What friction loss is and why every pipe run has it
Push a fluid through a pipe and some of its pressure disappears. It is not stored anywhere and you cannot recover it downstream — it is converted to heat by shear inside the fluid and at the wall. That loss is what the Darcy-Weisbach equation quantifies, and it is the single largest term in most pump-selection and pipe-sizing decisions.
The equation says the loss is proportional to three things. It rises in direct proportion to the length of the run, because every foot of wall does the same amount of shearing. It falls as diameter rises, because a fatter pipe carries the same flow at lower velocity and offers less wall per unit of volume. And it rises with the square of velocity, which is why doubling the flow through an existing pipe roughly quadruples the pressure drop rather than doubling it.
Darcy-Weisbach is the general form. Unlike Hazen-Williams, which is an empirical fit to water at ordinary temperatures, it works for glycol, fuel oil, brine and hot water alike, because the fluid's density and viscosity enter explicitly through the Reynolds number rather than being baked into a coefficient. If you are moving anything other than cool clean water, this is the equation to use.
Reading the formula term by term
Start with the ratio L/D. Both are lengths, so the ratio is dimensionless — it counts how many pipe diameters long the run is. A 100 ft run of 2-inch pipe is 600 diameters; the same 100 ft of 6-inch pipe is 200. That single ratio is why small pipe punishes you so heavily.
Next is V²/2g, the velocity head. It is the height a column of fluid would have to fall from to reach velocity V, and it is the natural energy unit for flow problems. At 4 ft/s it is 0.249 ft; at 8 ft/s it is 0.994 ft. Everything in pipe hydraulics — elbows, valves, sudden expansions — is eventually expressed as some multiple of the velocity head.
The friction factor f is the only part that requires work. Below a Reynolds number of about 2,000 the flow is laminar, moves in orderly layers, and f = 64/Re exactly. Wall roughness is irrelevant there, because the fluid nearest the wall is not moving. Above about 4,000 the flow is turbulent and f depends on both Reynolds number and relative roughness ε/D through the Colebrook-White equation. Colebrook-White is implicit — f appears on both sides — so this calculator solves it by fixed-point iteration to a tolerance far tighter than any roughness value you can actually know.
Between Re 2,000 and 4,000 lies the critical zone. Nothing predicts it well, because the flow itself is unstable and switches back and forth. If your design lands there, the calculator says so, and the correct response is to size for the turbulent value and accept the uncertainty.
Worked example: 100 gpm of 60 °F water through 100 ft of 4-inch steel
Take 100 gpm of water at 60 °F through 100 ft of 4-inch Schedule 40 steel pipe. The bore is 4.026 in, commercial steel roughness is ε = 0.0018 in, water density is 62.37 lb/ft³ and kinematic viscosity is 1.13 cSt.
- Flow area. D = 4.026 ÷ 12 = 0.3355 ft, so A = π/4 × 0.3355² = 0.08840 ft².
- Convert the flow. 100 gpm ÷ 448.831 = 0.22280 ft³/s.
- Velocity. V = 0.22280 ÷ 0.08840 = 2.520 ft/s.
- Kinematic viscosity in US units. 1.13 cSt × 1.07639 × 10⁻⁵ = 1.2163 × 10⁻⁵ ft²/s.
- Reynolds number. Re = 2.520 × 0.3355 ÷ 1.2163 × 10⁻⁵ = 69,520. That is comfortably turbulent.
- Relative roughness. ε/D = 0.0018 ÷ 4.026 = 0.000447.
- Friction factor. Solving Colebrook-White at Re = 69,520 and ε/D = 0.000447 gives f = 0.02100.
- Velocity head. 2.520² ÷ (2 × 32.174) = 6.3504 ÷ 64.348 = 0.09869 ft.
- Head loss. hf = 0.02100 × (100 ÷ 0.3355) × 0.09869 = 0.02100 × 298.06 × 0.09869 = 0.618 ft.
- Pressure drop. Δp = 62.37 × 0.618 ÷ 144 = 0.268 psi.
So the whole 100 ft run costs about a quarter of a psi. That is what a well-sized main should look like. Re-run the same numbers at 2-inch bore and the velocity jumps to roughly 10 ft/s and the loss to several psi per hundred feet — the arithmetic in step 3 and step 9 is doing all the work.
How to read the answer
Judge the result against loss per 100 feet, not total loss, because that is the number every published sizing table uses. In building water service, designers commonly work to somewhere between 2 and 8 psi per 100 ft of developed length, with the exact figure set by how much pressure you have to spend — see the water service pipe size calculator for the budget arithmetic. Long transmission mains are sized far tighter than that because pumping energy, not available pressure, is the binding constraint.
Velocity is the second check and often the governing one. Regardless of what the friction number says, most water piping is held to about 8 ft/s in cold service and lower in hot; the pipe water velocity calculator compares your figure to the usual material limits. Erosion-corrosion, noise and water hammer all scale with velocity, and none of them appears anywhere in Darcy-Weisbach.
Finally, look at where the friction factor sits. For water in ordinary pipe it lands between about 0.015 and 0.030 in nearly every case. If you get 0.4 you are in laminar flow, which for water means the flow is tiny; if you get 0.008 something is wrong with your viscosity entry. Use f as a sanity check on the inputs before you trust the head loss.
Absolute roughness of common piping materials
| Material | ε (in) | ε (mm) | ε/D at 4.026 in bore |
|---|---|---|---|
| Drawn tubing, copper, glass | 0.00006 | 0.0015 | 0.0000149 |
| PVC, CPVC, PE, PEX | 0.00006 | 0.0015 | 0.0000149 |
| Commercial steel, wrought iron | 0.0018 | 0.046 | 0.000447 |
| Galvanised steel | 0.006 | 0.15 | 0.00149 |
| Cast iron, uncoated | 0.010 | 0.26 | 0.00248 |
| Concrete, smooth finish | 0.012 | 0.30 | 0.00298 |
| Riveted steel | 0.035–0.35 | 0.9–9.0 | 0.0087–0.087 |
Roughness values are the classical Moody set reproduced in Crane Technical Paper 410. The ε/D column is computed as ε ÷ 4.026 for the example bore used on this page.
Mistakes that make the answer wrong
- Using nominal size as the bore. Nominal 4-inch Schedule 40 steel has a 4.026 in inside diameter and 4-inch Schedule 80 has 3.826 in. Because loss goes roughly as D⁻⁵ at fixed flow, a 5% diameter error is a 25% loss error.
- Forgetting equivalent length. Darcy-Weisbach as written covers straight pipe only. Elbows, tees, valves and the meter each add an equivalent length; on a short branch with many fittings, they can exceed the straight run. Add them to the length input.
- Entering dynamic viscosity where kinematic is wanted. The Reynolds number needs kinematic viscosity ν, in cSt or ft²/s. If your data sheet gives centipoise, divide by the specific gravity to get centistokes.
- Assuming new-pipe roughness for old pipe. Tuberculation in unlined ferrous mains raises ε by an order of magnitude and shrinks the bore at the same time. For an aged main, model the reduced bore explicitly rather than nudging the roughness.
- Applying it to a partly full gravity pipe. Darcy-Weisbach assumes a full pipe under pressure. For open-channel flow in a sewer or storm drain use the Manning's equation pipe flow calculator instead.
- Ignoring elevation. Friction is only one part of the pressure change. Static lift adds or removes 0.433 psi per foot for water; convert with the psi to feet of head converter.
Darcy-Weisbach against the alternatives
Three methods dominate practice, and they disagree in predictable ways. Darcy-Weisbach is the physically general one and the only one that handles a fluid other than water. Hazen-Williams is an empirical water-only fit whose roughness coefficient C is dimensionally inconsistent, which means its accuracy degrades outside the velocity and diameter range it was fitted to — roughly 3 to 10 ft/s in pipe 2 inches and larger. Manning's equation is for open-channel and partly full gravity flow and does not apply to pressurised pipe at all.
Fire protection engineering standardised on Hazen-Williams, and NFPA 13 hydraulic calculations are still performed with it, so a sprinkler submittal computed with Darcy-Weisbach will not match the reviewer's arithmetic. Process and mechanical engineering standardised on Darcy-Weisbach, and Crane Technical Paper 410 is the reference that made it the default. Use whichever one the governing document names, and use Darcy-Weisbach whenever nobody has named one.
The Moody diagram you may remember from a fluid mechanics course is nothing more than Colebrook-White plotted on log axes, with the laminar line f = 64/Re at the left. Solving the equation numerically, as this page does, is the same calculation without the chart-reading error. Once you have the head loss, convert it to a pump duty point with the sump pump capacity calculator or work backwards to a required bore with the Hazen-Williams tool.
Key terms
- Head loss
- Energy lost per unit weight of fluid, expressed as a height of that fluid. It is independent of density, which is why it is the preferred unit in pump work — the same pump develops the same head on any liquid.
- Reynolds number
- The ratio of inertial to viscous forces, VD/ν. It is dimensionless, so a value of 70,000 means the same flow structure whether the fluid is water in a 4-inch main or air in a duct.
- Relative roughness
- Absolute roughness divided by bore, ε/D. Only this ratio matters, not the absolute height of the bumps, because the turbulent boundary layer scales with pipe size.
- Velocity head
- V²/2g, the kinetic energy of the stream expressed as a height. Minor losses through fittings are catalogued as multiples of it.