Download Internal energy relaxation and chemical reactions.

Internal Energy Relaxation Models in
DSMC
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For a gas in thermal equilibrium, individual
molecules are constantly gaining and losing
energy through collisions, but the total change in
energy is zero.
Establishment of equilibrium takes some time.
We've looked at the DSMC modeling of
approach to equilibrium for a system of hard
sphere molecules initially all having the same
speed. The time it took the distribution function
to approach Maxwellian was several mean
collision times.
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Diatomic and polyatomic molecules possess internal
energy and this energy can also be exchanged in
collision in addition to translational energy.
In equilibrium, the total thermal energy is equipartioned:
each energy mode of a molecule on average has
energy proportional to the number of degrees of
freedom times temperature:

E= k T
2
Since molecules move in 3D physical space,
translational energy has 3 degrees of freedom tr =3 .
Diatomic molecules has 2 degrees of freedom for
rotational energy: rot =2 . Non-linear polyatomic
molecules have rot =3.
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Diatomic molecules approximated as harmonic oscillators
2 v /T
have
vibrational degrees of freedom.
vib=
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exp  v / T −1
The equation describing the time rate of approach to
equilibrium of a given mode of internal energy is of the form
d E E eq −E
=
dt

where  is the relaxation time (time required for the deviation
from equilibrium energy to fall to 1/e of its initial value).
Number of collisions needed for a particular internal mode to
approach equilibrium can be approximated as

Z=
col
where
col is the mean collision time.
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Rotational relaxation numbers were calculated for
diatomic molecules (see, e.g., J. G. Parker, “Rotational
and Vibrational Relaxation in Diatomic Gases”, Phys.
Fluids, Vol. 2, No. 4, 1959).
For nitrogen and oxygen at room temperature rotational
collision number Z ≈3−5.
rot
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Rotational collision number decreases with temperature.
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Vibrational collision numbers for diatomic molecules at
moderate temperatures (<3,000 K) can be
approximated as
Z vib=C 1 exp C 2 / T 1/ 3 
where C1, C2 are positive constants that depend on the
physical properties of the gas. The values of the
constants based on experiments of Millikan and White
are (Table A6, Bird): C1=9.1, C2=220.0 for N2.
Larsen-Borgnakke Model
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Larsen-Borgnakke model is used most often to model the
energy exchange between translational and internal modes in
DSMC.
This model assumes that only a fraction 1/ZDSMC of all
collisions are inelastic. If collision is regarded as inelastic, then
the post-collisional internal and translational energies of
molecules are sampled from the equilibrium distribution.
In order to reproduce the correct rates of approach to
equilibrium, the following collision numbers are used:
Z rot , LB =
●
 tr
 tr  rot
Z rot
Z vib ,LB =
 trrot
 tr  rot  vib
Z vib
Thus, the translational-rotational energy exchange is modeled
in a collision if
1
Rf
Z rot , LB
LB Energy Redistribution in Collision
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If the collision is to be modeled as inelastic, then
the post-collisional energy is chosen from the
equilibrium distribution corresponding to the total
collision energy of the pair.
The total collision energy is
E c= E tr E int
1
E tr = mr v2r
2
E int= E rot ,1 E rot ,2 E vib ,1E vib ,2
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For VHS model, the equilibrium distribution
function of translation energy of a colliding pair is
3/ 2−
f  E tr ∝ E tr
●
exp−E tr /kT 
Equilibrium df for internal energy of a colliding
pair is
 /2

f  E int ∝E int exp −E int / kT 
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Then the joint distribution function of translational
and internal energy in equilibrium is
3/2−
f  E tr , E int ∝E tr
∝E

 /2
E int exp− E tr E int/ kT 
3/2−
tr
 E c− E tr 

/ 2
exp−E c / kT 
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The post-collisional internal energy of the pair
can be sampled from this distribution by the
acceptance-rejection method.
Now this post-collisional internal energy needs to
be divided to two molecules.
The joint distribution function of the internal
energy in equilibrium for two molecules is
f E
●
*
int ,1
,E
*
int ,2
∝ E
*
 1/ 2
int , 1

E
*
 2/ 2
int ,2

*
int
exp−E / kT 
Finally, the post-collisional internal energy of
molecule 1 is sampled from the distribution
*
*
 1/ 2
f  E int ,1 ∝ E int ,1 
*
E int −E int ,1 
2 /2
Chemical Reactions
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A typical bimolecular reaction may be written as
AB C D
CD  A B
where A, B, C, D represent separate gas species.
Rate equation for the change of number density of species A
may be written
d nA
−
=k f T n A n B −k r T  nC n D
dt
The parameters kf(T), kr(T) are rate coefficients for forward
and reverse reaction, respectively, of the form
k T =T b exp−E a / kT 
where  ,b are constants and Ea is called activation energy.
Total Collision Energy (TCE) Model
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TCE model in DSMC allows to reproduce the Arrhenius
rate of a chemical reaction in equilibrium.
For a colliding pair of A and B molecules, the probability
of reaction is based on the total collision energy
E c= E tr E int
The probability of a reaction is set to zero, if Ec < Ea.
Else the probability of reaction is

Pr=C 1  E c− E a C 1−E a / E c  3/2−
2
where C1, C2 are related to Arrhenius rate parameters
1/ 2
1−

m
T




5
/2−

r
ref
C 2 =b−1
C=


1
2
 b 3/ 2 2k T ref
2 d ref  
k
b−1