# Flow Rate Calculated based on the Velocity-Area Principle

## Flow rate or discharge in an open conduit, channel or river can be calculated with the velocity-area principle

The velocity-area principle is based on velocity measurements in a open flow like a conduit, channel or river.

Velocities and depths across the stream are measured as indicated in the figure above. A partial discharge in a section of the stream can be calculated as

*q _{n} = v_{n} a_{n} (1)*

*where*

*q _{n} = flow rate or discharge in section n (m3/s, ft3/s)*

*v _{n} = measured velocity in section n (m/s, ft/s)*

*a _{n} = area of section n (m^{2}, ft^{2}) *

One simple way to express the section area is

*a _{n} = d_{n} (l_{n+1} - l_{n-1}) / 2 (2)*

The total flow in the stream can be summarized to

*Q = Σ _{1}^{n} v_{n} a_{n} (3)*

*where *

*Q = summarized flow rate or discharge in the conduit (m ^{3}/s, ft^{3}/s)*

The accuracy of estimate depends on the profile of the conduit and the number of measurements. For conduits with regular shapes like rectangular channels a limited number of measurements are required. For irregular shapes - like natural rivers or similar - higher accuracy requires more measurements both horizontal and vertical.

### Example - Computing Flow Rate in a Channel

From a conduit we have three measurements:

Measured Values | Calculated Values | ||||
---|---|---|---|---|---|

n | v(m/s) | d(m) | l(m) | a(m^{2}) | q(m^{3}/s) |

0 | 0 | 0 | 0 | ||

1 | 3 | 1 | 2 | 2 | 6 |

2 | 4 | 1.5 | 4 | 3 | 12 |

3 | 3 | 0.9 | 6 | 1.8 | 5.4 |

4 | 0 | 0 | 8 | ||

Summarized | 23.4 |

The section areas can be calculated like

*a _{1} = (1 m) ((4 m) - (0 m)) / 2*

* = 2 m ^{2} *

*a _{2} = (1.5 m) ((6 m) - (2 m)) / 2*

* = 3 m ^{2} *

*a _{3} = (0.9 m) ((8 m) - (4 m)) / 2*

* = 1.8 m ^{2} *

The flow rates can be calculated as

*q _{1} = (3 m/s) (2 m^{2})*

* = 6 m ^{3}/s*

*q _{2} = (4 m/s) (3 m^{2})*

* = 12 m ^{3}/s*

*q _{3} = (3 m/s) (1.8 m^{2})*

* = 5.4 m ^{3}/s*

The total flow can be summarized as

*Q = (6 m ^{3}/s) + (12 m^{3}/s) + (5.4 m^{3}/s)*

* = 23.4 m ^{3}/s*

Note - there are alternative ways to calculate the section flow rates:

### Simple Average Method

Using the simple average of two successive vertical depths, their mean velocity, and the distance between them can be expressed as

*q _{n to n+1 } = [(v_{n} + v_{n+1}) / 2] [(d_{n} + d_{n+1} ) / 2] (l_{n+1} - l_{n}) (4)*

### Midsection Method

With the midsection method, the depth and mean velocity are measured for each number of verticals along the cross section. The depth at a vertical is multiplied by the width, which extends halfway to the preceding vertical and halfway to the following vertical, to develop a cross-sectional area. The section flow rate can be expressed as

*q _{n} = v_{n} [((l_{n} - l_{n-1}) + (l_{n+1} - l_{n})) / 2] d_{n} (5)*

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