sharif university of technology school of mechanical engineering
TRANSCRIPT
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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-20
-10
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