## 3 Loss Modelling  
To predict axial turbine performance, various loss models have been previously introduced starting from [PERSON] [30], and then [PERSON] and [PERSON] [31]. This has been followed by modifications presented by [PERSON] and [PERSON] [32], [PERSON] [33], [PERSON] and [PERSON] [34], and finally off-design correlations proposed by [PERSON] et al. [35]. [PERSON]'s loss model accounts for the effect of profile and secondary flow losses, while tip clearance and trailing edge losses are ignored. Profile losses are calculated as a function of the flow deflection while the secondary losses are interpreted as function of the aspect ratio neglecting the effect of inlet boundary layer and blade geometry. While [PERSON]'s model is considered to be an oversimplified model where the effect of Mach number (Ma) and fluid non-dimensional parameters are neglected, it is considered to be satisfactory for preliminary design phase as it allows the loss within the stator and rotor to be estimated based on the amount of expansion that occurs within each passage [36].In the presented design framework, velocities approach the sonic speed and therefore [PERSON] model is considered to be more accurate for estimating the flow losses as the correlations were derived based on high Mach number data. In the absence of a tip clearance loss correlation in this model, the Ainley and Mathieson correlation has been used. It is worth mentioning that a comparative study was made between using Ainley and Mathieson correlations only and the combination of the two classes, and the same trends were observed with a deviation in the design point efficiency of approximately 1.65%. Equations (5)-(7) represent the loss coefficients predicted using Soderberg model and Equations (10)-(11) represent the tip clearance loss coefficient estimated using the Ainely and Mathieson model. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Design Parameter** & **Value** & **Design Parameter** & **Value** \\ \hline Turbine inlet temperature & 650\({}^{\text{o}}\)C [11] & Turbine inlet pressure [MPa] & 17 [22] \\ Expansion ratio [-] & 3.0 & Net power output [kW] & 100 \\ Rotational Speed [kRPM] & 150–250 & Flow coefficient [-] & 0.2–1.0 [25] \\ Degree of Reaction [-] & 0.0–0.5 [26] & Loading Coefficient [-] & 0.8–3.0 [25] \\ Ni-Cr-Co density [kg/m\({}^{3}\)] & 8000 & Inconel 718 equivalent stress [MPa] & 303 at 1073 K [27] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Input design specifications. In [PERSON]'s model, the losses are modelled as a function of the aspect ratio (\(H/l\)), the nominal loss coefficient \(\varepsilon^{*}\) and the blade deflection \(\varepsilon\). The nominal loss coefficient is obtained from the empirical correlation:  
\[\zeta^{*}=\ 0.04+0.06\Big{(}\frac{\varepsilon}{100}\Big{)}^{2}, \tag{5}\]  
where \(\varepsilon\) is the flow deflection angle, which corresponds to \(\varepsilon_{N}=\alpha_{1}+\alpha_{2}\) for the stator and \(\varepsilon_{R}=\beta_{2}+\beta_{3}\) for the rotor. The nominal loss coefficients are then obtained as follows:  
\[\zeta_{N}=\ \left(\frac{10^{5}}{Re}\right)^{1/4}\left[(1+\zeta^{*})\left(0.993 +0.075\frac{l}{H}\right)-1\right], \tag{6}\]  
\[\zeta_{R}=\ \left(\frac{10^{5}}{Re}\right)^{1/4}\left[(1+\zeta^{*})\left(0.975 +0.075\frac{l}{H}\right)-1\right], \tag{7}\]  
In [PERSON]'s model, the tip clearance coefficient \(Y_{k}\) is defined in terms of pressure drop as a function of the pitch to chord ratio \((s/c)\), average blade angle \(\beta_{m}\), the radial tip clearance (k), the average blade height (h) and a constant (B). \[\beta_{m}=\ \tan^{-1}[(\tan\ \beta_{3}-\tan\ \beta_{2}/2)], \tag{8}\]  
\[C_{L}=2\ \left(s/c\right)(\tan\ \beta_{2}+\tan\ \beta_{3})\cos\ \beta_{m}\, \tag{9}\]  
where B equals to 0.5 for radial tip clearance. Considering the manufacturing tolerances and uncertainties in thermal expansion during the operation of micro-scale turbines, the radial tip clearance for the un-shrouded blades is set to 0.1 mm [6].