## 6 Extrapolating Turbine Performance Parameters  
The method for extrapolating the turbine performance parameters of the double entry turbines is based on the models developed in Sections 3 and 5. Both turbine branches should be extrapolated simultaneously since the apparent efficiency model of each turbine branch shown in Equations (36) and (37) are dependent on the expansion ratios, efficiency, and total inlet temperature between there turbine inlet branches. Besides, the reduced mass flow and apparent efficiency equations of each turbine inlet branch are interrelated. As the apparent efficiency (Equations (36) and (37)) appears in the reduced mass flow (Equation (18)), and the effective equivalent nozzle area (Equation (8)) which comes from the reduced mass flow appears in the apparent efficiency equations. Therefore, a system with the reduced mass flow and the apparent efficiency equations of both turbine branches should be solved together using an iterative procedure for the extrapolation purpose. In Figure 25, the procedure used for extrapolating the performance maps of double entry turbine is presented. Following the flowchart, the extrapolation procedure starts with the input of the available map data of each turbine branch. By using this information, the equivalent nozzle area of each turbine branch can be solved from Equation (8) with the non-linear fitting procedure for finding the appropriate coefficient values (\"\(a\)\", \"\(b\)\", \"\(c\)\" and \"\(d\)\") of each turbine branch. Later, using the fitted \"\(d\)\" value, the next step will be estimating rotor inlet flow angle with the help of turbine geometry and also data from the maps. Later, using the information of efficiency maps of each turbine branch, the apparent efficiencies (Equations (36) and (37)) are fitted together to find the 11 fitting parameters that are discussed in Section 5. After obtaining all the necessary coefficients for the model, the system made of \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline
**Turbine** & \multicolumn{3}{c}{**\(\mathit{MFR_{1}}\)**} & \multicolumn{3}{c}{**Fitting Function (\(\boldsymbol{z^{\prime}}\))**} & \multicolumn{3}{c}{**\(\mathit{MFR_{0}}\)**} \\ \cline{2-11}
**Type** & \(\boldsymbol{b}_{\mathit{SHIV}}^{\prime}\) & \(\boldsymbol{c}_{\mathit{SHIV}}^{\prime}\) & \(\boldsymbol{f}_{\mathit{SHIV}}^{\prime}\) & \(\boldsymbol{b}_{\mathit{0}}^{\mathit{TEDV}}\) & \(\boldsymbol{k}_{\mathit{1}}^{\mathit{TEDV}}\) & \(\boldsymbol{k}_{\mathit{2}}^{\mathit{TEDV}}\) & \(\boldsymbol{k}_{\mathit{3}}^{\mathit{TEDV}}\) & \(\boldsymbol{k}_{\mathit{4}}^{\mathit{TEDV}}\) & \(\boldsymbol{b}_{\mathit{HISV}}^{\prime}\) & \(\boldsymbol{c}_{\mathit{HISV}}^{\prime}\) & \(\boldsymbol{f}_{\mathit{HISV}}^{\prime}\) \\ \hline twin-entry & 1.4471 & 0.1517 & 0.5767 & 4.0841 & 0.0753 & 0.2193 & 0.0582 & 6.721E-03 & 1.4164 & 0.1250 & 0.6643 \\ \hline dual-volute & 1.3997 & 0.1041 & 0.9309 & 4.0735 & 0.0779 & 0.1862 & 0.0241 & 2.568E-03 & 1.4074 & 0.1354 & 0.9088 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Final apparent efficiency fitting coefficients for both T#1 TE and T#2 DV turbochargers. Model constants for both turbines are found in a global fitting using both branches together and all the \(\mathit{MFR_{8}}\) data points. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Model**} & \multicolumn{3}{c}{**R Squred Adjusted \(\boldsymbol{R}_{adj}^{2}\)**} \\ \cline{2-5}  & \multicolumn{2}{c}{**Twin-Entry**} & \multicolumn{2}{c}{**Dual-Volute**} \\ \cline{2-5}  & **Shroud Branch** & **Hub Branch** & **Long Volute** & **Short Volute** \\ \hline Reduced mass flow & 0.9963 & 0.9964 & 0.9952 & 0.9984 \\ Apparent efficiency & 0.9333 & 0.9287 & 0.9572 & 0.9319 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Values of adjusted coefficient determination for both T#1 TE and T#2 DV. Equations (8), (36) and (37) can be solved with an iterative procedure to obtain the extrapolated values of equivalent nozzle area and apparent efficiencies of each turbine branch.