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 :

subject to:

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

or, in dimensionless terms:

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:

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

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:

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.

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