## 2 System Establishment and Modeling
### Modeling
#### 2.2.2 Aircraft Gas Turbine Engine Model  
To predict an aero-engine's performance, a theoretical model using mathematical models has been created. It is based on the continuity of mass flow method (CMF) [34] where the flight conditions and rotational speed of the compressors are known. Using this technique, known flight conditions and compressor rotational speeds allow for calculating inlet conditions for mass flow, pressure, and temperature. These inlet conditions derive from known flight Mach numbers and rotational speeds, enabling the computation of outlet thermodynamic values for any component. An initial guess for mass flow or pressure ratio, along with gas dynamics equations, assists in calculating subsequent component starting conditions. An energy balance determines combustion chamber outlet temperatures based on known fuel flow rates, while a pressure loss factor helps derive outlet pressures from inlet pressures. Turbine entry conditions and turbine characteristics, such as shaft speeds, aid in finding outlet conditions. The performance characteristics of the fan, low and high-pressure compressors, and turbines have been sourced from General Electric's (GE) collaborative program with NASA, denoted as E3 [35]. These performance maps provide specific pressure ratios and efficiencies corresponding to distinct mass flow rates and rotational speeds. The ensuing mathematical models are employed to encapsulate the attributes of the gas turbine engine, culminating in a comprehensive simulation model. * Ambient Conditions Evaluation As the aircraft is operating in different atmospheric conditions, the first thing to model is the environmental conditions using Equations (1) and (2). \[P_{ambient}=P_{std}\Big{(}1-2.25577\times 10^{-5}(Z)\Big{)}^{5.25588}\] (1) \[T_{ambient}=T_{std}+L_{a}Z\] (2) where \(Z\)--altitude, \(L_{a}\)--base temperature lapse rate per kilometer of geopotential altitude = 6.5 K/km, \(P_{std}\)--101,325 Pa, and \(T_{std}\)--288.15 K.
* Inlet Model Modeling the inlet diffuser of the aircraft SOFC/GT is based on the mathematical models shown in Equations (3) and (4). Its purpose is to convert the free stream inlet flow conditions to meet engine compressor air requirements. \[T_{inlet}=T_{amb}(1+\frac{\gamma-1}{2}M^{2})\] (3) \[P_{inlet}=P_{amb}(1+\frac{y-1}{2}M^{2})^{\frac{7}{\gamma-1}}\] (4) where the speed of the sound\"\(a_{0}=M\sqrt{\gamma_{a}R_{a}T_{a}}\) and speed of aircraft\"\(V_{a}=a_{0}M\)
* \(M\)--Mach number, \(\gamma_{a}\)--specific heat ratio for the air (1.4) and \(R\)--gas constant (J/kg \(\cdot\) K)
* Compression Stage Model To model the fan, as well as low and high-pressure compressors, the following mathematical models are utilized in this model from Equations (5)-(11). The map provides pressure ratio and mass flow rate and isentropic efficiency as a function of corrected mass flow and corrected rotational speed [36]. Each compressor in the compression stage has its own set of performance maps used to build this model. The data extracted from here are fed into the Simulink look-up tables, which are capable of outputting the required pressure ratio and efficiency for the corrected mass flow and rotational speed input. The fan, LPC and HPC utilize the same mathematical models from Equations (5)-(11). The inlet conditions of the component are the outlet conditions of the component before it. Corrected parameters used for interpolation methods are noted in Equations (5) and (6).