Question

1. Uncle Iroh’s just made a mass M of tea, but it comes off the kettle too hot at a temperature T_hot. Zuko’s asleep, but will scold his uncle if he sees him drinking, so Iroh wants to cool the tea to T_drink< T_hot as quickly as possible. He has a bunch of ice cubes with side length 2l at temperature T_ice<273 K< T_drink.

The mass-specific heat of the ice is c_{m ice}; the water and tea both have mass-specific heat c_{m water}. The density of ice is ρ in units of mass per volume, and the mass-specific latent heat of melting of the ice is ∆H_m.

a) Assume the tea and ice cubes are completely isolated from the environment. How

many ice cubes must Iroh put in the tea to get it to T_drink ?

b) Some physicist tells you that ice cools tea at a rate

P ∝ A, (1)

where A is the total exposed surface area of the ice. To speed up the process, you decide to cut the ice cubes each into 8 equally sized smaller cubes (by making two cuts on each large cube). If τ_{no cut} is the time it takes to cool the tea without splitting the cubes, and τ_cut is the time it takes using the split cubes, what is the ratio τ _cut/ τ_{no cut}?

c)Given your answers so far, can Iroh cool the tea using the ice much faster than

the environment could? What should his strategy be to cool the tea as quickly as

possible?

Answer 1