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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 www.me.utexas.edu/~lishi lishi@mail.utexas.edu. Drawbacks of Kinetic Theory.
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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 www.me.utexas.edu/~lishi lishi@mail.utexas.edu
Drawbacks 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 Transport Distribution Function of Particles: f= f(r,p,t) --probability of particle occupation of momentum p at location rand time t Equilibrium 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 Approximation t Relaxation time
Energy Flux q v Energy flux in terms of particle flux carrying energy: dk q k f Vector Integrate over all the solid angle: Scalar Integrate over energy instead of momentum: Density of States: # of phonon modes between e and e + de
Quasi-equilibrium Condition BTE Solution: Quasi-equilibrium Direction x is chosen to in the direction of q Energy Flux: Fourier Law of Heat 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 particle energy, e, then Kinetic theory:
