One end of a rod length `20cm` is inserted in a furnace at `800K`. The sides of the rod are covered with an insulating material and the other end emits radiation like a blackbody. The temperature of this end is `750K` in the steady state. The temperature of the surrounding air is `300K`. Assuming radiation to be the only important mode of energy transfer between the surrounding and the open end of the rod, find the thermal conductivity of the rod. Stefan constant `sigma=6.0xx10^(-1)Wm^(-2)K^(-4)` .

Explanation

Energy received per second at end 2
`I = (KA (theta_(1) - theta))/(L)`
Energy radiated per second at end 2
`mu = sigma A (theta^(4) - theta_(0)^(4))`
In steady state
`I = u`
`(KA (theta_(1) - theta))/(L) = sigma A (theta^(4) - theta_(0)^(4))`
`K = (sigma L (theta^(4) - theta_(0)^(4)))/((theta_(1) - theta))`
`= (6 xx 10^(-8) xx 0.2 [(600)^(2) - (300)^(4)])/((800 - 600))`
`(1.2 xx 10^(-8) xx (6^(4) - 3^(4)) xx 10^(8))/(200) `
`= 10^(-3) (1296 - 81)`
`= 1.215 W//m^(@)C`