A deep rectangular pond of surface area $A$, containing water (density $=\rho$ ), specific heat capacity $=\mathrm{s}$ ), is located in a region where the outside air temperature is at a steady value of $-26^{\circ} \mathrm{C}$. The thickness of the frozen ice layer in this pond, at a certain instant is $\mathrm{x}$. Taking the thermal conductivity of ice as $K$, and its specific latent heat of fusion as $L$, the rate of increase of the thickness of ice layer, at this instant, would be given by (1) $26 \mathrm{~K} / \rho \times(\mathrm{L}+4 \mathrm{~s})$ (2) $26 \mathrm{~K} / \rho \times(\mathrm{L}-4 \mathrm{~s})$ (3) $26 K /\left(\rho x^{2} L\right)$ (4) $26 \mathrm{~K} /(\rho \times \mathrm{L})$

Home » A deep rectangular pond of surface area , containing water (density ), specific heat capacity ), is located in a region where the outside air temperature is at a steady value of . The thickness of the frozen ice layer in this pond, at a certain instant is . Taking the thermal conductivity of ice as , and its specific latent heat of fusion as , the rate of increase of the thickness of ice layer, at this instant, would be given by (1) (2) (3) (4)

A deep rectangular pond of surface area $A$ , containing water (density $=\rho$ ), specific heat capacity $=\mathrm{s}$ ), is located in a region where the outside air temperature is at a steady value of $-26^{\circ} \mathrm{C}$ . The thickness of the frozen ice layer in this pond, at a certain instant is $\mathrm{x}$ . Taking the thermal conductivity of ice as $K$ , and its specific latent heat of fusion as $L$ , the rate of increase of the thickness of ice layer, at this instant, would be given by (1) $26 \mathrm{~K} / \rho \times(\mathrm{L}+4 \mathrm{~s})$ (2) $26 \mathrm{~K} / \rho \times(\mathrm{L}-4 \mathrm{~s})$ (3) $26 K /\left(\rho x^{2} L\right)$ (4) $26 \mathrm{~K} /(\rho \times \mathrm{L})$

Physics | Physics

A deep rectangular pond of surface area $A$ , containing water (density $=\rho$ ), specific heat capacity $=\mathrm{s}$ ), is located in a region where the outside air temperature is at a steady value of $-26^{\circ} \mathrm{C}$ . The thickness of the frozen ice layer in this pond, at a certain instant is $\mathrm{x}$ . Taking the thermal conductivity of ice as $K$ , and its specific latent heat of fusion as $L$ , the rate of increase of the thickness of ice layer, at this instant, would be given by (1) $26 \mathrm{~K} / \rho \times(\mathrm{L}+4 \mathrm{~s})$ (2) $26 \mathrm{~K} / \rho \times(\mathrm{L}-4 \mathrm{~s})$ (3) $26 K /\left(\rho x^{2} L\right)$ (4) $26 \mathrm{~K} /(\rho \times \mathrm{L})$

Answer (4)

Sol. Assume at any instant thickness of ice is $\mathrm{x}$ . And time taken to form additional thickness. $(\mathrm{dx})$ is $\mathrm{dt}$ .

$\mathrm{mL}=\frac{\mathrm{KA}[26-0] \mathrm{d} t}{\mathrm{x}}$
$(\mathrm{Adx}) \rho \mathrm{L}=\frac{\mathrm{KA}(26) \mathrm{dt}}{\mathrm{x}}$
$\frac{\mathrm{dx}}{\mathrm{dt}}=\frac{26 \mathrm{~K}}{\mathrm{x} \rho \mathrm{L}}$