How to read the result
- Velocity is the average speed of the water across the flow area (discharge ÷ area).
- Froude number compares the flow speed with the speed of a surface wave. Below 1 the flow is subcritical; above 1 it is supercritical. It does not apply to a pipe flowing exactly full, because there is no free surface.
- Maximum flow occurs below full. In this pipe the greatest discharge is 73.554 L/s at 93.8% depth, about 7.6% more than when the pipe is exactly full (68.378 L/s). Near the crown the wetted perimeter grows faster than the flow area.
- Two depths for one flow. Between the full-pipe flow and the maximum flow, two depths carry the same discharge. For example, 70.966 L/s in this pipe flows at 85.8% or 99.2% depth. The calculator reports both; which one occurs depends on the downstream conditions.
- Shear stress is the average drag of the water on the pipe wall (ρ·g·R·S).
This tool assumes steady, uniform flow. It does not check pressurised (surcharged) flow, inlet or junction losses, or any minimum or maximum velocity set by a design standard.
Worked example
A 300 mm concrete pipe (n = 0.013) is laid at a 0.5% slope (S = 0.005) and flows 70% full. Find the discharge and velocity.
- Central angle: θ = 2·arccos(1 − 2 × 0.7) = 3.9646 rad
- Flow area: A = D²(θ − sin θ) ÷ 8 = 0.3² × (3.9646 − sin 3.9646) ÷ 8 = 0.05285 m²
- Wetted perimeter: P = D·θ ÷ 2 = 0.5947 m
- Hydraulic radius: R = A ÷ P = 0.08887 m (88.870 mm)
- Discharge: Q = (1/n)·A·R2/3·S1/2 = (1/0.013) × 0.05285 × 0.088872/3 × 0.0051/2 = 0.05725 m³/s = 57.248 L/s
- Velocity: V = Q ÷ A = 1.083 m/s
These are the same numbers the calculator shows when it first loads.
Method and formulas
θ = 2·arccos(1 − 2y/D)
A = D²(θ − sin θ) ÷ 8 · P = D·θ ÷ 2 · R = A ÷ P · T = D·sin(θ/2)
Q = (1/n)·A·R2/3·S1/2 (SI units; in US units the factor is 1.486/n)
V = Q ÷ A · Fr = V ÷ √(g·A/T) · τ = ρ·g·R·S
- D
- Inside diameter (m)
- y
- Flow depth (m); y/D is the depth as a fraction of the diameter
- θ
- Angle at the pipe centre subtended by the water surface (radians)
- A, P, R, T
- Flow area (m²), wetted perimeter (m), hydraulic radius (m), top width of the water surface (m)
- n
- Manning roughness coefficient of the pipe wall
- S
- Pipe slope (m/m)
- g, ρ
- 9.80665 m/s² and 1,000 kg/m³
The required slope is found by rearranging Manning's equation for S. Normal depth and maximum flow are found numerically (bisection and golden-section search) to better than 1 part in a million.
Assumptions and limits
- Steady uniform (normal) flow.
- Circular pipe of constant diameter, slope and roughness.
- Manning n constant with depth.
- Not for pressurised (surcharged) flow; use a pressure-flow method.
- Between full-pipe and maximum flow two depths exist; both are reported, the user chooses with hydraulic controls in mind.
- Does not check minimum self-cleansing velocity or local design criteria.
Reference tables
| Pipe or conduit (wall) | Manning's n |
|---|---|
| Concrete Pipe (smooth) | 0.010–0.011 |
| Concrete Boxes (smooth) | 0.012–0.015 |
| Spiral Rib Metal Pipe (smooth) | 0.012–0.013 |
| Corrugated Metal Pipe, Pipe-Arch and Box: 2-2/3 by 1/2 inch (annular) | 0.022–0.027 |
| Corrugated Metal Pipe, Pipe-Arch and Box: 2-2/3 by 1/2 inch (helical) | 0.011–0.023 |
| Corrugated Metal Pipe, Pipe-Arch and Box: 6 by 1 inch (helical) | 0.022–0.025 |
| Corrugated Metal Pipe, Pipe-Arch and Box: 5 by 1 inch | 0.025–0.026 |
| Corrugated Metal Pipe, Pipe-Arch and Box: 3 by 1 inch | 0.027–0.028 |
| Corrugated Metal Pipe, Pipe-Arch and Box: 6 by 2 inch (structural plate) | 0.033–0.035 |
| Corrugated Metal Pipe, Pipe-Arch and Box: 9 by 2-1/2 inch (structural plate) | 0.033–0.037 |
| Corrugated Polyethylene (smooth) | 0.009–0.015 |
| Corrugated Polyethylene (corrugated) | 0.018–0.025 |
| Polyvinyl chloride (PVC) (smooth) | 0.009–0.011 |
Source: FHWA, Urban Drainage Design Manual (HEC-22), report FHWA-HIF-24-006, 4th edition, February 2024, Table 9.1 Manning's roughness coefficients for storm drain conduits.
For corrugated metal pipe, pipe-arch and box, Manning's n varies with barrel size. The manual adds: “HDS-5 (FHWA 2012a) documents laboratory-derived Manning's n values. Actual field values for culverts may vary depending on the effect of abrasion, corrosion, deflection, and joint conditions.”
The manual prints “Spiral Rip Metal Pipe”; this table corrects the label to spiral rib. The n values are unchanged.
| Depth (y/D) | Q / Qfull | V / Vfull |
|---|---|---|
| 10% | 0.021 | 0.401 |
| 20% | 0.088 | 0.615 |
| 30% | 0.196 | 0.776 |
| 40% | 0.337 | 0.902 |
| 50% | 0.500 | 1.000 |
| 60% | 0.672 | 1.072 |
| 70% | 0.837 | 1.120 |
| 80% | 0.977 | 1.140 |
| 90% | 1.066 | 1.124 |
| 93.8% (maximum flow) | 1.076 | 1.104 |
| 100% | 1.000 | 1.000 |
Source: Calculated with the EngiFormula Manning engine (method v1.1): circular-section geometry and Manning’s equation (FHWA HEC-22, 4th edition, 2024, equation 6.5). The ratios do not depend on diameter, slope or n.
Charts and diagrams
Show chart data as a table
| Flow depth (% of diameter) | Q / Qfull (discharge ratio) | V / Vfull (velocity ratio) |
|---|---|---|
| 0 | 0 | 0 |
| 5 | 0 | 0.26 |
| 10 | 0.02 | 0.4 |
| 15 | 0.05 | 0.52 |
| 20 | 0.09 | 0.62 |
| 25 | 0.14 | 0.7 |
| 30 | 0.2 | 0.78 |
| 35 | 0.26 | 0.84 |
| 40 | 0.34 | 0.9 |
| 45 | 0.42 | 0.95 |
| 50 | 0.5 | 1 |
| 55 | 0.59 | 1.04 |
| 60 | 0.67 | 1.07 |
| 65 | 0.76 | 1.1 |
| 70 | 0.84 | 1.12 |
| 75 | 0.91 | 1.13 |
| 80 | 0.98 | 1.14 |
| 85 | 1.03 | 1.14 |
| 90 | 1.07 | 1.12 |
| 95 | 1.07 | 1.09 |
| 100 | 1 | 1 |
Source: Calculated with the EngiFormula Manning engine (method v1.1): circular-section geometry and Manning’s equation (FHWA HEC-22, 4th edition, 2024, equation 6.5). The ratios do not depend on diameter, slope or n.
Common uses
- Storm drain capacity check: how much a pipe carries at a given depth and slope.
- Sanitary sewer partial-flow check: depth and velocity at the design flow.
- Culvert barrel flowing partly full: normal depth in the barrel (inlet and outlet control are not covered).
Frequently asked questions
How do you calculate flow in a partially full pipe?
Work out the flow area and wetted perimeter from the depth, divide them to get the hydraulic radius, then apply Manning's equation Q = (1/n)·A·R2/3·S1/2. The worked example above does this for a 300 mm pipe flowing 70% full: 57.248 L/s.
Why is maximum flow not at full pipe?
As the water nears the top of the pipe, the wetted perimeter grows faster than the flow area, so the hydraulic radius and the flow drop slightly. Maximum discharge occurs at about 93.8% depth and is about 7.6% higher than the full-pipe discharge.
What Manning's n should I use for concrete or PVC pipe?
The FHWA Urban Drainage Design Manual (HEC-22, 4th edition, 2024), Table 9.1, gives 0.010–0.011 for smooth concrete pipe and 0.009–0.011 for smooth PVC pipe. The roughness table on this page lists the other conduits. If your design manual or the pipe manufacturer gives a value, use that. This calculator uses whatever value you enter.
What is the difference between Manning's equation and Hazen-Williams?
Manning's equation is for gravity (open-channel) flow with a free surface, such as storm drains and sewers flowing partly full. Hazen-Williams is for water in pressure pipes flowing full. Use a pressure-flow method when the pipe is surcharged.
What is a good velocity for a gravity sewer?
Minimum and maximum velocities are set by the design standard that applies to your project, so check your local criteria. This calculator reports the velocity but does not judge it.
How do I calculate the flow rate (GPM) of a pipe?
For a gravity pipe, enter the diameter, Manning's n, slope and water depth; the calculator gives the discharge in ft³/s and US gallons per minute (1 ft³/s is about 448.8 US gal/min). The worked example above carries about 907 US gal/min. For a pressure pipe flowing full under pumping or mains pressure, the flow depends on the pressure available, so use a pressure-flow method instead.
How much flow can a 4 inch pipe carry?
It depends on slope and roughness. By gravity, a 4 in pipe with n = 0.011 (the upper end of the HEC-22 range for smooth PVC, 0.009–0.011) on a 1% slope carries about 101 US gal/min (6.4 L/s) flowing full, by Manning's equation. A steeper slope or a smoother pipe carries more; enter your own values in the calculator. In a pressure pipe the capacity depends on the pressure and pipe length, which needs a different method.
Sources and references
- U.S. Federal Highway Administration, Urban Drainage Design Manual, Hydraulic Engineering Circular No. 22 (HEC-22), report FHWA-HIF-24-006, 4th edition, February 2024, Section 6.1.4 (Manning's equation) and Table 9.1.
- U.S. Federal Highway Administration, Design of Roadside Channels with Flexible Linings, Hydraulic Engineering Circular No. 15 (HEC-15), 3rd edition, September 2005, equation 2.3 (mean boundary shear stress).
- Texas Department of Transportation, Hydraulic Design Manual, online edition (Manual Notice 2019-1), Chapter 6, Section 1, Equation 6-14 (Froude number with the hydraulic mean depth A/T).
- Circular-section area, wetted perimeter and top width: plane geometry, as set out in the Method section above.
- NIST Special Publication 811, Guide for the Use of the International System of Units (SI), 2008 edition, Appendix B (exact unit conversion factors).
About this calculator
Scope and disclaimer
Uniform gravity flow in circular pipes. Not a design of inlets, junction losses, surcharge or local criteria. It does not check minimum self-cleansing velocity or any local design criteria.
This calculator gives information and preliminary estimates only. It is not professional design advice and does not replace the governing code, product instructions or a qualified professional's judgement for your project.