(410g) Analysis of Transient Permeation and Conduction in Composites with External Transport Resistance

Laplace transform methods are known to provide ?Early-Time? series
solutions (with fewer significant terms as t®0)
to linear partial differential equations governing transient permeation
and heat conduction. These solutions tend to be particularly convenient
because, unlike some ?Long-Time? solutions, Early-Time solutions do not involve
eigenvalues defined by transcendental equations.

The Early-Time approach is applied here to the analysis of
permeation in two-layer composite membranes with external mass transfer
resistance. The governing equations are then :

                                                                                                                                (1)

subject to:

The goal is an expression for M(t), the time course of the cumulative
mass permeated per unit area, i.e.:

                                                                                                                                                (2)

or, in dimensionless terms:

                                  (3)

Notably, Sakai (1922) derived a
?Long-Time? series solution to Eqs. 1 in the general case of an arbitrary
number of layers, but with negligible external mass transfer resistance.

For purposes of deriving the Early-Time solution, Laplace-transform
operator is
defined as usual by:

                                                                                                                                    (4)

Eqs.  1 are thereby transformed to easily solved ordinary
differential equations. The end result is the following expression for the
transform of m:

            (5)

where:  .

Recovery of  and,
in turn, requires
inverse transformation of Eq. 5. Scientist^R numerical inversion software
(Micromath Inc.) provides essentially exact results with which those based on truncated
Early-Time(?ET?) series will be compared.

The latter series emerge from inverse transformation of  the
expression to which simplifies
when s is large. When only the lead terms are retained, the result is:

   (7)

Retaining additional terms extends the time over which the Early-Time
solution is accurate. In many cases of practical interest, only the lead terms
are necessary to accurately model essentially the entire non-steady-state
region.

Sakai, S. (1922) ?Linear conduction of heat through a series
of connected rods,? Sci. Rep. Tohoku Imperial Univ., Ser. I (Math, Phys.,
Chem.), 11, 351- 378.