Animals in cold climates often depend on two layers of insulation: a layer of bo

Question

Animals in cold climates often depend on two layers of insulation: a layer of body fat [of thermal conductivity 0.200 W/(m*K)] surrounded by a layer of air trapped inside fur or down. We can model a black bear (Ursus americanus) as a sphere 1.60 m in diameter having a layer of fat 4.10 cm thick. (Actually, the thickness varies with the season, but we are interested in hibernation, when the fat layer is thickest.) In studies of bear hibernation, it was found that the outer surface layer of the fur is at 2.80 degrees C during hibernation, while the inner surface of the fat layer is at 31.1 degrees C.

A) What should the temperature at the fat-inner fur boundary be so that the bear loses heat at a rate of 51.1 W?

B) How thick should the air layer (contained within the fur) be so that the bear loses heat at a rate of 51.1 W?

If you cut and paste formulas make sure that they show up correctly instead of a bunch of strange characters or my numbers so I can at least try to figure out what the formula was supposed to be by matching my numbers to the weird characters.

Explanation / Answer

a. dQ1/dt = KA(T2-T)/L

dQ2/dt = KA(T-T1)/L

here dQ2/dt = dQ1/dt

so equate RHS of above two equations, rearange and find T.

B. USING THIS T , FIND d

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