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Question 7.10: Consider again the schema of Fig. E.7.3 corresponding to the......

Consider again the schema of Fig. E.7.3 corresponding to the installa-tion of Example E.7.2. From the exergies of the flows, find expressions for:
(a) The exergy efficiencies of the components and construct the matrices K_{D} and H_{D}
(b) The total efficiency of the installation.
(c) Write the matrix equation JB = 0

لقطة الشاشة 2023-07-19 145826
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(a) With the fuel and product of each component defined, the exergy efficiency is obtained by applying Eq. (7.23), with the results in Table E.7.21.
The matrix H_{D} is shown in Table E.7.22 and K_{D} in Table E.7.23.
(d) The total efficiency of the installation is obtained by applying Eq. (7.44), which refers to the installation at its maximum level of aggregation.

\varphi_{j}={\frac{P_{j}}{F_{j}}}\qquad\qquad(7.23) \\ \varphi_{T}={\frac{P_{T}}{F_{T}}}=1-{\frac{I_{T}}{F_{T}}}=1-{\frac{\sum_{i=1}^{m}D_{i}+\sum_{i=1}^{m}L_{i}}{F_{T}}}=1-\sum_{i=1}^{m}y D,i-\sum_{i=1}^{m}y_{L,i}\qquad\qquad(7.44) \\ \varphi_{T}=\frac{\dot{B}_{21}+\left(\dot{B}_{15}-\dot{B}_{16}\right)+\dot{B}_{23}}{\dot{B}_{22}+\dot{B}_{24}+\dot{\Delta B}_{25}+\dot{\Delta B}_{26}}

(e) The matrix J is shown in Table E.7.24.
The matrix equation JB = 0 is equivalent to the following system of 10 equations:

  • {\bf S})\ \ {0}=\,-k_{1}\dot{{B}}_{1}+k_{1}\dot{{B}}_{2}+\dot{B}_{22_{}}-\stackrel{}{k_{1}}\dot{B}_{23}-\dot{B}_{27}
  • \\ {\bf CC})\,\,0\,=\,-k_{2}\dot{B}_{3}\,+\,k_{2}\dot{B}_{4}\,+\,\dot{B}_{24}\,-\,\dot{B}_{28}
  • \\{\bf C})\ \ 0=\dot{B}_{1}\:-k_{4}\dot{B}_{2}\:+\dot{B}_{3}\:-k_{4}\dot{B}_{4}\:-k_{4}\dot{B}_{5}\:+\dot{B}_{6}
  • \\{\bf CH})\ \ 0=\dot{B}_{5}\;-\dot{B}_{6}\;-k_{5}\dot{B}_{7}\;+k_{5}\dot{B}_{8}\;+\dot{B}_{25}
  • \\{\bf V1})\ \ 0=\dot{B}_{7}\,-\,\dot{B}_{8}\,-k_{6}\dot{B}_{9}\,+\,k_{6}\dot{B}_{10}\,-\,k_{6}\dot{B}_{17}\,+\,k_{6}\dot{B}_{18}
  • \\{\bf V}2)\ \ 0=\dot{B}_{9}\,-\,\dot{B}_{10}^{}\,-\,\dot{k}_{7}\dot{B}_{11}\,+\,k_{7}\dot{B}_{12}
  • \\ \mathbf{{H}}\mathbf{X})\ \ 0={\dot{B}}_{11}\;-{\dot{B}}_{12}\;-k_{8}{\dot{B}}_{13}\;+k_{8}{\dot{B}}_{14}
  • \\ \mathbf T)\ \ 0\;=\dot{B}_{13}\;-\dot{B}_{14}\;-k_{9}\dot{B}_{15}\;+k_{9}\dot{B}_{16}\;+\dot{B}_{26}
  • \\ \mathbf{V3})\ \ \mathrm{0}=\dot{B}_{17}\;-\dot{B}_{18}\;-k_{10}\dot{B}_{19}\;+k_{10}\dot{B}_{20}
  • \\ \mathbf{D})\ \ 0={\dot{B}}_{19}\;-{\dot{B}}_{20}\;-k_{11}{\dot{B}}_{21}\;

Table E.7.21 Exergy efficiencies of equipment.

\mathrm{Exergy~efficiency~}\varphi
S \left({\dot{B}}_{1}\,-\,{\dot{B}}_{2}\right)/{\dot{B}}_{22}
CC \left({\dot{B}}_{5}\,-\,{\dot{B}}_{4}\right)/{\dot{B}}_{24}
C (\dot{B}_{5}\,+\dot{B}_{2}\,+\dot{B}_{4})/(\dot{B}_{1}\,+\dot{B}_{3}\,+\dot{B}_{6})
HC \left({\dot{B}}_7\,-{\dot{B}}_{8}\,\,\,\right){\big/}\left({\dot{B}}_{5}\,-{\dot{B}}_{6}\right)
V1 \left[(\dot{B}_{9}\,-\,\dot{B}_{10})\,+\,\left(\dot{B}_{17}\,-\,\dot{B}_{18}\right)\right]/\left(\dot{B}_{7}\,-\,\dot{B}_{8}\right)
V2 (\dot{B}_{11}-\dot{B}_{12})/(\dot{B}_{9}-\dot{B}_{10})
HX (\dot{B}_{13}-\dot{B}_{14})/(\dot{B}_{11}-\dot{B}_{12})
T (\dot{B}_{15}\,-\dot{B}_{16})/\bigl[(\dot{B}_{13}\,-\dot{B}_{14})\,+\dot{B}_{26}\bigr]
V3 (\dot{B}_{19}-\dot{B}_{20})/(\dot{B}_{17}-\dot{B}_{18})
D {\dot{B}}_{21}/({\dot{B}}_{19}-{\dot{B}}_{20})

Table E.7.22 Matrix H_{D}.

H_{D}= \frac{{\dot B_1}-{\dot B_{2}}}{\dot B_{22}} _ _ _ _ _ _ _ _ _
_ \frac{{\dot B_3}-{\dot B_{4}}}{\dot B_{24}} _ _ _ _ _ _ _ _
_ _ \frac{\dot{B}_{5}+\dot{B}_{2}+\dot{B}_{4}}{\dot{B}_{1}+\dot{B}_{3}+\dot{B}_{6}} _ _ _ _ _ _ _
_ _ _ \frac{{\dot{B}}_{7}-{\dot{B}}_{8}}{{\dot{B}}_{5}-{\dot{B}}_{6}} _ _ _ _ _ _
_ _ _ _ \frac{\left(\dot{B}_{9}-\dot{B}_{10}\right)+\left(\dot{B}_{17}-\dot{B}_{18}\right)}{\dot{B}_{7}-\dot{B}_{8}} _ _ _ _ _
_ _ _ _ _ \frac{{\dot{B}}_{11}-{\dot{B}}_{12}}{{\dot{B}}_{9}-{\dot{B}}_{10}} _ _ _ _
_ _ _ _ _ _ \frac{{\dot{B}}_{13}-{\dot{B}}_{14}}{{\dot{B}}_{11}-{\dot{B}}_{12}} _ _ _
_ _ _ _ _ _ _ \frac{\dot{B}_{15}-\dot{B}_{16}}{\left(\dot{B}_{13}-\dot{B}_{14}\right)+\dot{B}_{26}} _ _
_ _ _ _ _ _ _ _ \frac{{\dot{B}}_{19}-{\dot{B}}_{20}}{{\dot{B}}_{17}-{\dot{B}}_{18}} _
_ _ _ _ _ _ _ _ _ \frac{{\dot{B}}_{21}}{{\dot{B}}_{19}-{\dot{B}}_{20}}

Table E.7.23 Matrix K_{D}.

K_{D}= \frac{{\dot{B}}_{22}}{{\dot{{B}}}_{1}-{\dot{B}}_{2}} _ _ _ _ _ _ _ _ _
_ \frac{{\dot{B}}_{24}}{{\dot{{B}}}_{3}-{\dot{B}}_{4}} _ _ _ _ _ _ _ _
_ _ \frac{\dot{B}_{1}+\dot{B}_{3}+\dot{B}_{6}}{\dot{B}_{5}+\dot{B}_{2}+\dot{B}_{4}} _ _ _ _ _ _ _
_ _ _ \frac{{\dot{B}}_{5}-{\dot{B}}_{6}}{{\dot{B}}_{7}-{\dot{B}}_{8}} _ _ _ _ _ _
_ _ _ _ \frac{\dot{B}_{7-}\dot{B}_{8}}{\left(\dot{B}_{9}-\dot{B}_{10}\right)+\left(\dot{B}_{17}-\dot{B}_{18}\right)} _ _ _ _ _
_ _ _ _ _ \frac{{\dot{B}}_{9}-{\dot{B}}_{10}}{{\dot{B}}_{11}-{\dot{B}}_{12}} _ _ _ _
_ _ _ _ _ _ \frac{{\dot{B}}_{11}-{\dot{B}}_{12}}{{\dot{B}}_{13}-{\dot{B}}_{14}} _ _ _
_ _ _ _ _ _ _ \frac{\left(\dot{B}_{13}-\dot{B}_{14}\right)+\dot{B}_{26}}{\dot{B}_{15}-\dot{B}_{16}} _ _
_ _ _ _ _ _ _ _ \frac{{\dot{B}}_{17}-{\dot{B}}_{18}}{{\dot{B}}_{19}-{\dot{B}}_{20}} _
_ _ _ _ _ _ _ _ _ \frac{\dot{B}_{19}-\dot{B}_{20}}{\dot{B}_{21}}

Table E.7.24 Matrix J.

J = -k_{1} k_{1} _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 1 -k_{1} _ _ _ -1 _
_ _ -k_{2} k_{2} _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 1 _ _ _ -1
1 -k_{4} 1 -k_{4} -k_{4} 1 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
_ _ _ _ 1 -1 -k_{5} k_{5} _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 1 _ _ _
_ _ _ _ _ _ 1 -1 -k_{6} k_{6} _ _ _ _ _ _ -k_{6} k_{6} _ _ _ _ _ _ _ _ _ _
_ _ _ _ _ _ _ _ 1 -1 -k_{7} k_{7} _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
_ _ _ _ _ _ _ _ _ _ 1 -1 -k_{8} k_{8} _ _ _ _ _ _ _ _ _ _ _ _ _ _
_ _ _ _ _ _ _ _ _ _ _ _ 1 -1 -k_{9} k_{9} _ _ _ _ _ _ _ _ _ 1 _ _
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 1 -1 -k_{10} k_{10} _ _ _ _ _ _ _ _
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 1 -1 -k_{11} _ _ _ _ _ _ _

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