Gas Flow in Complex Microchannels

Transcription

1 Gas Flow in Complex Microchannels Amit Agrawal Department of Mechanical Engineering Indian Institute of Technology Bombay TEQIP Workshop IIT Kanpur, 6 th October 2013

2 Motivation Devices for continual monitoring the health of 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

3 Motivation Acoustics Fan noise Exhaust Temperature (thermo limit) Graphics T j CPU T j Chassis Temperature Component T case *Slide taken from Dr. A. Bhattacharya, Intel Corp

4 Motivation: Electronics cooling Heat Flux > 100 W/cm 2* : 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 AMD.com Channel length 130 nm 90 nm 65 nm Die Size 0.80 cm cm cm 2 Heat flux 63 W/cm 2 78 W/cm W/cm 2

5 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

6 Introduction Knudsen number Kn = λ L λ is mean free path of gas (~ 70 nm for N 2 at STP) L is characteristic dimension Kn ~ 7x10-3 for 10 micron channel 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 inermolecular 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 regimeof our interest

7 Introduction How to determine the slip velocity at the wall? Maxwell s model (1869) u slip 2 σ du = λ σ dy wall σ is tangential momentum accommodation coefficient (TMAC) = fraction of molecules reflected diffusely from the wall Sreekanth s model (1969) 2 du 2 d u 1λ 2λ 2 dy wall wall uslip = C C dy C 1, C 2 : slip coefficients 7

8 Maxwell s Slip Model u slip 2 σ du = λ σ dy wall τ i τ r σ = τ i τ i = tangential momentum of incoming molecule τ r = tangential momentum of reflected molecule In practice, fraction of collisions specular, others diffuse σ is fraction of diffuse collisions to total number of collisions Specular reflection (τ r = τ i ; σ = 0) Diffuse reflection (τ r = 0; σ = 1)

9 Introduction New effects with gases Slip, rarefaction, compressibility Continuum assumptions valid with liquids Conventional wisdom should apply Similar behavior possible at nano-scales 9

10 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

11 Solution for gas flow in microchannel y, v x, u Governing Equations: Adp τ dx = d u da w p= ρrt pkn. = const 2 ( ρ ) Assumptions: Flow is steady, twodimensional and locally fully developed Flow is isothermal Integral momentum equation Ideal gas law From, μ λ = p πrt 2 11

12 Solution for gas flow in microchannel (contd.) Boundary conditions: u = u slip p = p Kn = 0 Kn 0 at y = 0, H } specified Solution procedure: at some reference location, x = x 0 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

13 Solution for gas flow in microchannel (contd.) Velocity profile: y/ H ( y/ H) + 2C Kn+ 8C Kn uxy (, ) = ux ( ) 1/6+ 2CKn+ 8 Pressure can be obtained from: CKn 2 2 p p 2 p 1+ 24CKn CKn 2 0 log + p0 p0 p0 p p p x x 2Re βχ { 12CKn CKn[ 1] log } = 48Reβ where p p po H 2ρuH 2 2 Re = ; β = Kn0 ; μ π 1 u χ = A = u /30+ 2/3CKn 1 + 8/3CKn 2 + 4C1Kn + 32C da (1 / 6 + 2CKn CKn 2 ) CKn + 64CKn

14 Solution for Velocity Streamwise velocity continuously increases flow accelerates Pressure drop non-linear Lateral velocity is non-zero (but small in magnitude)

15 Extension of Navier-Stokes to high Knudsen number regime 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,

16 Questions? Can we extend this approach to other cases? What are the values of slip coefficients? or tangential momentum accommodation coefficient (TMAC) C 1 = 2 σ σ Why does this approach work at all?

17 Extension of Navier-Stokes to high Knudsen number regime (contd.) Idea extended to - Gas flow in capillary (Agrawal & Dongari, IJMNTFTP, 2012) Conductance versus Kn - Cylindrical Couette flow (Agrawal et al., ETFS, 32, 991, 2008) Experimental Data and Curve Fit to Data by Knudsen (1950)

18 Extension of Navier-Stokes to high Knudsen number regime (contd.) e μ ( y) = e μλ ( y) λ Normalized volume flux versus inverse Knudsen number 1 Normalized Velocity Profile U/U centreline Kn = 1 Present model DSMC (Karniadakis et al. 2005) Pr: Pressure ratio y/h N. Dongari and A. Agrawal, Modeling of Navier-Stokes Equations to High Knudsen Number Gas Flow. International Journal of Heat and Mass Transfer, 55, , 2012

19 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 Recall Specular reflection (τ r = τ i ; σ = 0) Three main methods to determine TMAC: Rotating cylinder method Spinning rotor gauge method Flow through microchannels Diffuse reflection (τ r = 0; σ = 1) 19

20 TMAC versus Knudsen number for monatomic & diatomic gases Monatomic Diatomic: 0.7 σ = 1 log(1 +Kn )

21 Summary of Tangential Momentum Accommodation Coefficient TMAC = for monatomic gases irrespective of Kn and surface material Exception is platinum where σ measured = 0.55 and σ analytical = 0.19 TMAC = for all monatomic gases TMAC = for non-monatomic gases σ = 1 log 1 + Kn 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 ( 0.7 ) Agrawal et al. (2008) Survey on measurement of tangential momentum accommodation coefficient. Journal of Vacuum Science and Technology A, 26,

22 Boltzmann Equation O(Kn) Resulting Equations 0 Euler Equations 1 Navier Stokes Equations 2 Burnett Equations 3 Super Burnett Equations 22 5 October 2013

23 Generalized Equations 23 5 October 2013

24 Burnett Equations 24 5 October 2013

25 Higher Order Continuum Models Conventional Burnett Equations (1939) Super Burnett Equations Grad s Moment Equation BGK-Burnett Equations (1996,1997) Augmented Burnett Equations (1991) R13 Moments Equation Have been applied to Shock Waves Here, want to apply to Micro flows Ref: Agarwal et al. (2001) Physics of Fluids, 13: ; Garcia-Colin et al. (2008) Physics Report, 465: October 2013

26 Analytical Solution from Higher Order Continuum Equations 26 5 October 2013

27 Numerical Solution of Higher-Order Continuum Equations Agarwalet al. (2001): Planar Poiseuille flow; Convergent solution till Kn = 0.2; solution diverged beyond it Xueet al. (2003): Couette flow; Kn < 0.18 Bao & Lin (2007): Planar Poiseuille flow; Kn < 0.4 Uribe & Garcia (1999); Planar Poiseuille flow; Kn < October 2013

28 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 October 2013

29 Solution of the Equation 29 5 October 2013

30 Results Error (in %) versus Knudsen number as given by Burnett equation and Navier-Stokes at the wall (C 1 = 1.066, C 2 = 0.231; Ewart et al. (2007)) 30 5 October 2013

31 Slip Velocity 0.8 DSMC (Karniadakis et. al. 2005) Burnett Analytical Solution U s /U in Kn out = 0.20 Ma out = Pr = x/l 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

32 Complex Microchannels Test section for flow through sudden expansion / contraction Test section for flow through 90º bend Gas Nitrogen (Helium to a limited extent) Developing length is 10% greater than required from calculations 32

33 P/P o Lattice Boltzmann Simulation Microchannel with Sudden Contraction or Expansion y/w y/w x/w x/w x/l 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, , (2005). 33

34 Experimental Setup sccm mbar 3 2 Vacuum pump Rotary with speed of 350 lpm Maximum vacuum mbar To vacuum system To display 1 Gas cylinder 2 Pressure regulator 3 Particle filter 1 MFC 0-20 sccm sccm 4 Mass flow controller sccm Flow parameters range Re = Kn o = Inlet reservoir 6 Test section 7 Outlet reservoir 8 Absolute pressure transducer N 2 APT 0 1 mbar 0 100mbar 0 1 bar 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 N 2 at 300 K. 34

35 Validation (Flow in tube) different e11 gas-solid combinations; Kn range and Re range of Rf = Kn fre Sreekanth (Brass-N 2 ) Ewart ( Fused slilica- N 2 ) Ewart ( Fused slilica- He) Ewart ( Fused slilica- Ar) Brown ( Copper- Air) Brown ( Copper- H 2 ) Brown ( Iron- Air) Choi ( Fused slilica- N 2 ) Tang ( Fused silica- N 2 ) Tang ( Fused silica- He) Demsis (Copper- N 2 ) Demsis (Copper- O 2 ) Demsis (Steel- O 2 ) Demsis (Steel- Ar) } Present Verma et al. Journal of Vacuum Science and Kn fre for different combinations of gassurface material seems small as compared to experimental scatter. Technology A, 27(3) (2009). 35

36 Experimental setup and test section geometry for sudden expansion Figure 1a. Schematic diagram of the experimental set up Flow parameters range Re = Kn o = 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 Figure 1b: Test section geometry for sudden expansion 36

37 Flow through tube with sudden expansion 12 Static pressure variation along wall for area ratio = Sudden Expansion d = 4 mm, D = 32 mm AR = 64, Nitrogen Kn j =0.036 Kn j =0.007 Kn j =0.003 Kn j =0.001 No flow separation at the junction P/P o 6 Junction 4 2 P i P o X/L 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 37

38 Radial pressure variation Sudden Expansion AR = 64, Nitrogen 1 (P/P w ) static Kn j = 0.036, Just upstream of junction Kn j = 0.036, Just downstream of junction Kn j = 0.001, Just upstream of junction Kn j = 0.001, Just downstream of junction local radius/radius of smaller section (c) Radial static pressure variation just upstream and just downstream of the junction for AR =

39 Comparison with straight tube Sudden Expansion, AR = Re s =11.3,Kn j = , Nitrogen 1.3 Junction Straight SE P/P o L u L d X/L Static pressure variation (normalized with the outlet pressure) along the wall. 39

40 Variation in L u and L d Sudden Expansion Nitrogen Sudden Expansion Nitrogen AR = 3.74 AR = AR = 64 Curve fit L d /d = AR 1.35 /(400*Kn j 0.45 ) 20 5 L u /d 15 L d /d AR = 3.74 AR = AR = 64 Curve fit L u /d = 15.45(AR Kn j ) Variation in L u (normalized with smaller section tube diameter d) versus Knudsen number at junction (Kn j ). Kn j Variation in L d (normalized with smaller section tube diameter d) versus Knudsen number at junction (Kn j ). Kn j 40

41 Velocity profile Straight tube Sudden expansion (a) 41

42 Analysis of velocity distribution (p p )A τ π Ddx = [M M ] 1 2 w 2 1 p 1, p 2, known from experimental measurements, Need velocity distribution for calculating shear stress and momentum M 1 and M 2. Assuming velocity distribution as second order parabolic velocity profile u = a + The mass flow rate can be calculated as At r = R, velocity at the wall u = u s cr 2 m = R ρ2πrdru 0 42

43 Streamlines near sudden expansion junction Tube radius (m) Sudden Expansion d=9.5mm,d=33.5 AR = 12.43, Nitrogen Re s =2.8,Kn j =0.03 % of mass flow rate 99 % 95 % 90 % 70 % 50 % 30 % 10 % 5% 1% 0.5% X/L Streamlines near the junction for rarefied gas flow. 43

44 Streamlines near sudden expansion junction Varade et al.; JFM (under review) 44

45 Bend Microchannel (Lattice Boltzmann Simulations) Streamlines near the bend for Kn = Presence of an eddy at the corner for Re = 2.14!

46 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)

47 Experimental data for bend Test section for flow through 90º bend

48 Heat Transfer Coefficient (Experimental setup) To vacuum pump Port for Inlet Pressure gauge (P i ) Water Outlet (T ho ) Port for Outlet Temperature probe (T co ) Nitrogen Inlet Nitrogen Outlet Port for Inlet Temperature probe (T ci ) Hot Water Inlet (T hi ) Port for Outlet Pressure gauge (P o )

49 0.8 Nusselt number measurements (Circular tube with constant wall temperature BC) 0.7 Re range ( ) Gr / Re 2 < 10-4 Nu Nu versus Kn Kn m range Nu versus Re 0.1 Nu Kn m Re

50 Comparison against theoretical values (at Kn = 0.01) Source Larrode et al. [5] Ameel et al. [10] Tunc and Bayazitoglu [12] Aydin and Avci [20] Hooman [21] Present (Experimental) Nu Our experimental data predicts 3 orders of magnitude smaller value as compared to theoretical analysis!!

51 Validation & Repeatability Re Nu from Dittus- Boelter Correlation Nu from the present experiment Percentage difference Run 1 Run 2 Run 3 Run Nu Kn

52 Why is heat transfer coefficient so small? Rarefaction reduces ability of gas to remove heat from wall 2 σ T 2γ Temperature jump at wall Tg Tw = σ T Kn 1 + γ Pr T n w Radial convection may be important (i.e., term comparable to u T x ), but ignored in theoretical analysis v T r Reason for difference is still not fully understood

53 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 nonintuitive behavior

54 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

55 Effect of sudden expansion on velocity distribution Shear stress variation along the wall Momentum variation along the length Rate of increase in shear stress and momentum is more in sudden expansion than in straight tube towards the expansion junction Varade et al. (2012) (to be submitted) 55

56 Maxwell s Slip Model u slip 2 σ du = λ σ dy wall τ i τ r σ = τ i τ i = tangential momentum of incoming molecule τ r = tangential momentum of reflected molecule In practice, fraction of collisions specular, others diffuse σ is fraction of diffuse collisions to total number of collisions Specular reflection (τ r = τ i ; σ = 0) Diffuse reflection (τ r = 0; σ = 1)

57 DSMC Simulations (Couette flow) Slip from DSMC/ Slip from Maxwell model versus Knudsen number 57

58 DSMC Simulations (Couette flow) Slip from DSMC/ Slip from Maxwell model versus Knudsen number 58

59 Microchannel with Sudden Contraction or Expansion (Pressure distribution) 3 Kn 2.5 P/P o Kn x/l x/l

60 Microchannel with sudden Contraction or Expansion (Velocity distribution) U 0.05 U Kn x/l x/l

61 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?

62 TMAC versus Knudsen number 62

63 Flow Rate Vs Kn 6 Boltzmann Solution Exp. data (Pr = 5) 4 Exp. Data (Pr = 4) Exp. data (Pr = 3) G Burnett Analytical Solution (C1 = 1.466, C2 = 0.5) Kn 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). (C 1 = and C 2 = 0.5, Chapman & Cowling (1970) October 2013

64 Introduction (contd.) (Compressibility) 1 Dρ For flow to be compressible: = 0 (Gad-el-Hak, 1999) Dt i.e., both spatial and temporal variations in density are small as compared to the absolute density ρ Compressibility effect is important because of large pressure drop in microchannel

65 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

66 Velocity measurement in rarefied gas flow through tube Stagnation pressure correction coefficient Cp = P P o s ρu Schematic diagram of experimental set up For low Re the magnitude of viscous forces are comparable with inertia forces hence Bernoulli formula needs correction 29 - February Study of rarefied gas flow through microchannel 66

67 Velocity measurement in rarefied gas flow through tube Author Cp Re range (Re is based on Remark (Pitot tube pitot tube ID) geometry) Homann (1952) Hurd et al. (1953) Lester (1961) Chebbi and Tavoularis (1990) 1+[8/(Re+0.64Re 1/2 )] Cylinder 11.2/Re Circular blunt nosed 2.975/Re 0.43 Circular 4.2/Re Circular Experimental Experimental 1-10 Numerical < 1 Experimental 29 - February Study of rarefied gas flow through microchannel 67

68 Velocity measurement in rarefied gas flow through tube Variation in Cp with Re (based on Pitot tube ID) Variation in central velocity with Re (based on Pitot tube ID) 29 - February Study of rarefied gas flow through microchannel 68

69 Velocity measurement in rarefied gas flow through tube % deviation in central velocity with reference to Sreekanth (1969) 29 - February Study of rarefied gas flow through microchannel 69

70 Velocity measurement in rarefied gas flow through tube (a) (b) (c) (d) Comparison of experimental velocity measurement with Sreekanth (1969), Homann s correction 29 - February Study of rarefied gas flow through microchannel 70

71 Velocity measurement in rarefied gas flow through sudden expansion using Pitot tube Continuum flow 29 - February Study of rarefied gas flow through microchannel 71

72 Velocity measurement in rarefied gas flow through sudden expansion using Pitot tube Slip flow 29 - February Study of rarefied gas flow through microchannel 72