Halliday, Resnick & Walker — Temperature, Heat, and the First Law of Thermodynamics: Problem 78

Icicles. Liquid water coats an active (growing) icicle and extends up a short, narrow tube along the central axis (Fig. 18.32). Because the water–ice interface must have a temperature of \(0^{\circ}\text{C}\), 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 \(T_r\) can be below \(0^{\circ}\text{C}\). Take \(L = 0.12\text{ m}\) and \(T_r = -5^{\circ}\text{C}\). Assume that the central tube and the upward conduction path both have cross-sectional area \(A\). In terms of \(A\), what rate is (a) energy conducted upward and (b) mass converted 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 \(0.400\text{ W/m}\cdot\text{K}\), and the density of liquid water is \(1000\text{ kg/m}^3\).