## 2 Methods  
In order to prepare the computational model for analysis, a detailed examination of the provided turbine geometry was conducted. The centrifugal radial turbine's flow system is comprised of two fundamental components: the stationary stator (Figure 1) and the rotating rotor (Figure 2). Both components are of critical importance in defining the flow characteristics and overall performance of the turbine. The stator and rotor are composed of three distinct stages (Figure 3), each featuring a varying number of blades. The initial stage comprises 12, 18, and 24 blades, while the subsequent stage incorporates 12 and 15 blades. The final stage is characterized by 16 and 21 blades. The variation in blade count and blade geometry across stages is designed to manage the changing flow velocities and pressures throughout the turbine. A clearance of 0.2 mm has been defined for the tips situated in both the stator and rotor rows. In computational modeling, particular attention was paid to the quality of the mesh in order to guarantee accurate and stable simulation results. The mesh was designed in accordance with rigorous criteria for face angles, edge length ratios, and element volume ratios. In particular, the following limits were set for the mesh:  
* Maximum face angle: 165\({}^{\circ}\);
* Minimum face angle: 15\({}^{\circ}\);
* Maximum edge length ratio: 500;
* Connectivity number: 12;These parameters were selected with great care in order to optimize the quality of the mesh. The application of face angle limits serves to circumvent the potential for numerical inaccuracies that may arise from the presence of highly skewed elements. By establishing a maximum face angle of 165\({}^{\circ}\) and a minimum of 15\({}^{\circ}\), the mesh is configured to retain  
Figure 1: Radial turbine stator cross-section. Figure 3: Radial turbine blades geometry. Figure 2: Radial turbine rotor cross-section. element shapes that facilitate stable and accurate computations. The edge length ratio limit of 500 ensures that elements do not become excessively elongated, which can have a detrimental impact on convergence and solution accuracy. The volume ratio limits regulate the variation in element size between adjacent elements, facilitating a seamless transition in element sizes throughout the mesh. Setting the connectivity number to 12 helps maintain an optimal number of elements connected to a node, reducing computational errors associated with highly connected nodes. The mesh quality criteria were implemented with the objective of achieving a balance between computational efficiency and the accuracy of the simulation results, thereby ensuring reliable predictions of the turbine performance. The radial turbine flow path was calculated on a fine grid comprising approximately eight million elements. In the course of grid independence checks, grids of up to 12 million elements per whole passage were investigated. It was found that the maximum relative differences between corresponding values of isentropic efficiency for the finest investigated mesh of 12 million nodes (reference values), with respect to those of the mesh of 8 million nodes calculated, were less than 0.25% (Figure 4). The thermodynamic properties of the working fluid, R245 fa (Pentafluoropropane, HFC-245 fa), were obtained from the NIST Refprop database (Version 10). This database provides accurate real-fluid properties essential for simulating compressible flows with phase change potentials, ensuring reliable results under varying pressure and temperature conditions. The simulations were conducted using Ansys CFX (Version 2020 R2). The Reynolds-Averaged Navier-Stokes (RANS) equations were solved using a second-order spatial discretization scheme to achieve enhanced accuracy in capturing flow gradients and to minimize numerical diffusion. The Frozen Rotor interface was employed between the stationary stator and the rotating rotor domains (Figure 5).