Metabolism, Heat Loss, and Size

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The bigger the better?

Big Ideas: 
  • The body produces energy from food at a certain rate called the metabolic rate. Some of this energy is converted to heat.
  • Heat loss is proportional to surface area, while mass is proportional to volume.
  •  Heat loss per unit mass is smaller for larger animals than for smaller animals

When animals eat food, chemical reactions that occur while the foodis being digested release energy that is used to keep the animal  alive, mobile and warm.  The metabolic rate, $ \Gamma $, is the rate at which this energy is produced in the animal and it is related to the mass of the animal by

$ \Gamma = 4 M^{3/4} $ 

Such a relation where a physical property of an animal scales with mass to some power that is not one is called an allometric relation1.  It is obvious that an elephant which is large and consumes vast amounts of food will have a larger metabolic rate than a mouse that is much less massive and consumes much less. Considering, however, the metabolic rate per unit mass of these animals,

$ \dfrac{\Gamma}{M} = 4 M^{-1/4} $

we find to sustain itself an elephant needs less energy per unit mass than a mouse.  This explains why even though elephants eat a larger quantity of food compared to mice, mice actually need to eat more compared to their body weight than elephants.  Our goal is to understand how heat loss contributes to smaller animals having a higher metabolic rate per unit mass than larger animals.

As mentioned some of the energy generated inside an animal goes to heat which is radiated out through the surface area of the animal.  The rate of heat loss is proportional to the surface area, while the mass of an animal is proportional to its volume. The rate at which heat is radiated per unit mass then is proportional to surface area per unit volume.

For simplicity, if we take an animal to be spherical with radius, $ r $, then we can relate the heat loss per unit mass to size by

$ \dfrac{4 \pi r^{2}}{\dfrac{4}{3} \pi r^{3}} \propto \dfrac{1}{r} $  

As animals increase in size the rate of heat loss per unit mass decreases and less energy is needed to sustain the animal per unit mass.  It might appear an obvious advantage then to be a large animal. Large animals, however, still consume more energy on the whole than smaller animals so that they need more food. If food is not abundant larger animals can not sustain themselves1.

What we have learnt partially explains why children are more susceptible to developing hypothermia from cold temperatures than adults as they are smaller and have a higher heat loss per unit mass 2.

In conclusion, larger animals have a smaller heat loss per unit mass and require less food per unit mass even though they eat larger amounts than smaller animals.

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