# Thermoeconomic Analysis of Concentrated Solar Power Plants Based on Supercritical Power Cycles
## 2 Materials and Methods
### The Solar Receiver  
The solar receiver model describes the thermal behavior of the receiver when exposed to concentrated solar radiation [20]. This model has been implemented in Matlab [21], and the thermofluidodynamic properties of CO\({}_{2}\) are provided by the database NIST REFPROP [22]. The model is based on an energy balance in the direction of fluid heating and a thermal resistance model that characterises the temperature gradient across the panel thickness. The common iteration variable of both models is the temperature at the external surface of the absorber panel, as shown in Figure 5. \begin{table}
\begin{tabular}{c c c c} \hline
**State Points** & **P (bar)** & **T (\({}^{\circ}\)C)** & **h (kJ/kg)** \\ \hline
1 & 250 & 688 & 699.5 \\
2 & 86.2 & 545.1 & 531.4 \\
3 & 85.8 & 142.3 & 62.99 \\
4 & 85.4 & 85.38 & \(-\)12.13 \\
5 & 85 & 50 & \(-\)80.9 \\
6 & 120.3 & 77.05 & \(-\)66.64 \\
7 & 119.9 & 50 & \(-\)170.2 \\
8 & 251.2 & 80.18 & \(-\)147.1 \\
9 & 250.8 & 137.1 & \(-\)26.99 \\
10 & 250.8 & 136.3 & \(-\)28.48 \\
11 & 250.8 & 136.8 & \(-\)27.54 \\
12 & 250.4 & 482.8 & 440.9 \\ \hline Cycle power (MW\({}_{e}\)) & & 50 & \\ Source thermal power (\(MW_{th}\)) & & 103.42 & \\ Cycle efficiency (\%) & & 48.41 & \\ \hline \end{tabular}
\end{table} 
Table 1: Thermodynamic properties of the state points of simulated partial-cooling cycle (Source: [19]). Figure 4: Layout of the sCO\({}_{2}\) partial-cooling cycle, indirect coupling configuration, including the state points (1–12). T, Turbine; G, Generator; MC, Main Compressor; AC, Auxiliary Compressor; LTR, Low Temperature Recuperator; HTR, High Temperature Recuperator; PC, pre-cooler. To characterise the fluid heating along the flow direction, it is necessary to divide the channel into Heat Control Elements (_HCE_s), in each of which the fluid properties are assumed to be constant. For each of these _HCE_s, the energy balance is given by Equations (1)-(3). \[\dot{Q}_{solar}\Big{|}_{HCE}=\left.\dot{Q}_{abs}\right|_{HCE}+\left.\dot{Q}_{ loss,ref}\right|_{HCE} \tag{1}\]  
\[\left.\dot{Q}_{abs}\right|_{HCE}=\left.\dot{Q}_{conv,HTF}\right|_{HCE}+\left. \dot{Q}_{loss,conv}\right|_{HCE}+\left.\dot{Q}_{loss,rad}\right|_{HCE} \tag{2}\]  
\[\left.\dot{Q}_{loss}\right|_{HCE}=\left.\dot{Q}_{loss,rad}\right|_{HCE}+\left. \dot{Q}_{loss,ref}\right|_{HCE}+\left.\dot{Q}_{loss,conv}\right|_{HCE} \tag{3}\]  
As observed in Equation (3), the total heat losses are the sum of the radiation, reflection, and convection heat loss. Radiation heat loss is calculated using the simplified formula of Stefan-Boltzmann law [23], accounting for the view factor between the panel external surface and the aperture of the prism-shaped cavity. The reflection loss represents a percentage of the incident heat on the panel, depending on the solar reflectivity, and also includes the view factor. Convection losses are calculated using Siebers and [PERSON] correlation [24], which considers both forced convection and free convection. The thermo-fluid dynamic behavior of the CO\({}_{2}\) is characterised by the convection heat gain and pressure drop in the flow direction. Both effects depend on the fluid regime: laminar, turbulent, or transitional. The equations used for each case are summarised in Table 2.  
Figure 5: Scheme of the thermal models that characterise the solar receiver performance. In Table 2, \(Nu\) is the Nusselt number; \(Re_{Dh}\) is the Reynolds number based on the hydraulic diameter of the duct, \(D_{h}\); \(Pr\) is the Prandtl number at the fluid temperature; \(Pr_{sl}\) is the Prandtl number based on the duct inner surface temperature; \(L\) is the \(HCE\) length; \(D_{h}\) is the hydraulic diameter of the duct; \(\rho\) is the average fluid density; \(u\) is the average fluid velocity; and \(f_{D}\) is the Darcy friction factor, which is four times the Fanning friction factor, \(f_{F}\). The thermal resistance model across the panel thickness characterises the temperature gradient between parallel channels rows in the direction of concentrated solar flux.