ME 322: Instrumentation Lecture 39 April 27, 2015 Professor Miles Greiner Integral Control program, Proportional Control response model

Announcements/Reminders This week: Lab 12 Feedback Control HW 13 Will accept Wednesday HW 14 Due Friday, X3 (Last HW assignment) Review Labs 10, 11, and 12: Wednesday and Friday Open Lab Practice Session: Saturday and Sunday Lab Practicum Finals (Starting a week from today) Schedule on WebCampus Guidelines, http://wolfweb.unr.edu/homepage/greiner/teaching/MECH322Instrumentation/Tests/Index.htm Drop extra-credit LabVIEW Workshop and Lab 12.1 I will add 6 points to everyone’s Midterm 2 (1.5% of grade) Wind tunnel CTA We are working to adjust the CTA feedback system so they will work correctly for the final You are graded on LabVIEW programming, data acquisition, calculations, clear plots and tables, conclusions (not on equipment failures)

Modify Proportional Control Use Control-U to make block diagram clearer Shift register, input DTi Add 𝐹𝑇𝑂 𝑖 = 𝑇 𝑆𝑃 −𝑇 𝐷𝑇𝑖 to FTOp Display FTOi (bar and numerical indicators) Add 10log(DTi) and log(DTi) to plots Add Write to Measurement File VI Use next available file name No Headers One time column Microsoft Excel

Lab 12 Integral Control Block Diagram Modify proportional VI http://wolfweb.unr.edu/homepage/greiner/teaching/MECH322Instrumentation/Labs/Lab%2012%20Thermal%20Control/Lab%20Index.htm

Figure 1 VI Front Panel Plots help the user monitor the time-dependent measured and set-point temperatures T and TSP, temperature error T–TSP, and control parameters

Process Sample Data Add time scale in minutes Figure 3 Figure 4 http://wolfweb.unr.edu/homepage/greiner/teaching/MECH322Instrumentation/Labs/Lab%2012%20Thermal%20Control/Lab%20Index.htm Add time scale in minutes Calculate difference, general format, times 24*60 Figure 3 Plot T, TSP, DT and 10log(DTi) versus time Figure 4 Plot T-TSP, -DT, 10log(DTi) and 0 versus time Table 1 Determine time periods when behavior reaches “steady state,” and find 𝑇 𝐴 = 𝑇 and 𝑇 𝑅𝑀𝑆 = 𝜎 𝑇 during those times Be sure to use an integer number of cycles Figure 5 Plot 𝑇 𝑅𝑀𝑆 versus DT and DTi Figure 6 Plot steady state error e= 𝑇 𝐴 − 𝑇 𝑆𝑃 versus DT and DTi

Figure 3 Measured, Set-Point, Lower-Control Temperatures and DTi versus Time Data was acquired for 40 minutes with a set-point temperature of 85°C. Measure response to each setting for 10 minutes or less The time-dependent water temperature is shown with different values of the control parameters DT and DTi. Proportional control is off when DT = 0 Integral control is effectively off when DTi = 107 (10log(DTI) = 70)

Figure 4 Temperature Error, DT and DTi versus Time The temperature oscillates for DT = 0, 5, and 15°C, but was nearly steady for DT = 20°C Sometimes not steady until DT = 30°C May dependent on TC location and water level DTi was set to 100 from roughly t = 25 to 30 minutes, but the systems oscillated, and so it was increased to 1000. The controlled-system behavior depends on the relative locations of the heater, thermocouple, and side of the beaker, and the amount of water in the beaker. These parameters were not controlled during the experiment (Lab 12.1 investigates these rich behaviors).

Table 1 Controller Performance Parameters This table summarizes the time periods when the system exhibits steady state behaviors for each DT and DTi. During each steady state period TA is the average temperature TA – TSP is an indication of the average controller error The Root-Mean-Squared temperature TRMS is an indication of controller unsteadiness

TRMS is and indication of thermocouple temperature unsteadiness Figure 5 Controller Unsteadiness versus Proportionality Increment and Set-Point Temperature TRMS is and indication of thermocouple temperature unsteadiness Unsteadiness decreased as DT increased, and was not strongly affected by DTi.

The average temperature error Figure 6 Average Temperature Error versus Set-Point Temperature and Proportionality Increment The average temperature error Is positive for DT = 0, but decreases and becomes negative as DT increases. Is significantly improved by Integral control.

Proportional-Control Thermal Analysis TW Heater QIN = FTO(QMAX) QOUT = hA(TW -TEnv) TEnv TTC Is TW = TTC? Is TW uniform? Energy Balance for Water 𝑄−𝑊= 𝑄 𝐼𝑛 − 𝑄 𝑂𝑢𝑡 = 𝑑𝑈 𝑑𝑡 = 𝜌𝑐𝑉 𝑊 𝑑 𝑇 𝑊 𝑑𝑡 𝑄 𝑂𝑢𝑡 =ℎ𝐴( 𝑇 𝑊 − 𝑇 𝐸𝑁𝑉 ) For proportional control: 𝑄 𝐼𝑛 = 𝑄 𝑀𝑎𝑥 𝐹𝑇𝑂 = 𝑄 𝑀𝑎𝑥 𝑇 𝑆𝑃 − 𝑇 𝑇𝐶 𝐷𝑇 𝑄 𝑀𝑎𝑥 𝑇 𝑆𝑃 − 𝑇 𝑇𝐶 𝐷𝑇 −ℎ𝐴 𝑇 𝑊 − 𝑇 𝐸𝑛𝑣 = 𝜌𝑐𝑉 𝑊 𝑑 𝑇 𝑊 𝑑𝑡

For Large DT We observed that TTC is steady when DT is sufficiently large Under that condition, assume TTC = TW = 𝑇 𝑆𝑆 𝑄 𝑀𝑎𝑥 𝑇 𝑆𝑃 − 𝑇 𝑆𝑆 𝐷𝑇 −ℎ𝐴 𝑇 𝑆𝑆 − 𝑇 𝐸𝑛𝑣 = 𝜌𝑐𝑉 𝑊 𝑑 𝑇 𝑆𝑆 𝑑𝑡 𝑇 𝑆𝑃 𝑄 𝑀𝑎𝑥 𝐷𝑇 + 𝑇 𝐸𝑛𝑣 ℎ𝐴 = 𝑇 𝑆𝑆 𝑄 𝑀𝑎𝑥 𝐷𝑇 +ℎ𝐴 𝑇 𝑆𝑆 = 𝑇 𝑆𝑃 𝑄 𝑀𝑎𝑥 𝐷𝑇 + 𝑇 𝐸𝑛𝑣 ℎ𝐴 𝑄 𝑀𝑎𝑥 𝐷𝑇 +ℎ𝐴 Steady State Error 𝑒 𝑆𝑆 = 𝑇 𝑆𝑆 − 𝑇 𝑆𝑃 = 𝑇 𝑆𝑃 𝑄 𝑀𝑎𝑥 𝐷𝑇 + 𝑇 𝐸𝑛𝑣 ℎ𝐴 − 𝑇 𝑆𝑃 𝑄 𝑀𝑎𝑥 𝐷𝑇 +ℎ𝐴 𝑄 𝑀𝑎𝑥 𝐷𝑇 +ℎ𝐴 𝑒 𝑆𝑆 =− 𝑇 𝑆𝑃 − 𝑇 𝐸𝑁𝑉 𝑄 𝑀𝑎𝑥 𝐷𝑇 ℎ𝐴 +1 How does this prediction compare to measurements?

Measured Proportional-Control Steady-State Error 𝑒 𝑆𝑆 𝑇 𝑅𝑀𝑆 The temperature is steady (TRMS becomes small) once DT is sufficiently large Prediction: 𝑒 𝑆𝑆 = 𝑇 𝑆𝑆 − 𝑇 𝑆𝑃 =− 𝑇 𝑆𝑃 − 𝑇 𝐸𝑁𝑉 𝑄 𝑀𝑎𝑥 𝐷𝑇 ℎ𝐴 +1 Measurements show the error magnitude increases as 𝑇 𝑆𝑃 − 𝑇 𝐸𝑁𝑉 increases When 𝑇 𝑆𝑃 = 𝑇 𝐸𝑁𝑉 no need for control Steady state error decreases as 𝑄 𝑀𝑎𝑥 𝐷𝑇 ℎ𝐴 increases (DT decreases) But when DT is too small the temperature oscillates (TRMS) Why does this happen? ( 𝑇 𝑊 ≠ 𝑇 𝑇𝐶 )

Predict time-dependent TTC(t) for TC TW TTC QIN = hA(TW -TTC) TC Energy Balance (assuming 𝑇 𝑊 ≠ 𝑇 𝑇𝐶 ) 𝑄−𝑊= 𝜌𝑐𝑉 𝑇𝐶 𝑑 𝑇 𝑇𝐶 𝑑𝑡 𝑄= ℎ𝐴 𝑇𝐶 𝑇 𝑊 − 𝑇 𝑇𝐶 = 𝜌𝑐𝑉 𝑇𝐶 𝑑 𝑇 𝑇𝐶 𝑑𝑡 𝑯 𝑻𝑪 𝑻 𝑾 − 𝑻 𝑻𝑪 = 𝑴 𝑻𝑪 𝒅 𝑻 𝑻𝑪 𝒅𝒕 (Eqn. A) 𝐻 𝑇𝐶 = ℎ𝐴 𝑇𝐶 𝑀 𝑇𝐶 = 𝜌𝑐𝑉 𝑇𝐶 Constants Dynamic relationship between 𝑇 𝑇𝐶 (t) and 𝑇 𝑊 (𝑡)

Water Energy-Balance TW Heater QIN = FTO(QMAX) QOUT = hA(TW -TENV) TENV TTC 𝑄 𝐼𝑛 − 𝑄 𝑂𝑢𝑡 = 𝜌𝑐𝑉 𝑊 𝑑 𝑇 𝑊 𝑑𝑡 = 𝑀 𝑊 𝑑 𝑇 𝑊 𝑑𝑡 For proportional control: 𝑄 𝐼𝑛 = 𝑄 𝑀𝑎𝑥 𝑇 𝑆𝑃 − 𝑇 𝑇𝐶 𝐷𝑇 =𝐺 𝑇 𝑆𝑃 − 𝑇 𝑇𝐶 Where the controller gain is 𝐺= 𝑄 𝑀𝑎𝑥 𝐷𝑇 𝑄 𝑂𝑢𝑡 = ℎ𝐴 𝐴 𝑇 𝑊 − 𝑇 𝐸𝑁𝑉 + ℎ𝐴 𝑇𝐶 ( 𝑇 𝑊 − 𝑇 𝑇𝐶 ) 𝐺 𝑇 𝑆𝑃 − 𝑇 𝑇𝐶 − 𝐻 𝐴 𝑇 𝑊 − 𝑇 𝐸𝑁𝑉 − 𝐻 𝑇𝐶 ( 𝑇 𝑊 − 𝑇 𝑇𝐶 )= 𝑀 𝑊 𝑑 𝑇 𝑊 𝑑𝑡 𝐻 𝐴 = ℎ𝐴 𝐴 , 𝑀 𝑊 = 𝜌𝑐𝑉 𝑊 , 𝑇 𝐸𝑁𝑉 and 𝑇 𝑆𝑃 are constants In addition to 𝐺= 𝑄 𝑀𝑎𝑥 𝐷𝑇 , 𝐻 𝑇𝐶 = ℎ𝐴 𝑇𝐶 , 𝑀 𝑇𝐶 = 𝜌𝑐𝑉 𝑇𝐶

Collect Terms Couple with Eqn. A: 𝑯 𝑻𝑪 𝑻 𝑾 − 𝑻 𝑻𝑪 = 𝑴 𝑻𝑪 𝒅 𝑻 𝑻𝑪 𝒅𝒕 𝐺 𝑇 𝑆𝑃 − 𝑇 𝑇𝐶 − 𝐻 𝐴 𝑇 𝑊 − 𝑇 𝐸𝑁𝑉 − 𝐻 𝑇𝐶 ( 𝑇 𝑊 − 𝑇 𝑇𝐶 )= 𝑀 𝑊 𝑑 𝑇 𝑊 𝑑𝑡 𝑴 𝑾 𝒅 𝑻 𝑾 𝒅𝒕 + 𝑯 𝑻𝑪 + 𝑯 𝑨 𝑻 𝑾 = 𝑻 𝑻𝑪 𝑯 𝑻𝑪 −𝑮 + 𝑮 𝑻 𝑺𝑷 + 𝑯 𝑨 𝑻 𝐸𝑁𝑉 (Eqn. B) Dynamic relationship between 𝑇 𝑊 (𝑡) and 𝑇 𝑇𝐶 (𝑡) Couple with Eqn. A: 𝑯 𝑻𝑪 𝑻 𝑾 − 𝑻 𝑻𝑪 = 𝑴 𝑻𝑪 𝒅 𝑻 𝑻𝑪 𝒅𝒕 What do we have? Two, 1st-order, coupled, constant-coefficient liner-differential equations for 𝑇 𝑊 (t) and 𝑇 𝑇𝐶 (t)

System Solution Solve Eqn. A for 𝑇 𝑊 𝑇 𝑊 = 𝑀 𝑇𝐶 𝐻 𝑇𝐶 𝑑 𝑇 𝑇𝐶 𝑑𝑡 + 𝑇 𝑇𝐶 𝑑 𝑇 𝑊 𝑑𝑡 = 𝑀 𝑇𝐶 𝐻 𝑇𝐶 𝑑 2 𝑇 𝑇𝐶 𝑑 𝑡 2 + 𝑑 𝑇 𝑇𝐶 𝑑𝑡 Plug into Eqn. B and collect terms 𝑀 𝑊 𝑑 𝑇 𝑊 𝑑𝑡 + 𝐻 𝑇𝐶 + 𝐻 𝐴 𝑇 𝑊 = 𝑇 𝑇𝐶 𝐻 𝑇𝐶 −𝐺 + 𝐺 𝑇 𝑆𝑃 + 𝐻 𝐴 𝑇 𝐸𝑁𝑉 𝑀 𝑊 𝑀 𝑇𝐶 𝐻 𝑇𝐶 𝑑 2 𝑇 𝑇𝐶 𝑑 𝑡 2 + 𝑑 𝑇 𝑇𝐶 𝑑𝑡 + 𝐻 𝑇𝐶 + 𝐻 𝐴 𝑀 𝑇𝐶 𝐻 𝑇𝐶 𝑑 𝑇 𝑇𝐶 𝑑𝑡 + 𝑇 𝑇𝐶 = 𝑇 𝑇𝐶 𝐻 𝑇𝐶 −𝐺 + 𝐺 𝑇 𝑆𝑃 + 𝐻 𝐴 𝑇 𝐸𝑁𝑉

Collect Terms 𝐴 𝑑 2 𝑇 𝑇𝐶 𝑑 𝑡 2 +𝐵 𝑑 𝑇 𝑇𝐶 𝑑𝑡 + 𝐺+ 𝐻 𝐴 𝑇 𝑇𝐶 =𝐺 𝑇 𝑆𝑃 + 𝐻 𝐴 𝑇 𝐸𝑁𝑉 More Constant 𝐴= 𝑀 𝑊 𝑀 𝑇𝐶 𝐻 𝑇𝐶 B= 𝑀 𝑊 + 𝑀 𝑇𝐶 + 𝐻 𝐴 𝑀 𝑇𝐶 𝐻 𝑇𝐶 2nd order, linear, Constant-coefficient differential-equation, non-homogeneous (RHS= Constant) Solution 𝑇 𝑇𝐶 = 𝑇 𝑃 + 𝑇 𝐻 Particular Solution for RHS = Constant, 𝑇 𝑃 = 𝑇 𝑆𝑆 =𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝐺+ 𝐻 𝐴 𝑇 𝑆𝑆 =𝐺 𝑇 𝑆𝑃 + 𝐻 𝐴 𝑇 𝐸𝑁𝑉 𝑇 𝑆𝑆 = 𝐺 𝑇 𝑆𝑃 + 𝐻 𝐴 𝑇 𝐴 𝐺+ 𝐻 𝐴 = 𝑄 𝑀𝑎𝑥 𝐷𝑇 𝑇 𝑆𝑃 + ℎ𝐴 𝐴 𝑇 𝐸𝑁𝑉 𝑄 𝑀𝑎𝑥 𝐷𝑇 + ℎ𝐴 𝐴 Same as for simple 1st order system

Homogeneous Solution 𝐴 𝑑 2 𝑇 𝐻 𝑑 𝑡 2 +𝐵 𝑑 𝑇 𝐻 𝑑𝑡 + 𝐺+ 𝐻 𝐴 𝑇 𝐻 =0 𝐴 𝑑 2 𝑇 𝐻 𝑑 𝑡 2 +𝐵 𝑑 𝑇 𝐻 𝑑𝑡 + 𝐺+ 𝐻 𝐴 𝑇 𝐻 =0 Assumed solution 𝑇 𝐻 𝑡 =𝐶 𝑒 𝑏𝑡 , 𝑏= ? Characteristic Equation 𝐴 𝑏 2 +𝐵𝑏+𝐶 𝑒 𝑏𝑡 =0 𝑏= −𝐵± 𝐵 2 −4𝐴 𝐺+ 𝐻 𝐴 2𝐴 If DT is small enough, then G= 𝑄 𝑀𝑎𝑥 𝐷𝑇 will be large enough so that 𝐵 2 −4𝐴 𝐺+ 𝐻 𝐴 <0, b will be imaginary and 𝑇 𝐻 will be oscillatory 𝑇 𝐻 𝑡 =𝐷 𝑒 −𝑎𝑡 sin⁡(𝜔𝑡+𝜙) We observed oscillations for small DT

What is the largest minimum value of DT that will have a steady behavior? 𝐵 2 −4𝐴 𝐺+ 𝐻 𝐴 = 𝐵 2 −4𝐴 𝑄 𝑀𝑎𝑥 𝐷𝑇 + 𝐻 𝐴 <0 𝐵 2 4𝐴 − 𝐻 𝐴 < 𝑄 𝑀𝑎𝑥 𝐷𝑇 𝐷𝑇 < 𝐷𝑇 𝑀𝑖𝑛 = 𝑄 𝑀𝑎𝑥 𝐵 2 4𝐴 − 𝐻 𝐴 = 𝑄 𝑀𝑎𝑥 𝑀 𝑊 + 𝑀 𝑇𝐶 + 𝐻 𝐴 𝑀 𝑇𝐶 𝐻 𝑇𝐶 2 4 𝑀 𝑊 𝑀 𝑇𝐶 𝐻 𝑇𝐶 − 𝐻 𝐴 = 𝑄 𝑀𝑎𝑥 𝐻 𝑇𝐶 𝑀 𝑊 + 𝑀 𝑇𝐶 1+ 𝐻 𝐴 𝐻 𝑇𝐶 2 4𝑀 𝑊 𝑀 𝑇𝐶 − 𝐻 𝐴 𝐻 𝑇𝐶 𝐻 𝐴 = ℎ𝐴 𝐴 , 𝑀 𝑊 = 𝜌𝑐𝑉 𝑊 𝐻 𝑇𝐶 = ℎ𝐴 𝑇𝐶 , 𝑀 𝑇𝐶 = 𝜌𝑐𝑉 𝑇𝐶

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

Proportional Control 1st Law

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

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

1st Law TC

Water Heater HA HT MW Unknown: T, T­W

A B Homogenous Solutions:

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