Finite
disc.
enamelled
Para.
R
0.43Q/y,
X 2
Q^given by equation 3.6.5
Finite
disc. No
enamel
Para
R
0.54
X-l -> Q^-1.236
Finite
disc.
enamelled
Perp.
R
1.87Qi
X s
Qi given by equation 3.7.53
Finite
disc. No
enamel
Perp.
R
1.26
X-l -> Qj_-10/13
Sphere
with epoxy
—
R
1.04R (Qul+Q,J
(b+R)
Qslgiven by equation 3.9.13
Og2given by equation 3.9.14
Sphere
41
For a sphere, the hydrodynamic boundary layer separates from the sur
face at 109.6° from the stagnation point (Schlichting I960) and after this
point the diffusion layer is considered infinite, so that current contri
butions from beyond the point of separation are considered to be zero.
In practice there must be some small, but assumed negligible, contribution
from this r e g io n since it is still in contact with oxygenated fluid. So,
the in t e g r a tio n is only taken to the point of separation, but, for consis
tency the t o t a l metal surface area of the sphere is used in calculating the
average c ur rent density. That is, for all electrodes, when calculating
the average current density the total available reaction area is used,
whether or not it all contributes current. The sphere electrodes used
c l i n i c a l l y were partly covered by epoxy resin (as a mechanical support)
which decreased their available area.
3.6 The Calculation of The Current Response of an Enamelled Disc
P a r a l l e l to the Flow
The relevant geometry is shown in figure 3.6.1. The calculation for the
para llel disc follows directly from that for the parallel plate. This was
performed by Levich (1962) and is given in appendix A.I, where Schlichting's
(I960) form of the velocity components in the hydrodynamic boundary layer
is used rather than that used by Levich (the two forms are equivalent).
The geometry relevant to the parallel, enamelled plate is shown in figure
3.6.2.
For the enamelled parallel plate, Levich (1962) found the current
density to vary with position as
^ ’ u* tllat is there is no enamel, then equation 3.6.1 represents the P-ir.i 1 lei plate without enamelling. Equation 3.6.1 can easily be integrated
I've the output current for a plate, provided that edge effects in the irection are neclioiklp en that infoorafl nn in fhaf Hirontlnn i s finite
(3.6.1)
e<tion are negligible so that integration in that direction is finite,
)ut end
Figure 3.6.1 The usual form of enamelled disc parallel to the
42
x is f i n i t e and the integration is therefore curtailed. It is assumed
that the finite value of X, the extent of the plate in the x-direction,
has no e f f e c t on the concentration profile set up around the electrode.
Obtaining the total current for the parallel enamelled disc is more
d i f f i c u l t . The situation is illustrated in figure 3.6.3. Equation 3.6.1
can be i n t e g r a t e d to give the total current dl flowing from the elemental
strip shown in figure 3.6.3.
d l = 2 x 0.34 mec D
s
2/3 -1/6 ^ 1/2 (,3/4 _ u 3/4) 2/3
- h db (3.6.2)
The t o t a l current for the disc surrounded by the insulating enamel ring is XR
I - 2 x dl (3.6.3)
J0
with b » cR, and db * Rde, then
I - * x 0.34 mec D2'3 / ‘' S 1/2 R 3/2 « Q„ s v o s (3.6.4) where Qn “ X f r r p + y p ^ r p ] ^ _ 0
J
2/3 dc (3.6.5)The in t e g r a l Q„ is dimensionless and can be evaluated numerically for
various values of X. The computer generated variation of Q„ with x
is shown in figure 3.6.4.
The area of the electrode which is available for the reaction to take
place at i s uX2Rz and therefore the average current density J is
0.43 - 1/2 2/3 -1/6 1/2
---- Q„R ' mecg D ' v ' VQ '
"(XR)2 x
lor the enamelled parallel disc the shape fact
(3.6.6)
■ ■ ^ Q»R" 1/2 (3.6.7)
is. therefore, a function of the degree of enamelling. If X ■ I,
lat Is there is no enamel, then Q,, ■ 1.236 and the shape factor becomes
42
x is f i n i t e and the integration is therefore curtailed. It is assumed
that the f i n i t e value of X, the extent of the plate in the x-direction,
has no e f f e c t on the concentration profile set up around the electrode. Obtaining the total current for the parallel enamelled disc is more difficult. The situation is illustrated in figure 3.6.3. Equation 3.6.1 can be integrated to give the total current dl flowing from the elemental strip shown in figure 3.6.3.
The to ta l current for the disc surrounded by the insulating enamel ring is
The in teg ral Q„ is dimensionless and can be evaluated numerically for
various values o f X. The computer generated variation of Q„ with X
The area of the electrode which is available for the reaction to take dl = 2 x 0.34 mec D
s
'XR I = 2 x d l
J0
with b = cR, and db = Rde, then
(3.6.3) (3.6.4) where 0 X (3.6.5) is shown in figure 3.6.4.
place at is ttX2R2 and therefore the average current density J is
o
1/2
(3.6.6)
for the enamelled parallel disc the shape factor is
(3.6.7)
is. therefore, a function of the degree of enamelling. If X - 1,
at is there is no enamel, then Q„ ■ 1.236 and the shape factor that is there i
becomes
*■
I
Figure 3 ,6.3 The scheme o f i n t e g r a t i o n used f o r th e p a r a l l enam elled d i s c .
r
The variation of Q„ with x
3.7 The Calculation of the Current Response of an Enamelled Disc
Perpe n d i c u l a r to the Flow
For the plate perpendicular to the flow and the infinite disc (the
axisymmetric case) the solutions of the Laplace equation necessary to
find the potential flow are relatively easily obtained, and are shown
in table 3.7.1. The geometry relevant to perpendicular electrodes is
shown in figure 3.7.1.
Finding the potential flow distribution for finite electrodes is
not so easy. Complex variable theory is used to find the potential
flow f o r the finite perpendicular plate (Spiegel 1964). The complex
potential i ! ( z ) for fluid flow past the cylinder shown in figure 3.7.2 is
To convert this problem into that for the finite perpendicular plate it
is necessary to rotate the flow through 90°, achieved by multiplying (3.7.1)
where
(3.7.2)
Solution of the problem is expedited by transforming to elliptic co-ordinates u, v where
z “ X + iy = c cosh(u + iv) (3.7.3)
In the complex plane the complex potential, il(z)
S2(z) « <t>(x,y) + i ip (x,y) (3.7.4)
here M x -y) is the potential satisfying the Laplace equation and V(x-y) ''' t^le stream function. Therefore it is the real part of fi(z)
Figure 3.7.1a The geometry of perpendicular disc electrodes.
H-fcure 3.7.lh The perpendicular plate. For f i n i t e p l a t e s edge e f f e c t s in the z d i r e c t i o n a r e assumed n e g l i g i b l e .