1. Sharif University of Technology
School of Mechanical Engineering

2. THERMAL CONDUCTIVITY OF
&
GASES
LIQUIDS

3. PRESENTED BY
BEHRANG SADJADI
SUPERVISED BY
Dr. S. KAZEMZADEH

4. Contents Introduction Kinetic Theory of Gases
Thermal Conductivity of Dilute Gases
Thermal Conductivity of Dense Gases
Thermal Conductivity of Liquids
Conclusion

5. Introduction The equation of state and the equation of change
The transport coefficients
The object of theoretical attempts
Kinetic theory of dilute gases
Dense gases and liquids

6. Kinetic Theory of Gases
Ludwig Boltzmann
( )
James Clerk Maxwell
( )
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 W.
The volume of molecules is negligible.
Maxwell distribution function

7. Ultra-simplified Theory
Collision rate
Rate of collisions :
For Maxwellian distribution :
Mean free path :

8. Ultra-simplified TheoryB
If P is the property :
So :
For Maxwellian distribution :

9. Ultra-simplified Theory
For viscosity :
For thermal conductivity :
So it can be written :

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

11. Rigorous Kinetic Theory
Intermolecular potential function
Empirical intermolecular potential function
Rigid Impenetrable Spheres :
Lennard – Jones Potential :

12. Rigorous Kinetic Theory
Reduced mass :

13. Rigorous Kinetic Theory
Conservation of angular momentum :
Impact parameter
Conservation of energy :
The relation for r as a function of time :

14. Rigorous Kinetic Theory
Angle of deflection :
It can be written :
So the angle of deflection is obtained :

15. Rigorous Kinetic Theory
Boltzmann integro-differential equation
Assumptions of rigorous kinetic theory
Spherical molecules with negligible volume
Binary collisions
Small gradients

16. Rigorous Kinetic Theory
David Enskog
( )
Rigorous Kinetic Theory
Boltzmann equation
Enskog series
1st-order perturbation solution

17. Rigorous Kinetic Theory
Boltzmann equation
Distribution function
Flux vectors
Transport coefficients

18. Rigorous Kinetic Theory
Collision integrals (Omega integrals) :
Reduced mass :
Reduced initial velocity :

19. Rigorous Kinetic Theory

20. Rigorous Kinetic Theory
Thermal conductivity in terms of collision integrals :
Where :
Eucken correction factor :

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

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

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

24. Dense Gases Modified Boltzmann equation
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!
Change in the number of collisions per second
Collisional transfer of momentum and energy

25. Dense Gases Collisional transfer Dilute gases Dense gases
Flow of molecules
Dense gases
Flow of molecules +
Collisional transfer

26. Dense Gases 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.

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

28. Liquids Gas-like models Solid-like models Mixed models Cell model
Predvoditelev-Vargaftik :
Eyring :

29. Liquids Eyring’s theory Sound velocity : Adiabatic compressibility :
Henry Eyring
( )
Liquids
Eyring’s theory
Sound velocity :
Adiabatic compressibility :
For ideal gas :
But for most liquids c is greater than W by factors ranging from 5 to 10.

30. Liquids For liquids : From kinetic theory of gases :
With Jean’s correction factor :
So :
With
and
It can be written :
Which is similar to Bridgman empirical relation.

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

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

33. Empirical Correlations
Gases at atmospheric pressure :
Gases under pressure :

34. Empirical Correlations
Liquids at atmospheric pressure :
Liquids under pressure :

35. Generalized Charts
The principle of corresponding states

36. Further Discussion Non-spherical molecules Polar molecules
Rigid ovaloids
Rough spheres
Loaded spheres
Polar molecules
Stockmayer potential function :

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

38. 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

39. Thank You for Your Attention

40. Ultra-simplified Theory