You'd be unwise to try to design you own from scratch, in all likelihood. Probably better to go to one of the proprietary manufacturers, though BAF's a bit old hat now, and you can get a similar process footprint from deep activated sludge tanks.
There are a number of potential design problems with BAF which took quite a bit of trial and error to work out, the biggest one probably being prevention of air blinding of the bed.
If you did want to have a go, assuming you have a good characterisation of the effluent's quality and quantity, you need to know the specific surface area of your proposed medium. You use this to work out the amount of bugs in the system. This is used to design the biological side of things, just like a trickling filter. Hydraulic residence time would also usually feature in the design.
Air flow rates are calculated in a similar way to an activated sludge system design, bearing in mind you will probably be using coarse bubble aeration. The flowrates of air and water for bed fluidisation and backwashing cycles are designed more or less as you would any granular bed filter.
