Sixth International ASME Conference onNanochannels, Microchannels and Minichannels

23-25 June 2008, Darmstadt, Germany

DSMC Solution of Supersonic ScaleDSMC Solution of Supersonic Scaleto Choked Subsonic Flow in Microto Choked Subsonic Flow in Microto Choked Subsonic Flow in Microto Choked Subsonic Flow in Micro

to Nano Channelsto Nano Channels

Ehsan RoohiEhsan Roohi

to Nano Channelsto Nano Channels

PhD StudentDepartment of Aerospace Engineering

Sharif University of Technology

MasoudMasoud DarbandiDarbandiProfessor

Department of Aerospace Engineering

VahidVahid MirjaliliMirjaliliGraduate Student

Department of Aerospace EngineeringSharif University of Technology Sharif University of Technology

Sixth International ASME Conference on

Nanochannels, Microchannels and Minichannels

OutlineOutline IntroductionIntroduction

OutlineOutline

Flow RegimesBoltzmann Equation

N i l S hN i l S hNumerical SchemeNumerical SchemeDSMC MethodImplicit Boundary Treatment

Results and DiscussionResults and DiscussionSupersonic Flow Mixed Supersonic/Subsonic RegimeMixed Supersonic/Subsonic RegimeChocked Subsonic Flow

ConclusionConclusion2

ConclusionConclusion

Sixth International ASME Conference on

Nanochannels, Microchannels and Minichannels

Flow RegimesFlow RegimesFlow RegimesFlow Regimes

• Classification:K < 0 01 C ti

LKn /– Kn < 0.01 Continuum– Kn < 0. 1 Slip– Kn < 1 Transition

3

Kn 1 Transition– Kn > 10 Free Molecular

Sixth International ASME Conference on

Nanochannels, Microchannels and Minichannels

Boltzmann EquationBoltzmann EquationBoltzmann EquationBoltzmann Equation**

42 )()()()( dcdcffffnnfFnfcnf

Local rate of change

1110

)()(.)(.)( dcdcffffnnfc

Fnfr

cnft r

Binary Collision termInflux of molecules due

to external forceof number of molecules

to external force

Influx of molecules due

to convection

• Boltzmann Equation: temporal-spatial changes of number of molecules of class c

• Assumptions– i) Dilute Gas: Binary Collision

ii) Molecular Chaos: colliding particles are uncorrelated4

– ii) Molecular Chaos: colliding particles are uncorrelated

Sixth International ASME Conference on

Nanochannels, Microchannels and Minichannels

Direct Simulation Monte Carlo (DSMC)Direct Simulation Monte Carlo (DSMC)Direct Simulation Monte Carlo (DSMC)Direct Simulation Monte Carlo (DSMC)

Direct simulation of physical behavior of Direct simulation of physical behavior of molecules Particles move in physical space; Particles move in physical space; Particles possess microscopic properties, Collisions handled statistically; Collisions handled statistically; Movement handled deterministically. C ll i f th {u’, v’, w’

x y z Cell size ~ mean free path ; Time step ~ collision time;

{x, y, zerot, evib

5

Sixth International ASME Conference on

Nanochannels, Microchannels and Minichannels

DSMC Method Developed by Bird (1970’s)

DSMC Method Developed by Bird (1970’s) First applications: external hypersonic flow 1990s: MEMS, supersonic internal flow 2000s: MEMS, subsonic internal flow 2000s: Extension to very low speed

flow Information Preservation (IP) methodflow, Information Preservation (IP) method 2002: Developing hybrid continuum-

molecular schemes Current work: New Physical Simulation

6

Sixth International ASME Conference on

Nanochannels, Microchannels and Minichannels

Current Work: Physical StudyCurrent Work: Physical Study Supersonic flow with vacuum/specified back Supersonic flow with vacuum/specified back pressure pressure

Current Work: Physical StudyCurrent Work: Physical Studypressure pressure

Mixed flow regime Study of the choked subsonic flowStudy of the choked subsonic flow

Implicit I/O Boundary conditions: Wang et. al [2004]Role of Buffer zone

InletSymmetry

OutletGeometry with buffer zoneH/2

Wall

Geometry with buffer zone

7L

Sixth International ASME Conference on

Nanochannels, Microchannels and Minichannels

Simulated CasesSimulated CasesSimulated CasesSimulated Cases Ch l A t R ti 5 Channel Aspect Ratio: 5 Buffer: Either at the inlet or Outlet Inlet Mach: 4.15 Grid: 120-60 (W. Buffer), 100-60 (W.O. Buffer)Grid: 120 60 (W. Buffer), 100 60 (W.O. Buffer) Kn: Slip (0.062)/Transition (0.43) Mi d S i S b i Fl Mixed Supersonic-Subsonic Flow Subsonic Choked Flow

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Sixth International ASME Conference on

Nanochannels, Microchannels and Minichannels

Convergence HistoryConvergence HistoryConvergence HistoryConvergence History250 4

Inlet Mass Fluxm

s)

200

3

3.5Outlet Mass FluxInlet PressureOutlet Pressure

Rat

e(K

g/m

Pou

t

100

150

2

2.5

ass

Flow

R

P/P

50

100

1.5

2

M

00.5

1

9Print Cycle0 1000 2000 3000 4000

-50 0

Sixth International ASME Conference on

Nanochannels, Microchannels and Minichannels

Role of Knudsen NumberRole of Knudsen Number Kn=0.062: Oblique Shock,

Mach: 0 5 1 1 5 2 2 5 3 3 5 4 Kn=0.43: Normal Shocks, Mixed Regime

Mach: 0.5 1 1.5 2 2.5 3 3.5 4

4

X2E-05 4E-05 6E-05

3

3.5

Mach: 0.2 0.6 1 1.4 1.8 2.2 2.6 3

Mac

h

2

2.5

Kn=0.062

X2E-06 4E-06 6E-06

0 5

1

1.5 Kn=0.35

Kn=0.74

10X/L0 0.2 0.4 0.6 0.8

0.5

Balancing Effects of Shear stress/ Heat Transfer

Sixth International ASME Conference on

Nanochannels, Microchannels and Minichannels

Temperature FieldTemperature FieldTemperature FieldTemperature FieldT : 5 0 1 5 0 2 5 0 3 5 0 4 5 0 5 5 0

2 E -0 5 4 E -0 5 6 E -0 5

T : 5 0 1 5 0 2 5 0 3 5 0 4 5 0 5 5 0

Kn=0.062, Heating

X2 E -0 5 4 E -0 5 6 E -0 5

T: 450 650 850 1050

Kn=0.43, Cooling

X1E-06 2E-06

, g

11

Sixth International ASME Conference on

Nanochannels, Microchannels and Minichannels

Back Pressure EffectsBack Pressure EffectsBack Pressure EffectsBack Pressure Effects0.7

0.5

0.6

0.7

Current

M ach: 0.2 0.6 1 1.4 1.8 2.2 2.6 3

ress

ure

(Mpa

)

0.3

0.4

CurrentLe et. al.

X2E-06 4E-06

M=3.39, Vacuum

Pr

0.1

0.2 M ach : 0 .2 0 .6 1 1 .4 1 .8 2 .2 2 .6 3

X/L0.2 0.4 0.6 0.8 1

X2 E -0 6 4 E -0 6

12Validation M=3.39, Pb=0.5 Mpa

Sixth International ASME Conference on

Nanochannels, Microchannels and Minichannels

Choked Flow: Role of Buffer ZoneChoked Flow: Role of Buffer Zone

Mach: 0 1 0 25 0 4 0 55 0 7 0 85 1 1 15Mach: 0.1 0.25 0.4 0.55 0.7 0.85 1 1.15

a) Mach contours, without buffer zone, Non-physical solution (Mout > 1)

X5E-07 1E-06 1.5E-06

Mach: 0.1 0.25 0.4 0.55 0.7 1

b) Mach contours, with bufferzone, Correct physical

X5E-07 1E-06 1.5E-06

zone, Correct physical simulation

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Sixth International ASME Conference on

Nanochannels, Microchannels and Minichannels

Role of Buffer ZoneRole of Buffer ZoneRole of Buffer ZoneRole of Buffer Zone0.2

KnGLL D it with Buffer

0.15

KnGLL-Density with BufferKnGLL-Density without Buffer Once choking occurs, exit

pressure does not drop more,

Applying a lower back pressure

n GLL

-Den

sity

0.1

Applying a lower back pressure at the outlet results in physically incorrect prediction, (Wavy local Knudsen)

Kn

0.05

Fix pressure at the end of buffer and let the outlet pressure found during Simulation (Smooth local

0

during Simulation (Smooth local Knudsen)

14X/L

0 0.2 0.4 0.6 0.8 10

Sixth International ASME Conference on

Nanochannels, Microchannels and Minichannels

Velocity ProfilesVelocity ProfilesVelocity ProfilesVelocity ProfilesDSMCAnalytical (1st order)

d

0.2

0.4

X/L=0.3Kn=0.056 0.2

0.4

X/L=0.6Kn=0.0715 0.2

0.4 Analytical (2ndorder)Analytical (unified model)

X/L=0.9Kn=0.111

Y/H 0 Y/H 0 Y/H 0

-0.4

-0.2

-0.4

-0.2

-0.4

-0.2

U*0.4 0.6 0.8 1 1.2 1.4

U*0.4 0.6 0.8 1 1.2 1.4

U*0.4 0.6 0.8 1 1.2 1.4

Why DSMC predicts lower normalized velocity?

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Why DSMC predicts lower normalized velocity?Each velocity is normalized with its respective average

Sixth International ASME Conference on

Nanochannels, Microchannels and Minichannels

Computed DSMC mass flow rates and Computed DSMC mass flow rates and comparison with the analytical solutionscomparison with the analytical solutions

m0 (×105) (DSMC) 13.1

m0 (×105) Ref. [14] 1.42

m0 (×105) Ref. [16] 1.27

m0 (×105) Ref. [17] 1.33

DSMC gives much more mass flow rate

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Sixth International ASME Conference on

Nanochannels, Microchannels and Minichannels

Comparison of DSMC and analytical Comparison of DSMC and analytical p yp ypredictions for average velocitypredictions for average velocity

X/LX/L UUAnalytical Analytical (m/s)(m/s) UUDSMC DSMC (m/s)(m/s) UUDSMCDSMC /U/UAnalyticalAnalytical

0 30 3 69 769 7 98 4698 46 1 411 410.30.3 69.769.7 98.4698.46 1.411.41

0.60.6 90.990.9 125.4125.4 1.371.37

0.90.9 147.4147.4 205.7205.7 1.391.39

DSMC velocity (pressure gradient) is more than analytical’sDSMC velocity (pressure gradient) is more than analytical s,

Analytical solutions fail to predict correct properties of chokedflow due to high compressibility effects, not considered in derivation.

17

flow due to high compressibility effects, not considered in derivation.

Sixth International ASME Conference on

Nanochannels, Microchannels and Minichannels

Average Mach along the Centerline: Average Mach along the Centerline:

0 7

e age ac a o g t e Ce te ee age ac a o g t e Ce te eComparison with Fanno TheoryComparison with Fanno Theory

0.6

0.7

DSMCFanno

h

0.5 Good AgreementGood Agreement

Mac

h

0.4

0.3

18

X/L0.2 0.4 0.6 0.8 1

0.2

Sixth International ASME Conference on

Nanochannels, Microchannels and Minichannels

Conclusion

Knudsen Number greatly affects the flow behavior Knudsen Number greatly affects the flow behavior,

Imposing Back pressure leads to mixed regime,

Expansion cooling occurs while flow approaches vacuumed ambient,

Wrong solution for choked flow without buffer zone,

DSMC gives more mass flow rate, pressure gradient and DSMC gives more mass flow rate, pressure gradient and velocity for choked flow,

DSMC agrees with Fanno Theory.

19

g y

Sixth International ASME Conference on

Nanochannels, Microchannels and Minichannels

Further Work Simulating Nozzle Flow IP scheme for long channels

(L/H=100) (L/H=100)

)/PO

ut

0.04

M h N b 0 5 1 5 2 5 3 5

(P-P

Line

ar)

0.02

IP

Mach Number: 0.5 1.5 2.5 3.5

0 0 2 0 4 0 6 0 8 10

IPAnalytical, 1storderAnalytical, 2ndorder

5E-06 1E-05

20X/L

0 0.2 0.4 0.6 0.8 1

Sixth International ASME Conference on

Nanochannels, Microchannels and Minichannels

StepFlowp

Mach: 0.05 0.2 0.35 0.5 0.65 0.8 0.95

Y/S 1

2Mach: 0.05 0.2 0.35 0.5 0.65 0.8 0.95

Kn=0.01

X/S

Y

-2 0 2 4 6

Mach: 0.02 0.12 0.22 0.32 0.42 0.52

Y/S 1

2

Kn=0.10

21X/S

-2 0 2 4 6

Sixth International ASME Conference on

Nanochannels, Microchannels and Minichannels

StepFlowp

Mach: 0.05 0.15 0.25 0.35 0.45

Kn=1.0

Y/S 1

2Mach: 0.05 0.15 0.25 0.35 0.45

X/S

Y

-2 0 2 4 6

2Mach: 0.05 0.15 0.25 0.35 0.45

Y/S 1Kn=10.

22X/S-2 0 2 4 6

Sixth International ASME Conference on

Nanochannels, Microchannels and Minichannels

Validation by Analytical/NS solutionsValidation by Analytical/NS solutions

1.51.5

Kn = 0.5Current studyDSMC

Kn = 0.3

Uav

g

11

Uav

g

1

U/U

Analytic Sol.Navier-Stokes

Kn = 0.2

U/

0.5Navier StokesDSMC (coarse)DSMC (fine)

0.50.5

23y*0 0.1 0.2 0.3 0.4 0.5 y*

0 0.5y*

0 0.5

Sixth International ASME Conference on

Nanochannels, Microchannels and Minichannels

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