Math is not my best subject-what's the formula for figuring this out?
We're looking at the rate energy is being consumed, which is where the heat is coming from. This is power. If 150 lbs is dropping 3 ft per second, that is 450 foot pounds per second. This is the energy, per second, going into the friction zone. It is exactly the same power that would be required to RAISE 150 lbs 3 feet per second (or to raise 450 lbs 1 foot per second).
I remembered that one horsepower is about 550 ft pounds per second. I also remembered that one horsepower is about 750 watts. I then made a conservative estimate that 450 foot pounds per second is about 500 watts. It is actually over 600.
So imagine a 500 watt light bulb buried inside your knot. It won't take long for that knot to get hot.
I have oversimplified. Half the heat generated from descending actually starts in the rope, not the knot. Since the knot is much hotter than the rope, heat is flowing from the knot to the rope. The rope is carrying away not only its own half of the initial heat energy, but another unknown amount conducted from the knot.
Even so, the knot gets very hot. The lesson I take from all of this is that the knot/rope friction setup we use is such a poor dissipator of heat it cannot keep up with a steady input of 500 watts or so.
I have a rappel rack I use for descending SRT. It is the only friction element present. It also gets hotter than the rope. But the metal is a much better conductor and radiator of heat than a knot made of rope, so I find in practice it can easily keep up with much faster descents than I can manage with a prusik knot.