Sous vide: why do cylindrical / cylinder-like and spherical / sphere-like foods cook faster than slab-like?

Question

The Douglas Baldwin's "A Practical Guide to Sous Vide Cooking" has charts for sous vide cooking times. Unintuitive to me, cooking times are drastically shorter for spheres and cylinders than slabs. Does anybody have an explanation?

Answer 1

A picture really helps here:

There's your slab, cylinder, and sphere with the same thickness.

The time to cook does depend on thickness, since that's the distance the heat has to travel from the surface to the interior. But it also depends on the ratio of surface area to volume (and thus shape): the more surface area per volume you have, the more area you have to transfer heat compared to the amount of heat you need to transfer to fully heat the meat.

So unsurprisingly, at constant thickness, a slab takes longer than a cylinder, which takes longer than a sphere.

Of course, the typical thicknesses of the various shapes are different, so it's not all that meaningful to compare things at constant thickness. A slab might be a fish filet, a steak steak, or many smaller pieces of meat packaged together in a bag - nothing all that thick, unless it's a really full bag. A cylinder might be a pork tenderloin or a decent approximation of a chicken breast - still not that thick. A sphere might be a single steak medallion, or maybe an okay approximation of a large pork shoulder - a pretty wide range.

That said, in the end we often think about things in terms of the amount of meat, i.e. the volume. A 2 by 10 by 10 cm slab, a 5 cm diameter 10 cm long cylinder, and a 7.25 cm radius sphere all have about the same volume. According to that table, the slab needs 1.25 hours, the cylinder needs 2.75 hours, and the sphere needs 4 hours. So if you were thinking of similar quantities of meat, perhaps that matches your intuition better.

Answer 2

So, I'm expanding my comment into an answer.

I think it's possible that the chart was assuming different volumes for the different shapes, since only one dimension (thickness, which is usually the thinnest dimension) was listed on the chart.

So, for example, given a piece with a centimeter's thickness, the resulting sphere is a little nugget of one centimeter in all directions, with no wiggle-room, since spheres are symmetrical. A cylinder of the same thickness has its diameter defined, since width and thickness are the same in a cylinder, even if its length can be more variable - so a cylinder one centimeter thick, can easily be three centimeters long, and thus starts with about three times the volume of it's spherical "counterpart".

And then there's the slab. With only the thickness given, it can be considerably larger in both length and width. Looking at a one-centimeter thickness, we can start at three centimeters wide to match our previous three long, and three wide to make a flat slab instead of a strip, for nine times the volume of a one-centimeter sphere and three times the volume of a one-by-three centimeter cylinder.

As rumstscho's comment mentions, the difference must be rather visible for the shape to be considered a cylinder rather than a sphere, or actually a strip vs a cube (give or take irregularities and/or rounded edges), so it must have a longer length to qualify. Same for a slab rather than a cylinder - it must be visibly larger in two dimensions, or else it will look more like a strip or chunk (square-ish or spherical).

It is rather intuitive that a larger piece, with more volume and more mass, will take longer to cook, longer for the temperature to equalize form the edges in. Of course, that makes the "intuitive" problem the chart setup, which is comparing pieces of rather different sizes based on thickness alone - it might make a bit of sense for the rate of diffusion to be based on the thinnest dimension, but the results are, as you said, counter-intuitive since I didn't see anything mentioning that assumption. The time should be the same if the other variables like absolute size didn't matter, but clearly they have to be taken into account somehow. It might have made more sense to compare profiles for pieces of the same volume, or make a chart for each shape, or just mention the assumptions being made or something.

For the same volume, the same mass, I would expect a thin flat slab to heat up much faster, and a sphere or cube to heat slower, because of the ratio of surface area to volume. But once we're talking different amounts of mass in each shape, yes, I would expect the same volume of food to cook faster as diced small cubes, as opposed to cut into longer strips or left as fewer, wider planks, simply because the pieces are smaller and there are more cuts, therefore more surface area.

Answer 3

Physicist here. There's a very simple explanation.

Heat (as in energy) can only transfer through the surface of the food. Spherical objects have the same distance to the center of the sphere from all sides, but in a slab, the distance to the center is much shorter top and bottom than it is from the side. (For a 2" thick slab, it's 1" to the center from the top but might be 4" to the center from the side.) Because of this, all of the energy required to cook the center effectively comes from the top and bottom. Very loosely, you are eliminating about half the surface area through which energy can flow, so it takes about twice as much time to reach the same temperature.

Answer 4

As someone who took a course (years ago) specifically on heat transfer, I'm pretty sure Baldwin just used the standard modeling equations that assume an infinite slab and an infinite cylinder. That is, the model assumes that a horizontal slab only absorbs heat through the top and bottom, a cylinder only absorbs heat through its curved sides (not any "top" or "bottom"), and a sphere absorbs heat through all directions.

Or, to put it another way, what these "shapes" really mean is how many dimensions one assumes heat can be coming from. In the slab case, heat can only travel in one dimension through the slab. In the cylinder case, it travels in two dimensions. And in the spherical case, it travels in three dimensions, which is the reason for the shorter cooking times.

To those answers and comments speculating that Baldwin is assuming a larger volume for slabs and cylinders, I suppose that's correct, as the equations in general assume an infinite volume. (And the reason why the equations make these assumptions is because the math gets really hard, really fast if you want to model more complex shapes.) Obviously, real food doesn't have infinite volumes, so Baldwin's calculations should only be taken as an approximation.

There are methods using the equations to approximate other shapes, which Baldwin mentions briefly. Effectively, what one does is assume a non-integer number of dimensions for heat travel. For example, an infinite square prism (like a cylinder, only with square sides) would have more than the single dimension of heat travel assumed by the slab, but less than the equal travel from all directions in two dimensions assumed by the cylinder (because of the corners of the square, which means heat will take longer to travel toward the center). So, one can plug in "1.7 dimensions" and get something that's closer to a cylinder than a slab and get a number that's a reasonable approximation to how other shapes behave.