7.1.1 Hyperbolic Heat Equation CH 7. Non-equilibrium Energy Transfer in Nanostructures 7.1.1 Hyperbolic Heat Equation 2011. 12. 01 Taekkeun Kim

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

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.

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

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

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

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

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

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

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

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