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A steel pipe with foam insulation is embedded in a concrete wall. The steel pipe is carrying cold water and therefore gains heat from outside atmosphere. Heat transfers through the concrete wall, foam insulation and then to the pipe. Is it possible to calculate critical radius of insulation for this scenario?

(I understand that the critical radius = k/h can only be applied when the foam insulation is exposed to the atmosphere. We can't use heat transfer coefficient (h value) since convection is not possible between 2 solids (foam and concrete, in this case. In the above scenario, concrete wall becomes a part of the insulation.)

Is there another formula/methodology I can use to calculate critical radius for the above scenario?

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The concept of thermal resistance is especially useful when analyzing 1D steady-state heat conduction. For example, thermal resistances of adjacent materials can often be added, facilitating multilayer and composite analysis. The thermal resistance relates the heat flux in watts to a temperature difference.

The thermal resistance of a plane wall, for example, is $t/kA$, where $t$ is the thickness, $k$ is the thermal conductivity, and $A$ is the cross-sectional area. In other words, heat conduction is suppressed if the thickness is increased or the thermal conductivity or cross-sectional area is decreased.

The thermal resistance of an annulus (e.g., insulation wrapping a pipe) is $\ln(r_2/r_1)/2\pi Lk$, where $r_2$ and $r_1$ are respectively the outer and inner radii and $L$ is the height of the annulus.

The thermal resistance associated with convection is $1/hA$, where $h$ is the convective coefficient. For convection at the outside of an annulus, this is $1/2\pi hr_2L$.

The critical radius arises because the thermal resistance of an annulus (nominally of insulation around a pipe of radius $r_1$) exposed to external convection, $$\frac{\ln(r_2/r_1)}{2\pi Lk}+\frac{1}{2\pi hr_2 L},$$

has a minimum with respect to varying $r_2$.

To examine this critical radius parameter—and whether it even exists—for other simple geometries, we could assemble the appropriate thermal resistance. For two concentric annuli of thermal conductivity $k_1$ and $k_2$, for example, the thermal resistance would be $$\frac{\ln(r_2/r_1)}{2\pi Lk_1}+\frac{\ln(r_3/r_2)}{2\pi Lk_2}.$$

Does this feature a minimum for varying $r_2$ from $r_1$ to $r_3$?

For more complex geometries, finite element analysis may be more appropriate.

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