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About the book
Material and energy

Fluid-flow theory
Fluid-flow applications
Heat-transfer theory

separation processes


Size reduction
Index to Figures
Index to Examples
Useful links
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Grinding and cutting.
Energy Used in Grinding
New Surface Formed by Grinding
Grinding equipment.
Hammer mills
Fixed-head mills
Plate mills
Roller mills
Miscellaneous milling equipment

Raw materials often occur in sizes that are too large to be used and, therefore, they must be reduced in size. This size-reduction operation can be divided into two major categories depending on whether the material is a solid or a liquid. If it is solid, the operations are called grinding and cutting, if it is liquid, emulsification or atomization. All depend on the reaction to shearing forces within solids and liquids.




In the grinding process, materials are reduced in size by fracturing them. The mechanism of fracture is not fully understood, but in the process, the material is stressed by the action of mechanical moving parts in the grinding machine and initially the stress is absorbed internally by the material as strain energy. When the local strain energy exceeds a critical level, which is a function of the material, fracture occurs along lines of weakness and the stored energy is released. Some of the energy is taken up in the creation of new surface, but the greater part of it is dissipated as heat. Time also plays a part in the fracturing process and it appears that material will fracture at lower stress concentrations if these can be maintained for longer periods. Grinding is, therefore, achieved by mechanical stress followed by rupture and the energy required depends upon the hardness of the material and also upon the tendency of the material to crack - its friability.

The force applied may be compression, impact, or shear, and both the magnitude of the force and the time of application affect the extent of grinding achieved. For efficient grinding, the energy applied to the material should exceed, by as small a margin as possible, the minimum energy needed to rupture the material . Excess energy is lost as heat and this loss should be kept as low as practicable.

The important factors to be studied in the grinding process are the amount of energy used and the amount of new surface formed by grinding.

Energy Used in Grinding

Grinding is a very inefficient process and it is important to use energy as efficiently as possible. Unfortunately, it is not easy to calculate the minimum energy required for a given reduction process, but some theories have been advanced which are useful.


dE/dL = KLn                                                                                                             (11.1)

where dE is the differential energy required, dL is the change in a typical dimension, L is the magnitude of a typical length dimension and K, n, are constants.

Kick assumed that the energy required to reduce a material in size was directly proportional to the size reduction ratio dL/L. This implies that n in eqn. (11.1) is equal to -1. If

                       K = KKfc

where KK is called Kick's constant and fc is called the crushing strength of the material, we have:

                 dE/dL = KKfcL-1

which, on integration gives:

                       E = KKfc loge(L1/L2)                                                                        (11.2)

Equation (11.2) is a statement of Kick's Law. It implies that the specific energy required to crush a material, for example from 10 cm down to 5 cm, is the same as the energy required to crush the same material from 5 mm to 2.5 mm.

Rittinger, on the other hand, assumed that the energy required for size reduction is directly proportional, not to the change in length dimensions, but to the change in surface area. This leads to a value of -2 for n in eqn. (11.1) as area is proportional to length squared. If we put:

      K = KRfc

and so

dE/dL = KRfcL-2                                                                                                    

where KR is called Rittinger's constant, and integrate the resulting form of eqn. (11.1), we obtain:

      E = KRfc(1/L2– 1/L1)                                                                                          (11.3)

Equation (11.3) is known as Rittinger's Law. As the specific surface of a particle, the surface area per unit mass, is proportional to 1/L, eqn. (11.3) postulates that the energy required to reduce L for a mass of particles from 10 cm to 5 cm would be the same as that required to reduce, for example, the same mass of 5 mm particles down to 4.7 mm. This is a very much smaller reduction, in terms of energy per unit mass for the smaller particles, than that predicted by Kick's Law.

It has been found, experimentally, that for the grinding of coarse particles in which the increase in surface area per unit mass is relatively small, Kick's Law is a reasonable approximation. For the size reduction of fine powders, on the other hand, in which large areas of new surface are being created, Rittinger's Law fits the experimental data better.

              E = Ei (100/L2)1/2[1 - (1/q1/2)]                                                                            (11.4)

Bond defines the quantity Ei by this equation: L is measured in microns in eqn. (11.4) and so Ei is the amount of energy required to reduce unit mass of the material from an infinitely large particle size down to a particle size of 100 mm. It is expressed in terms of q, the reduction ratio where q = L1/L2.

Note that all of these equations [eqns. (11.2), (11.3), and (11.4)] are dimensional equations and so if quoted values are to be used for the various constants, the dimensions must be expressed in appropriate units. In Bond's equation, if L is expressed in microns, this defines Ei and Bond calls this the Work Index.

The greatest use of these equations is in making comparisons between power requirements for various degrees of reduction.

EXAMPLE 11.1. Grinding of sugar
Sugar is ground from crystals of which it is acceptable that 80% pass a 500
mm sieve(US Standard Sieve No.35), down to a size in which it is acceptable that 80% passes a 88 mm (No.170) sieve, and a 5-horsepower motor is found just sufficient for the required throughput. If the requirements are changed such that the grinding is only down to 80% through a 125 mm (No.120) sieve but the throughput is to be increased by 50% would the existing motor have sufficient power to operate the grinder? Assume Bond's equation.

Using the subscripts 1 for the first condition and 2 for the second, and letting m kg h-1 be the initial throughput, then if x is the required power
Now 88
mm = 8.8 x 10-6m , 125 mm = 125 x 10-6m , 500mm = 500 x 10-6m

     E1 = 5/m = Ei(100/88 x 10-6)1/2 [1 – (88/500)1/2]

     E2 = x/1.5m = Ei(100/125 x 10-6)1/2 [1 - (125/500)1/2]

E2/E1 = x/(1.5 x 5) =    (88 x 10-6)1/2[1 - (125/500)1/2
                                  (125 x 10-6)1/2[1 – (88/500)1/2]

x/(7.5) = 0.84 x (0.500/0.58)
          = 0.72

       x = 5.4 horsepower.

So the motor would be expected to have insufficient power to pass the 50% increased throughput, though it should be able to handle an increase of 40%.

New Surface Formed by Grinding

When a uniform particle is crushed, after the first crushing the size of the particles produced will vary a great deal from relatively coarse to fine and even to dust. As the grinding continues, the coarser particles will be further reduced but there will be less change in the size of the fine particles. Careful analysis has shown that there tends to be a certain size that increases in its relative proportions in the mixture and which soon becomes the predominant size fraction. For example, wheat after first crushing gives a wide range of particle sizes in the coarse flour, but after further grinding the predominant fraction soon becomes that passing a 250 mm sieve and being retained on a 125 mm sieve. This fraction tends to build up, however long the grinding continues, so long as the same type of machinery, rolls in this case, is employed.

The surface area of a fine particulate material is large and can be important. Most reactions are related to the surface area available, so the surface area can have a considerable bearing on the properties of the material. For example, wheat in the form of grains is relatively stable so long as it is kept dry, but if ground to a fine flour has such a large surface per unit mass that it becomes liable to explosive oxidation, as is all too well known in the milling industry. The surface area per unit mass is called the specific surface. To calculate this in a known mass of material it is necessary to know the particle-size distribution and, also the shape factor of the particles. The particle size gives one dimension that can be called the typical dimension, Dp, of a particle. This has now to be related to the surface area.

We can write, arbitrarily:

     Vp = pDp3                  


      Ap= 6qDp2.

where Vp is the volume of the particle, Ap is the area of the particle surface, Dp is the typical dimension of the particle and p, q are factors which connect the particle geometries.(Note subscript p and factor p)

For example, for a cube, the volume is Dp3 and the surface area is 6Dp2; for a sphere the volume is (p/6)Dp3 and the surface area is pDp2 In each case the ratio of surface area to volume is 6/Dp.

A shape factor is now defined as q/p = l (lambda), so that for a cube or a sphere l = 1. It has been found, experimentally, that for many materials when ground, the shape factor of the resulting particles is approximately 1.75, which means that their surface area to volume ratio is nearly twice that for a cube or a sphere.

The ratio of surface area to volume is:

                   Ap/Vp =( 6q/p)Dp = 6l/Dp                                                                      (11.5)

and so                
Ap= 6q Vp/pDp = 6l(VP/DP)                                        

If there is a mass m of particles of density rp, the number of particles is m/rpVP each of area Ap.

So  total area     At = (m/rpVP) x ( 6qVp/pDP) = 6qm/rp pDp
              = 6lm/rDp                                                                             (11.6)

where At is the total area of the mass of particles. Equation (11.6) can be combined with the results of sieve analysis to estimate the total surface area of a powder.

EXAMPLE 11.2. Surface area of salt crystals
In an analysis of ground salt using Tyler sieves, it was found that 38% of the total salt passed through a 7 mesh sieve and was caught on a 9 mesh sieve. For one of the finer fractions, 5% passed an 80 mesh sieve but was retained on a 115 mesh sieve. Estimate the surface areas of these two fractions in a 5 kg sample of the salt, if the density of salt is 1050 kg m-3 and the shape factor (
l) is 1.75.

Aperture of Tyler sieves, 7 mesh = 2.83 mm, 9 mesh = 2.00 mm, 80 mesh = 0.177 mm, 115 mesh = 0.125 mm.
Mean aperture 7 and 9 mesh      = 2.41 mm = 2.4 x 10-3m
Mean aperture 80 and 115 mesh = 0.151 mm = 0.151 x 10-3m
Now from Eqn. (11.6)

     A1 = (6 x 1.75 x 0.38 x 5)/(1050 x 2.41 x 10-3)
          = 7.88 m2
     A2 = (6 x 1.75 x 0.05 x 5)/(1050 x 0.151 x 10-3)
          = 16.6 m2.

Grinding Equipment

Grinding equipment can be divided into two classes - crushers and grinders. In the first class the major action is compressive, whereas grinders combine shear and impact with compressive forces.


Jaw and gyratory crushers are heavy equipment and are not used extensively in the food industry. In a jaw crusher, the material is fed in between two heavy jaws, one fixed and the other reciprocating, so as to work the material down into a narrower and narrower space, crushing it as it goes. The gyrator crusher consists of a truncated conical casing, inside which a crushing head rotates eccentrically. The crushing head is shaped as an inverted cone and the material being crushed is trapped between the outer fixed, and the inner gyrating, cones, and it is again forced into a narrower and narrower space during which time it is crushed. Jaw and gyratory crusher actions are illustrated in Fig. 11.1(a) and (b).

FIG. 11.1 Crushers: (a) jaw, (b) gyratory
Figure 11.1 Crushers: (a) jaw, (b) gyratory

Crushing rolls consist of two horizontal heavy cylinders, mounted parallel to each other and close together. They rotate in opposite directions and the material to be crushed is trapped and nipped between them being crushed as it passes through. In some case, the rolls are both driven at the same speed. In other cases, they may be driven at differential speeds, or only one roll is driven. A major application is in the cane sugar industry, where several stages of rolls are used to crush the cane.

Hammer mills

In a hammer mill, swinging hammerheads are attached to a rotor that rotates at high speed inside a hardened casing. The principle is illustrated in Fig. 11.2(a).

FIG. 11.2 Grinders: (a) hammer mill, (b) plate mill
Figure 11.2 Grinders: (a) hammer mill, (b) plate mill

The material is crushed and pulverized between the hammers and the casing and remains in the mill until it is fine enough to pass through a screen which forms the bottom of the casing. Both brittle and fibrous materials can be handled in hammer mills, though with fibrous material, projecting sections on the casing may be used to give a cutting action.

Fixed head mills

Various forms of mills are used in which the material is sheared between a fixed casing and a rotating head, often with only fine clearances between them. One type is a pin mill in which both the static and the moving plates have pins attached on the surface and the powder is sheared between the pins.

Plate mills

In plate mills the material is fed between two circular plates, one of them fixed and the other rotating. The feed comes in near the axis of rotation and is sheared and crushed as it makes its way to the edge of the plates, see Fig. 11.2(b). The plates can be mounted horizontally as in the traditional Buhr stone used for grinding corn, which has a fluted surface on the plates. The plates can be mounted vertically also. Developments of the plate mill have led to the colloid mill, which uses very fine clearances and very high speeds to produce particles of colloidal dimensions.

Roller mills

Roller mills are similar to roller crushers, but they have smooth or finely fluted rolls, and rotate at differential speeds. They are used very widely to grind flour. Because of their simple geometry, the maximum size of the particle that can pass between the rolls can be regulated. If the friction coefficient between the rolls and the feed material is known, the largest particle that will be nipped between the rolls can be calculated, knowing the geometry of the particles.

Miscellaneous milling equipment

The range of milling equipment is very wide. It includes ball mills, in which the material to be ground is enclosed in a horizontal cylinder or a cone and tumbled with a large number of steel balls, natural pebbles or artificial stones, which crush and break the material. Ball mills have limited applications in the food industry, but they are used for grinding food colouring materials.
The edge runner mill, which is basically a heavy broad wheel running round a circular trough, is used for grinding chocolate and confectionery.
Many types of milling equipment have come to be traditional in various industries and it is often claimed that they provide characteristic actions that are peculiarly suited to, and necessary for, the product.


Cutting machinery is generally simple, consisting of rotating knives in various arrangements. A major problem often is to keep the knives sharp so that they cut rather than tear. An example is the bowl chopper in which a flat bowl containing the material revolves beneath a vertical rotating cutting knife.


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