# Scaling of heat exchange in cooling eggs

Birds eggs vary widely in size, and this will affect the egg's heat exchanges in two ways.

The first is the variation of "thermal inertia" (more properly, thermal capacity): large eggs have more mass, and more energy is required to elevate their temperature by some amount than is the case in small eggs.

The second is the egg's surface area, which limits the heat fluxes between the egg and environment. Large eggs have larger surface areas than small eggs do, and large eggs of a given temperature will therefore lose more heat than will smaller eggs.

Both of these vary with egg size in a complex way. A large part of my research was to clarify just what these scaling relationships were.

## The Kendeigh equation

The foundation of incubation energetics was the so-called Kendeigh equation, named for the pioneering ecologist Charles Kendeigh. Kendeigh thought he could calculate energy cost of incubation from the cooling rate of eggs of different sizes. His reasoning was this:

One can calculate an egg's thermal conductance from its cooling rate, *k. *In an egg cooling in a uniform environment, *k *is the ratio of the thermal conductance, *K * and the thermal capacity, *C. C *is itself the product of the egg's specific heat and its mass. The thermal conductance is therefore the product of the cooling rate *k, *the egg's specific heat, and the egg mass.

## The scaling of thermal conductance

Eggs provide an interesting model system The foundation of incubation energetics was the so-called Kendeigh equation, named for the pioneering ecologist Charles Kendeigh. Kendeigh thought he could calculate energy cost of incubation from the cooling rate of eggs of different sizes. His reasoning was this:

One can calculate an egg's thermal conductance from its cooling rate, *k. *In an egg cooling in a uniform environment, *k *is the ratio of the thermal conductance, *K * and the thermal capacity, *C. C *is itself the product of the egg's specific heat and its mass. The thermal conductance is therefore the product of the cooling rate *k, *the egg's specific heat, and the egg mass.

Under conditions of cooling in air, Kendeigh showed that thermal conductance scales to the two-thirds power of egg mass. Kendeigh did not, however, account for the multiple modes of heat exchange, including radiative versus convective cooling, and heat conduction within the egg. When these things are properly accounted for, thermal conductance scales according to which mode of heat exchange prevails, which varies with body size. This has implications for how we think about thermoregulation in animals, including very large animals like dinosaurs.

## Selected publications

Turner, J. S. (1985). Cooling rate and size of birds' eggs - a natural isomorphic body. Journal of Thermal Biology 10: 101-104.

Turner, J. S. (1988). Body size and thermal energetics. How should thermal conductance scale? Journal of Thermal Biology 13: 103-117.