## 4 Efficiency Fitting Using VGT Model
\[\tan a_{3,i}^{j\text{ flow}}=\left(\frac{r_{0,i}^{j}\cdot 2\pi\cdot B_{h,i}^{j}}{ A_{0,i}^{j}}\right)\cdot\left(\frac{1+d_{i}^{j}\cdot\left(\Pi_{i,(0t,A)}^{j}-1 \right)}{\Pi_{i,(0t,A)}^{j}}\right)^{\frac{1}{7}} \tag{28}\]  
In Equation (28), \(A_{0,i}^{j}\) is the area at the tongue of each branch, \(B_{h,i}^{j}\) is the blade height of each branch, and \(r_{0,i}^{j}\) is the radius of the centroid of section \(A_{0,i}^{j}\), as shown in Figure 14. In this manner, the rotor inlet gas angle (\(\alpha_{3}^{\text{flow}}\)) for vaneless twin-entry or dual-volute turbines can be approximated. Figure 15 illustrates the variation of the rotor inlet flow angles with the total to static turbine expansion ratio of each turbine branch. Here, the results are corresponding to the twin-entry turbine T#ITE [12]. In Figure 15, it can be observed that as the flow in Shroud or Hub branch is increasing, the flow angle becomes lower. At _MFR_ 0.2 (yellow circled points) and _MFR_ 0.8 (black diamond points) conditions, the flow is lower in Shroud and Hub branches respectively. The flow angle values are higher and shows a continuous reduction of their values from lower to higher turbine expansion ratios. It is noteworthy that the \(K_{1}\) term in Equation (22) is dependent on the ratio of rotor outlet radius of each branch (\(r_{4,i}^{j}\)) and turbine wheel radius (\(r_{3}^{j}\)). For this, the geometry simplifications of twin-entry  
Figure 14: Geometrical description for calculating the flow angle in vaneless twin-entry/dual-volute turbines. Figure 15: Rotor inlet flow angles at different flow admission conditions for Shroud and Hub branch of T#ITE. and dual-volute turbines as demonstrated above for fitting the reduced mass flow model are also considered in fitting the apparent efficiency. In this way, the apparent efficiency model has been fitted separately for each branch of the twin-entry turbocharger using the data of whole turbine map to analyze the behavior of the coefficients for each turbine branch with _MFR_. Accordingly, a set of coefficients for each _MFR_ of the individual twin-entry turbine [12] branch is obtained, as shown in \[K_{1}=2\left(\frac{r_{4,i}^{j}}{r_{3}^{j}}\right)^{2} \tag{23}\]0 It is evident that the model does not show any clear trends with the mass flow ratios (e.g., the reduced mass flow fitting parameters), except that coefficients \(d_{i}^{jj}\) and \(e_{i}^{jj}\) tends to zero in all the _MFR_ for both branches. In addition, the value of the coefficient \(a_{i}^{jj}\) is close to zero for many _MFR_s. Figure 16 shows the model results of both branches at different _MFR_ by using the coefficients from \[K_{1}=2\left(\frac{r_{4,i}^{j}}{r_{3}^{j}}\right)^{2} \tag{23}\]1 The results are coherent with the experimental data of each branch with an average root mean square error of 0.02017 for both branches together (separate RMSE for each branch can be read at Figure 16). Furthermore, the model is sufficient to extrapolate until off-design conditions of each turbine branch map with there corresponding _MFR_ fitting constants from \[K_{1}=2\left(\frac{r_{4,i}^{j}}{r_{3}^{j}}\right)^{2} \tag{23}\]2 However, this model has a limitation that it is not potential to extrapolate to non-measured _MFR_s and reduced turbine speed as compared to the reduced mass flow model. This is because of the coefficients do not show any apparent trends with the _MFR_. Consequently, in this paper, the model has been improved to enable extrapolating to non-measured _MFR_s and speeds of twin-entry/dual-volute turbochargers. The procedure to refine the model is based on the analysis of twin-entry turbocharger experimental data. \[K_{1}=2\left(\frac{r_{4,i}^{j}}{r_{3}^{j}}\right)^{2} \tag{23}\]3 Comparing the apparent turbine efficiency results from VGT turbine model with the experimental values for T#ITE turbocharger. Each _MFR_ map is individually fitted for the respective turbine branch.