You are essentially describing a Chemostat— a continuous culturing device designed to continually replenish nutrients, while removing byproducts and waste, from biofermentation reactions.
When they work, they are great at keeping a steady stream of product flowing, while being very efficient at using available resources.
When they fail— and they do fail with quite some regularity- it is due to contamination with unwanted bacteria, or fungus. The culture can also crash due to instability in the nutrient supply, oxygenation or buildup of toxic products due to insufficient exfiltration. They also need constant parameter monitoring and are typically set up with Neptune apex style monitoring systems, except lab grade costing 10’s of 1000’s of $
If I were to do this— and, for the record, I wouldn’t— i would do a number of test runs first, before adding the display tank into the loop. You have to balance nutrient supply, flow, exfiltration (too fast and you waste resources before the phyto can utilize them) and it all has to be tailored to the multiplication rate of the organism in question.
With bacteria, which have a division time if 20-40 minutes under perfect lab conditions- setting up this system can be very difficult. And that’s with strains of bacteria adapted to optimal survival under these conditions.
Personally, I’d stick to doing batch culturing and performing decontamination on the materials on a routine basis. Batch culture is a lot easier, the yields can be easily scaled, and they don’t have to be monitored almost constantly. They also minimize your losses if a contamination event occurs.
Good idea though- good stuff