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Mass transfer for a modulated flow at a rotating disk electrode: asymptotic solutions
J Eiecrroanal Chem. 185 (1985) 171-176 Elsewer Sequoia S A, Lausanne - Plnted
171 m The Netherlands
Short communication MASS TR4NSFER FOR A MODLXATED ELECTRODE: ASYMF’TOTIC SOLUTIONS
C
DESLOUIS
Groupe Juweu,
AT A ROTATING
DlSK
and B TFUBOLLET
de Recherche 75230 Pans
(Received
FLOW
no 4 da C N R S , Physrque Cedex 05 (France)
17th September
1984.
des Lqurdes
1x1revrsed form Znd
Octobx
et Elecrrochrrnre
Tour 22, 5e etage.
4. place
1984)
The response analysis of an electrochenucal interface to a perturbatron of the flow IS a new tool for the study of electrochermcal ktnettcs, as well as mass transfer and hydrodynamtcs Bruckenstein et al [1,2] earher proposed to modulate smusotdally the angular velocity of a rotatmg dtsk electrode tn a newtontan solutton tn order to charactertze the dtffusrvrty of redox spectes m solution_ Some theoretrcal expresstons, either analytrcal [3] or numerical [4], of the mass transfer response to tins type of perturbatmg srgnal have been cstabhshed so far. However, these soluttons are not easy to use over the whole frequency range. To overcome this drawback, asymptottc behavrours of the mass transfer refor frttmg procedures, have been calcusoonse’,, which can be eastly implemented I&ted in the low and hrgh frequency domams respectively and are reported m thrs paper These soluttons were compared w1t.h the exact numerical solutions ANAL\
SK
The trme dependent concentration equation (for notation see ref. 4).
The quanttttes
distnbutron
111eqn (1) are tune dependent
is governed
by the mass balance
and defmed by:
1 +FRe{expItir}
!2=w ( c=F+JZe
AQ :ZexpJwt i CJ
(2)
1 >
0022-0728
/SS/SO3
30
8 1983 Elsewer
Sequoia S A
172
The dtffuston layer IS so thin for the usual liquids that the development hmited to the two first terms: V== -&+Pz3+
...
V== -?iz*+j?z3+
_._
of V_ may be (3) (4)
with a = 0.51023, Cr= a a3/2~-1/2, B= Q*v-’ /3 and from ref 4, & =p(o, s = 2p (the values of j’(o, p) are grven in ref. 4) Equatton (1) can be spht mto a steady and an unsteady part.
p)E/a,
In eqn. (6) the fluctuatmg quadrattc terms have been dropped smce An/a (< 1. By using the dtmensronless vanables and parameters B = (3/a4)‘j3, 5 = z/+b, $ = (30/E) ‘I3 = (3D/av)“3m, wtth p = w/H (the equrvalent 1s r(4/3)$).Equatrons (5) and (6) become-
K = w@/D
= (9/a2>‘/3
with
p SC”3
then K = (w/iiii)(9v/a2D)‘/3 Nemst diffusron layer thickness
(8) aC/a[,
the solutton
a?
of eqn. (5) IS well known
cca-i?(O)
% = ?(4/3)(1
+ 0.298
Sc’13)
exp
[5]
-5x+-&+
__.)
When w tends towards zero or mfnuty, some stmphfrcatrons arise: m partrcular the quanttty c = G/E is a functton of the dimenstonless frequency p( = w/Q) that had been determmed numencrlly in ref 4 and analytrcally calculated m refs 6 and 7 m the low and hrgh frequency ranges respectrvely: In the LF range,
l(p)-+~(0)=3/2 In the HF range,
l( p)
from ref. 7 comes.
= (JP)+yU
(the asymptouc
behavrours
of E( p)/r(O)
are reported
m Erg 1).
(I) LF Soiutron When K tends towards zero, a quasi steady state solutron can be inferred by assurmng that all the fluctuatmg quantitres ( - ) are Identical to the denvatrves of the relevant trme average quantrtres wrth respect to the angular velocrty.
173 10,
1
1
OS, -I-
I
-_a0 WW 02,
011 01
I 02
I 05
10
2@
50
10
20
P FIN 1 Normahzed amphtude vanauons of E versus the dlmenslonless calculation from ref 4 and asymptouc behawour from ref 7
The steady state solufion deduced from eqn. (9) as:
of the ccncel,tratlon
gradient
50
100
frequency
:,
(x)
at the mterface
numenul
can be
aF CCC-F(O) az I& = $lY(4/3)(1 + 0 298 SC-‘/~)
(10)
Then -- dF(0) dS1
dE
-az C-C aZ da at Op-+0
$r(4/3)(1
+ 0.298
2
-C(O)
-SC--‘/~ )
7
r(4,3)(C;;
0 298S~-“~)
(11) md:
-F(O)
ar aZ Op+O=
rl/r(4/3)(1
Smce 4 a E-lj3, one obtams J/q = -E/3a when p + 0. By pu:tmg J= D&F/& 10, ore obtams.
e(0)
JLF
-=-
J
c,
-F(O)
+1
(12)
+ 0 298Sc-1’3) = -e/3
and
therefore
= - l/2
03)
z
(II) HF solurlon As K tends towards homogeneous equation:
,,+)
mfuuty,
the convective
term
may
be lsregarded
in the
7
the solution
(14) 6 of which IS:
B=exp{-(/K)‘/ZS}
(15)
174
This expressron corresponds to the concentration response for an electrochenucal Impedance m the Warburg conditions Since the perturbanon frequency is large, the dtstance over whtch a concentratron wave proceeds IS small. then the term tn t3 may be dtsregarded wrth respect to the term u-r .$’ and &?/a< = &?/at 1o Equation (8) becomes:
(16) Then (17) From
the boundary
condrtron
a?/&$ ---, 0 when 5 -+ 00 one &tams-
(18)
Then.
-as = _(JKp2i.;) az
I
o
or wrth the same vanable
LF -= J
-
i;(O)
4~) E(~)
changes
T(4/3)(1
cc.3 -F(O)
(a) ^ J&J
-
9 o
(
,K
09)
y
as prevrously.
+ 0 298Sc-“3)(
/K)“”
9
+ (J#’
Two cases are of mterest, tron or flux at the wall. Concentrostatlc-
arl aZ
they are respectrvely
obtamed
E(P) ‘co)
for a constant
(20) concentra-
S(0) = 0
= l/2
(21)
(b) JHF/JLF
= (W(
JK)~“)(~
Intentiostattc-
PV~H
(22)
J = 0
(a)
E(O)LF=
C, - E(O) 2
(23)
175
Fig 2 Normahzed amphtude of the mass transfer response for the co~.centros~a~c case and for the mlenhostauc case
(b)
s.(O)HF -= 2.(o)LF r(4/3)(1
18 + 0.298St-“3)(,K)2
C(P) do)
As to the kugh frequency vmations, it is convement to define a cut-off frequency as the mtercept between the constant low frequency solution and *he power law h.@ frequency solution with a - 1.5 or - 2 exponent according to eqns. (22) and (24) and by setting l( 9)/e(O) equal to 1. Then: 11&/&II
= 3.06/(
PSC”~)~~
(25)
which leads to p cu1-d
z.(“hiF
-
@)LF
JP
= 2.11
= l-9/(
pSc’/‘)*(l
(26) + 0.298Sc-“3)
(27)
176
which leads to ~c”r_‘rSC”~=
138(1-0_149sc-"3)
(28)
In Fig. 2, eqns. (25) and (27) (for SC = cz)) are compared to the exact numerical solution grven in ref. 4. It can be seen that the low frequency solutrons (eqns 21 and 23) are m phase with the velocrty modulatron At high frequencies, the concentrostatic and mtentrostatic solutrons (eqns. 22 and 24) have phase lags wrth respect to the velocity modulatton of respectively 135” and 180” rf the effect of e(p) 1s not considered The Schmidt number correctron appears for the mtentrostatrc case (eqn 24) but not for the concentrostatic case. From the experimental standpoint, the hrgh frequency solutrons can be checked eastly by studymg the amplitude vanatrons with frequency REFERENCES 1 B Miller, M I Bella~axe and S Bruckenstem, Anal Ch-m , 44 (1972) 1983 2 K Tokuda, S Bruckewte?r~ &d B hUler, J Ehxtrochem Sot, 177 (1975) 1316 3 C Deslou~s, C Gabneill, Ph Samte-Rose Fan&me and B Tnbol!et. J Uectrochem 107 4 B TnboUet and J Newman, J Electrochem Sot, 130 (1983) 2016 5 J Newman J Phys Chem, 70 (1966) 1327 6 E M Sparrow and J L Gregg J Aerosp SCI , 27 (1960) 252 7 V P. Shamq Acta Mech , 32 (1979) 19
Sot , 129 (1982)
171 m The Netherlands
Short communication MASS TR4NSFER FOR A MODLXATED ELECTRODE: ASYMF’TOTIC SOLUTIONS
C
DESLOUIS
Groupe Juweu,
AT A ROTATING
DlSK
and B TFUBOLLET
de Recherche 75230 Pans
(Received
FLOW
no 4 da C N R S , Physrque Cedex 05 (France)
17th September
1984.
des Lqurdes
1x1revrsed form Znd
Octobx
et Elecrrochrrnre
Tour 22, 5e etage.
4. place
1984)
The response analysis of an electrochenucal interface to a perturbatron of the flow IS a new tool for the study of electrochermcal ktnettcs, as well as mass transfer and hydrodynamtcs Bruckenstein et al [1,2] earher proposed to modulate smusotdally the angular velocity of a rotatmg dtsk electrode tn a newtontan solutton tn order to charactertze the dtffusrvrty of redox spectes m solution_ Some theoretrcal expresstons, either analytrcal [3] or numerical [4], of the mass transfer response to tins type of perturbatmg srgnal have been cstabhshed so far. However, these soluttons are not easy to use over the whole frequency range. To overcome this drawback, asymptottc behavrours of the mass transfer refor frttmg procedures, have been calcusoonse’,, which can be eastly implemented I&ted in the low and hrgh frequency domams respectively and are reported m thrs paper These soluttons were compared w1t.h the exact numerical solutions ANAL\
SK
The trme dependent concentration equation (for notation see ref. 4).
The quanttttes
distnbutron
111eqn (1) are tune dependent
is governed
by the mass balance
and defmed by:
1 +FRe{expItir}
!2=w ( c=F+JZe
AQ :ZexpJwt i CJ
(2)
1 >
0022-0728
/SS/SO3
30
8 1983 Elsewer
Sequoia S A
172
The dtffuston layer IS so thin for the usual liquids that the development hmited to the two first terms: V== -&+Pz3+
...
V== -?iz*+j?z3+
_._
of V_ may be (3) (4)
with a = 0.51023, Cr= a a3/2~-1/2, B= Q*v-’ /3 and from ref 4, & =p(o, s = 2p (the values of j’(o, p) are grven in ref. 4) Equatton (1) can be spht mto a steady and an unsteady part.
p)E/a,
In eqn. (6) the fluctuatmg quadrattc terms have been dropped smce An/a (< 1. By using the dtmensronless vanables and parameters B = (3/a4)‘j3, 5 = z/+b, $ = (30/E) ‘I3 = (3D/av)“3m, wtth p = w/H (the equrvalent 1s r(4/3)$).Equatrons (5) and (6) become-
K = w@/D
= (9/a2>‘/3
with
p SC”3
then K = (w/iiii)(9v/a2D)‘/3 Nemst diffusron layer thickness
(8) aC/a[,
the solutton
a?
of eqn. (5) IS well known
cca-i?(O)
% = ?(4/3)(1
+ 0.298
Sc’13)
exp
[5]
-5x+-&+
__.)
When w tends towards zero or mfnuty, some stmphfrcatrons arise: m partrcular the quanttty c = G/E is a functton of the dimenstonless frequency p( = w/Q) that had been determmed numencrlly in ref 4 and analytrcally calculated m refs 6 and 7 m the low and hrgh frequency ranges respectrvely: In the LF range,
l(p)-+~(0)=3/2 In the HF range,
l( p)
from ref. 7 comes.
= (JP)+yU
(the asymptouc
behavrours
of E( p)/r(O)
are reported
m Erg 1).
(I) LF Soiutron When K tends towards zero, a quasi steady state solutron can be inferred by assurmng that all the fluctuatmg quantitres ( - ) are Identical to the denvatrves of the relevant trme average quantrtres wrth respect to the angular velocrty.
173 10,
1
1
OS, -I-
I
-_a0 WW 02,
011 01
I 02
I 05
10
2@
50
10
20
P FIN 1 Normahzed amphtude vanauons of E versus the dlmenslonless calculation from ref 4 and asymptouc behawour from ref 7
The steady state solufion deduced from eqn. (9) as:
of the ccncel,tratlon
gradient
50
100
frequency
:,
(x)
at the mterface
numenul
can be
aF CCC-F(O) az I& = $lY(4/3)(1 + 0 298 SC-‘/~)
(10)
Then -- dF(0) dS1
dE
-az C-C aZ da at Op-+0
$r(4/3)(1
+ 0.298
2
-C(O)
-SC--‘/~ )
7
r(4,3)(C;;
0 298S~-“~)
(11) md:
-F(O)
ar aZ Op+O=
rl/r(4/3)(1
Smce 4 a E-lj3, one obtams J/q = -E/3a when p + 0. By pu:tmg J= D&F/& 10, ore obtams.
e(0)
JLF
-=-
J
c,
-F(O)
+1
(12)
+ 0 298Sc-1’3) = -e/3
and
therefore
= - l/2
03)
z
(II) HF solurlon As K tends towards homogeneous equation:
,,+)
mfuuty,
the convective
term
may
be lsregarded
in the
7
the solution
(14) 6 of which IS:
B=exp{-(/K)‘/ZS}
(15)
174
This expressron corresponds to the concentration response for an electrochenucal Impedance m the Warburg conditions Since the perturbanon frequency is large, the dtstance over whtch a concentratron wave proceeds IS small. then the term tn t3 may be dtsregarded wrth respect to the term u-r .$’ and &?/a< = &?/at 1o Equation (8) becomes:
(16) Then (17) From
the boundary
condrtron
a?/&$ ---, 0 when 5 -+ 00 one &tams-
(18)
Then.
-as = _(JKp2i.;) az
I
o
or wrth the same vanable
LF -= J
-
i;(O)
4~) E(~)
changes
T(4/3)(1
cc.3 -F(O)
(a) ^ J&J
-
9 o
(
,K
09)
y
as prevrously.
+ 0 298Sc-“3)(
/K)“”
9
+ (J#’
Two cases are of mterest, tron or flux at the wall. Concentrostatlc-
arl aZ
they are respectrvely
obtamed
E(P) ‘co)
for a constant
(20) concentra-
S(0) = 0
= l/2
(21)
(b) JHF/JLF
= (W(
JK)~“)(~
Intentiostattc-
PV~H
(22)
J = 0
(a)
E(O)LF=
C, - E(O) 2
(23)
175
Fig 2 Normahzed amphtude of the mass transfer response for the co~.centros~a~c case and for the mlenhostauc case
(b)
s.(O)HF -= 2.(o)LF r(4/3)(1
18 + 0.298St-“3)(,K)2
C(P) do)
As to the kugh frequency vmations, it is convement to define a cut-off frequency as the mtercept between the constant low frequency solution and *he power law h.@ frequency solution with a - 1.5 or - 2 exponent according to eqns. (22) and (24) and by setting l( 9)/e(O) equal to 1. Then: 11&/&II
= 3.06/(
PSC”~)~~
(25)
which leads to p cu1-d
z.(“hiF
-
@)LF
JP
= 2.11
= l-9/(
pSc’/‘)*(l
(26) + 0.298Sc-“3)
(27)
176
which leads to ~c”r_‘rSC”~=
138(1-0_149sc-"3)
(28)
In Fig. 2, eqns. (25) and (27) (for SC = cz)) are compared to the exact numerical solution grven in ref. 4. It can be seen that the low frequency solutrons (eqns 21 and 23) are m phase with the velocrty modulatron At high frequencies, the concentrostatic and mtentrostatic solutrons (eqns. 22 and 24) have phase lags wrth respect to the velocity modulatton of respectively 135” and 180” rf the effect of e(p) 1s not considered The Schmidt number correctron appears for the mtentrostatrc case (eqn 24) but not for the concentrostatic case. From the experimental standpoint, the hrgh frequency solutrons can be checked eastly by studymg the amplitude vanatrons with frequency REFERENCES 1 B Miller, M I Bella~axe and S Bruckenstem, Anal Ch-m , 44 (1972) 1983 2 K Tokuda, S Bruckewte?r~ &d B hUler, J Ehxtrochem Sot, 177 (1975) 1316 3 C Deslou~s, C Gabneill, Ph Samte-Rose Fan&me and B Tnbol!et. J Uectrochem 107 4 B TnboUet and J Newman, J Electrochem Sot, 130 (1983) 2016 5 J Newman J Phys Chem, 70 (1966) 1327 6 E M Sparrow and J L Gregg J Aerosp SCI , 27 (1960) 252 7 V P. Shamq Acta Mech , 32 (1979) 19
Sot , 129 (1982)