With the current cold weather, I got curious — how should you stay warm while cycling?

Clearly, cycling faster requires more energy, heating you up. On the other hand, it also increases heat losses: the air flows around you faster, cooling you down. How does this scale? Does doubling your cycling speed increase or decrease your temperature over time?

It turns out the answer is easy. Let’s consider ourselves as a sphere. The Reynolds number, a dimensionless number estimating the degree to which a flow is turbulent, around a sphere with a particular radius, can be expressed as

We can make some rough approximations here. η is about 10⁻⁵ kg/m/s. ρ is about 1 kg/m³. We can assume *v* is about 5 m/s (so 18 km/h). The diameter *d* of a human is a bit questionable, but let’s say it’s about 0.8 meters. At that point, the Reynolds number is about 400.000 — pretty huge!

To actually calculate the heat loss, we can use a Nusselt correlation. We assume Re is large (which seems accurate) so that the constant term is irrelevant — it’s usually <10. *Pr* (Prandtl — always impossible to type) isn’t a function of the velocity. Therefore, heat losses scale as a function of Reynolds to α, which is 0.5 for a sphere.

Now, how much heat do we produce? This is a complicated topic, so we again result to scaling estimates. We have two types of losses: rolling friction and air resistance. Rolling friction is constant (it depends on the weight exerted — so unless you lose weight while cycling, it won’t change), so the dominant loss is air resistance, at high speeds.

As you can see, this scales quadratically with velocity. If we cycle twice as fast, we have twice as much drag, and therefore need to generate twice as much energy in our muscles. Assuming muscle efficiency is constant (it seems to be around 15–35%, with legs and arms on the lower side), this would also double the heat production.

Therefore, since convective heat losses scale with the square root of velocity, while heat production scales quadratically, doubling velocity while cycling will increase heat production by about 2.8 times.

If you want to feel warm while cycling in cold weather, there are two easy steps. 1. Turn into a sphere, then 2. cycle faster.