Mineral Liberation and the Batch Comminution …

1 1 Mineral Liberation and the Batch Comminution King and SchneiderComminution CenterUniversity of UtahSalt Lake City UT 84112 ABSTRACTThe Batch Comminution equation for multicomponent Mineral systems is a multidimensionalintegrodifferential equation0P(g,D;t)0t PR1PS(g1,D1)B(g,D g1,D1)p(g1,D1;t)dg1dD1that cannot be solved analytically. The phase space (g, D) is usually discretized and the equationis formulated asdpijdt Sijpij j 1l 1 K lk K lSklbijklpklSolutions to this discretized form of the equation are easy to generate provided that models for theselection function Sij and the multicomponent breakage function bijkl are known.

2 An assumption of random fracture provides an important simplification because, under the randomfracture assumption, the selection function is independent of the particle composition. The randomfracture assumption has often been used by researchers who developed models for the prediction ofmineral Liberation by Comminution and the solution of the Batch Comminution equation under thisassumption was found to bepij(t) Mjl 1 Aijle Sltwith the coefficients given byAijj pij(0) Mj 1i 1 AijlandAijn Mj 1l nSlM12k 1bijklAklnSj Sn2A general solution to the Batch Comminution equation that does not rely on the random fractureassumption is also derived in the paper and is given bypij Mjl 1M12k 1 ijkle Skltwith the coefficients given by ijmj 0ifigm ijij pij(0) Mj 1l 1M12k 1 ijkl ijmn Mj 1l nM12k 1 Sklbijkl klmnSij SmnThe new solution was compared with experimental data obtained from a Batch test on a difficult-to-liberate ore.

3 The model was found to be reliable and the derived Andrews-Mika diagram is derived models for the selection function and the Andrews-Mika diagram can be used tosimulate the effects of the non random fracture, and thus to assess their relative importance, and alsoto simulate the behavior of the ore in continuous milling and concentrating population balance equation for Batch milling0P(D;t)0t P DS(y)B(D y)p(y;t)dy(1)can be solved analytically (King, 1972, Nakajima and Tanaka, 1973) for a few specific selection andbreakage functions when it is not necessary to account for the Liberation of the Mineral phases duringcomminution.

4 Practical solutions are usually generated using a finite collection of ordinarydifferential equations that are generated by discretization of the size Sjpj Mj 1l 1 Slbjlpl(2)When the ore is heterogeneous the Liberation of the individual Mineral phases becomes importantand the Batch Comminution equation must be modified accordingly. The modifications that arerequired to apply this equation to binary ores were comprehensively investigated by Andrews andMika in 1975 who wrote the equation in terms of the particle assay and the particle mass (a, m)30F(a,m;t)0t PR1PS( ,))B(a,m ,))f( ,);t)d d)(3)In equation (3), F(a,m;t) is the fraction by mass in the mill charge at time t that consists of particleshaving individual masses less than or equal to m and individual assays less than or equal to a.

5 Thecumulative breakage function B(a,m , ) is the fraction by mass of the progeny with essay a andmass m that results from the fracture of a parent particle of mass and assay .Analytical solutions to equation (3) are not available except for systems that are severely andunrealistically restricted. Numerical methods are the only practical possibility for generating usefulsolutions. A convenient solution method is presented ANDREWS-MIKA DIAGRAM AND ITS SIGNIFICANCEThe double integral in equation (3) is taken over a region R1 in the particle mass, particlecomposition space.

6 This region contains all parent particles that can generate a progeny particle ofmass m and composition a. The region R1 does not comprise the entire half space ) m but isrestricted by mass conservation and mineralogical texture constraints. The significance of theserestrictions was pointed out by Andrews and Mika in 1975 and the determination of these constraintsis important when solutions to equation (3) are and Mika established the following conservative bounds for the region R1 m(4)and am(1 ) (1 a)m(5)These bounds reflect the obvious requirement that the parent particle must be at least as large as anyof its progeny and that the mass of Mineral in the parent particle must be at least as large as the massof Mineral in any progeny particle.

7 R1 is bounded above by the mass of the largest particle in the millfeed (mmax). The region R1 is illustrated in Figure 1 where point A represents the composition andmass of any progeny particle. The boundaries of R1 never intersect the vertical axes at a = 0 and a= because of the obvious fact that a completely liberated parent cannot produce any region R, complementary to R1, can also be defined. If point A in Figure 1 represents the mass andcomposition of a parent particle, region R is the attainable region for progeny that results when theparent is broken in the mill.

8 R 1 is called the feeder region for progeny at point A and R is called theattainable region from a parent at point A. The boundaries of R are continuations of the boundarylines of R1 and are obtained by reversing inequalities (4) and (5).4m (6)andam (1 a)m (1 ) (7)Unlike the boundaries of R1, those of R intersect the vertical axes a = 0 and a = 1 at points B and Crespectively. These intersection points reflect the conservation law: The mass of the largestpossible liberated progeny particle is equal to the mass of the Mineral phase contained in the parentparticle.

9 Point B is defined when this principle is applied to the gangue and point C when it isapplied to the Mineral phase. The segments BF and CG are attainable from the point A because aparticle of mass m and grade a can produce progeny consisting of pure gangue provided that themass of the liberated gangue particle is less than the mass of the gangue phase in the parent particle(equal to (1-a)m). Likewise a progeny particle consisting of pure Mineral can be produced from aparent particle at (a,m) if the mass of the liberated daughter is less then the mass of the Mineral phasein the practice the particle-mass, particle-assay coordinate system is not practical for population balancemodeling applications and the particle-size, particle-composition system is used instead.

10 Thispreference expresses the fact that it is much easier to classify particles on the basis of size than onthe basis of the mass of individual particles. When particle size is used rather than particle mass,equation (3) is written0P(g,D;t)0t PR1PS(g1,D1)B(g,D g1,D1)p(g1,D1;t)dg1dD1(8)The Andrews-Mika diagram in the particle-size particle-composition space has the same essentialcharacteristics as that in the mass-composition space. However, the inequalities (9) to (7) that definethe boundaries of the feeder and attainable regions cannot be written in precise form and thefollowing approximations are usedgD3 g1D13(9)for the Mineral and(1 g)D3 (1 g1)D13(10)for the fact these boundaries become fuzzy (in the sense of fuzzy set theory) because a population ofparticles of equal mass or volume have a distribution of sizes if the particles are of irregular fuzziness is neglected in this constraints expressed by inequalities (9)