§ 3.2  Module 3 — Head Loss: Friction

Darcy-Weisbach and the Moody Chart

Friction is the tax on every metre of pipe, and it is levied on velocity squared.Module 3

Module 2 wrote friction into the energy equation as a single symbol, h_f, and left it there. This lesson computes it. The equation has four numbers and three are on the drawing; the fourth, the friction factor, depends on how fast the water is going and how rough the wall is, and the Moody chart is that dependence drawn on one page. The part worth understanding properly is the chart's flat right-hand side, where the friction factor stops responding to flow at all — a pipe operating there has a head loss that is a pure quadratic, and nothing you do to the water will change it.

3.2.1Four numbers, one of which is not on the drawing

The friction loss in a length of full pipe running at steady flow is

hf = f · (L/D) · v²/2g

and each factor earns its place. L/D is dimensionless — a count of how many diameters long the pipe is, the only length a pipe cares about. v²/2g is the velocity head from §2.2, in metres of fluid. And f is dimensionless too, which is the reason to prefer this equation: with every factor either a ratio or a head, the same arithmetic runs in SI and in US customary units without a single coefficient changing — not true of the alternative in §3.4, and the reason Darcy-Weisbach is the method to compute in.

given   Q = 30 L/s (476 US gpm)   D = 150 mm (5.91 in)   L = 900 m (2 953 ft)
        unlined ductile iron, roughness e = 0.26 mm     water at 15 C

area          A = pi D^2/4        = 0.017671 m2
velocity      v = Q/A             = 1.698 m/s        (5.57 ft/s)
velocity head v^2/2g              = 0.1469 m         (0.4821 ft)
diameters     L/D                 = 6 000            (dimensionless)
Reynolds      Re = vD/nu          = 2.23 x 10^5      (nu = 1.139e-6 m2/s at 15 C)
rel roughness e/D                 = 0.001733
friction f    Colebrook-White     = 0.02344

SI            hf = 0.02344 x 6000 x 0.1469  =  20.66 m
US customary  hf = 0.02344 x 6000 x 0.4821 =  67.79 ft   = 20.66 m

hydraulic gradient                = 2.30 m per 100 m of pipe

The §1.2 station's force main, priced. Computed with headLossDarcy from src/core/hydraulics.js; the US customary column is the identical equation with g = 32.174 ft/s², and the agreement is a check on the arithmetic, not a coincidence.

Sit with that number. §1.2 measured this station's static lift between water surfaces at about 18 m and called the force main diameter the governing decision. The friction is 20.66 m — larger than the lift. The pump spends more of its work pushing water through a pipe than raising it, every hour it runs, for thirty years. Note what the figure excludes: the entrance, the check valve, the bends and the meter, all priced in Module 4.

That rearrangement is where the design intuition lives. At a flow the pump fixes, h_f goes as 1/D⁵ — the fifth power, because widening the pipe both slows the water (velocity as 1/D², and the loss as its square) and shortens the count of diameters. Hold this station's 30 L/s, its 900 m and its 0.26 mm wall, and change nothing but the bore:

Going from 150 mm to 200 mm divides the friction by 4.42. A pure 1/D⁵ law predicts (200/150)⁵ = 4.21; the extra 5% is f falling from 0.02344 to 0.02234 as the flow slows, which is the next section's subject. Read the table the other way too: the 100 mm main is 171 m of friction on an 18 m lift, a pipe no catalogue pump can serve. And the 300 mm main, the obvious winner on this page, runs at 0.42 m/s — below the 0.6 m/s §1.3 gave as common minimum practice for a sewage force main, so it silts. Friction is one term in the decision, not the decision.

3.2.2The friction factor is not a property of the pipe

The temptation is to look up f for ductile iron the way you look up a density. It does not work: the friction factor takes two arguments, the Reynolds number of the flow and the relative roughness ε/D. The same 0.26 mm wall is ε/D = 0.00260 in a 100 mm pipe and 0.000867 in a 300 mm one — the wall did not change, the pipe's opinion of it did. Two limits of that function have closed forms you can evaluate on paper.

Below Re ≈ 2000 the flow is laminar and f = 64/Re, exactly. Not a fit: substituting it into Darcy-Weisbach reproduces Hagen-Poiseuille, h_f = 32μLv/(ρgD²), the analytic solution for a round tube, in which roughness does not appear at all. Notice the exponent: with f ∝ 1/v, the loss goes as v, not v². Laminar is linear, turbulent is quadratic, which is why §3.1 classified the regime first. No municipal pipe lives here — this main would have to drop below 0.27 L/s.

At the other extreme, when the wall is rough enough, the Reynolds number drops out entirely and von Kármán's fully-rough law applies: 1/√f = 2·log₁₀(3.7D/ε) — a closed form with no iteration and no flow in it. For the rough concrete option on this station, ε/D = 3.0/150 = 0.0200, so 1/√f = 2·log₁₀(185) = 4.534 and f = 0.04864, a number you can produce with a log table and defend in a review. Between the extremes lies the transitional region, where f depends on both arguments and no closed form exists. §3.3 solves the implicit equation; this lesson reads the chart.

3.2.3Saying which region you are in

"Which region" has a physical answer, not merely a graphical one. Right at the wall turbulence cannot survive: a thin viscous sublayer clings there, its thickness set by the friction velocity u* = v·√(f/8). Taking that layer's conventional edge at y⁺ = 5 gives δ_v = 5ν/u*. Comparing the roughness height to that thickness is the physical question, but it is not quite the number everyone quotes: the standard criterion strips the y⁺ = 5 convention back out and measures ε against ν/u* alone. That is the roughness Reynolds number, k⁺ = ε·u*/ν — five times ε/δ_v, so a wall standing one sublayer tall is already k⁺ = 5. Nikuradse's sand-grain experiments put the boundaries at k⁺ < 5 hydraulically smooth, 5 to 70 transitional, k⁺ > 70 fully rough. Bumps shorter than the sublayer hide inside it; bumps standing clear are obstacles in a turbulent stream.

Put the station's main through it. At 1.698 m/s with f = 0.02344 the friction velocity is 0.0919 m/s, the sublayer 0.0620 mm thick, and the 0.26 mm roughness stands 4.2 times that. So k⁺ = 21: transitional, with f sitting 3.9% above the 0.02255 von Kármán's law would give it if it were fully rough. Now both ends of the same bore, same flow, same water: new PVC at ε = 0.0015 mm has k⁺ = 0.098 and roughness 2% of its sublayer — drowned, hydraulically smooth, h_f = 13.57 m. Rough concrete at ε = 3.0 mm has k⁺ = 349 and h_f = 43.04 m (141 ft). A three-fold spread, from the wall alone.

Set that against a piece of folklore. "Municipal pipes are always fully rough" is not true: for this main to reach k⁺ = 70 you would have to run it at 5.77 m/s — 102 L/s, three times the design flow — or give it a wall of at least 0.756 mm roughness. Aged cast iron at 1.5 mm does qualify, at k⁺ = 154 and within 0.7% of its plateau. The plateau is where a pipe ends up, not where it starts — the argument of §3.5.

3.2.4Why the right-hand side goes flat

The mechanism is a change in what the resistance is. In a smooth pipe the whole loss is viscous shear through the sublayer, and as the flow speeds up that layer thins relative to the bore, so the pipe grows proportionally slipperier — hence a smooth-pipe curve that falls right across the chart, from f = 0.0309 at Re = 10⁴ to 0.0180 at 10⁵, 0.0116 at 10⁶ and 0.00594 at 10⁸. Once the protrusions stand clear the water must go around them and they shed wakes. The loss is then pressure drag, going as ρv² with no viscosity in it at all — and a resistance already proportional to v² leaves f nothing to do.

The cleanest way to see it is to change the viscosity and watch which pipes notice. Warm the same 150 mm × 900 m main at a fixed 30 L/s from 4 °C to 45 °C, cutting kinematic viscosity from 1.569 × 10⁻⁶ to 0.602 × 10⁻⁶ m²/s and raising Reynolds from 1.62 × 10⁵ to 4.23 × 10⁵ without moving a litre per second:

• PVC, ε = 0.0015 mm — h_f falls from 14.43 m to 12.08 m, a change of −16.3%. Almost all of this pipe's loss is viscous, and warming the water genuinely removes some of it.
• Unlined ductile iron, ε = 0.26 mm — 20.93 m to 20.31 m, −3.0%. Partly exposed roughness, partly viscous loss, and the answer is in between.
• Rough concrete, ε = 3.0 mm — 43.09 m to 42.96 m, −0.30%. On the plateau the water's viscosity has stopped being relevant.

Three consequences, each used later. A pipe on the plateau has a friction loss that is exactly quadratic in flow — double the flow, quadruple the loss, to three figures — which is what makes a system curve a clean parabola above the static lift in Module 7; off the plateau the exponent is nearer 1.8, and a smooth pipe returns 3.53× for double the flow rather than 4.00×. Warming the fluid buys a rough main no discount, though §6.5 charges for the warmth at the suction. And on the plateau the only lever left is ε/D — which is why a design done on as-new roughness is a design for one year of a thirty-year asset.

3.2.5What this lesson has left open

Every friction factor between the two closed forms came out of Colebrook-White, which is implicit: f appears on both sides and must be iterated. That is why the chart existed at all, and why explicit approximations are still in every spreadsheet in the industry. The most common, Swamee-Jain, does well — for the station main the two differ by 0.71%, and a measured sweep in tools/verify-hydraulics.mjs puts the worst disagreement under 1.2% for Re ≥ 10⁵ across relative roughness 10⁻⁵ to 10⁻², rising to about 3% at the extreme corner of Re = 5 × 10³ with very rough pipe. The error is largest exactly where municipal work is not, and §3.3 maps it rather than repeating the folklore figure of 1%.

Three other debts. The roughness values above are design values for pipe in service, not laboratory values for pipe on a pallet; choosing them for year thirty is §3.5. The empirical formula most water utilities actually use, and its unit-dependent leading coefficient, is §3.4 — worth knowing precisely because it lacks the dimensional honesty this lesson opened with. And the pump remains a black box, assumed to deliver 30 L/s against whatever this pipe demands, which is what Module 6 and §7.2 take apart. What you can do now is the calculation itself, and the plateau check.