Figure 2 illustrates the transition in modes of spray formation as the flow rate increases, with the spinning disc operating at 1000 r·min^-1 and using a constant ethanol-water solution. At a low flow rate of 50 ml·min^-1, direct droplet formation is initially observed at the edge of the disc ( Figure 2A ). Driven by centrifugal force, the liquid spreads outward, forming a radial liquid film on the disc surface. Once the liquid film reaches the disc’s edge, it extends a finite distance radially, forming a liquid torus around the edge (Wang et al., 2015; Tan et al., 2019). Rayleigh-Taylor instability, induced by centripetal acceleration, destabilizes the torus, forming one or more bulges (Keshavarz et al., 2020; Eisenklam, 1964). Primary droplets form by elongating the bulge, followed by the creation of smaller satellite droplets from the liquid thread connecting the primary droplet with the bulge (Sahoo and Kumar, 2021a, 2021b). This mode is commonly employed in handheld and aerial rotary atomizers.

If the disc is oversupplied with liquid, this direct droplet formation mode is disrupted. When the flow rate increases to 200 ml·min^-1, a mixture of direct droplet formation and ligament formation is observed ( Figure 2B ). At this stage, ligaments are drawn out from the disc edge, but their distribution is chaotic and disordered. Some droplets still form directly from the disc edge, while others are formed from the breakup of ligaments (Peng et al., 2017). As the flow rate increases to 500 ml·min^-1, the mode transitions to ligament formation, with droplets formed directly being completely replaced by those formed from ligaments. Distinct, regular ligaments form with stable, orderly flight trajectories ( Figure 2C ). The ligaments break into droplets due to Plateau–Rayleigh instability, with disturbances of wavelength larger than the ligament diameter causing instability, eventually breaking up the ligaments and forming droplets (Rayleigh, 1879). This mode typically results in the formation of uniform droplets proportional to the disturbance wavelength (Li et al., 2018).

When the flow rate reaches 600 ml·min^-1, adjacent ligaments begin to merge, forming a liquid sheet ( Figure 2D ). At a flow rate of 880 ml·min^-1, the liquid film on the disc thickens, resulting in fully developed sheet formation ( Figure 2E ). In this mode, the liquid film leaves the disc as a free sheet and then breaks up irregularly (Prather et al., 2023; Li et al., 2018). Although the droplets formed in this mode are less uniform than those from ligament formation, overall, droplets from these three modes are more uniform than those produced by hydraulic nozzles (Salah et al., 2014; Yang et al., 2023).

Frost (1981) studied spinning disc atomization using aqueous solutions of glycerol and Agral surfactant. He established transition correlations for spray formation modes through dimensional analysis, see Appendix A .

Figure 3 presents the spray formation modes predicted by Frost’s transition correlations (9)-(11), with different symbols indicating various operating points. These modes were identified through visual inspection of recorded images and are depicted by the features shown in Figure 2 . Overall, the predicted spray formation modes closely matched the experimental results, except for ligament formation and ligament-sheet formation. The transition from ligament to sheet formation occurred earlier than predicted, indicating a more abrupt change. This discrepancy may result from subjectivity in defining the mode boundaries or because the equilibrium surface tension used in Frost’s calculations was lower than the actual surface tension. Frost (1981) used aqueous solutions of glycerol and Agral surfactant as test liquids, where the surface tension of the liquid after forming a new surface was equivalent to that of the glycerol solution. Due to the slow molecular dynamics of Agral surfactant, its delivery rate to the droplet surface is insufficient to reach equilibrium, causing the actual surface tension to be higher than the equilibrium surface tension (Kooij et al., 2018; Varghese et al., 2024).

D _V0.5 represents the droplet diameter at which half of the spray volume consists of droplets smaller than this size, and the other half consists of droplets larger than this size (Xiang et al., 2021). Frost (1981) proposed a correlation to predict the droplet size produced by a spinning disc atomizer, expressed as:

Under the ligament formation mode, Figure 4 shows that the D _V0.5 predicted by Frost’s correlation is significantly lower than the experimental results, with an error range from 0% to -60%. This underestimation may be due to the equilibrium surface tension used in Frost’s correlation being lower than the actual surface tension, as previously discussed. Therefore, to improve prediction accuracy, it is preferable to use test liquids with faster molecular dynamics when establishing the correlation.

Figure 5 illustrates that as the Re increased from 0.35×10⁶ to 1.50×10⁶, reflecting a rise in disc speed, the d* decreased across all conditions. In Figure 5A , with a nearly constant q ranging from 129.19 to 138.60, an increase in St from 0.46×10^-6 to 1.21×10^-6, reflecting a decrease in surface tension, led to a reduction in d*. Additionally, at St = 1.21×10^-6, Figure 5B demonstrates that the q (from 25.84 to 129.19) associated with flow rate had a negligible effect on d*.

The Re values range from 0.35×10⁶ to 1.50×10⁶, St values range from 0.46×10^-6 to 1.21×10^-6, and q values range from 25.84 to 138.60. The correlation was first fitted according to the Equation 3. However, the p-value for the coefficient γ associated with q is 0.108, being higher than 0.05, indicating this variable is statistically insignificant. This finding aligned with the results shown in Figure 5B . Therefore, q was removed from the dimensionless parameters of interest. By analyzing the coefficients in the Equation 3, a dimensionless droplet size correlation was obtained:

The magnitude of the coefficient associated with each dimensionless number gives some essential guidance on the phenomena’s prevalence in the spinning disc atomization process. The coefficient associated with Re and St in the Equation 8 corroborated the conclusions presented in Figure 5 . Figure 6 presents a comparison of experimental and predicted droplet sizes. The results indicate that the relative error between most experimental and predicted data is within ±15%, demonstrating the high accuracy and reliability of the Equation 8. This consistency also indicates the high regularity of the spinning disc atomization process. However, some larger droplet sizes exceeded the ±15% error range. This discrepancy may result from poor droplet uniformity at these operating points, affecting the measured D _V0.5 values.

Although this study was conducted in a simplified laboratory environment without considering field complexities such as wind speed, temperature, and humidity, which can indeed influence atomization efficiency and pesticide deposition in practical applications (Lin et al., 2021), the droplet size correlation proposed here holds substantial value for droplet size control. According to the theory of biological optimum droplet size, droplets of specific sizes are more likely to adhere to target organisms, thus enhancing pesticide adhesion and efficacy (Uk, 1977; Chen et al., 2022). Consequently, by adjusting disc speed or liquid surface tension under the guidance of the droplet size correlation, droplets can be generated that align with the biological optimum droplet size, thereby improving pest control effectiveness.

Under all operating conditions, D _V0.5 decreased as disc speed increased. Figure 7 illustrates the relationship between disc speed and D _V0.5. The blue solid line in Figure 7 indicates that at a flow rate of 100 ml·min^-1 and surface tension of 29.681 mN·m^-1, D _V0.5 decreases from 751 μm to 214 μm as disc speed increases from 1000 r·min^-1 to 4000 r·min^-1, a total reduction of 537 μm. Between 1000 r·min^-1 and 2500 r·min^-1, D _V0.5 decreased by 420 μm, accounting for 78.2% of the total reduction. This reduction may be due to the increased centrifugal force at higher speeds, which pushes the liquid to the disc edge, thins the liquid film, and forms smaller droplets (Mantripragada and Sarkar, 2017; Ahmed and Youssef, 2012). Moreover, Wu et al. (2018) noted that stronger shear effects within the liquid film and at the liquid-solid and gas-liquid interfaces contribute to smaller droplet sizes. However, as disc speed further increases, the slip velocity between the liquid film and the disc surface increases, slowing the reduction of liquid film thickness, which leads to a plateau in the decrease of D _V0.5 at higher disc speeds (Peng et al., 2018a).

The flow rate influences droplet size, but its effect is less significant than that of disc speed. Figure 8 depicts the effect of flow rate on D _V0.5. The solid line in Figure 8 shows that at low disc speeds, D _V0.5 heavily depends on the flow rate. At high disc speeds, as indicated by the dashed line, this dependence decreases significantly. At high disc speeds, D _V0.5 slightly increases with the rising flow rate. For instance, at a disc speed of 4000 r·min^-1, the orange dashed line in Figure 8 shows that as the flow rate increases from 100 ml·min^-1 to 500 ml·min^-1, D _V0.5 increases from 214 μm to 231 μm. This could be due to the spray formation mode showing more liquid ligament characteristics, where long ligaments extend from the disc edge. An increase in flow rate may lead to more liquid ligaments or their elongation, thus accommodating the increased liquid volume (Wang et al., 2016; Peng et al., 2018b). As shown in Figure 7 , when the disc speed exceeds 2500 r·min^-1, D _V0.5 is primarily controlled by disc speed, and the effect of flow rate becomes negligible. This phenomenon is supported by the study of the Micron ULVA rotary atomizer by Boize and Dombrowski (1976). At low disc speeds, D _V0.5 changes significantly with an increasing flow rate. For instance, the goldenrod solid line in Figure 8 shows that at a disc speed of 1000 r·min^-1 and a surface tension of 51.764 mN·m^-1, increasing the flow rate decreases droplet size. As the flow rate increases from 100 ml·min^-1 to 500 ml·min^-1, D _V0.5 decreases from 1158 μm to 792 μm. This might be due to a change in spray formation mode caused by the increase in flow rate. In the ligament formation mode, the droplet sizes produced are much smaller than those in the direct droplet mode at the same disc speed (Prather et al., 2023).

Figure 9 demonstrates the significant impact of surface tension on D _V0.5. Lower surface tension results in smaller droplets. The solid lines in Figure 9 indicate that at low disc speeds, D _V0.5 is strongly dependent on surface tension. In contrast, the dashed lines show that this dependence decreases significantly at high disc speeds. At high disc speeds, D _V0.5 decreases as surface tension decreases. The goldenrod dashed line in Figure 9 shows that at 4000 r·min^-1 and a flow rate of 500 ml·min^-1, D _V0.5 decreases from 376 μm to 231 μm as surface tension drops from 72.535 mN·m^-1 to 29.681 mN·m^-1. This is likely because surface tension represents the force resisting the formation of new surface areas (Lefebvre and McDonell, 2017). Lower surface tension allows droplets to form more quickly, resulting in smaller droplets (Kooij et al., 2018). At low disc speeds, D _V0.5 decreases significantly as surface tension decreases. The blue solid line in Figure 9 shows that at 1000 r·min^-1 and a flow rate of 100 ml·min^-1, D _V0.5 decreases from 1417 μm to 751 μm as surface tension drops from 72.535 mN·m^-1 to 29.681 mN·m^-1.