ManningHydraulicsOpen channelsSewersDrainageCivil 3D

Manning's Equation Explained: A Worked Example and an n Value Table (Concrete, PVC, Earth)

Manning's equation explained term by term, with a concrete channel solved by hand, a table of n values and the mistakes that show up most often.

Zeist·October 23, 2026·7 min read

Share on LinkedIn

Hydraulics · Channels and pipes

Manning's equation, term by term, with an example solved by hand

What each letter means, a concrete channel solved step by step in SI and US units, the n value table for concrete, PVC and earth, and the six mistakes that show up most in design calculations.

What does Manning's equation say?

SI:  V = (1/n) · R^(2/3) · S^(1/2)      V in m/s, R in m
US:  V = (1.49/n) · R^(2/3) · S^(1/2)   V in ft/s, R in ft
Q = V · A

V is the mean velocity.
n is the roughness coefficient of the material. It's the same number in both unit systems.
R is the hydraulic radius: R = A / P.
S is the bottom slope as a ratio. A 0.2 % slope is 0.002.
Q is the discharge, and A the flow area.

The 1.49 (more precisely 1.486) only converts units. Manning applies to uniform free-surface flow: channels, gutters, sanitary sewers and storm drains. It doesn't apply to pressurized pipes, which use Hazen-Williams or Darcy-Weisbach.

How do you solve an example step by step?

A rectangular concrete channel, 1.00 m (3.28 ft) wide, flowing 0.50 m (1.64 ft) deep, on a 0.2 % slope (S = 0.002), with n = 0.015. How much water does it carry?

Step	SI	Result
1. Flow area	A = 1.00 × 0.50	0.50 m²
2. Wetted perimeter	P = 1.00 + 2 × 0.50	2.00 m
3. Hydraulic radius	R = 0.50 / 2.00	0.25 m
4. Powers	R^(2/3) = 0.397 and S^(1/2) = 0.0447	—
5. Velocity	V = (1 / 0.015) × 0.397 × 0.0447	1.18 m/s
6. Discharge	Q = 1.18 × 0.50	0.59 m³/s

The same channel in US units: A = 5.38 ft², P = 6.56 ft, R = 0.82 ft, V = (1.49 / 0.015) × 0.82^(2/3) × 0.0447 = 3.89 ft/s, and Q = 20.9 cfs. That's the same 0.59 m³/s.

One more check: the Froude number, F = V / √(g · y) in a rectangular channel. Here F = 1.18 / √(9.81 × 0.50) = 0.53. It's below 1, so the flow is subcritical: calm and stable.

What n value should you use for concrete, PVC or earth?

These are the same values the Manning calculator uses:

Material	Design n	Typical range
PVC	0.010	0.009 – 0.011
Concrete pipe	0.013	0.011 – 0.015
Cast iron	0.013	0.010 – 0.014
Corrugated metal	0.024	0.021 – 0.030
Concrete-lined channel	0.015	0.013 – 0.016
Masonry	0.025	0.017 – 0.030
Clean earth	0.022	0.018 – 0.025
Earth with grass	0.027	0.022 – 0.033
Rock	0.035	0.025 – 0.040
Natural stream	0.030	0.025 – 0.033

The material matters a lot. The same channel dug in clean earth (n = 0.022) carries 0.40 m³/s, 32 % less. With grass (n = 0.027), 0.33 m³/s.

Use the design value, not the brand-new-pipe value: over the years you get sediment, joints and wear. If your agency or your state's design manual sets an n, use that one.

How does it work for a pipe that isn't flowing full?

With the pipe full, the geometry is simple: A = π · D² / 4, P = π · D, so R = D / 4. A 300 mm (about 12 in) concrete pipe (n = 0.013) on a 0.5 % slope:

R = 0.30 / 4 = 0.075 m
V = (1 / 0.013) × 0.075^(2/3) × 0.005^(1/2) = 0.97 m/s (3.2 ft/s)
Q = 0.97 × 0.0707 = 68 L/s (2.4 cfs)

With the pipe partly full, area and perimeter depend on the angle the water surface makes, and the math gets longer. One result surprises people: maximum discharge does not happen with the pipe full. It happens at 94 % of the diameter, and it's 7.6 % higher. As the pipe fills completely, the wetted perimeter grows faster than the area and the water slows down.

How do you find the depth if you know the discharge?

That's the reverse problem, and the most common one in design: you know how much water arrives and need to know how deep it will run. That depth is the normal depth.

Manning's equation can't be solved directly for depth, because area and perimeter both depend on it. You solve it by trial: guess a depth, compute the discharge, adjust until it matches. In the example channel, for 0.40 m³/s the normal depth is 0.375 m, at 1.07 m/s.

The calculator does this in Normal depth mode, for circular pipes and rectangular, trapezoidal and triangular channels.

What are the most common mistakes?

Slope as a percentage. Typing 0.2 instead of 0.002 multiplies the discharge by 10: the example channel would carry 5.92 m³/s.
Using the diameter as the hydraulic radius. For a full pipe, R is D/4. With R = D, the 300 mm pipe would carry 172 L/s instead of 68.
Mixing units. In feet the factor is 1.49. Leave it out and the discharge comes out 33 % low.
Picking the catalog n for new pipe. Design is for the service life, not day one.
Applying it to a pressurized pipe. Manning is for free-surface flow.
Assuming a full pipe is maximum capacity. The peak is at 94 % of the diameter. If more water arrives than that, the pipe surcharges and Manning no longer applies.

Sources

Ven Te Chow, Open-Channel Hydraulics, McGraw-Hill, 1959: the n value table. For PVC, common manufacturer values.
USGS: Roughness Characteristics of Natural Channels (Barnes, Water-Supply Paper 1849, 1967): n values for 50 natural channels, with photographs.
The example results were checked with Zeist's Manning calculator.

Data verified on October 23, 2026.

Skip the hand calculation: Manning calculator, for circular pipes and channels, in SI or US units.
From calculation to model: Civil 3D pipe networks: fittings, water and sewer.
3D without redrawing: a Civil 3D plugin that models in 3D.

Frequently asked questions about Manning's equation

What is Manning's equation?

It's the formula for the mean velocity of uniform free-surface flow: V = (1/n) · R^(2/3) · S^(1/2) in SI units, where n is the roughness of the material, R the hydraulic radius and S the slope. In US units the factor is 1.49 instead of 1. Discharge is Q = V · A. It's used for channels, sewers and storm drains.

What is Manning's n for concrete?

For concrete pipe, the usual design value is n = 0.013, within a range of 0.011 to 0.015. For a concrete-lined channel, n = 0.015, within 0.013 to 0.016. If your agency or design manual sets a different value, use that one.

What n value is used for PVC pipe?

The usual design value is n = 0.010, within 0.009 to 0.011 according to manufacturers. It's lower than concrete because the wall is smoother: at the same diameter and slope, a PVC pipe carries more water. Some design standards require a higher design n to account for aging.

Does Manning's equation work for pressurized pipes?

No. Manning applies to free-surface flow: channels and pipes that aren't running full, like sanitary sewers and storm drains. Pressurized pipes, like water mains, use Hazen-Williams or Darcy-Weisbach. If a sewer fills up and surcharges, it's no longer free-surface flow and Manning no longer applies.

Why does the slope go in as a ratio, not a percentage?

Because the equation uses slope as a ratio: a 0.2 % slope is entered as 0.002. Type 0.2 and the discharge comes out 10 times too high, because the square root of 100 is 10. It's the most common mistake in channel and sewer spreadsheets.

At what depth does a pipe carry its maximum flow?

At 94 % of the diameter, not when it's full. At that depth the discharge is 7.6 % higher than full flow, because as the pipe fills completely the wetted perimeter grows faster than the area and the water slows down. Zeist's Manning calculator shows it.