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I am designing a printed piece that will deflect air from an exhaust fan on a computer 90 degrees. My question is, what would be most efficient, a circular radius, parabolic radius, or hyperbolic radius in the ducting to change the flow of the air 90 degrees upward. I'm trying to minimize the resistance of the ducting to maximize my exhaust air flow.

I know very little in this domain, so please forgive me for not knowing even the terms to search for.

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A curved elbow will be considerably better than square elbow, because it reduces turbulence. The exact mathematical shape of the curve is not really important; usually a circular radius is chosen because it is easier to fabricate. It is important to keep the cross sectional area the same as the air moves through the turn; two circular arcs that share the same center point can accomplish this.

To improve efficiency beyond a radius you can also add turning veins. It is probably of less importance on a small pc fan than on a large 100+hp fan, but might be something to look into. These also need to follow the same profile as the elbow and have the same cross sectional area between them. Again this can be accomplished with circular arcs that share the same center point as your duct walls.

enter image description here Diagram sourced from this link

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Given the low speed and flow rate of the application then a simple angled flat plate will cause the airflow to change direction.

Of course the other answers are correct but for the application a flat plate will work, but a curved duct would be best.

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What you are referring to is minor loss

check out slide 32 and 35

Slide #32 Slide 32

Slide #35 Slide #35 you'll see that head loss is reduced in a curved pipe versus a right angle pipe

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