## 2 Materials and Methods
### Radial-Inflow Turbine Efficiency Prediction Model  
Basically, for the ORC system within the power range of 50-5000 kW, an axial turbine and a radial-inflow turbine are proposed as two options [23]. The radial-inflow turbine presents an excellent aerodynamic performance, as it can deal with large enthalpy drops with relatively low peripheral speeds, has higher single-stage expansion ratio, and good off-design performance [19; 20; 21]. In this paper, the heat source inlet temperature is at a relatively high level (523.15 K), the radial-inflow turbine was selected as the expander to deal with the high expansion ratio. A radial-inflow turbine efficiency prediction model is employed to calculate a more accurate turbine efficiency. Figure 3 shows the working fluid flow process in the radial-inflow turbine. State 0 represents the initial state of the working fluid at the radial-inflow turbine inlet (nozzle inlet), state 1 represents the state of the working fluid at the nozzle outlet or the rotor inlet, and state 2 represents the state at the rotor outlet. In the nozzle, the organic vapor from the evaporator expands and accelerates (from state 0 to state 1), with enthalpy dropping. Subsequently, the working fluid continues to expand and the kinetic energy and dropping enthalpy are converted into the mechanical energy of the rotor (form state 1 to state 2). Figure 4 shows the velocity triangle of the radial-inflow turbine, which is used to express the velocity distribution at the rotor inlet and outlet. The peripheral efficiency of the radial-inflow turbine can be expressed as [23]:  
\[\eta_{\rm u}=2\overline{u}_{1}[\varphi\cos\alpha_{1}\sqrt{1-\Omega}-\overline{D}_{ 2}^{2}\overline{u}_{1}+\overline{D}_{2}\psi\cos\beta_{2}\sqrt{\Omega+\varphi^{2 }(1-\Omega)+\overline{D}_{2}^{2}\overline{u}_{1}^{2}-2\overline{u}_{1}\varphi \cos\alpha_{1}\sqrt{1-\Omega}}]. \tag{13}\]  
According to reference [24], the velocity ratio \(\overline{u}_{1}\) and the degree of reaction \(\Omega\) have a greater impact on the peripheral efficiency, and the Lagrange multiplier method is used to optimize both parameters:  
\[\overline{u}_{1,\rm opt}=\frac{\psi}{\sqrt{\mu^{2}(\frac{\cos^{2}\beta_{2}}{m^{ 2}}-\psi^{2})+\frac{(1-m^{2}\psi^{4})\phi^{2}\cos^{2}\alpha_{1}}{m^{2}\psi^{2} (1-\phi^{2})}}}. \tag{14}\]  
\[\Omega_{\rm opt}=1-[\frac{(1-m\psi^{2})\phi\cos\alpha_{1}\overline{u}_{1}}{m \psi^{2}(1-\phi^{2})}]^{2}. \tag{15}\]  
where \(m\) is an intermediate variable and can be expressed as:  
\[m=\frac{1}{\psi^{2}}[1\pm\sqrt{\frac{\mu^{2}(1-\phi^{2})(1-\cos^{2}\beta_{2} \psi^{2})}{\cos^{2}\alpha_{1}\phi^{2}+\mu^{2}(1-\phi^{2})}}]. \tag{16}\]  
The efficiency of the radial-inflow turbine is defined as the ratio of the actual enthalpy drop \(\Delta h\) to the isentropic enthalpy drop \(\Delta h\)s [24]:  
\[\eta_{\rm tur}=\frac{\Delta h}{\Delta h_{\rm s}}=\frac{h_{0}-h_{2}}{h_{0}-h_{ 2\rm s^{\prime}}}. \tag{17}\]  
Five kinds of turbine loss are considered in the radial-inflow turbine efficiency prediction model, including nozzle passage loss, rotor passage loss, leaving velocity loss, friction loss, and leakage loss. The estimation of these losses is listed as follows:  
Figure 4: Velocity triangle of the radial-flow turbine. Figure 3: Working fluid flow process in the radial-inflow turbine.