ME 381R Lecture 8 & 9: Boltzmann Transport Equation & Thermal Conductivity Model

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ME 381R Lecture 8 & 9: Boltzmann Transport Equation & Thermal Conductivity Model. Dr. Li Shi Department of Mechanical Engineering The University of Texas at Austin Austin, TX 78712 Drawbacks of Kinetic Theory.
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ME 381R Lecture 8 & 9:Boltzmann Transport Equation& Thermal Conductivity ModelDr. Li ShiDepartment of Mechanical Engineering The University of Texas at AustinAustin, TX lishi@mail.utexas.eduDrawbacks of Kinetic Theory
  • Assumes single particle velocity and single mean free
  • path or mean free time.
  • Breaks down when, vg(w) or t(w)
  • Assumes local thermodynamics equilibrium: f = f(T)
  • Breaks down when L ; t  t
  • Cannot handle non-equilibrium problems
  • Short pulse laser interactions
  • High electric field transport in devices
  • Cannot handle wave effects
  • Interference, diffraction, tunneling
  • Boltzmann Transport Equation for Particle TransportDistribution Function of Particles: f= f(r,p,t)--probability of particle occupation of momentum p at location rand time tEquilibrium Distribution: f0, i.e. Fermi-Dirac for electrons, Bose-Einstein for phonons, Plank for photons, etc.Non-equilibrium, e.g. in a high electric field or temperature gradient:Relaxation Time ApproximationtRelaxation timeEnergy FluxqvEnergy flux in terms of particle flux carrying energy:dkqkfVectorIntegrate over all the solid angle:ScalarIntegrate over energy instead of momentum: Density of States: # of phonon modes between e and e + deQuasi-equilibrium ConditionBTE Solution: Quasi-equilibriumDirection x is chosento in the direction of qEnergy Flux: Fourier Law ofHeat Conduction:t(e) can be treated using Callaway (Phys. Rev. 113, 1046) or Holland model (Phys. Rev., 134, A471-A480)If v and t are independent of particleenergy, e, then Kinetic theory:
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