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A boundary element method for transient diffusion

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Abstract

Transient diffusion in two-dimensional geometries is considered. It is shown how the spatial variation of the mass transfer-limited flux of a minor species varies with time from initially uniform to the non-uniform, steady-state distribution. Flux distributions on sinusoidal electrode and on a line electrode embedded in an otherwise insulating plane are considered. A boundary-element method is used to solve the problem in Laplace transform space, and the results are subsequently inverted into the time domain.

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Abbreviations

A i,j :

fitting coefficients defined by Equation 15

c :

dimensionless concentration

c :

dimensional concentration (mol cm−3)

c :

bulk concentration (mol cm−3)

D :

diffusion coefficient (cm2 s−1)

f :

functions introduced in Equation 15

f :

Laplace transform of function introduced in Equation 15

g :

Green's function for modified Helmholz equation

K 0, K 1 :

modified Bessel functions of the second kind of order zero and one

n i :

number of node points used in BEM calculation

N :

number of values of s used in simulations

r :

dimensionless distance, defined by Equation 8

s :

Laplace transform variable, defined by Equation 5

q :

dimensionless flux normal to the electrode

q :

Laplace transform of the dimensionless flux normal to the electrode

t :

time (s)

y, z :

dimensionless cartesian coordinates

ž, tŷ :

dimensional cartesian coordinates (cm)

z :

generic interpolation function

α:

geometric parameter

βi :

curve-fitting parameter used in exponential functions

ε0 :

amplitude of sinusoidal roughness (cm)

Γ:

boundary of computational domain

λ:

wavelength of sinusoidal roughness (cm)

σ:

dimensionless arc length

τ:

dimensionless time

π:

3.1415926...

ζ:

path of integration

avg:

average

max:

maximum

min:

minimum

q:

point at which the concentration (or gradient) is determined

ss:

steady-state

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West, A.C., Matlosz, M. A boundary element method for transient diffusion. J Appl Electrochem 24, 261–267 (1994). https://doi.org/10.1007/BF00242894

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Keywords

  • Physical Chemistry
  • Spatial Variation
  • Boundary Element
  • Boundary Element Method
  • Flux Distribution