Non-Continuum Energy Transfer: Boltzmann Transport...

AME 60614 Int. Heat Trans.

D. B. Go Slide 1

Non-Continuum Energy Transfer: Boltzmann Transport

Equation


D. B. Go Slide 2

Phonons – What We’ve Learned •  Phonons are quantized lattice vibrations

–  store and transport thermal energy –  primary energy carriers in insulators and semi-conductors (computers!)

•  Phonons are characterized by their –  energy –  wavelength (wave vector) –  polarization (direction) –  branch (optical/acoustic) è acoustic phonons are the primary thermal

energy carriers

•  Phonons have a statistical occupation (Bose-Einstein), quantized (discrete) energy, and only limited numbers at each energy level –  we can derive the specific heat!

•  We can treat phonons as particles and therefore determine the thermal conductivity based on kinetic theory


D. B. Go Slide 3

Electrons – What We’ve Learned •  Electrons are particles with quantized energy states

–  store and transport thermal and electrical energy –  primary energy carriers in metals –  usually approximate their behavior using the Free Electron Model

•  energy •  wavelength (wave vector)

•  Electrons have a statistical occupation (Fermi-Dirac), quantized (discrete) energy, and only limited numbers at each energy level (density of states)

–  we can derive the specific heat!

•  We can treat electrons as particles and therefore determine the thermal conductivity based on kinetic theory

–  Wiedemann Franz relates thermal conductivity to electrical conductivity

•  In real materials, the free electron model is limited because it does not account for interactions with the lattice –  energy band is not continuous –  the filling of energy bands and band gaps determine whether a material is a

conductor, insulator, or semi-conductor


D. B. Go Slide 4

Gases – What We’ve Learned •  Gases can be treated as individual particles

–  store and transport thermal energy –  primary energy carriers fluids è convection!

•  Gases have a statistical (Maxwell-Boltzmann) occupation, quantized (discrete) energy, and only limited numbers at each energy level –  we can derive the specific heat, and many other gas properties using an

equilibrium approach

•  We can use non-equilibrium kinetic theory to determine the thermal conductivity, viscosity, and diffusivity of gases

•  The tables in the back of the book come from somewhere!


D. B. Go Slide 5

•  Phonons and electrons (and photons) possess wave-like characteristics –  to track waves we need to know amplitude, phase, direction è very

difficult! •  We already treat gases as particles and we also like to treat

phonons, electrons, and photons as particles as well è we’ve already applied kinetic theory to derive thermal conductivity! –  can’t capture phase coherence effects (interference, diffraction, etc.) –  can capture propagation, reflection, transmission, etc.

BTE – Particle Approach

free electron gas phonon gas

gas … gas


D. B. Go Slide 6

BTE – Transport Modeling •  To understand energy transfer, we must be able to model the

transport è scaling determines proper modeling approach –  based on physical dimensions in space and time –  compared to basic transport properties (wavelength, mean free path,

mean free time, collision time usually ~ps-fs) •  For very small time and length scales (on the order of a wavelength

and collision time) è quantum approaches must be used & wave behavior is significant –  Green’s Functions, molecular dynamics, density functional theory

•  For large time and length scales (greater than mean free path/time) è The macroscopic, continuum equations apply –  as the time scale increases è time-averaged equilibrium –  as the length scale increases è space-average equilibrium

•  The Boltzmann Transport Equation is a general transport equation that allows for non-equilibrium transport (on the order of the mean free path/time) –  derived for gases but applicable to any particle systems (phonons,

electrons and photons)


D. B. Go Slide 7

BTE – Equilibrium Distributions for a Gas At equilibrium, we can use Maxwell-Boltzmann statistics to determine the gas distribution for the relevant properties

f v( ) = m2πkBT!

"#

$

%&

32

exp −m vx

2 + vy2 + vz

2( )2kBT

(

)**

+

,--

velocity (Gaussian)

f v( ) = 4π

m2kBT!

"#

$

%&

32

v2 exp − mv2

2kBT(

)*

+

,-speed (Maxwellian)

v = 8kBTπm

f ε( ) = 2 ε

2 kBT( )3exp − ε

kBT"

#$

%

&'energy (Maxwellian)

ε =32kBT


D. B. Go Slide 8

BTE – Boltzmann Transport Equation Consider a packet of particles with a distribution f in time t, space r and momentum p

•  typically we consider equilibrium distributions (Maxwell-Boltzmann, Bose-Einstein, Fermi-Dirac) but here we are considering how the distribution changes in time

Look at the terms

€

r + d r = r + v dt p + d p = p +

F dtrecall

v = drdt

F = dm

vdt

=dpdt

time, t

f r, p, t( ) f r + dr, p+ dp, t + dt( )


D. B. Go Slide 9

BTE – Boltzmann Transport Equation Plugging in and expanding, we can show

€

f r + v dt, p + F dt,t + dt( ) − f r , p ,t( )

dt=∂f∂t

+ v ⋅ ∂f∂ r

+ F ⋅ ∂f

∂ p

= 0

Or in general mathematical terms …

€

∂f∂t

+ v ⋅ ∇ r f +

F ⋅ ∇ p f = 0

Notes: •  f here is a scalar (a distribution) and this equation describes the transport and time evolution of the distribution •  we can multiply this equation (called taking moments) by other quantities to determine the transport of those quantities

•  for example – multiply by momentum to determine the transport of momentum (Navier-Stokes!)

•  the F term is an external force (such as an electric field for electrons!)

What about collisions!


D. B. Go Slide 10

BTE - Collisions •  Thus far we’ve only described the transport of the distribution è

collisions between particles will modify the distribution –  in-scattering - a collision that increases the distribution of particles with

momentum p from momentum p’ –  out-scattering - a collision that increases the distribution of particles

with momentum p’ from momentum p

•  Collisions act as sources and sinks of the distribution function!

•  Relaxation time approximation

€

∂f∂t coll

= f r , # p ,t( )W # p , p ( )

# p ∑

in -scattering

− f r , p ,t( )W p , # p ( ) # p ∑

out -scattering

€

W " p , p ( ) ≡ probability of that a carrier with momentum " p will scatter into a state with momentum p

W p , " p ( ) ≡ probability of that a carrier with momentum p will scatter into a state with momentum " p

€

∂f∂t coll

=f0 − fτ relax

this implies that a small non-equilibrium distribution f will relax back to the equilibrium distribution f0 due to collisions in time τ


D. B. Go Slide 11

BTE – Control Volume Perspective

dx

€

f x,vx,t( )

x-direction in physical space, x

x-di

rect

ion

in v

eloc

ity s

pace

, vx

dvx

€

f x,vx + dvx,t( )

€

f x,vx,t( )

€

f x + dx,vx,t( )

collision (appear)

collision (disappear)


D. B. Go Slide 12

BTE – Boltzmann Transport Equation

The general Boltzmann Transport Equation

€

∂f∂t

+ v ⋅ ∇ r f +

F ⋅ ∇ p f =

∂f∂t coll

≈f0 − fτ relax

Notes: •  f here is a scalar (a distribution) and this equation describes the transport of the distribution •  we can multiply this equation (called taking moments) by other quantities to determine the transport of those quantities

•  for example – multiply by momentum to determine the transport of momentum (Navier-Stokes!)

•  the F term is an external force (such as an electric field for electrons!) •  the BTE is a PDE in 7 dimensions (3 space, 3 momentum, and 1 space) and the collision term is often treated as an integral so the BTE is a 7-dimensional integro-differential equation!


D. B. Go Slide 13

BTE – Using the Distribution Function •  The BTE describes the transport of the statistical distribution

function in time, space, and momentum •  We seek to determine macroscopic quantities (thermal conductivity,

viscosity) and macroscopic conservation equations (mass, momentum, energy) from this distribution

•  We can determine properties through simplifications of the general BTE and compare to other approaches

•  We can derive macroscopic conservation equations by taking moments of the BTE


D. B. Go Slide 14

BTE – Thermal Conductivity Consider heat conduction, there is no external force … per se

€

∂f∂t

+ v ⋅ ∇ r f =

f0 − fτ relax

Using the relaxation time approximation, the BTE simplifies to

Simplify further by assuming steady, 1-D transport

€

vx∂fx∂x

=f0 − fτ relax

Now assume that the distribution is near equilibrium such that

€

∂fx∂x

≈∂f0∂x

called local equilibrium

€

vx∂f0∂x

=f0 − fτ relax

€

f ≈ f0 − τvx∂f0∂x

We now have an equation for the distribution function f in local equilibrium!


D. B. Go Slide 15

BTE – Thermal Conductivity Consider a general description of the heat flux è the flow of internal energy

€

" " q x = εf ε( )D ε( )[ ]vxdε0

∞

∫

€

U = εf ε( )D ε( )0

∞

∫ dεrecall

energy occupation (distribution

function)

density of states

also recall that energy and momentum are related:

€

f r , p ,t( )↔ f r ,ε,t( )But for steady heat conduction, we have a form of the distribution function

€

f ≈ f0 − τvx∂f0∂x

= f0 − τvxdf0dT

dxdT

Plugging into the general description of heat flux

€

" " q x = f0 − τvxdf0dT

dTdx

%

& ' (

) * εD ε( )vxdε

0

∞

∫


D. B. Go Slide 16

BTE – Thermal Conductivity We can split this integral into independent expressions

Consider each term separately

€

" " q x = f0 − τvxdf0dT

dTdx

%

& ' (

) * εD ε( )vxdε

0

∞

∫ = f0εD ε( )vxdε0

∞

∫ − τvxdf0dT

dTdxεD ε( )vxdε

0

∞

∫

€

f0εD ε( )vxdε0

∞

∫ → f0εvxdvxvx

∫ = 0f0 is the equilibrium distribution and at equilibrium an equal amount of heat flows left as does right è no net heat flow

€

" " q x = − τvx2 df0

dTdTdxεD ε( )dε

0

∞

∫ = −13

τv 2 df0dT

εD ε( )dε0

∞

∫(

) *

+

, -

dTdx

€

qx = −k dTdx

Fourier’s Law

€

k =13

τv 2 df0dT

εD ε( )dε0

∞

∫

Thus if τ and v are

constant

€

k =13τv 2 df0

dTεD ε( )dε

0

∞

∫ =13τv 2 dU

dT=13τv 2C


D. B. Go Slide 17

BTE – Conservation Equations We can derive macroscopic conservation equations by taking moments of the BTE è multiplying by a scalar quantity (such as energy) and integrating over all states

consider a general scalar quantity Φ

€

Φ∂f∂t

+ v ⋅ ∇ r f +

F ⋅ ∇ p f = γ in − γ out

(

) * +

, - p∫ dp

€

∂f∂t coll

= f r , # p ,t( )W # p , p ( )

# p ∑

in -scattering

− f r , p ,t( )W p , # p ( ) # p ∑

out -scattering

= γ in − γ out

where

This can be written as

€

Φ∂f∂t

dpp∫ + Φ

v ⋅ ∇ r fdp

p∫ + Φ

F ⋅ ∇ p fdp

p∫ = Φ γ in − γ out( )dp

p∫

1 2 3 4


D. B. Go Slide 18

€

Φ v ⋅ ∇ r f( )dp

p∫ =

∇ r ⋅ Φ

v f( )dpp∫ − f ⋅

∇ rΦ v ( )dp

p∫ =

∇ r ⋅ Φ

v f( )dpp∫

= ∇ r ⋅ Φvn( )

BTE – Conservation Equations We define the local average as

Consider prior expression term-by-term

€

Φ =

Φfdpp∫

fdpp∫

=1n

Φfdpp∫

1

€

Φ∂f∂tdp

p∫ =

∂nΦ∂t

2

€

Φ F ⋅ ∇ p fdp

p∫ =

F ⋅ Φ

∇ p fdp

p∫ =

F ⋅

∇ pΦfdp

p∫ −

F ⋅ f

∇ pΦdp

p∫

= − F ⋅ f

∇ pΦdp

p∫ = −n

F ⋅ ∇ pΦ( )

3

€

Φ γ in − γ out( )dpp∫ = Γin −Γout4

(v is the bulk velocity)


D. B. Go Slide 19

BTE – Mass Conservation Equation

Consider mass conservation where Φ = m and ρ = nm

plugging in …

€

∂nm∂t

+ ∇ r ⋅ nvm( ) − n

F ⋅ ∇ pm( )[ ] = Γin −Γout

when reactions are not considered, these “source” & “sink” terms cancel out

We now have the following general scalar conservation equation

€

∂nΦ∂t

+ ∇ r ⋅ nvΦ( ) − n

F ⋅ ∇ pΦ( )[ ] = Γin −Γout

no external force affecting distribution of mass

€

∂nm∂t

+ ∇ r ⋅ nvm( ) =

∂ρ∂t

+ ∇ r ⋅ ρv( ) = 0

simplifying

this is our traditional mass conservation equation!

Non-Continuum Energy Transfer: Boltzmann Transport...

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