## 2 Material and Methods
### Mathematical-Formulation Part
#### 2.2.1 Modeling of the Solar System  
The solar thermal efficiency of the PTC (\(\eta_{\text{collector}}\)) is defined as the ratio of the useful heating (\(Q_{\text{u}}\)) to the incident solar direct beam irradiation (\(Q_{\text{sol}}\)):  
\[\eta_{\text{collector}}=\frac{Q_{\text{u}}}{Q_{\text{sol}}} \tag{1}\]  
The useful thermal product (\(Q_{\text{u}}\)) is expressed as follows:  
\[Q_{\text{u}}=m\cdot(h_{4}-h_{3}) \tag{2}\]  
It has to be said that the enthalpies are functions of pressure and temperature as below: \(h_{3}=f(p_{\text{l}},T_{\text{in}})\) and \(h_{4}=f(p_{\text{h}},T_{\text{out}})\). Moreover, an alternative expression of Equation (2) is by using the mean fluid specific heat capacity (\(c_{\text{p},\text{fm}}\)):  
\[Q_{\text{u}}=m\cdot c_{\text{p},\text{fm}}\cdot(T_{\text{out}}-T_{\text{in}}) \tag{3}\]  
The solar energy input in the system (\(Q_{\text{sol}}\)) is defined as:  
\[Q_{\text{sol}}=A_{\text{a}}\cdot G_{\text{b}} \tag{4}\]  
Figure 2: The investigated transcritical \(CO_{2}\) ORC. The thermal losses of the absorber (\(\rm Q_{loss,a}\)) are practically radiation thermal losses. It has to be said that the convection thermal losses are neglected due to the vacuum between cover and receiver:  
\[\rm Q_{loss,a}=A_{ro}\cdot\sigma\cdot\frac{\left(T_{r}^{4}-T_{c}^{4}\right)}{ \frac{1}{\varepsilon_{r}}+\frac{1-\varepsilon_{c}}{\varepsilon_{c}}\cdot\left( \frac{D_{ro}}{D_{ci}}\right)} \tag{5}\]  
The absorber emittance (\(\varepsilon_{r}\)) is a function of the mean receiver mean temperature [29]:  
\[\varepsilon_{r}=0.000327\cdot T_{r}-0.065971 \tag{6}\]  
The thermal losses from the cover to the ambient heat sink (\(\rm Q_{loss,c}\)) are radiation and convection thermal losses. \[\rm Q_{loss,c}=A_{co}\cdot\sigma\cdot\varepsilon_{c}\cdot\left(T_{c}^{4}-T_{ sky}^{4}\right)+A_{co}\cdot h_{out}\cdot\left(T_{c}-T_{am}\right) \tag{7}\]  
The sky temperature (\(\rm T_{sky}\)) is set to be 8 K lower than the ambient temperature [29]:  
\[\rm T_{sky}=T_{am}-8[K] \tag{8}\]  
The convection heat transfer coefficient for the cover tube (\(\rm h_{out}\)) can be estimated at 10 W/m\({}^{2}\)K by applying the following formula [34], for a wind speed (\(\rm V_{w}\)) at 1 m/s:  
\[\rm h_{out}=\frac{4\cdot V_{w}^{0.58}}{D_{co}^{0.42}} \tag{9}\]  
In steady-state conditions, the energy storage in the glass cover can be neglected and so it can be said [30] about the thermal losses (\(\rm Q_{loss}\)):  
\[Q_{\text{u}}=m\cdot(h_{4}-h_{3}) \tag{2}\]0  
The energy balance in the absorber shows that the absorbed solar energy (\(\rm Q_{abs}\)) is separated into useful production (\(\rm Q_{u}\)) and thermal losses (\(\rm Q_{loss}\)):  
\[Q_{\text{u}}=m\cdot(h_{4}-h_{3}) \tag{2}\]1  
The absorbed solar irradiation is found as the product of the optical efficiency (\(\rm\eta_{opt}\)) and of solar energy (\(\rm Q_{sol}\)):  
\[Q_{\text{u}}=m\cdot(h_{4}-h_{3}) \tag{2}\]2  
The optical efficiency is a function of the incident angle modifier (K) which is a function of the solar angle (\(\rm\theta\)). Therefore, the optical efficiency can be written as:  
\[Q_{\text{u}}=m\cdot(h_{4}-h_{3}) \tag{2}\]3  
Moreover, the incident angle modifier is calculated by using a reliable model that has been presented by [PERSON] and [PERSON] [35] and takes into account the sun position and the PTC end losses:  
\[Q_{\text{u}}=m\cdot(h_{4}-h_{3}) \tag{2}\]4  
In this study, the PTC axis is located in the south-north direction and there is a tracking mechanism in order to follow the sun-path in the east-west direction. The solar angle can be calculated for this tracking system as below [36]:  
\[Q_{\text{u}}=m\cdot(h_{4}-h_{3}) \tag{2}\]5where the solar time angle (\(\omega\)), the declination angle (\(\delta\)) and the zenith angle (\(\theta_{\rm z}\)) are used in the previous formula. The next part of the modeling regards the heat transfer between the absorber and the fluid.