Question

Two hollow spheres of different materials, one with double the radius and one-fourth wall thickness of the other, are filled with ice. If the times taken for complete melting of ice in the larger to the smaller one are in the ratio of $25: 16$, then their corresponding thermal conductivities are in the ratio

• A. $4: 5$
• B. $5: 4$
• C. $8: 25$
• D. $25: 8$

Answer 1

Answer: (C)

$\frac{\Delta Q}{\Delta t}=\frac{K A \Delta T}{\Delta x}$
$ \Rightarrow \Delta Q=K A\left(\frac{\Delta T}{\Delta x}\right) \Delta t$
Assuming the thickness of the spheres to be small, we have For smaller sphere:
(rate of heat flow) $($ time $)=($ volume of ice melted $)(\rho L)$
i.e., $ K_{1}\left(4 \pi r^{2}\right) \frac{\Delta \theta}{d} \cdot 16=\frac{4}{3} \pi r^{3} \rho L$ .....(i)
For larger sphere:
$K_{2}\left[4 \pi(2 r)^{2}\right] \frac{\Delta \theta}{d / 4} \cdot 25=\frac{4 \pi}{3}(2 r)^{3} \rho L$ .....(ii)
Dividing Eq. (ii) by Eq. (i),
$K_{2} / K_{1}=8 / 25$