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I can buy a Vane Anemometer cheaply but a Air Flow meter (with a cone) costs a lot more.

Can I just make my own cone?

What issues should I expect if I add a cone to a Vane Anemometer?

How do I convert the readings into liters per second?

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Damnit ... the one formula that we had to use for every problem in fluid dynamics, and our professor made us derive everything from it ...

  sum of ( area * flow rate in  * density ) 
- sum of ( area * flow rate out * density ) 
= change in mass

Effectively, for your problem, volume of air flow is going to be

sum of ( wind speed * area of the opening )

I say 'sum of', but it's really an integral as the speed's not going to be constant across the whole opening. The cone just forces the air to a smaller area so there's not as much variance, and it's of a fixed area to make the calculations easier.

I'd estimate it by looking at the opening as a 4x4 grid, and taking 16 air speeds, one for each grid cell, average them, then multiply by the opening size, and converting to dm^3/s (as a liter is a cube 0.1m per side). Or convert to m^3/s, then multiply by 1000.

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