Problems and Solutions on Thermodynamics and Statistical Mechanics

Therrnodynarnica 153

1154
The water at the surface of a lake and the air above it are in thermal
equilibrium just above the freezing point. The air temperature suddenly
drops by AT degrees. Find the thickness of the ice on the lake as a function
of time in terms of the latent heat per unit volume L/V and the thermal
conductivity A of the ice. Assume that AT is small enough that the specific
heat of the ice may be neglected.
(MITI
Solution:
Consider an arbitrary area AS on the surface of water and let h(t) be
the thickness of ice. The water of volume ASdh under the ice gives out
heat LASdhlV to the air during time dt and changes into ice. So we have

L AT

V h

ASdh - = A-ASdt

that is

Hence h(t) = [ ~ $3 1’2.

1155

A sheet of ice 1 cm thick has frozen over a pond. The upper surface of

the ice is at -20°C.

(a) At what rate is the thickness of the sheet of ice increasing?

(b) How long will it take for the sheet’s thickness to double?

The thermal conductivity of ice K is 5~

heat of ice L is 80 cal/g. The mass density of water p is 1 g/cm3

cal/cm. sec.OC. The latent

(SUNY, BufluIo)

Solution:

(a) Let the rate at which the thickness of the sheet of ice increases be

vl a point on the surface of ice be the origin of z-axis, and the thickness of

ice be z.

The heat current density propagating through the ice sheet is J’ =

-K ___ and the heat released by water per unit time per unit area

T - To

z