# What is the equation for temperature of water over time?

1. Apr 5, 2013

### chrisphd

Imagine I am trying to heat water from room temperature to 100 degrees Celsius, with a heating element of constant power.

One would expect that for low temperatures, Temperature vs. Time graph to be linear. When I did the experiment, the result however was linear up to about 60 degrees, but then it started to look more exponential. I expect this is due to Newton's law of cooling between the water and the surroundings, and more complicated factors as the water asymptotic to 100 degrees at boiling point.

How should I model heating water to the boil. Using T = 100 - Ae^-kt, did not fit my data well. What equations can used to model this phenomenon from 0 to 100 degrees.

PS. If you need to use multiple equations for different portions of the curve, that's fine by me.

2. Apr 5, 2013

### Omega0

So you are interested in curve fitting? Or are you thinking about the theory behind that? I mean what is a "heating element"? There is a big difference between a system which is slowly warmed up (dT/dt << 1) and say a immersion heater or a nuclear bomb - and this system depends from some factors.

3. Apr 6, 2013

### CWatters

Is the water being stirred?

4. Apr 6, 2013

### Mandelbroth

The relation $Q=mc\Delta T$ only gets you so far sometimes in the real world. If it is at constant power, P, we know that $P_{heating}=\frac{dE_{heating}}{dt}=\beta$, for some constant $\beta$, implying that $Q=E_{heat}=\beta\cdot x$. However, the relation assumes that c is constant and that no water leaves the system. If you are not mixing, then the water might heat unevenly, causing some water to be lost in the form of water vapor while other parts are left at a lower temperature. As a fun little aside, if we wanted to model heating without stirring, that would cause temperature to be a function of time AND position. Thus, we can use the Fourier heat equation, given by $\frac{\partial T}{\partial t}=\alpha\nabla^2T$. Looks pretty simple, right?

Since specific heat of water doesn't change much between 20°C and 100°C, my thought is that as time is passing, your water is losing mass, thus causing the rate of heat change to increase drastically. Remembering that temperature of a substance is related to the speed of its particles and that higher mass implies higher inertia, this kind of makes sense.

5. Apr 7, 2013

### QuantumPion

How are you heating the water? As you approach the boiling point, steam bubbles can form on the surface of the heating element, insulating the heater from the water and vastly reducing the heat transfer rate. This phenomenon is called nucleate boiling.