Question

When the air temperature is below 0 ?C, the water at the surface of a lake freezes to form a sheet of ice.

If the upper surface of an ice sheet 24.64cm thick is at -10.58?C and the bottom surface is at 0 ?C, calculate the time it will take to add 2.41mm to the thickness of this sheet.

Answer 1

            Consider a section of ice that has Area = A

            At time t, let the thickness be h.

            Short time interval t to t + dt.

            Let the thickness that freezes in this time be dh.

            The mass of the section that freezes in the time interval dt is dm = pdv = pAdh.

            The heat that must be conducted away from this mass of water to freeze it is

            dq = dmLr = (pALr)dh.

            H=dq/dt = kA(T/h)

            So the heat dq conducted in time dt troughout the thickness h that is already there is

            dq=kA(Tu-Tc/h)dt.

            pALrdh=kA(Tu-Tc/h)dt.

            hdh=(k(Tu-Tc)/pLr)dt.

            Integrate from t=0 to time t.

            At t=0 , h=0

            h = squareroot of(2k(Tu-Tc)/pLr)*squareroot of t

            The thickness after t is proportional to squareroot of t.

            t = h^2pLr/2k(Tu-Tc)

            = [(0.0024m)^2(920 kg/m3)(334*10^3J/kg)]/2(1.6w/mk)(0.c-(-10.58))

            t = 0.015 hr.or approximately 53 s.