Steam flowing through a long, thin-walled pipe maintains the...

Steam flowing through a long, thin-walled pipe maintains the pipe wall at a uniform temperature of $500 \mathrm{~K}$. The pipe is covered with an insulation blanket comprised of two different materials, A and B. The interface between the two materials may be assumed to have an infinite contact resistance, and the entire outer surface is exposed to air for which $T_{\infty}=$ $300 \mathrm{~K}$ and $h=25 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. (a) Sketch the thermal circuit of the system. Label (using the preceding symbols) all pertinent nodes and resistances. (b) For the prescribed conditions, what is the total heat loss from the pipe? What are the outer surface temperatures $T_{s, 2(\mathrm{~A})}$ and $T_{s, 2(\mathrm{~B})}$ ?

Steam flowing through a long, thin-walled pipe maintains the pipe wall at a uniform temperature of $500 \mathrm{~K}$. The pipe is covered with an insulation blanket comprised of two different materials, A and B.

The interface between the two materials may be assumed to have an infinite contact resistance, and the entire outer surface is exposed to air for which $T_{\infty}=$ $300 \mathrm{~K}$ and $h=25 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$.
(a) Sketch the thermal circuit of the system. Label (using the preceding symbols) all pertinent nodes and resistances.
(b) For the prescribed conditions, what is the total heat loss from the pipe? What are the outer surface temperatures $T_{s, 2(\mathrm{~A})}$ and $T_{s, 2(\mathrm{~B})}$ ?

Key Concepts

Interface (Contact) Resistance

Interface or contact resistance refers to the additional thermal resistance present at the interface between two different materials. In some cases, as when an infinite contact resistance is assumed, it represents a very poor thermal connection that significantly impedes heat flow, leading to abrupt temperature changes at the interface between materials.

Composite Insulation

Composite insulation involves the use of two or more different insulating materials in a layered configuration. Analyzing composite systems requires treating each layer's thermal resistance separately and accounting for potential discontinuities at material interfaces. This approach is essential for understanding how variations in material properties affect overall thermal performance.

Thermal Circuit Analysis

This concept involves representing the heat transfer path as an equivalent circuit made up of nodes and resistances. It allows the analysis of heat flow through different components by analogy to electrical circuits, where temperature differences drive heat transfer through thermal resistances, similar to voltage driving current through electrical resistors.

Conduction Resistance

Conduction resistance is a measure of a material's resistance to heat flow by conduction. In the context of insulated pipes, each layer of insulation provides a specific resistance based on its material properties and thickness. The temperature drop across a given insulation layer is proportional to the heat flux and the layer’s conductive resistance.

Convection Resistance

Convection resistance accounts for the resistance to heat loss from the surface of an object into a fluid, such as air. It is defined using the convective heat transfer coefficient and the effective surface area. In insulation problems, the outer surface's temperature drop relative to the ambient air is influenced by the convective resistance.

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