Tracking the Frank-Kamenetskii Problem

heat coefficient h

10

solution u versus α

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)

u

xx

+α

u

e

=0

for

0<x<1

,

u(x=0)=0

, and

u

x

x=1

+hu(x=1)=0

admits up to two solutions. Here,

u

is the dimensionless temperature and

h

is the heat transfer coefficient.

For

h=∞

and

α=e

, the BVP admits an analytical solution given by

u(x)=lncoshx-

1

2

θ

2

cosh

θ

4

, where

θ

is one of the two solutions of the transcendental equation

θ=

2e

cosh(θ/4)

(i.e.,

θ≈3.0362

and

θ≈7.1350

).

We use the homotopy continuation method and the Chebyshev orthogonal collocation technique (with

N+1=13

collocation points) to track the solutions,

u(x)

, in the

α

parameter space.

The plot of the norm of the solution

u=

N+1

∑

i=1

2

(u(

x

i

))

versus

α

clearly indicates that there can be up to two solutions. These two solutions are plotted in blue and magenta for

α=2.5

.

Details

In the discrete Chebyshev–Gauss–Lobatto case, the interior points are given by

y

j

=cos(jπ/N)

. These points are extremums of the Chebyshev polynomial of the first kind

T

N

(x)

.

The

(N+1)×(N+1)

Chebyshev derivative matrix at the quadrature points

D=

d

jk

,

0≤j

,

k⩽N

is given by

d

00

=

2

N

+1

6

,

d

NN

=-

2

N

+1

6

,

d

jj

=

-

y

j

21-

2

y

j



for

1≤j≤(N-1)

, and

d

jk

=

j+k

c

j

(-1)

c

k

(

y

j

-

y

k

)

for

0≤j

,

k⩽N

, and

j≠k

,

where

c

j

=1

for

1≤j≤(N-1)

and

c

0

=

c

N

=2

.

The matrix

D

is then used as follows:

v'=Dv

and

v''=

2

D

v

, where

v

is a vector formed by evaluating

u

at

y

j

,

j=0,…,N

, and

v'

and

v''

are the approximations of

u'

and

u''

at the

y

j

.

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.

Permanent Citation

Housam Binous, Brian G. Higgins

"Tracking the Frank-Kamenetskii Problem"
http://demonstrations.wolfram.com/TrackingTheFrankKamenetskiiProblem/
Wolfram Demonstrations Project
Published: May 29, 2013