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 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

Kinetic Theory of Gases The history of kinetic theory James Clerk Maxwell (1831-1879)

Assumptions of ultra-simplified theory  The molecules are rigid and non-attracting sphere.  All the molecules travel with the same speed Ω .  The volume of molecules is negligible.

Maxwell distribution function 1/ 2

 8kT  Ω=  π m  

Ludwig Boltzmann (1844-1906)

Ultra-simplified Theory 2σ

Collision rate

2Ω ∆t 1 1 n × π σ2 × 2Ω ∆t n × π σ2 × 2Ω ∆t Γ=6 +3 ∆t ∆t  Rate of collisions :

Γ = 1.276π σ2 nΩ = 1.276pσ 2 8π / mkT

 For Maxwellian distribution :  Mean free path :

L=

Γ = 2pσ 2 8π / mkT

Ω kT = Γ 2pπ σ2

Ultra-simplified Theory B

L

O

L

A  If P is the property :

 So :

 dP   dP  PA = PO − L  PB = PO + L   dz   dz 

1 1 πmkT  dP  ΨP = Ω( PA − PB ) = − ΩL  = −ξ 6 3  dz  nmπ σ2

 For Maxwellian distribution :

ξ=

8 3π 2

=

2 3π

Ultra-simplified Theory  For viscosity :

Pp = nmv ⇒ Ψp = −ξ

πmkT  dv  πmkT ⇒ η = ξ 2  dz  π σ π σ2

 For thermal conductivity :

πmkT c v  dT  πmkT c v Pq = nc vT ⇒ Ψq = −ξ ⇒λ=ξ 2 m  dz  π σ π σ2 m  So it can be written :

λ=

1 c vη m

λ = C vη λ = f ⋅ C vη

Ultra-simplified Theory Deviations of thermal conductivity of various gases calculated with ultra-simplified kinetic theory from experimental values. 0

100

200

300

0

Deviation (%)

-10 -20 -30 -40 -50 -60 -70 -80 -90 -100

Temperature (K) He

Ar

N2

400

500

Rigorous Kinetic Theory Intermolecular potential function ∞ ϕ(r ) = ∫ F( r )dr r

Empirical intermolecular potential function  Rigid Impenetrable Spheres :

∞ r < σ ϕ(r ) =  0 r > σ

 Lennard – Jones Potential :

 σ 12  σ 6  ϕ(r ) = 4ε   −     r    r 

Rigorous Kinetic Theory

d 2ri  Fi = mi 2  d2 Fi dt  ⇒ ( r − r ) = j 2 i 2  µ d rj  dt − Fi = m j 2  dt   Reduced mass :

1 1 1 = + µ mi m j

Rigorous Kinetic Theory  Conservation of angular momentum :

µbg = µr 2θ  Conservation of energy :

1 2 1 2 2 2 µg = µ( r + r θ ) + ϕ(r ) 2 2  The relation for r as a function of time :

1 2 1 2 1 2 2 2 µg = µr + µg ( b / r ) + ϕ(r ) 2 2 2

Impact parameter

Rigorous Kinetic Theory  Angle of deflection :

χ = π − 2θm  It can be written :

dr dr / dt  r 2  ϕ( r ) b2 = = 1− − 2 2   dθ dθ / dt  b  0.5µg r θm rm θm = ∫ dθ = − ∫ 0 ∞

(b / r 2 )dr 1 − [ϕ( r ) / 0.5µg 2 ] − (b 2 / r 2 )

 So the angle of deflection is obtained : ∞ χ( b, g ) = π − 2b ∫ rm

dr / r 2 1 − [ϕ(r ) / 0.5µg 2 )] − (b 2 / r 2 )

Rigorous Kinetic Theory

Boltzmann integro-differential equation Assumptions of rigorous kinetic theory  Spherical molecules with negligible volume  Binary collisions  Small gradients

Rigorous Kinetic Theory Boltzmann equation B(f , v, r, t , X, g, b) = 0

Enskog series f = f [0] + ξf [1] + ξ2f [ 2] + 

1st -order perturbation solution f (r, v, t ) = f [0] (r, v, t )[1 + φ(r, v, t )] Bφ (φ, v, r, t , X, b, g ) = 0

David Enskog (1900-1990)

Rigorous Kinetic Theory Boltzmann equation B(f , v, r, t , X, g, b) = 0

Distribution function f (r, v, t )

Flux vectors ( v , p, q )

Transport coefficients ( D, η, λ)

Rigorous Kinetic Theory Collision integrals (Omega integrals) : Ω

(l,s )

2πkT ∞ ∞ − γ 2 2s +3 = γ (1 − cosl χ)b db dγ ∫ ∫e µ 00

 Reduced mass :

m µ= 2

 Reduced initial velocity :

µ γ= g 2kT

Rigorous Kinetic Theory ϕ(r )

∞ χ(g , b) = π − 2 b ∫ rm

Ω

(l,s )

dr / r 2 b2

ϕ( r ) 1− 2 − r 0.5µg 2

2πkT ∞ ∞ − γ 2 2s +3 = γ (1 − cosl χ)b db dγ ∫ ∫e µ 00

Rigorous Kinetic Theory Thermal conductivity in terms of collision integrals : T/M 15 R λ = 0.0833 2 ( 2,2)∗ ∗ = η σ Ω (T ) 4 M  Where :

T∗ = kT / ε

Eucken correction factor : T/M  4 Cv 3  λ = 0.0833 2 ( 2,2)∗ ∗  +  σ Ω (T )  15 R 5 

Rigorous Kinetic Theory Deviations of thermal conductivity of various monoatomic gases calculated with rigid sphere model from experimental values. 100 80

Deviation (%)

60 40 20 0 -20 -40 -60 -80 -100 0

100

200

300

Temperature (K) He

Ar

N2

400

500

Rigorous Kinetic Theory Deviations of thermal conductivity of various monoatomic gases calculated with Lennard-Jones model from experimental values. 50 40

Deviation (%)

30 20 10 0 -10 -20 -30 -40 -50 0

200

400

600

Temperature (K) He

Ne

Ar

Kr

Xe

800

Rigorous Kinetic Theory Deviations of thermal conductivity of various polyatomic gases calculated with Lennard-Jones model from experimental values. 50 40

Deviation (%)

30 20 10 0 -10 -20 -30 -40 -50 0

100

200

300

Temperature (K) H2

O2

CO2

CH4

NO

400

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

Dense Gases Collisional transfer Dilute gases  Flow of molecules

Dense gases  Flow of molecules + Collisional transfer

Dense Gases  If Y is collisions frequency factor and y defines as :

2 ~ b y =  πnV Y =  ~ Y 3  V  It can be shown that :

~  pV  y=  −1  RT  η b 1  = ~  + 0.8 + 0.761y  η0 V  y  λ b 1  = ~  + 1.2 + 0.755 y  λ0 V  y   y is determined from experimental p-V-T data and b calculated from other properties like viscosity.

Dense Gases Deviations of thermal conductivity of nitrogen calculated with Enskog theory of dense gases from experimental values. 50 40

Deviation (%)

30 20 10 0 -10 -20 -30 -40 -50 0

500

1000

1500

Pressure (atm) 25 C

50 C

75 C

2000

2500

Liquids Gas-like models Solid-like models Mixed models  Cell model

 Predvoditelev-Vargaftik :  Eyring :

λ = 4.28 × 10−9 C p M −1 / 3ρ4 / 3

λ = 2.80kn 2 / 3γ −1 / 2c

Liquids Eyring’s theory  Sound velocity :

ν c=  β

0.5

 Adiabatic compressibility :

 For ideal gas :

Henry Eyring (1901-1981)

 RTγ  c=  M  

 1  ∂ν  β = −    ν  ∂p s 0.5

 8RT  Ω=  π M  

0.5

c≈Ω

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

Liquids 8kT  ν liq Ω =  πm  ν f

 For liquids :

1/ 3

  

L = (1 / n ) 1 / 3  From kinetic theory of gases :  With Jean’s correction factor :  So :

liq

λ

1 λgas = nc vΩL 3 1 1  λgas = nc vΩL  ( 9 γ − 5)  3 4 

 αc v  9 γ − 5  2 / 3 8 = c  n π γ  3  4 

 It can be written :

 With c v = 3k and γ = 4 / 3

λliq = 2.80kn 2 / 3γ −1 / 2c

 Which is similar to Bridgman empirical relation.

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

λ obs / λ cal

SUBSTAN

Methyl alcoho

Liquids Deviations of thermal conductivity of various liquids calculated with Eyring theory from experimental values. 50 40 30

Deviation (%)

20 10 0 -10 -20 -30 -40 -50

Empirical Correlations Gases at atmospheric pressure : n

T λ = λ 0    T0  λ log = a log Tr + b(log Tr ) 2 + c(log Tr )3 λ cr

Gases under pressure : λ = λ 0 + Bρn

Empirical Correlations Liquids at atmospheric pressure : λ = λ30 [1 − β( t − 30)] λ 2  ρ2  =   λ1  ρ1 

4/3

Liquids under pressure : m

λ 2  ρ2  =   λ1  ρ1  m = −2.94Tr + 3.77

Generalized Charts The principle of corresponding states

Further Discussion Non-spherical molecules  Rigid ovaloids  Rough spheres  Loaded spheres

Polar molecules  Stockmayer potential function :

 σ 12  σ 6  2µ 2 ϕ(r ) = 4ε   −    − 3  r   r  r 

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.

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.

Ultra-simplified Theory Ω rel = 0 Ω rel = 2Ω

Ω rel = 2Ω