A hollow spherical iron container with outer diameter 20 cm and thickness 0.2 cm is filled with iced water at 0°C. If the outer surface temperature is 5°C, determine the approximate rate of heat gain by the iced water in kW and the rate at which ice melts in the container. The heat of fusion of water is 333.7 kJ/kg. Treat the spherical shell as a plain wall, and use the outer area.
A hollow spherical iron container is filled with iced water at 0°C. The rate of heat gain by the iced water and the rate at which ice melts in the container are to be determined.
Assumptions 1 Steady operating conditions exist since the surface temperatures of the wall remain constant at the specified values. 2 Heat transfer through the shell is one-dimensional. 3 Thermal properties of the iron shell are constant. 4 The inner surface of the shell is at the same temperature as the iced water, 0°C. 5 Treat the spherical shell as a plain wall and use the outer area.
Properties The thermal conductivity of iron is k = 80.2 W/m?*C (Table A-3). The heat of fusion of water is given to be 333.7 kJ/kg.
Analysis This spherical shell can be approximated as a plate of thickness 0.4 cm and area
Then the rate of heat transfer through the shell by conduction is
Considering that it takes 333.7 kJ of energy to melt 1 kg of ice at 0°C, the rate at which ice melts in the container can be determined from
Discussion We should point out that this result is slightly in error for approximating a curved wall as a plain wall. The error in this case is very small because of the large diameter to thickness ratio. For better accuracy, we could use the inner surface area (D = 19.6 cm) or the mean surface area (D = 19.8 cm) in the calculations.
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