1. The quantity of flanges is completely unrelated to the number of holes.
2. There are no standard ANSI flanges with 6 bolt holes. Not sure about DIN flanges.
3. The pitch circle diameter is not generally calculated; it is looked up in tables.
4. For nonstandard flanges, calculations are by stress formulas and simple geometry.
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Not in DIN flanges eitrher, or any standard I'm aware of. Bolt holes are always in multiples of 4. Not sure whether there's a technical reason for that, or just convention.
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In Automobile and refrigeration industry, 2 bolt flanges and 3 bolt flanges with tongue and groove joints are also used. I am not sure of their standard but I had seen special valves and fittings in those two industries.
There may be some special service of specific equipment mounted fitting with 6 bolts. In gas turbines, this type of connections are made in low pressure services within the enclosers, where approach to the 8 bolts is not possible and 4 bolts leaves big space between the bolts, there 6 can be accomodated.
In 6 bolt flange PCD is twice the distance between two consecutive bolt centers.
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