This is a method of batch-continuous operation for treatment and separation of liquid-solid mixtures in a reactor with reactive solids comprising steps of continuously feeding the liquid in the reactor, contacting (possibly, mixing) liquid with the solids in at least a portion of the reactor, periodically separating the liquid from the solids in unmixed, or predominantly unmixed, portion of the reactor, discharging at least a portion of the separated liquid from the reactor, and retaining the separated solids in the unmixed portion of the reactor, and periodically resuspending the solids in the unmixed portion of the reactor and transferring the resuspended soilds into the portion of the reactor for contacting liquid with the solids. The contacting and / or mixing of the liquid with the solids in the mixed portion of the reactor can be continuous or periodic mixing. The unmixed zone differs from the quiescent zone in conventional clarifiers. It is a flow-through zone with a noticeably high velocity of flow as in aeration tanks of biological treatment systems. The retention time in unmixed zones is much shorter than in clarifiers. The method and apparatus can be used to treat water and wastewater, and in various chemical, biochemical, and pharmaceutical processing operations.