Question

Liquid water coats an active (growing) icicle and extends up a short, narrow tube along the central axis (see the figure below). 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. Take L 0.25 m and Tr-6.0°C. Assume that the central tubes and the upward conduction path both have cross-sectional area A. The thermal conductivity of ice is 0.400 W/m·K, and the density of liquid water is 1000 kg/m3. (Assume A has units of my Energy transfer L. Liquid coating (0° C) Liquid water o C) (a) In terms of A, what rate is energy conducted upward? A W (b) In terms of A, what rate is mass converted from liquid to ice at the top of the central tube? A kg/s (c) At what rate does the top of the tube move downward because of water freezing there? m/s

Answer 1

(a)

Rate of heat conduction upwards,

Q / t = kA $\Delta$ T / L

Q / t = 0.4*A*6 / 0.25

Q / t = 9.6A W

(b)

We know that,

Q = mL

Q / t = (m / t)*L

where,,L = heat of fusion = 333 kJ/kg

m/t = 9.6A / 333*10^3

m/t = 2.88*10^-5A kg/s

(c)

Volume flow rate = mass / density

V =  2.88*10^-5A / 10^3 =  2.88*10^-8A m^3/s

velocity v = volume flow rate / area

v = 2.88*10^-8A / A

v = 2.88*10^-8 m/s