Sharif University of Technology School of Mechanical Engineering.

Sharif University of Technology

School of Mechanical Engineering

Contents

Introduction

Kinetic Theory of Gases

Thermal Conductivity of Dilute Gases

Thermal Conductivity of Dense Gases

Thermal Conductivity of Liquids

Conclusion

Introduction

The object of theoretical attempts

Kinetic theory of dilute gases

Dense gases and liquids

The equation of state and the equation of change

The transport coefficients

Kinetic Theory of Gases

The history of kinetic theory

Assumptions of ultra-simplified theory

The molecules are rigid and non-attracting sphere. All the molecules travel with the same speed . The volume of molecules is negligible.

Ludwig Boltzmann(1844-1906)

James Clerk Maxwell(1831-1879)

Maxwell distribution function2/1

mkT8

2

t2

t

t2n31

t

t2n61 22

2p2

kTL

Mean free path :

mkT/8p276.1n276.1 22 Rate of collisions :

mkT/8p2 2 For Maxwellian distribution :

Ultra-simplified Theory

Collision rate

L

LO

A

B

32

238 For Maxwellian distribution :

dz

dPLPP OA

dzdP

LPP OB If P is the property :

2BAPnm

mkTdzdP

L31

)PP(61

So :

Ultra-simplified Theory

22ppmkT

dzdvmkT

nmvP

For viscosity :

mcmkT

dzdT

mcmkT

TncP v2

v2qvq

For thermal conductivity :

vcm1 So it can be written : vC

Ultra-simplified Theory

vCf

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0

0 100 200 300 400 500

Temperature (K)

Dev

iati

on (

%)

He Ar N2

Ultra-simplified Theory

Deviations of thermal conductivity of various gases calculated with ultra-simplified kinetic theory from experimental values.

Intermolecular potential function

r

dr)r(F)r(

Empirical intermolecular potential function

r0

r)r( Rigid Impenetrable Spheres :

612

rr4)r( Lennard – Jones Potential :

Rigorous Kinetic Theory

ji m1

m11

Reduced mass :

Rigorous Kinetic Theory

i

ji2

2

2j

2

ji

2i

2

ii

)(dt

d

dt

dm

dt

dm

Frr

rF

rF

)r()rr(21

g21 2222

Conservation of energy :

The relation for r as a function of time :

)r()r/b(g2

1r

2

1g

2

1 22222

Rigorous Kinetic Theory

2rbg

Conservation of angular momentum :Impact parameter

m2

Angle of deflection :

mr222

2

)r/b()]g5.0/)r([1

r/drb2)g,b(

So the angle of deflection is obtained :

Rigorous Kinetic Theory

2

2

2

2

r

b

g5.0

)r(1

br

dt/ddt/dr

ddr

m m

0

r

222

2

m)r/b(]g5.0/)r([1

dr)r/b(d

It can be written :

Rigorous Kinetic Theory

Boltzmann integro-differential equation

Assumptions of rigorous kinetic theory Spherical molecules with negligible volume Binary collisions Small gradients

]2[2]1[]0[ ffff

Enskog series

David Enskog(1900-1990)

Boltzmann equation0)b,g,,t,,,f(B Xrv

Rigorous Kinetic Theory

1st-order perturbation solution

0)g,b,,t,,,(B

)]t,,(1)[t,,(f)t,,(f ]0[

Xrv

vrvrvr

Distribution function)t,,(f vr

Flux vectors),p,( qv

Transport coefficients),,D(

Rigorous Kinetic TheoryBoltzmann equation

0)b,g,,t,,,f(B Xrv

0 0

l3s2)s,l( d db b)cos1(e kT2 2

2

m Reduced mass :

gkT2

Reduced initial velocity :

Collision integrals (Omega integrals) :

Rigorous Kinetic Theory

mr

22

2

2

g5.0

)r(

r

b1

r/drb2)b,g(

Rigorous Kinetic Theory

0 0

l3s2)s,l( d db b)cos1(e kT2 2

)r(

M

R

4

15

)T(

M/T0833.0

)2,2(2

Thermal conductivity in terms of collision integrals :

/kTT Where :

Eucken correction factor :

5

3

R

C

15

4

)T(

M/T0833.0 v

)2,2(2

Rigorous Kinetic Theory

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0

20

40

60

80

100

0 100 200 300 400 500

Temperature (K)

Dev

iati

on (

%)

He Ar N2

Deviations of thermal conductivity of various monoatomic gases calculated with rigid sphere model from experimental values.

Rigorous Kinetic Theory

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0

10

20

30

40

50

0 200 400 600 800

Temperature (K)

Dev

iati

on (

%)

He Ne Ar Kr Xe

Deviations of thermal conductivity of various monoatomic gases calculated with Lennard-Jones model from experimental values.

Rigorous Kinetic Theory

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0

10

20

30

40

50

0 100 200 300 400

Temperature (K)

Dev

iati

on (

%)

H2 O2 CO2 CH4 NO

Rigorous Kinetic Theory

Deviations of thermal conductivity of various polyatomic gases calculated with Lennard-Jones model from experimental values.

Dense Gases

By considering only two-body collisions and by taking into account the finite size of the molecules Enskog was able to graft a theory of dense gases onto the dilute theory developed earlier!

Modified Boltzmann equation

Change in the number of collisions per second

Collisional transfer of momentum and energy

Flow of molecules

Flow of molecules +Collisional transfer

Dense Gases

Collisional transfer

Dilute gases

Dense gases

Dense Gases

YV~b

YV~n32

y

y755.02.1y

1

V~b

0

y761.08.0y

1

V~b

0

1RT

V~py

If Y is collisions frequency factor and y defines as :

It can be shown that :

y is determined from experimental p-V-T data and b calculated from other properties like viscosity.

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0

10

20

30

40

50

0 500 1000 1500 2000 2500

Pressure (atm)

Dev

iati

on (

%)

25 C 50 C 75 C

Dense Gases

Deviations of thermal conductivity of nitrogen calculated with Enskog theory of dense gases from experimental values.

Liquids

Gas-like models

Solid-like models

3/43/1p

9 MC1028.4 Predvoditelev-Vargaftik :

ckn80.2 2/13/2 Eyring :

Mixed models

Cell model

Liquids

Henry Eyring(1901-1981)5.0

c

Sound velocity :

sp

1

Adiabatic compressibility :

Eyring’s theory

c

5.0

MRT

c

For ideal gas :

5.0

MRT8

But for most liquids c is greater than by factors ranging from 5 to 10.

Liquids

For liquids :

3/1

3/1

f

liq

n/1L

mkT8

c 8

n4

593c 3/2vliq

So :

Lnc31

vgas From kinetic theory of gases :

594

1Lnc

3

1v

gas With Jean’s correction factor :

With k3cv and 3/4

ckn80.2 2/13/2liq It can be written :

Which is similar to Bridgman empirical relation.

SUBSTANCE

calobs /

Methyl alcohol 0.95 Ethyl alcohol 0.94 Propyl alcohol 0.89 Isoamyl alcohol 1.05 Butyl alcohol 1.26 Acetone 1.10 Carbon disulfide 0.99 Ethyl bromide 0.94 Ethyl iodide 1.05 Water 1.16

Comparison between the thermal conductivity of various liquids calculated with Eyring theory and experimental values.

Liquids

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0

10

20

30

40

50

Dev

iati

on (

%)

Deviations of thermal conductivity of various liquids calculated with Eyring theory from experimental values.

Liquids

Empirical Correlations

Gases at atmospheric pressure :

3r

2rr

cr

n

00

)T(logc)T(logbTlogalog

TT

Gases under pressure :

n0 B

Empirical Correlations

3/4

1

2

1

2

30 )]30t(1[

Liquids at atmospheric pressure :

77.3T94.2m r

m

1

2

1

2

Liquids under pressure :

Generalized Charts

The principle of corresponding states

Further Discussion

Non-spherical molecules

Polar molecules

3

2612

r

2rr

4)r(

Stockmayer potential function :

Rigid ovaloids Rough spheres Loaded spheres

Conclusion

Transport properties of dilute gases can be predicted suitably for relatively simple molecules.

Transport properties of dense gases and liquids can be predicted just in limited cases.

The appropriate theory for transport phenomena of polar molecules has not yet been developed.

Experimental techniques are unavoidable in study of natural phenomena and theoretical approaches can just reduce the required experiences.

References

[1] Hirschfelder, J.O., Curtiss, C.F., Bird, R.B, Molecular theory of gases and liquids, John Wiley & Sons, 1954.

[2] Tsederberg, N.V., Thermal conductivity of gases and liquids, Translated by Scripta Technica, Edited by D. Cess, Cambridge: M.I.T. Press, 1965.

[3] Bridgman, P.W., The physics of high pressure, Dover Publications, 1970.

[4] Loeb, L.B., The kinetic theory of gases, Dover Publications, 1961.

[5] Kincaid, J.F., Eyring, H., Stearn, A.E., The theory of absolute reaction rates and its application to viscosity and diffusion in the liquid state, Chemical Reviews, 1941, Vol.28, pp.301-365.

0rel

2rel

2rel

Ultra-simplified Theory