For uniform flow, the Manning equation relates the discharge to the geometry of the section, the slope and the roughness of the lining. In SI units:
How it works
How to use the calculator
- Choose what you want to calculate: the normal flow depth for a given discharge, the capacity of a section you already know, or the optimal section.
- Select the shape of the section.
- Enter the inputs in the unit you prefer: each field has its own selector. You can type decimals with a point or a comma.
- Press “Calculate”. The results appear in the table; the dimensioned cross-section drawing is included in the calculation report.
Normal flow depth
The normal flow depth is the depth at which the discharge flows in uniform flow. The Manning equation cannot in general be solved explicitly for the flow depth, so the calculator solves it numerically: it looks for the depth at which the section conveys exactly that discharge.
In rectangular, trapezoidal and triangular sections the discharge always increases with the flow depth, so the solution is unique.
In the circular section the maximum discharge is not reached with the conduit flowing full, but at y/D ≈ 0.938, and it is 7.6% greater than the full-section value. For flows between those two values there are two solutions; the calculator gives the smaller one, which is the stable operating condition. If the discharge exceeds the maximum, the conduit cannot carry it with a free surface.
Hydraulically optimal section
For a given discharge, slope and roughness, the optimal section is the one that needs the smallest area and wetted perimeter, which reduces excavation and lining. The calculator applies these criteria:
- Rectangular: the bottom width is twice the flow depth (b = 2y), so that Rh = y/2.
- Trapezoidal: for the stated side slope z, b = 2y·(√(1 + z²) − z), which also gives Rh = y/2. It is half a hexagon when z = 1/√3.
- Triangular: for the stated side slope, the flow depth that conveys the discharge is obtained. The most efficient shape is z = 1 (walls at 45°).
- Circular: the diameter is chosen so that the discharge flows at y/D ≈ 0.81, the ratio that maximises the hydraulic radius.
Assumptions and limits
- Steady uniform flow: the flow depth is the normal one for the slope and roughness of the reach.
- Prismatic channel: section, slope and roughness constant along the reach.
- Small slope, so that the vertical depth and the depth perpendicular to the bed coincide.
- The circular section is calculated with a free surface or flowing full, non-pressurized; surcharged operation is not included.
- Acceleration due to gravity g = 9.81 m/s².
Formulas and method
The Manning equation
Q = (1/n) · A · Rh2/3 · S1/2, with Rh = A / P
- Q: discharge (m³/s)
- n: Manning roughness coefficient
- A: wetted area (m²)
- P: wetted perimeter (m)
- Rh: hydraulic radius (m)
- S: longitudinal bed slope (m/m)
Section geometry
The calculator obtains the wetted area, the wetted perimeter and the top width (T) from these relationships, where y is the flow depth, b the bottom width, z the side slope and D the diameter:
| Section | Area A | Perimeter P | Top width T |
|---|---|---|---|
| Rectangular | b·y | b + 2y | b |
| Trapezoidal | (b + z·y)·y | b + 2y·√(1 + z²) | b + 2z·y |
| Triangular | z·y² | 2y·√(1 + z²) | 2z·y |
| Circular | D²(θ − sin θ)/8 | D·θ/2 | D·sin(θ/2) |
In the circular section, θ = 2·arccos(1 − 2y/D) is the wetted central angle, in radians.
Froude number and flow regime
The Froude number compares the flow velocity with that of a surface wave: Fr = V / √(g·A/T), where A/T is the hydraulic depth. With Fr < 1 the flow is subcritical (tranquil) and with Fr > 1, supercritical (rapid). In channel design it is advisable to stay away from Fr = 1, because near the critical regime the water surface is unstable.
Typical values of the coefficient n
Normal values for channels in good condition according to Chow (1959), table 5-6. The appropriate value depends on the actual condition of the lining, vegetation and maintenance.
| Lining | n |
|---|---|
| Trowel-finished concrete | 0.013 |
| Float-finished concrete | 0.015 |
| Unfinished concrete | 0.017 |
| Shotcrete, regular section | 0.019 |
| Cemented rubble masonry | 0.025 |
| Corrugated metal | 0.025 |
| Earth, straight and uniform, clean | 0.022 |
| Earth with short grass and few weeds | 0.027 |
| Rock cut, smooth and uniform | 0.035 |
Frequently asked questions
Which units does the calculator work in?
You can type each input in the unit you prefer, metric or US customary (L/s, gal/min, feet, inches…), or change them all at once with the unit system selector. In “Preferences” you can set the unit of each quantity (discharge, flow depth, velocity…) for the inputs, the results and the checks. The calculation is always made in SI; the PDF report presents the inputs and results in your units and shows the conversion to SI at the start of the development.
Can I type decimals with a comma?
Yes. You can use a point or a comma as the decimal separator; do not use thousands separators.
What is the difference between the three calculation modes?
“Normal flow depth” starts from the discharge and the section, and obtains the water depth. “Capacity” starts from the section and the flow depth, and obtains the discharge. “Optimal section” starts from the discharge and proposes the dimensions with the smallest wetted perimeter.
Does it work for pipes flowing full?
It accepts the conduit flowing full, non-pressurized: with the flow depth equal to the diameter, it calculates the full-section discharge with the slope of the reach (the Froude number does not apply). It does not handle surcharged operation: pressure pipelines are calculated with head-loss formulas such as Darcy-Weisbach or Hazen-Williams.
References
- Chow, V. T. (1959). Open-Channel Hydraulics. McGraw-Hill.
- French, R. H. (1985). Open-Channel Hydraulics. McGraw-Hill.