Nonlinear Modeling of Electrostatic Atomization Processes
In many practical applications, there are significant
advantages to the use of electrostatic atomization techniques.
Today, this scheme is utilized in numerous
fields as diverse as agricultural and automotive sprays,
targeted drug delivery systems, space vehicle propulsion units,
liquid metal sprayers, ion sources, emulsifiers,
dust scavenging systems, and ink-jet printers.
In many applications, electrostatics are employed to provide
precise control of a single liquid jet emanating
from a circular orifice.
This is especially true in controlling droplet size;
electrostatics provide an independent
conventional mechanism for disrupting the jet into droplets.
Electric charging can also provide more spatially
dispersive and narrower in droplet diameter
distributions than what can be obtained using
conventional methods.
There are very few analyses of the charged liquid column (jet or ligament),
probably due to the fact that axial perturbations (as treated
in the liquid jet analyses) are of greater importance in most
electrostatic atomizers. Our motivation for examining this
geometry stems from recent work on multi-jet electrostatic
atomization. When the voltage is continuously
increased, the jet splits and forms multiple ligaments
around the end of a conventional orifice, as shown in Figure 1.
The number of ligaments
has been observed to increase with voltage.
Since the multi-jet mode can produce many small columns (or ligaments),
smaller drops can be obtained as
compared to those produced by a single-jet with the same total mass flow rate.
For a range of flow rates and fluid resistivities, a stable
multi-jet structure can be observed in which a number of
ligaments/jets appear anchored at regular intervals about the
circumference of the orifice exit. Due to electrostatic
repulsion, these jets diverge at small angles (see Figure 1 and 2) from the
centerline of the orifice to provide a highly structured and
coherent ``spray''. The interaction of the fluid columns
under the radial electrostatic repulsion motivates the liquid
column studies presented herein.
Figure 1: Multi-Jet Experimental Image by Epperson and Sojka
Figure 2: Schematic of Multi-Jet Atomization
In Figure 3, |E| (electric field distribution) is shown
for the single-jet when the direction of the electric field
(i.e., E_x and E_y) is plotted in Figure 4.
Similarly, |E| is shown
for the multi-jet in Figure 5, 6 and 7.
Figure 3: Electric Field Distribution for Single-Jet
Figure 4: Electric Field Direction in X-Y Components for Single-Jet
Figure 5: Electric Field Distribution for Multi-Jet
Figure 6: Electric Field Distribution for Multi-Jet (Zoomed View)
Figure 7: Electric Field Direction in X-Y Components for Multi-Jet
In Figure 8, a multi-jet configuration for Gamma_e=10.0,
varepsilon=0.05, n_l=4, and a=1.0 is shown.
As soon as the outer-most computational node
reaches a designated ground location
(i.e., b=13.0 for this case), the simulations stops.
Theoretically, it is known that a single-jet has fewer oscillations
for a higher charge level.
The results of the multi-jet simulations (see Figures 8, 9, and 10)
show similar trends.
Figure 8: Shape of the Multi-Jet for Gamma_e=10, varepsilon=0.05,
and a=1.0
Figure 9: Shape of the Multi-Jet for Gamma_e=5, varepsilon=0.2,
and a=1.0
Figure 10: Shape of the Multi-Jet for Gamma_e=1, varepsilon=0.25,
and a=1.0
When we applied no charging level (Gamma_e = 0.0), no diverging
phenomenon was observed (see Figure 11),
and the jet oscillates perpetually.
In Figures 12 and 13, more realistic jets were
simulated for a=0.5 and a=0.25 at Gamma_e = 1.0.
Note that we do have oscillations in Figure 13,
but they are not visible due to small value of a and varepsilon.
Figure 11: Shape of the Multi-Jet for Gamma_e=0, varepsilon=0.25,
and a=1.0
Figure 12: Shape of the Multi-Jet for Gamma_e=1, varepsilon=0.1,
and a=0.5
Figure 13: Shape of the Multi-Jet for Gamma_e=1, varepsilon=0.05,
and a=0.25