Rosie is an engineer working in the laundry factory of Fragrance Airlines to wash and clean fabrics. She is in charge of the hot water supply for the washing machine 24 hours a day. The hot water supply system is shown in Figure Q2 below with a rectangular tank of cross-sectional area 2 m x 1 m. At steady state, the inlet water supply has a temperature of 25°C and an outlet temperature of 90°C at a water level of 2 m to meet the demand of 5 liters of hot water for every kilogram of fabric being washed. The hot water temperature variation should be kept in the range of 87 - 93°C to ensure efficient washing without harming the fabrics. In normal operation, the factory is able to wash 300 kg of fabrics per hour, but the cleaning capacity can be increased when the flight frequency is at its peak.
This morning, Rosie received instructions from the manager that the factory needs to increase the cleaning capacity to 500 kg per hour at 2.00 pm. However, coincidentally, there is routine maintenance for the heater which will cause the shutdown of the heater for 45 minutes starting at 2.15 pm. Rosie would like to keep the hot water supply ongoing while the heater is shut down.
(Heat capacity, Cwater = 4200 J/kg·°C; Density of water, Ï = 1000 kg/m³)
Inlet water supply, Wi = 25°C
Hot water, W = 90°C
Heat supply, Q
Figure Q2
(a) Derive the model and transfer function for the temperature dynamics of the heating tank system temperature. (16 marks)