## 3 Reduced Mass Flow Fitting  
The approach of considering the two entries as individual turbines makes the resulting mass flow parameters show an explicit dependency of the flow behavior with the mass flow ratio (_MFR_, Equation (1)) (referring to Figure 6 and [12]). Accordingly, the model presented by [PERSON] et al. [30], which is briefly described in Appendix A.1, is re-designed to deal with twin-entry and dual-volute turbines. The model considers now two turbine entries as two separate equivalent nozzles, as shown in Figure 7, each as if it were a VGT. These turbine entries have their respective set of maps depending on the _MFR_ instead of the VGT position. The flow passage is divided into several regions, as shown in Figure 7 (station numbers are kept similar to the VGT turbine shown in Figure A1). The value of the effective area of two nozzles is calculated in the same way as in the case of a single entry VGT turbine, which uses the individual characteristic curves of expansion ratio versus reduced mass flow rate shown in Figure 6 for each branch. Considering the assumption of two individual turbines, it is now possible to express the effective equivalent nozzle area for each entry, as shown in Equation (8). \[A_{eff,i}^{j}=\frac{a_{i}^{j}\cdot A_{4,i}^{j\,geom}\cdot\sqrt{1+ \frac{\left(c_{i}^{j}\right)^{2}\cdot\left[\left(\frac{D_{4m,i}^{j}}{D_{3}^{j}} \right)^{2}-1\right]+b_{i}^{j}}{\overline{n}_{MFR_{k}(i/s)}^{2}}}}{\sqrt{1+ \left(c_{i}^{j}\cdot\frac{A_{4,i}^{j\,geom}}{A_{3,i}^{j\,geom}}\right)^{2} \cdot\frac{\left(\frac{1}{n_{i,(3A)}^{j}}\right)^{2}}{\left[1-\eta_{MFR_{k}(i/ s)}^{i}\cdot\left[1-\left(\frac{1}{n_{i,(3A)}^{j}}\right)^{\frac{2-1}{7}} \right]\right]^{2}}}} \tag{8}\]  
\(\Gamma_{i,(3A)}^{j}\) represents the pressure in the space between stator and rotor, estimated using Equation (9) with \(d_{i}^{j}\) as a fitting constant [30]. \[\Pi_{i,(3A)}^{j}=1+d_{i}^{j}\cdot\left(\Pi_{i,(0t,A)}^{j}-1\right) \tag{9}\]  
In Equation (8), the terms \(i\) and \(j\) represent the generic codes for the turbine inlet (shroud or hub; long volute or short volute) and the turbocharger (twin-entry/dual-volute), respectively. Therefore, for each turbine, there are two effective area equations (one for each turbine entry), dependent on their  
Figure 6: Turbine flow and efficiency maps of a dual-volute turbocharger (T#2 DV). corresponding turbine entry geometries, the measured data of apparent efficiency (\(\eta^{i}_{\text{MFR}_{x}(t/s)}\)), and the four fitting constants (\(a^{j}_{i}\), \(b^{j}_{i}\), \(c^{i}_{i}\), and \(d^{i}_{i}\)). The effective area of the nozzle shown in Equation (8) is dependent on the stator and rotor outlet geometrical area. In the case of variable geometry turbine, these areas are estimated based on the geometry of the turbine and VGT positions, as outlined in Appendix A.1 and [30]. Nevertheless, twin-entry and dual-volute turbocharger used in this work are vaneless turbines. Therefore, the stator nozzle area is estimated using the turbine rotor diameter (\(D_{3}\)) and blade height (\(Bh\)), as shown in Equation (10) (see Figure 8). In vaneless turbines, it is considered that the stator outlet area is the same as the rotor inlet area. \[A^{\text{geom}}_{3}=\pi\cdot D_{3}\cdot Bh \tag{10}\]  
For estimating the geometrical area at the rotor outlet, Equation (A4) can be used. In Equation (8), \(D_{4m}\) is an arithmetic mean diameter between turbine rotor shroud (\(D_{4}\)) and rotor hub (\(D_{nut}\)) diameters, as shown in Equation (11) (see Figure 8). \[D_{4m}=\left(\frac{D_{4}+D_{nut}}{2}\right) \tag{11}\]  
As aforementioned, twin-entry and dual-volute turbines have different solute designs and geometries. Subsequently, the stator and rotor geometrical areas are estimated in different ways for every turbine type, as further explained below. To have a clear explanation, firstly the method for determining the geometrical areas of twin-entry turbine is defined, and then for a dual-volute turbine. Figure 7: Vaneless double entry turbines station distribution and modeling approach.