Conduction, convection and radiation

Conduction

Conduction is the transfer of thermal energy through a material without the material itself moving.

Use the particle model. In the hot end of a metal spoon the particles vibrate with large amplitude. They knock against their neighbours, which then vibrate more strongly, and so on down the spoon. Energy is passed along, but the particles stay where they are.

Metals are far better conductors than wood or plastic. A metal also has free (delocalised) electrons that move through the whole solid and carry energy quickly. Wood has no free electrons, so it relies on vibrations alone and conducts poorly.

How fast does energy flow?

Take a bar of length $L$ and cross-sectional area $A$, with its ends at different temperatures. The rate of energy flow depends on four things:

• a bigger area $A$ gives a bigger flow (more particles take part)

• a larger temperature difference $\Delta T$ gives a bigger flow

• a longer bar $L$ gives a smaller flow (the energy has further to go)

• the material matters, and this is measured by its thermal conductivity $k$

$Q/t$ is in watts ($\text{W}$), $k$ in $\text{W m}^{-1}\text{K}^{-1}$, $A$ in $\text{m}^2$, $\Delta T$ in $\text{K}$ (or $^\circ\text{C}$, since it is a difference) and $L$ in $\text{m}$. A big $k$ means a good conductor. Copper has $k \approx 400$, while a good insulator such as mineral wool has $k \approx 0.04$.

Convection

Convection is the transfer of thermal energy by the movement of a fluid (a liquid or a gas). It cannot happen in a solid, because the particles in a solid are fixed in position and cannot flow.

Follow what happens over a radiator:

1. The air next to the radiator is heated. Its particles move faster and spread out.

2. The same mass of air now takes up a larger volume, so its density decreases.

3. The less dense warm air is pushed upwards by the denser cool air around it (it experiences an upthrust).

4. Cool air flows in to take its place, is heated in turn, and a convection current is set up.

The same idea explains why the hot water tank stores its hottest water at the top: the heated water is less dense, so it rises above the cooler water below.

Sea breezes

Land and sea receive the same sunshine, but the sea heats up much more slowly. Water has a specific heat capacity of about $4{,}180\ \text{J kg}^{-1}\text{K}^{-1}$, whereas dry land is about $800\ \text{J kg}^{-1}\text{K}^{-1}$. The same energy raises the temperature of the land much more. Also, the sea mixes by convection, so the energy is shared through a deep layer of water.

By day the land is warmer, so the air above it rises and cooler air flows in from the sea. This is the onshore breeze you feel at Salthill. At night the land cools faster than the sea. Now the air over the sea is warmer and rises, and the breeze reverses.

Radiation

Thermal radiation is energy carried by electromagnetic waves, mainly infrared. It needs no medium, so it crosses the vacuum between the Sun and the Earth. Every object above $0\ \text{K}$ emits it, and it travels at $3.0\times10^{8}\ \text{m s}^{-1}$.

What happens to radiation arriving at a surface depends on the surface:

A good absorber is also a good emitter. That is why a white house stays cooler in summer sunshine, and why a hot water bottle cover or a kettle is often shiny, to lose less energy by radiation.

The Stefan–Boltzmann law

The power radiated by a perfect emitter (a black body) depends strongly on its absolute temperature:

where $A$ is the surface area, $T$ is the temperature in kelvin, and $\sigma = 5.67\times10^{-8}\ \text{W m}^{-2}\text{K}^{-4}$. Doubling $T$ makes the power $2^4 = 16$ times bigger.

An object also absorbs radiation from its surroundings at temperature $T_s$. The net power lost is

Layers in series

A real wall or window has several layers. Energy must pass through every layer one after another. In the same way that resistors add in a series circuit, thermal resistances add.

For one layer of thickness $L$, area $A$ and conductivity $k$:

The unit is $\text{K W}^{-1}$. Rewriting the conduction formula gives $\dfrac{Q}{t}=\dfrac{\Delta T}{R}$. For layers in series, the same rate of flow passes through each layer and

A layer of poorly conducting material (like air or foam) has a large $R$ and dominates the total. That is why double glazing works: the trapped air layer provides most of the resistance. It also helps that a thin trapped layer of air cannot set up large convection currents.

Back to the start: the radiator in your classroom. The hot metal warms your hand by conduction. The air above it is heated and rises as a convection current. Your face is warmed by infrared radiation, which travels straight through the air.

Summary

• Conduction: energy passes from particle to particle in a solid. Metals are the best conductors because of free electrons. $\dfrac{Q}{t}=\dfrac{kA\Delta T}{L}$.

• Convection: energy is carried by a moving fluid. Heated fluid expands, becomes less dense and rises. It cannot happen in solids.

• Radiation: infrared and other electromagnetic waves. It needs no medium, and dark matt surfaces absorb and emit best. $P=\sigma A T^4$, with $T$ in kelvin.

• Net radiated power: $P_{net}=\sigma A(T^4-T_s^4)$.

• Layers in series: $R=\dfrac{L}{kA}$, $R_{total}=R_1+R_2+\dots$ and $\dfrac{Q}{t}=\dfrac{\Delta T}{R_{total}}$.

• Energy lost: $E = (Q/t)\times t$. To convert to kWh, use power in kW and time in hours.