Manning's Pipe Flow Calculator

What is Manning's equation?

Manning's equation is the standard empirical formula for the mean velocity of uniform, fully-developed turbulent flow in an open channel:

V = (k / n) · R^2/3 · S^1/2,   Q = V · A

where V is the cross-sectional mean velocity, n is Manning's roughness coefficient, R is the hydraulic radius (wetted area divided by wetted perimeter), S is the longitudinal slope of the pipe invert, and k is a unit prefactor: 1.486 for Imperial units (ft, ft/s, cfs) or 1.0 for SI units (m, m/s, m³/s).

What does this calculator solve for?

For a circular pipe, two solver modes are exposed:

• Given depth — specify the flow depth y and the calculator returns area, hydraulic radius, velocity, discharge, Froude number, and boundary shear stress.
• Given discharge — specify the design discharge Q and the calculator iterates (bisection on the rising branch of the partial-flow rating curve) to find the normal depth that satisfies Manning's equation.

What does the PDF report contain?

A fully white-label one-to-two-page engineering report: project header (your company, your logo, your client and job number), the design basis with a labelled cross-section diagram, a results table (depth, area, hydraulic radius, velocity, discharge, Froude number, regime, full-pipe capacity, critical depth, boundary shear stress), three figures (cross-section at design depth, hydraulic-elements curve, and a depth–discharge rating curve with the design point highlighted), engineer-facing notes and warnings, a sign-off and stamp block, and a standard disclaimer. The only Thurston attribution is one small line at the bottom of the report.

How do I read the hydraulic-elements curve?

The hydraulic-elements curve plots dimensionless ratios of area, hydraulic radius, velocity, and discharge against the relative depth y/D, normalised against the same pipe flowing just full at the same slope:

• V/V_full compares the mean velocity at depth y to the full-pipe velocity. Because Manning's equation makes V proportional to R^2/3, V/V_full peaks slightly above 1 at y/D ≈ 0.81 (V_max ≈ 1.14 V_full) before falling as the wetted perimeter grows faster than the area near the crown.
• Q/Q_full is the corresponding discharge ratio (Q = A·V). It peaks at y/D ≈ 0.938 (Q_max ≈ 1.08 Q_full); above that depth the small extra wetted area cannot offset the loss in hydraulic radius, so a pipe running just full actually carries slightly less than at the peak depth. This is why most sewer-design references recommend sizing for the peak depth rather than the geometric full condition (Camp 1946; FHWA HDS-3).
• A/A_full and R/R_full are purely geometric ratios (no slope or roughness dependence) and are shown as reference curves for shape comparison.

On the report, the horizontal dash-dot line marks the design depth ratio, the two coloured dots are the design Q and V ratios, and the faint dotted lines flag the depths at which V/V_full and Q/Q_full peak.

What are the calculator's limitations?

• Circular pipes only (no rectangular, trapezoidal, or arch-section conduits).
• Uniform, steady normal flow. The equation does not capture backwater, drawdown, hydraulic-jump, or unsteady transient effects.
• When the requested discharge exceeds the pipe's open-channel capacity, the report flags surcharge and Manning's equation is no longer applicable; switch to a pressure-flow method (Hazen-Williams or Darcy-Weisbach) for that regime.
• Manning's equation is least accurate at very shallow depths (y/D below ~0.10), where surface tension and sidewall effects dominate, and at very low Reynolds numbers (Re < ~2000) where the flow is no longer fully turbulent.
• Manning's n values are calibrated for clear water at typical environmental temperatures. For sediment-laden flow, slurries, or fluids with markedly different viscosity, use the Custom fluid option and treat the resulting n as approximate.

Suppose you need to check whether a 24 in. PVC storm drain at 0.50% slope can carry a peak design discharge of 9.5 cfs. The calculator settings are:

• Unit system: Imperial
• Solver mode: Given flow Q → compute y
• Inside diameter D = 24 in. (= 2.0 ft)
• Manning's n = 0.010 (PVC, smooth plastic)
• Slope S = 0.50% (= 0.005 ft/ft)
• Design discharge Q = 9.5 cfs
• Fluid: Water at 20 C (the default)

The calculator iterates on Manning's equation Q = (1.486 / n) · A · R^2/3 · S^1/2 on the rising branch of the partial-flow rating curve and converges on a normal depth of y ≈ 0.95 ft (about 11.4 in., or y/D ≈ 0.47). At that depth:

• Wetted area A ≈ 1.47 ft²
• Hydraulic radius R ≈ 0.48 ft
• Mean velocity V ≈ 6.5 ft/s (well above the 2 ft/s self-cleansing minimum)
• Velocity head h_v = V²/(2g) ≈ 0.66 ft
• Froude number Fr ≈ 1.33 — supercritical, so a hydraulic jump is possible at any downstream control or transition
• Boundary shear stress τ₀ = γRS ≈ 0.15 lb/ft²
• Reynolds number Re = V·4R/ν ≈ 1.2×10⁶ (fully turbulent)
• Q/Q_full ≈ 0.46, V/V_full ≈ 0.98, Q_full ≈ 20.8 cfs

The pipe sits at less than half its full-flow capacity, so there is plenty of structural margin, but the calculator flags the supercritical regime so the engineer can verify downstream controls (drop manholes, junction structures, or outlet headwalls) and estimate hydraulic-jump losses. Halving the slope to about 0.10% would push the flow into the subcritical regime; raising the design Q to 20 cfs at the same slope would put y/D above 0.94, where the calculator warns that the pipe is sitting beyond the partial-flow peak. Use the rating-curve plot in the PDF to see how much margin you have at adjacent depths.

1. V. T. Chow, Open-Channel Hydraulics. New York, NY, USA: McGraw-Hill, 1959, LCCN 58-13860.
2. ASCE and WEF, Gravity Sanitary Sewer Design and Construction, 2nd ed., ASCE Manual of Practice No. 60 / WEF Manual of Practice No. FD-5. Reston, VA, USA: American Society of Civil Engineers, 2007.
3. Federal Highway Administration, Design Charts for Open-Channel Flow, Hydraulic Design Series No. 3. Washington, DC, USA: U.S. Department of Transportation, 1961. FHWA PDF
4. Federal Highway Administration, Hydraulic Design of Highway Culverts, 3rd ed., Hydraulic Design Series No. 5, FHWA-HIF-12-026. Washington, DC, USA: U.S. Department of Transportation, 2012. FHWA PDF
5. B. R. Munson, D. F. Young, T. H. Okiishi, and W. W. Huebsch, Fundamentals of Fluid Mechanics, 6th ed. Hoboken, NJ, USA: John Wiley & Sons, 2009, ISBN 978-0-470-26284-9.
6. The Engineering ToolBox, "Water - Dynamic and Kinematic Viscosity vs. Temperature." engineeringtoolbox.com
7. T. R. Camp, "Design of sewers to facilitate flow," Sewage Works Journal, vol. 18, no. 3, 1946.
8. U.S. Bureau of Reclamation, Design of Small Canal Structures. Denver, CO, USA: U.S. Department of the Interior, 1978. USBR Manuals & Guidelines
9. ACI Committee 350, Code Requirements for Environmental Engineering Concrete Structures (ACI 350-20). Farmington Hills, MI, USA: American Concrete Institute, 2020. concrete.org