Journal of Applied Electrochemistry

, Volume 25, Issue 1, pp 41–47 | Cite as

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

  • Y. Ogata
  • T. Sakka
  • M. Iwasaki
Papers
Received:
Revised:

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.

Keywords

Hydrogen Physical Chemistry Single Layer Diffusion Equation Laplace Transformation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

List of symbols

a

approximation of parameter in the FILT

ai

concentration gradient in the ith layer at steady state

Ai

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

bi

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

Bi

dimensionless b i , b i /c s

ci

concentration in the ith layer

c0

initial concentration at the left-hand end of multilayer

cs

concentration at the left-hand end of multilayer

Ci

dimensionless concentration, c i /cs

Ci

the Laplace transformation of concentration C i

C0

dimensionless initial concentration, c0/cs

Di

diffusion coefficient in the ith layer

Dm

diffusion coefficient in a reference layer m

Ei

parameter, e√s/α i δ i

f(t)

function

F(t)

the Laplace transform of function f(t)

gi

coefficient determined by boundary conditions

hi

coefficient determined by boundary conditions

j

flux

j0

flux under the constant flux boundary condition

J

dimensionless flux, l m j/D m c s

J0

dimensionless flux under the constant flux boundary condition, l m j0/D m c s

ki

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

li

thickness of the ith layer

lm

thickness of a reference layer m

n

the number of layers

s

variable for the Laplace transformation

Si

parameter, √s/α i

t

time

xi

distance from the left-hand end of the ith layer

Xi

dimensionless distance, x i /l m

Greek symbols

αi

dimensionless diffusion coefficient, D i /D m

γ

large real number for the inversion of the Laplace transform

δi

dimensionless thickness, la/l m

τ

dimensionless time, D m t/l m 2

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Copyright information

© Chapman & Hall 1995

Authors and Affiliations

  • Y. Ogata
    • 1
  • T. Sakka
    • 1
  • M. Iwasaki
    • 1
  1. 1.Institute of Atomic EnergyKyoto UniversityUji, KyotoJapan

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