APPM 4350/5350
One of the simplest applications of the heat equation is heat conduction in a long circular rod. This problem is one-dimensional and the solution comes out in terms of sines and cosines. The problem of purely radial heat conduction is also one-dimensional in a sense, but the answer comes out in terms of Bessel functions instead of sines and cosines. In this lab, we will observe radial heat conduction and compare the process to what the heat equation predicts.
To solve the heat equation for a circular cylinder and compare your predictions with the actual temperature measurements made in the lab.
The heat conduction apparatus consists of a small brass cylinder fitted with temperature sensors at evenly spaced locations along its radius. The cylinder is heated in the center and cooled with water around the outside radius.
The instrumentation provided permits accurate measurement of temperature and power supply. Fast response temperature probes, with a resolution of 0.1°C, give direct digital readout. The power control provides a continuously variable electrical output of 0-100
Watts with direct readout. Figure 1 depicts the Heat Transfer Module with the necessary equipment (excluding cooler) to perform the experiment. You will be using the short, wide cylinder with the temperature probes in the radial direction. It looks like it is made out of plastic (because the outside is), but the cylinder on the inside is made of brass.

The first thing you need to do is to obtain access to the Heat Conduction Module . Look for a sign-up sheet on the ground floor of the ITLL in the southeast corner. If there is a sheet posted for this module, then you need to sign up for this lab ahead of time. To actually get the equipment when it's time to do the experiment, you need to page a lab assistant. There is a phone right next to the sign-up sheet along with extensive directions on how to check out labs. You will also need to get the recirculating cooler, which comes separate from the module. The lab module and the cooler you will need to perform the lab will probably be located on level 2B. If possible, use labstation 28 on level 2B, because it has a drain to catch drips from the cooler.
NOTES:
 Since two other experiments use the same apparatus as this one, you will want to be sure to perform the lab portion of this experiment before December. In addition to this other classes will also be using these labs and waiting until the end will make it very difficult to check out the equipment.
Setting-up the Apparatus:
This module consists of five basic elements: a control box, power regulator, selector box, a linear conduction apparatus with various samples, and a radial conduction apparatus. You will also need to checkout the water chiller that will provide the cooling for the apparatus.
1) Make sure the Control Box (Figure 1A) and the Power Regulator (Figure 1C) are turned OFF.
2) Check that the Control Box is plugged into the Power Regulator.
3) Connect the P-1 Military Plug to the appropriate jack on the labstation.
4) Connect the Heater Power Cable (Figure 1E) from the desired conduction apparatus
5) to the front of the Control Box.
B
A
D
C
E
Heater Knob OFF

Figure 1: Heat Conduction Controls
A: Control Box
B: Selector Box
C: Power Regulator
D: Heater Knob (counterclockwise is OFF)
E: Heater Power Cable (from either conduction apparatus)
6) If using the Linear Apparatus (Figure 2A), select and install the desired sample. The samples for the Radial Conduction Apparatus (Figure 2B) cannot be varied.
7) Determine which thermistors (Figure 2C) should be read by the computer. Connect these thermistor cables (Figure 2D) from the Conduction Apparatus to the connectors on the back of the Control Box labeled one through seven. The computer will read these thermistors if desired.
8) Connect thermistors 8 and 9 to the Control Box. thermistors 8 and 9 can only be read
9) manually on the front panel of the Control Box.

C
A
B
E

Figure 2: Heat Conduction Apparatuses
A: Linear Conduction Apparatus
B: Radial Conduction Apparatus
C: Thermistors
D: Thermistor cables
E: Tubing with quick disconnect fittings
10) Plug the Power Regulator into one of the outlets (either circuit #7 or #8) located next to the computer hard drives on the labstation.
11) Plug the Water Chiller into the remaining outlet located next to the computer hard drives on the labstation. You may want to begin cooling the water before you connect the Water Chiller to the Conduction Apparatus, in order to get the water at a steady state (this takes about 10 minutes). Instructions for the Water Chiller are indicated on the top of the unit. After the bucket is in place, make sure the cooler tubing is connected to itself with the valve open, plug in the cooler equipment to the lab station, and turn it on. There are four 120 V outlets down low on the end of each lab station (Sides A and B). Set the cooler to 5°C-10°C (record this value) and let it run for 10-20 minutes until the water temperature has stabilized at that temperature.
(If the cooler makes unhappy noises, it may be low on water; seek help.)
IMPORTANT: Do not plug both the Power Regulator and the Water Chiller into the same outlet. Use two different outlets that run off of separate circuit breakers.
Preparing to run an experiment:
12) Select the mode on the Selector Box (Figure 1B) to computer or manual. Remember that all thermistors can be read manually but the computer can only read thermistors
1-7.
13) Turn the heater knob (Figure 1D) on the Control Box fully counterclockwise (this is the OFF position).
14) If you intend to use the computer for data acquisition follow steps 17-20, “Using the
Heat Conduction Apparatus.vi,” at the end of this procedure.
15) Once you feel comfortable with the equipment, the water temperature has stabilized, and the probes are connected, you will start the experiment. Hit the play button on the
VI so it starts taking data. Let it go a minute or two to make sure everything looks alright. Plug in the heater cord from the insulated rod to the control box. Turn up the power on the heater to 15W or so (definitely less than 40W) and enter this value in the VI. Turn off the power on the cooler, connect the cooler tubes to the insulated rod tubes, and then turn the cooler back on. Note when you turned the cooler on, because this is when the experiment really started.
WARNING: If the heater temperature reaches 100º C the Control Box will shut down.
Monitor your temperature readings carefully.

16) The computer will take measurements of the temperature at each of the 7 probe locations every minute. Continue to take measurements until the experiment reaches steady state. You will do this for 10-20 minutes and should have data for each of the probes for the entire time.
17) When finished with the experiment disconnect everything and return the module to checkout.
18) Open the “Heat Conduction Apparatus.vi” using the following path:
H:\ITLL Documentation\ITLL Modules\Armfield Heat Conduction\Heat Conduction
Apparatus.llb
19) When the VI opens, enter the following information into the “User Inputs” dialogue box: a.
Indicate the temperature units desired b.
Specify the time interval between samples in minutes c.
Enter the watts applied for the experiment. (This is the same wattage that will be dialed in on the Control Box when running the experiment.) When you run
LabView (using the white arrow in the upper left-hand corner) the program will double check to see that the watts applied is entered by telling you to fill it in before pushing stop. When you press OK, LabView will begin taking data.
20) When you are ready to take data, apply power to the heater using the heater knob, turn ON the Water Chiller, and press OK on Labview.
21)
When you are finished taking data, hit “STOP” to quit and save the data to a file. The file will record the watts applied, the temperature units, and each data point. It is recommended to save with extension “.xls” so that Excel can open the file.

While the program is running, it will tell you how many samples it has taken, how long it has been running, and how long until the next sample.

The plot shows the various temperatures of each thermistor with a different color in a temperature versus time format.
The graph will remain blank until the second data point is taken because two data points are needed to create a line.
1.
Write the heat equation in two dimensions in cylindrical coordinates. Assume that the heat flux is purely radial, so that the temperature depends only on radial location and time, u(r,t). State explicitly the problem to be solved: the differential equation for u(r,t), the boundary conditions, and the initial conditions.

2.
Find the steady-state temperature distribution for this problem. Call this u s
(r).
Write the temperature as: u(r,t)=u s
(r) + w(r,t)
Find the equation that w(r,t) satisfies. You should be able to solve this problem by separation of variables, using Bessel functions. Solve it. Find the theoretical transient solution for the problem.
3.
Run the experiment and compare the theoretical solution with the experimental results. Discuss the discrepancies, if any.
4.
An important parameter in the problem is the thermal diffusivity of the material being tested. Find a way to infer this value from your measurements. Calculate this value and compare it to the value quoted for brass in a handbook.
For more information about Bessel functions, read section 5.5 of Powers book. In matlab, type help besselj for more information about numerically calculating values of Bessel functions of the first kind.
Dimensions
The cylinder is actually hollow with measurements as follows:



Inner radius: 5mm
Outer radius: 55mm
Height: 3mm
(??This may be inaccurate.??) The first temperature probe is located at the center of the cylinder, and the other probes are located at 10mm intervals. This means the 6 th probe is located at r=50mm. The cylinder is cooled by water running through a copper tube that runs around the outside of the cylinder and is directly in contact with the cylinder. Please see the top of Figure 2 for a detailed schematic.
Per Steve Stone with Armfield 12/11/02
The composition of the Brass specimen:
60 to 65% Copper
35 to 40% Zinc

K value of 117 W/m2
The composition of the Stainless Steel specimen:
.08% Carbon
2% Manganese
1% Silicon
16 to 18% Chromium
10 to 14% Nickel
2 to 3% Molybdenum
K value of 25 W/m2