# Biology of heat transfer

#### Prerequisites

In our example about thermal radiation, we saw quantitatively that radiation can play an important role in the transfer of energy between an organism and its environment.

So what are the implications of heat transfer?  Biological organisms that are endotherms want to maintain a constant temperature.  Depending on the surrounding temperature, they either have to keep their temperature above the background to stay warm or keep their temperature below the background to stay cool.  How might this work?

We are going to deal with a spherical animal for simplicity of the modeling.  Because the animal is generating heat through its metabolism, we can assume that the animal’s resting metabolic rate equals the heat transfer rate.  To calculate the heat transfer rate, we use equation (6).  We next have to assume a mechanism of heat transfer.  If we assume heat transfer by heat conduction alone, then the heat transfer coefficient for a spherical animal is 2κ / d where κ  is the thermal conductivity of the fluid and d is the sphere’s diameter (this comes from a comparison of equations 5 and 6). We also note that the surface area of a sphere is 4πr2 which is just πd2.  So eq 6 becomes

(8)

This heat transfer rate has to be equal to the resting metabolic rate.  The resting metabolic rate has been shown to vary with body mass to the ¾ power.  So we express this as

(9)

where M is the coefficient of resting metabolic rate and is a property of the kind of organism (e.g. bird, reptile, spiny lobster etc).  M has units of W / kg3/4.  The mass, m is just the volume times the density.  Since our animal is spherical, we can substitute

(10)

So we can now equate equations 8 and 10 and solve for how the temperature difference between the organism and the surroundings, ΔT, varies with M, κ and d.

(11)

What does this tell us?  First, it says that temperature differences will increase with the coefficient of resting metabolic rate, body density, and body diameter.  Therefore, animals such as mammals with larger metabolic rates will have larger temperature differences.  Also, animals that have greater density and larger diameters will also have bigger temperature differences.  Second, temperature differences will be smaller for media with larger thermal conductivities.  Since the thermal conductivity of water is 23 times bigger than that of air, the temperature difference will be much smaller in water.  This makes sense.  Water conducts away the heat much more efficiently than does air, and so organisms will tend to be closer in temperature to water.

Another thing we can glean from this model, is there will become an animal diameter at which the animal generates so much metabolic heat that the temperature difference becomes too great to be transferred just by conduction.  We can calculate at what size this occurs.  For air at normal temperatures, a temperature difference more than 30 K would become excessive.  For a typical mammal, M is 3.4 W/kg3/4 and ρb =1000 kg/m3 and for air, κ = 0.026 W/m K.  Therefore, the animal diameter at which this occurs is 2.3 cm.  Therefore, animals larger than that require additional cooling to prevent overheating. Essentially, large animals have a lower surface to volume ratio.  Therefore, they become limited by their surface area in expelling the heat generated by their larger internal volume.

What are some ways large animals might get around this?  First, they do not rely only on conduction for transfer heat.  Convection by air movement will carry away more heat than just conduction alone.  We can go through some similar modeling using equations for the convective heat transfer rate and find that for a spherical animal, it can be at least an order of magnitude bigger than an animal just relying on conduction.  Next, animals are typically not spherical.  They increase their surface area to provide more area over which to transfer the heat.  This is the role of elephant ears.  Elephants divert more blood flow to the ears when they need to increase the rate of heat transfer to cool down.  Increasing surface area for thermoregulation is also one possible explanation for the plates on the back of a Stegosaurus.  Next, large animals typically do not use any insulation.  Elephants and rhinoceroses do not use fur or feathers which might impede heat loss.  However, wooly mammoths, which needed quite large temperature differences to withstand the cold ice and snow did use the insulation for which they are named.  Such large animals could also function with a lower coefficient of metabolic rate, if they are willing to move slowly.  Finally, large animals can use an additional cooling mechanism.  They can evaporate water by either perspiring or panting.  This takes away a lot of heat (the heat of vaporization) to help keep a large animal cool.

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