School of Aerospace Engineering
Boltzmanns Relation
Relation between entropy and statistics
S = k ln
(I.15)
total number of unique ways to divide
total energy into given number of
molecules (N), each in a quantum state
If number of molecules (N) is lar
06/28/2012
School of Aerospace Engineering
Example: Nonequil. Normal Shocks
Assume tpg, 1d, steady, adiabatic, no body forces
In total derivative, there is still a
convective term
Mass
V.1
Mom.V.3
Species V.2
0 > steady
uj
D
Dt
0
xj
Du j
Dt
DYm
Dt
t
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Vibrational Nonequilibrium
Start by examining vibrational mode/energy
typically an important nonequilibrium mode
can develop analytic model for harmonic oscillator
Given
some vibrational population distribution, i.e., Nv=0,
School of Aerospace Engineering
Continuum (Classical) Gas Dynamics
NavierStokes Equations
Convection
viscous conservation (transport) equations
Mass/continuity
t
Rate of Change of
Momentum
Momentum
Physically pressure is
ux of momentum
pressure tenso
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Molecular Interpretation of
Transport Terms
Compare Molecular Gas Dynamic equations with
classical (NS) equations
MGD
NS
Mass
t
xj
cj
0
t
cj
uj
xj
0
uj
group velocity is fluid particle velocity
AE6050
Transport Property
For the derivation, begin by
a ssuming monatomic gases like Ar
with the Translational mode in
n onequilbrium
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Molecular Conservation Equations
Conservation equations must also hold for
molecules
i.e., flux through control
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Equilibrium Conditions
Consider gas in complete equilibrium (and no
external field applied, !"=0)
.
,# %&
%"
"
,# %&
%"
!" ,# %&
,2
# %& # '&
# %& # '& *(+% ()'
Then all LHS terms should be zero
#$(%&) not changing in s
School of Aerospace Engineering
Radiative Energy Transfer in Gases
Radiation significant source of heat transfer from (to) gases
at high temperature (and low density)
Radiation different than molecular collisions because gas
can exchange energy with oth
Once we have the distribution functions,
then we have Tau_ij and qj
School of Aerospace Engineering
Transport Models for NS Equations
From previous comparison of NS and molecular
conservation equations we had
!"
$!$ "
#
!
$ "$ "
"
%"
!"
,$ "
%&'
So to p
How to solve BE in the perturbation limit
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ChapmanEnskog Solution
Goal is to solve Boltzmann equation for small
departure from equilibrium velocity distribution
f close to fo
corresponds to weak gradients in flows
will
School of Aerospace Engineering
Radiative Energy Transfer in Gases
Radiation significant source of heat transfer from (to) gases
at high temperature (and low density)
Radiation different than molecular collisions because gas
can exchange energy with oth
School of Aerospace Engineering
Radiative Properties of Gases
Want state equations for radiative properties
Focus on absorption and emission
neglect scattering
If matter absorbs/emits radiation, it must change its
energy, e.g. internal energy
h
hc
ij
School of Aerospace Engineering
Chemical Nonequilibrium
Nonequilibrium chemistry, e.g., we can not use Kp to find
composition
need to find time rate of change of composition
Chemical reactions are still
collisions. Only electrons
a round a nucleus have
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Nonequilibrium Inviscid Flows
Need to incorporate (collision) rate equations to
inviscid equilibrium flow equations
Assumptions: to use the rate equations we
previously developed (e.g., for vibration and
chemical modes),
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Thermodynamic Properties
Start with Boltzmanns Relation (S Stat. Mech)
S = k ln
Q E
= Nk ln + 1 +
N T
Use results from classical thermodynamics that
relate other properties to entropy, e.g., Gibbs eqn.
S
S
dS =
dE +
E V
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Review of Compressible Flow
In continuum fluid mechanics, often start by
considering conservation or transport equations,
e.g.,
mass
momentum
In compressible flow, must include energy equation
kinetic energy of flow ca
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Kinetic Theory
Approach to understanding gas properties (both
equilibrium and rates) by examining
translational motions of molecules and
their interactions or collisions
Too many molecules to follow them all
individuall
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Partition Functions
Assume energy levels of molecule can be divided
into independent modes
= tr+rot+vib+elec+
Since partition function is defined as
Q = g i e i
kT
= (g i,tr g i,rot )e
=
( g
i, tr e
i, tr kT
( i, tr +
School of Aerospace Engineering
State Relations
We have showed that
TD properties can be calculated based on partition functions
we have models for partition functions of simple molecules
So, could now calculate thermodynamic properties, i.e., state
r
Equilibrium Composition of Air
Mole Fraction
1
0.1
N2
O2
1.0 atm
0.01
NO
0.001
N
O
0.0001
1000
2000
+
NO , e
3000
4000
Temperature (K)
5000

6000
Equilibrium Composition of Air
Mole Fraction
1
0.1
N2
O2
0.01 atm
0.01
NO
0.001
N
O
0.0001
1000
2000
+
NO ,
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Equil. Properties of Reacting Gas Mixtures
So far have looked at Statistical Mechanics results for
a single (pure) perfect gas
shown how to get gas properties (p, e, h, cv, s, , )
from partition function (Q)
For nonreact
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Dissociation of Homonuclear Gas
(N ) = (Q )
*A 2
N
A2
* A2
A2
Q
e D
T
(II.6)
A
Degree of dissociation N
*
m (Q A )
=A
e
1 * 2 V Q A
2
# of atoms of species A
NA
total # of A nuclei
Y A=
2
D
T
2
(II.7)
Characteristic dis
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Engineering
kf=CfT eK /RT
f
Air Mechanism1
Copyright 2001 by Jerry M. Seitzman. All
rights reserved.
f
Table from Anderson, Hypersonic and
High Temperature Gas Dynamics
AE6050
School of Aerospace Engineering
Example: High Temperature Air
Goal
determination of equilibrium composition of dry air
at elevated temperatures
Standard atmospheric (near sea level) composition
Gas
N2
Mole
Fraction
78.084
%
O2
Ar
20.95 0.934
%
%
CO 2
N
School of Aerospace Engineering
Nonequilibrium Processes
So far we have examined flows in two limits
<
coll >
coll
flow
flow
Equilibrium Flow
Frozen Flow
What happens in between?
coll
~ O(
flow)
Nonequilibrium Flow
properties not given by equilibrium stat
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Absorption Coefficient
Aji
Recall for single
molecular transition
j
i
e.g., LTE
All the transitions can occur
a t various frequencies.
Recall the integral of Phi
o ver all possible
frequencies is 1
The linestrength Sij is