# Mass Flow Scaling of Gas-Assisted Coaxial Atomizers
## 2 Theoretical Background
[PERSON] et al. [26] performed atomization experiments with \(d_{liq}\) = 4/6/8 mm and constant \(d_{gas}\) = 15 mm. The investigations focused on the instability frequencies, primary breakup morphology, and ligament length. By comparing the results at constant \(j~{}=~{}2.8\) and decreasing \(J\) (see Equation (4)), a significant increase in the primary breakup length was identified [26]:  
\[J~{}=~{}\frac{J_{gas}}{J_{liq}}~{}=~{}\frac{v_{gas}^{2}\cdot\rho_{gas}\cdot A _{gas}}{v_{liq}^{2}\cdot\rho_{liq}\cdot A_{liq}} \tag{4}\]  
The effect of an increase in the gas gap width from \(s_{gas}\) = 0.6-2 mm was investigated by [PERSON] et al. [27], which led to a decrease in the droplet size, and was explained by the free jet theory and Equation (5), with equivalent diameters of the gas orifice \(d_{eq}\) and axial distance \(z\)[28]:  
\[\frac{v(z)}{v_{gas}}~{}=~{}6.37\cdot\frac{d_{eq}}{z}\cdot\sqrt{\frac{\rho_{0 }}{\rho}} \tag{5}\]  
According to this theory, for increased gas gap width, the velocity of the gas phase exiting the nozzle orifice remains high over a longer distance, due to the decreased gas mass flow entrainment of the surrounding gas phase [28]. This effect results in a longer and more intense interaction between gas and liquid phase, which results in smaller droplet size. [PERSON] et al. [29] conducted investigations concerning the gas/liquid wall thickness. For increasing wall thickness, the interaction point between the emerging phases was shifted to a higher distance from the nozzle orifice, which results in a recirculation zone [29] and enhances the formation of flapping instabilities [30]. The effect of an increase in the gas channel angle was analyzed by several authors, leading to the conclusion that aerodynamic forces are enhanced and droplet size is reduced for low gas velocities [31; 32; 33]. Variations in operating conditions and exiting velocities (mostly \(\dot{M}_{gas}\) and corresponding \(v_{gas}\)) were extensively addressed in the literature by \(GLR\) variations (see (6)):  
\[GLR~{}=~{}\frac{\dot{M}_{gas}}{\dot{M}_{liq}} \tag{6}\]  
For increasing gas velocity or gas mass flow, a decrease in the resulting droplet size was detected by many authors for a variety of different liquid properties and nozzle geometries [34; 35; 36; 37]. At high gas velocities, the effect of further gas velocity increments on droplet size decreases. According to [PERSON] [8], an increase in the liquid jet velocity or liquid mass flow-applying one nozzle-leads to an increase in the resulting droplet size, mostly due to a decrease in the relative velocity between the two emerging phases. Even though there are many studies that deal with the effects of single parameters on the atomization process and spray characteristics, literature focusing on nozzle scaling is scarce. [PERSON] et al. [10] considered the effects of nozzle scaling for three primary jet thicknesses \(d_{liq}\) = 0.4/1/2 mm and gas gap widths \(d_{gas}\) = 3.5/6/8 mm. The authors choose the approach of comparing nozzles of different primary jet thickness with a constant \(Re_{liq}\) and momentum flow ratio \(J\) (see Equation (4)) with respect to the primary breakup morphology. For constant dimensionless parameters, different breakup morphology was detected, as the application of a small \(d_{liq}\) led to prompt atomization, whereas a large \(d_{liq}\) resulted in long primary ligaments and large droplets [10]. In a second study, [PERSON] et al. [11] performed droplet size measurements, and concluded, that droplet size is most affected by the primary breakup morphology. A scaling rule, concerning nozzle geometry or process parameters, for increased liquid mass flows, leading to constant droplet sizes for variable liquid mass flows, was not specified. The literature review reveals that many investigations were performed at lab-scale, concerning the influence of specific parameters as liquid properties, nozzle geometry, and operating conditions on primary jet breakup or resulting droplet size.