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. ScientistR 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.
See more of this Group/Topical: Engineering Sciences and Fundamentals