Agitation Background

We try to understand the physics of agitation so we can make better judgments about what works, and doesn’t, and why.

What does a propeller in a tank or chest do?  It creates a “jet” of flow.  In an infinitely large tank in the absence of friction or turbulence, the jet would continue on its way.  In reality, the jet diffuses, affecting the rest of the tank by friction or turbulence, or it being deflected by the tank walls.


At low power levels, for example in a midfeather chest, the jet does not diffuse, so the tank is not completely mixed.  The stock proceeds around in a circle, but stock on one side of the circle doesn’t ever see that on the other.  So any consistency variations are not evened out.


At higher power levels, enough energy is input to make enough turbulence to completely mix the chest.  This is what happens, hopefully, in a modern stock chest.  Stock is completely mixed, not just circulated, and consistency fluctuations damped out.  Power level and retention time dictate how well this is done.  Where the stock is in the process indicates how well this must be done.  A machine chest would need more agitation than a broke chest.

Side insert agitators produce a horizontal jet of flow.  As you can see, the jet will not affect the top of the chest if the stock level is too high, above around 0.8 of the chest diameter.  You can add power to try to mix higher, but a lot of power is required for little effect.  So a mixed “zone” is formed by a side insert agitator.  This mixed zone is where we completely mix for consistency control.  Fortunately, it is also where the pump suction is.


Figures 3 and 4 also tells you where to locate your pump suction and dilution lines.  The stock at floor level behind the prop has been “circulated” more.  Diluting above and behind the prop means dilution water will be immediately mixed in, and can help cope with consistency overloads.

So how are agitators sized?  Given a particular chest and conditions, how do we figure out how much agitation is required?  The standard method used by all manufacturers includes the following:

Basic Process Number – A number determined by tank diameter or width, which relates to how far the “jet” throws, and how strong it is.

Chest Size Factor – This relates how deep the stock level is, and in rectangular chests, how long versus how wide the chest is, to the simple “jet” geometry.

Stock Factor – A factor that accounts for different types of stock, such as hardwood or softwood, Kraft or Sulphite, bleached or refined or not, virgin fiber or recycle.

Consistency Factor – 6% stock is tough to mix, but 2% stock behaves like water.

Temperature Factor – The hotter the stock, the easier to pump or agitate.

Time Factor – The more time to mix completely, the less power required.

All these factors are multiplied together to get a “Process Requirement”, that is, how much agitator is needed to mix a chest.

This sounds pretty cookbook, and it is, up to a certain point.  But judgment in picking the factors can influence results by a factor of 2 or 3 in horsepower.  When different manufacturers come up with different sizing, this does not reflect agitator efficiency differences, but just assumptions that are different, or differing levels of experience.

So we now know how much agitator we need.  How do we pick a combination of propeller type, speed, diameter and horsepower to do the job?

Agitators are rated by how much momentum they produce.  Momentum is the product of mass flow rate and velocity of the jet, and is defined as

Momentum Number M = C x N^2 x D^4

where N = Propeller RPM
           D = Propeller Diameter
           C = Constant

You can understand this equation as follows.  N x D is velocity at the propeller tip, and N x D x (D x D) is the velocity times the propeller area.  The constant C takes care of the constants like pi having to do with area, variations in velocity from propeller center to tip, and conversion of units.  It also is a measure of propeller efficiency in creating momentum.

Sounds pretty straight forward, but it isn’t.  The “constants” we are measuring with this single number are very complex, like velocity distribution across the impeller, and very non linear, that is, they vary with velocity.  Also, the tests done to characterize velocities have to be done in water, a fluid where viscosity is predictable.  We are agitating stock, where apparent viscosity varies all over the map.

So the “science” we have is pretty good within the range we are used to working in.  It is good to remember our limitations, check against experience, and view precise engineering numbers on consistency, stock type and temperature with some suspicion.

After all, agitators last 20 years or more, and it is unusual for a unit to be operating on the same stock, at the same consistency and tonnage after 5 years.  Agitators don’t cost much compared to the rest of a mill, and increments of capability even less.  But poor agitation, either initially or as tonnage increases over the years, will show up in poor process control and wasted production. 

 

Mechanical Design of Agitators

An agitator is a propeller turning in stock.  What are the forces it needs to be constructed to resist?

In a side insert agitator, as shown schematically in Figure 1, forces are as follows:

  • Its own weight, that of the shaft and propeller.
     

  • Reaction from the force it imparts to the stock, or thrust in reaction to pushing out the “jet” or stock.
     

  • Torque involved in turning the propeller.
     

  • Fluid forces acting perpendicular to the shaft of the propeller created by radial flow from the propeller. This radial flow is not desirable for mixing (though it may be for dispersion of solids), but propellers can’t create perfect axial flow jets.
     

  • Imbalanced forces created by impeller imbalance.
     

  • Forces imposed by the driver, like V-belt drive tightening loads.
     

  • Impacts from lumps of stock, variations in consistency or tramp material, or the stock surface when level is below the top of the propeller.


The designer evaluates these forces, and combines them to get a total stress, at the various critical points on the shaft and propeller.  He then compares them with the materials available, and their chemical resistance, and sizes the components.  But this is not all there is to the picture.

Another design concern is stiffness, expressed in two criteria, critical speed and deflection.

Critical speed is the speed at which, if an agitator was turned, otherwise small imbalances or external forces would be additive and grow and grow until failure occurred.  Critical speed for an agitator is calculated based on shaft stiffness.

Designers calculate the critical speed of the shaft, and make sure the driving speed (the speed at which the small imbalance or external force operates, typically shaft RPM) is some fraction of the critical speed.

The fraction is called the critical speed ratio, and typically it is held to 0.2 or 0.3 of the agitator speed.  When the agitator is operated with stock level below the top of the propeller, the driving speed is the number of blades times the RPM, a much tougher problem.  So agitators operated in this manner must be much stiffer.

Another criteria is deflection.  This is important to keep packing and seal wear within reason.  For mechanical seals this must be carefully checked.  Unfortunately, the deflections that are important are caused by impacts and shocks, which cannot be predicted theoretically.  So we go by experience, evaluating differing requirements, from whitewater to high density storage.

What do we do at the James Brinkley Company to deal with these problems?

  • We use cast propellers and hubs.  The cast metal has better mechanical properties than welded and the rolled shapes absorb shock and fatigue.
     

  • We use steel shafts, protected with stainless steel sleeves and tubes for corrosion resistance.  Alloy steel shafts such as 4140 and Stressproof have 3 times their strength and better fatigue resistance than stainless steel.
     

  • We design shafts and bearings to take overhung V-belt loads, which are often the most critical loads.
     

  • We make sure drive loads, torque, vibration and impacts do not distort or misalign our structural steel frame.