Minerals Beneficiation - Energy-Size Reduction Relationships in Comminution

SEARCH for a consistent theory to explain the relationship between energy input and size reduction in a comminution process has accumulated, over the years, an enormous amount of plant and laboratory data. Although some correlation of these data has been possible for purposes of engineering design and for the advancement of research in fracture, there is still great need of a means of predicting behaviour of a solid when it is reduced in size by mechanical forces. The best known hypotheses proposed to describe the energy-size reduction relationships in crushing and grinding stem from a common origin. The present article analyzes problems of comminution in the light of the precepts of this origin. Its object is to reconcile points of difference between these well known hypotheses and to present relationships more widely applicable to comminution studies. Theoretical Considerations: Most existing relationships between energy and size reduction of a brittle solid stem from a single, simple, empirical proposition.&apos; Although this proposition can be demonstrated by observation and experiment, no theoretical derivation is yet possible. Mathematically, the proposition may be stated as follows: dE = -Cdx/xn [1] where dE = infinitesimal energy change, C = a constant, dx = infinitesimal size change, x = object size, and n = a constant. Eq. 1 states that the energy required to make a small change in the size of an object is proportional to the size change and inversely proportional to the object size to some power n. No stipulations are placed on the exponent n in either magnitude or sign. In 1867 Rittinger2 postulated that the energy required for size reduction of a solid would be proportional to the new surface area created during the size reduction. As far as can be determined there is as yet no physical basis for Rittinger&apos;s hypothesis. Rittinger&apos;s hypothesis can be stated mathematically as follows: E, = K(oa-a-0 . [2] Er = energy input per unit volume, K = a constant, <ti = initial specific surface, and o2 = final specific surface. In the size reduction of particles of size x, to particles of another smaller size, x2, Eq. 2 becomes the well known relation: ET = K&apos; {l/x2-l/xx) [3] where K&apos; is a constant. Eq. 3 may be arrived at from the proposition given in Eq. 1 by integrating and by assigning a value of 2 to the exponent n. J dE = J - C dx/x2 E = Kt (1x - 1x) where K&apos; = C. In 1885 Kick3 proposed the theory that equivalent amounts of energy should result in equivalent geometrical changes in the sizes of the pieces of a solid. For example, if one unit of energy reduced a number of equal-sized particles to particles of one half the size, then the same amount of energy applied to the particles resulting from the first test should result again in a size reduction of one half or a final size one quarter the original size. The Kick concept may be expressed as follows: Ek = K" log x1/x2 [41 K" = a constant and E, = energy per unit volume. The expression for Kick&apos;s law may be arrived at by again integrating Eq. 1 and in this case assigning a value of 1 to the exponent n. dE= J - C dx/x Eh = - C In {x/x2) = K" log (x,/x,) where K" = 2.3 C. Application of Kick&apos;s and Rittinger&apos;s laws to comminution has met with varied success. Gross and Zimmerley4 and Piret5 have shown that Rittinger&apos;s equation applies under certain conditions of experimentation. Walker and Shaw6 express the belief that in metal turning and shaping and in grinding of both metals and minerals the production of very fine particles (less than lP) follows Kick&apos;s hypothesis, whereas Rittinger&apos;s concept is valid for the size reduction of coarse particles. For the practical case of crushing and grinding, however, neither of the above hypotheses has received general acceptance. Bond&apos; has lately proposed that since neither Kick&apos;s nor Rittinger&apos;s hypotheses seem correct for plant design work, an energy-size reduction relationship somewhere between the two would be more applicable. The fundamental statement of Bond&apos;s work