gas flow in complex microchannelsteqipiitk.in/workshop/2013/pravartana13/slides...gas flow in...

Gas Flow in Complex Microchannels

Amit AgrawalDepartment of Mechanical EngineeringIndian Institute of Technology Bombay

TEQIP Workshop

IIT Kanpur, 6th October 2013

Motivation• Devices for continual monitoring the health

of a patient is the need of the hour• Microdevices are required which can

perform various tests on a person• Microdevice based tests can yield quick

results, with smaller amount of reagents, and lower cost

From internet 2 2

Motivation

CPUTj

GraphicsTj

ComponentTcase

ExhaustTemperature

(thermo limit)

Chassis Temperature

AcousticsAcousticsFan noiseFan noise

*Slide taken from Dr. A. Bhattacharya, Intel Corp

Motivation: Electronics cooling

• Heat Flux > 100 W/cm2*: IC density, operating frequency, multi-level interconnects

• Impact on Performance, Reliability, Lifetime

• Hot spots (static/dynamic) and thermal stress can lead to failure

• Strategy to deal with Very High Heat Flux (VHHF) Transients required

*2002 AMD analyst conference report @ AMD.com

Channel length 130 nm 90 nm 65 nmDie Size 0.80 cm2 0.64 cm2 0.40 cm2Heat flux 63 W/cm2 78 W/cm2 125 W/cm2

Applications

• Several potential applications of microdevices– Example: Micro air sampler to check for

contaminants– Breath analyzer, Micro-thruster, Fuel cells– Other innovative devices

• Relevant to space-crafts, vehicle re-entry, vacuum appliances, etc

5

Introduction

Flow regimes Kn Range Flow Model

Continuum flow Kn < 10-3 Navier-stokes equation with no slip B. C.

Slip flow 10-3 ≤ Kn < 10-1 Navier-stokes equation with slip B. C.

Transition flow 10-1≤ Kn < 10 Navier-stokes equation fails but iner-

molecular collisions are not negligible

Free molecular flow 10 ≤ Kn Intermolecular collisions are negligible as

compared to molecular collision to the

wall

Gas flow mostly in slip regime – regime of our interest

KnLλ

=Knudsen numberλ is mean free path of gas

(~ 70 nm for N2 at STP)L is characteristic dimension

Kn ~ 7x10-3 for 10 micron channel

Introduction

Maxwell’s model (1869)

Sreekanth’s model (1969)

2slip

wall

duudy

σ λσ−⎛ ⎞= ⎜ ⎟

⎝ ⎠

22

1 2 2λ λ= − −slipwall wall

du d uu C Cdy dy

How to determine the slip velocity at the wall?

σ is tangential momentum accommodation coefficient (TMAC) = fraction of molecules reflected diffusely from the wall

C1, C2: slip coefficients

7

Maxwell’s Slip Model

Specular reflection(τr = τi; σ = 0)

Diffuse reflection(τr = 0; σ = 1)

i r

i

τ τστ−

=

τi = tangential momentum of incoming molecule

τr = tangential momentum of reflected molecule

In practice, fraction of collisions specular, others diffuse

σ is fraction of diffuse collisions to total number of collisions

2slip

wall

duudy

σ λσ−⎛ ⎞= ⎜ ⎟

⎝ ⎠

Introduction

• New effects with gases– Slip, rarefaction, compressibility

• Continuum assumptions valid with liquids– Conventional wisdom should apply– Similar behavior possible at nano-scales

9

Outline and Scope• Flow in straight microchannel

– Analytical solution– Experimental data

• Applicability of Navier-Stokes to high Knudsen number flow regime?

• Flow in complex microchannels (sudden expansion / contraction, bend)– Lattice Boltzmann simulation– Experimental data

• Conclusions

10

Solution for gas flow in microchannelAssumptions:Flow is steady, two-

dimensional and locally fully developed

Flow is isothermalx, u

y, v

Governing Equations:2( )

.

wAdp dx d u dA

p RTp Kn const

τ ρ

ρ

− − =

==

∫ Integral momentum equationIdeal gas law

From,2RT

pμ πλ =

11

Solution for gas flow in microchannel(contd.)

Boundary conditions:

0

0

slipu u

p pKn Kn

=

=

=

at y = 0, H

} specified at some reference location, x = x0Solution procedure:1. Assume a velocity profile (parabolic in our case)

2. Assume a slip model (Sreekanth’s model in our case)

3. Integrate the momentum equation to obtain p

4. Obtain the streamwise variation of u

5. Obtain v from continuity12

Solution for gas flow in microchannel(contd.)

Velocity profile:2 2

1 22

1 2

/ ( / ) 2 8( , ) ( )1/ 6 2 8

y H y H C Kn C Knu x y u xC Kn C Kn

− + +=

+ +

Pressure can be obtained from:

{

22

1 0 2 00 0 0

2 2 20 0 01 0 2 0

20

2 2 2 21 2 1 1

1 24 1 96 log

2 Re 12 1 24 [ 1] log } 48Re

2 2Re ; ;

1/ 30 2 / 3 8 / 3 4 321

o

p p pC Kn C Knp p p

p p x xpC Kn C Knp p p H

where

uH Kn

C Kn C Kn C Kn Cu dAA u

βχ β

ρ βμ π

χ

⎛ ⎞ ⎛ ⎞ ⎛ ⎞− + − + +⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎛ ⎞⎛ ⎞ ⎛ ⎞ −

− + − − = −⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

= =

+ + + +⎛ ⎞= =⎜ ⎟⎝ ⎠∫

3 2 42 2

2 21 2

64(1/ 6 2 8 )

C Kn C KnC Kn C Kn

++ +

13

Solution for VelocityStreamwise velocity continuously increases –flow accelerates

Pressure drop non-linear

Lateral velocity is non-zero (but small in magnitude)

Normalized Volume Flux versus Knudsen Number

First-order slip-model fails to predict the Knudsen minima

Navier-Stokes equations seems to be applicable to high Kn with modification in slip-coefficients

Dongari et al. (2007) Analytical solution of gaseous slip flow in long microchannels. International Journal of Heat and Mass Transfer, 50, 3411-3421

Extension of Navier-Stokes to high Knudsen number regime

15

Questions?

• Can we extend this approach to other cases?

• What are the values of slip coefficients? – or tangential momentum accommodation

coefficient (TMAC)

• Why does this approach work at all?

12C σσ−

=

Extension of Navier-Stokes to high Knudsen number regime (contd.)

Idea extended to

- Gas flow in capillary (Agrawal & Dongari, IJMNTFTP, 2012)

- Cylindrical Couette flow (Agrawal et al., ETFS, 32, 991, 2008)

Conductance versus Kn

Experimental Data and Curve Fit to Data by Knudsen (1950)

Extension of Navier-Stokes to high Knudsen number regime (contd.)

N. Dongari and A. Agrawal, Modeling of Navier-Stokes Equations to High Knudsen Number Gas Flow. International Journal of Heat and Mass Transfer, 55, 4352-4358, 2012

( )( )e

e yy μλμλ

=Normalized volume flux versus inverse Knudsen number

Pr: Pressure ratio

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5

U/U

cent

relin

e

y/H

Present model

DSMC (Karniadakis et al. 2005)Kn = 1

Normalized Velocity Profile

Determination of Tangential Momentum Accommodation Coefficient (TMAC)

What does TMAC depend upon?Surface material - For same pressure drop, will flow rate in glass

& copper tubes be different?Gas - Will nitrogen and helium behave differently, in

the same material tube?Surface roughness and cleanlinessTemperature of gasKnudsen number

Three main methods to determine TMAC:Rotating cylinder methodSpinning rotor gauge methodFlow through microchannels

19



Recall

TMAC versus Knudsen number for monatomic & diatomic gases

Monatomic Diatomic: 0.71 log(1 )Knσ = − +

• TMAC = 0.80 – 1.02 for monatomic gases irrespective of Kn and surface material– Exception is platinum where σmeasured = 0.55 and σanalytical = 0.19– TMAC = 0.926 for all monatomic gases

• TMAC = 0.86 – 1.04 for non-monatomic gases – Correlation between TMAC and Kn– Different behavior for di-atomic versus poly-atomic gases?

• TMAC depends on surface roughness & cleanliness– Effect (increase/ decrease) is inconclusive

• Temperature affects TMAC– Effect is small for T > room temperature

( )7.01log1 Kn+−=σ

Summary of Tangential Momentum Accommodation Coefficient

Agrawal et al. (2008) Survey on measurement of tangential momentum accommodation coefficient. Journal of Vacuum Science and Technology A, 26, 634-645

Boltzmann Equation

5 October 201322

O(Kn) Resulting Equations0 Euler Equations1 Navier Stokes Equations2 Burnett Equations3 Super Burnett Equations

Generalized Equations

5 October 201323

Burnett Equations

5 October 201324

Higher Order Continuum Models

• Have been applied to Shock Waves • Here, want to apply to Micro flows

Conventional Burnett Equations (1939)

BGK-Burnett Equations (1996,1997)

Super Burnett Equations Augmented Burnett Equations (1991)

Grad’s Moment Equation R13 Moments Equation

5 October 201325

Ref: Agarwal et al. (2001) Physics of Fluids, 13:3061–3085; Garcia-Colin et al. (2008) Physics Report, 465:149–189

Analytical Solution from Higher Order Continuum Equations

5 October 201326

Numerical Solution of Higher-Order Continuum Equations

• Agarwal et al. (2001): Planar Poiseuille flow;Convergent solution till Kn = 0.2; solution diverged beyond it

• Xue et al. (2003): Couette flow; Kn < 0.18• Bao & Lin (2007): Planar Poiseuille flow; Kn <

0.4 • Uribe & Garcia (1999); Planar Poiseuille flow;

Kn < 0.1

5 October 201327

Present Approach• Start with the solution of the Navier-Stokes

equations; • Substitute the solution in the Burnett equations

and evaluate the order of magnitude of additional terms in the Burnett equations;

• Identify the highest order term and augment the governing equation with this additional term, and re-solve the governing equations;

• Repeat above two steps, till convergence is achieved.

5 October 201328

Solution of the Equation

5 October 201329

Results

Error (in %) versus Knudsen number as given by Burnett equation and Navier-Stokes at the wall (C1 = 1.066, C2 = 0.231; Ewart et al. (2007))

5 October 201330

Slip Velocity

0.2

0.4

0.6

0.8

0 0.2 0.4 0.6 0.8 1

Us/U

in

x/L

DSMC (Karniadakis et. al. 2005)

Burnett Analytical Solution

Knout = 0.20Maout = 0.212

Pr = 2.28

Slip velocity comparison with DSMC data (Karniadakis et al. 2005)

N. Singh, N. Dongari, A. Agrawal (2013) Analytical solution of plane Poiseuille flow within Burnett hydrodynamics, Microfluidics and Nanofludics, to appear

Complex Microchannels

32

Gas – Nitrogen (Helium to a limited extent)

Developing length is 10% greater than required from calculations

Test section for flow through sudden expansion / contraction

Test section for flow through 90º bend

Lattice Boltzmann SimulationMicrochannel with Sudden Contraction or Expansion

x/L

P/P o

0 0.25 0.5 0.75 11

1.5

2

2.5

3

x/w

y/w

46 48 50 52 540

1

2

x/w

y/w

46 48 50 52 540

1

2

1. Pressure drop in each section follows theory

2. Secondary losses are small

3. Limited transfer of information in microchannel

Possible to understand complex microchannels in terms of primary units?

Agrawal et al. "Simulation of gas flow in microchannels with a sudden expansion or contraction," Journal of Fluid Mechanics, 530,135-144, (2005). 33

34

Experimental SetupVacuum pumpRotary with speed of 350 lpmMaximum vacuum 0.001 mbar

MFC0 - 20 sccm0 - 200 sccm0 - 5000 sccm

APT0 – 1 mbar0 – 100mbar0 – 1 bar

1

23

4567

8

To vacuum system

To display 1 – Gas cylinder

2 – Pressure regulator

3 – Particle filter

4 – Mass flow controller

5 – Inlet reservoir

6 – Test section

7 – Outlet reservoir

8 – Absolute pressure transducer

The leakage is ensured to be less than 2 % of the smallest mass flow rate. The absolute

static pressure along the wall is measured for different mass flow rates of N2 at 300 K.

0.1 – 45 mbar

4 – 5000 sccm

N2Flow parameters range

Re = 0.2 – 837

Kno = 0.0001 – 0.0605

Validation(Flow in tube)

Kn

fRe

10-4 10-3 10-2 10-1 100 1010

10

20

30

40

50

60

70

Sreekanth (Brass-N2)Ewart ( Fused slilica- N2)Ewart ( Fused slilica- He)Ewart ( Fused slilica- Ar)Brown ( Copper- Air)Brown ( Copper- H2)Brown ( Iron- Air)Choi ( Fused slilica- N2)Tang ( Fused silica- N2)Tang ( Fused silica- He)Demsis (Copper- N2)Demsis (Copper- O2)Demsis (Steel- O2)Demsis (Steel- Ar)

Present} Verma et al. Journal of Vacuum Science and Technology A, 27(3) 584-590 (2009).

641 14.89

fKn

=+

Re

11 different gas-solid combinations; Knrange 0.0004-2.77 and Re range of 0.001-227

fRe for different combinations of gas-surface material seems small as compared to experimental scatter.

35

Experimental setup and test section geometry for sudden expansion

36

Figure 1a. Schematic diagram of the experimental set up

Figure 1b: Test section geometry for sudden expansion

Parameter Maximum uncertainty

Mass flow rate ± 2% of full scale

Absolute pressure ± 0.15 % of the reading

Diameter ± 0.1 %

Reynolds number ± 2 %

Knudsen number ± 0.5 %

Pressure loss coefficient ± 6 %

Temperature ± 0.3 K

Flow parameters rangeRe = 0.2 – 837Kno = 0.0001 – 0.075

Flow through tube with sudden expansion

37

Oliveira and Pinho (1997) noted separation at Re = 12.5 (AR = 6.76) for liquid flow

Goharzadeh and Rodgers (2009) noted separation at Re = 100 (AR = 1.56) for liquid flow

No flow separation at the junction

Static pressure variation along wall for area ratio = 64

X/L

P/P

o

0 0.25 0.5 0.75 10

2

4

6

8

10

12Knj = 0.036Knj = 0.007Knj = 0.003Knj = 0.001

Sudden Expansiond = 4 mm, D = 32 mmAR = 64, Nitrogen

PoPi

Junction

Radial pressure variation

38

local radius/radius of smaller section

(P/P

w) st

atic

0 1 2 3 4 5 6 7 80.95

0.96

0.97

0.98

0.99

1

1.01

1.02

Knj = 0.036, Just upstream of junctionKnj = 0.036, Just downstream of junctionKnj = 0.001, Just upstream of junctionKnj = 0.001, Just downstream of junction

Sudden ExpansionAR = 64, Nitrogen

(c)Radial static pressure variation just upstream and just downstream of the junction for AR = 64.

Comparison with straight tube

39

Static pressure variation (normalized with the outlet pressure) along the wall.

X/L

P/P

o

0 0.25 0.5 0.75 10.9

1

1.1

1.2

1.3

1.4

1.5

StraightSE

Sudden Expansion, AR = 12.43Res = 11.3, Knj = 0.0112, Nitrogen

Junction

Lu Ld

Variation in Lu and Ld

40

Variation in Ld (normalized with smaller section tube diameter d) versus Knudsen number at junction (Knj).

KnjL d

/d

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070

1

2

3

4

5

6

7

8

AR = 3.74AR = 12.43AR = 64Curve fit

Sudden ExpansionNitrogen

Ld/d = AR1.35/(400*Knj

0.45)

Variation in Lu (normalized with smaller section tube diameter d) versus Knudsen number at junction (Knj).

Knj

L u/d

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070

5

10

15

20

25

30

35

AR = 3.74AR = 12.43AR = 64Curve fit

Sudden ExpansionNitrogen

Lu/d = 15.45(AR×Knj)0.52

Velocity profile

41

(a)

Straight tubeSudden expansion

Analysis of velocity distribution

42

p1, p2, known from experimental measurements, Need velocity distribution for calculating shear stress and momentum M1 and M2 .

Assuming velocity distribution as second order parabolic velocity profile

The mass flow rate can be calculated as

At r = R, velocity at the wall u = us

1 2 w 2 1(p p )A Ddx [M M ]τ π− − = −

2crau +=

∫=R

rdrum0

2πρ

Streamlines near sudden expansion junction

43

X/L

Tube

radi

us(m

)

0.45 0.5 0.55 0.6 0.650

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

99 %95 %90 %70 %50 %30 %10 %5 %1 %0.5%

Sudden Expansiond = 9.5 mm, D = 33.5AR = 12.43, NitrogenRes = 2.8, Knj = 0.03

% of mass flow rate

Streamlines near the junction for rarefied gas flow.

Streamlines near sudden expansion junction

44

Varade et al.; JFM (under review)

Bend Microchannel(Lattice Boltzmann Simulations)

Streamlines near the bend for Kn = 0.202

Presence of an eddy at the corner for Re = 2.14!

Mass flow rate

Mass flux in bend to mass flux in straight

Mass flux in bend can be up to 2% more than that in straight under same condition

White et al. (2013) found same result using DSMC

Agrawal et al. Simulation of gas flow in microchannels with a single 90 bend. Computers & Fluids 38 (2009) 1629–1637

Experimental data for bend

Test section for flow through 90º bend

Heat Transfer Coefficient(Experimental setup)

To vacuum pump

2 3 4 5

1

6

7

Port for Inlet Temperature probe (Tci)

Hot Water Inlet (Thi)

Port for Outlet Pressure gauge (Po)

Nitrogen Outlet

Port for Outlet Temperature probe (Tco)

Water Outlet (Tho)

Port for Inlet Pressure gauge (Pi)

Nitrogen Inlet

Nusselt number measurements(Circular tube with constant wall temperature BC)

Knm

Nu

10-4 10-3 10-2 10-10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Re range (0.5-124.3)

Re

Nu

0 50 100 1500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Knm range 0.01-10-4

Nu versus Kn

Nu versus Re

Gr / Re2 < 10-4

Comparison against theoretical values (at Kn = 0.01)

0.00166Present (Experimental)8.58Hooman [21]3.55Aydin and Avci [20]3.55Tunc and Bayazitoglu [12]3.75Ameel et al. [10]3.6Larrode et al. [5]NuSource

Our experimental data predicts 3 orders of magnitudesmaller value as compared to theoretical analysis!!

Kn

Nu

10-3 10-2 10-110-4

10-3

10-2

10-1

Run 1Run 2Run 3Run 4

Validation & Repeatability

1125.028.084201238.334.310870

Percentage difference

Nu from the present

experiment

Nu from Dittus-Boelter

Correlation

Re

Why is heat transfer coefficient so small?

• Rarefaction reduces ability of gas to remove heat from wall

• Temperature jump at wall

• Radial convection may be important (i.e.,term comparable to ), but ignored in theoretical analysis

• Reason for difference is still not fully understood

wT

Twg n

TKnTT ⎟⎠⎞

⎜⎝⎛∂∂

⎥⎦

⎤⎢⎣

⎡+

−=−

Pr122γγ

σσ

rTv ∂∂

xTu ∂∂

Conclusions• Possible to derive solution for microchannel flow

for higher-order slip-model• Possible to extend the applicability of N-S

equations by modification of slip coefficients• TMAC of monatomic gas = 0.93 (independent of

temperature and surface and Kn)• Flow in complex microchannel exhibits non-

intuitive behavior

AcknowledgementsColleagues: Dr. S. V. Prabhu, Dr. A.M. Pradeep

Students: Nishanth Dongari, Abhishek Agrawal, Vijay Varade, Bhaskar Verma, Gaurav Kumar, Anwar Demsis

Funding: IIT Bombay, ISRO

Effect of sudden expansion on velocity distribution

55

Rate of increase in shear stress and momentum is more in sudden expansion than in

straight tube towards the expansion junction

Shear stress variation along the wall Momentum variation along the length

Varade et al. (2012) (to be submitted)

Maxwell’s Slip Model



i r

i

τ τστ−

=

τi = tangential momentum of incoming molecule

τr = tangential momentum of reflected molecule

In practice, fraction of collisions specular, others diffuse

σ is fraction of diffuse collisions to total number of collisions

2slip

wall

duudy

σ λσ−⎛ ⎞= ⎜ ⎟

⎝ ⎠

DSMC Simulations(Couette flow)

Slip from DSMC/ Slip from Maxwell model

versus

Knudsen number

57

DSMC Simulations(Couette flow)

Slip from DSMC/ Slip from Maxwell model

versus

Knudsen number

58

x/L0 0.25 0.5 0.75 1

Kn

Kn

Microchannel with Sudden Contraction or Expansion

(Pressure distribution)

x/L

P/P o

0 0.25 0.5 0.75 11

1.5

2

2.5

3

Microchannel with sudden Contraction or Expansion

(Velocity distribution)

x/L

U

0.48 0.49 0.5 0.51 0.52

0.02

0.03

0.04

0.05

0.06

x/L

U

0 0.25 0.5 0.75 10.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Kn

What do we learn?

1. Good agreement with theory

2. Pressure drop in each section follows theory

3. Secondary losses are small

4. Compressibility and rarefaction have opposite effects

5. Limited transfer of information in microchannel

Possible to understand complex microchannels in terms of primary units?

TMAC versus Knudsen number

62

Flow Rate Vs Kn

5 October 201363

0

2

4

6

0.01 0.1 1 10

G

Kn

Boltzmann Solution

Exp. data (Pr = 5)

Exp. Data (Pr = 4)

Exp. data (Pr = 3)

Burnett Analytical Solution (C1 = 1.466, C2 = 0.5)

Figure 2. Comparison of proposed sol. with the exp. data of Ewart et al.(2007) and the sol. of linearized Boltzmann equation by Cercignani et al.(2004). (C1 = 1.1466 and C2 = 0.5, Chapman & Cowling (1970).

Introduction (contd.)(Compressibility)

For flow to be compressible: (Gad-el-Hak, 1999)

i.e., both spatial and temporal variations in density are small as compared to the absolute density

Compressibility effect is important because of large pressure drop in microchannel

1 0DDtρ

ρ=

Gas versus liquid flow

• New effects with gases– Rarefaction, slip, compressibility

• Continuum assumptions valid with liquids– Conventional wisdom should apply– Similar behavior possible at nano-scales

Velocity measurement in rarefied gas flow through tube

29 - February - 2012 Study of rarefied gas flow through microchannel 66

For low Re the magnitude of viscous forces are comparable with inertia forces hence

Bernoulli formula needs correction

Schematic diagram of experimental set up

221 u

PPCp soρ−

=

Stagnation pressure correction coefficient



Author Cp

(Pitot tube

geometry)

Re range (Re is based on

pitot tube ID)

Remark

Homann (1952)1+[8/(Re+0.64Re1/2)]

Cylinder3.2 - 120 Experimental

Hurd et al. (1953)11.2/Re

Circular blunt nosed1.7 - 100 Experimental

Lester (1961)2.975/Re0.43

Circular1- 10 Numerical

Chebbi and

Tavoularis (1990)

4.2/Re

Circular< 1 Experimental



Variation in Cp with Re (based on Pitot tube ID) Variation in central velocity with Re (based on Pitot tube ID)



% deviation in central velocity with reference to Sreekanth (1969)



(a) (b)

(c) (d)Comparison of experimental velocity measurement with Sreekanth (1969), Homann’s correction

Velocity measurement in rarefied gas flow through sudden expansion using Pitot tube


Continuum flow

Velocity measurement in rarefied gas flow through sudden expansion using Pitot tube


Slip flow

gas flow in complex microchannelsteqipiitk.in/workshop/2013/pravartana13/slides...gas flow in...

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