1

A Project Presentation for

Applied

Computational Fluid Dynamics

By

Reni Raju

Finite Element Model of Gas Flow inside a Microchannel

MECH - 523

2

Objectives

Develop a 2D Finite Element Model for gas flow inside a microchannel.

Develop FE formulation for N-S equations.

Implement Slip and temperature boundary conditions.

Compare Numerical and Experimental Data for Microchannel.

3

Microchannel

Parameter Range or Mean ValueLength L 3000 m

Width W 40 m

Height H 1.2 m

Pressure Ratio 1.340, 1.680, 2.020, 2.361, 2.701

Inlet Temperature T0 314 K

Wall Temperature Tw 314 K

Knudsen Number, (Kn= l /L) 0.055

Abs. Viscosity 1.85 x 10-5 Ns/m2

Spec. Gas Constant (N2) R 296.7 J/kg K

Ratio of Spec. heats 1.4

Numerical data from

Chen et al (1998)

+H/2

-H/2

y

x

Experimental data from

Pong et. al (1994)

4

Governing Equations

Continuity

Momentum

Energy

Equation of State

0

y

v

x

u

t

2 2 2 2

2 2 2

10

3

u u u P u u u vu v

t x y x x y x x y

2 2 2 2

2 2 2

10

3

v v v P v v v uu v

t x y y x y y x y

2 2 222

2 2 03p

DT DP T T u v v u u vC k k

Dt Dt x x y y x y x y x y

RTP

5

Normalized Equations

Continuity

Momentum

Energy

Equation of State

0*

**

*

**

*

*

y

v

x

u

t

0**

*

*

*

3

1

*

*

*

*

Re

*

*

*

*

***

*

***

*

** 2

2

2

2

22

2

2

yx

v

x

u

y

u

x

u

x

P

y

uv

x

uu

t

u

0**

*

*

*

3

1

*

*

*

*

Re

*

*

*

*

***

*

***

*

** 2

2

2

2

22

2

2

yx

u

x

v

y

v

x

v

y

P

y

vv

x

vu

t

v

2222

2

22

2

2

*

*

*

*

3

2

*

*

*

*

*

*2

*

*2

Re

*

*

*

*

*

Pr.Re

*

*

*

*

***

y

v

x

u

y

u

x

v

y

v

x

uEc

y

T

x

Tk

Dt

DPEc

Dt

DTCp

2

***

M

TP

6

Flow Regimes

Kn=0.0001 0.001 0.01 0.1 1 10 100

Continuum Regime

Slip Flow Regime

Transition Regime

Molecular Regime

Knudsen Number Re2

Ma

LKn

Gad-el-

hak (1999)

7

Wall Conditions

Wall Slip

x

T

Ty

uuu

gaswV

Vwallgas

4

32

Maxwell (1879)

*

*2

*

*** Re)1(

2

32

x

T

Ec

Kn

y

uKnuu

wV

Vwallgas

wi

riV

8

Thermal Boundary Condition

Temperature Jump

wT

Twallgas y

TTT

Pr1

22

Von Smoluchowski (1898)

wT

T

y

TKnTT

wallgas

*

***

Pr1

22

wi

riT dEdE

dEdE

9

Finite Element Algorithm

Developed by the Computational Plasma Dynamics Laboratory at Kettering University (Roy, CMAME, v184, 87-98, 2000).

A Family of complex subroutines that can study macroscopic collisional plasmas. (Roy and Pandey, POP, v9, 4052-60, 2002).

Written in Fortran 77, use Cray-style Fortran pointers, and are designed for UNIX-type environment.

Two dimensional formulation (so far). Implemented Sub-Grid Embedded (SGM) FE for Coarse-grid

solution Stability,Accuracy and Tri-diagonal Efficiency (Roy and Baker, NHT-B, v33, 5-36, 1998).

Utilized to model Compressible flow through Electric Propulsion thrusters including Microchannels.

10

Numerical Details

WeakStatement

)()(),(

;),(),(),(;

tUxNtxu

txutxutxu

eljkjel

eljelj

hj

elel

h

and,

DiscreteApproximation

Nk is appropriate basis function; Chebyshev , Lagrange orHermite interpolation polynomials complete to degree k.

Terms Diffusion & Convective F & F

U Where,v

ii

T

vii

pTvu

FFt

UUL

},,,,{

0)()(

Problem

Statement

function.Test admissible any is Where,

w

dUwLe

0)(

11

FE Formulation

0Re.

)(

e

duP

uut

ux iji

ii

)()(),( tUxNtxu ejkje

0Re.

1

e

dUNPNUNUNt

UNNSWS e

Tke

Tke

Tke

Tk

ekke

h

0.Re.

1

e

dUNNPNNUNUNNt

UNNS e

Tkke

Tkke

Tke

Tkk

eTkke

MomentumEquation

Variable

Discretized Weak Statement

12

Discretization

0.Re.

1

e

dUNNPNNUNNt

UNNS e

Tkke

Tkke

Tkk

eTkke

eU

)1)(1(4

))(1)(1(2

))(1)(1(2

))(1)(1(2

))(1)(1(2

))(1)(1(

))(1)(1(

))(1)(1(

))(1)(1(

4

1)(

22

21

2221

2221

1221

1221

2121

2121

2121

2121

2

iN

e

Tie

eT

ie

YNy

XNx

)(

)(

2

2

FE Basis Cartesian Coordinate

1

4

2

3

7

8 6

5

1

9

13

Global/Local frame

yj

xi

N

y

Nj

x

NiN ii

i

2222

21.det. dddydxd e 1

ej

j

ej

j x

xJ

dUNNDh

eT

e

22Re.

eek

kkii

i

UddN

yj

xi

yj

xi

N21

21

1

21

1

det

eej

i

j

i

ki

Uddxx

NN21

1

1

221

1

det

0

eeee

ee UDUt

UMS

DiffusionTerm

MatrixForm

ElementJacobian

Differential

Element

14

Solution Procedure

ii

i

p

pn

iin

in

FUURtMU

whereUUUUU

)()/(

,

11

0

111

11

NRIteration

i

ii

U

UU 1ConvergenceCriteria = 10-4 for all integrated quantities.

; ( ) ( ) 0;

( )el

h hel el k el

el

el el

dUWS S N L U d M R U

dt

M S M

FEFormulation

0))1(()()( 11 nnnn RRtUUMUF TimeIntegration Form

15

Mesh

FE basis 2D-Quadratic 9-node 2D-Bilinear 4-node

Mesh 1369 nodes 324 elements

1

4

2

3

7

8 6

5

1

9

1

4

2

3

1

2

2

16

Code Formatdo i = 1,4 do j = 1,9 ii = loc_rho(i) jj = loc_v1(j) term = wt * rho * Nmat22(i) * DNmat33Dx(j) elemk_lin(ii,jj) = elemk_lin(ii,jj) + term jj = loc_v2(j) term = wt * epsi * rho * Nmat22(i) * DNmat33Dy(j) elemk_lin(ii,jj) = elemk_lin(ii,jj) + term enddoenddo

do i = 1,4 do j = 1,4 ii = loc_rho(i) jj = loc_rho(j) term = wt * velx * Nmat22(i) * DNmat22Dx(j) elemk_lin(ii,jj) = elemk_lin(ii,jj) + term jj = loc_rho(j) term = wt * epsi * vely * Nmat22(i) * DNmat22Dy(j) elemk_lin(ii,jj) = elemk_lin(ii,jj) + term enddoenddo

17

Boundary conditions At the Inlet

The Gas temperature Ti is specified as 314 K . The y-component of the velocity v = 0 . Inlet pressure, Pi is specified based on the corresponding Pressure

ratio.            

At the Outlet The pressure at the outlet, P0 is 100.8 KPa.

On the Walls (No-slip) For isothermal wall the wall temperature Tw is 314 K. u and v velocity components = 0.

18

x

y

0 1 2 3 4 50

0.5

1

1.5

2

2.5

3

3.5

P2.532192.363372.194562.025751.856941.688121.519311.35051.181691.012870.8440620.675250.5064370.3376250.168812

Pressure

x

y

0 1 2 3 4 50

0.5

1

1.5

2

2.5

3

3.5

V80.0027930.00260680.00242060.00223440.00204820.0018620.00167580.00148960.00130340.00111720.0009310010.0007448010.0005586010.0003724010.0001862

Microchannel Flow Contours

Velocity

Pressure ratios: 1.340, 1.680, 2.020, 2.361, 2.701

Channel Aspect Ratio: 2500

19

Slip/No-Slip Comparison

x/L

U/Uin

0 0.25 0.5 0.75 10

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

no-slip,Pr=1.340no-slip,Pr=1.680no-slip,Pr=2.020no-slip,Pr=2.361no-slip,Pr=2.701slip,Pr=1.340slip,Pr=1.680slip,Pr=2.020slip,Pr=2.361slip,Pr=2.701

Slip variation up to ~ +8%

Velocity

x/L

P/Pout

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11

1.25

1.5

1.75

2

2.25

2.5

2.75

no-slip,Pr=1.340no-slip,Pr=1.680no-slip,Pr=2.020no-slip,Pr=2.361no-slip,Pr=2.701slip,Pr=1.340slip,Pr=1.680slip,Pr=2.020slip,Pr=2.361slip,Pr=2.701

Slip variation up to ~ +4%

Pressure

20

Pressure Distribution(Numerical-Experimental Comparison)

x/L

P/Pout

0 0.25 0.5 0.75 11

1.25

1.5

1.75

2

2.25

2.5

2.75Pr-slip-1.340Pr-slip-1.680Pr-slip-2.020Pr-slip-2.361Pr-slip-2.701Experimental Data

x/L

P/Pout

0 0.25 0.5 0.75 11

1.25

1.5

1.75

2

2.25

2.5

2.75

Pr-slip-1.340Pr-slip-1.680Pr-slip-2.020Pr-slip-2.361Pr-slip-2.701Chen et al. -1.340Chen et al. -1.680Chen et al.- 2.020Chen et al.- 2.361Chen et al.- 2.701

Experimental validation within ~ 4 %

Numerical validation within ~ 1.3 %

21

Velocity Profile(Numerical-Numerical Comparison)

x/L

U/Uin

0 0.25 0.5 0.75 10

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035 Pr-slip-1.340Pr-slip-1.680Pr-slip-2.020Pr-slip-2.361Pr-slip-2.701Chen et al.- 1.340Chen et al.- 2.701

Numerical validation within ~ 2.5 %

22

Conclusion

For Microchannel- Finite Element Model compares well with

reported Numerical and Experimental results.

The slip conditions show higher flow rates.

+H/2

-H/2

y

x

23

Future Scope

Extend to the applicability of Slip-flow conditions to the Transition regime,

( 0.1 < Kn < 10 ).

Applicability to higher Knudsen number ranges (Nanoscale devices).

24

Acknowledgements

Dr. Subrata Roy. Dr. Birendra Pandey. Center of Nanotechnology, NASA

Ames. NSF/NPACI Supercomputer. Electric Propulsion Laboratory, NASA

Glenn Research Center.