# The Exergy Cost Theory Revisited

^{*}

## Abstract

**:**

## 1. Introduction

- Resources rule: In the absence of external assessment, the exergy cost of the flows entering the plant equals their exergy.
- Cost Conservation Rule: All cost generated by the production process must be included in the final product’s cost. In the absence of an external assessment, we have to assign a zero value for the cost of the losses of the plant.
- Unspent Fuel rule: If an output flow of a process is a part of the unspent fuel of this process, the unit exergy cost is the same as that of the input flow from which the output flow comes (also known as F rule).
- Co-products rule: If a process has a product composed of several flows, with the same thermodynamic quality, then the same unit exergy cost will be assigned to all of them (also known as P rule).

- Waste rule: The exergy cost of a waste stream leaving the system’s boundaries is allocated null (waste cost internalization). Its exergy cost, formed by adding the fuel required to produce it plus the fuel used to carry and dispose of it, is allocated to the processes, which have generated it.

## 2. Waste from a Thermodynamic Viewpoint

#### The Irreversibility Carrier Concept

_{2}, and NOx (plus water, oxygen, nitrogen, and other components of the incoming air, not considered waste). Carbon dioxide and water are produced by the combustion of carbonated fuel, while NOx and excess heat are produced in the combustion chamber. One can compare this combustion chamber to an ideal fuel cell. In such a case, only CO

_{2}and water would be produced. The remaining waste, CO

_{2}, and NOx are material streams that need additional dissipative units to get rid of them. Carbon dioxide requires carbon capture and storage techniques, while NOx requires special combustion chambers to prevent its formation. These abatement processes require further fuel and investment whose costs must be added to the plant product(s).

## 3. The Productive Structure Graph

- one or several inflows that provide exergy to the process and
- input and output flows enter into the process and leave it, after some exergy transfer to the process.

- one or several outflows produced by the process and
- output and input flows enter into the process and leave it, after some exergy increase.

#### 3.1. The Production Layer

#### 3.2. The Waste Layer

## 4. The Fuel–Product–Waste Cost Allocation Rules

- Resources rule: The exergy cost is relative to the system boundaries. In the absence of external assessment, the exergy costs of the flows entering the system equal their exergy.$${B}_{i}^{*}={B}_{i},\phantom{\rule{1.em}{0ex}}i\in {\mathcal{S}}_{h},\phantom{\rule{0.277778em}{0ex}}h\in {\mathcal{P}}_{0}$$
- Cost Conservation rule: The exergy cost is a conservative property.$${P}_{u}^{*}={F}_{u}^{*}+{R}_{u}^{*},\phantom{\rule{1.em}{0ex}}u\in \mathcal{V}$$This rule implies all the cost associated to a process, both the spent resources cost ${F}_{u}^{*}$ and cost of waste generated ${R}_{u}^{*}$, must be assessed as a production cost.
- Unspent fuel rule: All the outputs of a fuel group have the same unit cost equal to the unit cost of the input fuel streams:$${k}_{j}^{*}={\kappa}_{h}^{*}=\frac{{F}_{u,h}^{*}}{{F}_{u,h}},\phantom{\rule{1.em}{0ex}}j\in {\mathcal{S}}_{h},\phantom{\rule{0.277778em}{0ex}}h\in {\mathcal{F}}_{u}$$This rule implies that the process irreversibilities will be added to the cost of the output product streams.
- Co-production rule: All products of the same quality at the output of a system’s component have the same unit exergy cost.
- All the product groups of a process have the same unit exergy cost:$$\frac{{P}_{u}^{*}}{{P}_{u}}=\frac{{P}_{u,l}^{*}}{{P}_{u,l}},\phantom{\rule{1.em}{0ex}}\ell \in {\mathcal{P}}_{u}$$
- All the outputs streams of a product group have the same unit exergy cost:$${k}_{j}^{*}={\kappa}_{h}^{*},\phantom{\rule{1.em}{0ex}}j\in {\mathcal{S}}_{h},\phantom{\rule{0.277778em}{0ex}}h\in {\mathcal{P}}_{u}$$

The idea of products of the same quality needs a detailed aggregation level of the system. - Waste rule: The exergy cost of waste exiting the analyzed system must be charged to the product of the components that generate them.$$\begin{array}{cc}\hfill {\omega}_{0,h}^{*}& =0,\phantom{\rule{1.em}{0ex}}h\in \mathcal{R}\hfill \\ \hfill {R}_{u}^{*}& =\sum _{h\in \mathcal{R}}{\kappa}_{h}^{*}\phantom{\rule{0.166667em}{0ex}}{R}_{hu},\phantom{\rule{1.em}{0ex}}u\in {\mathcal{V}}_{P}.\hfill \end{array}$$This rule implies that the cost of waste flows, leaving the system boundaries, is zero, and then the cost of the non-recoverable exergy of these flows is assessed to the final products by means of the waste internalization.

#### 4.1. Fuel–Product–Waste Rules Matrix Representation

#### 4.2. Exergy Cost Computation

## 5. The Generalized Exergy Cost

## 6. Process Thermoeconomics

#### 6.1. The Fuel–Product Table

#### 6.2. The Dissipative Processes Table

#### 6.3. The Cost Equations for the Process Model

## 7. The Resource-Driven Model

#### 7.1. Cost Equations in the Resource-Driven Model

#### 7.2. Production Cost Decomposition

## 8. The Demand-Driven Model

#### Cost Equations in the Demand-Driven Model

## 9. The Irreversibility Cost Formula

_{2}.

_{2}emissions) for the unit cost of NG, and a value of 0.238 kW/kW NG for the cost of CO

_{2}abatement. Finally, we also include the depreciation, maintenance, and operating costs of the plant’s processes $\mathbf{Z}$. They have been calculated from the work in [33] and converted to energy units using a natural gas price of 20 MWh as the conversion factor. In this example, the costs of reducing CO

_{2}are considered as an external cost that is allocated to the stack process. These values are summarized in Table 10.

_{2}emissions.

_{2}emissions. The exergy cost of the flue gas is 12.8 times the exergy of the dissipated gases. This cost is redistributed to the final products of the plant and accounts for about 20% of the total cost.

## 10. Methods for Cost Allocation of Waste

- (a)
- The cost of waste must be charged to the productive units that have generated it. Here, it is essential to establish a conceptual difference between the exergy of waste and its cost. Costs also encompass exergy consumption to be involved in the elimination of the system waste. Therefore, it is the cost, not its exergy, that is of actual interest. This raises a need that is expressed in the second premise.
- (b)
- It is needed to identify the process of cost formation of waste. This process is parallel to the process of production cost formation because waste is eliminated in the dissipative components, but formed along with the productive processes.
- (c)
- The responsibility of a productive component for waste generation lies in its contribution to the formation cost of waste.

- Method A (Exergy distribution): It is directly based on the information provided by the FP table [12]. Then, we consider the values of the table $\left[\mathbf{D}\right]$ as the exergy provided by the component j that is disposed in the dissipative unit r:$${\psi}_{rj}={\displaystyle \frac{{E}_{ir}}{{F}_{r}-{E}_{0r}}},\phantom{\rule{1.em}{0ex}}r\in {\mathcal{V}}_{D},\phantom{\rule{0.166667em}{0ex}}j\in {\mathcal{V}}_{P}$$This method looks for the immediate cause and distributes the waste’s costs to the components that have produced it directly and in proportion to their exergy.The next methods are based on the contribution of the productive units to the direct exergy cost of waste. As the unit production cost has not been determined yet, we will consider the value of the unit cost due to irreversibilities ${c}_{P,i}^{e}$ and the values of the production cost operator $\left|\mathbf{P}\right.\u232a$ instead. In all cases, direct exergy cost is considered.
- Method B (Exergy cost distribution): This allocation method is similar to the previous one, but it distributes the cost in proportion to their exergy cost instead of exergy. Note that in many cases, the exergy of waste is not relevant but their cost. According to (52), if $r\in {\mathcal{V}}_{D}$ is a dissipative unit,$${C}_{F,r}^{e}={C}_{0r}+\sum _{j=1}^{n}{c}_{P,j}^{e}\phantom{\rule{0.166667em}{0ex}}{E}_{jr}$$Equation (52) means that the production cost is equal to the sum of the cost of exergy streams eliminated in this dissipative unit, and the waste cost must be allocated to the processes have produced them.$${\psi}_{rj}=\frac{{c}_{P,j}{E}_{jr}}{{C}_{F,r}^{e}-{C}_{0r}},\phantom{\rule{1.em}{0ex}}r\in {\mathcal{V}}_{R},\phantom{\rule{0.277778em}{0ex}}j\in {\mathcal{V}}_{P}$$
- Method C (Irreversibility–cost distribution): A more complex method considers the complete chain of production cost [34]. It is based on Equation (77):$${C}_{P,r}^{e}={P}_{r}+\sum _{j\in {V}_{P}}{I}_{j}{\pi}_{jr}^{*}$$where ${\pi}_{jr}^{*}$ is the corresponding element in matrix $\left(\right)$. Decomposing the exergy of waste by its individual contributions, we get$${C}_{P,r}^{e}={C}_{0r}+\sum _{j\in {V}_{P}}{E}_{jr}+{I}_{j}{\pi}_{jr}^{*}$$therefore, the waste cost distribution parameters could be defined as$${\psi}_{rj}=\frac{{E}_{jr}+{I}_{j}{\pi}_{jr}^{*}}{{C}_{P,r}^{e}-{C}_{0r}},\phantom{\rule{1.em}{0ex}}r\in {\mathcal{V}}_{D},\phantom{\rule{0.277778em}{0ex}}j\in {\mathcal{V}}_{P}$$This method combines the first one with the irreversibility cost formula, to distribute the cost according to the irreversibility carriers of waste streams. This method takes into account the full path of the waste cost formation.
- Method D (Resources distribution): The last proposed method is based on the idea of cost definition, i.e., the cost of waste is the amount of external resources spent to produce it, and then allocate the cost of waste proportionally to the resources consumed:$${C}_{P,r}^{e}=\sum _{j=1}^{n}{E}_{0j}\phantom{\rule{0.166667em}{0ex}}{\pi}_{jr}^{*}$$and the waste distribution parameters are defined as$${\psi}_{rj}=\frac{{E}_{0j}\phantom{\rule{0.166667em}{0ex}}{\pi}_{jr}^{*}}{{C}_{P,r}^{e}},\phantom{\rule{1.em}{0ex}}r\in {\mathcal{V}}_{R},\phantom{\rule{0.277778em}{0ex}}j\in {\mathcal{V}}_{P}$$This method allocates the waste cost to the overall system according to the proportion of each external resource used to produce the waste. This method is used in the TGAS example. It is equivalent to increasing the cost of natural gas by a factor of 1.0609 (6%), determined by calculation of ${\mathbf{C}}_{R}$, and therefore the cost of all streams is increased by the same factor.

## 11. Recycling Saving Accounting

## 12. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## Nomenclature

Scalars | |

n | Number of system processes |

m | Number of system flows |

ℓ | Number of productive groups |

r | Number of dissipative processes |

B | Exergy of a flow (kW) |

C | Generalized exergy cost |

c | Unit generalized exergy cost |

E | Exergy of a productive group (kW) |

F | Exergy of a fuel stream (kW) |

I | Internal irreversibilities of a process (kW) |

${I}^{R}$ | External irreversibilities of a process (kW) |

k | Unit consumption (kW/kW) |

L | Exergy of local Loses (kW) |

P | Exergy of a product stream (kW) |

q | Bifurcation ratio |

R | Exergy of waste allocated to a process (kW) |

${S}_{g}$ | Generated Entropy (kW/K) |

y | Bifurcation ratio |

Z | Cost associated to a process |

$\psi $ | waste allocation ratio |

$\rho $ | waste generation ratio |

$\sigma $ | Recycling ratio of waste |

Vectors and Matrices | |

${\mathbf{a}}_{0}$ | Adjacency matrix of system outputs $(\ell \times 1)$ |

$\left[{\mathbf{A}}_{E}\right]$ | Adjacency matrix of input flows of productive groups $(m\times \ell )$ |

$\left[{\mathbf{A}}_{F}\right]$ | Adjacency matrix of fuel streams of processes $(\ell \times n)$ |

$\left[{\mathbf{A}}_{P}\right]$ | Adjacency matrix of product streams of processes $(n\times \ell )$ |

$\left[{\mathbf{A}}_{R}\right]$ | Adjacency matrix of waste streams of processes $(\ell \times n)$ |

$\left[{\mathbf{A}}_{S}\right]$ | Adjacency matrix of output flows of productive groups $(\ell \times m)$ |

$\left[\mathbf{D}\right]$ | Dissipative process table |

$\left[\mathbf{E}\right]$ | Fuel–Product table |

$\langle \mathbf{A}\rangle $ | Productive matrix of the system $(\ell \times \ell )$ |

$\langle \mathbf{FP}\rangle $ | Matrix of bifurcation ratios $(n\times n)$ |

$\langle \mathbf{KP}\rangle $ | Matrix of unit consumptions $(n\times n)$ |

$\langle \mathbf{KR}\rangle $ | Matrix of waste generation ratios $(n\times n)$ |

$\langle \mathbf{PF}\rangle $ | Matrix of junction ratios $(n\times n)$ |

$\langle \mathbf{RP}\rangle $ | Matrix of waste allocation ratios $(n\times n)$ |

$\left(\right)$ | Generic base operator cost matrix for resources–driven model $(n\times n)$ |

$\left(\right)$ | Generic operator cost matrix for resources–driven model $(n\times n)$ |

$\left|\mathbf{X}\right.\u232a$ | Generic base operator matrix for demand–driven model $(n\times n)$ |

$\left(\right)$ | Generic operator matrix for demand–driven model $(n\times n)$ |

${\mathbf{u}}_{n}$ | Unitary vector $(n\times 1)$ |

${\mathbf{U}}_{n}$ | Identity matrix $(n\times m)$ |

${\mathbf{\nu}}_{0}$ | Vector of exergy of external resources $(\ell \times 1)$ |

${\mathbf{\omega}}_{0}$ | Vector of exergy of system outputs $(\ell \times 1)$ |

Sets | |

$\mathcal{B}$ | Set of system flows |

$\mathcal{L}$ | Set of system productive groups |

$\mathcal{R}$ | Set of system waste streams |

$\mathcal{V}$ | Set of system processes |

${\mathcal{E}}_{l}$ | Set of input flows of the productive group l |

${\mathcal{S}}_{l}$ | Set of output flows of the productive group l |

${\mathcal{F}}_{u}$ | Set of fuel streams of process u |

${\mathcal{P}}_{u}$ | Set of product streams of process u |

Subscripts, superscripts and accents | |

t | Transpose matrix or vector |

$\widehat{\phantom{x}}$ | Diagonal matrix |

* | Related to exergy costs |

0 | Related to environment |

e | Related to fuel |

P | Related to product |

R | Related to waste |

S | Related to system output |

T | Related to total or final production |

## Appendix A. Productive Matrices

- (a)
- $\mathbf{A}$ is productive.
- (b)
- There exists a positive diagonal matrix $\mathbf{D}$, such that ${\mathbf{D}}^{-1}\left(\right)open="("\; close=")">\mathbf{U}-\mathbf{A}$ is strictly diagonal dominant.
- (c)
- $\underset{k\to \infty}{lim}{\mathbf{A}}^{k}=0$, and $\left(\right)}^{\mathbf{U}}-1=\mathbf{U}+\sum _{k=1}^{\infty}{\mathbf{A}}^{k$
- (d)
- $\mathbf{U}-\mathbf{A}$ is inverse positive. That is, $\mathbf{L}={\left(\right)}^{\mathbf{U}}-1$ exists, and $\mathbf{L}>\mathbf{U}$.
- (e)
- The maximum of the moduli of the eigenvalues of $\mathbf{A}$, $\rho \left(\mathbf{A}\right)\le 1$

## Appendix B. The Woodbury Formula

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Flow | m (kg/s) | T (${}^{\circ}$C) | p (bar) | h (kJ/kg) | s (kJ/kg K) | B (kW) |
---|---|---|---|---|---|---|

0 | 37.53 | 20 | 1.013 | 0 | 0 | 0 |

1 | 0.7324 | 20 | 50,000 | 0 | 36,620 | |

2 | 37.53 | 291.4 | 8 | 272.4 | 0.0648 | 9512 |

3 | 38.26 | 1050 | 7.84 | 1205 | 1.170 | 32,990 |

4 | 38.26 | 598.2 | 1.05 | 676.5 | 1.264 | 11,707 |

5 | 38.26 | 137.4 | 1.013 | 137.3 | 0.394 | 835.3 |

6 | 10,225 | |||||

7 | 10,000 | |||||

8a | 7.6 | 20 | 1.013 | 83.9 | 0.296 | 0 |

8 | 7.6 | 212.4 | 20 | 2799 | 6.340 | 7168 |

9 | 38.26 | 137.4 | 1.013 | 137.3 | 0.394 | 835.3 |

Nr | Key | Process | Fuel | Product |
---|---|---|---|---|

1 | COMB | Combustor | ${B}_{1}$ | ${B}_{3}-{B}_{2}$ |

2 | COMP | Compressor | ${B}_{6}$ | ${B}_{2}-{B}_{0}$ |

3 | GTUR | Turbine | ${B}_{3}-{B}_{4}$ | ${B}_{6}+{B}_{7}$ |

4 | HRSG | HRSG | ${B}_{4}-{B}_{5}$ | ${B}_{8}-{B}_{8a}$ |

5 | STCK | Stack | ${B}_{5}$ | ${B}_{9}$ |

Id | Key | E (kW) | ${\mathit{E}}^{*}$ (kW) | ${\mathit{k}}_{\mathit{E}}^{*}$ (J/J) |
---|---|---|---|---|

1 | N1F1 | 36,620 | 36,620.0 | 1.0000 |

2 | N1P1 | 32,990 | 56,465.7 | 1.7116 |

3 | N2F1 | 10,225 | 18,417.3 | 1.8012 |

4 | N2P1 | 9512 | 18,417.1 | 1.9362 |

5 | N3F1 | 32,990 | 56,465.7 | 1.7116 |

6 | N3P1 | 20,225 | 36,429.3 | 1.8012 |

7 | N4F1 | 11,707 | 20,037.7 | 1.7116 |

8 | N4P1 | 7168 | 18,608.1 | 2.5960 |

9 | N5F1 | 835.3 | 1429.7 | 1.7116 |

10 | N5P1 | 835.3 | 1429.7 | 1.7116 |

11 | N0P1 | 36,620 | 36,620.0 | 1.0000 |

12 | N0F1 | 10,000 | 18,012.0 | 1.8012 |

13 | N0F2 | 7168 | 18,608.1 | 2.5960 |

14 | N0R1 | 835.3 | 1429.7 | 1.7116 |

Id | Process | F (kW) | P (kW) | I (kW) | k (J/J) | ${\mathit{k}}_{\mathit{P}}^{*}$ (J/J) |
---|---|---|---|---|---|---|

1 | COMB | 36,620 | 23,478 | 13,142 | 1.5598 | 1.6207 |

2 | COMP | 10,225 | 9512 | 713 | 1.0750 | 1.9362 |

3 | GTRB | 21,283 | 20,225 | 1058 | 1.0523 | 1.8012 |

4 | HRSG | 10,871.7 | 7168 | 3703.7 | 1.5167 | 2.5960 |

5 | STCK | 835.3 | 835.3 | 0.0 | 1.0000 | 1.7116 |

Total | 36,620 | 17,168 | 19,452 | 2.1330 | 2.1330 |

Id | From | To | B (kW) | ${\mathit{B}}^{*}$ (kW) | ${\mathit{k}}^{*}$ (J/J) |
---|---|---|---|---|---|

1 | N0P1 | N1F1 | 36,620 | 36,620.0 | 1.0000 |

2 | N2P1 | N1P1 | 9512 | 18,417.1 | 1.9362 |

3 | N1P1 | N3F1 | 32,990 | 56,465.7 | 1.7116 |

4 | N3F1 | N4F1 | 11,707 | 20,037.7 | 1.7116 |

5 | N4F1 | N5F1 | 835.3 | 1429.7 | 1.7116 |

6 | N3P1 | N2F1 | 10,225 | 18,417.3 | 1.8012 |

7 | N3P1 | N0F1 | 10,000 | 18,012.0 | 1.8012 |

8 | N4P1 | N0F2 | 7168 | 18,608.1 | 2.5960 |

9 | N5P1 | N0R1 | 835.3 | 1429.7 | 1.7116 |

Final Product | Process Resources | |||||||
---|---|---|---|---|---|---|---|---|

1 | ⋯ | j | ⋯ | n | Total | |||

External | ${E}_{01}$ | ⋯ | ${E}_{0j}$ | ⋯ | ${E}_{0n}$ | ${P}_{0}$ | ||

Resources | ||||||||

Process Products | 1 | ${E}_{10}$ | ${E}_{11}$ | ⋯ | ${E}_{1j}$ | ⋯ | ${E}_{1n}$ | ${P}_{1}$ |

⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | |||

i | ${E}_{i0}$ | ${E}_{i1}$ | ⋯ | ${E}_{ij}$ | ⋯ | ${E}_{in}$ | ${P}_{i}$ | |

⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | |||

n | ${E}_{n0}$ | ${E}_{n1}$ | ⋯ | ${E}_{nj}$ | ⋯ | ${E}_{nn}$ | ${P}_{n}$ | |

Total | ${F}_{0}$ | ${F}_{1}$ | ⋯ | ${F}_{j}$ | ⋯ | ${F}_{n}$ |

${\mathit{P}}_{\mathit{s}}$ | ${\mathit{F}}_{1}$ | ${\mathit{F}}_{2}$ | ${\mathit{F}}_{3}$ | ${\mathit{F}}_{4}$ | ${\mathit{F}}_{5}$ | Total | |
---|---|---|---|---|---|---|---|

${F}_{e}$ | 36,620.0 | 0.0 | 0.0 | 0.0 | 0.0 | 36,620.0 | |

${P}_{1}$ | 0.0 | 0.0 | 0.0 | 15,146.5 | 7737.1 | 594.5 | 23,478.0 |

${P}_{2}$ | 0.0 | 0.0 | 0.0 | 6136.5 | 3134.6 | 240.8 | 9512.0 |

${P}_{3}$ | 10,000.0 | 0.0 | 10,225.0 | 0.0 | 0.0 | 0.0 | 20,225.0 |

${P}_{4}$ | 7168.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 7168.0 |

${P}_{5}$ | 835.3 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 835.3 |

Total | 18,003.3 | 36,620.0 | 10,225.0 | 21,283.0 | 10,871.7 | 835.3 |

${\mathit{F}}_{0}^{*}$ | ${\mathit{F}}_{1}^{*}$ | ${\mathit{F}}_{2}^{*}$ | ${\mathit{F}}_{3}^{*}$ | ${\mathit{F}}_{4}^{*}$ | ${\mathit{F}}_{5}^{*}$ | Total | |
---|---|---|---|---|---|---|---|

${P}_{0}^{*}$ | 36,620.0 | 0.0 | 0.0 | 0.0 | 0.0 | 36,620.0 | |

${P}_{1}^{*}$ | 0.0 | 0.0 | 0.0 | 24,547.2 | 12,539.1 | 963.4 | 38,049.7 |

${P}_{2}^{*}$ | 0.0 | 0.0 | 0.0 | 11,881.4 | 6069.2 | 466.3 | 18,416.9 |

${P}_{3}^{*}$ | 18,011.7 | 0.0 | 18,416.9 | 0.0 | 0.0 | 0.0 | 36,428.6 |

${P}_{4}^{*}$ | 18,608.3 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 18,608.3 |

${P}_{5}^{*}$ | 0 | 1429.7 | 0 | 0 | 0 | 0 | 1429.7 |

Total | 36,620.0 | 38,049.7 | 18,416.9 | 36,428.6 | 18,608.3 | 1429.7 |

Process | ${\mathit{F}}^{*}$ (kW) | ${\mathit{R}}^{*}$ (kW) | ${\mathit{P}}^{*}$ (kW) | ${\mathit{k}}_{\mathit{P}}^{*}$ (kW/kW) | ${\mathit{k}}_{\mathit{P},\mathit{e}}^{*}$ (kW/kW) | ${\mathit{k}}_{\mathit{P},\mathit{r}}^{*}$ (kW/kW) |
---|---|---|---|---|---|---|

COMB | 36,620.0 | 1429.7 | 38,049.7 | 1.6207 | 1.5598 | 0.0609 |

COMP | 18,416.9 | 0.0 | 18,416.9 | 1.9362 | 1.8634 | 0.0728 |

GTRB | 36,428.6 | 0.0 | 36,428.6 | 1.8012 | 1.7335 | 0.0677 |

HRSG | 18,608.3 | 0.0 | 18,608.3 | 2.5960 | 2.4985 | 0.0975 |

STCK | 1429.7 | 0.0 | 1429.7 | 1.7116 | 1.6473 | 0.0643 |

COMB | COMP | GTURB | HRSG | STCK | |
---|---|---|---|---|---|

Z | 141.92 | 1216.38 | 1662.40 | 1076.51 | 8715.56 |

${\overline{C}}_{e}$ | 39623 | 0 | 0 | 0 | 0 |

COMB | COMP | GTURB | HRSG | STCK | |
---|---|---|---|---|---|

${c}_{e}^{*}$ | 1.0859 | 1.3628 | 1.2438 | 1.2647 | 11.5998 |

${I}_{1}$ | 0.6078 | 0.7262 | 0.6755 | 0.9736 | 0.6419 |

${I}_{2}$ | 0.0000 | 0.1516 | 0.0460 | 0.0663 | 0.0437 |

${I}_{3}$ | 0.0000 | 0.1038 | 0.0966 | 0.0454 | 0.0299 |

${I}_{4}$ | 0.0000 | 0.0000 | 0.0000 | 0.6535 | 0.0000 |

R | 0.4553 | 0.5439 | 0.5060 | 0.7293 | 0.4808 |

Total | 2.1490 | 2.8883 | 2.5679 | 3.7328 | 12.7961 |

Method A | Method B | Method C | Method D | |||||
---|---|---|---|---|---|---|---|---|

Process | ${\mathit{C}}_{\mathit{R}}$ | ${\mathit{c}}_{\mathit{P}}^{\mathit{r}}$ | ${\mathit{C}}_{\mathit{R}}$ | ${\mathit{c}}_{\mathit{P}}^{\mathit{r}}$ | ${\mathit{C}}_{\mathit{R}}$ | ${\mathit{c}}_{\mathit{P}}^{\mathit{r}}$ | ${\mathit{C}}_{\mathit{R}}$ | ${\mathit{c}}_{\mathit{P}}^{\mathit{r}}$ |

COMB | 1017.5 | 0.0433 | 951.3 | 0.0405 | 1130.4 | 0.0481 | 1429.7 | 0.0609 |

COMP | 412.2 | 0.1161 | 478.4 | 0.1231 | 278.0 | 0.1025 | 0.0 | 0.0728 |

GTUR | 0.0 | 0.0677 | 0.0 | 0.0677 | 20.9 | 0.0682 | 0.0 | 0.0677 |

HRSG | 0.0 | 0.0975 | 0.0 | 0.0975 | 0.0 | 0.0968 | 0.0 | 0.0975 |

STCK | 0.0 | 0.0643 | 0.0 | 0.0643 | 0.0 | 0.0638 | 0.0 | 0.0643 |

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## Share and Cite

**MDPI and ACS Style**

Torres, C.; Valero, A.
The Exergy Cost Theory Revisited. *Energies* **2021**, *14*, 1594.
https://doi.org/10.3390/en14061594

**AMA Style**

Torres C, Valero A.
The Exergy Cost Theory Revisited. *Energies*. 2021; 14(6):1594.
https://doi.org/10.3390/en14061594

**Chicago/Turabian Style**

Torres, César, and Antonio Valero.
2021. "The Exergy Cost Theory Revisited" *Energies* 14, no. 6: 1594.
https://doi.org/10.3390/en14061594