A hollow spherical iron container with outer diameter \(20 \mathrm{~cm}\) and thickness \(0.2 \mathrm{~cm}\) is filled with iced water at \(0^{\circ} \mathrm{C}\). If the outer surface temperature is \(5^{\circ} \mathrm{C}\), determine the approximate rate of heat loss from the sphere, in \(\mathrm{kW}\), and the rate at which ice melts in the container. The heat of fusion of water is \(333.7 \mathrm{~kJ} / \mathrm{kg}\).

To calculate the surface area of the sphere, we need to convert the outer diameter from centimeters to meters. Since there are 100 centimeters in a meter, we can do this by dividing by 100: Outer radius (in meters): \(r_o = \frac{20}{2} \cdot \frac{1}{100} = 0.1 \mathrm{~m}\) Now, the surface area of the sphere can be calculated using the formula: Surface area: \(A = 4 \pi r_o^2\) The thickness of the iron should also be converted to meters. Thickness (in meters): \(t = 0.2\times \frac{1}{100}= 0.002 \mathrm{~m}\) Now, we can calculate the surface area and the thickness of the sphere in meters.

In this exercise, there is a temperature difference between the inside and the outside of the sphere. Due to this difference, heat will flow through the container. To calculate the rate of heat transfer, we need to use the formula for heat flow through a sphere: \(Q = \frac{kA\Delta T}{r_o - r_i}\) where \(Q\) is the heat transfer rate, \(k\) is the thermal conductivity of the material, \(A\) is the surface area, \(\Delta T\) is the temperature difference, \(r_o\) is the outer radius of the sphere, and \(r_i\) is the inner radius of the sphere. The thermal conductivity of iron is approximately \(k = 80 \mathrm{~W/mK}\). The temperature difference is given: \(\Delta T = 5 - 0 = 5 \mathrm{~K}\). We can calculate \(r_i\) based on the known thickness and the outer radius: \(r_i = r_o - t = 0.1 - 0.002 = 0.098 \mathrm{~m}\) Now plug in the values to find the heat transfer rate: \(Q = \frac{80 \times 4 \pi (0.1)^2 \times 5}{0.1 - 0.098} \mathrm{~W}\)

Once we find the numerical value of the heat transfer rate (in watts), we convert it to kilowatts by dividing by 1000: \(Q \mathrm{~(in~kW)} = \frac{Q \mathrm{~(in~W)}}{1000}\)

To calculate the rate at which the ice melts within the container, we need to relate the heat loss rate (in kilowatts) to the heat of fusion of water (in kJ/kg): Rate of ice melting (\(m\) - mass in kg/s): \(m = \frac{Q \mathrm{(in~kW)}}{333.7 \mathrm{~kJ/kg}}\) After calculating the values from steps 1 to 4, we will have the approximate rate of heat loss from the sphere in kW and the rate at which ice melts in the container.