1. Yoon kichul Department of Mechanical Engineering Seoul National University Multi-scale Heat Conduction

2. Seoul National University Contents 1. What is the BTE? 2. Derivation of the BTE 3. Relaxation Time Approximation (RTA) 4. Equations from the BTE 1) General Hydrodynamic equation 2) Mass balance Equation 3) Momentum Equation 6) Fourier’s Law 4) Momentum Equation  Navier –Stokes Equation 5) Energy Equation 5. Summary

3. Seoul National University 1. What is the BTE? 1)2) - Takes account changes in caused by external forces and collisions 1)2) : Distribution function ∙ Simple Kinetic Theory - Based on local equilibrium ( relaxation time, mean free path) ∙ Formulated by Ludwig Boltzmann in investigation of gas dynamics - Extended to electron and phonon transport in solids and radiative transfer in gas ∙ Advanced Kinetic Theory (Based on the BTE) - Applied to non-equilibrium system ( relaxation time, mean free path)

4. Seoul National University 2. Derivation of the BTE Without collision  Distribution function does not change with time Assumption 1) By chain rule Liouville equation In the absence of body force : Substantial derivative In general, external forces and collisions exist Boltzmann Transport Equation Assumption 2)

5. Seoul National University 2. Derivation of the BTE (Continued..) 1) 2) : Scattering probability (  ) W : Nature of the scatters 1) Increased amount of particles that have : Source term 2) Decreased amount of particles that have : Sink term By collision, particles’ velocity changes Indicates change in with time by collision  Very complicated non-linear function

6. Seoul National University 3. Relaxation Time Approximation (RTA) Purpose of RTA use? Linear collision term  Easier way to solve the BTE f 0 : Equilibrium distribution τ(v) : Relaxation time When to be used? Under near equilibrium condition When τ(v) is independent of velocity Initial condition : f(t 1 ) at initial time t 1  Approximate t when near equilibrium is reached

7. Seoul National University 4. Equations from the BTE 1) General Hydrodynamic Equation : Molecular quantity 1)2)3) 1) Local average 2) 0

8. Seoul National University 4. Equations from the BTE 1) General Hydrodynamic Equation (Continued..) : Molecular quantity 1)2)3) 0 By substituting 1), 2), 3) = 0 When

9. Seoul National University 4. Equations from the BTE 2) Mass Balance Equation ( ) General formula By substituting 0 : Mass Balance Equation 0 Velocity : Bulk velocity, : Thermal velocity

10. Seoul National University 4. Equations from the BTE 3) Momentum Equation ( ) 1)2)3) 2) 0 1) 3) 00 v is independent variable : Stress tensor (covered in following page)

11. Seoul National University 4. Equations from the BTE 3) Momentum Equation ( ) (Continued..) By substituting 1), 2), 3) 1)2)3)4) 1) +4) = 2) +3) = By mass balance : Momentum Equation With substantial derivative and mass balance equation,

12. Seoul National University 4. Equations from the BTE 4) Stokes Relation : Relation of stress with flow property Summation of normal stresses ( ) : Stokes Hypothesis Including external pressure

13. Seoul National University 4. Equations from the BTE 4) Stokes Hypothesis, Momentum Eqn.  Navier-Stokes Equation : Momentum Equation : Navier-Stokes Equation

14. Seoul National University 4. Equations from the BTE 5) Energy Equation ( ) Only random motion  1) u : Mass specific internal energy Energy flux vector 2) 3) 1)2) 3) 4) 5)

15. Seoul National University 4. Equations from the BTE 5) Energy Equation ( ) (Continued..) 4) 0 Mass balance 5) 0 : Energy Equation By substituting 1), 2), 3), 4), 5)

16. Seoul National University 4. Equations from the BTE 6) Fourier’s Law ∙1-D Fourier’s Law (Under RTA and No External Force) Assumptions : f varies with only, Steady state, Constant 1) 2) Assumptions : f is near equilibrium  : Local Equilibrium 2) Heat flux Because f 0 is the equilibrium distribution  No heat flux

17. Seoul National University ∙3-D Fourier’s Law 4. Equations from the BTE 6) Fourier’s Law (Continued..) (Under RTA and No External Force) Assumptions : Steady state, Constant Assumptions : Local Equilibrium 3-D Fourier’s Law

18. Seoul National University 5. Summary ∙ BTE is an integro-differential equation of the ∙ RTA is used to simplify the collision term ∙ BTE includes the impact of external forces and collisions Change in distribution function ∙ BTE is applied to small length and time scale ( relaxation time, mean free path) ∙ General hydrodynamic eqn.  Mass balance, momentum, energy equations and Fourier’s Law ∙ Stokes relation, momentum eqn.  Navier-Stokes eqn.