The Wikipedia article has nice image showing how the Archimedes screw work:
As I understand, the red balls do not fall down because they are in minima caused by the screw.
Because of material optimization we would like the screw to be as short as possible. This will be achieved if the screw was vertical (parallel to gravity force). Then the red balls would move down because the surface of the screw is moving, and they "want" to keep their position (laws of dynamics). Only friction can stop the items (eg. cubes) from falling down.
Mostly, Archimedes screws are used to transport water and other liquids, for which the friction is much lower. Are there any calculations that would tell optimal slope for the screw, concerning the material use, so meeting these opposing criteria
- the screw would be as short as possible
- the pitch as large as possible, so the surface is smallest)
- it has as large efficiency as possible (ratio to water or liquid delivered to the upper side related to water that fell down is the biggest),
- other aspects I do not imagine?
Let us assume there is no need for distance transporting (the lower tank has infinite surface and we can use the screw at any place to pump to the upper tank).