Optimal Pipe Diameter

by Giovanni Quiel

Submitted : Spring 2010

This project focused on optimizing the diameter of a pipe connected to a pump. If the diameter of the pipe were too small, the pumping costs became excessive. If the diameter of the pipe were too large, the cost of piping became too large. The goal of the project was to find the piping diameter that will minimize the total cost. From here, it was mandated that there were two separate costs; the cost of pumping and the cost of constructing the pipe. Therefore, the total annual cost would be the sum of these two costs. This is how the total annual cost for pumping was determined. The next focus was to determine the diameter which would minimize the total cost. From calculus, the optimal pipe diameter can be found by the first derivative test, which determines where a relative minimum and maximum will occur in a function. The total annual cost is a cost function with respect to diameter, so a first derivative test would determine the precise diameter which would provide the utmost minimal cost. Using the first derivative test, with some help from Mathematica, it was determined that the total annual cost function has an absolute minimum where the diameter is equal to 0.02875 meters. Also, Mathematica was able to calculate the minimum total cost to be $243.13/yr to pump water at turbulent flow through a 50 meter pipe with a pressure drop of 885 Pa/m.

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