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What are the differences between a Notch and a Weir?

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Main difference: Notch is made with metal and Weir is made with concrete(Masonry)
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Notches are openings which is used to measure discharge through it. However a weir is a notch but in a very large scale.

A Notch is used to measure relatively smaller discharge(examples include discharge in a flume, water lining etc) while a weir is used to measure larger discharge(examples are discharge in a stream or river,strome discharge etc.)
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