*2.1. Machine Behaviour*

Experimental tests can provide the performance curves of a pump or a PAT at a certain rotational speed value or, alternatively, CFD techniques can be employed. Such curves express the relations of head ( *H*), mechanical power ( *P*) and efficiency (*η*) with the discharge ( *Q*), and can be expressed as follows:

$$\begin{aligned} H &= H(Q) \\ P &= P(Q) \\ \eta &= \eta(Q) \end{aligned} \tag{1}$$

Usually, a quadratic polynomial can be successfully used to fit the head data, while a cubic polynomial best explains the power data. The efficiency curve presents a maximum value *ηB* for the *QB* discharge. The working point ( *QB*, *HB*, *PB*, *ηB*) is called the best efficiency point (BEP). Affinity laws can be used to calculate the performance of two similar machines, having *D<sup>I</sup>* and *D<sup>I</sup> I* diameters, respectively, and rotating at different speeds. The performances of two homologous working points, (*Q<sup>I</sup>* , *H<sup>I</sup>* , *P<sup>I</sup>*) and ( *QI I*, *H<sup>I</sup> I*, *P<sup>I</sup> I*), respectively, are related to both the ratios of the respective speeds, *N<sup>I</sup>* and *N<sup>I</sup> I*, and the scale ratios, *D<sup>I</sup>* /*D<sup>I</sup> I*, as follows:

$$\begin{aligned} \frac{Q^I}{Q^{II}} &= \left(\frac{N^I}{N^{II}}\right)^1 \left(\frac{D^I}{D^{II}}\right)^3\\ \frac{H^I}{H^{II}} &= \left(\frac{N^I}{N^{II}}\right)^2 \left(\frac{D^I}{D^{II}}\right)^2\\ \frac{P^I}{P^{II}} &= \left(\frac{N^I}{N^{II}}\right)^3 \left(\frac{D^I}{D^{II}}\right)^5 \end{aligned} \tag{2}$$

The Equation (2) is generally used by pump manufacturers to predict the behaviour of their machines for different rotational speeds, even if some studies demonstrate that Equation (2) does not predict the real behaviour of turbomachines with a high accuracy in the whole range of rotational speeds [25,26]. Principally, according to affinity laws, the efficiency value of two homologous points is constant with speed. However, this result is not in agreemen<sup>t</sup> with the real behaviour of pumps. Simpson and Marchi [27] showed that the efficiency value at the BEP is attained only at a given optimal *Nmax* speed value, while a decrease is observed as the speed diverges. Furthermore, a recent experimental study [28] on the behaviour of PATs has demonstrated that the use of affinity laws could lead to significant errors in the prediction of machine performance, up to 30% in the prediction of head and 100% in the prediction of efficiency. The authors proposed a new semi-theoretical model for semi axial machines for a better prediction of characteristic PAT performance curves. *Nmax* was identified experimentally and related to geometrical pump parameters. Then, the BEP at a *N<sup>I</sup> I* rotational speed could be related to the BEP at *Nmax* by a relaxation of the affinity equations (RAE):

$$\begin{aligned} \frac{Q\_B^{II}}{Q\_B^{max}} &= f\_1 \frac{N^{II}}{N^{max}}\\ \frac{H\_B^{II}}{H\_B^{max}} &= f\_2 \frac{N^{II}}{N^{max}}\\ \frac{P\_B^{II}}{P\_B^{max}} &= f\_3 \frac{N^{II}}{N^{max}} \end{aligned} \tag{3}$$

By experimental estimation of the functions *f*1, *f*2 and *f*3 [28], a better agreemen<sup>t</sup> was found between the theoretical and experimental characteristic curves with a significant reduction in the prediction error.

### *2.2. Energy Recovery in Water Systems*

Hydropower production in a water supply system is a viable option where the available hydraulic power is fairly large and constant, with a grea<sup>t</sup> economic benefit and a small environmental impact. Examples of such scenarios are found along water transmission lines, from the water source to the distribution centre, where several small hydropower projects are already active [29]. Hydropower plants could also be used in water distribution networks, as a replacement of a PRV. Nevertheless, such a plant would face a large variability of both discharge and head, due to the hourly variability of water users' demand, as well as the small amounts of available power [30,31]. Furthermore, the turbine head curve exhibits a behaviour which is completely different from the head drop curve of a regulating valve: the former increases with the discharge, while the latter decreases as the flow rate increases [32]. All these difficulties, together with the small amount of power available, hinder the use of classic turbines, due to their high price and their long payback period. Some studies demonstrate that the use of PATs, instead of classic turbines, to substitute the dissipation valves could be a convenient practice even if the available power is low, due to the low price and the mechanical simplicity of such devices [33]. Several studies have shown that the best efficiency of a PAT could be greater than 70% [21–23]. Ramos et al. [34] showed that the cost of a PAT is much smaller if compared with the cost of a classic turbine, with unit costs ranging between 200 e/kW and 400 e/kW, where the turbine cost ranges between 300 e/kW and

800 e/kW. Carravetta et al. [35] showed that the payback period of a PAT energy recovery system could be really short, ranging between 6 months and 3 years, while Fecarotta et al. [36] showed that the coupling of the pressure control strategy with an energy recovery strategy within a water network could be convenient if the valves are replaced by PATs, with high 10 year net present values. Other studies propose different solutions for the regulation of the PAT plant to address the large variability of hydraulic characteristics and to match the turbine head and discharge with the needs of the network. Carravetta et al. [37] propose regulating the PAT either by a hydraulic or electric regulation system. In the former case, two hydraulic valves are placed in a series–parallel circuit to dissipate the excess head when the turbine head is too low or bypasses a part of the discharge when the flow rate is high. In the latter case, an electronic speed driver adjusts the rotational speed of the PAT to modify its performance. In 2014 , Carravetta et al. [38] compared the effectiveness of the different installation schemes of a PAT plant. Other preliminary studies have included an analysis of the optimal location of a PAT within a WDN [36,39,40] to maximize the power production.

### **3. Pump and PAT Characteristic Curves**

Experiments were performed in the Hydro Energy Laboratory (HELab) of the CESMA—University of Naples Federico II and in the qualification laboratory of Caprari Pumps Ltd. (Peterborough, UK). Pictures of the two laboratories are shown in Figure 1.

**Figure 1.** Pictures of the qualification laboratory of Caprari ltd (**a**); and of HELab of University of Naples (**b**).

Both laboratories are equipped with digital magnetic flow meters, piezometric pressure transducers and digital power meters. A fixed level water tank is located below ground level to feed the hydraulic circuit and receive the circulated water. A gate valve is used to regulate the flow. The testing conditions in both laboratories comply with ISO 9906 (Rotodynamic pumps—Hydraulic performance acceptance test) level 1, IEC 60034-2-1 (Rotating electrical machines) and EN 50598-2 (Efficiency classes of converters and drive systems). The maximum measurement uncertainties are reported in Table 1.


A two-stage pump unit, the Caprari model HMU, has been tested in the HELab. The specifications of the pump are reported in Table 2.


**Table 2.** Specification of the Caprari HMU pump, as tested in HELab.

The working conditions for different rotation speeds have been tested and the main hydraulic and electric parameters have been measured. Head and power curves have been determined experimentally. Test results in the normalized parameter are shown in Figure 2.

**Figure 2.** Experimental normalized head (**a**), power (**b**) and efficiency (**c**) of the HMU pump and regression curves.

Another pump, the Caprari model NC80, has been tested in inverse mode in the Caprari qualification laboratory. The head and efficiency as determined experimentally are shown in the normalized plot of Figure 3, while the specifications are reported in Table 3.

**Figure 3.** Experimental normalized head (**a**), power (**b**) and efficiency (**c**) of NC80 PAT and regression curves.

**Table 3.** Specification of Caprari NC80 PAT, as tested in Caprari laboratory.


### **4. PAT and Pump System (P&P) Modelling**

In the design of a pumping system or a hydro power plant, the rotational speed of the device is imposed by the grid frequency or by a variable frequency driver. Instead, in a P&P, the group is free to achieve any rotational speed. The PAT provides the power for the pump and the rotational speed is set by the combination of the performance curves of the two devices with the network characteristics.

*Water* **2017**, *9*, 62

Figure 4 shows a simplified scheme of a P&P plant. The whole water supply system can be considered as two separated network districts which are connected by the P&P plant. The residual head at the end of district 1 that can be turbined within the PAT is represented by the difference *HT* = *Hu* 1 − *H<sup>d</sup>* 1 − Δ *Hr* 1, where *Hu* 1 is the head measured at the end point of district 1, Δ *Hr* 1 is the head loss in the pipeline approaching the PAT and *H<sup>d</sup>* 1 is the head downstream of the PAT. The lengths of the pipelines are not representative of the real system and, if the values of *Hu* 1 and *H<sup>d</sup>* 2 are measured near the P&P system, the head losses Δ *Hr* 1 and Δ *Hr* 2 can be neglected. As a general case, the two values *H<sup>d</sup>* 1 and *Hu* 2 are considered different, but the outlet tank of the PAT and the inlet tank of the pump are often the same. The presence of four tanks, however, could even be unnecessary if the P&P system is inserted in a fully pressurized network, and the values of the four variables, *Hu* 1 , *H<sup>d</sup>* 1 , *Hu* 2 and *H<sup>d</sup>* 2 depend on time according to the network behaviour. *QT* is the flow rate available at the PAT inlet. The power produced by the PAT is *γ QTHTη<sup>T</sup>*, *ηT* being the PAT efficiency. Such power is transmitted to the pump by the shaft which connects the two machines. *QP* is the pumped flow rate while the total head required at the pump outlet is *HP* = *H<sup>d</sup>* 2 − *Hu* 2 + Δ *Hr* 2, where the meaning of the symbols is evident. The efficiency of the plant can be calculated as the ratio between the output hydraulic power at pump side and the input hydraulic power at PAT side. Thus, the plant efficiency *η* can be calculated as:

$$
\eta = \eta\_T \cdot \eta\_P \tag{4}
$$

*ηP* being the efficiency of the pump. If such a simplified scheme is used to model the water system, the design problem is defined by the following equations:

$$\begin{cases} H\_{T} = H\_{1}^{\mu} - H\_{1}^{\mu} - \Delta H\_{1}^{\mu} \\ H\_{P} = H\_{2}^{d} - H\_{2}^{u} + \Delta H\_{2}^{u} \\ \frac{H\_{T}}{N^{2}} = \left( a\_{T} \frac{Q\_{T}^{\gamma}}{N} + b\_{T} \frac{Q\_{T}}{N} + c\_{T} \right) n\_{t} \\ \frac{P\_{T}}{N^{3}} = \left( a\_{T} \frac{Q\_{T}^{\gamma}}{N} + \beta\_{T} \frac{Q\_{T}^{\gamma}}{N} + \gamma\_{T} \frac{Q\_{T}}{N} + \delta\_{T} \right) n\_{t} \\ \frac{H\_{P}}{N^{2}} = \left( a\_{P} \frac{Q\_{P}^{\gamma}}{N} + b\_{P} \frac{Q\_{P}}{N} + c\_{P} \right) n\_{P} \\ \frac{P\_{P}}{N^{3}} = \left( a\_{P} \frac{Q\_{P}^{\gamma}}{N} + \beta\_{P} \frac{Q\_{P}^{\gamma}}{N} + \gamma\_{P} \frac{Q\_{P}}{N} + \delta\_{P} \right) n\_{P} \\ P\_{T} = P\_{P} \end{cases} (5)$$

in the seven variables *HT*, *HP*, *QT*, *QP*, *PT*, *PP* and *N*, *PT* and *PP* being the mechanical power of the PAT and pump respectively, *nT* and *nP* the number of stages of the PAT and pump respectively and *aT*, *bT*, *cT*, *αT*, *β<sup>T</sup>*, *γT*, *δT*, *aP*, *bP*, *cP*, *αP*, *β<sup>P</sup>*, *γP*, and *δP* the experimental regression coefficients of the head curve and the power curve of the PAT and pump, respectively. A P&P node in the water system is described by the last five equations of Equation (5). The energy equations at the PAT intake and pump outlet balance the number of unknowns and equations. More generally, along the pipeline of the two districts, derivations, dissipation points, or additional supply points could be present. In this case, the design of the P&P plant is linked to the operating conditions of the two networks and the complexity of the design solution is connected to the complexity of the water system.

**Figure 4.** Simplified scheme of the P&P plant.

### *Behaviour of the P&P Plant*

The last five equations of Equation (5) can be rewritten if *qT* = *QT*/*N*, *qP* = *QP*/*N*, *hT* = *HT*/*N*<sup>2</sup> and *hP* = *HP*/*N*2:

$$\begin{cases} h\_T = (a\_T q\_T^2 + b\_T q\_T + c\_T) n\_T \\ h\_P = (a\_P q\_p^2 + b\_P q\_p + c\_P) n\_P \\ (a\_T q\_T^3 + \beta\_T q\_T^2 + \gamma\_T q\_T + \delta\_T) n\_T = (a\_P q\_P^3 + \beta\_P q\_P^2 + \gamma\_P q\_P + \delta\_P) n\_P \end{cases} \tag{6}$$

Equation (6) expresses the relationship between the four quantities *qT*, *qP*, *hT* and *hP*. A single curve relating *qP*/*qT* and *hP*/*hT* can be obtained if the number of stages of the pump and PAT are assigned. Because *qP*/*qT* = *QP*/*QT* and *hP*/*hT* = *HP*/*HT*, Equation (6) shows that the relationship between the ratio of the delivered discharge and the ratio of the delivered head is independent of the rotational speed. In Figure 5, the values of *HP*/*HT* are plotted versus *QP*/*QT* for the 1 to 5 pump stages with *nT* = 1. In Figure 6, the plant efficiency *η* is plotted versus *QP*/*QT*. The P&P working conditions are different depending on the number of pump stages. In the presence of a single stage, the pump head is slightly greater than the available head drop at the PAT and the flow rate ratio ranges between 0 and 0.38. With two pump stages, the pump head significantly increases (about 2.5 times the head drop in the flow range 0–0.15) and also the range of flow rate ratio is larger (up to 0.36). With an increasing number of pump stages, the range of flow rate ratio decreases (up to 0.3 with five stages), while the head ratio increases (up to 4.8 with five stages). The efficiency of the P&P system also depends on the number of stages. The lowest efficiency occurs for a single stage pump, when the P&P efficiency reaches its maximum value out of the range of the flow rate ratio. The efficiency significantly increases (from less than 0.35 up to more than 0.45) for a higher number of pump stages. Considering a P&P efficiency larger than 0.4, the range of working conditions that can be obtained with the selected pump is quite variable: up to 30% of the water coming from district 1 can be pumped into district 2 with a 50% increase in the pressure head, using a double stage pump, or, at the opposite extreme, with a four stage pump, 15% of the water coming from district 1 can be pumped into district 2 with a pressure head being three times larger than the residual head of district 1. The best efficiency occurs for a three stage pump where 20% of the incoming discharge can be pumped with a 220% increase in head.

**Figure 5.** Head ratio of the P&P plant for different numbers of pump stages with *nt* = 1.

**Figure 6.** Efficiency of the P&P plant for different pump stages with *nt* = 1.
