When the air temperature is below 0 C, the water at the surface of a lake freeze

Question

When the air temperature is below 0 C, the water at the surface of a lake freezes to form an ice sheet. The longer the air remains at this temperature, the thicker the ice sheet will become. Will a lake in Minnesota that is uniformly 40 m deep ever freeze solid? To answer this, calculate how long it would take to freeze all the water in the lake. Assume that the upper surface of the ice sheet is steady at -10 C and the bottom surface is at 0 C, and that all heat transfer occurs via conduction through the ice. (Hint: this is not a steady-state problem, and requires an integral.)

Explanation / Answer

Let at depth x there be an element dx which loses heat and freeze

let area of the lake be A

is density of water

s be latent heat

so heat lost by the element when it freezes => mass *latent heat = s**A*dx

now heat transfer through ice sheet in time dt = k*A*(T1-T2)/d *dt=> kAT/x (T1 = 0) (let T2 = T)

where k is coefficient of heat transfer of ice and d is thickness

Heat transfer = heat lost

s**A*dx = kAT/x*dt

sx(dx) = kT dt

integrating both side

sx2/2 = kTt

t = sx2/(2kT)

rest all are constant hence time is proportational to square of thickness. Yes sheet can freeze completely

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