Post on 29-Dec-2015
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
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-m3 at 25ºC and 1 atm)
statistic hypothesis
▪ long time laps: longer than mean-free time or relaxation time ▪ time average
simple kinetic theory of ideal molecular gases limited to local equilibrium based on the mean-free-path approximation
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
Distribution Function: particle number density in the phase space at any time
( , , )f r v t
6 ( , , ) x y zd N f r v t dxdydzdv dv dv 3 3( , , )f r v t d Vd
3 3, x y zd V dxdydz d dv dv dv
▪ number of particles per unit volume (integration over the velocity space)
63
3 ( , , )d N
f r v t dd V
33
3 ( , , )d N dN
f r v t dd V dV
( , )n r t
▪ number of particles in a volume element of the phase space in
,r dr v dv
▪ total number of particles in the volume V
( , ) ( , )r t m n r t
density:
3 3( ) ( , , )V
N t f r v t d Vd
In a thermodynamic equilibrium state, the distribution function does not vary with time and space.
( , , ) ( )f r v t f v
3( , , )dN
f r v t ddV
Local Average and Flux
( , , )r v t
: additive property of a single molecule such as kinetic energy and momentum▪ local average or simply average
(average over the velocity space)
( , )f d
r tfd
1
( , )f d
n r t
▪ ensemble average (average over the phase space)
V
V
f dVd
fdVd
1
( ) Vf dVd
N t
▪ flux of : transfer of 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
v
v dv
cosdV vdt dA
dA
vdt
v n̂
ˆv ndAdt
dt is so small that particle collisions can be neglected.
ˆ( , , ) ( , , )f r v t dVd f r v t v ndAdtd
ˆ( , , )f r v t v ndAdtd
dAdt
total flux of :
ˆJ fv nd
flux of within d
▪ particle flux:
1
ˆNJ fv nd
In an equilibrium state ( , , ) ( )f r v t f v
2 sind v d d dv ˆ( )NJ f v v nd
2 / 2 2
0 0 0( ) cos sin
vf v v v d d dv
2 / 2 3
0 0 0( ) cos sin
vf v v d d dv
3
0( )
vf v v dv
NJ
For an ideal gas: Maxwell’s velocity distribution 3/ 2 2
B B
( ) exp2 2
m mvf v n
k T k T
▪ average speed1
( , )r t f dn
2 3
0 0 0
1 1( ) sin
vv fvd f v v d d dv
n n
3
0
4( )
vf v v dv
n
3
0( )N
vJ f v v dv
4
nv
For an ideal gas: Maxwell’s velocity distribution
B8,
k Tv
m B
2N
k TJ n
m
▪ mass flux
ˆ( )4
m
vJ m f v v nd
ˆ , NJ fv nd nm
▪ kinetic energy flux
2
2
mv
25
KE0
ˆ ( )2 2
mv mJ fv nd f v v dv
KEJ
▪ momentum flux
, , 1, 2,3ij j iP mv fv d i j
1
( , )r t f dn
ij i jP m fv v d
1
i j i jmn fv v d mnv vn
i j ij ijv v P
11
The Mean Free Path
Mean Free Path :average distance between two subsequent collisions for a gas molecule.
dd 2d
m0 m1 m1 m2
Mean Free Path
12
number of collisions per unit time : (frequency)
ndV particles will collide with the moving particle.
dd 2d
vdt
2dV d vdt
2nd v
2
1v
nd
1
frequency
relative movement of particles
magnitude of the relative velocity :
rel rel relv v v
1 2 1 2( ) ( )v v v v
1 1 1 2 2 2= 2v v v v v v
1 1 1 2 2 22relv v v v v v v
1 2 0v v
Since and are random and uncorrelated,1v
2v
2 21 2relv v v
2relv v
14
relative movement of particles
: based on the Maxwell velocity distribution
- Ideal gas
relative 2 2 2
1 1 1
( 2 ) 2rel
v v vnd v nd v nd
PV nRT , AB A
N nR k N n
V B Ank N T
BP nk T
relative2
1
2B
Pd
k T
22Bk T
d P
15
: probability that a molecule travels at least between collisions
Probability for the particle to collide within an element
distance d:
probability to travel at least + d between collision
probability not to collide within + d
probability not to collide within d
( )p
d
( ) ( ) 1d
p d p
16
Probability density function (PDF) :
( ) ( ) 1d
p d p
( )( ) ( ) ( )
dp dp d p p
d
( ),
( )
dp d
p
( )
(0) 0
1 1p
pdp d
p
( )
(0)0
ln ( ) ,p
pp
ln ( ) ln (0)p p
/( )p e
/( ) 1( )
dpF e
d
(0) 1p
17
( )F is the mean free-path PDF.
0( ) 1F d
0( )F d
/
0
1 e d
/ /
00
1e e d
// 0
1lim e
e
2/ /
1 1lim lim 1
e e
18
: probability for molecules to have a free path less than
Free-path distribution functions
/
0 ( ) 1 ( ) ( ) 1p p F d e
Molecular gas at steady state
(Local equilibrium)
Average collision distance
Average Collision Distance
a a
0
Transport Eqs and Properties of Ideal Gases
(r, v, ) ( , v)f t f
2 / 2
0 02 / 2
0 0
cos cos
cos sina
dA d d
dA d d
2 233
dA
dA
21/( 2 )d n dAcos: projected area: coordinate along gradientcos: average projected length
Shear Force and Viscosity
Momentum exchange between upper layer and lower
layerAverage momentum of particles
Momentum flux across y0 plane
)(B y
Velo
cit
y in
y d
irecti
on
Flow direction, x
a0 y
a0 y
0yB0
area)unit per frequency (Collisionflux Molecular :4/n
speedmolecular Mean :/8ity,Bulk veloc :)(B mTky B
a
a
( ) ( )x Bp y m y
0
04B
P B a
y
dnJ m
dy
0
04B
P B a
y
dnJ m
dy
Net momentum flux : Shear force
Dynamic viscosity : Order-of-magnitude estimate
Dynamic viscosity from more detailed calculation and experiments
Simple ideal gas model → Rigid-elastic-sphere model
0 0
1
3B B
p p p yx
y y
d dJ J J
dy dy
2
/1 2
3 3Bmk T
d
1, , ( )P f T
weak dependence on pressure
22
1 1
2 2 2Bmk Tm
dd
Heat Diffusion
Molecular random motion →
Thermal energy
transfer
Net energy flux across x0 plane
Tem
pera
ture
, T
x direction
0xa0 x a0 x
0T
)( moleculeper energy thermalAverage : Tf
aa
E E E xJ J J q
0 0( ) ( )4 a a
nx x
0
1
3 x
dn
dx
Heat flux
T dependence
Thermal conductivity
0
1
3E E E xx
dJ J J q n
dx
0
1
3 vx
dTc
dx
0 0 0
vx x x
d d dT dTn n nmc
dx dT dx dx
2
1 2
3 3v
mkTc c
d
1 , & ( ) vc f T
Monatomic gas
Diatomic gas
< Tabulated values for real gases
≈ Tabulated values for real gases
Thermal conductivity versus Dynamic viscosity
Gas T(K) Pr (Eq.) Pr (Exp.)Air 273.2 0.74 0.73
Eucken’s formula:
0.667 vc
Pr / / 4 / 9 5pc k
Same aof momentum transport & energy transfer
9 5
4 vc
( 5 / 3) 2.5 1.25v vc c
( 1.4) 1.9 0.95v vc c
Mass Diffusion
Fick’s lawGas AnA = nnB = 0
Gas BnA = 0nB = n
x direction
nA(x) nB(x)
B andA between t coefficienDiffusion :ABD
rate transfer Mass : rate,transfer Molecular : AJmAJNAA mANA
AN
BN A
AN AB
dnJ D
dx
A
Am AB
dJ D
dx
B
BN BA
dnJ D
dx
B
Bm BA
dJ D
dx
distance, Central :2/)( BA ddd mass Reduced :)/( BABAr mmmmm
Diffusion coefficient
Net molecular flux
Uniform PA BN NJ J
0 0
A AA 0 a A 0 a( ) ( )
4 4ANx x
dn dnJ n x n x
dx dx
0
A1
3 x
dn
dx
AB 2r
1 3 1
3 8 2Bk T
Dnd m
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 potentialEmpirical expression (Lennard-Jones)
r
0
0r
Attractive
Repulsive
F F
Inte
rmol
ecul
ar p
oten
tial,
φ
r
diameter) (collisionlength sticCharacteri :0r
particlesth &th between Distance : jirijenergy, sticCharacteri :0
6 12
0 00( ) 4ij ij
ij ij
r rr
r r
Computer simulation of the trajectory of each moleculeMolecular dynamic is a powerful tool for dense phases, phase change
→ Not good for dilute gas → Direct Simulation Monte Carlo (DSMC)
Force between molecules
Newton’s law of motion for each molecule
13 7
0 0 0
0
r242 ij
ij ijij ij ij
r rF
r r r r
vF (r ,r , ) , 1,2, ...,i
ij i j ij
dt m i N
dt
: mean free path [m]u : energy density of particles [J/m3] : characteristic velocity of particles [m/s]
Taylor series expansion
heat flux in the z-direction
Thermal Conductivity
z +z
z
z - z
( )zu z
zq
cos ,z coszv v
1( ) ( )
2z z z zq v u z u z
and( ) ( )z z
duu z u z
dz ( ) ( )z z
duu z u z
dz
v
Averaging over the whole hemisphere of solid angle 2
1( ) ( )
2z z z z
du duq v u z u z
dz dz
2cosz z
du duv v
dz dz
2 / 2 2
0 0
1cos sin
2z
duq v d d
dz
zq
1
3
duv
dz
cos , cosz zv v
Assuming local thermodynamic equilibrium: u is a function of temperature
Fourier law of heat conduction
First term : lattice contribution
Second term : electron contribution
1 1 1
3 3 3z
du du dT dTq v v Cv
dz dT dz dz
z
dTq k
dz
1
3k C v
1
3l ek C v C v
v
r
F
r dr
v dv Volume
element in phase spacev
r
dVdr dr r vdt
Fv dv v adt v dt
m
Without collision, same number of particles in ,r dr v dv
( , , )dN f r v t dVd
( , , )f r dr v dv t dt dVd
( , , ) ( , , ) 0f r dr v dv t dt f r v t
v
m
r
The Boltzmann Transport Equation
( , , ) ( , , )f f f
f r dr v dv t dt f r v t dt dr dvt r v
( , , ) ( , , )f r dr v dv t dt f r v t f f dr f dv
dt t r dt v dt
0f f f
v at r v
ˆ ˆ ˆr
f f f ff i j k
r x y z
ˆ ˆ ˆvx y z
f f f ff x y z
v v v v
Liouville equationIn the absence of collision and body force
0Df f f
vDt t r
With collisions, Boltzmann transport equation
coll
f f f fv a
t r v t
coll
f
t
: number of particles that join the group in as a result of collisions
,r dr v dv
: number of particles lost to the group as a result of collisions
coll
( , ) ( , , ) ( , ) ( , , )v
fW v v f r v t W v v f r v t
t
( , )W v v: scattering probability the fraction of particles with a velocity that will change their velocity to per unit time due to collision
v
v
Relaxation time approximation
under conditions not too far from the equilibrium
0
coll ( )
f f f
t v
f0 : equilibrium distribution: relaxation time
Hydrodynamic EquationsThe continuity, momentum and energy equations can be derived from the BTE
The first termlocal average
( )f f f
d v d a d dt r v
1f d
n
( )f nd fd f d n
t t t t t
The second term
vf vf vf
fv v f
r
vf f v v f
Since velocity components are independent variables in the phase space, 0v
vf v f
( )n v nv
1f d
n
vf vf vf v f vf
v f vf vf
( )v f d vfd vf d
The third term
, ,
, ,
x x x
x x x
v v v
v v v
fa d a f f d na
v v v
( )F ra
m
Integrating by parts
( )( )
nn n v nv na
t t v
( ) ( )n n v n v at t v
( )d
m When
( ) ( ) 0m m
nm nmv n v m at t v
nm
B B( ) 0 0D
v vt Dt
or
Continuity equation
B R=v v v
Bv
Rv
: bulk velocity,
: random velocity
B R B R B=v v v v v v
( ) ( )n n v n v at t v
Momentum equation
When mv
: shear stress
( ) ( )n n v n v at t v
B R B R B B R R B R= 2vv v v v v v v v v v v
B B R R B R B B R R2v v v v v v v v v v
R Rv v
( ) ( ) 0mv mv
nmv nmvv n v mv at t v
BB B( ) ( )
vnmv v v
t t t t
B B R R B B R R( )nmvv v v v v v v v v
BB B B B B( ) ( ) = 0ij
vv v v v v P a
t t
B B B B ijv v v v P
0,mv
nt
0,nv mv
( ) ( ) 0mv mv
nmv nmvv n v mv at t v
mvna a
v
combination of all terms
applying the mass balance equation
BB B B B B( ) ( ) = 0ij
vv v v v v P a
t t
BB B B B( ) ( ) = 0ij
vv v v v P a
t t
B 1ij
DvP a
Dt
B
22 ,
3
,
i
i
ijji
j i
vP v i j
xP
vvi j
x x
Energy equation
: only random motion contributes to the internal energy
( ) ( )n n v n v at t v
2R
1
2mv
2 2B R R R B E
1 1( )
2 2n v v v v v uv J
u: mass specific internal energy
: energy flux vectorE RJ n fv d
2R
1( )
2n n mv u
t t t
( ) ( )n n v n v at t v
2R
12
0mv
n nt t
2R R B
1( - )
2nv nv mv v v v v
R B B( ) :ijv v v P v i
iji j j
vP
v
2R
R
12
0mv
na na a vv v
B E B( ) ( ) : 0iju uv J P vt
using the continuity equation
E B:ij
DuJ P v
Dt
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
0
( )
f ff f fv a
t r v v
0x
f ffv
x
0ff
x x
0
0 ,x
fv f f
x
0
0 ,x
ff f v
x
0
0 x
f dTf f v
T dx
0
( )
f ff fv
t r v
heat flux in the x direction
Under local-equilibrium assumption and applying the RTA
0E, 0x x x x x
f dTJ q f v d f v v d
T dx
0 0,xf v d
1
3xv v
x
dTq k
dx : 1-D Fourier’s
law
201
3
fk v d
T
: 3-D Fourier’s law
Eq J fv d k T
00 x
f dTf f v
T dx
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 coolingFluid flow inside nanostructures, such as nanotubes and nanojet
Micro/Nanofluidics and Heat Transfer
The Knudsen Number and Flow Regimes
ratio of the mean free path to the characteristic length
Knudsen Number
Knudsen number relation with Mach number and Reynolds number
KnL
Re ,L
L
a
Ma
,a RT 2 /RT
: ratio of specific heat
Re = 2 / Re 2 /
L
L
L L
RT RT
Ma Ma RTRT
= = 2 ReRe 2 / Re 2 / LL L
Ma RT MaKn
RT RT
Knudsen Number
Rarefaction or Continuum
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
BTE, DSMC
Flow Regimes based on the Knudsen Number
0.001Kn
0.001 0.01Kn
0.1 10Kn
10Kn
Flow regimes
1. 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
13
2
Velocity profilesTemperature profiles
by
( )x y
xy 1
2
3
( )T y
wT
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
13
2
Velocity profilesTemperature profiles
by
( )x y
xy 1
2
3
( )T y
wT
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.
centerline
13
2
Velocity profilesTemperature profiles
by
( )x y
xy 1
2
3
( )T y
wT
Velocity Slip and Temperature Jump
tangentialnorma
l
wall
Tangential momentum (or velocity):The sameNormal momentum(or velocity):Reversed
Specular reflection
No shear force or friction between the gas and the wall
rp m
ip m
Diffuse reflection
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.
Momentum accommodation coefficient
tangential components
normal components
w
(the incident)
(the reflected)
(the MVD corresponding to T )
i
r
w
For specular reflection
For diffuse reflection
i r
i w
p p
p p
i r
i w
p p
p p
p mv i: incident, r: reflectedw: MVD corresponding to Tw
0v v
1v v
Thermal accommodation coefficient
For specular reflection
For diffuse reflection
For monatomic molecules, T involves translational kinetic energy only which is proportional to the temperature (K).
i rT
i w
, i.e., 0i r T
, i.e., 1r w T
i rT
i w
T T
T T
For polyatomic moleculesTranslational, 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
i rT
i w
T T
T T
0.87 ~ 0.97T
< 0.02T
0.87 ~ 1v
0.75 ~ 0.85v
Velocity slip boundary condition
Temperature jump boundary condition
2( ) 3
8bb
v xx b
yv y
v R Tv y
y T x
2 ( )2 2( )
1 Pr 4b
x bTb w
T y
v yTT y T
y R
Poiseuille flow
Assume that W >> 2H, edge effect can be neglected.incompressible and fully developed with constant properties
When Kn = /2H < 0.1
wq
x
2HW
x
y
Navier-Stokes equations
2Dup u f
Dt
fully developed flow2
2
1xv dp
y dx
/y H Let
2
2
1
( )xv dp
H dx
2 2
2xv H dp
dx
or
Velocity slip boundary condition
2( ) 3
8bb
v xx b
yv y
v R Tv y
y T x
neglecting thermal creep
The symmetry condition
2( )
b
v xx b
v y
dvv y
dy
2( ) = v x
xv
dvv
H d
1
2( 1) v x
xv
dvv
H d
2 v
vv
Kn
Let
1
( 1) 2 xx v
dvv
d
0
0xdv
d
2 2
2xv H dp
dx
2
1xv H dp
Cdx
0
0xdv
d
1 0C
22
2
1( )
2x
H dpv C
dx
1
( 1) 2 xx v
dvv
d
2
2(1)2x
H dpv C
dx
2
2
1
2 xv
dv H dpC
d dx
2 2
2 22v
H dp H dpC
dx dx
2 12
2v
dp H
dx
bulk velocity
2 221 1
( ) 22 2x v
H dp dp Hv
dx dx
2
2( ) 4 12x v
dp Hv
dx
1 1 2
2
0 0
( ) 4 12m x v
H dpv v d d
dx
1
3
0
2 14
2 3v
H dp
dx
2
1 63v
H dp
dx
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
22
2
4 12( )
1 63
vx
mv
dp Hdxv
v H dpdx
23 1 4( )
2 1 6vx
m v
v
v
( 1)x
m
v
v
3 1 4 1 6
2 1 6 1 6v v
v v
2( ) 3 3(1 )
2 2x
m
v
v
Energy equation
2p
DTc k T p v
Dt
2
2p x
T Tc v
x y
thermally fully developed condition with constant wall heat flux
2
2x
m
v
v
2 ( )2 2( )
1 Pr 4b
x bTb w
T y
v yTT y T
y R
temperature jump boundary condition
2 2( )
1 Prb
Tb
T y
yy
22
2
3 3(1 )
2 2x
m
v
v
31
3 (1 )
2 2C
0
0d
d
1 0C
2 41 2
3 (1 )( )
4 8C C
1
( 1) 2 T
2 2
3 (1 ) 5( 1)
4 8 8C C
1
3 (1 )2 2 2
2 2T T T
2
52
8 TC
2 43 (1 ) 5( ) 2
4 8 8 T
dimensionless temperature
By boundary condition
: temperature-jump distance
bulk temperature
( ) w
w
T T
H q
( 1)w
w
T Tq
H
( 1) 2 T
2 2w w
wT T
T T T Tq
H H
2 T H
1
0
( )( )x
m mm
vd
v
2
2x
m
v
v
1 2
20
( )m m d
integration by parts 1 2
20
( )m d
1
0
1( )
0d
m
33 (1 ),
2 2
2 43 (1 ) 5
( ) 24 8 8 T
1
2 4 3
0
3 (1 ) 5 3 (1 )2 2
4 8 8 2 2T T
213
0
3 (1 )
2 2d
1 2 2 22 6 4
0
9 6 1 2 3 4
4 4 2d
2204 72 8
420
251 18 22
105m T
Nusselt number
hLNu
" 4 4h w
w m m
hD q HNu
T T
( )w w mq h T T 4 hD H( )m w
mw
T Tk
H q
2 2
4 140
68 24 (8 / 3) 280 17 6 (2 / 3) 70
140T T
Nu
Poiseuille flow
Poiseuille flow with one of the plate being insulated
circular tube of inner diameter D
2
140
17 6 (2 / 3) 70 T
Nu
2
140
26 3 (1/ 3) 70 T
Nu
2
48
11 6 48w
Dw m T
q DNu
T T k
Gas Conduction-from the Continuum to the Free Molecule RegimeHeat conduction between two parallel surfaces filled with ideal gases
1 2DF
T Tq
L
1T 2T
Lx
diffusion
jumpFree molecule
1T 2T
Lx
( )dT
q Tdx
9 5
4 vc
When Kn = /L << 1, diffusion regime
23/ 2 3 / 21 2
m,DF1 2
2
3
T TT
T T
2/ 3
3 / 2 3 / 2 3 / 21 1 2( )
xT x T T T
L
effective mean temperature and distribution
When Kn = /L >> 1, free molecule regime
Assume that T are the same at both walls.
1 21
(1 ),
2T
T
T TT
2 1
2
(1 )
2T
T
T TT
effective mean temperature
1 2m,FM 2
1 2
4T TT
T T
2 1FM
m,FM2 8
1T
T v
T Tq
RT
c P
net heat flux