# Modeling the Formation of Urea-Water Sprays from an Air-Assisted Nozzle
## 3 Results and Discussion
### Discussion of Droplet Sizes  
In addition to the droplet velocities, the diameter distribution of the spray is of interest for the assessment of the quality of the model. Therefore, the arithmetic diameter (D\({}_{10}\)), the Sauter mean diameter (D\({}_{32}\)) and diameter at the 90% volumetric quantile (D\({}_{\text{V90}}\)), depicting the characteristic parameters of a droplet spectrum, are evaluated in the radial direction at the first measurement position 50 mm downstream of the nozzle. This is conducted by means of a MATLAB(r) routine, as described in Section 3.2. The radial profiles of the characteristic parameters are plotted in Figure 13 for the three investigated operating points. The arithmetic diameter and the Sauter mean diameter calculated from the simulation data are in good agreement with the experimental data in the core region of the spray. In the outer area, starting from \(\pm 3\) mm, the arithmetic diameter is overestimated by the model. Figure 11: (**a**) Simulated and measured radial profiles of the turbulence kinetic energy calculated from the fluctuations of the corresponding velocity components of the droplets for the experiment and from the gas phase turbulence for the simulation. (**b**) Radial velocity fluctuations calculated from the fluctuations of the corresponding velocity components of the droplets for both the simulation and experiment. Figure 12: Stokes number of the simulated droplets as a function of the droplet diameter 50 mm (**a**) and 225 mm (**b**) downstream the nozzle exit averaged over the radial distance from \(-3\) mm to \(3\) mm. An explanation for this behavior is the absence of small droplets in this area as already presented in Figure 8. The Sauter mean diameter is in a range of 10 \(\upmu\)m to 14 \(\upmu\)m in the spray center for all three operating points, which confirms a continuously good atomization. The SMD is also correctly predicted by the model in the core zone, while it is overestimated in the outer area. Finally, the DV\({}_{90}\) from the simulation data is in a good agreement with the experiment over the whole radial distance for OP 1 (Figure 13a). For OP 2 (Figure 13b) and OP 3 (Figure 13c), there is a slight overestimation in the core area recognizable. The DV\({}_{90}\) depicts a sensitive parameter that can be strongly influenced by a small number of larger droplets. The correlation with the experimental data achieved by the model demonstrates that the droplet breakup and kinematics are predicted correctly. Large droplets emerging at the spray center are immediately broken up in the vicinity of the nozzle due to high relative velocities, while they persist in the outer area of the spray. This behavior can be analyzed by an evaluation of the Weber number of the droplets. The Weber number of the droplets at three different positions downstream of the nozzle exit is illustrated Figure 14. Further on, the Weber number is depicted in the spray center and the outer area of the spray at distances of 10 mm (Figure 14b) and 20 mm (Figure 14c) downstream the nozzle. Figure 14: Droplet Weber number as function of the droplet diameter for OP 2 in the vicinity of the nozzle (**a**) 10 mm (**b**) and 20 mm (**c**) downstream of the nozzle exit. Figure 13: Arithmetic diameter, Sauter mean diameter and diameter at the 90% volumetric quantile in the radial direction for OP 1 (**a**), OP 2 (**b**) and OP 3 (**c**). In the vicinity of the nozzle (Figure 14a), high Weber numbers up to 150 occur due to the high relative velocities of the droplets and the escaping gas jet. This results in an immediate breakup of the droplets that is also known from high-pressure injections. At the axial position of 20 mm (Figure 14c), the Weber numbers are smaller than 10, indicating that the main part of the secondary atomization is completed.