Animals in cold climates often depend on two layers of insulation: a layer of bo
ID: 2257797 • Letter: A
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 2.60?C during hibernation, while the inner surface of the fat layer is at 31.2?C.
Assume the surface area of each layer is constant and given by the surface area of the spherical model constructed for the black bear.
How thick should the air layer (contained within the fur) be so that the bear loses heat at a rate of 52.0W ?
Explanation / Answer
Rate of heat transfer
Q/ t = kA ( T0 - T') / L
Given Data
(Q/ t) = 52
K = 0.2W/mK
A = 4*3.14*r^2
L = 0.041m
GiVen T'=T0-[(Q/t)L/Ak]
=31.2-[(52)*(0.041)/(4*3.14*(0.8)^2*(0.2) ]
=29.87
T'=29.87
Thickness of fur is
L=KA( T0-T1)/(Q/t)
= (0.2)4*3.14*(0.8)^2*((31.2-2.6)/52)
=0.884m
=8.84cm
L=8.84