## 5 Proposed Model for Twin-Entry and Dual-Volute Turbines
### Mixed Flow Approach Corrected  
Based on the outcome of mixed outlet temperatures in all the admission conditions in Section 5.1, it is clear that a fitting parameter can be feasible for adjusting the error (\(\Delta T_{4}^{\mathit{MFR}_{x}}\)) generated by the mixed flow approach in the case of full and unequal flow admissions. In a way that is appropriate to reduce the error, more analysis was done with variables such as a blade to jet speed ratio (BSR), \(\mathit{MFR}_{x}\), and reduced turbine speed, as shown in Figure 22.  
If the error is examined against \(\mathit{MFR}_{x}\) parameter and turbine reduced speed, as shown in Figure 22a,b, respectively, the error shows a comparatively parabolic trend with reduced turbine speed and also the error decreases from full admission to partial admission conditions. Furthermore, if the error is observed with the blade to jet speed ratios of each branch for a different \(\mathit{MFR}_{x}\), as shown in Figure 22c, the error shows a logarithmic trend with \(\mathit{MFR}_{x}\) parameter and also moderately linear as BSR grows. Besides, if the error is compared with the similar reduced turbine speed of all the flow admission conditions, as shown in Figure 22d, the y-intercept values shows a parabolic dependency with the same reduced turbine speed for different \(\mathit{MFR}\)s and BSR. Moreover, in Figure 22, it is also clear that the observed trends can agree with \(\mathit{MFR}_{x}^{\prime}\), as this parameter shows that the error is symmetric, as shown in Figure 22b. It should be taken into account that the trend of the error in \(T_{4,TE}^{\mathit{MFR}_{x}}\) is the same than the trend of any correction coefficient designed to increase the efficiency at unequal and full admission conditions, in order to reduce error. By considering all these, a new fitting function (\(z^{\prime j}\)) is proposed, as shown in Equation(33), because of the noticed trends shown in Figure 22. Equation (33) is a term that is used to correct angle incidence losses, by reducing them in the light of new findings, in the flow oriented efficiency model of Equation (22). For \(z^{\prime j}\), five fitting coefficients are introduced, which are the same for both branches. It is worth noting that the term \(z^{\prime j}\) becomes zero at \(\mathit{MFR}_{x}\) = 0 or \(\mathit{MFR}_{x}\) = 1 since in this cases the error in temperature does not exist and efficiency is not under-predicted. \[z^{\prime j}=\left(\mathit{MFR}_{x}^{\prime}-0.5\right)\cdot\ln\left[\left( \mathit{MFR}_{x}^{\prime}\right)^{k_{0}^{j}}+k_{1}^{j}\cdot\sigma_{i}^{j} \right]\cdot\left[k_{2}^{j}+k_{3}^{j}\cdot\sigma_{i}^{j}\cdot\eta_{red,i}^{j} -\frac{k_{4}^{j}\left(\eta_{red,i}^{j}\right)^{2}}{\left(\mathit{MFR}_{x}^{ \prime}+\sigma_{i}^{j}\right)}\right] \tag{33}\]  
The proposed fitting function (\(z^{\prime j}\)) is added to the old \(z_{i}^{j}\) function (Equation (29)), as shown in Equation (34). \[z_{i}^{j}=-(b_{i}^{\prime j}\cdot\sigma_{i}^{j})+(c_{i}^{\prime j}\cdot\eta_{ red,i}^{j})+f_{i}^{\prime j}+z^{\prime j} \tag{34}\]  
It is important to note that the final \(z_{i}^{j}\) function, Equation (34), has 11 fitting coefficients in total for the two branches: three coefficients for each branch (\(b_{i}^{\prime j}\), \(c_{i}^{\prime j}\), and \(f_{i}^{\prime j}\)) obtained from data at \(\mathit{MFR}_{x}\) = 0 and \(\mathit{MFR}_{x}\) = 1 and further kept constant in the mixed flow approach and five coefficients (\(k_{0}^{j}\), \(k_{1}^{j}\), \(k_{2}^{j}\), \(k_{3}^{j}\), and \(k_{4}^{j}\)) of \(z^{\prime j}\) that are associated to correct the extra losses (not real) generated by the mixed flow approach. Now, Equation (34) is used in Equation (24) as well as in Equation (30). As a result, the final \(T_{4}^{\mathit{MFR}_{x}}\) (Equation (31)) is now calculated with \(z^{\prime j}\) for reducing losses (Equation (34)) apart from partial admission conditions.