§ 3.4  Module 3 — Head Loss: Friction

Hazen-Williams and Its Traps

An empirical formula is a promise about the conditions it was fitted in. This one keeps its promise, and almost nobody reads it.

Most water utilities in the English-speaking world size their mains with Hazen-Williams, and few of the engineers doing it could say what the number in front of the formula is for. It is a turn-of-the-century fit with one parameter describing the pipe, no fluid properties, and a leading constant that changes value between unit systems while the equation around it stays identical. That is the trap: a borrowed spreadsheet returns an answer wrong by a fixed factor of 0.4433 — 56% too little head loss — plausible-looking, and in the direction that undersizes the pump.

3.4.1The formula, and where its exponents come from

How much head does a full pipe of water lose to friction, if you would rather not compute a Reynolds number? Williams and Hazen answered that as a velocity, V = 0.849·C·R^0.63·S^0.54, with V in m/s, hydraulic radius R in m and S the friction slope. The US customary form is identical but carries 1.318, with V in ft/s and R in ft — and the difference is not physics: 1.318 × 0.3048^0.37 = 0.84918. Remember that artefact.

Substitute R = D/4 and Q = VπD²/4 for a full circular pipe and that becomes the discharge form everybody actually uses: hf = 10.67 · L · Q1.852 / (C1.852 · D4.8704), metres, with Q in m³/s and D, L in m. 1.852 is 1/0.54 rounded, 4.8704 is 2.63/0.54 rounded, and 10.67 is what 0.849 becomes under the same substitution, since (0.849 · π/4 · 4^−0.63)^−1/0.54 = 10.674. Each constant was rounded independently, which is why the two forms disagree by up to 0.19% here.

Doubling the flow multiplies the loss by 2^1.852 = 3.610, not by 4; Darcy-Weisbach with a nearly constant f gives close to 4, and that gap is §3.4.3. Doubling the diameter divides it by 2^4.8704 = 29.25; one nominal step, 250 to 300 mm, by 1.2^4.8704 = 2.43. PVC at C = 150 against forty-year cast iron at C = 80 is 3.20 times the loss. Length is linear.

Worked below on the pipe the instrument uses — 300 mm inside diameter, 1200 m, cast iron in good condition at C = 130, 60 L/s — the loss is 2.993 m at 0.849 m/s (2.79 ft/s). That velocity sits just under the 0.9 m/s where §3.4.3 starts caring about the gap to Colebrook, and the instrument says so on its face rather than hiding it: 60 L/s in a 300 mm main is an ordinary shoulder-of-the-day duty, and a formula is worth watching at the edge of its band and not only in the middle of it. Not yet accounted for: the valves, elbows and meter in the station, which Module 4 adds and which on station pipework routinely exceed this.

pipe            300 mm inside diameter, 1200 m, C = 130, Q = 60 L/s
area            pi x 0.300^2 / 4              =   0.070686 m^2
velocity        0.060 / 0.070686              =   0.8488 m/s   (2.79 ft/s)

SI form,  k = 10.67, Q in m3/s, D and L in m:
  Q^1.852       0.060^1.852                   =   0.0054593
  C^1.852       130^1.852                     =   8222.9
  D^4.8704      0.300^4.8704                  =   0.0028403
  hf            10.67 x 1200 x 0.0054593
                / (8222.9 x 0.0028403)        =   2.9929 m
  gradient      2.9929 / 1200 x 100           =   0.2494 m per 100 m

US form,  k = 4.73, Q in cfs, D and L in ft:
  Q             0.060 / 0.0283168             =   2.11888 cfs
  D             0.300 / 0.3048                =   0.984252 ft
  L             1200  / 0.3048                =   3937.008 ft
  Q^1.852       2.11888^1.852                 =   4.017436
  C^1.852       130^1.852                     =   8222.9      <- same C, same power
  D^4.8704      0.984252^4.8704               =   0.9256034
  hf            4.73 x 3937.008 x 4.017436
                / (8222.9 x 0.9256034)        =   9.8294 ft
  back to SI    9.8294 x 0.3048               =   2.9960 m   <- 0.10% from 2.9929

One pipe, both forms (src/core/hydraulics.js). Every intermediate is shown to the digits the next line uses, so the chain can be checked on a calculator. The cross-unit agreement is the rounding between 10.67 and 4.73 — exactly 0.105% on the unrounded constants, and 0.10% once these five-figure intermediates have been through it.

3.4.2The constant that silently breaks a borrowed spreadsheet

The equation is the same in both unit systems. Only the number in front changes: 10.67 with Q in m³/s and D, L in metres, returning metres; 4.73 with Q in cubic feet per second and D, L in feet, returning feet. Nothing in the formula's shape says which system a cell was written for, and nothing in the answer's magnitude says it either.

That the two describe one law is provable in a line. Head loss carries ft → m once and length m → ft once, and those cancel; flow carries cfs → m³/s to the power 1.852, diameter ft → m to the power 4.8704. So 4.73 × 35.3147^1.852 × 0.3048^4.8704 = 10.681, 0.105% from the conventional 10.67. The whole difference between the published constants is rounding.

Now the failure. Type the worked pipe into a US-built spreadsheet — 0.060, 0.300, 1200 and 130 into cells that expect cfs and feet, with 4.73 in the constant — and it returns 1.327 m where the truth is 2.993 m. That is 0.4433 of the truth, which is 4.73/10.67 exactly, at every flow, diameter, length and C. It does not grow with the pipe and does not vary with the duty point, so no sensitivity check finds it. And its sign is the bad one: friction is under-stated, the total dynamic head is short, and the pump chosen against that TDH does not deliver the design flow. The mirror-image mistake, 10.67 with feet and cfs, is 2.2558 times too high — which wastes capital and gets caught, because the head looks absurd.

The defence costs thirty seconds: compute the pipe in the other system, convert, and compare — 9.8294 ft × 0.3048 = 2.9960 m against 2.9929 m. Disagree by more than about a tenth of a percent and one of them has the wrong constant. It is the only test a wrong leading coefficient cannot survive.

3.4.3C is not roughness

C is tabulated by material and age, printed in the same handbook chapter as absolute roughness, and used as though the two described one wall interchangeably. Ask what roughness makes Colebrook agree with a given C and the question refuses to be answered without a velocity. That refusal is the finding.

Pin the two methods together at 1.5 m/s and 15 °C in the 300 mm pipe and the pairings land on real walls: C = 150 with ε = 0.016 mm, 130 with 0.195 mm, 120 with 0.401 mm, 100 with 1.471 mm. Plausible magnitudes, all four. Hold them against this course's own tables, though, and the map is well conditioned only at the rough end: 1.471 mm is within 2% of the 1.5 mm listed for aged cast iron, whose tabulated C is 100. The smooth end does not line up at all. The 0.0015 mm listed for PVC and HDPE is ten times smoother than the 0.016 mm paired with their tabulated C = 150 — and running that real 0.0015 mm back through the pairing returns C = 153. Tenfold in the wall, three points in C. That is not two scales describing the same objects; it is one scale that has run out of resolution, which is the harder version of this section's argument.

Now hold the wall fixed instead. ε = 0.26 mm in this pipe behaves like C = 135 at 0.14 m/s, 130 at 0.85 m/s, 125 at 1.70 m/s and 119 at 3.54 m/s. Nothing about the pipe changed. C drifted 12% because it is not a property of the pipe: it is the fitting parameter of a formula whose flow exponent is nailed down, so it absorbs whatever Reynolds-number dependence the real friction factor has at the calibration velocity.

As a claim about the Moody chart: since h_f ∝ Q^1.852 and f = 2gDh_f/(LV²), the friction factor it implies goes as V^−0.148 at every Reynolds number and every roughness: a tenfold change in flow moves it by exactly 10^−0.148 = 0.7112. That is a straight line of fixed slope; the real curves are not. At ε/D = 0.01 Colebrook gives 0.03796 at Re = 10⁶ and 0.03790 at Re = 10⁸, flat to 0.2% across two decades, which is §3.2's fully-rough plateau. Hazen-Williams cannot reach a plateau, because that slope is a constant of the formula and not a property of the pipe.

A field-calibrated C is the honest version: back it out of a pressure test at a measured flow and it is valid at that flow, stating its own conditions. A table value is valid at somebody else's. Thirty years of either is §3.5.

3.4.4The viscosity term it does not have

Hazen-Williams contains no temperature, no density and no viscosity. It contains water, implicitly, at about the temperature its gauged pipes ran at — usually quoted as 60 °F (15.6 °C). Darcy-Weisbach carries viscosity through the Reynolds number, and so does have a temperature.

The missing property is not density, which is the answer people reach for: head loss in metres of the fluid pumped does not depend on density at all. It is viscosity, and viscosity moves a great deal — water's kinematic viscosity is 1.733 × 10⁻⁶ m²/s at 1 °C against 6.580 × 10⁻⁷ at 40 °C, a factor of 2.63.

How much that costs depends on where the pipe sits on Figure 3.4.1. On the 300 mm × 1200 m pipe at 1.5 m/s (106 L/s), the rough pairing (C = 120, ε = 0.401 mm) gives 10.10 m at 1 °C and 9.85 m at 40 °C by Darcy, a spread of 2.6%, against 9.96 m at both temperatures by Hazen-Williams. The smooth pairing (C = 150, ε = 0.016 mm) gives 7.05 m and 6.10 m, a spread of 15.5%, against 6.59 m at both. The missing viscosity term is nearly harmless in an old rough main and worth 15% in new plastic — 19% if you use the 0.0015 mm the table actually lists for PVC rather than the 0.016 mm the C = 150 pairing implies — which is why the formula took hold in water distribution and never in process or slurry work.

Where it bites here is specific: a new HDPE or PVC force main carrying wastewater that is not at 15 °C. A station lifting 30 °C septic wastewater through new plastic is the ordinary sanitary case where the two methods select different pumps. Use Darcy-Weisbach.

3.4.5When it must not be used

Hazen-Williams has a stated domain. Every item below is a place practice puts it outside that domain.

1. Anything that is not water. No fluid property appears in the equation, so nothing changes when the fluid does. Sludge, polymer, glycol, oil: Darcy-Weisbach with a measured viscosity.
2. Laminar or near-laminar flow. There is no laminar branch. Below Re ≈ 2000 the exact answer is f = 64/Re, and this is not it.
3. Pipes under 2 in (50.8 mm), or velocities above 10 ft/s (3.048 m/s). Outside the sizes and velocities the fit was made on; the velocity bound is the firmer, and in a 300 mm pipe it is 215 L/s. Round it to 3 m/s and you have moved the limit to 212 L/s, which is a different pipe duty from the published one.
4. Far from the velocity C was fitted at. Pinned at 1.5 m/s and taken to 0.14 m/s, Darcy is 84% of Hazen-Williams; taken to 4.24 m/s, 115%. Two 15% errors in opposite directions, from one C.
5. Water far from about 16 °C in a smooth pipe. 15.5% between 1 °C and 40 °C above.
6. Any number you must defend to better than about 5%. Surge analysis, an energy claim, a capacity dispute, a thin pump margin.

The last item deserves a number, because the usual objection to Darcy-Weisbach is that it is fussier. Colebrook iterated against the explicit Swamee-Jain fit was measured for this course over a grid of Reynolds numbers and roughnesses: worst disagreement 0.81% for Re ≥ 10⁵ across ε/D from 10⁻⁵ to 10⁻² — and under 1% anywhere inside that box, not just on the grid points — rising to 2.8% only at the corner of Re = 5 × 10³ with very rough pipe (tools/verify-hydraulics.mjs, which asserts 1.2% and 3% as ceilings; §3.3 maps it). A force main lives above Re = 10⁵, so an explicit friction factor costs under 1% — a fifteenth of what operating away from a fitted C costs.

So use it where it is honest — water, ordinary temperature, a main inside the velocity band, a solver that wants a power law — and record what you did. Nothing here touches how C degrades with age (§3.5) or the fittings on top of pipe friction (Module 4). The habit to carry forward: "h_f = 3.0 m" is not a result. "h_f = 3.0 m, Hazen-Williams SI form, C = 130, 300 mm × 1200 m at 60 L/s, 15 °C" is a result, because somebody else can check it.