Question 8.3: We refer to the same facility as in Example E.8.1 but, in th......
We refer to the same facility as in Example E.8.1 but, in this case, the PF(R) formulation is used. That is to say, it is about a DHW production facility using a condensing boiler. The system consists of a boiler, a hydraulic compensator, a three-way valve acting according to the demand, a heat exchanger and an accumu-lation tank. In addition, there are three circulation pumps, one in each of the circuits, see the diagram of Fig. E.8.1. Using the PF(R) formulation, the questions are:
(a) Functional analysis of the installation flows.
(b) Construction of the {{\mathbb{J}}_{\mathrm{r}}} extended matrix and the {{\mathbb{Y}}_{\mathrm{s}}} vector.
(c) Symbolic expression of each flow exergy.
(d) Construction of the \langle P F\rangle,\langle K P\rangle{\mathrm{~and~}}\langle K{{R}}{\rangle} imatrixes.
(e) Symbolic expression of the total system’s efficiency.
(f) Symbolic expressions of unit exergy costs of the components’ product, associated to external resources and residues.
(g) Symbolic expressions of unit exergy costs of fuel components.
(h) Symbolic expressions of unit exergoeconomic costs of the components’ product, associated to external resources, components’ investment and residues.
(i) Symbolic expressions of unit exergoeconomic costs of components’ fuel.
The components considered for the analysis are shown in Fig. E.8.3, as well as the nomenclature assigned to the flows. In total, 15 flows are considered, with a product of the installation (\dot{B}_{11}) and one dissipative component, which is the chimney, with a loss flow (\dot{B}_{15}).
(a) Table E.8.24 shows the results of the functional analysis, as well as the unit exergy consumption and the total product of the plant. Compared to the previous examples, a dissi-pative component appears, which is the chimney. It has a loss flow (\dot{B}_{14}) associated with the combustion gases exiting through the chimney (\dot{B}_{15} is the corresponding exergy loss before being emitted to the atmosphere).
The changes due to the incorporation of the chimney have been highlighted. Once this functional analysis is done, the construction of matrices A_{F} and A_{P} is immediate.
(b) Table E.8.25 shows the {{\mathbb{J}}_{\mathrm{r}}} extended matrix (15.15) and the {{\mathbb{Y}}_{\mathrm{s}}} extended vector.
Note that the recirculation equations coincide with those of Table E.8.11; this hap-pens because the incorporation of the chimney adds an outgoing flow in the boiler (not an entrance) so that the recirculations are maintained. However, when the seventh component is added, two new flows are added, {\dot{B}}_{14}\ \mathrm{and}\ {\dot{B}}_{15}, so two additional equa-tions are needed. These additional equations are, on the one hand, the exergy balance in component 7 and, on the other hand, a new residual output flow \dot B_{s}^{R E S}=\dot{B}_{15}, so the \alpha_{s} matrix will now have (2,15) dimension. Table E.8.25 shows the \mathbb{J} extended matrix which is the result of composing the matrix J with \alpha_{r} and \alpha_{s} matrices.
(c) By solving Eq. (8.62), the symbolic expressions of the flow exergies are obtained. They are based on the unit exergy consumptions of the equipment, the recirculation parameters and the output flows of the installation. The expressions obtained are reflected in Table E.8.26.
(d) In order to obtain the exergy costs of components’ fuel and product, it is necessary to pre-viously build the \langle P F\rangle,\langle K P\rangle{\mathrm{~and~}}\langle K{{R}}{\rangle} matrices. This last matrix, of elements {}\,\rho_{r,i}, represents the amount of residues coming from the i-th equipment needed to produce a product unit. The matrices obtained are shown in Tables E.8.27, E.8.28 and E.8.29.
(e) Applying Eq. (8.86) the expression of the facility’s overall exergy efficiency is obtained, which expresses it in terms of the components’ unit consumptions, the recirculation param-eters and the system products. The obtained result is presented below
\varphi_{T}=\frac{{}^{t}u P_{S}}{ {}^{t}\langle F_{T}F\rangle|F\rangle P_{S}}\qquad\qquad(8.86) \\ \varphi_{T}=\frac{P_{T}}{F_{T}}=\frac{\left(\dot{B}_{11}-\dot{B}_{10}\right)}{\Delta\dot{B}_{12}+\dot{B}_{13}} = \frac{{\dot{B}}_{S_{11}}}{k_{1}\cdot k_{7}\cdot\dot{B}_{S_{15}}+k_{6}\cdot\frac{\dot{B}_{S_{11}}}{1+r_{6}}.\left[1\,+\,r_{6}\cdot k_{1}\cdot k_{5}\cdot\frac{k_{2}\cdot k_{3}\cdot\left(1\,-\,r_{2}\right)}{\left(1\,-\,k_{3}\cdot k_{4}\cdot r_{2}\right)}\right]}(f) From the previous matrices the unit product exergy costs are obtained. These costs are decomposed into those associated with external resources and those associated with residues (in this case, flow 14), hence, the total product exergy cost is the sum of both. The results are shown in Tables E.8.30 and E.8.31.
The \rho_{7,1} term represents the component of row 7 and column 1 of the\langle K R\rangle matrix that is,
\rho_{7,1}=\frac{\dot{B}_{15}}{(\dot{B}_{1}-\dot{B}_{12})+\dot{B}_{14}}(g) By means of Eq. (8.150), the unit exergy costs of components’ fuel are obtained. Table E.8.32 shows the expressions obtained for these costs
k_{F}^{*}=k_{e}^{*}+{}^{t}\langle P F\rangle\cdot k_{P}^{*}\qquad\qquad(8.150)(h) The exergoeconomic costs are associated with the following three sub-components: external resources, equipment investment and maintenance and the generated residues. By means of Eq. (8.151) the corresponding symbolic expressions are obtained. The results are shown in Tables E.8.33, E.8.34 and E.8.35.
c_{P}=c_{P}^{e}+c_{P}^{z}+c_{P}^{r}\qquad\qquad(8.151)(i) Finally, by applying Eq. (8.152), the exergoeconomic costs of the components’ fuel are calculated. The expressions obtained are reflected in Table E.8.36.
c_{F}=c_{e}+{}^{t}\langle P F\rangle\cdot c_{P}\qquad\qquad(8.152)Table E.8.24 Results of the functional analysis.
| n | Fuel | PROD. | Resid. | k | {{P}}_{s} | {{R}}_{s} | |
| ① | Boiler | {\dot{B}}_{13} | {\dot{B}}_{1}\,-{\dot{B}}_{2} | {\dot{B}}_{14} | \frac{\left(\dot{B}_{1}-\dot{B}_{2}\right)+\dot{B}_{14}}{\dot{B}_{13}} | 0 | – |
| ② | Hydraul. Compens. | {\dot{B}}_{1}\,-{\dot{B}}_{2} | {\dot{B}}_{3}\,-{\dot{B}}_{4} | – | \frac{{\dot{B}}_{3}-{\dot{B}}_{4}}{{\dot{B}}_{1}-{\dot{B}}_{2}} | 0 | – |
| ③ | Diverte | {\dot{B}}_{3} | {\dot{B}}_{5}\,+{\dot{B}}_{6} | – | \frac{{\dot{B}}_{5}+{\dot{B}}_{6}}{{\dot{B}}_{3}} | 0 | – |
| ④ | V3V | {\dot{B}}_{6}\,+{\dot{B}}_{7} | {\dot{B}}_{4} | – | \frac{\dot{B}_{4}}{\dot{B}_{6}+\dot{B}_{7}} | 0 | – |
| ⑤ | HX | {\dot{B}}_{5}\,-{\dot{B}}_{7} | {\dot{B}}_{8}\,-{\dot{B}}_{9} | – | {\frac{{\dot{B}}_{8}-{\dot{B}}_{9}}{{\dot{B}}_{5}-{\dot{B}}_{7}}} | 0 | – |
| ⑥ | Tank | ({\dot{B}}_{8}-{\dot{B}}_{9})+\Delta \dot B_{12} | {\dot{B}}_{11}\,-{\dot{B}}_{10} | – | \frac{\dot{B}_{11}-\dot{B}_{10}}{\left(\dot{B}_{8}-\dot{B}_{9}\right)+\Delta\dot{B}_{12}} | {\dot{B}}_{11} | – |
| ⑦ | Chimney | {\dot{B}}_{14} | {\dot{B}}_{15} | – | \frac{\dot{B}_{15}}{\dot{B}_{14}} | 0 | {\dot{B}}_{15} |
Table E.8.25 {{\mathbb{J}}} extended matrix’.
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | |
| J = | -k_{1} | k_{1} | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -k_{1} | 0 |
| 1 | -1 | -k_{2} | k_{2} | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
| 0 | 0 | 1 | 0 | -k_{3} | -k_{3} | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
| 0 | 0 | 0 | -k_{4} | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
| 0 | 0 | 0 | 0 | 1 | 0 | -1 | -k_{5} | k_{5} | 0 | 0 | 0 | 0 | 0 | 0 | |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | k_{6} | -k_{6} | 1 | 0 | 0 | 0 | |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -k_{7} | |
| \alpha_{s}= | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | |
| \alpha_{r}= | -{\frac{\dot B_2}{\dot B_{1}}} | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | -{\frac{\dot B_4}{\dot B_{3}}} | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
| 0 | 0 | 0 | 0 | 0 | -{\frac{\dot B_7}{\dot B_{6}}} | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | -{\frac{\dot B_9}{\dot B_{8}}} | 1 | 0 | 0 | 0 | 0 | 0 | 0 | |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -{\frac{\dot B_{10}}{\dot B_{11}}} | 0 | 0 | 0 | 0 | |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 0 | 0 | -{\frac{({\dot{B}}_{8}-{\dot{B}}_{9})}{\Delta \dot B_{12}}} | 0 | 0 | 0 | |
| {}^{t}\mathbb{Y_s = } | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
Table E.8.26 Symbolic expressions of flow exergies.
| \mathrm{Flow~exergies~symbolic~expressions} | |
| 1. | {\dot{B}}_{1}\,=\,r_{6}\cdot k_{2}\cdot k_{3}\cdot\frac{(1-r_{2})}{(1-r_{1})}.\frac{k_{5}.k_{6}}{(1-k_{3}\cdot k_{4}\cdot r_{2})}\cdot\frac{{\dot{B}}_{s_{11}}}{(1+r_{6})} |
| 2. | {\dot{B}}_{2}\,=r_1\cdot\,r_{6}\cdot k_{2}\cdot k_{3}\cdot\frac{(1-r_{2})}{(1-r_{1})}.\frac{k_{5}.k_{6}}{(1-k_{3}\cdot k_{4}\cdot r_{2})}\cdot\frac{{\dot{B}}_{s_{11}}}{(1+r_{6})} |
| 3. | {\dot{B}}_{3}\,=r_{6}\cdot k_{3} \frac{k_{5}.k_{6}}{(1-k_{3}\cdot k_{4}\cdot r_{2})}\cdot\frac{{\dot{B}}_{s_{11}}}{(1+r_{6})} |
| 4. | {\dot{B}}_{4}\,=r_2\cdot r_{6}\cdot k_{3} \frac{k_{5}.k_{6}}{(1-k_{3}\cdot k_{4}\cdot r_{2})}\cdot\frac{{\dot{B}}_{s_{11}}}{(1+r_{6})} |
| 5. | {\dot{B}}_{5}\,=\,r_{5}\cdot{\frac{[k_{3}\cdot k_{4}\cdot r_{2}-(1+r_{3})]\cdot\kappa_{d5}\cdot\kappa_{d6}}{(1+r_{3}\cdot\kappa_{4}\cdot r_{2})}}\cdot{\frac{{\dot{B}}_{s_{11}}}{(1+r_3)\cdot (1+r_{6})}} |
| 6. | \dot{B}_{6}\,=\,r_{2}\cdot r_{6}\cdot k_{3}\cdot k_{4}\frac{k_{5}\cdot k_{6}}{(1-k_{3}\cdot k_{4}\cdot r_{2})}\cdot\frac{\dot{B}_{s_{11}}}{(1+r_{3})\cdot(1+r_{6})} |
| 7. | \dot{B}_{7}\,=\,r_{3}\cdot r_{2}\cdot r_{6}\cdot k_{3}\cdot k_{4}\frac{k_{5}\cdot k_{6}}{(1-k_{3}\cdot k_{4}\cdot r_{2})}\cdot\frac{\dot{B}_{s_{11}}}{(1+r_{3})\cdot(1+r_{6})} |
| 8. | {\dot{B}}_{8}\,=\,r_{6}\cdot k_{6}\,{\frac{\dot B_{s_{11}}}{(1{-}r_{4})\cdot(1{+}r_{6})}} |
| 9. | {\dot{B}}_{9}\,=\,r_{4}\cdot k_{6}\cdot{\frac{{\dot{B}}_{s_{11}}}{(1{-}r_{4})\cdot(1{+}r_{6})}} |
| 10. | {\dot{B}}_{10}=r_{5}\cdot{\frac{{\dot{B}}_{s_{11}}}{1-r_{5}}} |
| 11. | {\dot{B}}_{11}={\frac{{\dot{B}}_{s_{11}}}{1-r_5}} |
| 12. | \Delta\dot{B}_{12}=k_{6}\cdot\frac{\dot{B}_{s_{11}}}{1+r_{6}} |
| 13. | {\dot{B}}_{13}=k_{1}\cdot k_{7}\cdot{\dot{B}}_{s_{15}}\,+r_{6}\cdot k_{1}\cdot\kappa_{5}\cdot k_{6}\,{\frac{k_{2}\cdot k_{3}\cdot(1{-r_{2}})}{(1-k_{3}\cdot k_{4}\cdot r_{2})}}\cdot{\frac{{\dot{B}}_{s_{11}}}{(1+r_{6})}} |
| 14. | {\dot{B}}_{14}=k_{7}\cdot{\dot{B}}_{s_{15}} |
| 15. | {\dot{B}}_{15}\,={\dot{B}}_{s_{15}} |
Table E.8.27 \langle{\mathrm{PF}}\rangle matrix.
| {}^{t}\langle F_{T}F\rangle\,= | 1 | 0 | 0 | 0 | 0 | \frac{1}{1+r_{6}} | 0 |
| \langle FP\rangle\,= | 0 | 1 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1\,-\,r_{2} | 0 | 0 | 0 | 0 | |
| 0 | 0 | 0 | 1 | 1 | 0 | 0 | |
| 0 | 0 | r_{2} | 0 | 0 | 0 | 0 | |
| 0 | 0 | 0 | 0 | 0 | \frac{r_6}{(1+r_{6})} | 0 | |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 |
Table E.8.28 \langle{\mathrm{KP}}\rangle matrix.
| \rm k_e^T = | \,k_{{{1}}} | 0 | 0 | 0 | 0 | \frac{k_{6}}{1+r_{6}} | 0 |
| \langle KP\rangle\,= | 0 | \,k_{{{2}}} | 0 | 0 | 0 | 0 | \,k_{{{7}}} |
| 0 | 0 | k_{3}\cdot(1-r_{2}) | 0 | 0 | 0 | 0 | |
| 0 | 0 | 0 | \,k_{{{4}}} | \,k_{{{5}}} | 0 | 0 | |
| 0 | 0 | k_{3}\cdot r_{2} | 0 | 0 | 0 | 0 | |
| 0 | 0 | 0 | 0 | 0 | \frac{k_{6}{\cdot{r}}_{6}}{(1+{{r}}_{6})} | 0 | |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 |
Table E.8.29 \langle{\mathrm{KR}}\rangle matrix.
| \langle KR\rangle\,= | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
| \frac{{\dot{B}}_{15}}{({\dot{B}}_{1}-{\dot{B}}_{2})+{\dot{B}}_{14}} | 0 | 0 | 0 | 0 | 0 | 0 |
Table E.8.30 Unit exergy costs of components’ fuel associated to external resources.
| \mathrm{Fuels}^{\prime}~u n i t~e x e r g y~\mathrm{costs} | |
| ① | k_{F_{1}}^{*}\,=1 |
| ② | k_{F_{2}}^{*}\,={\frac{k_{1}}{(1-\rho_{7,1}\cdot k_{d7})}} |
| ③ | k_{F_{3}}^{*}={\frac{(1-r_{2})}{(1-k_{3}\cdot k_{4}\cdot r_{2})}}\cdot{\frac{k_{2}\cdot k_{1}}{(1-\rho_{7,1}\cdot k_{d7})}} |
| ④ | k_{F_{4}}^{*}={\frac{(1-r_{2})\ \cdot k_3}{(1-k_{3}\cdot k_{4}\cdot r_{2})}}\cdot{\frac{k_{2}\cdot k_{1}}{(1-\rho_{7,1}\cdot k_{d7})}} |
| ⑤ | k_{F_{5}}^{*}={\frac{(1-r_{2})\ \cdot k_3}{(1-k_{3}\cdot k_{4}\cdot r_{2})}}\cdot{\frac{k_{2}\cdot k_{1}}{(1-\rho_{7,1}\cdot k_{d7})}} |
| ⑥ | k_{F_{6}}^{*}\,=\,1\,-\,{\frac{k_{6}{\cdot}r_{6}}{(1+r_{6})}}\,-\,{\frac{k_{6}{\cdot}r_{6}}{(1+r_{6})}}\cdot{\frac{{(1-r_{2})}\cdot k_{3}{}\cdot k_{5}}{(1-k_{3}{\cdot}k_{4}{\cdot}r_{2})}}\,{\frac{k_{2}{\cdot}k_{1}}{(1-\rho_{7,1}{\cdot}k_{d7})}} |
| ⑦ | k_{F_{7}}^{*}\,={\frac{k_{1}}{(1-\rho_{7,1}\cdot k_{d7})}} |
Table E.8.31 Unit exergy costs of components’ product associated to residues.
| \mathrm{P~unit~exergetic~cost~associated~with~residues} | |
| ① | k_{p_{1}}^{r,*}={\frac{\rho_{7,1}}{(1-\rho_{7,1}\cdot k_{7})}}\cdot k_{p_{7}}^{e,*} |
| ② | k_{p_{2}}^{r,*}=k_2\cdot {\frac{\rho_{7,1}}{(1-\rho_{7,1}\cdot k_{7})}}\cdot k_{p_{7}}^{e,*} |
| ③ | k_{p_{3}}^{r,*}\,=\,{\frac{(1-r_{2})\cdot k_{3}}{(1-k_{3}\cdot k_{4}\cdot r_{2})}}\cdot k_{2}\cdot{\frac{\rho_{7,1}}{(1-\rho_{7,1}\cdot k_{7})}}\cdot k_{p_{7}}^{e,*} |
| ④ | k_{p_{4}}^{r,*}\,=k_4\cdot \,{\frac{(1-r_{2})\cdot k_{3}}{(1-k_{3}\cdot k_{4}\cdot r_{2})}}\cdot k_{2}\cdot{\frac{\rho_{7,1}}{(1-\rho_{7,1}\cdot k_{7})}}\cdot k_{p_{7}}^{e,*} |
| ⑤ | k_{p_{5}}^{r,*}\,=k_5\cdot \,{\frac{(1-r_{2})\cdot k_{3}}{(1-k_{3}\cdot k_{4}\cdot r_{2})}}\cdot k_{2}\cdot{\frac{\rho_{7,1}}{(1-\rho_{7,1}\cdot k_{7})}}\cdot k_{p_{7}}^{e,*} |
Table E.8.31 Unit exergy costs of components’ product associated to residues.-\rm{cont'}{d}
| \mathrm{P~unit~exergetic~cost~associated~with~residues} | |
| ⑥ | k_{p_{6}}^{r,*}=\frac{k_{6}{}\cdot r_{6}}{(1+r_{6})}\cdot k_{5}\cdot\frac{(1-r_{2})\cdot k_{3}}{(1-k_{3}\cdot k_{4}\cdot r_{2})}\cdot k_{2}\cdot\frac{\rho_{7,1}}{(1-\rho_{7,1}\cdot k_{7})}\cdot k_{p_7}^{e,*} |
| ⑦ | k_{p_7}^{r,*}=\frac{\rho_{7,1}}{\left(1-\rho_{7,1}\cdot k_{7}\right)}\cdot k_{p_7}^{e,*} |
Table E.8.32 Unit exergy costs of components’ fuel.
| \mathrm{Fuels}^{\prime}~u n i t~e x e r g y~\mathrm{costs} | |
| ① | k_{F_{1}}^{*}\,=1 |
| ② | k_{F_{2}}^{*}\,={\frac{k_{1}}{(1-\rho_{7,1}\cdot k_{d_7})}} |
| ③ | k_{F_{3}}^{*}={\frac{(1-r_{2})}{(1-k_{3}\cdot k_{4}\cdot r_{2})}}\cdot{\frac{k_{2}\cdot k_{1}}{(1-\rho_{7,1}\cdot k_{d_7})}} |
| ④ | k_{F_{4}}^{*}={\frac{(1-r_{2})\ \cdot k_3}{(1-k_{3}\cdot k_{4}\cdot r_{2})}}\cdot{\frac{k_{2}\cdot k_{1}}{(1-\rho_{7,1}\cdot k_{d_7})}} |
| ⑤ | k_{F_{5}}^{*}={\frac{(1-r_{2})\ \cdot k_3}{(1-k_{3}\cdot k_{4}\cdot r_{2})}}\cdot{\frac{k_{2}\cdot k_{1}}{(1-\rho_{7,1}\cdot k_{d_7})}} |
| ⑥ | k_{F_{6}}^{*}\,=\,1\,-\,{\frac{k_{6}{\cdot}r_{6}}{(1+r_{6})}}\,-\,{\frac{k_{6}{\cdot}r_{6}}{(1+r_{6})}}\cdot{\frac{{(1-r_{2})}\cdot k_{3}{}\cdot k_{5}}{(1-k_{3}{\cdot}k_{4}{\cdot}r_{2})}}\,{\frac{k_{2}{\cdot}k_{1}}{(1-\rho_{7,1}{\cdot}k_{d_7})}} |
| ⑦ | k_{F_{7}}^{*}\,={\frac{k_{1}}{(1-\rho_{7,1}\cdot k_{d_7})}} |
Table E.8.33 Unit exergy costs of components’ product associated to external resources.
| \mathrm{P~unit~exergy~costs~associated~with~external~resources} | |
| ① | c_{p_{1}}^{e}=k_{1}\cdot c_{N G} |
| ② | c_{p_{2}}^{e}=k_{2}\cdot k_{1}\cdot c_{N G} |
| ③ | c_{p_{3}}^{e}={\frac{(1-r_{1})\cdot k_{3}}{(1-k_{3}\cdot k_{4}\cdot r_{2})}}\,\cdot k_{2}\cdot k_{1}\cdot c_{N G} |
| ④ | c_{p_{4}}^{e}=k_{4}{\cdot}\frac{(1-r_{2}){\cdot}k_{3}}{(1-k_{3}{\cdot}k_{4}{\cdot}r_{2})}\,k_{2}{\cdot}k_{1}\,{\cdot}c_{N G} |
Table E.8.33 Unit exergy costs of components’ product associated to external resources.-\rm{cont'}{d}
| \mathrm{P~unit~exergy~costs~associated~with~external~resources} | |
| ⑤ | c_{p_{5}}^{e}=k_{5}\cdot{\frac{(1{-}r_{2}){\cdot}k_{3}}{(1{-}k_{3}{\cdot}k_{4}{\cdot}r_{2}{)}}}\cdot k_2\cdot k_{1}\cdot c_{N G} |
| ⑥ | c_{p_{6}}^{e}=\frac{k_{6}.r_{6}}{(1+r_{6})}\cdot k_{5}\cdot\frac{(1{-}r_{2})\cdot k_{3}}{(1{-}k_{3}\cdot k_{4}\cdot r_{2})}\cdot k_{2}\cdot k_{1}\cdot c_{N G}\;+\frac{k_{6}}{(1{+}r_{6})}\cdot c_{w} |
| ⑦ | c_{p_7}^{e}=k_7\cdot k_{1}\cdot c_{N G} |
Table E.8.34 Unit exergy costs of components’ product associated to investment.
| \mathrm{P~unit~exergy~costs~associated~to~investment} | |
| ① | c_{p_{1}}^{z}\,=z_1 |
| ② | c_{p_{2}}^{c}=k_{2}\cdot z_{1}+z_{2} |
| ③ | c_{p_{3}}^{z}\,={\frac{z_{3}+k_{3}\cdot(1-r_{2})\cdot(k_{2}\cdot z_{1}+z_{2}+r_{2}\cdot z_{4})}{(1-k_{3}\cdot k_{4}\cdot r_{2})}} |
| ④ | c_{p_{4}}^{z}\,={\frac{z_{4}+k_{4}\cdot(1-r_{2})\cdot(k_{2}\cdot k_3\cdot z_{1}+k_{3}\cdot z_{2}+z_{3})}{(1-k_{3}\cdot k_{4}\cdot r_{2})}} |
| ⑤ | c_{p_{5}}^{z}\,=\,z_{5}\,+{\frac{\kappa_{5}\cdot(1-r_{2})\cdot(\kappa_{2}\cdot\kappa_{3}\cdot z_{1}+\kappa_{3}\cdot z_{2}+z_{3}+r_{2}\cdot\kappa_{3}\cdot z_{4})}{(1-k_{3}\cdot k_{4}\cdot r_{2})}} |
| ⑥ | c_{p_{6}}^{z}\;=\;z_{6}\ \,+\frac{\kappa_{6}\,.\,r_{6}}{(1+r_{6})}\cdot\frac{z_{5}+\kappa_{d_{5}}\cdot\left(1-r_{2}\right)\cdot\left(\kappa_{2}\cdot\kappa_{3}\cdot z_{1}+\kappa_{3}\cdot z_{2}+z_{3}+r_{2}\cdot\kappa_{3}\cdot z_{4}\right)}{(1-k_{3}\!\cdot k_{4}\cdot r_{2})}. |
| ⑦ | c_{p_{7}}^{z}\,=\,\kappa_7\cdot z_{1} |
Table E.8.35 Unit exergy costs of components’ product associated to residues.
| \mathrm{P~unit~exergy~costs~associated~to~residues} | |
| ① | c_{p_{1}}^{r}={\frac{\beta_{7,1}}{(1-\beta_{7,1}.k_{7})}}.k_{1}\cdot c_{N G} |
| ② | c_{p_{2}}^{r}=k_{2}\!\cdot\!{\frac{\beta_{7,{1}}}{(1-\beta_{7,{1}}.k_{7})}}\!\cdot\!k_{1}\!\cdot\!c_{N G} |
| ③ | c_{p_{3}}^{r}={\frac{(1-r_{2})\cdot k_{3}}{(1-k_{3}\cdot k_{4}\cdot r_{2})}}\cdot \kappa_{2}\cdot{\frac{\beta_{7,1}}{(1-\beta_{7,1}\cdot k_{7})}}\cdot k_{1}\cdot c_{N G} |
| ④ | c_{p_{4}}^{r}=k_4\cdot {\frac{(1-r_{2})\cdot k_{3}}{(1-k_{3}\cdot k_{4}\cdot r_{2})}}\cdot \kappa_{2}\cdot{\frac{\beta_{7,1}}{(1-\beta_{7,1}\cdot k_{7})}}\cdot k_{1}\cdot c_{N G} |
| ⑤ | c_{p_{5}}^{r}=k_5\cdot {\frac{(1-r_{2})\cdot k_{3}}{(1-k_{3}\cdot k_{4}\cdot r_{2})}}\cdot k_{2}\cdot{\frac{\beta_{7,1}}{(1-\beta_{7,1}\cdot k_{7})}}\cdot k_{1}\cdot c_{N G} |
| ⑥ | c_{p_{6}}^{r}=\frac{k_{6}\cdot r_{6}}{(1+r_{6})}\cdot k_{5}\cdot\frac{(1-r_{2})\cdot k_{3}}{(1-k_{3}\cdot k_{4}\cdot r_{2})}\cdot k_{2}\cdot\frac{\beta_{7,1}}{(1-\beta_{7,1}\cdot k_{7})}\cdot k_{1}\cdot c_{N G} |
| ⑦ | c_{p_{7}}^{r}={\frac{\beta_{7,1}}{(1-\beta_{7,1}.κ_{7})}}.k_{1}\cdot c_{N G} |
Table E.8.36 Unit exergoeconomic costs of components’ fuel.
| \mathrm{P~unit~exergy~costs~associated~to~residues} | |
| ① | c_{F_{1}}=c_{N G} |
| ② | c_{F_{2}}=c_{P_1} |
| ③ | c_{F_{3}}=(1-r_{2})\cdot c_{P_{2}}+r_{2}\cdot c_{P_{4}} |
| ④ | c_{F_{4}}=c_{P_3} |
| ⑤ | c_{F_{5}}=c_{F_3} |
| ⑥ | c_{F_{6}}=\frac{1}{(1+r_{6})}\cdot c_{w}\ +\frac{r_{\mathrm{s}}}{(1+r_{6})}\cdot c_{P_{5}} |
| ⑦ | c_{F_{7}}=c_{P_1} |
| Table E.8.11 Recirculation parameters | |||
| n | Recirculations | \alpha_r | |
| ① | r_{1,2} | r_1=\frac{\dot{B}_2}{\dot{B}_1} | \dot{B}_2-\dot{B}_1\cdot r_1=0 |
| ② | r_{3,4} | r_1=\frac{\dot{B}_4}{\dot{B}_3} | \dot{B}_4-\dot{B}_3\cdot r_2=0 |
| ③ | r_{6,7} | r_1=\frac{\dot{B}_7}{\dot{B}_6} | \dot{B}_7-\dot{B}_6\cdot r_3=0 |
| ④ | r_{8,9} | r_1=\frac{\dot{B}_9}{\dot{B}_8} | \dot{B}_9-\dot{B}_8\cdot r_4=0 |
| ⑤ | r_{10,11} | r_1=\frac{\dot{B}_{10}}{\dot{B}_{11}} | \dot{B}_{10}-\dot{B}_{11}\cdot r_5=0 |
| ⑥ | r_{8,9,12} | r_1=\frac{\dot{B}_8 -\dot{B}_9}{\dot{B}_{12}} | (\dot{B}_8-\dot{B}_8)-\dot{\Delta B_{12}}\cdot r_6=0 |
