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