Partial Pipe Flow Calculator

Partial Pipe Flow Calculator finds flow, depth, slope or pipe size for a part-full round pipe with Manning’s equation or Camp’s variable n, plus full-pipe capacity and travel time.

Using the Partial Pipe Flow Calculator

The Partial Pipe Flow Calculator finds the flow in a round pipe that runs with a free water surface, the way storm drains, culverts and sewers usually do. Solve For gives four answers: the flow at a known depth, the depth a known flow runs at, the slope needed to carry a flow, or the smallest pipe diameter that carries it.

Depth goes in whichever way you measured it. Enter it in inches, feet, millimeters, centimeters or meters, as a percent of the diameter, or as the central angle of the wetted arc. Slope takes percent, ft/ft, per thousand or feet per mile.

The Manning's n menu holds the pipe values from Caltrans Highway Design Manual Table 851.2, from precast concrete at 0.012 to 3 by 1 in corrugated metal at 0.028. The tool opens on corrugated-interior plastic pipe at 0.022, a common storm drain material.

The Roughness, Length and Units panel holds three extras. Roughness Model switches between a constant n and Camp's variable n. Pipe Run Length adds travel time and the invert drop over the run, and Result Units switches between US and SI outputs.

How to Calculate Flow in a Partially Full Pipe

A part-full pipe is an open channel with a curved bottom, so Manning's equation applies once you know the wetted area and perimeter. Both come from the central angle θ, the angle the water surface cuts at the pipe's center.

$$\theta =2\cdot {\cos }^{-1}\left(1-\frac{2y}{D}\right)$$

With θ in radians, the flow area and wetted perimeter follow from the circular segment under the water. Their ratio is the hydraulic radius R.

$$A=\frac{D^2}{8}\cdot (\theta -\sin \theta )$$

$$P=\frac{D\cdot \theta }{2}$$

$$R=\frac{A}{P}=\frac{D}{4}\cdot \left(1-\frac{\sin \theta }{\theta }\right)$$

Manning's equation then gives the flow, with k equal to 1.486 in US units and 1.0 in SI units. S is the pipe slope as a decimal, which stands in for the energy slope when flow is uniform. The Partial Pipe Flow Calculator runs all four steps and shows each one in its math panel.

$$Q=\frac{k}{n}\cdot A\cdot R^{2/3}\cdot S^{1/2}$$

Worked Example in a 12 in Pipe

The Partial Pipe Flow Calculator opens on a 12 in corrugated plastic pipe (n 0.022) at 0.85 percent slope, carrying water 4 in deep. That is one-third full, so θ is 141.1°, the flow area is 0.229 ft², the wetted perimeter is 1.231 ft and R is 0.186 ft.

$$Q=\frac{1.486}{0.022}\cdot 0.229\cdot 0.186^{2/3}\cdot 0.0085^{1/2}=0.465\text{cfs}$$

That is 208.8 gpm, or 0.301 MGD. The water moves at 2.03 ft/s, against 2.47 ft/s if the pipe ran full, and covers a 300 ft run in 2.5 minutes. The Froude number of 0.73 means subcritical flow, with critical depth at 3.39 in.

The flow uses 24.0 percent of the 1.94 cfs full-pipe capacity. The American Concrete Pipe Association's hydraulic elements data reads V/Vfull of 0.81 at one-third depth off its chart, which matches the 0.82 here.

Constant n vs Camp Variable n

Roughness in a round pipe does not stay constant as the depth changes. T. R. Camp proposed a curve for n at each depth compared with n flowing full, and it peaks at 1.29 times the full-pipe value between 20 and 30 percent depth.

A partially full pipe flow course by Harlan Bengtson, PhD, P.E. sets out Camp's curve as six straight-line pieces and calls it the preferred method for part-full pipes. The Partial Pipe Flow Calculator uses those same equations when you pick Camp variable n.

The gap is largest at shallow depth, where Camp's n cuts flow and velocity by about a fifth. In the 12 in example, switching to Camp raises n to 0.0282 at one-third depth and drops the flow from 0.465 to 0.363 cfs. Use Camp when low-flow velocity decides the design, such as a sewer that must keep solids moving.

The peaks move too. With constant n, flow peaks at 93.8 percent depth at 1.076 times full flow, and velocity peaks at 81.3 percent depth. With Camp n, flow peaks at about 96.4 percent depth at 1.051 times full, and velocity peaks near 92 percent.

Self-Cleansing Velocity

A sewer that runs too slowly drops solids that later block it. Illinois sets the usual rule in 35 IAC 370.320, which requires at least 2.0 ft/s (0.61 m/s) flowing full, based on Manning's equation with n 0.013.

The Partial Pipe Flow Calculator runs that check on every pipe you enter, using n 0.013 at full flow whatever material you picked. The 12 in example passes at 4.18 ft/s. It also tells you when the actual part-full velocity drops under 2.0 ft/s, since solids settle at low flows even in a pipe that passes the full-flow test.

That second check matters at shallow depth. At 10 percent depth, water moves at only 40 percent of its full-pipe speed with constant n, and 33 percent with Camp n. For the minimum slope behind the full-flow rule in each pipe size, the Mannings Flow Calculator page lists them, and the Sewer Line Slope Calculator turns a slope into fall over the run.

Maximum Flow Rate of a Gravity Pipe

The most a round gravity pipe can carry is its peak just below the crown, not its full-pipe flow. The Partial Pipe Flow Calculator shows both on its full-pipe card. In the 12 in example, the peak is 2.09 cfs at 93.8 percent depth, while the full pipe carries 1.94 cfs.

Pipe capacity for design is normally the full-flow figure, because the extra 7.6 percent disappears once the water seals the crown. When Flow Depth mode gets a flow above full-pipe capacity but under the peak, the tool still finds a depth and warns that any surge can seal the crown. Above the peak, it reports that the pipe would surcharge.

To size a storm drain, start from the peak runoff. The Storm Water Runoff Calculator gives the design flow, and the next section shows how this tool turns it into a pipe size.

Sizing a Pipe for a Flow Rate

Pipe Diameter mode finds the smallest pipe that carries a design flow at your slope and n without running deeper than you allow. Set that limit as a percent of D or a central angle, since the diameter is the unknown and a depth in inches has nothing to measure against.

The method rests on a scaling rule. At a fixed depth ratio, Manning flow grows with D to the 8/3 power, so the tool first finds the flow Q₁ in a 1 ft pipe at that ratio, then scales up.

$$D={\left(\frac{Q}{Q_1}\right)}^{3/8}$$

For 0.5 cfs on a 0.85 percent slope with n 0.022, the answer depends on how full you let it run. The table shows the minimum bore at three depth limits, so a looser limit buys a smaller pipe.

The result is a minimum bore, not a stock size, so round up to the next pipe your supplier makes and check its real inside diameter. Run that size back through Flow Rate mode to confirm the depth and velocity. Pipe Diameter mode works with Camp variable n too, and the math panel prints the Q₁ step and a matching Excel formula.

Flow Rate of 25 mm, 1 in and 100 mm Pipes

A pipe's flow rate depends on what drives the water. Under gravity it depends on slope and roughness, while under pump or main pressure it depends on how fast you let the water move. The table shows both for common small sizes, treating each size as its inside diameter. Enter any of them in the Partial Pipe Flow Calculator with a depth to see how far the gravity figure falls below full.

The gravity column is full flow at 1 percent slope with n 0.012 for smooth plastic. The pressure column uses 5 ft/s, the bottom of the Copper Development Association's 5 to 8 ft/s range for cold water.

For pressure flow at any velocity, the Pipe Flow Calculator multiplies bore area by speed. The Hazen Williams Equation Calculator adds the friction loss that limits how much a long pressure line delivers.

Flow from differential pressure

To get flow from DP, the pressure difference across an orifice plate, venturi or flow nozzle, use the meter equation. The discharge coefficient Cd and the throat area A come from the meter's data sheet, and ρ is the fluid density.

$$Q=C_d\cdot A\cdot \sqrt{\frac{2\cdot \Delta P}{\rho }}$$

Flow rises with the square root of the pressure difference, so doubling the flow takes four times the DP. That same square-root law is why a gravity pipe's flow rises only with the square root of its slope.

Pipe Schedule and Inside Diameter

A schedule number sets the wall thickness for a nominal size, while the outside diameter stays fixed. A thicker wall leaves a smaller bore, so the inside diameter is the outside diameter minus twice the wall.

$$ID=OD-2t$$

A 2 in pipe has a 2.375 in OD. Schedule 40 has a 0.154 in wall and a 2.067 in bore, while Schedule 80 has a 0.218 in wall and a 1.939 in bore. A 12 in pipe has a 12.750 in OD, giving an 11.938 in bore in Schedule 40 and 11.374 in in Schedule 80.

Always enter the inside diameter in the Partial Pipe Flow Calculator. For a full Schedule 40 list with walls and bores, see the sizing table on the Hazen Williams page. To find the cubic meters a pipe holds, multiply the bore area by the length, or use the Pipe Volume Calculator.

Flow Rate Unit Conversions

Partial pipe results come out in cfs, gpm and MGD in US units, or m³/s, L/s and m³/h in SI units. The Partial Pipe Flow Calculator also accepts a design flow in cfs, gpm, MGD, m³/s, L/s or m³/h, so check the factors below when a figure arrives in another unit.

So 1 CFM equals 1.699 m³/h, and 150 m³/h equals 88.3 CFM. For m³/h from a volume and a time, divide cubic meters by hours. For smaller flows in liters, the Liters per Minute Calculator converts across twelve units.

Partial Pipe Flow in Excel

The Step-by-Step Math panel prints the geometry and an Excel or Google Sheets formula built from your numbers. To keep a reusable sheet, put the diameter in A2, the depth in B2, n in C2 and the slope as a decimal in D2, all in feet for US units.

=LET(th,2*ACOS(1-2*B2/A2),ar,A2^2/8*(th-SIN(th)),pe,A2*th/2,(1.486/C2)*ar*(ar/pe)^(2/3)*SQRT(D2))

The formula returns cfs. For the 12 in example, enter 1, 0.3333, 0.022 and 0.0085 to get 0.465 cfs. Change 1.486 to 1 and use meters for an SI sheet that returns m³/s. LET works in Excel 365 and Google Sheets, and older Excel needs θ, area and perimeter in separate cells.

Print Results in the Partial Pipe Flow Calculator saves the result as a PDF, ready to attach to a drainage report or a sewer design file. Copy Results puts the same numbers on the clipboard as plain text.