A closed, cylindrical vessel is completely filled with water. The water is perfectly stirred. The volume of the stirrer may be neglected. The power delivered by the stirrer to the water may also be neglected. Initially, the temperature of the water is 100°C. The water loses heat to the surroundings. The temperature of the surroundings is Ts=20°C. The total heat loss is specified via the overall heat transfer coefficient, averaged over the total vessel area, is U= 25W/m2K, and the ratio of heat exchanging surface and liquid volume A/V=1.20m−1. Assume that the heat capacity of the water is independent of the temperature.

• What is the correct formulation of the energy balance of the water in the vessel?
• How long does it take for the water temperature to decrease to 50 °C?
• Subsequently, a heating element present in the vessel is switched on. The element has a power of ϕq=5000W/m3.
• How long does it now take for the water to reach a temperature of 100 °C again? (you may ignore any boiling effects)