University of California Berkeley MSE 200A Fall 2017 Heat Conduction in 3

# University of california berkeley mse 200a fall 2017

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University of California, Berkeley MSE 200A Fall, 2017 Heat Conduction in 3-Dimensions A temperature gradient produces a heat flux : T = ( T/ x) e x + ( T/ y) e y + ( T/ z) e z J Q = J Q x e x + J Q y e y + J q z e z How many thermal conductivities? In the most general case, 9 For a cubic or isotropic material, only need 1 For a cubic or isotropic material J Q = k T T t = k 2 T

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J.W. Morris, Jr. University of California, Berkeley MSE 200A Fall, 2017 Mechanisms of Heat Conduction Energy must be transported through the solid Electrons Lattice vibrations - phonons Light - photons (usually negligible) Conduction is by a gas of moving particles Particles move both to left and right (J Q = J Q + -J Q- ) Particle energy increases with T If T decreases with x, particles moving right have more energy Net flow of heat to the right (J Q > 0) v J = (1/2)nev J Q+ J Q- J Q = J Q+ - J Q- T + > T - J Q > 0
J.W. Morris, Jr. University of California, Berkeley MSE 200A Fall, 2017 Heat Conduction by Particles Transport of thermal energy Particle reaches thermal equilibrium by collisions Particle travels <l> = mean free path between collisions Particle transfers energy across plane Energy crossing plane reflects equilibrium <l x > upstream J Q <l x >

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J.W. Morris, Jr. University of California, Berkeley MSE 200A Fall, 2017 Mechanisms of Heat Conduction Particles achieve thermal equilibrium by collision with one another Particles that cross at x were in equilibrium at x =x-‹l x ‹l x › = mean free path J Q + = 1 2 nev x = 1 2 E v ( T ) v x = 1 2 E v [ T ( x l x )] v x E v T x l x ( ) [ ] = E v E v T dT dx l x = E v C v dT dx l x J Q + = 1 2 E v v x 1 2 C v v x l x dT dx
J.W. Morris, Jr. University of California, Berkeley MSE 200A Fall, 2017 Mechanisms of Heat Conduction Total heat flux : J Q + = 1 2 E v v x 1 2 C v v x l x dT dx J Q = J Q + J Q = C v v x l x dT dx = k dT dx k = C v v x l x In three dimensions: k = 1 3 C v v l = 1 3 C v v 2 τ ( l = v τ ) J Q = 1 2 E v v x + 1 2 C v v x l x dT dx

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J.W. Morris, Jr. University of California, Berkeley MSE 200A Fall, 2017 Conduction of Heat T t = k C V 2 T x 2 J Q = k dT dx k = 1 3 C v v l = 1 3 C v v 2 τ Thermal conductivity by particles: - v = mean particle velocity - <l> = mean free path between collisions - τ = mean time between collisions Particles include - electrons - phonons T 1 T 2 T 3 J Q 12 J Q 23
J.W. Morris, Jr. University of California, Berkeley MSE 200A Fall, 2017 Heat Conduction by Electrons Motion of electrons = electrical conductivity k = 1 3 C v v l = 1 3 C v v 2 τ Wiedemann-Franz Law k = LT ρ 0 L A high temperature low temperature ( ρ = ρ 0 + AT ) k = L σ T = LT ρ

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J.W. Morris, Jr.
• Fall '08
• Staff
• Thermodynamics, University of California, Phase transition, J.W. Morris

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