non-continuum energy transfer: gas dynamicssst/teaching/ame60634/lectures/ame...d.b.go slide18$$...

AME 60614 Int. Heat Trans.

D. B. Go Slide 1

Non-Continuum Energy Transfer: Gas Dynamics


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

•  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 (H2, O2, N2) è must account

for interactions between both atoms in the molecule –  polyatomic gasses are even more difficult

Gases – Individual Particles

free electron gas phonon gas

gas … gas


D. B. Go Slide 5

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) è position & velocity

•  Understanding a gas – statistically –  statistical mechanics helps us understand microscopic properties and

relate them to macroscopic properties –  statistical mechanics obtains the equilibrium distribution of the

particles

•  Understanding 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 è thermal conductivity!

€

mid v idt

= Fij r i, r j ,t( )

j=1

N−1

∑ ; i =1,2,3,....,N just not possible


D. B. Go Slide 6

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 Ni atoms

gas … gas

€

ε0,N0

€

ε1,N1

€

ε2,N2

€

εi,Ni

€

€

total atoms in the system:

€

N = Nii= 0

∞

∑ internal energy of the system:

€

U = εiNii= 0

∞

∑

•  We call each energy level εi with Ni atoms a macrostate

•  Each macrostate consists of individual energy states called microstates •  these microstates are based on quantized energy è related to the quantum mechanics è Schrödinger’s equation •  Schrödinger’s equation results in discrete/quantized energy levels (macrostates) which can themselves have different quantum microstates (degeneracy, gi) è can liken it to density of states


D. B. Go Slide 7

Gases – Statistical Mechanics •  There can be any number of microstates in a given macrostate è

called that levels degeneracy gi •  this number of microstates the is thermodynamic probability, Ω, of a

macrostate •  We describe thermodynamic equilibrium as the most probable

macrostate

•  Three fairly important assumptions/postulates (1) The time-average for a thermodynamic variable is equivalent to the

average over all possible microstates (2)  All microstates are equally probable (3) We assume independent particles

•  Maxwell-Boltzmann statistics gives us the thermodynamic probability, Ω, or number of microstates per macrostate

€

Ω = N! giNi

Ni!i= 0

∞

∏


D. B. Go Slide 8

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

€

ΩMB = N! giNi

Ni!i= 0

∞

∏

€

ΩBE =gi + Ni −1( )!gi −1( )!Ni!i= 0

∞

∏

€

ΩFD =gi!

gi − Ni( )!Ni!i= 0

∞

∏

Fermi-Dirac distribution

€

f ε( ) =1

exp ε −µkBT

$

% &

'

( ) +1

€

f ε( ) =1

exp ε −µkBT

$

% &

'

( ) −1

Bose-Einstein distribution

€

f ε( ) =1

exp ε −µkBT

$

% &

'

( )

Maxwell-Boltzmann distribution


D. B. Go Slide 9

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)

€

ΩAB =ΩAΩB€

SAB = SA + SB

How is the entropy related to the thermodynamic probability (i.e., microstates)? Only one mathematical function converts a multiplicative operation to an additive operation

€

S = kB lnΩ Boltzmann relation!


D. B. Go Slide 10

Gases – The Partition Function

€

Z = gie−ε i kBT

i= 0

∞

∑

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

€

Ni

N=

gie−ε i kBT

gie−ε i kBT

i= 0

∞

∑=gie

−ε i kBT

Zleads directly to Maxwell-

Boltzmann distribution


D. B. Go Slide 11

Gases – 1St Law from Partition Function

€

U = εiNii= 0

∞

∑

Heat and Work €

dU = εidNii= 0

∞

∑ + Nidεii= 0

∞

∑

adding heat to a system affects occupancy at each energy level

€

δQ = εidNii= 0

∞

∑

€

δW = − Nidεii= 0

∞

∑

a system doing/receiving work does changes the energy levels

dU = δQ+δW

First Law of Thermodynamics – Conservation of Energy!


D. B. Go Slide 12

Gases – Equilibrium Properties Energy and entropy in terms of the partition function Z

U = εiNii=0

∞

∑ =NZ

εigie−εi kBT

i=0

∞

∑ = NkBT2 ∂ lnZ( )

∂T$

%&

'

()V ,N

S = kB lnΩ = NkB 1+ lnZN"

#$

%

&'+T

∂ lnZ( )∂T

(

)*

+

,-V ,N

./0

10

230

40

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

chemical potential

€

µ = −kBT lnZN#

$ %

&

' (

Gibbs free energy

€

G = µN = −NkBT lnZN#

$ %

&

' (

Helmholtz free energy

€

A =U −TS = −NkBT 1+ ln ZN#

$ %

&

' (

)

* +

,

- .

pressure

€

P = −∂A∂V$

% & '

( ) T ,N= NkBT

∂ lnZ( )∂V

$

% &

'

( ) T ,N


D. B. Go Slide 13

Gases – Equilibrium Properties

enthaply

€

H =G + TS = −NkBT lnZN#

$ %

&

' ( + TNkB 1+ ln Z

N#

$ %

&

' ( + T

∂ lnZ( )∂T

*

+ ,

-

. / V ,N

0 1 2

3 2

4 5 2

6 2

€

H = NkBT 1+ T∂ lnZ( )∂T

#

$ %

&

' ( V ,N

) * +

, +

- . +

/ + =U + NkBT

€

H =U + PV =U + NkBT

but classically …

€

PV = NkBTideal gas law

the Boltzmann constant is directly related to the Universal Gas Constant

€

kB =nRN

=RNA


D. B. Go Slide 14

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

€

uRT

= T∂ lnZ( )∂T

#

$ %

&

' ( V

€

sRT

=1+ ln ZN"

# $

%

& ' + T

∂ lnZ( )∂T

)

* +

,

- . V

€

gRT

= −ln ZN#

$ %

&

' (

Helmholtz free energy

€

aRT

= − 1+ ln ZN#

$ %

&

' (

)

* +

,

- .

enthaply

€

hRT

=1+ T∂ lnZ( )∂T

#

$ %

&

' ( V

specific heat

€

cvR

=∂∂T

T 2∂ lnZ( )∂T

#

$ %

&

' (

)

* +

,

- . V

€

cpR

=1+∂∂T

T 2∂ lnZ( )∂T

#

$ %

&

' (

)

* +

,

- . V

Gases – Equilibrium Properties


D. B. Go Slide 15

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

€

Z = gie−ε i kBT

i= 0

∞

∑ ⇒ Ztr =2πmkBTh2

(

) *

+

, -

32V

we can plug this in to our previous equations

€

uRT"

# $

%

& ' tr

= T∂ lnZ( )∂T

)

* +

,

- . V

=32

€

sR"

# $

%

& ' tr

=52

+ ln2πm( )

32 kBT( )

52

h3P

)

* + +

,

- . .

€

cvR

"

# $

%

& ' tr

=32

cpR

!

"#

$

%&tr

=1+ cvR

!

"#

$

%&tr

=52

internal energy entropy

specific heat


D. B. Go Slide 16

Gases – Monatomic Gases

Where did P (pressure) come from in the entropy relation?

pressure

€

P = −∂A∂V$

% & '

( ) T ,N= NkBT

∂ lnZ( )∂V

$

% &

'

( ) T ,N

plugging in the translational partition function ….

P = NkBT∂ lnZ( )∂V

!

"#

$

%&T ,N

= NkBT

∂ ln 2πmkBTh2

'

()

*

+,

32V

'

())

*

+,,

∂V

!

"

#####

$

%

&&&&&T ,N

€

P = NkBT1V"

# $ %

& ' the derivative of the ln(C�V) is 1/V

€

PV = NkBTideal gas law


D. B. Go Slide 17

Gases – Monatomic Gases The electronic energy is more difficult because you have to understand the energy levels of electrons in atoms è not too bad for monatomic gases (We can look up these levels for some choice atoms)

Defining derivatives as

internal energy

€

uRT"

# $

%

& '

el

= T∂ lnZ( )∂T

)

* +

,

- .

V

=/ Z el

Zel

entropy

€

sR"

# $

%

& '

el

=( Z el

Zel

+ lnZel

specific heat

€

cv

R"

# $

%

& '

el

=cp

R"

# $

%

& '

el

=( ( Z el

Zel

−( Z el

Zel

"

# $

%

& '

2

€

Zel = gie−ε i

kBT

i= 0

∞

∑ ⇒ ' Z el =εi

kBTgie

−ε ikBT

i= 0

∞

∑ ⇒ ' ' Z el =εi

kBT(

) *

+

, -

2

gie−ε i

kBT

i= 0

∞

∑


D. B. Go Slide 18

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+119

€

εkBT

€

εkBT

geεkBT

€

geεkBT

€

εkBT

#

$ % %

&

' ( (

2

geεkBT

€

cp

R"

# $

%

& '

el

=( ( Z el

Zel

−( Z el

Zel

"

# $

%

& '

2

= 9.91×10−6

€

cpR

"

# $

%

& ' tr

=52

€

cp =52R =

52

2.077 kJkg - K

"

# $

%

& ' = 5.195 kJ

kg - K€

cpR

"

# $

%

& ' =

cpR

"

# $

%

& ' el

+cpR

"

# $

%

& ' tr


D. B. Go Slide 19

Gases – Monatomic Helium

from Incropera and Dewitt


D. B. Go Slide 20

Gases – A Little Kinetic Theory We’ve already discussed kinetic theory in relation to thermal conductivity è individual particles carrying their energy from hot to cold

G. Chen

€

k =13v 2τC =

13vC

The same approach can be used to derive the flux of any property for individual particles è individual particles carrying their energy from hot to cold

JΦ = −13v2τ dNΦ

dx= −13v dNΦ

dx

general flux of scalar property Φ


D. B. Go Slide 21

Gases – Viscosity and Mass Diffusion

Consider viscosity from general kinetic theory (flux of momentum) è Newton’s Law

τ yx = −µdvydy

= −13vdN mvy( )

dy= −13vNm

d vy( )dy

→ µ =13vρ

Consider mass diffusion from general kinetic theory (flux of mass) è Fick’s Law

JmA= −DAB

dρA

dy= −13vd N V( )dy

→DAB =13v

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


D. B. Go Slide 22

Gases – Average Speed

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

We can derive it based on assuming only translational energy, gi = 1 (good for monatomic gasses – recall that translation dominates electronic) €

f ε( ) =1

exp ε −µkBT

$

% &

'

( )

€

Ni

N=

gie−ε i kBT

gie−ε i kBT

i= 0

∞

∑=gie

−ε i kBT

Z

€

Ni

N=e−ε i kBT

Z=e−px2 + py

2 + pz2( )2mkBT

Z

€

ε =p2

2m

This is a ratio is proportional to a probability density function è by definition the integral of a probability density function over all possible states must be 1

€

f p ( ) =1

2πmkBT

#

$ %

&

' (

32

e−

px2 + py

2 + pz2( )2mkBT

probability that a gas molecule has a given momentum p


D. B. Go Slide 23

Gases – Average Speed

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

€

f v ( ) =m

2πkBT

#

$ %

&

' (

32

e−

m vx2 +vy

2 +vz2( )2kBT

€

f vx( ) =m

2πkBT#

$ %

&

' (

12

e−mvx

2

2kBT

€

f ε( ) = 2 ε

π kBT( )3e− ε kBT

€

εm =32kBT

€

f v( ) =4π

m2kBT#

$ %

&

' (

32

v 2e−mv

2

2kBT

€

vm =8kBTπm

; vmp =2kBTm


D. B. Go Slide 24

Gases – Mean Free Path The mean free path is the average distance traveled by a gas molecule between collisions è we can simply gas collisions using a hard-sphere, binary collision approach (billiard balls)

rincident

rtarget

incident particle

rincident

collision with target particle

d12

€

σ = πd2cross section defined as:

General mean free path Monatomic gas

€

=12nπd2

=kBT2πd2P

€

12 =m2

m1 + m2

"

# $

%

& '

12 1n2πd12

2


D. B. Go Slide 25

Gases – Transport Properties

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

€

k =2 mkBT3π 3 2d2

cvM#

$ %

&

' (

µ =2 mkBT3π 3 2d2

D =2 mkBT3π 3 2d2ρ

€

k = µcvM"

# $

%

& '

M is molecular weight

Recall, that

€

cvR

"

# $

%

& ' tr

=32

€

k =5232RM

"

# $

%

& ' µ

more rigorous collision dynamics model

€

k =52cvM"

# $

%

& ' µ


D. B. Go Slide 26

Gases – Monatomic Helium

from Incropera and Dewitt

€

k =154199 ×10−7( ) 8.3145 ×10

3

4.00$

% &

'

( )

€

k =155 ×10−3 Wm - K

€

k =5232RM

"

# $

%

& ' µ

only 2% difference!


D. B. Go Slide 27

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!

non-continuum energy transfer: gas dynamicssst/teaching/ame60634/lectures/ame...d.b.go slide18$$...

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