Question

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 gain by the iced water in \(\mathrm{kW}\) and the rate at which ice melts in the container. The heat of fusion of water is \(333.7 \mathrm{~kJ} / \mathrm{kg}\). Treat the spherical shell as a plain wall, and use the outer area.

Short answer

Answer: To find the rate of heat gain by the iced water, first calculate the thermal resistance (R) using the formula R = h / (kA). Then, calculate the heat transfer rate (q) using the formula q = ΔT / R. Finally, calculate the melting rate using the formula MeltingRate = q / L_f, where L_f is the heat of fusion of water. After completing the calculations, you will find the approximate rate of heat gain and the rate at which ice melts in the container.

Step-by-step solution

Find the thermal resistance of the wall

To find the heat transfer rate, we can use the formula: \(q = \frac{\Delta T}{R}\), where \(q\) is the heat transfer rate, \(\Delta T\) is the temperature difference, and \(R\) is the thermal resistance. The thermal resistance for a plain wall can be calculated as: \(R = \frac{h}{kA}\), where \(h\) is the wall thickness, \(k\) is the thermal conductivity of the wall material, and \(A\) is the outer surface area of the wall. In this case, the outer diameter is \(20 \mathrm{~cm}\), and the thickness is \(0.2 \mathrm{~cm}\), so the radius of the outer surface is \(10 \mathrm{~cm}\). Therefore, the outer surface area is: \(A = 4\pi r^2 = 4\pi (10 \mathrm{~cm})^2 = 400\pi \mathrm{~cm}^2\) The thermal conductivity of iron is approximately \(80 \mathrm{~W/(m \cdot K)}\). Using the thickness and thermal conductivity, the thermal resistance is: \(R = \frac{0.2 \mathrm{~cm}}{80 \mathrm{~W/(m \cdot K)} \times 400\pi \mathrm{~cm}^2} = \frac{0.002 \mathrm{~m}}{80 \mathrm{~W/(m \cdot K)} \times 4\pi (0.1 \mathrm{~m})^2}\)

Calculate the heat transfer rate

Now we can calculate the heat transfer rate using the temperature difference and the thermal resistance: \(q = \frac{\Delta T}{R} = \frac{5^{\circ}\mathrm{C}}{\frac{0.002 \mathrm{~m}}{80 \mathrm{~W/(m \cdot K)} \times 4\pi (0.1 \mathrm{~m})^2}}\) Find q in terms of \(\mathrm{W}\), then convert it to \(\mathrm{kW}\).

Calculate the rate at which ice melts

To find the rate at which ice melts, we will use the heat transfer rate and the heat of fusion of water. The formula for the melting rate is: \(MeltingRate = \frac{q}{L_f}\) where \(L_f\) is the heat of fusion of water. Given that \(L_f = 333.7 \mathrm{~kJ/kg}\), we have: \(MeltingRate = \frac{q}{333.7 \mathrm{~kJ/kg}}\) By completing the calculations, we will find the approximate rate at which ice melts in the container.