You are the Chief Engineer for a prominent hydraulics consulting company. Your client wants to pay you lots of money to design a system to deliver 1 m3/s from reservoir A at elevation 300 m to reservoir B at 500 m. The distance from A to B is 1000 km and the elevation between the two reservoirs is given by z (m) = 300 + 1.2x – 0.001x2, where x is the distance from A in km.

The system is to consist of a pipeline and a series of pumps. Commercial steel pipe with a roughness e = 0.05 cm and a pressure rating of 150 m head is available. The total cost of purchasing pipe and constructing a pipeline is listed below for various pipe sizes. Pumps that can deliver 1 m3/s at 100 m head cost $10,000,000 each (installed).

Determine the most economical design (i.e. pipe size and number and location of pumps). Hint: Vary your pipe diameter and consider pumps in series.

Pipe diameter (m) Total Cost ($/km)

0.9 80,000

1.0 100,000

1.1 120,000

1.2 150,000

1.4 200,000

Note 1: Theoretically, a pipeline may be designed to allow pressure to fall to the vapour pressure. In practice, however, water usually contains dissolved gasses that will vapourize well before the vapour pressure point is reached. Such gasses dissolve very slowly. They can move with the water in the form of large bubbles that disrupt the flow. Therefore, negative pressures shouldn’t be allowed to exceed about 70% of atmospheric pressure – use this figure in your design calculations.

Note 2: The pressure rating of a pipe is the maximum pressure that it can withstand.