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AP Physics 2
9.5 Specific Heat and Thermal Conductivity
AdvancedMCQMathematical14.6k
A cross-sectional view of two thin concentric circular rings representing insulating sleeves around a pipe. The inner circular region is labeled \(T_H\). The innermost ring has an average radius labeled \(r_1\), thickness labeled \(d_1\), and thermal conductivity labeled \(k_1\). The outer concentric ring has an average radius labeled \(r_2\), thickness labeled \(d_2\), and thermal conductivity labeled \(k_2\). The outer region surrounding the entire structure is labeled \(T_C\). Straight radial arrows point outward from the central region toward the exterior, indicating heat flow. No other labels, lines, text, or axes appear.
Cross-section of a pipe insulated by two thin concentric cylindrical layers.
A cylindrical pipe of length \(L\) carrying hot fluid at temperature \(T_H\) is insulated by two thin concentric cylindrical layers exposed to ambient air at temperature \(T_C\). The inner insulating layer has thermal conductivity \(k_1\), thickness \(d_1\), and average radius \(r_1\). The outer insulating layer has thermal conductivity \(k_2\), thickness \(d_2\), and average radius \(r_2\). Assuming steady-state radial heat conduction where the thickness of each layer is much smaller than its radius (\(d_1 \ll r_1\) and \(d_2 \ll r_2\)), which of the following expressions represents the rate of heat transfer \(H\) through the insulation?

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