Modeling Natural Convection in Copper Electrorefining: Describing Turbulence Behavior for Industrial-Sized Systems

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A computational fluid dynamics (CFD) model of copper electrorefining is discussed, where natural convection flow is driven by buoyancy forces caused by gradients in copper concentration at the electrodes. We provide experimental validation of the CFD model 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 large-scale systems have been thought to be turbulent by some workers and modeled accordingly with k-ε type turbulence models, but others have not considered turbulence effects in their modeling. We find that the turbulence model does not predict turbulence exists; however, we analyze carefully the fluctuation statistics predicted for a transient model, finding that most cases considered do exhibit a type of turbulence, an instability related to the interaction between velocity and copper concentration fields. We provide a comparison of the extent of turbulence for various electrode heights, and gap widths, and we emphasize industrial-sized electrorefining cells.

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B :

natural convection buoyancy force (N m−3)

C :

Copper concentration (kg m−3)

C ref :

average concentration of copper over the cathode (kg m−3)

D :

diffusion coefficient (m2 s−1)

F :

Faraday’s constant (A s mol−1)

F 2 :

blending function (-)

g :

Gravitational acceleration vector (m/s2)

Gr H :

Grashof number (-)

H :

Height of electrode (mm)

H 0 :

Parameter height of electrode (mm)

h :

width of electrode (mm)

h 0 :

parameter width of electrode (mm)

i :

current density (A m−2)

I :

turbulence intensity (-)

k :

kinetic energy (m2 s−2)

\( \mathop m\limits^{\bullet}{_{\text{Cu}}} \) :

flux of copper at the anode and cathode walls (kg m−2 s−1)

M Cu :

molecular weight of copper (g mol−1)

p :

pressure (Pa)

p’ :

modified pressure (Pa)

Ra H :

Rayleigh number (-)

S :

invariant measure of the strain rate (-)

Sc :

Schmidt number (-)

t + :

transference number (-)

v’ :

RMS turbulence velocity fluctuation (m s−1)

V average :

time averaged vertical velocity (m s−1)

v :

velocity vector (m s−1)

V :

velocity scale in dimensionless numbers (maximum near electrode velocity)

X :

X direction coordinate (m)

Y :

Y direction coordinate (m)

ΔY min :

Y minimum grid cell (mm)

ΔY min :

Y minimum grid cell (mm)

ΔΖ min :

Y minimum grid cell (mm)

ΔΖ min :

Y minimum grid cell (mm)

z :

valency (-)

Z :

Z direction coordinate (m)

α1 :

dimensionless turbulence parameter (-)

β :

coefficient of expansion (L g−1 s−1)

μ :

liquid laminar dynamic viscosity (kg m−1 s−1)

μ T :

turbulent dynamic viscosity (kg m−1 s−1)

υ T :

turbulent kinematic viscosity (m2 s−1)

ρ :

density (kg m−3)

σ T :

turbulence Schmidt number taken (-)

ω :

eddy frequency (s−1)


average used for V average




height H dimension


width h dimension









+ :



refers to parameter of h, e.g., h 0

T :



modified pressure p’ or average velocity u’


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The authors gratefully acknowledge funding from the AMIRA P705 sponsors. The authors also gratefully acknowledge Peter Witt, Graeme Lane, and Darrin Stephens. Mike Nicol is acknowledged for helpful discussions.

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Correspondence to Martin J. Leahy.

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Manuscript submitted October 31, 2010.

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Leahy, M.J., Phillip Schwarz, M. Modeling Natural Convection in Copper Electrorefining: Describing Turbulence Behavior for Industrial-Sized Systems. Metall and Materi Trans B 42, 875–890 (2011).

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  • Computational Fluid Dynamic
  • Turbulence Model
  • Copper Concentration
  • Computational Fluid Dynamic Model
  • Computational Fluid Dynamic Result