Need A Reminder On Transformer Temp Ratings

[EmagSamurai]

Can someone remind me why going from 55C to 65C transformer temp rise yields a 12% increase in capacity? That is, 55C to 65C is an 18% increase in temperature, but a 12% increase in capacity. I feel like I've seen a simple explanation of this somewhere along the way, however, either due to lack of sleep or lack of coffee, I can't remember what that was at the moment.

 

[jim dungar]

It is about heat transfer. The relationship between heat dissipation and ambient temperature is not linear.
The heat generated by a transformer is due to its losses. Winding (load) losses go up by the square of the increase (112% loading could yield 125% heat output)

 

[synchro]

Following through on what Jim said, the approximate nonlinear relationship of temperature rise in °C to watts dissipated in the relevant temperature range is shown at the link below:

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It shows that (temperature rise in °C) is proportional to (watts dissipated)⁰·⁸ .
After taking each of these two quantities to the 1.25 power to eliminate the 0.8 exponent on watts, we get:
Watts dissipated is proportional to (temp rise)¹·²⁵ .
So (watts @ 65°C) / (watts @ 55°C) = (65/55)¹·²⁵ =1.23
But as Jim said the watts dissipated in the windings is proportional to the square of the current.
So we take the square root of the watts increase to get the amount of increased current allowed at 65°C:
√ (1.23) = 1.11 or 11% higher output current from the transformer. Since the voltage has not changed the power from the transformer is also 11% higher, which is very close to the 12% that the OP mentioned.

 

[EmagSamurai]

The above is along the lines of what I remember, and it boils down to the heat being proportional to the power dissipated (which as you have both pointed out is related to the square of the current). I was thinking it had to be something relatively straightforward because the 112% seems to apply to all liquid-filled transformers regardless of the construction and all the other myriad variables that would impact heat transfer.

But maybe I'm getting this confused with something else.