Q. 14

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Given:
Temperature of the water: T1 = 0 °C
Temperature of the atmosphere: T2 = -10 °C
Change in temperature : ΔT = T1-T2 = 10 °C
Length of the ice formed : l = 10 cm = 0.1 m
Density of water: ρ = 1000 kg m–3
Latent heat of fusion of ice: L = 3.36 × 105 J kg–1.
Thermal conductivity of ice: K0. = 1.7 W
m–1 °C–1.
Formula used:
Rate of amount of heat flowing or heat current is given as: Here, Δθ is the amount of heat transferred, ΔT is the temperature difference, K is the thermal conductivity of the material, A is the area of cross section of the material and x is the thickness or length of the material.
Also,
Δθ = Q =L× m
Here, Q is the amount of heat absorbed or released, L is the Latent heat and m is the mass of the substance.
And,
Mass = Density × Volume
m = ρ V = ρAl
Here ρ is the density of water, A is the area and l is the length.
(a)
Let change in thickness be Δx,thus rate of change of thickness is:     Hence, the thickness of the ice increases at a rate of 5.05× 10-7 m/s.
(b)
Consider time required to form thin layer of ice dx is dt.
Using above formulations,
Mass of dx: dm = Adx × ρ
And heat absorbed by the thin layer,
dQ = dm× L
Now, rate of Heat transfer due to thin layer becomes:    Integrating on both sides and setting the limit of ice formed x: 0 to 0.1      Hence, it took 27.45 hours to form 10 cm thick ice.

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