Gas Flow in Complex Microchannels Amit Agrawal Department of - PowerPoint PPT Presentation

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

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

Motivation Acoustics Acoustics Fan noise Fan noise Graphics Exhaust T j Tem perature CPU ( therm o lim it) T j Chassis Tem perature Com ponent T case *Slide taken from Dr. A. Bhattacharya, Intel Corp

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 report @ AMD.com Channel length 130 nm 90 nm 65 nm Die Size 0.80 cm 2 0.64 cm 2 0.40 cm 2 Heat flux 63 W/cm 2 78 W/cm 2 125 W/cm 2

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

Introduction λ is mean free path of gas λ = Knudsen number Kn (~ 70 nm for N 2 at STP) L L is characteristic dimension Kn ~ 7x10 -3 for 10 micron channel Flow regimes Kn Range Flow Model Kn < 10 -3 Continuum flow Navier-stokes equation with no slip B. C. 10 -3 ≤ Kn < 10 -1 Slip flow Navier-stokes equation with slip B. C. Navier-stokes equation fails but iner- 10 -1 ≤ Kn < 10 Transition flow molecular collisions are not negligible Intermolecular collisions are negligible as 10 ≤ Kn Free molecular flow compared to molecular collision to the wall Gas flow mostly in slip regime – regime of our interest

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

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

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

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

Solution for gas flow in microchannel Assumptions: Flow is steady, two- y, v dimensional and locally fully developed x, u Flow is isothermal Governing Equations: ∫ − − τ = ρ 2 Adp dx d ( u dA ) Integral momentum equation w = ρ Ideal gas law p RT = μ π RT From, p Kn . const λ = p 2 11

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

Solution for gas flow in microchannel ( contd. ) Velocity profile: − + + 2 2 y H / ( / y H ) 2 C Kn 8 C Kn = 1 2 u x y ( , ) u x ( ) + + 2 1/ 6 2 C Kn 8 C Kn 1 2 Pressure can be obtained from: 2 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ p p p − + − + + 2 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 1 24 C Kn 1 96 C Kn log 1 0 2 0 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ p p p 0 0 0 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ − p p p x x { βχ − + − − = − β 2 2 2 ⎜ 0 ⎟ ⎜ 0 ⎟ ⎜ ⎟ 0 2Re 12 C Kn 1 24 C Kn [ 1] log } 48Re 1 0 2 0 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ p p p H o where ρ 2 uH 2 = β = 2 Re ; Kn ; μ π 0 2 + + + + + ⎛ ⎞ 2 2 2 3 2 4 1 u 1/ 30 2 / 3 C Kn 8 / 3 C Kn 4 C Kn 32 C C Kn 64 C Kn ∫ χ = = 1 2 1 1 2 2 ⎜ ⎟ dA + + ⎝ ⎠ 2 2 A (1/ 6 2 C Kn 8 C Kn ) u 1 2 13

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

Extension of Navier-Stokes to high Knudsen number regime First-order slip-model fails to predict the Knudsen minima Normalized Volume Flux versus Knudsen Number 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. 15 International Journal of Heat and Mass Transfer , 50, 3411-3421

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

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

Extension of Navier-Stokes to high Knudsen number regime (contd.) μλ e Normalized volume flux versus ( ) y μ = e ( ) y inverse Knudsen number λ Normalized Velocity Profile 1 0 .8 Pr: Pressure ratio U/U centreline Present m odel 0 .6 Kn = 1 DSMC ( Karniadakis et al. 2 0 0 5 ) 0 .4 0 0.1 0.2 0.3 0.4 0.5 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, 4352-4358, 2012

Determination of Tangential Momentum Accommodation Coefficient (TMAC) What does TMAC depend upon? Recall � Surface material - For same pressure drop, will flow rate in glass & copper tubes be different? � Gas - Will nitrogen and helium behave differently, in Specular reflection the same material tube? � Surface roughness and cleanliness ( τ r = τ i ; σ = 0) � Temperature of gas � Knudsen number Three main methods to determine TMAC: � Rotating cylinder method � Spinning rotor gauge method � Flow through microchannels Diffuse reflection ( τ r = 0; σ = 1) 19

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

Summary of Tangential Momentum Accommodation Coefficient • TMAC = 0.80 – 1.02 for monatomic gases irrespective of Kn and surface material – Exception is platinum where σ measured = 0.55 and σ analytical = 0.19 – TMAC = 0.926 for all monatomic gases • TMAC = 0.86 – 1.04 for non-monatomic gases ( ) σ = − + 0 . 7 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 Agrawal et al. (2008) Survey on measurement of tangential momentum accommodation coefficient. Journal of Vacuum Science and Technology A , 26, 634-645

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

Generalized Equations 5 October 2013 23

5 October 2013 Burnett Equations 24

Higher Order Continuum Models Conventional Burnett BGK-Burnett Equations Equations (1939) (1996,1997) Super Burnett Equations Augmented Burnett Equations (1991) Grad’s Moment Equation R13 Moments Equation • Have been applied to Shock Waves • Here, want to apply to Micro flows Ref: Agarwal et al. (2001) Physics of Fluids, 13:3061–3085; Garcia-Colin et al. (2008) Physics Report, 465:149–189 25 5 October 2013

Analytical Solution from Higher Order Continuum Equations 5 October 2013 26

Numerical Solution of Higher-Order Continuum Equations • Agarwal et al. (2001): Planar Poiseuille flow; Convergent solution till Kn = 0.2; solution diverged beyond it • Xue et al. (2003): Couette flow; Kn < 0.18 • Bao & Lin (2007): Planar Poiseuille flow; Kn < 0.4 • Uribe & Garcia (1999); Planar Poiseuille flow; Kn < 0.1 27 5 October 2013