Effects of barrel size on the coarse step of tumbling rocks. The step that takes a long time.
Since smaller diameter tumbling barrels were mentioned, consider smaller diameter high rpm machinery. NOT apples for apples, but related to a degree.
Back in the early 80's they were developing the storage of large amounts of energy in smaller diameter high rpm flywheels spinning at 20,000 to 50,000 rpm.
Fancy magnetic bearings, composite flywheels, in a vacuum, etc. For use in satellites for the most part.
Anyway, these devices were one of the most efficient ways of storing energy in the smallest, lightest device albeit expensive.
en.wikipedia.org/wiki/Flywheel_energy_storageLooking at our tumbling barrels averaging 4 to 11 inches in diameter at 70% barrel fill.
This would be a classic dimensional analysis equation.(a mathematical method used for solving a problem when too many variables are involved)
Assume the average tumbled rock size is 1.5 inches. Darn close...this average size is important.
Assume the average barrel diameters are 4,6,8,11 inches.
A 4" barrel is about 3 times the width of a 1.5" rock.
An 11" barrel is about 7 times the width of a 1.5" rock.
Now walk down to the local quarry and roll a 1.5 foot off of a 4 foot high steep slope. And then roll a 1.5 foot rock off of an 11 foot high equally steep slope.
The dynamics are totally different in each case. The terminal speed of the rock is much higher at 11 feet as opposed to 4 feet.
The barrel diameter to rock size ratio has a big effect since tumbler barrels generate internal slopes that are probably similar in steepness.
Consider our 1.5" rock moving at equal barrel surface speeds of 690 inches per minute - 20 rpm for the 11" barrel and 55 rpm for the 4 inch barrel creates a very different environment.
Or take it to the extreme, consider a 48 inch barrel for the 1.5" rock at the same 690 inches per minute surface speed, works out to 5 rpm.
So now this little 1.5" rock is constantly rolling off a very tall slope.(wow, a 1.5 foot rock off of a 48 foot slope)
If you have any sympathy (or want to avoid some very beat up 1.5" rocks) you better use either lots of smalls or some thick slurry in a 48 inch(or 11 inch barrel for that matter).
And we don't want to tumble 6 to 12 inch rocks for the (3 times for 4") to (7 times for 11") ratio in a 48 inch tumbling barrel because they would kill each other.(another mathematical reason).
So with all this in mind I set out to find the optimum speed and barrel diameter for coarse tumbling 1.5" rocks.
I came up with 6 inches inside diameter, 4 times the width of a 1.5" rock.
Why ?
Because there was a relatively short slope within, which allowed for higher speeds without damaging the rocks.
There is lots of time where the rock spent time organizing into an avalanche at the top to the midpoint of the avalanche(11 to 3 o'clock) where the rocks are moving/rubbing against each other.
Be aware that the rocks are connected to each other NOT GRINDING as they rotate from the bottom of the barrel to near top-dead-center over and over. The dead zone (6 to 11 o'clock).
High speeds of 80 to 100 rpm were doable before the rocks began to stick to the wall of the tumbler depending on the rock sizes and slurry conditions.
High speeds and fairly small diameter barrel means more cycles from 11 to 3 o'clock per minute. The main grinding zone IMO.
And here we go again, the use of slurry to stick the rocks together a bit extends the grind zone from 11 to 4 or 5 o'clock by keeping them from rolling over themselves.
If you change the average rock size to 1 inch then consider a 4 inch barrel at highest possible speed before rocks stick to barrel wall.
If you change the average rock size to 2.5 inches a different problem occurs, they need to be well protected in a (4 times) 10 inch barrel due to increased chance of impact bruising(math reasons due to kinetic energy)).
Changing the average rock to 2 inches use a (4 times) 8 inch barrel is doable with hard rocks like agates. Better to run a mix of sizes to avoid impact bruises.
Why do 2 to 2.5 inch and larger rocks bruise ? Because of their kinetic energy. Calculated by multiplying 1/2 the mass of the rock X the speed of the rock X the speed of the rock.
So the impact amount is effected by the rock's velocity multiplied twice times half the mass of the rock.
Consider a 1 inch rock hitting you on the head verses a 2.5 inch rock hitting you on the head thrown at the same speed. One will bruise, the other can kill.
Our rocks can only take so much surface impact before they bruise.
Granted, there are other grinding zones in a rotary, mainly the center zone which is not mentioned here.
But you will find it too is predominately in the 11 to 3 o'clock zone.