Question

A copper pipe with an outer radius of 0.011 m runs from an outdoor wall faucet into the interior of a house. The temperature of the faucet is 4.4° C and the temperature of the pipe, at 3.1 m from the faucet, is 24° C. In fourteen minutes, the pipe conducts a total of 249 J of heat to the outdoor faucet from the house interior. Find the inner radius of the pipe. Ignore any water inside the pipe.

Answer 1

Assume that the body of the pipe is heat insulated and heat is conducted from one end to the other.
dQ/dt = k•A• (dT/dx),
where the heat conduction rate
dQ/dt = 249/15•60 = 0.277 J/s,
k = 390 J/s•m•oC,
the cross-sectional area of the pipe
(R and r are the outer and inner radii of the pipe)
A = π• (R^2 - r^2),
the temperature gradient
dT/dx = (24°C - 4.4°C)/3.1 = 6.32 °C/m.
Working out the equation and
you should get r = 0.0127m.