**5. Procedure for Predicting the Performance of a Generic Pump**

This paragraph describes the global procedure for calculating the performance of a generic PAT, which uses both the pump/turbine fluid dynamic model and the geometric model. In the first phase, the geometry of the machine is built, as described in Section 3, starting from the data *Q*, *H*, *Hmo*, *h*2, *d*2, and *n* acquired from the catalog. The fluid dynamic model is then applied to calculate the performance of the machine in pump operation, and comparisons are made with the curves available in the catalog. At this point, two situations could occur: the first is that the two curves are very close, and the second is that the two curves do not coincide. In the second case, it is necessary to act on the geometric parameters calculated by the model, starting from those to which the model itself is most sensitive. The manual insertion of these parameters into the model which calculates the geometry (Section 3) is possible. When, after several attempts, the curves concur, the model can be applied in turbine operation. The procedure is shown in Figure 13.

To better illustrate what has just been described, an example carried out on the Caprari pump P65-250 is reported. This pump was supplied by the DIMEG Department of the University of Calabria, together with the five pumps previously described. The geometric model, first of all, calculates its geometry. Then, using these parameters, the fluid dynamic model derives the head and efficiency curve as a function of the flow rate, in pump operation, which will then be compared with the pump curve provided in the catalog (see Figure 14A).

**Figure 13.** Global procedure for determining the performance curves.

**Figure 14.** *Cont*.

**Figure 14.** (**A**) Head and efficiency for the PAT 65-250 (ns = 22.75) in pump operation. (**B**) Head and efficiency for the PAT 65-250 (ns = 22.75) in pump operation with z = 6, b = 0.038 m, b5 = 0.054 m, and cl = 3.5 <sup>×</sup> <sup>10</sup>−<sup>4</sup> m.

As can be seen from Figure 14A, the real head obtained from the model was too low if compared with the head reported in the catalog, as it is positioned below this curve. The geometric model provided a value of the number of blades equal to 5. However, having the real value of this parameter (*z* = 6) available, it was possible to replace it. The change of this parameter has modified the losses linked to the slip phenomenon, calculated using the Stodola formula [37,38], which depends on the value of the number of blades. To improve the characteristic curve, it was decided to act on the value of the width of the exit section of the volute (*b*), reducing it, to then obtain, using the geometric model, the corresponding value of the width of the final diffuser (*b*5). By decreasing the value of this parameter, the fluid passage section is reduced (*A*4), causing an increase in speed inside the volute (*c*4). The b-value initially obtained by the model was 0.048 m, and *b*<sup>5</sup> was equal to 0.064 m. Finally, as can be seen from the total efficiency curve calculated by the model, the efficiency is too high for low flow rates. This is linked to the assumptions imposed a priori and to having initially considered the volumetric efficiency, for any flow rate value, equal to its value at the BEP, set equal to 0.95. However, it was possible to improve the results by acting on the amplitude value of the seal, *cl*, through which the liquid leaks occur, which is therefore linked to the volumetric losses. By increasing the value of this parameter, the total efficiency curve is considerably lowered. After applying the changes illustrated above, the curves changed as shown in Figure 14B.

At this point, having obtained the conformity of the results for pump operation, it was possible to observe the behavior of the machine in turbine operation. Figure 15A,B shows the curves obtained from the model before and after the modifications to the geometric parameters analyzed. These curves, obtained from the fluid dynamic model relating to turbine operation, are compared, for simplification purposes, with the data obtained experimentally on the PAT. However, this method works even in the absence of experimental data.

**Figure 15.** (**left**) Head and efficiency for the PAT 65-250 (nst = 15.83) in turbine operation. (**right**) Head and efficiency for the PAT 65-250 with z = 6 in b = 0.038, b5 = 0.054 m, and cl = 3.5 <sup>×</sup> <sup>10</sup>−<sup>4</sup> m.

## **6. Conclusions**

The work carried out is part of a theoretical–experimental research context on centrifugal pumps used as turbines (PATs). Their convenience lies mainly in the lower costs incurred compared to a turbine with the same power and in the wide range of models available on the market. However, this advantage cannot be grasped if it is not possible to know the actual behavior of the machine when it is used as a turbine once a specific need has been recognized. The objective of this research is the development of a prediction model capable of obtaining the head–flow rate and efficiency–flow rate curves of the PAT, both in pump and turbine operation. The effort was the development of a series of fluid dynamic models that involve the pressure drops in the various components of the machine, as well as the slip phenomena at the inlet and outlet of the impeller. Furthermore, a geometric model was created for the reconstruction of the geometry of the machine, based on good design techniques, statistical data, and maps of good functioning, available in the literature. This analysis was necessary as the geometry is not provided by the manufacturer's catalog. These models were calibrated based on measurements made on the DIMEG hydraulic test bench on a sample of six machines, tested in both operating modes, whose geometric parameters were measured. The machines measured were six centrifugal pumps, namely five Ksb pumps (P40-335, P80-220, P40-250, P50-160, P100-200) and a Caprari pump (P80-160), which have a specific speed range from 9.05 to 43.48. For these pumps, the head flow rate and flow rate efficiency curves have been obtained in both operating modes. Generally, these curves, if compared with those obtained experimentally in the DIMEG test bench, show a good reliability, because they fall into error bands equal to +/−5%. To the right of the BEP, the efficiency curves are flat, which represents an advantage for those who use this technology, as the machine maintains good performance over a wide range of flow rates. Subsequently, a procedure was set up which envisages the use of the previously mentioned models, useful for calculating the performance curves of the machine both in pump and turbine operation, as well as for the reconstruction of the geometry. In the first phase, based on a few data available from the manufacturer's catalog, the model reconstructs the prototype geometry of the machine and calculates the performance curves of the machine in pump operation. If these curves match those present in the catalog, the geometry calculated by the geometric model is correct; otherwise, it is necessary to change some geometric parameters so that the predicted curves and those in the catalog coincide. For this purpose, a sensitivity analysis comes to the aid of the user, the purpose of which is to identify the parameters to which the model is most sensitive. The sensitivity analysis showed that the geometric parameters to which the model is most sensitive, in turbine operation, are the diameter of the hub (*d1m*), the height of the inlet blades (*b*1), the suction diameter (*d*0), and the dimensions of the volute (*hv*, *b*). Finally, once the appropriate parameters have been changed and the compliance of the performance curves with the catalog data has been obtained, it is possible to obtain the curves in turbine operation, which will certainly be reliable, given the adherence of the curves in pump operation. In conclusion, the article presented a flexible and interactive forecasting tool, with 95% reliability, which allows choosing the most suitable PAT model to exploit the available water resources. A simple model of general application was presented, useful for those who decide to rely on PAT technology.

**Author Contributions:** Conceptualization, S.B.; Data curation, S.B. and V.P.; Formal analysis, V.P. and M.A.; Investigation, S.B. and V.P.; Methodology, S.B.; Project administration, M.A.; Software, V.P.; Validation, V.P.; Writing–original draft, V.P.; Writing–review & editing, S.B. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Nomenclature**


