(410g) Analysis of Transient Permeation and Conduction in Composites with External Transport Resistance
AIChE Annual Meeting
2015
2015 AIChE Annual Meeting Proceedings
Engineering Sciences and Fundamentals
Mathematical Modeling of Transport Processes
Tuesday, November 10, 2015 - 4:45pm to 5:00pm
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. 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:
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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