## Radial Heat Flow Through the Sides of a Pipe

The hot water pipes in a house carry water from a boiler to the taps where the water is used. Heat is lost continuously through the sides of the pipes. If the pipes are lagged or insulated, this heat loss can be reduced.

Consider a pipe of inner radiusat temperatureand outer radiusat temperaturewhere

The rate of heat flow through the walls of the pipe iswhereis the cross sectional area of the pipe andis the rate of flow of heat at the point where the radius is

If the pipe is thin so thatthenis approximately constant and equal toandhence

If the pipe containskg of water at temperaturethen thenandso