how do you design grain augers?

Answer 1

the grain augers I am familiar with are basically a tube with a rotating helical blade (or screw) that elevates the grains, very much the way a rotating drill bit will throw chips up the hole and out.

inside the tube is an Archimedes Screw
check out the link

Designing one would require a list of constraints:
the limiting height to raise the grain
the average (and I mean all associated discrepancies) of the grain
the available power for running this auger
the acceptable amount of loss, damaged grain
the maintenance cycle plan, clogging
I am sure there are more (temp, humidity, etc) that accompany ever engineering problem
why do you ask?