# 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.

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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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