The melting of water initially at the fusion temperature, T_f = 0°C, was considered in Example 1.5. Freezing of water often occurs at 0°C. However, pure liquids that undergo a cooling process can remain in a super cooled liquid state well below their equilibrium freezing temperature, Tf' particularly when the liquid is not in contact with any solid material. Droplets of liquid water in the atmosphere have a super cooled freezing temperature, T_f,sc, that can be well correlated to the droplet diameter by the expression T_f,sc = - 28 + 0.87 In(D p ) in the diameter range 10^-7 < Dp < 10^-2 m, where T_f.sc has units of degrees Celsius and Dp is expressed in units of meters. For a droplet of diameter D = 50μm and initial temperature T_i = 10°C subject to ambient conditions of T∞ = - 40°C and h = 900 W/m² ∙ K, compare the time needed to completely solidify the droplet for case A, when the droplet solidifies at T_f = 0°C, and case B. when the droplet starts to freeze at T_f,sc, Sketch the temperature histories from the initial time to the time when the droplets are completely solid. Hint: When the droplet reaches T_f,sc in case B, rapid solidification occurs during which the latent energy released by the freezing water is absorbed by the remaining liquid in the drop. As soon as any ice is formed within the droplet, the remaining liquid is in contact with a solid (the ice) and the freezing temperature immediately shifts from T_f.sc to T_f = 0°C.