GATE 2026 MT – Question 59
A cylindrical furnace has an inner diameter of 0.1 m and height of 0.2 m. Its inner cylindrical wall $A_1$ and inner bottom $A_2$ are maintained at 1873 K. The sides and bottom are black bodies, well insulated and heated electrically. The top area $A_3$ is open to an atmosphere at 300 K, resulting in heat loss $q$.
Given $F_{13}=0.1175$ and $F_{23}=0.06$, where $F_{ij}$ is the fraction of radiation leaving surface $i$ intercepted by surface $j$. The Stefan–Boltzmann constant is $5.67\times10^{-8}\ \mathrm{W/(m^2K^4)}$.
The power needed to maintain the furnace at 1873 K is ________ W. (Approximate to the nearest integer.)
Practise this question in The GATE Grind →
Show answer and explanation
Correct answer: 5450 to 5510
Explanation
The wall and bottom are at equal temperatures, so their mutual exchange produces no net loss. The net radiation to the opening is $q=\sigma(A_1F_{13}+A_2F_{23})(1873^4-300^4)$.
Here $A_1=\pi Dh=\pi(0.1)(0.2)$ and $A_2=\pi D^2/4=\pi(0.1)^2/4$. The effective area is $A_1F_{13}+A_2F_{23}=0.00785398\ \mathrm{m^2}$.
Substitution gives $q\approx5476.94\ \mathrm W$. The required heater power is **5477 W**.