The volumetric mean flow velocity of the liquid fuel at the inlet of the nozzle passage, , can be calculated as

(6–1)

where is fuel mass flow rate, is liquid fuel density, A is nozzle cross-sectional area, and D is nozzle diameter.

The mean mass flow through the nozzle exit is always lower than that predicted by the Bernoulli equation, because of flow losses. The losses are caused by the acceleration and formation of the velocity profile at the inlet, the expansion after the vena contracta, and the wall friction. To quantify this difference, the discharge coefficient, , is defined as:

(6–2)

where p 1 and p 2 are pressures at location 1 and 2, respectively, in Figure 6.1: Flow through nozzle passage .

By using tabulated inlet loss coefficients (K inlet) and the Blasius or the laminar equation for wall friction, the discharge coefficient is given by:

(6–3)

with f defined as

(6–4)

and Re _D is the Reynolds number based on nozzle diameter.

A first estimation of the inlet pressure, p ₁ , for a turbulent flow yields:

(6–5)

Next Ansys Forte checks whether the flow under this condition is already cavitating. Assuming a flat velocity profile, and using Nurick’s expression for the size of the contraction (Figure 6.1: Flow through nozzle passage ), continuity gives the velocity at the smallest flow area:

(6–6)

(6–7)

In the case of cavitation the potential flow theory allows the application of the Bernoulli equation from point 1 to c without any losses:

(6–8)

If p _vena is lower than vapor pressure p _vapor, it is assumed that the flow must be fully cavitating and a new inlet pressure and discharge coefficient are calculated by:

(6–9)

(6–10)

In the cases of non-cavitating turbulent flows, the exit velocity of the droplets is set equal to the mean velocity, U _mean, and the diameter of the jet at the nozzle exit is equal to the nozzle diameter.

When the nozzle is cavitating, those values are calculated as:

(6–11)

(6–12)

(6–13)

The spray angle in the nozzle flow model is estimated using an aerodynamic model. This approach is based on Taylor’s analysis of high-speed liquid breakup due to the unstable growth of surface waves and the resulting mass shedding. In this approach, the spray angle, , is expressed as:

(6–14)

Where A is 3+0.28( L/D) and L/D is the length-to-diameter ratio of the nozzle. The function f(T) is approximated by

(6–15)

Instead of applying the nozzle-flow model, you can also specify empirical constant values for discharge coefficient and spray angle. The empirical discharge coefficient is then used to calculate the effective injection velocity and initial drop size at the nozzle exit.

Two options are provided for specifying the spray cone angle: one is user-specified constant value, the other is using a correlation. For sprays not involving flash boiling, the user-specified constant value is an acceptable option, and the default range 12° to 15° is adequate for typical solid cone sprays.

The cone angle correlation is generated based on spray simulation validation data over a wide range of conditions, especially flash boiling conditions. Since the flash-boiling injection condition is typically associated with a wildly expanding cone angle, the cone angle correlation option is recommended for simulating flash boiling conditions. The correlation takes the form of a quadratic function:

(6–16)

In which is a function of several other physical parameters:

(6–17)

, , and represent the pressure ratio, dimensionless surface tension, and fuel molecular weight:

(6–18)

(6–19)

where , , , represent fuel saturation pressure at injection temperature of , ambient gas pressure, surface tension, and Boltzmann constant, respectively. is the molecular surface area defined as , and is the liquid molecular volume.