A perfectly insulating cylindrical glass is partially full of water. At the top

Question

A perfectly insulating cylindrical glass is partially full of water. At the top of the water is a uniform layer of ice. The temperature of the water is 0 °C, and the temperature of the air above the ice is held fixed at –25.00 °C. If the initial thickness of the ice layer is 0.0550 m, and the depth of the water below the ice is 0.0650 m, how long, in hours, will it take for the rest of the water to freeze? Assume that there is no transfer of heat through the sides or bottom of the glass, and that the ice layer can slide freely up and down the glass. The latent heat of fusion of water is 3.33 × 105 J/kg and the thermal conductivity of ice is 2.180 W/m·C°. The density of water near freezing is 1000.0 kg/m3 and the density of ice is 917.00 kg/m3.

Explanation / Answer

tepmreture of water below ice is at 0oC, whereas Air tempreture above it is at - 25oC. As result, heat energy moves from water below the ice to Air above it. As water is at freezing point, any heat energy removed from it, changes water to ice. Rate of formation of ice depends upon rate of transfer of heat energy from water to air, through ice.

Rate of heat transfer through ice, at any instant when ice thickness is x , is given by
dQ/dt = KA (Tw - Ta)/ x ........1
K is thermal conductivity of ice.
A is area of cross section of ice or water.
Twis tempreture of water and Ta is tempreture of Air.

Amount of Ice formation, on transfer of heat dQ from water is given by
dm L = dQ where l is latent heat of fusion of water.
Also dm = A d (dx) where d is density of water, dx is thickness of ice added.

Hence equation can be written as
A d L dx/dt = K A (Tw - Ta)/ x
=> x dx = K (Tw - Ta) dt / d L
Integrating it we get
(x22 - x12) = 2 K(Tw - Ta)t / d L where x1 and x2 are intial and final thickness of ice.

(0.122 - 0.0552) = 2 x 2.18 x 25 t / 1000 x 3.33 x105
t = 34751 sec. = 9.65 Hrs

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