Spray drop size scales as We^{-0.31} (L/d0)^{-0.21} in pressure-swirl nozzles

The study quantifies how volume-median drop diameter decreases with nozzle-to-beam distance L as L^{-0.18} after ligament breakup. High-speed imaging of the conical sheet combined with systematic variation of pressure, orientation and standoff distance shows that the distance dependence persists into the far field. This distance correction accounts for part of the prefactor offset versus earlier far-field data and explains why laser-diffraction results reported at different stations cannot be compared directly. Industrial spray applications in fuel injection, dust capture and agricultural deposition require drop-size control at the actual target plane rather than at an arbitrary measurement station. The derived joint scaling allows nozzle performance to be predicted across operating pressures once the working distance is specified, offering a practical route to efficiency gains without hardware redesign. The work builds on Kooij et al. (J. Fluid Mech. 2018) by adding explicit L dependence and demonstrating that the We^{-1/3} collapse holds only at fixed distance. Remaining scatter after the joint fit suggests secondary influences from sheet thickness and air entrainment that future high-resolution imaging or DNS could isolate. Next steps include testing the scaling on hollow-cone and air-blast nozzles and embedding the distance-corrected relation into CFD spray models used by refining and coating industries.