Which would be most efficient, a round branch duct tapping out of the bottom of a round duct, or a round branch duct tapping out of the bottom of a rectangular duct? It's in a plenum, so looks do not matter. Cost doesn't matter either. I'm going for efficiency.
Well what do you have a square duct or a round one ?
If you have a round duct you can use a "T" in the main and reduce it for the size grill,flex or pipe, or you can have a lead in shoe or (boot Tap) made in round, if it's a square main you can tap in with a round collar (if you are using an 8" grill it is better to tap in with a 10" collar and reduce it) or you can make a boot tap and come off with a square duct (in this case the lead in is usually around 6".)
Either of these options are efficient.
to tap a round duct off the bottom you would need a sadlle tap
you could use a t-wye but then you have more conections to make.
as far as the efficiency, you are adding another line that wasnt load calced into the sytem and could change everything downstream.
you might want to verify with your ductulator size and location based on system esp. and total system design.