ME 381R Lecture 2

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ME 381R Lecture 2 Particle Transport Theory in
Thermal Fluid Systems Level 1Kinetic Theory
Dr. Li Shi Department of Mechanical Engineering
The University of Texas at Austin Austin, TX
78712 www.me.utexas.edu/lishi lishi_at_mail.utexas.
edu
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Nanotransistors
Ju and Goodson, APL 74, 3005
IBM SOI Chip
Lines BTE results
Hot spots!
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Microscopic Origins of Thermal Fluid
Transport --The Particle Nature
Materials Dominant energy carriers Gases
Molecules Metals
Electrons Insulators Phonons (crystal
vibration)
L
Hot
Cold
In micro-nano scale thermal fluid systems, often
L lt mean free path of collision of energy
carriers Fouriers law breaks down ?
Particle transport theories or molecular dynamics
methods
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Mean Free Path for Intermolecular Collision for
Gases
D
D
Total Length Traveled L
Average Distance between Collisions, ?mc L/(of
collisions)
Total Collision Volume Swept pD2L
Mean Free Path
Number Density of Molecules n
s collision cross-sectional area
Total number of molecules encountered in swept
collision volume npD2L
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Mean Free Path for Gas Molecules
kB Boltzmann constant 1.38 x 10-23 J/K
Number Density of Molecules from Ideal Gas Law
n P/kBT
Mean Free Path
Typical Numbers
Diameter of Molecules, D ? 2 Å 2 x10-10
m Collision Cross-section s ? 1.3 x 10-19 m2
Mean Free Path at Atmospheric Pressure
At 1 Torr pressure, ?mc ? 200 mm at 1 mTorr,
?mc ? 20 cm
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Effective Mean Free Path
Wall
?b boundary separation
Wall
Effective Mean Free Path
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Kinetic Theory of Energy Transport
Cold
Net Energy Flux
u(z?z)
z ?z
?
q
qz
z
through Taylor expansion of u
z - ?z
u(z-?z)
z
Hot
Solid Angle, dW sinqdqdf
See handout for detailed derivation
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Averaging over all the solid angles
Assuming local thermodynamic equilibrium u u(T)
Thermal Conductivity
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Thermal Conductivity of Gases
Heat Capacity
J/m3-K
Monoatomic gases
Diatomic gases
VxVsinqcosf VyVsinqsinf VzVcosq
Velocity
Vz
q
V
dW
Vy
f
Vx
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Maxwell-Boltzmann Distribution
Most probabale
Mean speed
Root-mean-square
Vrms
Vm
Vmp
Most probable speed
Mean Speed
Root-Mean-Square Speed
Used for thermal conductivity calculations
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T1
T2 gt T1
Increasing Temperature
Speed of helium atoms at 0 oC
Mass, m 1.66 x 10-27 (kg/proton) x 4 (protons)
6.65 x 10-27 kg
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Thermal Conductivity
y depends on the number of atoms in the
molecule
If mean free path is limited by intermolecular
collision
?thermal conductivity is independent of number
density and therefore independent of pressure
If mean free path is affected by boundary
scattering, thermal conductivity will depend on
pressure. (Saved as a future homework problem)
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Questions

• Kinetic theory is valid for particles can
  electrons and
• crystal vibrations be considered particles?

• If so, what are C, v, ? for electrons and
  crystal vibrations?