A computational fluid dynamics (CFD) model of copper electrorefining is discussed. Copper electrorefining takes place in a rectangular geometry, with two electrodes opposing each other, and a source and sink of copper ions at the respective electrodes. The resultant gradients in the copper concentration lead to buoyancy forces, and natural convection develops. The transport of copper ions is coupled to the Navier-Stokes equations in a CFD software package ANSYS CFX (version 11). Validation of the CFD model is provided for several cases varying in size, from a small laboratory scale to large industrial scale, including one that has not been compared with a CFD model previously. The larger scale systems are analysed in terms of the Rayleigh number, and we clarify that the important length scale for turbulence onset is the width of the cell, in addition to the cell height. Clarification of the appropriate turbulence model is given

Leahy, M. J

Schwarz, M. P.

University of Queensland eSpace

16th Australasian Fluid Mechanics Conference 
Crown Plaza, Gold Coast, Australia 
2-7 December 2007 
 
Computational Fluid Dynamics Modelling of  
Natural Convection in Copper Electrorefining 
 
M.J Leahy and M.P. Schwarz 
CSIRO Minerals, Clayton, Victoria, 3168 AUSTRALIA 
Abstract 
A computational fluid dynamics (CFD) model of copper 
electrorefining is discussed. Copper electrorefining takes place in 
a rectangular geometry, with two electrodes opposing each other, 
and a source and sink of copper ions at the respective electrodes. 
The resultant gradients in the copper concentration lead to 
buoyancy forces, and natural convection develops. The transport 
of copper ions is coupled to the Navier-Stokes equations in a 
CFD software package ANSYS CFX (version 11). Validation of 
the CFD model is provided for several cases varying in size, from 
a  small laboratory scale to large industrial scale, including one 
that has not been compared with a CFD model previously. The 
larger scale systems are analysed in terms of the Rayleigh 
number, and we clarify that the important length scale for 
turbulence onset is the width of the cell, in addition to the cell 
height. Clarification of the appropriate turbulence model is given. 
Introduction  
Copper plate electrorefining (ER) is a process used in industry 
for refining copper. Copper is dissolved from the anode plate into 
solution and is plated onto the opposing cathode, by means of the 
passage of current between the plates. Natural convection is well 
known to develop is these ER systems, as has been investigated 
on small scale laboratory systems as discussed by Eklund et al 
[3], with experimental validation of CFD models, and in the 
context of larger systems in Denpo et al [2] and Gurniki et al [5]. 
The large scale systems have been thought to be turbulent, with a 
k-ε turbulence model used in the CFD modelling by Gurniki et al 
[5] of an experimental setup of Ziegler [7]. In this work we 
develop a similar CFD model and compare the results with the 
experimental setup of Ziegler [7], and investigate the appropriate 
turbulence model, and indeed the necessity of a turbulence model 
at all. We also compare the CFD model to experimental data [6] 
from a much larger system, which has not previously been used 
for comparison with a CFD model. 
 
Figure 1. Schematic geometry, side and cross section views and cross 
section schematic mesh view on right. Actual mesh is not shown. 
 
CFD Model  
The CFD model is two-dimensional (2D) in a cross section of the 
cell, as shown in figure 1, with the assumption that the flow is 
negligible in the third dimension (X direction), parallel to the 
electrodes. The CFD model is set up within the ANSYS CFX 
framework [1]. The ER model is a single phase model which 
solves the Navier-Stokes equations, with an additional body force 
term to account for the buoyancy forces. A transport equation is 
solved for the concentration of the copper species (or other 
metal), with a source/sink at the appropriate anode and cathode 
boundary, based on Faraday’s Law. The equation of continuity is 
given by 
0)( =⋅∇ vρ       (1) 
and the momentum equation in steady state is given by 
Bvvvv +∇+∇+⋅+∇−∇=⋅∇ )]..)([(')( TTp µµρ   (2) 
where ρ
 
is the electrolyte density (assumed constant), v is the 
velocity, p’ is the (modified) pressure (including the hydrostatic 
part -ρg.x), and B is the natural convection buoyancy force, 
described below. The laminar viscosity is denoted µ (kg m-1 s-1), 
and µT (kg m-1 s-1) is the turbulent viscosity, described in 
equation (5). The buoyancy body force is given by 
)(
refCC −−= βρgB      (3) 
where g (m s-2) is the gravity vector, C (g L-1) is the 
concentration of copper, Cref (g L-1) is the average concentration 
of copper over the cathode, and β (L g-1) is the coefficient of 
expansion for the copper species. The steady state transport 
equation for the copper species is given by  
C
T
T SCDC +∇+⋅∇=⋅∇ )]()[()(
ρσ
µρv   (4) 
Where σT (-) is the turbulence Schmidt number taken as 0.9, SC 
(g L-1 s-1) is the source term, which describes the flux of copper 
at the anode and cathode, and D (m2 s-1) is the diffusion 
coefficient of copper (or other metal species). 
 
The turbulent viscosity µT in equations (2) and (4) is determined 
by solving transport equations for the turbulence model, i.e. k-ε 
or k-ω. In this work both models were tested, and the k-ω was 
found to have the best close-to-wall behaviour (good velocity 
profile prediction). The turbulent viscosity can be written in 
terms of the transported variables - kinetic energy k (m2 s-2) and 
eddy frequency ω (s-1) (or eddy dissipation if using k-ε) as  
ω
ρµ kT =      (5) 
Boundary Conditions 
The boundary conditions for the flux of copper at the anode and 
cathode walls 
copperm
•
  (kg m-2 s-1) are based on Faraday’s Law as 
follows: 
1000
)1( Cu
copper
M
zF
it
m +
•
−
=
    (6) 
where i (A m-2) is the current density, t+ (-) is the transport 
number, F (A s mol-1) is Faraday’s constant, z (-) the valency, 
and MCu (g/mol) is the molecular weight of copper. On the anode 
side a positive flux is applied, whilst on the cathode a negative 
flux of same size is applied. At all walls, no slip boundary 
conditions are applied, whilst at the top free surface a free slip 
(no friction) boundary condition is applied.  
112
 Parameter Eklund  Ziegler Konishi  
Current Density i (A m-2) 45.9 100  100 
Temperature (oC) 25 23 25 
Liquid laminar viscosity µ 
(kg m-1 s-1) 
0.9612 
x10-3 
1.91 
x10-3  
0.9612 
x10-3 
Liquid Density ρ (g L-1) 1045.4 1200 1045.4 
Coefficient of Expansion 
β (L g-1 s-1) 
0.0022 0.00159 0.0022 
Dimensions of cell 
H(mm) x h(mm) 
32 x 2  850 x 24  210 x 
145  
Rah   107  4x1011  1x1015  
RaH 5x1010 2x1016 3x1015 
Reh 3 450 3700 
ReH  50 1.6x104 5500 
Table 1. Table of parameters used in CFD validation cases. 
 
Mesh Size Eklund  Ziegler k-ε Ziegler k-ω Konishi  
 ∆Ymin (mm)  0.06  0.48  0.16  0.01  
∆Zmin (mm) 0.02 4.25 2.125 0.4 
 ∆Ymax (mm)  0.06  0.48  1  2.1  
∆Zmax (mm) 0.02 4.25 2.125 2.3 
Table 2. Mesh sizes for each case, minimum (near wall) and maximum 
(middle cell) in Y and Z directions. The Ziegler k-ω model uses a finer 
near wall mesh than the Ziegler k-ε  model (smaller ∆Ymin). 
 
CFD Model Validation and Discussion  
Three cases are discussed, a small laboratory sized cell [3], a 
large typical industrial size [7] and a large very wide cell [6]. All 
three cases provide data which is used for comparison with the 
CFD model. The Rayleigh number (Ra) from Gurniki [5], and the 
Reynolds number (Re) are defined as follows: 
Dυ
hmgRa 2
4
copper
h
•
=
β
  
Dυ
HmgRa 2
4
copper
H
•
=
β
  (7) 
υ
hVReh =     
υ
HVReH =     (8) 
where V (m s-1) is velocity scale, h (m) and H (m) are the width 
and height of the system, respectively, ν (m2 s-1) is kinematic 
viscosity. These dimensionless numbers are provided in table 1 
for each case, along with other important operating conditions. 
 
Small Scale Laminar Case 
This section compares the model with the small scale experiment 
of Eklund et al. [3, 4]. A fine uniform mesh (in both Y and Z) is 
used (see table 2), and the laminar CFD model is used since 
Reh~1 and Rah=107 are low (see table 1). There is a very good 
agreement between the CFD results and experimental data in 
figure 2. In figure 2(a), both CFD results and experimental data 
indicate that near the cathode and anode there is downflow and 
upflow respectively, due to the deposition and removal of copper 
to and from the respective plate.  The upflow and downflow 
occurs in the boundary layer where there is large concentration 
gradient (figure 2(c)), and the width of the boundary layer is 
predicted nicely compared to the data. CFD results (and 
experimental data) in figure 2 indicate that in a large portion of 
the middle of the cell there is very little flow. This is consistent 
with a natural convection recirculation zone, with the electrolyte 
moving downwards near the anode, and upwards near the 
cathode, due to the copper flux at each electrode. Copper 
stratification is clearly evident in the experimental data and the 
CFD prediction in figure 2(b), due to the lighter depleted copper 
electrolyte rising and the heavy metal laden electrolyte falling. 
 
  
(a)       (b) 
 
(c) 
Figure 2. Comparison between CFD results (solid lines) and experimental 
data (squares) from [3] (a) vertical velocity component (mm s-1) versus 
distance from cathode (mm) at mid cell height, (b) copper concentration 
(g L-1) at a mid-cell width versus distance (mm) from base, (c) copper 
concentration (g L-1) versus distance (mm) from cathode at mid cell 
height (triangles represent model predictions of Eklund et al. (1989)). 
 
Large Scale Thin Case 
This section shows the comparison of the model with 
experimental measurements of the vertical velocity of a large 
scale (thin) experiment from Ziegler [7]. The velocity is 
measured after 50 minutes has elapsed from start-up (figure 3 
(symbols). A turbulence CFD model is used despite the 
transitional nature: Reh~450 and Rah=4x1011 (see table 1). Figure 
3 shows the comparison between the experimental and CFD 
results of the vertical velocity profile at cell mid-height, 425mm 
from the base. The steady state CFD model results are shown for 
two CFD models: k-ε with coarse wall mesh 50 by 200 cells 
(figure 3 dotted line), and k-ω with fine near wall mesh using 50 
by 400 cells in Y and Z directions (figure 3 solid line). Both 
cases use a vertically uniform mesh (see table 2 for full mesh 
parameters). The two cases are compared because the k-ε (with 
coarse wall mesh) model was used by Gurniki [5], and we aim to 
determine the most appropriate numerical mesh and turbulence 
model. The predicted vertical velocity values for the k-ω model 
are in excellent agreement with the limited experimental data. 
The k-ε model comparison (figure 3 dotted line) is poor due to 
poor mesh resolution and k-ε model inaccuracy near the wall, 
resulting in a smearing of the velocity profile across the whole 
cross section. 
 
Figure 3. Comparison between CFD results k-ω with resolved wall mesh 
(solid lines) and k-ε with uniform mesh (dotted line) and experimental 
data (squares) from Ziegler [7]. Vertical velocity component (mm s-1) 
versus distance from cathode (mm) at a height of 425mm from base. 
113
In figure 4 we show details of the CFD results for the k-ω model 
(line plot of velocity shown in figure 3), with figure 4(a) showing 
a time snap shot of the contour plot of ratio of eddy to laminar 
viscosity, (b) the contours of the velocity scalar, (c) the vector 
plot and (d) the contour of the cadmium concentration. A natural 
convection recirculation zone is present, with the electrolyte 
moving downwards near the anode, and upwards near the 
cathode, due to the cadmium flux at each electrode. The 
electrolyte flow near the anode drags the highly metal-laden 
electrolyte downwards to the base, and the lighter metal-depleted 
electrolyte moves upwards near the cathode, causing a strong 
stratification in the cadmium concentration. In the middle of the 
cell, there is much slower flow, and small somewhat random 
fluctuations are present. Near the cathode and anode, there are 
small regions of high velocity (see figure 4(b)). The ratio of eddy 
to laminar viscosity in figure 4(a) is extremely low, indicating the 
turbulence level is insignificant; this suggests the system is not in 
the turbulent regime. Furthermore, the small vortexes in the 
middle of the geometry and the unsteady motion indicate the 
system is in the transitional regime. In figure 5 we show details 
of the CFD results for the k-ε model. This figure shows the poor 
description of the boundary layer (as also shown in figure 3), and 
dispersion of the boundary layer across the width of the gap, due 
to the smearing of the velocity profile by the k-ε model. The eddy 
viscosity remains high unlike in figure 4, for the k-ω model.  
 
We can conclude the system is not turbulent but is in the 
transitional regime, based on the following: the low Reynolds 
number Reh and the low Rayleigh number Rah, and the fact that 
the eddy viscosity becomes negligible with the experimentally 
validated k-ω model. We show in the next section that larger 
(wider) systems are more subject to turbulence, due to the extra 
width and space for eddies to develop. 
 
Large Scale Wide Case 
A large scale (wider) experiment by Konishi [6] has also been 
simulated, with a k-ω model and refined wall mesh (see table 2 
for mesh parameters), in addition to a refined mesh at the top and 
bottom of the geometry. A turbulence model is appropriate due to 
the large Reynolds and Rayleigh numbers: Reh~3700 and 
Rah=1x1015 (see table 1). This case is large being 210mm tall and 
very wide (145mm), compared with the previous case from 
Ziegler [7] which was 24mm wide and 850mm tall. The CFD 
prediction shown in figure 6(a) and (b) is very good. Figures 7 
and 8 show the CFD results, with figure 7 showing time snap 
shot of the velocity vector field and streamlines, and figure 8(a) 
showing the speed (log scale) and figure 8(b) the ratio of eddy 
and laminar viscosity. The streamlines with squares in figure 7 
represent clockwise recirculation, and those with circles represent 
anti-clockwise recirculation. The large recirculation zone 
(circles) in the middle is where the ratio of eddy to laminar 
viscosity is high, and this is moving in the opposite direction to 
that expected, counter to the recirculation near the electrodes. 
This unexpected result is due to the clockwise recirculation near 
the cathode and anode plates (streamlines with circles) and the 
vortex at the top (squares), which drags electrolyte in the 
opposite direction with anticlockwise direction (circles).  
 
The width of this Konishi [6] geometry is almost an order of 
magnitude larger than the Ziegler [7] case, and the eddy viscosity 
is now reasonably high, unlike the smaller (in width) Ziegler [7] 
case above (see figure 4), and this suggests the system is 
moderately turbulent. We can conclude that the width is a more 
important length scale than the height from the point of view of 
onset of turbulence, due to the fact that the eddies are restricted in 
the smaller gap width, and are free to evolve and move around in 
the larger scale system. 
 
 
Figure 4. CFD results used to compare with Ziegler [7] for k-ω model, (a) 
contour plot of ratio of eddy to laminar viscosity (-) (b) contour plot of 
speed scalar (log scale) (mm s-1), (c) velocity vectors coloured by speed 
(log scale), and (d) contour plot of cadmium concentration (kg m-3). 
Horizontal scale increased by a factor of 3 to view results. 
 
114
 Figure 5. CFD results used to compared with Ziegler [7] for k-ε model, 
(a) contour plot of ratio of eddy to laminar viscosity (-) (b) contour plot of 
speed scalar (log scale) (mm s-1), (c) velocity vectors coloured by speed 
(log scale), and (d) contour plot of cadmium concentration (kg m-3). 
Horizontal scale increased by a factor of 3 to view results. 
 
 
 
 
 
 
 
 
 
 
 
 
 
(a) 
 
(b) 
Figure 6. Comparison between CFD results (solid lines) and experimental 
data (squares) from Konishi [6]. (a) Vertical velocity component (mm s-1) 
versus distance from cathode (mm) and (b) copper concentration (mm s-1) 
versus distance from cathode (mm) at a height of 140mm from base. 
 
 
Figure 7. CFD results used to compare with Konishi [6], vector field 
coloured by speed and pink streamlines (squares – clockwise 
recirculation zone, circles - anti-clockwise recirculation zone). 
 
 
115
 Figure 8. CFD results used to compare with Konishi [6], (a) contour plot 
of speed scalar (log scale) and (b) contour plot of ratio of eddy viscosity 
to laminar viscosity (-).  
 
 
Conclusions 
A CFD model for ER has been developed and compared to 
several different sized experimental systems from the literature. 
The CFD results compared very well with the small scale 
experimental data, and were also very good for the larger scale 
experiments of Ziegler [7] and Konishi [6]. The k-ε model on a 
coarse wall mesh is unsatisfactory in comparison to the k-ω 
model with a fine mesh near the wall. The CFD model predicted 
unsteady behaviour for the experiment in [7]. The larger gap 
width (145mm) case in [6] led to a more unstable but not 
turbulent system, with eddies forming and moving into large 
eddies away from the electrodes. The width of the system has 
more of an effect on onset of turbulence rather than the height.  
 
Acknowledgments 
The authors gratefully acknowledge Mike Horne, Peter Witt, 
Darrin Stephens and Mike Nicol for helpful discussions. AMIRA 
P705A sponsors are acknowledged for permission to publish. 
 
References 
[1] ANSYS, (2007), CFX-11 Solver, ANSYS Inc., Canonsburg, 
USA, website address www.ansys.com/cfx 
[2] Denpo, K., Teruta, S., Fukunaka, Y. and Kondo, Y., 1983, 
Turbulent natural convection along a vertical electrode, 
Metallurgical Transactions B, 14B, 633-643. 
[3] Eklund, A., Alavyoon, F., Simonsson, D., Karlsson, R. and 
Bark, F. 1991. Theoretical and experimental studies of free 
convection and stratification of electrolyte in a copper 
refining cell, Electrochemica Acta, 36 (8), 1345-1254. 
[4] Eklund, A., Simonsson, D., Alavyoon, F., Karlsson, R., and 
Bark, F. 1991. Theoretical and experimental studies of free 
convection in a copper refining cell, I. Chem. E. Symposium 
Series No 112, 112, 47-58. 
[5] Gurniki, F. Bark, F. and Zahari, S., 1999. Turbulent free 
convection in large electrochemical cells with a binary 
electrolyte, J. Appl. Electrochem., 29, 27-34. 
[6] Konishi, Y., Tanaka, Y. Kondo, Y. and Fukunaka, Y., 2000. 
Copper dissolution phenomena along a vertical plane anode 
in CuSO4 solution. Electrochemica Acta, 46, 681-690. 
[7] Ziegler, D. and Evans, J., 1986. Mathematical modelling of 
electrolyte circulation in cells with planar vertical electrodes: 
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566.
 
116


Computational Fluid Dynamics Modelling of Natural Convection in Copper Electrorefining

M. J Leahy

M. P. Schwarz

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Computational Fluid Dynamics Modelling of Natural Convection in Copper Electrorefining

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