Hydrodynamics and Heat Transfer in Gaseous Microflows ... · PDF fileHydrodynamics and Heat Transfer in Gaseous Microflows: from Theory to Experiment ... air, standard conditions thermodynamic

Hydrodynamics and Heat Transfer in Gaseous Microflows:

from Theory to Experiment

Ecole thématique 2013 GDR CHANTPrapoutel les Sept Laux – January 10, 2013

S. Colin

2

Outline

1. Introduction : gas microflows applications

2. Gas microflows and rarefaction

3. Different regimes function of Kn

4. Importance of slip flow regime

5. Modelling of gaseous microflows

6. Examples of specific behaviours

7. Need for experimental data on gas microflows

8. Conclusions

3

Gas Microflows Applications

� Micro heat exchangers for cooling of electronic components or for chemical applications

� Micro nozzle for control of nano satellite position

� Micro gas analyzers – micro-chromatography

� Fluidic micro actuators for active control of aerodynamic flows

� Vacuum generators for extracting biological samples – Knudsen micropumps

� Mass flow and temperature micro-sensors

� Pressure gauges

� …Artificial lung

Micro-nozzle for

mass spectrometry

1. Introduction

4

Gas Microflows –Scales to Take Into Account

L

δλ

d

d : mean molecular diameter

δ : mean molecular spacing

λ : mean free path

L : characteristic length of the flow

2. Rarefaction

5

Limits of UsualContinuum Approach (N-S, classic BC…)

• sampling volume (SV) contains 10 000 molecules

� 1% of statistical fluctuations on macroscopic properties.

The mean volume occupied by a molecule is δ

4 310 22SV

l δ > ≈ 4 310L

δ≫

Negligible

Statistical Fluctuations

1L

λ≪ Thermodynamic Equilibrium

• local thermodynamic equilibrium: enough collisions inside the SV

� λ small compared with lSV

1SV

lλ ≪

• binary collisions

� enough molecular spacing

1d

δ≪ Dilute Gas



λ : mean free path

L : characteristic length of the flow

2. Rarefaction

6

Dilute gas

Thermodynamic equilibrium

Negligible statistical fluctuations

1d δ ≪

1Lλ ≪

4 / 310L δ ≫

Conventional Limits

δδδδ / d

101 100102

100

101

102

103

104

105

106

107

L / d

δδ δδ /

d == ==

7

L / δδδδ ==== 100

λλλλ / L

==== 0,1

dense gasdilute gas

thermodynamic

disequilibrium

significant

statistical

fluctuations

negligible

statistical

fluctuations

air, standard conditions

thermodynamic

equilibrium C

S

T

M



λ : mean free path

L : characteristic length

of the flow

BIRD G.A., Molecular gas dynamics and the

direct simulation of gas flows, Oxford,

Clarendon Press, 1998.

( ) ( )( )31 2 KnL d d Lλ πδ= =

( )3 22Hard Sphere

dλ δ π=

2. Rarefaction

7

( )2k RTλ µ ρ=

Reu Lρ

µ= Ma

u u

a RTγ= =

KnL

λ=

Knudsen number

2

MaKn

Rek γ=

δ / d101 100102

100

101

102

103

104

105

106

107

L d /K

n 10

=

-3

Standard conditions for He air SO2

Kn

10=

-1

Kn

10=

+1

(C)

(S)

(T)

(M)

δ=

/

7

d

Regimes as a function of Kn

3. Regimes

8

Kn

H2

H1

0,001 0,01 0,1 1 10

écoulement classique

Non raréfié

apparitionde glissementet d'un saut

de températureà la paroi

Légèrement raréfié

disparitionde la notion

de milieu continu

Modérèment raréfié

les chocs entre molécules

deviennent extrêmement

rares

Hautement raréfié

100 10 1 0,1 0,01

h(µm)

h

h

Regimes as a Functionof Hydraulic Diameter

Rarefaction increase

Continuum flow Slip flow Transition flow Free molecular flow

Classic BC

Continuum medium

classic flow negligible

intermolecular

collisions

continuum

hypothesis no

longer valid

slip velocity and

temperature jump

at the walls

Air in standard conditions

3. Regimes

9

Knudsen Layer and Slip at the Wall

� Thickness:

1 ou 2 λ

� Local

thermodynamic

disequilibrium

O (

)λ

Knu

dsen

Lay

er

s w allu u-

w allu

gasu

slipu

W all

G a s

n

s

4. Slip

10

10

2

sslip slip

uu u

nλ

∂ = + + ∂

sslip

uu

nλ

∂⇒ =

∂

• Basic explanation

~ λ

s

n

us(λ)

Slip at the Wall – Physical Phenomenon

4. Slip

11

Accommodation Coefficients

Specular reflection Diffuse reflexion

n

s

(a) (b)

1 − σu σu

Same behaviour for the temperature: σT

4. Slip

2u t

s wall

u

uu u

n

σλ

σ Γ

− ∂− =

∂

12

Slip Flow Regime: Some Examplesof Various Velocity Slip Boundary Conditions

� Initial form– Maxwell, J.C. (1879) Philosophical

Transactions of the Royal Society,170.

2 3

4

u t nt wall

u

u u R Tu u

n t p t

σ µλ

σ ΓΓ

− ∂ ∂ ∂ − = + +

∂ ∂ ∂

22 3 3

2 4

u tt wall

u

u T Tu u

n T s t T t

σ µ µλ

σ ρ ρΓΓ

− ∂ ∂ ∂− = − +

∂ ∂ ∂ ∂

2

1 *

u tt wall

u

Kn uu u

b Kn n

σ

σ Γ

− ∂− =

− ∂

( )2

tanh*

u tt wall

u

uu u Kn

n

σ

σ Γ

− ∂− =

∂

2 * ** *

* 2 *

u tt wall

u

u Kn pu u Kn Re

n t

σ

σΓ

− ∂ ∂ − = +

∂ ∂

2 2 22

2 2 2

2 92

16

u t t t tt wall

u

u u u uu u

n n s t

σ ∂ ∂ ∂ ∂λ λ

σ ∂ ∂ ∂ ∂Γ Γ

−− = − + +

� Curvature effects– Barber, R.W., et al. (2004) Vacuum, 76.

� Higher-order forms– Deissler, R.G. (1964) Int. Journal

of Heat and Mass Transfer, 7.

� Other hybrid dimensionless forms– Karniadakis, G.E. & Beskok, A. (2002) Microflows:

Fundamentals and Simulation, Springer-Verlag.

– Xue, H. & Fan, Q. (2000) Microscale ThermophysicalEngineering, 4.

– Jie, D., et al. (2000) Journal of Micromechanics and Microengineering, 10.

4. Slip

13

Slip Flow Regime: Examples of Temperature Jump Boundary Conditions

� Initial form– Smoluchowski, M. (1898) Annalen

der Physik und Chemie,64, 101-130.

� Additional term– Sparrow, E. M. & Lin, S. H.

(1962) Journal of Heat Transfer, 84, 363-369.

� Higher-order forms– Deissler, R. G. (1964) International

Journal of Heat and Mass Transfer,7, 681-694.

� Langmuir boundary condition– Myong, R. S., et al. (2006) International

Journal of Heat and Mass Transfer,49, 2502-2513.

2 2

1

Twall

T

TT T

Pr n

σ γ λ

σ γ Γ

− ∂− =

+ ∂

( )2 2 2 2

2 2 2

2 2

1

177 145 92

1 256

Twall

T

TT T

Pr n

T T T

n t s

σ γ λ

σ γ

γ λ

γ

Γ

Γ

− ∂− =

+ ∂

− ∂ ∂ ∂− + +

+ ∂ ∂ ∂

( ) ( )1wall nT aT a T λ=

= + −

( )2

2 2 4 1

1 1 2

t wallT T uwall

T T u p

u uTT T

Pr n c

σ γ λ γ σ σ

σ γ γ σ σΓ

−− ∂ −− = +

+ ∂ + −

4. Slip

14

Modelling Gas Microflows

Navier-Stokes equationswith no-slip B.C.

Navier-Stokes equations

with 1 -order slip B.C.st

QGD or QHD equations

with 1 -order slip B.C.

or

st

Navier-Stokes equations

with 2 -order slip B.C.nd

Burnett equationswith slip B.C.

DSMC or Lattice Boltzmann

Eu

ler

equ

atio

ns

Kn

1010-1

10-3

0

5. Modelling

15

Some Particularities of Gas Microflows:Thermal Creep

� Principle of Knudsen pump

2 3

4

u ss wall

u

u T

Tu u

n t

σλ

σ

µ

ρΓΓ

− ∂ − = +

∂

∂

∂

LT

LT

LT H

TH

TH

TL

T

stage 2 stage Nstage1flow flow

LT

continuum regime

slip or transition

regime

6. Specific Behaviours

16

Some Particularities of Gas Microflows:Thermal Creep

� Knudsen pump with curved parts

HT

HTHT

LT LT

HTLT HT

stage1

stage 2 stage N

HTLT LT

LT LT LT

LT LT

flow

flow


17

Thermal Creep

� Simulation using Fluent

+ specific UDFs

1.3

1.5

1.7

1.9

2.1

2.3

2.5

0 5 10 15 20 25

P1*

s*

N = 2N = 4 N = 8

N = 25

2 3

4

v

fluid wall

wallv wall

u v R Tu u

n s P s

σ µα λ

σ

− ∂ ∂ ∂ − = + +

∂ ∂ ∂

2 2

1

Tfluid wall

wallT

TT T

Pr n

σ γ λβ

σ γ

− ∂− =

+ ∂

( )1* av LP P RTρ=

HT

HTHT

LT LT

HTLT HT

stage1

stage 2 stage N

HTLT LT

LT LT LT

LT LT

flow


18

Heat Transfer in Slip Flow Regime

� Example

z

y

h−

h+

hϕ

Insulated wall

Uniform heat flux

( )

12

2

* 26 147 * 210 *

4 140 1 3 *Nu

ς ξ ξ

ξ

− + +

= + +

*wall

uu u

nξ

Γ

∂− =

∂

*w

TT T

nς

Γ

∂− =

∂Colin, S., Gas microflows in the slip flow regime: a critical review on

convective heat transfer, Journal of Heat Transfer, 134 (2), 2012.


19

Motivation for Experimental Analysis of Gas Microflows

� Wide literature on modelling and numerical simulation of gas microflows, in different rarefaction regimes

� However, few available experimental data

� Crucial need of smart experimental data, for example to:

– Help identifying the best BC to be used in slip flow regime and the limit of applicability of the associated analytical models

– Analyse the influence of surface, which may vary with � different materials – silicon, metals, polymers, glass and fused silica…

� different kind of manufacturing – wet chemical etching, reactive ion etching, laser etching, moulding, embossing, drilling, micromilling…

7. Experimentation

20

Experimental Analysis of Gas Microflows

� Example: gas flow in a microchannel

Main quantities of interest

Mass flowrate

GLOBAL data

Dijkstra, M., et al. (2008) Sensors and Actuators A: Physical,

143, 1-6.

Pressure

Temperature

Velocity

LOCAL data

Concentration

7. Experimentation

21

Flowrate: Accurate Measurement

p Vm

R T=

1 dV dp pV dTm p V

R T dt dt T dt

= + −

ɺ

Tank

(m, p, T, V)

mɺ

µsystem

Constant Pressure Method Constant Volume Method

Droplet Tracking Method

� For low flowrates: need of specific setups

� Basic principle based on the equation of state – suitable for dilute gases

with good

thermal insulation

7. Experimentation

22

Réservoir

amont

Réservoir

aval

MicrosystèmeUpstream

tank

Downstream

tank

Microsystem

( )P t

( )T t

Example of Constant Volume Method

7. Experimentation

23

ICA Mass Flowrate Setup

Pitakarnnop, J., et al. (2010) Microfluidics and Nanofluidics, 8, 57-72.

Reservoir A

Reservoir B

7. Experimentation

24

Kn010-210-1100101

Example of Results: Flowrates of He or Ar

Pitakarnnop, J., et al. (2010) Microfluidics and Nanofluidics, 8, 57-72.

Issue: difficult to separate

the roles of

• the model of BC

• the accommodation coefficient

Interest to access a direct

measurement of the velocity profile

7. Experimentation

25

Main Flowrates Data on Gas Microflows

� Droplet tracking method– Pong, K.-C., et al. (1994) FED-197, ASME, New York, pp. 51-56.

– Harley, J. C., et al. (1995) Journal of Fluid Mechanics, 284, 257-274.

– Zohar, Y., et al. (2002) Journal of Fluid Mechanics, 472, 125-151.

– Maurer, J., et al. (2003) Physics of Fluids, 15, 2613-2621.

– Colin, S., et al. (2004) Heat Transfer Engineering, 25, 23-30.

– Ewart, T., et al. (2006) Experiments in Fluids, 41, 487-498.

� Constant pressure method– Jousten, K., et al. (2002) Metrologia, 39, 519-529.

� Constant volume method– Arkilic, E. B., et al. (1998) Experiments in Fluids, 25, 37-41

– Arkilic, E. B., et aL. (2001) Journal of Fluid Mechanics, 437, 29-43.

– Ewart, T., et al. (2007) Journal of Fluid Mechanics, 584, 337-356.

– Pitakarnnop, J., et al. (2010) Microfluidics and Nanofluidics, 8, 57-72.

– Szalmás, L., et al. (2010) Microfluidics and Nanofluidics, 9, 1103-1114.(mixtures of gases)

7. Experimentation

26

Pressure Measurements – Discrete Data

� Shih, J. C et al. (1996), ASME DSC-59, pp. 197-203.– First local data for gas flows in microchannels

– Channel 4,000×40×1.2 µm3

– He & N2

� Zohar, Y. et al. (2002) Journal of Fluid Mechanics, 472, 125-151.– Detailed measurements

– Channels 4,000×40×(0.53 & 0.97) µm3

– He, Ar & N2

� Jang, J. & Wereley, S. T. (2004) Microfluidics and Nanofluidics, 1, 41-51. – Rectangular channels with higher aspect ratio (0.36)

– Channels ?×105×39 µm3

– Air

� Turner, S. E. et al. (2004) Journal of Heat Transfer, 126, 753-763.– Entrance effects analysed; influence of roughness 0.4 % to 6 %: insignificant

– Channel 30,000×1,000×(2.3 to 50) µm3

– Air

2(He) (N )0.16 or 0.055

oKn =

2(Ar) (N )0.20 or 0.067oKn =

0.0018o

Kn =

0.15oKn =

7. Experimentation

27

Pressure Measurement

� Integrated pressure sensors

Zohar, Y. et al. (2002) Journal of Fluid Mechanics, 472, 125-151.

• Microchannel• length 4000 µm

• width 40 µm

• depth 0.5 or 1 µm

• Capillary connection• width 4 µm

• depth 0.2 µm

7. Experimentation

28



Nitrogen flow• channel depth

0.97 µm

• outlet pressure

100 kPa

• accuracy 5 %

Model

• first order slip flow

• plane flow

• diffuse

accommodation

0.067o

Kn =

7. Experimentation

29



Shih, J. C et al. (1996), ASME DSC-59, pp. 197-203.

Argon flow• channel depth

0.53 µm

• outlet pressure

100 kPa

• accuracy 5 %

Model

• first order slip flow

• plane flow

• diffuse

accommodation

0.20o

Kn =

Helium flow, channel depth 1.2 µm

7. Experimentation

30

Measurements of Pressure Fieldsat the Wall

� Pressure-sensitive paints (PSP)– Luminescent molecules coated at the wall; once excited, emit at a longer

wavelength

– Luminescent intensity depends on O2 concentration (oxygen quenching phenomenon), related to pressure

– Non intrusive technique - High spatial resolution

– Need calibration and transparent side

– Cannot be used for oxygen free gases

– Too thick for use at microscale� Huang, C. et al. (2007) Journal of Microelectromechanical Systems, 16, 777-785.

� Pressure-sensitive molecular films (PSMF)– Technique developed at Nagoya University

� Mori, H. et al. (2005) Physics of Fluids, 17, 100610.

� Matsuda, Y. et al. (2007) Experiments in Fluids, 42, 543-550.

� Matsuda, Y., et al. (2009) Experiments in Fluids, 47, 1025-1032.

� Matsuda, Y., et al. (2011) Microfluidics and Nanofluidics, 10, 165-171.

7. Experimentation

31

Pressure-Sensitive Molecular Films (PSMF)

� Example of luminophore:

Pt(II) Mesoporphyrin IX

� Langmuir–Blodgett (LB)

deposition method

Matsuda, Y. et al. (2009) Experiments in Fluids, 47, 1025-1032.

7. Experimentation

32

From PSP to PSFM

� Relative luminescent intensity fields - 160 160 µm2 surface

Matsuda, Y. et al. (2011) Microfluidics and Nanofluidics, 10, 165-171.

PSP

Standard deviation 0.23

PSFM

Standard deviation 0.016

7. Experimentation

33

Experimental Results

� Micro-nozzle flow

– 2 configurations

– Pi = 10 kPa

– Po = 1 kPa


7. Experimentation

34

PSMF – Comparison with DSMC

Pressure distribution Pressure along the centerline


7. Experimentation

35

PSMF – Comparison with DSMC

Pressure distribution Pressure along the centerline


7. Experimentation

36

Temperature Measurement

� Various available techniques for measurement at the wall

– Thin film Resistance Thermo Detectors (RTD)

– Thin Film ThermoCouples (TFTC)

� 25x25 µm2 to 80x80 µm2 embedded junction in a 100-150 nm thick film.

20 °C – 900 °C

Zhang, X., et al. (2006) Journal of Micromechanics and Microengineering,

16, 900.

– Semiconducting Sensors (SC)

– Temperature sensitive paint (TSP)

� Promising new techniques for measurement within the flow

– Molecular Tagging Thermometry (MTT)

� Hu, H., et al. (2010) Measurement Science and Technology, 21,085401:1-14

7. Experimentation

37

Velocity Measurement

� Two recent techniques (for gas) currently under

investigation

– Micro Particule Image Velocimetry (µPIV)

– Micro Molecular Tagging Velocimetry (µMTV)

7. Experimentation

38

Principle of Classic PIV

http://www.dantecdynamics.com

7. Experimentation

39

Difference Between PIV and µPIV

http://www.dantecdynamics.com

Meinhart, C. D. et al. (2000)

Measurement Science and Technology, 11, 809-814.

7. Experimentation

40

µPIV for Air Flow in Square Microchannels

Yoon, S. Y. R. et al. (2006) Journal of Power Sources, 160, 1017-1025.

� Square sections

– 1 1 mm2

� Tracers

– smoke particles

– water droplets

� Re = 50 - 820

� spatial resolution

– 40 - 60 µm

� Not in rarefied regimes

7. Experimentation

41

µPIV for Nitrogenin Rectangular Channels

� Rectangular channels

– 1 mm 0.5 mm

� Tracers

– fluorescent oil droplets

– diameter 0.5 to 2 µm

� Re = 26 – 130

� Not in rarefied regimes

Sugii, Y. and Okamoto, K. (2006) In Proceedings of ICNMM2006, Limerick, pp. ICNMM2006-96216:1-6.

7. Experimentation

42

µPIV

• Particles seeding

• The whole test section is illuminated

• Resolution: defined by the optical depth

of field

• Limitations:

• Need of small particles (100- 300 nm)

=> sensitivity to Brownian motion

=> difficult particles pattern detection

• Background noise due to out-off focus

particles

µMTV

• Molecular tracers

• A line of molecules is tagged

• Resolution: defined by the laser beam

diameter

• Non intrusive technique

• Limitations to be clarified

Maynes and Webb, Exp. Fluids, 32(1), 2002

Microtube Ø 705 µm

Velocimetry at Small Scale

t0 t1 u

7. Experimentation

43

Molecular Tagging Velocimetry (MTV) Principle

� Direct UV tagging of specific molecules

– Once excited: immediate fluorescence

– After a delay: phosphorescence

� Efficient with liquids

– Supramolecules

� Only tested with gases in

unconfined flows

– Acetone & Biacetyl

7. Experimentation

44

Micro Molecular Tagging Velocimetry & Thermometry (µMTV-µMTT) in Liquids

� Current data on microflows

only for liquids. Examples:

– µMTV in a Hagen-Poiseuille flow in a fused silica microtube

– µMTV and µMTT in electro osmotic flow

Maynes, D. and Webb, A. R. (2002)

Experiments in Fluids, 32, 3-15.

∆t = 200 µs

Hu, H., et al. (2010) Measurement Science and Technology, 21, 085401:1-14.

∆t = 4.5 ms

7. Experimentation

45

µMTV with Gas: Delicate Choice of Material for the Walls

� Fluorescence images obtained in TSC3 (a) and Suprasil (b)

channels

Samouda, F., et al. (2011) Proceedings of GASMEMS11, Bertinoro, Italy.

(a) (b)

7. Experimentation

46

PEEK® channel

multilayer rectangular channel

1 x 5 mm2 cross-section, 20 cm length

Microchannel Fabrication

Samouda F. (2012) PhD Thesis University of Toulouse

7. Experimentation

47

PEEK® channel

Suprasil® lenses


7. Experimentation

48

PEEK® channel

Suprasil® lenses

Borofloat® window


7. Experimentation

49

PEEK® channel

Suprasil® lenses

Borofloat® window

O-ring sealing


7. Experimentation

50

PEEK® channel

Suprasil® lenses

Borofloat® window

Thermocouple

O-ring sealing


7. Experimentation

51

Suprasil® lenses

PEEK® channel

Borofloat® window

PEEK® plate

Thermocouple

O-ring sealing


7. Experimentation

52

PEEK® channel

Suprasil® lenses

Borofloat® window

O-ring sealing

Thermocouple

PEEK® plate

Swagelok® connectors


7. Experimentation

53

Camera access

Laser access


7. Experimentation

54

Inlet reservoir

Thermocouple

Outlet capacitive gauge

Visualization access

Suprasil® lens

Laser access

Flow

Suprasil® window

Thermocouples

Channel inlet Channel outlet

7. Experimentation

55

MTV SYSTEM

7. Experimentation

56

Laser guiding arm

CCD Camera

Intensified Relay Optics (IRO)

Argon

Seeding circuit

IRO Controller

Programmable Timing Unit

Computer

Channel

Laser

• Flow seeding

• Tagging

• Detecting

• Data processing

Velocity profiles measurement

7. Experimentation

57

m=200, n= 50 m=600, n= 50 m=1000, n= 50 m=60, n= 50

Post processing

time

12 s8 s2 s1 s

G

t0

Camera

Laser pulse

Image

acquisition

1 n

Final averaged

image

Average

G

t0

2 1 2 n

1 m

7. Experimentation

58

Data Processing

7. Experimentation

59

Standard averaging technique• t0 = 1 µs (35000 images) • t1 = 80 µs (50000 images)

Good agreement between theoretical and experimental data for the maximum

velocity

Experimental data

Theoretical

profile

Experimental flow conditions • Pmoy = Patm

• dp /dx = 1kPa/m• T= 25 °C• Re = 423

Comparison withTheory

7. Experimentation

60

MTV – First Results & Next Steps

� Preliminary steps in velocity field analysis– µMTV for confined flows: validated for continuum flows

– Towards rarefied flows: exploitable signal at lower pressure

� Reaching the slip flow regime: challenging for measurement

of slip at the wall

� Extensions of the technique to µMTT (thermometry)

– Lifetime and intensity of phosphorescence are a function of temperature

� Extensions of the technique to µMTM (manometry)

– Phosphorescence is quenched by O2

7. Experimentation

61

Conclusions – Future Needs

� A lot of theoretical and numerical investigations on gas

microflows and heat transfer

� No definitive consensus on some models, especially

concerning slip boundary conditions

� Need of smart local experimental data

– Velocity fields

– Temperature fields

– Data on heat transfer

62

Acknowledgements

� Funding from European

Community's 7th

Programme FP7/2007-

2013 under grant

agreement ITN

GASMEMS n°215504

� More information on

http://www.gasmems.eu

Hydrodynamics and Heat Transfer in Gaseous Microflows ... · PDF fileHydrodynamics and Heat Transfer in Gaseous Microflows: from Theory to Experiment ... air, standard conditions thermodynamic

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