Question 9.1: Inverse problem with an anomaly. Heating and DHW installatio......

The heating demand was obtained by TRNSYS v17, using Type 56 and 1 h intervals, considering a level of occupancy of the typical homes of the Basque Country. The demand for DHW was calculated using the Task 26 DHW [E1] program. In Fig. E.9.2 the profile of heating and DHW demand for 5 days of January are shown.

For the analysis, a total of 13 components and 24 flows were taken into account, as can be seen in Fig. E.9.3. Consider that two resources enter from the exterior: natural gas \left(\dot B_{20}\right) and the contribution due to the tank \left(\Delta\dot{B}_{21}\right), which is the difference between the initial and final exergy of the tank in the period under consideration. Both are represented by dotted arrows in Fig. E.9.3. The grated arrows indicate the final products generated by the installation, that is, heating \left(\dot B_{19}\right) and DHW \left(\dot B_{23}\right). The three circulation pumps were not considered in the analysis, due to their small power. If this were not so, it would be convenient to break down the physical exergy into its two thermal and mechanical components, which undoubtedly would make the analysis more precise, but we would considerably complicate the presentation of the example. Therefore, the calculations do not contain the effect of the pressure on the values of the physical exergy of the water flows.

Definition of the free condition.
For the reasons stated, the intervention of the control system must be cancelled to obtain the free condition. Fig. E.9.4 represents the process followed in TRNSYS v17. The REF index refers to the reference condition and FREE to the free condition (see FAULT incorporated in the lower part of the figure). It also shows the connections between components and the control system. For the operating conditions, we apply the same control strategy as in the reference conditions and in this way we define the free condition.
Effect of total production variation
There are two products from the installation: heating and DHW. The heating corresponds to the demand required by the users, so the heating produced for the free condition and the reference is the same \dot{Q}_{h e a t}=\dot{Q}_{h e a t}^{0}, heat, since both the environ-mental conditions and the internal comfort conditions for both states are the same and therefore \Delta P_{s, h e a t}=0. However, the DHW production corresponds to the water mass flow rate demanded (\dot m_{D H W}) at the specified temperature (T_{D H W}=55^{\circ}C). These conditions are obtained by means of the three-way valve V3, which mixes the cold water flow of the network (\dot{m}_{n w},T_{nw}) with the hot water from the storage tank (\dot{m}_{h w},T_{18}). Therefore, the DHW production is

where {\dot{H}}_{18} is the water flow enthalpy from the tank and {\dot{H}}_{24} that of the water flow from the network.
For obtaining the free condition, the control system forces valve V3 to act in the same way as in the reference condition and therefore {\dot{m}}_{hw}={\dot{m}}_{hw}^{0} and {\dot{m}}_{nw}={\dot{m}}_{nw}^{0}. However, since the hot water temperature at the tank outlet is a function of the tank temperature, that is, T_{18}=T_{18}(T_{21}) and due to the anomaly introduced in the installa-tion, the instantaneous tank temperature is different in the two conditions, that is, T_{18}\neq T_{18}^{0}. Therefore, since \dot{H}_{18}\neq H_{18}^{0}, there is a variation in the total DHW produc-tion and so \Delta P_{s,D H W}\neq0. In short, in this installation in which there are two products, only one of them can remain the same for the free and reference conditions, while the other varies.
Using TRSYS v17, the three conditions of the installation are simulated throughout the year: the real condition (with the anomaly introduced in the radiator system), the reference condition (without any anomaly) and the free condition (according to the comments made previously). The accumulated exergy values for each of the flows and the entire heating period are collected in Table E.9.1.

In order to demonstrate that the study is dynamic, Fig. E.9.5 shows the simulated exergy consumption of natural gas in the condensing boiler for 5 days in January. As shown, the blue dotted and red grated lines refer to the reference and free conditions (it is checked that the boiler is activated and deactivated in the same instants, see the horizontal axis); the full green line, on the other hand, corresponds to the operating condition (the boiler is activated when the control system intervenes).

Table E.9.2 shows the diagnosis results according to Eq. (9.32). The columns of the table show the final product variation \Delta P_{s}, the values of the malfunctions (M F_{0i}+M F_{i}), Eq. (9.33), where the malfunctions M F_{0i} correspond to the first term on the right of said equation and the dysfunctions (D F_{0i}+DF_{i}), Eq. (9.26), of each component. The sum of the last four columns corresponds to the irreversibility increase \Delta I, Eq. (9.34), that is, the part of the impact on fuel associated with the anomaly. The values of F_{T}^{0} and F_{T}^{fr e e} correspond to the total consumption of resources (fuel) in the reference state and in the free condition respectively. The difference between these values, as well as the sum of all the cells in Table E.9.2, shown in Eq. (9.32) matches the impact on fuel \Delta F_{T}^{a n o m}\,=3.010\,M J.

Analysing the results obtained, we can make the following comments:

• \Delta P_{s} in the RS component is zero, and there is a negative value in V3 for the total product variation [\Delta P_{s_{12}}\,=\,-105\:M\!J]. This result confirms what was previously affirmed, that is, that there is no variation in the heating production, while there is a reduction in the DHW production due to the presence of the anomaly in the free condition.
• If we refer to the MF malfunction vector, we observe that the malfunction affects mainly the RS component in which the anomaly exists [M F_{10}=382\,M J], but the malfunction is also significant in the boiler [M F_{1}=147\,M J]. For a clearer interpretation of this result, MF must be broken down into the malfunctions due to exergy consumption variation M F_{i} and to external resources variation, M F_{0,i}, Eq. (9.33). Thus, the boiler CB is affected mainly by the variation in consumption of external resources [M F_{0,1}=147\,M J].
  • The components located upstream of the anomaly (H C\ [M F_{2}=-12\,M J],\,M2[M F_{8}=3{9}\ M J],M3\,[M F_{9}=59\stackrel{}{M J}]) also exhibit malfunctions. In all cases, these are induced malfunctions.
  • The component most affected by the malfunctions of the other components is the first component in the energy transformation chain, that is, the boiler CB with [{ D}F_{1}=2390\,M J]. There are components with negative dysfunction values, such as \mathrm{M}3\;[D F_{5}=-9\;M J] or V3 [D F_{12}=-77\;M J] which are related to the decrease in the DHW production.

In order to delve into the origin of the dysfunctions, Table E.9.3 shows the dysfunc-tion matrix, as defined in Section 9.5.1.

With respect to this table, we can make the following comments:

•  The sum of each line \sum_{j}D F_{i j} corresponds to the total dysfunction generated in the i-th component which matches the corresponding element in Table E.9.2. For example, the sum of the first line (\left[{\sum_{j}{D F_{1j}}}=3640\,M J\right] are the dysfunctions that appear in CB) and is equal to the corresponding element of Table E.9.2 [{ D}F_{1}=3640\,M J].
• Using this table the effects induced in the i-th component caused by a malfunction in the j-th component can be visualized. For example, [{ D}F_{1,8}=204\,M J].corresponds to the CB dysfunction arising from M2.

Using Eq. (9.43) and Eq. (9.44), the malfunctions costs are obtained, see Table E.9.4. The last two columns show the values of the indices whose objective is the identification of the importance of the malfunction cost in relation to the impact on fuel, Eq. (9.45).

According to the values obtained, some comments can be made.

• The malfunction cost of the components that do not have malfunctions and which do not induce dysfunctions in other components is zero.
•  Some malfunction costs are negative \left[M{F}_{2}^{*}=-73\ M J\right] and \left[M{F}_{3}^{*}=-396\ M J\right]. It is because the dysfunctions induced by these components in the rest are negative, which means that they cause a decrease in local production in other components, compared with the refer-ence condition.
• The component with the largest index \psi_{MF^*_{i}} is RS \left[\psi_{MF^*_{10}}^{\phantom{l}}\,=89\%\right]. This result implies that 89% of the extra fuel consumption is due to the anomaly of the radiator system, which in turn causes an increase in consumption in other components [5% in CB, 8% in M2, 14% in M3 and 49% in V3] and a decrease in others [-2% in HC, -13% in D1], due to induced effects.
• Likewise, V3 involves a consumption reduction of 50 \left[\psi_{MF^*_{0,12}}^{\phantom{l}}\,=-50\%\right]. simply because a fraction of the additional consumption of fuel gives rise to a decrease in production in the free condition.

The results of Table E.9.2 and Table E.9.4 are represented graphically in Fig. E.9.6 and Fig. E.9.7, respectively. Fig E.9.6 shows the malfunctions, dysfunctions and the total product variation between the free and reference conditions. It can be seen that the boiler CB is the component with the greatest dysfunction. Fig E.9.7 shows the impact on fuel depending on the malfunctions costs and shows that RS is mainly responsible for the increase in fuel.

Finally, Table E.9.5 shows the results obtained for the exergoeconomic costs associated with the malfunctions, through Eqs. (9. 65) and (9. 66). With respect to this table, we can make the following comments:
• As might be expected, the RS component that contains the intrinsic anomaly is the one
that has the largest exergoeconomic cost associated with the cost of its malfunction [C_{M F_{11}}\,=\,147\,{€}].

• During the heating period, the economic  cost due to the presence of the anomaly goes up to [\Delta C_{F}\,=\,164\,{€}]. This cost is the sum of the contribution of each component, accord-ing to the index \psi_{MF^*_{}} , that is, €8 due to the induced malfunction in CB, -€22 in D1, €23 in M3, etc.