Ocean water is processed in a desalinization plant on the coast and transported by a pipeline to a city water plant 60 km inland as shown in Figure P6.23. Fresh water flows in the horizontal pipeline at a rate of 570 m3/h. The friction factor in the pipe can be assumed to be 0.019.

The initial cost of each pump can be expressed as a function of the pressure increase that must be provided to overcome the friction loss over a distance, L, the distance between the pumping stations, as

In this equation, a1 = 5000 $, and a2 = 0.00032 $/Pa1.2. The initial cost of the pipe can be expressed as a function of diameter by,

where a3 = 5,120,000 $/m1.5.

a. Formulate the objective function and constraint(s).

b. Using the method of Lagrange multipliers, solve the optimization problem formulated in Part

(a) and determine the number of pumps required to minimize the initial cost of the system.
The solution of Part

(b) is the mathematical optimum. Therefore, the number of pumps is likely not an integer, nor is the diameter of the pipe a standard diameter.
Assuming that schedule 40 commercial steel pipe is used,

c. Specify the nominal diameter (inches) and number of pumps for this application.
What is the total system initial cost for the number of pumps and pipe diameter you specified? How does this value compare with the total system initial cost determined in Part (b)?

Cpump = 1 +a,1.2