# Heat Temperature and my dinner

So today while revising for my QM midterm I decided to raid the cupboards for a well earned lunch, only to find nothing there but noodles and rice. So I put two metal pans of water on two pre-heated hobs (no idea what power) however one was smaller than the other in diameter (induction hobs). The rice pan had significantly more water in than the noodle pan and they both went on to the hobs within about 5 seconds of each other.

So I waited for the water to boil before putting any food in either of them.

I noticed that the noodle pan with the least water in (and on the smaller hob) formed lots of tiny bubbles first before the larger one formed any.

As time went on both started to form bubbles but the noodle pan had by far the most. All the while steam was leaving the system at an increasing rate "amount per second $g s^{-1}$.

I fanned it to see if changing the vapour pressure would change this rate but it didn't really have that much effect.

Then all of a sudden both pans at the same precise time boil simultaneously.

1) The size difference of the hobs and the volumes of the water should dictate that the larger one would take more heat since heat capacity is an extensive property. Therefore we should expect a slower rise in temperature no?

2) If the size of the hob does not matter then we would expect both pans to be receiving the same dose of heat per unit time and would then reach the same temperature a the same time (as per experimental results), also since temperature is an intensive variable the volumes of the water shouldn't matter.

• If it's an induction range, then "pre-heating" doesn't have any meaning (since the heating only occurs when there is a pan present) no? – Floris Nov 14 '14 at 14:01
• Hmm. To be honest I just come in and stick them on for like 20 minutes and then come back in and put the pan on. Never really checked to see if they are hot or not before. I guess this could save some time :p – user58536 Nov 14 '14 at 14:51
• Come to think of it ... How does the hob 'know" a pan is present? Because only certain types of pan will work! – user58536 Nov 14 '14 at 14:52
• The key concept is that in an induction hob, the heat is produced in the bottom of the pan itself, not in the hob. That's why induction hobs are more efficient than other types. – ntoskrnl Nov 14 '14 at 15:49

I agree with Andre that it was probably a coincidence that your two pans boiled at the same time; but I want to spend a little bit of time picking away at the physics, and specifically your assertions:

If the size of the hob does not matter then we would expect both pans to be receiving the same dose of heat per unit time

While you did not define "dose of heat", it would be reasonable to assume you mean "absorbed power" (J/s = W). If the hobs are designed to give constant power per unit area, this would mean the amount of power scales with the area of the pan; if the pans were filled to the same depth, then the rate of heating would be identical.

In this context it's worth describing the mechanism of an induction heater: a "high" frequency RF generator (I think the standard is 24 kHz) induces eddy currents in the bottom of the pan (which is why aluminum and copper based pans don't work well: the skin depth is too deep, and the currents flow through too much of the material and therefore don't experience enough resistance). Eddy currents scale with the size of the object - so power scales with the square (more current and more resistance). So if you produce the same rate of flux change in a hob, then pans with the same material properties (this is important!) will absorb the same heat per unit area.

and would then reach the same temperature at the same time (as per experimental results),

That follows from the above

also since temperature is an intensive variable the volumes of the water shouldn't matter.

No. The heat capacity scales with the volume of water. You don't put "temperature" into the pan, you put "heat". That is a very fundamental thermodynamic concept. You are not the first person to get confused by the use of intensive and extensive properties - and you won't be the last.

Finally - how does the heater "know" that there is a pan on the range? Well - when you apply an AC voltage to the induction coil in the absence of a pan, the current will be approximately in quadrature with the voltage (because of the inductance of the coil). This means that while power is stored in the magnetic field during peak current, no net power is lost. When you add a pan, the eddy currents induced act as a loss mechanism, and change the phase relationship between current and voltage - and so power is extracted. The hob doesn't need to know whether a pan is there (although it would be easy enough to measure the phase difference) - there is just no mechanism to extract heat (efficiently).

You should do the experiment for yourself some time: two identical pans, identical amounts of water. "Pre-heat" one hob, then place the two pans at the same time. See when the water boils. Repeat a few times. Convince yourself there is no statistically significant difference.

• 24 kHz is barely out of the audible spectrum – hardly RF. :) – ntoskrnl Nov 14 '14 at 15:50
• @ntoskrnl - long wave radio is considered to go down to 3 kHz - submarines use very low frequencies (to have a chance of getting through the highly conductive sea water). As long as it transmits, it's "radio". And my ears cut out around 8 kHz, not 20. The joys of old age. – Floris Nov 14 '14 at 16:08

I believe it's undisputed that bigger hobs emit more power.

Thus, both affirmations are wrong. As the volume of water increases, so does time it takes to boil, because, as you said in #1, more volume equals bigger heat capacity. However, a larger hob emits more heat, thus compensating a bit for that effect.

It's just a coincidence. You managed to compensate exactly for the stronger hob with a larger volume of water. Congratulations :)