# Development of a Predictive Tool for the Parametric Analysis of a Turbofan Engine
## 2 Materials and Methods
### Parametric Analysis  
Input and Output Variables  
The input variables that were considered for this study are the Mach number, turbine entry temperature, altitude, BPR, and FPR. They are described below:  
* Bypass ratio (BPR): The ratio of the fan output flow rate to the core mass flow rate. \(\dot{m_{c}}\) and \(\dot{m_{F}}\) are the mass flows that travel through the engine core and fan, respectively. The bypass ratio is represented by \(\alpha\) and computed as: \[\alpha=\frac{\dot{m_{F}}}{\dot{m_{c}}}\] (1)
* Turbine entry temperature (TET): The temperature of the output gases from the combustion chamber that enter the turbine. The turbine inlet or entry temperature is an essential variable that can impact the combustion cycle. Higher numbers can boost the cycle's net power production and efficiency.
* Fan pressure ratio: The ratio of fan discharge pressure to fan intake pressure. It is known that the ideal values of certain factors, such as the BPR and FPR, might greatly impact the values of the main performance metrics. The ideal FPR must be determined because, given all other factors, such as the overall pressure ratio, BPR, and turbine entrance temperature, the best FPR value assures maximum net thrust while minimizing specific fuel consumption (SFC).  
The specific thrust and SFC are the major metrics for performance in an on-design type of analysis. Propulsive efficiency and total fuel flow are also considered among the output parameters. Equations (2)-(4) were obtained from [PERSON]'s book, _Aircraft Engine Design_[27]. Steady, one-dimensional flow is assumed with the fluid behavior considered as a perfect gas. Specific thrust depends on the velocity ratio (\(V_{9}/a_{0}\)) and the overall static temperature ratio (\(T_{9}/T_{0}\)).
* Specific thrust (\(F/\dot{m}_{0}\)): Engine thrust per unit mass flow rate, as outlined below: \[\frac{F}{\dot{m}_{0}}=\frac{a_{0}}{g_{c}}\bigg{\{}\bigg{(}1+f_{0}-\frac{\beta}{ 1+\alpha}\bigg{)}\frac{V_{9}}{a_{0}}-M_{0}+\bigg{(}1+f_{0}-\frac{\beta}{1+ \alpha}\bigg{)}\frac{R_{9}}{R_{0}}\frac{T_{9}/T_{0}}{V_{9}/a_{0}}\frac{(1-P_{0 }/P_{9})}{\gamma_{0}}\bigg{\}}\] (2) where station 0 is at the inlet and station 9 is the exhaust, f is defined as the mass flow ratio of fuel to air in the combustion chamber: \(f=\frac{\dot{m}_{fuel}}{\dot{m}_{air}}\), \(\beta\) is the bleed air fraction: \(\beta=\frac{\dot{m}_{0}}{m_{c}}\), and \(f_{0}\) is the total fuel flow to engine inlet airflow ratio: \(f_{0}=\frac{\dot{m}_{f}+\dot{m}_{fAB}}{\dot{m}_{c}+\dot{m}_{F}}\).
* Specific fuel consumption (SFC): Fuel consumed by a vehicle for each unit of power output. \[SFC=\frac{f_{0}}{F/\dot{m}_{0}}\] (3)
* Propulsive efficiency (\(\mathbf{\eta_{p}}\)): Thrust ratio to engine gas flow rate of kinetic energy output. \[\eta_{P}=2\bigg{(}\frac{Fg_{c}}{\dot{m}_{0}V_{0}}\bigg{)}\bigg{/}\bigg{\{} \bigg{[}1+f_{0}-\frac{\beta}{1+\alpha}\bigg{]}\bigg{(}\frac{V_{9}}{a_{0}} \bigg{)}^{2}-1\bigg{\}}\] (4)  
The boundary conditions of the parametric analysis were the range of input variables used throughout this study. This range is shown in Table 1. Other boundary conditions include generic fuel type with a fuel heating value of 42 MJ/kg, and the overall pressure ratio was set to 32:1.