What is thermal conductivity?

Walking on bathroom tile in winter is annoying since it feels so much colder than the carpet. This is interesting, since the carpet and tile are usually both at the same temperature (i.e. the temperature of the interior of the house). The different sensations we feel is explained by the fact that different materials transfer heat at different rates. Tile and stone conduct heat more rapidly than carpet and fabrics, so tile and stone feel colder in winter since they transfer heat out of your foot faster than the carpet does.

In general, good conductors of electricity (metals like copper, aluminum, gold, and silver) are also good heat conductors, whereas insulators of electricity (wood, plastic, and rubber) are poor heat conductors. The figure below shows molecules in two bodies at different temperatures. The (average) kinetic energy of a molecule in the hot body is higher than in the colder body. If two molecules collide, an energy transfer from the hot to the cold molecule occurs. The cumulative effect from all collisions results in a net flux of heat from the hot body to the colder body. We call this transfer of heat between two objects in contact thermal conduction.

There are four factors ($k$k, $A$A, $\Delta T$delta, T, $d$d) that affect the rate at which heat is conducted through a material. These four factors are included in the equation below that was deduced from and is confirmed by experiments.

The letter $Q$Q represents the amount of heat transferred in a time $t$t, $k$k is the thermal conductivity constant for the material, $A$A is the cross sectional area of the material transferring heat, $\Delta T$delta, T is the difference in temperature between one side of the material and the other, and $d$d is the thickness of the material. These factors can be seen visually in the diagram below.

Image: Heat conduction occurs through any material, represented here by a rectangular bar, whether window glass or walrus blubber. (Image Credit: Openstax College Physics)

There's a lot to digest in the equation for thermal conduction $\dfrac{Q}{t}=\dfrac{kA\Delta T}{d}$start fraction, Q, divided by, t, end fraction, equals, start fraction, k, A, delta, T, divided by, d, end fraction. Let's look at what each factor means individually below.

$\dfrac{Q}{t}$start fraction, Q, divided by, t, end fraction: The factor on the left hand side of the equation $(\dfrac{Q}{t})$left parenthesis, start fraction, Q, divided by, t, end fraction, right parenthesis represents the number of $\text{joules}$start text, j, o, u, l, e, s, end text of heat energy transferred through the material per $\text{second}$start text, s, e, c, o, n, d, end text. This means the quantity $\dfrac{Q}{t}$start fraction, Q, divided by, t, end fraction has units of $\dfrac{\text{joules}}{\text{second}}=\text{watts}$start fraction, start text, j, o, u, l, e, s, end text, divided by, start text, s, e, c, o, n, d, end text, end fraction, equals, start text, w, a, t, t, s, end text.

$k$k: The factor $k$k is called the thermal conductivity constant. The thermal conductivity constant $k$k is larger for materials that transfer heat well (like metal and stone), and $k$k is small for materials that transfer heat poorly (like air and wood).

$\Delta T$delta, T: The heat flow is proportional to the temperature difference $\Delta T=T_{hot}-T_{cold}$delta, T, equals, T, start subscript, h, o, t, end subscript, minus, T, start subscript, c, o, l, d, end subscript between one end of the conducting material and the other end. Therefore, you will get a more severe burn from boiling water than from hot tap water. Conversely, if the temperatures are the same, the net heat transfer rate falls to zero, and equilibrium is achieved.

$A$A: Owing to the fact that the number of collisions increases with increasing area, heat conduction depends on the cross-sectional area $A$A. If you touch a cold wall with your palm, your hand cools faster than if you just touch it with your fingertip.

$d$d: A third factor in the mechanism of conduction is the thickness $d$d of the material through which heat transfers. The figure above shows a slab of material with different temperatures on either side. Suppose that $T_2$T, start subscript, 2, end subscript is greater than $T_1$T, start subscript, 1, end subscript, so that heat is transferred from left to right. Heat transfer from the left side to the right side is accomplished by a series of molecular collisions. The thicker the material, the more time it takes to transfer the same amount of heat. This model explains why thick clothing is warmer than thin clothing in winters, and why Arctic mammals protect themselves with thick blubber.

Materials with a high thermal conductivity constant $k$k (like metals and stones) will conduct heat well both ways; into or out of the material. So if your skin comes into contact with metal that is colder than your skin temperature, the metal can rapidly transfer heat energy out of your hand, making the metal feel particularly cold. Similarly, if the metal is hotter than your skin temperature, the metal can rapidly transfer heat energy into your hand, making the metal feel particularly hot.

This is why concrete will feel especially cold to our bare feet in winter (the concrete transfers heat out of our feet rapidly), and especially hot to our bare feet in summer (the concrete transfers heat into our feet rapidly).

A person wants to replace the window on her house, but she doesn't want her heating and cooling bills to change. The original window on the wall of the house has area $A$A and thickness $d$d, and is made out of glass that has a thermal conduction constant $k$k.

A single-paned window in your house is $0.65\text{ m}$0, point, 65, start text, space, m, end text wide, $1.25\text{ m}$1, point, 25, start text, space, m, end text tall, and has a thickness of $2\text{ cm}$2, start text, space, c, m, end text. The glass has a thermal conduction constant of $0.84 \dfrac{\text{J}}{\text{s}\cdot \text{m} \cdot ^o\text{C}}$0, point, 84, start fraction, start text, J, end text, divided by, start text, s, end text, dot, start text, m, end text, dot, start superscript, o, end superscript, start text, C, end text, end fraction. Assume the outside temperature of the glass is a constant $5^o\text{ C}$5, start superscript, o, end superscript, start text, space, C, end text and the inside temperature of the glass is a constant $20^o\text{ C}$20, start superscript, o, end superscript, start text, space, C, end text.

Solution:

$Q= 1.84 \times 10^6 \text{ J}\quad {\text{(calculate and celebrate)}}\quad$Q, equals, 1, point, 84, times, 10, start superscript, 6, end superscript, start text, space, J, end text, start text, left parenthesis, c, a, l, c, u, l, a, t, e, space, a, n, d, space, c, e, l, e, b, r, a, t, e, right parenthesis, end text