Question 7.2: An installation for heating and DHW consists of an alternati......

(a) This is an installation in which we have selected n = 10 components and m = 28 flows. The incidence matrix is represented in Table E.7.3.

(b) Multiplying the incidence matrix by the column vector M_{\mathrm{(25,1)}} gives the mass balances in each of the 10 components for the time-step considered.

• {\bf S})\,{0}=\,-\dot{ m}_{1}\,+\dot{m}_{2}
• \\ {\mathrm{CB)}}\,\mathrm{0}=\,-\dot m_{3}\,+\dot m_{4}
• \\ {\mathbf C})\;0={\dot{m}}_{1}\;-{\dot{m}}_{2}\,+{\dot{m}}_{3}\,-{\dot{m}}_{4}\,-{\dot{m}}_{5}\,+{\dot{m}}_{6}
• \\ {\mathbf{HC}})\;0={\dot{m}}_{5}\;-{\dot{m}}_{6}\,-{\dot{m}}_{7}\,+{\dot{m}}_{8}
• \\ {V1})\;0={\dot{m}}_{7}\;-{\dot{m}}_{8}\,-{\dot{m}}_{9}\,+{\dot{m}}_{10}\,-{\dot{m}}_{17}\,+{\dot{m}}_{18}
• \\ {\bf{V2}})\;0={\dot{m}}_{9}\;-{\dot{m}}_{10}\,-{\dot{m}}_{11}\,+{\dot{m}}_{12}
• \\ {\bf{HX}})\;0={\dot{m}}_{11}\;-{\dot{m}}_{12}\,-{\dot{m}}_{13}\,+{\dot{m}}_{14}
• \\ {\bf{T}})\;0={\dot{m}}_{13}\;-{\dot{m}}_{14}\,-{\dot{m}}_{15}\,+{\dot{m}}_{16}
• \\ {\bf{V3}})\;0={\dot{m}}_{17}\;-{\dot{m}}_{18}\,-{\dot{m}}_{19}\,+{\dot{m}}_{20}
• \\ {\bf{D}})\;0={\dot{m}}_{19}\;-{\dot{m}}_{20}

(c) Multiplying the incidence matrix by the column vector H_{(25.1)} gives the energy balances in each component. Flows 21 and 27 are heat transfer flows ({}{\dot{Q}}_{21},{\dot{Q}}_{27}), flow 23 is electrical energy ({\dot{E}}_{23}) exchanged during the time-step, flows 22 and 24 are external resources (\dot{H}_{22}={\dot{m}}_{22}HH{V}_{N G}\Delta t,\;\;\dot{H}_{24}={\dot{m}}_{24}HH{V}_{N G}\Delta{\bf t}) being the others mass flows \left(\dot{H}_{i}\,=\dot{m}_{i}h_{i}\Delta t\right).. We call flows 25 and 26 to the energy (exergy) variation in the inertia tank T and the hydraulic compensator HC respectively, during the time-step \Delta t considered.

• \\ {\bf S})\,0\,=\,-\dot H_{1}\,+\dot{H}_{2}\,+\dot{H}_{22}\,-\,\dot{E}_{23}\,-\,\dot{Q}_{27}
• \\ {\rm CB})\,0\,=\,-\dot H_{3}\,+\dot{H}_{4}\,+\dot{H}_{24}\,-\,\dot{H}_{28}
• \\ {\bf C})\,0=\dot{{H}}_{1}\,-\dot{{H}}_{2}\,+\dot{{H}}_{3}\,-\dot{{H}}_{4}\,-\dot{{H}}_{5}\,+\dot{H}_{6}\,
• \\ {\bf {HC}})\,0=\dot{{H}}_{5}\,-\dot{{H}}_{6}\,-\dot{{H}}_{7}\,+\dot{{H}}_{8}\,+\Delta U_{25}
• \\ {V1})\,0=\dot{{H}}_{7}\,-\dot{{H}}_{8}\,-\dot{{H}}_{9}\,+\dot{{H}}_{10}\,-\dot{{H}}_{17}\,+\dot{H}_{18}\,
• \\ {\bf V2})\,0=\dot{{H}}_{9}\,-\dot{{H}}_{10}\,-\dot{{H}}_{11}\,+\dot{{H}}_{12}
• \\ {\bf HX})\,0=\dot{{H}}_{11}\,-\dot{{H}}_{12}\,-\dot{{H}}_{13}\,+\dot{{H}}_{14}
• \\ {\bf {T}})\,0=\dot{{H}}_{13}\,-\dot{{H}}_{14}\,-\dot{{H}}_{15}\,+\dot{{H}}_{16}\,+\Delta U_{26}
• \\ {\bf V3})\,0=\dot{{H}}_{17}\,-\dot{{H}}_{18}\,-\dot{{H}}_{19}\,+\dot{{H}}_{20}
• \\ {\bf D})\,0=\dot{{H}}_{19}\,-\dot{{H}}_{20}\,-\dot{{Q}}_{21}

(d) Multiplying the incidence matrix by the column vector ^{}B_{(25,1)} and applying Eq. (7.7) gives the exergy destruction in each component. We know that for heat exchanged (\dot{B}_{Q,21,27} = {\dot Q}_{21,27}\cdot f_{h,21,27}), for mass flows (\dot{B}_{i}=\dot m_{i}b_{i}\Delta t) and for the external resources \left(\dot{B}_{22,24}=\dot{B}_{N G,22,24}^{c h}\Delta t\right), so we have

• \\ \mathbf{S})\;{\dot{D}}_{S}=\;-{\dot{B}}_{1}\,+{\dot{B}}_{2}\,+{\dot{B}}_{22}\,-{\dot{E}}_{23}\,-{\dot{B}}_{Q.27}
• \\ \mathbf{CC})\,\dot{D}_{C C}\,=\,-\dot{B}_{3}\,+\dot{B}_{4}\,+\dot{B}_{24}\,-\dot{B}_{28}
• \\ {\bf C})\;\dot{D}_{C}=\dot{B}_{1}\;-\dot{B}_{2}\,+\dot{B}_{3}\,-\dot{B}_{4}\,-\dot{B}_{5}\,+\dot{B}_{6}
• \\ \mathrm{CH})\;\dot{D}_{\mathrm{CH}}=\dot{B}_{5}\,-\dot{B}_{6}\,-\dot{B}_{7}\,+\dot{B}_{8}\,+\Delta A_{25}
• \\ {\bf V}{\bf1})\,\dot{D}_{V1}=\dot{B}_{7}\,-\dot{B}_{8}\,-\dot{B}_{9}\,+\dot{B}_{10}\,-\dot{B}_{17}\,+\dot{B}_{18}
• \\ \mathrm{V}2)\;\dot{D}_{\mathrm{V2}}\;=\dot{B}_{9}\;-\dot{B}_{10}\;-\dot{B}_{11}\,+\dot{B}_{12}\;
• \\ {\bf HX})\;\dot{D}_{\mathrm{HX}}=\dot{B}_{11}\,-\dot{B}_{12}\,-\dot{B}_{13}\,+\dot{B}_{14}
• \\ \mathrm{T})\,\dot D_{\mathrm{T}}=\dot B_{13}\,-\dot B_{14}\,-\dot B_{15}\,+\dot B_{16}\,+\Delta A_{26}
• \\ \mathrm{V3})\,\dot D_{\mathrm{V3}}=\dot B_{17}\,-\dot B_{18}\,-\dot B_{19}\,+\dot B_{20}
• \\ \mathrm{D})\,\dot D_{\mathrm{D}}=\dot B_{19}\,-\dot B_{20}\,-\dot B_{Q,21}

(e) At the maximum level of aggregation, the installation is a black box that exchanges re-sources and products with the exterior, see the scheme of Fig.7.4.

The components of the incidence matrix a_{(1,10)} with values different to 0 are shown in Table E.7.4.

The corresponding mass (for water), energy and exergy balances are reflected by the following equations