I worked at the Texas Medical Center, The Methodist Hospital, for the Facilities Department making sure the Hospital passed JCAHCO certification inspections. [https://www.jointcommission.org/]
We exhausted dirty laundry room air to the roof. Making the roof a very sketchy place to breathe.
In the case of a hood for our purposes, the fan should pack a filter, the filter might be emptied of particulate onto a sheet of white paper. I would exhaust outdoors, where Nature, I hope, does something useful with whatever remains. Nature has talents in this area.