A study of natural convection in an electrochemical system with a Rayleigh number of the order 10
10 is presented. Theoretical and experimental results for the unsteady behaviour of the concentration and velocity fields during electrolysis of an aqueous solution of a metal salt are given. The cell geometry is a vertical slot and the reaction kinetics is governed by a Butler-Volmer law. To reduce the effects of stratification, the flush mounted electrodes are located (symmetrically) in the middle parts of the vertical walls. It is demonstrated, both theoretically and experimentally, that a weak stratification develops after a short time, regardless of cell geometry, even in the central part of the cell. This stratification has a strong effect on the velocity field, which rapidly attains boundary layer character. Measured profiles of concentration and vertical velocity at and above the cathode are in good agreement with numerical predictions. For a constant cell voltage, numerical computations show that between the initial transient and the time when stronger stratification reaches the electrode area, the distribution of electric current is approximately steady.List of symbols
a
i
left hand side of equation system
-
b
i
right hand side of equation system
-
c
concentration (mol m
–3)
-
c
dimensionless concentration
-
c
i
concentration of species
i' (mol m
–3)
-
c0
initial cell concentration (300 mol m
–3)
-
c
0
dimensionless initial cell concentration
-
cwall
concentration at electrode surface (mol m
–3)
- dx
increment solution vector in Newton's method
-
D
i
diffusion coefficient of species
i (m
2 s
–1)
-
D
1
0.38 × 10
–9 m
2 s
–1
-
D
2
0.82 × 10
–9 m
2 s
–1
-
D
effective diffusion coefficient of the electrolyte (0.52 × 10
–9 m
2 s
–1)
-
x
unit vector in the vertical direction
-
y
unit vector in the horizontal direction
-
F
Faraday's constant (96 487 A s mol
–1)
-
g
acceleration of gravity (9.81 m s
–2)
-
i
dummy referring to positive (
i = 1) or negative (
i = 2) ion
- f
current density (A m
–2)
- f
dimensionless current density
-
i0
exchange current density (0.01 A m
–2)
-
J
ij
Jacobian of system matrix
-
L
length of electrode (0.03 m)
- N
i
transport flux density of ion
i (mol m
–2 s
–1)
- n
unit normal vector
-
p
pressure (Nm
–2)
-
p
dimensionless pressure
-
R
gas constant molar (8.31 J K
–1 mol
–1)
-
R
i
residual of equation system
-
Ra
Rayleigh number
gL
3
c
0/
D (2.54 × 101
10)
-
S
c
Schmidt number /
D (1730)
-
t
time (s)
-
t
dimensionless time
-
T
temperature (293 K)
-
velocity vector (m s
–1)
-
dimensionless velocity vector
-
U
characteristic velocity in the vertical direction
-
V
±
potential of anode and cathode, respectively
-
x
spatial coordinate in vertical direction (m)
-
x
dimensionless spatial coordinate in vertical direction
- x
solution vector for
c, and
-
y
spatial coordinate in horizontal direction (m)
-
y
dimensionless spatial coordinate in horizontal direction
-
z
i
charge number of ion
i
Greek symbols
symmetry factor of the electrode kinetics, 0.5
-
volume expansion coefficient (1.24 × 10
–4 m
3 mol
–1)
-
s
surface overpotential
-
constant in equation for the electric potential (–5.46)
-
s
diffusion layer thickness
-
scale of diffusion layer thickness
-
constant relating
c/
y to the Butler-Volmer law (0.00733)
-
kinematic viscosity (0.9 × 10
–6 m2 s
–1)
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