Rising hot air and a Baron

As I write the UK politics is in a bit of a mess. The referendum that kicked off this mess started in the actions of an Eton educated posh boy: David Cameron. But not all Eton educated posh boys have been a disaster for Britain. The picture above is of the partially-Eton-educated 3rd Baron Rayleigh, a brilliant late-Victorian scientist and genuine member of the aristocracy.

Baron Rayleigh studied many things, one of which is how, in fluids like air and water, heat moves from hot to cold. There are two ways heat can move from hot to cold, in fluids, these are: diffusion and convection.

We will start with heat diffusion, and let’s think about the air just above a hot plate. The hot plate will heat the air just above it. In this air, the molecules are constantly moving at random and colliding with each other. Collisions between lower (closer to the plate) and so on average hotter molecules, with higher and so colder molecules, transfers some of the heat energy of the slightly lower molecules to those higher up. Via this mechanism, heat diffuses upwards through the air above the plate.

In the absence of gravity, this is all that happens, but here on Earth there is gravity, and as air is heated it expands and becomes less dense. So the hot air at the bottom is lighter than the denser air above. Just as a hot-air balloon rises in colder air, the hot air at the bottom just above the hot plate will tend to rise. The hot air that rises, takes it heat with it, so this flow of air moves heat energy. This flow of air is an example of convection.

So, we have two ways for the heat to rise: diffusion and convection. Which one is most important, i.e., which of the two ways moves most of the heat? This is the question Baron Rayleigh answered. He showed that you should calculate a number, now known as the Rayleigh number to honour his contribution:

Rayleigh number = f l³g / να

for f the fractional change in the mass density of air, l the size of the volume of air we are interested in, g the acceleration due to gravity, and ν and α are the diffusion constants of momentum and heat, respectively, in air.

This looks a bit complex, but it isn’t so bad, we know that the factor g / να = 10¹¹ for air, and a difference of about 1 degree celsius changes the mass density by about 1% so then f = 0.01. Putting this all together, we get

Rayleigh number= 10⁹l³

Baron Rayleigh’s contribution back in 1916 was to show that convection kicks in and dominates the movement of heat when the Rayleigh number is above about 1000. With l = 1 mm, it is 1, while for l = 1 cm, the Rayleigh number is 1000.

So, over distances of millimetres or less, heat will move from hot air to cold air via diffusion, but over distances of centimetres or more, heat almost always moves via convection, air currants carry the heat energy from hot to cold. In a room 2 metres high and several metres across, there will always be convection — for example the heat of your body is more than enough to drive convection.

This is how heat gets about, but the air currants caused by convection also carry molecules with them. Convection above a plate of hot food doesn’t just cool the food, it also carries the smell of the food to you. In the last-but-one post, I pointed out that diffusion is not how molecules move across a room. Convection is one way that molecules do move across a room, they are carried along by (often only slightly) hotter air.