## 2 System Establishment and Modeling
### Modeling
#### 2.2.2 Aircraft Gas Turbine Engine Model
\[\overline{m}_{comp}=\frac{m_{comp}}{m_{DP}};\overline{n}=\frac{n}{n_{DP}}; \overline{p}_{in}=\frac{p_{in}}{p_{DP}};\overline{T}_{in}=\frac{T_{in}}{T_{DP}} \tag{5}\]  
\[\dot{n}=\pi/\sqrt{\tau_{in}};\dot{m}=\pi_{comp}/\sqrt{\tau_{in}} \tag{6}\]  
\[\pi_{comp}=f_{1}\big{(}\dot{m},\dot{n}\big{)} \tag{7}\]  
\[\eta_{comp}=f_{2}\big{(}\dot{m},\dot{n}\big{)} \tag{8}\]  
\[P_{out}=\pi_{comp}P_{in} \tag{9}\]  
\[T_{out}=T_{in}\!\!\left[1+\frac{\pi_{comp}^{\frac{\gamma_{2}-1}{\eta_{comp}}} -1}{\eta_{comp}}\right] \tag{10}\]  
\[T_{ambient}=T_{std}+L_{a}Z\]0where \(W\)--work, \(m\)--mass flow, \(cp\)--specific heat capacity, \(\pi\)--pressure ratio, \(\eta\)--efficiency, \(P\)--pressure, \(T\)--temperature, \(\overline{x}\)--corrected parameters, and \(DP\)--design point. * \(\bullet  
\[T_{ambient}=T_{std}+L_{a}Z\]1  
\[T_{ambient}=T_{std}+L_{a}Z\]2  
For a choked nozzle: The ratio of nozzle inlet pressure (\(P_{inlet}\)) to ambient pressure (\(P_{amb}\)) is greater than the ratio of nozzle inlet pressure to the choking critical pressure (\(P_{c}\)). Therefore:  
\[T_{ambient}=T_{std}+L_{a}Z\]3  
Then, nozzle exit conditions are:  
\[T_{ambient}=T_{std}+L_{a}Z\]4  
\[T_{ambient}=T_{std}+L_{a}Z\]5  
\[T_{ambient}=T_{std}+L_{a}Z\]6  
For an unchoked nozzle: It is the opposite. \[T_{ambient}=T_{std}+L_{a}Z\]7  
Then, nozzle exit conditions are:  
\[T_{ambient}=T_{std}+L_{a}Z\]8  
\[T_{ambient}=T_{std}+L_{a}Z\]9  
\[T_{inlet}=T_{amb}(1+\frac{\gamma-1}{2}M^{2})\]0  
* Thrust and Engine Performance Evaluation  
Finally, after these parameters are evaluated, the aeroengine thrust (\(F\)) can be calculated using Equation (33). \[T_{inlet}=T_{amb}(1+\frac{\gamma-1}{2}M^{2})\]1  
where the bypass and core fractions are calculated as:  
\[T_{inlet}=T_{amb}(1+\frac{\gamma-1}{2}M^{2})\]2  
\[T_{inlet}=T_{amb}(1+\frac{\gamma-1}{2}M^{2})\]3  
and the velocities for non-perfectly expanded nozzles are calculated as:  
\[T_{inlet}=T_{amb}(1+\frac{\gamma-1}{2}M^{2})\]4  
\[T_{inlet}=T_{amb}(1+\frac{\gamma-1}{2}M^{2})\]5  
To evaluate the aeroengine performance, the following performance evaluation parameters are adopted as shown. Thrust-specific fuel consumption: In the context of cycle analysis, the objective is to minimize thrust-specific fuel consumption (TSFC), aiming to achieve optimal efficiency in generating thrust while minimizing fuel consumption. \[T_{inlet}=T_{amb}(1+\frac{\gamma-1}{2}M^{2})\]6  
where \(\dot{m_{f}}\) is the GT fuel inlet flow rate. Thermal efficiency \(\eta_{th}\): This parameter evaluates how well an engine converts the thermal energy inherent in the fuel resulting in the net gain of the kinetic energy of the working medium. \[T_{inlet}=T_{amb}(1+\frac{\gamma-1}{2}M^{2})\]7  
where \(Q_{R}\) is the heating value of the fuel, \(\Delta KE\) is the change in kinetic energy. Propulsive efficiency \(\eta_{pr}\): The portion of the mechanical output of the engine converted to thrust power is called propulsive efficiency. \[T_{inlet}=T_{amb}(1+\frac{\gamma-1}{2}M^{2})\]8  
Overall efficiency \(\eta_{o}\): It is the product of thermal and propulsive efficiencies. \[T_{inlet}=T_{amb}(1+\frac{\gamma-1}{2}M^{2})\]9  
* Shaft Model  
Maintaining a power equilibrium between the compressors and turbine is of utmost importance during the operation of the GT. This task involves aligning the steady-state operation of the engine with the operational parameters of both the compressor and turbine. Any deviations from these operational parameters will lead to an unbalanced torque, which necessitates the establishment of a new steady-state equilibrium through dynamic adjustments (hence, Equation (42)). This is achieved by employing an iterative process, which iterates for a known rotational speed until the pressure ratio and efficiency of each turbine and compressor, as obtained from the performance maps, yields the power equilibrium. This power equilibrium is then utilized to perform steady-state calculations for the hybrid system.