mass and thermal diffusion in binary polyatomic gas mixtures · mass and thermal diffusion in...

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Mass and Thermal Diffusion in Binary Polyatomic Gas Mixtures

D. OMEIRI Laboratoire de Recherche sur la Physico-Chimie des Surfaces et Interfaces

Université de Skikda Route d'El Hadaik, BP 26 Skikda 21000

ALGERIA

D.E. DJAFRI Département de Physique

Université d’Annaba B.P. 12 El Hadjar Annaba 23200

ALGERIA

Abstract: - The mass diffusion coefficient and the thermal diffusion ratio are obtained by mean of Gross-Jackson type kinetic models for binary mixtures of polyatomic gases. These coefficients are functions of collisions integrals, simplified with physical assumptions. Explicit analytic expressions of these coefficients were obtained for Variable Hard Sphere (VHS) and Variable Soft Sphere (VSS) collision models. Finally the models were tested and evaluated for a variety of mixtures and applications to Ar-N2 mixtures are presented. Comparison is made with results derived from a Lennard Jones (12-6) potential model. The contribution of the rotational and vibrational energies of the molecules is also examined. Key-Words: mass, thermal, diffusion, polyatomic, mixtures, rotational, vibrational, VHS, VSS, potential. 1 Introduction The theoretical determination of the transport coefficients appearing in the Navier-Stokes equation governing gas flow is quite an old problem. Under classical conditions (elastic collisions, thermodynamic equilibrium and moderate speed and temperature), Chapman-Enskog type methods elaborated by Chapman and Cowling [1] and Hirshfelder [2] give results that are in good agreement with experiment. The extension of these methods to gases whose structure is more complex is however more difficult because of the inelastic collisions particularly when non equilibrium conditions prevail. Hirshfelder [3] Mason and Monchick [4] [5] obtained formal results for polyatomic gases and mixtures of polyatomic gases. Wang-Chang and Uhlenbeck [6] developed different methods. Models for the linearized collision operator of the Boltzmann equation were constructed first for elastic collisions by Gross-Jackson [7]. They made use of the asymptotic properties of the Maxwellian operator eigenvalues and they introduced a diagonalization constant at every approximation level. Then, for polyatomic gases, Hanson and Morse [8] introduced the W.C.U. polynomials in the development of Gross-Jackson. Brun and Zappoli [9] and Brun and Philippi [10] extended these models to nonequilibrium regimes and to mixtures

of polyatomic gases respectively. The mass and thermal diffusion coefficients are used in numerous studies of the flow of mixtures of polyatomic or polyatomic and monoatomic gases. Yet, little is known about them. The purpose of our work is to derive and test explicit relations expressing them. We choose to make use of previously developed models. Because of the complexity of the problem we used Monchick and Mason type approximations [11] to get simpler expressions. In the first part, the method of Gross-Jackson to build kinetic models is presented. N-order kinetic models corresponding to N-order hydrodynamic models are obtained. We get a closed system of equations for the moments distribution function enabling us to get the transport coefficients. These coefficients are functions of collision integrals that depend on the molecular collisions dynamic. Because the expressions are difficult to derive, simplifications had to be made. Among them, the collision integrals are expressed as functions of known or experimentally attainable macroscopic quantities. The second part of this work deals with these approximations. In this part, we are seeking expressions relating collision integrals to such macroscopic data as viscosity, self-diffusion coefficient and phenomenological relaxation time. Furthermore, internal

Proceedings of the 2006 WSEAS/IASME International Conference on Heat and Mass Transfer, Miami, Florida, USA, January 18-20, 2006 (pp78-84)

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energy terms are treated as a perturbation in order to separate internal parts from translational parts in the transport coefficients. Then, the coefficients are expanded in a descending series of powers of the corresponding relaxation time, retaining only first order terms. In the last part, the mass diffusion coefficient and the thermal diffusion factor are computed for different gas systems where the Variable Hard Sphere (VHS) of Bird [12] and Variable Soft Sphere (VSS) of Koura and Matsumoto [13] molecular potentials were used. As a check of the validity of our models, a comparison is made with theoretical results obtained from a Lennard Jones (12-6) potential model. 2 Wang-Chang and Uhlenbeck equation - 3rd order moments Let ),,,( tEff ipip rvp= be the distribution function where:

• pi designates the internal state of molecule p,

• pv is its velocity,

• ipE is its internal energy and r is its position at time t .

The W.C.U. equation for component p in the absence of external forces is:

( )

qpq

p

vg

rv

dddI

fffff

tf

klij

jkljqiplqkp

q

ipip

ϕχχsin×

−′′=∂

∂+

∂

∂∑ ∫∫∫∑ (1)

where ip and jq represent the internal states of molecules p and q before the collision and kp and lq their states after the collision, χ and ϕ are the polar and azimuthal angles which define the direction of pqg′

relative to pqg , klijI is the collision cross section and

pqpq vvg −= is the relative velocity of the molecules before collision. We are seeking a solution of the W.C.U. equation under the form )1()(

ipo

ipip ff Φ+= .Using the method of Gross-Jackson we get for the mass flow:

( ) [ ]

( )

[ ] ( )r

rr

jp

∂∂

−

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛−+⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

×

−=

=

==

TD

pmn

nn

nn

Dmmn

NT

pppp

NpqqpN

ln

ln

3

3

2

3

ρ

ρ

(2)

where n is number of particules per unit volume, pm

and qm are the masses of molecules p and q respectively and ρ is the density of the mixture. [ ]

3=NpqD is the third order mass diffusion coefficient in

the Gross-Jackson approximation and is written as a ratio of determinants 5544 ×× [10]. [ ] 3=N

TD is the third order thermal diffusion coefficient and is also written as a ratio of determinants 5555 ×× [10].

Monchick and Mason [5] found it to be a 6677 ×× ratio. The elements of these determinants are complicated functions of the collision integrals and can be found in Brun and Philippi [10]. When there are no inelastic collisions, these coefficients reduce to the second Chapman-Enskog approximation of the mass and thermal diffusion coefficients [1]. 3 Physical assumptions and evaluation of the collision integrals The introduction of the relaxation time, the internal energy diffusion coefficient and the properties of the pur components has enabled us to express the collision integrals (see appendix 6.3) as functions of known quantities. The expressions of the transport coefficients obtained on the basis of these hypotheses do not allow comparison with known results from elastic theory. Thus, we were lead to suppose that the inelastic terms are negligible compared with the elastic ones. These coefficients have been written as a sum of translationnal and internal contributions. The contribution of the translational energy arises from the elastic collisions and that of the internal energy arises from the rotation and vibration of the molecules. In our work, we made the following assumptions:

1. The third order kinetic model derived from a Gross-Jackson method has been used to calculate the coefficients [ ]

3=NpqD [ ] 3=NTD .

2. The dynamic viscosity coefficients pqη and pη

and the diffusion coefficients PPD and pqD were evaluated with the help of the first Chapman-Enskog approximation.

3. With the exception of resonant collisions, all "complex collisions" have been neglected.

4. In order to eliminate some collision integrals, we assumed that there is no correlation between internal and translational energy modes.

5. The analytic form of some collision integrals and the first order transport coefficient were


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obtained by mean of a variable soft sphere (VSS) potential model. This model takes into account the variation of the molecule diameter with the temperature. The VSS model is well suited for The Direct Simulation Monte Carlo (DSMC) method because of its simplicity and because it is a realistic intermolecular potential.

6. The internal energy diffusion coefficients intpqD and int

qpD are approximated by the ordinary

diffusion coefficient pqD and intppD is

approximated by the self diffusion coefficient ppD .

7. *pq

*pq B,A and *

pqC are calculated from the values corresponding to elastic collisions only.

8. For the rotational relaxation collision number rotpqZ and rot

ppZ , we used Parker's formula [14], [15]. The vibrational relaxation collision number

vibpqZ and vib

ppZ are evaluated from the experimental data of Millikan and White [16].

4 Calculations and Results The thermal diffusion factor Tα is defined as:

[ ] [ ] 112321

/ === NNT

T DDρρρα (3)

[ ] 112 =ND corresponds to the first order approximation of the binary diffusion coefficient in the Chapman-Enskog expansion, 1ρ , 2ρ and ρ are the densities of species 1, species 2 and of the mixture respectively. Tα can be put under the form:

intααα += transT (4)

transα is the contribution of the translationnal energy and

intα that of the internal energy (rotation and vibration): see appendix 6.1. Also:

TT XXK α21= (5)

1X and 2X are the mole fractions of species 1 and species 2: see appendix 6.2.The contribution of the internal energy to the mass diffusion coefficient is small (Monchick and Mason [5]) so [ ] 312 =ND will be approximated by [ ] 212 =ND defined in appendix 6.3 where only translationnal energy exchanges are accounted for. The transport coefficients are functions of collision integrals that depend on the molecular collision dynamics. We have calculated analytically these

integrals for the VHS and VSS potentials with the help of "Mathematica" software on a Pentium 4, 3.0GHz, 1Gb RAM. The computing times were very short and the analytic expressions thus obtained are given in appendix 6.4. We have also computed numerically these integrals for the Lennard-Jones (12-6) potential. The computing times turned out to be very long: 2000 seconds on the average for each value of the temperature. In the last part, we applied the theory to an Ar-N2 mixture for VSS and Lennard-Jones (12-6) potential models. Fig.1 shows [ ] 212 =ND as a function of the temperature in a 50% Ar-50% N2 mixture for VSS and Lennard-Jones potential models. We see that at low temperature the discrepancy between the two models is small (of the order of 3%) but increases with temperature and reaches 30% at 8000°K. Fig.2 shows the thermal diffusion ratio

TK at 300°K as a function of the mixture composition for the VSS and Lennard-Jones potential models. We see that the contribution of the internal energy is of the order of 12% and that the two models are in good agreement. Fig.3 represents TK as a function of the Ar mole fraction at high temperature (8000°K): this is the strong nonequilibrium case (SNE). There, the vibration energy contribution is important and even dominates at high temperature. We note that in this case, our kinetic models overestimate the vibration energy contribution and that there are important discrepancies between the two models. Fig.4 and Fig.5 show TK as a function of the temperature for a 50% Ar-50% N2 mixture for the VSS model. It is seen that the rotation energy contribution varies little with temperature and is of the order of 12% while in Fig.5 we note that the vibration energy contribution is negligible at low temperature (up to 3000°K) but dominates at high temperature. These results indicate that the weak nonequilibrium model (WNE) overestimates the vibration energy contribution. Finally in Fig.6, we have shown various plots of TK as a function of the temperature for the VSS and Lennard-Jones potentials. The difference transtot KK − is the internal energy contribution. The results obtained from the two models are in good agreement for transK . 5 Conclusion In this work we used the generalized method of Gross-Jackson to derive formal expressions of the thermal and mass diffusion coefficients taking into account the internal energy of the molecules for polyatomic mixtures. The main assumption that we made was to regard the internal energy as a perturbation. We also gave analytic expressions of the collision integrals for the VHS and VSS models. These coefficients were


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expressed in terms of measurable quantities such as the viscosity coefficient, the self diffusion coefficient of the constituents, the relaxation time and other various physical properties of the constituents. Numerical values of these coefficients have been computed for low and high temperature taking into account internal energy (rotation and vibration). We have shown that for the thermal diffusion coefficient, the internal energy represents an important part and cannot be neglected. As far as rotation is concerned, it is almost constant and is of the order of 10% at low and high temperature. Vibration is negligible at low temperature but dominates at high temperature. We have also shown that for the thermal diffusion coefficient, vibrational energy exchanges play an important role at high temperature. At very high temperature the Gross-Jackson kinetic model overestimates the contribution of the vibrational energy to this coefficient . In this model we have assumed a weak vibrational nonequilibrium [17] (one temperature model) In the case of strong nonequilibrium conditions (SNE), a multitemperature formalism should be used. 6 Appendix 6.1 Thermal diffusion factor The thermal diffusion factor Tα may be written as a sum of two terms, transα due to the translational energy and intα due to the internal energy (rotational and vibrational)

intααα += transT Where :

( ) ( )12212

221

21

221112 56

QXXQXQXSXSX

Ctrans ++−

−= ∗α

qP

pp MM

MM

+=

11122212221 PMEMMEMS ++−=

22122

121112 PMEMMEMS ++−= ( )11211 34 ηRTpDMP = , ( )21222 34 ηRTpDMP =

)1,1(163

pqpqpq n

kTDΩ

=µ

, )2,2(85

pqpq

kTΩ

=η

∗∗ −−= 121212 516

51211 ABE

∗∗ +−+= 12211222

21

2211 5

165

1265 AMMBMMME

∗∗ +−+= 1221122

122

2122 5

165

1265 AMMBMMME

2211 EPQ = , 1122 EPQ = ,

212

1222

21221112 PPEMMEEQ +−= ,

)1,1(

)2,2(

21

pq

pqpqA

Ω

Ω=∗ ;

( ))1,1(

)3,1()2,1(531

pq

pqpqpqB

Ω

Ω−Ω=∗ ,

( )

( )1,1

3,1

31

pq

pqpqC

Ω

Ω=∗

Let :

12212221

211 QXXQXQXB ++=

The contribution of the internal energy intα to the thermal diffusion factor is:

( )( ) ( )

( )⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛+

−−

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛ ++

+

−=

∗

∗

121

2

22

1

122

2

21

1

111

12

121

1

4223122

1

4123111

12int

56

56

BXXYR

YRV

YR

YRV

C

BXXB

VUVUV

BVUVU

VCα

where:

222212222122

21211 EMXXPMXEMMXXV −−=

11121112

112122212 EMXXPMXEMMXXV ++−=

1221222213 EMMPXEXV ++=

1221111124 EMMPXEXV ++=

RZC

ZC

ZC

ZC

MMAXXU

vib

vib

vib

vib

rot

rot

rot

rot 1

1564

21

2

12

1

21

2

12

1

21122112

⎟⎟⎠

⎞⎜⎜⎝

⎛+++×

⎟⎠⎞

⎜⎝⎛−= ∗

π

RZC

ZC

PXU vib

vib

rot

rot 138

11

1

11

11

211 ⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟

⎠⎞

⎜⎝⎛−=π

RZC

ZCPXU vib

vib

rot

rot 138

22

2

22

22

222 ⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟

⎠⎞

⎜⎝⎛−=π

11211 UUU += , 21222 UUU +=

⎟⎠⎞

⎜⎝⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+−= ∗

RZC

ZC

MAXXR vib

vib

rot

rot

518

12

1

12

11122112

⎟⎠⎞

⎜⎝⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+−= ∗

RZC

ZC

MAXXR vib

vib

rot

rot

518

21

2

21

22122121

⎟⎠

⎞⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+−=

RZ

C

Z

CPXRR

vib

vib

rot

rot 1

11

1

11

11

211211


5

⎟⎠⎞

⎜⎝⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+−=

RZC

ZC

PXRR vib

vib

rot

rot 1

22

2

22

22

222122

211

1211 X

DDXY += , 1

22

1222 X

DDXY +=

6.2 Mass diffusion coefficient The contribution of the internal energy to the diffusion coefficient is negligible. The coefficient [ ] 312 =ND can be replaced by the second Chapman-Enskog approximation [ ] 212 =ND .

[ ]∆−

== 112

212DD N

where :

( )( )1

12212221

21

12 56B

PXXPXPXC ++−=∆ ∗

and

1222

2122

2211

2112 2 EMMEMEMP −+=

The quantities 1P , 2P , and 1B are defined as before. 6.3 Collision integrals sl

qp,,Ω

pqpqls

pqpq

slpq dgQekT pq γγ

πµγ )(

2)(

0

32),( 2

∫∞

+−=Ω

Here ( ) 212 2kTg pqpqpq µγ = , pqg s the initial relative

speed of the colliding molecules, and ( )lQ is the total collision cross section:

( )bdbQ ll ∫∞

−=0

)( cos12 χπ

χ is the deflection angle:

( )( )∫

∞

−−

−=mr

pqpq g

rrb

rdr

bbg

22

2

2

21

12,

µ

ϕπχ

( )rϕ is the interaction potential, and b , the impact parameter, is the distance of closest approach in the absence of the potential ( )rϕ .

mr is the positive root of the equation:

( ) 021

21 22222 =−− bgrrrg pqpqpqpq µϕµ

and pqµ is the reduced mass of the colliding molecules.

6.3.1 Collision integrals for the intermolecular VSS potential The collision integrals sl

qp,,Ω for the VSS model were

calculated with "mathematica" software and they are given by :

( ) ( ) ( )( )

( )

( ) ( )

( )

( ) ( )( ) 1333

121

3

721

321,1

18

5

2

−

++−

++−

++

⎟⎟⎟

⎠

⎞

++−⎟⎟⎠

⎞⎜⎜⎝

⎛ +×

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛

+

++−=Ω

pqqp

qpqp

qppqrefq

qp

qp

qp

qppqrefprefpq

mmTk

mm

mmkTm

mmkTmm

mmmmkT

mdd

qp

qp

α

ωω

π

ωω

ωω

( ) ( ) ( )( )

( )

( ) ( )

( )( )

( ) ( )( )( ) 1

12124

2144

212112

121

21

211242

921

21

21

21

2141

221

2,212

218

75

2

21

21

−

++−

++−

+++×

⎟⎟⎟

⎠

⎞++−++−⎟⎟

⎠

⎞⎜⎜⎝

⎛ +

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛+

++=Ω

αα

ωωωωα

π

ωω

ωω

mmTk

mmmmkT

m

mmkTmm

mmmmkTmdd

ref

refref

( ) ( ) ( )( )

( )

( ) ( )

( )( )

( ) ( )( )( ) 1444

121

4

921

422,1

218

75

2

−

++−

++−

+++×

⎟⎟⎟

⎠

⎞++−++−⎟

⎟⎠

⎞⎜⎜⎝

⎛ +

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛

+

++=Ω

pqpqqp

qpqppqqp

qppqrefq

qp

qp

qp

pppqrefprefpq

mmTk

mm

mmkTm

mmkTmm

mmmmkT

mdd

qp

qp

αα

ωωωωα

π

ωω

ωω

( ) ( ) ( )

( )( ) ( ) ( )

( )( )( ))( ) ( )( ) 1555

121

5

1121

523,1

132

975

2

−

++−++−

++×

++−++−++−×

⎟⎟⎠

⎞⎜⎜⎝

⎛ +⎟⎟⎠

⎞⎜⎜⎝

⎛

+×

⎜⎜

⎝

⎛ ++=Ω

pqqp

qpqpqp

qp

qppqref

qqp

qp

qp

qppqrefprefpq

mmTk

mm

mmkTm

mmkTmm

mmmmkT

mdd

qpqp

α

ωωωωωω

π

ωωωω

6.3.2 Collision integrals for the intermolecular VHS potential For the VHS model the collision integrals are the same as in the VSS model except that pqα is set to 1. 6.4 Collision models 6.4.1 The Hard Sphere (HS) collision model The differential cross-section ΩId for the collision specified by the impact parameter b and ε is defined by εbdbdId =Ω .


6

Where εχχ ddd sin=Ω is the element of solid angle. So that ( ) χχ ddbbI sin= . Finally, the total collision

cross-section Q is:

χχπππ

dIIdQ ∫∫ =Ω=0

4

0

sin2

Consider a collision between a molecule of species p and one of species q . The effective diameters are

pd and qd . The collision becomes effective at

( )qppq ddd +=21

. For the hard sphere model, the

impact parameter b is related to the deflection angle χ by:

( ) ⎟⎠⎞

⎜⎝⎛== χθ

21cossin pqmpq ddb

So that 2

41

pqdI = and 2pqdQ π= .

6.4.2 The Variable Hard Sphere (VHS) collision model This model, developed by Bird in 1981 [12], uses cross sections that are functions of the relative velocity, but with a hard sphere (HS) scattering angle.

( )( )( )

21

21

2

252

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−Γ=

−

pq

pqpqrefpqrefpq

pqgkTdd

ωµ

ω

where:

( )qrefprefpqref ddd +=21

Γ is the Euler's Gamma function, prefd and qrefd are the diameters of molecules p and q at temperature

refT , and pqω is the viscosity temperature power law.

The pqd diameter in the VHS case is adjusted in such a way so as to produce a viscosity coefficient proportional to pqT ω ( 5.0=pqω for a hard sphere gas and 1= for a Maxwell gas). As in the HS model, the scattering angle χ is given by:

( )pqdbCos 1−=χ 6.4.3 The Variable Soft Sphere (VSS) collision model The VSS model, developed by Koura and Matsumoto [13] in 1991, is a generalization of the VHS model in

which the diameter varies in the same way as in the VHS model, but with a deflection angle χ given by:

( )[ ]pqpqdbCos αχ 112 −=

The parameter pqα is used to characterize the anisotropy of the scattering angle. It is set to 1 if the VHS model is used, and it is set to the appropriate value of the Schmidt number of the gas if the VSS model is used. References: [1] S. Chapman and T.G. Cowling. The Mathematical Theory of Non-Uniform Gases. Cambrige University Press, Cambridge England, 1970. [2] J.O. Hirchefelder, C.F. Curtiss, and R.B Bird. John Wiley, New York, 1954. [3] Hirchfelder J.O, editor. Proceedings of the joint conference on thermodynamic and transport properties of fluids. Institute of Mechanical Engineers London, 1958. [4] E.A. Mason and Monchick L. Heat Conductivity of polyatomic and polar gases. Journal of Chemical Physics, 36(6): 1622-1637, March 1962. [5] L. Monchick, K.S. Yun, and E.A. Mason. Heat conductivity of polyatomic gas mixtures. Journal of Chemical Physics, 39(3), 1963. [6] C.S. Wang Chang and G.E. Uhlenbeck. Studies in Statistical Mechanics, volume 5. North-Holland, 1970. The Kinetic Theory of Gases. The Dispersion of Sound in Monoatomic Gases. [7] E.P. Gross and E.A. Jackson. Kinetic model for binary gas mixture. Physics of Fluids, 2(4), 1959. [8] F.B. Hanson and T.F. Morse. Kinetic model for gases with internal degrees of freedom. The Physics of Fluids, 7(2):159-169 freedom. The Physics of Fluids, 7(2):159 [9] R. Brun and B. Zappoli. Model equations for vibrationally relaxing gas. The Physics of Fluids, 20(9):1441-1448, Sept. 1977. [10] P.C. Philippi and R. Brun. Kinetic modeling of polyatomic gas mixtures. Physica A, (105): 147-168, 1981. [11] L. Monchick, A.N.G. Pereira, and E.A. Mason. Heat conductivity and thermal diffusion in polyatomic gas mixtures. Journal of Chemical Physics, 42(9), 1965. [12] G. A. Bird. Monte Carlo simulation in an engineering context. Progress in Astronautics and Aeronautics, (74):239-255, 1981. [13] K. Koura and H. Matsumoto. Variable Soft sphere molecular model for air species. Physics of Fluids A, 4:1083-1085, 1992. [14] J.G. Parker. Rotational and vibrational relaxation in diatomic gases. Physics of fluids, 2:449-462, 1959. [15] J.D. Lambert. Vibrational and Rotational Relaxation in Gases. Clarendon, Oxford,1977.


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[16] D.R. Millikan, R.C.and White. Systematics of vibrational relaxation. Journal of Chemical Physics, 12(39):3209, 1963. [17] S. Pascal and R. Brun. Transport properties of nonequilibrium gas mixtures. Physical Review E, 47(5):3251-3267, May 1993.7.

Temperature (K)

Mas

sD

iffus

ion

(m2 s-1

)

2000 4000 6000 8000 100000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

0.011

0.012

[D12]2 VSS[D12]2 LJ

Ar-N2 mixture XAr =0.5

Fig.1: [ ] 212 =ND in Ar-N2 mixture, 5.0=ArX

Mole fraction of Ar (%)

Ther

mal

diffu

sion

ratio

(%)

0 0.25 0.5 0.75 1

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

Ktrans L.J.Ktot L.J.Ktrans VSSKtot VSS

Ar - N2 mixture T=300K

Fig.2: TK in Ar-N2 mixture, KT 300=

Mole fraction of Ar (%)

Ther

mal

diffu

sion

ratio

(%)

0 0.25 0.5 0.75 10

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0.11

0.12

Ktrans L.J.Ktot L.J.Ktrans VSSKtot VSS

Ar - N2 mixture T=8000K

Fig.3: TK in Ar-N2 mixture, KT 8000=

Temperature (K)

Ther

mal

diffu

sion

ratio

(%)

2000 4000 6000 8000 100000.013

0.0135

0.014

0.0145

0.015

0.0155

0.016

0.0165

0.017

0.0175

0.018

0.0185

0.019

0.0195

0.02

KTtransKTtrans + KTrot

Ar-N2 mixture XAr=0.5 VSS potential

Fig.4: TK in Ar-N2 mixture, 5.0=ArX VSS potential (translation and rotation only)

Temperature (K)

Ther

mal

diffu

sion

ratio

(%)

2000 4000 6000 8000 100000

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

KTtransKTtrans + KTrotKTtrans + KTrot + KTvib

Ar-N2 mixture XAr=0.5 VSS potential

Fig.5: TK in Ar-N2 mixture, 5.0=ArX VSS potential (translation, rotation and Vibration)

Temperature (K)

Ther

mal

diffu

sion

ratio

(%)

2000 4000 6000 8000 10000

0

0.05

0.1

0.15

0.2

0.25

0.3

KTtrans (LJ)KT tot (LJ)KTtrans (VSS)KT tot (VSS)

Ar-N2 mixture

Fig.6: TK in Ar-N2 mixture, 5.0=ArX VSS and L.J. 12-6 potentials


mass and thermal diffusion in binary polyatomic gas mixtures · mass and thermal diffusion in...

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