Surface charge representations and a finite-difference method for approximating the charge flow equation

Charge distribution in turbulent flow of charged liquid—Modeling and experimental validation

“…Several authors have published theoretical models of EHD flows or charge relaxation problems with free surfaces not accounting for surface charge 8,11,15 . This approach has led to good agreement with experimental data for the turbulent filling of large tanks with liquid volumes 9,10,16 while surface charge accumulation was found to be significant in smaller scale experiments 12 with negligible fluid velocities or after pumping stopped in experiments with intermediate size vessels 17 . In addition to surface charge accumulation, surface conduction at gas–liquid interfaces has been discussed as a relevant transport mechanism 17,18 …”

“…[8][9][10] Detailed computations of volume charge distributions are only established in the investigation of small scale electrohydrodynamic (EHD) flow phenomena. [11][12][13][14] These models only apply to the laminar flow regime or direct numerical simulations of turbulent flows. Strongly coupled equations of fluid flow and electrostatics make them computationally expensive and thus unsuitable for assessment of industrial scale applications.…”

“…Quantitative methods involving numerical simulations usually hinge on simplifying assumptions like homogeneous volume charge densities and neglect of surface charge 8–10 . Detailed computations of volume charge distributions are only established in the investigation of small scale electrohydrodynamic (EHD) flow phenomena 11–14 . These models only apply to the laminar flow regime or direct numerical simulations of turbulent flows.…”

“…Subsequently, the transport of volume charge, Equation(10), is solved in the liquid domain, as it does not depend on the electric field explicitly. In a last step, the coupled Equations(12) and (1b) for surface charge and electric potential are solved simultaneously on the entire geometry including liquid and gas domains. This numerical procedure allows the fluid flow and volumetric charge equations to be solved on a numerical mesh optimized for fluid flow.…”

“…Overall, the model validation yields an R 2 value of 0.8918 (0.8004).The proposed and validated model accounts for surface charge and includes advective effects. The advective term in Equation(12) dominates the behavior of surface charge as accumulation is counteracted by advection. When neglecting advection of surface charge and positioning the sensor near the liquid surface, surface charge accumulates beneath the sensor and increases the measured electric field.…”