This is one of these questions to which the answer is "You shouldn't start from here." In chemical engineering the design of reactor is often trivial compared with the design of the separation process, and the correct method is to design the reactor to facilitate separation, not to look for "end of pipe" solutions afterwards.

Feasible solutions will depend on the nature of the slurry and the liquid. If the slurry is crushed material having a wide particle distribution and many fines, then it will be if not impossible then close to it. If it is mainly (roughly) monosized beads, then you have a good chance for filters or dynamic separators such as cyclones. Note that even with pressure filtration the resulting cake is likely to contain more than 60% by volume of liquid. You will need to displace the liquid between the particles by blowing through or vacuum. How much is retained on the surface of the particles depends on the viscosity and surface tension.

For many downstream particle problems, the best solution is to change the particles, for example do not grind them so small, and sieve out the fines before putting them in the reactor. If this is possible you should seriously consider it rather than just fighting a losing battle with what you have already.

For this scale, batch treatment would be more appropriate than continuous. The most obvious is a centrifuge to separate the particles from the liquid. A small one could be used several times, e.g. a half gallon one used 30 times.

May you will use baffle, supporter (perforate distributor ) and sieve like materials with equal distance between each to increases contact area.

So you can take as input and integrate with your idea.

What is the solid density vs the liquid?! in wastewater treatment facilities a DAF unit is used with a scraper to take the solids out, other solution is to use a centrifuge for separation of the liquid and solids. There are a lot of options, but can you tell if the solid settles, float or mixed?

As Baru said: to filter in-situ, you must greatly increase the surface area of your filter and use appropriate supporting structures under the filter media.

If your slurry can be pumped and your reactor is open-topped, use a small drum filter above the reactor and scrape the filter cake directly back into the reactor.

This is one of these questions to which the answer is "You shouldn't start from here." In chemical engineering the design of reactor is often trivial compared with the design of the separation process, and the correct method is to design the reactor to facilitate separation, not to look for "end of pipe" solutions afterwards.

Feasible solutions will depend on the nature of the slurry and the liquid. If the slurry is crushed material having a wide particle distribution and many fines, then it will be if not impossible then close to it. If it is mainly (roughly) monosized beads, then you have a good chance for filters or dynamic separators such as cyclones. Note that even with pressure filtration the resulting cake is likely to contain more than 60% by volume of liquid. You will need to displace the liquid between the particles by blowing through or vacuum. How much is retained on the surface of the particles depends on the viscosity and surface tension.

For many downstream particle problems, the best solution is to change the particles, for example do not grind them so small, and sieve out the fines before putting them in the reactor. If this is possible you should seriously consider it rather than just fighting a losing battle with what you have already.

For this scale, batch treatment would be more appropriate than continuous. The most obvious is a centrifuge to separate the particles from the liquid. A small one could be used several times, e.g. a half gallon one used 30 times.