Formula and Method for Open Channel Flow (Manning's Equation)
Open channel flow describes water or another liquid moving through a canal, culvert, ditch, or river under gravity with a free surface exposed to the atmosphere — unlike pressurized pipe flow, which is fully enclosed. The standard engineering tool for analyzing this kind of flow is Manning's equation, an empirical formula published by Irish engineer Robert Manning in 1889 that relates flow velocity to channel shape, surface roughness, and bed slope: V = (k/n) · R^(2/3) · S^(1/2), where V is velocity, n is Manning's roughness coefficient, R is the hydraulic radius, S is the channel's bed slope, and k is a unit-conversion constant (1.0 for SI units, 1.486 for US customary units).
How the calculation works
This calculator models a trapezoidal channel, the general case that also covers rectangular channels (set side slope z = 0) and triangular channels (set bottom width b = 0). From the bottom width b, flow depth y, and side slope z (horizontal run per unit of vertical rise), it computes the cross-sectional area A = y(b + zy) and the wetted perimeter P = b + 2y√(1 + z²) — the length of channel boundary in contact with the water. The hydraulic radius R = A / P then feeds into Manning's equation to solve for velocity V, and multiplying by the area gives the discharge (flow rate) Q = V × A. Finally, the calculator finds the top width T = b + 2zy and the hydraulic depth D = A / T to compute the Froude number, Fr = V / √(g · D), which classifies the flow as subcritical, critical, or supercritical.
Common mistakes
- Using the wrong unit constant: k = 1.486 (sometimes rounded to 1.49) applies only when R is in feet and V comes out in ft/s; use k = 1.0 with meters and m/s. Mixing the two gives a velocity off by a factor of roughly 1.49.
- Entering slope as a decimal when the field expects percent (or vice versa): a 0.5% slope means S = 0.005 in the formula, not 0.5. This calculator's slope field accepts percent and converts it automatically.
- Picking an unrealistic roughness coefficient: Manning's n ranges from about 0.011 for smooth finished concrete to over 0.035 for rough, weedy earth channels; using a value from the wrong material meaningfully overstates or understates capacity.
- Applying Manning's equation where flow is not uniform: the equation assumes steady, uniform flow at constant depth; near drops, weirs, gates, or sharp bends, use gradually-varied-flow methods instead.
Real-world applications
- Sizing irrigation canals and roadside drainage ditches so they carry the design flow without overtopping.
- Designing storm sewers and culverts that run partially full, where Manning's equation predicts capacity at a given depth.
- Building discharge-versus-depth rating curves for flood studies and river hydraulics from repeated Manning's equation calculations.
- Checking whether a spillway or chute runs subcritical or supercritical, since that determines where hydraulic jumps and energy dissipators are needed.
Frequently Asked Questions
What is Manning's equation and why is it used for open channel flow?
Manning's equation, V = (k/n) R^(2/3) S^(1/2), is an empirical formula developed by Robert Manning in 1889 that relates flow velocity to a channel's hydraulic radius, surface roughness, and bed slope. It is the standard method for sizing and analyzing gravity-driven, free-surface flow in canals, rivers, storm sewers, and culverts running partially full.
What Manning's roughness coefficient (n) should I use?
Manning's n depends on the channel lining: about 0.011-0.013 for finished, trowel-smooth concrete, 0.013-0.017 for unfinished concrete or vitrified clay pipe, 0.022-0.026 for corrugated metal, 0.020-0.030 for straight, clean earth channels, and 0.035-0.045 for rocky or weedy channels. Pick the value that matches your channel's actual surface condition, not just its material.
What is the Froude number and why does it matter?
The Froude number, Fr = V / √(g × hydraulic depth), is the ratio of flow velocity to the speed of a small gravity wave in the channel. Fr < 1 is subcritical flow (deep and slow, controlled from downstream), Fr > 1 is supercritical flow (shallow and fast, controlled from upstream), and Fr = 1 is critical flow, which is unstable and best avoided in design because it produces standing waves.
Why does the constant in Manning's equation change between unit systems?
Manning's equation is empirical, not dimensionally homogeneous, so it needs a conversion constant k to work correctly in different unit systems: k = 1.0 when R is in meters and V comes out in m/s (SI), or k = 1.486 when R is in feet and V comes out in ft/s (US customary). Mixing units without swapping k gives a velocity that is off by roughly a factor of 1.49.
Common Tip Percentages
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| Food delivery | 10–20% | Consider extra for long distances |
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General Household Budget Ratios (50/30/20 Rule)
| Category | % of Net Income | Examples |
|---|
| Needs | 50% | Rent, groceries, utilities, insurance |
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Common Measurement Conversions for Everyday Use
| Quantity | US / Imperial | Metric Equivalent |
|---|
| 1 cup | 8 fl oz | 237 mL |
| 1 pint | 16 fl oz | 473 mL |
| 1 quart | 32 fl oz | 946 mL |
| 1 gallon | 128 fl oz | 3.785 L |
| 1 lb | 16 oz | 453.6 g |
| 1 ft | 12 in | 30.48 cm |