## 5 Proposed Model for Twin-Entry and Dual-Volute Turbines
### Mixed Flow Approach
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \(\boldsymbol{MFR_{x}}\) & **Branch** & \(\boldsymbol{a^{\prime TE}}\) & \(\boldsymbol{b^{\prime TE}}\) & \(\boldsymbol{c^{\prime TE}}\) & \(\boldsymbol{d^{\prime TE}}\) & \(\boldsymbol{e^{\prime TE}}\) & \(\boldsymbol{f^{\prime TE}}\) \\ \hline \multirow{2}{*}{0} & Sh & — & — & — & — & — & — \\  & H & 0.0205 & 1.4095 & 0.1379 & 0 & 0 & 0.6572 \\ \hline \multirow{2}{*}{0.2} & Sh & 0.1349 & 0.3864 & 0.4552 & 0 & 0 & \(3.850\times 10^{-14}\) \\  & H & \(2.313\times 10^{-14}\) & 1.064 & 0.0854 & 0 & 0 & 0.6612 \\ \hline \multirow{2}{*}{0.31} & Sh & 0.1443 & 0.0180 & 0.2638 & 0 & 0 & 0.3116 \\  & H & \(2.238\times 10^{-14}\) & 0.9210 & 0.0768 & 0 & 0 & 0.6227 \\ \hline \multirow{2}{*}{0.43} & Sh & 0.1380 & 0.0501 & 0.2382 & 0 & 0 & 0 \\  & H & \(1.496\times 10^{-13}\) & 0.8709 & 0.0678 & 0 & 0 & 0.7276 \\ \hline \multirow{2}{*}{0.53} & Sh & \(2.228\times 10^{-14}\) & 0.7188 & 0.1006 & 0 & 0 & 0.5424 \\  & H & \(3.994\times 10^{-14}\) & 0.7276 & 0.0551 & 0 & 0 & 0.8262 \\ \hline \multirow{2}{*}{0.57} & Sh & \(2.220\times 10^{-14}\) & 0.7060 & 0.0814 & 0 & 0 & 0.6297 \\  & H & \(0.1508\) & \(4.441\times 10^{-14}\) & 0.2012 & 0 & 0 & 0.1183 \\ \hline \multirow{2}{*}{0.67} & Sh & \(2.515\times 10^{-14}\) & 0.8564 & 0.1024 & 0 & 0 & 0.5150 \\  & H & \(0.1416\) & \(7.668\times 10^{-11}\) & 0.1990 & 0 & 0 & 0.3941 \\ \hline \multirow{2}{*}{0.8} & Sh & \(2.869\times 10^{-14}\) & 1.0488 & 0.1160 & 0 & 0 & 0.5362 \\  & H & \(0.0205\) & 0.5535 & 0.5070 & 0 & 0 & \(5.038\times 10^{-12}\) \\ \hline \multirow{2}{*}{1} & Sh & \(2.222\times 10^{-14}\) & 1.4958 & 0.1515 & 0 & 0 & 0.6301 \\  & H & — & — & — & — & — & — \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Apparent efficiency fitting coefficients from Equation (25) for each turbine branch of T#ITE [12]. Every _MFR_ map is fitted individually. \[z_{i}^{\text{TE}}=-(b_{i}^{\prime\text{TE}}\cdot\sigma_{i}^{\text{TE}})+(c_{i}^{ \prime\text{TE}}\cdot n_{red,i}^{\text{TE}})+f_{i}^{\prime\text{TE}} \tag{29}\]  
A methodology proposed by [PERSON] et al. [43] has been used to extrapolate the partial admission maps. Figure 18 shows that the accuracy is good and can be noted that the hub branch with a higher trimming (with a higher rotor inlet to outlet radius ratio) shows a significant difference in mass flow parameter between different reduced speeds for a given pressure ratio. It means a higher work in the centrifugal forces field than the Shroud branch with a lower radius ratio. Figure 18 also shows slightly higher peak efficiency at Hub branch, probably due to the higher trimming (higher specific work) and the lower tip leakage losses when the flow is concentrated in hub branch than in shroud [44; 45]. \begin{table}
\begin{tabular}{l l l l} \hline \hline \(\boldsymbol{MR_{x}}\) & \(\boldsymbol{b^{\prime TE}}\) & \(\boldsymbol{c^{\prime TE}}\) & \(\boldsymbol{f^{\prime TE}}\) \\ \hline
0 (H) & 1.4685 & 0.1493 & 0.6280 \\
1 (Sh) & 1.4278 & 0.1189 & 0.6895 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Fitting coefficients of partial admission conditions using apparent efficiency. Figure 17: Twin-entry/dual-volute radial turbines expansion process in partial admission conditions. For predicting the same turbine outlet temperature as in experiments at different flow admission conditions, a look-up table is generated with the extrapolated efficiency, blade to jet speed ratio, and reduced turbine speeds of \(MFR\) 0 and 1 separately, corresponding to hub and shroud entries in the turbine T#1 TE. Eventually, if every other flow admission condition ( \(MFR\)s other than 0 or 1) is merely a quasi-steady mass average of what happens at \(MFR\) 0 and \(MFR\) 1. It is possible to calculate the static turbine outlet temperature (\(T_{4,i}^{TE}\)) using Equation (30).