Critical radius of insulation — behaviour up to the critical point. Up to the critical radius of insulation on a cylinder, what is the effect of adding insulation on the heat loss to the surroundings?

Introduction / Context:
Unlike plane walls, cylinders can show counterintuitive heat-transfer behaviour when insulation is added. Up to a certain radius, adding insulation can actually increase heat loss because of the competing effects of conduction resistance and increased external surface area for convection.

Given Data / Assumptions:

• Long cylinder with outer convective coefficient h_o.
• Insulation of conductivity k applied concentrically.
• Steady, one-dimensional radial heat flow; radiation neglected.

Concept / Approach:
The critical radius r_c for a cylinder is r_c = k / h_o. For outer radius r_o below r_c, increasing r_o increases the external area faster than the conduction resistance grows, so overall heat transfer increases. Beyond r_c, further insulation increases conduction resistance sufficiently to reduce heat loss.

Step-by-Step Solution:
Total resistance R_total = R_cond + R_conv.R_cond = ln(r_o/r_i) / (2 * pi * k * l); R_conv = 1 / (h_o * 2 * pi * r_o * l).Differentiate Q with respect to r_o to find extremum; maximum Q occurs at r_o = r_c = k / h_o.Thus, for r_o < r_c, adding insulation (increasing r_o) increases Q; for r_o > r_c, it decreases Q.

Verification / Alternative check:
Plot Q versus r_o for typical k and h_o; the curve rises up to r_c and then falls, confirming the physical interpretation.

Why Other Options Are Wrong:
(b) holds only beyond the critical radius. (c) and (e) introduce unrelated comparisons; the governing competition is geometric area versus conduction resistance. (d) contradicts the known non-monotonic trend.

Common Pitfalls:
Assuming insulation always reduces heat loss regardless of geometry; that is true for plane walls but not necessarily for cylinders and spheres near small radii.

Final Answer:
added insulation will increase heat loss