Transcript Introduction to Modeling - Rose

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
electrical
input
Thermal
actuator
mechanical
output (motion)
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.
waste thermal
energy
TH
Q
TL
The three modes of heat transfer
TH
TL
Conduction
Convection (Conduction +
advection)
Radiation
Both
require a
material
medium
Does not require a material
medium
Working equations of the three modes
Conduction
T1  T2 T1  T2


Q
d
Rth
A
T1
T2
Q
T1
e1
T2
e2
A
TH
TL
T1  T2 T1  T2

Q

d
Rth
A
Material
Copper
Styrofoam
Silicon
i
e1  e2 e1  e2

d
R
A
κ (W/m·K = [ॐ·m]-1)
401
0.04
148
Working equations of the three modes
Q
Convection
T2
surface
area A at Ts
moving
fluid at T∞
T1  T2

Q  hA(T1  T2 ) 
Rth
T1
1
Rth 
hA
TH
TL
Convection
Fluid
Natural
Gas
Natural
Liquid
Forced
Gas
Forced
Liquid
h (W/m2·K =
[ॐ·m2]-1)
2-20
10-1000
20-250
100-20,000
Working equations of the three modes
Radiation
Q
Perfect blackbody
surface
area A at Ts
4
4



AT
Q  ATS
S
Non-ideal surface
TL
TH
Material
ε
Polished Al
0.04
0.98
0.90
0.67
Black paint
Skin
Si
Q
surrounding
surface at Tsurr
Small object completely surrounded by large surface
Q  A(TS4 T surr4)
Repaso del actuador térmico
Actuador térmico (brazo caliente)
~ 200 μm
Actuador térmico hecho de poli silicio
Actuador térmico
Como funciona el actuador
+
e
-
i
Modelo sencillo de actuador térmico
voltage
electrical
input
tip deflection
mechanical
output (motion)
Thermal
actuator
Our goal: ωtip = f(e)
waste thermal
energy
+
e
-
ωtip
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)
Modelo sencillo de actuador térmico
Perimeter, P
Ah
hot arm at Th
D
cold arm at Tc
L
side view
+
W
eh
Qconv  hA(Th  T )
-
+
e
-
Ac
I
-
ec
+
Qconv  hA(Tc  T )
Modelo sencillo de actuador térmico
Perimeter, P
Ah
hot arm at Th
D
cold arm at Tc
L
+
eh
side view
L
Rh  
Ah
-
+
e
-
i
-
ec
+
Rc  
L
Ac
Ac
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.
Rh
eh 
e
Rh  Rc
 ( L / Ah )

e
 ( L / Ah )   ( L / Ac )
Ac

e
Ah  Ac
ec 
Ah
e
Ah  Ac
Modelo sencillo de actuador térmico
Perimeter, P
Ah
hot arm at Th
L
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).
D
cold arm at Tc
side view
Ac
2
+
eh
eh

Qconv  hA(Th  T )  ieh 
Rh
-
+
W e
-
Te toca a ti
Q conv  hA(Th  T )
A = PL
2
Ah  Ac  2

 e  T
Th 
2 
hPh L  Ah  Ac 
i
-
ec
+
Qconv  hA(Tc  T )
2
Ac  Ah  2

 e  T
Tc 
2 
hPc L  Ah  Ac 
Modelo sencillo de actuador térmico
Hot arm thermal
stress σ
D
ωtip
Hot arm is initially at T∞, and is then
heated to Th. What is the thermal
strain?
 h   T (Th  T )
x
Induced bending
moment M ≈ DσAh
What about the cold arm?
 c   T (Tc  T )
Cold arm is much thicker than hot arm. So let’s assume both experience the same actual strain. Which one?
εboth = εh or εc ?
hot ______
arm experiences two pieces of strain – one due to thermal expansion and another extra piece
The ______
cold ______.
arm
due to the fact that it is hooked to the ______
 h   T (Th  T )   extra   c   T (Tc  T )
Modelo sencillo de actuador térmico
Hot arm thermal
stress σ
D
ωtip
Solve for this extra piece of strain,
εextra.
x
Induced bending
moment M ≈ DσAh
 extra   T (Tc  Th )
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?
  E
 E T (Tc  Th )
Modelo sencillo de actuador térmico
Beam bending relations
dx
x
ds  Rd
ds
θ
ω
For small deflections dx ≈ ds, hence dx ≈ Rdθ. So
d 1

dx R
dθ
R
For small deflections tan(θ) ≈ θ.
Can also show
1
M

R
EI
d

dx
d 2 1

2
R
dx
d 2
M


EI
dx2
Modelo sencillo de actuador térmico
Hot arm thermal
stress σ
D
This gives us an expression for the
deflection as a function of the
length, x.
ωtip
d 2
M
DAh


2
EI
EI
dx
x
Induced bending
moment M ≈ DσAh
Integrate this expression from x = 0 to x = L to get the tip deflection, ωtip.
DAh L2
tip  
2 EI
Finally, substitute expressions for σ and the temperatures to complete our model.
2

D T Ah e Ah  Ac  Ac
 
 
tip 
2h I  Ph  Ah  Ac  Pc

2
 Ah 


 Ah  Ac 
2



¡E no está!
Modelo un poco más complejo
Add a third resistor for the flexure:
Q conv
Q conv
Q conv
Modelo un poco más complejo
Electrical resistances of the arms given by
Allow for a temperature dependence of resistivity, ρ = ρ(T):
Modelo un poco más complejo
Comparison to the model of Huang and Lee (1999)
our model
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
Modelo un poco más complejo
Mechanical model
x
d 2 ( x)
M ( x)

2
dx
EI
1
h  
EI h
 M 0 Lh 2 VLh 3 



2
6


1
c 
EI f
 M f Lf 2

 2


Hc 
2
2
M

P

 L f  Lc   L f
f

2 
3
VL f  1  

2


6  EI c 
3
3
V L f  Lc   L f


6








Modelo un poco más complejo
Comparison to data
Lc = 120 μm
Lh = 240 μm
Lc = 180 μm
Lh = 240 μm
Modelo un poco más complejo
Comparison to data
Lc = 120 μm
Lh = 240 μm
Lc = 120 μm
Lh = 240 μm
E = 150 GPa
E = 10 Pa