# Thermoeconomic Analysis of Concentrated Solar Power Plants Based on Supercritical Power Cycles
## 2 Materials and Methods
### The Solar Receiver
The equivalent overall thermal resistance for a panel with a number of parallel channels rows equal to \(N_{c,rows}\) is given by Equation (4). The specific value of each of these resistances is given in Table 3.  
\[ \begin{split} R_{th,\ panel}=R_{p0,cond}\\ +\frac{N_{c,rows}}{2}\left[R_{p,cond}\right.\\ \left.+\Big{\{}R_{c,conv}\parallel\left(R_{f,cond}+\left(\left(R _{f,cond}+R_{c,conv}\right)\parallel R_{f,conv}\right)\right)\Big{\}}\right. \\ \left.+R_{HTF}\right]\end{split}  \tag{4}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Convection Heat Transfer:**\(\dot{Q}_{conv,\ HTF}=A_{channel}\cdot h_{conv}\cdot\left(T_{si}-T_{f}\right)\) \\ \hline Correlation & Validity \\ \hline \(Nu_{Dh}=\frac{(f/8)\cdot(Re_{Dh}-1000)\cdot Pr}{1+12\cdot 7\cdot\left(\sqrt{\frac{T}{6}} \right)\cdot(Pr^{2/3}-1)}\cdot\left(\frac{Pr}{Pr_{si}}\right)^{0.11}\) & \\ \(f=(1.82\cdot log_{10}(Re_{Dh})-1.64)^{-2}\) & \(5000\leq Re_{Dh}\) \\ \(Nu_{Dh}=4.089+\frac{Nu_{Channel}\parallel\left(Re_{Dh}-4.089\right)}{5000-2300} \cdot(Re-2300)\) & \(2300<Re_{Dh}<5000\) \\ \(Nu=4.089\) & \(Re_{Dh}\leq 2300\) \\ \hline
**Pressure drop:**\(\Delta P_{HTF}=\frac{1}{2}f_{D}\cdot\left(\frac{L_{HCE}}{D_{h}}\right)\cdot p \cdot u^{2}\), \(f_{D}=4\cdot f_{F}\) \\ \hline Correlation & Validity \\ \hline \(\frac{1}{f_{F}}=1.7372\cdot\ln\left[\frac{Re_{Dh}}{1.964\cdot ln(Re_{Dh})-3.82 15}\right]\) & \(10^{4}\leq Re_{Dh}\leq 10^{7}\) \\ \(f_{F}=f_{F,2300}+\frac{(f_{F}|_{Re-1000}-f_{F}|_{Re-2300})\cdot(Re-2300)}{1000 0-2300}\) & \(2300<Re_{Dh}<10^{4}\) \\ \(f_{F}=\frac{16}{Re_{Dh}}\) & \(Re_{Dh}\leq 2300\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Correlations for the convection heat transfer coefficient and friction factor of the fluid in each heat control element (Source: [3,10]). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Symbol** & **Description** & **Expression** \\ \hline \(R_{p,cond}\) & Thermal resistance due to conduction through the wall thickness of the & \(R_{p,cond}=\frac{t_{F}}{E_{rec}\cdot W_{HCE}\cdot L_{HCE}}\) \\ \(R_{c,conv}\) & Thermal resistance due to convection between the channel base and & \(R_{c,conv}=\frac{1}{R_{conv}\cdot W_{HCE}\cdot L_{HCE}}\) \\ \(R_{f,cond}\) & Thermal resistance due to conduction through the fin half length & \(R_{f,cond}=\frac{\left(\frac{t_{F}}{t}\right)}{E_{rec}\cdot t_  
In Table 3, \(t_{p}\) is the intermediate plate thickness (\(t_{p0}\) for frontal plate); \(t_{f}\) and \(l_{f}\) are the fin thickness and length, respectively; and \(L_{HCE}\) and \(W_{HCE}\) are the \(HCE\) length and width, respectively. These geometric parameters are also shown in Figure 5. Regarding the thermal parameters: \(\rho\), \(c_{p}\), and \(u\) are the average fluid density, specific heat, and velocity in each \(HCE\); \(k_{rec}\) is the absorber thermal conductivity; and \(h_{conv}\) is the convection heat transfer coefficient to the fluid. The solar receiver is sized to provide the thermal power required by the power cycle. The optimal CO\({}_{2}\) pressure at the inlet of the receiver is approximately 55 bar [25]. Assuming that the primary heat exchanger is balanced with a Terminal Temperature Difference (TTD) of 12 \({}^{\circ}\)C, and that the interconnection pipes are adiabatic, the temperature rise of the CO\({}_{2}\) in the solar receiver ranges from 494.8 \({}^{\circ}\)C to 700 \({}^{\circ}\)C.  
This solar receiver model has been validated [20] with data from a Thermal Resistance Model (TRM) and a Computational Fluid Dynamics (CFD) model and some limited experimentation [26].