## 2 Cycle Model and Thermodynamic Performance  
Figure 1 shows the \(T\)-\(s\) diagram of an irreversible variable temperature HR SFLC. Process \(1\to 2\) (\(3\to 1\)) is a constant volume (pressure) endothermic (exothermic) one, and process \(2\to 3\) is an irreversible expansion one (\(2\to 3s\) is the corresponding isentropic one). Assuming the cycle working fluid is an ideal gas, as well as the inlet (outlet) temperature of the hot- and the cold-side fluid are \(T_{Hin}(T_{Hout})\) and \(T_{Lin}(T_{Lout})\). The irreversible expansion efficiency (\(\eta_{e}\)) is defined as [41; 44; 46; 51]:  
\[\eta_{e}=\frac{T_{2}-T_{3}}{T_{2}-T_{3s}} \tag{1}\]  
Figure 1: Cycle \(T\)-\(s\) diagram. Assuming the heat transfer between the working fluid and HR obeys the law of Newton heat transfer, according to the ideal gas properties and the theory of HEX, the cycle heat absorption and heat release rates are, respectively:  
\[Q_{H}=C_{wf1}(T_{2}-T_{1})=C_{H\text{min}}E_{H1}(T_{Hin}-T_{1}) \tag{2}\]  
\[Q_{L}=C_{wf2}(T_{3}-T_{1})=C_{L\text{min}}E_{L1}(T_{3}-T_{Lin}) \tag{3}\]  
where \(C_{H}(C_{L})\) and \(C_{wf1}(C_{wf2})\) are heat source and working fluid TCRs (\(C_{wf1}=\dot{m}C_{v}\), \(C_{wf2}=\dot{m}C_{p}=kC_{wf1}\)), respectively, \(\dot{m}\) is the working fluid mass flow rate, \(C_{v}(C_{P})\) is the constant volume (pressure) specific heat, \(k\) is the specific heat ratio. \(E_{H1}\) and \(E_{L1}\) are the effectiveness of hot- and cold-side HEXs, respectively:  
\[E_{H1}=\frac{1-e^{-N_{H1}(1-C_{H\text{min}}/C_{H\text{max}})}}{1-(C_{H\text{ min}}/C_{H\text{max}})e^{-N_{H1}(1-C_{H\text{min}}/C_{H\text{max}})}} \tag{4}\]  
\[E_{L1}=\frac{1-e^{-N_{L1}(1-C_{L\text{min}}/C_{L\text{max}})}}{1-(C_{L\text{ min}}/C_{L\text{max}})e^{-N_{L1}(1-C_{L\text{min}}/C_{L\text{max}})}} \tag{5}\]  
where \(N_{H1}\) and \(N_{L1}\) are the heat transfer unit number of the two HEXs, \(C_{H\text{max}}(C_{H\text{min}})\) is the larger (smaller) of \(C_{H}\) and \(C_{wf1}\), and \(C_{L\text{max}}(C_{L\text{min}})\) is the larger (smaller) of \(C_{L}\) and \(kC_{wf1}\). Their expressions are, respectively:  
\[N_{H1}=\frac{U_{H}}{C_{H\text{min}}} \tag{6}\]  
\[N_{L1}=\frac{U_{L}}{C_{L\text{min}}} \tag{7}\]  
\[C_{H\text{max}}=\text{max}\Big{\{}C_{H},C_{wf1}\Big{\}},C_{H\text{min}}=\text {min}\Big{\{}C_{H},C_{wf1}\Big{\}} \tag{8}\]  
\[C_{L\text{max}}=\text{max}\Big{\{}C_{L},kC_{wf1}\Big{\}},C_{L\text{min}}=\text {min}\Big{\{}C_{L},kC_{wf1}\Big{\}} \tag{9}\]  
According to the second law of thermodynamics, one obtained:  
Figure 1:0  
From Equations (2) and (3), the expressions of \(T_{2}\) and \(T_{3}\) are, respectively:  
Figure 1:1  
Figure 1:2  
From Equations (1) and (10)-(12), the expression of \(T_{1}\) can be obtained as:  
Figure 1:3  
From to Equations (2), (3), and (11)-(13), the expressions of \(P\) and \(\eta\) can be obtained as:  
Figure 1:4Figure 1:5  
When \(\eta_{e}=1\), substituting into Equation (13), the expression of \(T_{1}\) for an endoreversible SFLC with variable temperature HR can be obtained as:  
Figure 1:6  
Combining Equations (4)-(9) and (14)-(16), by the numerical solution, the relationship between the \(P\) and \(\eta\) characteristics of the variable temperature HR endoreversible SFLC can be obtained. Substituting \(C_{H}=C_{L}\rightarrow\infty\) into Equations (4), (5) and (13)-(15) yields the expressions of the effectiveness of the two HEXs, \(P\), \(\eta\), and \(T_{1}\) for an irreversible SFLC with constant temperature HR [62]:  
Figure 1:7  
Figure 1:8  
Figure 1:9  
\[Q_{H}=C_{wf1}(T_{2}-T_{1})=C_{H\text{min}}E_{H1}(T_{Hin}-T_{1}) \tag{2}\]0  
\[Q_{H}=C_{wf1}(T_{2}-T_{1})=C_{H\text{min}}E_{H1}(T_{Hin}-T_{1}) \tag{2}\]1  
When \(\eta_{e}=1\) and \(C_{H}=C_{L}\rightarrow\infty\), all of the expressions become the results of an endoreversible SFLC with constant temperature HR [60].