Velocity Triangles

To evaluate the losses, it was first necessary to calculate the velocities in the various sections of the machine:


Concerning the phenomena of slip in turbine operation, the following correlations have been used:

For *ns* < 10, Stodola's formula was used (Equation (3)).

For *ns* > 10, the Stanitz formula [39,40] was used, expressed as follows:

$$h\_0 = 1 - 0.315 \left( \frac{2\pi}{z} \sin^{-1} \left( \frac{c\_{m2}}{\mu\_2} \right) \right) \tag{19}$$

This criterion is part of the tuning of the model by comparing the theoretical and experimental results.

For the same reason, for *ns* < 10, the tangential component of the absolute speed, *cu2*, is calculated by assuming the inlet angle *α*<sup>2</sup> as Worster suggests [46,51] and is used in other models [52]:

$$\alpha\_2 = \tan^{-1} \left[ \frac{b}{2\pi b\_2} \ln \left( 1 + \frac{2b}{d\_2} \right) \right] \tag{20}$$

Then,

$$\varepsilon\_{u2} = \frac{\varepsilon\_{m2}}{\tan(\alpha\_2)}\tag{21}$$

For *ns* > 10, *cu2* is calculated by assuming a free vortex distribution of velocities in the area between Section 3 and Section 2, i.e., between the volute and the impeller inlet. The expression used is as follows:

$$
\sigma\_{u2} = \sigma\_{u3} \frac{d\_3}{d\_2} \tag{22}
$$

## Hydraulic Losses

For the evaluation of hydraulic losses, also in this case, a distinction is made between friction losses and dynamic losses (Table 4). For the former, the formulas adopted are the same as those for direct operation (Table 1). The only difference lies in the evaluation of the speed *w*∞ which appears in the friction losses in the impeller, since in this case the slip also occurs in the inlet section, changing the value of the tangential component of the relative speed.

**Table 4.** Dynamic losses in turbine operation.

$$\begin{array}{ll} \text{Diffuser} & \text{Inlet losses} & h\_{dd} = \xi\_{d} \frac{c\_{1}^{2}}{2g} \\\\ & & \\\\ \text{Inlet losses} & h\_{dd} = 0.5 \cdot \left(1 - \frac{d\varphi\_{1}}{d\beta\_{1}}\right) \frac{c\_{13}^{2}}{2g} & (24) \\\\ \text{Inperlier} & & \\\\ \text{Shock losses}^{-1} & & h\_{dd} = \frac{[n\_{2}\varphi\_{1}(t)]^{2}}{2g} & (25) \\\\ \text{Instantaneous expansion losses} & h\_{d\xi} = (\xi\_{1} - 1)^{2} \frac{c\_{14}^{2}}{2g} & (26) \\\\ \text{Volate} & \text{Diffusion losses} & h\_{diff} = \frac{c\_{13}^{2} - c\_{24}^{2}}{2g} & (27) \\\\ \text{Outlet}^{2} & & h\_{inf} = 0.25 \left(\frac{Q}{\frac{\varphi\_{1}}{\varphi\_{2}}}\right)^{2} \frac{1}{2g} + \frac{c\_{14}^{2}}{2g} & (28) \end{array}$$

<sup>1</sup> The shock losses are computed considering the incidence angle *i* of the fluid at the runner entry. <sup>2</sup> The additional term *cu1*2/2*g* has been added to exhaust losses to consider the dissipative vortex generated by the presence of the tangential component *cu1*.

In the expression of the diffuser losses, following the flow direction of the fluid in turbine operation, this component is a converging duct. In the losses in the diffuser, *ξ<sup>d</sup>* is obtained as a function of the ratio between the final section volute width and the final section diffuser width, taken from [53] and reported in Table 5:

**Table 5.** Localized resistance coefficient as a function of the ratio *b*5/*b*.


The determination of the losses was carried out with reference to the recommendations of Idel'cick [54]. At this point, the engine head is calculated as the sum of the Eulerian work and the previously exposed hydraulic losses:

$$H\_m = H\_{\text{tll}} + \sum l \text{losses} \tag{29}$$

Even in turbine operation, the actual flow rate differs from that of the plant by an amount equal to the leakage of liquid from the clearances present between the impeller and the casing. The parameter that takes this phenomenon into account, that is, the volumetric efficiency, is determined as follows:

$$
\eta\_v = \frac{Q - Q\_s}{Q} \tag{30}
$$
