Question 7.12: Consider again the schema of the installation in Fig. 7.3 co......

Exergy Analysis and Thermoeconomics of Buildings Design and Analysis for Sustainable Energy Systems [1399314]

Consider again the schema of the installation in Fig. 7.3 corresponding to Example E.7.2, and determine:
(a) The exergy costs of flows.
(b) The exergy costs of the fuel and products of the equipment.

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(a) To obtain the exergy costs of the flows, we need to solve Eq. (7.111)

\mathbb{A}_{(m,{m})}B_{(m,{1})}^{\ast}=\mathbb{Y}_{e({m,{1}})}\qquad\qquad(7.111)

\\ \mathbb{A}_{(28,28)}\,\bf{B}_{\rm (28,1)}^{\ast}=Y_{\rm e(28,1)}

Therefore, we need to construct the extended matrix $\mathbb{A}{}=\begin{pmatrix}\\ \\ \frac{A}{\alpha_{e}}\\\ \overline{\alpha_{x}} \end{pmatrix}$ via the incidence matrix $A_{(10,28)},$ obtained from Table E.7.3 and the matrices that represent the economic structure $\alpha_{e}$ and $\alpha_{x},$ as well as the vector $Y_{e(28,1)}$ . Table E.7.29 defines the existing bifurcations for the construction of the distribution matrix $\alpha_{x(13,28)}$ ; in this case, the bifurcation parameters have also been named in the second column according to their conventional nomenclature $x_{ij}$ . Even so, for reasons of space, they have been renum-bered from 1 to 13. Table E.7.30 contains the incoming resources from outside and al-lows us to build the resources matrix $\alpha_{e(5,28)}$ .
In this installation, while there are no waste streams or sub-products, there are two loss flows, flows 27 and 28, thus arriving at the vector $Y_{e(28,1)}$ . 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. (7.112). The results obtained are shown in Table E.7.31.

B_{(m,1)}^{*}=\mathbb{A}_{(m,m)}^{-1}\mathbb{Y}_{e(m,1)}\qquad\qquad(7.112)

(b) The exergy costs of the fuel and products of the equipment are obtained from the exergy costs of flows, Table E.7.31 above, from the fuel and product matrices of the equipment and by applying Eqs. (7.117) and (7.118). The results obtained are shown in Table E.7.32.

F^{*}=A_{F}B^{*}\qquad\qquad(7.117)

\\ P^{*}=A_{P}B^{*}\qquad\qquad(7.118)

Table E.7.3 Incidence matrix A.

-1

Table E.7.29 Bifurcation equations.

		$\operatorname{Bifurcations}$	$\alpha_x$			$\operatorname{Bifurcations}$	$\alpha_x$
$1.$	$x_{1,2,23}$	${{{x_{1}\ ={\frac{\dot B_{23}}{\dot B_{1}-\dot B_{2}}}}}}$	$\dot{B}_{23}-(\dot{B}_{1}\,-\dot{B}_{2})\cdot x_{1}\,=0$	$8.$	$x_{7,8}$	$x_{8}\,=\,{\frac{\dot B_{7}}{\dot B_{8}}}$	${\dot{B}}_{7}-{\dot{B}}_{8}\cdot x_{8}\,=0$
$2.$	$x_{1,2,27}$	${{{x_{2}\ ={\frac{\dot B_{27}}{\dot B_{1}-\dot B_{2}}}}}}$	$\dot{B}_{27}-(\dot{B}_{1}\,-\dot{B}_{2})\cdot x_{2}\,=0$	$9.$	$x_{9,10}$	$x_{9}\,=\,{\frac{\dot B_{9}}{\dot B_{10}}}$	${\dot{B}}_{9}-{\dot{B}}_{10}\cdot x_{9}\,=0$
$3.$	$x_{3,4,28}$	${{{x_{3}\ ={\frac{\dot B_{28}}{\dot B_{3}-\dot B_{4}}}}}}$	$\dot{B}_{28}-(\dot{B}_{3}\,-\dot{B}_{4})\cdot x_{3}\,=0$	$10.$	$x_{11,12}$	$x_{10}\,=\,{\frac{\dot B_{11}}{\dot B_{12}}}$	${\dot{B}}_{11}-{\dot{B}}_{12}\cdot x_{10}\,=0$
$4.$	$x_{2,5}$	$x_{4}\,=\,{\frac{\dot B_{2}}{\dot B_{5}}}$	${\dot{B}}_{2}-{\dot{B}}_{5}\cdot x_{4}\,=0$	$11.$	$x_{13,14}$	$x_{11}\,=\,{\frac{\dot B_{13}}{\dot B_{14}}}$	${\dot{B}}_{13}-{\dot{B}}_{14}\cdot x_{11}\,=0$
$5.$	$x_{4,5}$	$x_{5}\,=\,{\frac{\dot B_{4}}{\dot B_{5}}}$	${\dot{B}}_{4}-{\dot{B}}_{5}\cdot x_{4}\,=0$	$12.$	$x_{16,15}$	$x_{12}\,=\,{\frac{\dot B_{16}}{\dot B_{15}}}$	${\dot{B}}_{16}-{\dot{B}}_{15}\cdot x_{12}\,=0$
$6.$	$x_{5,6}$	$x_{6}\,=\,{\frac{\dot B_{6}}{\dot B_{5}}}$	${\dot{B}}_{6}-{\dot{B}}_{5}\cdot x_{6}\,=0$	$13.$	$x_{17,18}$	$x_{13}\,=\,{\frac{\dot B_{17}}{\dot B_{18}}}$	${\dot{B}}_{17}-{\dot{B}}_{18}\cdot x_{12}\,=0$
$7.$	$x_{9,10,17,18}$	$x_7\ ={\frac{{\dot{B}}_{9}-{\dot{B}}_{10}}{{\dot{B}}_{17}-{\dot{B}}_{18}}}$	$(\dot{B}_{9}\,-\dot{B}_{10})\,-(\dot{B}_{17}\,-\dot{B}_{18})\cdot x_{7}\,=0$	$13.$	$x_{17,18}$	$x_{13}\,=\,{\frac{\dot B_{17}}{\dot B_{18}}}$	${\dot{B}}_{17}-{\dot{B}}_{18}\cdot x_{12}\,=0$

Table E.7.30 Input flows to the installation.

								$\mathrm{Entrances}$
$1.$								${\dot{B}}_{e1}\,={\dot{B}}_{22}$								$4.$									${\dot{B}}_{e4}\,={\Delta \dot{B}}_{26}$
$2.$								${\dot{B}}_{e2}\,={\dot{B}}_{24}$								$5.$									${\dot{B}}_{e5}\,={\dot{B}}_{16}$
$3.$								${\dot{B}}_{e3}\,={\Delta \dot{B}}_{25}$								$5.$									${\dot{B}}_{e5}\,={\dot{B}}_{16}$
$\mathrm{Y}\mathrm{t_{e}}$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	${\dot{B}}_{e1}$	${\dot{B}}_{e2}$	${\dot{B}}_{e3}$	${\dot{B}}_{e4}$	${\dot{B}}_{e5}$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$

Table E.7.31 Symbolic expressions of exergy costs of the flows.

	$\mathrm{Symbolic~expressions~of~exergy~costs~of~the~flows}$
$1.$	$B_{1}^{*}={\frac{1}{1-x_{6}}}\cdot\left({\frac{{\dot{B}}_{24}.x_{4}}{x_{3}+1}}+{\frac{{\dot{B}}_{22}.(x_{4}-x_{6}+1)}{x_{2}+1}}\right)$	$15.$	$B_{15}^{*}=\dot{B}_{16}\,+\frac{1}{x_7+1}\cdot\left(\frac{\dot{B}_{24}\cdot x_7}{x_{3}+1}+\frac{\dot{B}_{22}\cdot x_{7}}{x_{2}+1}\right)$
$2.$	$B_{2}^{*}={\frac{1}{1-x_{6}}}\cdot\left({\frac{{\dot{B}}_{24}\cdot x_{4}}{x_{3}+1}}+{\frac{{\dot{B}}_{22}\cdot x_{4}}{x_{2}+1}}\right)$	$16.$	$B_{16}^{*}=\dot{B}_{16}$
$3.$	$B_{3}^{*}={\frac{1}{1-x_{6}}}\cdot\left({\frac{\dot B_{24}\cdot(x_{5}-x_{6}+1)}{x_{3}+1}}\,+{\frac{\dot B_{22}\cdot x_{5}}{x_{2}+1}}\right)$	$17.$	$B_{17}^{*}={\frac{1}{(x_{7}+1){\cdot}(1-x_{12})}}\cdot\left({\frac{\dot{B}_{24}}{x_{3}+1}}+{\frac{\dot{B}_{22}}{x_{2}+1}}\right)$
$4.$	$B_{4}^{*}={\frac{1}{1-x_{6}}}\cdot\left({\frac{\dot B_{24}\cdot x_{5}}{x_{3}+1}}+{\frac{\dot B_{22}\cdot x_{5}}{x_{2}+1}}\right)$	$18.$	$B_{18}^{*}={\frac{x_{12}}{(x_{7}+1){\cdot}(1-x_{12})}}\cdot\left({\frac{\dot{B}_{24}}{x_{3}+1}}+{\frac{\dot{B}_{22}}{x_{2}+1}}\right)$
$5.$	$B_{5}^{*}={\frac{1}{1-x_{6}}}\cdot\left({\frac{\dot B_{24}}{x_{3}+1}}+{\frac{\dot B_{22}}{x_{2}+1}}\right)$	$19.$	$B_{19}^{*}={\frac{1}{(x_{7}+1){\cdot}(1-x_{13})}}\cdot\left({\frac{\dot{B}_{24}}{x_{3}+1}}+{\frac{\dot{B}_{22}}{x_{2}+1}}\right)$
$6.$	$B_{6}^{*}={\frac{1}{1-x_{6}}}\cdot\left({\frac{\dot B_{24}\cdot x_{6}}{x_{3}+1}}+{\frac{\dot B_{22}\cdot x_{6}}{x_{2}+1}}\right)$	$20.$	$B_{20}^{*}={\frac{x_{13}}{(x_{7}+1){\cdot}(1-x_{13})}}\cdot\left({\frac{\dot{B}_{24}}{x_{3}+1}}+{\frac{\dot{B}_{22}}{x_{2}+1}}\right)$
$7.$	$B_{7}^{*}={\frac{1}{1-x_{8}}}\cdot\left({\frac{\dot B_{24}}{x_{3}+1}}+{\frac{\dot B_{22}}{x_{2}+1}}\right)$	$21.$	$B_{21}^{*}={\frac{1}{x_{7}+1}}\cdot\left({\frac{{\dot{B}}_{24}}{x_{3}+1}}+{\frac{{\dot{B}}_{22}}{x_{2}+1}}\right)$
$8.$	$B_{8}^{*}={\frac{x_8}{1-x_{8}}}\cdot\left({\frac{\dot B_{24}}{x_{3}+1}}+{\frac{\dot B_{22}}{x_{2}+1}}\right)$	$22.$	$B_{22}^{*}=\dot{B}_{22}$
$9.$	$B_{9}^{\ast}=\frac{x_{7}}{(x_{7}+1)\cdot(1-x_{9})}\cdot\left(\frac{\dot{B}_{24}}{x_{3}+1}+\frac{\dot{B}_{22}}{x_{2}+1}\right)$	$23.$	$B_{23}^{*}\,={\frac{\dot B_{22}\cdot \,x_{1}}{(x_{2}+1)}}$
$10.$	$B_{10}^{\ast}=\frac{x_{7}\cdot x_9}{(x_{7}+1)\cdot(1-x_{9})}\cdot\left(\frac{\dot{B}_{24}}{x_{3}+1}+\frac{\dot{B}_{22}}{x_{2}+1}\right)$	$24.$	$B_{24}^{*}=\dot{B}_{24}$
$11.$	$B_{11}^{\ast}=\frac{x_{7}}{(x_{7}+1)\cdot(1-x_{10})}\cdot\left(\frac{\dot{B}_{24}}{x_{3}+1}+\frac{\dot{B}_{22}}{x_{2}+1}\right)$	$25.$	$B_{25}^{*}=\dot{B}_{25}$
$12.$	$B_{12}^{\ast}=\frac{x_{7}\cdot x_{10}}{(x_{7}+1)\cdot(1-x_{10})}\cdot\left(\frac{\dot{B}_{24}}{x_{3}+1}+\frac{\dot{B}_{22}}{x_{2}+1}\right)$	$26.$	$B_{26}^{*}=\Delta \dot{B}_{26}$
$13.$	$B_{13}^{\ast}=\frac{x_{7}}{(x_{7}+1)\cdot(1-x_{11})}\cdot\left(\frac{\dot{B}_{24}}{x_{3}+1}+\frac{\dot{B}_{22}}{x_{2}+1}\right)$	$27.$	$B_{27}^{*}\,={\frac{\dot B_{22}\cdot \,x_{2}}{(x_{2}+1)}}$
$14.$	$B_{14}^{\ast}=\frac{x_{7}\cdot x_{11}}{(x_{7}+1)\cdot(1-x_{11})}\cdot\left(\frac{\dot{B}_{24}}{x_{3}+1}+\frac{\dot{B}_{22}}{x_{2}+1}\right)$	$28.$	$B_{28}^{*}\,={\frac{\dot B_{24}\cdot \,x_{3}}{(x_{3}+1)}}$

Table E.7.32 Exergy costs of fuel and products.

	Fuel	Product
S	$F_{S}^{*}\,=\dot{B}_{22}$	$P_{S}^{*}={\frac{{\dot{B}}_{24}}{(1-x_{6}).(x_{3}+1)}}\,+{\frac{{\dot{B}}_{24}}{x_{2}+1}}$
CC	$F_{CC}^{*}\,=\dot{B}_{24}$	$P_{C C}^{*}={\frac{{\dot{B}}_{24}}{x_{3}+1}}+{\frac{{\dot{B}}_{24}}{(1-x_{6})\cdot(x_{2}+1)}}$

Table E.7.32 Exergy costs of fuel and products. $-\mathrm{cont}'\mathrm{d}$

	Fuel	Product
C	$F_{C}^{*}=\frac{(x_{4}+x_{5}+1)}{1-x_{6}}\cdot\left(\frac{\dot{B}_{24}}{x_{3}+1}+\frac{\dot{B}_{22}}{x_{2}+1}\right)$	$P_{C}^{*}=\frac{1+x_{4}+x_{5}}{1-x_{6}}\cdot\left(\frac{\dot{B}_{24}}{x_{3}+1}+\frac{\dot{B}_{22}}{x_{2}+1}\right)$
CH	$F_{CH}^{*}={\frac{\dot B_{24}}{x_{3}+1}}+{\frac{\dot B_{22}}{x_{2}+1}}$	$P_{C H}^{*}={\frac{\dot B_{24}}{x_{3}+1}}+{\frac{\dot B_{22}}{x_{2}+1}}$
V1	$F_{V1}^{*}={\frac{\dot B_{24}}{x_{3}+1}}+{\frac{\dot B_{22}}{x_{2}+1}}$	$P_{V1}^{*}={\frac{\dot B_{24}}{x_{3}+1}}+{\frac{\dot B_{22}}{x_{2}+1}}$
V2	$F_{V2}^{*}={\frac{x_{7}}{x_{7}+1}}\cdot\left({\frac{{\dot{B}}_{24}}{x_{3}+1}}+{\frac{{\dot{B}}_{22}}{x_{2}+1}}\right)$	$P_{V2}^{*}={\frac{x_{7}}{x_{7}+1}}\cdot\left({\frac{{\dot{B}}_{24}}{x_{3}+1}}+{\frac{{\dot{B}}_{22}}{x_{2}+1}}\right)$
HX	$F_{H X}^{*}=\frac{x_{7}}{x_{7}+1}\cdot \left(\frac{\dot B_{24}}{x_{3}+1}+\frac{\dot B_{22}}{x_{2}+1}\right)$	$P_{H X}^{*}=\frac{x_{7}}{x_{7}+1}\cdot \left(\frac{\dot B_{24}}{x_{3}+1}+\frac{\dot B_{22}}{x_{2}+1}\right)$
T	$F_{T}^{*}={\frac{x_{7}}{x_{7}+1}}\cdot\left({\frac{{\dot{B}}_{24}}{x_{3}+1}}+{\frac{{\dot{B}}_{22}}{x_{2}+1}}\right)\,+\,\Delta{\dot{B}}_{26}$	$P_{T}^{*}={\frac{1}{x_{7}+1}}\cdot\left({\frac{{\dot{B}}_{24}\cdot x_{7}}{x_{3}+1}}+{\frac{{\dot{B}}_{22}\cdot x_{7}}{x_{2}+1}}\right)$
V3	$F_{V3}^{*}={\frac{1}{x_{7}+1}}\cdot\left({\frac{{\dot{B}}_{24}}{x_{3}+1}}+{\frac{{\dot{B}}_{22}}{x_{2}+1}}\right)$	$P_{V3}^{*}={\frac{1}{x_{7}+1}}\cdot\left({\frac{{\dot{B}}_{24}}{x_{3}+1}}+{\frac{{\dot{B}}_{22}}{x_{2}+1}}\right)$
D	$F_{D}^{*}={\frac{1}{x_{7}+1}}\cdot\left({\frac{{\dot{B}}_{24}}{x_{3}+1}}+{\frac{{\dot{B}}_{22}}{x_{2}+1}}\right)$	$P_{D}^{*}={\frac{1}{x_{7}+1}}\cdot\left({\frac{{\dot{B}}_{24}}{x_{3}+1}}+{\frac{{\dot{B}}_{22}}{x_{2}+1}}\right)$