Flow through a pipe occurs by gravity or pressure. Flow may be a laminar (smooth) or turbulent (rough). The method used to determine flow, or flow rate, depends on the properties of the liquid and characteristics of the pipe channel. The simplest pipe flow rate to calculate is water flowing uniformly through a less than half-full pipe at a fixed slope with a fixed diameter under no additional pressure besides atmospheric. Uniform flow has a constant depth. A pipe flowing in this manner is considered an open channel and the Chezy-Manning equation can be used to calculate flow rate. All measurements should be converted to feet.

- Skill level:
- Challenging

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### Things you need

- Measuring tape or ruler depending on the size of the pipe.
- Slope measuring tool, or six-foot level
- Graph paper
- Pencil
- Scientific calculator with trigonometric functions

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## Instructions

- 1
Measure the depth of flow (d) in the pipe. This is the distance from the surface of the water to the bottom of a circular pipe at its lowest point. Keep the ruler or tape perpendicular to the slope of the pipe. Measure the inner diameter (D) of the pipe by keeping the end of the tape measure at the lowest point of the pipe and extend it to the opposite side of the circular cross-section. As an example using the following equations, we can use two feet for depth of flow (d) and five feet for inner diameter (D). Radius is half of the diameter, thus the radius of the pipe is 2.5 feet.

- 2
Determine the slope of the pipe by measuring the elevation drop over a distance. The elevation drop is the distance the flow travels in a vertical axis perpendicular to a level surface. A slope measurement tool could also be used, otherwise a six- foot construction level and measuring tape works well. Rest the end of the level on top of the pipe and line the level up with the pipe. Raise or lower the floating end of the level until the indicator reads level. Measure the distance from the bottom of the floating end of the level down to the top of the pipe keeping the tape measure perpendicular to the level. This measurement is called the "rise." Because a six-foot level was used, the "run" is six feet. Slope is calculated by dividing the "run" into the "rise." For example, if the rise is 0.5 feet, then the slope can be calculated:

S = "rise" / "run" = 0.5 feet / 6 feet = 0.05 = 5 per cent

The slope is 0.05. In percentage form, the slope of the pipe is 5 per cent.

- 3
Sketch the cross section of the pipe including the water flow and label your measurements. Calculate the wetted perimeter, P, of the pipe. This is the distance along the inside perimeter of the pipe that is underwater. This distance forms an arc, and can be solved for by multiplying the radius of the pipe by the angle it "subtends". Another way to think of the angle it "subtends" is the angle the arc spans. The equation for this distance looks like this: P = ...r where ... = the angle the arc subtends in radians, and r = the radius of the pipe.

- 4
Mark the centre of the pipe on your sketch, and draw two radius lines from the centre of the pipe to where the surface of the water meets the pipe wall. The angle, taken in radians, between these two radii lines can be used to calculate the wetted perimeter. This is the central angle. The central angle can be calculated using the following equation. This equation is different if the pipe is more than half full.

... = 2 arccos ((r - d) / r ) Where d = depth of flow, and r = radius of the pipe.

In the example, ... = 2 arccos ((2.5 feet - 2 feet) / 2.5 feet) = 2.74 radians. Remember that arc-cosine is the same as the inverse of cosine, or cos^-1.

- 5
Solve the equation for wetted perimeter (P) using the central angle and radius of the pipe. See this equation solved below for the example:

P = ...r = 2.74 radians * 2.5 feet = 6.85 feet

- 6
Mark the area of flow (A) on your sketch. This is all the area that is taken up by water. The geometric shape of this area is called a sector and the formula for finding this area is A = (r^2 (... - sin...)) / 2 where r = radius of the pipe, and ... = the central angle. See this equation solved below for the example:

A = (r^2 (... - sin...)) / 2 = (2.5^2 feet * (2.74 - sin(2.74))) / 2 = 7.34 square feet.

- 7
Calculate the hydraulic radius (R) or ratio between the area of flow and the wetted perimeter using the equation R = A / P where A = area of flow, and P = wetted perimeter. See this equation solved below for the example:

R = A / P = 7.34 square feet / 6.85 feet = 1.07 feet

- 8
Determine the Manning's roughness coefficient for the pipe material. The rougher the material, the more friction on the water, thus the slower the flow. This coefficient takes this into account in the calculation for flow. The Manning Roughness Coefficient can be found in most civil engineering reference manuals. Some common values are 0.013 for concrete, 0.009 for PVC and 0.024 for corrugated steel. For example, a value of 0.013 indicates a concrete pipe.

- 9
Use the Chezy-Manning equation to solve for flow. This equation is a combination of the Chezy equation and Manning equation. It is used on a regular basis in civil engineering to calculate the flow rate through a pipe. The value of flow rate (Q) is in cubic feet per second.

Q = (1.49/n) (A) (R^(2/3)) (√S)

Where Q = flow rate, n = Manning's roughness coefficient, A = area of flow, R = hydraulic radius and S = slope of the pipe. See this equation solved below for the example:Q = (1.49/n) (A) (R^(2/3)) (?S) = (1.49/0.013) * (7.34 square feet) * (1.07^(2/3) feet) (?0.05) = 196.8 cubic feet per second.

- 10
Check your maths using a calculator. The flow rate (Q) is in cubic feet per second. In the example this means 196.8 cubic feet of water is coming out of the pipe every second.

#### Tips and warnings

- Velocity can be calculated from the flow rate and area using the equation V = Q/A.
- For large distances, elevation change should be measured using a surveyor's level and rod.
- The Chezy-Manning equation is slightly different for metric measurements: Q = (1.00/n) (A) (R(2/3)) (?S) [m^3/s]