# Mechanisms and Laws Governing Size Reduction

The mechanism of size reduction depends upon the nature of the material and each material requires separate treatment. Generally, fracture occurs along the lines of weakness. During size reduction, fresh surfaces are created or existing cracks and fissures are opened up, wherein the former requires more energy. There may be a tendency that after processing agglomerates of particles are formed. Size reduction is an energy-inefficient process because a small amount of energy is required in subdividing the particles. A lot of the energy is spent in overcoming the friction and inertia of machine parts and the friction between particles and deforming the particles without breaking them. This energy is released as heat.

## Laws Governing Size Reduction Process

One of the mechanisms of size reduction called grinding is very inefficient and thus it is important to use energy as efficiently as possible. It is not easy to calculate the minimum energy required for a given size reduction process. Fortunately, certain theories are useful in approximately calculating energy requirements. Although several theories have been put forth to predict the energy requirements, none give accurate results.

The theories of size reduction and estimation of energy requirement depend upon the basic assumption that the energy required to produce a change dL in a particle of a typical size dimension L, is a simple power function of L:

\frac{dE}{dL}=KL^n ...(1)

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

### Kicks Law:

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 equation (1) is equal to −1.

If,

K=K_kf_c  ...(2)

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

\frac{dE}{dL}=K_kf_cL^{-1} ...(3)

On integrating equation (3) gives:

E=K_kf_c\log_e\frac{L_1}{L_2} ...(4)

Equation (4) is a statement of Kick's Law. It states that the specific energy required to crush a material, for example, is from 10 cm to 5 cm. The same energy is required to crush the same material from 5 mm to 2.5 mm. Thus, in simple terms Kicks law can be stated as the energy required to reduce the size of a given quantity of material is constant for the same reduction ratio regardless of the original size.

### Rittinger’s Law:

Rittinger assumed that the energy required for size reduction is directly proportional to the change in surface area. This leads to a value of −2 for n in equation (1) as the area is proportional to the length squared. If we put:

K=K_Rf_c ...(5)

Then,

\frac{dE}{dL}=K_Rf_cL^{-2} ...(6)

where KR is called Rittinger's constant. On integrating equation (6), we obtain:

E=K_Rf_c\left(\frac1{L_2}-\frac1{L_1}\right) ...(7)

Equation (7) is known as Rittinger's Law.

The specific surface of a particle (the surface area per unit mass) is proportional to 1/L. Rittinger's Law states 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 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. Thus in simple terms, Rittinger's law can be stated as the energy used for particulate size reduction is directly proportional to the new surface produced.

### Griffith theory:

The Griffith theory states that the amount of force to be applied depends on the crack length and focus of stress at the atomic bond of the crack apex.

### Bond’s law:

Bond's law states that energy used to reduce particle size is proportional to the square root of the diameter of the particle produced.

For the grinding of coarse particles wherein the increase in surface area per unit, mass is relatively small, Kick's Law is a reasonable approximation. For size reduction of fine powders where large areas of new surfaces are being created better fits Rittinger's Law.

Size reduction of pharmaceutical products involves a reduction mechanism consisting of deforming the material pieces until it breaks or tear. This deformation may be achieved by applying diverse forces. The types of forces commonly used in the size reduction process are compression, impact, attrition or shear, and cutting Fig.1. In this operation, more than one type of force is usually acted. Table.1 summarizes these types of forces and examples of some of the mills commonly used in the pharmaceutical industry.

Table.1: Types of Forces Used In Size Reduction

 Force Principle Example of equipment Approximate particle size (µm) Compression Nutcracker Roller mill Pestle-Mortar Crushing rolls 50 - 10,000 Impact Hammer Hammer mill disintegrator 50 - 8000 Attrition File Colloidal mill roller mill 1 - 50 Cutting Scissors Scissors shears cutter mill rotary knife cutter 100 - 80,000 Combined impact and attrition Ball Ball mill 1 - 2000

### Compression

In this mechanism, the material is crushed by the application of pressure. Compressive forces are used for the coarse crushing of hard materials. Coarse crushing implies reduction to a size of about 3 mm.

## Impact

Impact occurs when the material is more or less stationary and is hit by an object moving at high speed or when the moving particle strikes a stationary surface. In both cases, the material is crushed into smaller pieces. Usually, both will take place, since the substance is hit by a moving hammer and the particles formed are then thrown against the casing of the machine. Impact forces can be regarded as general-purpose forces and may be associated with the coarse, medium, and fine grinding of a variety of materials.

### Attrition

In attrition, the material is subjected to pressure as in compression, but the surfaces are moving relative to each other, resulting in shear forces that break the particles. Shear or attrition forces are applied in fine pulverization when the size of products can reach the micrometer range. Sometimes a term referred to as ultra-fine grinding is associated with processes in which the sub-micron range of particles is attained.

### Cutting

Cutting reduces the size of solid materials by mechanical action (sharp blade/s) by dividing them into smaller particles. Cutting is used to break down large pieces of material into smaller pieces and definite shapes suitable for further processing, such as in the preparation of powders and granules.

 Fig.1: Mechanisms of Size Reduction

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