# Modeling heat build-up in air-cooled outrunners

I'm creating a thermal model for heat build-up in my outrunner motors. So far, I haven't found any online resources that discuss this. Here is the basic equations for forced convective transfer:

Q = h * A * delT, where Q is the heat transferred, A is the exposed area, delT is the temperature differential, and h is the heat transfer coefficient.

For forced air, h ranges from 1 to 1000, so it's easy to see why the devil is in this detail. A three-order of magnitude difference in cooling efficiency is huge, to put it mildly. Typical published values for forced convection are in the 10-100 range, but those seem ludicrously low for the amount of heat that a typical multirotor motor needs to shed.[*]

There are ways to calculate h from Reynolds and Prandtl relations, but without extensive modeling those are exceedingly hard to get in the turbulent flow regime around a spinning motor.

Although this particular application is a fixed-wing with a cowled motor, the basics still apply and only the airstream speed and volume changes.

Can anyone shed any insight?

[*] If you have a 100W motor that's 80% efficient, that's 20W of heat energy. To put things in perspective, it's like a 20W incandescent bulb, only much smaller. So if the heat weren't being dumped very quickly into the airstream, the motor would get excruciatingly hot. Ergo, we can conclude from all available evidence that the cooling coefficient is quite high.

• Ooh, I love this kind of stuff but fear it's too much complex physics (CFD and thermodynamics) for it to be well-answered here. IMO this might be better on Physics.SE, but I don't want to flag this.
– ifconfig
May 11 '20 at 17:09
• Eh, my masters was in heat transfer, so I'm good to take this on theoretically. ;) What the Physics.SE guys don't have experience with is the particularity of a drone's setup and outrunner motors. And, to be honest, I don't either. There are almost certainly some good rules of thumb which are broadly applicable to the generalized , and yet peculiar, case of a spinning heated element with forced airflow, but I don't know what they would be. (BTW, if you're into that kind of stuff, my thermo prof is one of the guys who found a closed-form N-S solution. I think at the time only 13 were known?) May 11 '20 at 17:36
• Huh! Well, it seems that you know what you're talking about here. I just don't know whether or not we have any more like you on the site yet. :)
– ifconfig
May 11 '20 at 17:42
• I'd also say you're probably better off in the Physics.SE. Maybe you have to set a bounty - if it is worth that much to you (but you sound like it is) - and if you have enough reputation. Maybe you can also post a link to your question here, so if there is someone here he/she might be able to help in some respects over there. May 11 '20 at 19:03
• That's good advice, although I would be very surprised if any of my friends or colleagues who are CFD peeps would have any intuition about this. Heat transfer is scalable, but specific. If you haven't done something similar, then your intuitions are off. Like, I can tell you all kinds of things about LN2, boiling, and evaporative heat transfer but I don't know anything at all about general orders of magnitude for outrunners. It really comes down to lots of flow interactions which you determine empirically or through complex modeling. My hope is to connect w/drone peeps who have done it. May 11 '20 at 23:04

My experience is that motor cooling does change enormously depending on how they are installed, but not three orders of magnitude.

My Funfighter motor initially got hot enough that a drop of water would sizzle off the outside, yet when flown without the cowl it could be merely warm at the end of a flight. Similarly I have a helicopter where the standard motor was too hot to touch at the end of a flight but a replacement with a built-in fan stayed cool. I've also noticed that you can put 100W through a 2204 sized motor on a pylon racer but not on a 3D model that prop-hangs a lot.

It's also worth considering that our flights are fairly short. Using a heat capacity of 0.4 J/g-C as an average of copper and iron, a 50g, 100W, 80% efficient motor would heat up 20*0.4/50 = 0.16 degrees C per second, or 10 degrees per minute. Even with no cooling, that would only raise the motor from cold to warm in a 3-4 minute flight. It's not "excruciatingly hot" because it has so much more mass than a light bulb and takes longer to heat up.

I guess this variability is bad news for anyone trying to estimate how much cooling their motor will experience, but it's good news for your average modeller who can just improve the airflow over an over-heating motor to fix the problem.

FWIW my rule of thumb for fixed wings is 50W for a 2204 motor, 100W for a 2208, 150 for a 2212 (based on the stator size, not HobbyKing's overall size numbers).

For quads, the peak power can be a lot higher because you only use it for short periods but the average power over the whole flight should be similar. 4s1500mAh battery over 5 minutes is 200W, or 50W per motor - just right for 2204 motors.

• Your math checks out, but keep in mind airflow is significantly different in a multirotor. A full punchout from hover can exceed 100 amps burst current, at 22.2V that's 500W per motor, with no (instantaneous) cooling. I'm no mathemetician - did you consider the decreased efficiency of hotter copper? Seems like a positive feedback loop that may contribute to temperature increase Sep 14 '21 at 13:27
• @BrydonGibson - I don't think we need to worry about hot copper, as the motors don't have time to heat up. Even in your extreme case, the spike in power is normally very brief and the heat capacity of the motor smooths it out. Sep 14 '21 at 13:45
• Emperically, motors land quite hot, especially when they are receiving lost of varying inputs due to gyro noise. 60-70deg C is not unheard of. It's not good (and indicates tuning issues), but it's absolutely possible Sep 14 '21 at 18:21