#### Variables in Ball Mill Operation

Ball mill operation is often regarded as something of a mystery for several reasons. Ball milling is not an art - it’s just physics. The first problem will ball mills is that we cannot see what is occurring in the mill. The other problem is that many of the independent variables are non-linear or have paradoxical effects on the end results.

In ball milling of dry solids the main independent variables are mill diameter, mill speed, media size, solids loading and residence time. For most companies the production mill already exists so we can ignore mill diameter and focus on the other variables.

Mill Speed is one variable that can often be easily changed with a variable frequency drive (VFD). The starting point for mill speed calculations is the critical speed. Critical speed (CS) is the speed at which the grinding media will centrifuge against the wall of the cylinder. Obviously no milling will occur when the media is pinned against the cylinder so operating speed will be some percentage of the CS.

The formula for critical speed is CS = 1/2π √(g/(R-r) where g is the gravitational constant, R is the inside diameter of the mill and r is the diameter of one piece of media. This reduced to CS = 265.45/√(R-r).

Dry mills typically operate in the range of 50%-70% of CS and most often between 60%-65% of CS. Below 50% CS too little energy is imparted to fracture particles and above 70% CS the media will start to cataract or be thrown from the top of the pile of media with each piece of media making a single energetic impact at the bottom of the mill. Too high a speed can lead to poor milling, increased milling time and accelerated media and shell wear. The movement of media at different percentages of critical speed is illustrated below.

Media Size - The size of the media is an important variable but not one that can be easily experimented with on a production level. The paradoxical aspect of media size is that the larger the media diameter the more energetic will be each impact, but the larger the media the fewer pieces of media in the mill and consequently fewer impacts per revolution. By using a larger than necessary media, finer particle size may well be achieve, but more slowly because there are fewer pieces of media and fewer impacts per revolution. Similarly smaller media will be less energetic, but there will be many more impacts per revolution of the mill. Consider one cubic foot of 1” spherical media. Easy enough to calculate that 12x12x12 will be 1,728 pieces of media. If we reduce this to ½” then there will be 24x24x24 or 13,824 pieces of media. An 8 fold increase.

The first issue to resolve then is how large, or small, does the media need to be to product sufficiently energetic impacts to fracture the particles and achieve the final particle size desired. In general, using the smallest media possible will give the shortest milling time because there are many more pieces of media and consequently many more impacts with every rotation of the mill.

Solids loading is another variable that is critical to optimal mill performance. The principle of mill operation is the impingement of the solids between pieces of media. The majority of media motion and milling occurs on the surface of the bed of media where the media is cascading. If there are too few solids, then media will strike media with little milling effect but with accelerated wear of the media and mill shell. If there is too much solids product the impacts will be buffered by the bed of solids. The interstitial space between perfect spheres is 26%. There will be less free space between other shapes of media such as spheroids, cylinders or mixed sizes of media.

Free space can easily be checked by pouring water into a known volume of media in a known size container and measuring the amount of water needed to just cover the media. Considering a typical mill with a 50% media charge there will be 13% free space as a percentage of the total mill volume. About 10% more solids is added to assure that there is solids between the pieces of media available to be impacted as the media cascades. Generally a 50% media charge will require about a 25% solids charge.

A properly loaded mill will have solids available between the pieces of media to absorb the energy of media to media impact.

An underloaded mill will have sub-optimal milling performance and accelerated media wear.

Scale Up - Experimentation on a small laboratory scale jar rolling mill is possible as long as scale-up is considered. Ball milling is one of the few unit operation that improves with increasing mill diameter. If acceptable results can be obtained with a 12” jar mil, then good result will certainly be obtained with a 72” diameter mill. Of course there is the possibility of over milling, obtaining a different particle size distribution or a need to change media size or mill speed. A general rule of thumb for mill scale-up is √(𝑑/𝐷), where d is the size of the smaller mill and D the inside diameter of the larger mill.

One way to side step mill scale-up issues while maintaining relatively small experimental batch sizes is to use a Slice Mill. Mill performance is based on mill diameter and length only increases or decreases capacity. The Slice Mill is simply a mill of the same diameter as the production mill but much shorter. A Slice Mill of 72” diameter by 12” wide would replicate the result of a normal production, mill 72” in diameter as 120” long.

A Slice Mill is the same diameter as the production mill but shorter in length.

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