Non-Continuum Energy Transfer: Boltzmann Transport Equation

AME 60614 Int. Heat Trans.

D. B. Go Slide 1

Non-Continuum Energy Transfer: Boltzmann Transport

Equation


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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) 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 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 very

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

phonons, electrons, and photons as particles as well 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


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

transport 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) 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) The macroscopic, continuum equations apply – as the time scale increases time-averaged equilibrium– as the length scale increases 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 GasAt equilibrium, we can use Maxwell-Boltzmann statistics to determine the gas distribution for the relevant properties

velocity (Gaussian)

speed (Maxwellian)

energy (Maxwellian)


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BTE – Boltzmann Transport EquationConsider 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

recall

time, t


D. B. Go Slide 9

BTE – Boltzmann Transport EquationPlugging in and expanding, we can show

Or in general mathematical terms …

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

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 approximationthis 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

x-direction in physical space, x

x-di

rect

ion

in v

eloc

ity s

pace

, vx

dvx

collision(appear)

collision(disappear)


D. B. Go Slide 12

BTE – Boltzmann Transport Equation

The general Boltzmann Transport Equation

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 ConductivityConsider heat conduction, there is no external force … per se

Using the relaxation time approximation, the BTE simplifies to

Simplify further by assuming steady, 1-D transport

Now assume that the distribution is near equilibrium such that

called local equilibrium

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


D. B. Go Slide 15

BTE – Thermal ConductivityConsider a general description of the heat flux the flow of internal energy

recall

energyoccupation (distribution

function)

density of states

also recall that energy and momentum are related:

But for steady heat conduction, we have a form of the distribution function

Plugging into the general description of heat flux


D. B. Go Slide 16

BTE – Thermal ConductivityWe can split this integral into independent expressions

Consider each term separatelyf0 is the equilibrium distribution and at equilibrium an equal amount of heat flows left as does right no net heat flow

Fourier’s LawThus

if τ and v are constant


D. B. Go Slide 17

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

consider a general scalar quantity Φ

where

This can be written as

1 2 3 4


D. B. Go Slide 18

BTE – Conservation EquationsWe define the local average as

Consider prior expression term-by-term

1

2

3

4

(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 …

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

We now have the following general scalar conservation equation

no external force affecting distribution of mass

simplifying

this is our traditional mass conservation equation!

Non-Continuum Energy Transfer: Boltzmann Transport Equation

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