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Airplane - Heat Problem

Please view attached heat problem. I am having trouble understanding the cylinder part/ and should be used to find the temperature. any help would be appreciated. thanks

Assume that the body of an airplane is considered as a 20-m long hollow cylindrical structure made up of unknown composite material. The plane body has inside and outside diameters of 6 and 8 meters respectively. Assume that the thermal conductivity of the composite structure varies with radius according to the relation, K(r) = 2/r2. If the inside surface of the plane is exposed to convection coefficient of 20 W/m2 K and ambient temperature of 30 oC and the outside surface of the plane is exposed to a constant temperature of −50 oC, determine the inside surface temperature of the plane. How much heat the passenger, sitting by the window, will loose or gain assuming that the passenger's body temperature is 36 oC. Assume negligible radiation heat transfer inside or outside the plane.

Note: the thermal conductivity of cylindrical wall is a function of r so that the temperature distribution in the cylinder is 1/r d/dr(k(r) r dt/dr) = 0

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Solution Preview

We assume cylindrical symmetry and so neglect the heat transfer via the bases of the cylinder and along the length of its walls.

Consider a thin cylindrical layer of length L, radius r, thickness dr, tempearature differential dT and conductivity K(r) = k/r^2, where k = 2 W m/K. The heat flux through it is

Q = -2πrLK(r)dT/dr (1)

As the flux is the same for all layers, we have

dT = -Qdr/[2πrLK(r)] = -[Q/(2πkL)]*rdr (2)

with Q constant, and ...

Solution Summary

Assume that the body of an airplane is considered as a 20-m long hollow cylindrical structure made up of unknown composite material. The plane body has inside and outside diameters of 6 and 8 meters respectively. Assume that the thermal conductivity of the composite structure varies with radius according to the relation, K(r) = 2/r2. If the inside surface of the plane is exposed to convection coefficient of 20 W/m2 K and ambient temperature of 30 oC and the outside surface of the plane is exposed to a constant temperature of −50 oC, determine the inside surface temperature of the plane. How much heat the passenger, sitting by the window, will loose or gain assuming that the passenger's body temperature is 36 oC. Assume negligible radiation heat transfer inside or outside the plane.

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