Critical thickness of insulation
Consider a single layer of insulation which is put around a cylindrical pipe of length  . The inner temperature of the insulation is fixed at temperature  and the outer surface is exposed to an environment temperature 
Equation 10 for a single layer of insulation is
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(11) |
As  increases  increases which means there is an increasing resistance to radial conduction. Increase in increases outer furnace area as well which means  decreases. This dual effect suggests that there exists a particular value of for which heat loss is maximum. For a given  , the particular value of can be determined by putting 
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(12) |
Solving equation 12, we get
 ,
where  is critical radius of insulation at which heat loss is maximum. This suggests that heat loss does not decrease always with the increase in insulation thickness. Heat loss could increase by increasing the thickness of the insulation beyond because outer surface area increases and hence heat losses due to convection increases.
References:
1) J.P. Holman .Heat Transfer, P 346.
2) R. Schuhmann: Metallurgical Engineering, Volume 1 Engineering Principles
3) D.R.Poirier and G.H.Geiger: Transport phenomena in materials processing
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