## Theory of Screening

The purpose of screening is to separate granular substance particles that are smaller than the screen opening from those that are larger. This is not as simple as it sounds, and the difficulties compound as the opening becomes smaller.

For example, if a sample of a crushed mineral ore containing 50% by weight of particles smaller than 1/8” is dropped on a static test sieve, most of the undersize will remain on the shaker screen, with only a trickle passing through. If the sieve is shaken with a reciprocating motion in the horizontal plane, or a gyratory motion with both vertical and horizontal components, the minus 1/8″ particles will begin to pass through the screen at a decreasing rate until all but the particles closest to the opening size have been separated out. The time required for the shaking to achieve this stage will be roughly equal to the amount of sample deposited on the test sieve, which defines the depth of the static material bed prior to the shaking beginning.

The most generally used metric of screen efficiency is the cumulative weight of material that has passed the screen in any time interval, given as a percentage of the total weight of undersize in the feed. When oversize the product is to be recovered, efficiency is defined as the weight percent of the material in the screened oversize fraction relative to the total weight of oversize in the feed.

The probability (p) that any particle will pass a square opening in a woven wire screen is governed by the difference between its average diameter (d) and the opening dimension (L), and the wire diameter (t). A Swedish inventor, Dr. Fredrick Mogensen, predicts the probability p of a particle passing a square mesh sieve opening, if it approaches at 90 deg. to the plane of the opening, and does not touch a boundary wire, as:

from which it can be seen that the probability of an undersized particle passing the opening will diminish exponentially as its diameter approaches the opening dimension, and increase exponentially as the wire diameter (t) approaches zero. It may also be noted that, if the particle is removed (d=0), the equation equals the percent open area of a square mesh wire screen÷100. Thus if p is proportional to capacity, in a square mesh wire screen capacity must be proportional to the percent open area, a relationship that is made use of later in deriving the capacity correction factor F for the ratio L/t.

When the screen, supporting a static bed of material of extended size range, is shaken, a phenomenon called “trickle stratification”2 causes the particles to stratify from finer at the bottom to coarser at the top. The shaking motion may be in the horizontal plane of the screen, circular or reciprocating, or with a vertical component, or it may be a vibration applied directly to the screen wires.3 In the example above, the particles in the fraction smaller than 1/8” that reach the screen surface have a chance of passing an opening that is expressed by the Mogensen probability function. Then ideally, for an average particle diameter less than 1/8”, the number of particles of diameter d that will pass in a unit of time is the product of the probability function times the number of times a single particle is presented to an opening (without touching a boundary wire).

This ideal is confounded by unpredictable uncertainties. The necessary turbulence in the material bed caused by the motion of the screen causes interparticle interference and affects the angle at which a particle approaches an opening. The possibility for a particle to pass the opening without touching a boundary wire, a condition of the Mogensen function, is nil. Impact forces from contact with the boundary wires act as impedances to the force of gravity, the only force causing the particle to fall through the opening.

So the motion of the shaker screen, necessary for it to work, also can have the effect of limiting its capacity, in terms of the rate of passage of undersize per unit of area. Different kinds of motion are employed in the design of screening machines, and each has its special characteristics. Most modern screening machines can be sorted into four separate categories4. Each is subdivided into a variety of individual differences, but the following example will assign operating parameters typical of its category.

- The Gyratory Screen: 285 rpm, 2-1/2” horizontal circle dia.
- B. The Shaking Screen: 475 rpm, 1” stroke, zero pitch, 6 deg. slope.
- C. The Inclined Vibrating Screen: 1200 rpm, 1/4” vertical circle dia.
- D. The Horizontal Vibrating Screen: 840 rpm, 1/2” stroke at 45°.

Each has a .063” dia. wire screen with 1/8” clear opening, moving under a particle travelling at an assumed 20 fpm, for A, 40 fpm for B., 80 fpm for C, and 60 fpm for D. Omitting details of the calculations, the approximate number of openings presented to the particle per second is A. 200; B. 64; C. 98; D.50. The time available for the particle to fall through the opening, in sec.x10-3, is A. 5.0; B. 15.6; C. 10.2; D. 20.0 If it is assumed that the probability of passage of a single undersize particle is inversely proportional to the number of openings per second passing underneath, owing to interference with the boundary wires, the relative probabilities in each case are the same as the time available. Then, on the premise stated previously that the probabilities are in direct proportion to the number of opportunities (openings) per second, the product of the two probabilities is exactly the same for each case.

The time for this theoretical particle to pass the opening, from an approach at 90° and without touching a boundary wire, is 3.3-sec x 10-4. The ratio of time available in each case to the time required is A. 15.2; B. 47.3; C. 30.9; D. 60.6, which leads again to the same conclusion as before.

Should this oversimplified example lead to a conclusion that there is no inherent difference in relative performance among these four categories of motion?The answer is no, because such a conclusion would be overwhelmed by the realities of differences, to name a few, in turbulence, interparticle and boundary wire interference, depth of bed, slope of screen surface, relative velocities between particle and surface, displacement normal to the surface, and acceleration patterns. The correct conclusion is that performance claims favoring any particular design, whether Category A, B, C, or D to be valid, must be based on demonstrated comparative test results.