Hydraulic Losses

The model evaluates the hydraulic losses by referring to the individual components and are classified into friction losses (Table 1) and dynamic losses (Table 2).

**Table 1.** Friction losses in pump operation.

$$\text{Impeller} \tag{5}$$

$$h\_{f\:\%} = \lambda \frac{w\_{\text{co}}^2}{2g} \left(\frac{l}{d\_{\text{eq}}}\right) \tag{6}$$

Vaneless diffuser *hf c* <sup>=</sup> *<sup>λ</sup>*

$$h\_{fc} = \frac{\lambda}{2g} \frac{1}{D\_{h3}} \frac{c\_3^2}{\text{sen}\left(\alpha\_2'\right)} \frac{d\_3}{d\_2} \left(\frac{d\_3 - d\_2}{2}\right) \tag{6}$$

$$\text{Volute} \tag{7} \\ \text{Volume} \qquad \qquad h\_{f\upsilon} = \sum\_{j=1}^{18} \lambda\_j \frac{\xi\_4^2}{2\xi} \\ \frac{\left(\Delta S\_{\subset l} + \Delta S\_{\text{lim}} + \Delta S\_{\text{exp}}\right)\_j}{A\_{\theta m\text{j}}} \\ \frac{Q\_j}{Q} \tag{7}$$

$$\text{Final diffusivity} \tag{8}$$

$$h\_{fd} = \frac{\lambda}{8\epsilon n(a\_d)} \left[ 1 - \left(\frac{A\_4}{A\_5}\right)^2 \right] \frac{\mathcal{L}\_4^2}{2\mathcal{g}} \tag{8}$$

In evaluating these losses, the value of the friction coefficient was obtained using the Colebrook–White formula [36], in transition and turbulent conditions. The speed *w*∞ which appears in the expression of the losses in the impeller (Equation (5)) represents the average value of the relative speeds calculated between the inlet section and the outlet section of the impeller. The choice to use this expression is the result of an in-depth experimental analysis. Regarding the friction losses inside the volute, the approach recommended by Worster was followed [46]. It is hypothesized that the velocities inside the volute have a purely tangential direction and that a free vortex velocity distribution exists in the volute. The analysis is carried out by dividing the component into 18 sectors and evaluating the friction losses in each of them. The purpose is to analyze the volute considering the variations in both dimensions and speeds. It was considered more correct to evaluate these losses by referring to the average speed inside the sections rather than to that at the outer edge of the volute in the exit section. This last approach would have led to a trend of decreasing friction losses as the flow rate increased. Table 2 shows the dynamic losses obtained for the related machine components.

Inlet *hinlet* = 0.25 ⎛ ⎝ *Q πd*<sup>2</sup> 0 4 ⎞ ⎠ 2 1 <sup>2</sup>*<sup>g</sup>* (9) Impeller Shock losses *hshock* <sup>=</sup> [*w*1*sen*(*i*)] 2 <sup>2</sup>*<sup>g</sup>* (10) Wake losses *hdg* <sup>=</sup> (*ξ*<sup>2</sup> <sup>−</sup> <sup>1</sup>) 2 *c*2 *m*2 <sup>2</sup>*<sup>g</sup>* (11) Vaneless diffuser Instantaneous expansion losses *hdc* <sup>=</sup> *<sup>c</sup>*<sup>2</sup> *m*2 2*g* <sup>1</sup> <sup>−</sup> *<sup>A</sup>*2*<sup>r</sup> A*<sup>3</sup> 2 (12) Volute Mixing losses *hdv* <sup>=</sup> *<sup>c</sup>*<sup>2</sup> *m*3 <sup>2</sup>*<sup>g</sup>* (13) Final diffuser Diffusion losses <sup>1</sup> *hdd* <sup>=</sup> *<sup>ξ</sup><sup>d</sup> c*2 4 <sup>2</sup>*<sup>g</sup>* (14)

**Table 2.** Dynamic losses in pump operation.

<sup>1</sup> *ξ<sup>d</sup>* represents the localized resistance coefficient, and its value was obtained as a function of the ratio *c*, reported in Table 3, and taken from [47].

**Table 3.** Localized resistance coefficient as a function of the parameter *c*.


The shape of the final diffuser was assumed as that of a diverging duct with a gradual widening of the section. The value of parameter *c* is obtained from the following equation, as a function of the input (*b*) and output (*b*5) sections of the component and its length (*Ld*):

$$z = \frac{b\_5 - b}{2 \ L\_d} \tag{15}$$

In addition to the losses in Table 2, another loss is detected, due to the vortex, which arises above all at low flow rates. The model, being one-dimensional, is unable to consider bidimensional phenomena such as vorticity. This observation was confirmed by the experimental investigations carried out by Van der Braembussche [48]. For overcoming this critical issue, some expressions based on experimental observations were proposed.

For *Q* < *Qbep*,

$$h\_{diff} = \frac{c\_{u3}^2 - c\_{u3bpp}^2}{2g} \tag{16}$$

For *Q* > *Qbep*,

$$h\_{diff} = 0\tag{17}$$

Once the theoretical head and hydraulic losses have been evaluated, it is possible to obtain the real head by calculating the difference between the theoretical head and the hydraulic losses.

$$H\_m = H\_{\rm th} - \sum l \text{losses} \tag{18}$$
