Head flowmeters in open conduits

Weirs are commonly used to measure the flow rate of liquids in open conduits. The theory is based on the Bernoulli equation for frictionless flow. From equation 1.13 with Ah = 0 and hf= 0 

Consider any point in the liquid at a height z vertically above the weir 

Flow of liquid over a sharp-crested weir 

Equation 8.19 is based on the following assumptions the approach velocity v is uniform and parallel, the streamlines are horizontal above the weir, there is atmospheric pressure under the nappe, and the flow is frictionless. 

Equation 8.20 gives the point velocity at a height z above the weir crest. If the width of the conduit is ft at this point, the volumetric flow rate through an element of cross-sectional flow area of height dz is 

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