ME 368First Order Systems and Time Constants Laboratory 9

Laboratory 9

First Order Systems and Time Constants

Equipment needed

  • Stagnant (unstirred) ice water bath, made from tap water and ice, contained in an insulated container. A liquid-to-ice ratio of at least 10:1 works best for this lab.
  • Thermo bath set to a temperature equal to the room air temperature + 40 degrees C.
  • Thermo bath set to a temperature equal to the room air temperature + 20 degrees C.
  • Fast thermocouple: Omega TMQSS-032G-12 (0.032” diameter)
  • Slow thermocouple: Omega TMQSS-125G-12 (0.125” diameter, 4x larger)
  • myDAQ / PC / LabVIEW
  • thermocouple extension wire and female type T thermocouple jack, used to attached either of the thermocouples to the myDAQ.
  • optional soft-sided zipper-top cooler and thermal mass to stabilize temperature of myDAQ (& TC cold junction)

Goals and Objectives

  • Understand the similarities between 2 first-order systems: the physical thermometer and the RC low pass filter
  • Predict the time constant of both of these first order systems
  • Measure and compare the time constants of both of these first order systems using step forcing
  • Understand the phase lag and magnitude reduction suffered by both of these (and all other) first order systems when sinusoidal forcing is applied
  • Understand how finite time constants limit the use of first-order sensors in transient time-domain applications
  • Understand how the measurement environment can impact the time response of a physical thermometer
  • Predict the measurement error that will occur for both step and sinusoidal forcing of a first order system

Background resources useful for understanding this lab include Dunn Chapter 4 and the Labs_9-10_background.docx file on the course website.

1.Experimental setup

1.1Preparing the heated baths

Prepare the two Thermo baths to operate at 20 and 40 degrees C above room temperature, respectively.

1.2Noise

Because thermocouple voltages are small, it is important to manage noise and interference in this lab. When you connect your thermocouple extension wire into the myDAQ, you may find it helpful to jumper ai0- to AGND.

1.2 Setting up a LabView VI with sample compression

It is possible for a small thermocouple immersed in water to have a time constant below 0.1 s. Thus, in order to be able to capture the transient behavior of the thermocouple, the data acquisition system should sample the temperature at a rate significantly higher than 10 Hz.

  • Configure your myDAQ to acquire thermocouple measurements at 200 kHz for 8 seconds, then use sample compression to achieve an effective sample rate of 200 Hz. Use N samples rather than continuous samples. Note that the 200 kHz rate for 8 seconds results in 1.6 M data points before compression. If these 1.6 M data points bog down the computer, reduce the acquisition rate from 200 kHz, while retaining the 8-s record length and the 200 Hz effective sample rate after compression.
  • Prepare the ice bath.
  • Transition your thermocouple from the ice bath to the room air or vice versa while you record data. It will take your code 8 seconds to display the output each time it loops or each time you run it.
  • Set your TC cold junction temperature within the DAQ Express block, and include a calibration if needed so that your TC temperatures are within 3 degrees C of the correct values for the ice bath and the hottest Thermo bath. 3 degrees C is good enough for this lab since we are most interested in dynamics and not so concerned about temperature accuracy in this lab.

2.0 Time Constant Measurements

  • TEST1: Start with the small thermocouple in the room air. Once the temperature is very stable near room temperature, start your VI, thentransition the thermocouple intothehotter Thermo bath. Just submerge the thermocouple, don’t stir it. Some potential problems with such transitions that you should try to avoid are:
  • moving the thermocouple too slowly through the gas immediately above the water. This gas is hotter than the room and can start heating your TC prematurely.
  • moving the thermocouple too quickly through the water. This is like stirring the thermocouple in the water and results in a higher heat transfer coefficient than intended.
  • touching metal near heater elements. This shouldn’t be too much of a problem for the Thermo baths but sometimes heater elements in such baths make the metal locally hotter than the bulk water temperature.

You must capture the full transition from room to high temperature on your graph on your front panel. Export this data to Excel or similar so you have it for future analysis.

  • TEST2: Repeat TEST1, but for a transition from room temperature air to the cooler Thermo bath.
  • TEST3: Repeat TEST1, but for a transition from room temperature air to the ice water bath.
  • TEST4: Reverse TEST1, using the transition from the hotter Thermo bath to room air. Just hold it stationary in the air, don’t shake it.
  • TEST5: Repeat TEST1, but this time using the large thermocouple.

completion a: Plot the TEST1, TEST2, and TEST 3 data on the same temperature [°C] vs time [s] plot. By deleting initial points from two (or all) datasets, align the data so the last pre-transition data points in the three curves occur at the same time.

completion b: Convert your “completion a” plot into one of magnitude ratio [-] vs time [s] (see eq. 4.20 in your sheet) Find the three associated time constants. Discuss.

report 1: Plot the TEST1, TEST4, and TEST5 data on the same temperature [°C] vs time [s] plot. By deleting initial points from two (or all) datasets, align the data so the last pre-transition data points in the three curves occur at the same time.

report 2: Convert your “report 1” plot into one of magnitude ratio [-] vs time [s]. Find the three associated time constants, and list them in a table in order of ascending time constant.Add another column of the table that lists the ratio of each time constant to the shortest one you measured. Describe your time constant computation approach. Explain why the last 3 values in your ratio column are > 1, one-by-one (provide 3 explanations).

report 3: {note: this is basically a homework problem related to this week’s lab.} Consider a hypothetical experiment in which an inch of liquid water at 0 °C is heated to boiling in a hot pot as rapidly as possible. Assume the temperature of the water is uniform and water motion is negligible. Based on a simple energy balance, the temperature rise rate for this situation is expected to be 0.50 °C / s. Based on your TEST1 and TEST5 results, estimate the bias in the measurement [°C] for your small and large thermocouple at the instant the actual water temperature reaches 50 °C. Hint: start your solution with eq 4.15.

3.0 The RC lowpass filter analog to your small thermocouple

  • Configure ao0 to produce a square wave and record it on ai0:
  • Physically connect ao0 directly (no filters) to ai0.
  • Simulate a 0 to 10 V square wave using the simulate signal Express block. Start with the following settings but change them as necessary:
  • square wave frequency =0.5 Hz
  • sample rate = 1 kS/s
  • signal duration = 2 s
  • Feed the simulated square wave to a DAQ Assistant block for ao0. Set the sample rate and durationon ao0 to match the parameters of the simulated signal.
  • Setup ai0 to record data for the same duration as the ao0 output duration. Set the effective sample rate, after compression, to the same sample rate as ao0. Set the actual ai0 sample rate as high as possible.
  • Test your setup to see if you can record square waves.
  • Build a low-pass filter from the resistance and capacitance substitution boxes that has a time constant similar to that of your small thermocouple for TEST1. Insert the low-pass filter physically between ao0 and ai0.
  • Record the filtered square wave. It should closely resemble the time response of the small thermocouple. Export this data to Excel so you have it for future analysis.

report 4: Plot the TEST1 data and the data you just obtained on the same magnitude ratio [-] vs time [s] plot. Align the time axes as before. Comment (quantitatively) on how similar the two curves are.

4.0 Sinusoidal forcing of a first-order system

In addition to step forcing, sinusoidal forcing is another fundamental forcing function of interest.

It is somewhat difficult experimentally to present sinusoidal forcing to the small thermocouple, but easy to apply sinusoidal forcing to the electrical analog you have built. Therefore, we will in this part study sinusoidal forcing of the electrical filter, and thereby understand how the small thermocouple would respond if sinusoidal temperature forcing were applied.

  • Setup an ao“sine wave generator” – to – ai“data acquisition” system:
  • Physically connect ao0 directly (no filters) to ai0. Also physically connect ao1 directly to ai1.
  • Simulate a -10 to 10 V sine wave using the simulate signal Express block. Start with the following settings but change them as necessary:
  • sample rate = 1 kS/s
  • signal duration = 2 s
  • Feed the simulated sine wave to both aochannels using a single output DAQ assistant using the architecture shown:
  • Setup both ai channels to record data. Set the effective sample rate, after compression, to the same sample rate as above. Set the actual sample rate as high as possible.
  • Test your setup to see if you can record sine waves. Plot both sine waves on a single graph on the LabVIEW front panel.

Physically insert the filter between ao0 and ai0. Set the sine wave frequency to a value about half the cutoff frequency of the filter. Take a screenshot of the sine wave plot. Repeat for a frequency approximately equal to the cutoff frequency. Repeat for a frequency about double the cutoff frequency.

completion c: Present your 3 screenshots. Measure the magnitude ratios from the 3 plots. Quantitatively compare the measured magnitude ratios to the theoretical values (see Dunn Chapter 4 material). Measure the phase shift from the 3 plots in [s]. Quantitatively compare the measured phase shifts to the theoretical βvalues for each case.

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