Hyperbolic Heat Equation

1. 1-D BTE or assume (4.64) (4.63a) or • Hyperbolic Heat Equation Integrate over velocity space assuming that t is an averaged relaxation time

2. Energy equation without convection In 3-D Cattaneo equation Maxwell-Cattaneo equation

3. Decay in amplitude: hyperbolic equation Without internal heat generation speed of thermal wave Damped wave equation For an insulator:

4. semi-infinite solid under a constant heat flux at the surface propagation speed : vtw pulse wavefront: x1 = vtwt1, x2 = vtwt2 short pulse long pulse In the case of a short pulse, temperature pulse propagates and its height decays by dissipating its energy to the medium as it travels. Hyperbolic heat equation : valid strictly when tp > t

5. Entropy Local energy balance Local entropy balance Entropy production rate Hyperbolic heat equation sometimes predicts a negative entropy generation. (Energy transferred from lower-temp. region to higher-temp. region) Negative entropy generation is not a violation of the 2nd law of thermodynamics Because the concept of “temperature” in the hyperbolic heat equation cannot be interpreted in the conventional sense due to the lack of local thermal equilibrium

6. When When • Dual-Phase-Lag Model Finite buildup time after a temperature gradient is imposed on the specimen for the onset of a heat flow: on the order of the relaxation time t Lag between temperature rise and heat flow electrons → lattice phonons → temperature rise In analogy with the stress-strain relationships of viscoelastic materials with instantaneous elasticity → Cattaneo eq.

7. By assuming that : effective conductivity (heat diffusion) : elastic conductivity (hyperbolic heat) retardation time

8. : intrinsic thermal properties of the bulk material dual-phase-lag model First order approximation

9. When When The requirement may cause . The heat flux depends not only on the history of the temperature gradient but also on the history of the time derivative of the temperature gradient. → Cattaneo eq. → Fourier law

10. Solid-fluid heat exchanger immersed long thin solid rods in a fluid inside a sealed pipe insulated from the outside rod diameter d, number of rods N inner diameter of pipe D average convection heat transfer coefficient h total surface area per unit length total cross-sectional area of the rods total cross-sectional area of the fluid

11. assume Due to the initial temperature difference between the rod and the fluid , a local equilibrium is not established at any x inside the pipe until after a sufficiently long time.

12. Two-Temperature Model electrons → electron-phonon interaction → phonons Under the assumption that the electron and phonon systems are each at their own local equilibrium, but not in mutual equilibrium (valid when t > t) e : electron, s : phonon C : volumetric heat capacity = rcp G : electron-phonon coupling constant

13. (5.25) (5.55b) keq : thermal conductivity at or (5.10) Assume the the lattice temperature is near or above the Debye temperature so that electron-electron scattering and electron-defects scattering are insignificant compared with electron-phonon scattering.

14. tq : not the same as the relaxation time t due to collision Thermalization time: thermal time constant for the electron system to reach an equilibrium with the phonon system

15. or Heat Conduction Across Layered Structures • Equation of Phonon Radiative Transfer (EPRT) S0: electron-phonon scattering • Phonon BTE in 1-D system

16. T1 q q + forward direction (m> 0) T2 - forward direction (m< 0) phonon intensity under non-equilibrium distribution function : energy transfer rate in a direction from a unit area, per unit frequency, per unit solid angle phonon Intensity: summation over the three phonon polarizations

17. Radiative Transfer Equation (RTE) neglecting scattering optical thickness of the medium acoustical thickness of the medium corresponds to Equation of Phonon Radiative Transfer (EPRT)

18. Equilibrium intensity, Bose-Einstein statistics analogy to blackbody intensity

19. : Phonon Stefan-Boltzmann constant va: average phase velocity of the 2 transverse and 1 longitudinal phonon modes integrating over all frequency → total intensity for all 3 phonon modes

20. At temperature higher than Debye temperature high frequency limit because the shortest wavelength of the lattice wave should be on the order of atomic distances, the lattice constants energy flux at phonon equilibrium: I* invariant with direction particle flux

21. Dulong-Petit law: energy carried by a phonon = energy flux / particle flux energy density = energy carried by a phonon X number of phonons per unit volume At low temperature At high temperature

22. radiative equilibrium total quantities

23. T1 q q T2 • Solution to EPRT steady state • Two-flux method in planar structures boundary conditions: gray medium, diffuse and gray walls

24. Solutions to EPRT • Heat Flux (spectral)

25. For diffuse surface total heat flux

26. Heat flux at very low temperature Heat flux between blackbodies with small T difference Thermal conductivity

27. acoustically thick limit

28. Effective Conductivity of the Film

29. Thermal Resistance Network effective thermal conductivity of the whole layered structure

30. Internal thermal resistance with Fourier’s law Thermal resistance at the interface Thermal resistance at the boundaries

31. 1 2 3 • Thermal Boundary Resistance (TBR) • Thermal Contact Resistance • due to incomplete contact between two materials • thermal resistance between two bodies, usually with very rough surfaces, srms > 0.5 mm TH TL • Thermal Boundary Resistance • due to the difference in acoustic properties of adjacent materials • Acoustic Mismatch Model • specular reflection of phonons • similar to geometric optics • no scattering or diffusion • Diffuse Mismatch Model • diffusive reflection of phonons • no information (except for energy) retained after a scattering

32. Snell’s law • Acoustic Mismatch Model (AMM) For longitudinal phonon modes with linear dispersion in a Debye crystal (equilibrium, isotropic, perfect contact) transmission coefficient

33. Let : hemispherical transmission

34. Snell’s law critical angle: qC

35. For longitudinal phonon modes with linear dispersion in a Debye crystal (equilibrium, isotropic, perfect contact)

37. In the low temperature limit, For all 3 phonon modes Thermal boundary resistance

38. a: lattice constant When the temperature difference is small, TBR by AMM is proportional to T-3. AMM assumes perfect specular reflection which is valid only when the characteristic wavelength of the phonons is much larger than the surface roughness. (Lmp >> srms) → DMM (Diffuse Mismatch Model)

39. Diffusion Mismatch Model (DMM) In DMM, all phonons striking the interface are scattered once, and are emitted into the adjoining substances elastically (Dw= 0) with a probability proportional to the phonon density of states (DOS) in the respective substances. Distribution of the emitted phonons is independent of the incident phonon, whether it is from side 1 or 2, longitudinal or transverse. integration over the solid angle

40. by DMM (G12) with the assumption at low temperature

41. Measured Data TBR between indium and sapphire[2]. X, normal indium; ● super conducting indium. Data B are for roughened sapphire surface. A and C are for smooth sapphire surfaces with different indium thicknesses. AMM predicts a flat line at 20.4 cm2K4/W