0.1 Temperature measurement and first-order dynamic response  (Page 4/5)

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

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

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:

Method 3–Analyze acquired data

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

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 initialtemperature to the final temperature.

7. Change the file path to prevent overwriting data. (You can either change the name of the file after it iswritten, or you can change the name of the path that the next filewill 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.

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 thedata gathered. If the error fraction plot is linear, then the first-order model is a good approximation of the dynamic behaviorof the system. (See pages 78–79 of Figliola and Beasley for a more detailed discussion of the error fraction and its use to calculatethe 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 whenthe 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.

1. Using Excel, compute a value for tau so that the square of the error between the theory and yourexperimental 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 modelfits the data.