A method for separating a feed into a plurality of products, comprising: passing a feed comprising a plurality of components into a first divided volume on a first side of a dividing wall in a distillation column, the feed comprising i) at least one of a target component and a target boiling range and ii) one or more components or boiling ranges that are substantially adjacent to the target component or the target boiling range, the first divided volume being in fluid communication with a top common volume and in fluid communication with a bottom common volume; operating the distillation column to separate the feed into a plurality of products; withdrawing, during the operating, an intermediate stream from an intermediate withdrawal location on the first side of the dividing wall into an external loop, the intermediate stream comprising the at least one target component or target boiling range and at least one substantially adjacent component or substantially adjacent boiling range, the intermediate stream comprising a greater volume percentage of the at least one target component or target boiling range than the feed, the intermediate stream comprising fewer components than the feed; passing the intermediate stream into a separation stage in the external loop to form a separation stage effluent and a separation stage rejection stream, wherein the separation stage comprises a membrane separator and/or an adsorber, wherein the separation stage effluent comprises a greater weight percentage of the at least one target component or target boiling range than the intermediate stream, and wherein the separation stage rejection stream comprises a greater weight percentage of the at least one substantially adjacent component or substantially adjacent boiling range than the intermediate stream; passing the separation stage effluent into a second divided volume on an opposite side of the dividing wall in the distillation column; and withdrawing a product stream comprising the at least one target component or target boiling range from the second divided volume, the product stream comprising a greater weight percentage of the at least one target component or target boiling range than the separation stage effluent. passing a feed comprising a plurality of components into a first divided volume on a first side of a dividing wall in a distillation column, the feed comprising i) at least one of a target component and a target boiling range and ii) one or more components or boiling ranges that are substantially adjacent to the target component or the target boiling range, the first divided volume being in fluid communication with a top common volume and in fluid communication with a bottom common volume; operating the distillation column to separate the feed into a plurality of products; withdrawing, during the operating, an intermediate stream from an intermediate withdrawal location on the first side of the dividing wall into an external loop, the intermediate stream comprising the at least one target component or target boiling range and at least one substantially adjacent component or substantially adjacent boiling range, the intermediate stream comprising a greater volume percentage of the at least one target component or target boiling range than the feed, the intermediate stream comprising fewer components than the feed; passing the intermediate stream into a separation stage in the external loop to form a separation stage effluent and a separation stage rejection stream, wherein the separation stage comprises a membrane separator and/or an adsorber, wherein the separation stage effluent comprises a greater weight percentage of the at least one target component or target boiling range than the intermediate stream, and wherein the separation stage rejection stream comprises a greater weight percentage of the at least one substantially adjacent component or substantially adjacent boiling range than the intermediate stream; passing the separation stage effluent into a second divided volume on an opposite side of the dividing wall in the distillation column; and withdrawing a product stream comprising the at least one target component or target boiling range from the second divided volume, the product stream comprising a greater weight percentage of the at least one target component or target boiling range than the separation stage effluent.