The Frank-Kamenetskii Problem Using Arc-Length Continuation
The Frank-Kamenetskii Problem Using Arc-Length Continuation
The Frank–Kamenetskii problem relates to the self-heating of a reactive solid. When the heat generated by a reaction is balanced by conduction in a one-dimensional slab of combustible material, the nonlinear boundary value problem (BVP) +α=0 for , , and +hu(x=1)=0 admits up to two solutions. Here, is the dimensionless temperature and is the heat transfer coefficient.
u
xx
u
e
0<x<1
u(x=0)=0
du
dx
x=1
u
h
For and , the BVP admits an analytical solution , where is one of the two solutions of the transcendental equation (given by and ).
h=∞
α=e
u(x)=lncoshx-cosh
1
2
θ
2
θ
4
θ
θ=
2e
cosh(θ/4)θ≈3.0362
θ≈7.1350
We use the arc length continuation method and the Chebyshev orthogonal collocation technique (with collocation points) to track the solutions, , in the parameter space.
N+1=21
u(x)
α
The plot of the norm of the solution versus clearly indicates that there can be up to two solutions. These two solutions are plotted in blue and magenta for .
u=
N+1
∑
i=1
2
u()
x
i
α
α=2.5