Critical insulation thickness for spheres For a spherical insulation on a hot or cold body, the critical radius (or thickness) is:

Introduction / Context:Adding insulation does not always reduce heat loss immediately for cylindrical and spherical geometries. The concept of a critical radius explains why heat loss can first increase before decreasing as insulation thickness increases.

Given Data / Assumptions:

• External convection with heat-transfer coefficient h0 at the insulation outer surface.
• Insulation thermal conductivity k is uniform.
• Steady, radial heat transfer; contact resistances neglected.

Concept / Approach:For spheres, the critical outer radius r_crit is 2k/h0. If the actual outer radius is below r_crit, adding insulation can increase heat loss because the increase in outer area (boosting convection) outweighs the added conduction resistance. Beyond r_crit, further insulation reduces heat loss.

Step-by-Step Solution:Write total thermal resistance R_total(r_o) = R_cond,insulation + R_conv,outer.Differentiate heat loss Q with respect to r_o and set dQ/dr_o = 0 for extremum.Solve condition to obtain r_crit = 2k / h0 for a sphere (compare with k / h0 for a long cylinder).

Verification / Alternative check:Dimensional check: k (W/m·K) divided by h0 (W/m^2·K) gives metres; multiplying by 2 gives a length, as required.

Why Other Options Are Wrong:(A) is the cylindrical result, not spherical; (C) and (D) invert units; (E) has incorrect dimensions.

Common Pitfalls:Forgetting geometry matters: cylinder → k/h0; sphere → 2k/h0. Also, “critical thickness” is often described via the outer radius, not the material thickness alone.

Final Answer:2k / h0