A hollow spherical conducting sheel of inner radius `R_(1) = 0.25 m` and outer radius `R_(2) = 0.50 m` is placed inside a heat reservoir of temperature `T_(0) = 1000^(@)C`. The shell is initially filled with water at `0^(@)C`. The thermal conductivity of the material is `k =(10^(2))/(4pi) W//m-K` and its heat capacity is negligible. The time required to raise the temperature of water to `100^(@)C is 1100K ln.(10)/(9) sec`. Find `K`. Take specific heat of water `s = 4.2 kJ//kg.^(@)C`, density of water `d_(w) = 1000 kg//m^(3) pi = (22)/(7)`

Answer

5

Explanation

For any general moment `1000^(@)C`.
`i_(H) = (1000 -T)/(R_(eq)) = (dQ)/(dt)`
where `R_(eq) = int (dx)/(k(4pix^(2))) = (1)/(4pik) ((x^(-2+1))/(-2+1))_(R_(1))^(R_(2))`
`R_(eq) = (1)/(4pik) {(1)/(R_(1))-(1)/(R_(2))} = (1)/(50)`
Now, mass of water inside cavity
`M = rho xx (4)/(3) pi R_(1)^(3)`
`(dQ)/(dt) = MS (d theta)/(dt) = (1000 -T)/(R_(eq)) (d theta =dT)`
`overset(100^(@))underset(0^(@))int (dT)/((1000-T)) = (1)/((R_(eq)MS)) overset(t)underset(0)int dt`
`t = R_(eq) MS xx {ln((1000)/(900))}`