Non-Continuum Energy Transfer: Gas Dynamics - PowerPoint Presentation

Slide1

Non-Continuum Energy Transfer: Gas DynamicsSlide2

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 theorySlide3

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-conductorSlide4

We will consider a gas as a collection of individual particles

monatomic gasses are simplest and can be analyzed from first principles fairly readily (He, Ar, Ne)

diatomic gasses are a little more difficult (H

2

, O

2, N2)  must account for interactions between both atoms in the moleculepolyatomic gasses are even more difficult

Gases – Individual Particles

gas … gasSlide5

Gases – How to Understand One

Understanding a gas – brute force

suppose we wanted to understand a system of

N

gas particles in a volume V (~

1025 gas molecules in 1 mm3 at STP)

 position & velocity

Understanding a gas – statisticallystatistical mechanics

helps us understand microscopic properties and relate them to macroscopic propertiesstatistical mechanics obtains the equilibrium distribution of the particlesUnderstanding a gas – kinetically

kinetic theory

considers the transport of individual particles (collisions!) under non-equilibrium conditions in order to relate microscopic properties to macroscopic transport properties  thermal conductivity!

just not possibleSlide6

Gases – Statistical Mechanics

If we have a gas of

N

atoms, each with their own kinetic energy

ε

, we can organize them into

“

energy levels

” each with

N

i atoms

gas … gas

total atoms in the system:

internal energy of the system:

We call each

energy level

ε

i

with

N

i

atoms a

macrostate

Each

macrostate

consists of individual energy states called

microstates

these microstates are based on quantized energy



related to the quantum mechanics



Schrödinger

’

s equation

Schrödinger

’

s equation results in

discrete/quantized

energy levels (

macrostates

) which can themselves have different quantum microstates (degeneracy,

g

i

)



can liken it to density of statesSlide7

Gases – Statistical Mechanics

There can be any number of

microstates

in a given

macrostate

 called that levels degeneracy g

i

this number of microstates the is thermodynamic probability, Ω,

of a macrostateWe describe

thermodynamic equilibrium

as the most probable macrostate

Three fairly important assumptions/postulatesThe time-average for a thermodynamic variable is equivalent to the average over all possible microstates

All microstates are equally probableWe assume independent particles

Maxwell-Boltzmann statistics gives us the thermodynamic probability, Ω

, or number of microstates per macrostateSlide8

Gases – Statistics and Distributions

The thermodynamic probability can be determined from basic statistics but is dependant on the

type of particle

. Recall that we called phonons

bosons

and electrons

fermions

. Gas atoms we consider

boltzons

boltzons:

distinguishable particles

bosons:

indistinguishable particles

fermions:

indistinguishable particles and limited occupancy (Pauli exclusion)

Maxwell-Boltzmann statistics

Bose-Einstein statistics

Fermi-Dirac statistics

Fermi-Dirac

distribution

Bose-Einstein

distribution

Maxwell-Boltzmann

distributionSlide9

Gases – What is Entropy?

Thought Experiment

: consider a chamber of gas expanding into a vacuum

A

B

A

B

This process is irreversible and therefore

entropy increases

(additive)

The

thermodynamic probability

also increases because the final state is more probable than the initial state (multiplicative)

How is the entropy related to the thermodynamic probability (

i.e.

, microstates)? Only one mathematical function converts a multiplicative operation to an additive operation

Boltzmann relation!Slide10

Gases – The Partition Function

The partition

function

Z

is

an useful

statistical

definition quantity that will be used to describe macroscopic thermodynamic properties from a microscopic representation

The

probability of atoms in energy level

i is simply the ratio of particles in i to the total number of particles in all energy levels

leads directly to Maxwell-Boltzmann distributionSlide11

Gases –

1

St

Law from Partition Function

Heat and Work

adding heat to a system affects occupancy at each energy

level

a system doing/receiving work does changes the energy levels

First Law of Thermodynamics – Conservation of Energy!Slide12

Gases – Equilibrium Properties

E

nergy

and entropy in terms of the partition function

Z

Classical definitions & Maxwell Relations then lead to the statistical definition of other properties

chemical potential

Gibbs free energy

Helmholtz free energy

pressureSlide13

Gases – Equilibrium

Properties

enthaply

but classically …

ideal gas law

t

he

Boltzmann constant is

directly

related to the Universal Gas ConstantSlide14

Recalling that the specific heat is the derivative of the internal energy with respect to temperature, we can rewrite intensive properties (per unit mass

) statistically

internal energy

entropy

Gibbs free energy

Helmholtz free energy

enthaply

specific heat

Gases – Equilibrium

PropertiesSlide15

Gases – Monatomic Gases

In diatomic/polyatomic gasses the atoms in a molecule can vibrate between each other and rotate about each other which all contributes to the internal energy of the

“

particle

”

monatomic gasses are simpler because the internal energy of the particle is their kinetic energy and electronic energy (energy states of electrons)

an evaluation of the quantum mechanics and additional mathematics can be used to derive

translational

and

electronic partition functions

consider the translational energy only

we can plug this in to our previous equations

internal energy

entropy

specific heatSlide16

Gases – Monatomic Gases

Where did

P

(pressure) come from in the entropy relation?

pressure

plugging in the translational partition function ….

the derivative of the

ln

(

C



V

)

is

1/

V

ideal gas lawSlide17

Gases – Monatomic Gases

The electronic energy is more difficult because you have to understand the energy levels of electrons in atoms



not too bad for monatomic

gases

(We can look up these levels for some choice atoms)

Defining derivatives as

internal energy

entropy

specific heatSlide18

Gases – Monatomic Helium

Consider monatomic hydrogen at 1000

K …

I can look up electronic degeneracies and energies to give the following table

level

g

1

0

0

0

0

0

2

3

229.9849711

2.282E+100

5.2484E+102

1.207E+105

3

0

239.2234393

0

0

0

4

8

243.2654669

3.564E+106

8.67E+108

2.1091E+111

5

3

246.2119245

2.5445E+107

6.2648E+109

1.5425E+112

6

3

263.622928

9.2705E+114

2.4439E+117

6.4427E+119Slide19

Gases – Monatomic Helium

from Incropera and DewittSlide20

Gases – A Little Kinetic Theory

We

’

ve already discussed kinetic theory in relation to thermal conductivity



individual particles carrying their energy from hot to cold

G. Chen

The same approach can be used to derive the flux of any property for individual particles



individual particles carrying their energy from hot to cold

general flux of scalar property

ΦSlide21

Gases – Viscosity and Mass Diffusion

Consider viscosity from general kinetic theory (flux of momentum)



Newton

’

s Law

Consider mass diffusion from general kinetic theory (flux of mass)



Fick

’

s Law

Note that all these properties are related and depend on the average speed of the gas molecules and the mean free path between collisionsSlide22

Gases – Average Speed

The average speed can be derived from the Maxwell-Boltzmann distribution

We can derive it based on assuming

only translational energy,

g

i

= 1

(good for monatomic gasses – recall that translation dominates electronic)

This is a ratio is proportional to a probability density function



by definition the integral of a probability density function over all possible states must be 1

probability that a gas molecule has a given momentum

pSlide23

Gases – Average Speed

From the Maxwell-Boltzmann momentum distribution, the energy, velocity, and speed distributions easily follow Slide24

Gases – Mean Free Path

The

mean free path

is the average distance traveled by a gas molecule between collisions



we can simply gas collisions using a

hard-sphere, binary collision approach (billiard balls)

r

incident

r

target

incident particle

r

incident

collision with target particle

d

12

cross section defined as:

General mean free path

Monatomic gasSlide25

Gases – Transport Properties

Based on this very simple approach, we can determine the transport properties for a monatomic gas to be

M

is molecular weight

Recall, that

more rigorous collision dynamics modelSlide26

Gases – Monatomic Helium

from Incropera and Dewitt

only 2% difference! Slide27

Gases – What We

’ve Learned

Gases can be treated as individual particles

store

and

transport thermal energyprimary 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!