E12 Physical Systems Analysis: Lab #1

Graphs & analysis due 1 week after lab date

Final report due 2 weeks after lab date

Overall Objectives

This lab serves multiple purposes. First, it introduces you to the PCI-1200 data acquisition system. Second, it gives you some initial experience with C programming. Finally, it will provide an introduction to thermal systems and how we can model those systems using tools you already know.

Lab groups for this semester will consist of three people. We will mix up the lab groups every other lab so that you get the experience of working with a number of different people.

Grading

In general, each lab will have a combination of required and optional parts. If your team does only the required aspects of a lab and writes a clear and concise writeup, your team will receive a B+ on that lab. If you complete one or more of the optional parts, or extend the lab in a direction of your own choosing and do it well, then your team will earn some flavor of A.

The quality of your writeup will be reflected in the grade it receives. For some labs you will do a formal writeup, and for some the requirements will be less stringent. For the more informal writeups, the lab page will give you an idea of what should be in them, but it is up to you to present the information in an organized, clear, and concise manner. Before you hand in a lab, read through it yourself and make sure it conveys clearly the process you followed, the results you obtained, and any insights you might have gained.

Introduction

Heat Transfer and Thermodynamics

We will model the system as a simple first-order system with heat flowing in and out, and energy accumulation evidenced by increased temperature. The rate of energy accumulation in the box is equal to the rate of energy flowing into the box minus that flowing out. This system is analogous to an electrical system.

In an electrical system current (Amperes) is analogous to heat flow (Watts) in a thermal system, so charge (Coulombs) is analogous to energy (Joules). Voltage (Volts) is analogous to temperature (Kelvins or °C), electrical resistance (Ohms) to thermal resistance (Kelvin/Watt), and electrical capacitance (Farads) to thermal capacitance (Watt-Seconds/Kelvin, also called thermal mass). There is no thermal analog for inductance.

For the electrical system shown, we can write the equation:

If the output of the circuit is connected to a resistor to ground, we have:

and

For the thermal system, our model leads to the equations:

Rate of Energy Accumulated = Rate of Energy in - Rate of Energy out
Rate of Energy Accumulated = Heat in - Heat out,

or in terms of the systems physical properties.

where m, c_p an T are the mass, specific heat and temperature of the air in the box, P is the power (heat) supplied by the hair dryer, To is ambient temperature, R is the thermal resistance of the box and t is time. This model is based upon the assumption that the only energy leakage is due to a thermal resistance, and the energy accumulation is due to the heating of the air that is inside the box. If we let Theta = T - T_o, and C = m · c_p the equation can be rewritten:

From this equation the analogy between electrical systems becomes clear. Heat (current) flows into the system. As the temperature (voltage) increases the heat (current) flow to ambient temperature (ground) increases proportionately due to resistive losses. It is the difference between the temperature (voltage) of the house and ambient (ground) that drives the heat (current) flow. Energy is stored in the thermal (electrical) capacitance. The greater the temperature (voltage) differential between the house and the ambient (ground) the greater the amount of energy stored. We assume that an arbitrarily large amount of heat (current) can be absorbed by the ambient (ground) without changing its temperature (voltage). You'll find that this is not exactly true, the lab is noticeably warmer at the end of the afternoon than at the beginning-- though not enough to be a very significant cause of error

From this information it should be possible to find Theta as a function of time if P is a step input (i.e., it is initially off, and is turned on a t=0).

Setting Up

Equipment

• a simulated house with 40W light bulb,
• a thermocouple with amplifier
• a computer with data acquisition equipment

The thermocouple is connected to a thermocouple amplifier whose output decreases 10 mV for every 1°C rise in temperature. Use a measurement of the current air temperature in the classroom at the beginning of the lab to determine a reference temperature (i.e. look at the thermostate for the room).

Software

If you want to be able to understand the code you will be using to gather the data, read the C reference and Data acquisition reference documents before coming to lab.

As described in the DAQ reference, create a project for your team on one of the computers in the E12 lab. You might want to create it in your personal folder on the server so that you can access it from any of the E12 lab computers.

Once you have created a project, do the following to make sure everything is working correctly.

1. Connect the power for the thermocouple and ammplifier circuit.
2. Attach a voltmeter to the circuit so that you can see what voltage the thermocouple is producting.
3. Attach the thermocouple to one of the analong inputs.
4. Compile and run the default program that comes with your project.
5. Make sure the voltages the computer is reading are close to the voltages on the voltmeter.

Tasks

1. Connect the output of the thermocouple amplifier to the voltmeter and connect the power for the thermocouple circuit.
2. Connect the output of the thermocouple to one of the analog inputs on the DAQ board.
3. Without changing any of the input conditions, collect 100 data points from the thermocouple. Using this data, calculate the mean and standard deviation of the measurement system (assuming a zero-mean Gaussian distribution of measurement noise).
4. Use the data collection program to collect data from an analog input for as long as it takes for the system to reach equilibrium (plus an additional 10 seconds for initial data). You will need to determine a good sampling rate for the task before starting to collect data. Collect data at a rate faster than really necessary so that you can average the results to get more accurate estimates.

   Note that the program in the NI-DAQ project template is very similar to the data collection program in the Data Acquisition handout but includes some additional lines to save the data in a file. Feel free to modify the program if you want additional functionality (like converting the data into temperature automatically).

5. Use the program to perform the following experiment (with the lid of the box free to open):
   1. Start your data acquisition program running at an appropriate sampling rate, and set it to run for 3 minutes. 10 seconds after starting the program, turn on the light bulb. Monitor the box, and don't let the light bulb get too hot.
   2. When the data collection is finished, open the box to let it cool off.
   3. Look at your data in a graphing program and see if the system reached equilibrium. If not, re-run this experiment for a longer time period, but be sure to monitor the light bulb so that it does not get too hot.
6. Repeat the previous step with the holes, seams, and cracks covered as tightly as possible and a weight on the lid to keep it shut.
7. After the second trial, do not open the box. Instead, we will collect data from the box to track the cooling down process. Start up the data acquisition as soon after turning the light off as possible and let it run for at least 10 minutes. Set an appropriate sampling rate so that your don't have an overload of data.

Calculations

1. In a spreadsheet, calculate the mean and standard deviation of your measurements of the constant temperature case.

Also in the spreadsheet, use a moving Gaussian filter to try and reduce the noise in your signal. Select a time range that is reasonable for the data (within which the temperature is appoximately constant). When you fit exponentials to the data, compare using both the raw data and the filtered data.

2. Calculate the expected R of the system using the fact that the R-value of the Tuff-R insulation is 7.2 R/inch and the conversion ratio to get it into SI units is 0.1761 (Cm² / W) * (R)^-1. Note that, to get the thermal resistance (C/W), you will need to calculate and use the surface area of the box.
3. Determine the resistance R of the system for the heating runs from the final steady state temperature (estimate it with a best fit exponential if the box does not reach steady state).
4. To calculate thermal capacitance, you will need the density and specific heat of the air in the box. Values for 50 C are 1.21 kg/m³ and 1 kJ / kg K, respectively. Measure the box to find its volume, and use this value to calculate its mass. Use the mass and specific heat to calculate the thermal capacitance C of the box.
5. Determine C from the time constant of the system from both the heating and cooling runs. You should find the time constant using a curve fitting program (like the one in Kaleidograph), don't simply look for 67% of the final value. See me if you need help with this.

Extensions

1. For fun, try heating the box with something with a little more juice. Try putting a hair dryer completely inside the box and closing up the hole as best you can. Repeat the heating experiment and compare and contrast your results for R and C. (The hair dryer will cut off after about 20-30 seconds, so be prepared to grab data quickly at a high time resolution.)
2. If you block off some portion of the box (like half of it) and redo the experiment, how does this affect your C value?
3. What happens if you put the thermocouple inside a box inside the box? Try to use something for which you can estimate the R-value.
4. Try a more sophisticated method of filtering the data (see the prof).

Lab Report

1. Plots of your data using incremental temperature (Theta) vs. time. Take time = 0 to be the time that the light bulb is turned on. (You don't need to plot t < 0). For the cooling data, time = 0 should be the first data point.
2. Report all of the values of the thermal resistance that you have calculated. Is it higher or lower with the box sealed tightly. Is this what you expected? How does the value of R during cooling compare with the value during heating?
3. Report your measured and calculated values of C for each data run. Use a table to show the values side-by-side. How well do the theoretical and measured values match? How do they compare to the value of C from the cooling run?
4. Be sure to put all of your theoretical and measurement-derived R and C values in a single table for easy comparison. Put any other relevant information in the table as well.
5. If you modified the program at all, include a copy of the program as an appendix. Your modifications should be well commented, and you should include a description of them in the writeup.
6. In your discussion, consider any problems with the model we used. Discuss, and try to explain any discrepancies between the theoretical and measured values.