2.1.1. Pump Operation

The fluid dynamic model relating to pump operation is shown, referring to the procedure described in Figure 1.

### Velocity Triangles

To evaluate the hydraulic losses, the model calculates the flow speed in the various sections of the machine. Initially, this calculus is done at the inlet and outlet of the impeller, given the direct influence on the theoretical head (Eulerian work) estimation. The hypothesis made for the inlet section is that the fluid reaches the impeller in a direction perpendicular to the passage area, therefore with an angle *α*<sup>1</sup> equal to 90◦. The tangential component of the absolute inlet speed is therefore equal to zero (*cu1* = 0), as can be seen from the velocity triangles represented in Figure 3.

**Figure 3.** Velocity triangles at the inlet and outlet of the impeller in pump operation under design conditions (BEP) with slip deviation.

A smaller tangential component of the absolute speed (*cu2*) is obtained, and thus a lower theoretical head is obtained compared to that determined in the one-dimensional design. It represents an inability of the impeller of the machine, having a finite number of blades, to transfer all energy to the fluid. To account for this loss of performance, the slip speed (Figure 4), *vs*, has been calculated:

$$
\upsilon\_s = (1 - h\_0) \cdot \mu\_2 \tag{1}
$$

**Figure 4.** Impeller input–output speed triangles in turbine operation with slip deviation.

The tangential component of the absolute velocity, corrected with the slip, is obtained from the following expression:

$$
\mathfrak{c}\_{\mathfrak{u}2}^{\*} = \mathfrak{c}\_{\mathfrak{u}2} - \mathfrak{v}\_{\mathfrak{s}} \tag{2}
$$

In Equation (1), *h*<sup>0</sup> represents the slip factor.

Many correlations were proposed for assessing the *h*<sup>0</sup> parameter:


Stodola's formula, however, proved to be the best option as it provided results more consistent with the characteristic curves of the machines supplied by the manufacturers. By following Stodola's formula,

$$h\_0 = 1 - \frac{\pi}{z} \sin(\beta\_{2p}) \tag{3}$$

For some models, the slip phenomenon was considered also at the impeller inlet: the inlet is not axial and a value of the angle *α*<sup>1</sup> different from 90◦ has been obtained. Considering the slip phenomena, calculated, also in this case, using Stodola's formula (Equation (3)), the tangential component of the absolute velocity becomes *cu1*<sup>∗</sup> and the theoretical head is calculated as follows:

$$H\_{th} = \frac{1}{\mathcal{S}} \left( u\_2 c\_{u2}{}^\* - u\_1 c\_{u1}{}^\* \right) \tag{4}$$

This solution has led to an improvement in the results achieved, allowing the head curve obtained from the model to be brought closer to that provided by the catalog.
