# Transient Heat Conduction with Temperature-Dependent Thermal Conductivity

Transient Heat Conduction with Temperature-Dependent Thermal Conductivity

Consider transient heat conduction in a finite slab with temperature-dependent thermal conductivity. The phenomenon is governed by the following dimensionless equation:

∂θ

∂τ

∂θ

2

∂x

2

∂θ

∂x

2

where is the dimensionless thermal diffusivity and is an empirical parameter to be chosen by the user. The variables , , and are the dimensionless time, temperature, and position. The initial condition and boundary conditions are: , , and .

f(θ,δ)

δ

τ

θ

x

θ(x,τ=0)=0

θ(x=1,τ)=1

=0

∂θ

∂x

x=0

This Demonstration plots the dimensionless temperature versus the dimensionless position in the slab for , , , and (the orange, green, red, and cyan curves, respectively). The colored dots correspond to the solution using orthogonal collocation. You can specify the number of collocation points for the calculation as well as the functional dependency for the thermal diffusivity through the parameter . The colored curves correspond to the dimensionless temperature obtained using the Mathematica built-in function NDSolve. The solid curves correspond to ; the dashed curves correspond to , and the dotted curves are obtained using . The calculations show that for small values of the linearized form for the thermal diffusivity is adequate for short time periods, but less so for large time periods.

τ=0.01

0.05

0.2

0.5

δ

f(θ,δ)=exp(δθ)

f(θ,δ)=1+δθ

f(θ,δ)=1

δ