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Temperature Measurement and First-Order Dynamic Response

Module by: Luke Graham. E-mail the author

Summary: In this lab exercise, you will determine how various temperature measurement systems respond to different inputs. This analysis will include a thermometer, a thermocouple, and a thermistor.

Introduction:

In this lab exercise, you will determine how various temperature measurement systems respond to different inputs. This analysis will include a thermometer, a thermocouple, and a thermistor.

Teaching Objectives:

The purpose of this laboratory is for each student to understand and do the following:

1. Instrument calibration.

2. Measurements with thermocouples and thermistors.

3. Determination of time constants and comparisons of the dynamic responses of three measuring systems.

4. Application of first-order dynamic behavior to dynamic temperature measurement problems.

Preparatory Reading:

Figliola and Beasley

Measurement System Behavior, First Order Systems: pp. 72–87

Thermistors: p. 295

Thermoelectric Temperature Measurement: pp. 300-313

Overview:

In this lab, we will use first-order models to approximate the response of a thermometer, a thermocouple, and a thermistor to temperature inputs. Each of these temperature sensors measure temperature in a different way.

  • A thermometer senses a change in temperature as a change in the density of a fluid.
  • A thermocouple consists of two wires of different metals joined at one end (the junction). When a voltage is applied across the free ends of the two wires, the differing properties of the wires create an induced voltage that it proportional to the temperature change at the junction.
  • A thermistor is a thermally sensitive resistor. When the temperature of the thermistor changes, a proportional change in resistance occurs.

Procedure:

Part 1: Modify a VI for temperature measurements

In this lab, we will build on the skills that were developed in Introduction to Benchtop Equipment and Data Acquisition. It is assumed that the student understands the concepts of a Front Panel, a Block Diagram, the Control Palette, and the Functions Palette. It is also assumed that the student can search the control and functions palette for necessary features.

1. Connect a thermocouple to channel 0 of the SCXI-1112 module.

2. Open LabVIEW 8.0.

3. Open the VI that you developed for Introduction to Benchtop Equipment and Data Acquisition.

4. Click File>>Save As…>>Substitute Copy for Original to save your VI with a new name.

5. On the Block Diagram, double click the DAQ Assistant to open the configuration dialogue box.

Figure 1: DAQ Assistant Configuration Box
Figure 1 (Graphic1.png)

6. Configure the DAQ Assistant as shown above.

    1. Click Voltage and press Delete to delete the channel you used in the first lab.
    2. A dialog box will ask if you want to delete the selected channel. Click Yes.
    3. Add a new measurement channel.
      1. Click Add Channels>>Voltage.
      2. Select the channel that the thermocouple is connected to (SC1Mod2>>ai0).
      3. Click OK.
    4. The input range should be 80mV max to -80mV min. (Thermocouples have a low voltage output.)
    5. The Terminal Configuration should be Differential. (The ground terminal is not used.)
    6. Change Acquisition Mode to Continuous.
    7. Click OK.
    8. A dialog box will open asking if you would like a While Loop to be made around the DAQ Assistant. Click No. (You will place a While Loop in the next step that includes the subVIs that are already on the Block Diagram.

7. Place a While Loop around the entire Block Diagram as shown below. LabVIEW While Loops are similar to While Loops in any text-based programming language. The routine inside the loop will run repeatedly until a true value is passed to the loop condition, represented by a red button at the bottom right corner of the loop.

Figure 2: Block diagram with while loop
Figure 2 (Graphic2.png)

8. A toggle button can be used to end the While Loop routine. Right click on the loop condition icon and click Create>>Control.

9. Double click the Write to Measurement File ExpressVI to open the configuration box. Under If a file already exists, select Append to File. Click OK. (Because the Write to File function now occurs in a loop, the data must be appended to the end of the file, rather than overwriting the data from the previous loop.)

10. On the front panel, place a Thermometer.

11. Right Click the Thermometer. Click Visible Items>>Digital Display.

12. On the Block Diagram, inside the While Loop, place a Time Delay Express VI. When the Configuration Box opens, change the Time Delay (seconds) to 0.001. (Without a delay in the While Loop, the program will repeat as quickly as it can and consume processor power.)

13. Your Block Diagram should resemble Figure 3.

14. Click File>>Save As…>>Copy>>Create Unopened Disk Copy. Save the VI with “Lab3” in the file name. (You will use the VI as it is now as a starting point in Operational Amplifiers and Signal Conditioning.)

15. Save a second copy with “Lab4” in the file name. (You will also use the VI as it is now as a starting point in Strain Gages and Force Measurement.)

Figure 3: Completed block diagram
Figure 3 (Graphic3.png)

Part 2: Calibration Curve

You will create a static calibration curve to correlate voltage measurements with temperature readings. We will use the thermometer as the standard. (Note: We use the thermometer as a standard for convenience. Do not assume that an alcohol-in-glass thermometer is a temperature measurement device with superior accuracy.)

2.1 Calibrate the thermocouple

1. Take temperature readings with the thermometer in ice water and boiling water. Be careful not to allow the devices to come into contact with the hot plate or the bottom of the beaker.

2. Record the temperatures measured.

3. Use the thermocouple to measure both temperatures. Again, be careful not to allow the devices to come in contact with the hot plate or the bottom of the beaker.

4. In Excel:

    1. Plot the temperatures measured with the known standard values on the y-axis and the ADC output (voltage) resulting from the thermocouple input on the x-axis.
    2. Create a linear curve fit using the two data points.

The linear fit results in an equation that relates the ADC output obtained from the thermocouple to the known temperature from the thermometer:

T=k(Vadc) + b

where “T” is the temperature, “k” is the gain (°C/V), “Vadc” is the voltage read in to the computer, and “b” is the zero-voltage offset. The gain “k” is the inverse of the static sensitivity, which has units of V/°C.

5. Repeat the calibration procedure for the thermistor.

2.2 Modify your VI for calibration

Figure 4: Modified block diagram
Figure 4 (Graphic4.png)

In this section, you will multiply the data by the conversion scalar “k” that you found experimentally. You will also add an offset “b”.

1. Delete the Tone Measurement ExpressVI. Delete the Frequency and Amplitude indicators.

2. Delete the wires that connect the DAQ Assistant to the thermometer, the Waveform Graph, and the Write to File ExpressVI. Press Ctrl+b to remove any broken wire remnants.

3. Place a Multiply and an Add on the block diagram.

4. Create a control and connect it to the y terminal of the Multiply icon. Rename the control “k”.

5. Create a control and connect it to the y terminal of the Add icon. Rename the control “b”.

6. Wire your block diagram as shown in Figure 4.

7. On the front panel, enter the values you found experimentally for the “k” and “b” values.

8. When you run the VI, your measurements should now be displayed as temperatures.

Part 3: Time Constants

The calibration curves you found represent static calibrations. In the following section, you will consider the dynamic behavior of the sensors. The dynamics of all the temperature sensors used in this lab can be modeled accurately as first-order systems. The natural response of a first-order system can be characterized by a single parameter — the time constant.

Time constants can be found by observing the response of the instrument to a step input in temperature. Note that the second medium governs the physics of the transient response. For each of the sensors, you will determine the time constant for transfers from ambient air to ice water and from hot air to ambient air.

An ice water bath will be provided for your tests. A heat gun will be used to generate hot air. Caution: Keep all temperature sensors (particularly the thermometer and the thermistor) at least 9 inches away from the heat gun. Holding them closer to the heat gun will result in temperatures outside the operating range of the sensors.

Four methods will be used to calculate various time constants. Table 1 summarizes the tests to be performed. The testing methods are described below.

Table 1. Time constant calculation methods.

Table 1
  thermometer thermocouple thermistor
ambient air to ice water 1, 2 3, 4 3, 5
hot air to ambient air 1 3 3
ice water to ambient air - 3 -

Method 1- Time Elapsed at 63.2% Sensor Accuracy

Thermometer: Perform this test for ambient air to ice water. Repeat for hot air to ambient air.

1. Record the temperature of the initial medium, T0, and the temperature of the final medium, T_inf.

2. Calculate the intermediate reference temperature, Tr that is 63.2 percent of the way from T0 to T_inf.

3. With the instrument at steady state (reading T0) in the initial medium, move it quickly to the final medium.

4. The time from when the step in temperature began to when the instrument reads Tr is the time constant, . This time can be measured using a stopwatch.

Method 2 - Mathematic Definition of Tau

Thermometer: Perform this test moving from ambient air to ice water.

1. Record the temperature of the initial medium, T0, and the temperature of the final medium, T_inf.

2. Choose an arbitrary reference temperature, Tr that is between 60 and 80 percent of the way from T0 to T_inf.

3. With the instrument at steady state (reading T0) in the initial medium, move it quickly to the final medium.

4. Record the time trthat it takes to reach the reference temperature Tr.

5. The following expression can be used to calculate the time constant tau:

Figure 5: equation (1)
Figure 5 (Graphic5.png)

Method 3 – Analyze Acquired Data

Thermocouple: Perform this test for ambient air-to-ice water, hot air-to-ambient air, and ice water-to-ambient air.

Thermistor: Perform this test for ambient air-to-ice water, and hot air- to-ambient air.

1. Run the VI.

2. Insert the thermocouple into the initial medium.

3. Click the Enable toggle control as you transfer the thermocouple to the final medium.

4. When the sensor comes to equilibrium, stop the VI.

5. Open your measurement file in Excel.

6. Determine the time required for the sensor to complete 63.2 percent of the transient from the initial temperature to the final temperature.

7. Change the file path to prevent overwriting data. (You can either change the name of the file after it is written, or you can change the name of the path that the next file will be written to.) You will use the saved data for subsequent calculations.

Method 4- Error Fraction

Thermocouple: Complete this step using the data from the ambient air-to-ice water transfer.

Thermistor: Repeat this step using the data from the ambient air-to-ice water transfer.

Equation 2 below gives an expression for the error fraction.

Figure 6: equation (2)
Figure 6 (Graphic6.png)

1. Calculate the error fraction at each instant of time for the data you have gathered.

2. Take the natural log of the error fraction data.

3. Plot the natural log of the error fraction versus time.

4. Find the negative reciprocal of the slope of this curve. This is the time constant.

The error fraction versus time curve provides an easy way to judge the accuracy of a first-order model for the data gathered. If the error fraction plot is linear, then the first-order model is a good approximation of the dynamic behavior of the system. (See pages 78–79 of Figliola and Beasley for a more detailed discussion of the error fraction and its use to calculate the time constant.) Are the thermocouple and thermistor in water adequately modeled by the first-order system approximation?

Method 5- Least Squared Error

Thermistor: Complete this step using the ambient air-to-ice water data.

Equation (3) below gives the theoretical expression for the temperature, T(t), as a function of time when the temperature of the environment is suddenly changed from T0 to T∞. The thermocouple is assumed to be at temperature T0 when the change occurs at t = 0.

Figure 7: equation (3)
Figure 7 (Graphic7.png)

1. Using Excel, compute a value for tau so that the square of the error between the theory and your experimental result is minimized. This is called a least square error approach and is very common in engineering.

2. Plot both the test data and the results from Equation (3) on the same axes to compare how well this model fits the data.

Part 4: Additional Tests

Once you have performed all of the tests outlined above, consider the following tests:

•Transfer the thermocouple from ice water to room temperature air.

How does the time constant compare with the one obtained from the air-to-ice-water transition? Is it different? Why or why not?

•When calculating the time constant for the thermocouple in transition from air to ice water, you measured the time to go 63.2 percent of the way from the initial temperature (around 21 C) to the final temperature (around 0 C). Using the same graph, recalculate the time constant using14 C as the starting temperature (i.e., what is the time required to go 63.2 percent of the way from 14 C to 0 C?). Is this time constant value significantly different from the initial value you computed? Why or why not?

Part 5: Thought Questions

Document your response to these questions in your lab book.

1. How do the time constants of the thermometer, thermocouple, and thermistor compare? Why is the thermometer “slow” compared to the other devices?

2. How do the time constants obtained for air and water compare? Are they significantly different? If so, explain why.

3. Does the instrument need to be at steady state before transferring it to the final medium?

4. How accurate (qualitatively) is the first-order model approximation for the different sensors and media?

Part 6: Frequency Response

From the measured time responses of the thermometer, the thermocouple, and the thermistor, it can be seen that the response time of these sensors is not instantaneous. From this we can infer that these measurement systems may have some difficulty following inputs that change quickly. Frequency response is a means of characterizing the dynamic response capabilities of a system.

The frequency response of a system can be determined by calculating the response of a system to sinusoidal inputs at different frequencies. Two important characteristics are:

  • the magnitude ratio between the output and input waveforms
  • the phase difference between the output and input waveforms

For a first-order system, we can create the frequency response plot using only our knowledge of the system time constant. The magnitude ratio can be calculated from the following expression.

Figure 8: equation (4)
Figure 8 (Graphic8.png)

The phase difference is determined from expression 5.

Figure 9: equation (5)
Figure 9 (Graphic9.png)

In Excel, create a frequency response plot for the thermometer, thermocouple, and thermistor using the time constant from the ice water tests. Plot the magnitude ratio on a log-log scale and the phase difference on a semilog scale (log scale of frequency axis). Consider frequencies from 0.1 rad/s to 1000 rad/s. Your plots will look nice if you use ten points per decade of frequency (i.e., 0.1, 0.2, 0.3, … ,1, 2, 3, … , 10, 20, 30, …). Overlay the three magnitude responses on one plot and the three phase responses on another. How do the frequency response characteristics of the thermometer, thermocouple, and thermistor compare?

Part 7: Dynamic Temperature Measurement Problem

Include the analysis for this problem in your lab book.

Assume that the thermocouple you used is placed in a water stream at a point where the temperature varies sinusoidally between 60 C and 80 C at a frequency of 40 rad/s. Assuming that the thermocouple time constant is the same as the value you determined in the ice water case, what will be the range of sinusoidal variation in the temperature readings? Repeat the calculations for a temperature frequency of 2 rad/s. Now assume the thermocouple is in air instead of water. Determine the range of the variation using input frequencies of 10 rad/s and 1 rad/s.

Clean Up:

Please straighten up your lab work area. Turn off the data acquisition hardware and leave your work area clean. Your help in keeping the lab neat is appreciated.

Report:

For this lab, you are to write up the Method section of a full formal report. Guidelines are given in the example report in the Undergraduate Guide. Avoid presenting results in the Methods section. Your writing assignment is relatively short. Take the time to do a good job. Since you will not be submitting Results or Discussion of Results in the report, you will need to document your results in your lab book. Also, be sure to answer the questions from the lab handout in your lab book.

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