Question

In the figure shown, $AB$ is a rod of length $30\text{cm}$ and area of cross-section $1.0{\text{cm}}^2$ and thermal conductivity 336 S.I. units. The ends $A$ & $B$ are maintained at temperatures ${20}^{\circ }\text{C}$ and ${40}^{\circ }\text{C}$ respectively. A point $C$ of this rod is connected to a box $D$, containing ice at $0^{\circ }\text{C}$, through a highly conducting wire of negligible heat capacity. Find the initial rate (in $\text{mg/s}$) at which ice melts in the box. [Assume latent heat of fusion for ice $L_f=80\text{cal/gm}$]

Answer 1

Point C acts like a junction.


Hence, thermal current received by the ice
$i_{net}=i_1+i_2$

According to Fourier's law,
Thermal current $i_{th}=\frac{\Delta T}{R_{th}}$
Thermal resistance $R_{th}=\frac{L}{kA}$

So, $R_{AC}=\frac{10\times {10}^{-2}}{336\times {10}^{-4}}=\frac{{10}^3}{336}$
$R_{BC}=\frac{20\times {10}^{-2}}{336\times {10}^{-4}}=\frac{2\times {10}^3}{336}$

$i_{n}et=i_1+i_2=\frac{\Delta T_{AC}}{R_{AC}}+\frac{\Delta T_{BC}}{R_{BC}}=\frac{20\times 336}{{10}^3}+\frac{40\times 336}{2\times {10}^3}=\frac{4\times 336}{100}J/s$

For melting of ice,
$\Delta Q=\Delta mL$
$\Rightarrow \frac{\Delta Q}{\Delta t}=\frac{\Delta m}{\Delta t}{L}_f$
$\Rightarrow i_{net}=\frac{\Delta m}{\Delta t}{L}_f$

$\Rightarrow \frac{\Delta m}{\Delta t}=\frac{i_{net}}{L_f}=\frac{4\times 336}{100\times 80\times 4.2}=0.04g/s=40mg/s$