## 2 Automated Preliminary Design Methodology  
Radial inflow turbines comprise of mainly a volute, nozzle, and rotor. Figure 1 illustrates the principal components and stations. Because almost all energy loss produces in nozzle and rotor parts, the design and optimization of radial inflow turbines will focus on those two components in this paper. According to References [26; 28], loading coefficient \(\psi\) and flow coefficient \(\phi\) are defined as Equations (1) and (2), respectively. In addition, they can be selected from Figure 2:  
\[\psi=\frac{\Delta H_{u}}{U_{4}^{2}}, \tag{1}\]  
\[\phi=\frac{C_{m6}}{U_{4}}. \tag{2}\]  
It can be seen from Figure 2 that the maximum efficiency occurs when the loading coefficient is in the range of 0.9-1.0, and the flow coefficient is between about 0.2-0.3. However, in many cases, designers have to use values significantly different from those optima, and this is mainly because the application restrains the design. For example, turbochargers have to use higher flow coefficient to lower outlet blade height to result in a more compact design. Therefore, it becomes difficult to determine the maximum efficiency that the design can achieve under specified constraints in a short time. Figure 1: Components of a radial inflow turbine. Figure 3 shows the conventional flowchart of the mean line design process. It starts with a design vector \((\psi,\phi,n)\) and an initial guess of the total-to-static efficiency \(\eta_{ts}\), then calculates the geometric and flow features for both nozzle and rotor, and after that iterates the efficiency using well-established loss models until a convergence criterion has been achieved. The feasibility check makes sure that the design output satisfies the application nature and constraints. It should be noted that the total-to-static efficiency is initially estimated based on Figure 2 after the loading coefficient and the flow coefficient are set, but the initial total-to-static efficiency cannot influence the final result, but only the iteration steps. Therefore, the final result is only decided by the loading coefficient, the flow coefficient, and the rotor rotational speed. In addition, there are two kinds of calculations in the process of stator and rotor calculation in Figure 3. The first one is the calculation of main geometry size and aerodynamic parameter according to the loading coefficient, flow coefficient, and rotor rotational speed. The second one is the calculation of various energy losses according to the result of the first step. For radial inflow turbines, seven kinds of losses are considered in the mean line design procedure, and they are nozzle loss, rotor incidence loss, rotor passage loss, rotor tip clearance loss, rotor trailing  
Figure 3: Flowchart of the mean line design process. Figure 2: Correlation of loading and flow coefficients for radial inflow turbine [26; 28]. edge loss, rotor backface windage loss, and rotor exit energy loss. All of these loss models are well established by some scholars and widely used in numerous open literature reports. The loss happened in the nozzle is evaluated by the model presented by [PERSON] [7], as Equation (3):  
\[\Delta h_{n}=\frac{C_{4}^{2}}{2}\frac{0.05}{\text{Re}^{0.2}}\left[\frac{3\tan \alpha_{4}}{s/c}+\frac{s\cos\alpha_{4}}{b}\right]. \tag{3}\]  
The incidence loss that happened in rotor inlet is caused by the difference between flow direction and rotor blade direction, and it is the energy used to turn the flow to the correct direction. The model presented by NASA [6] is adopted, as Equation (4):  
\[\Delta h_{i}=\frac{1}{2}W_{4}^{2}\sin^{2}\left(\beta_{4}-\beta_{4,opt}\right). \tag{4}\]  
The rotor passage loss is a summation of different loss items that happened in the rotor passage. It mainly concerns the cross flow and the secondary flow.