Electrostatic transport equation for turbulent flow

Journal of Electrostatics, 17 (1985) 29--45 Elsevier Science Publishers B.V., Amsterdam -- Printed in The Netherlands

29

ELECTROSTATIC TRANSPORTEQUATION FOR TURBULENT FLOW J. C. GIBB!NGS

Dept. Mechanical Engineering, The U n i v e r s i t y of Liverpool (England)

ABSTRACT The equations f o r the conservation of charge are extended to embody terms r e s u l t i n g from turbulence in a f l u i d f l o w . Account is taken of the f l u c t u a t i o n of the i n t e r n a l f i e l d and of f l u c t u a t i o n effects on the d i s s o c i a t i o n and recombination terms and on the conduction and d i f f u s i o n currents. INTRODUCTION Phenomena in which e l e c t r o s t a t i c effects i n t e r a c t with f l u i d motion have attracted much a t t e n t i o n in recent times.

These i n t e r a c t i o n s are dominant i n

the gas flow in e l e c t r o s t a t i c p r e c i p i t a t o r s , in the flow of l i q u i d s developing e l e c t r o s t a t i c streaming currents, i n the motion of l i q u i d e l e c t r o l y t e s in such d i f f e r i n g things as c o n d u c t i v i t y c e l l s and large transformers, in blood flows, in l i q u i d sprays and i n atmospheric motion. In the great majority of real flows turbulence is present in the motion and many workers have considered the attendant e f f e c t s .

Unfortunately some workers

on e l e c t r o s t a t i c s have confused turbulence with the flow due to a steady set of eddy flows:

turbulence is by d e f i n i t i o n t h a t part of the motion which is

random in time and space and of a scale that is general~small compared with the size of the flow system.

In the present study a " s t a t i s t i c a l l y

stationary"

flow is considered which, when an average with respect to time is defined as, =

Lt T-~

1 ~'F

has # # ¥ ( t ) .

I T+t

~(t')dt'

,

-T+t That is ensemble averages are not considered here (Ref. I ) .

When turbulence is present in a flow the influence of the associated terms in the governing equations can completely dominate those due to molecular d i s s i p a t i o n phenomena such as v i s c o s i t y and d i f f u s i o n .

Standard texts that

discuss these e f f e c t s of turbulence regard body forces as uninfluenced by the turbulence (Ref. 2) and so do not consider t h e i r influence upon the i n t e r n a l e l e c t r o s t a t i c f i e l d forces:

nor do they consider the d i s s o c i a t i o n and

recombination terms. Following on the extensive l i t e r a t u r e on mass t r a n s f e r in t u r b u l e n t f l o w ,

30 previous workers have made use of analogous transfer c o e f f i c i e n t s and eddy diffusivities

(Refs. 3,4).

A more detailed approach to the influence of

turbulence has been made by Dumarge and his colleagues (Ref. 5).

In t h e i r

work an assumption was made concerning the correlation between the f l u c t u a t i o n of v e l o c i t y and of charge concentration. In t h i s present paper the existing analytical studies are extended to f i l l the gaps mentioned and to resolve some present uncertainties and also to develop some basic relationships.

I t follows a previous review which assessed

the then current experimental data (Ref. 6). THE BASIC EQUATIONS The governing equations to be used here have been set forth elsewhere (Ref. 7) and a physical discussion to j u s t i f y t h e i r choice has also' been given (Ref. 6).

They are advanced as being equally v a l i d for the flow of gases

as for that of l i q u i d s .

They are as follows, the notation appearing at the

end of this paper. The conservation of ions for e i t h e r of two species is expressed by, Dc+ ~= D+ V2C_+ ¥

k± v.(Ec+_)

+ ~cm -6c+c_

(I)

The current density is given by j±

=

Izl Fk+ c± E - D+ z+ Fv c±

(2)

the total net current density being, j

= j+

+ j_

(3)

The charge density is related to the ion concentration by, o

=

IzlF

(4)

( c + - c_)

and Poisson's law is o

= v.(cE)

(5)

The conductivity is related to the m o b i l i t y by, =

IzlF

(k+ C+ +

k_ c_)

and the potential is defined by,

(6)

31 Dc+

B

-

i

__

(

B

ui)

+

( ~ )

(13)

T

Because the ion concentration has a f l u c t u a t i n g component, Eqn. 4 is w r i t t e n as + o'

:

IzlF (T+

+ c+' - ~_ - c_')

(14)

giving the f l u c t u a t i o n in charge density. A p o s s i b i l i t y of the d i e l e c t r i c c o e f f i c i e n t being influence by turbulence i s now considered.

This c o e f f i c i e n t arises from alignment in the e l e c t r o l y t e

of the dipole molecules and ions by the e l e c t r i c f i e l d (Ref. 8).

Because the

concentration of ions w i l l be very much smaller than those of the e l e c t r o l y t e molecules then the P e r m i t t i v i t y e f f e c t , shown in Eqn. 5 is associated solely with the l a t t e r .

The turbulence in the flow, being formed by the presence

of rotating v o r t i c e s , (Ref. 9), then the dipoles might conceivably be rotated, by a f r i c t i o n a l torque, against the f i e l d lines thus reducing the polarising field.

However, there is experimental evidence* that the major portion of

such a flow is in the form of i r r o t a t i o n a l vortex flows in which the angular v e l o c i t y of an element is zero. The length of a dipole is t y p i c a l l y + three orders of magnitude less than that of a small turbulent vortex and so only the supposed very small proportion of the dipoles that are in the small rotational flow close to the axes of vortices would be rotated against the field.

Thus on these physical grounds, i t is not expected that turbulence

would have a s i g n i f i c a n t e f f e c t upon the d i e l e c t r i c c o e f f i c i e n t . With then ~ = constant, Eqn. (5) becomes, + o

,

~

= c~Ti

(-Ei + Ei

,)

(15)

This r e l a t i o n shows the p o s s i b i l i t y of turbulence causing local fluctuations in the force f i e l d .

This p o s s i b i l i t y is not covered in the e x i s t i n g l i t e r a -

ture where a so-called 'body force' is regarded as being impressed from outside the flow region and hence, as with g r a v i t y f o r example, having no fluctuating component (Refs. 2,10).

In contrast the case considered here has

f i e l d forces arising from the charges that are w i t h i n the flow.

To be the subject of a separate paper +This is not so for the long-chain polymer molecules used in solution in water to suppress turbulence; then the sizes are comparable.

32 E

=

(7)

-v$

With

DC, Dt=at

ac+

ac, +

'iaxi

then the use of the continuity

equation

for an incompressible

flow leads to

(Ref. 6),

D‘, Dt=at

ac,

+ v$

REPRESENTATION

(8)

u)

OF THE TURBULENCE

The foregoing

governing

equations

at any time in the turbulent "statistically

stationary"

as a sum of the time-mean

ui

=

cy+

'i

uniform

fluctuate.

c

DC,

flow the velocity

components

value plus a fluctuating

flow and so apply

limitation

to

are written,

as usual,

part; that is, at any point,

(9)

by this turbulent

concentration

motion

does not exist,

This concentration,

at any point,

so that similarly,

if a per-

then the ion concentration is written

will

as,

(10)

=c,+c+’

Substitution

m

With the previous

’

Ions will be transported fectly

are valid for unsteady

motion.

=

a?, +

of Eqns.

+

(9) and (10) into Eqn. (8) results

ac,' at +

*

('++'+')

1

(q+'i')

1

ac, ac,’

=at+at+axi

Taking

the ensemble

a

q

q

+

[ time average

27

_

'i

of this equation

in

'

t

q

gives

C,'

+

c+'

ui'

1

(11)

(Ref. l),

(12) This is the time average

of the left-hand

case we take the time average

side of Eqn. (1).

to give (Ref. 1):

For the stationary

33

The right hand side of Eqn. (I) will now become, a2-~+

a2c+,

(z_ + c_+') (-Ei + Ei' )

¥ L_ ~

+ ~(~

+ Cm')-

B(~+ + c+') (~_ + c_')

(16)

For this " s t a t i s t i c a l l y steady" flow having constant time mean values, a time average of this is now constructed. Doing this reduces the f i r s t term of (16), which is the diffusion term, to, (Ref. II)

D+ _

2

(17)

axi

The second term of (16), written for the positive ions, and using Eqns. and (15), can be written (E+ + c+ ') (~i

+ El')

= [~TF

(~+ o')+

:

-~i

"7

~

-Ei

+ [

~_Ei

+ c _ ' l (-El + Ei'

+ Ei ) + ~

~~

ax---~

+

14)

(-Ei + Ei')

+ c_')

+ Ei~

~xi

~xi

+ ~_ Ei'

+ c-'-Ei

+ c- ' Ei' ]

Constructing the time average of this reduces this second term of (16) to,

--~

(Ei 2)

+

~xi

2

+

The term for the negative ions, in a similar way, becomes,

-

'

(18)

34 92 + C+' Ei' )

~xi

(19)

I t is seen that two fluctuation or turbulence terms are introduced in (18) and (19). One is Ei '2 which is a measure of the field fluctuation and the other is c+' Ei ' a cross-product fluctuation. The time mean value of the third term of Eqn. (16) is simply ~ m

(20)

Thus there is no influence of turbulence upon the dissociation term. The special case of "dissociation" in liquids of low dielectric coefficient needs consideration. Following a previous discussion (Ref. 7) this "dissociation" is represented by either c+_ c+

(21)

or by

c+_ 2

(22)

or indeed by both. The time mean values of these two terms are respectively,

[

#+_

C--~ + C+_

]

c+

(23)

and

~ [ ~+_2 + c+_,2 ]

(24)

and so these contain turbulence terms. Representation of the dissociation term by either Eqn. 23 or 24 was originally advanced for the case when c+_ >> c_+ (Ref. 7), a typical value being c+

= 0 [ lO6 c+ ]

A " d~ssociation" "

of part of c+_ of ion-pairs results in the creation of an

equal number of ions of c+ and c_. -

6C+_

=

(25)

6C+

Noting Eqn. 25 gives

Thus,

85

-

c+_

-

0

10 - 6

-

(26)

T+

As physically @c+/c+ <- 1 and i f ~c+_ and ac+_ are representative of the orders of c+_' and c+' then i t is seen that the second terms in both Eqn. (23) and (24) are negligible: again, as with Eqn. (20) there is no turbulence term in the dissociation term. The above discussion refers to a point in a flow that is wholly turbulent. In contrast, at the edge of a jet of electrolyte flowing into a region containing only the solvent then the value of c+_ could fluctuate between zero and a finite value. However, the relation of Eqn. 26 is s t i l l valid. The fourth term of Eqn. 16 has the mean value of

-~[~+~

+ c+--~_ ]

(27)

A heuristic argument is that as a particle of f l u i d is moved by the turbulent motion over a short distance the recombination within i t remains in balance with the dissociation. From the argument just presented that there is no turbulence component upon the dissocation i t would follow that there is none upon the recombination. This postulated balance is supported by the fact that the frequency of kinetic collisions are of much smaller time scales than those of turbulent fluctuations. The diffusion component of the current is given by ~D + JD'

= -'zlF [ D+ vc+ - D vc_ ]

=-IzlF

[ D+ (v~+ + v c + ' ) -

D_ (v~_

+ vc_')]

The time mean value of this equation becomes, ~D :

-Iz'F

ID+ ~ + -

(2B)

D v~- ]

so that JD' = 0 and there is no influence of turbulence upon the diffusion component of the current. However, the turbulence will have an effect upon the value of vc-Z_+because this latter quantity is obtained by solution of the later Eqn. 33 which shows a strong influence of turbulence. The conduction component of the current is expressed by,

Jc + Jc

:

I zlF (g + E')

[

k+ (~+ + c+')

+ k

(~

]

+ c ')

36 A time mean value of this i s ,

~c =

IzlF

k+~E~+

+E'c+'

I

÷ k_

~

+ E'c '

which suggests a c o n t r i b u t i o n of turbulence terms to the c o n d u c t i v i t y c u r r e n t . However, from Eqn. (15) we have t h a t a'

:

~Ei , c --

and so Ei '

(30)

~Xi

is obtained by an i n t e g r a t i o n over a l l

throughout the flow.

the charges d i s t r i b u t e d

I t is a f e a t u r e o f turbulence t h a t f l u c t u a t i o n s are

random both in space and time (Ref. 2).

An instantaneous p i c t u r e o f the charge

d e n s i t y might be as shown in Fig. l ( a ) .

I n t e g r a t i o n of t h i s according to

Eqn. (30) might give the r e s u l t of Fig. l ( b ) .

0

As with a time i n t e g r a l , the

=,

~bl

Fig. 1 space i n t e g r a l r e s u l t s in E' + 0 f o r i n t e g r a t i o n over a s u f f i c i e n t

distance.

But t h i s answer is independent o f the p o i n t where the i n t e g r a t i o n is s t a r t e d . Thus, E'

= 0

is a general r e s u l t .

(31) I t then f o l l o w s from Eqn. (29) t h a t (32)

i n d i c a t i n g w i t h Eqn. 28 t h a t there is no i n f l u e n c e of turbulence upon the t o t a l

3? mean current.

I t is important to recall that i t can only apply when the

scale of the flow is large compared with the typical size of the turbulent eddies.

For example, i t may not be valid for the flow of a thin turbulent

boundary layer. This result received experimental confirmation from an experiment by Saluja (Ref. 12).

In his experiment kerosene was pumped through a stainless-

steel tube of length:diameter ratio of 1500. A small platinum wire electrode was aligned along the tube axis inside the tube at the e x i t .

The ratio of

the measured current, from this electrode to the tube, to the applied potent i a l difference is shown plotted in Fig. 2 against the Reynolds number of the I

i

I

i

~-.i 1015S l+ 0 0

I

I

i

I

2

3

i

R.. 10-3

Fig. 2 flow.

The two lowest values of the Reynolds number shown corresponded to

laminar flow along the tube; the highest one corresponded to turbulent flow. This measure of the current for constant boundary conditi.ons i s seen to be closely uninfluenced by the onset of the turbulence there being no discontinuity at transition between the two flow regimes. Another experimental result also confirms the absence of any influence on current by an eddying flow which contains vortices that now are comparable with the size of the flow pattern.

This is for the electrohydrodynamic flow

in a closed c e l l . Once flow convection has set in, the current i n i t i a l l y oscillated but about a mean value that stayed constant as the flow b u i l t up (Ref. 13).

A careful study showed again that the current was almost unchanged

by the onset of the eddying convection (Refs. 14,15). As a result of the preceding discussion Eqn. (1) becomes, a

a

~2~± _

+

+

a

]

°±

+

zm-

c+ c_

+

F

~

~EEi 2 +

(33)

38 The only influence of turbulence upon t h i s equation f o r the time-mean value of the ion concentration is the second term on the left-hand side.

In mass

t r a n s f e r phenomena i t is commonly combined with the d i f f u s i o n term as ~2~±

D+ -

~xi

2

~xi

(34)

(~+-?~i)

The l a t t e r of these two terms, through a convection term, is referred to as the turbulent diffusivity

term.

THE EDDY DIFFUSIVITY Turbulent motion alone does not mix the ion concentration (Ref. 16).

The

e f f e c t of the turbulence v e l o c i t y is to transport a portion of f l u i d of one ion concentration into an ajoining region of d i f f e r e n t concentration. two consequences then markedly increase the molecular d i f f u s i o n :

In doing so firstly,

the

vortex filaments of the motion are stretched so that the surface area across which ions can d i f f u s e is g r e a t l y increased and secondly, the local gradient of charge concentration is also g r e a t l y increased. In order to make practical c a l c u l a t i o n s i t is common to w r i t e (Ref. 2),

- c±'

u i'

~

~yi

(~+/~xi)

The q u a n t i t y ~yi is called the eddy d i f f u s i v i t y . justification,

(35)

There is no real physical

in the form of Eqn. (35), f o r r e l a t i n g the t u r b u l e n t d i f f u s i v i t y

term to the mean concentration gradient and, indeed, only in some very special flow patterns does experimental observation show that ~yi is anything l i k e a constant.

Though in most flows ~yi > 0 there are cases where the analogous

q u a n t i t y f o r heat t r a n s f e r has a negative value (Ref. 17); s i m i l a r r e s u l t s are observed in the flows through e l e c t r o s t a t i c p r e c i p i t a t o r s (Ref. 18).

However,

because of the complexity of turbulence d i f f u s i o n , Eqn. (34) is a common basis for calculation. As an example, the equivalent momentum e d d y - v i s c o s i t y , Em, on the centrel i n e of a t u r b u l e n t pipe flow is t y p i c a l l y of the value (Ref. 2 ) , P~m _

p

2.10-2.Rel/2

which f o r a Reynolds number of 107 gives P~m IJ

63

39

Experiment also shows that, for a i r , (Ref. 2) EmlEY = 0.625 so that p~y/u

= lO0

The typical value of the Schmidt number for molecular diffusion in a gas is ~/pD = 0.75 so that for this flow, ~y/D = 75 The turbulent d i f f u s i v i t y is thus seen to dominate the transport and eventual molecular diffusion, of ions. and c±' is small:

Even so the degree of correlation between ui '

t y p i c a l l y with

I/2

I/2 /Uo = 0.06

and

{c±-~

/c+

= 0.02

then

Uo-

± :

0

1o-2.

c+

APPLICATION TO THE ELECTROSTATIC PRECIPITATOR In an extensive essay on the electrostatic precipitator (Ref. 19), Robinson stressed the great importance of the mean flow velocity distribution and also of i t s turbulence level. A simple model of such flow is shown in Fig. 3(a). The phenomenon is simplified as being two-dimensionalwi~ two parallel surfaces, as shown, one being an electrode the other a collecting surface. An approximate solution is given as i l l u s t r a t i v e of the prior analysis. For the stationary turbulent flow, eqn. (33) for negatively charged particles, gives,

40

[oLLe{fing surfo~e

'l /

1

(}

'~

":°' \ Electrode

(a.2

~,L_

(b_) Fig. 3

~#_ -C -~-

I ~2#-_ S2~-_I

~ +

k_s

--~x (c_'--~-'7)

~2Ex2

~

(c_-~)

+

= - D

7]

a%21

(36)

+ ~y2 ]

Also eqn. (35) becomes,

~x (c_-3---~-) + ~

(37)

(~)

To obtain the approximate solutions eqns. (36) and (37) are s p l i t in two. This is not altogether unreasonable for across the flow both f i e l d and space charge densities do not vary greatly in the y-direction (Ref. 19). Thus for variations in the x-direction we have, dc_

~

d2~

-~

~

d2c_

: D_ ~

k_ s

--~-r

d2 Ex

~

2

Jw

+ T~Cw

(38)

where now the last term will supply continuity of charge and will represent the remaining terms of Eqn. (36). The last term is, from Eqns. (28) and (32) given by,

41

Jw

=

IzIF

D_

As a f u r t h e r

[

~-~I

=

y :w

+

-E yw

k

c

(39)

-w

a p p r o x i m a t i o n we p u t ,

- ~_/w

y=w and a l s o = -w

so t h a t Eqn. (38) w i l l

refer

t o some mean v a l u e o f c

= T_(y).

From Eqns.

(14) and (15) we again a p p r o x i m a t e by ~-Ey/ay = 0 so t h a t dE x

IzlF c.

(40)

Using these a p p r o x i m a t i o n s w i t h Eqns. (38) and (39) r e s u l t s

(~ +D_) ~

= (u-k_gx) ~

k_ IT~-J +~

in dEx

-k_gYw ~ -

(41) A n u m e r i c a l example has been d e r i v e d f o r the f o l l o w i n g =

1.46.10 -3

m2.s - I

D

=

4 , 1 0 -6

m2,s -1

u

=

0.5

m.s

k

=

1 . 8 . 1 0 -7

m2 v -1 s -1

w

=

0.1

m

=

-2.10 5

V.m -1

yw

Using t h i s A

z

B

-

C

-

data results

u + D

-I

in =

3.415.102

- ~z = w(~y + D.)

2.462.102

(D_/w)

-

k

E

- k _ / ( E y + D_)

=

- 1 . 2 3 0 . 1 0 -4

sample d a t a :

42 I n s p e c t i o n of these values suggests a f i r s t d3E dx 3

A d2E ~

-

dE B ~-~

-

B

=

a p p r o x i m a t i o n t o eqn. (41) to be

0

(42)

or

d2E dx2

A

dE dx

- -

=

D

where now E z Ex/Exo. The solution required is E

=

e -ax

(43)

where a

= ~

- A + {A 2 + 4B} I / 2

(44)

0.7193

This is shown plotted in Fig. 4.

I t is noted that eqn. (44) is closely approx-

imated, with the present numerical values, by a

=

B/A

=

0.7208

and t h i s gives a

=

(D_/w) - k_ E Y W.

U

showing a result that does not involve the turbulence effect due to ~y. Numerical solution of eqn. 41 gave the two curves shown also in Fig. 4 for two values of Exo" These show an increasing rate of decay with increase of Exo. Figure 5 shows the i n i t i a l slope of the field and hence the i n i t i a l charge density. A singularity appears for a value of the i n i t i a l field just above lO.6V.m- l for the present numerical example. CONCLUDING REMARKS A d e t a i l e d study o f the t r a n s p o r t e q u a t i o n has shown t h a t t h e r e i s no contribution

o f the t u r b u l e n c e t o e i t h e r the c o n d u c t i v i t y c u r r e n t o r the d i f f u s i o n

current.

However, the appearance of a very l a r g e t u r b u l e n t d i f f u s i v i t y

i n the

e q u a t i o n f o r c o n s e r v a t i o n o f ion c o n c e n t r a t i o n can show a marked e f f e c t upon the mean charge d i s t r i b u t i o n .

This would have to be accounted f o r in r e a l i s ~ c

4:3

1-0 '-

EQ. ~3

0~.

E

0.1

0"0~.

0"01

i

i

i

i

10

20

30

~0

50

x/~w Fig. 4

12

1.1 dE

d (XlW~) 1"C

0"9

0-8

0.7 10~.

t

i

~,.10~'

10s

i

~.-10s

#

Fig. 5 calculation of charge distribution in such as electrostatic precipitators and pipe flow with charge absorption at the walls leading to a streaming current. In contrast the flow through a fine-pore f i l t e r would be laminar and so the effect would be absent. REFERENCES

l 2.

P. Bradshaw, An Introduction to Turbulence an'd i t s Measurement, Pergamon, Oxford, 1971. J. O. Hinze, Turbulence, McGraw H i l l , New York, 1975 (2nd Ed.).

44 3.

4. 5. 6. 7. 8. 9. I0. II. 12. 13. 14. 15. 16. 17. 18. 19.

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NOTATION Cm

Molecular concentration

C+

Ion concentration

C+_

Ion-pair concentration

D, D+ Diffusion coefficient E

Field

F

Faraday constant

J_+

Ion current

45

Jc JD JW

Wall current

k_+

Ion mobility

t,T

Time

ui

Conduction current Diffusion current

Component velocity

W

Width of precipitator channel

Z

Valency

C~

Dissociation factor Recombination factor Permittivity

em

Y

Momentum eddy diffusivity Mass eddy diffusivity

x

Conductivity

la

Solvent viscosity Solvent density

(J

Charge density Electrical potential Function

I

[Dash] denotes fluctuation component