See also also bike pic for other bike-related technical discussions and more failed parts.
Bicycles occasionally have sudden tire blow-off failures when the rim gets hot. This is especially a problem with tandems, which have weight approaching double that of a single bike, but air drag much less than double — so the brakes have to work harder, because less energy is going to air drag.
When you go down a hill, your potential energy (the work you did going up!) is converted to kinetic or "speed" energy. The faster you go down, the faster you get it back, so in principle the more work your brakes have to do, and the hotter they get.
On the other hand, the faster you go, the more energy gets dumped in to air drag, and a tornado cools your brakes better than a whispering breeze.
So from the standpoint of brake heat, what is the safest speed to go down a hill? I do not have test data, but here are some calculations...
Consider the power which is generated by the weight of bike+riders descending at a steady rate. The power is related to how fast their altitude is changing.
Grades are often stated as a "percent" grade, where the grade is the ratio of (vertical distance traveled) to (horizontal distance traveled). A 45-degree angle is a 100% grade.
"Grade" is convienient but also a little tricky for some calculations, as we usually measure (and think about) road speed, but grade is about the horizontal and vertical distances you travel. If you think of a righ triangle, road speed is your speed on the hypotenuse of the triangle
For a gentle grade, horizontal speed and road speed are nearly the same, and the vertical rate is thus close to the road speed times the percent grade. For example, 100 kph on a 0.5% grade is 100*0.5/100 = 0.5 kph vertical speed. On the other hand, 100 kph on a 50% grade works out to about 45 kph vertical and not 50 kph.
The following is printed in hectowatts as the numbers print better...
Call this "grade power": the power you get from going down a given grade at a given speed. The power of pedaling, braking, and air drag are not included in grade power.
For example, going down an 8% grade at 25 kph, a 150 kgf single rider or tandem team gets 8 hectowatts, which is 800 Watts. In comparison, most riders in steady riding on the flat put oat 1-2 hectowatts or 100-200 Watts.
Grade Power 150 kgf: Hectowatts (100's of Watts), kph vs. grade % \% 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 kph +----+---+---+---+---+---+---+---+---+---+---+---+---+---+---+ 5| 0 1 1 2 2 2 3 3 4 4 5 5 5 6 6 10| 1 2 2 3 4 5 6 7 7 8 9 10 11 12 13 15| 1 2 4 5 6 7 9 10 11 13 14 15 16 18 19 20| 2 3 5 7 8 10 12 13 15 17 18 20 22 24 26 25| 2 4 6 8 10 12 14 17 19 21 23 25 27 30 32 30| 2 5 7 10 12 15 17 20 22 25 28 30 33 36 39 35| 3 6 9 11 14 17 20 23 26 29 32 35 38 42 45 40| 3 7 10 13 16 20 23 26 30 33 37 40 44 48 51 45| 4 7 11 15 18 22 26 30 34 38 41 45 49 54 58 50| 4 8 12 16 21 25 29 33 37 42 46 50 55 60 64 55| 4 9 13 18 23 27 32 36 41 46 51 56 60 66 71 60| 5 10 15 20 25 30 35 40 45 50 55 61 66 71 77 65| 5 11 16 21 27 32 38 43 49 54 60 66 71 77 83 70| 6 11 17 23 29 35 40 46 52 58 64 71 77 83 90 75| 6 12 18 25 31 37 43 50 56 63 69 76 82 89 96
The power lost to air and rolling drag can be estimated, for example http://en.wikipedia.org/wiki/Bicycle_performance#Power_required.
Unfortunately, they don't say much about the rider position, and that makes a big difference. Let us assume still air and apply a corrective factor to the k2 air drag term. Let's say 1.5 to account for some combination of more bike/rider skin drag and sitting upright to lose more power.
(That is, a rider trying to go fast on the flat will crouch to reduce air drag, while a rider trying to go slow down a steep hill will sit up in order to increase air drag. The following assumes the calculation is for "trying to go fast" so sit up to try and slow down, and the air drag might be 1.5 times greater.)
Rolling and air drag depend on speed, but do not depend on grade, so drag power is the same at every grade. Let's call this "steady power", the power to just match air and rolling drag, and thus maintain a steady speed.
Steady Power 150 kgf: Hectowatts (100's of Watts), kph vs. grade % \% 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 kph +----+---+---+---+---+---+---+---+---+---+---+---+---+---+---+ 5| 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 10| 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 15| 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 20| 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 25| 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 30| 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 35| 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 40| 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 45| 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 50| 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 55| 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 60| 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 65| 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 70| 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 75| 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27
The German magazine "Bike" tested brakes to failure many years ago; their brake tests report rim brakes have 1100W capacity at 25 kph. I'm going to pretend they simulated a 25 kph wind over the rim and that capacity is increased by more wind and decreased by less wind, and that brake capacity is linearly related to the air speed over the rim. This is a big "if", of course, but it makes sense to me. If you want to try different assumptions the calcuation is easy enough.
Further, assume two 1100W brakes, and assume the cyclist can evenly distribute power between the two brakes. (I know I cannot get them the same,., but this is a theoretical calculation!) Again, cooling depends on speed so capacity depends on speed, but capacity does not depend on grade. Let us call this "brake capacity power" — that is, how much power can the brake dump.
Brake Capacity Power, 2200W@25kph nominal brake capacity 150 kgf: Hectowatts (100's of Watts), kph vs. grade % \% 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 kph +----+---+---+---+---+---+---+---+---+---+---+---+---+---+---+ 5| 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 10| 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 15| 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 20| 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 25| 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 30| 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 35| 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 40| 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 45| 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 50| 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 55| 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 60| 53 53 53 53 53 53 53 53 53 53 53 53 53 53 53 65| 57 57 57 57 57 57 57 57 57 57 57 57 57 57 57 70| 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 75| 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66
Now we can build a table that compares
That is, take grade power, and subtract steady power; to keep going the same speed, the brakes have to do the rest (I assume nobody is pedaling down the hill...), so compare to brake capacity power. If grade less steady is below the brake capacity, then you have capacity to spare. If grade less steady is above the brake capacity, the brakes will fail (perhaps by an overheat that blows the tire off the rim).
For example, at 8% grade and 25 kph, there's 8 hectowatts coming in, steady power is 1 hectowatt, so the brake has to dump 8-1=7 hectowatts if you want to keep going 25 kph. At the same grade and speed, the brake capacity power is 22 hectowatts, so the brake excess capacity is 22-7=15 hectowatts.
There is at least one other case to consider: if grade power is small and steady power is large, then even with no brakes you will slow down (again: assuming no pedaling). In the following table "<<" is where hill power is less than the steady power, so even with no braking the bike slows down. Otherwise, positive numbers indicate the amount of excess capacity the brakes have, and negative numbers indicate how much the brake is overloaded.
Brake Excess Capacity (- == overload), 22 hW @ 25 kph brake capacity 150 kgf: Hectowatts (100's of Watts), kph vs. grade % \% 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 kph +----+---+---+---+---+---+---+---+---+---+---+---+---+---+---+ 5| 4 4 3 3 2 2 2 1 1 0 -0 -1 -1 -1 -2 10| 8 7 7 6 5 4 3 2 2 1 -0 -1 -2 -3 -4 15| 13 11 10 9 8 6 5 4 3 1 -0 -1 -3 -4 -6 20| 17 15 14 12 10 9 7 5 4 2 0 -2 -3 -5 -7 25| 21 19 17 15 13 11 9 7 5 3 0 -2 -4 -6 -9 30| 26 24 21 19 16 14 11 9 6 4 1 -2 -4 -7 -10 35| << 28 26 23 20 17 14 11 8 5 2 -1 -4 -8 -11 40| << 33 30 27 23 20 17 13 10 7 3 -1 -4 -8 -11 45| << 39 35 31 28 24 20 16 12 8 5 1 -3 -8 -12 50| << << 40 36 32 28 24 19 15 11 6 2 -2 -7 -12 55| << << 46 41 37 32 28 23 18 14 9 4 -1 -6 -11 60| << << 52 47 42 37 32 27 22 17 12 6 1 -5 -10 65| << << << 54 48 43 37 32 26 21 15 9 3 -2 -9 70| << << << 61 55 49 43 37 31 25 19 13 7 0 -6 75| << << << << 62 56 49 43 37 30 24 17 10 3 -4
The above is scary in at least one way: it is usually beneficial to "go as fast as possible" so you dump power in to the air directly, rather than heating the brakes. However, for very steep grades, going faster means you get a lot more grade power but not a lot more brake capacity, so going faster (at some speeds) actually hurts more than it helps. This is, however, only for very steep grades.
Here is the same basic calculations, but with with 3300 W braking capacity. Note that the bigger the brake capacity, the more it goes up with high-speed cooling. So while the 3300W brake gives you barely more capacity at low speeds, it dramatically more at high speeds, due to better cooling.
Brake Excess Capacity (- == overload) 33 hW @ 25 kph brake capacity 150 kgf: Hectowatts (100's of Watts), kph vs. grade % \% 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 kph +----+---+---+---+---+---+---+---+---+---+---+---+---+---+---+ 5| 6 6 5 5 5 4 4 3 3 3 2 2 1 1 0 10| 13 12 11 10 9 9 8 7 6 5 4 3 2 2 1 15| 19 18 17 15 14 13 12 10 9 8 7 5 4 2 1 20| 26 24 22 21 19 17 16 14 12 11 9 7 5 3 2 25| 32 30 28 26 24 22 20 18 16 14 11 9 7 5 2 30| 39 37 34 32 30 27 25 22 19 17 14 12 9 6 3 35| << 44 41 38 35 32 29 26 23 20 17 14 11 8 5 40| << 51 48 44 41 38 34 31 28 24 21 17 13 10 6 45| << 58 55 51 47 44 40 36 32 28 24 20 16 12 8 50| << << 62 58 54 50 46 41 37 33 28 24 20 15 10 55| << << 70 66 61 57 52 47 43 38 33 28 23 18 13 60| << << 79 74 69 64 59 54 49 43 38 33 27 22 16 65| << << << 82 77 71 66 61 55 49 44 38 32 26 20 70| << << << 91 86 80 74 68 62 56 50 44 37 31 24 75| << << << << 95 89 82 76 70 63 57 50 43 36 29
Brake Excess Capacity (- == overload), 22 hW W@ 25 kph brake capacity 200 kgf: Hectowatts (100's of Watts), kph vs. grade % \% 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 kph +----+---+---+---+---+---+---+---+---+---+---+---+---+---+---+ 5| 4 3 3 2 2 1 1 0 -0 -1 -2 -2 -3 -3 -4 10| 8 7 6 5 4 3 1 0 -1 -2 -3 -4 -6 -7 -8 15| 12 11 9 7 6 4 2 1 -1 -3 -5 -6 -8 -10 -12 20| 16 14 12 10 8 5 3 1 -1 -4 -6 -8 -11 -13 -16 25| 21 18 15 13 10 7 4 2 -1 -4 -7 -10 -13 -16 -19 30| 26 22 19 16 12 9 6 2 -1 -4 -8 -12 -15 -19 -22 35| 31 27 23 19 15 11 7 3 -1 -5 -9 -13 -17 -21 -26 40| << 31 27 23 18 14 9 5 0 -4 -9 -14 -18 -23 -28 45| << 37 32 27 22 17 12 7 1 -4 -9 -14 -20 -25 -31 50| << 42 37 31 26 20 14 9 3 -3 -9 -14 -20 -27 -33 55| << 48 42 36 30 24 18 11 5 -1 -8 -14 -21 -27 -34 60| << << 48 41 35 28 21 14 8 1 -6 -13 -21 -28 -35 65| << << 54 47 40 33 25 18 11 3 -4 -12 -20 -28 -36 70| << << 61 53 46 38 30 22 14 6 -2 -10 -19 -27 -36 75| << << << 60 52 44 36 27 19 10 1 -8 -17 -26 -35
With a 3300 W nominal brake:
Brake excess capacity (- == overload) -- Hectowatts (100s of Watts) 200 kgf, kph vs. grade % -- aero factor: 1.5; brake: 33 hW @ 25 kph \% 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 kph +----+---+---+---+---+---+---+---+---+---+---+---+---+---+---+ 5| 6 6 5 5 4 3 3 2 2 1 1 0 -1 -1 -2 10| 12 11 10 9 8 7 6 5 4 2 1 0 -1 -2 -4 15| 19 17 16 14 12 11 9 7 5 4 2 0 -2 -3 -5 20| 25 23 21 19 17 14 12 10 8 5 3 1 -2 -4 -7 25| 32 29 26 24 21 18 15 13 10 7 4 1 -2 -5 -8 30| 39 36 32 29 26 22 19 16 12 9 5 2 -2 -6 -9 35| 46 42 38 34 31 27 23 19 15 11 7 3 -2 -6 -10 40| << 49 45 40 36 31 27 22 18 13 9 4 -1 -6 -11 45| << 56 51 46 41 37 31 26 21 16 11 6 0 -5 -11 50| << 64 59 53 48 42 36 31 25 19 13 8 2 -5 -11 55| << 72 66 60 54 48 42 36 29 23 17 10 3 -3 -10 60| << << 74 68 61 54 48 41 34 27 20 13 6 -1 -9 65| << << 83 76 68 61 54 47 39 32 24 17 9 1 -7 70| << << 92 84 77 69 61 53 45 37 29 21 12 4 -5 75| << << << 93 85 77 69 60 52 43 34 25 16 7 -2
Even assuming the assumptions are correct, there's lots the above does not say -- for example, what is the heat induced in a cold brake in a hard stop? My guess is even a well-cooled 200g steel disk rotor has little thermal inertia compared to a 600g aluminum rim, and that both heat up quickly enough that when run hot, there is little extra margin from starting cool.
Another day.