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How would I calculate the time it takes to heat a given volume of water to a given temperature with a given temperature T1 when it is submerged in a volume of water with a temperature T2

Estimation is ok, as I know the warm body of water will lose heat to the surroundings, so some formula or explanation covering a closed system is okay.

Example: A bottle of 100ml water at 5C is submerged in a bowl of water holding 95C. How much time would it take to raise the temperature from 5C to 20C

Thanks!

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up vote 2 down vote accepted

There isn't a simple answer to your question.

Suppose you had a solid bottle rather than one containing liquid water, then you can solve the heat equation to describe the change in temperature with time. You'd probably need to do this numerically as only special cases like spheres would allow an analytical solution. However even then you can't answer a question like "how long does it take to heat to xx degrees" because the object isn't at a uniform temperature. There would be a temperature gradient from the surface to the middle.

With a bottle of water life is even more complicated because the water inside the bottle will develop convection currents as it starts to heat and these make your calculation even more difficult.

However you can probably simplify the problem if you assume that the water in the bottle is vigorously stirred and the water outside is vigorously stirred so that all the water is the same temperature. In that case the heat flow will simply be controlled by the thickness of the bottle walls and the thermal conductivity of the bottle walls.

If we simplify the system slightly and assume the heat flow through the bottle wall to be one dimensional then we can use Fourier's law for heat flow:

$$ \dot{q} = k \frac{T_w - T_b}{d} $$

where $T_w$ is the temperature of the water bath, $T_b$ is the temperature of the water in the bottle, $k$ is the thermal conductivity of the bottle wall and $d$ is the thickness of the bottle wall. To solve this we take the external temperature $T_w$ to be constant, and note that the change in the bottle temperature $T_b$ is equal to the heat transfered divided by the total specific heat of the bottle contents. Without going through all the details, solving this equation gives:

$$ T_w - T_b = A \space e^{-Bt} $$

where $A$ and $B$ are constants that contain contributions from things like the thermal conductivity of the bottle wall, the wall area, the wall thickness and the heat capacity of the fluid in the bottle. I've rolled everything into the two constants because to be honest calculating from first principles is hard and we'd usually just do a few experiments to calibrate the system. Once you've measured the constants you can use them to calculate the behaviour of the system for any initial conditions.

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Excellent answer! I'll attempt to measure the constants using the equation you provided, and see I'm able to get at least an approximation of the time required :) –  Yngve B. Nilsen Aug 21 '12 at 9:47
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