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Time Scale of Thermal Conduction Xiaoshan Xu the Importance of Thermal Conduction

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Time Scale of Thermal Conduction Xiaoshan Xu the Importance of Thermal Conduction Time scale of thermal conduction Xiaoshan Xu The importance of thermal conduction • Energy consumption: Hot reservoir • transfer through conduction • Energy generation: Δ푉 • Thermoelectric effect (Seebeck effect) Cold reservior • Spin Seebeck effect Basics of thermal conduction Example of irreversible process: Heat conduction Heat Conductivity in Solids (an example for irreversibility) Remember: Heat is an energy transferred from one system to another because of temperature difference T1 > T2 System System 1 2 Heat Q flows from 1 to 2 Example of irreversible process: heat conductions as a non-equilibrium process: *(in the textbook T >T ) T1 > T2 2 1 Heat reservoir Heat reservoir 1 2 L T(x) T1 T2 0 L x A Heat transfer per time interval through homogeneous solid object: Q K K: thermal conductivity of the rod (T1 T2 )A where L t L A: cross-section of the rod Diffusion (time and space dependence of matter/energy) T(x) T1 T2 퐿 → ∆푥 푇2 − 푇1 → ∆푇 0 L x Q T K At x Fourier’s law Define flux: 푄 휕푇 푥, 푡 푆 = lim 푆 푥, 푡 = −퐾 Q Δ푡K→0 퐴∆푡 휕푥 (T T )A t L 1 2 Diffusion (time and space dependence of matter/energy) A 푺: flux (energy flow per unit area per unit 푆(푥, 푡) 푢(푥, 푡), 푈 푆(푥 + ∆푥, 푡) time, similar to electric current) 풖: energy density (energy per unit volume) 푥 푥 + ∆푥 How would energy change in the system? 퐴 푆 푥 + ∆푥, 푡 − 푆 푥, 푡 ∆푡 = −[푢(푥, 푡 + ∆푡) − 푢(푥, 푡)]퐴∆푥 푆 푥 + ∆푥, 푡 − 푆 푥, 푡 푢 푥, 푡 + ∆푡 − 푢 푥, 푡 = − ∆푥 ∆푡 Continuity equation: 휕푢 푥, 푡 휕푆 푥, 푡 = − 휕푡 휕푥 Diffusion (time and space dependence of matter/energy) A 푆(푥, 푡) 푢(푥, 푡), 푈 푆(푥 + ∆푥, 푡) Flux across a boundary 푥 푥 + ∆푥 휕푇 푥, 푡 푆 푥, 푡 = −퐾 휕푥 Energy change in a system 휕푢 푥, 푡 휕푆 푥, 푡 = − 휕푡 휕푥 Diffusion (time and space dependence of matter/energy) A 푺: flux (energy flow per unit area per unit 푆(푥, 푡) 푢(푥, 푡), 푈 푆(푥 + ∆푥, 푡) time, similar to electric current) 풖: energy density (energy per unit volume) 푥 푥 + ∆푥 휕푢 푥, 푡 휕푆 푥, 푡 To study temperature distribution. = − Continuity equation: 휕푡 휕푥 휕푈 푈 = 푈 + ∆푇 = 푈 + 푀푐푀∆푇 0 휕푇 0 푉 푉 푈(푡) 푈 푀푐푀∆푇 푢 = = 0 + 푉 = 푢 + 푐푀∆푇 푉 푉 푉 0 푉 푀 푢 − 푢0 = ∆푢 = 푐푉 ∆푇 휕푆 푥, 푡 휕푇(푥, 푡) 휕푢 푥, 푡 푀 휕푇(푥, 푡) 푀 = 푐 = −푐푉 휕푡 푉 휕푡 휕푥 휕푡 Diffusion (time and space dependence of matter/energy) A 푺: flux (energy flow per unit area per unit 푆(푥, 푡) 푢(푥, 푡), 푈 푆(푥 + ∆푥, 푡) time, similar to electric current) 풖: energy density (energy per unit volume) 푥 푥 + ∆푥 휕푆 푥,푡 휕푇(푥,푡) From continuity equation = −푐푀 (1) Diffusion equation 휕푥 푉 휕푡 휕푇(푥, 푡) 휕2푇 푥, 푡 From Fourier’s law = 퐷 2 휕 휕푡 휕푥 휕푇 푥, 푡 Take 휕푆 푥,푡 휕2푇 푥,푡 퐾 휕푥 = −퐾 (2) 푆 푥, 푡 = −퐾 휕푥 휕푥2 퐷 = 푀 diffusivity 휕푥 푐푉 Application of Diffusion Equation (Decay of a hot spot) Diffusion equation 푇(푥, 푡) 휕푇(푥, 푡) 휕2푇 푥, 푡 = 퐷 휕푡 휕푥2 푇 Describes how the spatial distribution evolves with time. One of the solution (example): 푥 퐴 푥2 푇 푥, 푡 = 푇0 + exp − 푡 + 푡0 4퐷 푡 + 푡0 퐴 푥2 Gaussian peak. 푇 푥, 0 = 푇0 + exp(− ) 푡0 4퐷푡0 Application of Diffusion Equation (Decay of a hot spot) Spatial dependence at 푡: 푇(푥, 푡) 퐴 푥2 푇 = 푇 + exp(− ), 푤 푡 = 4퐷(푡 + 푡 ) 0 푤 푤2 푡 0 푇 푡0 Spatial evolution with time: 1) Peak becomes broader 푡1 푡2 2) Total area is conserved (energy conservation 푑푈 = 푑푇). 푡3 푥 휕푇(푥, 푡) 휕2푇 푥, 푡 = 퐷 휕푡 휕푥2 Scaling rule of the decay 퐴 푥2 푇 푥, 푡 = 푇0 + exp − 푡 + 푡0 4퐷 푡 + 푡0 퐴 푥2 = 푇 + exp − 0 푡 푡 푡0 1 + 4퐷푡0 1 + 푡0 푡0 푡 If we use 푡0 as the unit, and let 휏 = , we get a universal relation 푡0 퐴 푥2 푇 푥, 푡 = 푇0 + exp − 1 + 휏 4퐷 1 + 휏 So the decay scales with 푡0 Scaling rule of the decay 퐴 푥2 푇 푇 푥, 0 = 푇0 + exp(− ) 푡0 4퐷푡0 2 = 2퐷푡0 2 푥 2 Since the decay scales with 푡0, it scales with the . Another view using analogy to circuits Thermal conduction is like discharging a capacitor, the time constant is 휏 = 푅퐶 For thermal conduction, we use the same formula 휏 = 푅퐶 푑 푅 = 퐴 퐶 = 푐푀푉 = 푐푀퐴푑 : thermal conducitivity 푑: thickness 퐴: area hot cold : density 푐푀: specific heat So 푑 푑 푐 휏 = 푅퐶 = 푐 퐴푑 = 푀 풅ퟐ 퐴 푀 Again, this scales with width2. Some hands-on data A simple experiment 1.0 dT more dT less 0.8 0.6 Less 0.4 normalized dT 0.2 0.0 0.00 0.02 0.04 0.06 0.08 dt (day) More Time scale in thin films Laser pulse Film Substrate 20 nm LFO 20 nm 1.0 LFO YbFO YbFO 50 nm 1.0 50 nm BFO 0.8 35 nm 0.6 dc/c 0.5 dc/c 0.4 0.2 0.0 -2 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 0.000 0.005 tdelay t (ns) 푡/푑2 (ns/nmtdelay 2) For the previous water cooling experiment: 흉/풅ퟐ = ퟎ. ퟎퟎퟑ (ns/nm2) 휏 = 1080s 푑 = 2 cm Conclusion • We discussed the scaling rule of the thermal conduction, which is proportional to the thickness^2. • A few examples (microscopic and macroscopic) are given, which can be unified using the scaling rule..
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