Fundamentals of Physics 8th Edition: Jearl Walker Ch.18: #65 A tank of water has

Question

Fundamentals of Physics 8th Edition: Jearl Walker Ch.18: #65
A tank of water has been outdoors in cold weather, and a slabof ice 5.0cm thick has formed on the surface. The air abovethe ice is at -10o C.  Calculate the rate of iceformation (in centimeters per hour) on the ice slab. Takethermal conductivity of ice to be 0.0040 cal/s x cm xCo and its' density to be 0.92 g/cm3. Assume no energy transfer through the tank walls orbottom. Ch.18: #65
A tank of water has been outdoors in cold weather, and a slabof ice 5.0cm thick has formed on the surface. The air abovethe ice is at -10o C.  Calculate the rate of iceformation (in centimeters per hour) on the ice slab. Takethermal conductivity of ice to be 0.0040 cal/s x cm xCo and its' density to be 0.92 g/cm3. Assume no energy transfer through the tank walls orbottom.

Explanation / Answer

let us assume that h as the thickness of the slab and Ais its area then the rate of heat flow
   through the slab is given by
   Pcon = k A (TH -TC) / h
   where k is the thermal conductivity of ice
   TH = 0oC
   TC = - 10oC
   as the heat leaving the water freezes it the heatrequired to freeze mass m of water will be
   Q = LF m
   where LF is the latent heat of fusion ofwater
   on differentiating with respect to time we get
   dQ / dt = Pcond
   Pcond = LF dm / dt
   we know that density = mass / volume
   so the mass is given by
   m = A h
   dm / dt = A (dh / dt)
   Pcond = LF Adh / dt
   on equating we get
   dh / dt = k (TH - TC) /LF h
             = ....... m / s
             = ....... cm / s

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