Since you run the beads side by side, overlapping even, to layer or clad and other applications, it would seem there is not really a minimum.  But I also guess that would depend upon certain applications.  Two seperate Tee joints.  How close can you get them depending upon their size and still get the proper fillet welds in place?  There are probably many things that would make that call for you but I don't believe there is actually anything in D1.1 that says there must be x amount of distance between two parallel weld beads.

Have a Great Day,  Brent

The same question has been posted a lot of times in this forum. Click on Search, write "Minimum distance" and click again on Search. A lot of postings will show up.
Giovanni S. Crisi
Sao Paulo - Brazil