Qualitative analysis of the minimum flow rate of a cone-jet of a very polar liquid

Abstract

Electrostatic atomization of a liquid of finite electrical conductivity in the so-called cone-jet regime relies on the electric shear stresses that appear in a region of the liquid surface when a meniscus of the liquid is subjected to an intense electric field. An order of magnitude analysis is used to describe the flow induced by these stresses, which drive the liquid of the meniscus into a jet that issues from the tip of the meniscus and breaks into droplets at some distance from it. When the dielectric constant of the liquid is large, the electric shear stresses extend into the jet and cause a depression that sucks liquid from the meniscus. The induced flow rate is estimated and shown to represent approximately the minimum flow rate at which a cone-jet can be established. It is argued that the meniscus can be stabilized by the electric field that the charge of the jet induces on it. This stabilizing mechanism weakens when the flow rate supplied to the meniscus decreases, and its failure may determine an alternative minimum flow rate for the cone-jet regime. The instability of the jet and existing scaling laws for the size of the spray droplets are discussed.

The minimum flow rate of electrosprays in the cone-jet mode

“…Contrary to existing theories (Fernández de la Mora & Loscertales 1994; Gañán-Calvo et al. 1997, 2013; Higuera 2017), the dimensionless minimum flow rate for either liquid is not constant but increases with the electrical conductivity. In terms of the Reynolds number, the experimental data are well fitted by the power laws for tributyl phosphate and for propylene carbonate.…”

“…A possible explanation of the apparent disagreement with Gañán-Calvo et al. (2013) and Higuera (2017) is that the values investigated may not be large enough for the inviscid limits (1.3) and (1.5) to apply. Furthermore, the Reynolds number for these solutions (and probably for most electrosprayable solutions) is never too large or too small, which limits the accuracy that can be expected from asymptotic estimations for very large or very small Reynolds numbers.…”

“…A similar inference results from our simulations of tributyl phosphate, for which we find solutions at flow rates below the minimum experimental one. On the other hand Higuera (2017) argues that the minimum flow rate is an intrinsic value that the electric shear stress can pull under important charge relaxation effects, and that no solution may exist at lower flow rates. In this regard we note that convergence in our algorithm becomes increasingly difficult at decreasing flow rate near the minimum one, especially for the more polar liquid propylene carbonate.…”

“…Thus, the minimum flow rate criteria proposed by Gañán-Calvo et al. (2013) and Higuera (2017) seem unable to explain the experimental relationship because most investigated solutions have large values and would need to follow the Reynolds-independent laws (1.3) and (1.5) respectively. The disagreement is especially prominent for the propylene carbonate solutions due to their very large : for these very large values of the viscous stress should be much smaller than the polarization force in the case of Gañán-Calvo’s criterion, much smaller than inertia in Higuera’s criterion, and the Reynolds number should drop from the relationship contrary to the experimental data in figure 3.…”

“…This balance combined with scaling laws for the forces (Gañán-Calvo et al. 2013) yield the following criterion: Higuera (2017) has identified the minimum flow rate condition as the characteristic flow that the electric shear stress can pull when balanced by either inertia or viscous stresses, in the presence of important relaxation effects. Higuera derives the criteria for polar liquids (large dielectric constant), and for apolar liquids (dielectric constant near one).…”