An icebox almost entirely filled with ice at 0^ C is submerged in a large volume of water at 20^ C . The walls of the box have a surface area of 2400 cm ^2 , a thickness of 2.0 mm , and a thermal conductivity of 0.06 W m ⁻¹ ^ C⁻¹ . Determine the rate at which the ice melts inside the box. (Latent heat of fusion of ice = 3.4 10^5 J kg ⁻¹ )

Step-by-step solution

The rate of heat transfer through the walls of the icebox is given by Fourier's law of heat conduction: dQ dt = kA(T₂ - T₁) d Substituting the given values: k = 0.06 W m ⁻¹ ^ C ⁻¹ A = 2400 cm ^2 = 0.24 m ^2 T₂ - T₁ = 20 - 0 = 20^ C d = 2.0 mm = 2 10⁻³ m dQ dt = 0.06 0.24 20 2 10⁻³ = 144 J s ⁻¹ The rate at which the ice melts is given by: dm dt = 1 L dQ dt Substituting L = 3.4 10^5 J kg ⁻¹ : dm dt = 144 3.4 10^5 kg s ⁻¹ To find the rate in kg h ⁻¹ , multiply by 3600 s : dm dt = 144 3600 3.4 10^5 kg h ⁻¹ dm dt = 5184