Worked example: a supply line with an in-line fitting

Pipe friction is only part of the pressure a supply line loses. Every bend, valve, tee and reducer adds its own loss, and on a short line those minor losses can add up to more than the friction in the straight pipe. They are easy to leave out of a quick hand calculation and easy to get wrong, because the loss across a fitting depends on the velocity squared, not on length. This worked example isolates the effect: a single supply line with one in-line elbow, so the fitting loss sits alongside the pipe friction and the elevation term.

The setup

A reservoir supplies a fixed demand through a single line that contains one in-line fitting:

• Reservoir at 40 m head.
• First leg: 250 m of 150 mm pipe at 0.15 mm roughness, up to the fitting.
• The fitting: a standard 90 degree elbow with a loss coefficient K of 0.9, on a 150 mm bore, sitting at 5 m elevation.
• Second leg: 250 m of 150 mm pipe at 0.15 mm roughness, down to the outlet.
• Outlet: a fixed demand of 0.02 cubic metres per second (20 L/s) at 5 m elevation.

The elbow and the outlet both sit 5 m above the reservoir datum, so elevation pulls the gauge pressure down as well, which is the third effect on show.

The physics and the method

The straight pipe loses head to Darcy-Weisbach friction with a Churchill (1977) friction factor. The elbow loses head as a fixed-K minor loss, K times the velocity head, following the Crane TP-410 approach, where the velocity head is the velocity squared over twice gravity. Because the flow is fixed by the demand boundary, the velocity in the 150 mm bore is set, so the friction loss, the fitting loss and the velocity head are all determined directly. The reservoir head, minus the two friction losses, minus the elbow loss, minus the rise in elevation, gives the head and hence the gauge pressure at the outlet.

Fluid Network Studio assembles the line, including the elbow as a minor-loss term on the node, and solves heads and flow together with the Global Gradient Algorithm (Todini and Pilati, 1988). The conservation residual is reported on every solve, and this incompressible mode is cross-checked against EPANET, part of the published verification cases. Separating the elbow out as its own element lets you see how much head it costs in its own right, rather than burying it in an equivalent length.

The solved result

The demand fixes the flow at 20 L/s, so the velocity in the 150 mm bore is 1.132 m/s and every loss follows from it. Here is the head accounted for from the reservoir to the outlet:

The elbow costs 0.0589 m and the straight pipe costs 4.662 m, so on a 500 m line one bend is very nearly a rounding error. What actually eats the delivered pressure is the 5 m of elevation, which is worth more than the whole friction budget. Reverse the geometry and the lesson reverses with it: shorten the two legs to 5 m each and the same elbow at the same 1.132 m/s still costs 0.0589 m, but the friction almost vanishes and the fitting becomes the dominant loss. Minor losses are only minor relative to the length of pipe they sit in.

The numbers above come from the solver rather than a spreadsheet, and they are reproducible. Open the example, press Solve, and check the head budget yourself.

What you learn

Solving the example gives you the velocity in the line, the friction loss in each leg, the head lost across the elbow, and the gauge pressure at the outlet. You can see how the outlet pressure is built from the reservoir head by subtracting friction, fitting loss and the elevation rise. The useful experiment is to change the fitting: a sharper bend or a partly closed valve carries a much higher K, and re-solving shows how quickly one bad fitting eats into the delivered pressure.

This example runs on the free plan. The glossary explains minor losses, the loss coefficient K and the difference between head and gauge pressure, and the how it works page covers the solver method.

Open this example in FNS and change the fitting's K value to see how much a single bend costs.