# Feasibility of a Helium Closed-Cycle Gas Turbine for UAV Propulsion
## 3 The Helium Closed-Cycle Gas Turbine UAV Propulsion
### Engine Model Development
#### 3.1.1 Turbomachinery Model  
Dimensionless parameters describe the behaviour of the turbomachinery component. The performance characteristics are usually plotted as pressure ratio against dimensionless or corrected mass flow (\(CMF\)), corrected speed (\(CS\)) and corrected enthalpy (\(CH\)). The following expressions represent the dimensionless parameters:  
\[CMF=\ \left(\frac{W\sqrt{\theta}}{\delta}\times\sqrt{\frac{R}{\gamma}}\right),\ \ CS=\ \left(\frac{N}{\sqrt{\theta R\gamma}}\right),\ \ CH=\ \left(\frac{\Delta H}{\sqrt{\theta R \gamma}}\right) \tag{1}\]  
where,  
\[\theta=\ \left(\frac{T}{T_{ref}}\right),\ \ \delta=\frac{P}{P_{ref}} \tag{2}\]  
The temperature at the exit of the compressor is obtained from the isentropic relationship between inlet temperature, isentropic efficiency, pressure ratio and the ratio of specific heats, given by the expression:  
\[T_{Cout}=T_{Cin}+\frac{T_{Cin}}{\eta_{c}}\Bigg{[}\bigg{(}\frac{P_{Cout}}{P_{ Cin}}\bigg{)}^{(\frac{\gamma-1}{\gamma})}-1\Bigg{]} \tag{3}\]  
where the compressor discharge pressure is derived from the given pressure ratio:  
\[PR_{c}=\ \frac{P_{cout}}{P_{cin}}=f\ (CMF,\ CS) \tag{4}\]  
The Non-Dimensional Compressor Work (\(NDCW\)) is derived as:  
\[NDCW=\ \frac{CW}{\delta\sqrt{\theta}}=f\ (CMF,\ CS) \tag{5}\]  
where the Compressor Work (\(CW\)), is a product of the mass flow rate, specific heat capacity at constant pressure and the overall temperature rise in the compressor.  
\[CW=WC_{part}T_{cout}-WC_{P_{in}}T_{c_{in}} \tag{6}\]To solve Equations (1)-(5), the input data is obtained from the design point inlet conditions and compressor performance map.  
Similar to the compressor, the performance characteristics of the turbine is described by several dimensionless parameters. The non-dimensional parameters defined for the compressor is similar to the turbine. The input data at the design point are the turbine inlet conditions (mass flow rate, temperature, and pressure), and component efficiency.  
The temperature at the outlet of the turbine is obtained from the following expression:  
\[T_{tout}=T_{itin}-(T_{itin}\times\eta_{t})\left[1-\left(\frac{P_{tout}}{P_{itin} }\right)^{(\frac{\gamma-1}{\gamma})}\right] \tag{7}\]  
The Non- Dimensional Turbine Work (_NDTW_) can be obtained from:  
\[NDTW=\ \frac{TW}{\delta\sqrt{\theta}}=f\left(\left(\frac{P_{tout}}{P_{itin}} \right),\ CS\right) \tag{8}\]  
where _TW_ is the turbine work, which is expressed as:  
\[TW=WC_{P_{out}}T_{tout}-WC_{P_{in}}T_{tin} \tag{9}\]  
Implementing system components pressure losses will mean that the turbine pressure ratio becomes:  
\[PR_{t}=\ \frac{P_{cat}}{P_{cin}} \tag{10}\]  
The overall compressor and turbine design assumed a free vortex velocity distribution with constant outer diameter annulus configuration. The annulus area for the turbo-set inlet and outlet geometry is given as:  
\[A=\ \frac{W\sqrt{T}}{Q_{o}K_{b}P}=\frac{\pi D_{m}^{2}}{4} \tag{11}\]  
where,  
\(K_{b}\) = blockage factor  
\(Q_{0}\) = Non-dimensional mass flow  
The number of stages is given by:  
\[Number\ of\ Stages=\ \frac{C_{p}\Delta T}{\Delta h} \tag{12}\]  
The stage loading and flow coefficient are given as:  
\[Stage\ load=\ \frac{C_{p}\Delta T}{U_{m}^{2}} \tag{13}\]  
\[Stage\ flow\ coefficient=\ \frac{V_{a}}{U_{m}} \tag{14}\]  
The length of the turbo-set is calculated using Sagerser empirical relationship such that compressor length is given as [33]:  
\[L_{c}=\ D_{m}\times[(0.2+[(0.234-(0.218*hub\_tip\_ratio))]\times\ N_{S})] \tag{15}\]  
And the turbine length is given as:  
\[\begin{array}{c}L_{T}=\ N_{S}[\left(\frac{\overline{D}_{t}-\overline{D}_{t} }{0.96*hub\_tip}\right)_{rotor}+\left(\frac{\overline{D}_{t}-\overline{D}_{b} }{0.96*hub\_tip}\right)_{ stator}\\ +(2N_{S}-1)\left(0.4\times\frac{\overline{D}_{t}-\overline{D}_{b}}{0.96*hub\_tip }\right)_{rotor}]\end{array} \tag{16}\]