1 Given thermocouple data datatxt starting when the thermoco

1. Given thermocouple data (data.txt) starting when the thermocouple is put into a constant temperature water bath. Find the thermocouple’s time constant and rise time using a linear curve fit. The thermocouple has a K = 50 µV/°C. What is the temperature of the water bath? What is the initial temperature of the thermocouple? (Hint: Refer to slides on graphical determination of time constant (tau) of a first order system) 2. For the instrument in problem one subject to a sinusoidal temperature (like a thermal wave, they exist but not at room temperature) with an amplitude of 5°C, what would be the maximum frequency that can be measured with an error of 5%. Plot the ideal signal (the actual temperature) and the temperature as sensed by the thermocouple at the maximum frequency calculated.

Solution

Procedure 1. Locate a copy of the time-temperature data file, temp.dat. 2. Load or import the data into Excel and create a plot of Temperature vs. Time. 3. Get a hardcopy of the graph you created by printing this figure. 4. Estimate the initial condition T0 and the steady-state value TSS. Record these values on the lab worksheet. 5. Use the graph to estimate an average time constant using values at approximately , 2, 3, and so forth. Show your work on the graph, by hand. Which time constant do you think is most accurate? Record this value on the worksheet. Fig. 3 Plot of thermocouple data file, temp.dat. Method 2: Time constant from the log-incomplete response plot Discussion You are to manipulate the data using the Excel spreadsheet so that you will be able to plot the incomplete response curve and use it to find the time constant, , of the ES205 Lab 7 6 of 9 © 2003 RHIT thermocouple system. From a plot of Z(t) vs. time you will be able to determine the slope of the linear-least-squares curve, from which you can obtain an estimate of the time constant. Recall that the incomplete response only uses data points when time is less than 4 . Once the incomplete response curve is found this will be compared to the actual data set. Procedure 1. Identify the initial value of the Temperature response, T0. 2. Identify the steady state value of the temperature response file, Tss. 3. Identify what time corresponds to 4 . 4. Set up a column in Excel to calculate the incomplete response 0 ( ) ( ) ln m SS SS Tt T t Z t T T = = Use only the data points which fall below 4 . 5. Create a plot of Z(t) vs. t and use a least squares fit to determine the slope and intercept of the line. Select the option which forces the curve to pass through the origin. 6. From the slope determine the estimate of the time constant, . 7. Explore the consequences of varying various parameters such as T0 and Tss. Can you obtain a better curve-fit? When the linear least-squares curve fit is as good as you can get it (by comparing the R2 values), print the resulting figure and record the resulting value of time constant, , on the worksheet. Also record the final values used for T0 and Tss. Fig 4. Incomplete response plot with best fit. ES205 Lab 7 7 of 9 © 2003 RHIT Method 3: Time constant using a cost function Discussion A value for the time constant may also be found by computing and minimizing the value of a cost function, J, which is based on the sum of errors squared. This method compares the known form of the analytical solution using different values of with the experimental data until the cost function has reached a minimum. The cost function is given by = [ ] f t t m J T t T t 0 ( ) ( ) ( , ) , 2 where Tm(t) is the experimental temperature set and ( ) / 0 (, ) t Tt T T T e SS SS =+ Procedure 1. Select an initial guess for the value time constant, . Set it up as a variable in the Excel spreadsheet. 2. Set up a new column in Excel which calculates the temperature predicted by the analytical solution to the DE. 3. Set up another column which computes each individual term of the cost function. [ ] 2 ( ) ( , ) for 1 to T t Tt i N m = 4. Compute the sum of this column to get the cost function, J(). 5. Use the “Solver” function to minimize J with respect to . 6. Use the “Solver” function to minimize J with respect to , T0 and Tss. 7. Create plots of the experimental data, the tuned analytical temperature fit ( only) and the tuned analytical fit adjusting , T0 and Tss versus time. Comparing results of three methods of determining the time constant 1. On the lab worksheet, record your best estimate of the time constant for each of the three methods. 2. Create one last plot which shows the temperature vs. time plot for the three different line fits provided by each of the different time constants ( 1, 2 and 3) that were found. These are to be all shown on the same graph along with the original temperature data set. Adhere to the graphics standards, add a legend, and use different line-types (not colors). 3. RHIT ESTIMATING THE CONVECTION COEFFICIENT Assume the copper-constantan thermocouple bead has the following properties: density = 8920 kg/m3 , specific heat Cv = 410 J/kg·K at 100ºC, diameter d = 0.5 mm, volume/area V / A = d/6. Using equation (2) and your range of best estimates of the time constant, compute a range of values for the convection coefficient h. Show your calculations on the worksheet and record your values of h. For free convection in liquids, the convection coefficient h is generally in the range of 50 to 1000 W/m2 ·K. Compare your results to these published values.