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(b) An iceberg of frozen water is towed from Antarctica to Melbourne for use as a reservoir of fresh water. The initial temperature of the iceberg in Antarctica at the start of the journey is T₁ = -50°C = 223 K. During the journey and upon arrival in Melbourne the iceberg is immersed in sea water at an average temperature Twater = +15°C = 288 K.) About 1/10, by volume, of the iceberg is above sea level, the remainder is below (see diagram). The iceberg may always be approximated as a uniform sphere during the melting process with a radius of Ri = 100 m.) You may assume the density of both water and ice is a constant of value p = 1.0 × 10³ kgm ³. Heat is transferred into the iceberg by sunlight on the area fraction above sea level at an average continuous rate of: qsun = (1 - Alb). 340 Wm-² where the albedo of ice is Alb= 0.90 and heat is transferred from the sea water at an average rate given by: qwater h(Twater-To) Wm-2 where To = 273 K is the freezing temperature of the ice and h is a constant for a slowly moving iceberg of value h = 460 WK-¹m-² and note that 1 W = 1 Js-¹. qwater Water, Twater Sunlight, qsun R₁ = 100 m Iceberg, T (t) PHYC20012 2022 Exam ⒸUniversity of Melbourne
(i) With this approximation, the amount of heat, dQ, transferred from the sea water bath to an PHYC20012, Semester 1, 2022 page 3 of 17 Show that qsun « qwater and hence the direct contribution from the Sun is negligible. You may therefore assume A is the entire surface area of the iceberg and the rapid heat conduction through the ice heats the entire mass uniformly so at any time the entire mass is at the same temperature T(t). iceberg of surface of area A in time dt can be approximated as: dQ = qwater. A. dt (ii) Define the concepts of heat capacity, specific heat capacity and latent heat capacity. (iv) The specific heat capacity of ice for T ≤ 273 K may be modeled by: Cp = 0.0067.T+0.251 kJkg-¹K-¹ where T is in K. The latent heat capacity of the ice-water transition at T = 273 K has the value Lice-water = 334 kJkg-¹. Determine the amount of heat needed to raise the temperature of the frozen iceberg from its initial temperature of T₁ = 223 K to T = 273 K Hence, being mindful that the radius of the iceberg is approximately constant for this process, determine the time, t₁, required for the iceberg to absorb the heat. By considering the gradual absorption of heat by the un-melted ice (see diagram below where R; > R≥ 0) show that the time taken to melt the iceberg from ice to liquid at T = 273 K is given by: (v) Lice-water-P.R h. (Tw-To) Hence determine the total time, trotal = t₁ + t₂, it takes to melt the entire iceberg from its initial temperature T₁ = 223 K to the final temperature of liquid water at T, = 273 K. qwater Water, Twater t₂ = R₁ = 100 m PHYC20012 022 Exam ⒸUniversity of Melbourne AN dR Liquid at 273 K (grey), ice at 273 K (white).