Critical radius of insulation (cylindrical) — A 10 cm outside-diameter steam pipe at 180°C is insulated with a material of k = 0.6 W/m·°C. Ambient air is at 30°C, and the external convective coefficient is h = 0.8 W/m²·°C. Neglect pipe-wall and steam-side resistances. If 2 cm of insulation is added, the rate of heat loss from the pipe will be:

Introduction:
For cylinders and spheres, adding insulation can paradoxically increase heat loss until the outer radius reaches a critical value. This arises because insulation increases conductive resistance but also increases the surface area available for convection. The balance is captured by the critical radius of insulation concept.

Given Data / Assumptions:

• Pipe outside radius r1 = 0.05 m (10 cm diameter).
• Insulation conductivity k = 0.6 W/m·°C.
• External convection coefficient h = 0.8 W/m²·°C.
• Ambient 30°C; surface radiation neglected; steady state.

Concept / Approach:
The critical radius for a cylinder is r_crit = k / h. If the insulated outer radius r2 is less than r_crit, adding insulation increases heat loss. If r2 > r_crit, insulation decreases heat loss as usually expected.

Step-by-Step Solution:
Compute r_crit = k / h = 0.6 / 0.8 = 0.75 m.With 2 cm insulation, r2 = r1 + 0.02 = 0.05 + 0.02 = 0.07 m.Compare: r2 = 0.07 m is far less than r_crit = 0.75 m.Conclusion: adding 2 cm insulation increases the heat loss relative to the bare pipe.

Verification / Alternative check:
Qualitatively, the added area for convection (circumference grows with r2) dominates the small increase in conduction resistance when r2 < r_crit. Numerical evaluation of the composite resistance R_total = ln(r2/r1)/(2πkL) + 1/(2πh r2 L) shows R_total decreasing for small r2 increments here, confirming higher heat loss.

Why Other Options Are Wrong:

• (b) and (c) contradict the r_crit criterion for these values of k and h.
• (d) Although 5 cm insulation (r2 = 0.10 m) still yields r2 < r_crit and thus even greater loss than 2 cm, the question asks comparison to the bare pipe, making (a) the correct choice.
• (e) Radiation omission does not drive heat loss to zero; convection remains.

Common Pitfalls:
Forgetting that the critical-radius effect is significant only when h is small or k is high; assuming insulation always reduces heat loss for any geometry.

Final Answer:
Greater than that for the uninsulated steam pipe