Abstract
A mathematical model of mass transfer processes during autocatalytic dissolution of metallic copper in oxygen-containing ammonia solutions using the rotating disc technique is presented. The model is based on the equations of steady state convective diffusion with volumetric mass generation terms and boundary conditions of the third kind, in more generalized form, at the disc surface and of the first kind in the bulk solution. The boundary value problem was solved numerically using the finite difference method with variable mesh spacing. Comparison of calculated and experimental results indicates that the model quantitatively represents the measurements. The rate of the reaction Cu(II)+Cu→2Cu(I) determines the overall rate of the process.
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Abbreviations
- A :
-
rotating disc surface area, (cm2)
- B :
-
dimensionless constant,B=k 3 c 01 ω−1
- c i :
-
concentration of speciesi, c i=c i(y) (mol cm−3)
- c 0i :
-
concentration of species i in the bulk of solution,c 0i =c 0i (t) (mol cm−3)
- c i, 0 :
-
concentration of species i at the disc surface,c i,0=c i (y=0) (mol cm−3)
- C i :
-
concentration ratio,C i=c i/c 0i ,C i=C i(ξ)
- C 0i :
-
concentration ratio (in the bulk of solution),C i=c 0i /c 0i
- C i,0 :
-
concentration ratio (at the disc surface),C i,0=c i,0/c 0i
- D i :
-
molecular diffusivity of species i (cm2 s−1)
- h :
-
space increment,h=Δξ=(ω/v)1/2Δy, dimensionless
- j i :
-
mass flux of species i (mol cm−2 s−1)
- k i :
-
first-order reaction rate constant (cm s−1 or cm3 mol−1 s−1)
- K i,j :
-
diffusivity ratio,K i,j=D i/D j, dimensionless
- M :
-
number of space increments
- n i :
-
total number of moles of Cu(II) entering the bulk of solution referred to the unit disc surface area (mol cm−2)
- \(\dot r_i \) :
-
rate of production of species i by the chemical reaction (mol cm−3 s−1)
- Sc i :
-
Schmidt number,Sc i=v i/D i
- t :
-
time, (s)
- Δt :
-
time increment (s)
- v :
-
fluid velocity vectorv=(u, v, w) (cm s−1)
- V :
-
volume of solution (cm3)
- W 1,W 2 :
-
dimensionless group,W 1=(K 3,2/D 1) (v/ω)1/2,W 2 = (K 1,2/D 2(v/ω)1/2
- x 1 :
-
coordinates,l=1, 2, 3
- y :
-
axial coordinate (perpendicular to the disc surface)
- Δy :
-
space increment (cm)
- ▿:
-
nabla operator
- ν:
-
kinematic viscosity of solution (cm2 s−1)
- μi :
-
stoichiometric coefficients
- ω:
-
disc angular velocity (s−1)
- ζ:
-
dimensionless axial coordinate, ζ=(ω/v)1/2 y
- Δζ:
-
dimensionless space increment, Δζ=(ω/v)1/2Δy
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Zembura, Z., Piotrowski, A. & Kolenda, Z. A mass transfer model for the autocatalytic dissolution of a rotating copper disc in oxygen saturated ammonia solutions. J Appl Electrochem 20, 365–369 (1990). https://doi.org/10.1007/BF01076042
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DOI: https://doi.org/10.1007/BF01076042