Transcript What is Heat Transfer?

AME 60614
Int. Heat Trans.
Non-Continuum Energy
Transfer: Gas Dynamics
D. B. Go
Slide 1
AME 60614
Int. Heat Trans.
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 2
AME 60614
Int. Heat Trans.
•
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 3
AME 60614
Int. Heat Trans.
Gases – Individual Particles
• 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)  must account
for interactions between both atoms in the molecule
– polyatomic gasses are even more difficult
phonon gas
free electron gas
gas … gas
D. B. Go
Slide 4
AME 60614
Int. Heat Trans.
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
N -1
dv i
mi
= å Fij ( ri , rj ,t ); i = 1,2,3,...., N
dt j=1
just not possible
• 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  thermal conductivity!
D. B. Go
Slide 5
AME 60614
Int. Heat Trans.
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
ei,N i
gas … gas
e2,N 2
e1,N1
e0,N 0
¥
total atoms in the system:
N = å Ni
¥
internal energy of the system: U
i= 0
= åe i N i
i= 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  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, gi)  can liken it to density of states
D. B. Go
Slide 6
AME 60614
Int. Heat Trans.
Gases – Statistical Mechanics
• There can be any number of microstates in a given macrostate 
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
D. B. Go
Slide 7
AME 60614
Int. Heat Trans.
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
Maxwell-Boltzmann statistics
boltzons:
distinguishable particles
bosons:
indistinguishable particles
fermions:
indistinguishable particles
and limited occupancy
(Pauli exclusion)
D. B. Go
Maxwell-Boltzmann
distribution
f (e) =
Bose-Einstein statistics
Fermi-Dirac statistics
1
æe - mö
expç
÷
è kB T ø
Bose-Einstein
distribution
f (e) =
1
æe - mö
expç
÷ -1
k
T
è B ø
Fermi-Dirac
distribution
f (e) =
1
æe - mö
expç
÷ +1
k
T
è B ø
Slide 8
AME 60614
Int. Heat Trans.
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)
SAB = SA + SB
• 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!
D. B. Go
Slide 9
AME 60614
Int. Heat Trans.
Gases – The Partition Function
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
-e i
kB T
-e i
Ni
gie
gie kB T
= ¥
=
e
i
N
Z
kB T
g
e
åi
leads directly to MaxwellBoltzmann distribution
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
¥
Z = å gie
ei
-
kB T
i= 0
D. B. Go
Slide 10
AME 60614
Int. Heat Trans.
Gases – 1St Law from Partition Function
First Law of Thermodynamics – Conservation of Energy!
¥
U = åe i N i
i= 0
¥
¥
i= 0
i= 0
dU = åei dN i + å N i dei
dU = dQ + dW
Heat and Work
adding heat to a system affects occupancy at each energy level
¥
dQ = åei dN i
i= 0
a system doing/receiving work does changes the energy levels
¥
dW = -å N i dei
i= 0
D. B. Go
Slide 11
AME 60614
Int. Heat Trans.
Gases – Equilibrium Properties
Energy and entropy in terms of the partition function Z
é
ù
-ei
N ¥
k BT
2 ¶ ( ln Z )
U = åei Ni = åei gi e
= NkBT ê
ú
Z i=0
ë ¶T ûV,N
i=0
¥
ìï
æ Z ö é¶ ( ln Z ) ù üï
S = kB lnW = NkB í1+ ln ç ÷ + T ê
ú ý
è N ø ë ¶ T ûV,N ïþ
ïî
Classical definitions & Maxwell Relations then lead to the statistical definition
of other properties
chemical potential
Gibbs free energy
æZö
m = -kB T lnç ÷
èNø
æZö
G = mN = -Nk B T lnç ÷
èNø
Helmholtz free energy
é
æ Z öù
A = U - TS = -NkB Tê1+ lnç ÷ú
è N øû
ë
D. B. Go
pressure
é¶(ln Z ) ù
é ¶A ù
P = -ê ú = NkB Tê
ú
ë¶V ûT ,N
ë ¶V ûT ,N
Slide 12
AME 60614
Int. Heat Trans.
Gases – Equilibrium Properties
enthaply
ìï
æZö
æ Z ö é¶ (ln Z ) ù üï
H = G + TS = -Nk B T lnç ÷ + TNkB í1+ lnç ÷ + Tê
ú ý
ïî
èNø
è N ø ë ¶T ûV ,N ïþ
ìï
é¶ (ln Z ) ù üï
H = NkB T í1+ Tê
ú ý = U + Nk B T
ïî
ë ¶T ûV ,N ïþ
but classically …
ideal gas law
H = U + PV = U + NkB T
PV = NkB T
the Boltzmann constant is directly related to the Universal Gas Constant
kB =
D. B. Go
nR R
=
N NA
Slide 13
AME 60614
Int. Heat Trans.
Gases – Equilibrium Properties
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
é¶ (ln Z ) ù
u
= Tê
ú
RT
¶
T
ë
ûV
æ Z ö é¶ (ln Z ) ù
s
= 1+ lnç ÷ + Tê
ú
è
ø
RT
N
ë ¶T ûV
enthaply
Gibbs free energy
é¶ (ln Z ) ù
h
= 1+ Tê
ú
RT
ë ¶T ûV
æZö
g
= -lnç ÷
èNø
RT
Helmholtz free energy
é
æ Z öù
a
= -ê1+ lnç ÷ú
è N øû
RT
ë
specific heat
c v é ¶ æ 2 ¶ (ln Z ) öù
= ê çT
÷ú
R ë¶T è
¶T øûV
D. B. Go
é ¶ æ 2 ¶(ln Z ) öù
cp
= 1+ ê çT
÷ú
R
¶T øûV
ë¶T è
Slide 14
AME 60614
Int. Heat Trans.
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 = å gie
e
- i
3
kB T
i= 0
æ 2pmkB T ö 2
Þ Z tr = ç
÷ V
2
è h
ø
internal energy
é¶(ln Z ) ù
æ u ö
3
ç ÷ = Tê
ú =
è RT øtr
ë ¶T ûV 2
we can plug this in to
our previous equations
entropy
3
5 ù
é
2
2
æ sö
2
p
m
k
T
5
(
)
(
)
B
ú
ç ÷ = + lnê
3
è R øtr 2
h P
êë
úû
specific heat
D. B. Go
æ cv ö
3
ç ÷ =
è R ø tr 2
æ cp ö
æ cv ö 5
ç ÷ =1+ ç ÷ =
è R øtr 2
è R øtr
Slide 15
AME 60614
Int. Heat Trans.
Gases – Monatomic Gases
Where did P (pressure) come from in the entropy relation?
pressure
é¶(ln Z ) ù
é ¶A ù
P = -ê ú = NkB Tê
ú
ë¶V ûT ,N
¶
V
ë
ûT ,N
plugging in the translational partition function ….
é¶ ( ln Z ) ù
P = NkBT ê
ú
¶
V
ë
ûT ,N
the derivative of the
ln(CV) is 1/V
D. B. Go
é
ææ 2p mk T ö3 2 ö ù
B
ê¶ ln çç
÷ú
V
÷
2
çè h
÷ú
ø
ê
è
ø
= NkBT ê
ú
¶V
ê
ú
ê
ú
ë
ûT ,N
é1ù
P = NkB Tê ú
ëV û
ideal gas law
PV = NkB T
Slide 16
AME 60614
Int. Heat Trans.
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
¥
Z el = å gie
e
- ik T
B
i= 0
¥
ei
gie
k T
i= 0 B
Þ Z el¢ = å
internal energy
e
- ik T
B
æ ei ö2 -e i k T
Þ Z el¢¢ = åç
÷ gie B
k T
i= 0 è B ø
¥
entropy
é¶ (ln Z ) ù
æ u ö
Z el¢
ç ÷ = Tê
ú =
è RT øel
ë ¶T ûV Z el
æ sö
Z el¢
+ ln Z el
ç ÷ =
è R øel Z el
specific heat
æ cv ö æ c p ö
Z el¢¢ æ Z el¢ ö
-ç ÷
ç ÷ =ç ÷ =
è R øel è R øel Z el è Z el ø
D. B. Go
2
Slide 17
AME 60614
Int. Heat Trans.
Gases – Monatomic Helium
Consider monatomic hydrogen at 1000 K
…
I can look up electronic degeneracies and energies to give the following table
e
e
level
g
1
0
2
3
229.9849711 2.282E+100
3
0
239.2234393
4
8
5
6
kB T
ge
0
0
e
kB T
ge
kB T
æ
ö2 e
çç e ÷÷ ge kB T
è kB T ø
0
0
5.2484E+102
1.207E+105
0
0
243.2654669 3.564E+106
8.67E+108
2.1091E+111
3
246.2119245 2.5445E+107
6.2648E+109
1.5425E+112
3
263.622928 9.2705E+114
2.4439E+117
6.4427E+119
2
æ
ö
æcp ö
Z el¢¢
Z el¢
-6
=
=
9.91´10
ç
÷
ç ÷
è R øel Z el è Z el ø
D. B. Go
kB T
e
0
æcp ö
5
ç ÷ =
è R ø tr 2
æcp ö æcp ö æcp ö
ç ÷ =ç ÷ +ç ÷
è R ø è R øel è R øtr
5
5æ
kJ ö
kJ
c p = R = ç2.077
=
5.195
÷
2
2è
kg - K ø
kg - K
Slide 18
AME 60614
Int. Heat Trans.
Gases – Monatomic Helium
from Incropera and Dewitt
D. B. Go
Slide 19
AME 60614
Int. Heat Trans.
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
1
1
k = v 2tC = v C
3
3
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
1
dNF
1 dNF
JF = - v 2t
=- v
3
dx
3
dx
general flux of scalar property Φ
D. B. Go
Slide 20
AME 60614
Int. Heat Trans.
Gases – Viscosity and Mass Diffusion
Consider viscosity from general kinetic theory (flux of momentum)  Newton’s
Law
d ( vy )
1 dN ( mvy )
1
1
t yx = -m
=- v
= - v Nm
®m = v r
dy
3
dy
3
dy
3
dvy
Consider mass diffusion from general kinetic theory (flux of mass)  Fick’s Law
J mA
drA
1 d (N V)
1
= -DAB
=- v
® DAB = v
dy
3
dy
3
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 21
AME 60614
Int. Heat Trans.
Gases – Average Speed
The average speed can be derived from the Maxwell-Boltzmann distribution
-e i
-e i
kB T
Ni
gie
gie
= ¥
=
e
i
N
Z
kB T
å gie
f (e) =
kB T
1
æe - mö
expç
÷
è kB T ø
i= 0
We can derive it based on assuming only translational energy, gi = 1 (good for
monatomic gasses – recall that translation dominates electronic)
e
- i
N i e kB T e
=
=
N
Z
p2
e=
2m
p x2 + p y2 + p z2 )
(
-
2mkB T
Z
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
æ 1 ö 2 -( p x + p y + p z ) 2mk T
B
f ( p) = ç
÷ e
è 2pmk B T ø
3
D. B. Go
2
2
2
probability that a gas
molecule has a given
momentum p
Slide 22
AME 60614
Int. Heat Trans.
Gases – Average Speed
From the Maxwell-Boltzmann momentum distribution, the energy, velocity, and
speed distributions easily follow
f (e) = 2
e
p ( kB T )
3
e
3
æ m ö f (v ) = ç
÷ e
è 2pkB T ø
2
-e
(
m v x2 +v y2 +v z2
3
3
2
em = kB T
kB T
)
1
2kB T
2
4 æ m ö 2 2 - mv 2kB T
f (v ) =
ç
÷ ve
p è 2kB T ø
D. B. Go
æ m ö 2 - mv x2 2k T
B
f (v x ) = ç
÷ e
è 2pkB T ø
8kB T
2kB T
vm =
; v mp =
pm
m
Slide 23
AME 60614
Int. Heat Trans.
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)
incident
particle
rincident
cross section defined as:
s = pd 2
General mean free path
1
æ m2 ö 2 1
÷
12 = ç
2
m
+
m
n
p
d
è 1
2ø
2
12
D. B. Go
collision with
target particle
rincident
d12
rtarget
Monatomic gas
=
1
kB T
=
2npd 2
2pd 2 P
Slide 24
AME 60614
Int. Heat Trans.
Gases – Transport Properties
Based on this very simple approach, we can determine the transport properties for
a monatomic gas to be
2 mk B T æ c v ö
k=
÷
32 2 ç
3p d è M ø
2 mk B T
m=
3p 3 2 d 2
2 mk B T
D= 32 2
3p d r
æ cv ö
k = mç ÷
èMø
more rigorous collision
dynamics model
5æ c
ö
k = ç ÷m
2è M ø
v
M is molecular weight
Recall, that
æ cv ö
3
=
ç ÷
è R øtr 2
D. B. Go
5æ3 R ö
k= ç
÷m
2è2 M ø
Slide 25
AME 60614
Int. Heat Trans.
Gases – Monatomic Helium
from Incropera and Dewitt
5æ3 R ö
k= ç
÷m
2è2 M ø
only 2% difference!
3ö
æ
15
-7 8.3145 ´10
k = (199 ´10 )ç
÷
4
4.00
è
ø
-3
k = 155 ´10
D. B. Go
W
m- K
Slide 26
AME 60614
Int. Heat Trans.
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 27