### Optimal Pipe Diameter for Pumping a Viscous Fluid

## by Crystal D'Orville

Submitted : Fall 2011

The goal of this project is to find the optimal pipe diameter for pumping a viscous fluid through, while minimizing the cost of the pipe as well as pumping costs. The total cost function is represented by adding the pumping costs to the pipe costs. The electricity cost to run the pump is in $/W hr so the power requirement is equal to the volumetric flow rate (in m^{3}/s) times the pressure drop. We are given the volumetric flow rate, the viscosity, and the density of the fluid. To make the pressure drop dependent on one variable we can substitute the Reynolds number into the friction factor for laminar flow, which can then be substituted into the pressure drop equation, and finally substituted into the power requirement equation. The only variable not given is the diameter. The power requirement is multiplied by the cost per watt hour and the number of hours of operation per year.

The piping costs per year equal $759 x length of the pipe x (diameter)^{1.5}. We are given the length of the pipe so the piping costs are also just dependent one variable, the diameter. If we add the two cost functions together, we get the total cost function. The next step is to find at what point we minimize costs. To do this we must take the derivative of the cost function with respect to the diameter and set it equal to zero. When we solve for the diameter, this will be the optimal pipe diameter to minimize costs.

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