2.3 Thickness of Insulation

We know that by adding more insulation to a flat wall always decreases heat transfer, because of the added resistance. The thicker the insulation, the lower the heat transfer rate and hence the heat loss. This is expected, since the heat transfer area \(A\) remains constant, and adding insulation always increases the total resistance for heat transfer.

Adding insulation to a cylindrical piece or a spherical shell, however, is a different matter. The additional insulation increases the conduction resistance of the insulation layer but decreases the convection resistance of the surface because of the increase in the outer surface area for convection.

The heat transfer from the pipe may increase or decrease, depending on which effect dominates.

Consider the heat transfer from the cylindrical pipe of outside radius \(r_i\), which is insulated up to \(r\) (\(r > r_i\)), with an insulating material of thermal conductivity \(k\). The heat transfer coefficient of ambient air is \(h\). The heat transfer resistance for unit length of cylinder, for this condition is given by \[R = R_{\text{insulation}} + R_{\text{ambient}}\] i.e., \[R = \frac{\ln(r/r_i)}{2\pi k} + \frac{1}{2\pi r h}\] It can be seen from the above formula, that with increase in \(r\), resistance due to insulation increases, and resistance to the ambient decreases. Therefore, \(R\) goes through a minimum; and heat loss goes through a maximum. The value of \(r\) for minimum \(R\) is obtained by using the equation \(dR/dr = 0\). This radius (\(r_{oc}\)) is called the critical radius of insulation. \[\frac{dR}{dr} = \frac{1}{2\pi k}\frac{1}{r} - \frac{1}{2\pi r^2 h} = 0 \qquad (\text{at $r=r_{oc}$})\] Solving the above, we get \(r_{oc}=k/h\). Similarly, for the spherical surface, it can be shown that \(r_{oc}=2k/h\).

  • The critical thickness of insulation \((=r_{oc}-r_i)\) corresponds to the condition when the sum of conduction and convection resistances is a minimum.

  • For a given temperature difference, critical thickness results in a maximum heat transfer rate, and the critical radius \(r_{oc}\) is given by \[\begin{align*} & \text{for cylinder:} & \qquad r_{oc} &= \frac{k}{h} \tag*{(25)} \\ & \text{for sphere:} & r_{oc} &= \frac{2k}{h} \tag*{(26)}\end{align*}\] where \(k\) is the thermal conductivity of the insulation and \(h\) is the convective heat transfer coefficient of the fluid outside the insulation.

If the insulating material is chosen in such a way that \(r_{oc}\le r_i\), then any addition of insulation leads to decrease in heat loss.