I'm going to assume that you're utilizing the liquid phase based on your terminology. It sounds as though you're utilizing 100% liquid formic acid, and attempting to dissolve "X", a solid or powder into it. I additionally am assuming that you're trying to achieve a continuous production flow as opposed to a batch reaction - If these are correct, then I have a few recommendations.
1) Is it possible to utilize a common solvent of both "X" and Formic acid to continue reacting in the liquid phase? Hypothetical: perhaps the solubility of "X" in water is considerably greater than in formic acid, and a 50/50 mixture of formic acid and water may be able to hold considerably more of "X". You mentioned that your desired product is an insoluble precipitate (in formic acid). You may be able to find a common solvent that still renders the product solid as insoluble. Even if this isn't the case, you may be able to get around this by increasing the reaction extent until reaching the solubility limit, at which point the precipitation will occur. There may be other hurdles to get around at this point (side reactions with solvent/selectivity, undesirable solubility limits of product etc).
2) A second option would possibly be a fluidized bed reactor. This would be more complicated than a simple re-circulation fluidized bed, because your product precipitate would become intermingled with your feed reactant, "X". A way to get around this would be to have an extended fluidized bed reactor as shown below (excuse my crude drawing)
Vessel #1 would be utilized as a "saturating vessel" - the formic acid would be loaded up to the solubility limit with reagent "X".
At point #2, an optional additional CSTR could be added to achieve some significant retention time if needed.
At point #3, a solids-liquid separator of some sort could be utilized to quickly remove the desired product and return an "X" depleted stream directly back to the saturation vessel.
A few notes: If you're only attempting to do a batch reaction, this is much more complicated than necessary. This idea is intended for continuous production. Additionally, based on your 10 hour reaction time, it's highly likely that an additional CSTR would be needed. There will also be considerable engineering hurdles such as how to maintain an "X" bed, what is the optimum exit concentration of "X" in the depleted stream (it will likely be very high to increase rate of reaction, but that depends on the reaction mechanism and order). Additionally, depending on mass transfer rates, "X" particle mechanical recalcitrance and reaction rates, it may be entirely overkill to fluidize the bed in vessel #1, and a static solids contacting bed may be enough. Lastly, you may run into issues of product precipitation inside of vessel #1 - the critical factor here will be particle size & density differences of reagent "X" and the product. If the Stokes velocity/settling velocity of the product is greater than the reagent "X", this design will not work as shown. It would require considerable changes to entrap "X" in a sort of "fixed bed plug flow solubilizer" (not that this can't be done). Additionally, this was my "gut reaction" to your scenario. With more time and thought, a more elegant solution may present itself.
The primary benefit that I see from this design and added complexity is that by adjusting retention time, you can control the concentration of "X" in the depleted stream. In general, retaining high concentrations of all reagents in the reaction medium will increase the rate of reaction.