I figure that the thing on the floor in engineering in TOS was the reactor. We've seen it house the dilithium in "Elaan of Troyus." It is located at the rear of the saucer section. the two large round things are the mattar and anti-matter injectors. Tmatter and anti-matter are drawn up fromtheir storage areas inthe secondary hull. They are sent through the intermix chamber and then the drive plasma is sent back down to the engineering hull. The whole thing would occupy the same space as seen in TMP. This explainationjust fits the best for me.
JNG perhaps that piece in TAS is related to this in TWOK:
It also reminds me of that depth tube on the bridge of the Nautilus from 20,000 Leagues Under the Sea.