Question 7.9: Let us refer once again to the schema in Fig. E.7.1 correspo......
Let us refer once again to the schema in Fig. E.7.1 corresponding to the installation of Example E.7.1. From the exergies of the flows, find expressions for:
(a) The exergy efficiency 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
(a) Once the fuel and product of each component have been defined, the respective exergy efficiencies are obtained through Eq. (7.23), giving Table E.7.17.
\varphi_{j}={\frac{P_{j}}{F_j}}\qquad\qquad(7.23)The diagonal matrix of efficiencies H_{D(8,8)}=d i a g(\varphi_{1},\varphi_{2},\ldots,\varphi_{8}) and the diagonal matrix of unit consumption K_{D(8,8)}=d i a g(k_{1},k_{2},\cdots k_{8}) are shown in Tables E.7.18 and E.7.19 respectively
(b) The total efficiency of the installation is the relationship between the product P_{T}and the total fuel F_{T}, Eq. (7.44), that is, it is the relationship between the product and the fuel at the high-est level of aggregation, Fig. E.7.2.
\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{\left(\dot {B}_{11}-\dot {B}_{13}\right)+\left(\dot {B}\prime_{12}-\dot {B}\prime_{14}\right)+\left(\dot {B}_{17}-\dot {B}_{18}-\dot {B}_{20}\right)}{\dot {B}_{21}+\dot {B}_{22}+\Delta\dot {B}_{23}+\Delta\dot {B}_{24}}(c) According to Eq. (7.34) we construct the matrix J, as presented in Table E.7.20. The matrix equation JB = 0 equals the system of the following eight equations:
J=A_{F}-K_{D}A_{P}\qquad\qquad(7.34)- ①\ \ 0\,=\,-k_{1}\dot{B}_{1}\,+k_{1}\dot{B}_{2}\,+\dot{B}_{21}
- ②\ \ 0\,=\,-k_{2}\dot{B}_{7}\,+k_{2}\dot{B}_{8}\,+\dot{B}_{22}
- ③\ \ 0=\dot{B}_{1}\:-\:\dot{B}_{2}\:-k_{3}\dot{B}_{3}\:+\:k_{3}\dot{B}_{4}\:-\:k_{3}\dot{B}_{5}\:+\:k_{3}\dot{B}_{6}
- ④\ \ 0=\dot{B_{5}}\,-\dot{B_{6}}\,-k_{4}\dot{B_{9}}\,+k_{4}\dot{B}_{10}
- ⑤\ \ 0=\dot{B}_{9}\,-\dot{B}_{10}\,-k_{5}\dot{B}_{15}\,+k_{5}\dot{B}_{16}\,+k_{5}\dot{B}_{19}\,+\dot{B}_{24}
- ⑥\ \ 0=\dot{B}_{7}\,-\dot{B}_{8}\,-k_{6}\dot{B}_{19}\,+k_{6}\dot{B}_{20}\,+\dot{B}_{20}
- ⑦\ \ 0=\dot{B}_{3}\,-\dot{B}_{4}\,-k_7\dot{B}_{11}\,-k_7\dot{B}_{12}\,+k_7\dot{B}_{13}\,+k_7\dot{B}_{14}
- ⑧\ \ 0=\dot{B}_{15}\,-\dot{B}_{16}\,-k_{8}\dot{B}_{17}\,+k_{8}\dot{B}_{18}
Table E.7.17 Exergy efficiencies of components.
| Exergy efficiency \varphi | |
| ① | \left(\dot B_1-\dot{B}_{2}\right)/\,\dot B_{21} |
| ② | \left(\dot B_7-\dot{B}_{8}\right)/\,\dot B_{22} |
| ③ | \left[\left(\dot B_{3}\,-\dot B_{4}\right)\,+\,\left(\dot B_{5}\,-\dot {B}_{6}\right)\right]/\left(\dot {B}_{1}\,-\dot {B}_{2}\right) |
| ④ | (\dot B_{9}-\dot {B}_{10})\,/(\,\dot {B}_{5}\,-\dot {B}_{6}) |
| ⑤ | \left(\dot B_{15}-\dot B_{16}\,-\dot B_{19}\right)/\left[\left(\dot B_{9}\,-\dot B_{10}\right)\,+\dot B_{23}\right] |
| ⑥ | \left(\dot B_{19}-\dot B_{20}\right)/\left[\left(\dot B_{7}-\dot B_{8}\right)\;+\;\dot B_{24}\right] |
| ⑦ | \left[\left(\dot B_{11}-\dot B_{13}\right)\,+\,\left(\dot B_{12}-\dot B_{14}\right)\right]/\left(\dot B_{3}-\dot {B}_{4}\right) |
| ⑧ | \left(\dot B_{17}-\dot B_{18}\right)/\left(\dot B_{15}-\dot B_{16}\right) |
Table E.7.18 Matrix H_{D}.
| H_{D}= | \frac{\dot B_{1}-\dot B_{2}}{\dot B_{21}} | _ | _ | _ | _ | _ | _ | _ |
| _ | \frac{\dot B_{7}-\dot B_{8}}{\dot B_{22}} | _ | _ | _ | _ | _ | _ | |
| _ | _ | \frac{\left(\dot B_{3}-\dot B_{4}\right)+\left(\dot B_{5}-\dot B_{6}\right)}{\dot B_{1}-\dot B_{2}} | _ | _ | _ | _ | _ | |
| _ | _ | _ | \frac{\dot B_{9}-\dot B_{10}}{\dot B_{5}-\dot B_{6}} | _ | _ | _ | _ | |
| _ | _ | _ | _ | \frac{\dot B_{15}-\dot B_{16}-\dot B_{19}}{(\dot B_{9}-\dot B_{10})+\dot B_{23}} | _ | _ | _ | |
| _ | _ | _ | _ | _ | \frac{\dot B_{19}-\dot B_{20}}{(\dot B_{7}-\dot B_{8})+\dot B_{24}} | _ | _ | |
| _ | _ | _ | _ | _ | _ | \frac{\left(\dot B_{11}-\dot B_{13}\right)+\left(\dot B_{12}-\dot B_{14}\right)}{\dot B_{3}-\dot B_{4}} | _ | |
| _ | _ | _ | _ | _ | _ | _ | \frac{\dot B_{17}-\dot B_{18}}{\dot B_{15}-\dot B_{16}} |
Table E.7.19 Matrix K_{D}.
| K_{D}= | \frac{\dot B_{21}}{\dot B_{1}-\dot B_{2}} | _ | _ | _ | _ | _ | _ | _ |
| _ | \frac{\dot B_{22}}{\dot B_{7}-\dot B_{8}} | _ | _ | _ | _ | _ | _ | |
| _ | _ | \frac{\dot{B}_{1}-\dot{B}_{2}}{\left(\dot{B}_{3}-\dot{B}_{4}\right)+\left(\dot{B}_{5}-\dot{B}_{6}\right)} | _ | _ | _ | _ | _ | |
| _ | _ | _ | \frac{\dot{B}_{5}-\dot{B}_{6}}{\dot B_{9}-\dot{B}_{10}} | _ | _ | _ | _ | |
| _ | _ | _ | _ | \frac{\left(\dot{B}_{9}-\dot{B}_{10}\right)+\dot{B}_{23}}{\dot{B}_{15}-\dot{B}_{16}-\dot{B}_{19}} | _ | _ | _ | |
| _ | _ | _ | _ | _ | \frac{\left({\dot{B}}_{7}-{\dot{B}}_{8}\right)+{\dot{B}}_{24}}{{\dot{B}}_{19}-{\dot{B}}_{20}} | _ | _ | |
| _ | _ | _ | _ | _ | _ | \frac{{\dot{B}}_{3}\!-\!{\dot{B}}_{4}}{\left({\dot{B}}_{11}\!-\!{\dot{B}}_{13}\right)+\left({\dot{B}}_{12}\!-\!{\dot{B}}_{14}\right)} | _ | |
| _ | _ | _ | _ | _ | _ | _ | \frac{\dot{B}_{15}-\dot{B}_{16}}{\dot B_{17}-\dot{B}_{18}} |
Table E.7.20 Matrix J.
| J= | -k_{1} | k_{1} | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | 1 | _ | _ | _ | _ |
| _ | _ | _ | _ | _ | _ | -k_{2} | k_{2} | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | 1 | _ | _ | _ | |
| 1 | -1 | -k_{3} | k_{3} | -k_{3} | k_{3} | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | |
| _ | _ | _ | _ | 1 | -1 | _ | _ | -k_{4} | k_{4} | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | |
| _ | _ | _ | _ | _ | _ | _ | _ | 1 | -1 | _ | _ | _ | _ | -k_{5} | k_{5} | _ | _ | k_{5} | _ | _ | _ | 1 | _ | _ | |
| _ | _ | _ | _ | _ | _ | 1 | -1 | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | -k_{6} | k_{6} | _ | _ | _ | 1 | _ | |
| _ | _ | 1 | -1 | _ | _ | _ | _ | _ | _ | -k_{7} | -k_{7} | k_{7} | k_{7} | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | |
| _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | 1 | -1 | -k_{8} | k_{8} | _ | _ | _ | _ | _ | _ | _ |
