## 2 Power Cycle Modeling
### Mathematical Model
#### 2.1.2 Recuperator Model  
There have been different approaches adopted in the literature to model the heat exchangers and recuperators. Most common methodologies are based on either U.A. values or effectiveness values \(\epsilon\). Recuperator [27; 32] modeling techniques based on the effectiveness of the heat exchanger are computationally less costly. Thus, the NTU method that utilizes the heat exchanger's effectiveness was adopted for the current study, as given by the following equations.  
\[\epsilon_{HTR}=\frac{C_{h,HTR}(T_{8}-T_{9})}{C_{min,HTR}(T_{8}-T_{4})} \tag{4}\]  
\[\epsilon_{HTR}=\frac{C_{h,HTR}(T_{8}-T_{9})}{C_{min,HTR}(T_{8}-T_{4})} \tag{5}\]  
The definitions of \(C_{min,LTR}\) and \(C_{min,HTR}\) are given by Equations (7) and (8).  
\[C_{min,HTR}=\min(Cp_{h,HTR},Cp_{c,HTR}) \tag{6}\]  
\[C_{min,LTR}=\min(Cp_{h,HTR},xCCp_{c,HTR}) \tag{7}\]  
It was assumed that both recuperators, i.e., the high-temperature recuperator (HTR) and low-temperature recuperator (LTR), are entirely insulated. Therefore, the resulting energy balance equations can be presented in the form provided below.  
\[h_{9}=h_{8}-(h_{6}-h_{4}), \tag{8}\]  
\[h_{10}=h_{9}-x(h_{3}-h_{2}), \tag{9}\]  
where \(x\) is the fraction of mass flowing toward the recompressor, termed the split mass fraction. It can be calculated using the following relationship:  
\[x=\frac{\dot{m}_{c}}{\dot{m}},\;\text{where}\;\dot{m}=\dot{m}_{c}+\dot{m}_{rc}. \tag{10}\]  
Considering ideal mixing, the mixing value was modeled using Equation (10).  
\[h_{4}=x\;h_{3}+(1-x)h_{5}. \tag{11}\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Effectiveness, HTR** & \(\epsilon_{HTR}\) & **0.9** \\ \hline Effectiveness, LTR & \(\epsilon_{LTR}\) & 0.9 \\ Efficiency, Turbine & \(\eta_{T}\) & 0.90 \\ Efficiency, Compressor & \(\eta_{C}\) & 0.85 \\ Relative Pressure loss & \(f\) & 0.01 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Component parameters used for the cycle simulation [20].  
To estimate the pressure losses, Equation (12) was considered. The definition of the cycle efficiency used in the current work was provided by Equation (13).  
\[p_{out}=p_{in}(1-f). \tag{12}\]  
\[\eta_{thermal}=\frac{w_{T}\eta_{m}-(w_{c}+w_{Rrc})/\eta_{m}}{q_{in}} \tag{13}\]  
Definitions of \(w_{t}\), \(w_{c}\), _and_\(w_{rc}\) are given by the following equations:  
\[w_{t}=(h_{7}-h_{8}), \tag{14}\]  
\[w_{c}=x(h_{2}-h_{1}), \tag{15}\]  
\[w_{rc}=(1-x)(h_{5}-h_{10}). \tag{16}\]  
The cycle's selected conditions listed in Table 2 are from the literature [8]. Cycle calculations under selected boundaries (Table 2) were computed using a Matlab code for various values of \(x\). The values in Table 2 were approximated from the literature [4; 7; 33]. Thermophysical properties were obtained by linking Matlab code with Refprop. Results of the cycle simulations are plotted in Figure 2.  
Figure 2: Flowchart of the Matlab code.  
As shown in Figure 2, the code initializes with known boundary conditions (Table 2). The turbine and main compressor were solved first using Equation (1) and Equation (3), respectively, and later HTR was solved based on an assumed value of \(T_{6}\) based on which \(T_{4}\) was computed. Finally, \(T_{4}\) was used to solve recompressor and LTR. The above-mentioned process keeps iterating in a loop until \(T_{6}\) is converged.