Suppose that a pipe carrying a hot fluid with an external temperature of Ti and outer radius ri is to be insulated with an insulation material of thermal conductivity k and outer radius ro. Show that if the convective heat transfer coefficient on the outside of the insulation is h and the environmental temperature is T?, the addition of insulation can actually increases the rate of heat loss if ro < k /h and that maximum heat loss occurs when ro = k/h . This radius, rc, is often called the critical radius.

Question

GIVEN

An insulated pipe External temperature of the pipe = Ti Outer radius of the pipe = ri Outer radius of insulation = ro Thermal conductivity = k Ambient temperature = T? Convective heat transfer coefficient = h



FIND

Show that (a) The insulation can increase the heat loss if ro < k/h (b) Maximum heat loss occurs when ro = k/h

ASSUMPTIONS

The system has reached steady state The thermal conductivity does not vary appreciably with temperature Conduction occurs in the radial direction only

SKETCH

Answer 1

Radial conduction for a cylinder of length L is given



Convection from the outer surface of the cylinder is given



The outer wall temperature, To, is an unknown and must be eliminated from the equation

Solving for Ti – To



Substituting this into the convection equation



Examining the above equation, the heat transfer rate is a maximum when the term



which is greater than zero at ro = k/h, therefore ro = k/h is a true minimum and the maximum heat loss occurs when the diameter is ro = k/h. Adding insulation to a pipe with a radius less than k/h will increase the heat loss until the radius of k/h is reached.