Vol. II · Heat Plate 2 of 11square law, thickness, cooking time, diffusion, potato, scalingHT

Twice as Thick, Four Times as Long.

Double the thickness and heat has twice as far to go — through layers that are each slower to warm. The time goes up fourfold.

Cooking times do not scale with size the way intuition says. A potato twice as wide does not take twice as long to boil; it takes about four times as long. The reason is in conduction: heat must travel twice as far, and the temperature difference that drives it is spread over twice the distance, so the flow is half as strong. Twice the distance at half the speed is four times the time.

This is the most useful single rule in cooking. It says why small pieces cook fast (stir-frying, IV.5), why a whole bird takes hours, why a thick steak needs gentler heat, and why a recipe's time cannot simply be multiplied when the food is larger. The workshop boils potatoes of different sizes and fits a line to their times.

Below: minutes for the centre of a potato to reach 90 °C in boiling water, against its diameter, on logarithmic axes; the slope of the fitted line is the exponent of the law.

What you will learn

State the square law for heating time.
Read a power law from a log–log plot.
Estimate a new cooking time when size changes.
Explain why small pieces cook quickly.

Workshop

Potatoes treated as cylinders of the given diameter, from 20 °C into boiling water; minutes for the centre to reach 90 °C (thermal bench). Grey line: a slope of exactly 2.

Why the square

A 20 mm potato reaches 90 °C at the centre in 5.4 min; a 40 mm one in 20 min, 3.7 times as long. Across the five sizes the fitted exponent is 1.92, close to the square law's 2. It falls a little short because a small piece also spends a larger share of its time simply having its surface warmed — but for any piece more than a couple of centimetres across, the square rules.

Use it both ways. To cut cooking time, cut the food: halving a potato's width cuts its boiling time to about a quarter, which is why diced potatoes boil in ten minutes and whole ones in forty. To scale a recipe for a larger roast, scale the time with the square of the thickness, not the weight — and then check with a thermometer (I.4), because shape, bone and starting temperature all move the number.

At the stove

Boil a small and a large potato together, starting from cold water so both start the same. Test each with a skewer every five minutes from the boil. Note the times; the ratio will be close to the square of the ratio of their widths.

The square law is the signature of diffusion: the same mathematics sets how far a drop of ink spreads in still water in a minute, how long salt takes to soak into ham (III.8), and how far a smell drifts in a still room.

Terms

Square law: Heating time to the centre grows with the square of thickness.
Power law: One quantity varying as another raised to a fixed power; a straight line on log–log axes.
Log–log plot: A graph with both axes logarithmic, on which a power law is a straight line whose slope is the exponent.
Characteristic length: The distance heat must travel to reach the centre: half the thickness of a slab, the radius of a cylinder or sphere.

Cooking time grows with the square of the thickness: twice as thick, about four times as long.

Sources

Y. Choi and M. R. Okos, "Effects of temperature and composition on the thermal properties of foods", in Food Engineering and Process Applications, vol. 1, Elsevier, 1986, pp. 93–101.