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GATE 2024 PI – Question 14

General Engineering · Thermal and Fluids Engineering · 1 mark · Multiple choice

An electrically heated thin copper wire is covered with insulation thickness t. Under steady conditions heat loss per unit length is Q. Which statement is true?

  1. Q increases monotonically with t.
  2. Q decreases monotonically with t.
  3. Q first increases, then decreases as t increases.
  4. Q first decreases, then increases as t increases.

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Correct answer: (C) Q first increases, then decreases as t increases.

Explanation

The intended critical-insulation model holds wire-surface and ambient temperatures fixed. Cylindrical thermal resistance is $\ln(r_o/r_i)/(2\pi k)+1/(2\pi hr_o)$, minimized at critical radius $r_o=k/h$. For a wire initially thinner than this radius, heat loss first rises and then falls: **C**. This assumes the wire radius is below the critical radius. At fixed electrical power with no other heat path, steady heat loss instead equals the input and wire temperature adjusts; the source leaves these operating assumptions implicit.

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