# Is there a formula that describes the relationship between the rate of change of temperature and surface area?

i.e. $$\frac{dT}{dt}$$

I'm doing an experiment in which I've changed the exposed surface area of a certain volume of boiling hot water and measured the temperature for over a period of 5 minutes as it cools. I got a linear relationship between rate of change of temperature and surface area but I can't find a formula that confirms (or rejects) my results.

• You might be looking for thermal conductivity? Commented Dec 6, 2020 at 3:36
• Perhaps? I don't know the power though. Do you think the work done on the water would remain constant ? Commented Dec 6, 2020 at 3:45
• In the case of cooling water, most of what you're looking for in terms of energy transfer will be heat, not work. You don't really need to know power if all you're looking for is a proportionality statement, in which case Fourier's Law will suffice. Commented Dec 6, 2020 at 4:02

$$\frac{\mathrm{d}T}{\mathrm{d}t} = -\frac{hA}{C} (T_{obj} - T_{env})$$.
Where $$A$$ is the surface area of the object exposed to the cooler temperature, $$h$$ is a constant, and $$C$$ is the heat capacity of the object.