Gas Flow in Complex Microchannels Amit Agrawal Department of - - PowerPoint PPT Presentation

Gas Flow in Complex Microchannels

Amit Agrawal Department of Mechanical Engineering Indian Institute of Technology Bombay

TEQIP Workshop IIT Kanpur, 6th October 2013

Motivation

• Devices for continual monitoring the health
• f 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

CPU Tj Graphics Tj Com ponent Tcase Exhaust Tem perature ( therm o lim it) Chassis Tem perature

Acoustics Acoustics Fan noise Fan 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 nm Die Size 0.80 cm2 0.64 cm2 0.40 cm2 Heat 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

Kn L λ =

Knudsen number

λ 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) 2

slip wall

du u dy σ λ σ − ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠

2 2 1 2 2

λ λ = − −

slip wall wall

du d u u C C dy 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 τ σ τ − = 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 2

slip wall

du u dy σ λ σ − ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠

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 microchannel

Assumptions: Flow is steady, two- dimensional and locally fully developed Flow is isothermal

x, u y, v

Governing Equations:

2

( ) .

w

Adp dx d u dA p RT p Kn const τ ρ ρ − − = = =

∫

Integral momentum equation Ideal gas law From,

2 RT p μ π λ =

11

Solution for gas flow in microchannel

(contd.)

Boundary conditions:

slip

u u p p Kn Kn = = =

at y = 0, H

}

specified at some reference location, x = x0 Solution 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 continuity

12

Solution for gas flow in microchannel

(contd.)

Velocity profile:

2 2 1 2 2 1 2

/ ( / ) 2 8 ( , ) ( ) 1/ 6 2 8 y H y H C Kn C Kn u x y u x C Kn C Kn − + + = + +

Pressure can be obtained from:

{

2 2 1 2 2 2 2 1 2 2 2 2 2 2 1 2 1 1

1 24 1 96 log 2Re 12 1 24 [ 1] log } 48Re 2 2 Re ; ; 1/ 30 2 / 3 8 / 3 4 32 1

• p

p p C Kn C Kn p p p p p x x p C Kn C Kn p p p H where uH Kn C Kn C Kn C Kn C u dA A u βχ β ρ β μ π χ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ − + − + + ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ − − + − − = − ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ = = + + + + ⎛ ⎞ = = ⎜ ⎟ ⎝ ⎠

∫

3 2 4 2 2 2 2 1 2

64 (1/ 6 2 8 ) C Kn C Kn C Kn C Kn + + +

13

Solution for Velocity

Streamwise 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?

1

2 C σ σ − =

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

y y μλ μ λ =

Normalized volume flux versus inverse Knudsen number

Pr: Pressure ratio

0 .4 0 .6 0 .8 1

0.1 0.2 0.3 0.4 0.5 U/Ucentreline y/H

Present m odel DSMC ( Karniadakis et al. 2 0 0 5 )

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 cleanliness Temperature of gas Knudsen number Three main methods to determine TMAC: Rotating cylinder method Spinning rotor gauge method Flow through microchannels

19

Specular reflection (τr = τi; σ = 0) Diffuse reflection (τr = 0; σ = 1)

Recall

TMAC versus Knudsen number for monatomic & diatomic gases

Monatomic Diatomic:

0.7

1 log(1 ) Kn σ = − +

• 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 .

1 log 1 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 2013 22

O(Kn) Resulting Equations Euler Equations 1 Navier Stokes Equations 2 Burnett Equations 3 Super Burnett Equations

Generalized Equations

5 October 2013 23

Burnett Equations

5 October 2013 24

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 2013 25

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 2013 26

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 2013 27

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 2013 28

Solution of the Equation

5 October 2013 29

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 2013 30

Slip Velocity

0 .2 0 .4 0 .6 0 .8 0 .2 0 .4 0 .6 0 .8 1

Us / Uin x/ L

DSM C ( Karniadakis et. al. 2 0 0 5 ) Burnett Analytical Solution Kn out = 0 .2 0 Ma out = 0 .2 1 2 Pr = 2 .2 8

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 Simulation

Microchannel with Sudden Contraction or Expansion

x/L P/Po

0.25 0.5 0.75 1 1 1.5 2 2.5 3

x/w y/w

46 48 50 52 54 1 2

x/w y/w

46 48 50 52 54 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 Setup

Vacuum pump

• Rotary with

speed of 350 lpm

• Maximum

vacuum 0.001 mbar

MFC

• 0 - 20 sccm
• 0 - 200 sccm
• 0 - 5000 sccm

APT

• 0 – 1 mbar
• 0 – 100mbar
• 0 – 1 bar

1 2 3 4 5 6 7 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 N2 Flow parameters range Re = 0.2 – 837 Kno = 0.0001 – 0.0605

Validation

(Flow in tube)

Kn f Re

10

• 4

10

• 3

10

• 2

10

• 1

10 10

1

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).

64 1 14.89 f Kn = + R e

11 different gas-solid combinations; Kn range 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 range Re = 0.2 – 837 Kno = 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/Po

0.25 0.5 0.75 1 2 4 6 8 10 12

Knj = 0.036 Knj = 0.007 Knj = 0.003 Knj = 0.001 Sudden Expansion d = 4 mm, D = 32 mm AR = 64, Nitrogen

Po Pi Junction

Radial pressure variation

38 local radius/radius of smaller section (P/Pw)static

1 2 3 4 5 6 7 8 0.95 0.96 0.97 0.98 0.99 1 1.01 1.02 Knj = 0.036, Just upstream of junction Knj = 0.036, Just downstream of junction Knj = 0.001, Just upstream of junction Knj = 0.001, Just downstream of junction Sudden Expansion AR = 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/Po

0.25 0.5 0.75 1 0.9 1 1.1 1.2 1.3 1.4 1.5 Straight SE Sudden Expansion, AR = 12.43 Res = 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).

Knj Ld/d

0.01 0.02 0.03 0.04 0.05 0.06 0.07 1 2 3 4 5 6 7 8 AR = 3.74 AR = 12.43 AR = 64 Curve fit Sudden Expansion Nitrogen Ld/d = AR

1.35/(400*Knj 0.45)

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

Knj Lu/d

0.01 0.02 0.03 0.04 0.05 0.06 0.07 5 10 15 20 25 30 35 AR = 3.74 AR = 12.43 AR = 64 Curve fit

Sudden Expansion Nitrogen Lu/d = 15.45(AR×Knj)

0.52

Velocity profile

41 (a)

Straight tube Sudden 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 − − = −

2

cr a u + =

∫

=

R

rdru m 2π ρ

Streamlines near sudden expansion junction

43

X/L Tube radius (m)

0.45 0.5 0.55 0.6 0.65 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 Expansion d = 9.5 mm, D = 33.5 AR = 12.43, Nitrogen Res = 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

• 1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Re range (0.5-124.3)

Re Nu

50 100 150 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)

0.00166 Present (Experimental) 8.58 Hooman [21] 3.55 Aydin and Avci [20] 3.55 Tunc and Bayazitoglu [12] 3.75 Ameel et al. [10] 3.6 Larrode et al. [5] Nu Source Our experimental data predicts 3 orders of magnitude smaller value as compared to theoretical analysis!!

Kn Nu

10

• 3

10

• 2

10

• 1

10

• 4

10

• 3

10

• 2

10

• 1

Run 1 Run 2 Run 3 Run 4

Validation & Repeatability

11 25.0 28.0 8420 12 38.3 34.3 10870 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

w T T w g

n T Kn T T ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + − = − Pr 1 2 2 γ γ σ σ

r T v ∂ ∂

x T u ∂ ∂

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

Acknowledgements

Colleagues: 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

Specular reflection (τr = τi; σ = 0) Diffuse reflection (τr = 0; σ = 1)

i r i

τ τ σ τ − = 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 2

slip wall

du u dy σ λ σ − ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠

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/L

0.25 0.5 0.75 1

Kn Kn

Microchannel with Sudden Contraction

• r Expansion

(Pressure distribution)

x/L P/Po

0.25 0.5 0.75 1 1 1.5 2 2.5 3

Microchannel with sudden Contraction

• r 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

0.25 0.5 0.75 1 0.01 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 2013 63 2 4 6 0 .0 1 0 .1 1 1 0

G Kn

Boltzm ann Solution

• Exp. data ( Pr = 5 )
• Exp. Data ( Pr = 4 )
• Exp. data ( Pr = 3 )

Burnett Analytical Solution ( C1 = 1 .4 6 6 , 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 D Dt ρ 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

2 2 1

u P P Cp

s

• ρ

− =

Stagnation pressure correction coefficient

Velocity measurement in rarefied gas flow through tube

29 - February - 2012

Study of rarefied gas flow through microchannel

67

Author Cp (Pitot tube geometry) Re range (Re is based on pitot tube ID) Remark

Homann (1952) 1+[8/(Re+0.64Re1/2)] Cylinder 3.2 - 120 Experimental Hurd et al. (1953) 11.2/Re Circular blunt nosed 1.7 - 100 Experimental Lester (1961) 2.975/Re0.43 Circular 1- 10 Numerical Chebbi and Tavoularis (1990) 4.2/Re Circular < 1 Experimental

Velocity measurement in rarefied gas flow through tube

29 - February - 2012

Study of rarefied gas flow through microchannel

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

Velocity measurement in rarefied gas flow through tube

29 - February - 2012

Study of rarefied gas flow through microchannel

69 % deviation in central velocity with reference to Sreekanth (1969)

Velocity measurement in rarefied gas flow through tube

29 - February - 2012

Study of rarefied gas flow through microchannel

70 (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

29 - February - 2012

Study of rarefied gas flow through microchannel

71

Continuum flow

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

29 - February - 2012

Study of rarefied gas flow through microchannel

72

Slip flow