# 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 temperatur

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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 1100 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)

by (90.1k points)

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))}