Water Resources HubTheory explainer
Manning vs Hazen–Williams vs Darcy–Weisbach: Which Equation, When?
Three equations, one decision. Candidates lose more points picking the wrong friction equation than solving the right one wrong — this page gives you the decision rule, the validity limits, and one fully worked, independently verified example per equation.
Last reviewed: 2026-10-03. All worked examples were independently re-solved by hand; the calculator pages cross-check each answer.
Manning's equation — open-channel flow with a free surface
V = (k/n) · R2/3 · S1/2 Q = A·V
| k | 1.486 for US customary units (ft/s), 1.0 for SI (m/s) — the classic exam trap |
| R = A/P | hydraulic radius = cross-sectional area / wetted perimeter |
| n | Manning's roughness coefficient (empirical, material + condition) |
| S | longitudinal bed slope (dimensionless, ft/ft or m/m) |
Manning's equation is empirical — it describes uniform flow (constant depth along the channel) in gravity-driven channels with a free surface. Uniform flow means the energy grade line, water surface, and channel bed are all parallel. It does not apply to pressurized pipes, rapidly varied flow (jumps, weirs), or backwater curves — it is the uniform-flow equation, not the general open-channel equation.
Typical Manning's n values (US customary)
| Finished concrete (formed) | 0.012–0.014 |
| Concrete pipe | 0.013 |
| Clean earth channel | 0.022 |
| Earth channel with weeds/light brush | 0.025–0.035 |
| Natural stream, clean and straight | 0.030 |
| Natural stream with pools and vegetation | 0.040–0.050 |
| Corrugated metal pipe | 0.022–0.024 |
| PVC / smooth plastic | 0.009–0.011 |
Exam problems almost always give you n. The table above is for the rare qualitative item — and for sanity-checking answers (a concrete channel solved with n = 0.13 instead of 0.013 is off by a factor of ~4.6 in velocity).
Worked example Trapezoidal channel discharge (US units)
Given: a trapezoidal channel with bottom width b = 6 ft, flow depth y = 2 ft, side slopes z = 2 (H:V), Manning's n = 0.014, bed slope S = 0.0016. Find Q under uniform flow.
Solution:
- Area: A = (b + z·y)·y = (6 + 2×2)×2 = 10 × 2 = 20.0 ft²
- Wetted perimeter: P = b + 2y√(1+z²) = 6 + 4√5 = 6 + 8.9443 = 14.9443 ft
- Hydraulic radius: R = A/P = 20.0/14.9443 = 1.3383 ft
- Velocity: V = (1.486/0.014) × (1.3383)2/3 × √0.0016 = 106.143 × 1.2144 × 0.04 = 5.156 ft/s
- Discharge: Q = A·V = 20.0 × 5.156 = 103.1 cfs
Answer: Q ≈ 103 cfs. Sanity check: 5.2 ft/s is a plausible (if brisk) channel velocity, and R = 1.34 ft is comfortably less than the 2-ft depth — the classic R-vs-depth trap avoided.
Verify it yourself on our Manning's equation calculator — it must return the same numbers.
Hazen–Williams — pressurized water pipes, typical distribution ranges
Velocity form: V = 1.318 · C · R0.63 · S0.54 (US: V in ft/s, R in ft)
Head-loss form: hf = 4.727 · L · Q1.852 / (C1.852 · D4.87) (US: Q in cfs, D in ft, L in ft)
| C | Hazen–Williams roughness coefficient — higher = smoother (not a friction factor, not interchangeable with f) |
| R = D/4 | for a full circular pipe, hydraulic radius is D/4 |
| S = hf/L | friction slope (energy grade line slope) |
Hazen–Williams is empirical and narrow: water only, turbulent flow, typical water-distribution pipe sizes and velocities. It is the workhorse of distribution-system problems (pipe networks, Hardy Cross, service pressures) — and the wrong answer for oil, gas, slurries, laminar flow, or very large/small diameters. Those belong to Darcy–Weisbach. The SI velocity constant is 0.849 (not 1.318) — another unit trap.
Typical Hazen–Williams C factors
| PVC / polyethylene (any age) | 140–150 |
| New ductile iron, cement-lined | 130–140 |
| New steel, lined | 130–140 |
| Aged ductile iron (10–20 yr) | 110–120 |
| Old cast iron (30+ yr, tuberculated) | 80–100 |
| Concrete pipe | 120–130 |
Exam trap: pick C for the pipe's age and condition, not its material alone. A 40-year-old cast-iron main at C = 90 loses dramatically more head than the same pipe new at C = 130.
Worked example Head loss in a water main (US units)
Given: Q = 3.0 cfs through L = 2,500 ft of 12-inch (D = 1.0 ft) ductile-iron pipe, C = 120 (aged but not tuberculated). Find the friction head loss hf, and cross-check with the velocity form.
Solution (head-loss form):
- hf = 4.727 · 2,500 · (3.0)1.852 / [ (120)1.852 · (1.0)4.87 ]
- (3.0)1.852 = 7.612; (120)1.852 = 7,055.6; (1.0)4.87 = 1.0
- hf = 4.727 × 2,500 × 7.612 / 7,055.6 = 89,962 / 7,055.6 = 12.75 ft
Cross-check (velocity form):
- Velocity: V = Q/A = 3.0 / (π·1.0²/4) = 3.0/0.7854 = 3.820 ft/s
- Friction slope: S = hf/L = 12.75/2,500 = 0.00510
- V = 1.318 · 120 · (0.25)0.63 · (0.00510)0.54 = 158.16 × 0.4185 × 0.0577 = 3.818 ft/s ✓ — matches the continuity velocity
Answer: hf ≈ 12.8 ft. The velocity-form cross-check agreeing to 0.05% is your signal that the exponents and constants landed in the right places.
Verify it yourself on our Hazen–Williams calculator.
Darcy–Weisbach — the universal one (needs f)
hf = f · (L/D) · (V2/2g)
| f | Darcy friction factor — dimensionless; from the Moody chart / Colebrook equation (turbulent) or f = 64/Re (laminar) |
| V²/2g | the velocity head — Darcy–Weisbach is built on it, so keep the arithmetic tight |
| L/D | dimensionless length ratio — the equation is dimensionally consistent in any unit system |
Darcy–Weisbach is physically grounded, not empirical: it works for any Newtonian fluid, laminar or turbulent, any pipe size. The price is the friction factor — the exam will either give you f outright (a speed item: just plug in), give you roughness ε and expect Moody/Colebrook, or set up laminar flow where f = 64/Re. Note it is the Darcy f here, not the Fanning friction factor (which is f/4) — a favorite distractor in fluids stems.
Worked example Same pipe, Darcy–Weisbach — and where they agree
Given: the same scenario as the Hazen–Williams example — Q = 3.0 cfs, D = 1.0 ft, L = 2,500 ft of water pipe — but the stem gives a Moody-chart friction factor f = 0.0225 instead of C. Find hf.
Solution:
- Velocity (as before): V = 3.820 ft/s
- Velocity head: V²/2g = (3.820)²/(2 × 32.2) = 14.592/64.4 = 0.2266 ft
- hf = 0.0225 · (2,500/1.0) · 0.2266 = 56.25 × 0.2266 = 12.75 ft
Answer: hf ≈ 12.8 ft — agreeing with Hazen–Williams to the decimal.
That agreement is not a coincidence: on a water-distribution pipe inside Hazen–Williams' valid range, the two equations are calibrated against the same physical reality. The exam exploits this by making the stem decide: given C → Hazen–Williams; given f or ε → Darcy–Weisbach. Where they diverge is outside HW's range — oil at Re = 424 has f = 64/Re = 0.151 and Hazen–Williams simply has nothing to say about it.
How the exam tells you which one it wants
| Given C (a number like 120, 140) | Hazen–Williams. C is HW's coefficient; it appears nowhere else. |
| Given f, or ε (roughness height), or "use the Moody chart" | Darcy–Weisbach. f is DW's friction factor. |
| Open channel, bed slope S, and Manning's n | Manning's. Uniform flow with a free surface. |
| Fluid is oil, gas, slurry — or flow is laminar | Darcy–Weisbach, even if the pipe looks like a water main. HW is water-only. |
| Pipe flowing partially full (storm sewer, culvert) | Manning's — a free surface makes it open-channel flow, using the filled portion's A and P. |
| Asked for the friction factor itself | Darcy–Weisbach by definition; expect a Colebrook/Moody step first. |
Frequently asked questions
Can I use the Manning equation for pressurized pipe flow?
No. Manning's equation describes uniform gravity-driven open-channel flow with a free surface. For a full, pressurized pipe the exam expects Hazen–Williams (water distribution, typical sizes) or Darcy–Weisbach (general case). Manning in a pressurized pipe is the classic exam trap.
When does the exam expect Darcy–Weisbach instead of Hazen–Williams?
When the stem gives you a friction factor f or a roughness height (or asks you to use the Moody chart), when the fluid is not water, when flow may be laminar or transitional, or when the pipe is outside Hazen–Williams' typical distribution ranges. Hazen–Williams is for water, turbulent flow, typical distribution pipe sizes.
What is the difference between the Hazen–Williams C factor and the Darcy–Weisbach f factor?
They are not interchangeable and do not convert directly. C is an empirical roughness coefficient (higher C = smoother pipe, typically 100–150); f is a dimensionless friction factor from the Moody chart or Colebrook equation. A stem giving you C is telling you to use Hazen–Williams.
Do Hazen–Williams and Darcy–Weisbach give the same answer?
On the same water-distribution pipe they agree within a few percent — in the worked comparison above, both give 12.75 ft of head loss. They diverge outside Hazen–Williams' valid range (non-water fluids, extreme diameters, laminar flow), which is exactly when the exam expects Darcy–Weisbach.
Which equation do I use for a partially full storm sewer?
Manning's. A free surface makes it open-channel flow even inside a pipe — use the filled portion's area and wetted perimeter with the pipe material's Manning's n.
What are the unit traps in these three equations?
Manning's 1.486 constant is US-only (1.0 for SI). The Hazen–Williams velocity form uses 1.318 in US units (0.849 in SI), and the 4.727 head-loss constant assumes Q in cfs with D in feet. Darcy–Weisbach is dimensionally consistent in any unit system — but f is dimensionless while C is not, so never swap them.