# Analysis of Nanofluids Behavior in Concentrated Solar Power Collectors with Organic Rankine Cycle
## 2 Mathematical Model
### CSP Solar Collector Model  
The power absorbed from solar radiation by the CSP solar collector can be determined by the following [3; 4; 5; 6; 23];  
\[P_{abs}=DIN*cos\;\theta*\;\eta_{opt}*IAM*\;R_{s}*\;E_{L}*\;SF_{Atail} \tag{1}\]  
where:  
* DNI: Solar radiation (w/m\({}^{2}\))
* \(\Theta\): Angle of incidence
* \(\eta_{opt}\): Optical efficiency
* _IAM_: Incidence angle modifier. * \(R_{s}\): Row shadow
* \(E_{I}\): End losses
* \(SF_{Atail}\): Solar field available  
The CSP collected power is given by;  
\[P_{\textit{collector}}=P_{abs}-P_{\textit{losscol}}-P_{\textit{losspip}} \tag{2}\]  
where:  
* \(P_{abs}\): Collector power absorbed by CSP defined in Equation (1)
* \(P_{\textit{losscol}}\): Collector thermal losses of the CSP. * \(P_{\textit{losspip}}\): solar field piping losses. Figure 2: P-H diagram for the mixture R134a, R245 fa, R125, R236 fa. The collected power calculated in Equation (2) is transferred to the heat transfer fluid, thermal oil, and nanofluids loop. It is also assumed that this thermal energy will be accumulated in the thermal storage tank that supplies the thermal energy to drive the ORC cycle. The CSP collector thermal losses are calculated by Equations (3)-(5);  
\[P_{losscol}=L_{c1}+L_{c2} \tag{3}\]  
\[L_{c1}=a_{2}\Delta\mathrm{T}_{1}^{2}+a_{1}\Delta\mathrm{T}_{1}-a_{0} \tag{4}\]  
\[L_{c2}=\left(b_{2}\Delta\mathrm{T}_{1}^{2}+b_{1}\Delta\mathrm{T}_{1}-b_{0} \right)\left(\frac{DNI}{900}\right)\cos\theta \tag{5}\]  
Readers interested in the detailed calculation of the collector thermal losses are advised to consult references [3; 4; 5; 6; 23]. On the other hand, the piping losses are determined from the following equation:  
\[P_{losspip}=L_{p1}\Delta\mathrm{T}_{1}-L_{p2}\Delta\mathrm{T}_{1}^{2}+L_{p3} \Delta\mathrm{T}_{1}^{3} \tag{6}\]  
Readers interested in the detailed calculation of the collector piping losses are advised to consult references [3; 4; 5; 6; 23]. Finally, the power of the CSP collector is  
\[P_{collector}=P_{abs}-P_{losscol}-P_{losspip} \tag{7}\]  
where Equations (3) and (6) define the collector thermal losses \(P_{losscol}\) and piping losses, \(P_{losspip}\), respectively. The mass flow rate of the basic heat transfer fluid, thermal oil, can be given by the following:  
\[m_{oil}=\frac{P_{collectedA_{aperture}}}{C_{p_{in}}\left(T_{field_{autle}}-T_{ field_{inlet}}\right)} \tag{8}\]  
where  
\(A_{aperture}\): is the total aperture (m\({}^{2}\))  
\(T_{field\ outlet}\) and \(T_{field\ int}\) are the outlet and inlet temperature of the thermal oil in the solar field. The Dowtherm fluid was used in this CSP loop since it offers good thermal stability at temperatures up to 625 \({}^{\circ}\)F (330 \({}^{\circ}\)C). The maximum recommended film temperature is 675 \({}^{\circ}\)F (360 \({}^{\circ}\)C). Between 550 \({}^{\circ}\)F and 600 \({}^{\circ}\)F (290 \({}^{\circ}\)C and 315 \({}^{\circ}\)C), its stability is 15 to 30 times greater than that of a typical hot oil. Where the specific heat of the thermal oil is [37]:  
\[C_{p_{all}}=1509+2.496T_{oil}+7.887*10^{-4}T_{oil}^{2} \tag{9}\]  
where \(T_{oil}\) is the thermal _oil_ temperature. The thermal heat accumulated in the thermal storage tank during the charging process is:  
\[P_{\textit{collector}}=P_{abs}-P_{\textit{losscol}}-P_{\textit{losspip}} \tag{2}\]0  
where \(m_{oil}\) is the oil flow rate obtained from Equation (8) and \(C_{poil}\) is the specific heat of thermal oil in the base fluid.