Water pipe optimization

[nh3simman]

We talk theory about optimization, so let's see how it works in practice.


In South Africa, we have a power supply problem. A pumped storage scheme was proposed some time ago to pump water to a dam in the Drakensberg mountains. This saves water and produces hydro-electric power during off-peak times. It is apparently a very successful project.


An ideal (and relatively simple) target for optimization is the stage pipe diameter.


The total cost of the stage can be simplified to (1) installed pipe and (2) the pumping cost as follows:

Installed cost of the pipe = 800 x diameter
Pumping cost = 100x10^12 / (diameter)^5

What diameter would give the least cost?