Tracking the Frank-Kamenetskii Problem
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The Frank–Kamenetskii problem relates to the self-heating of a reactive solid. When the heat generated by reaction is balanced by conduction in a one-dimensional slab of combustible material, the nonlinear boundary value problem (BVP) for
,
, and
admits up to two solutions. Here,
is the dimensionless temperature and
is the heat transfer coefficient.
Contributed by: Housam Binousand Brian G. Higgins (May 2013)
Open content licensed under CC BY-NC-SA
Snapshots
Details
In the discrete Chebyshev–Gauss–Lobatto case, the interior points are given by . These points are extremums of the Chebyshev polynomial of the first kind
.
The Chebyshev derivative matrix at the quadrature points
,
,
is given by
,
,
for
, and
for
,
, and
,
where for
and
.
The matrix is then used as follows:
and
, where
is a vector formed by evaluating
at
,
, and
and
are the approximations of
and
at the
.
References
[1] P. Moin, Fundamentals of Engineering Numerical Analysis, Cambridge, UK: Cambridge University Press, 2001.
[2] L. N. Trefethen, Spectral Methods in MATLAB, Philadelphia: SIAM, 2000.
[3] B. G. Higgins and H. Binous, "A Simple Method for Tracking Turning Points in Parameter Space," Journal of Chemical Engineering of Japan, 43(12), 2010 pp. 1035–1042. doi:10.1252/jcej.10we122.
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