**1. Introduction**

Over the years, world energy consumption has drastically increased, with different trends in various countries based on the degree of wealth and development and the availability of raw materials and resources. Consequently, as regards the water sector, the idea of using inverted flow pumps, also known as PAT [1,2], has begun to make its way into the market. The first researchers who realized the actual potential in exploiting a pump used as a turbine were Thoma and Kittredge [3]. They began experimenting in laboratories on this technology around the 1930s. There are also traces of the use of this technology in the 1970s; however, it failed to play an important role as energy was cheap and there was still no sensitivity towards recycling and saving. It was therefore more convenient to buy energy directly from the grid, and few tended to invest in new plants to produce a small number of kilowatt-hours. However, in the 1980s, some factors prompted a re-evaluation of the use of this technology, favoring its development and its establishment on the market, including the following:

**Citation:** Barbarelli, S.; Pisano, V.; Amelio, M. Development of a Predicting Model for Calculating the Geometry and the Characteristic Curves of Pumps Running as Turbines in Both Operating Modes. *Energies* **2022**, *15*, 2669. https:// doi.org/10.3390/en15072669

Academic Editors: Luis Hernández-Callejo, Sergio Nesmachnow and Sara Gallardo Saavedra

Received: 28 February 2022 Accepted: 31 March 2022 Published: 6 April 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).


PATs are mainly used as pressure-reducing valves (PRVs) [4,5] or in the pico/microhydroelectric sector [6]. Regarding the first application, the typical examples are aqueducts: water distribution systems that must always be under pressure. The pressure must remain within certain values: a minimum value, necessary to reach the highest altitudes, but at the same time a maximum value that must not be exceeded, as losses increase and cause problems of operability. It is therefore necessary to adjust the pressure, as the excess energy would be lost, for this reason, it is preferable to use an inverted flow pump. The pico-hydroelectric sector, on the other hand, is mainly linked to self-production. Even small water sources, commonly neglected for economic reasons, are therefore exploited. The aim is to satisfy one's energy needs, without selling the energy or feeding it into the power grid. It is therefore a question of withdrawing as little as possible from the watercourse, which is then released, reducing the impact on the resource. PAT can also be used by harnessing tidal energy [7], i.e., harvesting its height range in natural bays and estuaries or in artificial barrages, or extracting the kinetic energy from the tidal currents across natural and artificial channels [8]. The main feature of PAT pumps is represented by their reversibility, as it is possible, by reversing the direction of the fluid path inside the machine, to produce energy. They are generally marketed as monobloc electric pumps [9,10] in which the motor is of the asynchronous type. It can play two opposite roles: motor in direct operation and electric generator in reverse operation.

The main advantages of the use of PAT technology are represented by the following:


On the other hand, since pumps are not designed to operate as a turbine, they have the following disadvantages:


PATs can therefore be used as a replacement for traditionally employed turbines (Francis, Kaplan, Pelton) and turbines discussed in other articles [11,12]. However, it is necessary to make sure that the PAT operation is well adapted to the characteristics of the system where the PAT will be installed. It is necessary to establish, knowing the characteristic curves of operation of a pump, the characteristic curves of the machine that operates as a turbine. The aim of the present paper is to develop a combined procedure for assessing both the geometry (which is not given by manufacturers) and the fluid dynamic performances of a generic PAT. For this purpose, many fluid dynamics models calculating losses, head, and efficiencies by changing the flow rate of the machine were calibrated on a sample of six PATs. Then, a geometrical model capable of reconstructing a prototypal geometry, by a rough sizing of the PAT, was developed, involving charts, maps, good design rules, statistical correlations, and so on. Compared to the past approaches, in this work, the possibility of refining the calculus of geometrical parameters of the geometry model is given. The deviation between the curves given by the manufacturer's catalog and the ones foreseen in the pump operation can be reduced or annulled by changing one or more geometrical parameters, calculated by the model. A sensitivity analysis was conducted for this purpose, in such a way as to understand which parameters to act on. The objective of this research is therefore to provide a flexible tool, which allows calculating the performance of any PAT, in both modes of operation, starting from little information available from the catalog provided by the manufacturer. This would make it possible for anyone who decides to approach this technology to facilitate the choice of the PAT that guarantees the best efficiency according to the energy resources available. As already mentioned, only the behavior of the machine when operating as a pump is known, but not the behavior of the machine when operating as a turbine.
