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I have a rather odd question that may just boil (sorry) down to a fairly simple problem I can't solve.

The scenario is this: A person is testing the intensity of a super-focused beam of sunlight in a solar forge by placing a jar of water in its focal point and checking the time it takes to boil. Assuming the jar contains 0.5 liter of water and the focal point of the forge is intended to bring steel to forging temperature (1000-1200 degrees celsius) in one minute, is there a way to determine how long it will take for the water in the jar to boil?

I did quite a bit of Googling but all I found was lots of people asking if cold water boiled faster than warm, which naturally wasn't helpful.

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    $\begingroup$ Is this a textbook problem (where we can make lots of assumptions), or is this a real issue? If textbook, just look at the "specific heat equation" and solve it twice for the two materials assuming same power delivered to both. If this is real, then you have to worry about power delivery efficiency and radiative heat loss which will make it much more complex. $\endgroup$
    – BowlOfRed
    Commented Apr 16, 2015 at 8:17
  • $\begingroup$ It is not. I'm a writer, and I just wanted to make sure whatever amount of time I had the character find be plausible. That said, I'm not helpless when it comes to math, so I will look for this equation and see if I can find an answer. $\endgroup$
    – user2712865
    Commented Apr 16, 2015 at 12:14
  • $\begingroup$ It doesn't appear that the specific heat equation includes units of time, which is what I'd like to find. $\endgroup$
    – user2712865
    Commented Apr 16, 2015 at 12:22
  • $\begingroup$ As an additional plausibility check, MIT's mirror array capable of melting steel is 12-ft wide. $\endgroup$
    – pentane
    Commented Apr 17, 2015 at 12:54

2 Answers 2

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As a physics problem in a textbook, you could get somewhat close. Both the water and the steel have a heat capacity that relates the amount of temperature rise that would accompany an input of energy. If you make a few assumptions, you can relate the two.

The problem with a real-world application is that those assumptions may be far from valid. The two biggest ones I see are whether or not the energy input to both substances is similar. Any part of the sunlight that reflects or goes through the substance does not contribute to heating. So if the two materials behave differently (glass jar? shiny metal?), then the energy input could be quite a bit different.

The other factor is the energy loss from the material. As uninsulated materials heat up, they release greater amounts of heat energy. That allows the temperature to drop and makes the calculations harder. As an example, a simple stovetop can boil a pan of water, but would never be able to melt steel. Instead all the energy it puts into a hot piece of metal would just be radiated away and the temperature would stop climbing. Your standard solar oven isn't very fast and they only have to reach a delta of a couple hundred K. Making a solar forge would be much more difficult, but if you get enough power, you can almost overwhelm the energy loss. A MIT Solar dish is able to heat things to very high temperatures.

Now, if you want to ignore those effects (say you have very, very good insulation so that you can ignore cooling for a bit), then you can just take the heat capacity equation and fill in some numbers. See for example Hyperphysics Specific Heat

$$Q = mC \Delta T$$

$Q$ is total energy, but here you want to relate the power of the beam over a period of time. So that can be changed to $$P \text{ }t = mC\Delta T $$

You don't say how much steel can be melted in a minute, so we can't plug in the numbers for the mass ($m$) (I hope it's not too much), but the rest of it you have. The change in temperature for the steel will be around $1000 K$, $t$ is less than $60s$ and $C$ for iron is about $450 \frac{J}{kg \text{ }K}$. That gets you the (minimum) power of the forge when you solve for $P$.

Turn it around and solve for $t$ with the jar of water, but there the water heat capacity is $4182 \frac{J}{kg \text{ }K}$ and the $\Delta T$ will be about $75K$ (assuming it starts at around room temperature).

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  • $\begingroup$ +1 nice answer, but I want to note that a solar forge capable of melting steel is certainly plausible and in fact was built at MIT in 2008. $\endgroup$
    – pentane
    Commented Apr 16, 2015 at 18:49
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Since the heat capacity of water is 10x that of steel, the same energy input to equal masses of steel and water will raise the temperature of the steel 10x more. So to raise a certain mass of water from 0 °C to 100 °C (boil it) and raise the same mass of steel from 0 °C to 1000 °C (melt it) will take approximately the same amount of time, assuming the same rate of energy input to both and no energy loss.

These assumptions are realistic for your solar forge if the water and the steel absorb the same fraction of the incoming energy (which you could attain by placing them in a black container and heating the container instead of the material directly) and if the masses are small enough that the heating happens quickly (~10 seconds) so energy loss is minimal. Just as an example, if the forge was capable of boiling 1 kg of water in 10 seconds it could melt 1 kg of steel in 10 seconds and (assumptions holding) 6 kg in 60 seconds.

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