PROPOSAL FOR A SOLOMONIC SETTLEMENT BETWEEN THE THEORIES OF VON RITTINGER, KICK, AND BOND

This paper presents a preliminary analysis of the fundamental relationship between the net energy used and the respective product s i ze throughout the entire range of s i zes covered by crmshing and grinding, and an attempt to find a sensible correlation between the existing theories.

by R. T. Hukki

alker, Lewis, McAdams, and ~ i l l i l a n d ' have given w the following differential equation of a general form for comminution:

where E is the net energy required per unit weight in a certain process of comminution; x is the factor indicating the fineness of the product; n is the ex- ponent indicating the order of the process, and C is a constant related with the material, units chosen, etc.

I£ exponent n in the above equation is replaced by numerical figures 2, 1, and 1 1/2, the integrated form of the general equation leads to the well known fundamental theories represented by the law of von ~ i t t i nge r , ' law of ~ i c k , ~ and the third theory of comminution by ~ o n d , ~ respectively.

The total net energies ( E ~ ) from infinite feed size to a product of s ize x a r e a s follows:

1 Rittinger: E t = C x k Wh/ t [21

Kick: E~ = -C ln: k ~ h / t [31

Bond : 1

Et = 2C k Wh/t [41

The corresponding net energies (E) required to reduce product xl to product x2 a r e

Rittinger: E = C (-& - $) k w h / t [51

Kick: E = -C lnX1 k ~ h / t X 1

[61

Bond: 1 1

E = 2C (- - -) kWh/ t . Jx, Jxl PI The net energy required in a certain process of

comminution is proportional to the new surface de- veloped according to the law of von Rittinger, to the

R . T. HUKKl i s Professor of Mineral Dressing, Inst. of Technology, Helsinki, Finland. TP59B239. Manuscript, Nov. 9, 1959. Discussion of this paper, submitted in dupl i - cate prior to July 1, 1962, will appear in AlME Trans- actions (Mining), 1962, vol. 223.

weight o r s ize of the bodies treated according to the law of Kick, and to the length of the new cracks formed which initiate breakage according to the theory and explanation by Bond.

On a logarithmic paper, where particle size is presented on the abscissa and energy consumption on the ordinate (see Fig. 21, all three relationships a r e represented by straight lines. The slope m of the line according to the law of von Rittinger is equal to -1.0; of Kick, 0; and of Bond, -0.5.

Experimental evidence in favor of the law of Rittinger has been presented, e.g., by Gross and zimmerley5 on quartz crushed in a drop weight crusher and evaluated for surface by the method of dissolution, by Deans on magnetite crushed by a s imilar method and evaluated for surface by the de- termination of coersive force, by piret7 and co- workers on a group of minerals crushed again by a similar method a s well a s by compression and evaluated for surface by permeability and gas ad- sorption methods, and by schellinger8 on a group of minerals ground in a calorimetric ball mill and evaluated for surface by gas adsorption.

Experimental evidence in favor of the law of Kick seems to be scant in the field of comminution. On the other hand, in the field of mechanical engineer- ing Kick's law seems to be of fundamental nature in processes such a s cutting, pressing, shaping, and rolling of metallic substances.

Experimental evidence in favor of the third theory has been provided by ~ o n d . ~ " To a large extent, data a r e based on the vast amount of grindability tests performed in the laboratories of Allis-Chal- mers Manufacturing Co.

In addition to the devoted proponents of one o r the other of the basic theories listed above, certain in- vestigators have indicated that one of the theories might be applied for a certain range of sizes, while another theory might be used for other sizes. In a discussion of a paper by Bond," Dobie" presented a statement a t the International Mineral Dressing Congress in London (1952) indicating that 1) for large particles, the law of Kick was approximately correct; 2) for finer particles, von Rittinger's sug- gestion was nearer to the truth; and 3) Bond's new

1p lop l oop Imm 10mm 40omm Irn lorn log Particle Size

Fig. 1 -Illustration of the basic reduction steps in comminution with a tabulation of the energy con- sumption between the various steps based on the theories of von Ritting-er, Bond, atzd Kick.

theory was a unification of the two older suggestions. To the question raised by ~ o b i e " concerning the limits of particle s ize over which the Bond theory could be used, Bond answers by stating that the third theory of comminution appears to have no s ize range limitations. Contrary to the opinion expressed by Dobie, Walker and shawl2 have indicated that the law of Kick might be applicable in s izes below l p , while the law of von Rittinger might suit for s izes coarser than 1p. charles13 some time ago derived another general equation for comminution and proposed that the hypotheses of Kick, von Rittinger, and Bond may be derived a s special cases of his equation. He presents corresponding experimental evidence in favor of Bond and Rittinger, but not in favor of Kick. In the summary of his paper Charles states that Kick's hypothesis would be approximately valid for production of extremely fine material, a conclusion that is in agreement with the views of Walker and shaw.I2 Recently, Svensson and ~ u r k e s ' ~ have claimed that, while none of the ear l ier theories will correspond to the results of their extensive grinding experiments. sti l l another empirical re- lationship developed by them would offer a solution of general validity.

It is somewhat startling to note that the actual s ize range covered by dependable experimental evidence presented in the referred investigations is always relatively narrow. In spite of that, conclusions of general applicability have been presented.

CLASSIFICATION OF BASIC REDUCTION STEPS IN COMMINUTION

Most conventional processing of mineral raw materials includes three major steps: 1) explosive shattering of the ore o r rock, 2) crushing, and 3) grinding. No exact boundaries between them can be given. In spite of that, a somewhat crude, yet quali- tatively satisfactory classification of the successive steps of comminution may be based on the very con- venient metric decimal system a s follows: 1) Explosive shattering: f rom infinite size to -1 m. 2) Primary crushing: from -1 m to -100 mm. 3) Secondary crushing: from - 100 rnrn to - 10 mm. 4) Coarse grinding: from - 10 rnrn to - 1 rnrn. 5) Fine grinding: from - 1 rnrn to - 1 0 0 ~ . 6) Very fine grinding: from -100p to - lop . 7) Superfine grinding: from - l o p to -1p.

Fig. 1 shows seven parallel straight lines repre- senting the foregoing classification on a customary logarithmic paper. Size is shown on the abscissa and cumulative percentage passing a certain size (screen) on the ordinate. This method of plotting follows the Gates-Gaudin-Schuhmann relationship with slope m = 1.0, considered to represent the ideal size distribution in comminuted products.

Pr imary and secondary crushing may consist in practice of more than two successive steps because the reduction ratio of most conventional crushers is less than 10. Explosive shattering may also be carried out to a finer end product than indicated. Coarse grinding as shown in Fig. 1 would roughly correspond to grinding in rod mills and fine grinding to grinding in ball mills. Other features of the se- quence shown in Fig. 1 should be self-explanatory.

It i s not unusual that a total of 10 k wh / t of energy is used in all steps of conventional crushing and grinding combined. As an example it i s now assumed that a total of 10 k w h / t is used in reduction steps 2 through 5 a s shown in Fig. 1. For the time being, no attention is paid to the mechanical efficiencies of the respective crushers and mills. Distribution of the total energy available between the four steps based on Eqs. 5 through 7 i s tabulated in Fig. 1. As in- dicated in the preceding, the data corresponding to the law of von Rittinger form a straight line of slope rn = -1.0 on a logarithmic paper, showing par- ticle size on the abscissa and energy consumption on the ordinate; those corresponding to the law of Kick, a horizontal straight line with slope m = 0; and the data representing the theory of Bond, a straight line of slope m = -0.5. If the three ser ies of data a r e compared with the respective figures obtained in industrial practice, it is not difficult to see that only those following the theory by Bond will be of any reasonable value.

If, however, this imaginary s ize reduction is car- ried further from - 100 to - lop-a tenfold step again-it will be easily conceived that the energy consumption figure of 21.8 k Wh/ t by Bond seems too small to satisfy the requirements of conventional practice. It i s well known, for example, that to grind cement to a fineness of 92 pct -325 mesh (44p), about 35 k w h / t of energy is needed. Such a product is much coarser than the - l op product now under study. The respective Rittinger figure of 90 k w h / t should be more reasonable. The situation becomes still more clear by analysing the imaginary size re- duction from - 10 to -1p. The tabulated figure for Rittinger appears to be reasonable, very doubtful for Bond, impossible for Kick.

On the the other hand, if the coarse end of the sequence shown in Fig. 1 is studied, Rittinger figures seem to be fully unacceptable, Bond figures questionable, but the figures for Kick more and more reasonable. As a crude example in favor of the law of Kick it might be reasoned that, if a certain amount of net power i s needed to crush a block weighing one ton in a big jaw crusher, twice that amount of net power would be needed to cause analogous changes of co~zfiguration in a block weigh- ing two tons.

The outcome of this simple analysis is shown qualitatively in Fig. 2. It indicates that, within a proper relatively narrow size range, each one of the three theories may be correct within a very narrow margin of e r ror . In Fig. 2, the basic curve shown is based on the following net energy consump- tion figures : 1) Explosive shattering: from infinite s ize to - lm:

unknown. 2) Primary crushing: from -1 to -100 mm: 0.35

k w h / t . 3) Secondary crushing: from - 100 mm to - 10 mm:

0.6 k wh/t . 4) Coarse grinding: from - 10 mm to -1 mm: 1.6

k Wh/t . 5) Fine grinding: from -1 mm to -0.1 mm: 10k Wh/t.

The respective cumulative curve shown by the dotted line represents the total net energy used in the various steps of crushing and grinding combined. It is apparent that the cumulative curve follows very closely the respective differential curve with the ex- ception of the crushing range. The exact position of the cumulative curve in this range is difficult to evaluate, since no information usually exists about the energy used in explosive shattering.

Extension of the basic curve beyond 0.1 mm to the range of fine sizes is open to imagination, since no dependable experimental data exist. However, even within this little known range, the unit crystal of the solid substance forms an ultimate limit which can- not be exceeded by mechanical methods of size re- duction. Accordingly, the basic reduction character- istic for quartz, for example, should be a curve which a t the coarse end of comminution deviates asymptotically from a horizontal line, has a slope of gradually increasing negative value with in- creasing fineness of the product, and approaches finally in the extreme reduction range asymptotically a vertical line drawn through the point representing the size of the unit crystal of quartz. Because it might be impossible to reduce quartz by mechanical means of comminution to a powder substantially all of the size of unit crystals, the respective imaginary energy consumption should be infinitely large. In Fig. 2, a number of possible extensions of the basic curve is indicated a t an expenditure of 1 million k w h / t

The basic curve shown in Fig. 2 seems to be in good accord with the existing experimental evidence. Furthermore, it may even give partial explanation for certain peculiarities found in a number of ear- l ier investigations. Thus, for example, the energy- s ize curve shown has a slope greater than -1.0 in the extremely fine sizes. It may be recalled that while studying higher energy concentrations than those used by Gross and ~ i m m e r l e y , ~ Piret and his coworkers7 observed gradual deviation from a straight line energy-surface relationship in the di- rection of lacking surface production. Similar ob- servations were made by Svensson and ~ u r k e s . ' ~ If true, then the Rittinger slope of -1.0 would be valid only for a certain limited, relatively fine s ize range.

Should the basic reduction characteristic in a general form follow the features described above,

Fig. 2-Imaginary example of the basic reduction characteristic plotted on logarithmic paper

then Eq. 1 will not be the correct differential equa- tion of general validity for comminution. In it ex- ponent n is not a constant but a variable whose value is a function of fineness x of the product. In a revised form, Eq. 1 may be written as

EXPERIMENTAL

The experimental investigations carried out in connection with this subject had three principal ob- jectives:

1) In all cases, the power readings were to be re- corded in such a way that the net energy consump- tion could be evaluated with a fair accuracy.

2) The investigations were designed to cover a s wide a s ize range a s possible with the equipment a t out disposal, a t least a 1000-to-1 size reduction performed in three o r more successive steps.

3) In plotting the results, conclusive evidence was sought of the general shape of the reduction charac- teristics.

Two machines only were used for experimental in- vestigations. For the crushing range, a jaw crusher was accepted a s a representative machine. For the range covered normally by rod and ball mills, a se t of rolls was used. Basically, the rolls should cor- respond well to a rod mill where the rods act largely the same way a s a multiple set of long rolls. A rod mill can be used-although seldom is-to produce mineral powders of fineness comparable to those obtained in conventional ball mill circuits. By a substitution of these mills with the rolls many in- definite variables inherent to rod and ball mills of any conventional design were eliminated in the ex- perimental part of this work.

For s ize reduction from 10 to 1 cm, a 7 X12- in. Kue-Ken jaw crusher was applied either in one step (set 10 mm) o r in two successive steps (sets 35 and 10 mm, respectively). Samples passing a grizzly with 10 cm wide longitudinal openings and weighing a t least 100 kg each were used. A study of the crushed products indicated that the third dimension, the thickness, was in all particles less than the s e t

Quartz - Fe/dspor -x- O r e A - - Ore 6 --b

Ore C

4r

' . '. .. 01 10 10 100

log Particle Size, mm

Fig. 3-Experimental reduction characteristics of quartz, feldspar, and ores A , B , and C plotted on logarithmic paper. The ordinate shows the cumu- lative net energy consumption and the abscissa the square screen opening through which 70 pct of the product will pass.

of jaw crusher. It was observed that with some samples substantial differences could exist between the maximum and minimum net energy consumption figures obtained by crushing, e.g., ten successive 100 kilo samples. However, in such cases a test sample weighing a t least one ton was used, and the final net energy consumption figure obtained should now be considered fairly dependable.

For size reduction from 10 to 1 mm, a special precision se t of rolls, 25x 12 cm, was built. Either roll was driven by a separate motor. The rolls were mounted on oversize shafts supported on oversize precision roller bearings; no springs were used. The set of the rolls could be adjusted by means of solid spacer plates a t 1 mm with a great accuracy. Similar precision adjustment was also attempted a t 0.1 mm. However, i t was soon discovered that the rate of feeding a s well a s the hardness of the mineral crushed had an effect on, for example, the 0.1 mm se t which, although very small in actual dimensions, in relative terms produced an un- pleasant source of e r ror . Thus the simple idea that the set of the machine could be used a s a prac- tical size-controlling variable did not yet fulfill the expectations in the very fine range. Consequently, all results reported here a r e based on conventional screen analyses of the various products.

Fig. 3 shows the cumulative reduction character- istics of five large samples including quartz, feld- spar , and three separate ores. The abscissa of each point plotted corresponds to the square screen open- ing through which 70 pct of the product will pass. The f i rs t points on the right a t about 10 mm repre- sent the net energy consumption values in s ize re- duction from 10 to 1 cm.

The results of these tests indicate that the general reduction characteristic on a logarithmic paper is a curve rather than a straight line. Each brittle solid substance seems to have its own characteristic which follows the general features outlined in the

preceding. Thus these tests give further evidence, perhaps difficult to invalidate, that the unifying theory of energy-size relationship presented in this paper is correct.

Regarding the peculiar experimental data on o re C, i t should be mentioned that this is a most un- usual case. The o re itself is a massive sulfide ore of high specific gravity where hard crystals of pyrite represent the bricks and other softer sulfides the plaster of a masonry. In crushing and coarse grind- ing, the plaster offers but little resistance. However, the net energy consumption increases a t a very rapid rate a s soon a s the individual crystals of pyrite a r e broken.

Other preliminary experiments indicate that the same net energy is used in reducing a certain mineral product from a certain feed s ize to the same fineness independently of the number of separate successive reducing steps used a t least in one and the same machine. Furthermore, s imilar com- parisons performed between a jaw crusher and a gyratory gave surprisingly consistent results.

The very important question of the mechanical efficiency and of the idling power consumption of all crushing and grinding machinery thus once again is brought to the attention of a l l those having an active interest in the advancement of the scientific and technical aspects of comminution.

GRIND LIMIT

Grind limit is meant to be a certain particle size value within the fine s ize range a t which it is as- sumed that size reduction by mechanical means of comminution such a s crushing and grinding termi- nates mainly because of structural imperfections in the crystal lattice which a r e supposed to prevent effectively the formation of stil l finer particles. While the grind limit may vary from mineral to mineral, ~ o n d , ' has indicated that according to his calculations it should lie within the limits of 0.200 to 0.050p, 0.100p being a satisfactory average value.

To the best knowledge of the author, the concep- tion and evaluation of the grind limit is more a matter of theory and speculation than fact. To a certain extent, data on such a limit a r e based only on mathematical solutions by trial and e r r o r to satisfy this o r that theory of comminution rather than on even the slightest experimental evidence.

In view of the fact that accurate information on the grind limit should be of fundamental theoretical importance, the author has made an attempt to establish dependable experimental evidence con- cerning it. The test procedure follows.

While crushing quartz in a se t of rolls, some of the resulting dust was sucked through a vacuum bottle filled with alcohol. From the fine suspension obtained, the finest fraction was separated in Model SB centrifuge built by International Equipment Co., run for 14 min a t 1500 rpm. From the resulting final sample, proper preparations for electron micros- copy were made by M. Sulonen, Dept. of Metallurgy, Inst. of Technology, Helsinki, Finland, by a technique

known to specialists in this field. From the many pictures taken, two a r e reproduced in Fig. 4. In Fig. 4a, magnification is X15,OOO; in Fig. 4b, X100,OOO. The maximum magnification used was X150,OOO.

From the resulting unretouched photographs, it is obvious that in this sample of quartz, particles do exist whose length and breadth fall between 0.001 and 0 . 0 1 ~ (10Wand 100A), but whose thickness a s the smallest dimension seems to be only a few multi- ples of the s ize values of the corresponding unit crystal. With the greatest magnification used, it also appears that the edges and corners of the particles a r e somewhat rounded, a feature indicative of some action of dissolution in the alcohol and water solutions used in the preparation of the samples.

This preliminary investigation reveals that while it has not been possible to a s se r t experimentally the final grind limit for quartz, the s izes of parti- cles observed by means of an electron microscope provide convincing proof that the grind limit values such a s those reported by ~ o n d ~ a r e a t least ten times too large. Moreover, i t i s apparent that the entire question of the grind limit needs re-evalua- tion, taking into account the length, breadth, and thickness of the particles related with the mineral- ogical characteristics of the respective minerals.

OUTLINE FOR CRUSHABILITY AND GRINDABILITY INVESTIGATIONS

From the preceding discussion it should be ob- vious that grindability tests such a s those performed by Bond a r e approximately valid for a relatively narrow size range only. Extrapolation of the re- sulting data for s ize range covered by coarse and fine crushing as well a s by very fine and superfine grinding offers little-if any-dependable informa- tion a t all.

It may be argued that crushing, even with all the conventional crushing steps combined, requires s o little energy (usually of the order of 1 k ~ h / t ) that even precise information is of questionable practical value in the design of a crushing plant. In a search for truth, such arguments should be disregarded.

To obtain truly dependable experimental informa- tion on the various steps of s ize reduction, the tests should be performed in accord with the basic prin- ciples of the respective industrial methods of size reduction. Crushing should be investigated with a machine such a s a jaw crusher, coarse grinding (rod milling) in a s e t of rolls, and fine grinding in a ball mill. The fineness of the feed and product in each step should be defined in clear and simple terms. Each s tep should represent (for example) a 10 to 1 reduction in product size x . In all cases, the net power should be evaluated and the net energy con- sumption calculated.

By a procedure outlined previously, three indices would be obtained: one for crushing (from 10 to 1 cm), one for coarse grinding (from 1 cm to 1 mm), and one for fine grinding (from 1 to 0.1 mm). Each of the indices would be characteristic for its re- spective s ize range. On the basis of this information,

Fig. 4-Electron microscope studies of quartz dust. a) -(Top) X15,OOO; original scale, l p or 15 m m to 19/32 in. b) -(Below) X100,OOO; original scale, 0 . 1 ~ or 10 m m to 13/32 in. Reduced approximately 5 pct for reproduction.

a cumulative net energy-size curve could be drawn. Each mineral, rock, and ore will have its own characteristic size reduction curve. From this curve the net energy required for any conventional crushing and grinding operation can be very closely obtained. By knowing the mechanical efficiency of the machinery and the tonnage treated, s izes of the machinery and motors can now be selected a t a good accuracy.

CONCLUSIONS

The preceding discussion leads now to the follow- ing conclusions:

1) In comminution of brittle solids, the net energy necessary for equal reduction ratios increases with increasing fineness of the material treated.

2) On logarithmic paper, the net energy-size re- lationship is presented by a hyperbolic o r near hyperbolic curve. For different solids, position and shape of the curve will vary.

3) The well known theories of Kick, Bond, and Rittinger a r e represented by tangents to this curve having slopes of 0, -1/2, and -1, respectively. At

best, each one of them constitutes an approximation valid for a relatively narrow size range only.

4) The net energy required in comminution for a specified size reduction is independent of the num- ber of successive steps involved. In conventional practical applications, however, the mechanical efficiency of the machinery and the efficiency of the actual reduction process itself have resulted in multiple step processes rather than in the one-step process.

5) The net energy consumption in producing ex- tremely fine mineral powders (such a s all -0.1~)

has become already so enormous that economic application of the present principles of comminution becomes questionable.

ACKNOWLEDGMENTS

The crushing experiments reported in this paper were performed by J. Tanila in the Mineral Dressing Laboratory of the State Inst. for Technical Research, Helsinki, Finland. The electron microscope photo- graphs were taken by M. Sulonen a t Helsinki Uni- versity.

REFERENCES

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'P. R. von Rittinger: Lehrbuch der Aufbereitungskunde, Berlin, 1867. 'F. Kick: Das G e s e t z der Proportionalen Widerstande und s e i n e

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'R. S. Dean: Magnetite a s a Standard Material for Measuring Grinding Efficiency, Transactions AIME, 1939, vol. 134, pp. 324-326.

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tion, Trunsactions AIME, 1957, vol. 208, pp. 80-88; MINING ENGI- NEERING, January 1957.