During a winter day, the atmospheric temperature falls to -10^ C , and ice begins to form on the surface of a lake. Determine the total time required for the formation of 10 cm of ice. Assume the temperature of the entire body of water reaches 0^ C prior to the onset of ice formation. Take the density of water as 1000 kg m ⁻³ , the latent heat of fusion of ice as 3.36 10^5 J kg ⁻¹ , and the thermal conductivity of ic

Step-by-step solution

Let x be the thickness of ice at time t . The rate of heat transfer through the ice layer is given by: dQ dt = KA T x The heat released when a small thickness dx of ice forms is: dQ = dm L = ( A dx) L Equating the two expressions for the rate of heat transfer: A L dx dt = KA T x ₀^ x x dx = ₀^ t K T L dt x^2 2 = K T L t t = L x^2 2 K T Substituting the given values ( = 1000 kg m ⁻³ , L = 3.36 10^5 J kg ⁻¹ , x = 0.1 m , K = 1.7 W m ⁻¹ ^ C ⁻¹ , T = 10^ C ): t = 1000 3.36 10^5 (0.1)^2 2 1.7 10 t = 3.36 10^6 34 s t 988