# Performance Analysis of the Supercritical Carbon Dioxide Re-compression Brayton Cycle
## 2 Methodology
### Total Heat Transfer Area Coefficient and Total Size Parameter Value (\(SP_{Total}\))  
\(UA_{Total}\) and \(SP_{Total}\) are useful parameters for estimating the cost of the system. \(UA_{Total}\) indicates the total area required for the heat exchangers in the \(sCO_{2}\) re-compression Brayton cycle, and thus the cost associated with the investment and maintenance of the heat exchangers. The hypothesis is that differences between the heat transfer coefficients in the \(sCO_{2}\) cycle heat exchangers are not significant. The total heat transfer area in the heat exchangers increases as a result of increasing \(UA_{Total}\), thereby increasing the investment and maintenance costs of the heat exchangers, and thus the system [21]. A smaller value of \(UA_{Total}\) is desirable for a cost-effective design. For a heat exchanger in general, we have  
\[UA=\frac{Q}{\Delta T_{M}}, \tag{21}\]  
\[\Delta T_{M}=\frac{\Delta T_{max}-\Delta T_{min}}{\ln\left(\frac{\Delta T_{max }}{\Delta T_{min}}\right)}, \tag{22}\]  
where \(\Delta T_{M}\) is logarithmic mean temperature difference and \(\Delta T_{max}\) and \(\Delta T_{min}\) are the maximum and minimum temperature differences, respectively, at the two ends of the heat exchanger. However, in the case of supercritical carbon dioxide Brayton cycle (\(sCO_{2}\)-\(BC\)), properties of the working fluid change swiftly, and the definition of the \(\Delta T_{M}\) is not applicable in this case. In order to cope with the situation, the length of the heat exchanger was divided into N number of segments. The length of each segment was kept small enough that the variation within a particular small segment can be ignored, as shown in Figure 3. The value of the number of segments (N) depends upon the operation region of the heat exchanger. For the pre-cooler and LTR, the numbers of segments will be large, as these two heat exchangers operate close to the critical point, and properties of \(sCO_{2}\) change at high rates. For HTR, in contract with LTR and pre-cooler, but operating away from the critical point where variation in the properties is small, a lesser number of segments would be required. Further details on the model can be found in the previous studies [8; 12; 22; 23]. Based on the discretized model definition of the logarithmic mean temperature difference \(\left(\Delta T_{M}\right)^{i}\), the \(i^{th}\) segment is given by the Equation (23)  
\[\begin{split}\left(\theta_{2}\right)^{i}=\left(\text{T}^{\text{h }}\right)^{i}-\left(\text{T}^  
Thus, UA value through the \(i^{th}\) segment could be computed as given by the following equation.  
\[(UA)^{i}=\frac{(dq)i}{\left(\Delta T_{M}\right)^{i}}. \tag{24}\]  
And UA value associated with the whole heat exchanger can be calculated using the following relation. \[UA=\sum_{i=1}^{N}(UA)^{i}. \tag{25}\]  
Later, combining the \(UA\) values for the \(sCO_{2}\) system, we obtain  
\[UA_{Total}=UA_{HX}+UA_{HTR}+UA_{LTR}+UA_{GC}. \tag{26}\]  
\(SP\) indicates the size of the turbomachinery in the system, and a high \(SP_{Total}\) means comparatively large sizes of the turbine and compressors. Consequently, the investment and maintenance costs of the turbomachinery will increase, which, in turn, will increase the total cost of the system. Because of this, a small \(SP_{Total}\) is desirable. [PERSON] and [PERSON] [14] used the turbine \(SP\) to determine the expander size, which for the \(sCO_{2}\) re-compression Brayton cycle, is given by  
\[SP_{T}=\frac{\sqrt{V_{8s}}}{\sqrt[4]{\Delta t_{s}}}=\frac{\sqrt{m_{CO_{2}}v_{8 s}}}{\sqrt[4]{\Delta t_{7}-h_{8s}}}, \tag{27}\]  
where \(V_{8s}\) is the isentropic value of the volume flow rate of the working fluid at the turbine exit, and \(\Delta t_{s}\) is the isentropic specific enthalpy drop. The main compressor and re-compression compressor \(SP\) values can be respectively defined as  
\[SP_{MC}=\frac{\sqrt{m_{CO_{2}}xv_{1}}}{\sqrt[4]{h_{2s}-h_{1}}}, \tag{28}\]  
Figure 3: Discretized model of the heat exchanger to capture the effect of variation in the properties of \(sCO_{2}\).