Have you ever wondered why a teaspoon dipped in a hot drink is instantaneously warm to the touch while a burning match can be held in your fingers until it almost burns out completely? While playing with fire is never a good idea, it seems counterintuitive for your afternoon tea to burn you before the actual fire would. Turns out, the ability of the spoon and the matchstick to transfer heat from their source is the main reason behind this. This property is known as the thermal conductivity of a material. As you can imagine, quantifying this property for different materials allows us to control heat transfer on varying occasions, ensuring that handles of hot things like pots and spoons remain cold while metal radiators stay as hot as possible. In this article, we'll look at thermal conductivity, its properties, and applications.
Thermal Conductivity Definition
Heat transfer occurs when heat moves from a hotter object to a colder one. Depending on the material these objects are made of, the rate of heat transfer will vary. This rate can be described using thermal conductivity.
Thermal conductivity is the measure of a material’s ability to transfer thermal energy.
It's a property of matter on an atomic scale. Not to be confused with thermal conduction, which is one of the three ways in which heat can be transferred.
Thermal conduction is the transfer of heat from an object with a higher temperature to that with a lower temperature through direct contact.
Thermal conduction and conductivity are intertwined concepts, as one relies on the other. If thermal conductivity is increased, the rate of heat transfer (conduction) will also increase, meaning they are directly proportional.
Thermal Conductivity Formula
Thermal conductivity is one of the parameters used to calculate the rate at which heat is transferred between two objects with different temperatures, as visualized in Figure 1 below.
Mathematically, that can be expressed using the thermal conduction equation
$$ \frac{Q}{\Delta t} = k\frac{A\Delta T}{L}$$
where \(\frac{Q}{\Delta t}\) is the rate of heat transfer, \(A\) is the cross-sectional area, \(\Delta T\) is the temperature difference, \(L\) is the plane thickness, and \(k\) is the thermal conductivity of the material. By simply rearranging this equation, we can obtain the formula for thermal conductivity
$$ k=\frac{Q}{\Delta t}\frac{L}{\Delta T A}$$
which has the units of \(\frac{\mathrm{W}}{\mathrm{m} \, \mathrm{K}}\). In other words, thermal conductivity is the heat capacity that a material with a cross-sectional area of \(1\,\mathrm{m}^2\) can transmit at a distance of \(1 \, \mathrm{m}\), if a temperature difference of \(1\, \mathrm{K}\) has occurred in the material. The thermal conductivity of a specific material will vary with temperature and is usually determined experimentally.
Thermal Conductivity Example Values
Some commonly used materials and their thermal conductivity values, \(k,\) are compiled in the table below.
Table 1 - Thermal conductivity values for metals, solids, and gases.
| Material | \(k\) \( \left(\frac{\mathrm{W}}{\mathrm{m} \, \mathrm{K}}\right) \) |
| Aluminum | 205.0 |
| Brass | 109.0 |
| Copper | 385.0 |
| Lead | 34.7 |
| Mercury | 8.3 |
| Silver | 406.0 |
| Steel | 50.2 |
| Concrete | 0.8 |
| Cork | 0.04 |
| Felt | 0.04 |
| Fiberglass | 0.04 |
| Glass | 0.8 |
| Ice | 1.6 |
| Rock wool | 0.04 |
| Styrofoam | 0.027 |
| Wood | 0.12 - 0.04 |
| Air | 0.024 |
| Argon | 0.016 |
| Helium | 0.14 |
| Hydrogen | 0.14 |
| Oxygen | 0.023 |
High Thermal Conductivity Materials
Being aware of the thermal conductivity of various materials is important in everyday life and construction. For instance, heat should be conducted well by the surfaces of heating radiators, which are usually made of steel or aluminum. Those are examples of high thermal conductivity materials.
High thermal conductivity materials allow for heat to pass through them easily.
The best heat conductors, just like electric current conductors, are metals. That is due to their atomic structure consisting of free electrons. As heat is supplied to a metal, the electrons vibrate more intensely and therefore energy gets passed through the material quickly.
Despite diamonds having the highest thermal conductivity among all solid materials, they are great insulators.
Heat pipes are devices that transfer heat at a very high efficiency. They are closed evaporator condenser systems, where a working fluid gets vacuum sealed in a metal shell. The fluid is the key component determining the thermal conductivity of a specific heat pipe. Depending on its purpose, the pipe can be filled with various liquids like water, ammonia, or sodium.
Fig. 2 - Heat pipes are used in laptops to divert the heat from the processing units and prevent the device from overheating.
To put it in perspective, copper is a great thermal conductor with \( k=385 \, \frac{\mathrm{W}}{\mathrm{m} \, \mathrm{K}}\). Heat pipes, however, take it to the next level with thermal conductivity ranging from \(10 \,000 \) to \(100 \,000\, \frac{\mathrm{W}}{\mathrm{m} \, \mathrm{K}} \). They are commonly used in spacecraft, computers, and nuclear power cells as cooling systems.
Low Thermal Conductivity Materials
The opposite of high thermal conductivity materials is those with low thermal conductivity.
Low thermal conductivity materials don't allow for heat to pass through them easily.
These are especially important when we need thermal control and energy efficiency in manufacturing. For instance, during the winter, the walls, roof, and windows of a building must retain the heat in the rooms as much as possible and therefore conduct the heat as poorly as possible. That's why special microporous insulating materials and double-glazed windows are made because air (and all other gases) are poor heat conductors. Non-metals and many organic substances conduct heat much worse. The case of a burning match that we mentioned in the beginning, is a great example of a poor thermal conductor.
Fig. 3 - When a match is ignited, the person holding it won't burn their fingers from touching the wood, as it's a material with low thermal conductivity.
Materials that have low thermal conductivity are great insulators. Wood, for instance, has many air pockets within its structure containing the heat rather than passing it on. Plastic, on the other hand, has low thermal conductivity due to the lack of free electrons, which are responsible for passing the heat in the case of metals. Other examples of low thermal conductivity materials are visible in Table 1, including concrete, glass, and gasses.
Aerogel is the material with the lowest thermal conductivities among all solids \( \left (0.013 \, \frac{\mathrm{W}}{\mathrm{m} \, \mathrm{K}}\right ) \), which is even less than that of argon gas. It's a very fragile solid, mostly made up of air, and is used in laser experiments, thermal insulation, and even managing waste.
Fig. 4 - Aerogel is the most effective thermal insulator ever made, with the lowest thermal conductivity among solids.
Thermal Conductivity Examples
Let's look at some examples of problems utilizing the concept of thermal conductivity.
One end of an insulated metal rod is held at \(120\,\mathrm{C^{\circ}}\), while the other end is placed in an ice bath keeping its temperature \(0\,\mathrm{C^{\circ}}\). It takes half an hour for \(6810 \, \mathrm{J}\) of energy to get transferred from one end of the rod to the other, melting half the ice in the bath. Determine the type of metal this rod is made of, if it's \(80 \, \mathrm{cm}\) long, and has a diameter equal to \(0.5 \, \mathrm{cm}\).
Solution
We can use the heat transfer equation
$$ \frac{Q}{\Delta t} = k\frac{A\Delta T}{L}$$
to calculate the thermal conductivity \(k\) of the rod and therefore determine the material.
A rod is shaped like a long cylinder, so its cross-sectional area will be a circle. Considering we are given the diameter \(D\), in SI units the area will then be
\begin{align} A&=\pi r^2 \\ A&=\pi \left (\frac{D}{2} \right )^2 \\A&=\pi \left (\frac{0.005 \, \mathrm{m}}{2} \right )^2 \\ A&=2\times10^{-5} \, \mathrm{m}^2. \end{align}
One end of the rod has a temperature of \(0\,\mathrm{C^{\circ}}\), so the temperature difference is simply equal to \(120\,\mathrm{C^{\circ}}\), which in kelvins is equal to
$$ \Delta T = 120\,\mathrm{C^{\circ}} + 273 = 393 \, \mathrm{K}. $$
Finally, we are told that the heat transfer occurs for \(30 \, \mathrm{min}\) which can be expressed in seconds as
$$ \Delta t = 30 \, \cancel{\mathrm{min}} \cdot \frac{60 \, \mathrm{s}}{1 \, \cancel{\mathrm{min}}}= 1800\, \mathrm{s}.$$
Now we simply plug in all the values into the rearranged heat transfer equation
\begin{align} k&=\frac{Q}{\Delta t}\frac{L}{\Delta T A} \\ k&=\frac{(6810 \, \mathrm{J})}{(1800 \, \mathrm{s})}\frac{(0.8 \, \cancel{\mathrm{m}})}{(393 \, \mathrm{K})(2\times10^{-5} \, \mathrm{m}^\cancel{2})} \\ k&=385 \, \frac{\mathrm{W}}{\mathrm{m} \, \mathrm{K}}. \end{align}
We can compare this result to the thermal conductivity values compiled in Table 1 above and conclude that this rod is likely made from copper.
Thermal Conductivity - Key takeaways
- Conduction is the transfer of heat from an object with a higher temperature to that with a lower temperature, through direct contact.
- Thermal conductivity is the measure of a material’s ability to transfer thermal energy.
- Mathematically, thermal conductivity can be found using the thermal conduction equation \[k=\frac{Q}{\Delta t}\frac{L}{\Delta T A}.\].
- High thermal conductivity materials allow for heat to pass through them easily, metals being a common example.
- Low thermal conductivity materials don't allow for heat to pass through them easily, for example, wood and plastic.
References
- Fig. 1 - Thermal conduction diagram, StudySmarter Originals.
- Table 1 - Thermal conductivities, Young, Hugh D., University Physics with Modern Physics, 14th Ed., Addison-Wesley, Boston, 2015.
- Fig. 2 - Copper heat pipes (https://unsplash.com/photos/4D9PHPZ5U_c) by Vishnu Mohanan (https://unsplash.com/@vishnumaiea) on Unsplash is licensed by Unsplash License (https://unsplash.com/license).
- Fig. 3 - Burning match (https://commons.wikimedia.org/wiki/File:Burning_match.jpg) by Осадчая Екатерина licensed by CC BY-SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/deed.en).
- Fig. 4 - Aerogel hand (https://commons.wikimedia.org/wiki/File:Aerogel_hand.jpg) by Courtesy NASA/JPL-Caltech licensed by Public Domain.
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