# Flow Characteristics and Loss Models of the Transonic Radial Turbine
###### Abstract  
A hydrogen turbo hybrid power system has the significant advantages of zero carbon emissions, high efficiency and high reliability. The need to increase the power density of hydrogen turbo hybrid power systems and improve the adaptability of turbines over a wide range of expansion ratios has encouraged the study of transonic turbines. This paper is aimed at analyzing the flow characteristics and developing the loss models of a transonic turbine. The main losses for a subsonic radial turbine are usually divided into four parts: incidence loss, passage loss, tip clearance loss and trailing edge loss. Nevertheless, when the expansion ratio of a turbine is greater than about 2.6, the turbine will choke and work in transonic conditions. A shock wave will occur at the trailing edge, which will cause a lot of losses. The loss caused by the shock wave at the trailing edge is ignored by previous loss models. This paper develops a shock wave-induced loss model to predict the performance in transonic conditions more accurately. With the developed shock wave-induced loss model, the predicted efficiency deviation in transonic conditions decreases from 10% to 3.5% maximally.  
hydrogen turbo hybrid power system; transonic turbine; loss model; shock wave +
Footnote †: journal: Applied sciences  
loss models can provide design results accurately and quickly [5; 6; 7]. These loss models reflect the irreversibility of turbomachinery and cause a decrease in efficiency as a result [8]. Nevertheless, these loss models are mainly based on subsonic conditions. Modern turbomachine systems are expected to operate in high expansion ratio transonic conditions. Performance prediction is a fundamental part of the preliminary design process of radial inflow turbines and is effectuated using analytical loss models. Analytical loss models are based on theoretical and experimental analyses, which have been developed and refined by many authors. [PERSON] and [PERSON] proposed the typical formulation of the incidence loss model [9]. [PERSON] and [PERSON] conducted flow visualizations for various inlet angles and discovered the optimum relative angle at rotor inlet should be set in the region of \(-20^{\circ}\)\(-40^{\circ}\)[10]. [PERSON] and [PERSON] developed the approach for estimating the rotor passage loss [11], which considers the meridional component of kinetic energy at the inlet and the whole kinetic energy at the exit. [PERSON] deduced the leakage flow model through the gap between the tip and shroud by modeling the flow through an orifice and a shear flow. [PERSON] and [PERSON] developed the trailing edge loss model based on the total enthalpy loss [12]. A well-known summary of selected loss models was provided by [PERSON] and [PERSON] [6]. The works performed by [PERSON] and [PERSON] are most comprehensive, and the results of these are implanted in many radial inflow turbine design tools.  
It is clear from the literature review that the present loss models of the radial turbine are not comprehensive because transonic conditions are not considered. This paper mainly focuses on the comparison of flow field characteristics between subsonic conditions and transonic conditions of a radial inflow turbine. According to the flow field characteristics of the transonic turbine, a shock wave-induced loss model is developed to predict the performance of the transonic turbine more accurately.  
Figure 1: Schematic of hydrogen turbo hybrid power systems. (**a**) Hydrogen is oxidized in the SOFC stack. (**b**) Hydrogen/ammonia is burned in the ICE cylinder. (**c**) Hydrogen/ammonia is burned in the gas turbine combustor.