Micro/Nanofluidics and Heat transfer

27 Oct 2011

In Joo Hwang

Contents

1. The Knudsen number and flow regimes 1. The Knudsen number and flow regimes

2. Velocity slip and temperature jump 2. Velocity slip and temperature jump

3. Gas conduction from the continuum to the free molecule regime

3. Gas conduction from the continuum to the free molecule regime

1. The Knudsen number and flow regimes 1. The Knudsen number and flow regimes

The Knudsen number and flow regimes

Lc : Characteristic dimension

Λ : Mean free path

Lc ~ Λ Lc < Λ

Not valid for continuum model

Ex) low pressure (rarefied gases) micro or nano channel

The Knudsen number and flow regimes

Kn ≡ ─ : Knudsen number

Kn : determining the degree of deviation from the continuum assumption and method of calculation

Λ

L

Kn : The ratio of the mean free path to the characteristic length

Regime Method of calculation Kn range

ContinuumN-S equation and energy equation

with no-slip/ no-Jump b.c. Kn ≤ 0.001

Slip flowN-S equation and energy equation

with slip/ Jump b.c. DSMC 0.001 < Kn ≤ 0.1

Transition BTE, DSMC0.1 < Kn ≤ 10

Free molecule BTE, DSMC Kn > 10

The Knudsen number and flow regimes

Tw

T(y)vx(y)

yb

y

xCenterline

Velocity profiles vx(y) Temperature profiles T(y)

Number Kn Boundary condition

1 Kn < 0.001 flow adjacent = wall

2 0.001 < Kn ≤ 0.1 slip flow, temperature jump

3 Kn > 10 Boundary scattering

1

2

3

2. Velocity slip and temperature jump 2. Velocity slip and temperature jump

Velocity slip and temperature jump

Momentum accommodation coefficient

||

wi

riv pp

pp

wi

riv pp

pp For tangential components

For normal components

Thermal accommodation coefficient

Specular reflection :

Diffuse reflection :

0 vv

1 vv

wi

riT

Monatomic molecules

Kinetic energy K∝

wi

riT TT

TT

Often extended to polyatomic molecule

Velocity slip and temperature jump

Velocity slip boundary condition

R

yv

y

TTyT bx

yT

Twb

b4

)(

Pr1

22)(

2

bbyy

x

v

vbx x

T

T

R

y

vyv

8

32

)(

Temperature jump boundary condition

thermal creep due to the temperature gradient

viscous dissipation caused by the slip velocityusually negligibly small

Velocity slip and temperature jump

2HW

xy

vx

wq

W ≥ 2H

Kn = ─ Λ2L

Poiseuille flow with heat transfer

dx

dPH

d

vd x

2

2

2

1

2)1(

d

dvv x

vx Knv

vv

2

v

v

m

x

v

v

61

41

2

3)( 2

1

0

2

3)61()(

dx

dPHdvv vxm

2

2

)1(3

2

3)(

m

x

v

v

Hy / 0/ 0 ddvx

)61/(6/)1( vvmx vv

bulk velocity

defining velocity slip ratio

velocity slip condition

velocity distribution

Velocity slip and temperature jump

m

x

v

v

2

2

2142

8

1

4

3)( CC

1

2)1(

d

dT

Pr1

22 Kn

T

TT

H

HyTT

y

Tq

T

w

Hy

w

2

)(

1

0)(

)( d

v

v

m

xm

mmw

wh H

TT

qhDNu

44

2

2

2

2

y

T

x

T

x

Tvc xp

ww qTTH /))(/()(

01 C TC 28/)5(2

temperature – jump distance

Nusselt number , HDh 4

energy equation

temperature jump condition

3. Gas conduction from the continuum to the free molecule regime

3. Gas conduction from the continuum to the free molecule regime

Gas conduction from the continuum to the free molecule regime

T1

T2

T1

T2

L Lx x0 0

Diffusion

Jump

Free molecule

Kn = ─ << 1 ΛL

diffusion Fourier’s law

L

TTqDE

21

vc4

59

2

21

2/32

2/31

, 3

2

TT

TTT DFm

Effective mean temperature

Temperature distribution

3/2

2/32

2/31

2/31)(

L

xTTTxT dxdTTq /by integrating

Gas conduction from the continuum to the free molecule regime

Kn = ─ >> 1 ΛL

• collide with the wall > collide with each other

• mean free path > actual distance • neglect the collisions between molecules• heat transfer by the molecules

T

T TTT

2

)1( 211

Thermal accommodation coefficients : T

T

T TTT

2

)1( 122

Flux temperatures

Effective mean temperature in the free molecule regime

221

21,

4

TT

TTT FMm

Net heat flux

Pc

RT

TTq

vT

FMmT

FM

1

82 ,

21

heat flux P∝

independent of L

DFm

FMm

T

T

FM

T

TKnL

TTq

,

,

21

1592

1

assumption 21 TTT