On a winter day when the atmospheric temperature drops to -10^(@)C, ice form on the surface of a lake. (a) Calculate the rate of increases of thickness of the ice when 10 cm of ice is already formed. (b) Calculate the total time taken in forming 10 cm of ice. Assume that the temperature of the entire water reaches 0^(@)C before the ice starts froming. Density of water = 1000 kg//m^(3), latent heat of fusion of ice = 3.36 xx 10^(5) J//kg and thermal conductivity of ice = 1.7 W//m-^(@)C. Neglect the expansion of water on freezing

On a winter day when the atmospheric temperature drops to -10^(@)C, ic

1.	On a winter day when the atmospheric temperature drops to -10^(@)C, ice form on the surface of a lake. (a) Calculate the rate of increases of thickness of the ice when 10 cm of ice is already formed. (b) Calculate the total time taken in forming 10 cm of ice. Assume that the temperature of the entire water reaches 0^(@)C before the ice starts froming. Density of water = 1000 kg//m^(3), latent heat of fusion of ice = 3.36 xx 10^(5) J//kg and thermal conductivity of ice = 1.7 W//m-^(@)C. Neglect the expansion of water on freezing
Answer» Solution :(a) Let a time `t`, thickness of ice formed is `x` Let in time `dt`, thickness of ice formed is `dx`. RATE of heat flow through ice `(dQ)/(dt) = (KA [0 - (- theta)])/(x)` Let in time `dt`, mass of ice formed is `dm` `dQ = dmL = rho A dx L` From (i) and (ii) `(rho A dx L)/(dt) = (KA theta)/(x)` `(dx)/(dt) = (K theta)/(rho LX) = (1.7 xx 10)/(1000 xx 3.36 xx 10^(5) xx 0.1) = 5 xx 10^(-7) m//s` (b) `int dt = (rho L)/(2 K theta) int_(0)^(x) x d x` `t = (rho L x^(2))/(2 K theta) = (1000 xx 3.36 xx 10^(5) xx (0.1)^(2))/(2 xx 1.7 xx 10)` `= 976605 = 2.71 hr`