Chapter 5, Problem 5.42P

The melting of water initially at the fusion temperature, $T_f=0°\text{C,}$ was considered in Example 1.6.Freezing of water often occurs at $0°\text{C}\text{.}$ However, pure liquids that undergo a cooling process can remain in a supercooled liquid state well below their equilibrium freezing temperature, $T_f,$ particularly when the liquid is not in contact with any solid material. Droplets of liquid water in the atmosphere have a supercooled freezing temperature, $T_{f,sc},$ that can be well correlated to the droplet diameter by the expression $T_{f,sc}=-28+0.87\ln \left(D_p\right)$ in the diameter range ${10}^{-7}<D_p<{10}^{-2}\text{m,}$ where $T_{f,sc}$ has units of degrees Celsius and $D_p$ is expressed in units of meters. For a droplet of diameter $D=50\mu \text{m}$ and initial temperature $T_i=10°\text{C}$ subject to ambient conditions of $T_{\infty }=-40°\text{C}$ and $h=900{\text{W/m}}^{\text{2}}\cdot \text{K,}$ compare the time needed to completely solidify the droplet for case A, when the droplet solidifies at $T_f=0°\text{C,}$ 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 to $T_f=0°\text{C}\text{.}$