Icicles. Liquid water coats an active (growing) icicle and extends up a short, n

Question

Icicles. Liquid water coats an active (growing) icicle and extends up a short, narrow tube along the central axis (figure 18-54). Because the water - ice interface must have a temperature of 0?C, the water in the tube cannot lose energy through the sides of the icicle or down through the tip because there is no temperature change in those directions. It can lose energy and freeze only by sending energy up (through distance L) to the top of the icicle, where the temperature Tr can be below 0?C. Assume that the central tube and the upward conduction path both have cross-sectional area A. Take L = 0.135 m, A = 0.000319 m2 and Tr = -5.0?C. What rate is (a) energy conducted upward and (b) mass converted (in kg/s) from liquid to ice at the top of the central tube? (c) At what rate does the top of the tube move downward because of water freezing there? The thermal conductivity of ice is 0.400 W/m-K, heat of fusion for water is 333 kJ/kg and the density of liquid water is 1000 kg/m3.

Explanation / Answer

(a) Rate of heat Transfer = KAT/l
Hence in this case Q/t = 0.4*0.000319*5/ 0.135 = 0.00472592593 W

(b) Now Heat of fusion = 333 kJ/kg; And rate at which there is loss of heat =  0.00472592593 J/sec
Hence rate at which mass is converted = 0.00472592593/333000 = 1.41919698 × 10-8 kg/sec

(c)Now we have rate of change of volume = Mass/density = 1.41919698 × 10-8/103 = 1.41919698 × 10-11 m3/sec

Hence change in height = Rate of change of volume / Area = 1.41919698 × 10-11/0.000319 = 4.44889335 × 10-8 m/sec

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