Think about what happens inside a fixed tubesheet shell and tube exchanger, where both ends of every tube are welded rigidly to the shell. The tube bundle and the outer shell usually sit at different average temperatures because one carries the hot fluid path and the other carries the cold fluid path, so they naturally try to stretch or shrink by different amounts. Because both tube ends are locked in place, this mismatch cannot be absorbed by the tubes sliding or shifting, and it shows up instead as build up of internal thermal stress in the tubes, the tubesheet welds, and the shell itself. An expansion bellow is just a corrugated flexible section built into the shell, and corrugations can stretch or compress easily under small axial loads. By letting the shell itself change length a little without much force, the bellow absorbs the difference in expansion between shell and tubes, which keeps the stress from building up to a damaging level. So the real job of the bellow is to relieve the thermal expansion stresses in the exchanger, which is option 2.