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MICRO FLOWS: AN INTRODUCTION

Michael Shusser

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SIZE RANGES OF MACRO, MICRO, AND NANO DEVICES

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FLUID FLOW AND HEAT TRANSFER IN SINGLE-PHASE

FLOW OF A NEWTONIAN FLUID IN A MICRO-CHANNEL

• NO MULTIPHASE FLOW• NO POLYMERS OR BIO-FLUIDS• NO COMPLEX GEOMETRIES• NO ELECTRO-KINETIC FLOWS

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IS EVERYTHING

DIFFERENT OR JUST

SMALLER?

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IS THE CONTINUUM APPROXIMATION VALID?

POSSIBLE NON-CONTINUUM EFFECTS:

• SLIP AT THE BOUNDARY• STRESS/RATE OF STRAIN

RELATION IS NONLINEAR• CONTINUUM APPROXIMATION

FAILS

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FOR THE TIME BEING WE ASSUME THAT THE

CONTINUUM THEORY IS VALID

• LIQUIDS

• GASES FOR L > 5 μM

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MANY OF STUDIES OF BASIC HEAT AND FLUID FLOW PROBLEMS IN

BASIC GEOMETRIES FOUND LARGE DEVIATIONS FROM EXPECTED

RESULTS

• FRICTION FACTOR f

• NUSSELT NUMBER Nu

• CRITICAL REYNOLDS NUMBER ReC

5.3f

f5.0

MACRO

MICRO

16Nu

Nu2.0

MACRO

MICRO

43.0Re

Re13.0

MACRO,C

MICRO,C

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LAMINAR FLOW OF AN INCOMPRESSIBLE FLUID WITH CONSTANT PROPERTIES IN A

CIRCULAR PIPE

• FRICTION FACTOR

• REYNOLDS NUMBER

• POISEUILLE NUMBER

dx

dp

r

ur

rr

1

Du

Re mD

2u

Ddx

dp

f2m

DRefPo

9

DRe

64f 64Po

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SCALING EFFECTS• THE EFFECTS THAT CAN BE

NEGLECTED IN MACRO SCALES BUT ARE IMPORTANT IN MICRO SCALES ARE CALLED SCALING EFFECTS

• PROVIDED THE CONTINUUM APPROXIMATION REMAINS VALID, ALL THE DISCREPANCIES BETWEEN MICRO AND MACRO FLOWS CAN BE EXPLAINED AS SCALING EFFECTS

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• ENTRANCE EFFECTS

• VISCOUS HEATING

• TEMPERATURE- AND PRESSURE DEPENDENT PROPERTIES

• WALL ROUGHNESS

• COMPRESSIBILITY

• CONJUGATE HEAT TRANSFER

• AXIAL HEAT CONDUCTION

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ENTRANCE EFFECTS

FOR LAMINAR FLOW IN A CIRCULAR PIPE

Dhyd,fd Re05.0

D

X PrRe05.0

D

XD

therm,fd

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WATER FLOW IN A 2D CHANNEL – CFD/EXPERIMENT

ReD

xx

h

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• ENTRANCE EFFECTS ARE NOT ALWAYS NEGLIGIBLE IN MICRO FLOWS

• DEVELOPING FLOW IS STRONGLY INFLUENCED BY THE INLET VELOCITY PROFILE

• THERE IS NOT ENOUGH DATA ON ENTRANCE EFFECTS FOR VARIOUS CROSS-SECTIONS

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VISCOUS HEATING

ENERGY EQUATION FOR FLOW IN A PIPE

VISCOUS HEATING

(VISCOUS DISSIPATION)

Tk

uBr

2m

2

2

2

dr

du

PrRe

Br

r

Tr

rr

1

x

T

PrRe

1

x

Tu

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BRINKMAN NUMBER• THE IMPORTANCE OF THE VISCOUS

HEATING TERM IS DETERMINED BY THE BRINKMAN NUMBER

• FOR EXAMPLE, FOR CONSTANT HEAT FLUX

• IN MACRO FLOWS VISCOUS HEATING IS IMPORTANT ONLY FOR VERY VISCOUS FLUIDS OR VERY HIGH VELOCITIES

Br229.0

1

Br4811

48Nu

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• IN MICRO FLOWS BRINKMAN NUMBER IS USUALLY VERY SMALL

• WATER: μ = 8.55·10-4 kg(m·s) k = 0.613 W/(m·K)

ΔT = 1 ºC um = 0.1 m/s Br ≈ 1.4·10-5

• AIR: μ = 1.846·10-5 kg(m·s) k = 0.0263 W/(m·K)

ΔT = 1 ºC um = 1 m/s Br ≈ 7·10-4

• THE INFLUENCE OF VISCOUS HEATING ON HEAT TRANSFER IN MICRO FLOWS IS USUALLY NEGLIGIBLE

Tk

uBr

2m

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VISCOUS HEATING CAN BE IMPORTANTDUE TO VERY STRONG DEPENDENCE OFLIQUID VISCOSITY ON TEMPERATURE

WATER T = 300 K ν = 8.576·10-7 m2/s T = 310 K ν = 6.999·10-7 m2/s

TEMPERATURE RISE OF 10 K CAUSES18% DECREASE IN KINEMATIC VISCOSITYWHICH RESULTS IN CORRESPONDINGINCREASE OF THE LOCAL Re NUMBERAFFECTING THE FRICTION FACTOR

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THERMAL EXPLOSIONTHE MOMENTUM AND ENERGY

EQUATIONS FOR FULLY DEVELOPED

FLOW IN A CIRCULAR PIPE ARE

FOR EXPONENTIAL DEPENDENCE OF

LIQUID VISCOSITY ON THE

TEMPERATURE

dr

dur

dr

d

r

1

dx

dp

20

0

000 RT

TTEexp

RT

Eexp

RT

Eexp

0dr

du

kdr

dTr

dr

d

r

12

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INTRODUCING NEW VARIABLES

THE ENERGY EQUATION REDUCES TO

IT HAS NO SOLUTION FOR

NO FULLY DEVELOPED FLOW!

20

2

r

r

20

0

RT

TTE

0ed

d1

d

d2

2

01 0d

d

0

const

2

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ISOPROPANOL FLOW IN A SQUARE MICRO CHANNEL

• L = 11.4 cm; D = 74.1 μm; (L/D = 1543)

• FOR Re ≈ 300 Tin - Tout =6.2 oC

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EXAMPLE OF A CFD RESULT

• INLET CONDITIONS

D= 20 μm; T = 300 K ν = 8.576·10-7 m2/s

Re = 2000 V = 85.76 m/s !

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• VISCOUS HEATING HAS USUALLY NO INFLUENCE ON HEAT TRANSFER IN MICRO FLOWS

• ITS INFLUENCE ON FRICTION FACTOR CAN BE IMPORTANT DUE TO VERY STRONG DEPENDENCE OF LIQUID VISCOSITY ON TEMPERATURE, ESPECIALLY FOR LONG CHANNELS

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VARIABLE PROPERTIES• DUE TO LARGE GRADIENTS IN MICRO

FLOWS THE DEPENDENCE OF PROPERTIES ON PRESSURE AND TEMPERATURE IS IMPORTANT

• LIQUIDS SHOULD BE MODELED AS INCOMPRESSIBLE WITH TEMPERATURE-DEPENDENT VISCOSITY

• SOMETIMES PRESSURE-DEPENDENCE OF VISCOSITY SHOULD ALSO BE TAKEN INTO ACCOUNT

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LIQUID FLOW AT 30 MPa

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COMPRESSIBILITY EFFECTS

• THE FRICTION-INDUCED PRESSURE DROP PER TUBE LENGTH COULD BE LARGE IN FLOW THROUGH A NARROW CHANNEL

• COMPRESSIBILITY EFFECTS CAN BE IMPORTANT IN GAS FLOWS EVEN FOR LOW MACH NUMBERS

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PRESSURE AND DENSITY VARIATIONS ALONG THE TUBE AT DIFFERENT INLET

MACH NUMBERS

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WALL ROUGHNESS• ROUGHNESS LEADS TO INCREASING

FRICTION FACTOR AT THE SAME Re NUMBER AND DECREASING VALUE OF THE CRITICAL Re NUMBER (EARLIER TRANSITION FROM LAMINAR TO TURBULENT FLOW)

• THE INFLUENCE OF THE ROUGHNESS IS DETERMINED BY ITS GRAIN SIZE ks AND FRICTION VELOCITY v* (OR WALL SHEAR STRESS τw)

w*v

0rrw r

u

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FLOW REGIMES FOR ROUGH PIPES

HYDRAULICALLY SMOOTH

LAMINAR

TURBULENT

TRANSITION TURBULENT

COMPLETELY

ROUGH

TURBULENT

Reff 5vk

0 *S

70vk

5 *S

70vk *S

Re,Re

kff s

Re

kff s

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• FOR LOW Re (D < 100 μm) SOME EXPERIMENTS OBSERVED DEVIATIONS FROM THE CLASSICAL THEORY INCLUDING THE INFLUENCE OF ROUGHNESS IN LAMINAR FLOW

• ONE POSSIBLE REASON FOR THE DISCREPANCY IS NON-UNIFORMITY OF THE ROUGHNESS

• THERE IS NOT ENOUGH DATA ON INFLUENCE OF ROUGHNESS ON HEAT TRANSFER

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CONJUGATE HEAT TRANSFER• IN MICRO FLOWS THE RELATIVE

THICKNESS OF THE CHANNEL WALL s/Dh IS USUALLY MUCH LARGER THAN IN MACRO FLOWS

• THEREFORE CONVECTIVE HEAT TRANSFER IN THE FLUID AND HEAT CONDUCTION IN THE WALL MUST BE ACCOUNTED FOR SIMULTANEOUSLY

• THIS CONJUGATED HEAT TRANSFER IS USUALLY NEGLIGIBLE FOR MACRO FLOWS

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EXPERIMENT

• LAMINAR FLOW Re ≈ 50 L/D ≈ 160

• CONSTANT WALL HEAT FLUX

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THEORETICAL SOLUTION

• WALL TEMPERATURE

• BULK TEMPERATURE

• NUSSELT NUMBER

constcm

q

dx

dT

p

ww

constcm

q

dx

dT

p

wm

36.411

48

k

D

TT

qNu

mw

w

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EXPERIMENT - RESULTS

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CFD – CONJUGATE HEAT TRANSFER INCLUDED

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AXIAL CONDUCTION NUMBER

• THE IMPORTANCE OF THE CONJUGATE HEAT TRANSFER IS GIVEN BY THE AXIAL CONDUCTION NUMBER M

VceL

ek

Mf

ss

conv

//cond

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• THE NUMBER M IS USUALLY VERY LOW FOR MACRO CHANNELS (HIGH V, SMALL eS/ef, LARGE L) BUT CAN BE LARGE FOR MICRO CHANNELS (LOW V, eS/ef IS NOT SMALL, SMALL L)

• FOR LARGE M THE WALL HEAT FLUX BECOMES STRONGLY NON-UNIFORM: MOST OF THE HEAT IS TRANSFERRED TO THE FLUID NEAR THE ENTRANCE TO THE CHANNEL

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AXIAL HEAT CONDUCTIONENERGY EQUATION FOR FLOW IN A PIPE

AXIAL HEAT CONDUCTION

• AXIAL HEAT CONDUCTION CAN USUALLY BE NEGLECTED UNLESS PECLET NUMBER IS VERY LOW

r

Tr

rr

1

x

T

Pe

1

x

Tu

2

2

50PrRePe

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• OILS: Pr >>1 LIQUIDS: Pr ~ 5

GASES: Pr ~ 0.7 LIQUID METALS: Pr << 1

• IN MACRO FLOWS THE AXIAL HEAT CONDUCTION IS NEGLIGIBLE EXCEPT LIQUID METAL FLOWS

• IN MICRO FLOWS THE AXIAL HEAT CONDUCTION SOMETIMES MUST BE TAKEN INTO ACCOUNT

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TURBULENCE IN MICRO FLOWS

• MICRO FLOWS ARE USUALLY LAMINAR (Re < 2000)

• MOST EXAMPLES OF TURBULENT FLOW ARE USUALLY FOR RELATIVELY LARGE DIAMETERS (D > 300 μm)

• FOR LARGE PRESSURE DIFFERENCE, GAS FLOWS CAN BE TURBULENT EVEN FOR SMALL DIAMETERS

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CFD: PIPE FLOW• D = 50 μm; PIN ≈ 20 atm; POUT ≈ 2 atm

• VISCOUS COMPRESSIBLE TURBULENT FLOW

• INLET: VX ≈ 125 m/s Re ≈ 25,000

• DO STANDARD TURBULENCE MODELS (LIKE k-ε) WORK IN THIS CASE?

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NON-CONTINUUM EFFECTS - GASES

• THE FLOW IS RAREFIED FOR GASES AND THE WALLS “MOVE”

• TO A CERTAIN DEGREE THE SITUATION IS SIMILAR TO LOW-PRESSURE HIGH-ALTITUDE AERONAUTICAL FLOWS

• HOWEVER, REYNOLDS AND MACH NUMBERS ARE MUCH LOWER

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MOLECULAR MAGNITUDES• NUMBER DENSITY OF MOLECULES n

• MEAN MOLECULAR SPACING δ

• MOLECULAR DIAMETER dDILUTE GAS: δ/d > 7 AIR:

THE DATA FOR p = 1 atm; T = 0 ºC

Tk

pn

B

325m1069.2n

3/1n m103.3 9

m107.3d 10

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MEAN FREE PATH• THE DISTANCE TRAVELED BY THE

MOLECULES BETWEEN COLLISIONS IS KNOWN AS MEAN FREE PATH λ

AT p = 1 atm; T = 25 ºC

GAS AIR N2 CO2 O2 He Ar

λ, nm 61.1 60.4 40.2 65.0 176.5 64.4

2nd

12

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KNUDSEN NUMBER

• THE KEY DIMENSIONLESS PARAMETER IS THE KNUDSEN NUMBER Kn

Kn < 0.01 CONTINUUM

0.01 < Kn <0.1 SLIP FLOW

0.1 < Kn < 10 TRANSITIONAL FLOW

Kn > 10 FREE-MOLECULAR FLOW

Re

M

2LKn

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LIMITS OF APPROXIMATIONS

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NON-CONTINUUM EFFECTS - LIQUIDS

• FOR SUFFICIENTLY HIGH STRAIN RATE THE STRESS/RATE OF STRAIN AND HEAT FLUX/TEMPERATURE GRADIENTS RELATIONS BECOME NONLINEAR

HERE τ IS THE MOLECULAR TIME-SCALE

• THE CRITICAL VALUE IS VERY HIGH FOR ORDINARY LIQUIDS BUT NOT SO FOR COMPLEX FLUIDS

2

y

u

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FUTURE DIRECTIONS OF RESEARCH

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CONCLUSIONS• PROVIDED THE CONTINUUM

APPROXIMATION REMAINS VALID, ALL THE DISCREPANCIES BETWEEN MICRO AND MACRO FLOWS CAN BE EXPLAINED AS SCALING EFFECTS

• THE MAIN SCALING EFFECTS ARE VARIABLE PROPERTIES, COMPRESSIBILITY, CONJUGATE HEAT TRANSFER

• SOME INFLUENCE OF ENTRY LENGTH, VISCOUS HEATING, AXIAL HEAT CONDUCTION AND ROUGHNESS IS ALSO POSSIBLE

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REFERENCES1. Bayraktar & Pidugu, Int J Heat Mass Trans, 20062. Cui et al, Phys Fluids, 20043. Gad-el-Hak, Int J Heat Mass Trans, 20034. Gamrat et al, Int J Heat Mass Trans, 20055. Guo & Li, Int J Heat Mass Trans, 20036. Herwig & Hausner, Int J Heat Mass Trans, 20037. Herwig, ZAMM, 20028. Hetsroni et al, Int J Heat Mass Trans, 2005, p. 19829. Hetsroni et al, Int J Heat Mass Trans, 2005, p. 558010. Judy et al, Int J Heat Mass Trans, 200211. Karniadakis & Beskok, Micro Flows, 200212. Koo & Kleinstreuer, Int J Heat Mass Trans, 200413. Maranzana et al, Int J Heat Mass Trans, 2004

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THANKS!