1 A Project Presentation for Applied Computational Fluid Dynamics By Reni Raju Finite Element Model...
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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).