Frozen Solid

Estimate how long it will take for a small pond of average depth D = 0.5 m to freeze completely in a very cold winter, when the temperature is always below the freezing point of water (see Figure 1.1). Take the thermal conductivity of ice to be κ = 2.2 W/m K, the latent heat of fusion q = 3.4 × 10⁵ J/kg, and the density ρ = 0.9 × 10³ kg/m³. Take the outside temperature to be a constant T₀ ~ 263 K.

Figure 1.1

SOLUTION

If the ice does not freeze too fast (which is usually the case with lakes), we can assume that the temperature is distributed linearly across the ice. Suppose that the thickness of the ice at a time t is x. Then the heat balance can be written in the form

(1)

where T_m is the melting temperature of ice. The left side represents the flow of heat through one square meter of ice surface due to the temperature gradient, and the right side the amount of heat needed to melt (freeze) an amount of ice ρ dx. Integrating (1), we obtain

(2)

where t₀ and α are integration constants. If we assume that there is no ice (x = 0) initially (t₀ = 0), then α = 0, and we find that the time t to freeze solid is

(3)