## Appendix A  
_Meanline Velocity Triangle Analysis Equations_[17]  
\[r_{tip,3}=\frac{D_{max}}{2}\] (A1)  
\[r_{mean}=\frac{r_{tip,3}(1+HTR)}{2}\] (A2)  
\[U_{m}=\frac{r_{mean}\pi N}{30}\] (A3)  
\[\Delta Tt=\eta_{ts}T_{t1}\left[1-(\frac{P_{3}}{P_{t1}})^{\frac{\gamma-1}{ \gamma}}\right]\] (A4)  
\[\psi=\frac{2C_{p}\Delta Tt}{{U_{m}}^{2}}\] (A5)  
\[\beta_{3}=arctan\left(\tan\left(\alpha_{3}\right)+\frac{1}{\phi}\right)\] (A6)  
\[R=\phi\beta_{3}-\frac{\psi}{4}\] (A7)  
\[\beta_{2}=arctan\left(\frac{1}{2\phi}(\frac{\psi}{2}-2R)\right)\] (A8)  
\[\alpha_{2}=arctan\left(\tan\left(\beta_{2}\right)+\frac{1}{\phi}\right)\] (A9) \begin{table}
\begin{tabular}{c c c c} \hline
**Parameter** & **MATLAB** & **CFD** & **\% Difference** \\ \hline \(M_{1}\) & 0.26 & 0.26 & 0.87 \\ \(M_{2}\) & 0.62 & 0.61 & \(-\)1.35 \\ \(M_{3}\) & 0.36 & 0.36 & \(-\)0.35 \\ \(\rho_{1}\) (kg/m\({}^{3}\)) & 0.76 & 0.76 & \(-\)0.28 \\ \(\rho_{2}\) (kg/m\({}^{3}\)) & 0.64 & 0.65 & 0.50 \\ \(\rho_{3}\) (kg/m\({}^{3}\)) & 0.45 & 0.46 & 1.83 \\ \(\eta_{ts}\) (\%) & 74.94 & 73.98 & \(-\)1.29 \\ \(\eta_{tt}\) (\%) & 84.20 & 83.46 & \(-\)0.89 \\ Power (kW) & 21.51 & 20.30 & \(-\)5.96 \\ Torque (Nm) & 1.29 & 1.22 & \(-\)5.86 \\ \hline \end{tabular}
\end{table} 
Table 7: _Cont._\[V_{a}=\ \phi U_{m} \tag{10}\] \[V_{2}=\frac{V_{a}}{\cos\left(\alpha_{2}\right)}\] (11) \[V_{1}=V_{a}\] (12) \[V_{3}=\frac{V_{a}}{\cos\left(\alpha_{3}\right)}\] (13) \[W_{2}=\frac{V_{a}}{\cos\left(\beta_{2}\right)}\] (14) \[W_{3}=\frac{V_{a}}{\cos\left(\beta_{3}\right)} \tag{15}\]  
_Meanline Fluid Property Analysis Equations [15; 17]_  
\[T_{1}=T_{t1}-\frac{{V_{a}}^{2}}{2C_{p}} \tag{16}\]  
\[P_{1}=P_{t1}\left(\frac{T_{1}}{T_{t1}}\right)^{\frac{\gamma}{\gamma-1}} \tag{17}\]  
\[T_{2}=T_{t1}-\frac{{V_{2}}^{2}}{2C_{p}} \tag{18}\]  
\[V_{a}=\ \phi U_{m} \tag{10}\]0  
\[V_{a}=\ \phi U_{m} \tag{10}\]1  
\[V_{a}=\ \phi U_{m} \tag{10}\]2  
\[V_{a}=\ \phi U_{m} \tag{10}\]3  
\[V_{a}=\ \phi U_{m} \tag{10}\]4  
\[V_{a}=\ \phi U_{m} \tag{10}\]5  
\[V_{a}=\ \phi U_{m} \tag{10}\]6  
\[V_{a}=\ \phi U_{m} \tag{10}\]7  
\[V_{a}=\ \phi U_{m} \tag{10}\]8  
_Equations for Calculation of Meanline Thermodynamic Loss Components_  
\[V_{a}=\ \phi U_{m} \tag{10}\]9  
\[V_{2}=\frac{V_{a}}{\cos\left(\alpha_{2}\right)}\]0  
\[V_{2}=\frac{V_{a}}{\cos\left(\alpha_{2}\right)}\]1\[V_{2}=\frac{V_{a}}{\cos\left(\alpha_{2}\right)}\]2 (A30)  
Rotor Meanline Loss [15]  
\[V_{2}=\frac{V_{a}}{\cos\left(\alpha_{2}\right)}\]3 (A31)  
\[V_{2}=\frac{V_{a}}{\cos\left(\alpha_{2}\right)}\]4 (A32)  
_Free Vortex Theory Equations (for stagnation points I = 1,2,3 and span layer j = hub, tip) [15; 17]_  
\[V_{2}=\frac{V_{a}}{\cos\left(\alpha_{2}\right)}\]5 (A33)  
\[V_{2}=\frac{V_{a}}{\cos\left(\alpha_{2}\right)}\]6 (A34)  
\[V_{2}=\frac{V_{a}}{\cos\left(\alpha_{2}\right)}\]7 (A35)  
\[V_{2}=\frac{V_{a}}{\cos\left(\alpha_{2}\right)}\]8 (A36)  
\[V_{2}=\frac{V_{a}}{\cos\left(\alpha_{2}\right)}\]9 (A37)  
\[V_{1}=V_{a}\]0 (A38)  
\[V_{1}=V_{a}\]1 (A39)  
_Blade Geometry Estimation Equations_  
Note Equations (A40) to (A47) were obtained from [PERSON] et al. [15], Equations (A48) to (A53) were obtained from [PERSON] [16] and Equations (A54) to (A55) were obtained from [PERSON] [18]. \[V_{1}=V_{a}\]2 (A40)  
\[V_{1}=V_{a}\]3 (A41)  
\[V_{1}=V_{a}\]4 (A42)  
\[V_{1}=V_{a}\]5 (A43)  
\[V_{1}=V_{a}\]6 (A44)  
\[V_{1}=V_{a}\]7 (A45)  
\[V_{1}=V_{a}\]8 (A46)  
\[V_{1}=V_{a}\]9 (A47)  
\[V_{3}=\frac{V_{a}}{\cos\left(\alpha_{3}\right)}\]0 (A48)\[V_{3}=\frac{V_{a}}{\cos\left(\alpha_{3}\right)}\]1 \[V_{3}=\frac{V_{a}}{\cos\left(\alpha_{3}\right)}\]2 (50) \[V_{3}=\frac{V_{a}}{\cos\left(\alpha_{3}\right)}\]3 (51) \[V_{3}=\frac{V_{a}}{\cos\left(\alpha_{3}\right)}\]4 (52) \[V_{3}=\frac{V_{a}}{\cos\left(\alpha_{3}\right)}\]5 (53) \[V_{3}=\frac{V_{a}}{\cos\left(\alpha_{3}\right)}\]6 (54) \[V_{3}=\frac{V_{a}}{\cos\left(\alpha_{3}\right)}\]7  
_Equations for Turbine Stage Performance Prediction_  
Note Equations (A56) to (A60) were obtained from [PERSON] and [PERSON] [13] and Equations (A61) to (A72) were obtained from [PERSON] [6] and [PERSON] et al.