A 0307mthick sheet of ice covers a lake The air temperature

A 0.307-m-thick sheet of ice covers a lake. The air temperature at the ice surface is -19.6 °C. In 3.40 minutes, the ice thickens by a small amount. Assume that no heat flows from the ground below into the water and that the added ice is very thin compared to 0.307 m. Calculate the number of millimeters by which the ice thickens. Do not enter unit.

Solution

Consider an area of one square meter. You\'ve got water at 0 degrees on one side of the ice and a temperature of -18 degrees on the other. The thermal conductivity of ice k is about 2.2 J / s m K.

Q = k A T / d = ( 2.2 ) ( 1 ) ( 19.6 ) / .307

Q = 140.45 J / s

Now, how much 0-degree water can you turn into ice in five minutes if you pull energy out at a rate of 140.45 J/s? The heat of fusion of water is about 334 kJ / kg. We have ( 140.45  J / s ) ( 3.40minutes ) ( 60 s / minute ) =

140.45*3.4*60 = 28651.8 J,

freeze water = 28651.8)/(334000)

which will freeze 0.0857 kg or 85.7 grams of water. Water occupies about 1 cubic centimeter per gram, so we have about 85.7 cubic centimeters of water.

85.7 cm^3 ( 1 m / 100 cm )^3 = .0000857 m^3. That volume is 1 meter by 1 meter by 0.0857 cm. So we add about 0.0857 cm of thickness to the ice in five minutes.