GATE 2025 CH – Question 53
An electrical wire of 2 mm diameter and 5 m length is insulated with a plastic layer of thickness 2 mm and thermal conductivity $k = 0.1$ W/(m K). It is exposed to ambient air at 30 °C. For a current of 5 A, the potential drop across the wire is 2 V. The air-side heat transfer coefficient is 20 W/(m$^2$ K). Neglecting the thermal resistance of the wire, the steady state temperature at the wire-insulation interface is ____ °C (rounded off to 1 decimal place).
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Correct answer: 38.61 to 38.99
Explanation
The heat generated in the wire is $Q = VI = 2 \times 5 = 10$ W and it all flows out through the insulation to the air. The wire radius is 1 mm and the outer radius is 3 mm. The insulation resistance is $\frac{\ln(3/1)}{2\pi \times 0.1 \times 5} = 0.3497$ K/W and the air film resistance is $\frac{1}{20 \times 2\pi \times 0.003 \times 5} = 0.5305$ K/W. The interface temperature is $30 + 10 \times (0.3497 + 0.5305) = 38.8$ °C.