## Appendix A Design Steps  
From the thermodynamic specification of the turbine and an assumed turbine efficiency the enthalpy drop across the entire turbine \(\Delta I_{0}\) can be readily found. The blade speed U and axial velocity component \(C_{a}\) then follow from the loading coefficient \(\psi\) and flow coefficient \(\phi\):  
\[U=\sqrt{\frac{2\Delta h_{os}}{\psi}} \tag{1}\]  
\[C_{a}=U\phi \tag{2}\]  
The blade angles for the stator and rotor are determined as a function of the blade speed U, the loading coefficient \(\psi\), flow coefficient \(\phi\) and degree of reaction \(\wedge\):  
\[\tan\ \beta_{2}=\frac{1}{2\phi}\left(\frac{\psi}{2}-2\wedge\right) \tag{3}\]  
\[\tan\ \alpha_{2}=\tan\ \beta_{2}+\frac{1}{\phi} \tag{4}\]  
\[\tan\ \beta_{3}=\frac{1}{2\phi}\left(\frac{\psi}{2}+2\wedge\right) \tag{5}\]  
\[\tan\ \alpha_{3}=\tan\ \beta_{3}-\frac{1}{\phi} \tag{6}\]  
From the calculated angles, the absolute and relative velocities can be found and thus the thermodynamic properties at the inlet and outlet of each stage can be determined.  
\[C_{2}=\frac{Ca}{\cos\ \alpha_{2}} \tag{7}\]  
\[C_{3}=\frac{Ca}{\cos\ \alpha_{3}} \tag{8}\]  
\[V_{2}=\frac{Ca}{\cos\ \beta_{2}} \tag{9}\]  
\[V_{3}=\frac{Ca}{\cos\ \beta_{3}} \tag{10}\]Following from an initial estimate for nozzle loss coefficient the isentropic enthalpy \((h^{{}^{\prime}}_{2})\) can be calculated:  
\[h_{2}^{{}^{\prime}}=h_{2}-\left[\frac{1}{2}\lambda_{N}C_{2}{}^{2}\right] \tag{11}\]  
where 1, 2 and 3 subscripts correspond to stator inlet, rotor inlet and rotor exit conditions respectively.  
Using the the axial velocity, the area required to pass the specified mass-flow rate can be found. The passage area of the nozzle and rotor at different planes can be obtained as a function of the density \(\rho\) and axial velocity component.  
\[A=\frac{\dot{n}}{\rho C_{a}} \tag{12}\]  
The blade height \(b\) and mean radius \(r_{m}\) can be calculated using the blade speed and rotational speed \(N\):  
\[r_{m}=\frac{U}{2\pi N} \tag{13}\]  
\[b=\frac{AN}{U} \tag{14}\]  
The stator and rotor passage losses can be obtained using the enthalpy loss coefficients. These coefficients are expressed as a function of the enthalpy difference between an isentropic and real expansion and the kinetic energy of the flow.  
\[\lambda_{N}=\frac{h_{2}-h_{2}{}^{\prime}}{C^{2}/2} \tag{15}\]  
\[\lambda_{R}=\frac{h_{3}-h_{3}{}^{\prime\prime}}{V^{3}/2} \tag{16}\]  
\[W=h_{03}-h_{01} \tag{17}\]  
From the estimated loss coefficients, the total-to-total isentropic efficiency can be determined.  
\[\eta_{tt}=\left[1+\left(\frac{\lambda_{R}\frac{V^{3}}{2}+\frac{C^{2}}{2} \lambda_{N}\frac{T_{3}}{T_{2}}}{h_{0_{1}}-h_{0_{3}}}\right)\right]^{-1} \tag{18}\]  
Following the above blade calculations, detailed dimensions for the rotor radius at the hub and tip can be obtained by applying the free vortex theory. Additionally, the number blades and blade profile can be obtained through estimating the optimum pitch to chord ratio and pitch to throat ratio as a function of the blade angles [28].  
To check the design consistency with the permissible level of stress within the rotor blades the centrifugal stress applied on the blade and the gas bending stress can be calculated using the following equations assuming a tapered blade shape:  
\[\sigma_{max}=\frac{4}{3}\pi N^{2}\rho_{b}A \tag{19}\]  
\[\sigma_{bending}=\frac{\dot{m}Ca\left[\tan\alpha_{2}+\tan\alpha_{3}\right]}{n_ {R}}\times\frac{h}{2}\times\frac{1}{zc^{3}} \tag{20}\]  
where \(\rho_{b}\) is the density of blade material, \(n_{R}\) is the number of rotor blades, \(h\) is the mean blade height, \(A\) is the mean blade area and \(z\) is a constant obtained from a graph by [PERSON] [26].