A thin hollow sphere of radius R is filled with water. The temperature of atmosphere is − T 0 which is below 0 ∘ C . Now water reached at 0 ∘ C and start freezing from the outer surface towards cent

Question

A thin hollow sphere of radius R is filled with water. The temperature of atmosphere is − T 0 ​ which is below 0 ∘ C . Now water reached at 0 ∘ C and start freezing from the outer surface towards centre. Assume conductivity of the sphere is very high. Find out time taken in complete freezing of water present in the sphere. Take density of water ρ , latent heat of fusion L and conductivity of ice k .

TestHub · Accepted Answer

For an unfrozen core of radius r, the ice-shell resistance is:
$R_{t h} = \frac{R - r}{4 πk R r}$
Heat conduction freezes a thin layer:
$H = \frac{T _{0}}{R _{t h}}$
$H = - 4 π r^{2} ρ L \frac{d r}{d t}$
$d t = - \frac{ρ L}{k R T _{0}} (r R - r^{2}) d r$
Integrating from r=R to r=0:
$t = \frac{ρ L}{k R T _{0}} \int_{0}^{R} (r R - r^{2}) d r$
$t = \frac{ρ L}{k R T _{0}} (\frac{R ^{3}}{2} - \frac{R ^{3}}{3})$
$t = \frac{ρ L R ^{2}}{6 T _{0} k}$

Physics - Heat & Thermo Question with Solution | TestHub

Options:

Doubts & Discussion