Grille/Register Face Velocity and Free-Area Sizing
Face velocity V = CFM/(gross area x free-area ratio), or the required gross size from a target velocity. 400 CFM to a 500 fpm supply -> 154 in^2 (a 12x14); returns run slower so they size larger. The manufacturer free-area and throw data govern.
Formula and source
A_free = A_gross x ratio; V_face = CFM / A_free; A_gross_req = CFM / (V_target x ratio).
The grille/register free-area face-velocity relation from ASHRAE / SMACNA air-distribution practice, by name.
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