# Mass Flow Scaling of Gas-Assisted Coaxial Atomizers
## 2 Theoretical Background  
As previously noted, various studies investigated the effects of liquid properties, nozzle geometries, and operating conditions on the primary jet breakup of coaxial gas-assisted atomizers (see Figure 1) at lab-scale. In the following, several relevant studies addressing the atomization of liquid jets with high-velocity gas streams, which describe jet breakup and parameters influencing spray characteristics, are summarized. A classification of different primary breakup regimes was performed by [PERSON] for different nozzle geometries at \(d_{liq}\) = 1-1.5 mm [13]. The jet breakup morphologies were classified using the dimensionless numbers \(Re_{liq}\) and \(We_{aero}\), in accordance with Equations (1) and (2), in which the liquid jet diameter \(d_{liq}\), velocity \(v\), density \(\rho\), dynamic viscosity \(\eta\), and surface tension \(\sigma\) were used for the calculations. The subscripts \(gas\) and \(liq\) represent the gas and liquid phase, respectively:  
\[Re_{liq}\ =\ \frac{d_{liq}\cdot v_{liq}\cdot\rho_{liq}}{\eta_{liq}} \tag{1}\]  
\[We_{aero}\ =\ \frac{\left(v_{gas}-v_{liq}\right)^{2}\cdot\rho_{gas}\cdot d_{ liq}}{\sigma} \tag{2}\]  
The Rayleigh-type breakup leads to the disintegration of a liquid jet into large droplets close to the center line of the spray, and it occurs at \(We_{aero}<25\). For \(25<We_{aero}<70\), the membrane-type breakup is detected. In this regime, gas-filled membranes near the nozzle orifice are formed, which disintegrate into small droplets and an accumulated liquid rim. For \(We_{aero}>100\), the fiber-type breakup is divided into two submodes; namely, pulsating and superpulsating. In the pulsating submode, small fibers are peeled off the liquid jet near the nozzle orifice, and the liquid jet is atomized into small liquid fragments. Superpulsating results in droplet number density fluctuations in the resulting spray, while the liquid jet is atomized immediately after the nozzle discharge [13]. In subsequent investigations, [PERSON] and [PERSON] [14] used the momentum flux ratio \(j\), presented in Equation (3), to distinguish between the fiber-type breakup submodes:  
\[j\ =\ \frac{j_{gas}}{j_{liq}}\ =\ \frac{v_{gas}^{2}\cdot\rho_{gas}}{v_{liq}^{2} \cdot\rho_{liq}} \tag{3}\]  
Figure 1: Schematic of a gas-assisted coaxial atomizer with central liquid jet (blue) and annular gas stream (green). As the dynamic viscosity \(\eta_{liq}\) significantly affects the primary jet breakup, due to the damping effects of the liquid, investigations on primary jet breakup of high-viscosity liquids were performed inter alia by [PERSON] et al. [15] and [PERSON] et al. [16]. The resulting spray after the primary and secondary breakup is characterized in the literature concerning influencing parameters such as liquid properties, nozzle geometry, and operating conditions. Most investigations of liquid properties regarding spray formation have focused on changes in liquid viscosity by either the application of Newtonian liquids [17] or shear-thinning fuels [18; 19; 20; 21] at increased viscosities. A common result when utilizing high-viscosity liquids is an increased droplet size, primary ligament length, and spray angle. [PERSON] et al. [22] performed investigations intended to specify the influence of particles on the resulting droplet size by comparing pure liquids and suspensions at constant viscosity. An increase in droplet size was reported in the presence of particles, which could be explained by the tensile strength approach reported by [PERSON] et al. and [PERSON] [23; 24]. The effect of nozzle geometry on the resulting droplet size can be structured in studies of liquid jet diameter, gas gap width, gas/liquid wall thickness, and gas channel angle. Liquid jet diameter between \(d_{liq}\) = 2-17 mm at \(A_{gas}\) = 248 mm\({}^{2}\) was investigated by [PERSON] et al. [25], which revealed a nonmonotonic trend on the resulting droplet size with a minimum that moves for small \(GLR\approx 0.27\), from \(d_{liq}\) = 2 mm to \(d_{liq}\) = 10 mm at \(GLR\) = 5.48.