Question 8.1: Let us consider a DHW production facility based on a condens......
Let us consider a DHW production facility based on 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 674 Exergy Analysis and Thermoeconomics of Buildings diagram of Fig. E.8.1. Using the FP formulation and without considering residues, the following have been addressed:
(a) Functional analysis of installation flows.
(b) Construction of the extended matrix \mathbb{J}_e and the vector \mathbb{Y}_e.
(c) Symbolic expressions of each flow exergy.
(d) Construction of the matrix \langle F P\rangle.
(e) Symbolic expressions of components’ fuel.
(f) Symbolic expressions of components’ product.
(g) Symbolic expressions of components’ irreversibility.
(h) Symbolic expression of the total system efficiency.
A total of m = 13 flows and n = 6 components have been considered, see Fig. E.8.2. Circulation pumps have not been taken into account due to their small po-wer. Therefore, the effects of pressure on the water flow exergy values are not consid-ered in the analysis.
(a) The result of the functional analysis is shown in Table E.8.1, which also represents each component unit’s consumption. There is one product on the installation, flow 11, which corresponds to the DHW flow, with associated exergy {\dot{B}}_{11} during the time-step considered.
Once this functional analysis is done, the construction of matrices A_{\mathrm{F}} and A_{\mathrm{P}} is immediate. Note that the tank accumulated exergy increase during the analysis period (\Delta\dot{B}_{12}) is considered as fuel, that is, the exergy difference between the initial and final moment of the time-step.
(b) The extended matrix \mathbb{{{J}}}_{\mathrm{{x}}} and the vector \mathbb{{{Y}}}_{\mathrm{{e}}} are constructed according to Eq. (8.4) and Eq. (7.110). They are shown below
The matrix J, which is part of the extended matrix \mathbb{{{J}}}_{\mathrm{{x}}} is defined according to J=A_{F}-K_{D}A_{P}. It is presented in Table E.8.2.
The \textstyle\alpha_{x} bifurcation matrix relates the flow exergies going out of the components. In Table E.8.3 bifurcation parameters are presented (as many as output flows of each compo-nent minus 1); due to space reasons, in the second column they are named according to the nomenclature x_{ij}, but in the third and fourth columns, and throughout the Example, they are numbered from 1 to 4.
In Table E.8.4 the matrix associated with the bifurcation parameters \alpha_{x(4,13)} is shown.
The (3.13) dimension matrix \alpha_{e} contains the coefficients of the incoming flows; besides, the vector \mathbb{Y}_e (in fact, its transpose) contains the total external resources. Both are reflected in Table E.8.5.
(c) Once the external fuel (\dot{B}_{\mathcal{e}}), the exergy unit consumptions ({k}_{\mathcal{i}}), and the bifurcation parameters that define the physical system are known, the symbolic expressions of components fuel are obtained. Eq. (8.8) is used for the resolution and the results are collected in Table E.8.6.
B={\mathbb{J}}_{x}^{-1}\mathbb{Y}_{e}\qquad\qquad(8.8)(d) The previous matrices are used for the construction of the matrix \langle F P\rangle,{\mathfrak{}} taking into account the relation \langle F P\rangle=A_{F}A_{P}^{(-1)}, see Table E.8.7.
It is verified that the boiler product is entirely transformed into the fuel of the discharge manifold and something similar happens with the hydraulic compensator and the three-way diverter. From the total product of the diverter, however, the portion {\frac{x_{2}+x_{3}}{x_{2}+1}} goes out to V3V, and the remaining part goes to the exchanger. The total product of the heat exchanger is also transformed into the tank fuel. It can also be seen that the total product of the tank becomes entirely part of the total product of the installation (last term of vector {\mathbf{P}_T\mathbf{P}}).
(e) From Tables E.8.3 and E.8.6, and by using the \mathbf{F}={\mathbf{A}_F\mathbf{B}} relation, the symbolic expressions of components’ fuel are obtained, see Table E.8.8.
(f ) Similarly, using the \mathbf{P}={\mathbf{A}_p\mathbf{B}} relationship, symbolic expressions of the components’ product are obtained, see Table E.8.9.
(g) The corresponding expressions for the components’ irreversibility are obtained subtracting the previous expressions, since I = F-P. The results are shown in Table E.8.10.
(h) The total efficiency of the installation is
\varphi_{T}={\frac{P_{T}}{F_{T}}}={\frac{\left({\dot{B}}_{11}-{\dot{B}}_{10}\right)}{\Delta{\dot{B}}_{12}+{\dot{B}}_{13}}} = \frac{\frac{\dot{B}_{e12}}{k_{6}}-\dot{B}_{e13}\cdot k_{4}\cdot\frac{1-x_{3}}{k_{6}\cdot k_{1}\cdot k_{2}\cdot[x_{2}+x_{3}-k_{3}\cdot k_{4}\cdot k_{5}(1+x_{2})]}}{\dot{B}_{e12}+\dot{B}_{e13}}Table E.8.1 Functional analysis.
| n | \rm Fuel | \rm Prod. | k_{D} | p_{s} | |
| ① | Boiler | {\dot{B}}_{13} | {\dot{B}}_{1}\,-{\dot{B}}_{2} | \frac{\left({\dot{B}}_{1}-{\dot{B}}_{2}\right)}{\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}}{\stackrel{}{\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.8.3 Bifurcation parameters.
| n | x_{ij} | \rm Bifurcations | \alpha_{X} |
| ① | x_{1,2} | x_{1}\,=\frac{\dot{B}_{2}}{\dot B_{1}} | \dot{B}_{2}-\dot{B}_{1}\cdot x_{1}\,=0 |
| ② | x_{5,6} | x_{2}\,=\frac{\dot{B}_{6}}{\dot B_{5}} | \dot{B}_{6}-\dot{B}_{5}\cdot x_{2}\,=0 |
| ③ | x_{5,7} | x_{3}\,=\frac{\dot{B}_{7}}{\dot B_{5}} | \dot{B}_{7}-\dot{B}_{5}\cdot x_{3}\,=0 |
| ④ | x_{8,9} | x_{4}\,=\frac{\dot{B}_{9}}{\dot B_{8}} | \dot{B}_{9}-\dot{B}_{8}\cdot x_{4}\,=0 |
Table E.8.4 \alpha_{x} bifurcation matrix.
| \alpha_{x}= | -{\frac{{\dot{B}}_{2}}{\dot{B}_{1}}} | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | -{\frac{{\dot{B}}_{6}}{\dot{B}_{5}}} | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
| 0 | 0 | 0 | 0 | 0 | -{\frac{{\dot{B}}_{7}}{\dot{B}_{5}}} | 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 |
Table E.8.5 \alpha_e input matrix and \mathbb{Y}_e vector.
| \alpha_{e}= | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | |
| {}^{t}\mathbb{Y}_{e}= | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | B_{e10} | 0 | \Delta B_{e12} | B_{e13} |
Table E.8.6 Symbolic expressions of flows. FP representation.
| \mathrm{Flows~symbolic~expressions~}\{k,{\bf x},{\bf B}_{\mathrm{e}}\} | |
| 1. | {\dot{B}}_{1}\,={\dot{B}}_{e13}/[k_{1}\cdot(1\,-x_{1})] |
| 2. | {\dot{B}}_{2}={\dot{B}}_{e13}.x_{1}/[k_{1}\cdot(1\,-x_{1})] |
| 3. | \dot{B}_{3}\,=\,-\dot{B}_{e13}{\cdot}k_{3}{\cdot}k_{4}{\cdot}(1\,+x_{2})/[k_{1}{\cdot}k_{2}{\cdot}[x_{2}\,+x_{3}\,-k_{3}{\cdot}k_{4}{\cdot}(1\,+x_{2})]] |
| 4. | \dot{B}_{4}\,=\,-\dot{B}_{e13}{\cdot}(x_{2}\,+x_{3}){\Big/}\,[k_{1}{\cdot}k_{2}{\cdot}[x_{2}\,+x_{3}\,-\,k_{3}{\cdot}k_{4}{\cdot}(1\,+x_{2})]{\Big]} |
| 5. | \dot{B}_{5}\,=\,-\dot{B}_{e13}{\cdot}k_{4}/[k_{1}{\cdot}k_{2}{\cdot}[x_{2}\,+x_{3}\,-\,k_{3}{\cdot}k_{4}{\cdot}(1\,+x_{2})]] |
| 6. | {\dot{B}}_{6}\,=\,-{\dot{B}}_{e13}{\cdot}k_{4}{\cdot}x_{2}/[k_{1}{\cdot}k_{2}{\cdot}[x_{2}\,+x_{3}\,-\,k_{3}{\cdot}k_{4}{\cdot}(1\,+x_{2})]] |
| 7. | \dot{B}_{7}\,=\,-\dot{B}_{e13}{\cdot}k_{4}{\cdot}x_{3}/[k_{1}{\cdot}k_{2}{\cdot}[x_{2}\,+x_{3}\,-k_{3}{\cdot}k_{4}{\cdot}(1\,+x_{2})]] |
| 8. | \dot{B}_{8}\,=\dot{B}_{e13}\!\cdot\!k_{4}\!\cdot\!(1\,{\stackrel{}{-}}\,x_{3})/[(x_{4}\,-1)\!\cdot\!k_{1}\!\cdot\!k_{2}\!\cdot\![x_{2}\,+x_{3}\,-k_{3}\!\cdot\!k_{4}\!\cdot\!k_{5}(1\,+x_{2})]] |
| 9. | \dot{B}_{9}\,=\dot{B}_{e13}{\cdot}x_{4}{\cdot}k_{4}{\cdot}\left({1\ -x_{3}}\right)/[(x_{4}\,-1){\cdot}k_{1}{\cdot}k_{2}{\cdot}[x_{2}\,+x_{3}\,-k_{3}{\cdot}k_{4}{\cdot}k_{5}(1\ +x_{2})]] |
| 10. | \dot{B}_{10}=\dot{B}_{e10} |
| 11. | \dot{B}_{11}\,=\dot{B}_{e10}\,+\dot{B}_{e12}/\kappa_{6}\,-\dot{B}_{e13}\cdot k_{4}\cdot(1\,-x_{3})/k_{6}\!\cdot k_{1}\cdot k_{2}\cdot[x_{2}\,+x_{3}\,-k_{3}\cdot k_{4}\cdot k_{5}(1\,+x_{2})] |
| 12. | \Delta\dot{B}_{12}\,=\dot{B}_{e12} |
| 13. | {\dot{B}}_{13}\,={\dot{B}}_{e13} |
Table E.8.7 \langle F P\rangle matrix.
| {}^{t}{\langle\mathbf{P}_T\mathbf{P}\rangle=} | 0 | 0 | 0 | 0 | 0 | 1 |
| {\langle\mathbf{F}\mathbf{P}\rangle=} | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | |
| 0 | 1 | 0 | 1 | 0 | 0 | |
| 0 | 0 | {\frac{x_{2}+x_{3}}{x_{2}+1}} | 0 | 0 | 0 | |
| 0 | 0 | {\frac{1-x_{3}}{x_{2}+1}} | 0 | 0 | 0 | |
| 0 | 0 | 0 | 0 | 1 | 0 |
Table E.8.8 Symbolic expressions of components’ fuel. FP representation.
| {\mathrm{Fuels~symbolic~expression}} | |
| ① | F_{1}={\dot{B}}_{e13} |
| ② | F_{2}\,=\frac{\dot{B}_{e13}}{k_{1}} |
| ③ | F_{3}\;=\;-\dot{B}_{e13}\cdot k_{3}\cdot k_{4}\cdot\frac{1+x_{2}}{k_{1}\cdot k_{2}\cdot\left[x_{2}+x_{3}-k_{3}\cdot k_{4}\cdot\left(1+x_{2}\right)\right]}\; |
| ④ | F_{4}\ =\ -\dot{B}e13\cdot k_{4}\cdot\frac{(x_{2}+x_{3})}{k_{1}\cdot k_{2}\cdot\left[x_{2}+x_{3}-k_{3}\cdot k_{4}\cdot(1+x_{2})\right]} |
| ⑤ | F_{5}\;=\;-\dot{B}_{e13}\cdot k_{4}\cdot\frac{1-x_{3}}{k_{1}\cdot k_{2}\cdot|x_{2}+x_{3}-k_{3}\cdot k_{4}\cdot(1+x_{2})|}\; |
| ⑥ | {F}_{6}\,=\,\dot{B}_{e13}\cdot k_{4}\cdot\frac{1-x_{3}}{k_{1}\cdot k_{2}\cdot[x_{2}+x_{3}-k_{3}\cdot k_{4}\cdot k_{5}(1+x_{2})]}\,+\dot{B}_{e12} |
Table E.8.9 Symbolic expressions of components’ product. FP representation.
| {\mathrm{Products}}^{\cdot}\ {\mathrm{symbolic}}\ \rm expressions | |
| ① | P_{1}\,=\frac{\dot{B}_{e13}}{k_{1}} |
| ② | P_{2}\,=\,-\dot{B}_{e13}\cdot\frac{k_{3}\cdot k_{4}\cdot(1+x_{2})-(x_{2}+x_{3})}{k_{1}\cdot k_{2}\cdot[x_{2}+x_{3}-k_{3}\cdot k_{4}\cdot(1+x_{2})]} |
| ③ | P_{3}\,=\,-\dot{B}_{e13}{\cdot}k_{4}{\cdot}\frac{1{+}x_{2}}{k_{1}{\cdot}k_{2}{\cdot}[x_{2}+x_{3}{-}k_{4}{\cdot}(1{+}x_{2})}) |
| ④ | P_{4}\,=\,-\dot{B}_{e13}\cdot\frac{x_{2}+x_{3}}{k_{1}{\cdot}k_{2}{\cdot}[x_{2}+x_{3}{-}k_3\cdot k_{4}{\cdot}(1{+}x_{2})} |
| ⑤ | P_{5}=\dot{B}_{e13}\cdot k_{4}\cdot\frac{1-x_{3}}{k_{1}\cdot k_{2}\cdot[ x_{2}+x_{3}- k_3 \cdot k_{4}\cdot(1+x_{2})]}) |
| ⑥ | P_{6}={\frac{{\dot{B}}_{e12}}{k_{6}}}\,-\,{\dot{B}}_{e13}\cdot{{k}}_{4}\cdot{\frac{1-x_3}{{{k}}_{6}\cdot{{k}}_{1}\cdot{{k}}_{2}\cdot[x_{2}+x_{3}-{{k}}_{3}\cdot{{k}}_{4}\cdot{{k}}_{5}(1+x_{2})]}} |
Table E.8.10 Irreversibility symbolic expressions. FP representation.
| {\mathrm{Irreversibilities}}^{,}\ {\mathrm{Symbolic}}\ {\mathrm{expressions}} | |
| ① | I_{1}={\dot{B}}_{e13}\cdot{\left(1-{\frac{1}{k_{1}}}\right)} |
| ② | I_{2}=\dot{B}_{e13}\cdot\left[\frac{1}{k_1}-\frac{k_{3}\cdot k_{4}\cdot(1+x_{2})-(x_{2}+x_{3})}{k_1\cdot k_{2}\cdot(x_{2}+x_{3}-k_{3}\cdot k_{4}\cdot(1+x_{2})]}\right] |
| ③ | I_{3}\,=\,-\dot{B}_{e13}\cdot\kappa_{4}\cdot{\frac{1+x_{2}}{[k_{1}\cdot k_{2}\cdot[x_{2}\cdot(+x_{3}-k_{3}\cdot k_{4}\cdot(1+x_{2})]]}}\cdot\left(k_{3}\,-\,1\right) |
| ④ | I_{4}\,=\,-\dot{B}_{e13}\,\cdot\frac{(x_{2}+x_{3})}{k_{1}\cdot k_{2}\cdot[x_{2}+x_{3}-k_{3}\cdot k_{4}\cdot(1+x_{2})]]}\cdot (k_{4}\,-\,1) |
| ⑤ | I_{5}\,=\,-\dot{B}_{e13}\cdot k_{4}\cdot\frac{1-x_{3}}{k_{1}\cdot k_{2}}\cdot\left(\frac{1}{[x_{2}+x_{3}-\kappa_{3}\cdot \kappa_{4}\cdot(1+x_{2})|}\,-\,\frac{1}{[x_{2}+x_{3}-k_{3}\cdot k_{4}\cdot k_{5}(1+x_{2})]}\right) |
| ⑥ | I_{6}\,=\,\left[\dot{B}_{e13}\cdot\kappa_{4}\cdot\frac{1{-}x_{3}}{k_{1}\cdot k_{2}\cdot\left[x_{2}+x_{3}-k_{3}\cdot k_{4}\cdot(1+x_{2})\right]}\,+\dot{B}_{e12}\right]\cdot\left(1~-\frac{1}{k_{6}}\right) |
