Boltzmann Transport Equation

Yoon kichulDepartment of Mechanical EngineeringSeoul National University

Multi-scale Heat Conduction

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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

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1. What is the BTE?

coll

f f f f

t t

v a

r v

1) 2)

- Takes account changes in caused by external forces and collisions ( , , )f tr v1) 2)

: Distribution function( , , )f tr v

∙ 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)

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2. Derivation of the BTE

Without collision Distribution function does not change with time

Assumption 1)

( , , ) ( , , ) ( , , )0 0

df t f dt dt t dt f t

dt dt

r v r v v a r v

By chain rule( , , )

0df t f d f d f f f f

dt t dt dt t

r v r v

v ar v r v

Liouville equation

In the absence of body force 0f f

t

v

r

Df

Dt : Substantial derivative

In general, external forces and collisions exist

coll

f f f f

t t

v a

r v Boltzmann Transport Equation

Assumption 2)

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2. Derivation of the BTE (Continued..)

coll

( , ) ( , , ) ( , ) ( , , )f

W f r t W f r tt

v

v v v v v v1) 2)

: Scattering probability ( )( , )W v v v v

: Scattering probability ( )vv( , )W v vW : Nature of the scatters

1) Increased amount of particles that have : Source term v2) Decreased amount of particles that have : Sink term v

coll

f

t

By collision, particles’ velocity changes

Indicates change in with time by collision( , , )f tr v

Very complicated non-linear function

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3. Relaxation Time Approximation (RTA)

Purpose of RTA use? Linear collision term Easier way to solve the BTE

0

coll ( )

f ff

t v

f0 : Equilibrium distribution

τ(v) : Relaxation time

When to be used? Under near equilibrium condition

When τ(v) is independent of velocity

*0 0

0

ln( ) exp( )df dt t t

f f C f f Cf f

Initial condition : f(t1) at initial time t1

* *1 1 11 0 1 0 0 1 0( ) exp( ) [ ( ) ]exp( ) ( ) [ ( ) ]exp( )

t t t tf t f C C f t f f t f f t f

Approximate t when near equilibrium is reached

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4. Equations from the BTE

1) General Hydrodynamic Equation

( )f f f

d d d dt

v a

r v : Molecular quantity

1) 2) 3)

1) ( )f n

d fd f d nt t t t t

1

f dn

Local average

2) ( ) ( ( )f d fd f d n n v v v) v v

0( ) ( ) ( ) , ( )f f f v v v v v + v

( )f

f ft t t

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4. Equations from the BTE

1) General Hydrodynamic Equation (Continued..)

( )f f f

d d d dt

v a

r v : Molecular quantity

1) 2) 3)

3) , ,

, ,( ) x x x

x x x

v v v

v v v

fd f f d n

a a av v v

0

u v f a f

u v

a v v

uv uv u v f f a a v

By substituting 1), 2), 3)

( ) ( )n n nt t

v v av

= 0 Whenj v

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4. Equations from the BTE

2) Mass Balance Equation ( )m

( ) ( ) 0n n nt t

v v avGeneral formula

By substituting m

D

Dt t

v( ) B B Bv v + v

, N Nm

n nmV V

( ) ( ) 0m m

nm nm n mt t

v v av

0

( ) 0 0D

t Dt

B Bv v : Mass Balance Equation

B R B Rv = v v v = v + v 0Velocity : Bulk velocity, : Thermal velocityBv

Rv

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4. Equations from the BTE

3) Momentum Equation ( )m v

( ) ( ) 0m m

nm nm n mt t

v vv vv v v a

v

1) 2) 3)

2) ( ) [ ( )( )] [ ( 2 )]nm B R B R B B B R R Rvv v v v v v v v v v v0

( ) ( ) ( )nmt t t

Bv v v1)

3) m m

n mt

v vv v a a

v

0 0

v is independent variable

B Rv = v v ( ) ( ) B B B B B Bv v v v v v ( ) ijP R Rv v

: Stress tensor (covered in following page)ijP

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4. Equations from the BTE

3) Momentum Equation ( ) (Continued..)m v

By substituting 1), 2), 3)

( ) ( ) ijPt t

B

B B B B B

vv v v v v a = 0

1) 2) 3) 4)

1) +4) = ( )D

t Dt

B BB B

v vv v

2) +3) = [ ( )] 0t

B Bv v By mass balance

: Momentum Equation 1ij

DP

Dt Bv

a

With substantial derivative and mass balance equation,

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4. Equations from the BTE

4) Stokes Relation : Relation of stress with flow property

2 ( )xx

u u v w

x x y z

( )xy yx

u v

y x

2 ( )yy

v u v w

y x y z

( )yz zy

v w

z y

2 ( )zz

w u v w

z x y z

( )zx xz

w u

x z

Summation of normal stresses ( ) : Stokes Hypothesis 0xx yy zz

2 ( ) 3 ( ) 0xx yy zz

u v w u v w

x y z x y z

2

3

Including external pressure ijP

22 ,

3i

i

vP i j

x

Bv

, ji

j i

vvi j

x x

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4. Equations from the BTE

4) Stokes Hypothesis, Momentum Eqn. Navier-Stokes Equation

: Momentum Equation 1ij

DP

Dt Bv

a

ˆˆ ˆ( ) ( ) ( )ji ij jj kj jkii ki ik kkij

P P P P PP P P PP i j k

x y z x y z x y z

2 2 2 2 2 2

2 2 2 2

2ˆ : ( )3

P u u v w u v u w ui

x x x x y z x x y y z x z

2 2 2 2 2 2

2 2 2 2

2ˆ : ( )3

P v u v w v u v w vj

y y y x y z y y x x z y z

2 2 2 2 2 2

2 2 2 2

2ˆ : ( )3

P w u v w w u z v wk

z z z x y z z x z x y z y

ˆˆ ˆui vj wk Bv

2( )3

D p

Dt

BB B

vv v a : Navier-Stokes Equation

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4. Equations from the BTE

5) Energy Equation ( )21

2m Rv Only random motion

1) 21

( ) ( )2

nm ut t

Rv u : Mass specific internal energy

Energy flux vector

2) 2 2 21 1 1( ) ( ) ( ) ( )2 2 2 Enm u R R B R R Bv v v v v v v J

3)

21( ) ( )2 0

nm u

t t

Rv

2 2

2 2 2

1 1( ) ( )1 1 12 2( ) ( ) ( ) 0

2 2 2

nm mnm nm n m

t t

R R

R R R

v vv v v v v a

v

1) 2)

3)

4)

5)

B Rv = v v

21( ) ( ), ( )

2u u n

m m

RR R R R

vv v v v

( )E f d n

R RJ v v

E f d

RJ v

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4. Equations from the BTE

5) Energy Equation ( ) (Continued..)21

2m Rv

4) 2 21 1 1( ) 2 [ ( )] ( )2 2 2

n m R R R R R B R Bv v v v v v v v v v v v v v

, B Rv v v 0 v

( ) ( ) : :ijP R B R R B Bv v v v v v v

E( ) ( ) J : 0iju u Pt

B Bv v

0( ) ( ) [ ] [ ( )]

u u Duu u u u u

t t t t Dt

B B B Bv v v v

Mass balance

5) 21

( )2 0

mn

R

R

va a v

v

0

EJ :ij

DuP

Dt Bv : Energy Equation

By substituting 1), 2), 3), 4), 5)

: ( )iij ij

i j j

vP P

x

Bv

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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 x1)

0 0 00

( ) x x

f f f f df f f df dTv f f v

t dx dT dx

vr v

1) 2)

Assumptions : f is near equilibrium : Local Equilibrium 0df df

dx dx2)

0, 0Jx E x x x x

df dTq f v d f v v d

dT dx

Heat flux

0 0xf v d

Because f0 is the equilibrium distribution No heat flux

2 2 2 2 2 2 2 2 21, ,

3x y z x y z xv v v v v v v v v

20 01

3x x x

df dfdT dT dTq v v d v d k

dT dx dT dx dx

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∙3-D Fourier’s Law

4. Equations from the BTE

6) Fourier’s Law (Continued..)

(Under RTA and No External Force)

0 0 ( )

f f f f f f f

t

v vr v r

Assumptions : Steady state, Constant

0 0 0 00

f f f f ff f f T T Tu v w f f u v w

x y z T x T y T z

Assumptions : Local Equilibrium0 0 0, , f f ff f f

x x y y z z

x y z k T q q q q 3-D Fourier’s Law

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5. Summary

∙ BTE is an integro-differential equation of the ( , , )f tr v

∙ 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.