## 4 Efficiency Fitting Using VGT Model  
In the case of efficiency fitting, the model described in Appendix A and [30] is re-designed in a similar way to the reduced mass flow fitting procedure. The efficiency equation (Equation (22)) (hereafter called apparent efficiency) is fitted independently for each entry of the turbine. As aforementioned, the turbochargers used in this study are vaneless turbines; accordingly, the VGT model is modified. The stator flow outlet angle (\(\varphi_{2}^{\text{flow}}\)) is no longer guided by blades as in VGT turbine model; accordingly, the \(K_{2}^{*}\) term is expressed as shown in Equation (24). However, in \(z_{i}^{j}\) (Equation (25)), the set of each turbine maps depending on the _MFR_ are provided. It is a first and straightforward adaption of Equation (A15) and the model explained in [30]. \[\eta_{i_{{}_{i}}\ (t/s)}^{j}=-K_{1}\cdot\left(\sigma_{i}^{j}\right)^{2}+K_{2}^{ *}\cdot\left[1-\frac{K_{3}}{\left(\sigma_{i}^{j}\right)^{2}}\right]^{\frac{1} {r-1}}\cdot\left(\sigma_{i}^{j}\right) \tag{22}\]  
\[K_{1}=2\left(\frac{r_{4,i}^{j}}{r_{3}^{j}}\right)^{2} \tag{23}\]  
\[K_{2}^{*}=2\frac{A^{j}_{eff,i}}{A^{j\;gem}_{0,i}}\left(z_{i}^{j}\cdot\tan \alpha_{3,i}^{j\;\text{flow}}+\sqrt{\frac{K_{1}}{2}}\cdot\tan\beta_{4,i}^{j\; metal}\right) \tag{24}\]  
\[z_{i}^{j}=-(a_{i}^{\prime j}\cdot n_{red,i}^{j}+b_{i}^{\prime j})\cdot\sigma_{ i}^{j}+(c_{i}^{\prime j}\cdot n_{red,i}^{j}+d_{i}^{\prime j}\cdot MFR^{j} ^{2}+e_{i}^{\prime j}\cdot MFR^{j}+f_{i}^{\prime j}) \tag{25}\]  
Nevertheless, in vaneless turbines, the volute or casing will take over the function of flow guidance at the entry to a rotor in addition to flow distribution and acceleration [13]. Therefore, a physical definition based on the geometry for a compressible flow has been considered according to [PERSON] and [PERSON] [13], to calculate the rotor inlet gas angle (\(\alpha_{3}^{\text{flow}}\)) for vaneless turbines, as shown in Equation (26). \[\tan\alpha_{3}^{\text{flow}}=\frac{r_{0}\cdot\rho_{3}\cdot 2\pi\cdot B_{h} \cdot C_{D\cap 3}}{A_{0}\cdot\rho_{0}\cdot C_{D0}} \tag{26}\]  
Just for calculating this flow angle, it was assumed that the discharge coefficient at the tongue (\(C_{D0}\)) (station 0 in Figure 14) is similar to the radial discharge coefficient (\(C_{D\cap 3}\)) and, therefore, their ratio is very close to 1. For estimating the density ratios shown in Equation (26), an iterative procedure should be used; to avoid this, an isentropic expansion between stations 0 and 3 was assumed. Then, the density ratio can be expressed as a function of pressures ratio (\(p_{3}/p_{0}\)). However, the pressure ratio between stations 0 and 3 is not available in the turbine maps, thus an additional assumption is made at this point for this particular purpose. Equation (9) can be used to express the isentropic expansion between stations 0 and 3, as shown in Equation (27). It can be noted that coefficient \"\(d\)\" in Equation (9) comes from the reduced mass flow fitting and it represents the pressure ratio in the rotor with respect to the total turbine pressure ratio. \[\frac{\rho_{3}}{\rho_{0}}=\left(\frac{p_{3}}{p_{0}}\right)^{\frac{1}{7}}=\left( \frac{1+d_{i}^{j}\cdot\left(\Pi_{i,(0t,A)}^{j}-1\right)}{\Pi_{i,(0t,A)}^{j}} \right)^{\frac{1}{7}} \tag{27}\]  
Therefore, Equation (26) can now be expressed as Equation (28) for both branches and it should be calculated with their corresponding turbine entry geometry and expansion ratios according to the assumption of two turbines modeling.