# Performance Analysis of the Supercritical Carbon Dioxide Re-compression Brayton Cycle
## 2 Methodology
### Energy and Exergy Analysis  
To simplify the analysis, the following assumptions are made:  
1. Pressure drops in the heat exchangers and pipes are neglected. 2. Steady-state operation is assumed for all the devices in the system. 3. Heat loss from the components is negligible. The following set of governing equations represent the energy analysis for the \(sCO_{2}\) cycle:  
\[W_{T} = m_{CO_{2}}(h_{7}-h_{8}), \tag{1}\] \[W_{MC} = m_{CO_{2}}x(h_{2}-h_{1}),\] (2) \[W_{RC} = m_{CO_{2}}(1-x)(h_{5}-h_{10}), \tag{3}\] \begin{table}
\begin{tabular}{c c c} \hline
**Parameter** & **Symbol** & **Value** \\ \hline \(CO_{2}\) mass flow rate, kg/s & \(m_{CO_{2}}\) & 10 \\ Main compressor inlet temperature, \({}^{\circ}\)C & \(T_{1}\) & 35 \\ Main compressor inlet pressure, MPa & \(P_{1}\) & 7.8 \\ Main compressor outlet pressure, MPa & \(P_{2}\) & 24 \\ Main compressor isentropic efficiency & \(\eta_{\text{MC}}\) & 0.85 \\ Re-compressor isentropic efficiency & \(\eta_{RC}\) & 0.85 \\ Turbine isentropic efficiency & \(\eta_{T}\) & 0.93 \\ Effectiveness of the HTR & \(\varepsilon_{HTR}\) & 0.90 \\ Effectiveness of the LTR & \(\varepsilon_{LTR}\) & 0.90 \\ Ambient temperature (other than seasonal energy analysis), \({}^{\circ}\)C & \(T_{0}\) & 24.5 \\ Ambient pressure, MPa & \(P_{0}\) & 0.1 \\ Water inlet temperature in the gas cooler (other than seasonal exergy analysis), \({}^{\circ}\)C & \(T_{w,in}\) & 23.7 \\ Water outlet temperature in the gas cooler, \({}^{\circ}\)C & \(T_{w,out}\) & 40 \\ \hline \end{tabular}
\end{table} 
Table 1: Design parameters used in \(sCO_{2}\) cycle simulations. Figure 2: Ambient and water temperature data for the city of Busan [19]. \[Q_{in}=m_{\text{CO}_{2}}(h_{7}-h_{6}), \tag{4}\] \[Q_{GC}=m_{\text{CO}_{2}}x(h_{10}-h_{1}),\] (5) \[W_{net}=W_{T}-W_{MC}-W_{RC},\] (6) \[\eta_{th}=\frac{W_{net}}{Q_{in}}. \tag{7}\]  
Thermal efficiency (\(\eta_{th}\)) is a parameter that takes into account how much of the energy input from the heat source is converted into the net shaft work by the turbine, but it does not reflect the irreversibilities that are involved in the \(sCO_{2}\) re-compression Brayton cycle. The exergy analysis of the system is significantly useful, since energy is conserved but exergy is destroyed by the irreversibility [20]. The second law (exergy) efficiency of the system is an indicator of the maximum theoretical work that can be generated as the system is brought to equilibrium with the environment. The specific exergy of flow at any state in the system is given by  
\[e=h-h_{0}-T_{0}(s-s_{0}). \tag{8}\]  
By applying exergy balance over each component, the exergy destruction in the different components of the system can be defined as follows. **Primary heat exchanger:**  
\[I_{HX}=m_{\text{CO}_{2}}(e_{6}-e_{7})+m_{\text{s}}(e_{si}-e_{\infty}), \tag{9}\]  
where \(m_{\text{s}}\) is the mass flow rate of the heat transfer fluid. **Main compressor:**  
\[I_{MC}=m_{\text{CO}_{2}}x(e_{1}-e_{2})+W_{MC}. \tag{10}\]  
**Re-compressor:**  
\[I_{RC}=m_{\text{CO}_{2}}(1-x)(e_{10}-e_{5})+W_{RC}. \tag{11}\]  
**Gas cooler:**  
Figure 2:0  
**Turbine:**  
Figure 2:1  
**High-temperature recuperator:**  
Figure 2:2  
**Low-temperature recuperator:**  
Figure 2:3  
The total exergy destruction in the system is given by  
Figure 2:4  
The heat rejection from the system in the gas cooling process, which can be utilized for district heating purposes, is determined from the following equation:  
Figure 2:5By balancing the exergy throughout the whole system, the exergy input to the system is  
Figure 2:6  
The exergy efficiency of the system is given by  
Figure 2:7  
The total irreversibility ratio (\(IR\)) of the system can be written as  
Figure 2:8