Quasi 1-d Modeling of an Electrostatically Acuated … | 2 υ2 ρ υ ρ x momentum: ... – Release...

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Modeling an Electrostatically Actuated MEMS Diaphragm

Pump – Part IJames Nabity9 Mar 2004

Submitted in partial requirements of Fluid-Structures Interactions Class (ASEN 5519-006)

TDAResearchTDAResearch

Acknowledgements

• ONR SBIR Phase II “Liquid Fuel Atomizer” sponsored by Dr. Chris Brophy

• ANSYS simulations performed by Mr. Gopi Krishnan under AFOSR grant sponsored by Dr. Mitat Birkan


Outline

• Background & Motivation• Objective• What have others done ?• Modeling & Simulation

– 1-d Model– ANSYS

• Conclusions• Future Plans


BackgroundSeiko Epson TM-8000J configuration

Can inkjet technology be applied to other applications???


Motivation

• Can inkjet technology be applied to other applications???– Yes…optics, automotive & aerospace

• Analysis tools required for design, BUT commercial CFD/FEA software packages are usually difficult to learn and use

• Comprehensive models difficult to develop• THUS, simple and reasonably accurate

model(s) are required to quickly evaluate design potential


Micropump for Aerospace

micropump chamber

diaphragm supports at exit port

fluid check valves


Objective

• Develop, validate and exercise a simplequasi 1-d control volume based model

– Capture important physics– Apply to liquid fuels

• Simulate micropump w/ ANSYS


What have others done?

• Control-volume or lumped-mass models– L.S. Pan, et al, “Analytical solutions for the dynamic analysis

of a valveless micropump—a fluid-membrane coupling study,” Sensors and Actuators A 93 (2001) pp 173-181

– Anders Olsson, et al, “A numerical design study of the valveless diffuser pump using a lumped-mass model,” J. Micromech. Microeng. 9 (1999) pp 34-44

• Numerical modeling– Nam-Trung Nguyen, “Numerical Simulation of Pulse-Width-

Modulated Micropumps with Diffuser/Nozzle Elements”– Gopi Krishnan, PhD candidate, U of Colorado-Boulder


Physical Model Comparison

• Nabity

• Pan et al

• Olsson et al

• Nguyen et al

L = w

1 2 e

-md2ydt2

Fe Fk

yP-+

fillcycle

expulsioncycle

∞

CV

h0

τw

LL = w

1 2 e

-md2ydt2

Fe Fk

yP-+

fillcycle

expulsioncycle

∞

CVCV

h0

τw

L


1-D Model Developent


Model Formulation

( ) ( ) eeffCV

pAAtApAtAhLdV

dtdApAt ||||)(|12|| 2

222

112 +=+=⋅−

++ ∫∫∫ υρυρυµυρυρ vvvv

L = w

1 2 e

-md2ydt2

Fe Fk

yP-+

fillcycle

expulsioncycle

∞

CV

h0

τw

L

∑ ∫∫∫

++−+==

CVkey dV

dtyd

dtdgmApFFF

vvvvv

ρ0

( ) ( )dtdVolAtAtm ee ρυρυρ −== 1|| vv

&Continuity:

( )∫∫∫∫∫∫∫∫ ++

++−

++=∆

f

i

y

ykeoutinCV dyFFdmpudmpuEdmvv

|2

|2

|22 υ

ρυ

ρ

x momentum:ΣF of diaphragm:

inertial terms << loads and may be neglectedEnergy: adiabatic, unreacting flow


Assumptions

• 1-d adiabatic, incompressible flow• Uniform velocity profile• Unsteady, but assume quasi-static

Hagen-Poiseuille flow during each time step

• Perfect check valve• Diaphragm displaced uniformly


Poiseuille Flow(parallel plates)

h

u(y)

dyyduw

dyd

dxdP

lam)(/0 µττ

=+−=

dxdP

dyud=2

2µ

Integrate twice and apply no slip boundary conditions at the wall

dxdPhcc

µ8&0

2

21 −==

From x-momentum

( ) ( )22 481 yhdxdPyu −−=

µ

Now, flow rate can be found for large plate width

( )∫−

−=⋅=2

2

3

12

h

hdxdPwhwdyyuQ

µ


Flat Plate Assumptions

• thin flat plates (t/w < ¼) of uniform thickness and of homogeneous isotropic material; actual t/w = 0.005

• fixed edges with a uniformly distributed load

• the maximum deflection under load must be very small; y < t/2 (y = t possible for our pump)

• all forces are normal to the plate• the diaphragm is nowhere stressed beyond

the elastic limit


Simplified Equations

• continuity

• x-momentum (CV1-2 for example)

• ∑Fy = 0

v 2

w2 ∆y∆t⋅

w h⋅:=

P 1 w2⋅ F e F k+

∆M ρ JP10 v 12

⋅ P 1+

w⋅ G f⋅ ρ JP10 v 2⋅ Q disp⋅+

12 µ f⋅ L star⋅ w⋅ v 2⋅

Gf− ρ JP10 v 2

2⋅ +

−:=

ve

w2 ∆y∆t⋅

wn hn⋅:=

P 2 w⋅ Gf⋅


Forces on Diaphragm

• Pressure

• Electrostatic

• Spring

ke FFwP +=⋅ 21

F e w2

12ε o⋅ ε f⋅ ε d

2⋅ Vi

2⋅

ε d h 0 y−( )⋅ ε f Gd⋅+ 2

:=

F ky E⋅ t d

3⋅

α w4⋅

− w2⋅:=

Roark’s Formulas for Stress and Strain, McGraw-Hill Publishing Co, 7th Edition, 2002


Energy Equation

∆E system ∆E 1 ∆E 2− ∆E e+ ∆E k− ∆E CV−:=

∆Ei uiPi

ρ fuel+

vi2

2+

ρ fuel⋅ Qi⋅ ∆t⋅:= ui boundary work + KE

∆E e

y i

y f

y

12ε o⋅ ε f⋅ ε d

2⋅ Vi

2⋅

ε d h 0 y−( )⋅ ε f Gd⋅+ 2

A diaphragm

⌠⌡

d:=

∆E k

y i

y f

yy E⋅ t d

3⋅

α w4⋅

A diaphragm

⌠⌡

d:=

electrostatic

spring

∆E CV u 1v 2

2

2+

P 1 P 2+( )2ρ JP10

+

− ρ JP10⋅ Q 2 Q 1−( )⋅ ∆t⋅:= work inside deforming control volume


SS302 Diaphragm Mat’l Properties

• Density (8.0 gm/cc)• Yield Strength (276 MPa)• Modulus of Elasticity (20,700 MPa)• Poisson’s Ratio (0.3)


Thermally Grown Oxide Dielectric Layer

• Limits the Maximum Applied Voltage– Dielectric Constant (3.8)– Voltage Breakdown Strength (1000 V/µm)– MAX thickness about 3 µm


JP-10 Fuel Properties@ 25°C

• Density (0.938 gm/cc)• Viscosity (0.003 kg/m-s)• Specific Heat (1.55 kJ/kg-K)• Surface Tension (0.031 N/m)• Dielectric Constant (2.46)


MathCAD Solution Methodology #1

• Solution entails one-half cycle– Electrostatic pull-in– Release to neutral position

Neutral position Release to neutral position

Electrostatic Actuation



• Setup geometry– Chamber length & height– Exit port length & height

• Define parameters– Voltage– Time step– and more

• Define initial conditions (displacement)



• Calculate:– Electrostatic force– Spring force– Upstream pressure– Simultaneously solve for:

• Displacement• Volumetric flow rate• Downstream Pressure

– Check conservation of mass, momentum & energy– Repeat for next time step


Initial ANSYS Results

L*y(t)

w

0

5

10

15

20

0 2 4 6

t, msec

y(t),

um

1-d modelANSYS

010

2030

4050

6070

80

0 2 4 6

t, msec

P, p

si

L* = wL* = w/20 < L* < wANSYS

w = 10000umh = 50um

Gopi Krishnan, Mesh and Time Dependence Study (ANSYS results) Apr 2002


Additional ANSYS Results

volumetric flowrate

0102030405060708090

0 2 4 6

t, msec

V, c

c/m

in

L* = wL* = w/20 < L* < wANSYS

.

velocity

0

1

2

3

0 2 4 6

t, msec

vexi

t, m

/s

L* = wL* = w/20 < L* < wANSYS

Discrepancy largely due to difference in the diaphragm displacement profile (+20% at 1 msec) for ANSYS case


Parametric Analysis

• Characteristic length for Poiseuille pressure loss ( & nominally )

• Actuation voltage & frequency• Pump Size• Supply pressure

wL t <∆f

t GwywL

⋅∆⋅

=∆

2


Actuation Voltage

0

10

20

30

40

50

0 1 2 3 4 5

t, msec

P, p

si

1000V5000V

0

1

2

3

4

5

6

0 1 2 3 4 5t, msec

vexi

t, m

/s

1000V5000V

0

25

50

75

100

125

150

175

0 1 2 3 4 5

t, msec

V, c

c/m

in

1000V5000V

0

5

10

15

20

25

30

35

0 1 2 3 4 5

t, msec

y(t),

um

1000V5000V

.

w = 10000umh = 50um


Actuation Frequency

0

5

10

15

20

25

30

35

0 1 2 3 4 5

t, msec

y(t),

um

130 Hz290 Hz

w = 10000umh = 50umV = 1000V

0

25

50

75

100

125

150

175

200

0 1 2 3 4 5

t, msec

V, c

c/m

in

130 Hz290 Hz

0

1

2

3

4

5

6

7

0 1 2 3 4 5t, msec

vexi

t, m

/s

130 Hz290 Hz

0

10

20

30

40

50

0 1 2 3 4 5

t, msec

P, p

si

130 Hz290 Hz


Pump Size

0

5

10

0 1 2 3 4 5

t, msec

y(t),

um

10000um x 10000um x 50um

5000um x 5000um x 25um

0

0.5

1

1.5

2

2.5

3

0 1 2 3 4 5

t, msec

P, p

si

10000um x 10000um x 50um

5000um x 5000um x 25um

V = 1000V

0

1

2

0 1 2 3 4 5t, msec

vexi

t, m

/s

10000um x 10000um x 50um

5000um x 5000um x 25um

0

5

10

15

20

25

30

0 1 2 3 4 5

t, msec

V, c

c/m

in

10000um x 10000um x 50um5000um x 5000um x 25um


Scaling Law

• Flowrate proportional to– displaced volume (i.e. atomizer size)– actuation voltage– actuation frequency

V fVQ disp ⋅⋅α&

Proportionality constant is 0.0025


Conclusions

• A MathCAD model developed for parametric evaluation of electrostatically actuated diaphragm pumps– MathCAD not well suited to iterative problems– predictions appear qualitatively correct– quantitative accuracy, yet to be validated

• Weaknesses– steady 1-d Poiseuille flow– uniform diaphragm deflection


Next Class Period

• Comparison with ANSYS linear solution results for a model problem

• Future work

TDAResearch


Pump – Part II

James Nabity11 Mar 2004



Acknowledgements




ANSYS Model Development


Model Pump with Passive Valves

Diaphragm

Plenum

Outlet

Inlet Works because flow resistance is less in outlet direction


Numerical Methods

• ANSYS Multi-physics finite element code is used to carry out simulation:– Weak sequential algorithm to couple structural

and fluid dynamics (FSI module),– Linear structural solver,– Arbitrary Lagrangian-Eulerian formulation solves

for the fluid flow with moving boundaries, – Fluid dynamics is solved using ANSYS FLOTRAN

(Flow treated as incompressible),– Non-linear pressure velocity coupling via

SIMPLEF scheme


Linear vs Non-linear Solver

0

5

10

15

20

25

30

0 10 20 30 40 50 60

Pressure ( KPa )

Defle

ctio

n ( m

icro

ns )

linear non linear


Mesh

Issues:-contact: mesh thickness can’t be zero-mesh refinement: educational version

limited to 32,000 elements

Fluid: FLOTRAN FLUID142 elementsStructure: SOLID 54 elements


ANSYS Pump GeometryTime Dependent 3-D Simulation

Diaphragm

Plenum

Outlet

Inlet Ldiaph = 1000 µm

tdiaph = 10 µm

tplenum = 100 µm

tpassages = 100 µm

Lpassages = 330 µm

Winpassages = 66.7 µm

αvalve = 5 degrees

Simulation uses vertical symmetry plane

Calculations for 100-900 Hz


1-d Model Procedure

• Setup ANSYS model geometry• For each time step:

– Match ANSYS displacement– Calculate the flow rate, pressures, and

velocity


Displacement500 Hz

NOTE: Max displacement ~ 3X the diaphragm thickness. Non-linear structural solver should be used.

-30-20-10

0102030

Dis

plac

emen

t (m

icro

ns)

108642

Time (msec)

suct

ion

stro

ke

pressure stroke


ANSYS Flow FieldMid of Suction

Stroke

Top of Suction Stroke

t = 1 msec

Bottom of Pressure Stroke

t = 2.5 msec

t = 1.5 msec


Flow Rates700 Hz

-10

-5

0

5

10

Vol

ume

Flow

(mm

3 /sec

)

2.82.62.42.22.01.81.6

Time(msec)

Inlet Outlet Net


Net Flow Rate

012345678

0 200 400 600 800 1000actuation frequency, Hz

volu

met

ric fl

ow ra

te, c

c/m

in ANSYS net flow

1-d constantdeflection


Discussion

• Very large discrepancy in results…What does it mean?– Constant displacement of diaphragm vs

non-linear 3-d deflection (factor of 3)– Perfect check valves vs high leakage

valves (factor of 4)– Other? (factor of 2)


Flow Datadiaphragm & check valve corrections

012345678


volu

met

ric fl

ow ra

te, c

c/m

in ANSYS net flowANSYS total flow1-d constant deflection

pyramidal deflection profile

?


Pressure at Diaphragm Center500 Hz

30x103

20

10

0

Pres

sure

(Pa)

1086420

Time (ms)

Baseline 1-d model peak pressure

Corrected for diaphragm displacement & fluid properties


Discussion

• Still a large discrepancy (2.0 vs 0.9 cc/min)

• What else?– What is the ANSYS diaphragm displacement

profile?– What will the actual displacement look like?

higher mode response

pyramidalconstant


Electrostatic Displacement of Diaphragm

Electrostatic-structural coupling only


Conclusions

• While both models appear to simulate the micropump, there is an unexplained discrepancy between the models.

• Flow is complex, but the simple 1-d model can guide design efforts.


Future Plans

• Identify source of discrepancy between the ANSYS and 1-d models

• Improve the 1-d model for known deficiencies– Stokes time-dependent flow vs steady Poiseuille

flow• fluid inertial force proportional to actuation frequency• introduces phase shift

– improved software or solution methodology– 3-d pyramidal diaphragm displacement

Add stokes equation

TDAResearch


Pump – Part IIIJames Nabity27 Apr 2004



Acknowledgements




Outline

• Recent Accomplishments• Review 1-D and ANSYS Models• Problem Found !!!• Some Simulation Results• Conclusions


Recent Accomplishments

Discrepancy between the ANSYS and 1-d models found1-d model improvements

improved MathCAD solution methodology using EXCEL files to close iteration loop and store data3-d pyramidal diaphragm displacement implemented


1-D Model

( ) ( ) eeffCV

pAAtApAtAhLdV

dtdApAt ||||)(|12|| 2

222

112 +=+=⋅−

++ ∫∫∫ υρυρυµυρυρ vvvv

L = w

1 2 e

-md2ydt2

Fe Fk

yP-+

fillcycle

expulsioncycle

∞

CV

h0

τw

L

∑ ∫∫∫

++−+==

CVkey dV

dtyd

dtdgmApFFF

vvvvv

ρ0

( ) ( )dtdVolAtAtm ee ρυρυρ −== 1|| vv

&Continuity:

( )∫∫∫∫∫∫∫∫ ++

++−

++=∆

f

i

y

ykeoutinCV dyFFdmpudmpuEdmvv

|2

|2

|22 υ

ρυ

ρ

x momentum:ΣF of diaphragm:

inertial terms << loads and may be neglectedEnergy: adiabatic, unreacting flow


ANSYS MODEL

Diaphragm

Plenum

Outlet

Inlet


Flow Data Comparisondiaphragm & check valve corrections

012345678


volu

met

ric fl

ow ra

te, c

c/m



???

Kd = 1

Kd = 1/3


Recall: ANSYS Model

Only ½ of the pump was modeled.

Therefore, calculated flow was only ½ of the total pump capacity !!!


Flow DataANSYS flow corrected for symmetry b.c.

012345678


volu

met

ric fl

ow ra

te, c

c/m




Micropump performance

• Now what is the performance of a micropump ?

• Given:– 1cm x 1cm x 50um micropump chamber– 50um thick metal diaphragm– Jet fuel


Diaphragm Displacement ( ½ cycle )

0.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

0 1 2 3 4 5 6 7

time, msec

h, u

m

-2.0

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

18.0

0 1 2 3 4 5 6 7

time, msec

y, u

m

electrostatic – spring force during compression

electrostatic + spring force during release

neutral position

neutral position

compression release

compression

release

f = 83 Hz


Micropump Flowrate( ½ cycle)

0.00

5.00

10.00

15.00

20.00

25.00

0 1 2 3 4 5 6 7

time, msec

Q, c

c/m

in

compression

release

f = 83 Hz

Recall that Q is proportional to gap height cubed


1-D model micropump performance

0.00

2.00

4.00

6.00

8.00

10.00

12.00

14.00

0 50 100 150 200 250 300

f, Hz

Q, c

c/m

in

Optimal frequency likely near knee of curve.

Experimental tests needed to confirm.


Scaling Law

• Flowrate proportional to– displaced volume (i.e. atomizer size)– actuation voltage– actuation frequency

V fVoltsQ disp ⋅⋅α&

Proportionality constant still 0.0025, if displaced volume now equal to Kd*(y*w2)


Conclusions

• 1-D Micropump Model Developed for Parametric Studies

• Approximately 15% Error in Results


What’s Left ?

• Implement Stokes time-dependent flow to solve for velocity

• fluid inertial force proportional to actuation frequency

• introduces phase shift

• Unfortunately, problem reformulation likely

uptu vvv 2∇+−∇=∂∂ µρ

Quasi 1-d Modeling of an Electrostatically Acuated … | 2 υ2 ρ υ ρ x momentum: ... – Release...

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