1. 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.

2. 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 Thermal actuator THTH TLTL

3. The three modes of heat transfer Conduction THTH TLTL Convection (Conduction + advection) Radiation Both require a material medium Does not require a material medium

4. Working equations of the three modes THTH TLTL Conduction A T1T1 T2T2 T1T1 T2T2 e1e1 e2e2 Material κ (W/m·K = [ ॐ ·m ] -1 ) Copper Styrofoam Silicon 401 0.04 148

5. Working equations of the three modes THTH TLTL Convection Fluid h (W/m 2 ·K = [ ॐ ·m 2 ] -1 ) NaturalGas NaturalLiquid ForcedGas ForcedLiquid 10-1000 20-250 100-20,000 surface area A at T s moving fluid at T ∞ T1T1 T2T2 2-20

6. Working equations of the three modes THTH TLTL Radiation Material ε Polished Al Black paint Skin Si 0.98 0.90 0.67 surface area A at T s 0.04 Perfect blackbody Non-ideal surface Small object completely surrounded by large surface surrounding surface at T surr

7. Repaso del actuador térmico Actuador térmico (brazo caliente) Actuador térmico hecho de poli silicio ~ 200 μm

8. Actuador térmico Como funciona el actuador i +e-+e-

9. Modelo sencillo de actuador térmico mechanical output (motion) voltage waste thermal energy Thermal actuator electrical input tip deflection +e-+e- ω tip ω tip = f(e) Our goal:

10. 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)

11. Modelo sencillo de actuador térmico + e h - - e c + I +e-+e- L hot arm at T h cold arm at T c AcAc AhAh D side view Perimeter, P

12. Modelo sencillo de actuador térmico + e h - - e c + i +e-+e- L hot arm at T h cold arm at T c AcAc AhAh D side view Perimeter, P Te toca a ti Find the voltage drops across the hot arm and cold arm ( e h and e c ) in terms of the input voltage ( e ), the resistivity of the actuator material ( ρ ), and its geometry.

13. Modelo sencillo de actuador térmico + e h - - e c + i +e-+e- L hot arm at T h cold arm at T c AcAc AhAh D side view Perimeter, P Te toca a ti Find the temperatures the hot arm and cold arm ( T h and T c ) in terms of the input voltage ( e ), the resistivity of the actuator material ( ρ ), its geometry, and the heat transfer coefficient ( h ). A = P L

14. Modelo sencillo de actuador térmico ω tip Hot arm thermal stress σ D Induced bending moment M ≈ DσA h x Hot arm is initially at T ∞, and is then heated to T h. 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? ε both = ε h or ε c ? 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 arm cold arm

15. Modelo sencillo de actuador térmico ω tip Hot arm thermal stress σ D Induced bending moment M ≈ DσA h x Solve for this extra piece of strain, ε extra. 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 ?

16. Modelo sencillo de actuador térmico dx ds x ω R θ dθ For small deflections dx ≈ ds, hence dx ≈ Rdθ. So For small deflections tan(θ) ≈ θ. Can also show Beam bending relations

17. Modelo sencillo de actuador térmico ω tip Hot arm thermal stress σ D Induced bending moment M ≈ DσA h x This gives us an expression for the deflection as a function of the length, x. Integrate this expression from x = 0 to x = L to get the tip deflection, ω tip. Finally, substitute expressions for σ and the temperatures to complete our model. ¡ E no está!

18. Modelo un poco más complejo Add a third resistor for the flexure:

19. Modelo un poco más complejo Electrical resistances of the arms given by Allow for a temperature dependence of resistivity, ρ = ρ(T):

20. Modelo un poco más complejo 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

21. Modelo un poco más complejo Mechanical model x

22. Modelo un poco más complejo Comparison to data L c = 120 μm L h = 240 μm L c = 180 μm L h = 240 μm

23. Modelo un poco más complejo Comparison to data E = 150 GPa E = 10 Pa L c = 120 μm L h = 240 μm L c = 120 μm L h = 240 μm