Worked example: a single-loop network
Written by Haimi Jordaan, MEng (Mechanical), University of Pretoria. Seven years in a specialist engineering analysis and design group across CFD, FEA and DEM. He wrote the solver behind Fluid Network Studio.Published . Last updated .
Once a pipe network contains a loop, it stops being something you can work out branch by branch. A loop gives the flow more than one path between the same two points, and the split is not free to choose: it must settle so the head loss is the same whichever way you go around the loop. Get that balance wrong and the heads at the shared nodes would not agree, which is impossible. This worked example is the smallest network that shows the effect: one loop, two parallel branches, and a flow that divides itself between them.
The setup
A single reservoir feeds the network, which splits into two parallel branches and recombines at a fixed-pressure boundary. Every pipe has a roughness of 0.5 mm:
- Supply: 500 m of 300 mm pipe from the reservoir at 50 m head to the first junction.
- Upper branch: 600 m of 200 mm pipe, then another 600 m of 200 mm pipe to the outlet.
- Lower branch: 600 m of 150 mm pipe, then another 600 m of 150 mm pipe to the outlet.
- Outlet: a fixed-pressure boundary at 245 000 Pa (245 kPa gauge).
The two branches are deliberately different sizes. The wider 200 mm branch offers less resistance than the narrower 150 mm one, so it should carry the larger share of the flow, but exactly how the total divides is the question the solver answers.
The physics and the method
Each pipe carries a Darcy-Weisbach head loss with a Churchill (1977) friction factor. The governing condition is that the head loss along the upper branch must equal the head loss along the lower branch, because both connect the same junction to the same outlet pressure. Since head loss rises with the square of velocity and falls steeply with diameter, the narrower branch builds head loss far faster per unit of flow, so it carries proportionally less. The flow split is whatever makes those two head losses equal while continuity holds at every node.
Fluid Network Studio solves this without you having to identify the loop or apply the Hardy Cross method by hand. The Global Gradient Algorithm (Todini and Pilati, 1988) works directly from the incidence matrix, with no loop detection step, and solves all the nodal heads and pipe flows at once. The fixed-pressure boundary anchors the downstream end, the reservoir anchors the upstream end, and the solver finds the internal heads that satisfy both continuity and the head-loss relation on every pipe. The conservation residual is reported each solve, and this incompressible mode is cross-checked against EPANET, part of the published verification cases.
The solved result
The reservoir delivers 78.43 L/s, and it divides 53.47 L/s down the 200 mm branch against 24.96 L/s down the 150 mm one:
| Pipe | Diameter (mm) | Flow (L/s) | Velocity (m/s) | Head loss (m) |
|---|---|---|---|---|
| L0, supply | 300 | 78.43 | 1.110 | 2.414 |
| L1, upper branch first half | 200 | 53.47 | 1.702 | 11.28 |
| L3, upper branch second half | 200 | 53.47 | 1.702 | 11.28 |
| L2, lower branch first half | 150 | 24.96 | 1.413 | 11.28 |
| L4, lower branch second half | 150 | 24.96 | 1.413 | 11.28 |
The heads from the reservoir to the outlet:
| Node | Head (m) | Gauge pressure (kPa) |
|---|---|---|
| Reservoir | 50.00 | fixed boundary |
| J1, where the loop splits | 47.59 | 465.8 |
| J2, upper branch midpoint | 36.31 | 355.4 |
| J3, lower branch midpoint | 36.31 | 355.4 |
| Outlet boundary | 25.03 | 245.0 |
The loop condition is visible straight off the table: each branch loses 22.56 m in total, so the two paths from J1 to the outlet agree to the digits shown, and the energy residual around the loop is 9.3 x 10^-10 m. The split is the interesting part. The wider branch is only a third larger in diameter, yet it carries 68 per cent of the flow, because at a fixed head loss the capacity of a pipe climbs roughly as the two-and-a-half power of its diameter (head loss itself falls close to D^-5 at fixed flow). Widening a branch a little buys a lot of flow, which is why looped systems redistribute so sharply when one main is upsized.
These values are reproducible. Open the example, press Solve, and check them against your own hand balance of the loop.
What you learn
Solving the network tells you how the total flow divides between the two branches, the velocity in each, and the head at the two interior junctions, and you can confirm that the head loss around the loop balances. The instructive move is to change one branch diameter and re-solve: widen the narrow branch and watch it pick up more flow, or narrow it further and watch the wider branch take over almost entirely. That is the core behaviour of every looped distribution system, shown at its simplest.
This example runs on the free plan. The glossary explains head loss and the hydraulic grade line, and the how it works page describes the Global Gradient Algorithm in more detail.
Open this example in FNS and change a branch diameter to see the flow split shift.