Question

A $100\, cm$ long cylindrical flask with inner and outer diameters $2\, cm$ and $4\, cm$ respectively is completely filled with ice as shown in the figure-4.48. The constant temperature outside the flask is $40^{\circ} C$. (Thermal conductivity of the flask is $\left.0.693\, W / m ^{\circ} C , L_{\text {ice }}=80\, cal / gm \right)$.

• A. Rate of heat flow from outside to the flask is $80\, \pi J / s$
• B. The rate at which ice melts is $\frac{\pi}{4200} Kg / s$
• C. The rate at which ice melts is $100\, \pi Kg / s$
• D. Rate of heat flow from outside to flask is $40\, \pi J / s$

Answer 1

Answer: (A), (B)

Integrating we have
$H=\frac{K A d \theta}{d r}=K(2 \pi r L) \frac{d \theta}{d r}$
$\int\limits_{r_{1}}^{r_{2}} \frac{d r}{r}=\frac{2 \pi K L}{H} \int\limits_{\theta_{1}}^{\theta_{2}} d \theta$
$\Rightarrow \frac{d Q}{d t}=H=\frac{2 \pi K L\left(\theta_{2}-\theta_{1}\right)}{\text{In}\left(\frac{r_{2}}{r_{ t }}\right)}=80\, \theta$
$\Rightarrow \frac{d m}{d t} L =80\, \pi$
$\Rightarrow \frac{d m}{d t} =\frac{8 \pi}{L}+\frac{80 \pi}{80 \times 4200}$
$=\frac{\pi}{4200} Kg / s$