A fully coupled finite element analysis (FEA) technique was developed for analyzing the discharge phenomena and dielectric liquid flow while considering surface charge density effects in dielectric flow guidance. In addition, the simulated speed of surface charge propagation was compared and verified with the experimental results shown in the literature. Recently, electrohydrodynamics (EHD) techniques have been widely applied to enhance the cooling performance of electromagnetic systems by utilizing gaseous or liquid media. The main advantage of EHD techniques is the noncontact and lownoise nature of smart control using an electric field. In some cases, flow can be achieved using only a main electric field source. The driving sources in EHD flow are ionization in the breakdown region and ionic dissociation in the subbreakdown region. Dielectric guidance can be used to enhance the speed of discharge propagation and fluidic flow along the direction of the electric field. To analyze this EHD phenomenon, in this study, the fully coupled FEA was composed of Poisson’s equation for an electric field, charge continuity equations in the form of the Nernst–Planck equation for ions, and the NavierStokes equation for an incompressible fluidic flow. To develop a generalized numerical technique for various EHD phenomena that considers fluidic flow effects including dielectric flow guidance, we examined the surface charge accumulation on a dielectric surface and ionization, dissociation, and recombination effects.
1. Introduction
Electrical insulation is the most important consideration in electrical power devices such as power transformers, vacuum interrupters, circuit breakers, and transmission lines
[1

8]
. In power electrical systems, most failures of the insulation systems are associated with the breakdown of solid insulators
[9]
. One of the key issues in the analysis of solid insulating materials is electrical breakdown prediction, which considers electric discharge processes including the surface tracking stages. To analyze surface tracking accurately, the discharge mechanism should be well understood, including electric field dependent charge dynamics and surface charge density.
In addition, to analyze fluidic flow effects such as flow electrification and pumping, engineers should examine the distributions of flow vectors considering solid insulating materials. For these types of problems, the surface charge density on the dielectric surface plays an important role as it affects flow behavior and velocity. To analyze this electrohydrodynamics (EHD) phenomenon, in this study, the fully coupled finite element analysis (FEA) was composed of Poisson’s equation for an electric field, charge continuity equations (in the form of Nernst–Planck equations for electrons and ions), and the Navier–Stokes equation for an incompressible fluidic flow.
It is well known that there are four basic phenomena in EHD: flow, electric field dynamics, energy variation, and charge dynamics
[10]
; these four phenomena are considered in this paper.
Fig. 1
shows the schematic diagram of the multiphysics analysis method for EHD phenomena coupled with electric discharge.
Schematic diagram of multiphysics analysis method for EHD phenomena coupled with electric discharge.
2. Fully Coupled Governing Equation for Generalized EHD Formulation
 2.1 Governing equations in liquid region
The generalized hydrodynamic driftdiffusion equations combined with Poisson’s equation have been widely employed for analyzing discharge analysis in dielectric liquids as follows
[6

10]
:
where the subscripts +, −, and
e
indicate the positive ions, negative ions, and electrons, respectively,
ε
is the dielectric permittivity,
V
is the electric scalar potential,
ρ
is the charge density,
t
is the time,
G_{I}
(
E
) is the electricfielddependent molecular ionization source term,
G_{D}
(
E
,
T
) is the electric field and the temperature dependent ionic dissociation source term,
e
is the electron charge,
R_{xy}
is the recombination rate of
x
and
y
carriers,
v
is the velocity of fluidic medium,
τ_{a}
is the electron attachment time constant,
T
is the temperature,
ρ_{l}
and
c_{v}
are the mass density and the specific heat capacity, respectively,
k_{T}
is the thermal conductivity,
E
⋅
J
represents the electrical power dissipation term in the fluidic medium, η is the dynamic viscosity,
p
is the pressure, and
F
is the body force density including the electric force density. The formulation of
G_{I}
(
E
) and
G_{D}
(
E
,
T
) was developed in
[9]
, in this study, we adopt the same setup for discharge analysis.
 2.2 Governing equations in solid region
The governing equation in solid insulation is Gauss’s law with zero space charge, as follows:
where the current density
J
_{SD}
is zero because the solid insulator has zero conductivity.
ρ_{SD}
is the mass density,
c_{SD}
is the specific heat capacity, and
k_{SD}
is the thermal conductivity in the solid insulator region. COMSOL Multiphysics software implementing a detailed numerical setup was used to solve these coupled equations.
 2.3 Electric and buoyant body force density
To examine body forces, we included the Coulomb force for net charge, the Kelvin force for the electric field gradient, and the Boussinesq buoyancy force for the temperature gradient as
[10
,
11]
where
P
is the polarization vector,
β
is the thermal expansion coefficient,
T_{ref}
is the buoyancy reference temperature, and
g
is the acceleration due to gravity.
 2.4 Calculation of surface charge accumulation at dielectric liquidsolid interface
The surface charge density
σ_{s}
at the dielectric liquid–solid interface can be calculated as
where
J_{i}
=
ρ_{i}μ_{i}
E
and
n
is the outward normal unit vector from solid to liquid. To determine the surface charge density in Poisson’s equation, the surface charge density can be calculated using Gauss’s law as
3. Space Charge Propagation with LiquidSolid Interfaces by the Applied Step Voltage
Tipplate electrodes with two different dielectric liquid–solid interfaces are shown in
Fig. 2
. Case I has a diskshaped dielectric solid located on the cathode, which forms a vertical liquidsolid interface, and Case II has a parallel interfaces representing the direction of the liquidsolid tubeshaped dielectric solid between the anode and the cathode, which forms a parallel liquidsolid interface. The vertical and parallel interfaces are defined with respect to the direction of the electric field. The order of the initial electric field intensity around the tip was approximately 10
^{8}
V/m, which is sufficient for the initiation of streamers
[9]
.
Tipplate electrode model with two different liquid–solid interfaces. Case I has a diskshaped vertical dielectric solid and Case II has a tubeshaped parallel dielectric solid (with respect to the direction of the electric field).
Figs. 3
and
4
show the electric field distributions from 100 ns to 2.15 μs in Case I and from 100 ns to 325 ns in Case II, respectively. In Case I, there are three sections for the discharge propagation of the electric field wave:
Fig. 3(a)(c)
for the pure oil region with a propagation speed of 6.7 km/s;
Fig. 3(c)(e)
for the top surface with a propagation speed of 1.6 km/s; and
Fig. 3(e)(f)
for the side surface with a propagation speed of 0.23 km/s. In
Fig. 3(e)(f)
, the velocity significantly decreases due to the small electric field aligned to the surface slowing down the charge development. In Case II, the propagation speed of the electric field wave was 11.1 km/s, which is much faster than that in Case I.
Electric field distributions from 100 ns to 2.15 μs with the vertically aligned insulator in Case I. The propagation velocity of the electric field wave into the space in (a)(c): 6.7 km/s, (c)(e): 1.6 km/s, and (e)(f): 0.23 km/s.
Electric field distributions from 100 ns to 325 ns with the parallel insulator in Case II. The propagation velocity of the electric field wave into space in (a)(f) is 11.1 km/s.
Fig. 5
shows the temporal dynamics of the electric field and surface charge density with a step voltage of 20 kV and 100 ns rising time for Case I and Case II. With the tubeshaped interface, the propagating speed was approximately 11.1 km/s. In the literature
[12
,
13]
, the experimental propagating speed is approximately 12 km/s. Our numerical result has good agreement with that of the experimental results. The high accuracy of our numerical setup produced the velocity profile depicted in
Fig. 6
. The velocity was almost equal to the propagation speed in the breakdown region.
Temporal dynamics of the electric field and surface charge density with a step voltage of 20 kV and 100 ns rising time for Case I and Case II.
Distributions of normalized vector fluidic flow (as arrows) and temporal electric field (as surface plots) for different time steps.
Comparisons of breakdown velocity with tubeshaped interface
Comparisons of breakdown velocity with tubeshaped interface
4. Conclusion
A fully coupled FEA was successfully developed for electrohydrodynamics (EHD) analysis of dielectric liquid flow, including the surface charge density effects on dielectric flow guidance. The simulated speed of surface charge propagation was compared through experimental results shown in the literature. To analyze this complicated multiphysics problem, the generalized hydrodynamic driftdiffusion equations were composed of Poisson’s equation, the NernstPlank equation, the NavierStokes equation, continuity equations, and the energy balance equation. With the parallel guidance with respect to the direction of the electric field, the speed of discharge propagation was significantly enhanced and had good agreement with that of experiments.
Acknowledgements
This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF2013R1A1A2013111) and Kyungpook National University Research Fund, 2012.
BIO
HoYoung Lee was born in Korea in 1981. He received B.S. degrees in Department of Electronic Engineering from Keimyung University, Daegu, Korea, in 2009. He received his M.S. and Ph.D. degrees in electrical engineering from Kyungpook National University, Daegu, Korea, in 2011 and 2014, respectively. Presently, He is on a postdoctoral course in the School of Electronics Engineering at Kyungpook National University. His research interests are electromagnetic multiphysics system designs for Electric discharge, High voltage, MEMS, NEMS, energy harvesting, power and nano devices.
In Man Kang He received the B.S. degree in electronic and electrical engineering from School of Electronics and Electrical Engineering, Kyungpook National University (KNU), Daegu, Korea, in 2001, and the Ph.D. degree in electrical engineering from School of Electrical Engineering and Computer Science (EECS), Seoul National University (SNU), Seoul, Korea, in 2007. He worked as a teaching assistant for semiconductor process education from 2001 to 2006 at Interuniversity Semiconductor Research Center (ISRC) in SNU. From 2007 to 2010, he worked as a senior engineer at Design Technology Team of Samsung Electronics Company. In 2010, he joined KNU as a fulltime lecturer of the School of Electronics Engineering (SEE). Now, he has worked as an assistant professor. His current research interests include CMOS RF modeling, silicon nanowire devices, tunneling transistor, lowpower nano CMOS, and IIIV compound semiconductors. He is a member of IEEE EDS.
SeeHe Lee received his B.S. and M.S. degrees in electrical engineering from Soongsil University, Seoul, Republic of Korea, in 1996 and 1998, respectively. He received his Ph.D. degree in electrical and computer engineering from Sungkyunkwan University in 2002. He performed postdoctoral research at Massachusetts Institute of Technology (MIT) and worked for the Korea Electrotechnology Research Institute (KERI) before joining the faculty of Kyungpook National University in the School of Electrical Engineering and Computer Science in 2008. His research interests focus on analysis and design for Electromagnetic Multiphysics problems spanning the macro to the nanoscales.
Chang J. S.
,
Kelly A. J.
,
Crowley J. M.
1995
Handbook of Electrostatic Processes
Marcel Decker, Inc.
Kim H. K.
,
Chong J. K.
,
Lee S. H.
2014
"Analysis of SLF Interruption Performance of SelfBlast Circuit Breaker by Means of CFD Calculation,"
Journal of Electrical Engineering & Technology
9
254 
258
DOI : 10.5370/JEET.2014.9.1.254
Ahn H. M.
,
Kim J. K.
,
Oh Y. H.
,
Song K. D.
,
Hahn S. C.
2014
“Multiphysics Analysis for Temperature Rise Prediction of Power Transformer,”
Journal of Electrical Engineering & Technology
9
(1)
114 
120
DOI : 10.5370/JEET.2014.9.1.114
Mizutani T.
2006
“PulseSequence Analysis of Discharges in Air, Liquid and Solid Insulating Materials,”
Journal of Electrical Engineering & Technolgy
1
(4)
528 
533
DOI : 10.5370/JEET.2006.1.4.528
Lee S. H.
,
Lee S. Y.
,
Park I. H.
2007
“Finite Element Analysis of Corona Discharge Onset in Air with Artificial Diffusion Scheme and under FowlerNordheim Electron Emission Condition,”
IEEE Trans. Magn.
43
(4)
1453 
1456
DOI : 10.1109/TMAG.2007.892469
Lee H. Y.
,
Lee S. H.
2011
“Hydrodynamic modeling for discharge analysis in dielectric medium with the finite element method under lightning impulse,”
Journal of Electrical Engineering & Technology
6
(3)
397 
401
DOI : 10.5370/JEET.2011.6.3.397
Lee H. Y.
,
Jung J. S.
,
Kim H. K.
,
Park I. H.
,
Lee S. H.
2014
“Numerical and Experimental Validation of Discharge Current with Generalized Energy Method and Integral Ohm’s Law in Transformer Oil,”
IEEE Trans. Magn.
50
(2)
7006204 
George Hwang J.
2010
“Effects of Nanoparticle Charging on Streamer Development in Transformer Oil Based Nanofluids,”
J. Appl. Phys.
107
(1)
014310 
DOI : 10.1063/1.3267474
O’Sullivan F. M.
2007
A Model for the Initiation and Propagation of Electrical Streamers in Transformer Oil and Transformer Oil Based Nanofluids, Ph.D dissertation
Massachusetts Inst. of Tech.
Cambridge, MA, USA
Lee H. Y.
,
Kim Y. S.
,
Lee W. S.
,
Kim H. K .
,
Lee S. H.
2013
“Fully Coupled Finite Element Analysis for Cooling Effects of Dielectric Liquid Due to Ionic Dissociation Stressed by Electric Field,”
IEEE Trans. Magn.
49
(5)
1909 
1912
DOI : 10.1109/TMAG.2013.2246551
Melcher J. R.
1981
Continuum Electromechanics
MIT press
Cambridge, Massachusetts
Massala G.
1998
“Positive Streamer Propagation in Large Oil gaps: Experimental Characterization of Propagation Modes,”
IEEE Tran. Diel. Elec. Insu.
5
(3)
360 
370
DOI : 10.1109/94.689425
Massala G.
,
Lesaint O.
1998
“Positive Streamer Propagation in Large Oil gaps: Electrical Properties of Streamers,”
IEEE Tran. Diel. Elec. Insu.
5
(3)
371 
381
DOI : 10.1109/94.689426