## 2 Automated Preliminary Design Methodology
The CETI (Concepts NREC LLC, White River Junction, VT, USA) passage loss model presented by [PERSON] [8] is selected here, as Equation (5):  
\[\Delta h_{p}=K_{p}\Bigg{\{}\left(\frac{L_{H}}{D_{H}}\right)+0.68\Bigg{[}1- \left(\frac{\tau_{5}}{r_{4}}\right)^{2}\Bigg{]}\frac{\cos\beta_{5}}{(b_{5}/c) }\Bigg{\}}\frac{1}{2}\left(W_{4}^{2}+W_{5}^{2}\right) \tag{5}\]  
in which,  
\[\psi=\frac{\Delta H_{u}}{U_{4}^{2}}, \tag{1}\]0  
\[\psi=\frac{\Delta H_{u}}{U_{4}^{2}}, \tag{1}\]1  
The rotor tip clearance loss is evaluated by the model presented by [PERSON] [29], which considers the interaction of axial clearance flow and radial clearance flow, as Equation (8),  
\[\psi=\frac{\Delta H_{u}}{U_{4}^{2}}, \tag{1}\]2  
where, in most cases, \(K_{a}=0.4,\ K_{r}=0.75,\ K_{a,r}=-0.3\), results in good agreement with experimental data:  
\[\psi=\frac{\Delta H_{u}}{U_{4}^{2}}, \tag{1}\]3  
\[\psi=\frac{\Delta H_{u}}{U_{4}^{2}}, \tag{1}\]4  
The trailing edge loss refers to the loss due to the sudden expansion when fluid passes the trailing edge. The model presented by [PERSON] [30] is adopted as Equation (11):  
\[\psi=\frac{\Delta H_{u}}{U_{4}^{2}}, \tag{1}\]5  
The windage loss happens between rotor backface and casing, and the model presented by [PERSON] et al. [31] is used to evaluate this loss, as Equation (12):  
\[\psi=\frac{\Delta H_{u}}{U_{4}^{2}}, \tag{1}\]6  
\[\psi=\frac{\Delta H_{u}}{U_{4}^{2}}, \tag{1}\]7\[\psi=\frac{\Delta H_{u}}{U_{4}^{2}}, \tag{1}\]8  
The rotor exit energy loss is defined as Equation (15):  
\[\psi=\frac{\Delta H_{u}}{U_{4}^{2}}, \tag{1}\]9  
The feasibility check comprises a set of parameters that need to be met and then the turbine design could be considered feasible. These parameters include three aspects, the structural constraints, the flow feature constraints and geometric constraints. The structural constraints guarantee the elastic stress (Equation (2) of Reference [32]) lower than the material yield stress by a safety factor of 10%. The flow feature constraints include a rational range (\(-20^{\circ}\)--\(40^{\circ}\)) for the rotor incidence angle and a maximum Mach number (\(M<1\)) at the nozzle outlet to prevent supersonic flow loss. The geometric constraints contain a minimum inlet-outlet tip radius ratio of 1.3, a minimum outlet hub-tip radius ratio of 0.3, and a minimum blade height to inlet radius ratio of 0.3 to ensure the structure rationality. More constraints can be specified for certain consideration according to a specific application. In the conventional mean line design method of radial inflow turbines, the design result is decided by three parameters, loading coefficient, flow coefficient, and rotor rotational speed. That is to say, once the three parameters are set, the main geometrical size and aerodynamic performance of turbines are specified. Thus, these three parameters are very important for a radial inflow turbine with higher aerodynamic performance, and it is hard to choose them, especially for an engineering designer without the full experience. Therefore, in this paper, an automated radial inflow turbine preliminary design approach is developed using the Genetic Algorithm [33]. With the help of GA, the proposed approach explores the whole design space automatically and independently without violating the specified constraints. Figure 4 illustrates the fundamental procedure of GA. The evolution usually starts from a population of randomly generated individuals. In each generation, the fitness of every individual in the population is evaluated. The more fit individuals are stochastically selected from the current population, and each individual's genome is modified by crossover and mutation operation to form a new generation. The new generation of candidate solutions is then used in the next iteration of the algorithm. When either a satisfactory fitness level has been reached, or a maximum number of generations have been produced, the algorithm terminates. \[\phi=\frac{C_{m6}}{U_{4}}. \tag{2}\]0 Genetic algorithm (GA) procedure.