problem 1 steady state concentration profile across a thin film with d

Search for question

Question

Problem 1. Steady state concentration profile across a thin film with DB(c)
Learning Objective: Use of shell balances and Fick's law to derive a 1D concentration profile in a material with a
concentration-dependent diffusion coefficient.
We have previously derived concentration profiles for steady state diffusion across a thin film assuming constant diffusivity.
However, it is common for solvent diffusivity in polymers to have an exponential dependence upon concentration:
DAB Doe, where Do and b are empirical constants.
Derive the concentration profile for steady diffusion across a planar film of area A and thickness L for a solute with this
function dependence of DAB. Show the following steps:
a) Write a shell balance on a differential volume element in the film, and derive the 1-D species equation of continuity
in terms of diffusion flux jA.
b) Incorporate Fick's law into the species equation of continuity using DAB= Doela,
c) Integrate the equation in b) in terms of integration constants.
=
=
d) Evaluate the integration constants for the boundary conditions CACAO at z = 0 and CA - CAL at z = L.
e) Write the final form of the concentration profile ca(z).