Two-layer insulation on a steam pipe A steam pipeline is to be insulated using two concentric layers made of different materials with different thermal conductivities. For minimum heat transfer from the hot pipe to the surroundings, where should the better (lower-k) insulation be placed?

Introduction / Context:
Composite cylindrical insulation is common in steam distribution. This question tests understanding of radial conduction and how arranging layers of different thermal conductivity affects overall thermal resistance and thus heat loss.

Given Data / Assumptions:

• Two insulation layers of fixed inner radius r1 (pipe outer surface) and fixed outer radius r3 (overall insulation OD).
• Layer conductivities: k1 and k2, with one being the better insulator (lower k).
• Steady one-dimensional radial conduction; contact resistance neglected.

Concept / Approach:
For a hollow cylinder, radial conduction resistance is R = ln(r_out/r_in) / (2 * pi * k * L). For two layers, total resistance is the sum of individual resistances. For minimum heat transfer (maximum resistance), we choose the layer ordering that maximizes the sum.

Step-by-Step Solution:
Let r2 be the interface radius. Then R_total = ln(r2/r1)/(2pik1L) + ln(r3/r2)/(2pik2L).Maximize R_total with respect to r2 for fixed r1 and r3. The derivative with respect to r2 is proportional to (1/k1 − 1/k2)/r2.If k1 < k2 (layer 1 is the better insulator), then 1/k1 > 1/k2, so increasing r2 increases R_total. The maximum occurs when layer 1 occupies the larger radial span, i.e., it is placed inside adjacent to the pipe.Therefore, the lower-k material should be nearer the hot surface to maximize total resistance and minimize heat transfer.

Verification / Alternative check:
Numerical check with r1 = 1, r3 = 2, k_better = 0.05, k_poorer = 0.1 and r2 = 1.5 shows R_total is larger when the lower-k layer is the inner layer, confirming the derivative result.

Why Other Options Are Wrong:

• The better insulation outside: reduces the logarithmic leverage of the low-k layer; gives lower total resistance.
• Either arrangement same: false; cylindrical geometry breaks this symmetry.
• Depends primarily on steam temperature: placement effect is geometric and conductivity-driven, not temperature-driven (within property limits).
• Higher-k inside to avoid stresses: not a heat-loss minimization argument.

Common Pitfalls:
Assuming series layers behave like plane walls (where order does not matter); forgetting the ln(r) dependence in cylinders.

Final Answer:
The better insulation must be put inside (closer to the pipe wall).