# Residue Cost Formation of a High Bypass Turbofan Engine
## 4 Exergy Analysis  
### Exergy Forms  
Exergy is defined as the maximum useful work that can be produced by a system interacting with a specified reference environment (known as the dead state and denoted by a zero subscript). In the absence of influences such as nuclear effects, magnetism, electricity, and surface tension, the exergy of a stream of a substance comprises its kinetic (\(\dot{E}^{K}\)), potential (\(\dot{E}^{P}\)), physical (\(\dot{E}^{PH}\)), and chemical (\(\dot{E}^{CH}\)) exergies. \[\dot{E}_{t}=\dot{E}^{K}+\dot{E}^{P}+\dot{E}^{PH}+\dot{E}^{CH} \tag{6}\]  
The kinetic exergy is equal to the kinetic energy when the velocity is considered relative to the surface of the Earth, and the potential exergy is equal to the potential energy when considered with respect to the average surface level of the Earth in the environment of the process under consideration [23]. \[\dot{E}^{K}=\frac{\dot{m}u^{2}}{2}\ \ \text{and}\ \ \ \dot{E}^{P}=\dot{m}gz \tag{7}\]  
where \(u\) and \(z\) are respectively the velocity and elevation relative to coordinates of the environment and \(g\) is the gravity. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**State**} & **Mass** & **Temperature** & **Pressure** \\  & **Flow Rate** & & \\ \hline \(a\) & \(\dot{m}_{a}\) & \(T_{a}\) & \(P_{a}\) \\ \(0a\) & \(\dot{m}_{a}\) & \(\dfrac{T_{ta}}{T_{a}}=1+\dfrac{\gamma_{a}-1}{2}M^{2}\) & \(\dfrac{P_{ta}}{P_{a}}=\left(1+\dfrac{\gamma_{a}-1}{2}M^{2}\right)^{\frac{ \gamma_{a}}{\gamma_{a}-1}}\) \\ \(01\) & \(\dot{m}_{a}\) & \(T_{t1}=T_{ta}\) & \(\dfrac{P_{t1}}{P_{a}}=\left(1+\eta_{D}\dfrac{\gamma_{a}-1}{2}M^{2}\right)^{ \frac{2\gamma_{a}}{\gamma_{a}-1}}\) \\ \(02\) & \(\dot{m}_{a}\) & \(\dfrac{T_{t2}}{T_{a}}=1+\dfrac{1}{\eta_{F}}\left(\pi_{F}^{\frac{\gamma_{a}-1}{ \gamma_{a}}}-1\right)\) & \(\dfrac{P_{t2}}{P_{t1}}=\pi_{F}\) \\ \(03\) & \(\dfrac{\dot{m}_{a}}{1+\beta}\) & \(\dfrac{T_{t3}}{T_{t2}}=1+\dfrac{1}{\eta_{C}}\left(\pi_{C}^{\frac{\gamma_{a}-1}{ \gamma_{a}}}-1\right)\) & \(\dfrac{P_{t3}}{P_{t2}}=\pi_{C}\) \\ \(04\) & \(\dot{m}_{cg}\) & \(T_{t4}\) & \(\dfrac{P_{t4}}{P_{t3}}=1-\Delta P_{b}\) \\ \(05\) & \(\dot{m}_{cg}\) & \(T_{t5}=T_{t4}-\dfrac{T_{t3}-T_{t2}}{1+(1+\beta)far}\) & \(\dfrac{P_{t5}}{P_{t4}}=\left[1-\dfrac{1}{\eta_{HPT}}\left(1-\dfrac{T_{t5}}{T_{ t4}}\right)\right]^{\frac{\gamma_{qt}}{\gamma_{a}-1}}\) \\ \(06\) & \(\dot{m}_{cg}\) & \(T_{t6}=T_{t4}-\dfrac{(1+\beta)\left(T_{t2}-T_{t1}\right)}{1+(1+\beta)far}\) & \(\dfrac{P_{t6}}{P_{t5}}=\left[1-\dfrac{1}{\eta_{LPT}}\left(1-\dfrac{T_{t6}}{T_{ t5}}\right)\right]^{\frac{\gamma_{qt}}{\gamma_{a}-1}}\) \\ \(7\) & \(\dot{m}_{cg}\) & \(\dfrac{T_{7}}{T_{t6}}=1-\eta_{N}\left[1-\left(\dfrac{P_{t}}{P_{t6}}\right)^{ \frac{\gamma_{qt}}{\gamma_{a}-1}}\right]\) & \(P_{7}=P_{a}\) \\ \(8\) & \(\dfrac{\dot{m}\dot{m}_{a}}{1+\beta}\) & \(\dfrac{T_{8}}{T_{t2}}=1-\eta_{FN}\left[1-\left(\dfrac{P_{a}}{P_{t2}}\right)^{ \frac{2\gamma_{a}}{\gamma_{a}-1}}\right]\) & \(P_{8}=P_{a}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Thermodynamic states of the turbofan engine. The physical exergy is the maximum work obtainable by taking the substance through reversible physical processes from its initial state to the state determined by \(T_{0}\) and \(P_{0}\)[23]. It is defined as \(dE^{PH}=dH-T_{0}(dH/T-V/TdP)\) and for a mixture of ideal gases yields:  
\[\dot{E}^{PH}=\dot{m}\left[c_{Pm}\left(T-T_{0}-T_{0}\ln\frac{T}{T_{0}}\right)+R _{m}T_{0}\ln\frac{P}{P_{0}}\right] \tag{8}\]  
where \(c_{Pm}\) is the heat capacity at a constant pressure and \(R_{m}\) is the specific gas constant of the mixture. The chemical exergy is the work that can be obtained by taking a substance at \(T_{0}\) and \(P_{0}\) to chemical equilibrium with the environment [23]. The molar chemical exergy of a mixture of ideal gases can be determined from the knowledge of the standard molar chemical exergies (\(\epsilon_{0,i}^{CH}\)) and molar compositions (\(x_{i}\)) of the species in the mixture: [24].