Preliminary observations on bulk electroconvection in electrically stressed liquid insulants

Journal of Electrostatics, 9 (1980) 1--14

1

© Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands

PRELIMINARY OBSERVATIONS ON BULK ELECTROCONVECTION IN ELECTRICALLY STRESSED LIQUID INSULANTS PART II: THEORETICAL INVESTIGATION

J.M.H. PETERS

Department of Mathematics, Faculty of Engineering, Liverpool Polytechnic, Liverpool L69 3BX (Gt. Britain) J.L. SPROSTON

Department of Mechanical Engineering, The University of Liverpool, Liverpool L69 3BX (Gt. Britain) and G. WALKER

Department of Applied Mathematics and Theoretical Physics, The University of Liverpool, Liverpool L69 3BX (Gt. Britain) (Received March 8, 1979; accepted in revised form October 1, 1979)

Summary Electroconvective motions which occur in the bulk of insulating liquids under electric stress are considered. From an examination of the theoretical structure of the problem, it is suggested that dielectrophoretic forces, acting both in the liquid bulk and on the polyethylene particles used to visualise the motions are negligible and that the motions are induced by the Coulomb force acting on free charge in the liquid. The governing equations were solved using a finite difference scheme and the results give reasonable support for the suggested shearing model which was presented in Part I.

Notation a

Cp d

E

f,(D f(p)

fe J L P qE

Radius of conducting sphere Specific heat at constant pressure Distance used in eqn. (8) Electric field vector Force in bulk of liquid Force on a particle Force of electrostriction in the liquid Current density vector Reference length Dynamical pressure Charge in eqn. (6)

qv q v (p)

qc R

Rw t tc T U U U Z

6 e0 ep el

# P CO (.Oz

¢ T G

p II g

Volumic charge density Volumic charge density on particle Contact charge Reynolds number or gas constant Reynolds number at wall Time Contact time Temperature Reference speed x-component of velocity y-component of velocity Valency of charge species Viscous coefficient in eqn. (5) Diffusivity Permittivity Permittivity of free space Permittivity of particles Permittivity of liquid Dynamical viscosity Stream function Kinematic viscosity Vorticity vector z-component of vorticity Electrostatic potential Relaxation time Conductivity Density See eqn. (14) Zeta potential Mobility

Introduction In a previous paper [ 1], photographs of the bulk motions induced in liquid insulants under electrical stress from external fields of the Order of 20--100 kV/m were presented and discussed. In the present paper consideration is given to those mechanisms which may have provided a driving force for these motions and those which may have influenced the flow visualisation particles in the liquid. Two effects related to electromechanical phenomena are magnetism and Joule heating. The role of the former in electroconvective processes in insulating liquids was shown to be negligible by Reynolds [2]. Its dismissal means that the electric field vector E is irrotational. The low conductivity, o, of liquid insulants also precludes Joule heating as an effective driving force,

since the rate of change of temperature is given by (see for example [3] ): (1)

d T / d t = oE~/pCp

where p and Cp are the density and specific heat at constant pressure of the liquid. Substitution of values typical to the present investigation into eqn. (1) gives: d T / d t = 1.3 × 10 -7 K/s

which is negligible. The total volumic force f o ) of electrical origin acting in the bulk of a fluid dielectric of permittivity e, under electrical stress from an external field E, is given by (see for example [~] ):

: qvE-

1

rE 2 [ae~ "l

E2Ve + v/Tp

jT j

(2)

The first term on the R.H.S. of (2) represents the Coulomb force which acts on charged ions in the fluid. The second term is the polarisation force which is due to the presence of dipoles in the liquid. The third term accounts for the force of electrostriction. For non-polar liquids such as carbon tetrachloride and kerosine (used in [1] ), which satisfy the Clausius--Massotti relation, the force of electrostriction becomes: 1 [(e-%)(e

+ 2e0)] V2E

(3)

where e0 is the permittivity of free space. This means that with liquid mixtures, components with different permittivities m a y separate with the consequent formation of an interface between them. It is likely that such an interface would be highly unstable in the presence of steady fields because of the accumulation of electric charge there. Such instabilities have been investigated by Reynolds [2] and Taylor [5] for cylindrical and plane interfaces respectively stressed by steady and low frequency electric fields. The fact that no observable motion occurred in any of the electroconvective cells described in [ 1], when the liquid they contained was stressed with a.c. fields of 50 Hz and up to 100 kV/m in strength, suggests that interfaces did n o t occur in the liquid because of electrophoretic separation. If t h e y had f o r m e d in spite of this, then their presence had no detectable effect. Taking the curl of eqn. (2) gives, together with the result that the electric field g is irrotational, vxf°)

1 = VqvXE+~V eX

VE 2

(4)

which means that a rotational body force depends upon a spatial distribution of charge or permittivity in the liquid. If there is no temperature gradient, both the permittivity and the conductivity remain uniform in the liquid bulk. Hence there can be no rotational force contribution from the polarisa-

tion force. Also if the conductivity is constant, the volumic charge vanishes from the liquid bulk in a very short time even though the liquid may be moving (see Melcher [6] ). We conclude from this that for the motions described in [1], the rotational effect was due to forces acting at the boundary of the liquid and n o t in the bulk. There is the possibility that the polyethylene particles used to follow the liquid motion may have been influenced by the external electric field. The volume force acting on a particle is given in [7] : f(P) = qv (p) E + ~ ( e p - e l ) A- 1 V E 2 + ~ U

(5)

where qv Cp) is the volumic charge density on the particle and ep and el are permittivities of the particle and liquid respectively. The remaining parameters are described below. The first term on the R.H.S. of (5) represents the Coulomb force which acts on a charged particle. In discussing its nature, attention is confined to spherical particles even though the polyethylene particles were n o t spherical. It can be shown that the charge qE acquired b y a small conducting sphere of radius a, in contact with a plane electrode in a medium of permittivity e, subjected to an electric field of strength IEI, is given b y [8] : 2

(6)

qE = ~ II3ea2 IEI

Inserting typical present values gives qE ~ 0.2 pC. The charge q acquired by an insulating particle in contact with a plane electrode for a time to, is given by: q = [1- exp(-tc/r)]qE

(7)

where r is the relaxation time of the insulating material of the particle. Since the time of contact tc was negligible compared with the relaxation time r cff the polyethylene (21.5 days), it is clear that in view of the smallness of qE, for all practical purposes, the flow visualisation particles acquired no charge from the electrode by this process. A second charging mechanism which operates when a particle touches an electrode, even in the absence of an electric field, is that of contact charging [9]. The contact charge acquired by a conducting sphere of radius a, with its centre at a distance d from a conducting infinite plane in a medium of permittivity e, can be shown to be oo

qc = 4~ae[~¢p) - ~b~e)] sinha ~

cosech na

(8)

nffil

where cosh a = d/a, and ~(p) and ~bce) are the electric potentials corresponding to the work functions of the materials of the particle and the electrode respectively. The work function for steel is 4.4 eV and Davies [10] has suggested 4.7 eV for polyethylene. Thus for a conducting sphere of diameter 70 pm, with the same work function as polyethylene, at a distance of 1 A

from a conducting earthed plane, the contact charge would be of the order of 0.04 pC. However, since the polyethylene is a good insulator, the actual contact charge it gained is likely to have been much less than this (see eqn. (7)). The second term on the R.H.S. of eqn. (5) represents the polarisation force acting on the particles. For spherical particles the dyadic quantity A is given

by A = [(2 eI + ep)/3 e l ] [

(9)

I being the unit dyad, [7]. This gives the total polarisation force acting on a dielectric sphere of radius a, as: F

=

2~a3el(eP

-

el) VE 2

(10)

2el + ep Again, introducing present values into this gives a polarisation force equivalent to 10 -~ dynes, with a corresponding speed (according to Stokes' law) of 7 × 10 -4 mm/s. This is negligible compared with the recorded speeds which were of the order of millimetres per second. The final term on the R.H.S. of eqn. (5) accounts for the interaction between the external electric field and the diffuse double layer which surrounds a particle in a liquid. This interaction causes the particle to move with an electrophoretic speed U. The constant ~ is included in eqn. (5) to account for viscous drag. The electrophoretic speed U has been calculated for thick double layers surrounding perfectly insulating spheres by Henry [ 11 ]. He obtained the formula, U = 2 eE~/3#

(11)

where ~ is the zeta potential. With the previous values and a zeta potential of 170 mV [12], eqn. (11) yields an electrophoretic speed of 0.2 mm/s. Since the typical measured speeds at this field strength were of the order of 6 mm/s, this result accounts for 3% of the motion. In summary, it is suggested that dielectrophoretic forces acting in the liquid bulk and on the polyethylene particles were negligible and that liquid motion was induced by the Coulomb force acting on free charge in the liquid. It is also concluded that the speed of the particles as they followed the motion was in error by at most 3% of their measured speed, this being due to the electrophoretic interaction between the external electric field and the diffuse part of the double layer surrounding the particles. The governing equations The electrostatic interactions between liquid insulants and electric fields under isothermal conditions, are described by the laws of conservatiQn of mass, momentum and electric charge, together with Maxwell's electrostatic equations and Gauss' law. The conservation of momentum requires the inclu-

sion of effects due to the Coulomb force which acts on free charge in the liquid and which is transmitted to the liquid mass by friction, and the electrostrictive force which modifies the pressure distribution. Since the bulk permit~ tivity is constant, there is no bulk polarisation force. The conservations of mass and m o m e n t u m for incompressible flow in electrically stressed liquid insulants are respectively expressed by: V'v = 0

(12)

and p Dv/Dt

= - V I I - pg]e + p V2v + q v E

(13)

where p, # and v are the density, viscosity and velocity of the liquid and

~1 E2p(ae/Op)T

II = P -

(14)

accounts for the dynamical pressure P and the effects of electrostriction. Also included in eqn. (13) is the gravitational volume force, in which g is the gravitational acceleration and k is a unit vector directed vertically upwards. Since magnetic effects are neglected, it therefore follows from Maxwell's electrostatic equations that: V X E

=

0

(15)

and therefore:

(16)

E = -V¢

Gauss' law relates the electric field vector E to the volumic charge density qv. In differential form it is: V" ( e E ) = qv

(17)

When the permittivity is constant, eqns. (16) and (17) m a y be combined to give Poisson's equation. Finally the conservation of charge is described by: = 0

Oq~/~t+V.J

(18)

where the current density vector ] includes the electric currents of Ohmic conduction, convection and diffusion. It is given by: J = oE + qvv-6Vqv

(19)

in which 6 represents a mean diffusivity associated with ionic diffusion and in which the contribution to the conductivity o of each charge species is the product of its mobility ~ and the associated volumetric charge density qv. For a given species, the ratio of diffusion and conduction currents is 8[Vqvl/olgl

~ ~ /LiKEI

(20)

in terms of a reference length L. The diffusivity 5 is related to the mobility K by Einstein's relation: /K = R T / z F

(21)

where R and F are the universal gas constant and the Faraday respectively, T is the absolute temperature and z is the valency (algebraic sign included) of the charge species. At an ambient temperature of 18°C in the presence of a field of l 0 s V/m, eqns. (20) and (21) give: ~]V qv]/o[EI ~ 2.5 X l O - 7 / L I z l .

For low conductivity liquids with a non-uniform conductivity, gradients in the liquid bulk are relatively weak, so that L is likely to be much larger than 10 -7 m, in which case diffusion currents in the bulk away from electrical boundary layers (diffuse charge layers) m a y be neglected. In view of this, the remaining terms in eqn. (19) may be substituted into eqn. (18) to yield: (22)

D q v / D t + V" (oE) = 0

With eqn. (17) this reduces to: D q v / D t + qv/r = 0

where r = e/o is the liquid relaxation time which is taken to be constant. When this equation is multiplied by qv and integrated over the volume of the liquid and use is made of the divergence theorem, it is found that 1

2

21 ot fvqv dv+ fs q" o.as

+1

fvqv dV = o

where vn is the normal c o m p o n e n t of liquid velocity across the element of surface dS. At the liquid boundaries Vn will be zero and hence the above surface integral will vanish. Subsequent integration then yields:

fvqv 2 d v = exp(-2tl,) fvqv2(O)dV where q.(O) denotes the volumic charge distribution at the instant the electric field is applied. The relaxation times of the liquids under investigation were of the order of 3 seconds. Thus for t >> 1.5 s, we may conclude that the charge in the liquid bulk, away from any boundaries, has vanished in spite of the fact that the liquid moves; so in the case of constant conductivity, there is no bulk volume charge after times of electric stress measured in minutes. This is in accord with the findings of several authors (see for example [13,14] ). The boundary conditions are obtained in the usual way by integrating the bulk equation across a boundary. However, they are n o t applied here owing to the difficulty of prescribing the charge at the edge of the electrical boundary layer. Hence, as a first approximation, it was decided that the bulk conductivity of the liquid be treated as constant so that there would be no bulk volume charge. This does n o t preclude the existence of charge layers at the boundaries, the thickness of which would be of the order of the distance covered by injected charge during the relaxation time r. We thus propose to replace the electrical boundary condition with a dynamic one related to electrical shear at the wall.

8 Proposed model The electroconvective motions observed in the bulk of the liquid sample (as described in [ 1] ) remained essentially two-dimensional. Furthermore, the laminar flow regimes were steady, so that the streaklines of the particles coincided with streamlines in the liquid. Following Cade and Row [15], the problem can thus be formulated in terms of the two-dimensionai distributions associated with a stream function ~ and a vorticity function co, the motion being generated by an electric shear at the wall. For two-dimensional flow, the components of velocity m a y be expressed in terms of the stream function ~ by: u = a~/ay

(23)

v = -a$/ax

(24)

Taking the curl of both sides of the m o m e n t u m equation (13), gives for a two-dimensional motion: (25)

D t ~ / D t = v V26o

where v is the kinematic viscosity and 6~ is the curl of the velocity, viz: 60 = V × v

(26)

Introducing (23) and (24) into (25) and (26), and using the fact that in two dimensions = ¢o~/~

(27)

give s: \ax 2 and, a2¢, ~)X2

ay 2 ]

ay ax

ax ay /

a2~ +

ay 2

-

~

(29)

With the dimensionless variables defined as follows, Xo = x / L ,

Y0 = y / L ,

Co = $ / U L

and

U~o = L c o z / U

L being a reference length based on the cell dimensions and U a reference speed based on the m a x i m u m shearing speeds observed in the region of the cell walls, the eqns. (28) and (29) become, on dropping the zero subscripts, a2¢o - -

ax 2

and

a2¢o +

~y2

( a ~ au~ -

R

ax

a~ a ~ ) ax

(30)

a2 $

a2 +

ax 2

-

co

(31)

ay 2

where (32)

R = UL/v

is the Reynolds number. It was decided to solve eqns. (30) and (31) for flow in a square cavity, numerically, subject to boundary conditions on ~ consistent with the presence of prescribed shearing velocities at the walls of the cavity. These follow from the condition of no slip and no normal flow, together with eqns. (23) and (24). T h e y are: a~P lan = Uo

(33)

and = 0

(34)

at the walls of the cavity, where n denotes the variable with preferred direction normal to the wall and IU01 < 1 is the prescribed dimensionless shearing speed. Numerical

results

Equations (30), (31), (33) and (34) were solved with a numerical procedure suggested by Greenspan [16]. Here, the analytic equations and their region of definition were approximated by difference equations on a rectangular mesh. The resulting finite difference scheme was solved iteratively with a modification of the corresponding computer program listed in Greenspan's text. The experimental investigations described in Part I [1] suggest that in the split-electrode short cell, a shearing force acted in the region between the electrodes, from the earthed to the positive one. The Reynolds number based on speeds measured close to the wall (Rw) was about 150. The shear model used to approximate the flow is indicated in Fig. 1 and the streamline picture corresponding to the numerical solution is shown in Fig. 2. The shear model proposed for the flow occurring in the parallel-electrode cell with the electrode edges outside the cell is illustrated in Fig. 3. Here a unit shearing speed was imposed at the top wall of the cavity while that at the b o t t o m wall was maintained at a fraction of this. In the streamline picture presented in Fig. 4, this fractional dimensionless velocity was 0.25. The wall Reynolds n u m b e r was 75. The flow patterns observed experimentally in [1] in the spindle-electrode cell were reproduced numerically with the shear schemes depicted in Figs. 5 and 7. Here a shearing of unit velocity was assumed at each wall o f the cavity with the direction o f flow at the top and right-hand walls depending on

10 10

08

U=I

O0

0'2

014

0.6

0 ~'8

10

Fig. 1. Split-electrode short cell; shear flow model; R w = 150. Fig. 2. Split-electrode short cell; streamline picture.

10

0-8 U=I 0-6

04

,II--

02

U=.25

0.0

f

,

O0

02

'

0L

Fig. 3. Parallel-electrode cell, edges outside; shear flow m o d e l ; R w = 75. Fig. 4. Parallel-electrode cell, edges outside; streamline picture.

'

0.5

O-8

10

11

10 f J J 0"6

U = -1

0'4 Uffil

U= --]

0"2

J 0"0 0"0

i=

U=I

012

04

0-6

0"8

1"0

ols

~o

Fig. 5. Spindle-electrode cell; shear flow m o d e l for l o w voltages; R w ffi 25.

Fig. 6. Spindle-electrode cell; low voltage streamline picture.

1"0

0"8

U='~ 0"6

O-L,

U=I

U=I

0'2

U=l

O'C

oo

o!2

o'~

o16

Fig. 7. Spindle-electrode cell; shear flow m o d e l for high voltages; R w = 75. Fig. 8. Spindle-electrode cell; high-voltage streamline picture.

12 whether the voltage was low or high. The reproductions are shown in Figs. 6 and 8. Unfortunately convergence could only be obtained with wall Reynolds numbers lower than those actually observed. However, further investigation with the c o m p u t e r program revealed that the streamline picture corresponding to a given shearing scheme remained essentially the same for all Reynolds numbers for which convergence could be obtained. The wall Reynolds numbers for the t w o patterns in Figs. 6 and 8 were respectively 25 and 75, whereas the measured ones were of the order of 80 and 125. The resemblance between the predicted streamline patterns and the corresponding photographs of the streaklines gives reasonable support for the suggested shearing model. It is concluded therefore, that charge injection from the electrodes and its subsequent relaxation to the liquid boundary where it interacted with the electric field, was responsible for the well-defined flow patterns. In the split-electrode short cell, it is believed that electrons injected from the cathode and their movement in the region between the electrodes close to the wall, produced sufficient shear to cause the laminar flow patterns in the photographs in Figs. 5 and 6 of Part I [1]. This is indicated by the way in which the centre of the vortex was thrown downwards, which shows that the shearing force was also directed downwards, away from the cathode. There is also the possibility that the liquid became positively charged at the anode due to electron depletion there. However, the photographs suggest that the shear force arising from this source was dominated by that associated with electron injection since again the overall shear direction was away from the cathode. Also further investigation with the computer program revealed that with oppositely directed shearing forces at a single wall of the cavity, meeting at the mid-point of the wall, t w o large vortices occur in the top and b o t t o m half of the cavity, which does n o t correspond to the photographs of this case. The streamline picture in Fig. 4 bears a resemblance to the streakline pat~ tern in the photographs of Figs. 9 and 10 of [1], the liquid close to the upper wall of the cavity was repelled from the cathode due to the presence of negative charge there. The streamlines in Fig. 4 are also instructive because they show h o w the position of the main vortex is displaced from the centre line in the same direction as the shear. Note, however, that Greenspan's work indicates that the degree of this displacement is reduced as the wall Reynolds number increases. The best support for the suggested shearing model is provided by the streamline pictures in Figs. 6 and 8. Here it is suggested that at low fields the liquid close to the glass walls of the cavity becomes positively and negatively charged at the anode and cathode respectively. The resulting shearing forces would then be directed as suggested in Fig. 7. Note that Mackey [17] observed liquid streaming away from both positively and negatively charged electrodes. At higher field strengths, negative or positive charge could have become dominant in the wall region and the resultant interaction would have caused the alternative shearing mechanism depicted in Fig. 8.

13 Conclusions

Evidence has been presented which suggests that the basic driving force for electroconvection in the bulk of insulating liquids under electric stress from steady electric fields, is provided by the interaction between free charge, present in the liquid, and the external electric field. The characteristics of the motion depend upon the distribution of this charge in the liquid. This distribution is affected by the electrical relaxation time of the liquid and charge injection from the electrodes. The similarity between the streakline photographs of the steady, well defined, electroconvective motions at relatively low field strengths and the streamlines corresponding to the two-dimensional flow induced by given shearing velocities at the walls of the square cavity, suggests that in certain instances bulk electroconvection is generated by electrical shear stresses in the liquid adjacent to the non-conducting walls of its container. From this it is concluded that when the rate of charge-injection from the electrodes is less than that due to charge-relaxation, the injected charge is swept from the liquid bulk by the electric field, to the liquid boundaries, where it reinforces the diffuse charge in the double layer. The overall movement of this charge, in a direction parallel to the liquid boundary, by the similarly directed component of the electric field, then produces sufficient shear for the motions in the bulk of the liquid. To explain the alternative flow-regimes which occurred at relatively high field strengths, it is suggested that when the rate of charge-injection exceeds the charge-relaxation rate, its electrically induced movements in the liquid bulk creates vorticity there. Acknowledgment The authors wish to acknowledge the help and advice given by Dr. J.C. Gibbings throughout the present investigation. References 1 2 3 4 5 6 7 8 9 10 11

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