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New Model for Electro-Hydrodynamics of Semi-Conductive Fluids

This article highlights a new modeling capability being developed for the next release of FLOW-3D.

It has long been understood that strong electric fields can disrupt liquid surfaces. One particularly useful application of this observation has been the development of electro-spray-ionization (ESI) systems. The basic concept is to eject liquid from a nozzle connected to a voltage source that has a relatively high electric potential compared to its surroundings. When adjusted for certain operating conditions, a thin jet of liquid is ejected from the nozzle that subsequently breaks up into charged droplets having a relatively uniform size. There are many useful industrial applications for a system that produces small droplets of specified size; particularly if the droplets don't coalesce because of their electrical repulsion. Having a charge also means that these drops can be electrically deflected toward a target. This technology, for example, has been advantageously applied to paint spraying, atomization of fuels, printing, mass spectroscopy and a variety of spray drying processes.

An extension to the FLOW-3D to include a mechanism for simulating the electro-hydrodynamic behavior of semi-conducting fluids code is completed and ready for inclusion in a future release.

The new capability has been validated by application to a simple laboratory bench test and to the formation of Taylor cones, the principal component in ESI systems.

FLOW-3D leaky dialectric model simulation results
Figure 1

Comparison of theoretical and experimental results
Figure 2

FLOW-3D simulation of a cross section of the computed Taylor cone
Figures 3a & 3b

A close-up of the flow distribution in a Taylor cone using FLOW-3D
Figure 4: Flow distribution
in cone region

Charge distribution in a Taylor cone simulated with FLOW-3D
Figure 5: Charge
distribution on surface

Validation Test

The Leaky-Dielectric model defined by Melcher and Taylor has been found useful for a variety of important applications including electro-spraying. In their proposed model, a fluid is assumed to be weakly conducting to the extent that a bulk charge density does not exist, but a surface charge density can develop at fluid interfaces. Under such conditions the application of an electric field having a component tangential to an interface will result in tangential stresses acting on the fluid.

To validate this model, simulations have been performed of an experiment by Melcher and Taylor (J.R. Melcher and G.I. Taylor, “Electrohydrodynamics: A Review of the Role of Interfacial Shear Stresses,” Ann. Rev. Fl. Mech., 111, 1969).The test consisted of a shallow, conducting liquid in a container with electrodes at either end. At one end the electrode was extended above the liquid surface to form a tilted cover, which generated a tangential component of the electric field at the liquid surface.This field in conjunction with conductivity in the liquid generated a tangential surface stress that drives a counter-clockwise circulatory flow in the liquid pool, Figs.1 and 2. Figure 2 shows a comparison of computations (solid dots and squares), theory (solid line) and experimental data (other symbols and dashed line). Plot shows the product of maximum electrode spacing and the interfacial velocity as a function of the applied potential. Figure taken from Melcher-Taylor paper cited in text; FLOW-3D data are plotted as solid symbols. Solid squares used a conductivity close to that reported for the experiment.

Application to Taylor Cones

In a classic paper, G.I. Taylor (“Disintegration of Water Drops in an Electric Field,” Proc. R. Soc. Lon. A 280, 383 (1964) described how electric fields can distort liquid drops and in some circumstances produce a thin liquid jet emanating from a drop. Now referred to as Taylor cones, these jet flows have found uses for a wide variety of applications that require a stream of more-or-less uniform droplets having an electric charge.

By way of demonstration, an axisymmetric arrangement was used that consisted of a nozzle of inside diameter 7.0e-4m charged to 6000 volts. Downstream 25.0e-4m there is located a grounded plate perpendicular to the axis of the nozzle with a central hole of diameter 3.0e-4m. A flow of liquid into the bottom of the nozzle has a fixed axial velocity of 0.02m/s. The properties of the liquid are: density=827.0km/m3, viscosity 0.0081kg/m/s, dielectric constant=10.0, surface tension=0.0235N/m, and conductivity=8.05e-6S/m.

The computational grid selected for this test consisted of 30 radial cells and 220 axial cells. A minimum radial cell size of 0.8e-5m was used at the axis of symmetry. The total time simulated was 0.03s. Initially liquid was even with the top of the nozzle. The jet forms explosively at about t=0.017s and then slowly evolves into a smaller diameter jet as it approaches a near steady conditions.

Figure 3a shows the cross section of the computed Taylor cone. The jet has a maximum axial velocity of approximately 4.34m/s and exhibits small perturbations in diameter that move up and down its length. It appeared that the jet might be trying to reduce its mean diameter, but it was pushing the resolution of the grid since the minimum jet radius observed is only about 1.5 grid cells in thickness.

A close-up of the flow distribution in the cone (where only every other velocity vector in each direction has been plotted) is given in Fig.4, while Fig.5 shows the charge distribution. There is no recirculation in this case, contrary to observations by some researchers (e.g., Barrero, et al, J.Electrostatics, 47, 13 (1999)). Instead, at the exit of the nozzle the velocity is largest at the axis, as would be expected with Poiseuille flow in the nozzle. However, approximately one nozzle radius downstream from the exit there is almost no radial variation in axial velocity. Further downstream the profile is reversed with the minimum axial velocity at the axis and the largest at the surface of the liquid. It is the rapid increase in surface velocity that leads to the drawing out of liquid into a jet.

A repeat of the simulation with the potential of the nozzle raised from 6kv to 8kv produced the Taylor Cone shown in Fig.3b. Higher field strengths produce a shorter cone with a larger cone angle at the base of the jet. This is in agreement with observations by Hayati, et al (I. Hayati, A.I. Bailey and Th.F. Tadros, J. Colloid Inter. Sci. 117, 205 (1987)).

Acknowledgement: Flow Science was assisted in the pursuit of this development by Dr. Jun Zeng of Coventor.