A cylinder of radius R is surrounded by a cylindrical shell of inner radius R and outer radius 2R. The thermal conductivity of the material of the inner cylinder is K₁ and that of the outer cylinder is K₂. Assuming no loss of heat, the effective thermal conductivity of the system for heat flowing a…

Question

A cylinder of radius R is surrounded by a cylindrical shell of inner radius R and outer radius 2R. The thermal conductivity of the material of the inner cylinder is K₁ and that of the outer cylinder is K₂. Assuming no loss of heat, the effective thermal conductivity of the system for heat flowing along the length of the cylinder is

Accepted Answer

(4) (K₁+3K₂)/4. Idea: the heat runs along the axis, so the core and the shell are two paths side by side between the same two ends — a parallel combination, in which conductances add. Core area A₁=π R²; shell area A₂=π(2R)²-π R²=3π R². Both have the same length ℓ, so (K_eff(A₁+A₂))/ℓ=(K₁A₁)/ℓ+(K₂A₂)/ℓ. K_eff=(K₁π R²+K₂ 3π R²)/(4π R²)=(K₁+3K₂)/4. It is an area-weighted average, so it lies between K₁ and K₂, closer to K₂ because the shell is the larger cross-section.

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