1. Particle Transport Theory in Thermal Fluid Systems:
ME 381R Lecture 2
Particle Transport Theory in Thermal Fluid Systems:
Level 1—Kinetic Theory
Dr. Li Shi
Department of Mechanical Engineering
The University of Texas at Austin
Austin, TX 78712

2. Nanotransistors Hot spots! Ju and Goodson, APL 74, 3005 IBM SOI Chip
Lines: BTE results
Hot spots!

3. 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 < mean free path of collision of energy carriers & Fourier’s law breaks down
 Particle transport theories or molecular dynamics methods

4. 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

5. Mean Free Path for Gas Molecules
kB: Boltzmann constant
1.38 x 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 m2
Mean Free Path at Atmospheric Pressure:
At 1 Torr pressure, mc  200 mm; at 1 mTorr, mc  20 cm

6. Effective Mean Free Path
Wall
b: boundary separation
Wall
Effective Mean Free Path:

7. Kinetic Theory of Energy Transport
Cold
u(z+z)
Net Energy Flux
z + z q  qz z
through Taylor expansion of u
z - z
u(z-z) z q x y f dW
Hot
Solid Angle, dW = sinqdqdf
See handout for detailed derivation

8. Averaging over all the solid angles
Assuming local thermodynamic equilibrium: u = u(T)
Thermal
Conductivity

9. Thermal Conductivity of Gases
Heat Capacity
Monoatomic gases:
[J/m3-K]
Diatomic gases:
Velocity:
Vx=Vsinqcosf
Vy=Vsinqsinf
Vz=Vcosq Vz q V dW Vy f Vx

10. Maxwell-Boltzmann DistributionV
Most probabale
Mean speed
Root-mean-square
Vmp Vm
Vrms
Most probable speed:
Mean Speed:
Root-Mean-Square Speed
Used for
thermal conductivity
calculations

11. V T1
T2 > T1
Increasing
Temperature
Speed of helium atoms at 0 oC
Mass, m = 1.66 x (kg/proton) x 4 (protons) = 6.65 x kg

12. 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)

13. 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?