How does the number of blades in a fan affect the flux of air? I read that 3 blades are the best option but some companies uses more blades because there's a misconception among people that more blades generates more air.
[Physics] Number of blades in a fan
aerodynamics
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I think that such a deep filter will eliminate the need for a screen.
My guess/gut feeling is that (due to high pressure loss and flow through a porous medium) the air leaves the filter in a laminar fashion (even if it was turbulent before), so what kind of fan you use shouldn't matter if e.g. the volume flux (i.e. cfm) achieved is the same.
Everything hinges on the pressure loss across the filter (i guess pretty high) and the resulting volume flux of the whole system (depending on the characteristic/power of your fan). With this you get the flow velocity, and you can build a Reynolds number, and if this is higher than a critical number, you'll get turbulent flow.
The choice of length scale is important here. Do you pick the width at the end of the small channels in your filter? (but they are triangular/have a non-constant diameter, and flow through the walls to boot). The width of the fume hood? (But this is only ~one width long, so hardly counts as a "channel"...)
The problem is to find the critical Reynolds number that applies to your situation, which is not exactly standard. Critical Re for e.g. (round) pipe flow is ~2300 (but you have rectangular channels, could maybe pick the hydraulic diameter), and for a free jet I found a number of 1300, but take that with a grain of salt.
All in all, you're back to gut feelings. You could just build it though, and make experiments: Hold an incense stick or cigarette near the filter (downstream) and observe the smoke - is it smooth for the first couple centimeters (->laminar), or already jagged (->turbulent)?
I wont give you precise formulas, but one can calculate this. The most efficient theoretical rotor has only one blade. Obviously 1 blade would cause problems due to misplaced center of mass. That is why we use at least 2. Then to get more thrust you need long blades and you need to spin them very fast. There are two issues that would force you to have more blades. First is the blade tip must remain subsonic. Second is the practical size of the rotor. Very large rotors would require large areas to land and maneuvering would be more difficult. So basically more blades mean more power but with less efficiency and they take up less space. A particular helicopter design should use as little blades as possible given specific requirements.
Best Answer
3 blades would be indeed close to optimum for power efficiency and require less material during manufacturing.
But typical engineering goal for fans is different: maximum performance / noise. More blades $\implies$ slower rotation for same performance $\implies$ less turbulence $\implies$ less noise.
Also, check out my older relevant question: Wind generators - why so few blades?