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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. Kinetic Description of Dilute Gases.

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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


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


  • 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


number of particles in a volume element of the

phase space in

  • Distribution Function

: particle number density in the phase

space at any time

▪ number of particles per unit volume

(integration over the velocity space)


density:

▪ total number of particles in the volume V

In a thermodynamic equilibrium state, the distribution function does not vary with time and space.


: additive property of a single molecule

such as kinetic energy and momentum

  • Local Average and Flux

▪ local average or simply average

(average over the velocity space)

▪ ensemble average

(average over the phase space)


number of particles with velocities between and that passes through the area dA in the time interval dt

q

dA

vdt

flux of y within

▪ flux of y : transfer of y across an area element dA

per unit time dt per unit area

dt is so small that particle collisions can be neglected.

total flux of y :


that passes through the area particle flux:

In an equilibrium state

For an ideal gas: Maxwell’s velocity distribution


that passes through the area average speed

For an ideal gas: Maxwell’s velocity distribution

▪ mass flux


that passes through the area kinetic energy flux

▪ momentum flux


d that passes through the area

2d

d

  • The Mean Free Path

  • Mean Free Path :

average distance between two subsequent collisions for a gas molecule.

m0

m1

m1

m2

Mean Free Path

11


d that passes through the area

2d

d

  • ndV particles will collide with the moving particle.

  • number of collisions per unit time : (frequency)

12


  • magnitude of the relative velocity :

Since and are random and uncorrelated,


- Ideal gas

: based on the Maxwell velocity distribution

14


: probability that a molecule travels at least that passes through the area x

between collisions

Probability for the particle to collide within an element

distance dx :

probability not to collide within dx

probability to travel at least

x + dx between collision

probability not to collide within x + dx

15


16


is the mean free-path PDF. that passes through the area

17


: probability for molecules to have a free path less than that passes through the area x

Free-path distribution functions

18


Transport Eqs and Properties of Ideal Gases that passes through the area

  • Average Collision Distance

Molecular gas at steady state

(Local equilibrium)

Average collision distance

dAcosq : projected area

x: coordinate along gradient

Lcosq : average projected length


Momentum exchange between upper layer and lower layer

Average momentum of particles

Momentum flux across y0 plane

Velocity in y direction

Flow direction, x


Net momentum flux : Shear force that passes through the area

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


Thermal energy transfer

Molecular random motion →

Net energy flux across x0 plane

Temperature,T

xdirection


Heat flux that passes through the area

Thermal conductivity

T dependence


Thermal conductivity that passes through the area 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


Fick’s law

Gas A

nA = n

nB = 0

Gas B

nA = 0

nB = n

nA(x)

nB(x)

x direction


Uniform that passes through the area P

Net molecular flux

Diffusion coefficient


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


Force between molecules that passes through the area

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)


and that passes through the area

  • 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



Assuming local thermodynamic equilibrium: that passes through the area

u is a function of temperature

Fourier law of heat conduction

First term :latticecontribution

Second term :electroncontribution


The Boltzmann Transport Equation that passes through the area

Volume element in phase space

Without collision, same number of particles in


Liouville equation that passes through the area

In the absence of collision and body force


: number of particles that join the group in that passes through the area

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

With collisions, Boltzmann transport equation


Relaxation time approximation that passes through the area

under conditions not too far from the equilibrium

f0 : equilibrium distribution

t : relaxation time


The continuity, momentum and energy equations can be derived from the BTE

The first term

local average

The second term



The third term phase space,

Integrating by parts


When phase space,

: bulk velocity,

: random velocity

  • Continuity equation


When phase space,

  • Momentum equation

: shear stress




: energy flux vector phase space,

  • Energy equation

: only random motion contributes to the

internal energy

u: mass specific internal energy



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


: 1-D Fourier’s law phase space,

: 3-D Fourier’s law

heat flux in the x direction

Under local-equilibrium assumption and applying the RTA


Micro/Nanofluidics and Heat Transfer phase space,

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


The knudsen number and flow regimes
The Knudsen Number and Flow Regimes phase space,

  • Knudsen Number

ratio of the mean free path to the characteristic length

  • Knudsen number relation with Mach number and

    Reynolds number

g : ratio of specific heat


Knudsen Number phase space,

Rarefaction or Continuum

Flow Regimes based on the Knudsen Number


centerline phase space,

3

1

1

3

2

2

Velocity profiles

Temperature profiles

  • Flow regimes

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


centerline phase space,

3

1

1

3

2

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).


centerline phase space,

3

1

1

3

2

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.


Velocity slip and temperature jump
Velocity Slip and Temperature Jump phase space,

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


For diffuse reflection, phase space,

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


tangential components

normal components

i: incident, r: reflected

w: MVD corresponding to Tw

For specular reflection

For diffuse reflection


For specular reflection

For diffuse reflection

For monatomic molecules,

aT involves translational kinetic energy only which is proportional to the temperature (K).


For polyatomic molecules phase space,

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


Velocity slip boundary condition phase space,

Temperature jump boundary condition


When Kn = L/2H < 0.1

Assume that W >> 2H, edge effect can be neglected.

incompressible and fully developed with constant properties


Navier-Stokes equations phase space,

fully developed flow

Let

or

Velocity slip boundary condition


neglecting thermal creep phase space,

Let

The symmetry condition


bulk velocity phase space,


velocity distribution in dimensionless form phase space,

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


Energy equation phase space,

thermally fully developed condition with constant wall heat flux

temperature jump boundary condition


Let phase space,

The symmetry condition


dimensionless temperature phase space,

By boundary condition

: temperature-jump distance

bulk temperature




Poiseuille flow phase space,

Poiseuille flow with one of the plate being insulated

circular tube of inner diameter D


diffusion phase space,

Free molecule

jump

  • Gas Conduction-from the Continuum to the Free

    Molecule Regime

Heat conduction between two parallel surfaces filled with ideal gases

  • When Kn = L/L << 1, diffusion regime


effective mean temperature and distribution phase space,

  • When Kn = L/L >> 1, free molecule regime

Assume that aT are the same at both walls.

effective mean temperature

net heat flux



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