# Modeling the Formation of Urea-Water Sprays from an Air-Assisted Nozzle
## 2 Experimental and Numerical Methods
### Computational Method
#### 2.2.4 Initialization of the Liquid Phase
corrections have opposite effects on the spatially averaged droplet diameter distribution (see Figure 4c). Finally, the combination of both corrections is applied in order to derive a droplet size distribution for the initialization of the parcels in the simulation. Every time step, nine droplet size classes are chosen randomly on the probability axis of the distribution and are distributed over 15 randomly selected radial and circumferential positions of the parcel release area. This procedure results in a total number of 135 parcels per time step and ensures an appropriate discretization of the spray. The cone angle of the spray is set to 15\({}^{\circ}\) which is characteristic for the studied injector (see Figure 3a). The initial temperature of the droplets is set equal to the initial temperature of the gas jet by means of the assumption that the heat exchange inside the mixing chamber reaches an equilibrium state following the procedure of a previous study [17]. [PERSON] et al. reported [20] that the initial velocity of the gas phase and liquid phase are in the same range directly upstream of the discharge orifice. In the present study, however, this assumption was found to result in an overestimation of the experimentally observed droplet velocities at the first measurement position downstream of the atomizer. Therefore, the two-phase flow inside the atomizer is investigated analytically in order to estimate an appropriate initial velocity of the parcels directly downstream of the nozzle exit. The main assumption of the analytical investigation is that the liquid phase downstream of the mixing chamber may be treated as droplets that are dispersed in the gas flow due to the high gas-to-liquid ratio of the two-phase flow. In the first step, the velocity of the gas phase downstream of the mixing chamber \(\mathrm{v_{mix}}\) is estimated based on the main flow parameters shown in Figure 5. The mass flow of the gas phase \(\mathrm{\dot{m}_{air}}\) and the liquid phase \(\mathrm{\dot{m}_{UWS}}\), as well as the temperature of the final mixture \(\mathrm{T_{mix}}\), are known from the experimental investigation [17]. However, the pressure directly downstream of the mixing chamber \(\mathrm{p_{mix}}\) is unknown. Figure 5: Estimation of the velocity of droplets of different size at the atomizer exit. In order to address this issue, the supercritical nozzle flow is investigated under theoretical considerations. A choked flow through the discharge orifice is assumed, and pressure losses due to polytropic expansion, the mixing process, wall friction, and changes of the cross section are taken into account. Using experimental data from a pressure tapping upstream of the mixing chamber \(\mathrm{p_{air}}\) and the thermodynamic parameters of the surrounding hot gas \(\mathrm{T_{hg}}\) and \(\mathrm{p_{hg}}\), the desired velocity \(\mathrm{v_{mix}}\) is determined. Furthermore, the velocity directly upstream of the discharge orifice \(\mathrm{v_{orf}}\) is calculated based on the reduction of the cross section from a diameter of 3 mm to a diameter of 2 mm. The main results of the estimated gas flow are summarized in Table 3.  
Based on the calculation of the gas flow inside the atomizer, the droplet velocity at the nozzle exit is estimated. Thereby, droplets with different diameters representing the size classes of the initial droplet spectrum are initialized at the exit of the mixing chamber. The velocity of the droplets is assumed to be zero at this position. Starting from this point, the droplets are accelerated by the gas flow inside the channel, according to Equation (6). The resulting droplet velocities are exemplarily plotted for OP 2 over the channel length in Figure 5. For the droplet initialization in the CFD model, the calculated exit velocities are averaged over the whole droplet spectrum, considering their influence on the mean value by weighing the velocity of each size class by its corresponding volume probability.