A pipe of radius R is maintained at a uniform temperature T. To reduce heat loss from the pipe, it is lagged by an insulating material of thermal conductivity κ. The lagged pipe has radius r>R. Assume that all surfaces lose heat according to Newton's law of cooling J=hΔT, where h=∣h∣ can be taken to be a constant. Show that the heat loss per unit length of pipe is inversely proportional to
hr1+κ1ln(Rr),(10.82)
and hence show that thin lagging doesn't reduce heat loss if R<κ/h.