# Mass Flow Scaling of Gas-Assisted Coaxial Atomizers
## 4 Results and Discussion
### Empirical Model for Liquid Mass Flow Scaling  
To derive a liquid mass flow scaling principle for gas-assisted coaxial nozzles, the droplet size results were plotted over the gas momentum flow \(J_{gas}\) as presented in Figure 6 (left).  
The diagram shows that for an increase in the liquid mass flow, an offset in droplet size to higher values occurs. This implies that achieving a constant droplet size with an increased liquid mass flow requires increased gas momentum flows. The gas momentum flow is defined by the factors of gas velocity, gas density, and the gas orifice area of the nozzle. As gas density is a typical process condition, a potential nozzle scaling approach must enable the calculation of the gas velocity and gas orifice area. With this objective for a liquid mass flow scaling approach, a potential fit with the dependence of \(\dot{M}_{liq}\) and \(J_{gas}\) was selected, according to Equation (9), due to the shape of the Saut  
Figure 5: Integral Sauter mean diameter for varying liquid mass flow (or liquid tube diameter) (**left**) and aerodynamic Weber number (or GLR) (**right**); symbols: \(\square\) stands for data from PDA, \(\triangle\) represents data from high-speed camera.  
Figure 6: Integral Sauter mean diameter for varying gas momentum flow as symbols for measurements and lines as calculation of proposed scaling model (**left**); parity plot for deviation observation between measured and calculated droplet sizes via proposed model (**right**). Symbols: \(\square\) stands for data from PDA; \(\triangle\) represents data from high-speed camera.  
\[ID_{32,m}\ =\ A\Big{(}\dot{M}_{liq}\Big{)}\cdot e^{-\frac{lgs}{B\big{(}\dot{M}_{liq} \big{)}}}\ +C\Big{(}\dot{M}_{liq}\Big{)} \tag{9}\]  
For varying liquid mass flows \(\dot{M}_{liq}\), three mass flow dependent parameters \(A\), \(B\), and \(C\) were applied and correlated via the least-square method. The dependence on liquid mass flows was kept linear to maintain the model's simplicity, as is reflected in Equations (10)-(12):  
\[A\Big{(}\dot{M}_{liq}\Big{)}\ =\ 4.6\cdot\dot{M}_{liq}+91 \tag{10}\]  
\[B\Big{(}\dot{M}_{liq}\Big{)}\ =\ 0.006\cdot\dot{M}_{liq}+0.03 \tag{11}\]  
\[C\Big{(}\dot{M}_{liq}\Big{)}\ =\ 0.67\cdot\dot{M}_{liq}+45 \tag{12}\]  
The parity plot in Figure 6 (right) depicts the accuracy of the model, as the measured and calculated droplet sizes are compared. For all measurement conditions, good agreement between the measured and calculated values was achieved. The maximum percentage deviation with 13.2% (\(\Delta D_{32}\ =\ 17.6\) um) was observed at \(\dot{M}_{liq}\ =\ 100\) kg\(\cdot\)h\({}^{-1}\) and \(We_{aero}\ =\ 250\). In contrast, the maximum deviation in absolute droplet size was \(\Delta D_{32}\ =\ 55.5\) um (3.3%) at \(\dot{M}_{liq}\ =\ 500\) kg\(\cdot\)h\({}^{-1}\) and \(We_{aero}\ =\ 250\).  
Thus, the model approach presented in Equation (9) is considered adequate for the mass flow scaling of gas-assisted nozzles with central liquid jets in the range of \(\dot{M}_{liq}\) = 20-500 kg\(\cdot\)h\({}^{-1}\) for \(v_{liq}\) = 1.7 m\(\cdot\)s\({}^{-1}\).  
The principle must be applied as follows: (i) using Equation (9), the applied liquid mass flow, requested droplet size, and necessary gas momentum flow can all be calculated; (ii) \(GLR\) must be specified as process condition and gas mass flow calculation; (iii) as the liquid velocity remained constant in the investigation, the liquid tube diameter can be determined; and (iv) with the definition of the gas mass and momentum flows, the gas velocity and gas gap width are provided.