Question

A tank of water has been outdoors in cold weather until a 5.00 cm thick slab of ice has formed on its surface. The air above the ice is at -12.0 degrees C. Calculate the rate of formation of ice (in centimeters/hour) on the bottom surface of the ice slab. Take the thermal conductivity of ice to be 0.0040 cal/s-cm-degree C, the density to be 0.92 g/cm^3, and the heat of fusion to be 80 cal/g. Assume that no heat enters or leaves the water through the walls of the tank.

Answer 1

The water at the boundary between the water and the ice will be at a temperature of 0^o C. Now the heat transported through the ice of thickness L = 5 cm with a temperature difference of , area A and thermal conductivity k = 0.0040 cal/s-cm-degree- C is

Now the heat of fusion of ice is 80 cal/g. Suppose a mass m of water is transformed each second into ice. This produces a total heat equal to

As no heat enters or leaves the water, these two heats must be equal. So

A mass m of ice (covering an area A) will have a thickness d, where d is given by

So the rate of ice growth is