The relative merits of chevron vs cup and cone PTFE packing ? Why do the two co-exist, are there applications one is better ?
TFE Chevron packing is directional.  Cup and cone works in both directions.  If you experience vacuum it would be necessary to invert TFE Chevron, or if the vaccum with butterfly valves is not continuous,to use an upright set, lantern ring, and inverted packing set.  Or just throw in a C&C set and don't worry about it. The chances of having a C&C set installed properly after the first rebuild are substantialy greater.  
The performance of cup and cone packing is then equivalent.  Both are virgin TFE or some loaded variation of TFE, Both are energized to seal by compression of the gland into the stuffing box( compression causes the id to be reduced and the OD to be expanded) , with supplemental process loading that assists sealing by providing additional compression.  TFE chevron has reasonably delicate sealing edges that are a bit more subject to dirt and or poor-stem-finish abrasion, and TFE Chevron and C&C both must be assembled and installed as a packing SET in a particular sequence, not just random rings loaded into the stuffing box until it's full.

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