# Dynamic Investigation of a Solar-Driven Brayton Cycle with Supercritical CO2
## Material and Methods
### Modeling of the Brayton Cycle  
The thermodynamic cycle is modeled as a regenerative Brayton cycle (RBC), where the working fluid is set to be the sCO\({}_{2}\). This fluid's level of safety is the highest possible [36]. The main advantage of CO\({}_{2}\) is the low critical temperature, allowing even low-temperature sources to be exploited [36]. Furthermore, the use of supercritical fluid enables an avoidance of the complexity of the phase-changing process that occurs for every working fluid [28; 36]. Working with low pressure near the critical pressure of the fluid implies high C\({}_{\mathrm{p}}\) values. Additionally, in high temperatures, the thermodynamic properties of CO\({}_{2}\) are identifiedwith the ideal gas properties [28]. Finally, the size of the turbomachinery can be significantly reduced compared to a steam turbine in a conventional ORC [29]. The heat input in this cycle is provided by a heat exchanger, which is the component of the total system that connects the solar thermal system with the thermodynamic cycle. This heat exchanger is modeled using the number of transfer units method (NTU). The heat of the hot stream (\(\rm{Q_{hot,s}}\)), i.e., the molten salt, is calculated by using the following equation [37]. \[\rm{Q_{hot,s}}=\dot{m}_{s}\times\ C_{P,s,avg}\times(T_{s,in}-T_{s,out}) \tag{23}\]  
The heat of the cold stream (\(\rm{Q_{cold,CO2}}\)), i.e., \(\rm{CO_{2}}\), is calculated using the equation below. These two heat values, the \(\rm{Q_{hot,s}}\) and the \(\rm{Q_{cold,CO2}}\), must be equal [37]. \[\rm{Q_{cold,CO_{2}}}=\dot{m}_{CO_{2}}\times\ C_{P,CO_{2}}\times\left(T_{CO_{2}, out}-T_{CO_{2},in}\right) \tag{24}\]  
The effectiveness of the heat exchanger (\(\epsilon\)) is calculated using the following equation, where the heat exchanger is chosen to be counter-current [37]. \[\epsilon=\frac{1-\exp\Bigl{(}-\left(1-\frac{\rm{CF_{min}}}{\rm{CF_{max}}} \right)\times\frac{\rm{U_{0}}\times\rm{A_{\rm{ex}}}}{\rm{CF_{min}}}\Bigr{)}} {1-\frac{\rm{CF_{min}}}{\rm{CF_{max}}}\times\exp\Bigl{(}-\left(1-\frac{\rm{CF _{min}}}{\rm{CF_{max}}}\right)\times\frac{\rm{U_{0}}\times\rm{A_{\rm{ex}}}}{ \rm{CF_{min}}}\Bigr{)}} \tag{25}\]  
The equation below calculates the product (CF) of the mass flow rate (\(\dot{m}_{f}\)) with the mean heat capacity (\(\rm{C_{f}}\)) of the fluid. This product is calculated for both streams of the heat exchanger. In the current study, these two streams are the molten salt stream and the \(\rm{CO_{2}}\) stream. Its maximum (\(\rm{CF_{max}}\)) or minimum (\(\rm{CF_{min}}\)) value refers to the comparison between these product values for the molten salt and \(\rm{CO_{2}}\) as fluids [37]. \[\rm{CF}=\dot{m}_{f}\times\rm{C_{f}} \tag{26}\]  
The outlet temperature of the hot stream (\(\rm{T_{s,out}}\)) is finally calculated using the equation below, where \(\dot{m}_{s}\) is the mass flow rate of the molten salt [37]. \[\epsilon=\frac{\dot{m}_{s}\times\ C_{P,s,avg}\times(T_{s,in}-\ T_{s,out})}{ \rm{CF_{min}}\times\left(T_{s,in}-\ T_{CO_{2},in}\right)} \tag{27}\]  
The total power output of the turbine (\(\rm{P_{turb}}\)) is calculated based on the following equation, where \(\rm{n_{m,turb}}\) is the mechanical efficiency of the turbine, \(\dot{m}_{CO_{2}}\) is the mass flow rate of the carbon dioxide and \(\rm{h_{in,turb}}\) and \(\rm{h_{out,turb}}\) are the specific enthalpies of the carbon dioxide at the inlet and the outlet of the turbine, respectively [30]:  
\[\rm{P_{turb}}=\eta_{m,turb}\times\dot{m}_{CO_{2}}\times(h_{in,turb}-h_{out,turb}) \tag{28}\]  
The thermodynamic properties of \(\rm{CO_{2}}\) are used according to the table of properties of the fluid [38]. Because all the thermodynamic properties of \(\rm{CO_{2}}\) in the inlet of the turbine are known, the entropy is also known.