GATE 2024 ME – Question 50
Consider a hemispherical furnace of diameter $D = 6$ m with a flat base. The dome of the furnace has an emissivity of 0.7 and the flat base is a blackbody. The base and the dome are maintained at uniform temperature of 300 K and 1200 K, respectively. Under steady state conditions, the rate of radiation heat transfer from the dome to the base is ________ kW (*rounded off to the nearest integer*).
Use Stefan-Boltzmann constant = $5.67 \times 10^{-8}$ W/(m$^2$ K$^4$)
Practise this question in The GATE Grind →
Show answer and explanation
Correct answer: 2727
Explanation
The dome area is $A_d = 2\pi r^2 = 2\pi(3)^2 = 56.55$ m$^2$ and the base area is $A_b = \pi r^2 = 28.27$ m$^2$. The base sees only the dome, so $F_{b-d} = 1$. The radiation network has the dome surface resistance $\frac{1 - \varepsilon}{\varepsilon A_d} = \frac{0.3}{0.7 \times 56.55} = 0.00758$ and the space resistance $\frac{1}{A_bF_{b-d}} = 0.03537$ (the base is black, so it has no surface resistance), a total of 0.04295 m$^{-2}$. The driving difference is $\sigma(1200^4 - 300^4) = 117\,114$ W/m$^2$. So $Q = \frac{117\,114}{0.04295} = 2.727 \times 10^6$ W, which is 2727 kW.