A numerical simulation of non-stationary electrohydrodynamic processes in weakly condcting liquids

Journal of Electrostatics, 23 (1989) 431-439 Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands

431

NUMERICAL SIMULATION OF NON-STATIONARY ELECTROHYDRODYNAMIC PROCESSES IN WEAKLY CONDUCTING LIQUIDS

A

A.A. VARTANYAN I , V.V. GOGOSOV 1 , V.A. POLYANSKY I and G.A. SHAPOSHNIKOVA 2 I Institute of Mechanics, Mos©ow Unlve~sity, 119899 Moscow (USSR) 2Institute of Pine Chemical Technology, 119435 Moscow (USSR) SUMMARY The non-stationary phenomena in weakly-conducting liquids, immersed in the electric field, are studied. The system of equations is proposed, which simulate the weakly-conducting liquid by a three-component medium consisting of a non-charged liquid and both positive and negative ions. The model takes into account the volume and surface electrochemical reactions, the concentration diffusion of ions and their drift in the electric field. The main objective of simulation consists in: explaining the available experimental data on the non-stationary passage of electric current, :peculiaritles of currentvoltage characteristics and the formation of near-electrode structures of a volume electric charge, obtained for various weakly-conducting liquids in cells with planar electrodes; the development of methods for determining the properties of liquids, such as the conduction, the coefficients of mobility of ions, the constants of volume and surface electrochemical reactions from the experimental data on a non-stationary passage of current and current-voltage characteristics, INTRODUCTION The processes occurring in weakly-conducting liquids under an effect of electric fields are considered. The examples of liquids under study are oil, liquid hydrocarbons, transformer oil, organic solvents, etc. The experimental study of electrohydrodynamic parameters of such liquids is often carried out in electrolytic cells with plane-parallel electrodes to which the constant or variable-in-time difference of electric potential is applied from externalscu~ces. The measurements include the determination of time dependence of electric current through a cell, the dependence of current on the applied difference of electrical potentials (the current-voltage characteristic), the distribution of electric field intensity in the inter-electrode gap. Depending on properties of a liquid and on the material of electrodes the qualitatively different time behaviour of current is observed during 0304-3886/89/$03.50

© 1989 Elsevier Science Publishers B.V.

432

pulse applying the potential difference: the current may either grow, or drop, or non-monotonously change with time. The non-linear relation between the current and applied potential difference is also observed. When the difference of electrical potential changes, the hysteresis of current is observed in some liquids. The distribution of volume charge density in the near-electrode regions can be found from measured distributions of the electric field strength near the electrode.

In this case, along with usual

Debye screening layers, when the "unipolar" volume charge layer, whose sign is opposite to the sign of an electrode, is formed nea2 the electrode, the charge layers have also formed (ref.1), whose sign coincided with the sign of charge at an electrode. Some more complicated "bipolar" structures also take place (ref.1), when the volume charge layer of the same sign, as at an electrode, is formed directly near the electrode, and then follows the charge layer of opposite sign. To interprete the listed experimental data and to develop the methods for determining the properties of weakly-conducting liquids and parameters characterizing the interaction of liquid with the surface of electrodes in strong electric fields, the numerical simulation methods are used in the paper. In these methods the real liquids containing, generally, many types of positive and negative ions (technical liquid dielectrics) are simulated by the medium consisting of a carrier liquid and a small admixture of two sorts of ions with effective number densities n+. This simulation implies that the kinetic and transport properties of any sort of positive and, respectively, negative ions are close to each other in their values, and one may introduce the effe~ tive coefficients of mobility

b~

and diffusion

D± , the effec-

tive rate W of volume ionization and the effective coefficient o~ of volume recombination of ions, as well as the effective parameters of surface electrochemical reactions. The used simulation is also valid, when the medium contains two prevailing sorts of positive and negative ions, and the contribution of other sorts of ions into the current value and into the volume charge is small. The latter situation is specially created in some experimental studies by solving in a liquid some specified amount of admixture of iodine, butyl alcohol type, etc.

433 S TA~.NT

0F THE PROBLEM

S~stem of equations Let the liquid Be moveless in a cell with plane-pa~allel trodes and temperature

T

elec-

be constant throughout the liquid bulk.

The distance L between electrodes is assumed to be much less than the characteristic linea~ dimension of electrodes and, as a result, all variables depend on time and on a single space coordinate

x

only, axis

x

being perpendicular to an electrode.

Write down the system of equations which are satisfied in onedimensional non-stationary case by distributions of densities of ions E

n± , of electric potential

~

and electric field strength

(ref.2) :

2~-n~o'+ = W - ~ n + n _

;

(1)

8~ = 4~e (~_ + n+),

2"---~

8~

8~e e n÷

-

7.,"e

=

- ~.~- OO'n± a~

-

W=

N

if:an_+

-'

~_+ Q ~ T / e , =

~

o8~o ~x

~'~-

,~

=

(2)

" '

(3)

•.

8~

~ r e (#÷ -,- #_) / e

"

;

¢5)

(6)

Equations of continuity (I) take into account the volume reactions of dissociation of weak electrolyte molecules into positive and negative ions and the recombination; in strong electric fields the effective ionization rate W may depend on the field strength (ref.3) in accordance with (6), where Iz~ is the density of neutral molecules of a weak electrolyte, g is the dielectric permeability, e is the proton charge, k is the Boltzmann constant. Coefficient ~ equals I or 0 depending on the fact, whether the W(E) dependence is taken into account or not, respectively. Current I , measured by an instrument, is determined by the full current density is' I = is.S, where S is the electrode area. The expression for the total current density Js (4) includes conductivity currents J+ and the displacement current which may be essential in non-stationary processes.

434 Boundary conditions We

shall suppose that

at the electrode-liquid interfaces the

surface electrochemical reactions of the first order take place. During these reactions of A + + e ~-~A, B- - e ~ - B type the ions and neutral molecules birth and die. These processes will be simulated by specifying at electrodes for

x = O, T, the den-

sities of ion fluxes which are defined by some specified effective parameters • O,L describing the birth and death of a+O,L , ~± ions, respectively. These parameters may depend on the field strength. Besides, the electric potential is specified at electredes. The boundary conditions are

x=O,

~-- ~ w f t )

x=L,

9~ = 0

,

,

o

o

n+ ~÷ = a_+ - ~ : n~

;

L

n ~

=-as+_ + ~ n + _

(7)

(8)

In solving the non-stationary problems the distribution of ion densities n~o (t o , x) is specified at the initial time moment t o. As n~o one can take either the constant equilibrium value of a quasineutral ion density neq = (We/ so )I/2 in the absence of applied field, or the stationary distribution numerically obo tained for some small value ~ w ~e ~ / ~ T ~ < I' ~). The numerical solution method The problem governed by equations (I)-(8) is numerically solved. The implicit-in-time difference scheme with a variable step in x is used for this purpose. The non-linear system of difference equations is linearized with using iteration, the linear three-diagonal system obtained is solved by Gauss method. Dimensionless parameters The system of equations and boundary conditions (I)-(8) is written in the dimensionless form, where the following quantities are taken to be characteristic ones: the interelectrode distance L , the equilibrium density neq , the chemical equilibrium establishment time ~ ch = (We ~o )-I/2, the characteristic diffu sion flux neqD/L and the thermal potential ~ t = kT/e. Besides, one may introduce into the problem the characteristic diffusion time ~ d' the ion drift time ~ e ' the charge relaxation time W r' the equilibrium Debye distance r d and the liquid conduction

~

= Z 2

~

.9.

: = L___4__

" ~

~oEo

=

" %

~

~'~

~_

" %

~7"

~-"ue~ ~ 9

(9)

435

,

,

=

The equality

~

= ~

%

=

(9)

is met when the Langevene formula (5)

is used for the characteristic value of recombination coefficie nt

~o The normalization in equations and boundary conditions gives

rise to the following dimensionless parameters:

~.

=

,z-~

~'~

_

~ d

2Z ~

G

-

~'~ ~'~

~'ch

'

~'e

' (io)

Per liquids with conduction 10 -15 meter

~

<

~o < 10 -12 Sm/cm para-

is of the order of 10 -3 • 10 -5 (for L = 0.1 cm, D O --

= 10 -5 cm2/s). This parameter characterizes narrow nearelectrode diffusion layers of ions whose thickness is of the order of ~ Parameter

G

is the ratio of charge relaxation time (or chemi-

cal equilibrium establishment time) to the time of ions drift through the interelectrode gap under the field action. The value of this parameter depends on the applied field and may vary within a rather wide range. S OLUTION RESULTS The values of parameter

G

<<

I

and

G

~

I

result in qua-

litatively different pictures of distributions of ion densities and electric field strength in the interelectrode gap. In the first case the densities of ions throughout the liquid bulk, except narrow near-electrode layers, are close to equilibrium ones, and the field strength only

slightly differs f~em the

strength E o = ~ / L . The current-voltage characteristics, constructed for various time moments after applying the voltage, are practically linear, only slightly differ from each other and may be used for determining the liquid conduction with using formula ~o = IL/ ~ w S. In the narrow nea~-electrode layers the distributions of ion concentrations and field strengths essentially depend on the relation between parameters of surface and volume processes, i.e. on the value of parameter rameter is Just the ratio

O

. This pa-

of densities of a characteristic flux

of ions, injected from the electrode surface, to the characteris-

436

tic drift flux of ions in the applied field E oFor ~ < f the layer of negative chaxge of thickness of the order of G I/2 is formed near the anode, and the layer of positive charge is formed near the cathode. As parameter

~

grows up

to the values of ~ ~ ~crit 7 1, the sign of a volume cha~ge is near-electrode regions changes: the charge becomes positive nea~ the anode and negative near the cathode. For intermediate values I ~ ~ < ~crit the two types of bipolar structures of volume charge may form near the electrodes. In the first type of structures, which was observed in the experiment (ref.1), the charge of the same sign, as the electrode, is formed directly near the electrode surface and then, when moving from the electrode, the charge sign changes to opposite one. In the second type of structures the oppositely charge layer is adjacent to the electrode, and then follows the layer with the same sign of charge as the electrode. The last type of structures has not been experimentally observed so far. The evolution of distributions of volume charge's dimensionless density q = n+ - n _ ( n = n /n eq_) nea~ the anode with changing parameter ~ for low G (G = 0.132) is shown in Fig.1. The .

calculations were carried out for the following values of liquid's parameters: ~ = 6.88.10 -3 Sm/cm, b~ = 3.7"10 -5 cm2/V s, W = = 2.3.1011 cm-3s -I, ~ = 6.10 -11 cm3/s, neq = 0.5.1011 c~ -3, 6 = 2.23, L = 0.1 cm, T = 300 K. These paxameters correspond to a liquid caxbon tetrachloride. The data are presented for the applied potential difference

~ w = I03.4V, which produces the ap-

plied field of 1.O34 kV/cm. The time dependence of current is given in Fig.2. For a weak injection of ions f~om the electrode surface ( 8 = 0.25) the current drops in time after applying the potential difference and in the case of strong injection ( ~ = 3) the current grows. In this case the initial value of current density differs from the stationary value (at t --~ ~ ) within the range of 10-15%. The liquid's conduction value can be determined with such an error from the slope of a current-voltage characteristic constructed for each moment of time. Note that the conduction value ~o is dete~mlned by the value of current Js at initial time moment

(

~o ffi jsT,/

~ w )"

For G > > I the concentrations of ions in the bulk of weaklyconducting liquid may essentially differ f~om equilibrium ones both due to violation of chemical equilibrium unde~ an effect of

437

a

0,8

?. ~o"

•

8

"x"o'4

-

.

-2"

,

-0,~

,

0,2

-~

-0~8

0,~

0,00t

Distance, x/L

0,00;~

-e

Distance, x/L

Fig. I. The evolution of stationery distributions of a volume charge density near the anode w~th parameter variation for G = 0.132, ~v~= I03.4V; I - v = 0.25, 2 ~ = 0.75, 3 - ~ = 1.75, 4 - ~ = 3; a) general view, b) detailed picture of nea~-anode layers.

b

8,5

7 0 '

o

'

'

'

i

Time, t / ~ch

,

'

~ 8,0~

'

'

Time, t /

~ch

Fig. 2. The non-statione.~y v e ~ i a t i o n of a f u l l current density ~s e~e~ applying the p o t e n t i ~ d i f f e r e n c e at G = 0.132, ~w = --.I03.4V for various values of ~ : a) 8 = 0.25, b) ~ = 3, Js = Js/Jo ' Jo = kTboneq/L"

438

electric field, and due to flow (injection) of ions born at the surface of electrodes. Depending on parameter

G

the values of

current, passed through the cell, may either increase ~ 8 > #~ or decrease ~ @ -~ i~ this case the

after applying the potential difference. In

Js v s t

dependences may be non-monotonous. The

time of establishment of stationary distribution of parameters in the cell may several times exceed the establishment time for the case

G < < I.

A large series of calculations within a wide range of values of applied potential difference has shown that the current-voltage characteristics in weakly-conducting liquids can be essentially non-linear in the absence of convective motion in a cell only in the case, when either the volume rate of ion formation or the surface reaction parameters are dependent on the field strength. Within the framework of the model considered one can explain the experimental data on the non-monotonous time

variation of

a current passed in a cell with the stepwise change of electrodes polarity (ref.4). Fig. 3 shows the time dependences of current for two qualitatively different cases: a) G = 1.2,

0 = 0.55,

~ w = 1810V; b) G = 1.32, ~ = 2.25, ~ w = 2069V. At the initial time moment (t~ = O) some constant potential difference is applied to electrodes. Then, depending on the ~

W

parameter va-

lue, the current either decreases ~ O < # ) , or grows ~ 8 > I ~ to some stationary value. At some dimensionless time moment

up t I -= 2.5 the stepwise change of electrodes polarity occurs. I~ is seen that after such a change in the applied potential difference in the first case ~ @ < i~ the current grows in magnitude and then drops down to a stationary value, which is equal in magnitude, but opposite in sign to a previous stationary value. In the strong injection [ 0 >

I)

the character of variation of current

is opposite. Hence, these experimental data allow to evaluate qualitatively the contribution of surface reactions into the current passed in a cell. Note that the non-monotonous time variation of current is typical for weak electrolyte solutions only. The numerical analysis also shows that for calculating the coefficient of mobility of ions from the experimental data on the current reverse in a cell one should use the total time of achieving the new stationary value of current, rather than the time of achieving the current extremum, as it is sometimes applied

439

(~ef.5). This is due to the fact that the time of achieving the current extremum corresponds to the time of re-formation of narrow diffusion layers near electrodes, which may be considerably less than the ion flight time through the interelectrede distance. 8-

a

_

/,--

b

o

o .

•

m

O-

D

O-

.r-i

a)

1.1 r,...i

-8 0

I

I

,,

I

2

I

l

6

Time, t/ ~'ch Time, t/ t~'ch Fig. 3. Non-stationary profiles after applying the potential difference (at t~ = O) and after ste/owise cha~ge of electrodes p~larity (at t~ = 2.5): a) G = 1.2, ~ = 0.55, ~ w = 1810 V; b) = = 1,32,

8 ,= 2,.25,

~ w = 20691/'.

REFERENCES

I 2

3 4

5

Yu.K. Stishkov, Yu.~. Rychkov, The electric field densi%y and the volume charge in technical "liquid dielecrics',' Kolloidny Zhurnal, 6 (1978) 1204-1206 (in Russian). V.V. Gogosov, V.A. Polyansky, G.A.Shaposhnikova and Yu.D.Shikhmurzaev, Determination of the constants of electrochemical reactions and the ion mobility coefficients in weakly conducting liquids from an analysis of the current-voltage characteristics, Fluid Dynamics (Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza), 6 (1984) 956-963. H.J. Plumley, Conduction of electricity by dielectric liquids at high field strengths, Phys. Rev., 59 (1941) 200-207. R. Romanets, G. Karasev, Yu. Kostenko and E. Frankovich, Transient currents in dielectric liquids, in: G. •olinari (Ed.), Conf. Record of 8th Conference on Conduction and Breakdown in Dielectric Liquids, Pavia, Italy, July 24-27, 1984, N.Y., 1984, pp. 135-139. S. Yasufuku, T. Umemura, T. Tanii, Electric conduction phenomena and carrier mobility behavior in dielectric fluids, IEEE Trans. Elec. Insul. E 1-15 2 (1980) 149-152.