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In the classic explanation of a venturi tube there is a wide part, a narrow part (throat) and then another wide part.

Can an accurate measurement also be obtained if there is only a wide and narrow part?
Is the second wide section needed?

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The measurement device doesn't care what happens downstream.

The idea behind a venturi is a temporary reduction in cross-section to accelerate the flow while allowing larger pipes to be used to prevent head (pressure) loss. A venturi will reduce flow but not nearly to the same extent as a smaller pipe.

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The purpose of the widening part, called a diffuser when the flow is subsonic, which is usually the case when using this device for flow measurement, is to recover the kinetic energy of the flow at the upstream throat (the place where the flow area is minimum) and bring the exit pressure up close to the pressure at the inlet of the venturi.

So the answer to the question is, yes, with or without a diffuser, the static pressure at the throat allows you to calculate the flow velocity there, if you know the inlet conditions. But the pressure at the throat is less than that at the inlet, and without a diffuser - and if the venturi is meant to be an in-line flow meter, you lose a lot of pressure, or flow energy. The kinetic energy of the throat flow becomes dissipated or wasted.

So the broader answer to the question is, no, the venturi would not work as a flow meter if the downstream connection were at the same cross flow area as the throat. There would be a large pressure drop because of the venturi and this would most likely significantly decrease the flow. So what good would it be, if it's meant to be an instrument to only measure and not modify?

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