# Residue Cost Formation of a High Bypass Turbofan Engine
## 5 Exergoeconomic Cost Analysis  
### Residue Formation Cost  
The formation costs of residues must be rationally allocated to the functional system products. This analysis is based on the premise that residues are formed within the productive components and released to the environment from the dissipative components, where they are treated with additional \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline  & & & & & & & & & & & & & & \\ \multicolumn{12}{c}{**Resources of Productive Components**} & \multicolumn{6}{c}{**Dissipative Components**} \\  & & & & & & & & & & & & & \\ \multicolumn{12}{c}{**R\({}^{C}\)**} & \(\mathbf{\hat{R}}^{RS}\) & \(\mathbf{\hat{R}}^{RS}\) & \(\mathbf{\hat{R}}^{RS}\) & \\  & & & & & & & & & & & & \\ \multicolumn{12}{c}{**F\({}_{0}\)**} & \(\mathbf{\hat{F}}_{D}\) & \(\mathbf{\hat{F}}_{C}\) & \(\mathbf{\hat{F}}_{C}\) & \(\mathbf{\hat{F}}_{HPT}\) & \(\mathbf{\hat{F}}_{LPT}\) & \(\mathbf{\hat{F}}_{N}\) & \(\mathbf{\hat{F}}_{ES}\) & \(\mathbf{\hat{F}}_{HS}\) & \(\mathbf{\hat{F}}_{chS}\) \\ \hline \(\hat{P}_{0}\) & & & & & & & & & & & \\ \(\hat{P}_{D}\) & & & & & & & & & & & & \\ \(\hat{P}_{F}\) & & & & & & & & & & & & \\ \(\hat{P}_{FN}\) & \(\hat{P}_{FN}\) & & & & & & & & & & & \\ \(P_{C}\) & & & & & & & & & & & & \\ \(\hat{P}_{CC}\) & & & & & & & & & & & & \\ \(\hat{P}_{HPT}\) & & & & & & & & & & & & \\ \(\hat{P}_{LPT}\) & & & & & & & & & & & & \\ \(\hat{P}_{N}\) & \(\hat{P}_{N}\) & & & & & & & & & & & \\ \hline \hline \end{tabular} \(\mathbf{F}_{e}=\{\hat{E}_{0i}\}\): Energy of the external resources used by the \(i\)th component. \(\hat{P}_{T}=\sum\limits_{i\in\mathcal{P}}\hat{E}_{0j}\)  
\(\mathbf{P}_{s}=\{\hat{E}_{0i}\}\): Energy of the final product obtained from the \(i\)th component. \(\hat{P}_{T}=\sum\limits_{i\in\mathcal{P}}\hat{E}_{i0}\)  
\(E_{ij}\): Product of the \(i\)th component used as resource in the \(j\)th component.  
\(\mathbf{R}^{\prime}=\{\hat{E}_{kk}\}\): Product of the \(i\)th component and dissipated in the \(r\)th dissipative component. \(\hat{P}_{k}=\sum\limits_{i\in\mathcal{P}}\mathbf{R}_{i}^{k}=\sum\limits_{i\in \mathcal{P}}\hat{E}_{ik}\).  
Recircirculation coefficients  
\(r_{CD}=\frac{\hat{P}_{CD}}{\hat{P}_{CD}+\hat{P}_{CF}},r_{CF}=\frac{\hat{P}_{CF} }{\hat{P}_{CD}+\hat{P}_{CF}},\;\;\;r_{CD}+r_{CF}=1\)  
\(r_{hD}=\frac{\hat{P}_{hD}}{\hat{P}_{hD}+\hat{P}_{hF}+\hat{P}_{C}+\hat{P}_{CC}^{ BH}},\;r_{HF}=\frac{\hat{P}_{hF}}{\hat{P}_{hD}+\hat{P}_{hF}+\hat{P}_{C}+\hat{P}_{CC}^{ PH}}\),  
\(r_{C}=\frac{\hat{P}_{C}}{\hat{P}_{hD}+\hat{P}_{hF}+\hat{P}_{C}+\hat{P}_{CC}^{PH}},r _{CC}^{PH}=\frac{\hat{P}_{CC}^{PH}}{\hat{P}_{hD}+\hat{P}_{hF}+\hat{P}_{C}+\hat{P} _{CC}^{PH}}\);  
\(r_{hD}+r_{HF}+r_{C}+r_{CC}^{PH}=1\); \(r_{CC}^{CH}=\frac{\hat{P}_{chS}}{\hat{F}_{chS}}=1\)  
\end{table} 
Table 4: FPR table and recirculation coefficients for the turbofan. resources, and whose abatement costs are charged to the productive components. The exergoeconomic cost of a residue (\(\dot{\Pi}_{\dot{R}_{r}}\)) has three contributions: the residue formation cost that corresponds to the exergy in a residual exergy flow, which is the resource of a dissipative component (\(\dot{\Pi}_{\dot{F}_{r}}\)); the cost of additional exergetic resources employed for its elimination (\(\dot{\Pi}_{\dot{F}_{ar}}\)); the non-exergetic abatement costs required for its elimination (\(\dot{Z}_{r}\)). The cost balance for the \(r\)th dissipative component takes the form [13]:  
\[\dot{\Pi}_{\dot{R}_{r}}=\dot{\Pi}_{\dot{F}_{r}}+\dot{\Pi}_{\dot{F}_{ar}}+\dot{Z }_{r},\ r\in\mathcal{D} \tag{15}\]  
The residues of the turbofan engine are exergy losses because they leave the stacks without any further use, so their exergy and exergoeconomic costs are zero. For all the flight phases, \(\dot{\Pi}_{\dot{R}_{r}}=0=\dot{R}_{r}^{*}\) for all \(r\in\mathcal{D}\). Because the stacks are imaginary dissipative components, there are no abatement costs for these losses (i.e., \(\dot{Z}_{r}=0\) for all \(r\in\mathcal{D}\)).