I would suppose that a negative value of the critical radius would mean that no matter what the radius of the pipe is, it is greater than the critical radius, and hence the heat transfer would always decrease with additional lagging.
can anyone tell me what a negative value of the critical radius of insulation would tell me?
from differentiation is giving as
x = klag/hout
where x = radius of lagged pipe, k is the thermal conductivity of the lagging material and h is the heat loss out side of the lagging.
an explanation is that if the critical radius is less than the radius of the unlagged pipe then the heat transfer would decrease with any increase in lagging - with a minimum at the critical density.
if the radius of the unlagged pipe is less than the critical density the opposite is true