A method for separating a ternary mixture comprising a light key component, a heavy key component, and at least one intermediate key component, wherein said mixture has a distillation boundary on a distillation region diagram that intersects a heterogeneous, minimum boiling binary azeotrope between said intermediate key component and said heavy key component that has a higher boiling point than the light key component, said method comprising: (a) introducing a feed stream comprising said mixture into a thermally integrated distillation system comprising a rectification zone, a first stripping zone, a second stripping zone, and a decantation zone; (b) withdrawing first and second overhead vapor streams from said first and said second stripping zones respectively; (c) introducing said first and said second overhead vapor streams into said rectification zone; (d) withdrawing a rectification liquid stream from said rectification zone; (e) introducing said rectification liquid stream into said decantation zone; (f) separating said rectification liquid stream in said decantation zone into a first liquid phase enriched in said heavy key component and a second liquid phase enriched in said intermediate key component, wherein said first and said second liquid phases have compositions that are in different distillation regions of said distillation region diagram; (g) introducing a first liquid stream comprising at least a portion of said first liquid phase into one of said first stripping zone and said second stripping zone; (h) introducing a second liquid stream comprising at least a portion of said second liquid phase into the other of said first stripping zone and said second stripping zone; (i) recovering a first bottoms product stream enriched in said heavy key component from said first stripping zone; (j) recovering a second bottoms product stream enriched in said intermediate key component from said second stripping zone; and (k) recovering a first overhead product stream enriched in said light key component from said rectification zone. (a) introducing a feed stream comprising said mixture into a thermally integrated distillation system comprising a rectification zone, a first stripping zone, a second stripping zone, and a decantation zone; (b) withdrawing first and second overhead vapor streams from said first and said second stripping zones respectively; (c) introducing said first and said second overhead vapor streams into said rectification zone; (d) withdrawing a rectification liquid stream from said rectification zone; (e) introducing said rectification liquid stream into said decantation zone; (f) separating said rectification liquid stream in said decantation zone into a first liquid phase enriched in said heavy key component and a second liquid phase enriched in said intermediate key component, wherein said first and said second liquid phases have compositions that are in different distillation regions of said distillation region diagram; (g) introducing a first liquid stream comprising at least a portion of said first liquid phase into one of said first stripping zone and said second stripping zone; (h) introducing a second liquid stream comprising at least a portion of said second liquid phase into the other of said first stripping zone and said second stripping zone; (i) recovering a first bottoms product stream enriched in said heavy key component from said first stripping zone; (j) recovering a second bottoms product stream enriched in said intermediate key component from said second stripping zone; and (k) recovering a first overhead product stream enriched in said light key component from said rectification zone.