Steady State Diffusion Video Lecture | Mass Transfer - Chemical Engineering

let's develop an expression for steady
state diffusion this could be for either
gases or liquids for example in solids
so we will use a ficks law of diffusion
that states that m dot a divided by a
equals minus da b times d CA over DX now
in a previous tutorial we saw all these
terms m dot a is the mass of component a
that is diffusing is a mass flow rate so
as kilograms per second a is the area in
square meters da B is the diffusivity of
a in B and D CA is the concentration
gradient in Direction X so if we
separate the variables in this
differential equation we have m dot a
divided by a times DX equals minus da B
times D CA since X and the concentration
are the two variables and we have
separated them in this expression next
we can integrate and we have m dot a
divided by a and then we integrate DX
from some location X 1 to X 2 and that
equals minus da B and we have the
integral from C a 1 that is the
concentration at location X 1 which is C
a1 and then concentration C a2 and in
the integral we have DCA so we evaluate
these integrals
these are simple integrals so we get m
dot a divided by a
and in parentheses we have X 2 minus X 1
and that equals minus da B and in
parentheses we have CA 2 minus CA 1 so
we can write down then our integrated
expression as m dot a divided by a
equals D a B and in parentheses we have
CA 1 minus C a2 divided by X 2 minus X 1
so this gives us an expression to
determine the mass flux and due to
molecular diffusion when there is a
certain concentration gradient of course
this was for rectangular coordinates if
you have a cylindrical shape we can also
obtain an expression we will leave it up
to you to carry out that simple
integration the expression you will get
is m dot a equals D a B times 2 pi L
where L will be the length of the
cylinder and in parentheses C a1 minus C
a2 and divided by natural log of r2
divided by r1 where R 1 and R 2 are the
two radii of the cylinder the inside
radius R 1 and outside radius R 2 note
that we have followed the steady state
conditions for these solutions that
means that the concentrations at the
boundaries are constant with time that
the diffusion is limited to molecular
motion within the solid so we have a
porous solid and it's either gas or
liquid that is diffusing but it is
diffusing due to molecular motion not by
convection
and that diffusivity is not influenced
by concentration and because we used a
constant term for diffusivity at d a.b
and also that there are no temperature
gradients within the solid so those are
some of the assumptions that are used to
carry out this solution