A Moving Bed Biofilm MBBR reactor consists of a tank with submerged but floating plastic (usually HDPE, polyethylene or polypropylene) media having specific gravity less than 1.0. The large surface area of the plastics provides abundant surface for bacterial growth. Biomass grows on the surface as a thin film whose thickness usually varies between 50-300 microns. The size depends upon community and can be easily transportation. Each individual bio carrier increases productivity through providing protected surface area to support the growth of heterotrophic and autotrophic bacteria within its cells. It is this high-density population of bacteria that achieves high-rate biodegradation within the system, while also offering process reliability and ease of operation. Medium or coarse bubble diffusers uniformly placed at the bottom of the reactor maintains a dissolved oxygen (DO) concentration of > 2.5-3 mg/L for BOD removal. Still higher DO concentrations are maintained for nitrification. To retain the media flowing out of the tank, screens are placed on the downstream walls. A clarifier or a DAF is placed downstream of the MBBR tank to separate the biomass and the solids from the wastewater. No sludge recycle is required for this process.
MBBR is effectively IFAS without a return sludge line.
If you search for IFAS design guidelines you will find more material - there are several PhDs and MScs that provide detail on the *process* design. Newer 'design'-based papers tend to be adverts for companies, rather than allowing you to recreate their algorithms. The presentation that Kim Helleshoej Soerensen has provided a link for gives you some detail on process design, but not hydraulics, and is valuable for being more specific to MBBR systems than the usual IFAS material.
Hydraulic aspects are usually ignored - as they are in most material on activated sludge design. The material I have seen on hydraulic design is proprietary, around the design of the media separators and, to a lesser extent, to ensure adequate mixing/cleaning of media. I have not seen anything in the open literature on the hydraulic aspects. If you do find anything, please share!