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Mass-transfer rates in the presence of an oscillating boundary layer
Ctiical Engineering Science. Vol. 41, No. Printed in Great Britain.
7, pp. 1735-1741.
1986.
OOO%-2509/86 $3.00 + 0.00 Fkrgamon Journals Ltd.
MASS-TRANSFER RATES IN THE PRESENCE OSCILLATING BOUNDARY LAYER PETER S. FEDKIW,t
JAMES
M. POTENTE
and DANIEL
OF AN
R. BROUNS
Department of Chemical Engineering, North Carolina State University, Raleigh, NC 27695, U.S.A. (Received 25 February 1985) Abstract-The boundary-layer form of the convective-diffusion equation has been solved under a spatially uniform but sinusoidally oscillating wall-concentration boundary condition. An analytical solution for the stationary-state local wall flux is found for two cases: (i) the wall moves at a constant velocity, and (ii) the fluid flows past a static wall. The flux in the leading-edge region is in phase with the wall concentration oscillation and decays in the same manner as the steady-state flux for each case. In both cases at large dimensionless distance From the leading edge (corresponding to a high oscillation frequency), the time-dependent contribution to the tlux becomes independent of streamwise position; but the time-averaged wall flux follows the normal decay of the L&^eque solution when the time-averagedwallconcentrationis usedas the boundary condition. A spatially uniform, but time-dependent, boundary layer thickness is established in marked contrast to the increasing growth under steady-state conditions. The results presented here may be used as a basis for calculating the mass-transfer rate in the uresence of an arbitrary wallconcentration oscillation
INTRODUCTION
The classical L&tZque solution to the boundary-layer form of the convective-diffusion equation for mass transfer assumes a spatially uniform and timeindependent reactant wall concentration; for example, for a mass-transfer-limited reaction, the wall concentration is zero. In electrochemical reactors, however, it is possible to manipulate the reaction rate by modulating the electrode voltage, and hence, the surface concentration of the reactant. Electroplaters commonly practice “pulse plating”, in which the current is varied periodically from a high to a low level. In this manner, the metal is found distributed, and, in addition, may occur.
to be more uniformly morphological benefits
Cheh and co-workers (Viswanathan et al., 1978; Viswanathan and Cheh, 1979; Pesco and Cheh, 1984) have examined both theoretically and experimentally the oscillation of the reactant’s surface concentration on a rotating-disk electrode when a periodic current is imposed on a mass-transfer-limited reaction. They found that at high oscillation frequency, the perturbation imposed on the concentration profile was confined to a region very near the surface. Consequently, a simple diffusion model with no convective transport of the reactant could be used to calculate the mass-transfer rate. In addition, since the surface of the rotating disk is uniformily accessible to a mass-transfer-limited reagent, the oscillating component of the resulting concentration profile is independent of radial position. Chin and co-workers (Chin, 1980,1983, Cheng and Chin, 1984) have studied the eff&t
of oscillating
spherical
electrode.
‘To
current
In contrast
on a rotating
hemi-
to the rotating-disk
whom correspondence should be addressed.
electrode, the reaction distribution for a mass-transfer limited reaction is nonuniform. The thickness of the reactant depletion layer grows from zero at the pole to its largest value at the equator. Chin (1971) used a perturbation analysis of the boundary-layer form of the convective-diffusion equation to calculate the steady-state reaction distribution. Newman (1972) showed how the Lighthill transformation could be applied to the conservation equation to calculate the local mass-transfer rate. In doing so, he determined that the perturbation solution overestimated local reaction rates; or, alternatively, the boundary-layer thickness predicted was too thin over most of the surface. Chin and co-workers (Chin, 1980,1983; Cheng and Chin, 1984) used the Nemst diffusion-layer concept to solve the transient form of the diffusion equation with the boundary-layer thickness obtained from his perturbation solution at steady-state conditions. A spatially uniform reaction rate along the sphere’s surface was assumed as a boundary condition. Because of the inherent nonuniformity of masstransfer-limited reactions on a sphere, this boundary condition can be rigorously applied only at large oscillation frequencies. Deslouis et al. (1983) have presented an approximate analytical solution to the convective-diffusion equation without invoking the stagnant diffusion-layer concept for both the rotatingdisk and hemispherical electrode subject to an oscillating surface concentration and current, respectively. In this paper, the solution to the boundary-layer form of the convective-diffusion equation is presented when the surface concentration oscillates sinusoidally in time, but remains spatially uniform. An analytical expression is derived for the local, instantaneous wall flux for two cases: (i) the wall moves at a constant velocity V, and (ii) the fluid flow past a static wall. The results presented here may be used as the basis to
1735
PETER S.FEDKIW
1736
calculate the mass-transfer rate for an arbitrary waiiconcentration oscillation by use of a Fourier decomposition coupled with a superposition integral. An analytical solution to the transient Lev^eque problem for a step change of the wail concentration in time-but spatially uniform-has been presented previously (Hudson and Bankoff, 1964, Soiiman and Chambre, 1967). In addition, a solution using a Levin transformation to approximatethe solution in Lapiace transform space and experimental verification of the result have also been given (Compton and Daly, 1984). In principle, the convolution integral could be used with these results to calculate the reaction rate when the wall concentration oscillates sinusoidally. In practice, however, the resulting integrals are unwieldy and are not presented here. The present solution procedure uses the property that a linear system which is perturbed in a periodic manner will-after transient responses have decayed-respond in a like fashion.
MODEL
et al,
subject to the following boundary conditions:
e,(x = 0,~) = 0
eM(X,y +
The solution to eq. (3) may be found by standard techniques as 8,=
C,(t, x) = (C,}
+AC,cos
The oscillating component of the concentration may be found as the real part of a complex function:
(2)
In writing eq. (2) it is assumed that the Schmidt number is so large that the concentration boundary layer is confined to a region near the plate such that tr, = V and v,, = 0. A similar formulation of the convective-diffusion equation for mass transfer from the electrolyte to a wire electrode pulled through a plating bath has been given by Alkire and Varjian (1977). Chin et al. (1977) have shown how the convective-diffusion equation may be solved in the more general case for the moving-plane problem. It is relevant to note that the steady-state wail flux predicted from the solution of eq. (2) [given by eq. (6)] is equivalent to that presented by Chin et al. (1977) in the limit of large Schmidt number. The stationary-state solution to eq. (2) is sought subject to eq. (1) and the uniform concentration Co both at x = 0 and y -B co. The solution may be written as the sum of a steady-state and periodic function. The governing equation for the steady-state component is 3
Daze, dX
=dy”
Y) = “(2
y, = Re{F(X,
Y)eiT}.
(7)
0 The
governing dimensionless equation for Br is de, ae, ar+ax=-
se, i3Y2
18)
and is subject to the boundary conditions:
e,(x = 0,~) = 0
vat
D&T ax=avzs
D nvx J- --
ycov
(1)
where (C, ) is the average wail concentration, AC, is the amplitude and w is the frequency of the oscillation. Far from the wall, the concentration of the reactant is uniform at Co. The boundary-layer form of the convective-diffusion equation is assumed to apply:
g+-
&f].
Irs=
0,(X,
(wt)
*erfc[
The steady-state contribution to the local dimensionless wall flux is calculated from differentiation of eq. (5):
DEVELOPMENT
Case 1: moving wall Assume that a planar wall moves at a constant velocity V through a fluid which contains a reactant which is consumed at the wail’s surface. The reactive section begins at the leading edge where x = 0. The surface concentration is assumed to be spatially uniform but sinusoidally oscillates in time as
00) = 0.
(3)
e,(x,Y = 0) = heir e,(x,
Y+
(9)
00)= 0.
After substituting eq. (7) and applying the Lapiace transform in the X coordinate to eq. (8), an ordinary differential equation for F( Y, s) is generated which is easily solved: F( Y, s) = fe-Js+j y_ (10) S
Elementary properties of Lapiace transforms and a standard Lapiace inversion table (Oberhettinger and Radii, 1973) can be used to invert the above: F(X,
Y) = ~[eyi”‘erfc +e-ri”‘erfc
(&+
@)
-_
J-11
Y
iX
( 2fi
_
(11)
The oscillating component of the local dimensionless wall flux can be calculated by evaluating the real portion of the normal derivative of eq. (7) to find cos(r-XX) J’=$ACW=
m
+ JZ[(cosr)S(J~) -
(sin T)C ( &&G)]
where C(U) and S(U) are the (Abramowitz and Stegun, 1970).
(12) Fresnei
functions
Mass-transfer rates Case
2: stationary
wall
The fluid flows past a stationary wall on which the reactant concentration varies sinusoidally in time but is uniform in position. According to boundary-layer theory, the convective-diffusion equation reduces to
ac
Da2c
=+Gyg=
(13)
ay2
subject to the boundary conditions
1737
In order to invert the above, the following identity was utilized (Oberhettinger and Badii, 1973): 2-r
{&s-r)}
= JI f(u)J&2,/+du
where f(X) =LF- s {f(s)>. (21) Thus the problem has been reduced to determining the inverse Laplace transform of
C(x = 0, y) = co C(x,
y =
+AC,cosd
0) = (C,)
(14)
(22)
C(x, y -+ cc) = c,. The stationary-state solution to eq. (13) is sought and is found-as in case l-by writing the solution as the sum of a steady-state component and the real part of an oscillating complex function. The steady-state solution is the LRvi?que solution to the boundary-layer form of the convective-diffusion equation, and, for completeness, is reproduced below along with the steady-state local wall flux.
ud J,
= ~
Y
O”
l--(4/3) s 1
=
jss
=
G&=
e-c’dt; 1
Y
_
(1%
l-(4/3)
(9X’)“3
-
(16)
The inverse of eq. (22) was calculated by integration in the complex plane according to the inversion formula LF-‘{f(s)}
(17)
0,(X, B,(X,
00) = o.-
(18)
Laplace transform with respect to X can be applied to the above-after assuming a solution in the form of eq. (7)-to find the solution for F(s, Y) as The
=312
>
= nm
Ai(
(24)
has an infinite number of zeros (yk) along the negative real axis: = 0,
k =
1,2, . . . .
(25)
the integrand approaches ill2 as s -P cc. The inversion of eq. (22) thus consists of three contributions: a delta function from the integration as s + co; an infinite series from the sum of the residues at the poles; and an integral contribution from the integration along the branch cut. In evaluating the integral along the branch cut, the analytical continuation formula for the Bessel functions must be applied (Abramowitz and Stegun, 1970). After considerable manipulation, the oscillating contribution to the local dimensionless wall flux can be written as
L=
A
The inverse of P (s, Y) was not calculated; however, the time-dependent contribution to the local dimensionless wall flux was found by evaluating the real component of the normal derivative at the wall of the Laplace inverse of F(s, Y ):
= cos(r+x,4)-3k~m
2k=,
Iy 1-3’2 ’
xexp[-2x12yk1-3’2] xcos[r-~/4-2x12yk1-3’2] m e-Xf? K(r, r)dr +2rro s * 1
(26)
where
where 1dP -=A bdY I r=o
’
(23)
Also,
Y = 0) = scosot Y +
2 _ ( 3
Ai
= 0, Y) = 0
esxf(s)ds.
The Airy function Ai
subject to the boundary conditions 0,(X
s”i”
The integrand is multivalued, therefore, in order to apply analytic function theory, a branch cut was introduced along the negative real axis. In addition, the integrand contains an infinite number of poles, as can be seen from the relationship (Abramowitz and Stegun, 1970) Kll3
a2e, at-2
= &
y-ice
The transient solution is governed by
ae, yae, ar+-=ax
= f(x)
K,,,
(zs-l
i3”)
K,,,
(fs-~
i3,2)-
Ai(-
and ‘20b’
K(r, r) =
(N2/3/N1/3)COS(T-_2/3 -
NUM/DENOM
IYrl) = 0 +&1/3
-55rc/i2)
PErER8
1738
NUM
= N,,,N,/,ws
=+&~/a
-42/J
FEDICIW et ul. The stationary-state contribution to the wall flux for the fixed wall was also calculated by this method. The series expansion for the modified Bessel function in both the numerator and denominator of eq. (20b) was used to rewrite the equation as a ratio of two infinite complex series:
-+$ >
M,,,cos
-xN2,3
z+B,,3-'$2,3-~x >
(
+nM,,,N,,,cos
T+&,3-e2,3++ >
(
K2455-53f2) -
n2M2,3
M,,,cos
r+&,,--2,3-E
=
N:,,
+7r2M:,3
xcos
K
>
(
DENOM
-2xN,,3
113
M,,3
t
P3’+
)
x (a0 + alefisv2”
3
(
_fis,,3 \ =e
/.T
+ u2eds-4/3
(bO + b,efis-2/3
~1,3-%3+5Iz. >
+ . - -)
+ b2,$is-4/J + . . . )
(29)
where 1
1 b3n = 32”-“3n!F(n+2/3)
a3n = 32n-2’3n ! F(n + l/3)
-1 ajn+l
-0 -
a3"+2
=
-1
3’“+ ‘13n ! lY(n + 5/3)
b 3n+1
=32m+ff3n!r(n+4/3)
b 3n+2
-0 -
n = 0, 1, 2 _ . _ which after the division is performed, results in a single infinite series:
where N,,(z)~-“.(~) = ker, (I) -i kei, (z) MV(~)e-e.(z) = The
lxx,.(z)
-
i bei,(
argument of M,, NV, 0, and 4, is $ r.
Alternative series solution It is possible to exploit the properties of the Laplace transform to develop for both cases an alternative, but equivalent, series solution for the local wall flux. For the moving wall case, a binomial expansion for the normal derivative of eq. (IO) at the wall leads to the following: i(l
+i/s)‘j2
A
(!i)(+!)(+!).
= s- 112+~is-912+~s-si2+
-.
+ c,enfis- 243 + . . . } CO
=
aolbo
C”
=
a,-
c
Jt=
(_
The above may be inverted term-by-term and then multiplied by eiz. Taking the real portion of the resulting expression leads to the series:
L +
“E, (-
n 2
1.
cos r
4n-2 3
l)“c,.-;nx’+~“’ V0
[ RESULTS
It can be shown that eqs (12) and (28) are equivalent by introducing the series expansion for cos (X), sin(X) and the Fresnel functions in eq. (12).
/bo
( )
r-
i
+ ..-- (27)
>
~)n+lC2n_2X(4n-5)/3
“=l
n!
bjcm-j
The above expression after multiplication by - iI/2 IS was used in eq. (20a) and inverted term-by-term to find: m
. . (~-n)ins-@+ll3
5 j=l
(30)
AND
1 1
sinr*
(31)
DISCUSSION
Since the behaviour of the steady-state flux is well known for both cases, results shall be presented only for the time-dependent contribution to the masstransfer rate. Figure 1 presents Jt for the moving-wall case calculated from eq. (12) or (28) for various cycle times. Figure 2 is the similar result but for the stationary-wall case calculated from eq. (26) or (31). (Note: Care must be taken in evaluating eq. (26)
Mass-transfer rates
1739
6
X Dimensionless
Streamwise
Coordinate
Fig. 1. Timedependent contribution to the stationary-statemass-transferflux at various cycle times: moving-wallcase.
because the integral and the series are both divergent, but in a compensating manner such that their sum is finite. See Appendix.) From the series solution, it is seen that near the leading edge of the plate the mass-transfer rate is in phase with the wall-concentration oscillation and decays as X - ‘/= and X -‘I3 for the moving-wall and stationary-wall case, respectively. This leading-edge behaviour is identical to the steady-state contribution to the mass-transfer rate, and hence the oscillations in this vicinity only change the effective wall concentra-
tion. On the other hand, far from the leading edge, the dimensionless flux becomes independent of X and approaches cos (r + n/4) for both cases, as is clear from an asymptotic analysis of eqs (12) and (26). The total dimensionless flux may be written as the sum of the steady-state and stationary terms.
i
co&=
= J,(X’)
+ S
m
J,(r,
X’(o/G)“=).
(32) Since the steady-state contribution approaches zero
PETER S. FEDKIW
el al.
G
normal gradient of streamwise velocity at the wall, du,/dy, sP 1
i i JS
J-1 mass-transfer flux at wall, mol/cm2.
J, Jv W K&4
i,l(C0
kei,(x) ker, (x) W(x) N, (x1 t V
DImensionless
Streamwise Coardinfite
Fig. 2. Time-dependentcontributionto the stationary-state mass-transferfluxat variouscycletimes:stationary-wallcase.
far from the leading edge, the mass-transfer rate is dominated by the time-dependent term, which is proportional to the amplitude of the oscillation and the square root of its frequency; furthermore, the greater the time-constant ratio, w/G, the nearer to the leading edge the stationary-state contribution becomes the more dominant term. In terms of an effective boundary-layer thickness, the oscillating wall concentration has established a position-independent (but time-dependent) boundary layer far from the leading edge. Thus, at high oscillation frequency, the surfacewhich at steady state is nonhomogenous from a masstransport viewpointPbecomes uniformily accessible. An analogous behaviour was also predicted by Chin on the hemispherical electrode, and is, in fact, the anticipated behaviour at high oscillation frequency for any surface under boundary-layer flow conditions. The results presented here can be used as a basis to calculate the mass-transfer rate in a more general, periodic oscillation. A Fourier decomposition may be used to perform the calculation for any periodic wallconcentration function; and if the wall concentration is not independent of position (as is most likely to occur), a superposition integral may be formed with the flux response calculated here serving as the kernel function.
; X’ Y Y
i,l@C, &) Bessel function of order v modified Bessel function of order Y Kelvin function, Im (i-‘K,(xi-“2)) Kelvin function, Re {i-YK,(xi1’2)} of Kelvin function, [bery’(x) modulus + bei: (x)]l12 of Kelvin function, [ker:(x) modulus + kei: (x)]“’ time, s velocity of moving plate, cm/s streamwise boundary layer coordinate, cm (moving wall) 03”x/(G ,,6) ox/v (stationary wall) x$W normal boundary layer coordinate, cm Y&Z
Greek
letters
kth root of Airy function AC,&, Y/m [9x m]
e 0” (x)
1’3
dimensionless concentration phase angle of Kelvin function,
NOTATION
Ai
bei, (x) ber, (x) C AC, D
Airy function Kelvin function, Im { J, (xi 312)> Kelvin function, Re ( J,(xi 3/2)} concentration, mol/cm3 amplitude of wall-concentration oscillation, mol/cm3 average of wall-concentration oscillation, mol/cm3 diffusivity, cm2/s
arctan
[bei, (x)/ber,.(x)I
cot iW y CO
phase angle of Kelvin function, CkeiY(x)/kerV(x)l ( - CO)/C, oscillation frequency, rad/s
arctan
Subscripts inlet conditions 0
ss :,
steady state time-dependent wall
REFERENCES
Abramowitz, M. and Stegun, X.
Acknowledgement-This work was supported by the National Science Foundation under grant CPE 8414166.
s
m)
A., 1974 Handbook of Dover, New York. Alkire, R. and Varjian, R., 1977, Moving resistive wire electrodes. J. Electrochem. Sot. 124, 388-395. Cheng, C. Y. and Chin, D. T., 1984, Mass transfer in ac electrolysis: Part 1. Theoretical analysis using a lilm model for sinusoidal current on a rotating hemispherical eleu trode. A.1.Ch.E. J. 30, 757-764. Chin, D. T., 197 1, Convective diffusion on a rotating spherical electrode. J. Electrochem. Sot. 118. 1434-1438. Chin, D. T., 1980, Sinusoidal ac modulation of a rotating hemispherical electrode. J. Electroehem. See. 127, 2162-2166. Chin, D. T., 1983, Mass transfer and current-potential relation in pulse electrolysis. J. Electrochem. Sot. 130, 1657-1667. Chin, D. T., Viswanathan, K. and Gutowski, R.. 1977, Mass Mahematical
Functions.
Mass-transfer rates transfer to a continuous moving surface. J. Electrochem. Sot. 124, 713717. Compton, R. G. and Daly, P. J., 1984, Current transients at a channel electrode produced by a potential step. J. electroanal. Chem. 178,45-52. Deslouis, C., Gabriel& C. and Tribollet. B.. 1983, An analytical solution of the nonsteady convective diffusion equation for rotating electrodes. J. Electrochem. Sot. 130, 20442046. Hudson, J. L. and Bankoff, S. G., 1964, An exact solution of unsteady heat transfer to a shear flow. Chem. Engng Sci. 19, 59 l-598. Newman, J., 1972, Mass transfer to a rotating sphere at high Schmidt numbers. J. Electrochem. Sot. 119, 69-71. Oberhettinger, F. and Badii. L., 1973, Tables of Laplace Transforms. Springer, New York. Pesco, A. M. and Cheh, H. Y., 1984, Mass transfer in alternating current electrolysis. J. Electrochem. Sot. 131, 2259-2266. Soliman, M. and Chambrb, P. L.. 1967, On the timedependent L&?que problem. Int. J. Heat Mass Transfer 10, 169-180. Viswanathan, K. and Cheh, H. Y., 1979, Mass transfer aspects of electrolysis by periodic currents. J. Electrochem. Sot. 126,398-401. Viswanathan, K., Farrell Epstein, M. A. and Cheh, H. Y., 1978, The application of pulsed current electrolysis to a rotating-disk electrode system. J. Electrochem. Sot. 125, 1772-1776.
1741
integrate accurately e - =“K (r. r)/r near r = 0, the asymptotic behaviour indicated above is added and subtracted from the integrand with the use of analytic integration formulae. For the sake of convenience, the resulting integral is then divided into three intervals: (0, rL), rL, rU) and (rU, 03). where rL and rU are arbitrary lower and upper limits of integration but with rU constrained such that K(2r”/3, T) = 2~0s (T -n/4). With rU defined in the above manner, the integral from rU to co of the modified integrand can be calculated analytically. The resulting expression for Jt(X, T) may now be written as: = cos (r + x/4) -i
[y + In X + E, (X/r,)]
XcoS(T--/4)+ --
3e-2XIz~.l-3’f 2
IY*I=
+ $(1,3,
cos (5 - x/4)
2 L=l
krr
co~(r-rr/4--2X~2y,~-~‘~)
X/r,)
(2X/3)-
1’3
+ $[(2r,/3)“Je-x~r~ - (2X/3)1’3F(2/3, x [K(2r/3,
X/r,)]
rt_e-X/r 1 + 2x s O r
~)--A(s)(2r/3)-~‘~
APPENDIX
solution for Jt for the stationary wall is analytically correct; but, in its present form, it is numerically inconvenient to utilize. The infinite series and integral individually diverge but their sum is convergent. The divergent nature of the series arises because as k + co, the series approaches Z (l/k); whereas, the kernel function K(2r/3, x) becomes 2~0s (r -x/4) as r + co, which results in the integral becoming infinite in a logarithmic manner. After appropriate addition and subtraction of these divergent terms, the resulting new series and integral are both convergent. Numerical integration of the modified integrand must still be performed. An asymptotic analysis of the kernel shows that as r + 0, K (2r/3, 7) * A(r)($)“3 + B(r)&)“-’ + O(r). Hence, in order to The
x [K(2r/3, - 2 ~0s
(7
T) - A(~)(2r/3)‘~/~ -
x/4)]
dr
where y is Euler’s constant; F(o, z) is the incomplete gamma function; and E,(z) is the exponential integral.
A(7)
=
B(7)
=
cos
1.053721967485932*
7
sinr.
7, pp. 1735-1741.
1986.
OOO%-2509/86 $3.00 + 0.00 Fkrgamon Journals Ltd.
MASS-TRANSFER RATES IN THE PRESENCE OSCILLATING BOUNDARY LAYER PETER S. FEDKIW,t
JAMES
M. POTENTE
and DANIEL
OF AN
R. BROUNS
Department of Chemical Engineering, North Carolina State University, Raleigh, NC 27695, U.S.A. (Received 25 February 1985) Abstract-The boundary-layer form of the convective-diffusion equation has been solved under a spatially uniform but sinusoidally oscillating wall-concentration boundary condition. An analytical solution for the stationary-state local wall flux is found for two cases: (i) the wall moves at a constant velocity, and (ii) the fluid flows past a static wall. The flux in the leading-edge region is in phase with the wall concentration oscillation and decays in the same manner as the steady-state flux for each case. In both cases at large dimensionless distance From the leading edge (corresponding to a high oscillation frequency), the time-dependent contribution to the tlux becomes independent of streamwise position; but the time-averaged wall flux follows the normal decay of the L&^eque solution when the time-averagedwallconcentrationis usedas the boundary condition. A spatially uniform, but time-dependent, boundary layer thickness is established in marked contrast to the increasing growth under steady-state conditions. The results presented here may be used as a basis for calculating the mass-transfer rate in the uresence of an arbitrary wallconcentration oscillation
INTRODUCTION
The classical L&tZque solution to the boundary-layer form of the convective-diffusion equation for mass transfer assumes a spatially uniform and timeindependent reactant wall concentration; for example, for a mass-transfer-limited reaction, the wall concentration is zero. In electrochemical reactors, however, it is possible to manipulate the reaction rate by modulating the electrode voltage, and hence, the surface concentration of the reactant. Electroplaters commonly practice “pulse plating”, in which the current is varied periodically from a high to a low level. In this manner, the metal is found distributed, and, in addition, may occur.
to be more uniformly morphological benefits
Cheh and co-workers (Viswanathan et al., 1978; Viswanathan and Cheh, 1979; Pesco and Cheh, 1984) have examined both theoretically and experimentally the oscillation of the reactant’s surface concentration on a rotating-disk electrode when a periodic current is imposed on a mass-transfer-limited reaction. They found that at high oscillation frequency, the perturbation imposed on the concentration profile was confined to a region very near the surface. Consequently, a simple diffusion model with no convective transport of the reactant could be used to calculate the mass-transfer rate. In addition, since the surface of the rotating disk is uniformily accessible to a mass-transfer-limited reagent, the oscillating component of the resulting concentration profile is independent of radial position. Chin and co-workers (Chin, 1980,1983, Cheng and Chin, 1984) have studied the eff&t
of oscillating
spherical
electrode.
‘To
current
In contrast
on a rotating
hemi-
to the rotating-disk
whom correspondence should be addressed.
electrode, the reaction distribution for a mass-transfer limited reaction is nonuniform. The thickness of the reactant depletion layer grows from zero at the pole to its largest value at the equator. Chin (1971) used a perturbation analysis of the boundary-layer form of the convective-diffusion equation to calculate the steady-state reaction distribution. Newman (1972) showed how the Lighthill transformation could be applied to the conservation equation to calculate the local mass-transfer rate. In doing so, he determined that the perturbation solution overestimated local reaction rates; or, alternatively, the boundary-layer thickness predicted was too thin over most of the surface. Chin and co-workers (Chin, 1980,1983; Cheng and Chin, 1984) used the Nemst diffusion-layer concept to solve the transient form of the diffusion equation with the boundary-layer thickness obtained from his perturbation solution at steady-state conditions. A spatially uniform reaction rate along the sphere’s surface was assumed as a boundary condition. Because of the inherent nonuniformity of masstransfer-limited reactions on a sphere, this boundary condition can be rigorously applied only at large oscillation frequencies. Deslouis et al. (1983) have presented an approximate analytical solution to the convective-diffusion equation without invoking the stagnant diffusion-layer concept for both the rotatingdisk and hemispherical electrode subject to an oscillating surface concentration and current, respectively. In this paper, the solution to the boundary-layer form of the convective-diffusion equation is presented when the surface concentration oscillates sinusoidally in time, but remains spatially uniform. An analytical expression is derived for the local, instantaneous wall flux for two cases: (i) the wall moves at a constant velocity V, and (ii) the fluid flow past a static wall. The results presented here may be used as the basis to
1735
PETER S.FEDKIW
1736
calculate the mass-transfer rate for an arbitrary waiiconcentration oscillation by use of a Fourier decomposition coupled with a superposition integral. An analytical solution to the transient Lev^eque problem for a step change of the wail concentration in time-but spatially uniform-has been presented previously (Hudson and Bankoff, 1964, Soiiman and Chambre, 1967). In addition, a solution using a Levin transformation to approximatethe solution in Lapiace transform space and experimental verification of the result have also been given (Compton and Daly, 1984). In principle, the convolution integral could be used with these results to calculate the reaction rate when the wall concentration oscillates sinusoidally. In practice, however, the resulting integrals are unwieldy and are not presented here. The present solution procedure uses the property that a linear system which is perturbed in a periodic manner will-after transient responses have decayed-respond in a like fashion.
MODEL
et al,
subject to the following boundary conditions:
e,(x = 0,~) = 0
eM(X,y +
The solution to eq. (3) may be found by standard techniques as 8,=
C,(t, x) = (C,}
+AC,cos
The oscillating component of the concentration may be found as the real part of a complex function:
(2)
In writing eq. (2) it is assumed that the Schmidt number is so large that the concentration boundary layer is confined to a region near the plate such that tr, = V and v,, = 0. A similar formulation of the convective-diffusion equation for mass transfer from the electrolyte to a wire electrode pulled through a plating bath has been given by Alkire and Varjian (1977). Chin et al. (1977) have shown how the convective-diffusion equation may be solved in the more general case for the moving-plane problem. It is relevant to note that the steady-state wail flux predicted from the solution of eq. (2) [given by eq. (6)] is equivalent to that presented by Chin et al. (1977) in the limit of large Schmidt number. The stationary-state solution to eq. (2) is sought subject to eq. (1) and the uniform concentration Co both at x = 0 and y -B co. The solution may be written as the sum of a steady-state and periodic function. The governing equation for the steady-state component is 3
Daze, dX
=dy”
Y) = “(2
y, = Re{F(X,
Y)eiT}.
(7)
0 The
governing dimensionless equation for Br is de, ae, ar+ax=-
se, i3Y2
18)
and is subject to the boundary conditions:
e,(x = 0,~) = 0
vat
D&T ax=avzs
D nvx J- --
ycov
(1)
where (C, ) is the average wail concentration, AC, is the amplitude and w is the frequency of the oscillation. Far from the wall, the concentration of the reactant is uniform at Co. The boundary-layer form of the convective-diffusion equation is assumed to apply:
g+-
&f].
Irs=
0,(X,
(wt)
*erfc[
The steady-state contribution to the local dimensionless wall flux is calculated from differentiation of eq. (5):
DEVELOPMENT
Case 1: moving wall Assume that a planar wall moves at a constant velocity V through a fluid which contains a reactant which is consumed at the wail’s surface. The reactive section begins at the leading edge where x = 0. The surface concentration is assumed to be spatially uniform but sinusoidally oscillates in time as
00) = 0.
(3)
e,(x,Y = 0) = heir e,(x,
Y+
(9)
00)= 0.
After substituting eq. (7) and applying the Lapiace transform in the X coordinate to eq. (8), an ordinary differential equation for F( Y, s) is generated which is easily solved: F( Y, s) = fe-Js+j y_ (10) S
Elementary properties of Lapiace transforms and a standard Lapiace inversion table (Oberhettinger and Radii, 1973) can be used to invert the above: F(X,
Y) = ~[eyi”‘erfc +e-ri”‘erfc
(&+
@)
-_
J-11
Y
iX
( 2fi
_
(11)
The oscillating component of the local dimensionless wall flux can be calculated by evaluating the real portion of the normal derivative of eq. (7) to find cos(r-XX) J’=$ACW=
m
+ JZ[(cosr)S(J~) -
(sin T)C ( &&G)]
where C(U) and S(U) are the (Abramowitz and Stegun, 1970).
(12) Fresnei
functions
Mass-transfer rates Case
2: stationary
wall
The fluid flows past a stationary wall on which the reactant concentration varies sinusoidally in time but is uniform in position. According to boundary-layer theory, the convective-diffusion equation reduces to
ac
Da2c
=+Gyg=
(13)
ay2
subject to the boundary conditions
1737
In order to invert the above, the following identity was utilized (Oberhettinger and Badii, 1973): 2-r
{&s-r)}
= JI f(u)J&2,/+du
where f(X) =LF- s {f(s)>. (21) Thus the problem has been reduced to determining the inverse Laplace transform of
C(x = 0, y) = co C(x,
y =
+AC,cosd
0) = (C,)
(14)
(22)
C(x, y -+ cc) = c,. The stationary-state solution to eq. (13) is sought and is found-as in case l-by writing the solution as the sum of a steady-state component and the real part of an oscillating complex function. The steady-state solution is the LRvi?que solution to the boundary-layer form of the convective-diffusion equation, and, for completeness, is reproduced below along with the steady-state local wall flux.
ud J,
= ~
Y
O”
l--(4/3) s 1
=
jss
=
G&=
e-c’dt; 1
Y
_
(1%
l-(4/3)
(9X’)“3
-
(16)
The inverse of eq. (22) was calculated by integration in the complex plane according to the inversion formula LF-‘{f(s)}
(17)
0,(X, B,(X,
00) = o.-
(18)
Laplace transform with respect to X can be applied to the above-after assuming a solution in the form of eq. (7)-to find the solution for F(s, Y) as The
=312
>
= nm
Ai(
(24)
has an infinite number of zeros (yk) along the negative real axis: = 0,
k =
1,2, . . . .
(25)
the integrand approaches ill2 as s -P cc. The inversion of eq. (22) thus consists of three contributions: a delta function from the integration as s + co; an infinite series from the sum of the residues at the poles; and an integral contribution from the integration along the branch cut. In evaluating the integral along the branch cut, the analytical continuation formula for the Bessel functions must be applied (Abramowitz and Stegun, 1970). After considerable manipulation, the oscillating contribution to the local dimensionless wall flux can be written as
L=
A
The inverse of P (s, Y) was not calculated; however, the time-dependent contribution to the local dimensionless wall flux was found by evaluating the real component of the normal derivative at the wall of the Laplace inverse of F(s, Y ):
= cos(r+x,4)-3k~m
2k=,
Iy 1-3’2 ’
xexp[-2x12yk1-3’2] xcos[r-~/4-2x12yk1-3’2] m e-Xf? K(r, r)dr +2rro s * 1
(26)
where
where 1dP -=A bdY I r=o
’
(23)
Also,
Y = 0) = scosot Y +
2 _ ( 3
Ai
= 0, Y) = 0
esxf(s)ds.
The Airy function Ai
subject to the boundary conditions 0,(X
s”i”
The integrand is multivalued, therefore, in order to apply analytic function theory, a branch cut was introduced along the negative real axis. In addition, the integrand contains an infinite number of poles, as can be seen from the relationship (Abramowitz and Stegun, 1970) Kll3
a2e, at-2
= &
y-ice
The transient solution is governed by
ae, yae, ar+-=ax
= f(x)
K,,,
(zs-l
i3”)
K,,,
(fs-~
i3,2)-
Ai(-
and ‘20b’
K(r, r) =
(N2/3/N1/3)COS(T-_2/3 -
NUM/DENOM
IYrl) = 0 +&1/3
-55rc/i2)
PErER8
1738
NUM
= N,,,N,/,ws
=+&~/a
-42/J
FEDICIW et ul. The stationary-state contribution to the wall flux for the fixed wall was also calculated by this method. The series expansion for the modified Bessel function in both the numerator and denominator of eq. (20b) was used to rewrite the equation as a ratio of two infinite complex series:
-+$ >
M,,,cos
-xN2,3
z+B,,3-'$2,3-~x >
(
+nM,,,N,,,cos
T+&,3-e2,3++ >
(
K2455-53f2) -
n2M2,3
M,,,cos
r+&,,--2,3-E
=
N:,,
+7r2M:,3
xcos
K
>
(
DENOM
-2xN,,3
113
M,,3
t
P3’+
)
x (a0 + alefisv2”
3
(
_fis,,3 \ =e
/.T
+ u2eds-4/3
(bO + b,efis-2/3
~1,3-%3+5Iz. >
+ . - -)
+ b2,$is-4/J + . . . )
(29)
where 1
1 b3n = 32”-“3n!F(n+2/3)
a3n = 32n-2’3n ! F(n + l/3)
-1 ajn+l
-0 -
a3"+2
=
-1
3’“+ ‘13n ! lY(n + 5/3)
b 3n+1
=32m+ff3n!r(n+4/3)
b 3n+2
-0 -
n = 0, 1, 2 _ . _ which after the division is performed, results in a single infinite series:
where N,,(z)~-“.(~) = ker, (I) -i kei, (z) MV(~)e-e.(z) = The
lxx,.(z)
-
i bei,(
argument of M,, NV, 0, and 4, is $ r.
Alternative series solution It is possible to exploit the properties of the Laplace transform to develop for both cases an alternative, but equivalent, series solution for the local wall flux. For the moving wall case, a binomial expansion for the normal derivative of eq. (IO) at the wall leads to the following: i(l
+i/s)‘j2
A
(!i)(+!)(+!).
= s- 112+~is-912+~s-si2+
-.
+ c,enfis- 243 + . . . } CO
=
aolbo
C”
=
a,-
c
Jt=
(_
The above may be inverted term-by-term and then multiplied by eiz. Taking the real portion of the resulting expression leads to the series:
L +
“E, (-
n 2
1.
cos r
4n-2 3
l)“c,.-;nx’+~“’ V0
[ RESULTS
It can be shown that eqs (12) and (28) are equivalent by introducing the series expansion for cos (X), sin(X) and the Fresnel functions in eq. (12).
/bo
( )
r-
i
+ ..-- (27)
>
~)n+lC2n_2X(4n-5)/3
“=l
n!
bjcm-j
The above expression after multiplication by - iI/2 IS was used in eq. (20a) and inverted term-by-term to find: m
. . (~-n)ins-@+ll3
5 j=l
(30)
AND
1 1
sinr*
(31)
DISCUSSION
Since the behaviour of the steady-state flux is well known for both cases, results shall be presented only for the time-dependent contribution to the masstransfer rate. Figure 1 presents Jt for the moving-wall case calculated from eq. (12) or (28) for various cycle times. Figure 2 is the similar result but for the stationary-wall case calculated from eq. (26) or (31). (Note: Care must be taken in evaluating eq. (26)
Mass-transfer rates
1739
6
X Dimensionless
Streamwise
Coordinate
Fig. 1. Timedependent contribution to the stationary-statemass-transferflux at various cycle times: moving-wallcase.
because the integral and the series are both divergent, but in a compensating manner such that their sum is finite. See Appendix.) From the series solution, it is seen that near the leading edge of the plate the mass-transfer rate is in phase with the wall-concentration oscillation and decays as X - ‘/= and X -‘I3 for the moving-wall and stationary-wall case, respectively. This leading-edge behaviour is identical to the steady-state contribution to the mass-transfer rate, and hence the oscillations in this vicinity only change the effective wall concentra-
tion. On the other hand, far from the leading edge, the dimensionless flux becomes independent of X and approaches cos (r + n/4) for both cases, as is clear from an asymptotic analysis of eqs (12) and (26). The total dimensionless flux may be written as the sum of the steady-state and stationary terms.
i
co&=
= J,(X’)
+ S
m
J,(r,
X’(o/G)“=).
(32) Since the steady-state contribution approaches zero
PETER S. FEDKIW
el al.
G
normal gradient of streamwise velocity at the wall, du,/dy, sP 1
i i JS
J-1 mass-transfer flux at wall, mol/cm2.
J, Jv W K&4
i,l(C0
kei,(x) ker, (x) W(x) N, (x1 t V
DImensionless
Streamwise Coardinfite
Fig. 2. Time-dependentcontributionto the stationary-state mass-transferfluxat variouscycletimes:stationary-wallcase.
far from the leading edge, the mass-transfer rate is dominated by the time-dependent term, which is proportional to the amplitude of the oscillation and the square root of its frequency; furthermore, the greater the time-constant ratio, w/G, the nearer to the leading edge the stationary-state contribution becomes the more dominant term. In terms of an effective boundary-layer thickness, the oscillating wall concentration has established a position-independent (but time-dependent) boundary layer far from the leading edge. Thus, at high oscillation frequency, the surfacewhich at steady state is nonhomogenous from a masstransport viewpointPbecomes uniformily accessible. An analogous behaviour was also predicted by Chin on the hemispherical electrode, and is, in fact, the anticipated behaviour at high oscillation frequency for any surface under boundary-layer flow conditions. The results presented here can be used as a basis to calculate the mass-transfer rate in a more general, periodic oscillation. A Fourier decomposition may be used to perform the calculation for any periodic wallconcentration function; and if the wall concentration is not independent of position (as is most likely to occur), a superposition integral may be formed with the flux response calculated here serving as the kernel function.
; X’ Y Y
i,l@C, &) Bessel function of order v modified Bessel function of order Y Kelvin function, Im (i-‘K,(xi-“2)) Kelvin function, Re {i-YK,(xi1’2)} of Kelvin function, [bery’(x) modulus + bei: (x)]l12 of Kelvin function, [ker:(x) modulus + kei: (x)]“’ time, s velocity of moving plate, cm/s streamwise boundary layer coordinate, cm (moving wall) 03”x/(G ,,6) ox/v (stationary wall) x$W normal boundary layer coordinate, cm Y&Z
Greek
letters
kth root of Airy function AC,&, Y/m [9x m]
e 0” (x)
1’3
dimensionless concentration phase angle of Kelvin function,
NOTATION
Ai
bei, (x) ber, (x) C AC,
Airy function Kelvin function, Im { J, (xi 312)> Kelvin function, Re ( J,(xi 3/2)} concentration, mol/cm3 amplitude of wall-concentration oscillation, mol/cm3 average of wall-concentration oscillation, mol/cm3 diffusivity, cm2/s
arctan
[bei, (x)/ber,.(x)I
cot iW y CO
phase angle of Kelvin function, CkeiY(x)/kerV(x)l (
arctan
Subscripts inlet conditions 0
ss :,
steady state time-dependent wall
REFERENCES
Abramowitz, M. and Stegun, X.
Acknowledgement-This work was supported by the National Science Foundation under grant CPE 8414166.
s
m)
A., 1974 Handbook of Dover, New York. Alkire, R. and Varjian, R., 1977, Moving resistive wire electrodes. J. Electrochem. Sot. 124, 388-395. Cheng, C. Y. and Chin, D. T., 1984, Mass transfer in ac electrolysis: Part 1. Theoretical analysis using a lilm model for sinusoidal current on a rotating hemispherical eleu trode. A.1.Ch.E. J. 30, 757-764. Chin, D. T., 197 1, Convective diffusion on a rotating spherical electrode. J. Electrochem. Sot. 118. 1434-1438. Chin, D. T., 1980, Sinusoidal ac modulation of a rotating hemispherical electrode. J. Electroehem. See. 127, 2162-2166. Chin, D. T., 1983, Mass transfer and current-potential relation in pulse electrolysis. J. Electrochem. Sot. 130, 1657-1667. Chin, D. T., Viswanathan, K. and Gutowski, R.. 1977, Mass Mahematical
Functions.
Mass-transfer rates transfer to a continuous moving surface. J. Electrochem. Sot. 124, 713717. Compton, R. G. and Daly, P. J., 1984, Current transients at a channel electrode produced by a potential step. J. electroanal. Chem. 178,45-52. Deslouis, C., Gabriel& C. and Tribollet. B.. 1983, An analytical solution of the nonsteady convective diffusion equation for rotating electrodes. J. Electrochem. Sot. 130, 20442046. Hudson, J. L. and Bankoff, S. G., 1964, An exact solution of unsteady heat transfer to a shear flow. Chem. Engng Sci. 19, 59 l-598. Newman, J., 1972, Mass transfer to a rotating sphere at high Schmidt numbers. J. Electrochem. Sot. 119, 69-71. Oberhettinger, F. and Badii. L., 1973, Tables of Laplace Transforms. Springer, New York. Pesco, A. M. and Cheh, H. Y., 1984, Mass transfer in alternating current electrolysis. J. Electrochem. Sot. 131, 2259-2266. Soliman, M. and Chambrb, P. L.. 1967, On the timedependent L&?que problem. Int. J. Heat Mass Transfer 10, 169-180. Viswanathan, K. and Cheh, H. Y., 1979, Mass transfer aspects of electrolysis by periodic currents. J. Electrochem. Sot. 126,398-401. Viswanathan, K., Farrell Epstein, M. A. and Cheh, H. Y., 1978, The application of pulsed current electrolysis to a rotating-disk electrode system. J. Electrochem. Sot. 125, 1772-1776.
1741
integrate accurately e - =“K (r. r)/r near r = 0, the asymptotic behaviour indicated above is added and subtracted from the integrand with the use of analytic integration formulae. For the sake of convenience, the resulting integral is then divided into three intervals: (0, rL), rL, rU) and (rU, 03). where rL and rU are arbitrary lower and upper limits of integration but with rU constrained such that K(2r”/3, T) = 2~0s (T -n/4). With rU defined in the above manner, the integral from rU to co of the modified integrand can be calculated analytically. The resulting expression for Jt(X, T) may now be written as: = cos (r + x/4) -i
[y + In X + E, (X/r,)]
XcoS(T--/4)+ --
3e-2XIz~.l-3’f 2
IY*I=
+ $(1,3,
cos (5 - x/4)
2 L=l
krr
co~(r-rr/4--2X~2y,~-~‘~)
X/r,)
(2X/3)-
1’3
+ $[(2r,/3)“Je-x~r~ - (2X/3)1’3F(2/3, x [K(2r/3,
X/r,)]
rt_e-X/r 1 + 2x s O r
~)--A(s)(2r/3)-~‘~
APPENDIX
solution for Jt for the stationary wall is analytically correct; but, in its present form, it is numerically inconvenient to utilize. The infinite series and integral individually diverge but their sum is convergent. The divergent nature of the series arises because as k + co, the series approaches Z (l/k); whereas, the kernel function K(2r/3, x) becomes 2~0s (r -x/4) as r + co, which results in the integral becoming infinite in a logarithmic manner. After appropriate addition and subtraction of these divergent terms, the resulting new series and integral are both convergent. Numerical integration of the modified integrand must still be performed. An asymptotic analysis of the kernel shows that as r + 0, K (2r/3, 7) * A(r)($)“3 + B(r)&)“-’ + O(r). Hence, in order to The
x [K(2r/3, - 2 ~0s
(7
T) - A(~)(2r/3)‘~/~ -
x/4)]
dr
where y is Euler’s constant; F(o, z) is the incomplete gamma function; and E,(z) is the exponential integral.
A(7)
=
B(7)
=
cos
1.053721967485932*
7
sinr.