ME 322: Instrumentation Lecture 39

1. ME 322: InstrumentationLecture 39 April 28, 2014 Professor Miles Greiner

2. Announcements/Reminders • This week: Lab 12 Feedback Control • HW 13 Due now • HW 14 Due Wednesday, 4/30/2014 • X3 (On Web, Last HW assignment) • Review Labs 9, 10, 11, and 12; • 4/30 and 5/2 • Open Lab Practice • May 2-4 • Lab Practicum Finals (May 6-13) • Guidelines, Schedule • http://wolfweb.unr.edu/homepage/greiner/teaching/MECH322Instrumentation/Tests/Index.htm • Extra Credit Lab 12.1 • See Lab 12 instructions (study effect of DT, DTi, TSP, heater and TC locations) • Check out Lab-in-a-Box for DeLaMare Library • Only 0.5% of grade

3. Proportional-Control Thermal Analysis Heater QIN = FTO(QMAX) TEnv TTC TW QOUT = hA(TW -TEnv) • Is TW = TTC? Is TW uniform? • Energy Balance for Water • For proportional control:

4. For Large DT • We observed that TTC is steady when DT is sufficiently large • Under that condition, assume TTC = TW= • Steady State Error • How does this prediction compare to measurements?

5. Measured Proportional-Control Steady-State Error • The temperature is steady (TRMS becomes small) once DT is sufficiently large • Prediction: • Measurements show the error magnitude increases as increases • When no need for control • Steady state error decreases as increases (DT decreases) • But when DT is too small the temperature oscillates (TRMS) • Why does this happen? (

6. How to predict TC temperature TTC(t)? QIN= hA(TW -TTC) TTC TW • TC Energy Balance • (Eqn. A) • Constants • Dynamic relationship between (t) and

7. Water Energy-Balance Heater QIN = FTO(QMAX) TENV TTC TW QOUT = hA(TW -TENV) • For proportional control: • Where the controller gain is • , , and are constants • In addition to , ,

8. Collect Terms • (Eqn. B) • Dynamic relationship between and • Couple with Eqn. A: • What do we have? • Two, 1st-order, coupled, constant-coefficient liner-differential equations for (t) and (t)

9. System Solution • Solve Eqn. A for • Plug into Eqn. B and collect terms

10. Collect Terms • More Constant • 2nd order, linear, Constant-coefficient differential-equation, non-homogeneous (RHS= Constant) • Solution • Particular Solution for RHS = Constant, • Same as for simple 1st order system

11. Homogeneous Solution • Assumed solution • Characteristic Equation • If DT is small enough, then • will be large enough so that • , • b will be imaginary and • will be oscillatory • We observed oscillations for small DT

12. What is the largest minimum value of DT that will have a steady behavior? • , • ,

13. Problem X3 • Problem X3: A 200-Watt heater and a 1.5-mm-diameter thermocouple are placed in a water-filled beaker of diameter 3 cm and height 5 cm. If the heat transfer coefficient between the beaker and air is 5 W/m2K, and between the water and thermocouple is 1000 W/m2K, estimate the lowest proportional control temperature increment DTMin, for which the control system will be steady (not oscillatory). Assume the thermocouple properties to be that of iron, and evaluate water properties at 30°C. • Heater: Q = 200 W • Beaker: D = 3 cm, H = 5 cm, hAir = 5 W/m2K • TC: D = 1.5 mm, hTC= 1000 W/m2K, iron

17. Proportional Control 1st Law

18. Proportional Control Find Want DT to be small, but that leads to oscillation.

19. Proportional Control No way for a 1st order constant coefficient deferential equation to give oscillation. How to predict/model oscillations? (Twater ≠ TTC) Better Model

20. 1st Law TC

21. Water Heater HA HT MW Unknown: T, T­W

22. A B Homogenous Solutions:

23. Characteristic Equation If Then set complex If DT is small enough then get oscillations.