Transcript A study on the effect of micro voids on permeability

Micro/Nanofluidics and
Heat transfer
27 Oct 2011
In Joo Hwang
Contents
1. The Knudsen number and flow regimes
2. Velocity slip and temperature jump
3. Gas conduction
from the continuum to the free molecule regime
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
Λ
Kn ≡ ─
L
: Knudsen number
Kn : The ratio of the mean free path to
the characteristic length
Regime
Method of calculation
Kn range
Continuum
N-S equation and energy equation
with no-slip/ no-Jump b.c.
Kn ≤ 0.001
Slip flow
N-S equation and energy equation
with slip/ Jump b.c. DSMC
0.001 < Kn ≤ 0.1
Transition
BTE, DSMC
Free molecule
BTE, DSMC
0.1 < Kn ≤ 10
Kn > 10
Kn : determining the degree of deviation from the
continuum assumption and method of calculation
The Knudsen number and flow regimes
x
Centerline
1
y
2
vx(y)
T(y)
yb
3
Velocity profiles vx(y)
Tw
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
2. Velocity slip and temperature jump
Velocity slip and temperature jump
Momentum accommodation coefficient
 v 
pi  p r 

pi  p w  
For tangential components
v 
pi  pr 

pi  pw ||
For normal components
Specular reflection :  v   v  0
Diffuse reflection
: v  v  1
Thermal accommodation coefficient
  
 T  i r 
i   w 
Monatomic molecules
Kinetic energy ∝ K
T 
Ti  Tr 

Ti  Tw 
Often extended to polyatomic molecule
Velocity slip and temperature jump
Velocity slip boundary condition
v x ( yb )  
2  v
v
 v 
R  T 
 x   3



y
8

T

x

 yb

 yb
thermal creep due to
the temperature gradient
Temperature jump boundary condition
v x2 ( yb )
2   T 2   T 

 
T ( yb )  Tw  
 T   1 Pr  y  yb
4R
viscous dissipation caused
by the slip velocity
usually negligibly small
Velocity slip and temperature jump
qw
y
vx
d 2vx H 2 dP

d 2
 dx
 dv 
vx (  1)  2 v  x 
 d  1
vx ( ) 3 1  4 v  2

vm
2 1  6 v
W
Λ
2L
x
Kn = ─
vx ( ) 3   3(1   ) 2



vm
2
2
W ≥ 2H
  y/H
dvx / d  0  0
v 
2  v
v
Kn
velocity slip condition
velocity distribution
H 2  dP 
vm   vx ( )d  (1  6 v )


0
3  dx 
1
Poiseuille flow with
2H heat transfer
bulk velocity
defining velocity slip ratio
  vx (  1) / vm  6v /(1  6v )
Velocity slip and temperature jump
  2T  2T 
T
c p vx
   2  2 
energy equation
x

x

y


2
  vx

( )  ( / H )(T  Tw ) / qw
 2 vm
3   2 1  4
 
  C1  C2
4
8
( ) 
 d 

(  1)  2T 
 d  1
2   T 2 Kn
T 
T   1 Pr
 T 
T T(y  H)

qw   
 w
2T H
 y  y  H
C1  0 C2  (  5) / 8  2T
temperature jump condition
temperature – jump distance
vx ( )
( )d
0 v
m
1
m  
Nu 
hDh


qw 4 H
4

Tw  Tm 
m
Nusselt number , Dh  4H
3. Gas conduction
from the continuum to the free molecule regime
Gas conduction from the continuum to the free molecule regime
Diffusion
T1
Jump
T1
T2
0
x
Free molecule
L
Λ
L
0
Kn = ─ << 1
 
qDE
diffusion
T1  T2
L
Effective mean temperature
Tm, DF
 2 T13 / 2  T23 / 2 

 
3
T

T
1
2


x
T2
L
Fourier’s law
9  5


cv  
 
4


2
Temperature distribution


x

T ( x)  T13 / 2  T13 / 2  T23 / 2 
L

2/3
by integrating q   T dT / dx
Gas conduction from the continuum to the free molecule regime
• collide with the wall > collide with each other
• mean free path
> actual distance
• neglect the collisions between molecules
• heat transfer by the molecules
Λ
L
Kn = ─ >> 1
Thermal accommodation coefficients :  T
Flux temperatures
T1 
T1  (1   T )T2
2  T
T2 
T2  (1   T )T1
2  T
assumption T  T1  T2
Effective mean temperature in the free molecule regime
Tm, FM 
T
4T1T2
1
 T2

2
Net heat flux
 
qFM
T1  T2
2  T  8RTm,FM
T   1cv P
heat flux∝ P
independent of L
 
qFM
 T1  T2 

2   T 9  5 Tm , FM
L1  Kn

 T   1 Tm , DF




