Maximum Brake Temperatures

To simplify this a bit, I'm going to ignore rolling resistance and aerodynamic drag, so all the work is done by braking. Also, when you brake hard, the front brake does almost all the work, so ignoring the rear brake to simplify the scenario.

Stopping

Mass of bike + rider = 100kg

Speed = 10 m/s (36km/h, about 22.4mph)

kinetic energy = 0.5 m v^2 = 5000J

A basic 180mm steel brake rotor weighs about 150g and the specific heat of steel is about 0.5 J/gK (Joules per gram-degree), so converting 5kJ of kinetic energy into heat in the rotor should increase the rotor temperature by 67 degrees (C), assuming none of the energy is absorbed by the brake pad, caliper body or brake fluid (not realistic, but useful to identify an upper bound for the rotor).

Descents

To maintain constant speed on an even slope, the change in gravitational potential energy must equal the heat energy extracted by braking (plus other sources of friction and drag that we're ignoring).

Descending a 1 in 10 slope for a 100m distance gives a height change of 9.95m (not 10m, because the distance is along the hypotenuse).

Gravity on earth is about 9.8 m/s^2.

Energy change = m g h = 100 * 9.8 * 9.95 = 9751J

This would increase the rotor temperature by 130 degrees.

Heat Dissipation and the limits of my knowledge

The two calculations above don't have a duration because I haven't allowed for heat loss during the event; here are some of the complicating factors:

• The friction creates a film of pad material on the rotor. This film insulates the rotor, but also is necessary for cohesive friction, which generates heat without consuming the pad material, but gives way to destructive friction as pressure increases.
• The heat is generated at two patches on either side of the rotor, not uniformly throughout the rotor material, so as the rotor turns through the contact patch, heat penetrates from the surface to the core of the rotor, then as the rotor leaves the contact patch, the surface is cooled proportionally to the surface-air temperature difference, and once the surface is colder than the core, heat moves from core to surface. Heat is also conducted parallel to the surface, from hotter to colder parts of the core.
• The airflow path length across the rotor is not consistent.
• The airflow around the rotor is turbulent due to the disruption caused by the leading part of the wheel and, for the upper aft part of the rotor, the fork leg and brake caliper.

Modelling heat in rotors is sufficiently complicated that it is a topic used in Engineering project assignments for university students, and the software typically used to perform the calculations is very expensive.

"Experimental and Numerical Thermal Analysis of Formula Student Racing Car Disc Brake Design", Manthan Vidiya1 and Balbir Singh, Manipal Institute of Technology. Published in Journal of Engineering Science and Technology Review 10(1)(2017)138-147

Example model "Heat Generation in a Disc Brake" for COMSOL Multiphysics:

As this pops up #1 on "mtb brake temperature" search, let us continue the answer of @Emyr. On descents the potential energy will be converted into the heat of the braking system, and then dissipated into the air. There are two main physical processes going on:

• accumulation of (thermal) energy in the brakes, governed by P * t = C * M * DT, where P=power, t=time, C=heat capacity (of steel/aluminium, or whatever), M=mass of the disc, DT = change of the disc temperature from the ambient temperature, and

• dissipation of heat into the air, governed by P = k * S * DT, where S is the area of contact, k is the convective heat transfer coefficient; so for given DT, there is a given power that will be radiated away,

Now the heat transfer coefficient is actually a "capability" depending on many things (like geometry), but also - and mainly, mostly linearly - on the airspeed. There are different values you can find, but I include some typical ones below, so let's say it is 35 W/(K * m^2) for 16kmh and 70 W/(K * m^2) for 32kmh (and let's ignore the dependence of these coefficients on the temperature).

Now these processes play as follows. On a long descent the heat capacity will quickly be saturated, and all the potential energy lost will have to be radiated away, i.e. the temperature of the disc will go up until the point where the disc radiates away as much as is generated from the descent. Assuming 800W generated (typical biker on ~16kmh descent on somewhat steep slope, and both brakes working), and a disc brake of (2-side) surface (both sides together) 5 dm^2 = 0.05 m^2 disc area, we get to

DT = 400 / (35 * 0.05) = 228 °C

Of course, should the slope be less steep, and allow for same potential energy generated at twice the speed, we would get 114°C. That's why it is the steep slopes that "kill" the brakes.

Now before the steady state is achieved, the disc will store the energy as heat. Assuming the disc is 2mm in diameter, so 0.05dm^3 in volume, so 400g of steel [it is usually much less due to perforations], with heat capacity of 500 J / (kg * °K) we get roughly

DT = 2°C * t [s],

so without the convective losses the disc would warm up by 2°C per second. (Incidently, if your disc is worn out, say to 1mm, it will warm up twice as fast.)

One can now play with different "braking strategies" etc. but almost for sure the following will hold:

• increasing the radiating area will help; S → 2S leads to DT → DT/2 (use both brakes, and/or brakes that have somewhat greater area from which heat is dissipated),
• you may increase the radiated power if you allow for a greater temperature (even for limited period of time), but only if the airspeed is the same; it is almost certainly a bad idea to get to "pulsate" by getting to greater speeds and brake to much lower ones (as at the point after the heat is generated, the airspeed will become low, and hence the transfer will be lower).
• also notice that for a given slope, the generated power (potential energy lost), and the radiative heat transfer are both proportional to the velocity; hence the steady state temperature of the brakes will be independent of the (steady) velocity.

Let us finish with a view of braking/working disc, and the source of these simulations, taken from "Thermal/Mechanical Measurement and Modeling of Bicycle Disc Brakes", ISEA 2018, Ioan Feier and Robin Redfield.