If I record the time it takes to boil a known amount of water with heat at a known temperature, can I extrapolate for an unknown temperature by comparing how long it takes to boil the same amount of water? That is, if the water boils in half the time, is the temperature twice as high?

For example, I have a cast iron skillet and a thermometer which goes up to 400F. I apply temperature to the skillet such that it’s stable at 300F. I put a pot of water on the skillet and record that it takes S1 seconds to boil the water completely away. I turn the heat up under the skillet to where it is very hot and is over the max of what my thermometer can read. I then repeat the timing of boiling away a pot of water and record that the time to boil with the higher heat is S2. Can I use the ratio of S1 and S2 along with the known skillet temperature from the first trial of 300F to determine the temperature of the skillet in the second trial?

It’s not going to be exactly proportional to the absolute temperature. But, there is Fourier’s law of heat conduction, which says you can expect the rate at which heat flows to be proportional to the difference in temperature between the hot skillet and the water. Of course there are simultaneously other effects at work but you can start to work it out from that.

The short answer is no, at least not so easily. Of course you could estimate temperature of the heat source based on time to boil, but the relationship isn’t going to be linear. If memory serves, heat loss or heat gain requires a logarithmic function to solve for. So even if you do double the temperature of the heat source, well, first of all, Fahrenheit is not an absolute scale (zero Fahrenheit isn’t actually “absolute zero”), but even discounting that (and considering it’s actually the temperature difference between the heat source and the heat sink that matters anyway, so the scale doesn’t really matter), the point is the relationship to solve for time to heat or cool using something like Newton’s equation is logarithmic. Which means that doubling the temperature difference doesn’t necessarily halve the time.

Yeah, the complication here is that it depends on the temperature difference, but that will vary with time, as the water heats up. Like, if your source is at 300ºF, and the water starts off at 70ºF, then that’ll initially be a 230ºF difference, but by the time the water reaches a boil, it’ll only be an 88ºF difference. We could instead use a heat source at 530ºF, which is initially twice as hot, but that one would end at a temperature difference of 318º, over three times as hot.

It’ll be close to linear if we have a much hotter source, say in the thousands of degrees, because then the change in temperature of the water won’t be as significant. But it still won’t quite be completely linear.

The boiling water is essentially at a fixed temperature. Since it’s sitting on the skillet, the skillet is essentially at a fixed temperature. By measuring the time it takes from when the water starts boiling, to when it boils dry, you can estimate how much energy you are using (electricity, gas), how fast you are using it, and how much it is costing you.

Lets first calculate the heat stored in the skillet

1> Lets assume a standard 12 inch skillet which weighs 8 lbs (3.63 kg)
2> Lets take a temperature drop from 400 F to 212 F (204.5 C to 100 C) (skillet temp to boiling water temp)
3> The heat released by the skillet is 540x3.63x104.5= 204,841 J or 194 BTU (Specific heat of cast iron = 540 J/kg.C)

Assuming that the water pot was already at boiling temperature, this heat will boil 204,841/2,260,000 = 0.09 kg or 0.2 lbs of water (Latent heat of water = 2,260,000 J/kg)

Lets assume you have a quart of water in the pot which makes it 2 lb. All it will take is 10% of the water to boil over before the skillet will reach 212 F (100 C)

If your skillet was at 588 F (309C), it will need 0.18 or 0.4 lbs of water to evaporate before the skillet gets to 212 F.

So if you can measure the weight of water evaporated until the skillet gets to 212 F, it will be a good indication of the initial temperature (assuming that you put an already boiling pan of water on it). In place of measuring weight, you could also measure the level drop in the pan. Double the level drop = double the temp difference of the skillet from 212 F.

Adding energy to the mix from the burner will just complicate the equations

Heat capacity doesn’t seem to matter here. There is a heat source, which is continually maintaining the temperature of the skillet. It’s not like the skillet is being heated and then left to cool off by boiling away water. It seems like a pretty straight use of Newton’s equation for heat transfer is appropriate.

In terms of heat transfer and the time to bring water to boil, how does one account for the shape of the vessel, if at all?

From my experience in the kitchen, if I put a liter of liquid in a wide, open skillet over high heat, it comes to a boil much faster than if the same amount of liquid is in a comparatively taller and narrower vessel. I assume this reflects the direct contact between the liquid and the bottom of the pan, and the corresponding amount of possible heat transfer. The bottom of the skillet is larger, therefore more direct contact with the liquid, and more heat transfer from the metal.

I presume that there is some inverse factor in terms of heat transfer from the stovetop to the vessel that mitigates the effect — i.e. a gas burner will take longer to heat a large skillet than a small narrow pot, ergo the available amount of heat to be transferred from vessel to liquid is reduced — but how closely do these typically track?

TLDR: Based on my time in the kitchen, there’s more to determining boiling time than simply the amount of water and the level of heat under it; the shape of the vessel matters too.

Edit to add: My current kitchen has an induction hob, which heats the cooking vessel much more quickly and efficiently than either gas or electric; this may also be a factor in my experience.

A lot of this is due to the size of the burner / hob vs the size of the pan.

For a gas hob, fire up the side of the pan is much less effective than fire applied to the bottom of the pan. So a small diameter pan will be limited to a small diameter fire. Whereas a large diameter pan can be placed over a larger diameter fire. Since the fire diameter is connected to how much you turn up the gas flow, the pan diameter becomes a limitation on the flux heat applied.

A non-induction electric stove has similar considerations. Each burner is a different diameter and each has a different electrical wattage at the maximum setting. A burner larger than the pan wastes the majority of the heat that’s emitted around the pan. And a burner smaller than the pan may lose heat to the surrounding unheated surface if the stove is a flat-top. And in general the smaller burners will be the lower-wattage ones.

Many complications.

Back to the OP: It sounds like you have a problem you want to solve, then you decided how to solve it, then asked us for advice on implementing your proposed solution. I wonder if you might do better to tell us the actual top level problem you’re trying to solve and maybe we can offer a different better solution. This is commonly called an XY problem.

The impetus for my question was from wondering if there was an at-home way I could approximate the temperature of a very hot skillet used to sear steak on a BBQ grill. That got me thinking about maybe using the relative time of boiling water on different heat levels as a rough way to estimate the temperature of the very high temperature skillet. Then that got me wondering about the general question about if the time to boil water is linear depending on the temperature of the heat source. It sounds like that’s not the case and would take some complex calculations to even get a very rough estimate of the 2nd temperature source. But it’s not like the answer is needed to cook my steak properly or anything. It was really just idle curiosity that got me here.

You could get an empirical, tabulated solution to your problem, without having to do calculations: That is, you rent or borrow a standard temperature-measuring device, like an infrared thermometer, for a day so you can do experiments. Then you record “When the pan is at 500º, it takes 10 minutes to boil the water, and when the pan is at 550º, it takes 8 minutes”, and so on. Then, if the water takes 9 minutes to boil, you know the pan is somewhere in between 500º and 550º.

But that probably still wouldn’t be a good solution to your problem, because after the water boils, what you know is what temperature the pan was at. But now it’s probably cooler, and you can’t go back in time and put your steak on the pan before you cooled it with all of the water.

You can mitigate this by using a smaller amount of water, so the change will be small. And one common method for gagueing the temperature of a pan is to add a few droplets of water to it: When it’s hot enough, the droplets will “dance” on the surface on top of a vapor barrier. That’s often used as a standard for when a pan is hot enough to properly sear steaks.

I have seen crepe makers get a good feel of the temperature of the pan by wiping the pan with a wet cloth and seeing if the cloth glides on the pan or how quickly the water vaporizes.

It’s the Leidenfrost (sp?) in effect which happens around 380F. This is also the temperature at which all pans become “non-stick”.

I have seen blacksmiths get a sense of the metal temperature by sprinkling a little water on it too.

Yeah, for cooking it’s generally straightforward to have some tricks to know when the surface at the right heat for what you’re cooking, but that doesn’t necessarily convey the actual temperature. Through experience you may know that when the water dances on the pan a certain way it’s ready for use, but you don’t necessarily know what temperature the pan is at. The same with the skillet on my grill. I crank the heat to get the pan as hot as possible by letting it sit over the burners for a while. After some minutes it’s at the max temperature and won’t get any hotter, but I don’t actually know what that temperature is. For cooking I don’t really have a need to know what the exact temperature of the pan is, but I’d still like to know to satisfy my idle curiosity. I may see if I can get an infrared thermometer just to find out for my own satisfaction.

I think the water boiling approach is off; that is, the water will boil at its boiling temperature (about 100 C), and if you know a temperature somewhere else in the system, what you have is an index of the resistance to heat transfer.

An infrared thermometer would be easy to use, and I bought one at Home Depot for maybe $29. I also like naita’s suggestion.

You might get an idea by trying to melt a lead weight, a bit of solder, or similar reference substances in your pan. You’ll want to clean them away thoroughly, as most such things are toxic, but they’re not so toxic that we don’t make plumbing and pewter dishes out of them.