Over the past few weeks I had to study fluid mechanics again, so I thought that for this week post I could share an example of that.
Problem description
Given a network of pipes, where the length and diameter of each section is known, calculate the flow of water in the network (the Bernoulli boundary values must also be known, details later). Lets solve for the network and data that follows (the construction material is PVC).
| Section | Length[m] | Diameter[m] |
|---|---|---|
| 1 | 2800 | 0.25 |
| 2 | 1300 | 0.18 |
| 3 | 800 | 0.18 |
| 4 | 1300 | 0.13 |
Theory
The Bernoulli equation says that the energy difference between two nodes is due to the friction losses by fluid flow:
The friction losses (h) are proportional to the length of the section (L) and the adimensional head loss (J). Several models are available for J, for water the Hazen-Williams correlation is:
Where the coefficient C is specific for the material. There are four unknown flow rates (Q), we can write three Bernoulli differences between pairs of boundary nodes, and the last equation is the flow rate balance at the central node.
This kind of problems are solved easily using Excel's solver, giving as result Q1=92[l/s], Q2=15[l/s], Q3=55[l/s] and Q4=23[l/s]. The Excel sheet is available on this link.
