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Boltzmann Transport Equation

Yoon kichul Department of Mechanical Engineering Seoul National University. Boltzmann Transport Equation. Multi-scale Heat Conduction. Contents. 1. What is the BTE?. 2. Derivation of the BTE. 3. Relaxation Time Approximation (RTA). 4. Equations from the BTE.

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Boltzmann Transport Equation

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

  2. 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 4) Momentum Equation Navier –Stokes Equation 5) Energy Equation 6) Fourier’s Law 5. Summary

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

  4. 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 Assumption 2) : Substantial derivative In general, external forces and collisions exist Boltzmann Transport Equation

  5. 2. Derivation of the BTE (Continued..) 2) 1) By collision, particles’ velocity changes : Scattering probability (  ) : 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 Indicates change in with time by collision  Very complicated non-linear function

  6. 3. Relaxation Time Approximation (RTA) f0 : Equilibrium distribution τ(v) : Relaxation time When to be used? Under near equilibrium condition Purpose of RTA use? Linear collision term  Easier way to solve the BTE When τ(v) is independent of velocity Initial condition : f(t1) at initial time t1  Approximate t when near equilibrium is reached

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

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

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

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

  11. 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 With substantial derivative and mass balance equation, : Momentum Equation

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

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

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

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

  16. 4. Equations from the BTE 6) Fourier’s Law ∙1-D Fourier’s Law (Under RTA and No External Force) 1) 2) 1) Assumptions : f varies with only, Steady state, Constant 2) Assumptions : f is near equilibrium  : Local Equilibrium Heat flux Because f0 is the equilibrium distribution  No heat flux

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

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

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