## 1 Introduction
It is worth highlighting that the flow conditions between the entries of a double entry radial inflow turbine can be divided into three different categories as follows:  
* Full/equal admission have the same mass flow rate in each entry of the turbine. * Unequal admission have different inlet temperature, pressure, and mass flow levels between the entries of a double entry radial inflow turbine. Figure 1: Difference in double entry radial turbine geometries due to their designs. (**a**) Twin-entry turbine (T#1 TE) [12], (**b**) Dual-volute turbine (T#2 DV). Partial admission block one of the turbine inlets and pass the flow through the open channel and vice versa. In normal engine operation conditions, the twin-entry or dual-volute turbochargers operate under different flow admission conditions due to the pulsating exhaust gas natures coming from the engine. However, it should be noted that the partial admission condition is not possible to appear continuously under regular engine operating conditions. Many researchers have also reported that the engine exhaust pulsatile flows significantly influence the performance of a double entry turbine powered turbocharger [13; 14; 15]. Even though partial admission is not often found in automotive applications, it is a quite important operation regime in other energy conversion and management processes. These kinds of radial turbines are used for obtaining electric power from renewable energy sources, such as geothermal energy [16; 17], solar energy [18], and compressed air energy storage systems [19]. The Organic Rankine Cycle (ORC) is widely used for these types of energy conversions with an appropriate working fluid because it converts low-grade thermal energy into useful energy [20]. In ORCs, the available thermal energy as a renewable heat source is not supplied continuously due to the variations of available thermal energy [21; 22; 23]. In this case, the regulation of radial inflow turbines based on partial admission operations is standard practice. The model proposed in this work would also help to better simulate such essential regulation maneuvers in radial turbines. Nowadays, automotive manufacturers are focusing on a wide range of engine operating conditions which are different from traditional full load operation. To achieve an optimum matching between the turbocharger and internal combustion engines, automotive manufacturers are relying on one-dimensional engine cycle simulation tools to predict and study the effect of various parameters on engine performance. 1D simulation codes make possible the calculation of gas dynamics engine behavior at low computational costs. Furthermore, the method also shows a critical approach to simulate the unsteady performance of the turbine [24]. Moreover, the exhaust system is essential for meeting the emissions limits across the operating range of the engine. As asserted in [13], during the engine part loads and transient conditions for the internal combustion engines, the turbocharger turbine works at off-design conditions. It is not viable to capture this behavior by the standard turbine maps provided by the manufacturers, as they only offer a narrow range of data that they could measure in the gas stand. Therefore, the turbocharger turbine models should be capable of simulating the real-life conditions, so that the prediction of exhaust gas mass flow, the pressure drop across the turbine and the energy transfer to the compressor is essential. Turbine map extrapolation tools are necessary when using one-dimensional modeling tools to predict the system's behavior outside of the turbine design operative conditions [25]. Numerous researchers were able to succeed in modeling the extrapolation tools for conventional single entry fixed and variable geometry turbines [26; 27; 28; 29; 30]. Some of these models are more physical and are able to calculate the losses generated in the turbine while extrapolating the efficiency maps to off-design conditions [31]. Mainly, turbocharger turbine models are based on steady flow maps.