A continuous evaporator is being used to concentrate a scale-forming solution of sodium sulfate in water. The overall coefficient of heat transfer decreases according to the following expression:

Where U = overall coefficient of heat transfer, Btu/(h)(ft2)(°F), and θb, = time in operation, h.

The only factor which affects the overall coefficient is the scale formation. The liquid enters the evaporator at the boiling point, and the temperature and heat of vaporization are constant. At the operating conditions, 990 Btu are required to vaporize 1 lb of water, the heat-transfer area is 400 ft2, and the temperature-difference driving force is 70°F. The time required to shutdown, clean, and get back on stream is 4 h for each shutdown, and the total cost for this cleaning operation is $100 per cycle. The labor costs during operation of the evaporator are $20 per hour. Determine the total time per cycle for minimum total cost under the following conditions:

(a) An overall average of 65,000 lb of water per 24-h day must be evaporated during each 30-day period. (b) An overall average of 81,000 lb of water per 24-h day must be evaporated during each 30-day period.