1 MICRO FLOWS: AN INTRODUCTION Michael Shusser

2 SIZE RANGES OF MACRO, MICRO, AND NANO DEVICES

3 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

4 IS EVERYTHING DIFFERENT OR JUST SMALLER?

5 IS THE CONTINUUM APPROXIMATION VALID? POSSIBLE NON-CONTINUUM EFFECTS: SLIP AT THE BOUNDARY STRESS/RATE OF STRAIN RELATION IS NONLINEAR CONTINUUM APPROXIMATION FAILS

6 FOR THE TIME BEING WE ASSUME THAT THE CONTINUUM THEORY IS VALID LIQUIDS GASES FOR L > 5 μM

7 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 Re C

8 LAMINAR FLOW OF AN INCOMPRESSIBLE FLUID WITH CONSTANT PROPERTIES IN A CIRCULAR PIPE FRICTION FACTOR REYNOLDS NUMBER POISEUILLE NUMBER

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10 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

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

12 ENTRANCE EFFECTS FOR LAMINAR FLOW IN A CIRCULAR PIPE

13 WATER FLOW IN A 2D CHANNEL – CFD/EXPERIMENT

14 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

15 VISCOUS HEATING ENERGY EQUATION FOR FLOW IN A PIPE VISCOUS HEATING (VISCOUS DISSIPATION)

16 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

17 IN MICRO FLOWS BRINKMAN NUMBER IS USUALLY VERY SMALL WATER: μ = 8.55·10 -4 kg(m·s) k = 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 = 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

18 VISCOUS HEATING CAN BE IMPORTANT DUE TO VERY STRONG DEPENDENCE OF LIQUID VISCOSITY ON TEMPERATURE WATER T = 300 K ν = 8.576·10 -7 m 2 /s T = 310 K ν = 6.999·10 -7 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

19 THERMAL EXPLOSION THE MOMENTUM AND ENERGY EQUATIONS FOR FULLY DEVELOPED FLOW IN A CIRCULAR PIPE ARE FOR EXPONENTIAL DEPENDENCE OF LIQUID VISCOSITY ON THE TEMPERATURE

20 INTRODUCING NEW VARIABLES THE ENERGY EQUATION REDUCES TO IT HAS NO SOLUTION FOR NO FULLY DEVELOPED FLOW!

21 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

22 EXAMPLE OF A CFD RESULT INLET CONDITIONS D= 20 μm; T = 300 K ν = 8.576·10 -7 m 2 /s Re = 2000 V = m/s !

23 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

24 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

25 LIQUID FLOW AT 30 MPa

26 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

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

28 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 )

29 FLOW REGIMES FOR ROUGH PIPES HYDRAULICALLY SMOOTH LAMINAR TURBULENT TRANSITIONTURBULENT COMPLETELY ROUGH TURBULENT

30 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

31 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

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

33 THEORETICAL SOLUTION WALL TEMPERATURE BULK TEMPERATURE NUSSELT NUMBER

34 EXPERIMENT - RESULTS

35 CFD – CONJUGATE HEAT TRANSFER INCLUDED

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

37 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

38 AXIAL HEAT CONDUCTION ENERGY EQUATION FOR FLOW IN A PIPE AXIAL HEAT CONDUCTION AXIAL HEAT CONDUCTION CAN USUALLY BE NEGLECTED UNLESS PECLET NUMBER IS VERY LOW

39 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

40 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

41 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?

42 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

43 MOLECULAR MAGNITUDES NUMBER DENSITY OF MOLECULES n MEAN MOLECULAR SPACING δ MOLECULAR DIAMETER d DILUTE GAS: δ/d > 7 AIR: THE DATA FOR p = 1 atm; T = 0 ºC

44 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 GASAIRN2N2 CO 2 O2O2 HeAr λ, nm

45 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

46 LIMITS OF APPROXIMATIONS

47 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

48 FUTURE DIRECTIONS OF RESEARCH

49 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

50 REFERENCES 1.Bayraktar & Pidugu, Int J Heat Mass Trans, Cui et al, Phys Fluids, Gad-el-Hak, Int J Heat Mass Trans, Gamrat et al, Int J Heat Mass Trans, Guo & Li, Int J Heat Mass Trans, Herwig & Hausner, Int J Heat Mass Trans, Herwig, ZAMM, Hetsroni et al, Int J Heat Mass Trans, 2005, p Hetsroni et al, Int J Heat Mass Trans, 2005, p Judy et al, Int J Heat Mass Trans, Karniadakis & Beskok, Micro Flows, Koo & Kleinstreuer, Int J Heat Mass Trans, Maranzana et al, Int J Heat Mass Trans, 2004

51 THANKS!