The lack of laundry facilities in orbit was brought up recently and it occurred to me that this lack seems to be more for lack of trying than real difficulty. The lack of trying that I suspect could be because it is more trouble than it is worth to wash clothes on the ISS when it is a relatively minor mass and volume is to just throw away the dirty and ship up more from time to time. Eventually with enough human activity in LEO and BEO though, the choices will be wash, wear dirty until worn out, make more on board, or ship increasingly large quantities of disposable clothing from Earth. I think option one will be desirable at some point.
The problems with washing clothes in microgravity as I understand it are more to do with the chemicals normally used in detergents than with mechanical problems. The chemicals from the detergents and most soaps are very unwelcome in a closed environment as any environmental contamination is a major issue. It’s not like the air conditioner is going to vent problems outside the laundry room.
To me, the likely solution will be more mechanical with plain water than the soaps and softeners we use here on the ground. Something along the lines of pressure cleaners allied with old style wringer washing methods.
In this cartoon, he dirty laundry is fed into the rollers on the left. The rollers seal the entrance and feed the clothing into porous chain that travels through the cleaning area where it is hit from both sides by staggered nozzles that are calibrated to have enough pressure to knock dirt loose from the clothing but not enough to damage it. The clothing exits the chain through a second set of rollers that squeezes out most of the excess water and seals the exit from the rest of the ship.
During this the fluids in the wash area are continuously recycled through a liquid air separator/dirt filter/pump so that the same water is used indefinitely. This is represented by the oval in the lower section of the box. Second function of the filter/pump/separator is to maintain a lower pressure inside the machine than in the rest of the station so that there can be no contamination escape to the living area.