Friction, heat, time & surface area

[Langmarais]

Would it be possible to calculate the time it would take for heat to reach a given temperature between two rubbing surfaces and would the size/ area of contact influence the time if the co-efficient of friction and applied force remains constant. E.g:Let's say you have two wheel brake assemblies with the same drum size, same friction material and same brake application force, with the brake shoes in one assembly half the size of those in the other assembly (as the only variable), would there be a difference in the time it would take for these assemblies to reach the same, given temperature?

 

[Q_Goest]

Hi Langmarais, welcome to the board.

Would it be possible to calculate the time it would take for heat to reach a given temperature between two rubbing surfaces and would the size/ area of contact influence the time if the co-efficient of friction and applied force remains constant. E.g:Let's say you have two wheel brake assemblies with the same drum size, same friction material and same brake application force, with the brake shoes in one assembly half the size of those in the other assembly (as the only variable), would there be a difference in the time it would take for these assemblies to reach the same, given temperature?

I think your example is, given two, identical cars in which the only difference is the size of their brake pads and drums (drum brakes), will the smaller set of brakes heat up hotter or faster given they were both braking equally?

If the cars are both braking equally (an equal amount of energy is going into the brakes of both cars which must be rejected as heat) then why might one heat up to a higher temperature or more quickly than the other?

 

[Langmarais]

Thank you Q_Goest for your reply. Would the fact that the same amount of work (friction) is done in a smaller area not affect the rate at which the heat increases. Why would trucks have larger brake shoes/ -pads if surface area is not important. Is there a formula one can use to calculate the time it would take for to surfaces to reach a given temperature that requires co-efficient of friction and surface area?

Thanx!

 

[Q_Goest]

Hi Langmarais.

Would the fact that the same amount of work (friction) is done in a smaller area not affect the rate at which the heat increases. Why would trucks have larger brake shoes/ -pads if surface area is not important.

As I'm sure you're aware, the frictional force produced is independent of area. Also, the amount of work that has to be dissipated as heat is also equal for the two cars used in your example. If the energy dissipated is the same, but the contacting area is different, then the heat generated per unit area is higher for the smaller brake pads. That heat increases the temperature at the pad, and that higher temperature allows heat to flow into the surrounding area. That heat is translated mostly by conduction and convection. So if the path the heat has to take to get away from the brake pad is reduced, that path restricts the flow of heat. Since the rate of heat being rejected is the same for the small and large drum brakes in the cars, the car that has a smaller brake pad will get hotter. It will also get hotter faster if the thermal mass is smaller. To answer your question then, smaller brakes will get hotter and get hotter faster than larger brakes disipating the same amount of heat. Yes, that is why brakes on larger vehicles must be larger - they have to accommodate the larger heat load.

Regarding how to calculate this, because the geometry is generally fairly complex, it gets difficult. The easiest way to do this is to assume for a drum brake, that the braking surface gets uniformly hot and dissipates heat radially to where the fins on the outside of the drum reject that heat to atmosphere. So there's thermal conduction through the metal, followed by a convective heat transfer to the air. Generally, one can neglect radiation heat transfer for these kinds of things as it's often insignificant. The conduction part can be very difficult to do because it isn't generally a neat, uniform area and the convection part can be very difficult because of the highly variable air velocity acting on the outer surface of the drum.

 

[Langmarais]

Thanx Q_Goest

I have been dealing with a case where the brakes on a bus, traveling down a mountain pass, severely overheated and the bus ran out of control, resulting in a fatal collision. The incident was the result of "brake fade", caused by an obvious significant reduction in co-efficient of friction within the foundation brakes. Poor maintenance on the braking system resulted in partial drum to shoe exposure (+-50% surface contact) with full brake application. The force with which the shoes were pushed against the drums would not have been compromised. Would it then be safe to assume that the brakes would have overheated significantly sooner as a result of this condition?

 

[Q_Goest]

Hi Lang'. That's a different question altogether. It sounds like you need some legal advice, not internet advice. That said, I suppose learning a bit about brakes so you can understand the technicalities would be of benefit, so I'll see if I can help in that regard.

Would it then be safe to assume that the brakes would have overheated significantly sooner as a result of this condition?

The short answer is it's unlikely.

Here's the long answer: The heat generated by friction is a heat 'flux'. That's a heat flow divided by an area. In this case, the magnitude of the heat flow is simply the amount of energy being dissipated. That heat flow is the same regardless of how much brake area there is. What you're interested in is the area part of this heat flux. The area is the circumferential area on the ID of the drum. In other words, the heat generated by braking goes into the drum at the inner surface (area) of the drum. That heat goes through the drum and is dissipated into the air at the outer surface of the drum only. The amount of heat that can be rejected by the brake pad itself is small and can be neglected because there's essentially no air flow inside a drum, and what little air flow there is has very little surface area to exchange heat with the brake shoe. So to a high degree of accuracy, you can assume all the heat has to be transferred to the inner surface (area = circumference times contact width) of the drum. From there, heat has to move through thermal conduction to the outer surface of the drum where the air flowing over the outer surface can heat up and take that heat away through a process called convective heat transfer.

In the case of a damaged or malworn brake pad, there are a couple of different scenarios.
1. The brake pad has a thickness, and when that wears out the metal beneath will contact the drum. From what you've said, that doesn't seem to be the issue here.
2. The brake pad has a length in the direction of rotation, and I'm assuming this length is what was compromised. If that length is compromised, the area through which heat is rejected (as discussed above) by the drum hasn't changed. Note the area is still a cylindrical area equal to the width of the brake pad times the circumference of the drum.
3. The brake pad also has a width, which if the brake pad were narrower for example, the heat flux around the inner surface of the drum would be reduced, increasing the heat flux and the brake would theoretically get hotter. It would get hotter because the area is smaller. The area is smaller because the area is equal to the inner circumference (which hasn't changed) times the width (which HAS changed). The width is narrower, so the thermal conductivity through the brake drum now has a smaller area to get through. Note however, that cutting the area in half does not cut the thermal conductivity in half, so it won't get twice as hot, it will simply be hotter. However, brakes generally can't wear like that because they are essentially hinged so if one side of the pad were not touching, there would be a moment created, forcing it to rotate and to come into contact across the full width. So I'm having a hard time imagining the width of the brake pad being compromised, but figured I should point this out anyway.

To summarize, I'm assuming it was the length of the brake pad that was shortened, and if that's the case, there's no significant thermal difference. There would be a difference however in the surface shear stress. Half the area means twice the shear stress at the brake shoe surface, which may cause physical damage as opposed to heat damage. But brake pads can be expected to wear unevenly to some degree and the manufacturer of the bus should have brakes large enough so there's some margin of safety above and beyond what might be expected.

You might want to look a bit deeper into the brake issue though. If there was unusual wear, it may be indicative of some other problem. Also, brake fade isn't just an issue of brake pad surface temperature, there is also the issue of drum expansion and potential boiling of dissolved water or gasses in the brake fluid. There's a pretty decent article on it here:
http://en.wikipedia.org/wiki/Brake_fade

Brake fade is a real issue, but from what I understand of the situation, there isn't sufficient information to suggest it was a maintenance issue yet.