1. 7.1.1 Hyperbolic Heat Equation
CH 7. Non-equilibrium Energy Transfer in Nanostructures
7.1.1 Hyperbolic Heat Equation
Taekkeun Kim

2. Contents 1. Motivations Derivation of Hyperbolic Heat Equation
Derivation of Modified Fourier Equation

3. Hyperbolic Heat Equation
Motivations
Limitation of Classical Fourier Law
The local-equilibrium condition breaks down at the micro-scale when
the characteristic length L is smaller than a mechanistic length scale,
such as the mean free path
Non-equilibrium energy transfer refers to the situation when the assumption
of local equilibrium does not hold.
The effective temperature distribution cannot be described by the heat
diffusion theory derived from Fourier’s law using the concept of equilibrium
temperature without considering the temperature jumps at the boundaries.
The micro-structural interaction effect, such as phonon-electron interaction
or phonon scattering, may significantly enhance heat transfer in short times.

4. Hyperbolic Heat Equation
Motivations (Cont.,)
hot wall at Th
hot wall at Th
Local Equilibrium does not hold !
L <<
t <<
L >>
t >>
adiabatic wall
adiabatic wall
adiabatic wall
adiabatic wall
cold wall at Tc
cold wall at Tc
(a) Kn << 1
(b) Kn >> 1
Fig. 1 Comparison of Temperature profile according to Knudsen Number
Classical Heat Conduction Equation
Parabolic Equation
For example,
We consider 1D transient heating of a semi-infinite medium.
Governing Equation

5. Hyperbolic Heat Equation
Motivations (Cont.,)
Boundary Conditions
Semi-infinite & homogeneous medium
Initially at a uniform temperature
The wall at is heated with a constant heat flux
at and insulated at
Solutions of Temperature distribution T(x,t) t
material of
heat
specific : c
density ty
conductivi
thermal y
diffusivit /
heating
step
the
width p r k a t =
where

6. Hyperbolic Heat Equation
Derivation of Hyperbolic Heat Equation
Modified Fourier equation (Cattaneo equation)
Energy equation for heat conduction
Divergence of
modified Fourier equation
Time derivative of energy equation ① ②

7. Hyperbolic Heat Equation
Derivation of Hyperbolic Heat Equation (Cont.,) ②
Hyperbolic Heat Equation
Hyperbolic Equation

8. Hyperbolic Heat Equation
Derivation of Hyperbolic Heat Equation (Cont.,)
No heat generation
telegraph equation or damped wave equation
Damping, diffusion term
The solution of the hyperbolic heat equation : propagating wave
: The speed of temperature wave in the high-frequency limit or the short-time limit
Amplitude of temperature wave decays according to 8

9. Hyperbolic Heat Equation
Derivation of Hyperbolic Heat Equation (Cont.,)
A semi-infinite solid under a constant heat flux at the surface
Assume
Propagation speed :
Predicted by Classical Heat Conduction Equation
Pulse wave-front :
A short pulse, (b) A long pulse,
Illustration of the solution of the hyperbolic equation at short timescales.
The solid curves are the solutions of the hyperbolic heat equation.
The dash-dotted and dashed curves are the solutions of the parabolic heat equation.
In the case of a short pulse, temperature pulse propagates
and its height decays by dissipating its energy to the medium as it travels.
The parabolic heat equation predicts a continuous temperature distribution. 9

10. Hyperbolic Heat Equation
Derivation of Modified Fourier Equation
The transient 1-D BTE under the relaxation time approximation :
Assuming :
Not far from local equilibrium
Multiplying the above equation by ,
and then integrating each term over the momentum space,
In the other hand,
Eq. (4.63a)
Eq. (4.64) 10

11. Hyperbolic Heat Equation
Derivation of Modified Fourier Equation (Cont.,)
Therefore, or
: 1D case
: 3 D case
Modified Fourier Equation or Cattaneo Equation 11