thermal actuators give qualitative and quantitative descriptions of the three modes of heat...

Thermal actuators Give qualitative and quantitative descriptions

of the three modes of heat transfer. Explain the behavior of a hot arm actuator,

both qualitatively and quantitatively, based on our simplified lumped element model.

A generic thermal actuator

Inside the actuator energy has been converted to the thermal mode.

Thermal energy moves from regions of high temperature to low temperature. This is called heat transfer.

electrical input

mechanical output (motion)

waste thermal energy

Thermalactuator

TH TLQ

The three modes of heat transfer

Conduction

TH TL

Convection (Conduction + advection)

Radiation

Both require a material medium

Does not require a material medium

Working equations of the three modes

TH TL

Conduction

A

T1

T2

Q

Q

thR

TT

A

dTT

Q 2121

R

ee

A

dee

i 2121

T1 T2e1

e2

Material κ (W/m·K = [ॐ·m]-1)

Copper

Styrofoam

Silicon

401

0.04

148

Ad

TT

21

thR

TT 21


TH TL

Convection

Q

Convection Fluid h (W/m2·K = [ॐ·m2]-1)

Natural Gas

Natural Liquid

Forced Gas

Forced Liquid

10-1000

20-250

100-20,000

)( 21 TThAthR

TT 21

surface area A at Ts

moving fluid at T∞

Q

T1

T2

thRhA

1

2-20


TH TL

Radiation

Q

Material ε

Polished Al

Black paint

Skin

Si

0.98

0.90

0.67

4

SAT

surface area A at Ts

Q

0.04

Perfect blackbody

4

SAT

Non-ideal surface

)( 44surrS TTAQ

Small object completely surrounded by large surface

Q surrounding surface at Tsurr

Repaso del actuador térmico

Actuador térmico (brazo caliente)

Actuador térmico hecho de poli silicio

~ 200 μm

Actuador térmico

Como funciona el actuador

i

+e-

Modelo sencillo de actuador térmico

mechanical output (motion)

voltage

waste thermal energy

Thermalactuator

electrical input

tip deflection

+e-

ωtip

ωtip = f(e)Our goal:

Te toca a ti

Ideas on modeling

List some ideas about how you might create such a model. What physical concepts would you use? What simplifications would you make?

• Assume actuator has only two arms (hot arm and cold arm) each with only

one temperature

• The actuator is at steady state with a continuous electrical input being

dissipated in the two electrical resistances created by the hot arm and the cold

arm.

• All the stress is initially experienced by the hot arm, which can be calculated in a

way similar to thermal mismatch stress.

• The hot arm stress causes a bending moment in the cold arm, the deflection of

which can be calculated using standard beam bending theory. (Bernoulli beam

bending)

)( TThA cconv

)( TThA hconv


+ eh -

- ec +

I+e-

L

hot arm at Th

cold arm at Tc

Ac

Ah

D

side view

Q

Q

W

Perimeter, P


+ eh -

- ec +

i+e-

L

hot arm at Th

cold arm at Tc

Ac

Ah

D

side view

Perimeter, P Te toca a ti

Find the voltage drops across the hot arm and cold arm (eh and ec) in terms of the input voltage (e), the resistivity of the actuator material (ρ), and its geometry.

eRR

Re

ch

hh

eALAL

AL

ch

h

)/()/(

)/(

h

h A

LR

eAA

A

ch

c

eAA

Ae

ch

hc

cc A

LR


+ eh -

- ec +

i+e-

L

hot arm at Th

cold arm at Tc

Ac

Ah

D

side view

Perimeter, P Te toca a ti

Find the temperatures the hot arm and cold arm (Th and Tc) in terms of the input voltage (e), the resistivity of the actuator material (ρ), its geometry, and the heat transfer coefficient (h).

Q

Q

W

)( TThA hconv

)( TThA cconv

hieh

h

R

e 2

TeAA

A

Lh

AT

ch

c

h

hh

2

2

2P

)( TThAQ hconv

A = PL

TeAA

A

Lh

AT

ch

h

c

cc

2

2

2P


ωtip

Hot arm thermal stress σ

D

Induced bending moment M ≈ DσAh

x

Hot arm is initially at T∞, and is then heated to Th. What is the thermal strain?

What about the cold arm?

Cold arm is much thicker than hot arm. So let’s assume both experience the same actual strain. Which one?

h )( TThT

c )( TTcT

εboth = εh or εc ?

The ______ ______ experiences two pieces of strain – one due to thermal expansion and another extra piece due to the fact that it is hooked to the ______ ______.

hot armcold arm

h )( TThT extra c )( TTcT


ωtip


D


x

Solve for this extra piece of strain, εextra.

How would you model the stress/strain in the hot arm? What would the relation for strain be, then? Is the arm in tension or compression?

)( hcT TT extra

E

)( hcT TTE


dx

ds

x

ω

R

θ

dθ

Rdds

Rdx

d 1

For small deflections dx ≈ ds, hence dx ≈ Rdθ. So

For small deflections tan(θ) ≈ θ.

Rdx

d 12

2

dx

d

EI

M

dx

d

2

2EI

M

R

1

Can also show

Beam bending relations


ωtip


D


x

This gives us an expression for the deflection as a function of the length, x.

EI

AD

EI

M

dx

d h2

2

Integrate this expression from x = 0 to x = L to get the tip deflection, ωtip.

EI

LAD htip 2

2

222

2 ch

h

c

c

ch

c

h

hhTtip AA

AA

AA

AA

Ih

eAD

PP

Finally, substitute expressions for σ and the temperatures to complete our model.

¡E no está!

convQ

convQ convQ

Modelo un poco más complejo

Add a third resistor for the flexure:


Electrical resistances of the arms given by

Allow for a temperature dependence of resistivity, ρ = ρ(T):


Comparison to the model of Huang and Lee (1999)

Q. A. Huang and N. K. S. Lee, “Analysis and design of polysilicon thermal flexure actuator,: J. Micromech. Microeng., vol. 9, pp. 64–70, 1999

our model


Mechanical model

EI

xM

dx

xd )()(2

2

x

62

1 320 hh

hh

VLLM

EI

6

22

1

62

1

33

22

32

fcf

fcfc

f

c

fff

fc

LLLV

LLLH

PM

EI

VLLM

EI


Comparison to data

Lc = 120 μmLh = 240 μm

Lc = 180 μmLh = 240 μm


Comparison to data

E = 150 GPa E = 10 Pa

Lc = 120 μmLh = 240 μm

Lc = 120 μmLh = 240 μm

thermal actuators give qualitative and quantitative descriptions of the three modes of heat...

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