Frequency response of small electrodes to hydrodynamic or to potential perturbations

Ekcrmchimka An, Vd. 35, No. 14, pp. 1847-1856, 1993 Printed in arimim

T 0013-4686193$6.00+0 O v 1993 . Pergamop Pmp

Gnat

Ltd

FREQUENCY RESPONSE OF SMALL ELECTRODES TO HYDRODYNAMIC OR TO POTENTIAL PERTURBATIONS C. Destouis, O . GIL and B . TRl130LLer UPR 15 du CNRS, "Physique des Liquides et Electrochimie", Associe: 3 I'Universite P. et M . Curie, 4 Place Jussieu, 75252 Paris Cedex 05, France (Received 10 March 1993)

Abstract-Small electrodes functioning under convection-diffusion control can be used as hydrodynamic sensors as they are sensitive to flow variations . Analysis of the statistics of the flow fluctuations requires the knowledge of the transfer function between the mass flux and the velocity gradient . Calculation of these functions in the streamwise and in the spanwise directions with respect to the average flow direction is presented here. An experimental measurement of the transfer function in the streamwise direction with a cone-and-plate system confirms the theoretical predictions . As an application, the measurement of the power spectrum density of the wall velocity gradient fluctuations of the turbulent flow in a rectangular duct in the streamwise and in the spanwise directions are reported. Key words :

electrochemical probes, transfer function, hydrodynamics, mass transport, convection-

diffusion .

INTRODUCHON In order to take into account mass transport in the analysis of electrochemical and engineering problems, hydrodynamic electrodes[l, 2] have been widely used in the recent years . The best known is the rotating disk electrode[3] which has the advantage of the uniform accessibility, the other systems commonly used are the tubular or channel electrode, walljet (impinging jet) electrodes and other configurations . Hydrodynamic electrodes and impedance techniques, though more complex to analyze theoretically, are very powerful, but only the rotating disk electrode had been extensively studied[4, 5]. The present paper is concerned with non-accessible electrodes, as channel electrodes, the forced convection being employed as a variable, by moving the solution over a stationary electrode embedded in the wall. These electrodes has been investigated mainly with a view to analytical applications[6] or to hydrodynamic measurements[7, 8] . Impedance techniques corresponding to a perturbation of potential (or current) or of hydrodynamic will be considered. In the latter case, the first experimental or theoretical studies appear in the early fifties, the goal being to achieve probes sensitive to the local wall velocity gradient value a of the fluid . The well known property of those probes is that the limiting diffusion current is proportional to a 313[7] in steady-state conditions. For use in electrochemical engineering, an increasing interest is now focussed on the non-steady-state behaviour of those small electrodes, namely in conditions of fluctuating velocity gradient a(t). Theoretical developments show that it is possible to deduce hydrodynamic informations from the limiting current measurement either in quasi-steady-state where 1(t) x ar 13 (t), or at higher frequency, in terms of spectral analysis[9, 10] . In the latter case, obtaining the velocity spectra is possible EA

3arl4-F

from the mass transfer spectra, where the transfer function between the mass transfer rate and the velocity perturbation is known . However, in most cases, charge transfer is not infinitely fast and the analysis requires to know also the convective diffusion impedance, ie the transfer function between a concentration modulation at the interface and the resulting flux of mass in steady-state convection . The knowledge of this convective diffusion impedance is also required for the analysis of the usual electrochemical impedance . The electrodes are qualified "small" in that the flow velocity components parallel to the wall in its vicinity are only functions of the coordinate normal to the wall, y. In addition, the diffusion regime is not established so that the extension of the diffusion boundary layer is small with respect to y, and thus, the problem will be reduced locally to a twodimensional one, the coordinate x corresponding to the mean flow direction. Such a simplification enables us to ignore the main flow geometry and gives the result of the calculation a wider relevance . In this paper, a mathematical treatment is presented on one hand for the diffusion impedance influenced by convection, when diffusion-convection becomes prevailing and on the other hand for the transfer function between the mass transfer rate and the velocity perturbation ; this part is followed by an experimental verification . The response of rectangular, circular and bi-circular electrodes will be analyzed, the latter one having a greater practical importance in hydrodynamics. ANALYSIS Let us consider a small electrode (in any direction parallel to the interface plane), embedded in an insulating wall, on which occurs a fast electrochemical reaction with a diffusion controlled kinetics (c = c, )l.

1847



C. Destouts et al .

1848

The length of this electrode in the mean flow direction is small enough to keep the diffusion layer thickness 6 very small and thus minimizing the effect of the normal velocity component varying as y 2 , with respect to the longitudinal velocity component proportional to y, the coordinate normal to the wall . We considered situations such that xX 2 /D (X being the electrode dimension in the mean flow direction x) larger than 5000 in order to ignore molecular diffusion along the same coordinate, ie azc/axz, as stated by Ling[lI] for steady-state conditions. The steady-state Ling's criterion is likely to be not sufficient especially at extremely high frequencies where it would be necessary to consider this longitudinal diffusion term. The term azc/8zz was also dropped since it is, at most, of the order of azc/3x 2. This should not be so for potentials being more extended in the mean flow direction than in the transverse one . When choosing a local frame of reference (x, y, z) attached to the electrode, the mass balance equation governing the concentration distribution c of a species transported by convection and diffusion is in boundary layer approximation : ac ac ac L c ar+ayax+6Ya=Day„

(1)

where V, = ay, V = sy with # = 0, since there is no mean flow in the transverse direction . The overbar designates a time average value. For simplification, in this equation, we consider that the electrode is sufficiently small so that any flow is uniform in the diffusion layer, then a and $ are assumed independent of the space coordinates . The time average solution for the concentration distribution has been given earlier by Lbv2que[12] : r/d C C = 323['(4/3) 13C c exp( - n 3 /9)dn + c.,

For a ring electrode we have to use AR instead of I and 2aR instead of L. On a circular electrode we have : R

J«

R+J(R2-z21

-R

= 0.84

imc- D

4320 ae as +Ya -ya = 2 a y= ax ax

0 cc

(6)

fory=Qxzx„ for y=0,x
Y-'00 .

For a flow perturbation a and fi are different from zero and the complete equation (6) must be considered with the following boundary conditions : c=0 fory=o,x2x 1 , ac=0 ay

On the rectangular electrode the steady state flux cJD2 130,11312/3L (3) 2['(4/3)

=0

2=0

is : 31/3(cm

as

For a = # = 0, the equation is homogeneous and correspond to the condition of the convective diffusion impedance, with the boundary conditions :

0 for a rectangular probe.

0,e

(5)

Re means the real part and the symbol - represents a complex quantity . The perturbations must be small enough, these conditions can be written as a/a 1 4 1 and ac/Oz 4 SSJSz for minimizing the quadratic terms (a ac/ax and fi 3E/az) and, therefore, for satisfying the linearity conditions . Then, the non-steady part of the mass balance equation may be written as :

ay

D ay dx =

(4)

V, = Re{$y exp iwt} .

where c, is the species concentration in the bulk and 6(x, z) _ ((D/&(x - x (4)]r/ 3 . The local value of the diffusion layer is 3 2/ 3f(4/3) times 6(x, z). x,(z) represents the position of the leading edge of the probe with the conventions of Fig. 1 .

Ja =L

1/3(c . - c )D 2/3 a 113d 3/3 2F(4/3)

V = ay + Re{ay exp irut}

ac"

-

3

LOCAL SOLUTION We consider a sine wave perturbation of the concentration field or of the flow, small enough in order to satisfy the linearity conditions . The instantaneous concentration and velocity components are defined as : c = c + Re{c exp i(ot)

(2)

x 1(z)

aY

R-'1W- .+t

c=Ac

x,(z) = R - ,/R' - z 2 for a circular probe .

as

D - dx

dz

fory=O,x
c"=0 fory-.w . By introducing the dimensionless concentration function h defined by h - Ac/0 for the convective diffusion impedance and by : 1 =1h

c=(c m -cJ(a-

for the flow perturbation and with the dimensionless variables : 1/3 a n D(X 0

Fig 1 . Scheme of the circular electrode . The coordinate y is in the direction perpendicular to the plane of the electrode (x, z)

-

XI)

and ~-

~ (x-x1) 2\u3 at(\

D0,

j /

Hydrodynamic electrodes which correspond to n = y/S and Equation (6) then becomes : 2

10

+3

n

= o8 2 (x)/D .

ah n2 8h 8h 001), at - 3 an - n t =

a

1849

sion impedance is proportional to the quantity Z o and : 00I (7)

with g(n) = 1' exp(-n'/9)/3'''f (4/3), in the case of the flow perturbation and g(n) = 0 for the convective diffusion impedance . In this type of problem, one has generally to consider a low and high frequency solution with respect to i; parameter : as ~ contains a dependence on the space coordinates, it will be necessary to firstly derive the local solution . Low frequency solution As suggested in refs [13-15], we seek a solution in the form : h= E Rrh"(nn . (8) "-o In fact, since only the interface flux is the observable quantity, we need only to know the ah/aq 1o expression : ah dh" (9) an o "=o do o By using Newman's method of integration of partial differential equations[16, 17], the functions h"(n) had been calculated and the derivatives of first order at n = 0 had been deduced. Those values are listed for m 5 59 in ref . [18] for the case of the convective diffusion impedance and in ref. [15] for the case of the frequency response to an hydrodynamical perturbation.

Zo 1

= f-

0~

axaz=-

I

o o

~

z) dz (13)

and on the other hand the frequency response to a flow perturbation is :

r a p dx, ah dx . (14) ,~,~ Dc"(& & x dz ) dnl o 3(x, z) dz For a rectangular electrode dx,/dz is equal to zero and for a circular one dx,/dz is an even function in z and then disappears in the integration . In each case the frequency response is proportional to f j ahlan to dx/S(x, z) dz the definition of Oh/an to being given by the expression (9) in the low frequency range with the values of dh,_/dnl o corresponding to each case and by expressions (11) or (12) in the high frequency range . For a bi-circular electrode, on the opposite of what happened on a circular electrode the difference between the overall fluxes measured on each half electrode is considered, the term containing & in equation (14) being an even function in z, makes no contribution and the quantity corresponding to the difference of the fluxes is proportional to if dx,/dz ah/an o d,, dz/d(x, z). _

I

Rectangular electrode For a rectangular small electrode, by using the dimensionless frequency o' = w(l 2/Da 2)1 " :

ah dx al2 113 High frequency solution = LDH(a'), (15) an 0 S(x) dz D Since the concentration modulation is rapidly damped close to the wall at high frequencies, the with convective term can be disregarded[19] and equaI dl tion (7) becomes : H(a) 2a, 0 dl: . z J, do i~h ( 10) s(n) . aqh = In the low frequency range, the expression of H(a) is obtained from the series expansion (9) with the With the previous boundary conditions, the analyticorresponding values for dhjdn lo : cal solution is : ah 3 `° (1o)- dh" . =-1/(it;)(11) H(a') _ (16) Fo 2 Fo m + 1 do o for the diffusion impedance and corresponds to a In the high w frequency range, the integration Warburg impedance. must be split in two parts since the leading edge of a real electrode will be always under a low': frequency 2 1 ah (12) regime. Indeed the local thickness of the diffusion o = 3513r(4/3) x (g)312 Fit layer, equal to 3 2j3 r(4/3)3(x) and thus proportional to xu3, is very small at the leading edge and for the flow perturbation . always remains small even for high o)/27r values . For 6 5 ~ 5 13, the high frequency solution and the low frequency one present a satisfying overlap , ( l either for the convective diffusion impedance or for dt+ dc] . (17) H(e)=-2o, o do O J an 0 the frequency response to a flow perturbation . The first integral corresponds to the low frequency regime and the second one to the high frequency FREQUENCY RESPONSE OF A SMALL regime where expressions (11) must be used for the ELECTRODE convective diffusion impedance : The dimensionless frequency response of a small 0.25i electrode can be defined by summing the effects of H(d) _-+ 0 91 12 (18) a the local frequency responses . On one hand the diffu-

ff

C. Daw .outs et at.

1850 and I Zo(e) 1 _ 0.80755 Qu2 JZD(0)I

In the high frequency range, by using the same splitting previously described, the expression of H, for a circular electrode in the case of the convective diffusion impedance can be written as :

0.1768) +

312

\

with 0 = -arctan I -

0.25i H,(a)=--+n/4(ia)"2 a

0.35361 3(2

and

As a consequence, a fair overlap between equations (16) and (18) is obtained for 6 5 a' 5 13 . For the frequency response to a flow perturbation, the expression (9) must be used in the low frequency regime and expression (12) in the high one : H(d) - 3 .715 3 .99 H(0) - 767' - (Id)3/2~

(19)

As a consequence, a fair overlap between equations (16) and (19) is also obtained for : 6 5 d 5 13 . Circular microelectrode R+J(R2-22) dh R-JU2_22) dp

•R

rdo

dx dz S(x, z)

dz -R

dx 5(x, z)

o

(20) Interchanging the order of integration, equation (20) may be written as :

ff

dq

dx dz = 3(x, z)

0

( o(D2d5)"3H .(a),

4

I

113

Dal

H=(a)=-

dpi 7 0 d~

B = -arctan

a' -

o

.

(22)

H,(c) _ 4.416 H,(0) go

r (iFrde) dp t

(i

)m d

Jl - x 3 x'dx

(t aa)mI_= )mI_

f

_ -

3 2

I

I

I

30

20

W

0 .3 J 6

D1

m (iar"f,„ moo

a

4

m+1 f~n ` ) r 3 3m+1 ' 3 ) - V O rl3 + 161 ) `

where 13 and r are, respectively, the beta and gamma functions[20] . H 2(a)

I

10

0

1

I

0

3

_

I

a

= (tar"



I

,O

P

a

x N 'a N •

(26)

. C o

and .i ,/a3 - F3

5.3 (ia)3f2

Bi-circular electrode For a bi-circular electrode, with the opposite of what happened on a circular electrode, the difference so

(23)

a s)2

0.451 1 -i . a

I In Fig. 2, the variations of Z a(d) or Zax(a) in amplitude and phase shift look identical in a first approximation and are shifted in frequency by a factor 1 .14. Fortune and Hanratty[21] had already justified this fact by showing that a circular electrode of diameter d yields the same current as a rectangular one of length 0 .82d and width d and therefore, the factor 1 .14 is consistent with (0 .82) -2)3 . In the same way, the expression of the frequency response to a flow perturbation for a circular probe can be written, in good agreement with the expression obtained by Nakoryakov et al .[14] as :

b

(J r

0.225) 9 312

with

Then in the low frequency range :

m=0

7112 ( 1 +

1 Zox(0)

In Fig . 3, the variations of H(d) and H,(a) in amplitude and phase shift are plotted versus the dimensionless frequencies . The two curves are also shifted by a factor of about 1, 14 .

and

sf2

Z Dx(a) _ 0.865

1

(21)

with d:

(25)

"' W

(24) 0

U

Zoos B o,

I

0 .3 3 1 10 30 DIMENSIONLESS FREgUENQ(roro-'1

Fig . 2. Overall normalized diffusion impedance for the rec angular or circular electrodes vs. a=md613 1D1,3 a3f3 or a 'a ad2/300 2 2l3



Hydrodynamic electrodes

1851

Table 1 . Expressions of the convective diffusion impedance and of the frequency response to a flow perturbation Convective diffusion impedance Rectangular for a' 5 4 : 1ZD(a) _ (1 + 0.433a'z - 0.0084a's) - i 1 z I Zdo) 0 = -arctan{0.55275a(t - 0.071x' 2 Zda') 1 = 0.80755 for a' > 6 : 1 o "z I Zd0) I

+

0.0023x' 4)}

+ 0.1768 ~1

0.3536, B = -arctan 1 - Q i -111-f Circular ZDZ(a)I = (1 + 0.339, 2 - 0.0058a')""2 for a' S 4 : I I Zox(0) I 0 = -arctan{0.48567a(t - 0.060 2

+

0.0018a`)}

_ 0.865 0.228 for d > 6 : IZox(a)I 1 + 3a D1/2 a Z,x(0) I \ 0=-artan 1-

01 4 ~ a

Frequency response to a flow perturbation Rectangular H(a) I for a'!-, 6 : I = (1 + 0.056x' 2 1 H(0) I

+

0.00126a'*) 112

0 = -arctan[0.276a (1 + 0 .020' 2 - 0.00026x'4)] for a'> 6

H(a') : H(0)

3.715 id

3.99 (1,) 312

Circular for a 5 6 : I = (1 + 0.049, 2 H,(0) l

+

0.0006x 4) -12

0 = -arctan[0.242a(1 + 0.0124x 2 - 0.00015x`)] for a > 6 :

HAa)

H,(0)

4.416 to

5.3 (ia)31~

Bi-circular for a 5 6 :

H,(a)

= 0 + 0.0352, 2 + 0.000246x4) -1 l 2

1 H,(0) I 0 = -arctan[0.1974a(1 + 0.00321a 2)] for a>61

HJa)

~(O)

6.19

10

between the overall fluxes measured on each half electrode is considered, the term containing 8 in equation (17) being an even function in z, makes no contribution and the quantity corresponding the difference of the fluxes is proportional to f f dx 2 /dz 8h/3q 1 o dx dz/3(x, z)

(air: _ 4312) ~~ o

(27) (o d

a

mt

In the low frequency range (a c 13), any point of the microelectrode area is governed by the lowfrequency regime, so expression (9) can be applied : (to). I 4 °

then to H,(a) with : H.(a) = 2as1 :

.29 - 4.641 0

wa 16

(m+lKm+2)

(28) d7 o

If a is larger than 13, the integration of HJa) must be split into two parts since the trailing portion of the microelectrodes is characterized by the high frequency behaviour. The first integral corresponds to a



1852

C . D st outs et at.

1 REAL PART (4f)

0 .5

1

2 .5

5

10

25

50

Fig. 4. Experimental diffusion impedance diagram for a circular electrode : diameter 80 pin ; flow velocity U a = 5 .94 cm s - ' .

100

REDUCED FREQUENCY Toro- ,

Fig. 3 . Dimensionless transfer functions for the rectangular (H), circular (Ho or bicircular (H,) probes vs. a or a' . low frequency range and the second one to the high frequency range where expressions (9) and (12) must be used, respectively : HJa) _ 6 .19

H,(0)

fa

10 9.29 - 4.64i (ta 3 2 a°I2

( 29 )

This equation is valid for a > 6, as a consequence a fair overlap between equations (28) and (29) is obtained for 6 5 a 5 13 . The variations of H,(a) are also plotted in Fig . 3 . As the low frequency solutions are not easy to use in the form of series developments, we sought an algebraic representation of H or Z functions providing an accuracy to within 1% in the worst case as compared to the exact solution . The different expressions obtained by a fitting technique, for the low frequency solution are given in Table 1 . EXPERIMENTAL VERIFICATION FOR THE CONVECTIVE DIFFUSION IMPEDANCE The theoretical framework presented above does not require a specific flow field but only that the wall velocity gradient a is known, that the electrode is small enough for ensuring a constant value over its surface and also a negligible influence of a possible velocity component normal to the electrode plane . The chosen system is the flow in a circular pipe with circular platinum small electrodes embedded flush with the wall . The experimental set-up had been described in ref. [18] . The electrochemical reaction used in the reduction of triiodide to iodide (13 + 2e - U 31 - ) and the measurements were performed around 25°C, where D = 1 .1310 -5 cm 2 s' and v=0 .8910 -2 Cm 2 s - ' The experimental diagrams, obtained at the half limiting current value, follow the shape of the theoretical diagram as shown, for example, in Fig . 4 for an electrode with d = 80µm and a flow velocity Co of 5.94cms' . The dimension of our electrode is 10 times smaller than the electrode used in ref. [22] and so in our case the low frequency behaviour can be accurately described. For synthesizing the measurements on different microelectrodes of various diameters (from 80 to 300µm) and for different flow rates, the data were reduced by use of the dimensionless frequency a = al(d2/Da l )'" . We also chose to represent the

2-

0.5 0.4 0 .3

a 0 .2

z o.i F

H 0 .05

I

O

0 .3

1

3

10

DIMENSIONLESS FREOUENCY a=U(

30 60

d

' 1/3

Fig. S. Comparison of the experimental and theoretical imaginary part of the diffusion impedance obtained for different flow rates and different electrode diameters d=80µm : Ua =5.94cms - '(x) ; 9-17=s-'(0) ; and 20.4cms-'(+) d=92pm : UD =8.5cms - '(O) d = 300 µm : U a - 8.5 cm s - ' (A). imaginary component-which yields more important relative variations-for frequencies below and above the characteristic frequency corresponding to its maximum, and normalizing with respect to the diffusion resistance (le the value of the diffusion impedance at zero frequency) . For showing the generality of the theoretical expressions, laminar of turbulent flow conditions have been imposed. All the experimental data are reported in Fig . 5 with a = ar, or d r, and are compared to the theoretical curve (----), showing a very good agreement .

EXPERIMENTAL VERIFICATION FOR THE FREQUENCY RESPONSE TO A FLOW PERTURBATION A well defined periodic flow is very difficult to obtain in a pipe or a channel, and the previous attempts with those flows[21, 23, 24] were not completely successful. The flow generated by a rotating disc, the angular velocity of which is sinusoidally modulated, is accurately known for both amplitude and phase . The first quantitative measurements were obtained by use of a modulated rotating disk electrode[15] . However, with this flow geometry, the instantaneous direction of the velocity vector not being aligned with the time average one except for very low frequencies[5], the comparison between theory and experiment was limited to circular microelectrodes . For extending the validity of the above theoretical





Hydrodynamic electrodes predictions and checking the response of rectangular probes, a similar experimental study with a modulated flow in a cone-and-plate system was carried out[25] . In the absence of secondary flow, this system provides indeed a one-dimensional velocity field in the circumferential direction. First, an analysis of the sinusoidally modulated laminar flow in small oscillations for this system is reported . The resulting effect on mass transfer is then deduced and the relevant variations with frequency further compared to the experimental data relative to circular or rectangular probes.

The equations of motion for the cone-and-plate system are adequately expressed in spherical coordinates with the symbols given in Fig . 6. In the present case, the cone is rotating at an angular velocity n(t) such that :

=0 for =8=n/2,

Vs

v, = nr cos

a'

for B=2+Ze .

vj

& =

= Y o

tan Zy

.

(32)

This classical result implies a constant gradient value, whatever the position on the plate . The modulated velocity gradient at the wall is given by : a8, = -e 1E - An~ cos Z" (33) a - ay o raA o sin{x* f'} ' with ur 2 The asymptotic behaviour of & when w - 0 identifies with the values of & (see equation 32) since fe . 1. From equation (33), an hydrodynamic transfer function Zan between the velocity gradient and the angular velocity of the cone, ie &/An, can be defined. Its variations with f e are presented in amplitude and phase shift for different Ar values in Fig. 7.

(30)

The plate is immobile which provides the following set of boundary conditions : v, v,

the wall can be determined :

f+ =1-iv

VELOCITY DISTRIBUTION IN MODULATED FLOW

n(t) = 0 + AQRe{exp jwt} .

1853

EXPERIMENTAL To be consistent with the previous section, two cone angles A' of 2° or 3° were chosen . The cone was rotated at 0 and the plate was immobile . Circular or rectangular platinum microelectrodes were embedded flush with the plate plane. Both cone and plate were machined from plexiglas. A large Pt grid, placed outside the region between the cone and plate, was used as counter electrode. The electrochemical reaction used was the reduction step of a very rapid redox system-2 potassium ferri-ferrocyanide 10 M-with KCI

Therefore, in so far as primary flow is concerned lie v, = 0 and vs = 0 for any r, B and (p), for both the steady-state and the fluctuating quantites, the steady-state solution is readily obtained from ref [25] by assuming small A values : v,

= f}r

Sin .t ,, tan Z

(31)

where r sin a . z rA represents also the normal distance to the wall y, so that the velocity gradient at

I

V 0 .001 500

I

a!= 5

2_6 2-5 f'

. a

,o?

Fig. 6. General scheme of the cone-and-plate system : a point M(r, B, rp) between the cone and the plate is at a distance y = r sin d from the plate and at a distance r cos d from the axis .

l

0 x>'

ttF

Dimensionless Frequency

Fig . 7 . Variation of the transfer function Zao =a/Afl vs . the dimensionless frequency w' = oxs/v according to expression (33).



1854

C. Destouts et al.

(0.7 M) as supporting electrolyte. This solution was used either as prepared for low viscosity measurements or with an admixture of glycerol (50 or 70%) so as to increase viscosity . Viscosity was measured with a Couette viscometer (CONTRAVES) and diffusivity determined by measuring the Schmidt number with a modulated rotating disk electrode by using the EHD impedance technique[26] . The sine wave modulation of the angular velocity was obtained by superimposing a sine wave voltage of low amplitude to the constant voltage used as reference for defining the average angular velocity of the do motor. Due to inertia limitations, the upper frequency limit is w/2x x 100 Hz. The experimental transfer function l/An was measured by means of a two channel transfer function analyzer (TFA SOLARTRON 1250) which delivers the potential signal for flow modulation : two responses are sent to the channels of the TFA, one from the optical encoder yields a voltage signal proportional to the instantaneous angular velocity and the other from the electrochemical interface a voltage signal proportional to the instantaneous current. Correlation performed by the TFA removes the time average value .

10 - ' cm2 s -1 , d = 0.032 cm, . ) = 50 rpm) are in the ranges of those used for the condition of Hstudy, Zaa being in quasi-steady-state . As can be seen in Fig. 8, the relative positions of theoretical curves for Zn, and ff verify this assumption . The overall impedance data (l/AS]) which corresponds to open circles, corrected from the effect of Z„D, provide the solid circles which show a very fair agreement with the curve in full line representative of the theoretical prediction of Hs given in Table 1 . In particular, the high frequency behaviour of the amplitude as a power law w -1 is observed over more than one decade, a result which is very important for psd analysis . Rectangular probe So far, no experimental verification of the validity of the theoretical expression of function H in the case of a rectangular electrode has been successfully performed. Two guard electrodes of the same width as the working one were placed at each side of it . The impedance data are displayed in Fig. 9 with the same set of parameters as in Fig . 8. Due to the rather large value of the microelectrode length (Ar z 0.26 cm) the correction effected by Z„ D required an integration :

RESULTS Circular probe The experimental data for a (70-30) glycerolwater mixture are reported in Fig . 8. The parameters values (r = 1 .14 cm, v = 0.2 cm2 s'', D=4 a

t

ZAr +& ZDa dr

H = J'

Here again, the agreement between the corrected data (symbols in solid) and the theoretical curve in full line is excellent.

ID ID_ 9 .3

9 s i

U

Fig . 8 . Theoretical transfer function H, for a circular microelectrode plotted vs . the dimensionless frequency according to the expressions given in Table I(-). The theoretical hydrodynamic transfer function Zi a is also plotted in there coordinates with r = 1 .14 cm, v - 0 .2 cm 2 s -1 , d = .032 0 cm, D = 4 x 10' cm' s - ', A = 2°, O = 50 rpm . The experimental points (0) are corrected by Z . in order to get the experimental H, values (0).

PREPARATION OF SMALL ELECTRODES BY PHOTOLITHOGRAPHY dx 1 /dz is an odd function in z and the term containing & is an even function in z only if the two parts of the bi-circular electrode are really identical . This necessity of a well defined probe geometry requires the use of the photolithography technique[5] . The microelectrode designs are drawn at scale 100 . A first reduction of 10 is taken on a high grade film on which the electrical contacts are added, and after a second reduction of 10, the photographic mask at scale I is obtained . On a GaAs substrate doped with chromium (resistivity : 105 f cm), a photosensitive resin layer is spread with a constant thickness of 0.8 µm. The photographic mask, which is an emulsion on a flexible material, is placed against the GaAs wafer with the resin. The resin is then bathed with a UV mercury lamp and finally developed, leaving bare the GaAs portions corresponding to the electrode area and the electrical contacts. Before the chemical deposition of gold, the bare surface of GaAs is activated with a solution of palladium chloride in acidic medium. The gold layer thickness is 1500A . The sun-bath technique with uv is used for a second level of masking in order to cover with insulated resin, the electrical contacts and to leave only the active part of the microelectrodes (Fig . 10) in contact with solution .



Hydrodynamic electrodes

1855

Fig . 9. Photograph of a bi-circular electrode obtained by the photolithography technique . The insulating gap and the electrical contact for the electrodes polarization are shown.

0

a

b

a Y

1

f

I

t

0 .3

I soc D p• { iLEH D 50n tn p' -

0

c

I 0 .5

~ I 1 I 1 I 1

2

5

10 20

50 t00

125

s a

100

d

75

t

50

a

N

25 a 0 0 .2 0 .5 1 2 5 10 20 Dirrwnsionless Frequency

50 100

(r '

Fig . 10. Theoretical transfer function H for a rectangular electrode plotted vs. the dimensionless frequency A(o' = w(72/Dfl)"') according to the expressions given in Table 1 . The theoretical hydrodynamics] transfer functions Zaa are plotted in these coordinates for each rotation speed with r=1 .04 cm, v=0.2cm2 s -1 , I=0.026 cm, D - 4 x 10"cm 2 s - ', d4 = 2' . The experimental points (open circle for f1 = 50 rpm and open triangle for 11 =30 rpm) are corrected by Zaa in order to get the experimental H, (black points) .

APPLICATION TO HYDRODYNAMICS Experimental set-up The experiments were conducted on a fully turbulent flow in a two-dimensional rectangular channel . This channel (2 .85m long, 12cm wide and 1 .2cm high) was placed between two tanks (=2001) . The flow through the channel was achieved by imposing a pressure in one tank, the other one being open to the atmosphere pressure . The electrochemical section was located at 2 m downstream in one tank so as to ensure fully developed flow conditions. The measurement was, therefore, performed in one direction, the fluid being recirculated toward the upstream tank after each pass . The pressure was controlled by a pneumatic feedback device in order to stabilize the flow rate value within ±0 .5% . The controlled voltage of the microelectrode sets to zero the concentration value of the reacting species at the wall . The diffusion current in the electrolysis cell was measured by means of a current follower . The spectrum analyzer used (Hewlett Packard 5451 C) allows the signal to be analyzed in a wide frequency range (10`-5 x 10 4 Hz). In these experiments, the useful range was limited to 10 - '10 -3 Hz : the higher limit was imposed by the instrumentation background noise and the lower limit by the tank volume imposing the acquisition time for a given flow rate . Time correlations : analysis of the power spectrum density As shown previously, by measuring simultaneously the currents r, and 7, on each part of the bi-circular microelectrode (see Fig. 10), we are able



1956

C . Dau.otns et al. ,

REFERENCES

,o'"-

1

Re_19100

I I I ,0r +d 10

+

FREQUENCY (Hz)

Fig. 11. Power spectrum density of the sum T, + l, and the difference T, - Ts vs . the frequency.

r,

r, - 2

to analyze the sum + rz and the difference T (Fig . 11). So, the power spectral density of the longitudinal velocity gradients fluctuations W can be obtained through the relationship :

H°

IHJf)I2

W1'+ I,

and, the power spectral density, of the transverse velocity gradient fluctations W' can be obtained through :

J = I H1f)I 2 Wt+-,2,

W

(35)

where H,(f)=FDHjf) and HAD =FDH,(f), respectively. The dsp W, and W, are plotted in Fig . 12 as an example for a Reynolds number of 19,000 . The linear portion of W. and Wp in the high frequency range are parallel with a slope of -1 .65 . An analysis of the coherence function between r, shows no correlation and so the + T, and velocity gradient fluctuation in the longitudinal and transverse directions are not correlated .

r, - rr 10

Re

1 9100 1

1 01

,0'

10

FREQUENCY(Hz)

Fig. 12 . Power spectrum density of the wall velocity gradients fluctuations in the mean flow direction W, and in the transverse W.-

. C . M . A . Brett and A. M . C . F. Oliveira Brett, Hydrody1 namic Electrodes in Comprehensive Chemical Kinetics Vol . 26 (Edited by C. H . Bamford and R . G. Compton . pp.355-441(1986) . 2 . P. R . Unwin and R . G . Compton, The Use of Channel Electrodes In the Investigation of Interfacial Reaction Mechanisms in Comprehensive Chemical Kinetics Vol . 29 (Edited by R . G . Compton), pp . 173-296 (1989). 3 . V . G . Levich, Physiochemical Hydrodynamics. Prentice Hall, New Jersey (1962) . 4 . B. Tribollet and J. Newman, J. eectrochem . Soc . 130, 822 (1983) . 5 . C. Deslouis and B. Tribollet, Flow Modulation Techniques In Electrochemistry in Advances in Electrochemicat Science and Engineering Vol . 2 (Edited by H . Gerischer and C . W . Tobias), pp. 205-264 (1991). 6. W. J. Blaedel and G. W. Schiaffer, Anal. Chem . 49, 49 (1977). 7. T. J. Hanratty and J. A. Campbell, Measurement of Wall Shear Stress in Fluid Mechanics Measurements (Edited by Goldstein). Hemisphere, Washington (1983) . 8. B. M. Grafov, S. A. Martemjanov and L. N. Nekrasov, The Turbulent D(fusion Layer in Electrochemical Systems . Nauka (in Russian) (1990). 9. C. Deslouis, B. Tribollet and L Viet, Ann. NY Acad. Sci . 404, 471(1983). 10. V. E. Nakoryakov, 0. N . Kashinsky and B. K. Kozmenko, Electrochemical Method for Measuring Turbulent Characteristics of Gas-Liquid Flows, Measuring techniques in Gas-Liquid Two-Phase Flows IUTAM Symposium, Nancy (France), pp. 695-721(1983) . 11 . S. C. Ling, Trans. Am. Soc . mech . Engrs . Series C, J . Heat Transfer 85, 230 (1963) . 12. M . A. LevPque, Ann. Mines 13, 283 (1928). 13. T. J . Pedley, J . Fluid Mech . 55, 329 (1972). 14. V. E. Nakoryakov, A . P. Burdokov, 0 . N. Kashinsky and P. 1 . Geshev, Electrodiffusion Method of Investigation into the Local Structure of Turbulent Flows (Edited by V. E. Gasenko) Novosibirsk (1986) . 15 . C . Deslouis, 0 . Gil and B. Tribollet, J . Fluid Mech . 215.85 (1990). 16 . J . Newman, Ind. Engng Chem. 7, 514 (1968). 17. J. Newman, Electrochemical Systems, 2nd Ed . Prentice Hall, New Jersey (1991). 18. C. Deslouis, B. Tribollet and M . A . Vorotyntsev, J . electrochem. Sm. 138, 2651 (1991) . 19. C . Deslouis and B . Tribollet, J , electroanal . Chem . 185, 171 (1985). 20 . M . Abramowitz and I . A . Stegun, Handbook of Mathematical Functions, p. 446. Dover, New York (1970) . 21 . G . Fortuna and T. J. Hanratty, Int. J. Heat Mass Transfer 14,1499 (1971). 22 . R . G . Compton, M. E. Laing and P. R Unwin, J. eleciroanal . Chem. 207, 309 (1986) . 23 . A . Ambari, C . Desloui; and B. Tribollet, Int. J . Heat Mass Transfer 29,35 (1986) . 24 . L. Talbot and J . J . Steinert, Trans. ASME K : J. Biomech Engng 109, 60 (1987). 25 . C . Deslouis, 0. Gil and B. Tribollet, Int. J . Heat Mass Transfer 33, 2525 (1990). 26 . B . Robertson, B . Tribollet and C. Deslouis, J . electrothem Soc. 135, 2279 (1988) .