I've stumped myself on what (I think) should be a very easy problem. I'm trying to come up with an expression for the temperature of a small amount of water as heat is added at a constant rate. The water is partially exposed to open air, so can lose energy to the surroundings. The total temperature change is 20 degrees or so over many minutes, so I do need to keep track of the loss.
The power is coming from a resistor, so the instantaneous energy change should be (I^2)R-k(T_water-T_air), where k accounts for exposed surface area, etc, etc and which I can determine as the water cools back down. This should be directly proportional to the instantaneous change in temperature, right?
I've baffled myself thoroughly enough that I'm currently just trying to determine two things. First, If I've described the instantaneous power in and out, should I be able to solve the differential equation easily by way of integration? Second, is it reasonable to try to extract the specific heat of water from a setup like this? That is, is there any way to fit a heating/cooling curve from a setup like this having only the specific heat as a free parameter without having very detailed information about surface area, airflow, etc?
If that was coherent enough to make sense of, I'd be extremely grateful for a kick in the right direction