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

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

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**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**Ju and Goodson, APL 74, 3005 IBM SOI Chip Lines: BTE results Hot spots!**Microscopic Origins of Thermal Fluid Transport**--The Particle Nature MaterialsDominant 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:**q**dW y f x Kinetic Theory of Energy Transport Cold Net Energy Flux u(z+z) z + z q qz z through Taylor expansion of u z - z u(z-z) z 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 [J/m3-K] Monoatomic gases: Diatomic gases: Vx=Vsinqcosf Vy=Vsinqsinf Vz=Vcosq Velocity: Vz q V dW Vy f Vx**Maxwell-Boltzmann Distribution**V Most probabale Mean speed Root-mean-square Vrms Vm Vmp 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?