Question

A metallic cylindrical vessel whose inner and outer radii are $r_1$ and $r_2$ is filled with ice at $0^{\circ }$. The mass of the ice in the cylinder is m. Circular portions of the cylinder is sealed with completely adiabatic walls. The vessel is kept in air. Temperature of the air is ${50}^{\circ }C$. if time taken to for ice to melt completely is $t=mLin\left(r_2/r_1\right)/X\pi kl$.Thermal conductivity of the cylinder is k and its length is $l$. Latent heat of fusion is L.find the value of X

Answer 1

Top view of cylinder

We cut a cylinder of radius r

$H=\frac{dQ}{dt}-\frac{KAd\theta }{dr}$

$A=\pi rl$ (length will be same)

$\frac{dQ}{dt}=\frac{KAd\theta }{dr}=2\pi rl\frac{kd\theta }{dr}$

$\frac{H}{2\pi lK}{\int }_{{}_{r_1}}^{r_2}\frac{dr}{r}={\int }_0^{{50}^0}d\theta$

$\frac{H}{2\pi lK}ln\left(\frac{r_2}{r_1}\right)=50H=\frac{100\pi k}{ln\left(r_2/r_1\right)}$

$Ht=mL$ (latent heat)

$t=\frac{mL}{H}=\frac{mLln\left(r_2/r_1\right)}{100\pi Kl}$