*2.8. Non-Dimensional Coefficients*

To allow the comparison among the holes, the results of the nozzle CFD model have been lumped in non-dimensional parameters that qualify the flow at each outlet section [45]. In the adopted scheme, Equation (16), the liquid-phase mass flow rate ( *mr*) passes through an effective area ( *Ae*) with a uniform velocity (*ve*); thus, vena contracta and cavitation affect the flow reducing the geometrical area ( *Ag*) to the effective one ( *Ae*). Considering an ideal flow, the theoretical velocity *vt*, Equation (17), is defined by means of a Bernoulli equation, assuming negligible velocity at the inlet section. The theoretical mass flow passing through the geometrical nozzle outlet section ( *A*g) is given by Equation (18). The first non-dimensional parameter is the discharge coefficient, *C*D defined as follows (Equation (19)):

$$
\Delta m\_{\mathbf{r}} = \rho\_{\mathbf{l}} A\_{\mathbf{e}} v\_{\mathbf{e}\prime} \tag{16}
$$

$$v\_t = \sqrt{\frac{2\left(P\_{\text{inj}} - P\_{\text{back}}\right)}{\rho\_l}},\tag{17}$$

$$m\_{\mathbf{t}} = \rho\_{\mathbf{l}} A\_{\mathbf{g}} v\_{\mathbf{t}\prime} \tag{18}$$

$$\mathbf{C}\_{\rm D} = \frac{m\_r}{m\_t}.\tag{19}$$

In order to isolate the area from the velocity contribute, the discharge coefficient is viewed as the product of two further non-dimensional parameters *C*V and *C* A, Equation (20):

$$\mathbb{C}\_{\rm D} = \mathbb{C}\_{\rm A}. \mathbb{C}\_{\rm V} \tag{20}$$

$$\mathbf{C}\_{\rm V} = \frac{\upsilon\_{\rm eff}}{\upsilon\_{\rm t}} \,\prime \tag{21}$$

$$\mathcal{C}\_{\rm A} = \frac{A\_{\rm eff}}{A\_{\rm g}}.\tag{22}$$
