# Residue Cost Formation of a High Bypass Turbofan Engine
## 5 Exergoeconomic Cost Analysis
### Production Costs  
In this study, the symbolic thermoeconomic methodology [17] was used to calculate the production costs. The FPR table (Table 4) was used as the starting point to numerically calculate the costs of the system flows. This methodology is derived from the application of the extended cost propositions of the ECT, one of which is that the cost of residues is generated by the productive components from which they originate [10; 13].  
* The costs of the external resources are known: \(\dot{\Pi}_{E_{f}}=c_{f}\dot{E}_{f}\) and \(\dot{\Pi}_{E_{a}}=c_{a}\dot{E}_{a}\).
* The production cost of the \(i\)th productive component (\(\dot{\Pi}_{P_{i}}\)) is the sum of the cost of resources required to obtain it (\(\dot{\Pi}_{F_{i}}\)), the formation cost of residues allocated to this component (\(\dot{\Pi}_{F_{i}R_{i}}\)) (Equation (18)), and the purchase cost of the component (\(\dot{Z}_{i}\)). \(\dot{\Pi}_{P_{i}}=\dot{\Pi}_{F_{i}}+\dot{\Pi}_{F_{i}R_{i}}+\dot{Z}_{i}\), \(i\in\mathcal{P}\).
* If a component has a product comprising several flows, the same unit exergoeconomic cost is assigned to them. In fact, even if two or more products can be identified in the same component, their formation processes are inseparable at the considered level of aggregation.  
Symbolic thermoeconomics has two ways to determine the production costs of a system, including the effects of residues: the PFR and FPR models (see Table 10). In the PFR model, the thermoeconomic variables are functions of the recirculation coefficients, system products, and unit consumption of the components. This is commonly applied when the total production is fixed and the resource consumption varies. The FPR model is the most suitable for numerically calculating the production costs because it relates any exergoeconomic production cost with the external resources through a linear function of the distribution coefficients:  
\[\mathbf{\Pi}_{P}=\langle\mathbf{\mathsf{P}}^{*}|(\mathbf{\Pi}_{e}+\mathbf{\mathsf{ Z}})\text{, where }\langle\mathbf{\mathsf{P}}^{*}|=(\mathbf{\mathsf{U}}_{D}-\langle\mathbf{\mathsf{F}}\mathbf{\mathsf{P} }\rangle-\langle\mathbf{\mathsf{R}}\mathbf{\mathsf{P}}\rangle)^{-1} \tag{20a}\] \[\mathbf{\Pi}_{F}=\mathbf{\Pi}_{e}+\langle\mathbf{\mathsf{F}}\mathbf{\mathsf{P}} \rangle\mathbf{\Pi}_{P}\] (20b) \[\mathbf{\Pi}_{F_{R}}=\langle\mathbf{\mathsf{R}}\mathbf{\mathsf{P}}\rangle\mathbf{ \Pi}_{P} \tag{20c}\]  
where \(\langle\mathbf{\mathsf{F}}\mathbf{\mathsf{P}}\rangle\) and \(\langle\mathbf{\mathsf{R}}\mathbf{\mathsf{P}}\rangle\) are square matrices of \(n\times n\). Their elements are distribution coefficients defined respectively by \(y_{ij}=\dot{E}_{ji}/\dot{P}_{j}\) and \(\psi_{rj}=\dot{R}_{jr}/\dot{P}_{j}\) and satisfy \(\sum\limits_{i\in\mathcal{P}}\left(y_{ij}+\psi_{ij}\right)=1\). This model can decompose the exergoeconomic cost of products into three contributions: energy resources (\(\mathbf{\Pi}_{P}^{e}\)), component costs (\(\mathbf{\Pi}_{P}^{Z}\)), and residue costs (\(\mathbf{\Pi}_{P}^{r}\)) (Equation (21)).  
\[\mathbf{\Pi}_{P}=\langle\mathbf{\mathsf{P}}^{**}|(\mathbf{\Pi}_{e}+\mathbf{\mathsf{ Z}}+\mathbf{\Pi}_{R})=\mathbf{\Pi}_{P}^{e}+\mathbf{\Pi}_{P}^{Z}+\mathbf{\Pi}_{P}^{r} \tag{21}\]  
Here, \(\langle\mathbf{\mathsf{P}}^{**}|=(\mathbf{\mathsf{U}}_{D}-\langle\mathbf{\mathsf{F}}\mathbf{ \mathsf{P}}\rangle)^{-1}\) indicates the proportion at which the cost increases owing to the internal and external irreversibilities of each component. The most important application of calculating the costs of internal streams is to assess the impact of the additional fuel to compensate component malfunctions (inefficiences).