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Chemical Engineering, Department of
1-1-1991
A Mathematical Model of Electrochemical
Reactions Coupled with Homogeneous Chemical
Reactions
Ken-Ming Yen
Texas A & M University - College Station
Taewhan Yeu
Texas A & M University - College Station
Ralph E. White
University of South Carolina - Columbia, [email protected]
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Journal of the Electrochemical Society, 1991, pages 1051-1054.
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DOI: 10.1149/1.2085714
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A Mathematical Model of Electrochemical ReactionsCoupled
with HomogeneousChemical Reactions
Ken-Ming Yin,* Taewhan Yeu,* and Ralph E. White**
Chemical Engineering Department, Texas A & M University, College Station, Texas 77843-3122
T h e z i n c / b r o m i n e (Zn/Br2) flow battery has r e c e i v e d considerable a t t e n t i o n in r e c e n t years [e.g., (2-4)]. A l t h o u g h it
is agreed that t h e solution c h e m i s t r y is i m p o r t a n t in the
system, m o s t of t h e w o r k that has b e e n d o n e is concentrated on t h e d e s i g n variables. In this n o t e the basic mass
transfer-solution and surface kinetics are studied to furnish a b e t t e r u n d e r s t a n d i n g of the system. The results pres e n t e d are for e l e c t r o c h e m i c a l reaction
2Br- ~ Br2 * 2e
[1]
gle electrochemical-chemical
reactions, A one-dimensional model is used here, i.e., the model is strictly valid at
the center of the RDE. Additional assumptions used are:
(i) Dilute solution theory applies, i.e., the driving force for
the flux of species is related to the ion-solvent friction. Ionion interactions are not considered. (it) Double-layer
charging is not considered. (iii) The solution is isothermal.
(iv) The physical and transport parameters are constant
within the diffusion layer.
c o u p l e d w i t h t h e h o m o g e n e o u s c o m p ] e x a t i o n reaction
Governing equations.--Steady-state mass c o n s e r v a t i o n
of species i can be written as (Ref. (5), p. 218)
kf
B r - + Br2 ~ Br3[2]
kb
on a rotating d i s k e l e c t r o d e (RDE) (4). A detailed discussion of the m i g r a t i o n effect is included.
T h e m i g r a t i o n effect for cases w i t h o u t the i n t e r f e r e n c e
of h o m o g e n e o u s c h e m i c a l reactions has b e e n d i s c u s s e d
t h o r o u g h l y by N e w m a n (Ref. (5), Chap. 19) and other researchers [e.g., Ref. (6-8)]. Also, H a u s e r and N e w m a n (9)
p o i n t e d out t h e possibility of an interesting potential minim u m w i t h i n the diffusion layer w h e n the s u p p o r t i n g electrolyte participates in t h e e l e c t r o d e reaction w i t h o u t hom o g e n e o u s c h e m i c a l reactions involved.
E l e c t r o c h e m i c a l m e t h o d s h a v e b e e n u s e d for the determ i n a t i o n of h o m o g e n e o u s reaction rate constants (10-13).
F o r e x a m p l e , t h e l i m i t i n g c u r r e n t d e n s i t y d e p e n d s on the
rate of a h o m o g e n e o u s c h e m i c a l reaction; Kouteck:~ and
L e v i c h (12) analytically d e r i v e d a l i m i t i n g c u r r e n t density
e x p r e s s i o n for an e l e c t r o c h e m i c a l r e a c t i o n on a R D E
c o u p l e d w i t h c h e m i c a l reactions of t h e t y p e s A ~ B and 2A.
B. Also, an e x p e r i m e n t a l d e t e r m i n a t i o n of the dissociation rate of acetic acid has b e e n d o n e on t h e R D E by A1b e r y and Bell (Ref. (11), p. 132).
A l t h o u g h analytic analysis has b e e n done, it is s h o w n in
this c o m m u n i c a t i o n that a c o m p r e h e n s i v e n u m e r i c a l analysis is r e q u i r e d to s t u d y t h e s e systems. F o r e x a m p l e , the
b e h a v i o r b e l o w the l i m i t i n g c u r r e n t w i t h h o m o g e n e o u s
c h e m i c a l reactions c a n n o t be p r e d i c t e d by any analytic approach, b e c a u s e t h e e l e c t r o d e kinetics n e e d to be incorporated. A m a t o r e and S a v e a n t (14) n u m e r i c a l l y investigated
the e l e c t r o c h e m i c a l , chemical, e l e c t r o c h e m i c a l (ECE) and
the d i s p r o p o r t i o n a t i o n m e c h a n i s m s . A l t h o u g h e l e c t r o d e
kinetics are i n c l u d e d , t h e y forcefully set one of the reactant c o n c e n t r a t i o n s to zero at the interface, e v e n at the
n o n l i m i t i n g c u r r e n t condition, w h i c h is not correct. Yen
and C h a p m a n (15) u s e d t h e o r t h o g o n a l collocation m e t h o d
for t h e E C E m e c h a n i s m and s h o w e d that t h e value of the
h o m o g e n e o u s rate c o n s t a n t modifies t h e limiting current
d e n s i t y t r e m e n d o u s l y . T h e m e t h o d t h e y u s e d m a y save
c o m p u t a t i o n time, b u t a c c u r a c y is sacrificed b e c a u s e the
c h e m i c a l r e a c t i o n occurs close to t h e interfacial region in
w h i c h no collocation points are allocated. A d a n u v o r et al.
(16) c o n s i d e r e d t h e s a m e s y s t e m as that s t u d i e d here w i t h
the m i g r a t i o n effect; h o w e v e r , the b u l k c o n c e n t r a t i o n s
t h e y u s e d do n o t satisfy the e q u i l i b r i u m condition, w h i c h
leads to q u e s t i o n a b l e p r e d i c t e d c u r r e n t densities. In this
note, this d i s c r e p a n c y is r e m o v e d . R e c e n t l y , H a u s e r and
N e w m a n (17) s t u d i e d t h e c o m p t e x a t i o n reaction rate of cup r o u s ions by the singular perturbation method and demonstrated a strategy for the data analysis by lumping relevant variables.
Theory
For generality, the mode] equations are developed for
m u l t i p l e reactions, a l t h o u g h the s t u d i e d s y s t e m is for sin* Electrochemical Society Student Member.
9* E l e c t r o c h e m i c a l S o c i e t y A c t i v e M e m b e r .
0 = - V 9N, + Ri
[3]
w h e r e Ni is the m o l a r flux of species i and R, is the net generation rate of i by h o m o g e n e o u s c h e m i c a l reactions. F o r
the h o m o g e n e o u s c h e m i c a l reaction s h o w n in Eq. [2], RBr= RBr2 = --RBr3- = - k f CBr-CSr2 ~- kbCBr3-. Ni i n c l u d e s migration, diffusion, and c o n v e c t i o n (Ref. (5), p. 301)
Ni = -ziu~Fc~VdP
- -
[4]
DiVc~ + VC,
w h e r e the m o b i l i t y u, can be a p p r o x i m a t e d by DJRT acc o r d i n g to the N e r n s t - E i n s t e i n relation (Ref. (5), p. 229).
F o r a o n e - d i m e n s i o n a l model, Eq. [3] b e c o m e s
0 = D , ~ y 2 - v, = - + z i u y l c i
dy
\ dy 2
dy dy /
+ R,
[5]
T h e n o r m a l v e l o c i t y at a small d i s t a n c e from t h e d i s k surface can be e x p r e s s e d b y the first t e r m of a p o w e r series
approximation
v~ = -a12
y2
[6]
w h e r e a is a c o n s t a n t w i t h the v a l u e 0.51023262 (Ref. (5),
p. 282), f~ is t h e rotation s p e e d (rad/s), and v is t h e kinetic
viscosity. Also, the solution c o n c e n t r a t i o n s of the various
species m u s t satisfy the e l e c t r o n e u t r a l i t y c o n d i t i o n
nion
0 -= ~ ciz,
[7]
i=l
B o u n d a r y conditions.--The ionic c o n c e n t r a t i o n s app r o a c h t h e u n i f o r m b u l k c o n d i t i o n s after a certain characteristic d i s t a n c e or the diffusion layer t h i c k n e s s (5) f r o m
t h e interface
=\a~/
\~/
[8]
w h e r e D R r e p r e s e n t s t h e diffusivity of t h e limiting species
(Br-). It is c o n v e n i e n t to set the b u l k b o u n d a r y conditions
at y = 2~
cL(2~) = Ci,b~tk
[9]
w h e r e t h e b u l k ionic c o n c e n t r a t i o n s satisfy t h e equilibr i u m c o n d i t i o n (Keq = kr/kb = CB,-3.b~Ik/CB,..bu~kCm2.bu,k)and
the e l e c t r o n e u t r a l i t y c o n d i t i o n (Eq. [7]). T h e solution potential at 25 can be assigned an arbitrary c o n s t a n t for conv e n i e n c e ; since t h e v a l u e of cP(25) is immaterial, it only
serves as a basis of c o m p u t a t i o n (18). Note that these
b o u n d a r y c o n d i t i o n s do n o t i n c l u d e the e x a c t position of
the r e f e r e n c e e l e c t r o d e (YRE) b e c a u s e t h e o h m i c drop bet w e e n 25 and YRE can be easily c o m p e n s a t e d if the applied
c u r r e n t d e n s i t y and specific c o n d u c t i v i t y of the solution
d. Electrochem. Soc., Vol. 138, No. 4, April 1991 9 The Electrochemical Society, Inc.
1051
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J. Electrochem. Soc., Vol. 138, No. 4, April 1991 9 The Electrochemical Society, Inc.
1052
are k n o w n b y a s s u m i n g t h a t O h m ' s law (/T = --~ ddp/dy)
applies b e t w e e n 25 and YRET h e b o u n d a r y c o n d i t i o n s at t h e e l e c t r o d e surface are
that the flux of e a c h ionic species is e q u a l to the associated
surface e l e c t r o c h e m i c a l reactions
nr
,
i-
-- Z siz~j = Ni
[10]
2.0
. . . .
,
. . . .
|
. . . .
, . . . .
kt=10 ?
k~lOs
^ke'0~
"'
~
/~
t=lO
,
. . . .
9
/
/ X
L6
j=l n y
w h e r e Ni = - ziuiFc~ dcP/dy - D~ d c / d y at t h e surface a n d s,j
is t h e s t o i c h i o m e t r i c coefficient of species i in electroc h e m i c a l r e a c t i o n j. T h e partial c u r r e n t d e n s i t y ij is exp r e s s e d b y t h e B u t l e r - V o l m e r e q u a t i o n (18, 19)
kt_~lOH
I.e
- -
with migration
............ without migration
exp
(c 0t-
r
c~a'jnjF "'" -
- Uj ,'ef)l
J
1.0
. . . .
o.o
'
o.,
. . . .
'
. . . .
o.z
t
. . . .
o.',
,
. . . .
o.4
o.~
i T (A/am e)
[11]
Fig. 1. Polarization curves with different rote constants: kf -< lO 3,
= | 0 s, = 10 7, = | 0 9, -> | 0 u, from lower to higher current densities.
T h e e x c h a n g e c u r r e n t d e n s i t y i0j.~efis b a s e d on the c h o s e n
r e f e r e n c e c o n c e n t r a t i o n s ; p,.j and qij are t h e reaction orders for a n o d i c and c a t h o d i c reactions. The potentiald e p e n d e n t term, Va - (P0 - U~.~,f,is t h e o v e r p o t e n t i a l at the
interface for r e a c t i o n j, w h i l e a,,j and (~c4 are the corresp o n d i n g transfer coefficients for a n o d i c and c a t h o d i c reactions. T h e e x p r e s s i o n U j ~ is s h o w n in Eq. [8] of Ref. (18).
Finally, the e l e c t r o n e u t r a l i t y c o n d i t i o n (Eq. [7]) and t h e exp r e s s i o n of total a p p l i e d c u r r e n t d e n s i t y
Results and Discussion
F i g u r e 1 s h o w s t h e s i m u l a t e d a n o d i c polarization c u r v e s
w i t h different h o m o g e n e o u s rate constants. T h e migrat i o n - e x c l u d e d case is i n c l u d e d for c o m p a r i s o n , w h i c h is
d o n e by t a k i n g out t h e m i g r a t i o n t e r m in Eq. [5] and by rem o v i n g the e l e c t r o n e u t r a l i t y c o n d i t i o n (Eq. [7]). F o r kf _<
103, the p e r t u r b a t i o n of the h o m o g e n e o u s c h e m i c a l react i o n can b e n e g l e c t e d so that t h e polarization curves are
t h e s a m e as t h o s e calculated by letting Ri = 0. It is s h o w n
that for kf -> 107, t h e p r e d i c t e d l i m i t i n g c u r r e n t densities inc l u d i n g m i g r a t i o n are significantly larger t h a n t h o s e without i n c l u d i n g t h e m i g r a t i o n effect. Obviously, the migrat i o n effect is m a g n i f i e d by i n c r e a s i n g kf (or kb). The
c o n c e n t r a t i o n profiles w i t h i n t h e diffusion layer are s h o w n
in Fig. 2a. T h e m u c h larger Br2 c o n c e n t r a t i o n at t h e interface w h e n c o n s i d e r i n g m i g r a t i o n reflects t h e larger limiting c u r r e n t d e n s i t y in Fig. 1. T h e generally larger predicted c u r r e n t densities w h e n m i g r a t i o n is i n c l u d e d can be
e x p l a i n e d b y t h e c o n c e n t r a t i o n profiles of B r - and Br3b e i n g v e r y close to t h e e l e c t r o d e surface as s h o w n in
Fig. 2b. At t h e interface CN,+ = CB~3-,since e l e c t r o n e u t r a l i t y
m u s t be p r e s e r v e d w h e n m i g r a t i o n is considered, this
causes a h i g h e r CBr3- n e a r the surface. In o t h e r words, t h e
a n o d e attracts m o r e n e g a t i v e Br3- ions to the electrode,
w h i c h t h e n releases m o r e r e a c t a n t B r - by t h e c h e m i c a l reaction [2] n e a r t h e e l e c t r o d e c o m p a r e d to the case w i t h o u t
migration. B e c a u s e Br3- p l a y e d t h e role as t h e s u p p l i e r of
B r , B r - has a h i g h e r c o n c e n t r a t i o n n e a r t h e e l e c t r o d e
w h e n c o n s i d e r i n g migration, and, c o n s e q u e n t l y , has a
h i g h e r g r a d i e n t at t h e interface at t h e l i m i t i n g c u r r e n t condition. F r o m t h e l i m i t i n g c u r r e n t d e n s i t y at kf = 109 in
Fig. 1, it s h o u l d be clear t h a t t h e g r a d i e n t is a b o u t t w i c e as
m u c h w h e n c o n s i d e r i n g m i g r a t i o n c o m p a r e d to the one
w i t h o u t c o n s i d e r i n g migration.
A n i n t e r e s t i n g b e h a v i o r is o b s e r v e d for t h e electric field
(E = - d , b / d y ) . As s h o w n in Fig. 3a, the electric field is al-
l ~l
\Ci,ref/
exp
L
~
(Y d -
(~I)0 -
,
i T = ~ ij
[12]
j=l
s e r v e as t h e last t w o e q u a t i o n s for the interface solution
p o t e n t i a l q)0 a n d t h e e l e c t r o d e p o t e n t i a l ltd. In this currentc o n t r o l l e d m o d e l , total c u r r e n t d e n s i t y (iT) rather t h a n t h e
a p p l i e d p o t e n t i a l (Vd - 4PRE)is set.
Solution Technique
T h e g o v e r n i n g e q u a t i o n s and related b o u n d a r y conditions are cast in t h e finite difference form. T h e s e e q u a t i o n s
are s o l v e d u s i n g N e w m a n ' s B A N D s u b p r o g r a m w i t h a matrix solver M A T I N V (Ref. (5), p. 419). T h e u n k n o w n s determ i n e d are ionic c o n c e n t r a t i o n s , q(y), solution potential,
(I)(y), and e l e c t r o d e potential, Vd. T h e e l e c t r o d e potential,
Vd, w h i c h has o n l y a single v a l u e at t h e electrode, is treated
as an u n k n o w n c o n s t a n t in B A N D (6, 7, 19).
Parameters
T h e e x a m p l e c h o s e n is t h e Br-/Br2 e l e c t r o d e reaction
c o u p l e d w i t h the t r i b r o m i d e c o m p l e x reaction in the
a q u e o u s solutions. T h e c h e m i c a l s i n t r o d u c e d into the b u l k
solution are Br 2 (0.3 molfliter) and N a B r (0.3 mol/liter).
N o t e the e q u i l i b r i u m b u l k c o n c e n t r a t i o n s s h o u l d be calculated b a s e d on t h e e q u i l i b r i u m c o n s t a n t and t h e a m o u n t of
c h e m i c a l s t h a t are added. T h e associated c h e m i c a l and
e l e c t r o c h e m i c a l reactions are s h o w n in Table I. T h e
n e e d e d t r a n s p o r t data are listed in Table II.
Table II. Mass transport data.
Table I. Electrochemical and chemical reactions.
Electrochemical reaction
2/~r- - Br2 --> 2e
(~a.,
0.5
(x~.~
0.5
Dal • I 0 s
U~a (V)
1.087b
Electrochemical kinetic parameters
n~
i0j.~ef(A/cm2)
2
0.001861 ~
(~)
Species
U~.,'~f(v)
1.0466d
Chemical equilibrium
Equilibrium constant (cm3/mol)
kf
Br2+Br ~Br3kb
17 x 10~
a Although the example is for a single electrochemical reaction,
the subscript j is kept for consistency.
b Ref. (3).
c The exchange current density corresponds to c,.r,.rin Ref. (3).
d Calculated from Eq. [8] in Ref- (18).
zt
Cbi,bulk X 103
c%,.,,r • I0 :r
(mol~
(mol~
\~m3]
\ c ma/
Na +
~r-
1
- 1
1.334
2.084
0.3
0.106647
1.0
1.08
Br2
Br3-
0
-1
1.31
1.31
0.106647
0.193353
0.05
0.92
v=0.0123cm2/s T - 2 9 8 K
=2.23• 10 scm
po=l.0g/cm 3 12= 104.7rad/s
a DNa%DBr from Ref. (5), p. 230; Din2, DBr3 from Ref. (3).
b The equilibrium bulk concentrations are calculated according
to the introduced NaBr and Br2 concentrations.
c The concentrations are from Ref. (3); they correspond to ir =
0.001861 A/cm2.
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J. Electrochem. Soc., Vol. 138, No. 4, April 1991 9 The Electrochemical Society, Inc.
$SO.O
.
.
.
1053
,
.
. . . .
,
. . . .
,
. . . .
10.0
~.O.O
9.0 ~
&O
. . . .
'
. . . .
'
'
kf = 10 It
. . . .
kf = 109
180.0
k
zo ~
A
n
. . . .
with migration
s.O4.o~.0""..."
....~
. . .'..,
. ~ . x ~ ~,
k f ffi 1 0 7
........................
kf = 105
kf = 103
............ without migration
~.o ........- : : i x' : : : . , ~ ~ , ~
z.o
S40.O
............
200.0
ISO.O
+
_ __
+i
B.r~-
1
,o
0.5
0.0
0.0
~
O.O
,
0.5
,
I
!.5
,
1.0
1,0
dimensionless
I.w
distance
S.O
(~')
2,0
dimensionless distance (0
800.0
....
,
. . . .
,
. . . .
,
. . . .
,
. . . .
,
....
~.0.0
SSO.O
25.0
Na +
uo.o
20.0
200.0
~-
~
with. migration
......... without migration
1o.o
~
100,0
-
l'a.O
k r = lolO~~________....~
/
/
120,0
k f ffi I 0 O
............
6O.O
.
~
~
k f = 107
40.0
5,0
Br-
0.0
o.o ~;'"":':'?::'7."7.'"I'F"7"."':'"':':'II
0.000
0.000
0.010
0.015
II
I
O.OBQ
dimensionless distance (0
Fig. 2. (a, top) The concentration profiles within the diffusion !ayer
under limiting current density for kt ~- !0 9. (b, bottom) Concentration
profiles of Na +, Br-, and Br3- close to the electrode under limiting current density for kf = 10 9.
most constant near the outer region of the diffusion layer
where the concentrations are close to the bulk concentrations. However, higher homogeneous rate constants correspond to higher E values because of the larger limiting
current densities. E increases rapidly when approaching
the electrode because the ionic species are depleted there.
There is a m a x i m u m in E close to the electrode as can be
seen more clearly in Fig. 3b. Note that the m a x i m u m for kf
= 10~ occurs at ~ ~ 0.04, as shown in Fig. 3a. Such phenomenon can be explained by the significant difference in corn
centration gradients in two regions, i.e., the outer diffusion
region and the i n n e r chemical reaction region. Figure 4
shows the concentration profiles of Br- within the diffu~
sion layer. It is clear that for kf ~ 107, a significant difference occurs between the concentration gradients in the
two regions. The larger the r~te constant, the narrower the
inner region (see Fig. 2b for the inner region at kf = 10~).
The total current density, which is constant across the diffusion layer, can be e~;pressed by the respective contributions from Ohm's law and the diffusion current (Ref. (5),
p. 221)
dcP
dci
iT -~ -K---dy - F~. ziD~ d Y
....
' .....
' ....
' ....
o.~1o
o.oso
O.OZO
dimensionless distance (0
o.o~
o.oso
Fig. 3. The potential gradient profiles (a, top) within and (b, bottom)
near the diffusion layer at different rate constants under limiting current conditions.
in iDiffafter a maximum, then iDiff increases rapidly near
the surface. The drop in ibm.m u s t be due to the decrease in
the concentration gradients (dc,/dy, especially for Br-), i.e.,
concentrations in this region are somewhat flattened. The
flattening should come from the continuous release of Brfrom Br3-. The flattened concentration region is reflected
by the m i n i m u m of iDiff in Fig. 5b. It should now be clear
that iDiff will rise again at the interface because of the larger
dCBr-/dy there due to the Br- concentration passing
through a flattened region and then suddenly dropping to
zero at the limiting current condition. Note that for kr =
1011, the diffusion current increases after a m i n i m u m at ~ <
0.005 which cannot be seen in Fig. 5b. The flattened concentration region and the latter larger concentration gradient at interface characterize the increase and decrease of
the potential gradient in Fig. 3b.
10.0
[13]
where K = F~z~uic~. As shown in Fig. 5a and b, Ohm's law
(iOhm = --K dq)/dy) applies well for ~ > 1.5 until the diffusion
current ( i D i f f = ~ FZiziDi dcJdy) becomes important closer to
the interface. For kf -< 103, when the chemical reaction can
be neglected, the diffusion current density (Fig. 5b) increases monotonically toward the electrode because concentration gradients increase steadily when approaching
the electrode. As kr increases, the diffusion-migration
m e c h a n i s m induced diffusion current is important when
-< 1.5. Unlike the no-chemical reaction case, there is a drop
' " " ' ' ' ....
o.ooo o.ooe
///:/
S.O
----
-
k / . l l O 11
kf=lO u
.L
o
4.o
kt=lO ?
kr=lO 5
~o
kt-~lOs
O.O
I
O.O
0.6
dimensionless
1.0
distartce
1.6
,
,
,
,
2.0
(0
Fig. 4. The Br- concentration profiles within the diffusion layer at
various homogeneous rate constants.
Downloaded 14 Jun 2011 to 129.252.106.20. Redistribution subject to ECS license or copyright; see http://www.ecsdl.org/terms_use.jsp
J. Electrochem. Soc., Vol. 138, No. 4, April 1991 9 The Electrochemical Society, Inc.
1054
0.50
.
.
.
.
I
.
.
.
.
I
.
.
.
.
I
.
.
.
.
0.45
0.40
0.35
}...................................................................................................................
0.00
O.25
2
........
k f = l O 11
...............
kf=lO ?
0.20
kr=lO 3
0.10
0.I0
0.00
,
O.OO
0.o
,
,
.
, . . . .
0.0
dimensionless
, . . . .
l.O
distance
, . . . .
1.$
(~)
2.0
0.10
0.14
0.15
0.12
O.II
O.IO
O.OS
i
0.06
0.07
........
k f = l O 11
...............
kf=lO ?
~, O.OS
kt=lO 3
0.05
0.04
0.03
O.OS
0.01
O.OO
o.o
0.0
I.O
I~
2.0
d i m e n s i o n l e s s distance (~)
Fig. 5. (a, top) The contribution from ohmic current within the diffusion layer at different rate constants under the limiting current conditions. (b, bottom) The contribution from diffusion current within the diffusion layer at different rate constants under the limiting current
conditions.
i0j.re~ exchange current density at reference concentrations for reaction j, A/cm 2
iT
total current density, A/cm 2
kb
backward h o m o g e n e o u s rate constant, 1/s
kf
forward h o m o g e n e o u s rate constant, cm3/s mol
Keq equilibrium constant, cm3/mol
nlon n u m b e r of ionic species
nr
n u m b e r of electrochemical reactions (equal to 1
here)
nj
n u m b e r of electrons transferred in reaction j
N1
flux vector of species i, mol/cm 2 9 s
Ni
flux of species i in y direction, mol/cm 2 - s
p,j
anodic reaction order of ionic species i in reaction j,
dimensionless
qi4
cathodic reaction order of species i in reaction j, dimensionless
R
universal gas constant, 8.3143 J/mol 9K
Ri
the h o m o g e n e o u s reaction rate of species i,
mol/s 9c m 3
sij
stoichiometric coefficient of ionic species i in electrochemical reaction j, dimensionless
T
absolute temperature, K
ui
mobility of species i (=Di/RT), tool - cm2/J 9 s
Uj.r~f theoretical open-circuit potential of reaction evaluated at reference concentrations, V
U~
standard electrode potential for reaction j, V
v
solution velocity vector, cm/s
vy
normal solution velocity, cm/s
Vd
potential of the working electrode, V
y
normal coordinate, cm
YRE position of reference electrode, cm
zi
charge n u m b e r of species i
Greek letters
~a,j anodic transfer coefficient for reaction j, dimensionless
~c.j cathodic transfer coefficient for reaction j, dimensionless
characteristic layer thickness according to Eq. [8],
cm
K
v
po
~0
Summary
A one-dimensional model of electrochemical reactions
coupled with h o m o g e n e o u s chemical reactions is given.
An e x a m p l e of the oxidation of Br- coupled with Br + Brz
Br3- shows the importance of including the migration
effect. The h o m o g e n e o u s chemical reaction causes a maxim u m in the electric field within the reaction layer.
1.
2.
3.
4.
Acknowledgments
The authors are grateful for the support of this project
given by Sandia National Laboratories, the Texas Advanced Technology and Research Program, and the Texas
A&M University Board of Regents through the Available
University Fund.
5.
Manuscript submitted J u l y 23, 1990; revised manuscript
received Nov. 30, 1990.
9.
10.
Texas A & M University assisted in meeting the publication costs of this article.
11.
6.
7.
8.
12.
L I S T OF SYMBOLS
a
c~
cj~
0.51023262
concentration of species i, mol/cm 3
concentration of species i at the solid-solution interface, mol/cm 3
ci.b~m bulk solution concentration of species i, mol/cm 3
ci.~r reference concentration of species i, mol/cm 3
D~
diffusion coefficient of species i, cm2/s
DR
diffusion coefficient of the limiting species (Br-),
cm2/s
E
electric field (= -d(P/dy), V/cm
F
Faraday's constant, 96,487 C/mol
ij
partial current density due to reaction j, A/cm ~
iD~,~ diffusion current density (= -F~iziD i dci/dY), A/cm ~
ioh~ ohmic contribution to total current density (= -K
d~P/dy), A/cm 2
13.
14.
15.
16.
17.
18.
19.
solution conductivity, 1/~ 9cm
kinematic viscosity, cm2/s
dimensionless distance (=y/g)
pure solvent density, g/cm 3
solution potential, V
solution potential adjacent to electrode surface, V
disk rotation speed, rad/s
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