# Dynamic Investigation of a Solar-Driven Brayton Cycle with Supercritical CO2
## Material and Methods
### Modeling of the Storage Tanks  
Thermal energy (TES) storage tanks are essential for STPs. Solar thermal power plants can store thermal energy, in contrast to other RES that can only store electrical energy [32]. Using a thermal storage tank allows for the readjustment of power production according to electricity consumption peak hours rather than solar irradiance peak hours [33]. In STPs, two thermal storage tanks are used, one for the hot stream of the HTF and one for the cold stream, right after the outlet of the heat exchanger between the solar thermal system and the thermodynamic cycle. The HTF is charged through the central receiver due to the absorbed solar heat, stored in the tank and then discharged to feed the coupled thermodynamic cycle through the heat exchanger [34]. For STPs, the HTF chosen is molten salt because of the higher temperature rate in which it can be used, in comparison with other HTF\({}_{\mathrm{s}}\) such as water and thermal oils [17]. Molten salt is a chemical compound that under normal conditions has the structure of a solid salt, but at high temperatures it liquefies [35]. The three main thermodynamic properties of interest are the melting point (\(\mathrm{T_{m}}\)), the density (\(\mathrm{\rho_{s}}\)) and the heat capacity under constant pressure of the molten salt (\(\mathrm{C_{p,s}}\)). The density of the molten salt (\(\mathrm{\rho_{s}}\)) is calculated using the equation below, where \(\mathrm{R_{s}}\) is the initial density at 0 \({}^{\circ}\mathrm{C}\) and \(\mathrm{B_{s}}\) is the density variation according to temperature [35]. \[\rho_{\mathrm{s}}=\mathrm{R_{s}}-\mathrm{B_{s}}\times(\mathrm{T_{s}}-273.15) \tag{15}\]  
The TES is modeled based on mass and energy balance. The mass balance equation is shown below. The stored mass flow rate (\(\dot{\mathrm{m}}_{\mathrm{st}}\)) is the difference between the inlet (\(\dot{\mathrm{m}}_{\mathrm{in}}\)) and outlet (\(\dot{\mathrm{m}}_{\mathrm{out}}\)) mass flow rate of the molten salt [32]. \[\dot{\mathrm{m}}_{\mathrm{st}}=\dot{\mathrm{m}}_{\mathrm{in}}-\dot{\mathrm{m}} _{\mathrm{out}} \tag{16}\]  
The stored heat (\(\mathrm{Q_{st}}\)) is calculated using the energy balance equation for the thermal storage tank, given by the following equilibrium [26]. \[\mathrm{Q_{st}}=\mathrm{Q_{in}}-\mathrm{Q_{loss}}-\mathrm{Q_{out}} \tag{17}\]  
The storage heat input (\(\mathrm{Q_{in}}\)) is calculated using the equation below, where the inlet temperature of the salt (\(\mathrm{T_{s,in}}\)) is taken to be the outlet temperature from the central receiver, where \(\dot{\mathrm{m}}_{\mathrm{s,in}}\) is the mass flow rate that enters the TES [26]. \[\mathrm{Q_{in}}=\dot{\mathrm{m}}_{\mathrm{s,in}}\times\ C_{\mathrm{p,s,avg}} \times(\mathrm{T_{s,in}}-\mathrm{T_{amb}}) \tag{18}\]  
The storage heat output (\(\mathrm{Q_{out}}\)) is calculated with the following equation, where \(\dot{\mathrm{m}}_{\mathrm{s,out}}\) is the mass flow rate that exits the TES [26]. \[\mathrm{Q_{out}}=\dot{\mathrm{m}}_{\mathrm{s,out}}\times\ C_{\mathrm{p,s,avg}} \times(\mathrm{T_{s,st}}-\mathrm{T_{amb}}) \tag{19}\]  
The thermal losses from the storage surface to the ambient are calculated using the equation below. The overall heat transfer coefficient of the tank (\(\mathrm{U_{t}}\)) depends on the structural materials of the tank's shell and needs to be as low as possible to reduce thermal losses to the ambient. The area of the storage used to calculate the total thermal losses (\(\mathrm{A_{t}}\)) is the sum of the collateral surface of the cylindrical-shaped storage and the surface on top of it, which is considered a circle [32]. \[\mathrm{Q_{loss}}=\mathrm{U_{t}}\times\ A_{t}\times(\mathrm{T_{st}}-\mathrm{ T_{amb}}) \tag{20}\]  
Finally, the temperature of the stored molten salt (\(\mathrm{T_{st}}\)) is calculated using the equation below, where the volume (\(\mathrm{V_{s}}\)) is considered to be the instantaneous volume of the molten salt and \(\mathrm{\rho_{s}}\) the instantaneous density of it [14; 26].