Question 7.11: Consider again the schema of the installation in Fig. 7.1 co......
Consider again the schema of the installation in Fig. 7.1 corresponding to Example E.7.1. Determine:
(a) The exergy costs of the flows.
(b) The exergy costs of the fuel and products of the equipment.
(a) To obtain the exergy costs of the flows, we need to solve Eq. (7.111) we show below
\mathbb{A}_{(m,{m})}B_{(m,{1})}^{\ast}=\mathbb{Y}_{e({m,{1}})}\qquad\qquad(7.111) \\ \mathbb{A}_{(25,25)}B_{(25,1)}^{\ast}=Y_{e(25,1)}Therefore, we need to construct the extended matrix \mathbb{A}{}=\begin{pmatrix}\\ \\ \frac{A}{\alpha_{e}}\\\ \overline{\alpha_{x}} \end{pmatrix} via the matrix A_{(8,25)} of Table E.7.1 and the matrices that represent the economic structure \alpha_{e} and \alpha_{x}, as well as the vector Y_{e(25,1)}. Table E.7.25 defines the existing bifurcations for the construction of the distribution matrix \alpha_{x(9,25)}, defined in P3. It should be noted that, for reasons of space, although the relationship between the exergies of the flows i and j is defined with the nomenclature x_{ij}, the bifurcation parameters have been named according to numbering from \underline 1 to 9. However, the second column represents the respective x_{ij}.
Table E.7.26 contains the incoming resources from outside that allows the resources matrix \alpha_{e(8,25)} and the vector Y_{e(25,1)} to be built. There are no sub-products or waste, with the only flow of losses in the 25th row, see Table E.7.5. Since there is no external assessment, we have that B_{e}^{\ast}=B_{e}. Next, the matrix equation is solved, inverting the extended matrix and clearing the vector of the exergy costs of the flows, Eq. (7.106). The results obtained are shown in Table E.7.27.
(b) The exergy costs of the fuel and products of the equipment are obtained from the exergy costs of the flows of Table E.7.27, and from the fuel and product matrices, which are re-flected in Tables E.7.6 and E.7.7. Using the matrix Eqs. (7.111) and (7.112) we get the re-sults that are shown in Table E.7.28.
The results shown in these Tables are the symbolic expressions of the exergy costs of the flows and those of fuels and products of the equipment. For a given thermody-namic state of the installation, both the bifurcation factors and the exergies of the flows will take specific numerical values. This Example and the next one serve as an intro-duction to the Symbolic Thermoeconomics that we develop in the next chapter.
Table E.7.1 Incidence matrix A.
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | ||
|
|
-1 | 1 | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | 1 | _ | _ | _ | -1 | |
|
|
_ | _ | _ | _ | _ | _ | -1 | 1 | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | 1 | _ | _ | _ | |
|
|
1 | -1 | -1 | 1 | -1 | 1 | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | |
|
|
_ | _ | _ | _ | 1 | -1 | _ | _ | -1 | 1 | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | |
|
|
_ | _ | _ | _ | _ | _ | _ | _ | 1 | -1 | _ | _ | _ | _ | -1 | 1 | _ | _ | 1 | _ | _ | _ | 1 | _ | _ | |
|
|
_ | _ | _ | _ | _ | _ | 1 | -1 | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | -1 | 1 | _ | _ | _ | 1 | _ | |
|
|
_ | _ | 1 | -1 | _ | _ | _ | _ | _ | _ | -1 | -1 | 1 | 1 | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | |
|
|
_ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | 1 | -1 | -1 | 1 | _ | _ | _ | _ | _ | _ | _ |
Table E.7.25 Bifurcation equations.
| {\mathrm{Bifurcations}} | \ \alpha_{X} | ||
| 1. | x_{1,2,25} | x_{1}=\frac{\dot{B}_{25}}{\dot{B}_{1}-\dot{B}_{2}} | {\dot{B}}_{25}\,-({\dot{B}}_{1}\,-{\dot{B}}_{2})\cdot x_{1}\,=0 |
| 2. | x_{5,6.3.4} | x_{2}=\frac{\dot{B}_{5}-\dot{B}_{6}}{\dot{B}_{3}-\dot{B}_{4}} | \left(\dot{B}_{5}\,-\dot{B}_{6}\right)\,-\left(\dot{B}_{3}\,-\dot{B}_{4}\right)\cdot x_{2}\,=0 |
| 3. | x_{1,2} | x_{3}=\frac{\dot{B}_{2}}{\dot{B}_{1}} | {\dot{B}}_{2}-{\dot{B}}_{1}\cdot x_{3}=0 |
| 4. | x_{5.6} | x_{4}=\frac{\dot{B}_{6}}{\dot{B}_{5}} | {\dot{B}}_{6}-{\dot{B}}_{5}\cdot x_{4}=0 |
| 5. | x_{9.10} | x_{5}=\frac{\dot{B}_{10}}{\dot{B}_{9}} | {\dot{B}}_{10}-{\dot{B}}_{9}\cdot x_{5}=0 |
| 6. | x_{7.8} | x_{6}=\frac{\dot{B}_{8}}{\dot{B}_{7}} | {\dot{B}}_{8}-{\dot{B}}_{7}\cdot x_{6}=0 |
| 7. | x_{12.14.11.13} | x_{7}=\frac{\dot{B}_{12}-\dot{B}_{14}}{\dot{B}_{11}-\dot{B}_{13}} | \left(\dot{B}_{12}\,-\dot{B}_{14}\right)\,-\left(\dot{B}_{11}\,-\dot{B}_{13}\right)\cdot x_7\,=0 |
| 8. | x_{4.3} | x_{8}=\frac{\dot{B}_{4}}{\dot{B}_{3}} | {\dot{B}}_{4}-{\dot{B}}_{3}\cdot x_{8}=0 |
| 9. | x_{15.16} | x_{9}=\frac{\dot{B}_{16}}{\dot{B}_{15}} | {\dot{B}}_{16}-{\dot{B}}_{15}\cdot x_{9}=0 |
Table E.7.26 Entrance flows to the installation.
| \mathrm{Entrances} | |||||||||||||||||||||||||
| 1. | {\dot{B}}_{e1}\,={\dot{B}}_{21} | 5. | {\dot{B}}_{e5}\,={\dot{B}}_{13} | ||||||||||||||||||||||
| 2. | {\dot{B}}_{e2}\,={\dot{B}}_{22} | 6. | {\dot{B}}_{e6}\,={\dot{B}}_{14} | ||||||||||||||||||||||
| 3. | {\dot{B}}_{e3}\,={\dot{B}}_{23} | 7. | {\dot{B}}_{e7}\,={\dot{B}}_{18} | ||||||||||||||||||||||
| 4. | {\dot{B}}_{e4}\,={\dot{B}}_{24} | 8. | {\dot{B}}_{e8}\,={\dot{B}}_{20} | ||||||||||||||||||||||
| \mathrm{Y}\mathrm{t_{e}} | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | {\dot{B}}_{e1} | {\dot{B}}_{e2} | {\dot{B}}_{e3} | {\dot{B}}_{e4} | {\dot{B}}_{e5} | {\dot{B}}_{e6} | {\dot{B}}_{e7} | {\dot{B}}_{e8} | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
Table E.7.5 Classification of flows in F, P and L.
| Fuel | Product | Losses | |
| ① | {\dot{B}}_{21} | {\dot{B}}_{1}-{\dot{B}}_{2} | {\dot{B}}_{25} |
| ② | {\dot{B}}_{22} | {\dot{B}}_{7}-{\dot{B}}_{8} | _ |
| ③ | {\dot{B}}_{1}-{\dot{B}}_{2} | \left(\dot{B}_{3}\,-\dot{B}_{4}\right)\,+\,\left(\dot{B}_{5}\,-\dot{B}_{6}\right) | _ |
| ④ | {\dot{B}}_{5}-{\dot{B}}_{6} | {\dot{B}}_{9}-{\dot{B}}_{10} | _ |
| ⑤ | \left(\dot{B}_{9}\;-\;\dot{B}_{10}\right)\;+\;\dot{B}_{23} | \dot{B}_{15}\,-\dot{B}_{16}\,-\,\dot{B}_{19} | _ |
| ⑥ | \left(\dot{B}_7\;-\;\dot{B}_{8}\right)\;+\;\dot{B}_{24} | {\dot{B}}_{19}-{\dot{B}}_{20} | _ |
| ⑦ | {\dot{B}}_{3}-{\dot{B}}_{4} | \big[{}~{(\dot{B}_{11}~-\dot{B}_{13})}~+{}~{(\dot{B}_{12}~-\dot{B}_{14})}] | _ |
| ⑧ | {\dot{B}}_{15}-{\dot{B}}_{16} | {\dot{B}}_{17}-{\dot{B}}_{18} | _ |
Table E.7.6 Matrix A_{F}.
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | ||
| A_{F}= | ① | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | 1 | _ | _ | _ | _ |
| ② | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | 1 | _ | _ | _ | ||
| ③ | 1 | -1 | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | |
| ④ | _ | _ | _ | _ | 1 | -1 | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | |
| ⑤ | _ | _ | _ | _ | _ | _ | _ | _ | 1 | -1 | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | 1 | _ | _ | |
| ⑥ | _ | _ | _ | _ | _ | _ | 1 | -1 | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | 1 | _ | |
| ⑦ | _ | _ | 1 | -1 | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | |
| ⑧ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | 1 | -1 | _ | _ | _ | _ | _ | _ | _ | _ | _
|
Table E.7.7 Matrix A_{P}.
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | ||
| A_{P}= | ① | 1 | -1 | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ |
| ② | _ | _ | _ | _ | _ | _ | 1 | -1 | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | |
| ③ | _ | _ | 1 | -1 | 1 | -1 | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | |
| ④ | _ | _ | _ | _ | _ | _ | _ | _ | 1 | -1 | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | |
| ⑤ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | 1 | -1 | _ | _ | -1 | _ | _ | _ | _ | _ | _ | |
| ⑥ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | 1 | -1 | _ | _ | _ | _ | _ | |
| ⑦ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | 1 | 1 | -1 | -1 | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | |
| ⑧ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | 1 | -1 | _ | _ | _ | _ | _ | _ | _ |
Table E.7.27 Exergy cost expressions of the flows.
| \mathrm{Symbolic~expressions~of~the~costs~of~the~flows} | |||
| 1. | B_{1}^{*}\,=\,{\frac{-B_{21}}{[(x_{1}+1)\cdot(x_{3}-1)]}} | 14. | B_{14}^{*}\,=B_{14} |
| 2. | B_{2}^{*}\ ={\frac{-B_{21}\cdot x_{3}}{[(x_{1}+1)\cdot(x_{3}-1)]}} | 15. | B_{15}^{\ast}\,=\,\left(\frac{-B_{21}.x_{2}}{(x_{1}+1)\cdot(x_{2}+1)}\,+B_{20}\,+B_{22}\,+B_{23}\,+B_{24}\,\right)\cdot\frac{-1}{(x_{9}-1)} |
| 3. | B_{3}^{\ast}\ =\frac{B_{21}}{[(x_{1}+1)\cdot(x_{2}-1)\cdot(x_{8}-1)]} | 16. | B_{16}^{\ast}=\left(\frac{-B_{21}.x_{2}}{(x_{1}+1)\cdot(x_{2}+1)}\,+B_{20}\,+B_{22}\,+B_{23}\,+B_{24}\right)\cdot\frac{-x_{9}}{(x_{9}-1)} |
| 4. | B_{4}^{\ast}\ ={\frac{B_{21}\,{\cdot}x_{8}}{[(x_{1}+1)\cdot(x_{2}-1)\cdot(x_{8}-1)]}} | 17. | B_{17}^{\ast}=B_{18}\,+B_{20}\,+B_{22}\,+B_{23}\,+B_{24}\,+\frac{B_{21}\,\!\cdot\!x_{9}}{[(x_{1}+1)\cdot(x_{2}+1)]} |
| 5. | B_{5}^{\ast}\ =\frac{B_{21}\cdot x_{2}}{[(x_{1}+1)\cdot(x_{2}-1)\cdot(x_{8}-1)]} | 18. | B_{18}^{*}\,=B_{18} |
| 6. | B_{6}^{\ast}\ =\frac{B_{21}\cdot x_{2}\cdot x_{8}}{[(x_{1}+1)\cdot(x_{2}-1)\cdot(x_{8}-1)]} | 19. | B_{19}^{\ast}=B_{20}\,+B_{22}\,+B_{24} |
| 7. | B_{7}^{\ast}\,=\,\frac{-B_{22}}{(x_{6}-1)} | 20. | B_{20}^{*}\,=B_{20} |
| 8. | B_{8}^{\ast}\,\equiv\,\frac{-B_{22}\cdot x_6}{(x_{6}-1)} | 21. | B_{21}^{*}\,=B_{21} |
| 9. | B_{9}^{*}\ ={\frac{B_{21}\cdot x_{2}}{[(x_{1}+1)\cdot(x_{2}-1)\cdot(x _5-1)]}} | 22. | B_{22}^{*}\,=B_{22} |
| 10. | B_{10}^{*}\;=\frac{B_{21}.x_{2}.x_{5}}{[(x_{1}+1)\cdot(x_{2}-1)\cdot(x_{5}-1)]} | 23. | B_{23}^{*}\,=B_{23} |
| 11. | B_{11}^{*}\,=B_{13}\,+\frac{B_{21}}{[(x_{1}+1)\cdot(x_{2}+1)\cdot(x_7+1)]} | 24. | B_{24}^{*}\,=B_{24} |
| 12. | B_{12}^{*}=B_{14}\ +\frac{B_{21}.x_7}{[(x_{1}+1)\cdot(x_{2}+1)\cdot(x_7+1)]} | 25. | B_{25}^{*}\,\ =\,\frac{B_{21}\,{\cdot}{x_1}}{{(x_{1}+1)}} |
| 13. | B_{13}^{\ast}\,\equiv B_{13} | ||
Table E.7.28 Exergy costs of fuel and products.
| \mathrm{Fuel} | {\mathrm{Product}} | |
| ① | F_{①}^{\ast}\;=B_{21} | P_{①}^{*}\;=\frac{B_{21}}{x_{1}+1} |
| ② | F_{②}^{\ast}\;=B_{22} | \textstyle{P}_{②}^{\ast}\;=\frac{B_{21}}{(x_{1}+1)} |
| ③ | F_{\,_{③}}^{\ast}\;=\frac{B_{21}}{x_{1}+1} | P_{③}^{\ast}\;=B_{22} |
| ④ | F_{④}^{*}\ =\frac{-B_{21}\cdot x_{2}}{[(x_{1}+1\cdot(x_{2}-1)]} | {P}_{④}^{\ast}\;\;{=}\;\frac{-B_{21}\cdot x_{2}}{[(x_{1}+1)\cdot(x_{2}-1)]} |
| ⑤ | F_{⑤}^{*}={\frac{-B_{21}\cdot x_{2}}{[(x_{1}+1)\cdot(x_{2}-1)]}}+B_{23} | P_{⑤}^{\ast}=\frac{-B_{21}\cdot x_{2}}{(x_{1}+1)\cdot(x_{2}+1)}\,+B_{23} |
| ⑥ | { F}_{⑤}^{*}={ B}_{22}+{ B}_{24} | {P}_{⑤}^{*}={ B}_{22}+{ B}_{24} |
| ⑦ | F_{⑥}^{*}\,=\frac{-B_{21}}{[(x_{1}+1)\,\cdot(x_{2}-1)]} | P_{⑥}^{*}\,=\frac{B_{21}}{[(x_{1}+1)\,\cdot(x_{2}+1)]} |
| ⑧ | F_{⑧}^{*}=\frac{-B_{21}.x_{2}}{(x_{1}+1).(x_{2}+1)}+B_{20}\,+B_{22}\,+B_{23}\,+B_{24} | P_{⑧}^{*}=B_{20}\,+B_{22}\,+B_{23}\,+B_{24}\,+{\frac{B_{21}\cdot x_9}{[(x_{1}+1)\cdot(x_{2}+1)]}} |