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 0.200W/(m?K) ] surrounded by a layer of air trapped inside fur or down. We can model a black bear (Ursus americanus) as a sphere 1.60m in diameter having a layer of fat 4.10cm 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?Cduring hibernation, while the inner surface of the fat layer is at 31.1?C.

Part A

What should the temperature at the fat-inner fur boundary be so that the bear loses heat at a rate of 51.7W ?

Part B

How thick should the air layer (contained within the fur) be so that the bear loses heat at a rate of 51.7W ?

Explanation / Answer

First, you need to get the square meters of the surface of the idealized "bear-sphere".

Second, you need to convert the fat layer thickness to meters.

Third, Heat rate = thermal conductivity * area * /T / thickness, solve for delta-T, and you know that the high temp side is 31.1

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