Discussion of Papers - Feedback Process Control of Mineral Flotation, Part I. Development of a Model for Froth Flotation

T. S. Mika (Department of Mineral Technology, University of California, Berkeley, Calif.) - Dr. Cooper's attempt to establish a correlation between process behavior and operational variables on the basis of a statistical analysis after imposing a reasonable process model is a very commendable improvement on the use of standard regression techniques. However, it must be recognized that the imposition of a model has the potential of yielding a poorer representation if its basic assumptions or mathematical formulation are invalid. It appears that at least two aspects of his treatment require some comment. First, the limitations on the kinetic law where xta represents a hypothetical terminal floatable solids concentration (cf. Bushell1), should be mentioned. Most current investigations2-9 appear to utilize the concept of a distribution of rate constants rather than a single unique value, k, to describe flotation kinetics. A distributed rate constant is certainly a more physically meaningful concept than that of a terminal concentration. The study of Jowett and safvi10 strongly indicates that xta is merely an empirical parameter, whose actual behavior does not correspond to that expected from a true terminal concentration. Rather than being a strictly mineralogical variable, as Dr. Cooper's treatment implies, it apparently represents the hydromechanical nature of the test cell as well as the flotation chemistry. The extension of batch cell kinetic results to full-scale continuous cell operation is a suspect procedure if the effect of such nonmineralogical influences on x,, remain unevaluated. There is evidence that introduction of a terminal concentration is necessitated by the inherent errors which arise in batch testing and are eliminated by continuous testing methods.' Possible lack of validity of the author's use of Eq. 1 is indicated by two unexpected results of the statistical analysis of his batch data. The first is the apparent corroboration of the assumption that the rate constant, k, is independent of particle size, i.e., of changes in the size distribution of floatable material. This assumption directly contradicts numerous results 2,4,11-l8 for cases where first order kinetics prevailed and ignores the phenomenological basis for the analysis of flotation in terms of a distribution of k's. It must be recognized that, if the rate constant is size dependent, the lumped over-all k would be time dependent; Eq. 1 would then no longer be valid. Cooper's x,, is determined by batch flotation of a distribution of sizes for an arbitrary period of time. If the size dependence of k is artificially suppressed, x,, will become a function of the experimental flotation time used in its determination. Upon reviewing the rather extensive literature concerning batch flotation kinetics, there appear to be few instances where constant k and x,, adequately adsorb variations in floatability due to particle size. The second surprising result is the low values of the distribution modulus, n, determined. Contrary to Cooper's assertion, most batch grinding (ball or rod mill) products yield values of n > 0.6, which increase as the material becomes harder.'' It is likely that the values of n = 0.25 and n = 0.42 for Trials 1 and 2, respectively, are completely unreasonable, and even the value n = 0.54 obtained for Trial 3 is unexpectedly low. Possibly, this indicates inherent flaws in the three trial models considered, in particular the assumed particle size independence of the rate constant, k. The above does not necessitate that Eq. 1 (and the terminal concentration concept) is invalid; it could constitute a good first approximation. However, the qualitative arguments used by Dr. Cooper in its justification are somewhat frail and require verification, particularly since much of the flotation kinetics literature is in opposition. Apparently, no effort was made to test these hypotheses on the actual data; in fact, since they pertain to a single batch test time, his data cannot be utilized to evaluate the kinetics of flotation. To evolve a control algorithm on the basis of this infirm foundation seems a questionable procedure. Another difficulty in his analysis arises in consideration of the froth concentrating process. As Bushel1 ' notes, for Eq. 1 to be valid it is necessary that the rate of recycle from the froth be directly proportional (independent of particle size) to the rate of flotation transport from the pulp to the froth, a restrictive condition." Harris suggests that it is more realistic to assume that depletion occurs in proportion to the amount of floatable material in the pertinent froth phase volume (treating that volume as perfectly mixed).12,21,22 The physical implications of