The distinction is between partitioning reusable fuel constituents from fission-product waste, and subsequently partitioning uranium from plutonium. In the first operation, chemical selectivity transfers selected actinides into a different stream while most fission products remain in the waste stream. In the second, differences in chemical speciation allow the two recovered elements to follow different streams. Neither operation separates uranium isotopes: uranium enrichment and chemical element separation are different tasks.
At a conceptual level, solvent extraction exploits unequal affinities of dissolved species for immiscible liquid phases. complexation changes those affinities, while a change in oxidation state can change an element's preferred phase without changing its identity. This explains why a stream bearing actinides can first be separated from much of its accompanying waste, and why uranium and plutonium need not subsequently remain together. Dissolved and undissolved residues, contaminated solvents and off-gases remain waste-management obligations; a recovery percentage alone is not a complete material balance.
Equipment must provide phase contact and phase disengagement within shielded, confined systems. Remotely maintained contactors and settlers are equipment categories serving those functions; sealed transfer systems, shielded cells and monitored off-gas handling limit contamination and exposure. Cooling manages residual decay heat, and nuclear material accountancy follows inventories and losses. criticality control in radioactive waste is essential because solvent extraction can redistribute fissile constituents. Scoped limitation: this answer gives the chemical principles and equipment functions; operational fissile-separation recipes, reagent sequences, contactor design details and process conditions are intentionally not supplied.

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