A perfectly stirred reactor is one in which the characteristics of the contents at any point in the vessel is homogeneous at all times.  The characteristics may change with time, but they will change uniformly and instantaneously throughout the vessel so they are ALWAYS constant at all points in the vessel.  From a practical point of view, theoretically perfect conditions are impossible to achieve.  No vessel will ever change instantaneously so characteristics everywhere in the vessel are always perfectly constant.  But many vessels and processes are very nearly perfect, and from a practical point of view, they are perfectly stirred reactors.  The assumption of theoretical perfection should be applied to guide thought processes to predict results in these vessels.

For example, standard tumbling ball mills are practical examples of perfectly stirred reactors.  The contents within such mills at all times are homogeneously uniform (this is true theoretically, and for the most part practically as well.)  Does the shape of the mill matter?  No, not really.  In typical ball mills, which have lengths somewhat greater than their diameters, samples taken from any location within the mill will be generally uniform.  Even in mills which are long cylinders with small diameters, media and particles at either end of the mill are free to move from one end to the other.  It may take a while for them to do so, but overall, the contents of the mill will be generally uniform.  Mills which are relatively short with large diameters will best achieve perfectly stirred reactor conditions.

There is no net flow in perfectly stirred reactors.  Perfect recirculation and mixing occurs within the vessels.  Powders and media are free to move anywhere with the vessels.  There is no standing flow pattern from one end to another.  Properties within each cross-section and properties in all cross-sections are similar.  Any particle or media ball can move from any one spot to any other spot within the vessel with ease and without bias.

Plug Flow Conditions

At the opposite extreme of the spectrum from the perfectly stirred reactor is a vessel in which plug flow conditions exist.  Homogeneity is lacking along the length of the mill, and flow proceeds from one point (usually from one end of the vessel) to another.  An example of plug flow is the extruded column that exits the die of an extruder.  Flow is in one direction only (out of the extruder) but relative flow between particles in extruded columns is zero.  Particles aren't changing positions relative to one another, but the whole column is flowing as a plug in a single direction.

Plug flow conditions also occur in some reaction vessels which are run continuously, with feed materials entering one end and product materials flowing from the other.  Plug flow conditions are most severe when vessels are long with narrow diameters.  Unlike the extrusion example, particles and media are free to move around, but within a cross-section of the mill, conditions are generally uniform. 

Many stirred ball mills function essentially as plug flow devices.  Stirred ball mills are employed for fine milling operations.  They use relatively fine media sizes (1/8" and smaller) and since the small media do not have sufficient masses to tumble and acquire momentum properly (as in tumbling ball mills), stirred ball mills contain shafts with perpendicular stirrer arms to provide the comminution energy.    Many of these mills are long with relatively small diameters.  Feed suspensions enter one end;  product suspensions exit the other.  As the length to diameter ratios of stirred ball mills increase, conditions are more and more perfectly plug flow.  Media and powder properties anywhere within a single cross-section will be essentially uniform, but media and powder properties at different locations along the length of the unit will differ substantially.

Why Should We Care?

We should care and pay attention to such differences because we frequently think that a mill is a mill; a continuous mill is no different from a batch mill; a lab tumbling ball mill is equivalent to a production continuous tumbling ball mill; a lab batch stirred ball mill is equivalent to a production continuous stirred ball mill; a production batch mill is equivalent to a semi-continuous production batch mill; a lab stirred ball mill is equivalent to a production tumbling ball mill, etc.

Most problems occur when procedures are tested in the lab for application and use in the plant -- especially for scale-up purposes.  Many lab mills are small batch mills.  Production mills are large.  Some are continuous.  Some are semi-continuous -- that is they run continuously on part of a large batch.

Laboratory batch mills should scale up reasonably well to production batch mills.  Laboratory continuous mills should scale up reasonably well to production continuous mills.  But lab batch mills do not easily scale up to production continuous mills.  Similarly, laboratory stirred ball mills should scale up reasonably well to production stirred ball mills.  Lab batch tumbling ball mills should scale up reasonably well to production batch tumbling ball mills.  Etc.  But crossing the scale-up lines from stirred to tumbling or from batch to continuous or vice versa can cause problems.

          Batch Tumbling Ball Mills

Small lab batch mills are essentially equivalent to large production batch mills.  Media sizes and the media bed depths, however, usually differ.  Most mills are filled half full of media, but this corresponds to a 6" bed depth in small lab mills, and to several feet of bed depth in large production mills.  The static load on particles in the bottom of a mill will be much greater in large mills than in small mills.  Also, when mills are run at rpms that launch the balls off the top of the stack to fall and impact the bed near the bottom of the mill (a.k.a. cataracting mode), the balls in the larger mill will have more momenta when they impact the bed.  Larger momentum means greater stresses and impact energies during comminution events.  Greater impact energies can produce smaller progeny particles.  The weakest particles in a batch are the biggest ones;  the strongest particles are the smallest ones;  so greater impact energies can translate into smaller progeny fines (smaller Ds values) as well as faster milling of the large particles.

Scaling from small batch mills to large batch mills should be possible (with these caveats considered.)  We will not go into the whole subject of scaling in this article, however, because (1) it is a huge subject unto itself, and (2) it is beyond the scope of this article.

          Batch Stirred Ball Mills

Most stirred ball mills are continuous, and essentially plug flow from end to end.  Scaling is somewhat easier from small to large stirred ball mills because it's possible to have the same media size (for example, 1/8" ) in lab mills as well as in production mills.  Since many of these mills are oriented horizontally, chamber diameter and media depth don't change as dramatically during scale-up as they do for tumbling ball mills.  Constant residence times in milling chambers and similarity of milling conditions are much easier to scale from lab stirred ball mills to production stirred ball mills than from lab batch tumbling mills to production batch tumbling mills.

The least expensive and easiest-to-use lab stirred ball mills, however, are batch mills.  Set a standard jar mill (with the lid off) on a drill press table;  chuck a stirrer bar into the drill press;  fill the jar with small media, and voilà -- you have a stirred ball mill.  Some companies sell lab-sized stirred ball mills of this design.  BUT NOTE -- these are batch stirred ball mills.

Results from this type of mill, nevertheless, scale up okay because all particles see constant residence times in the lab mill, and (due to the plug flow in the continuous mills) all particles in the production continuous mill also see essentially constant residence times.  Just remember, however, that this type of lab mill is batch while most production mills are continuous -- and there may be some unobvious differences between the two.

          Continuous Stirred Ball Mill PLUS One or Two Mixing Tanks

Sometimes, a small production stirred ball mill is used semi-continuously by placing a large batch tank beside it.  Then, suspension from the tank is pumped continuously through the mill, and product from the mill is returned to the same mixing tank.  There is a closed loop between the mixing tank, to the stirred ball mill, and back to the mixing tank.

The stirred ball mill in this example is described as a 'small' one because this arrangement is frequently used to mill larger batches than the capacity of the stirred ball mill can easily handle.  If the batch size is large enough, and according to the time constraints of the process and the maximum capacity of the stirred ball mill, it may not be possible to pass all of the batch through the mill as the batch is pumped from the holding tank to the next step in the process.  That might simply take too long to accomplish.  So if suspension is pumped continuously through the stirred ball mill and returned to the tank, and then eventually, the tank contents are pumped directly to the next step in the process, the suspension will have seen at least some milling.

But this is a very different condition than when the production stirred ball mill is large enough to continuously handle the contents of one tank and deliver the product to a second tank (or to the next process step.)  Why?  Because stirred ball mills are plug flow devices and a good mixing tank is a perfectly stirred reactor.  When you use a closed loop to pump from a tank, through a stirred ball mill, and back to the same tank, only some suspension from the tank will see stirred ball milling.  Since the mixing tank is a perfectly stirred reactor, when used in this way, some particles in the mixing tank will never pass through the stirred ball mill, while other particles in the mixing tank will pass through the stirred ball mill again and again and again.

The effect this will have on the product particle size distribution is to broaden it.  Comminution products from particles that pass through the stirred ball mill multiple times can be considerably smaller than they would have been had they passed only once through the mill.  Then too, the particles that never pass through the stirred ball mill will not be reduced in size at all.

So a small stirred ball mill that runs semi-continuously (in a recirculation loop from a single batch tank) will produce a broader particle size distribution (more finer fines plus some large feed particles remaining in the mixing tank), than will be produced in a larger, properly sized stirred ball mill through which ALL of the suspension passes ONCE.  When all particles pass through a stirred ball mill, the largest particles in the distribution (D_L) will be reduced from the original feed size of the batch.  When only some particles pass through a stirred ball mill, the D_L of the distribution will remain equal to the D_L of the feed size.

To intentionally broaden a size distribution, set up a stirred ball mill, feed it with a continuous loop from a holding tank, and return the product to the tank.  To maintain a narrow particle size distribution and/or when ALL particles must see some comminution, pump from the feed tank, through the stirred ball mill, to a second product tank. 

          Batch versus Continuous High Intensity Dispersion (Batch HID vs CHID)

This same problem exists for HID systems.  Most lab HID devices are batch mixers (for example, a milkshake mixer).  Instead of installing large batch production HID blungers, some companies install CHID systems beside their production blunger tanks.  To achieve full benefits from HID, residence times in HID blungers must be relatively long.  When residence times are translated to CHID devices, residence times must also be relatively long -- which means CHID flow rates must be really slow (read:   slooooowwwwwwww!!). 

In most cases, required CHID flow rates are too slow to pass all suspension through the CHID as the batch is pumped from one tank to the next process step.  To compensate, the semi-continuous mode of CHID is frequently employed.  A CHID is installed next to the holding tank and suspension from the tank is pumped continuously through the CHID and back to the tank.  Relatively speaking, this is usually a small CHID.  Since most holding tanks are perfectly stirred reactors, only some of the batch will ever see the CHID.  Much (most??) of the batch will not see HID.  This is the parallel case to the semi-continuous mode of stirred ball milling discussed above.  (In fact, a CHID can be piped in series with a stirred ball mill in a recirculation loop like this.)

Particles that pass through the CHID will see some HID.  Those particles that never pass through the CHID will not see any HID.  In such cases, the CHID step will only affect some of the particles in the tank.  If a batch contains lots of agglomerates, only some of them can be deagglomerated in a closed loop system like this. 

The closed loop CHID which applies some HID to some of the batch is a very different arrangement than when the whole batch passes through the CHID -- that is, when some HID is applied to ALL of the batch -- or when the whole batch is subjected to HID in an HID blunger. 

          Batch versus Continuous Tumbling Ball Mill

Laboratory-sized continuous tumbling mills are expensive, and therefore uncommon.  Almost all lab tumbling mills are batch mills.  What happens when lab mills are batch, but production mills are continuous?  Continuous tumbling ball mills revolve on open-centered trunnions through which feed and product powders and suspensions pass.

All feed particles in a batch tumbling ball mill will see the whole duration of milling.  Milling for an hour in a batch mill means all particles will see one hour's worth of milling.  Batch mills are perfectly stirred reactors, but there's no place for the particles to go during milling, other than to circulate around and be milled. 

Large continuous ball mills are also perfectly stirred reactors, but there are places for particles to go in such mills.  That is, some feed particles will almost instantaneously pass through the mill and exit with the product WITHOUT having seen much comminution.  This includes all sizes of feed particles.  Some fines and some middles and some coarse will all pass through continuous tumbling mills almost intact, without being exposed to many comminution events.  Then too, some feed materials will remain in continuous tumbling ball mills practically forever -- being exposed to comminution event after event after event.  Once again, this will broaden product distributions to produce many small progeny particles.  The total effect is to have particularly broad product distributions from continuous tumbling ball mills -- containing some large feed particles and some extremely fine particles.  The product distributions from batch tumbling ball mills will be narrower -- simply because the largest product particles (D_L) will be smaller than the largest feed particles.

This discussion of continuous tumbling ball milling presents the perfect case.  Obviously, large feed particles don't pass through such mills instantaneously, and some particles won't really remain in the mill forever.  But residence times of particles in continuous tumbling ball mills will vary from very short times to very long times and product size distributions will be broadened as a result.  This is a major difference from batch mills, in which all particles are milled for exactly the same lengths of time.

Conclusions

Many times, laboratory conditions are not similar to production conditions.  This is especially true when batch lab results are used to predict continuous production results. 

The distinctions between perfectly stirred reactors and plug flow reactors must be considered upon scale-up and, generally, whenever changing from one process device to another.  To scale up processes, many production groups transition from batch to semi-continuous to continuous operations without completely considering the differences between these methods in terms of perfectly stirred vs plug flow reactors.  "A ball mill is a ball mill," is simply not true.  This also applies to HID.  Size matters.  Batch versus continuous matters.  Tumbling versus stirred matters.  Perfectly stirred versus plug flow matters.

Continuous stirred ball mills scale up reasonably well.  When the whole batch passes through a continuous stirred ball mill, the particle size distribution remains relatively narrow and predictable.  When continuous stirred ball mills are run in a semi-continuous mode (a side loop on a batch holding tank), the product particle size distribution of the whole batch will be broadened compared to simple batch results.  The unpredictable part of this process is the PSD of the perfectly stirred suspension in the holding tank (from which feed suspension is taken and to which product is returned.)

Batch tumbling ball mills scale up reasonably well, as long as size and depth of media variations are considered.  But batch mill results are not easily scaled up to predict continuous tumbling ball mill results.  All particles in batch tumbling ball mills see exactly the same milling times.  Continuous tumbling ball mills, however, allow a range of milling times for particles -- which produces both coarser and finer product particles than are produced by batch milling.

Finally, high intensity dispersion (HID) results are also affected by batch versus semi-continuous versus continuous operations.  Batch operations expose whole suspensions to fixed HID times.  Semi-continuous operation (a side loop on a holding tank) exposes some of the suspension to HID, some to multiple passes through the HID, and some to no HID at all.  True continuous HID where 100% of each batch passes through the CHID will work on ALL of the batch.  The size of the CHID and the flow rate determines how much or how little HID each suspension will see.  Here again, exposing all suspension to HID (batch or true continuous operations) allows potential deagglomeration of all agglomerates -- which can produce finer particle size distributions in the product (fewer coarse agglomerates and many individual fine particles).  Exposing only some of each batch to HID (semi-continuous mode) potentially deagglomerates only some of the suspension.  Resulting particle size distributions are once again broader (many coarse agglomerates remain, and some individual fine particles will have been produced).

When changing from HID to CHID, or changing from one type of mill to another, or scaling up HID or milling operations, all of these characteristics should be considered:  types of blungers or mills, durations, nature of the process (batch versus semi-continuous versus continuous) and character of the process (perfectly stirred versus plug flow) should all be considered.  All of these variables will affect product particle size distributions.

Miscellany

Suggested topics for future issues of this E-zine .... Please continue to send your ideas or questions for future topics.  Thanks.  Until next time ...

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