The expression  is known as logarithmic mean area. Consider a thin walled pipe or a duct with a wall thickness less than inside diameter  , the area ratio  is less than 2 and the logarithmic mean area is substantially equal to the arithmetic mean area  . However, if is greater than 2 which could be a thick-walled pipe or thick insulation of small pipes or tube furnaces, use of arithmetic mean area would predict higher heat losses than the logarithmic mean area.
Figure 4 is the construction of a composite cylindrical wall having radius  and  measured from the centre of the cylinder of radius  such that  and  . The thermal conductivity of the refractory material of thicknesses  and  is  and  respectively. The length of the cylinder is  . As in the case of flat wall  and  are the heat transfer coefficients that determine surface temperatures  and  . The composite wall is placed in between furnace temperature  and environment temperature  .
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| Figure 29.4: |
Temperature distribution in a composite cylindrical wall of different diameters at steady state. |
As no heat is produced in the composite wall, steady state heat flow for the length of the cylinder is
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(9) |
Adding thermal resistance in series
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(10) |
The equation 10 determines the heat flow in a composite wall. The temperature profile is as shown in the figure by the solid line.
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