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Diffusion through a multilayered phase in electrochemical systems: an approach by numerical inversion of the Laplace transform

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Abstract

Diffusion through a multilayered material is analysed by means of the Laplace transformation. An algorithm using a new method for numerical inversion of the Laplace transform is successfully developed for solving the diffusion equations. The procedure is applied to an analysis of hydrogen permeation through a simple mulilayered material related to electrochemical testing. The problem appears simple, but the exact analytical solution is difficult; the present technique makes it possible to solve this problem while retaining a part of the advantage of the analytical method. The results are compared with results obtained by the conventional analytical method, which is based on diffusion through a single layer. The applicability and limit of use of the conventional analytical method is also investigated.

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Abbreviations

a :

approximation of parameter in the FILT

a i :

concentration gradient in the ith layer at steady state

A i :

dimensionless concentration gradient at steady state, l m a i /c s

b i :

concentration at the left-hand end of the layer i at steady state

B i :

dimensionless b i , b i /c s

c i :

concentration in the ith layer

c 0 :

initial concentration at the left-hand end of multilayer

c s :

concentration at the left-hand end of multilayer

C i :

dimensionless concentration, c i /c s

C i :

the Laplace transformation of concentration C i

C 0 :

dimensionless initial concentration, c 0/c s

D i :

diffusion coefficient in the ith layer

D m :

diffusion coefficient in a reference layer m

E i :

parameter, e√s/α i δ i

f(t) :

function

F(t) :

the Laplace transform of function f(t)

g i :

coefficient determined by boundary conditions

h i :

coefficient determined by boundary conditions

j :

flux

j 0 :

flux under the constant flux boundary condition

J :

dimensionless flux, l m j/D m c s

J 0 :

dimensionless flux under the constant flux boundary condition, l m j 0/D m c s

k i :

distribution coefficient at the ith interface, c i +1|x i +=0/c i |x i =δ i

l i :

thickness of the ith layer

l m :

thickness of a reference layer m

n :

the number of layers

s :

variable for the Laplace transformation

S i :

parameter, √s/α i

t :

time

x i :

distance from the left-hand end of the ith layer

X i :

dimensionless distance, x i /l m

αi :

dimensionless diffusion coefficient, D i /D m

γ:

large real number for the inversion of the Laplace transform

δ i :

dimensionless thickness, la/l m

τ:

dimensionless time, D m t/l m 2

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Authors and Affiliations

  1. Institute of Atomic Energy, Kyoto University, 611, Uji, Kyoto, Japan

    Y. Ogata, T. Sakka & M. Iwasaki

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  1. Y. Ogata
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  2. T. Sakka
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  3. M. Iwasaki
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Ogata, Y., Sakka, T. & Iwasaki, M. Diffusion through a multilayered phase in electrochemical systems: an approach by numerical inversion of the Laplace transform. J Appl Electrochem 25, 41–47 (1995). https://doi.org/10.1007/BF00251263

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  • DOI: https://doi.org/10.1007/BF00251263

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