There have been several threads about Flare Stack and Angel's Petrochem not running together, ending in the assumption, that this was an intentional change from Angel's side.
Since I couldn't find any specific reference in Angel's mods, I looked further and it turns out it's not as simple...
Angel disables the vanilla "oil-processing" technology on which Flare Stack relies and replaces it with "angels-oil-processing". By changing all references to "angels-oil-processing" in this mod, I got it to work properly alongside Angel's.
A proper fix will obviously have to include conditional logic, but shouldn't be too hard to do.
As a point of note, Angel's clarifier and flare stack are significantly faster than these at the cost of more space and slightly more power. Flare Stack does support a few more fluids though.