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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

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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

qn = vn an              (1)

where

qn = flow rate or discharge in section n    (m3/s, ft3/s)

vn = measured velocity in section n   (m/s, ft/s)

an = area of section n   (m2, ft2

One simple way to express the section area is

an = dn (ln+1 - ln-1) / 2                              (2)

The total flow in the stream can be summarized to

Q = Σ1n vn an                         (3)

where

Q = summarized flow rate or discharge in the conduit   (m3/s, ft3/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 ValuesCalculated Values
nv
(m/s)
d
(m)
l
(m)
a
(m2)
q
(m3/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

a1 = (1 m) ((4 m) - (0 m)) / 2

     = 2 m2   

a2 = (1.5 m) ((6 m) - (2 m)) / 2

     = 3 m2  

a3 = (0.9 m) ((8 m) - (4 m)) / 2

     = 1.8 m2  

The flow rates can be calculated as

q1 = (3 m/s) (2 m2)

    = 6 m3/s

q2 = (4 m/s) (3 m2)

    = 12 m3/s

q3 = (3 m/s) (1.8 m2)

    = 5.4 m3/s

The total flow can be summarized as

Q = (6 m3/s) + (12 m3/s) + (5.4 m3/s)

   = 23.4 m3/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

qn to n+1  = [(vn + vn+1) / 2] [(dn + dn+1 ) / 2] (ln+1 - ln)                  (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

qn = vn [((ln - ln-1) + (ln+1 - ln)) / 2] dn                 (5)

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