**5. Case Study**

The operating conditions in district 1 exhibit a daily pattern determined by the variation in user demand. Therefore, the *QT* and *HT* values depend on both the time and system configuration. The operating conditions in district 2 could be similar to those of district 1 if direct pumping in the network is provided. More frequently, the pumping station supplies water to an elevated tank, with fairly constant values of both *QP* and *HP*. As a case study, a simplified system is analyzed. The discharge approaching the PAT is imposed and the head losses along the pipe downstream of the pumping station are completely neglected. Two different supply conditions have been considered for the feeding of district 2:


Equation (6) has been solved in order to find the three unknowns (*HT*, *QP*, *N* for scenario 1 and *HT*, *HP*, *N* for scenario 2). The daily pattern of a measuring station of the urban water distribution system of Pompei (Campania—Italy) has been scaled to obtain the approaching discharge *QT*. Figure 7 shows the non-dimensional 4 days pattern of flow rate *cq*, i.e., the instantaneous discharge *QT*, scaled with the average discharge. For both supply condition, two cases have been considered:

(a) A high difference in ground elevation between district 1 and district 2 and large turbined discharge (b)Alowdifferenceingroundelevationbetween district1anddistrict2andsmallturbineddischarge

**Figure 7.** Discharge pattern.

The average turbined discharge, (*QT*), is 40 L/s in case *a* and 25 L/s in case *b*. In the scenarios 1*a* and 1*b* the value of *HP* is assigned and set to 50 m in case *a* and 15 m in case *b*. In the scenarios 2*a* and 2*b*, the pumped discharge is assigned and equal to 4.4 L/s in case *a* and 6.4 L/s in case *b*. The first case corresponds to a four stage pump, allowing a maximum flow rate in district 2 equal to about 0.3*QT* with the head being 4.25*HT* maximum. In the second case, a two stage pump is considered, allowing a maximum flow rate in district 2 equal to about 0.38*QT* with the head being 2.5*HT* maximum. Table 4 shows the configuration of the four different scenarios.



In Figure 8, the turbined head is shown for the four scenarios, as well as the pumped head for scenarios 1*a* and 1*b* and the pumped flow rate for scenarios 2*a* and 2*b*. The results show the performance of the P&P system in the assigned conditions specified in Table 4. Figure 8I,III,V,VII shows the values of the turbined head depending on time. The calculated values of *Ht* should be less than the available residual pressure while an additional valve could be placed to dissipate the excess head, if necessary. When the pumping head is fixed (scenarios 1*a* and 1*b*) the pumped discharge is variable, according to Equation (5), as shown by Figure 8II,IV. If the average values of *Qp* comply with the needs of district 2, the elevated tank could compensate for the variable discharge. Otherwise, the district should be provided with a second pumping system to supply the missing discharge. Figure 8VI,VIII shows the pumped head when the pumped discharge is fixed. Such a scenario can happen, for example, when a direct pumping is performed and the district is provided with an end-line tank to receive the excess discharge or feed the network during the high demand hours. In such a case, the pumped pressure should be contained within a certain range: a minimum pressure should be guaranteed by the pump to supply the end users, and a maximum value should not be exceeded to avoid any structural or leakage problems. When the average turbined discharge is higher (scenario 2*a*), Figure 8VI shows that the pumped head is always higher than 40 m and can also exceed 130 m in certain moments. In such a case, a dissipation valve could be placed to reduce the head when the pressure exceeds the upper limit. Instead, when the turbine discharge is lower (scenario 2*b*), the pumped head is always lower than 40 m and its minimum value is lower than 10 m. In this last case, if the minimum pressure is not guaranteed, a second ordinary pump can be installed to increase the pressure when necessary.

**Figure 8.** Behaviour of the P&P system for scenario 1*a* (**I**,**II**); 1*b* (**III**,**IV**); 2*a* (**V**,**VI**); and 2*b* (**VII**,**VIII**).

Table 5 shows the values of the hydraulic power involved in the turbining and in the pumping at the P&P station. Both the average and the maximum values demonstrate that the power values of case *a* are higher than case *b*. Scenario 1*a* corresponds to the highest power values while scenario 1*b* exhibits the lowest power. The ratio between the average pumped power and the average turbined power represents the average efficiency of the P&P station. It can be considered satisfactory in all the four cases, since it is quite close to the maximum values of Figure 6. The last column of Table 5 shows the annual energy saving range, considering an average efficiency of the pumping group ranging between 0.4 and 0.8. Such a saving is always relevant and it is obviously larger in scenarios 1*a* and 2*a*, where the pumped power is larger.


**Table 5.** Hydraulic power values of the different scenarios.
