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AME 60614 Int. Heat Trans.
D. B. Go Slide 1
Non-Continuum Energy Transfer: Boltzmann Transport
Equation
AME 60614 Int. Heat Trans.
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
AME 60614 Int. Heat Trans.
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
AME 60614 Int. Heat Trans.
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!
AME 60614 Int. Heat Trans.
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
AME 60614 Int. Heat Trans.
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)
AME 60614 Int. Heat Trans.
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
AME 60614 Int. Heat Trans.
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( )
AME 60614 Int. Heat Trans.
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!
AME 60614 Int. Heat Trans.
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
AME 60614 Int. Heat Trans.
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)
AME 60614 Int. Heat Trans.
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!
AME 60614 Int. Heat Trans.
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
AME 60614 Int. Heat Trans.
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!
AME 60614 Int. Heat Trans.
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
∞
∫
AME 60614 Int. Heat Trans.
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
AME 60614 Int. Heat Trans.
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
AME 60614 Int. Heat Trans.
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)
AME 60614 Int. Heat Trans.
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!