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Suppose you have two otherwise-identical pots of different radii on two identical burners, both large enough to heat the entire bottoms of both pots. If you want exactly 1L of boiling water (say, 373.15K) in each pot, to which pot should you add more water?

Essentially, my question is about which effect is greater: the increased evaporation for a larger surface area vs. the increased rate of heat transfer over that surface area in the bigger pot (decreasing the time required to boil the water, at which point measurement ends). Would a pot large enough to heat each molecule of water at the same time be less water-efficient than a pot 1 molecule wide, which can only allow 1 molecule of water to evaporate at a time?

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  • $\begingroup$ Hint: Which one provides greater contact surface area between the heating element (or flame) and the pot? $\endgroup$
    – The Photon
    Jul 19, 2017 at 14:34
  • $\begingroup$ It probably makes more sense (and is a bit simpler) to demand that the two pots have identical mass rather than volume, but this is an interesting question $\endgroup$
    – AGML
    Jul 19, 2017 at 14:54
  • $\begingroup$ Mass would be simpler, but I was also curious specifically about the volume aspect. I'd accept a mass-only answer as well. $\endgroup$ Jul 19, 2017 at 15:00

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I came up with the following rough qualitative argument. Throughout I assume that both pots are surrounded by identical air, and in particular, identically humid air. This last assumption could be problematic, but without it the answer will depend strongly on the details of the room etc.

The evaporation rate (kg / time) will depend linearly on the number of molecules exposed to the air (moles), which will itself be linear in the surface area exposed to the air (m^2): $M/t \propto A_\mathrm{top}$, $M$ being the total mass of the water, $A_\mathrm{top}$ being the area of the top of the pot.

To bring the pot to boiling we need to deliver a certain amount of energy (Joules) to the pot from the burner. The total energy needed is linear in the total mass (kg) of the water. With temperature held constant the mass is linear in the volume (m^3): $Q \propto M \propto V$.

The heat transfer rate (Joules/s) obeys $Q/t \sim A_\mathrm{bot} \Delta T$, where $A_\mathrm{bot}$ is the surface area of the bottom of the pot and $\Delta T$ the temperature difference between the burner and the water. I assume the pot is slowly stirred so that convective transfer through the water is essentially instantaneous. Dropping this assumption complicates things considerably, but I'll try and relax it later.

It will take about $t_\mathrm{boil} \sim Q/ A_\mathrm{bot} \Delta T$ seconds to bring the pot to boiling. The mass of water lost during this time will then follow $M \propto A_\mathrm{top} t_\mathrm{boil} \sim A_\mathrm{top} Q / A_\mathrm{bot} \Delta T$. For a cylindrical pot with $A_\mathrm{top} = A_\mathrm{bot}$ the area should thus drop out, and both pots should require equal amounts of water to have equal volume at the boiling point (at the boiling point, as at any constant temperature, the volume of water in the pot should depend on the mass in the same way for both pots).

Roughly speaking, the rate of convective heat transfer between two fluid elements with an interface area $A$ is also linear in $A$, so this argument should hold to low order even if convection starts to be important. I'm not sure what will happen once say convective currents and asymmetries start to be the dominant effect, so this won't be true of any two pots probably. For example the ocean would probably evaporate differently than a column of water the size of your pinky finger with the volume of the ocean.

Edit:

I failed to take into account heat loss from the tops of the pots. Given that both pots are surrounded at all times by "the same air", this changes the heat transfer rate to $Q/t = A_\mathrm{bot} \Delta T_1 - A_\mathrm{top} \Delta T_2$, where $\Delta T_1$ ($\Delta T_2$) is between the burner and the water (between the air and the water), and so the area will still drop out of the quotient. It seems to me it still will, even if the possibly time-dependent humidity and temperature of the air are taken into account.

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  • $\begingroup$ I think the convection aspect is an important difference between the pots, especially for unidirectional heat (consider the extreme cases I mentioned). Also, would the turbulence that makes stirring an effective way to increase the speed of energy transfer also increase evaporation by disturbing the surface and increasing its area? $\endgroup$ Jul 19, 2017 at 16:38
  • $\begingroup$ You could conceivably stir the pot slowly enough, or from the bottom, that the surface remains effectively unbroken. I agree that the convection probably matters here in practice but I don't think there's a simple answer to this question once it is taken into account. I also failed to take into account the heat loss from the top of the pots; I'm adding an addendum to account for that. $\endgroup$
    – AGML
    Jul 19, 2017 at 17:31
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Very qualitative. Assuming liquid/vapour equilibrium is not reached, that is air is sufficiently dry, in a given time more water molecules will escape from the larger pot. This effect should take place independently from heating the container or not. Bottom counts only if the pot are not cylinder but bottle shaped. I would conclude that you need less energy to finally attain 1 l boiling water if the recipient is small. I assume no heat loss as well as burner are identical in term of heat transfer (electrical plate at same T and fitting the bottom ).

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  • $\begingroup$ The boiling aspect provides the time limit, so it's not irrelevant: the bigger pot takes less time to boil the same amount of water, so there is less time for evaporation to occur. I am not taking energy into account for efficiency, only water. $\endgroup$ Jul 19, 2017 at 16:22
  • $\begingroup$ @Tranquilled. So let me say you want to minimise the initial amount of water, arrive at 1 l boiling water doesn't matter how much energy you need? I ll try to think about it, is interesting :) $\endgroup$
    – Alchimista
    Jul 19, 2017 at 18:03
  • $\begingroup$ I'm thinking of fixing the system (under the assumptions of the two answers plus additional ones such as constant specific heat). One fix is that the energy of the final state is the same. Problem is the continuously changing amount of water. Cannot find a simple system of equations. In case you are still thinking about I MENTION here a term missing in the previous discussions: At the top heat is removed by the evaporation of the x grams of water. Latent heat of water is quite high. May be hint is useful. I am tempted to say that, for a given pair of pot, one bigger and one smaller, faster $\endgroup$
    – Alchimista
    Jul 19, 2017 at 21:52
  • $\begingroup$ you heat closer you get (or eventually bigger is the water saving) to your goal of loosing less water in the big pot. $\endgroup$
    – Alchimista
    Jul 19, 2017 at 21:54

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