I hope when you mentioned you put the sludge back in the water, you didn't put it in the water you already filtered...but if I take what I think you mean as trying to re-run the sludge back through again to optimize your final product. There's no problem with that, but always use a new container to catch the second or every other subsequent filtration so exactly what happened doesn't happen to perfectly well-done product. I do a CWE virtually every other day at least, if not every day, and what I have found to work quicker, better, and easier, is take one or two cotton balls (depending on size) and squeeze them into a funnel (make sure the cotton balls come out the other side of the hole slightly). And before you send your liquid through, run some tap water through to wet the cotton balls so you don't lose your solution that initially wets the cotton. Then, depending on the size of your funnel and cup, just rest it on the top of your container and pour your liquid through the funnel. But when you almost get to the end of pouring it through, and you come to the separated Tylenol/APAP concentration at the bottom, get another glass and pour the rest through. You can see the result of the second glass and it's consistency will be more cloudy as what you poured through the first glass and will have the highest concentration of APAP (it's near impossible to filter ALL the APAP out the first run). So rerun that little bit of solution through again using fresh wet cotton balls into your first glass. And if you are attempting to eliminate more than 10 grams of APAP--by doing a large batch--do all of the steps again...of course starting with heating it back up and cooling it down slowly.
But to answer your question directly, if you've filtered the liquid well enough that it's a sludge, don't do anything with it but throw it away. If you want to rerun your product again get a new filter. BTW, coffee filters are super cheap and work moderately well...and makes more sense than a T-shirt, although I've never used a T-shirt, so maybe I'm wrong and you may be onto something.
Any more questions just reply here. I know of other extractions that don't even require a filter, but gravity and porous strands of material...like a T-shirt lol. But be safe and what you did by coming on BL and asking is exactly what you should have done. I hope you didn't drink it, but rather saved it for replies from here.
