## 5 Proposed Model for Twin-Entry and Dual-Volute Turbines
### From Mixed Flow Corrected Approach to Apparent Efficiency Merit Functions  
In Section 5.2, it is shown that the mixed turbine outlet temperatures can be well predicted by adding the suggested calibrated function in Equation (29), as shown in Equation (34). However, most of the time, the double entry turbine maps will be in terms of apparent efficiency either adiabatic or adiabatized from standard gas stand maps [46]. Therefore, one must find the relation between real efficiencies calculated with the corrected mixed flow approach and the apparent efficiencies measured in the gas stand, instead of that with measure outlet mixed flow temperatures. With this approach, it is now possible to use the final \(T_{4}^{\text{MFR}_{x}}\) (Equation (31)) in Equation (35) to obtain total temperature and later on in apparent efficiency definition in Equation (5) to estimate the efficiency values similar  
\[\eta_{MFR_{x}\ (t/s)}^{H_{x}/SV_{x}} = (1-MFR_{x})\cdot\eta_{H_{x}/SV_{x}}^{TE/DV}+\frac{MFR_{x}}{\left(1- \left(\Pi_{0t,A}^{H_{x}/SV_{x}}\right)^{\frac{1-\gamma}{7}}\right)} \tag{37}\] \[\cdot\left[1+\left(\eta_{Sh_{x}/LV_{x}}^{TE/DV}\left(1-\left(\Pi_ {0t,A}^{Sh_{x}/LV_{x}}\right)^{\frac{1-\gamma}{7}}\right)-1\right)\frac{T_{0t }^{Sh_{x}/LV_{x}}}{T_{0t}^{H_{x}/SV_{x}}}\right]\]  
If both branches have the same temperature at the inlet (as could be the case in some gas-stand tests), Equations (36) and (37) are significantly simplified and Equations (38) and (39) are obtained. \[\eta_{MFR_{x}\ (t/s)}^{Sh_{x}/LV_{x}}=MFR_{x}\cdot\eta_{Sh_{x}/LV_{x}}^{TE/DV}+(1- MFR_{x})\cdot\eta_{H_{x}/SV_{x}}^{TE/DV}\cdot\frac{\left(1-\left(\Pi_{0t,A}^{H_{x}/ SV_{x}}\right)^{\frac{1-\gamma}{7}}\right)}{\left(1-\left(\Pi_{0t,A}^{Sh_{x}/ LV_{x}}\right)^{\frac{1-\gamma}{7}}\right)} \tag{38}\]  
\[\eta_{MFR_{x}\ (t/s)}^{H_{x}/SV_{x}}=(1-MFR_{x})\cdot\eta_{H_{x}/SV_{x}}^{TE/DV}+MFR _{x}\cdot\eta_{Sh_{x}/LV_{x}}^{TE/DV}\cdot\frac{\left(1-\left(\Pi_{0t,A}^{Sh_{ x}/LV_{x}}\right)^{\frac{1-\gamma}{7}}\right)}{\left(1-\left(\Pi_{0t,A}^{H_{x}/ SV_{x}}\right)^{\frac{1-\gamma}{7}}\right)} \tag{39}\]  
\(\eta_{Sh_{x}/LV_{x}}^{TE/DV}\) and \(\eta_{H_{x}/SV_{x}}^{TE/DV}\) are the actual apparent efficiencies of individual branches that come from Equation (22) with the new \(z_{i}^{j}\) fitting function (Equation (34)) inside the \(K_{2}^{*}\) term. However, the merit function for fitting the coefficients are based on apparent efficiencies (Equations (36) and (37)) since they are what can be experimentally obtained. The \(z^{j}\) coefficients are the same for the actual apparent efficiencies of two branches (\(\eta_{Sh_{x}/LV_{x}}^{TE/DV}\) and \(\eta_{H_{x}/SV_{x}}^{TE/DV}\)) as the mixed turbine outlet temperature is common for both apparent efficiency definitions as expressed beforehand. In summary, both apparent efficiency equations (36) and (37)) have 11 fitting coefficients together and they are globally fitted using a non-linear fitting procedure for a given double entry turbine using the whole data points from the maps of the two branches (all available \(MFR\)s). For initial values, the fitting constants of mixed turbine outlet temperature model shown in Table 5 were used. A statistically based function, as shown in Equation (40), is used for minimizing the overall root mean square error of the apparent efficiencies of both branches, and also not to have very different errors between them. \[\varepsilon_{RMS_{\text{overall}}}=\left(\varepsilon_{RMS_{\eta_{\text{Skr}}}}+ \varepsilon_{RMS_{\eta_{\text{Hz}}}}\right)+\left|\varepsilon_{RMS_{\eta_{ \text{Skr}}}}-\varepsilon_{RMS_{\eta_{\text{Hz}}}}\right| \tag{40}\]  
In Figure 24a, the level of correlation between the experimental and modeled apparent efficiency values (Equations (36) and (37)) of twin-entry turbocharger is shown. It can be observed that modeled efficiency values of Shroud and Hub branch are well predicted with the experimental data, which is evidenced by the overall RMSE and individual branch RMSE values.