Question 7.1: In Fig. E.7.1 the diagram of a heating and DHW installation ......

(a) The incidence matrix A(n,m) is constructed with n = 8 rows (one for each component) and m = 25 columns (one per flow). As we have said, in each row (equipment) there is (1) for the incoming flows to that equipment and (-1) for the outgoing, see Table E.7.1.
(b) Mass balances in matrix form satisfy equation A_{(8,25)}M_{(25,1)}=0_{(8,1),} where M_{(25,1)} is the column vector (25,1) of the mass flow rates. The resulting mass balances in the set of the eight components are

• ①\ \ 0\,=\,-\dot m_{1}\,+\,\dot m_{2}
• \\ ②\ \ 0\,=\,-\dot m_{7}\,+\,\dot m_{8}
• \\ ③\ \ 0={\dot{m}}_{1}\,-\dot m_{2}\,-{\dot{m}}_{3}\,+{\dot{m}}_{4}\,-{\dot{m}}_{5}\,+{\dot{m}}_{6}
• \\ ④\ \ 0={\dot{m}}_{5}\,-\dot m_{6}\,-{\dot{m}}_{9}\,+{\dot{m}}_{10}
• \\ ⑤\ \ 0={\dot{m}}_{9}\,-\dot m_{10}\,-{\dot{m}}_{15}\,+{\dot{m}}_{16}\,+{\dot{m}}_{19}
• \\ ⑥\ \ 0={\dot{m}}_{7}\,-\dot m_{8}\,-{\dot{m}}_{19}\,+{\dot{m}}_{20}
• \\ ⑦\ \ 0={\dot{m}}_{3}\,-\dot m_{4}\,-{\dot{m}}_{11}\,-{\dot{m}}_{12}\,+{\dot{m}}_{13}\,+{\dot{m}}_{14}
• \\ ⑧\ \ 0={\dot{m}}_{15}\,-\dot m_{16}\,-{\dot{m}}_{17}\,+{\dot{m}}_{18}

(c) The matrix equation that represents the energy balances in the components is A_{(8,25)}H_{(25,1)}=0_{(8,1)}, where H_{(25,1)} is the vector of energy flows, being \left({\dot{H}}_{i}=\dot m_{i}h_{i}\Delta t\right) for the mass flows, and \left({\dot{H}}_{21}=\dot{m}_{21}H H V_{N G}\Delta t,\;\;\dot{H}_{22}=\stackrel{}{(G\,A)}_{22}\Delta t\right) for the external re-sources. As the energy balances refer to the time-step considered \Delta t, we call flows 23 and 24 to the energy (exergy) variation in the energy storage tanks during the time-step.

The resulting equations for the set of the eight components are:

• ①\ \ 0=\,-\dot{H_{1}}+\dot{H_{2}}+\dot{H_{21}}-\dot{H_{25}}
• \\ ②\ \ 0=\,-\dot{H_{7}}+\dot{H_{8}}+\dot{H}_{22}
• \\ ③\ \ 0=\dot{H}_{1}\,-\dot{H}_{2}\,-\dot{H}_{3}\,+\dot{H}_{4}\,-\dot{H}_{5}\,+\dot{H}_{6}
• \\ ④\ \ 0=\,\dot{H_{5}}-\dot{H_{6}}-\dot{H_{9}}+\dot{H}_{10}
• \\ ⑤\ \ 0=\dot{H}_{9}\,-\dot{H}_{10}\,-\dot{H}_{15}\,+\dot{H}_{16}\,+\dot{H}_{19}\,+\Delta U_{23}
• \\ ⑥\ \ 0=\dot{H}_{7}\,-\dot{H}_{8}\,-\dot{H}_{19}\,+\dot{H}_{20}\,+\Delta U_{24}
• \\ ⑦\ \ 0=\dot{H}_{3}\,-\dot{H}_{4}\,-\dot{H}_{11}\,+\dot{H}_{12}\,+\dot{H}_{13}\,+\dot{H}_{14}
• \\ ⑧\ \ 0=\,\dot{H_{15}}-\dot{H_{16}}-\dot{H_{17}}+\dot{H}_{18}

(d) The equation for exergy balance in the set of the eight components is in matrix form A_{(8,25)}B_{(25,1)}=D_{(8,1)}, where B_{(25,1)} represents the exergy of the flows; thus, (\dot{B}_{i}=\dot{m}_{i}b_{i}\Delta t) for the mass flows, and \left(\dot{B}_{21}=\dot{B}_{N G}^{c h}\Delta t,\,\,\,\dot{B}_{22}=\dot{B}_{r,22}\Delta t\right) for the external resources. For its part D_{(8,1)} is the exergy destruction vector in the set of eight components. The resulting exergy balances are reflected by the following equations

• ①\ \ \dot{D}_{①}=-\dot B^{ch}_{NG}+\dot{B}_{2}+\dot{B}_{21}\,-\dot{B}_{25}
• \\ ②\ \ \dot{D}_{②}=-\dot B_{7}+\dot{B}_{8}+\dot{B}_{22}
• \\ ③\ \ \dot{D}_{③}=\dot{B}_{1}-\dot{B}_{2}-\dot{B}_{3}+\dot{B}_{4}-\dot{B}_{5}+\dot{B}_{6}
• \\ ④\ \ \dot{D}_{④}=\dot B_{5}-\dot{B}_{6}-\dot{B}_{9}\,+\dot{B}_{10}
• \\ ⑤\ \ \dot{D}_{⑤}=\dot{B}_{9}\,-\dot{B}_{10}\,-\dot{B}_{15}\,+\dot{B}_{16}\,+\dot{B}_{19}\,+\Delta A_{23}
• \\ ⑥\ \ \dot{D}_{⑥}=\dot{B}_{7}\,-\dot{B}_{8}\,-\dot{B}_{19}\,+\dot{B}_{20}+\Delta A_{24}
• \\ ⑦\ \ \dot{D}_{⑦}=\dot{B}_{3}-\dot{B}_{4}-\dot{B}_{11}-\dot{B}_{12}+\dot{B}_{13}+\dot{B}_{14}
• \\ ⑧\ \ \dot{D}_{⑧}=\dot B_{15}-\dot{B}_{16}-\dot{B}_{17}\,+\dot{B}_{18}

(e) At the maximum level of aggregation, the complete installation appears as a single compo-nent, see Fig. E.7.2, in which the inflows and outflows are the flows exchanged with the external environment, that is, external resources, final products and loss flows. The inci-dence matrix is a_((1.25)) and is shown in Table E.7.2. where only the columns with values 1 or -1 are drawn.

The corresponding mass, energy and exergy balances are

• 0=-{\dot{m}}_{11}-{\dot{m}}_{12}+{\dot{m}}_{13}+{\dot{m}}_{14}-{\dot{m}}_{17}+{\dot{m}}_{18}\,+{\dot{m}}_{20}\,-{\dot{m}}_{25}
• 0\,=\dot{m}_{21}{H}H V_{N G}+(G{A})_{22}+{\Delta}U_{23}+{\Delta}U_{24}-\dot{H}_{11}-\dot{H}_{12}+\dot{H}_{13}\,+\dot{H}_{14}\,-\dot{H}_{17}\\ +\dot H_{{\mathrm{18}}}+\dot H_{\mathrm{20}}-\dot H_{\mathrm{25}}
• \dot{{D}}_{T}=\dot{B}_{N G}^{c h}+\dot{B}_{r,22}+\Delta A_{23}\,+\Delta A_{24}\,-\dot{B}_{11}\,-\dot{{B}}_{12}\,+\dot{{B}}_{13}\,+\dot{{B}}_{14}\,-\dot{{B}}_{17}\\ +\dot{B}_{18}+\dot{B}_{20}-\dot{B}_{25}