Thermal Conductivity Meter: Lee’s Disc Method

1. Introduction to Lee’s Disc Method

Little is known of the origins of Lee and Charlton’s method of thermal conductivity determination, the earliest mention of their work was found to be in the Philosophical Magazine’s 41st Volume, 1896⁽¹⁾. This method, commonly known as Lee’s Disc method, is a simple and effective way to calculate the thermal conductivity of materials with low thermal conductivities. With the use of easy-to-find parts, anyone can assemble and operate a Lee’s disc apparatus with the goal of thermal conductivity calculation. Ultimately Lee’s Disc method will provide a better understanding of how to calculate thermal conductivity.

2.0 Background Information and Relevant equations

2.1 Background Information

In order to obtain accurate readings, the brass discs should be wrapped in a thermal insulator such as cotton, felt or plastic. Loss of heat by conduction and convection with the environment (air) can skew the results obtain thus using an insulating material where appropriate is important in order to obtain proper results using Lee’s Disc method.

2.2 Relevant equations and constants

Heat Flow Rate (Q) = k ⋅ A ⋅ (T₁-T₂) / d

• k = Thermal Conductivity in Watts/meter-Kelvin
• A = Area (m²)
• T₁ and T₂ = Temperatures in Kelvin
• d = Distance which the heat travels in meters
• Q = Heat Flow Rate in Watts

Cross-sectional Area = πD² / 4 OR πr²

Rate of heat loss (Q) = m ⋅ c ⋅ (dT/dt)

• m = mass of a given sample in Kilograms
• c = specific heat of a given material in Joules/Kilogram-Kelvin
• (dT/dt) = Rate of cooling in °Kelvin/Second
• Q = rate of heat loss in Watts

Specific heat capacity of Brass: 377 Joules/Kilogram-Kelvin

Steady-State = Heat entering the system is the same as the heat exiting the system OR no change in temperature (by more than 0.5°C) for 10 minutes.

3.0 Materials, Experimental Set up and Procedure

3.1 Materials

*Metals with high thermal conductivity include Aluminum, Copper and Silver. Others can be found in the material database*

3.2 Lee’s Disc Method Experimental Setup

1. Machine 4 holes in one of the brass discs with each pair facing each other and separated by ¼ of the circumference.
2. Place 1 bolt in each hole leaving enough space to tie some cord around the protruding end of the bolt

1. Machine 1 hole in both brass discs with diameter equal to the diameter of the thermometer (or thermocouple).
2. Should air pockets be present, fill the hole with thermal grease.
3. Set-up the lab stand with the ring clamp just below to top of the stand.
4. Attach 4 cords of same length to the protruding end of the bolts and attach to the ring clamp.
5. Use the Vernier caliper to measure the thickness (OR mean thickness) of the sample and record it as d.
6. Place the insulating material of choice on the suspended brass disk followed by the second brass disc.
7. Place the Steam Chamber on the brass disc.

3.3 Procedure

1. Turn on the steam generator.
2. Allow the entire system to reach steady-state (no change in temperature by more than 0.5 degrees in 10 minutes).
3. Record the Temperature T₁ and T₂ once steady state is reached.
4. Once temperatures are recorded, remove the sample material.

1. Place the steam chamber on top of the lower brass disc and allow its temperature to rise above its stead-state temperature (T₂) by 10-15°C.
2. Remove the steam chamber and turn off the steam generator.
3. Quickly place some insulating material on top of the brass disc covering the area which was covered by steam chamber.

1. Allow the lower brass disc to cool down.
2. When the brass disc approaches T₂, begin recording the temperature at regular time intervals (5-10 seconds).
3. Record the temperatures for 5°C above and below T₂.
4. Plot a Temperature vs Time plot by hand or use any available program.
5. Calculate the slope of the tangent line to T₂ on the cooling curve of the lower brass disc by hand or using any program at hand (such as excel).

4.0 Calculations & Comparisons

4.1 Calculations

The Rate of heat transfer at steady state through a given material is given by the following equation

• Q = k ⋅ A ⋅ (T₁ – T₂) / d
  (1)

The rate of cooling of at steady state temperature is given by this equation

• Q = m ⋅ c ⋅ (dT/dt)
  (2)

Since at steady state, the amount of heat entering a system is the same as the amount of heat leaving a system, the following equation may be produced

• k ⋅ A ⋅ (T₁ – T₂) /d = m ⋅ c ⋅ (dT/dt)
  (3)

The Thermal Conductivity constant may be isolated to produce the following

• k = [m ⋅ c ⋅ (dT/dt) ⋅ d] / A ⋅ (T₁ – T₂)
  (4)

4.2 Experimentally determine Thermal Conductivity values for comparison

Once all the data has been collected and the thermal conductivity of your sample calculated, the results may be compared to data obtained from highly specialized labs.

*Thermal Conductivity values are taken from the Thermtest thermal properties database, for more Thermal Conductivity values visit: https://thermtest.com/materials-database

5. Conclusion

Opposite to the Searle’s Bar experiment, Lee’s disc method is designed to calculate the thermal conductivity of primarily thermal insulators such as polystyrene. A thin, large surface area sample is used in the determination of thermal conductivity in order to quickly reach steady state. With a budget of between 150-250$, anyone can test the thermal conductivity of insulating materials.

Web Resources for further information:
https://thermtest.com/thermal-resources/build-your-own/searles-apparatus
https://thermtest.com/history-6-lees-disc-method/
http://iosrjournals.org/iosr-jmce/papers/vol3-issue1/F0315360.pdf?id=2586
To make your own simulation:
https://thermtest.com/thermal-resources/energy2d-heat-transfer-simulations/

Reference

• (1) Charles H. Lees, J. D. Chorlton; LIV. On a Simple apparatus for determining the thermal conductivity of cements and other substances used in the Arts; Philosophical Magazine; Vol 41; Issue 253, 1896.

6.0 Save a few dollars

For the more technological savvy, certain components of this experiment may be made at home. These components include the steam chamber, steam generator and two channel thermometer.

6.1 Steam chamber

In order to make a steam chamber all that is required is some metal cylinder with thin walls. The metal used should have a relatively high thermal conductivity (safe bets: Aluminum, Copper). The height of said cylinder should be shorter than the area available above the upper brass disc and should cover the cross-sectional area of the upper brass disc. Two holes must be made in either side of the cylinder (one of the steam inlet and one for the steam outlet).

6.2 Steam Generator

In order to make a steam generator all that is required is a hot plate, metal container full of water and an air tight lid with a hole large enough for tubing. It is important to note that the airtight seal will produce a closed system and pressure could build with faulty tubing. The tubing will be connected to the steam inlet hole in the steam chamber.

6.3 Two Channel Thermometer

Below is an appropriate design of a two channel thermometer capable of auto-recording the temperature.

The metal plates temperature signal from thermocouples could be read by ADC chips like ADS1248 or MCP3424, and then the digital value would be sent to a CPU like ARM cortex 0. Do remember the thermocouples need cold joint compensation, this could be completed by a thermistor or IC temperature sensor like DS18B20 at thermocouples’ cold joint. The CPU can record the temperatures along with time and then, display, store, or send the recorded data to a computer.