# Residue Cost Formation of a High Bypass Turbofan Engine
## 5 Exergoeconomic Cost Analysis
### Residue Formation Cost
The aircraft losses come from the residual physical exergies of the air and exhaust gases (\(\dot{F}_{cS}=\dot{E}_{8}^{PH}\) and \(\dot{F}_{hS}=\dot{E}_{7}^{PH}\)) formed in the productive components of \(\mathcal{P}_{cS}\) and \(\mathcal{P}_{hS}\), whereas the chemical exergy of the exhaust gases forms only in the CC (\(\dot{F}_{c1S}=\dot{E}_{7}^{CH}=\dot{E}_{4}^{CH}\)). According to Equation (15), the cost balance for the cold, hot, and chemical stacks is:  
\[\dot{\Pi}_{\dot{F}_{r}}=-\dot{\Pi}_{\dot{F}_{ar}}>0,\ r\in\mathcal{D}=\{\text{ cS, hS, chS}\} \tag{16}\]  
In this case study, the dissipative components were not part of the energy system. If it is desired that these components are charged with the elimination costs, Equation (16) indicates that a negative value must be allocated to the cost of \(\dot{F}_{r}\) as an output stream of the system [31]. This is only a case of internalizing the costs taking place downstream of the main process. These costs must be allocated to the productive components involved in the formation cost of each residue. In this manner, the formation cost of a residue (\(\dot{\Pi}_{\dot{F}_{r}}\)) is decomposed into several costs corresponding to the originating components. \[\dot{\Pi}_{\dot{F}_{r}}=\sum_{i\in\mathcal{P}_{r}}\dot{\Pi}_{\dot{F}_{ri}},\ r \in\mathcal{D} \tag{17}\]  
where \(\mathcal{P}_{r}\) is the set of productive components that generate the residue dissipated in the \(r\)th dissipative component (\(\mathcal{P}_{cS}\), \(\mathcal{P}_{hS}\), and \(\mathcal{P}_{chS}\)) and \(\dot{\Pi}_{\dot{F}_{ri}}\) is the formation cost of the residue dissipated in the \(r\)th component that has been generated by the \(i\)th productive component. The formation cost of the residues charged to the \(i\)th productive component is given by:  
\[\dot{\Pi}_{\dot{F}_{hi}}=\dot{\Pi}_{\dot{F}_{cS,i}}+\dot{\Pi}_{\dot{F}_{hS,i}}+ \dot{\Pi}_{\dot{F}_{chS,i}} \tag{18}\]  
To determine the values of \(\dot{\Pi}_{\dot{F}_{ri}}\), thermoeconomics defines the residue cost allocation ratios. For a given residue dissipated in the \(r\)th dissipative component, the residue cost allocation ratio associated with the \(i\)th productive component \(\left(\mu_{i}^{r}\right)\) is the fraction of the residue cost allocated to this component. \[\dot{\Pi}_{\dot{F}_{ri}}=\mu_{i}^{r}\dot{\Pi}_{\dot{F}_{r}},\ \ \text{with}\ \sum_{i\in\mathcal{P}_{r}}\mu_{i}^{r}=1,\ r\in\mathcal{D} \tag{19}\]  
These ratios determine how the cost of a residue should be decomposed into several costs. However, there is no definitive method to determine them. A frequently used criterion is to allocate the cost of residues proportionally to entropy changes during the process [32; 33]. The criterion used in this work was proposed by [PERSON] et al. [13] to make the residue cost allocation ratios proportional to the exergy flow produced in the \(i\)th component that is processed and dissipated in the \(r\)th component according to the productive structure of the plant. According to the thermoeconomic theory, these ratios are known as recirculation coefficients. They can be obtained directly from the information provided by the productive structure and FPR table [34; 35]. For the turbofan engine, they are defined in Table 4.