calculated based on the energy balance in the fluid volume, using the equation below, where (\(\dot{\text{m}}_{\text{s}}\)) is the molten salt's mass flow rate.