## Lesson 14. Energy and Power Requirement for Size Reduction

In the breakdown of hard and brittle food solid materials, two stages of breakage are recognized: (a) initial fracture along existing fissures within the structure of the material; and (b) formation of new fissures or crack tips followed by fracture along these fissures. It is also accepted that only a small percentage of the energy supplied to the grinding equipment is actually used in the breakdown operation. Thus, grinding is a very inefficient process, perhaps the most inefficient of the traditional unit operations. Much of the input energy is lost in deforming the particles within their elastic limits and through inter-particle friction. A large amount of this wasted energy is converted as heat which, in turn, may be responsible for rise in temperature and damage of nutrients of biological materials.

In the breakdown of hard and brittle food solid materials, two stages of breakage are recognized: (a) initial fracture along existing fissures within the structure of the material; and (b) formation of new fissures or crack tips followed by fracture along these fissures. It is also accepted that only a small percentage of the energy supplied to the grinding equipment is actually used in the breakdown operation. Thus, grinding is a very inefficient process, perhaps the most inefficient of the traditional unit operations. Much of the input energy is lost in deforming the particles within their elastic limits and through inter-particle friction. A large amount of this wasted energy is converted as heat which, in turn, may be responsible for rise in temperature and damage of nutrients of biological materials.

Elastic and inelastic properties of a given food material often vary considerably with moisture content and the distribution of water in the material. Further complications arise because these properties are often strongly anisotropic, with various layers or parts having extremely different mechanical resistances. Furthermore, the properties of materials can vary with the rate with which the stress is applied; some materials are plastic and ductile if the stress is applied slowly, but can be elastic or brittle if the stress is applied by impact. Consequently, it is not possible, at present, to describe a food material or furnish the parameters needed to design a size reduction operation. These parameters must be determined experimentally. The energy needed to cause rupture is the work needed to deform the material plus the energy needed to form the new surface. The latter is given by:

E = ∆ (σ A(4.1)

Where σ is the interfacial energy of the surface and A is the surface area. The minimum work of distortion can be measured by placing a sample in tension (or compression) in a machine (such as an Instron testing machine) that simultaneously measures both the applied force and the elongation up to the breaking point. By plotting force vs. elongation and measuring the area under the curve between zero elongation and the elongation (∆x)max at rupture, one can evaluate the energy needed for breaking the piece:

E =$\int\limits_0^{{{(\Delta x)}_{\max}}}{F\,dx}$   (4.2)

In an actual grinding machine, the particles undergo many elastic or inelastic deformations that do not exceed the breaking stress and therefore do not cause breakage. These deformations require work, however, which is entirely wasted except for newly formed cracks that facilitate breakage on subsequent impacts. In fact, only about 1% of the energy used in grinding is used to create new surface. The remaining energy is dissipated as heat in the product and equipment, and high temperatures may result. The cost of power is the major expense in crushing and grinding operation. Thus accurate estimation of the energy required is important in the design and selection of size reduction equipment.

During size reduction, the solids particles are first distorted and strained, work required to strain them is stored temporarily in the solids as mechanical energy of stress. By applying additional force, the stressed particles are distorted beyond their ultimate strength and suddenly rupture into fragments. Thus, new surface is generated. As a unit area of solid has a definite amount of the surface energy, the generation of new surface requires work, which is given by the release of the stress when the particles break. The energy of stress in excess of the new surface energy created appears as heat.

The energy E required to reduce the size of particulate solids depends upon the energy absorbed by the solids and the mechanical efficiency of the process ηm which takes account of frictional losses.

Therefore by definition

${\eta _m}$=$\frac{{energy\,absorbed\,by\,solid}}{E}$    (4.3)

The expression for E can be expanded to give

$E$=$\frac{{{e_s}\,\left({{A_p}-\,{A_f}}\right)}}{{{\eta _m}\,{\eta _c}}}$ (4.4)

where es is the surface energy per unit area, Ap and Af are the surface areas per unit mass of the product and feed, respectively, and ηc is a crushing efficiency. The later is likely to be very small, of the order of 1%. Using the definitions of specific surface and sphericity, the surface area per unit mass, for non-spherical particles, becomes

$A$=$\frac{6}{{\varphi \,{\rho _s}\,x}}$    (4.5)

where ρs is the density of the solids and x is the particle size. Now, across n size fractions of the particle size distribution, the total mass specific surface is

$A$=$\sum\limits_{i = \,1}^{i = \,n} {\frac{{6\,{w_i}}}{{\varphi \,{\rho _s}\,{x_i}}}}$  (4.6)

and assuming that sphericity and density are constant for all size fractions

$A$=$\frac{6}{{\varphi \,{\rho _s}\,{x_{p,f}}}}$ (4.7)

Substituting into Eq. (13.4) gives the energy input as

$E$=$\frac{{{e_s}}}{{{\eta _m}\,{\eta _c}}}\frac{6}{{\varphi \,{\rho _s}}}\left( {\frac{1}{{{x_p}}} - \frac{1}{{{x_f}}}\,} \right)$ (4.8)

where xf and xp are the surface-volume mean particle sizes of the feed and product, respectively.

Equation (13.8) suggests that energy input is a function of the initial and final size of the particles.

Theoretical considerations suggest that the energy input dE required to produce a small change in the size dx of unit mass of material can be expressed as a power function of the size of the material:

$\frac{{dE}}{{dx}}$=$-\frac{C}{{{x^n}}}$   (4.9)

where dE is the change in energy, dx is the change in size, C is a constant, n is the power value and x is the particle size.

Equation (13.9) is often referred to as the general law of comminution and has been used by a number of researchers to derive more specific laws depending on the application. It has been developed based on the energy needed for causing deformation, creating new surface, or enlarging cracks.

It is not possible to estimate accurately the power requirement of crushing and grinding equipment to effect the size reduction of a given material, but a number of empirical laws have been put forward e.g., Rittinger’s law, Kick’s law and Bond’s law.