In most closed circuit operations, a steady state condition is attained and most of the literature deals with the mill and classifier performances at various steady state conditions. Most circuits are complex in the ore-dressing industries but in small units simpler circuits are frequent. The mill in an open-circuit process must meet the specifications for the product by its grinding processes alone; in close-circuit operations, the mill and its classifier together meet the specifications for the output. Failure of either machine may lead to drastic alterations of the performance of the other; however, in the main, authors have tended to be critical of classifiers rather than mills. With the more efficient classifiers (and none appear to be really efficient at removing all the finest particles), the capacity of a closed circuit system to produce an output specified by a limiting upper size is greater than that of an open circuit mill. If the size grading of the output is important, and it generally is, closed circuit grinding is not always advantageous; for example, cement must contain an appreciable proportion of “flour” and closed circuit mills tend to produce too coarse a product.

The mathematical treatments by Mineral processing engineers of closed circuit operations as exemplified are incomplete : to a large degree this incompleteness derives from the failure of these methods to analyses the grinding processes. Some published results give make-up rates, recirculation ratios and size analyses of output. Whilst the results showed that a particular make-up rate was associated with the maximum and finest output, it is impossible to tell from them whether, for example, an alteration to the classifier would have produced better results. Some formulated equations fully describing steady-state closed circuit operation for the basic circuit shown in Fig. 4; these equations may be extended to any other circuit and permit an evaluation of the relative effect of mill and classifier performance. (The equations for the reversed closed circuit Fig. 8, are given in Appendix II.) Such equations should prove useful in many circumstances, of which two examples may be mentioned. First, suppose a mill has been tested in open circuit so that the matrix defining the breakage process has been determined—we shall call this matrix D. The elements of the D' matrix may vary with, say, feed rate and size grading of the feed. It is proposed to couple the mill with a classifier whose efficiency curves are known for various, say, input rates and size distributions; denote this by another matrix C. It is then possible with the equations mentioned above to compute the output rate and size grading for given make-up rates and size, gradings. Secondly, such equations will show whether accumulated plant data are or are not sufficient for an analysis of a mill circuit.

Closed circuit tests have revealed some very important features of milling operations that are related to characteristics of the feed. It appears that some ball mills grind more efficiently when presented with feed that is closely graded and free of much very fine material. The cause seems to be two-fold: first, the optimum grinding media are more' easily established, and secondly, the material can pass through the mill at a greater rate and the fraction broken can be relatively large. However, quite often a particular supply of ore tends to build up a quantity of a “hard-to-grind” size fraction. There are a number of possible reasons for this. The most commonly accepted explanations attribute unfavorable physical characteristics to the particular size fraction; but, for the available published data, it is equally tenable to attribute the increase in the amount in a particular size grade to poor classification, especially if large grinding media were used.

It is not proposed to discuss further closed circuit grinding, except to suggest that the fuller recording of results and- a considerably more detailed examination of results than is common may well lead to considerable improvements if coupled with matrix analyses of the milling and classification processes. '

Some of the major topics that concern the practical and theoretical characterization of tumbling mills have been discussed. Emphasis has been laid on the problem of analyzing the breakage processes of these mills. For the studies discussed in this paper, the particles of all materials have been presumed to break to a product with a similar size distribution, given by the, function B(x/y). The strength of the particles and the peculiarities of the machine breaking them are concealed in the selection function defining the proportions of each size that break. The mathematics used for the analysis of breakage processes are closely allied to the characteristically discontinuous nature of size distributions, and although developed particularly for materials breaking in a homogeneous fashion can be extended to heterogeneous materials. Thus far, however, simple breakage models have provided adequate descriptions of the breakage processes of several mills and a number of dissimilar materials.

**Closed-Circuit Equations**

The basic closed-circuit illustrated by Fig. 4 shows the make-up or input material entering the system after size classification of the product from the mill and being fed together with the recirculated particles into the mill. The equations defining this circuit which have been given elsewhere 'will not be presented here. Another common type of closed-circuit involves the make-up material entering the system before the classifier. This system is illustrated by Fig. 8. The equations defining this circuit are given below as a supplement to those of the earlier papers. Let m and M represent the vector describing the size grading of the make-up material and the rate at which make-up is added, respectively, and let the size gradings and rates for the output, the feed, the product and the load or blend entering the classifier be represented by q and Q, f and F, p and P, 1 and L, respectively. It is convenient to represent the matrix describing the breakage process by D; and hence p = Df (7).

If f, p and B are known, it is possible to estimate ir. Generally, the observations of f and p are subject to sampling and size analysis errors and thus it is preferable to estimate 71-, the proportion of the feed broken, by some least squares method. Now, if the ^-breakage model was proposed as a description of the breakage process but was not a good one, then by calculating p from equation (1) using the estimated value of 7r, a comparison of the calculated and observed product size analyses will reveal major discrepancies. This comparison is the basic check that any breakage model is a realistic one. Breakage processes may be more severe than is predicted by the breakage matrix B. In some cases the breakage process may be represented.

Where k is an integer and k = 2 suggests that representative fraction was first broken according to B and then the broken product broken a second time. This process is termed tr, K-breakage, e.g. the modified conical mill had a breakage process given here.

Processes with a more complex relationship between the probability with which particles of the feed are selected for breakage and the size of the particles may be represented by a selection function, or a selection matrix, S. S is the n2 matrix in which fractions representing the proportion of each size grade in the feed selected for breakage, appear as the elements of the main diagonal and all other elements are zero: thus the selection matrix where C is the n2 matrix with cx, c2, cn as the elements of the main diagonal and representing the proportion of each size grade of the classifier feed returned to the mill for further grinding and en+x is the fraction of particles smaller than a,n recirculated. Frequently cn+1 is not zero.