Recycle, bypass and purge

Real plants have many units joined together. You can draw a boundary round the whole plant (the overall balance), round each unit, or round a mixing or splitting point, and each boundary gives its own equations. Start with the boundary that has the fewest unknowns, often the overall one.

Reactions rarely convert all the feed in one pass, so the unreacted part is separated from the product and sent back: a recycle. Suppose fresh A comes in at 100 mol/h, the reactor converts 20% of what enters it each pass, and a separator recycles every bit of unreacted A. Overall, all 100 mol/h of A must leave as product, so the reactor must react 100 mol/h. At 20% per pass it needs 500 mol/h of A going in: 100 fresh plus 400 recycled.

That's the trade-off: a low conversion per pass can mean a big recycle, and so big pumps, pipes and separators. Recycle also saves raw material and energy that would otherwise be thrown away. Overall conversion (fresh feed in to product out) can be close to 100% even when single-pass conversion is low.

A bypass sends part of a stream round a unit and blends it back after. It's used to hit a target exactly: if a process makes water purer than needed, bypassing some untreated water saves money and still meets the spec.

If the feed carries something that doesn't react and isn't separated out, an inert like argon in the air fed to an ammonia plant, a recycle loop would let it build up forever. So a small purge stream is bled off the recycle. At steady state the purge must remove exactly as much inert as comes in, which sets how much valuable material is lost with it.