Transcript
PROPOSAL FOR A SOLOMONIC SETTLEMENT BETWEENTHE THEORIES OF VON RITTINGER, KICK, AND BOND This paper presents a preliminary analysis of the fundamental relationshipbetween the net energy used and the respective product size throughout theentire range of sizes covered by crmshing and grinding, and an attempt tofind a sensible correlation between the existing theories. by R. T. Hukki alker, Lewis, McAdams, and ~illiland' ave given w he following differential equation of a generalform for comminution:where E is the net energy required per unit weightin a certain process of comminution; x is the factorindicating 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 bynumerical figures 2, 1, and 1 1/2, the integratedform of the general equation leads to the well knownfundamental theories represented by the law of von~ittinger,' aw of ~ick,~nd the third theory ofcomminution by ~ond,~espectively.The total net energies (E~)rom infinite feed sizeto a product of size x are as follows: 1 Rittinger: Et = C xk Wh/t [21Kick: E~ = -C ln: k ~h/t [31Bond : 1 Et = 2C k Wh/t [41The corresponding net energies (E) required toreduce product xl to product x2 areRittinger: E = C (-& - $) kwh/t [51Kick: E = -C lnX1 k~h/t X 1 [61Bond: 1 1 E = 2C (- - -) kWh/t. Jx, Jxl PI The net energy required in a certain process ofcomminution is proportional to the new surface de-veloped according to the law of von Rittinger, to the R . T. HUKKl is Professor of Mineral Dressing, Inst. ofTechnology, Helsinki, Finland. TP59B239. Manuscript,Nov. 9, 1959. Discussion of this paper, submitted indupl - cate prior to July 1, 1962, will appear in AlME Trans-actions (Mining), 1962, vol. 223. weight or size of the bodies treated according to thelaw of Kick, and to the length of the new cracksformed which initiate breakage according to thetheory and explanation by Bond. On a logarithmic paper, where particle size ispresented on the abscissa and energy consumptionon the ordinate (see Fig. 21, all three relationshipsare represented by straight lines. The slope m ofthe 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 ofRittinger has been presented, e.g., by Gross andzimmerley5 on quartz crushed in a drop weightcrusher and evaluated for surface by the methodof dissolution, by Deans on magnetite crushed by asimilar 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 asimilar method as well as by compression andevaluated for surface by permeability and gas ad-sorption methods, and by schellinger8 on a group ofminerals ground in a calorimetric ball mill andevaluated for surface by gas adsorption.Experimental evidence in favor of the law of Kickseems to be scant in the field of comminution. Onthe other hand, in the field of mechanical engineer-ing Kick's law seems to be of fundamental nature inprocesses such as cutting, pressing, shaping, androlling of metallic substances.Experimental evidence in favor of the third theoryhas been provided by ~ond.~ o a large extent,data are based on the vast amount of grindabilitytests performed in the laboratories of Allis-Chal-mers Manufacturing Co. In addition to the devoted proponents of one or theother of the basic theories listed above, certain in-vestigators have indicated that one of the theoriesmight be applied for a certain range of sizes, whileanother theory might be used for other sizes. In adiscussion of a paper by Bond, Dobie presented astatement at the International Mineral DressingCongress in London (1952) indicating that 1) forlarge particles, the law of Kick was approximatelycorrect; 2) for finer particles, von Rittinger's sug-gestion was nearer to the truth; and 3) Bond's new 1p lop loop Imm 10mm 40omm Irn lorn log Particle Size Fig. -Illustration of the basic reduction steps incomminution with a tabulation of the energy con-sumption between the various steps based on thetheories of vonRitting-er, Bond, atzd Kick. theory was a unification of the two older suggestions.To the question raised by ~obie oncerning thelimits of particle size over which the Bond theorycould be used, Bond answers by stating that the thirdtheory of comminution appears to have no size rangelimitations. Contrary to the opinion expressed byDobie, Walker and shawl2 have indicated that the lawof Kick might be applicable in sizes below lp, whilethe law of von Rittinger might suit for sizes coarserthan 1p. charles13 some time ago derived anothergeneral equation for comminution and proposed thatthe hypotheses of Kick, von Rittinger, and Bond maybe derived as special cases of his equation. Hepresents corresponding experimental evidence infavor of Bond and Rittinger, but not in favor of Kick. In the summary of his paper Charles states thatKick's hypothesis would be approximately valid forproduction of extremely fine material, a conclusionthat is in agreement with the views of Walker andshaw.I2 Recently, Svensson and ~urkes'~aveclaimed that, while none of the earlier theorieswill correspond to the results of their extensivegrinding experiments. still another empirical re-lationship developed by them would offer a solutionof general validity.It is somewhat startling to note that the actual sizerange covered by dependable experimental evidencepresented in the referred investigations is alwaysrelatively narrow. In spite of that, conclusions ofgeneral applicability have been presented.CLASSIFICATION OF BASIC REDUCTION STEPSIN COMMINUTIONMost conventional processing of mineral rawmaterials includes three major steps: 1) explosiveshattering of the ore or rock, 2) crushing, and 3)grinding. No exact boundaries between them can begiven. In spite of that, a somewhat crude, yet quali-tatively satisfactory classification of the successivesteps of comminution may be based on the very con-venient metric decimal system as follows:1) Explosive shattering: from 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 - 100~. 6) Very fine grinding: from -100p to -lop.7) Superfine grinding: from -lop to -1p.Fig. 1 shows seven parallel straight lines repre-senting the foregoing classification on a customarylogarithmic paper. Size is shown on the abscissaand cumulative percentage passing a certain size(screen) on the ordinate. This method of plottingfollows the Gates-Gaudin-Schuhmann relationshipwith slope m = 1.0, considered to represent the ideal size distribution in comminuted products.Primary and secondary crushing may consist inpractice of more than two successive steps becausethe reduction ratio of most conventional crushers isless than 10. Explosive shattering may also becarried out to a finer end product than indicated.Coarse grinding as shown in Fig. 1 would roughlycorrespond to grinding in rod mills and fine grindingto grinding in ball mills. Other features of the se-quence shown in Fig. 1 should be self-explanatory.It is not unusual that a total of 10 k wh/t of energyis used in all steps of conventional crushing andgrinding combined. As an example it is now assumedthat a total of 10 kwh/t is used in reduction steps 2 through 5 as shown in Fig. 1. For the time being, noattention is paid to the mechanical efficiencies of therespective crushers and mills. Distribution of thetotal energy available between the four steps basedon Eqs. 5 through 7 is tabulated in Fig. 1. As in-dicated in the preceding, the data corresponding tothe law of von Rittinger form a straight line ofslope rn = -1.0 on a logarithmic paper, showing par-ticle size on the abscissa and energy consumptionon the ordinate; those corresponding to the law ofKick, a horizontal straight line with slope m = 0; andthe data representing the theory of Bond, a straightline of slope m = -0.5. If the three series of dataare compared with the respective figures obtainedin industrial practice, it is not difficult to see thatonly those following the theory by Bond will be of anyreasonable value. If, however, this imaginary size reduction is car-ried further from - 100 to - op-a tenfold stepagain-it will be easily conceived that the energyconsumption figure of 21.8 k Wh/t by Bond seemstoo small to satisfy the requirements of conventionalpractice. It is well known, for example, that to grindcement to a fineness of 92 pct -325 mesh (44p),about 35 kwh/t of energy is needed. Such a productis much coarser than the-lop product now understudy. The respective Rittinger figure of 90 kwh/t should be more reasonable. The situation becomesstill more clear by analysing the imaginary size re-duction from - 10 to -1p. The tabulated figure forRittinger appears to be reasonable, very doubtful forBond, impossible for Kick.On the the other hand, if the coarse end of thesequence shown in Fig. 1 is studied, Rittingerfigures seem to be fully unacceptable, Bond figuresquestionable, but the figures for Kick more and morereasonable. As a crude example in favor of the lawof Kick it might be reasoned that, if a certainamount of net power is needed to crush a blockweighing one ton in a big jaw crusher, twice thatamount of net power would be needed to cause analogous changes of co~zfiguration n a block weigh-ing two tons. The outcome of this simple analysis is shownqualitatively in Fig. 2. It indicates that, within aproper relatively narrow size range, each one ofthe three theories may be correct within a verynarrow margin of error. In Fig. 2, the basic curveshown is based on the following net energy consump-tion figures : 1) Explosive shattering: from infinite size to - m:unknown.2) Primary crushing: from -1 to -100 mm: 0.35 k wh/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 thedotted line represents the total net energy used inthe various steps of crushing and grinding combined.It is apparent that the cumulative curve follows veryclosely the respective differential curve with the ex-ception of the crushing range. The exact position ofthe cumulative curve in this range is difficult toevaluate, since no information usually exists aboutthe energy used in explosive shattering.Extension of the basic curve beyond 0.1 mm to therange of fine sizes is open to imagination, since nodependable experimental data exist. However, evenwithin this little known range, the unit crystal of thesolid 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 curvewhich at the coarse end of comminution deviatesasymptotically from a horizontal line, has a slopeof gradually increasing negative value with in-creasing fineness of the product, and approachesfinally in the extreme reduction range asymptoticallya vertical line drawn through the point representingthe size of the unit crystal of quartz. Because itmight be impossible to reduce quartz by mechanicalmeans ofcomminution to a powder substantially allof the size of unit crystals, the respective imaginaryenergy consumption should be infinitely large. In Fig. 2, a number of possible extensions of the basiccurve is indicated at an expenditure of 1 million k wh/t The basic curve shown in Fig. 2 seems to be ingood accord with the existing experimental evidence.Furthermore, it may even give partial explanationfor certain peculiarities found in a number of ear-lier investigations. Thus, for example, the energy-size curve shown has a slope greater than -1.0 inthe extremely fine sizes. It may be recalled thatwhile studying higher energy concentrations thanthose used by Gross and ~immerley,~iret and hiscoworkers7 observed gradual deviation from astraight line energy-surface relationship in the di-rection of lacking surface production. Similar ob-servations were made by Svensson and ~urkes.'~ f true, then the Rittinger slope of -1.0 would be validonly for a certain limited, relatively fine size range.Should the basic reduction characteristic in ageneral form follow the features described above, Fig. 2-Imaginary example of the basic reductioncharacteristic 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 whosevalue is a function of fineness x of the product. In a revised form, Eq. 1 may be written asEXPERIMENTALThe experimental investigations carried out inconnection 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 aswide a size range as possible with the equipment atout disposal, at least a 1000-to-1 size reductionperformed in three or more successive steps.3) In plotting the results, conclusive evidence wassought of the general shape of the reduction charac-teristics.Two machines only were used for experimental in-vestigations. For the crushing range, a jaw crusherwas accepted as a representative machine. For therange covered normally by rod and ball mills, a setof rolls was used. Basically, the rolls should cor-respond well to a rod mill where the rods act largelythe same way as a multiple set of long rolls. A rodmill can be used-although seldom is-to producemineral powders of fineness comparable to thoseobtained in conventional ball mill circuits. By asubstitution of these mills with the rolls many in-definite variables inherent to rod and ball mills ofany conventional design were eliminated in the ex-perimental part of this work.For size reduction from 10 to 1 cm, a 7 X12- in.Kue-Ken jaw crusher was applied either in one step(set 10 mm) or in two successive steps (sets 35 and10 mm, respectively). Samples passing a grizzlywith 10 cm wide longitudinal openings and weighingat least 100 kg each were used. A study of thecrushed products indicated that the third dimension,the thickness, was in all particles less than the set Quartz - e/dspor -x- OreA - Ore 6 --b Ore C 4r ' '. . 01 10 10 100 log Particle Size, mm Fig. 3-Experimental reduction characteristics ofquartz, feldspar, and ores A, B, and C plotted onlogarithmic paper. The ordinate shows the cumu-lative net energy consumption and the abscissa thesquare screen opening through which 70 pct of theproduct will pass. of jaw crusher. It was observed that with somesamples substantial differences could exist betweenthe maximum and minimum net energy consumptionfigures obtained by crushing, e.g., ten successive100 kilo samples. However, in such cases a testsample weighing at least one ton was used, and thefinal net energy consumption figure obtained shouldnow be considered fairly dependable.For size reduction from 10 to 1 mm, a specialprecision set of rolls, 25x 12 cm, was built. Eitherroll was driven by a separate motor. The rolls weremounted on oversize shafts supported on oversizeprecision roller bearings; no springs were used.The set of the rolls could be adjusted by means ofsolid spacer plates at 1 mm with a great accuracy.Similar precision adjustment was also attempted at0.1 mm. However, it was soon discovered that therate of feeding as well as the hardness of themineral crushed had an effect on, for example, the0.1 mm set which, although very small in actualdimensions, in relative terms produced an un-pleasant source of error. Thus the simple ideathat the set of the machine could be used as a prac-tical size-controlling variable did not yet fulfill theexpectations in the very fine range. Consequently,all results reported here are based on conventionalscreen 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 eachpoint plotted corresponds to the square screen open-ing through which 70 pct of the product will pass.The first points on the right at about 10 mm repre-sent the net energy consumption values in size re-duction from 10 to 1 cm.The results of these tests indicate that the generalreduction characteristic on a logarithmic paper is a curve rather than a straight line. Each brittle solidsubstance seems to have its own characteristicwhich follows the general features outlined in thepreceding. Thus these tests give further evidence,perhaps difficult to invalidate, that the unifyingtheory of energy-size relationship presented in thispaper is correct.Regarding the peculiar experimental data on ore C, it should be mentioned that this is a most un-usual case. The ore itself is a massive sulfide oreof high specific gravity where hard crystals of pyriterepresent the bricks and other softer sulfides theplaster of a masonry. In crushing and coarse grind-ing, the plaster offers but little resistance. However,the net energy consumption increases at a very rapidrate as soon as the individual crystals of pyrite arebroken.Other preliminary experiments indicate that thesame net energy is used in reducing a certainmineral product from a certain feed size to the samefineness independently of the number of separatesuccessive reducing steps used at least in one andthe same machine.Furthermore, similar com-parisons performed between a jaw crusher and agyratory gave surprisingly consistent results.The very important question of the mechanicalefficiency and of the idling power consumption of allcrushing and grinding machinery thus once again isbrought to the attention of all those having an activeinterest in the advancement of the scientific andtechnical aspects of comminution.GRIND LIMIT Grind limit is meant to be a certain particle sizevalue within the fine size range at which it is as-sumed that size reduction by mechanical means ofcomminution such as crushing and grinding termi-nates mainly because of structural imperfections inthe crystal lattice which are supposed to preventeffectively the formation of still finer particles.While the grind limit may vary from mineral tomineral, ~ond,' as indicated that according to hiscalculations it should lie within the limits of 0.200to 0.050p, 0.100p being a satisfactory averagevalue.To the best knowledge of the author, the concep-tion and evaluation of the grind limit is more amatter of theory and speculation than fact. To acertain extent, data on such a limit are based onlyon mathematical solutions by trial and error tosatisfy this or that theory of comminution ratherthan on even the slightest experimental evidence. In view of the fact that accurate information onthe grind limit should be of fundamental theoreticalimportance, the author has made an attempt toestablish dependable experimental evidence con-cerning it. The test procedure follows.While crushing quartz in a set of rolls, some ofthe resulting dust was sucked through a vacuumbottle filled with alcohol.From the fine suspensionobtained, the finest fraction was separated in ModelSB centrifuge built by International Equipment Co.,run for 14 min at 1500 rpm. From the resulting finalsample, proper preparations for electron micros-copy were made by M. Sulonen, Dept. of Metallurgy,Inst. of Technology, Helsinki, Finland, by a technique
