Question 8.2: We refer to the same installation considered in Example E.8.......
We refer to the same installation considered in Example E.8.1, that is to say, a DHW production facility with a condensing boiler. The system consists of a condensing boiler, a hydraulic compensator, a three-way valve acting according to the demand, a heat exchanger and an accumulation tank. In addition, there are three circulation pumps, one in each of the circuits, see the diagram of Figure E.8.1. Using the PF formulation and without considering residues, the factors considered are:
(a) Functional analysis of the flows.
(b) Construction of the {\mathbb{J}}_{r} extended matrix and the {\mathbb{Y}}_{s} vector.
(c) Symbolic expression of each flow exergy.
(d) Construction of the \langle P F\rangle matrix.
(e) Symbolic expressions of components’ fuel.
(f ) Symbolic expressions of components’ product.
(g) Symbolic expressions of components’ irreversibilities.
(h) Symbolic expression of the total system’s efficiency.
(i) Symbolic expressions of unit exergy costs of components’ fuel and unit exergy cost of components’ product.
( j) Symbolic expressions of unit exergoeconomic costs of components’ fuel and unit exergoe-conomic costs of components’ product.
(k) Expression of the facility’s total cost.
(a) The facility’s functional analysis is done in Example E.8.1 and shown in Table E.8.1.
For the construction of the {\mathbb{Y}}_{s} vector and the {\mathbb{J}}_{r} extended matrix, Section 8.3.1 for-mulas are taken into account, such as
The J matrix, defined according to the following equation J=A_{F}-K_{D}A_{P}, is the same as that in Table E.8.2. The \alpha_{r} recirculation matrix relates the exergies of the entering flows to the equipment. In Table E.8.11, the component recirculation param-eters are shown for those components having more than one inflow (there are as many parameters as inputs to the equipment minus 1), and in Table E.8.12, the recirculation matrix with dimension (6,13) is presented.
The \alpha_{s}(1,13) matrix, which contains the coefficients of the outgoing flows, and the {}^{t}\mathbb{Y}_{s} transpose vector, which contains the exergy of the total product flows, are reflected in Table E.8.13.
(b) If the exergy of the output flows ({\dot{B}}_{s}), the exergy unit consumptions ({{k}}_{i}) and the recirculations that define the system structure ({{r}}_{r}) are known, the symbolic expressions of the flows are obtained. For that, Eq. (8.62) reproduced below should be solved
B={\mathbb{J}}_{r}^{-1}\mathbb{Y}_{s}\qquad\qquad(8.62) \\ \dot{B}=\left[\begin{array}{c}{{J}}\\ {{\alpha_{s r}}}\end{array}\right]^{-1}\left[\begin{array}{c}{{0(6,1)}}\\ {{\omega_{S}(1,1)}}\\ {{0(6,1)}}\end{array}\right](c) The symbolic expressions of the exergy of each flow are presented in Table E.8.14.
(d) The matrix \langle P F\rangle with dimension (6.6) depends exclusively on the recirculation parameters r and is presented in Table E.8.15.
It can be verified that the sum of each column \sum_{j=1}^{n}(P F_{i j}+F_{T}F_{j}), is equal to the unit. When analysing the \langle PF\rangle matrix, it is seen that the fuel of the hydraulic compensator comes entirely from the product of the boiler. However, the fuel of the diverter, equip-ment 3, consists of the products of two different components: the fraction F_{3}\cdot(1-r_{2}) comes from the product of the hydraulic compensator (P_{2}) and the remaining fraction, F_{3}\cdot r_{2}, from the product (P_{4}) of the three-way valve V3V. It may seem strange since the output of the V3V component is not physically directed to the separator; however, at the production level, it works as a recirculation of equipment input 3. The product of the diverter supplies the fuel (F_{4}) to V3V and the heat exchanger (F_{5}). The fraction F_{6}\cdot{\frac{r_{6}}{1+r_{6}}} of the fuel in the tank is the product P_{5} of the heat exchanger, and the remain-ing fraction \langle F_{T}F\rangle_{6}=F_{6}.{\frac{1}{1+r_{6}}} comes from the outside, which corresponds to the input cold-water network. On the other hand, in the boiler, all the fuel is supplied from outside \langle F_{T}F\rangle_{1}=1, which justifies why the first column of matrix \langle PF\rangle is zero.
Therefore, the PF representation offers the information of the required resources for obtaining the final product of the components, while the FP representation enables the data of the components’ product obtained from the incoming resources.
As seen, both representations are closely related, and it can be passed from one to the other through the \langle FP\rangle and \langle PF\rangle matrices. According to Eq. (8.102)
where P_{D} and F_{D} are the diagonal matrices (6.6) that contain the values of the product and fuel of the component. This equation allows relating the recirculation and bifur-cation parameters in both representations. With q_{ij} the elements of the \langle PF\rangle matrix and y_{ji} the elements of the \langle FP\rangle matrix, both matrices are related according to the previous equation, that is
q_{i j}={\frac{P_{i}}{F_{j}}}.y_{j i}For this specific example, the relations are
y_{21}={\frac{F_{2}}{P_{1}}}\cdot q_{12} \\ y_{32}={\frac{F_{3}}{P_{2}}}\cdot q_{23} \\ y_{43}={\frac{F_{4}}{P_{3}}}\cdot q_{34} \\ y_{53}={\frac{F_{5}}{P_{3}}}\cdot q_{35} \\ y_{34}={\frac{F_{3}}{P_{4}}}\cdot q_{43} \\ y_{65}={\frac{F_{6}}{P_{5}}}\cdot q_{56}(e) The symbolic expressions of the components’ fuel are obtained by means of equation F=A_{F}{\cdot}B and the results are shown in Table E.8.16.
(f) The symbolic expressions of the components’ product are obtained by means of equation P=A_{P}{\cdot}B and the results are shown in Table E.8.17
(g) The symbolic expressions for the components’ irreversibility are obtained by means of equation I = F – P and the results are shown in Table E.8.18.
(h) The facility’s whole efficiency is obtained from Eq. (8.85), the example being
\varphi_{T}=\frac{{}^t u P_{S}}{{}^t {\langle F_{T}F\rangle F}}\qquad\qquad(8.85) \\ \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_{6}\cdot{\frac{\dot{B}_{s_{11}}}{1+r_{6}}}\cdot\left[1+r_{6}\cdot k_{1}\cdot k_{5}\cdot\frac{k_{2}\cdot k_{3}\cdot(1-r_{2})}{(1-k_{3}\cdot k_{d4}\cdot r_{2})}\right]}}(i) The symbolic expressions of the unit exergy costs of the components’ fuel are obtained from Eq. (8.94) and those of the components’ product through Eq. (8.91). The results are shown in Tables E.8.19 and E.8.20 respectively.
k_{P}^{*}={}^{t}|P\rangle k_{e}^{*}\qquad\qquad(8.91) \\ k_{F}^{*}={}^{t}|F\rangle k_{e}^{*}\qquad\qquad(8.94)(j) The exergoeconomic unit costs of the components’ fuel are obtained by Eq. (8.100) and shown in Table E.8.21.
c_{F}={}^{t}|F\rangle(c_{e}+H_{DZP})-z_{F}\qquad\qquad(8.100)(k) The exergoeconomic costs of the components’ product, as seen, can be broken down into two components: those due to external resources, c_{P}^{e}\equiv{}^{t}|P\rangle c_{e} and the capital cost rate of the components c_{P}^{Z}\,={}^{t}|P\rangle H_{D} z_ P. The corresponding symbolic expressions are obtained by applying Eq. (8.98). The results are shown in Tables E.8.22 and E.8.23 respectively.
c_{P}={}^{t}|P\rangle(c_{e}+H_{D} z_ P)\qquad\qquad(8.98)(l) Finally, the cost of the final product of the installation, which in this case is DHW, is given by the C_{P T}={}^{t}{c}_{P}P_{S} expression, or applying Eq. (8.101), resulting in
{ C}_{P T}={}^{t}c_{e}F+{}^{t}u Z\qquad\qquad(8.101) \\ C_{P_{D HW}}=P_{6}\cdot c_{F6}=\dot{B}_{s_{11}}\cdot\frac{1}{\left(1+r_{6}\right)}\cdot c_{w}+\frac{r_{6}}{\left(1+r_{6}\right)}\cdot c_{P5}Table E.8.1 Functional analysis.
| n | Fuel | Prod. | {{k}}_{D} | {{P}}_{s} | |
| ① | Boiler | {\dot{B}}_{13} | {\dot{B}}_{1}\,-{\dot{B}}_{2} | \frac{({\dot{B}}_{1}-{\dot{B}}_{2})}{\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 | Diverter | {\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 | \left(\dot{B}_{8}\ -\dot{B}_9\right)\ +\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} |
Table E.8.2 J matrix definition.
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | |
| J = | -k_{1} | k_{1} | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | -1 | -k_{2} | k_{2} | 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 | -k_{4} | 0 | 1 | 1 | 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 | 1 | -1 | k_{6} | -k_{6} | 1 | 0 |
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_{2}\ =\frac{{\dot B_{4}}}{\dot{B}_{3}} | \dot{B}_{4}-\dot{B}_{3}\cdot r_{2}\,=0 |
| ③ | r_{6,7} | r_{3}\ =\frac{{\dot B_{7}}}{\dot{B}_{6}} | \dot{B}_{7}-\dot{B}_{6}\cdot r_{3}\,=0 |
| ④ | r_{8,9} | r_{4}\ =\frac{{\dot B_{9}}}{\dot{B}_{8}} | \dot{B}_{9}-\dot{B}_{8}\cdot r_{4}\,=0 |
| ⑤ | r_{10,11} | r_{5}\ =\frac{{\dot B_{10}}}{\dot{B}_{11}} | \dot{B}_{10}-\dot{B}_{11}\cdot r_{5}\,=0 |
| ⑥ | r_{8,9,12} | r_{6}\ =\frac{{\dot B_{8}-\dot{B}_{9}}}{\Delta\dot{B}_{12}} | \left(\dot{B}_{8}\,-\dot{B}_{9}\right)\,-\dot{\Delta B}_{12}\cdot r_{6}\,=0 |
Table E.8.12 \alpha_{r} recirculations matrix.
| \alpha_{r} | -\frac{{\dot{B}}_{2}}{{\dot{B}}_{1}} | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 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 | -\frac{{\dot{B}}_{7}}{{\dot{B}}_{6}} | 1 | 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 | 1 | -\frac{{\dot{B}}_{10}}{{\dot{B}}_{11}} | 0 | 0 | |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 0 | 0 | -\frac{\left(\dot{B}_{8}-\dot{B}_{9}\right)}{\Delta\dot{B}_{12}} | 0 |
Table E.8.13 \alpha_{s} output matrix and transpose of \mathbb{Y}_{s} vector.
| \alpha_{s}= | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| {}^{t}\mathbb{Y}_s= | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
Table E.8.14 Symbolic expressions of the flows exergy.
| \mathbf{Flows~symbolic~expressions~}\{\mathbf{k},\mathbf{r},\mathbf{B}_{s}\} | |
| 1. | {\dot{B}}_{1}\,=\,r_{6}\cdot k_{2}\cdot k_{3}\cdot\frac{(1-r_{2})}{(1-r_{1})}.\frac{k_{5}\cdot k_{6}}{(1-k_{3}\cdot k_{4}\cdot r_{2})}.\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}\cdot k_{6}}{(1-k_{3}\cdot k_{4}\cdot r_{2})}.\frac{\dot B_{s_{11}}}{(1+r_{6})} |
| 3. | {\dot{B}}_{3}\,=r_{6}\cdot k_{3}\frac{k_{5}\cdot k_{6}}{(1-k_{3}\cdot k_{4}\cdot r_{2})}.\frac{\dot B_{s_{11}}}{(1+r_{6})} |
| 4. | {\dot{B}}_{4}\,=r_{2}\cdot r_{6}\cdot k_{3}\frac{k_{5}\cdot k_{6}}{(1-k_{3}\cdot k_{4}\cdot r_{2})}.\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 k_{5}\cdot k_{6}}{(1-k_{3}\cdot k_{4}\cdot r_{2})}.\frac{\dot{B}_{s_{11}}}{(1+r_{3})\cdot(1+r_{6})} |
Table E.8.14 Symbolic expressions of the flows exergy. -\mathbb{c o n t}'\,\mathrm{d}
| \mathbf{Flows~symbolic~expressions~}\{\mathbf{k},\mathbf{r},\mathbf{B}_{s}\} | |
| 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\, \,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}\,\cdot\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}=r_{6}\cdot k_{1}\cdot k_{5}\cdot k_{6}\cdot\frac{k_{2}\cdot k_{3}\cdot(1-r_{2})}{(1-k_{3}\cdot k_{4}\cdot r_{2})}.\frac{\dot{B}_{s_{11}}}{(1+r_{6})} |
Table E.8.15 \langle P F\rangle matrix.
| {}^{t}\langle F_{T}F\rangle\,= | 1 | 0 | 0 | 0 | 0 | \frac{1}{1+r_{6}} | |
| \langle PF\rangle\,= | 0 | 1 | 0 | 0 | 0 | 0 | |
| 0 | 0 | 1 – r_2 | 0 | 0 | 0 | ||
| 0 | 0 | 0 | 1 | 1 | 0 | ||
| 0 | 0 | r_2 | 0 | 0 | 0 | ||
| 0 | 0 | 0 | 0 | 0 | \frac{r_{6}}{(1+r_{6})} | ||
| 0 | 0 | 0 | 0 | 0 | 0 | ||
Table E.8.16 Symbolic expressions of the components’ fuel.
| \mathrm{Fuels~symbolic~expressions} | |
| ① | {F}_{1}\,=\,k_{1}\cdot k_7\cdot\dot{B}_{s_{15}}\,+\,r_{6}\cdot k_{5}\cdot k_{6}\,\cdot\frac{k_{2}\cdot k_{3}\cdot(1-r_{2})}{(1-k_{3}\cdot k_{4}\cdot r_{2})}.\frac{\dot B_{\mathrm{s}_{11}}}{(1+r_{6})} |
| ② | F_{2}=r_{6}\cdot k_{2}\cdot k_{3}\cdot(1\,-\,r_{2})\cdot{\frac{k_{s}\cdot k_{6}}{(1-k_{3}\cdot k_{4}\cdot r_{2})}}\cdot{\frac{\dot B_{\mathrm{s}_{11}}}{(1+r_{6})}} |
| ③ | {F}_{3}\,=\,r_{6}\cdot k_{3}\frac{k_{5}\cdot k_{6}}{(1-k_{3}\cdot k_{4}\cdot r_{2})}\cdot\frac{\dot B_{s_{11}}}{(1+r_{6})} |
| ④ | {F}_{4}\,=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_{6})} |
| ⑤ | F_{5}=\left(r_{5}\cdot\left[k_{3}\cdot k_{4}\cdot r_{2}\ -\left(1\,+\,r_{3}\right)\right]\,-\,r_{3}\cdot r_{2}\cdot r_{6}\cdot\kappa_{3}\cdot\kappa_{4}\right)\cdot\left[\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})}\right] |
| ⑥ | F_{6}=(r_{6}\,-\,r_{4})\cdot k_{6}\!\cdot\!\frac{\dot{B}_{s_{11}}}{(1{-}r_{4})\cdot(1{+}r_{6})}+k_{6}\!\cdot\!\frac{\dot{B}_{s_{11}}}{1{+}r_{6}} |
Table E.8.17 Symbolic expressions of the components’ product.
| \mathrm{Products~symbolic~expressions} | |
| ① | P_{1}=r_{6}\cdot k_{2}\cdot k_{3}\cdot(1\,-\,r_{2})\cdot{\frac{k_{5}\cdot k_{6}}{(1-k_{3}\cdot k_{4}\cdot r_{2})}}\cdot{\frac{\dot{B}_{s_{11}}}{(1+r_{6})}} |
| ② | P_{2}=r_{6}\cdot k_{3}{\frac{k_{\mathrm{5}}\cdot k_{6}}{(1-k_{3}\cdot k_{4}\cdot r_{2})}}.{\frac{\dot{B}_{\mathrm{s}_{11}}}{(1+r_{6})}}.(1\ -r_{2}) |
| ③ | P_{3}=(r_{5}\cdot[k_{3}\cdot k_{4}\cdot r_{2}\,-\,(1\,+\,r_{3})]\,+\,r_{2}\cdot r_{6}\cdot \kappa_{3}\cdot\kappa_{4})\cdot\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})} |
| ④ | P_{4}=r_{2}\cdot r_{6}\cdot k_{3}\frac{k_{8}\cdot k_{6}}{(1-k_{3}\cdot k_{4}\cdot r_{2})}\cdot\frac{\dot{B}_{s_{11}}}{(1+r_{6})} |
| ⑤ | P_{5}=(r_{6}\,-\,r_{4})\cdot k_{6}\cdot{\frac{\dot{B}_{s_{11}}}{(1-r_{4})\cdot(1+r_{6})}} |
| ⑥ | P_{6}=\dot{B}_{s_{11}} |
Table E.8.18 Symbolic expressions of the components’ irreversibility.
| {\mathrm{Irreversibilites~symbolic~expressions}} | |
| ① | I_{1}\,=\,k_{1}\cdot k_7\cdot\dot{B}_{s_{15}}\,+\,r_{6}\cdot k_{1}\cdot k_{5}\cdot k_{6}\cdot \frac{k_{2}\cdot k_{3}\cdot(1-r_{2})}{(1-k_{3}\cdot k_{4}\cdot r_{2})}.\frac{\dot{B}_{s_{11}}}{(1+r_{6})}\cdot\left(k_{1}\;-1\right) |
| ② | I_{2}\,=\,r_{6}\cdot k_{3}\frac{k_{5}\cdot k_{6}}{(1{-}k_{3}\cdot k_{4}\cdot r_{2})}{\cdot}\frac{\dot{B}_{s_{11}}}{(1{+}r_{6})}\cdot\left(k_{2}\,-1\right) |
Table E.8.18 Symbolic expressions of the components’ irreversibility.-\mathbb{c o n t}'\,\mathrm{d}
| {\mathrm{Irreversibilites~symbolic~expressions}} | |
| ③ | I_{3}=\frac{k_5{\cdot}k_{6}}{(1\!-\!k_{3}{\cdot}k_{4}{\cdot}r_{2})}{\cdot}\frac{\dot{B}_{\mathrm{s}_{11}}}{(1\!+\!r_{6})}{\cdot}\!\left(r_{6}\cdot k_{3}\cdot\left(1\,-\!r_{2}\cdot k_{4}\right)-r_{5}\cdot\left[k_{3}{\cdot}k_{4}{\cdot}r_{2}\,-\left(1\,+\!r_{3}\right)\right]\right) |
| ④ | I_{4}=r_2 \cdot r_6\cdot k_3\cdot \frac{k_5{\cdot}k_{6}}{(1\!-\!k_{3}{\cdot}k_{4}{\cdot}r_{2})}{\cdot}\frac{\dot{B}_{\mathrm{s}_{11}}}{(1\!+\!r_{6})}{\cdot}(k_4 -1) |
| ⑤ | I_{5}=(r_{5}\cdot[k_{3}\cdot k_{4}\cdot r_{2}\,-(1\,+\,r_{3})]\,-\,r_{3}\cdot r_{2}\cdot r_{6}\cdot k_{3}\cdot k_{4})\cdot\left[{\frac{k_{s}\cdot k_{6}}{(1-k_{3}\cdot k_{4}\cdot r_{2})}}{\frac{\dot{B}_{s_{11}}}{(1+r_{3})\cdot(1+r_{6})}}\right] -(r_{6}-r_{4})\cdot k_{6}\cdot\frac{\dot{B}_{s_{11}}}{(1-r_{4})\cdot(1+r_{6})} |
| ⑥ | I_{6}=(r_{6}\,-\,r_{4})\cdot k_{6}\cdot\frac{\dot B_{s_{11}}}{(1-r_{4})\cdot(1+r_{6})}\,+\,k_{6}\cdot\frac{\dot{B}_{s_{11}}}{1+r_{6}}-\dot{B}_{s_{11}} |
Table E.8.19 Unit exergy costs of components’ fuel.
| \mathrm{Fuels'~u n i t~e x e r g y~\mathrm{costs}} | |
| ① | k_{F_{1}}^{*}=1 |
| ② | k_{F_{2}}^{*}=k_{1} |
| ③ | k_{F_{3}}^{\ast}={\frac{(1-r_{2})}{(1-k_{3}\cdot k_{4}.r_{2})}}\cdot k_{2}\cdot k_{1} |
| ④ | k_{F_{4}}^{\ast}={\frac{k_3\cdot (1-r_{2})}{(1-k_{3}\cdot k_{4}.r_{2})}}\cdot k_{2}\cdot k_{1} |
| ⑤ | k_{F_{5}}^{\ast}={\frac{k_3\cdot (1-r_{2})}{(1-k_{3}\cdot k_{4}.r_{2})}}\cdot k_{2}\cdot k_{1} |
| ⑥ | k_{F_{6}}^{\star}\,=\,1\,-\,{\frac{k_{6}\,{r_{6}}}{(1+{r_{6}})}}\,-{\frac{k_{6}\cdot r_{6}}{(1+r_{6})}}\cdot{\frac{k_{5}\cdot k_{3}\cdot(1-r_{2})}{(1-k_{3}\cdot k_{4}\cdot r_{2})}}\cdot k_{2}\cdot k_1 |
Table E.8.20 Unit exergy costs of components’ product.
| \mathrm{Products'~u n i t~e x e r g y~\mathrm{costs}} | |
| ① | k_{p_{1}}^{*}=k_1 |
| ② | k_{p_{2}}^{*}=k_{2}\cdot k_1 |
| ③ | k_{p_{3}}^{\ast}={\frac{(1-r_{2})}{(1-k_{3}\cdot k_{4}.r_{2})}}\cdot k_{2}\cdot k_{1} |
| ④ | k_{p_{4}}^{\ast}=k_4\cdot {\frac{(1-r_{2})}{(1-k_{3}\cdot k_{4}.r_{2})}}\cdot k_{2}\cdot k_{1} |
| ⑤ | k_{p_{5}}^{\ast}=k_5 \cdot{\frac{(1-r_{2})}{(1-k_{3}\cdot k_{4}.r_{2})}}\cdot k_{2}\cdot k_{1} |
| ⑥ | k_{p_{6}}^{\ast}\ =\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 k_{1}\ \ +\frac{k_{6}}{(1+r_{6})} |
Table E.8.21 Unit exergoeconomic costs of the components’ fuel.
| \mathrm{Fuels'~u n i t~ exergoeconomic~\mathrm{costs}} | |
| 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_{6}}{(1+r_{6})}{\cdot}c_{P_{5}} |
Table E.8.22 Product unit exergoeconomic costs associated to the external resources.
| Product unit exergoeconomic costs associated to 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_{2})\!\cdot\!k_{3}}{(1\!-\!k_{3}\!\cdot\!k_{4}\!\cdot\!r_{2})}\!\cdot\!k_{2}\cdot k_1\cdot\!c_{NG} |
| ④ | c_{p_{4}}^{e}= k_{4}\!\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_{NG} |
| 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_{NG} |
| ⑥ | c_{p_{6}}^{e}\ =\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 k_{1}\cdot c_{N G}\ \ +\frac{k_{6}}{(1+r_{6})}\cdot c_{w} |
Table E.8.23 Unit exergoeconomic costs of the products associated to the capital cost rate.
| Product unit exergoeconomic costs associated with investment | |
| ① | c_{p_{1}}^{z}=z_{1} |
| ② | c_{p_{2}}^{z}=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{k_{5}\cdot(1-r_{2})\cdot(k_{2}\cdot k_{3}\cdot z_{1}+k_{3}\cdot z_{2}+z_{3}+r_{2}\cdot k_{3}\cdot z_{4})}{(1-k_{3}\cdot k_{4}\cdot r_{2})}} |
| ⑥ | c_{p_{6}}^{z}=z_{6}\ +{\frac{k_{6}\cdot r_{6}}{(1+r_{6})}}\cdot \frac{z_5+k_{5}\cdot(1-r_{2})\cdot(k_{2}\cdot{k}_{3}\cdot{z}_{1}+k_{3}\cdot z_{2}+z_{3}+r_{2}\cdot k_{3}\cdot z_{4})}{(1-k_{3}\cdot k_{4}\cdot r_{2})} |