A coffee related physics question.

I’ve got a traditional style espresso maker (the kind you put on the hob).

Anyway, given that I can get the exact temperature of the hob (I have an IR thermometer to hand) and I can measure the amount of water and the surface area of the device, is there an equation I can use to estimate how long it’ll take for the water to start boiling so I don’t have to keep an eye on it to avoid the pot boiling dry?

Also would depend on atmospheric pressure and the pressure inside the machine.

This sort of measurement is best done experimentally.

Not exactly. The temperature of the hob is not a useful guide to its energy input capability. The area under the pot will be cooled by the pot’s presense. The hob won’t be a constant temperature device (even it it is thermostatically controlled) - at least not enough to be useful in such an estimate.

What you need is the power of the hob and an estimate of the energy loss (convection mostly) in the system. Condutivity of the hob to pot is probably not an important issue, but in principle should also be facored in. Then you need to work out the specific heat of the system - which allows you to work out when it reaches boiling point. So, mass of the pot, knowledge of the material the pot is made of (probably aluminium) plus volume of water. This still isn’t brilliant because the pot won’t heat evenly. Because the specific heat of the water will dominate anyway, you could probable get a good estimate by simply working out the power input and the specfic heat of the water. Easy. But not exactly accurate.

In truth the simplest, and by far and away most accurate, thing to do is to time it once. That takes into account everything. You are already assuming that from one use to the next the system is reasonably constant.

What is a “hob”? Is it a “stove burner”?

If it’s electrical, its energy output will vary as the square of the change in your supply voltage, which could certainly change by 10%. This would cause your time to boil to vary by about 20%. I am using the approximation that for x<<1, (1+x)^2 = 1 + 2x. This would mean a voltage meter near the stove would be a big help.

The part of a food preperation unit (typically an oven) that you put something on top of rather than into or below. Also known as the stove.

Do a number of experiments and plot Time to Boil vs. Water Volume on an Excel graph, and then find a trendline equation with an R-squared > 0.98 for a confident fit; et voila!
If you really want to be fancy, factor in water starting temperature, air temperature, air pressure etc. and make a pivot table.

Alternatively, do what some of my colleagues do, which is fill a kettle right to the top from the hot tap (fed from a roof tank with dead pigeons in, probably), set it to boil, and then f*&@ off for 20 minutes.