I was wondering if the TriPhase LED would give a decent effect without a fog machine? If not, what fog machine would be good for home use in a 20' x 20' area?

Depends, what type of effect are you looking for? If you are looking to just add color to the room on the floor, walls, objects in the room, etc, you don't need fog. If you are looking for air effects and that cloud look, then you need fog.