1. KINETIC THEORY AND MICRO/NANOFLUDICS
Kinetic Description of Dilute Gases
Transport Equations and Properties of Ideal Gases
The Boltzmann Transport Equation
Micro/Nanofludics and Heat Transfer

2. Kinetic Description of Dilute Gases
simple kinetic theory of ideal molecular gases limited to local equilibrium based on the mean-free-path approximation
Hypotheses and Assumptions
molecular hypothesis
▪ matter: composition of small discrete particles
▪ a large number of particles in any macroscopic
volume (27×106 molecules in 1-mm3 at 25ºC and 1 atm)
statistic hypothesis
▪ long time laps: longer than mean-free time or
relaxation time
▪ time average

3. kinetic hypothesis
▪ laws of classical mechanics: Newton’s law of motion
molecular chaos
▪ velocity and position of a particle: uncorrelated
(phase space)
▪ velocity of any two particles: uncorrelated
ideal gas assumptions
▪ molecules: widely separated rigid spheres
▪ elastic collision: energy and momentum conserved
▪ negligible intermolecular forces except during collisions
▪ duration of collision (collision time) << mean free time
▪ no collision with more than two particles

4. Distribution Function
: particle number density in the phase
space at any time
▪ number of particles in a volume element of the
phase space in
▪ number of particles per unit volume
(integration over the velocity space)

5. density:
▪ total number of particles in the volume V
In a thermodynamic equilibrium state, the distribution function does not vary with time and space.

6. Local Average and Flux : additive property of a single molecule
such as kinetic energy and momentum
▪ local average or simply average
(average over the velocity space)
▪ ensemble average
(average over the phase space)

7. ▪ flux of y : transfer of y across an area element dA
per unit time dt per unit area
number of particles with velocities between and that passes through the area dA in the time interval dt dA
vdt q
dt is so small that particle collisions can be neglected.
flux of y within
total flux of y :

8. ▪ particle flux:
In an equilibrium state
For an ideal gas: Maxwell’s velocity distribution

9. ▪ average speed
For an ideal gas: Maxwell’s velocity distribution
▪ mass flux

10. ▪ kinetic energy flux
▪ momentum flux

11. average distance between two subsequent collisions for a gas molecule.
The Mean Free Path
Mean Free Path :
average distance between two subsequent collisions for a gas molecule. m0 m1 m1 m2
Let’s talk about second subject, “the mean free path”. The mean free path is very important concept and defined as the average distance
At first, let’s assume that a particle of diameter d moves at an average velocity v bar and all other particles are at rest. As you can see, I described respectively some situations that a particle makes a collision. If these situations are normalized through arrangement of particles’ center-lines, we can imagine the area where collisions could be happened.
Mean Free Path d 2d 11 11

12. ndV particles will collide with the moving particle.
During a time interval dt, the volume swept by moving particle is dV=
ndV particles will
Therefore the number of We can call it also frequency.
The mean free path 람다 is a kind of length so we can obtain the mean free path by multiply the average velocity by time between two subsequent collisions 타우. It only depends on the particle size and the number density. 타우 is the inverse of the frequency of collision. It is also called a relaxation time.
number of collisions per unit time : (frequency) 12 12

13. relative movement of particles
magnitude of the relative velocity :
Now let’s think about relative movement of particles. In previous pages, we assumed all particles are stuck except a moving particle. But actually, all particles are moving and make collisions. That’s why we should consider relative movement of particles.
Particles are moving at random, so we should try to obtain the relative velocity between 2 particles. This figure describes the vector of relative velocity.
That is the way how to obtain the magnitude of the relative velocity. That is a root of inner product of relative velocities.
The average of relative velocity could be calculated like that.
And this average value should be zero because velocities are random and uncorrelated.
So we can obtain the relation between the average relative velocity and the average velocity.
Since and are random and uncorrelated, 13

14. relative movement of particles
- Ideal gas
So there is the mean free path of moving particles.
If it is a ideal gas, the mean free path could be modified slightly like this and it is based on the Maxwell velocity distribution.
: based on the Maxwell velocity distribution 14 14

15. : probability that a molecule travels at least x between collisions
Probability for the particle to collide within an element
distance dx :
Because particles are too hard to observe their movement, It is necessary to talk about free path by using probability. Let’s p 크시 is probability
And we can say it is probability for the
And there is a equation. The left one means probability to as I told you before. And look at the right part. This part means probabilty not to collide And this whole right one means probability no to Therefore, the both meanings are same.
probability not to collide within dx
probability to travel at least
x + dx between collision
probability not to collide within x + dx 15 15

16. Probability density function (PDF) :
Through the equation, we can see the quantity of increment for p 크시, -d 크시 over capital 람다.
Integrating from 0 to 크시, it is given by like this.
So we can express The probability density function for the free path like this.
Probability density function (PDF) : 16 16

17. is the mean free-path PDF.
We can obtain this one because the integrating of PDF is always 1.
And if we use the local average which we studied several previous pages, the average of free paths is obtained.
There is a integration process, and the solution is capital 람다. It means f 크시
is the mean free-path PDF. 17 17

18. : probability for molecules to have a free path less than x
The probability for molecules to have a free path less than 크시 is given as this and this figure shows that. And there is a relation with p less than 크시 and p 크시.
Free-path distribution functions 18 18

19. Transport Eqs and Properties of Ideal Gases
Average Collision Distance
Molecular gas at steady state
(Local equilibrium)
Average collision distance
dAcosq : projected area
x: coordinate along gradient
Lcosq : average projected length

20. Shear Force and Viscosity
Momentum exchange between upper layer and lower layer
Average momentum of particles
Momentum flux across y0 plane
Velocity in y direction
Flow direction, x

21. Net momentum flux : Shear force
Dynamic viscosity : Order-of-magnitude estimate
weak dependence on pressure
Dynamic viscosity from more detailed calculation and experiments
Simple ideal gas model → Rigid-elastic-sphere model

22. Heat Diffusion
Thermal energy transfer
Molecular random motion →
Net energy flux across x0 plane
Temperature, T
x direction

23. Heat flux
Thermal conductivity
T dependence

24. Same Laof momentum transport & energy transfer
Thermal conductivity versus Dynamic viscosity
< Tabulated values for real gases
Same Laof momentum transport & energy transfer
Eucken’s formula:
Gas
T(K)
Pr (Eq.)
Pr (Exp.)
Air
273.2
0.74
0.73
≈ Tabulated values for real gases
Monatomic gas
Diatomic gas

25. Mass Diffusion Fick’s law nA(x) nB(x) x direction Gas A nA = n nB = 0
Gas B
nA = 0
nB = n
nA(x)
nB(x)
x direction

26. Uniform P
Net molecular flux
Diffusion coefficient

27. Rigid-elastic-sphere model → Not actual collision process
Intermolecular Forces
Rigid-elastic-sphere model → Not actual collision process
Attractive force (Van der Waals force)
Fluctuating dipoles in two molecules
Repulsive force
Overlap of electronic orbits in atoms
Intermolecular potential
Empirical expression (Lennard-Jones)
Repulsive
Attractive
Intermolecular potential, φ

28. Force between molecules
Newton’s law of motion for each molecule
Computer simulation of the trajectory of each molecule
Molecular dynamic is a powerful tool for dense phases, phase change
→ Not good for dilute gas → Direct Simulation Monte Carlo (DSMC)

29. Thermal Conductivity
L : mean free path [m]
u : energy density of
particles [J/m3]
: characteristic velocity
of particles [m/s]
z +Lz L  z
z - Lz
heat flux in the z-direction
Taylor series expansion
and

30. Averaging over the whole hemisphere of solid angle 2

31. Assuming local thermodynamic equilibrium:
u is a function of temperature
Fourier law of heat conduction
First term : lattice contribution
Second term : electron contribution

32. The Boltzmann Transport Equation
Volume element in phase space
Without collision, same number of particles in

33. Liouville equation
In the absence of collision and body force

34. With collisions, Boltzmann transport equation
: number of particles that join the group in
as a result of collisions
: number of particles lost to the group as a
result of collisions
: scattering probability
the fraction of particles with a velocity that
will change their velocity to per unit time
due to collision

35. Relaxation time approximation
under conditions not too far from the equilibrium
f0 : equilibrium distribution
t : relaxation time

36. Hydrodynamic Equations
The continuity, momentum and energy equations can be derived from the BTE
The first term
local average
The second term

37. Since velocity components are independent variables in the phase space,

38. The third term
Integrating by parts

39. Continuity equation
When
: bulk velocity,
: random velocity

40. Momentum equation
When
: shear stress

41. combination of all terms

42. applying the mass balance equation

43. Energy equation
: only random motion contributes to the
internal energy
u: mass specific internal energy
: energy flux vector

44. using the continuity equation

45. Fourier’s Law and Thermal conductivity
BTE under RTA
Assume that the temperature gradient is in the only x-direction, medium is stationary
local average velocity is zero, distribution function with x only at a steady state
If not very far away from equilibrium

46. heat flux in the x direction
: 1-D Fourier’s law
Under local-equilibrium assumption and applying the RTA
: 3-D Fourier’s law

47. Micro/Nanofluidics and Heat Transfer
Microdevices involving fluid flow : microsensors, actuators, valves, heat pipes and microducts used in heat engines and heat exchangers
Biomedical diagnosis (Lab-on-a-chip), drug delivery, MEMS/NEMS sensors, actuators, micropump for ink-jetprinting, microchannel heat sinks for electronic cooling
Fluid flow inside nanostructures, such as nanotubes and nanojet

48. The Knudsen Number and Flow Regimes
ratio of the mean free path to the characteristic length
Knudsen number relation with Mach number and
Reynolds number
g : ratio of specific heat

49. Flow Regimes based on the Knudsen Number
Rarefaction or Continuum
Flow Regimes based on the Knudsen Number
Regime
Method of calculation
Kn range
Continuum
Navier-stokes and energy equation with no-slip /no-jump boundary conditions
Slip flow
Navier-stokes and energy equation with slip /jump boundary conditions/DSMC
Transition
BTE, DSMC
Free molecule

50. Flow regimes centerline Velocity profiles Temperature profiles1 3 2
Velocity profiles
Temperature profiles
Continuum flow (Kn < 0.001)
The Navier-Stokes eqs. are applicable.
The velocity of flow at the boundary is the same as that of the wall
The temperature of flow near the wall is the same as the surface
temperature.
Conventionally, the flow can be assumed compressibility.
If Ma < 0.3, the flow can be assumed incompressible.
Consider compressibility : pressure change, density change

51. 2. Slip flow (0.001 < Kn < 0.1)
centerline 1 3 2
Velocity profiles
Temperature profiles
2. Slip flow (0.001 < Kn < 0.1)
Non-continuum boundary condition must be applied.
The velocity of fluid at the wall is not the same as that of the wall
(velocity slip).
The temperature of fluid near the wall is not the same as that of the wall (temperature jump).

52. 3. Free molecule flow (Kn > 10)
centerline 1 3 2
Velocity profiles
Temperature profiles
3. Free molecule flow (Kn > 10)
The continuum assumption breaks down.
The “slip” velocity is the same as the velocity of the mainstream.
The temperature of fluid is all the same : no gradient exists
The BTE or the DSMC, are the best to solve problems in this regime.

53. Velocity Slip and Temperature Jump
Tangential momentum (or velocity):
The same
Normal momentum(or velocity):
Reversed
No shear force or friction between the gas and the wall
tangential
normal
wall
Specular reflection

54. For diffuse reflection,
the molecule is in mutual equilibrium with the wall.
For a stream of molecule,
the reflected molecules follow the Maxwell velocity distribution at the wall temperature.
Diffuse reflection

55. Momentum accommodation coefficient
tangential components
normal components
i: incident, r: reflected
w: MVD corresponding to Tw
For specular reflection
For diffuse reflection

56. Thermal accommodation coefficient
For specular reflection
For diffuse reflection
For monatomic molecules,
aT involves translational kinetic energy only which is proportional to the temperature (K).

57. For polyatomic molecules
Translational, rotational, vibrational degrees
Lack of information: neglect those degrees of freedom
Air-aluminum & air-steel:
He gas-clean metallic
(almost the specular reflection)
Most surface-air
N2 , Ar, CO2
in silicon micro channel

58. Velocity slip boundary condition
Temperature jump boundary condition

59. Poiseuille flow
When Kn = L/2H < 0.1
Assume that W >> 2H, edge effect can be neglected.
incompressible and fully developed with constant properties

60. Navier-Stokes equations
fully developed flow
Let or
Velocity slip boundary condition

61. neglecting thermal creep
Let
The symmetry condition

63. bulk velocity

64. velocity distribution in dimensionless form
Define the velocity slip ratio
: the ratio of the velocity of the fluid at the
wall to the bulk velocity
velocity distribution in terms of slip ratio 64

65. Energy equation
thermally fully developed condition with constant wall heat flux
temperature jump boundary condition

66. Let
The symmetry condition

68. dimensionless temperature
By boundary condition
: temperature-jump distance
bulk temperature

69. integration by parts

70. Nusselt number

71. Poiseuille flow
Poiseuille flow with one of the plate being insulated
circular tube of inner diameter D

72. Gas Conduction-from the Continuum to the Free Molecule Regime
Heat conduction between two parallel surfaces filled with ideal gases
diffusion
jump
Free molecule
When Kn = L/L << 1, diffusion regime

73. effective mean temperature and distribution
When Kn = L/L >> 1, free molecule regime
Assume that aT are the same at both walls.
effective mean temperature
net heat flux

74. For intermediate values of Kn,