Critical Thickness of Insulation

Consider insulation around a circular pipe as shown in the Details section. The inner temperature of the pipe is fixed at . The pipe length is taken equal to .
The heat losses per unit length of the pipe are given by:
where is the radius of the pipe, is the radius of the insulation, is the temperature of the convection environment, is the thermal conductivity of the insulation, and is the heat transfer coefficient of the convection environment.
This Demonstration plots the heat losses per unit length of the pipe versus the dimensionless radius of the insulation, .
For sufficiently small values of , heat loss may increase with the addition of insulation. This is a result of the increased surface area available for losses by convection.
There is a critical radius, shown by the red dot in the figure, above which heat losses start to decrease. This critical radius is obtained by setting . The magenta region gives the values of the dimensionless radius, , where the insulation is effective in preventing heat losses. The heat loss for a pipe without insulation is shown by the cyan dot in the figure.


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Here is the insulation around the pipe.
The resistances to heat transfer due to insulation and environment are given by and , respectively. Their sum is effectively in the denominator of the right-hand side of the equation for in the caption.
[1] J. P. Holman, Heat Transfer, 10th ed., New York: McGraw-Hill, 2010.
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