# Residue Cost Formation of a High Bypass Turbofan Engine
## 3 Energy Analysis  
The energy balances of the turbofan components are presented in Table 1. \(\dot{H}_{t}\) is the stagnation enthalpy flow, which is the sum of the flow rates for the static enthalpy (\(\dot{H}\)) and kinetic energy (\(\dot{m}u^{2}/2\)). Figure 1: Schematic diagram of a turbofan engine. The fuel-to-air ratio (\(far\)) is obtained from the energy balance for the CC (Table 1) and is expressed as:  
\[far=\frac{\dot{m}_{f}}{\dot{m}_{a}}=\left(\frac{1}{\beta+1}\right)\frac{h_{t4}-h_ {t3}}{\eta_{b}LHV-h_{t4}} \tag{2}\]  
where \(h_{t4}\) and \(h_{t3}\) are the stagnation enthalpies of air and combustion gases in States 3 and 4, respectively, and \(\eta_{b}\) is the adiabatic efficiency of the CC. Table 2 lists the temperatures and pressures of each state, as derived from the energy balance. \(T_{t}\) and \(P_{t}\) are the stagnation temperature and pressure, respectively. If the nozzles are not choked, then the pressure at the exit of each nozzle is equal to the specified back pressure (\(P_{7}=P_{a}=P_{8}\)), as shown in Table 2. The resulting velocities are expressed as follows. \[u_{7}=\sqrt{\frac{2\gamma_{cg}}{\gamma_{cg}-1}R_{cg}T_{t6}\eta_{N}\left[1- \left(\frac{P_{a}}{P_{6}}\right)^{\frac{\gamma_{cg}}{\gamma_{cg}-1}}\right]} \text{ and }u_{8}=\sqrt{\frac{2\gamma_{a}}{\gamma_{a}-1}R_{a}T_{t2}\eta_{ FN}\left[1-\left(\frac{P_{a}}{P_{t2}}\right)^{\frac{\gamma_{bg}}{\gamma_{a}-1}} \right]} \tag{3}\]  
In this study, the thrust is the net force acting on the \(x\)-surface of the control volume in the \(x\)-direction. If the density is constant at the entrance and exit surfaces, the exhaust plane pressure is atmospheric and \(u_{a}=M\sqrt{\gamma_{a}R_{a}T_{a}}\). The thrust per unit airflow is then given as:  
\[\frac{\tau}{\dot{m}_{a}}=\frac{\left[1+\left(1+\beta\right)far\right]u_{7}-u_{ a}+\beta\left(u_{8}-u_{a}\right)}{1+\beta} \tag{4}\]  
The energy balance of the turbofan presented in Table 1 can be expressed as:  
\[\dot{Q}_{b}=\underbrace{\frac{\left[1+\left(1+\beta\right)far\right]u_{7}^{2} -u_{a}^{2}+\beta\left(u_{8}^{2}-u_{a}^{2}\right)}{1+\beta}\dot{m}_{a}}_{\text {Propulsive kinetic energy (useful energy)}}+\underbrace{\frac{\left[1+\left(1+\beta\right)far\right]h_{7}- h_{a}+\beta\left(h_{7}-h_{a}\right)}{1+\beta}\dot{m}_{a}}_{\text{Thermal energy (unuseful energy)}} \tag{5}\]  
The energy performance indicators are the thermal efficiency, propulsion efficiency, and overall efficiency. According to Equation (5), the thermal efficiency \(\eta_{th}\) is the relation between the propulsive \begin{table}
\begin{tabular}{c c c} \hline \hline
**Component** & **Energy Balance** & **Adiabatic Efficiency** \\ \hline D & \(\dot{H}_{t1}-\dot{H}_{a}=\frac{1}{2}\dot{m}_{a}u_{a}^{2}\) & \(\eta_{D}=\frac{h_{t1s}-h_{a}}{h_{t1}-h_{a}}\) \\ F & \(W_{F}=\dot{H}_{t2}-\dot{H}_{t1}\) & \(\eta_{F}=\frac{h_{t2s}-h_{t1}}{h_{t2}-h_{t1}}\) \\ C & \(\dot{W}_{C}=\dot{H}_{t3}-\dot{H}_{t2}\) & \(\eta_{C}=\frac{h_{t3s}-h_{t2}}{h_{t3}-h_{t2}}\) \\ CC & \(\dot{Q}_{b}=H_{t4}-H_{t3}\) & \(\eta_{b}=\frac{q_{b}}{LHV}\) \\ HPT & \(\dot{W}_{HPT}=H_{t4}-H_{t5}=\dot{W}_{C}\) & \(\eta_{HPT}=\frac{h_{t4}-h_{t5}}{h_{t4}-h_{t5s}}\) \\ LPT & \(\dot{W}_{LPT}=\dot{H}_{t5}-\dot{H}_{t6}=\dot{W}_{F}\) & \(\eta_{LPT}=\frac{h_{t5}-h_{t6}}{h_{t5}-h_{t6s}}\) \\ FN & \(\dfrac{\beta\dot{H}_{t2}-\left(1+\beta\right)\dot{H}_{8}}{1+\beta}=\frac{\beta \dot{m}_{a}}{2\left(1+\beta\right)}u_{8}^{2}\) & \(\eta_{FN}=\frac{h_{t2}-h_{8}}{h_{t2}-h_{t8s}}\) \\ N & \(\dot{H}_{t6}-\dot{H}_{7}=\dfrac{1}{2}\dot{m}_{tcg}u_{7}^{2}\) & \(\eta_{N}=\frac{h_{t6}-h_{7}}{h_{t6}-h_{t7s}}\) \\ \hline Turbofan & \(\dot{H}_{ta}+\dot{Q}_{b}=\dot{H}_{7}+H_{t8}\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Energy balance and adiabatic efficiency of the turbofan components. kinetic energy and total heat input \(\dot{Q}_{b}\). The propulsion efficiency \(\eta_{prop}\) relates the thrust power \(\dot{\tau}=\tau u_{a}\) required by the turbofan to the propulsive kinetic energy.