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 3

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 0 .

5  f MICRO f MACRO  3 .

5 • NUSSELT NUMBER Nu 0 .

2  Nu MICRO Nu MACRO  16 • CRITICAL REYNOLDS NUMBER Re C 0 .

13  Re C , MICRO Re C , MACRO  0 .

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LAMINAR FLOW OF AN INCOMPRESSIBLE FLUID WITH CONSTANT PROPERTIES IN A CIRCULAR PIPE  1 r   r • FRICTION FACTOR r  u  r • REYNOLDS NUMBER • POISEUILLE NUMBER  f dp dx dp D  dx Re  u D 2 2  m  u  m D Po  f  Re D 8

f  64 Re D Po  64 9

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 10

• ENTRANCE EFFECTS • VISCOUS HEATING • TEMPERATURE- AND PRESSURE DEPENDENT PROPERTIES • WALL ROUGHNESS • COMPRESSIBILITY • CONJUGATE HEAT TRANSFER • AXIAL HEAT CONDUCTION 11

ENTRANCE EFFECTS

FOR LAMINAR FLOW IN A CIRCULAR PIPE X fd , hyd  0 .

05 Re D D X fd , therm  0 .

05 Re D Pr D 12

WATER FLOW IN A 2D CHANNEL – CFD/EXPERIMENT x   x D h Re 13

• 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 14

VISCOUS HEATING

ENERGY EQUATION FOR FLOW IN A PIPE u  T  x  1 Re Pr     2 T  x 2  1 r   r    r  T  r        Br Re Pr    du dr    2 VISCOUS HEATING (VISCOUS DISSIPATION) Br   u 2 m k  T 15

BRINKMAN NUMBER

• THE IMPORTANCE OF THE VISCOUS HEATING TERM IS DETERMINED BY THE BRINKMAN NUMBER • FOR EXAMPLE, FOR CONSTANT HEAT FLUX Nu  11  48 48 Br  1 0 .

229  Br • IN MACRO FLOWS VISCOUS HEATING IS IMPORTANT ONLY FOR VERY VISCOUS FLUIDS OR VERY HIGH VELOCITIES 16

• • IN MICRO FLOWS BRINKMAN NUMBER IS USUALLY VERY SMALL Br   u 2 m k  T

WATER:

μ = 8.55·10 -4 kg(m·s) k = 0.613 W/(m·K) ΔT = 1 ºC u m = 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 u m = 1 m/s Br ≈ 7·10 -4 • THE INFLUENCE OF VISCOUS HEATING ON HEAT TRANSFER IN MICRO FLOWS IS USUALLY NEGLIGIBLE 17

VISCOUS HEATING CAN BE IMPORTANT DUE TO VERY STRONG DEPENDENCE OF LIQUID VISCOSITY ON TEMPERATURE WATER T = 300 K

ν

T = 310 K

ν

= 8.576·10 -7 = 6.999·10 -7 m 2 /s m 2 /s TEMPERATURE RISE OF

10 K

CAUSES

18%

DECREASE IN KINEMATIC VISCOSITY WHICH RESULTS IN CORRESPONDING INCREASE OF THE LOCAL Re NUMBER AFFECTING THE FRICTION FACTOR 18

THERMAL EXPLOSION

THE MOMENTUM AND ENERGY EQUATIONS FOR FULLY DEVELOPED FLOW IN A CIRCULAR PIPE ARE dp dx  1 r d dr     du dr    r 1 d dr    r dT dr      k    du dr    2  0 FOR EXPONENTIAL DEPENDENCE OF LIQUID VISCOSITY ON THE TEMPERATURE    0 exp    E RT      0 exp    E RT 0    exp     E  T  RT 0 2 T 0     19

INTRODUCING NEW VARIABLES   r 2 r 0 2   E  T  T 0  RT 0 2 THE ENERGY EQUATION REDUCES TO d 2  d  2  1  d  d    e   0   const     0 d  d    0  0 IT HAS NO SOLUTION FOR   2 NO FULLY DEVELOPED FLOW!

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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 T in - T out =6.2 o C 21

EXAMPLE OF A CFD RESULT • INLET CONDITIONS D= 20 μm; T = 300 K

ν

= 8.576·10 -7 m 2 /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 23

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 24

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 26

PRESSURE AND DENSITY VARIATIONS ALONG THE TUBE AT DIFFERENT INLET MACH NUMBERS 27

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 k s AND FRICTION VELOCITY v * (OR WALL SHEAR STRESS

τ w

) v *    w  w    u  r r  r 0 28

FLOW REGIMES FOR ROUGH PIPES HYDRAULICALLY SMOOTH 0  k S v *   5 TRANSITION 5  k S  v *  70 f  f LAMINAR TURBULENT f  f    k Re s , Re    TURBULENT COMPLETELY ROUGH k S  v *  70 f  f    k Re s    TURBULENT 29

• 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 30

CONJUGATE HEAT TRANSFER • IN MICRO FLOWS THE RELATIVE THICKNESS OF THE CHANNEL WALL s/D h 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 31

EXPERIMENT

• LAMINAR FLOW Re ≈ 50 L/D ≈ 160 • CONSTANT WALL HEAT FLUX 32

THEORETICAL SOLUTION

• WALL TEMPERATURE dT w dx  q w  c p  const • BULK TEMPERATURE dT m dx  q w  c p  const • NUSSELT NUMBER Nu  T w q  w T m D  k 48 11  4 .

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EXPERIMENT - RESULTS

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CFD – CONJUGATE HEAT TRANSFER INCLUDED 35

AXIAL CONDUCTION NUMBER

• THE IMPORTANCE OF THE CONJUGATE HEAT TRANSFER IS GIVEN BY THE

AXIAL CONDUCTION NUMBER

M M    cond conv //  k s e  ce f L s V 36

• THE NUMBER M IS USUALLY VERY LOW FOR MACRO CHANNELS (HIGH V, SMALL e S /e f , LARGE L) BUT CAN BE LARGE FOR MICRO CHANNELS (LOW V, e S /e f 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 37

AXIAL HEAT CONDUCTION

ENERGY EQUATION FOR FLOW IN A PIPE u  T  x  1 Pe    2 T  x 2  1 r   r r  T  r   AXIAL HEAT CONDUCTION • AXIAL HEAT CONDUCTION CAN USUALLY BE NEGLECTED UNLESS PECLET NUMBER IS VERY LOW Pe  Re Pr  50 38

• 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 39

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 40

CFD: PIPE FLOW • D = 50 μm; P IN ≈ 20 atm; P OUT ≈ 2 atm • VISCOUS COMPRESSIBLE TURBULENT FLOW • • INLET: V X ≈ 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

AND THE WALLS

“MOVE”

FOR GASES • TO A CERTAIN DEGREE THE SITUATION IS SIMILAR TO LOW PRESSURE HIGH-ALTITUDE AERONAUTICAL FLOWS • HOWEVER, REYNOLDS AND MACH NUMBERS ARE MUCH LOWER 42

MOLECULAR MAGNITUDES

•

NUMBER DENSITY OF MOLECULES n

• n  p k B T n  2 .

69  10 25 m  3

MEAN MOLECULAR SPACING δ

  n  1 / 3   3 .

3  10  9 m •

MOLECULAR DIAMETER d

DILUTE GAS: δ/d > 7 AIR: d  3 .

7  10  10 m THE DATA FOR p = 1 atm; T = 0 ºC 43

MEAN FREE PATH

• THE DISTANCE TRAVELED BY THE MOLECULES BETWEEN COLLISIONS IS KNOWN AS

MEAN FREE PATH λ

  1  d 2 n 2 AT p = 1 atm; T = 25 ºC GAS AIR N 2 CO 2 O 2 He Ar λ, nm 61.1

60.4

40.2

65.0 176.5 64.4

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KNUDSEN NUMBER

• THE KEY DIMENSIONLESS PARAMETER IS THE

KNUDSEN NUMBER Kn

Kn   L   M 2 Re Kn < 0.01

0.01 < Kn <0.1

0.1 < Kn < 10 Kn > 10 CONTINUUM SLIP FLOW TRANSITIONAL FLOW FREE-MOLECULAR FLOW 45

LIMITS OF APPROXIMATIONS

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NON-CONTINUUM EFFECTS LIQUIDS

• FOR SUFFICIENTLY HIGH STRAIN RATE   HEAT FLUX/TEMPERATURE GRADIENTS RELATIONS BECOME NONLINEAR     u  y  2  HERE

τ

IS THE MOLECULAR TIME-SCALE • THE CRITICAL VALUE IS VERY HIGH FOR ORDINARY LIQUIDS BUT NOT SO FOR COMPLEX FLUIDS 47

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 49

REFERENCES 1.

2.

3.

4.

5.

6.

Bayraktar & Pidugu, Int J Heat Mass Trans, 2006 Cui et al, Phys Fluids, 2004 Gad-el-Hak, Int J Heat Mass Trans, 2003 Gamrat et al, Int J Heat Mass Trans, 2005 Guo & Li, Int J Heat Mass Trans, 2003 Herwig & Hausner, Int J Heat Mass Trans, 2003 7.

8.

Herwig, ZAMM, 2002 Hetsroni et al, Int J Heat Mass Trans, 2005, p. 1982 9.

Hetsroni et al, Int J Heat Mass Trans, 2005, p. 5580 10. Judy et al, Int J Heat Mass Trans, 2002 11. Karniadakis & Beskok,

Micro Flows

, 2002 12. Koo & Kleinstreuer, Int J Heat Mass Trans, 2004 13. Maranzana et al, Int J Heat Mass Trans, 2004 50

THANKS!

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