# Free and Hindered Settling

In his paper entitled, Development of Hindered-Settling Apparatus, Dr. Richards has related the history of the development of the hindered-settling classifier and given illustrations of the several types of this apparatus which he has designed. An earlier paper, Close Sizing Before Jigging, shows the nature of the hindered-settling phenomenon and the results which it produces. In the first-mentioned paper Dr. Richards describes the method of getting hindered settling and shows the relation that the constriction bears to the classifier. The question of proper ratios between the area of the sorting column and the area of the constriction opening Dr. Richards leaves open, aside from certain statements as to the ratios which have been found to give good results in practice.

While working with a hindered-settling classifier of the type now in use at the Boston & Montana Reduction Plant of the Anaconda Copper Mining Co. a number of interesting facts were brought out which have thrown considerable light on the question of proper ratio between the area of the sorting column and the area of the constriction and which will be made the subject of this paper.

## Free and Hindered Settling

If a mineral particle be dropped into a vessel containing still water, the particle will sink in accordance with the laws of free settling in water, which are as follows:

1. Of two grains having the same specific gravity but differing in size, the larger will settle in still water at the greater velocity.
2. Of two grains having the same size but differing in specific gravity, the heavier grain will settle at the greater velocity.

From these laws it becomes evident that the velocity of settling depends primarily on the specific gravity and the size of the mineral grain. The fracture of the mineral may exert a very considerable influence on the settling velocity. Rittinger has expressed the settling velocity of mineral grains by the formula, V = K√D (G – 1), where V = velocity in millimeters per second, K is a constant dependent in part on the shape of the mineral grain as determined by its characteristic fracture, D is the diameter of the grain in millimeters, and G its specific gravity. Richards has shown that this general formula holds true only within a specified range of sizes; in the case of quartz down to 0.2 mm.

If the mineral grain be subjected to the action of a rising current of water, as in a hydraulic classifier, we find that the settling velocity is exactly equal to the velocity of the rising current of water which will keep the grain poised, neither causing it to rise nor allowing it to sink.

If, instead of a single mineral grain, we drop a considerable number of grains into the vessel containing still water, we find that some of the grains are prevented from settling at their normal velocity by reason of the movement of neighboring grains. The grains are hindered in their efforts to settle, and this I shall term the hindered-settling effect. The same effect is observed to an even greater extent when the grains are subjected to the sorting action of a rising current of water, as in a hydraulic classifier. The action is similar to that which we might expect were the grains settling through a medium of higher specific gravity than that of water. The constriction which is placed in the sorting column of the hindered-settling classifier causes to be maintained in the sorting column a bed of grains, each grain of which is poised freely in the water. Dr. Richards has likened these grains to the grains of sand sometimes observed in similar motion in a boiling spring. This quicksand column acts as a filter, through which the other grains must make their way in order to be able to pass out of the apparatus through the spigot. The smaller grains of lighter material are unable to penetrate this bed to any depth and are consequently turned back, thus giving a spigot product with the high diametral ratios between average light mineral and average heavy mineral particles which are characteristic of hindered-settling classifiers.

Thus hindered settling may take place in a free-settling classifier when the classifier is overfed. The hindered-settling classifier differs from the free-settling classifier in that it is designed to make the best possible use of this hindered-settling effect and at the same time to furnish means of controlling and maintaining these conditions.

## The Constriction

The constriction is the essential part of the hindered-settling classifier. Without some form of constriction at the base of the sorting column or teeter chamber no classifier can properly be classed as a hindered-settling classifier.

The constriction may consist of a plate or diaphragm with a central circular opening of less diameter than that of the sorting column or with a series of smaller circular openings, or of a plate in which a series of short pipes or tubes have been inserted. The latter form of constriction has been found to give the best results when the classifier is designed to overflow quartz grains with a diameter of 0.75 mm. or larger. The former is sufficient in the case of deslimers.

The effect produced by the constriction is as follows: It is evident from Fig. 1, which represents the type of classifier in use at the concentrator of the Boston & Montana Reduction Works of the Anaconda Copper Mining Co., that with a given volume of water rising through the constriction, C, the velocity through this constricted area will be greater than the velocity of the rising water in the teeter chamber, T; the velocities varying inversely as the respective cross-sectional areas.

Such being the case, grains of sand, the free-settling velocities of which are greater than the rising velocity in the teeter chamber, will sink and find their way into the teeter chamber, T. Such of these grains as have a free-settling velocity greater than the rising velocity through the constricted area will pass through the opening in the constriction plate and out the spigot. Grains which are able to sink into the teeter chamber but are not able to sink through the swifter rising current in the constricted area accumulate in the teeter chamber and remain there poised freely in the water, moving among themselves and forming a quicksand bed which is in its effect somewhat analogous to a liquid having a higher specific gravity than water.

When grains have thus accumulated in the teeter chamber until a point is reached such that, in any cross-section taken through the sorting column or teeter chamber, the area occupied by sand is to the area occupied by water as the constricted area is to the area of the sorting column, then a condition of equilibrium is reached, which the classifier, if properly designed, will maintain. The classifier may now be said to have become bedded. The grains which now enter the classifier, and which are able, by reason of their higher settling velocity, to penetrate the teetering mass of grains in the teeter chamber, tend to upset this state of equilibrium and will either pass through the teeter chamber and out the spigot, or, if they have not a settling velocity sufficiently great, will, by increasing the density of the teetering column, cause a sufficient amount of material to be discharged through the spigot to restore the condition of equilibrium above mentioned. The bed or teetering mass of grains in an open-spigot classifier may be considered as composed of grains of a definite size; i. e., they are the grains which will sink into the teeter chamber and which are not able under free-settling conditions to pass through the opening in the constriction plate.

Observation of classifiers under operating conditions has shown that for each size grain there is a certain density that can be maintained in the teeter chamber, and that the ratio existing between the area of the sorting column and the area of the constriction opening depends primarily upon this permissible density. Dr. Richards points out that when the ratio is too large full teeter will take place in the sorting column, but no sand will be discharged through the spigot, while if the ratio is too small there will be no bed; in other words, the classifier will act by free settling.

With a closed-spigot classifier, such as Dr. Richards used in his experiments, this statement holds true. In the case of an open-spigot classifier it appears that if the ratio between the area of the sorting column and the area of the constriction opening be too large the result is a decrease in the capacity of the classifier and tendency to banking. There is a ratio which gives the maximum capacity obtainable from the classifier, and this ratio is the one arrived at by designing the classifier to maintain a teetering column of maximum permissible density.

The permissible density which can be maintained with any given size of grains is that density which exists when all of the grains in the sorting column or teeter chamber are poised in the water, free to move up or down among themselves. This condition Dr. Richards has very appropriately called “ full teeter.” The density at full teeter varies, being greater when the bed of grains in the classifier is composed of large grains than when it is composed of finer particles. Experiments which will be described later seem to indicate that this permissible density varies inversely as the square root of the total surface exposed in a given weight of grains. This law apparently holds within the range of sizes that are usually handled by classifiers. There is an upper limit to the density that it is theoretically possible to maintain, and when this has been reached the theoretical ratio of the area of the sorting column to the area of the constriction opening becomes unity. Beyond this point we are unable to maintain a bed of teetering grains and the classifier acts by free settling. As this point would be reached in a classifier with which we wished to overflow 2.35-mm. material, it is of no practical consequence as far as the problem of hydraulic classification is concerned.

## Permissible Density

In order to determine what the full-teeter densities were the following experiment was tried. A quantity of tailings material was screen sized and the separate sizes treated in a sorting tube with closed spigot. This sorting tube was similar to the one used by Dr. Richards in his experiments. The weight of the grains of each size contained in a given volume at full teeter was determined. The densities which were obtained in this way formed a series varying from 4,618 g. to the gallon in the case of grains having an average diameter of 2.19 mm. to 866 g. to the gallon in the case of grains 0.07 mm. in size. A considerable number of relations were tried, but it was finally found that the relation that seemed to hold most nearly true was that the densities varied inversely as the square root of the total surface exposed in 1 g. of grains of each of the given sizes. The assumption was made that the particles were spheres with diameters equal to the average screen size. On this assumption the volume of one grain, the weight in grams of one grain, the number of grains in 1 g., the surface of one grain in square millimeters, and the total surface exposed in 1 g. of grains of each size were computed. Table I. shows the experimental results side by side with the computed results.

In Table I. the line marked “ Density at Full Teeter ” gives the densities that were determined by experiment and are in fact the average of several results which agreed closely. The density in the case of grains 0.841 mm. in diameter being practically what we have found in practice, was assumed to be correct, and the values given in the line marked “ Permissible Density ” were computed on the assumption that the densities varied in the manner stated in the preceding paragraph. As will be seen, this relation seems to hold approximately true.

The ratio of area of sorting column to area of constriction opening may be computed from the formula R = K/√D — 1, where K is a constant approximately equal to 3.

We have so far considered the bed as composed of quartz grains of the size which would not ordinarily be able to penetrate the current rising through the constriction opening and which would not overflow under free-settling conditions. As a matter of fact, there must be some middlings grains and some heavy mineral grains in the classifier bed. This would not affect the results to any considerable extent. Wherever we have had occasion to use constrictions with these ratios in classifiers the classifiers have carried a bed and have given satisfactory spigot products. The curve, Fig. 2, covers the range of sizes with which we are ordinarily concerned in designing classifiers. The size or diameter in millimeters of the maximum quartz grain to be overflowed is plotted as abscissa and the corresponding ratio between the area of the sorting column and the area of the constriction opening is plotted as ordinate. Thus in the case of a four-spigot classifier in which we wish to make the first spigot deliver 2 to 1.25 mm. quartz; the second spigot, 1.25 to 0.75 mm.; the third spigot, 0.75 to 0.35 mm., and the fourth spigot 0.35 to 0.10 mm. quartz, the ratio should he 1.8, 2.6, 4.2, and 8.9, respectively. To work properly, each pocket would have to be designed with reference to the tonnage of each of these sizes which the classifier might be expected to handle.

Having established the proper ratios for area of sorting column to area of constriction, it may be of interest to see how the other computations are made.

## Size of Constriction Opening

In designing a hindered-settling classifier the size of the constriction opening is the factor that determines the capacity of the classifier. It is, therefore, the part of the classifier which must first receive attention.

The minimum size of constriction opening depends upon the amount of spigot product that the classifier is to be called upon to deliver. If we know the average diameter of quartz grain which is to be in the spigot and the maximum tonnage of spigot product to be delivered, we would proceed as follows: From a table of settling velocities find the average settling velocity of the average quartz grain above referred to. Now, from the tonnage calculate the volume that must pass the constriction in a given time; say, in one second. Divide the volume found in this way by the free-settling velocity of the average quartz grain, the former being in cubic inches and the latter in inches per second, and we have the area that would be required for the opening in order that it should just pass this volume of material. We have found from experience that this area must be multiplied by about 3, for reasons that will be stated later. This latter statement refers only to classifiers which have a rising current of water in the sorting column. The area found in this manner must now be multiplied by the proper ratio, as shown on the curve, Fig. 2, in order to obtain the area of sorting column that will be necessary.

## Design of Classifier Top

The top of the classifier should be so designed that the rising current at the top due to the combined effect of fresh water and feed water is equal or nearly equal to the rising current through the constriction or to the free-settling velocity of the largest quartz grain which the classifier is designed to overflow. If this is done, then the finer particles which we wish to keep out of the spigot, and which are not able to penetrate the bed of teetering grains in the sorting column, overflow at once. When the classifier is designed with a conical top or pocket and when the feed water quantity varies within wide limits it is practically impossible to so design a classifier as to enable it to do uniformly good work. When, however, the classifier is to treat the spigot of a deslimer and the volume of feed is kept constant the top can be so designed as to do very accurate sizing. In practice it has been found advantageous to feed a classifier of this type through a central feed cone so designed that the annular space at the bottom will give a rising current equal to the rising current through the constriction, with the annular space at the top approximately 25 per cent, greater than that at the bottom. In this way the full effect of the feed water may be utilized for the work of sorting.

## Method of Designing a Classifier

It may be of interest to outline the method that is used in designing a classifier of the general type illustrated in Fig. 1 so as to satisfy a given set of conditions. We will suppose a case in which we want a single-pocket classifier capable of desliming 500 tons per 24 hours, the material being the undersize of 5-mm. trommels. We will further assume that we have 8 ft. of head room, that we wish the deslimed product to have a density of 1,200 g. to the gallon, and that the feed to the classifier contains 10 per cent, of material finer than 200 mesh which we wish to overflow as slime. The feed water amounts to 300,000 gal. per 24 hours.

## Size of Spigot

In order that we may obtain a spigot discharge with a density of 1,200 g. to the gallon, the spigot discharge must amount to 340,200 gal. per 24 hours, 450 x 2,000 x 453.6/1,200 = 340,200. This is equivalent to 3.938 gal. per second, or 910 cu. in. per second. With a 2.25-in. spigot, which has an opening equal to 3.97 sq. in., we would require a velocity of 910 ÷ 3.97, or 229 in. per second in order to obtain the above-mentioned spigot discharge. From the formula V² = 2gh, we can compute the head room theoretically required for the classifier. This comes out equal to about 68 in., or 5 ft. 8 in. This would be satisfactory, as we have allowed ourselves 8 ft. of head room. To allow for friction we would probably so construct the classifier as to have the overflow about 6 ft. higher than the spigot opening. This may be further modified when we come to design the classifier top.

### Area of Constriction Opening

If we are to make 450 tons per 24 hours of spigot product then we shall have:

450 x 2,000/86,400 = 10.42 lb. per second, or

10.42 x 1,728/64.5 x 3.1 = 90.05 cu. in. per second.

This calculation is based on the assumption that the specific gravity of the material is 3.1.

We will assume that we have made a sizing test on the feed after first screening out the through-200-mesh material, and that we have obtained results as follows :

The average screen size figures out 1.7 mm., and this grain has a settling velocity of 136 mm. per second, or, say, 5.35 in. per second. In order that 90.05 cu. in. of sand may pass the constriction with an average velocity of 5.35 in. per second the constriction opening must have an area of at least 16.8 sq. in. To allow for variations in feed we will make the constriction opening 5 in. in diameter, giving us an area of 19.6 sq. in.

## Size of Sorting Column

If we were making the calculations for a classifier which was to be operated with a rising current and intended to overflow 200-mesh material we would multiply the constriction area which we have just found by 10.56, the ratio obtained from the curve, Fig. 2. This would necessitate a sorting column 16 in. in diameter. For a deslimer we have found that this does not seem to be necessary and that a 12-in. sorting column will answer the purpose. In the case of a deslimer the sorting column should be made at least 4 in. in length. The coarser the material which we wish to overflow, the longer should the sorting column be. A sorting column 8 in. long is, however, sufficient in most cases.

## Area of Classifier Top

A grain of quartz 0.078 mm. in diameter settles at the rate of 4 mm. per second. We will therefore make the rising velocity at the top of the classifier 3.5 mm. per second, in order to be sure that we overflow the least possible amount of on-200-mesh material. In a deslimer of this sort we need use only enough fresh water to supply the spigot, and all the water which will overflow is the water which comes to the deslimer with the feed. This we have stated amounts to 300,000 gal. per 24 hours, or 300,000/86,400 = 3,472 gal. per second, or 802 cu. in. per second.

If our rising velocity is to be 3.5 mm. per second, or 0.138 in. per second, we must have a top with an area of 5,812 sq. in. or a top with diameter of 86 in. If we make our cone 48 in. in vertical height, the classifier will have to be somewhat more than 6 ft. in vertical height, in order to allow for the sorting column, pressure chamber, and spigot discharge. About 7 ft. of head room would be required. As we have 8 ft. of head room, this will work out nicely. The classifier designed in this way will deliver its overflow with a density of 150 g. to the gallon.

As was stated previously, a classifier in which there is to be a rising current would be designed in a similar manner; the only difference being that when the size of the constriction opening has been computed the area found should be multiplied by about three. The reason for this is that with a given rising velocity through the constriction, and a definite volume of sand sinking through the constriction at a known average velocity, if the area of the constriction opening is made double the sand area the rising velocity is doubled and the classifier will bank or choke, since the rising velocity becomes greater than the average rate of settling of the grains in the spigot discharge. We will suppose a case in which the average settling velocity of the grains going into the spigot discharge is 170 mm. per second and where we have a rising velocity of 100 mm. per second.

In this case, if the sand area were doubled to get the constriction area the effective rising velocity would become 200 mm. per second. If the sand area be multiplied by three then the constriction area when sand is passing is reduced one-third and the rising current becomes 150 mm. per second, or may have this value momentarily. Now, the average settling velocity is 170 mm. per second, and experience has shown that under these conditions the classifier will operate satisfactorily and will handle its rated tonnage. It might in some cases be advisable to make the constriction opening a little larger in order to be on the safe side, but in order that we may make the most economical use of water the classifier should be made only sufficiently large for the work that it is called upon to do.

If the classifier is made too large for the amount of feed which it is to handle there will not be enough material coming into the classifier to maintain a bed and the classifier will tend to act by free settling. Classifiers designed in the manner outlined have been found to operate satisfactorily and to be extremely efficient as regards water consumption.

In the mills we have two variables which affect the operation of classifiers; namely, variations in tonnage and variations in density of feed. For this reason a classifier, to do its best work, should be especially designed to meet the conditions under which it is to operate. This is, however, impracticable from the point of view of the manufacturer of milling machinery, who is obliged to standardize the parts and confine himself to a few standard sizes. Of the two variables, the one which is the more serious as regards operation of the classifier is the variation in volume of feed. This can be made negligible in classifiers of this type by desliming the pulp previous to classification, the classifier receiving its feed directly as the spigot product of a deslimer. In this way a few standard sizes may be designed so as to satisfy practically any condition which may arise in practice.