# Modeling the Formation of Urea-Water Sprays from an Air-Assisted Nozzle
## 2 Experimental and Numerical Methods
### Computational Method
#### 2.2.3 Spray Modeling  
The spray model is based on a discrete droplet approach whereby a sufficient amount of parcels is introduced in order to provide statistically significant spray data. Every parcel consists of physically identical, noninteracting droplets and represents one discrete droplet size. The trajectory of the parcels is tracked through the simulation domain in a Lagrangian manner. The droplet kinematics is mainly driven by the drag force \(F_{D}\) acting on the droplets, which can be described by Equation (6) [30]. \[\rho_{l}V_{d}\overrightarrow{F}_{D}=\frac{1}{2}\rho_{g}C_{d}A_{d}*\left| \overrightarrow{u}-\overrightarrow{v}\right|*\left(\overrightarrow{u}- \overrightarrow{v}\right) \tag{6}\]  
The droplet motion is, apart from the drag force, influenced by the interaction with the turbulent structures. To characterize this interaction, the dimensionless Stokes number is evaluated, relating the time scales of the particle to the surrounding flow. The Stokes number \(St\) is defined by Equation (7), where \(\tau\) represents the particle relaxation time and \(t_{e}\), the eddy lifetime [30]. \[St=\frac{\tau}{t_{e}} \tag{7}\]  
Bigger droplets are accompanied by high relaxation times and result in higher Stokes numbers. Those droplets are less influenced by the turbulent eddy motion, while smaller droplets are characterized by small Stokes numbers and tend to follow the local gas flow. To take this behavior into account, a turbulent dispersion model was considered. This model is based on the approach of [PERSON] [31] that adds a fluctuating velocity to the relative velocity in the equation of droplet \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Standard Value** \\ \hline \(C_{\mu}\) & 0.09 \\ \(C_{\varepsilon 1}\) & 1.44 \\ \(C_{\varepsilon 2}\) & 1.92 \\ \(C_{\varepsilon 3}\) & 0.8 \\ \(C_{\varepsilon 4}\) & \(-\)0.373 \\ \(\sigma_{k}\) & 1 \\ \(\sigma_{\varepsilon}\) & 1.3 \\ \(\sigma_{p}\) & 0.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Standard values of the parameters of the k-\(\varepsilon\) model. kinematics. The fluctuating components of the gas velocity are randomly chosen based on the turbulence kinetic energy of the gas phase calculated by the k-\(\varepsilon\) model. The fluctuating velocity is updated if either the eddy lifetime or the time the droplet needs to pass through the eddy is expired. A well-known deficiency of 2-equation turbulence models is the assumption of isotropic turbulence. Therefore, the different components of the fluctuation velocity are only determined by the random numbers chosen by the model. The temporal evolution of the droplet sizes is determined by the droplet breakup, evaporation, and droplet collision phenomena. The droplet breakup is modeled using the WAVE breakup model [32], which is based on the assumption that growing oscillations on the droplet surface due to high relative velocities lead to instabilities and droplet breakup. The oscillations are characterized by the maximum growth rate \(\Omega\) and its corresponding wavelength \(\Lambda\), which are dependent on the physical properties of the injected liquid and surrounding gas. The radius of the stable product droplet r and the characteristic breakup time \(\tau\), calculated by Equations (8) and (9), determine the continuous change of the radius of the unstable parent droplet, according to Equation (10). \(B_{0}\) and \(B_{1}\) represent model constants that can be used to tune the model, providing an effective method to fit the simulation to experimental results. A more detailed description of the WAVE model, including the Rayleigh breakup for low relative velocities, can be found in [32]. \[r=B_{0}*\Lambda \tag{8}\]  
\[\tau=\frac{3.726B_{1}r_{p}}{\Lambda\Omega} \tag{9}\]  
\[\frac{dr_{p}}{dt}=-\frac{\left(r_{p}-r\right)}{\tau} \tag{10}\]  
The evaporation of the urea-water droplets is modeled according to an approach of Birkhold [33].