## 1 Introduction
Moreover, the model also requires a significant amount of experimental data for finding the values of the fitting constants. [PERSON] et al. [36] studied the pulse flow modeling of a twin-scroll turbine under pulse flow operating conditions and pointed out that the overall performance prediction of the model can be enhanced by taking into account the partial admission characteristics during modeling. [PERSON] et al. [37] proposed a mean-line model for asymmetrical twin-entry turbine by specifying different flow conditions at each turbine inlet and static pressure at the outlet of a turbine. The methodology is to solve each turbine passage from inlet to the splitter location separately, and from the volute splitter to the rotor inlet, two streams mix into one uniform flow. The model further considers the swirl energy loss from the divider wall to the rotor inlet in the form of pressure loss. The results demonstrate that the mass flow parameters are well predicted for full and partial admission flow conditions and efficiencies were over-predicted in the case of lower turbo speeds and pressure ratios. [PERSON] et al. [38] proposed a more complex model for twin-entry turbine using a loss parameters calibration criteria and the modeling was based on approximation of real physics. [PERSON] et al. [39] implemented three significant changes in the single-entry model of [PERSON] and [PERSON] [27] for predicting the variables of asymmetrical twin-entry turbine. Firstly, they added the second entry model having the calculation procedure being identical to that of the single entry model. Secondly, they added inlet duct models each one of the entries to resemble the flow passage from the turbine inlet to the tongue. Third, they included the interspace model to obtain the resulting flow angle and the flow properties after the mixing. The model was validated with three flow conditions: equal pressure, equal mass flow, and unequal flow admissions. It was concluded that the performance of the model is best at equal-pressure conditions and becomes worse as flow imbalance between the branches is increasing. The root mean square analysis showed that the model was able to predict all the parameters within 10% error concerning the experimental data for similar pressure conditions and, in unequal admission flows, the difference in mass flow parameter between the model and experimental was about 20%. The current paper demonstrates the methodology for modeling both twin-entry and dual-volute turbines to extrapolate the performance maps into off-design conditions for all the flow admission conditions. The approach for modeling this is the same for both types of turbines, and it is mainly based on systematizing the characteristic maps of the turbine, which is explained in [12]. The paper is divided into five main sections. In Section 2, an experimental method to obtain the performance maps of a double entry turbine at different mass flow ratios is summarized. In Sections 3 and 4, the VGT model of [PERSON] et al. [30] (which is summarized in Appendix A) is re-designed and adapted to work for double entry turbines. The re-design procedure is done using data from one of the two types of double entry turbines analyzed in this work (twin-entry). The results of modeling are discussed by comparing with the experimental data for both types of double entry turbines (twin-entry and dual-volute). The main variables are reduced mass flow and apparent adiabatic efficiency. This apparent efficiency is a magnitude proposed and defined in this paper to achieve the proposed objectives. In Section 5, a new model for double entry turbines efficiency fitting and calculation is detailed with a procedure based on a merit function pointing to the defined apparent efficiency to predict the behavior of every turbine branch under all the tested \(MFR\) and BSR conditions. The proposed model is further validated with a different type of double entry radial-inflow turbine, namely a dual-volute asymmetric radial-inflow turbine.