Particle Transport Theory in Thermal Fluid Systems: ME 381R Lecture 2 Particle Transport Theory in Thermal Fluid Systems: Level 1—Kinetic Theory 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

Nanotransistors Hot spots! Ju and Goodson, APL 74, 3005 IBM SOI Chip Lines: BTE results Hot spots!

Microscopic Origins of Thermal Fluid Transport --The Particle Nature Materials Dominant energy carriers Gases: Molecules Metals: Electrons Insulators: Phonons (crystal vibration) L Hot Cold In micro-nano scale thermal fluid systems, often L < mean free path of collision of energy carriers & Fourier’s law breaks down  Particle transport theories or molecular dynamics methods

Mean Free Path for Intermolecular Collision for Gases D D Total Length Traveled = L Average Distance between Collisions, mc = L/(#of collisions) Total Collision Volume Swept = pD2L Mean Free Path Number Density of Molecules = n s: collision cross-sectional area Total number of molecules encountered in swept collision volume ~ npD2L

Mean Free Path for Gas Molecules kB: Boltzmann constant 1.38 x 10-23 J/K Number Density of Molecules from Ideal Gas Law: n = P/kBT Mean Free Path: Typical Numbers: Diameter of Molecules, D  2 Å = 2 x10-10 m Collision Cross-section: s  1.3 x 10-19 m2 Mean Free Path at Atmospheric Pressure: At 1 Torr pressure, mc  200 mm; at 1 mTorr, mc  20 cm

Effective Mean Free Path Wall b: boundary separation Wall Effective Mean Free Path:

Kinetic Theory of Energy Transport Cold u(z+z) Net Energy Flux z + z q  qz z through Taylor expansion of u z - z u(z-z) z q x y f dW Hot Solid Angle, dW = sinqdqdf See handout for detailed derivation

Averaging over all the solid angles Assuming local thermodynamic equilibrium: u = u(T) Thermal Conductivity

Thermal Conductivity of Gases Heat Capacity Monoatomic gases: [J/m3-K] Diatomic gases: Velocity: Vx=Vsinqcosf Vy=Vsinqsinf Vz=Vcosq Vz q V dW Vy f Vx

Maxwell-Boltzmann Distribution V Most probabale Mean speed Root-mean-square Vmp Vm Vrms Most probable speed: Mean Speed: Root-Mean-Square Speed Used for thermal conductivity calculations

V T1 T2 > T1 Increasing Temperature Speed of helium atoms at 0 oC Mass, m = 1.66 x 10-27 (kg/proton) x 4 (protons) = 6.65 x 10-27 kg

Thermal Conductivity y depends on the number of atoms in the molecule If mean free path is limited by intermolecular collision thermal conductivity is independent of number density and therefore independent of pressure If mean free path is affected by boundary scattering, thermal conductivity will depend on pressure. (Saved as a future homework problem)

Questions Kinetic theory is valid for particles: can electrons and crystal vibrations be considered particles? If so, what are C, v,  for electrons and crystal vibrations?