## 2 Turbine Design Methodology
The process starts with the choice of the non-dimensional parameters, namely the flow coefficient (\(\phi\)), and loading coefficient (\(\psi\)), which are defined as:  
\[\phi=C_{a}/U, \tag{1}\]  
\[\psi=\frac{2\Delta h_{os}}{U^{2}}, \tag{2}\]  
Figure 1: Cycle schematic drawing. Figure 2: Entropy versus temperature (\(T\)-s diagram) for the thermodynamic cycle in Figure 1.  
where \(\Delta h_{os}\) is the enthalpy drop across the stage, \(U\) is the blade velocity and \(C_{a}\) is the axial velocity. Alongside these, the degree of reaction \(\Lambda\) and an initial estimate for the total-to-total isentropic efficiency are also specified. The values of the mass flow rate, turbine inlet temperature and pressure are taken from the cycle analysis. The selection of the dimensionless parameters is made with the aim of maximising turbine efficiency; the first two parameters, \(\phi\) and \(\psi\), according to typical values taken from the Smith chart [25], while the degree of reaction is assumed to be 0.5 [26]. Then, the loss coefficients are calculated using the selected loss model; where they are chosen to obtain an accurate estimation for the profile, secondary and tip clearance losses at the design operating conditions as detailed in Section 3. Following this, the design is re-iterated using the calculated loss coefficients and the estimate for the turbine efficiency is updated. Once the design geometry is obtained, the blade heights and mean diameter are calculated as a function of the area, mean velocity and rotational speed, and hence the feasibility of the design can be verified in comparison to the feasible manufacturing values for the inlet blade height and blade diameter [11]. Furthermore, pressure and centrifugal loads are examined where Ni-Cr-Co alloy (Inconel 718) is assumed for the turbine blades; it has been considered as a suitable material in the past [26]. The loads are calculated, assuming tapered blades with an average density of 8000 kg/m\({}^{3}\) and a maximum equivalent stress of 303 MPa, using Equations (A19) and (A20) [26; 27]. The centrifugal load is calculated at the mean blade height and mean rotor area. To calculate the bending load, the number of rotor blades are defined as function of the blade pitch (s) at the mean radius (\(r_{m}\)):  
\[n_{R}=2\pi r_{m}/s, \tag{3}\]  
The optimum pitch to chord ratio is obtained as a function of the flow angles and the chord length is obtained assuming an aspect ratio of 1 [28]. The design steps and methodology are summarised in Appendix A and in the flow chart (Figure 3). Figure 3: Design flow chart. In principle, the optimal axial turbine efficiency occurs within a certain range of specific speed, typically \(0.1<N_{\text{s}}<1.0\)[29]. For the specified specific speed range, the rotational speed ranges varies between 64 and 640 kRPM. Running the turbine at higher specific speed imposes high level of stresses on the rotor blades and could also result in rotordynamic instabilities. Consequently, in the present study the turbine design will be evaluated at three different shaft speeds, namely 150k, 200k and 250k RPM, corresponding to specific speed ranging from 0.23 to 0.39 rad, to achieve an overall turbine efficiency greater than 80%. The shaft speed is determined using Equation (4), by assuming that the specific speed is within the optimal range for axial turbines:  
\[N_{\text{s}}=\frac{\omega\sqrt{\mathcal{Q}}}{\Delta{t_{\text{os}}}^{3/4}}, \tag{4}\]  
In Equation (4), \(\omega\) is the rotor rotational speed in rad/s, \(\dot{Q}\) is the volumetric flow rate at the rotor outlet in m\({}^{3}/s\) and \(\Delta{t_{\text{os}}}\) is the isentropic enthalpy drop across the turbine, in J/kg. To examine the effect of the various design parameters on the turbine performance and the feasibility of the design, the flow coefficient and loading coefficients have been varied over the range of 0.2 to 1 and 0.8 and 3 respectively [25]. Also, the the degree of reaction is varied from from 0-0.5 [26].