Modernist-Cuisine-Vol.-1-Small

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For more on the thermal characteristics of
common kitchen materials, see From Pan
Bottom to Handle, page 280.
THE HISTORY OF
Fourier and the Heat Equation
cook turns the burner up or down yet not be so
sensitive that it fails to hold a stable temperature
despite minor fluctuations in the heat source. To
put it in scientific terms, the heat capacity of the
material is just as important as the conductivity of
the cookware. Manufacturers don’t advertise the
heat capacity of their wares, unfortunately, and it’s
a little tricky to calculate because you need to
know the thickness of the bottom, the specific heat
of the material it’s made from, and its density.
Density is surprisingly important. Consider
aluminum, which has the highest specific heat of
any material commonly used in cookware. That
means you must pump a lot of energy per unit of
mass into aluminum to raise its temperature. Yet
aluminum is famously fast to heat. Why? The
reason, in large part, is that the metal is lightweight;
it has a low density and thus a relatively
small amount of mass to heat.
In the early 19th century, the French mathematician Jean
Baptiste Joseph Fourier developed a formula that describes
how heat travels through solids by conduction. Now known
simply as the heat equation, Fourier’s elegant discovery has
contributed to advances in modern physics, chemistry,
biology, social science, finance—and now cooking.
The heat equation helps to answer a question chefs often
ask themselves: is it done yet? What we want to know, in more
technical terms, is how the heat is distributed in the food
we’re cooking. The answer is
∂T ∂t =α 2T
In this equation, ∂T ∂t represents the rate at which temperature
is changing with time, 2 T is the temperature gradient in
the food, and α is the thermal diffusivity of the food (a measure
of how fast heat spreads in that particular food at
a particular temperature).
The heat equation tells us that the steeper the temperature
gradient between the inside and the outside of the food, the
faster heat will flow to its interior. Our instincts tell us that,
too—but our instincts don’t tell us the actual temperatures in
specific parts of the food at exact times. Fourier’s model does.
Or rather it could if the complexity of food did not defy our
ability to model it mathematically. Solid foods typically
Cast iron, in contrast, has a low specific heat,
half that of aluminum. By that measure, you might
expect it would be easy to heat. Instead a cast-iron
skillet warms slowly and delivers remarkably even
heat because it’s so dense and thus heavy.
Fortunately, there is a single measure that takes
into account all three of the properties that matter
in cookware: conductivity, specific heat, and
density. It’s called diffusivity. Diffusivity indicates
how fast a material transmits a pulse of heat.
This all-encompassing trait gives rise to the
macroscopic behavior that we praise or condemn in
our pots, pans, and utensils. People say that copper
cookware “conducts” heat well, and in fact copper
is an excellent conductor. But what they actually
mean is that its high conductivity and low specific
heat are balanced by considerable density. They
mean that it heats not only quickly but also evenly.
They mean, in a word, that it has high diffusivity.
consist of an elaborate assemblage of different substances
with different heat-transfer properties. Heat moves differently
in muscle, bone, and fat, for example. And each piece of food
has its own unique patterning of components. It would take
extraordinary effort to represent those individual patterns in
a heat-transfer model. Fortunately, even simplified models
that provide approximate figures can be very useful to cooks.
Conduction in Food
Conduction is the slowest form of heat transfer.
It’s especially slow in food, in which the structure
of cells thwarts the movement of heat. The thermal
diffusivity of food is typically 5,000 to 10,000
times lower than that of copper or aluminum!
Hence conduction, more than any other means of
transferring heat, is the rate-limiting step that
determines the cooking time for solid food.
For that reason, it’s a good idea to understand
how the geometry of food affects the conduction
of heat. Yes, we said “geometry:” the rate of heat
flow in a solid food depends not only on the size of
the food but also on its shape.
Generally, when cooking, you want to move
heat to the core of the foodor at least some
distance into the interior. And you’re usually able
to apply heat only directly to the surface of the
food. Heat conducts inward slowly, so the outside
warms faster and sooner than the inside.
Most chefs and home cooks develop an intuition
for how long a given cut of meat, say, needs to
sizzle in the pan. Trouble arises when a cook tries
to use that intuition to estimate a cooking time for
a larger or smaller cut, however, because conduction
scales in counterintuitive ways. A steak 5 cm
/ 2 in thick, for example, will take longer to cook
than a cut that is only 2.5 cm / 1 in thick. But how
much longer? Twice as long?
That’s a good guessbut a wrong one. In fact,
the thicker cut will take roughly four times as
long to cook. This scaling relationship comes from
a mathematical analysis of an approximation to the
Fourier heat equation (see previous page).
So the general rule for estimating cooking times
for flat cuts is that the time required increases by
the square of the increase in thickness. Two times
thicker means four times longer; three times
thicker means nine times longer.
This scaling rule breaks down, however, when
the thickness of a food begins to rival its other
dimensions, as when foods are more cube-shaped
or cylindrical. (Think of a roast, for example, or
a sausage.) Then the heat that enters through the
sides does contribute significantly to conduction.
We have done extensive computer simulations
that demonstrate that when the length and width
are five times the thickness (for a block of food) or
when the length is five times the diameter (for a
cylinder), then the simple scaling rule works well.
Outside these boundaries, however, the situation
is more complicated. The heat equation still works,
but the result has to be calculated individually for
each shape.
No general rules apply across all varieties of
“three-dimensional” foods. Mastering this kind of
cooking is a matter of judgment informed by
experience and experimentation. Rest assured
that you won’t be the first chef (or physicist) to dry
out a thick chop waiting for conduction to heat the
center.
Clearly, it’s important to consider shape as well
as size when you’re buying meats. Bear in mind
that, like many other worthy endeavors, cooking it
will probably take longer than you think.
When Hot Particles Move
Convection is the second most commonly used
mode of heat transfer in cooking. In liquids and
gases such as air, molecules are not locked in place
as they are in solidsthey move. So hot molecules
in fluids do not have to collide with adjacent,
cooler molecules to transmit energy as heat. They
can simply change position, taking their energy
with them. That process is convection, the movement
of hot particles.
A potato impaled with aluminum rods cooks more quickly because the metal helps to conduct heat to the interior
of the food. This principle inspired the “fakir grill,” a Modernist device named for the Near Eastern mystics who lie
on beds of nails. The analogy is imperfect, of course, because the spikes of the grill are meant to stab the overlying
food, whereas the recumbent mystics remain unscathed.
278 VOLUME 1 · HISTORY AND FUNDAMENTALS
HEAT AND E NERGY 279