*2.1. Fluid Dynamic Model*

As already mentioned in the introduction, the proposed fluid dynamic models can predict the performance of the PATs, both in pump and turbine operation. The model provides head, efficiency, and loss variations versus flow rates changes. Figure 1 shows a flow chart summarizing the main steps of the development.

**Figure 1.** Flow chart summarizing the main steps of the development.

The evaluation of the losses inside a hydraulic machine cannot ignore the knowledge of its geometry, as it is indispensable for doing the calculations. It is convenient to adopt a prototype of a centrifugal pump, already described in another article [35], with characteristics common to most of the pumps available. Figure 2 shows a schematic representation of it, highlighting the passage sections and the symbols used to identify them.

**Figure 2.** Reference geometry.

Regarding the main components that make up the machine, the simplifications adopted are as follows:

Aspiration (Section 0): A conical axial section was adopted, and the hypothesis was also assumed that the inlet diameter *d*<sup>0</sup> coincides with the blade tip diameter *d1p* of the inlet section of the impeller.

Impeller (Sections 1–2): As regards the inlet section, a truncated conical surface was considered, having upper and lower base diameters *d1m* and *d1p* and laterally delimited by the width of the blades. As a representative diameter [36], the one in correspondence with the average current line has been adopted, which divides the section into two equal parts.

Volute (Sections 3–4): Given the complexity of its geometry, some simplifications have been adopted regarding the section and its evolution. A volute with a square terminal section is assumed, with sides *b* and *hv*. The height of the volute varies linearly along the peripheral direction, until *hv* is reached. It is also assumed that the terminal section has a normal along the tangential direction.

Final diffuser (Section 5): It is assumed that the dimensions vary linearly moving from the inlet to the outlet section according to a reflection equal to the tangent of an angle *αd*, set equal to 3.5 degrees.

In the next sections, pump and turbine operations are separately analyzed.
