R. R. KLIMPEL
L. G. AUSTIN THE
I N A D E Q U A C Y OF E X I S T I N G GRINDING LAWS^* AND THE ENORM O U S LITERATURE O N THE "ART"
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A UNIT OF GRINDING M A Y B E VISUALIZED AS TWO, SIMULTANEOUS PROCESSES A PORTION OF EACH S I Z E RANGE IN THE CHARGE 15 PASSED THROUGH A GRINDER (GI TO YIELD A N IWTERMEDIATE SIZEW E I G H T DISTR1!3UTION A S E C O N D P O R T I O N I S PASSED, U N G R D U N D , DIRECTLY TO THE PRODUCT S U C H A M O D E L IS CHARACTERIZED BY A FUNCTION FOR EACH PROCESS ONE FUNCTION DETERMINES THE INTERMEDIATE OlSTRlSUTlONS F O R M E 0 BY GRINDING EACH SIZE RANGE O F T r t E CHARGE, ANOTHER SELECTS THE FRACTION OF THE CHARGE TO B E GROUNO
i.s a unit operation which has no sound underlying theory comparable to that which exists for other unit operations, and the design of a grinding machine for a given duty is an art based on accumulated experience of the manufacturers (49). It is not for lack of either interest or investigation that a quantitative unit operations theory of grinding does not exist. Grinding is an important industrial operation, and one bibliography of grinding (72) lists over 1600 pertinent references. Must it then be assumed that the process of grinding is too complex to hope for such a theory? It is our contention that recent work (7, 75, 26, 27) shows considerable promise of eventually leading to a satisfactory basis for quantitative design. In essence, the design of a mill for a specified duty has to answer five major questions:
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-What type of machine should be selected? - 9 t maintenance will be required or how can wear be reduced to tolerable levels? -What size machine is required? -What power requirement is to be specified? -What will be the performance of the mill under varying feed conditions and what are the optimum operating conditions? Questions 1 and 2 are closely related and their answers depend on a number of factors which are known from previous experience (49). Questions 3,4, and 5 are also related and it is to answer them that a grinding theory is needed. In the following sections the inadequacy of existing “laws” is discussed and a brief review of present understanding of fracture is given as a necessary preliminary to the newer approaches to the problem.
Energy “Laws” of Comminution
Two early theories of comminution have received much study and have led to considerable controversy between their respective proponents. Rittinger’s (44) law, proposed 1857, states that the energy input to a size reduction process is used to produce fresh surface, where the specific surface energy is a characteristic of the material. At first sight this seems a reasonable hypothesis and measurements of increases in surface area compared to energy inputs were used to estimate specific energies. However, a critical examination of the law, with the benefit of the many years of research since the law was proposed, shows that it is a great over-simplification. I t assumes, that the energy input is completely transferred to the charge being ground. This is not true, and indeed the fraction of the energy transferred to the charge will vary for different types of machine and different operating conditions. Again, when a solid is fractured by slow load it has been found that the strain energy is converted to several different types of energy other than surface energy-for example, propagated stress wave energy, kinetic energy of the breakage fragments, plastic deformation energy, etc. The fraction of the total energy converted to surface energy will be extremely variable, depending on the conditions of fracture. As a crude example, consider the fracture of a helical spring, loaded until it breaks across one part of the material. The two resultant halves will contain much of the applied energy as strain energy and, by contraction after the break, the energy will be released as heat. There is considerable evidence to show that in nearly all grinding operations the fraction of energy input used to produce fresh surface is less than a few per cent. Rose (46) is experimentally determining detailed energy balances on brittle materials ground in a ball mill, and has shown that the fraction of energy converted to fresh surface is less than 370 and is probably much less than this (experimental error providing the uncertainty as to the exact figure). Kick’s (39) law, proposed in 1883, states that for geometrically similar size reduction, the energy per unit volume is constant. This again might appear to be a reasonable hypothesis, from similarity considerations. If we break a large particle into ten equal fragments, then break each of these fragments into ten equal fragments, the energy for each of the second breakages is one-tenth that of the first, since the scale is reduced by ten. However, since there are ten of these breakages in the second step, the over-all energy used per unit volume is constant. Although this seems reasonable, it is clear that it does not agree with Rittinger’s law. I n mathematical form Rittinger’s law can be expressed as : dE/dp = -kik/p2
(11
E is the energy, per unit volume of charge, required to produce size reduction, p is size, k l is the surface energy, and k is a shape factor. Equation 1 integrates to
E = kl x (Final surface area per unit volume initial surface area per unit volume) 20
INDUSTRIAL A N D ENGINEERING CHEMISTRY
which is Rittinger’s law. as :
Kick’s law may be expressed
d E / d p = -kz/2.3p (2) For unit volume of size p1, broken to unit volume of size pz, this integrates to E = kz log (pu/pz) Thus, kz is the energy per unit volume required to produce a 10-fold reduction in size. When kz is independent of initial size, this is Kick’s law. It is clear that Equations 1 and 2 cannot both hold a t the same time. The similarity argument behind Kick’s law is extremely tenuous ; in a given machine, smaller particles will often have much greater forces per unit area applied to them than larger particles, the number of particles subjected to crushing in a grinding zone will 1-ary with size, and, considering the Griffith crack theory (see below), one might expect smaller particles to have statistically fewer weak flaws and thus be much stronger than larger material. It is not surprising that this “law” is of little practical utility in grinding operations. Bond (13, 14) has proposed a law which falls between the two previous laws. Mathematically it is d E / d p = --k3/2p1J (3) which integrates to E = k&/dL - V d Z ) Therefore, ka is the energy per unit volume required to reduce from infinite size to unit size. As applied in practice, any weight versus size distribution is given a mean size defined by the sieve size at which 80% by weight passes and 20% is retained. This mean size would only be proportional to the correct, weighted, root mean
p
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fractional volume
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if all distributions used had geometrical similarity-that is, if the weight per cent versus fractional size (size as a fraction of largest size present) were the same for all the distributions. This is not so in general. E is found to be a machine property a s well as a material property and, therefore, it has little theoretical significance. Bond’s law must be considered as an empirical rule of thumb which is principally useful as a means of codifying a mass of industrial experience so that interpolation and limited extrapolation can be made for known materials and equipment. A common failing of all of the “laws” is that they do not give information on the size-throughput relations of a mill or on optimum operation conditions. A good analogy, perhaps, might be the calculation of the pumping power/cu. ft. of gas absorbed in an absorption tower. It would not be accepted that this pumping power is a fundamental property of the gas being absorbed, or that power/cu. ft. is a complete design parameter for the tower. The power required to obtain a given throughput of product in a mill is an indirect mechanical consequence of the breakage processes occurring in the mill and if the mill could be designed from a satisfactory theory, the power requirement would follow by relatively simple dynamic calculations.
Physical Concepts of Fracture
I t would seem that in this age of scientific and technical progress it should be possible to produce more detailed theories of size reduction than those discussed above. Partly to this end many workers have studied the fracture of single specimens of various materials. The basis of this work was the pioneering study of Griffith (32), who used the mathematical stress analysis tools provided by Inglis (34). Inglis had shown, by solving the differential stress-strain equations, that the presence of flaws in a material could lead to stress concentration in a solid under stress. As a geometry which approximated a crack and which was amenable to mathematical treatment, Griffith supposed the flaws in brittle materials to be elliptical in two dimensions with a narrow minor axis and constant in the third dimension. The tips of the flaws were small so that chemical bonds existed in a range of states from unstressed to fully stretched and at the point of breaking. Essentially this assumes that the crack tip is in existence in the flaw. Therefore, stress is not required to provide force to stretch bonds to the breaking point, since such bonds exist already. Stress is required, however, to provide the energy necessary for continued breakage and production of new surface. Since the crack tip exists before stressing and since it is assumed that this tip propagates unchanged in shape, the surface energy change per unit width of crack is 4yAc, y being specific surface energy and Ac the change in crack half-length. By equating the differential of strain energy (due to the crack) to the differential of surface energy, with respect to c, the half length of the initial crack, the well known Griffith criterion is obtained:
(4)
P is the critical tensile stress perpendicular to the crack, distant from the crack, and E is Young’s modulus. This form applies when the load is maintained constant or when the “fixed grip” (constant extension) condition is considered (35). The crack propagates along the major axis of the ellipse, in both directions. Equation 4 relates the tensile stress at fracture to the size of the flaw. The Griffith flaw theory is undoubtedly correct in its conception of crack propagation from tiny flaws in the material. The mathematical analysis is incomplete due to the use of an energy balance which neglected the kinetic energy of the stress wave moving with the propagating crack. Much work has since been done on the propagation of tensile stresses and the energetics of crack propagation (7, 4, 5, 19, 21, 24, 31, 36-38, 43, 57-53). I t is easily shown that a simple treatment of stress and energy gives the maximum tensile stress of a material as
(5) a is the interatomic distance. Materials fractured in this manner would disintegrate equally at all planes perpendicular to the stress. Experiment has shown that the presence of Griffith cracks leads to failure stresses which are normally 1/1000 to 1/100 of the maximum
stress. Mott (41) has introduced the kinetic energy of the stress field propagating at the tip of the crack. By appropriate integration of the kinetic energy, plus an energy balance, he showed that the crack propagation velocity increases rapidly as the crack propagates, to a maximum given by
urn is the maximum crack velocity, uL is the longitudinal velocity of sound propagation in the solid, and k represents a dimensionless number which is the integral of the stress field in the zone around the crack tip, in reduced form. k is constant if the stress symmetry does not change during propagation. Roberts and Wells (45) showed, from calculations of the stress field, that is about 0.4. This gives a urn/uL ratio roughly in accord with experimental evidence. As Anderson (3) has pointed out, a number of effects disturb this analysis. The true geometry of cracks is not known and the above theories are two-dimensional and not three-dimensional. The stress field will undoubtedly change as fracture proceeds; for constant loading the stress may increase due to the finite length of the crack compared to the plate dimensions. Where hackled regions are produced in the path of the crack, fresh surface area is produced which is not included in the primary crack analysis. Rumpf (48) has pointed out that energy of chemisorption may be a significant part of the energy balance when the material is fractured in certain gases. In addition, the assumption of isothermal crack propagation is certainly open to question. Stress relief in a fragment completely enclosed by fracture surfaces will give rise to internal shock waves, where the energy may cause further fracture or may dissipate as heat of internal friction. Complete yield, of course, removes the applied stress and only internally stored strain energy and the kinetic energy of any propagating cracks or waves are available to supply the energy for the production of new surface. The nearer the failure stress is to the ideal stress of Equation 5 , then the more catastrophic the fracture will be, leading to many fragments in the path of the primary cracks. The above theories can only apply to purely brittle materials-that is, those which fracture before the stress reaches values sufficient to give slip under shear or plastic flow. It is worth noting that the terms brittle or ductile apply to a material in a certain state, and material does not necessarily remain brittle or ductile over a range of conditions. For example, if we consider the failure of metals under simple tensile stress, the failure may be brittle or ductile depending on the conditions. At low temperatures, molecular movements have a lower probability of occurrence and the mobility of defects or dislocations is small. Therefore, slip and plastic deformation are hindered, leading to high values of the shear yield stress, and brittle failure is enhanced. At high temperatures, plastic deformation can more readily occur and failure by ductile fracture is enhanced. For example, Cottrell (20) has described a mechanism of ductile failure under tension which involves the mobility
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VOL. 5 6
NO. 1 1
NOVEMBER 1 9 6 4
21
Grinding operations are too complex for other than statistical treatment
of dislocations under stress. The dislocations pile up at grain boundaries, twin interfaces, or slip interfaces and nucleate a cavity. Alternatively a cavity may start from a foreign inclusion. Once formed the cavity will be stretched under tension, while the “neck” of the ductile specimen reduces the solid cross-sectional area. A number of such cavities will coalesce to form a fibrous crack perpendicular to the tension. I t is possible that once sufficient coalescence has occurred to give a critical crack length, then crack propagation occurs, rapidly joining existing cavities. The fracture then has something of the character of brittle failure. Allen (2) has discussed the energetics of crack propagation, under conditions where plastic deformation can occur at the crack tip. I n addition to the terms discussed above, it is necessary to include an energy term for plastic deformation. This energy will be a function of crack length. Thus a crack may not spread when increased tension is applied, or it may spread only slowly because the elastic strain energy is dissipated as energy of plastic deformation. Due to work hardening, the yield stress of the plastic material ahead of the crack increases. This means that further deformation and dissipation of energy becomes more difficult. This, of course, may lead to an accelerating Griffith crack and brittle failure. Once the stress pattern at the crack tip has reached high velocities, the energy for plastic deformation becomes small, since as the plastic effect is one of mobility, it requires a definite time to develop. The Griffith theory requires that tensile stress exists across the crack in order to open it out. A uniform compressive loading can only close up a crack. It is found that compact materials can withstand very high equal triaxial compressive forces without fracture or permanent deformation. O n the other hand, uniaxial compression leads to failure. I t is generally realized (77, 42) that tests of compressive loading are not so amenable to theoretical treatment as tensile tests. Because of the restriction of material movement between the pressing plattens and the cube or cylinder of test material, tensile forces exist in compression (78). However, the tensile stresses are much less than the applied compressive stress and, as expected, brittle materials fracture under much higher compressive stress than under direct tensile stress. The more ductile materials, however, fail a t almost the same stress in both cases. This appears to be due to the initiation of failure being caused by slip, which, being principally a shear effect, is fairly independent of whether the uniaxial force is tensile or compressive. I t is probable that the slip under compression produces tensile forces by a wedging action and these tensions initiate fracture. I t should be noted that materials which are brittle in tensile or bending tests may be partially ductile in compression tests, due to the higher loading required for fracture. 22
INDUSTRIAL A N D ENGINEERING CHEMISTRY
Great advances have been made in the theory of fracture in recent years, and the application of these theories already has been of great value in the space program and industry (especially the metallurgical and ceramic industries). However, whereas the materials scientist interested in fracture stress concerns himself principally with the initiation and propagation of failure, the scientist investigating comminution needs to know the size distribution of the fragments of fracture. The forces applied in most industrial grinding processes are complex and usually include impactive compressive forces which are not readily amenable to mathematical analysis. I n practical instances of grinding of irregularly shaped particles, it would appear impossible to predict the amount and size distribution of products with our present knowledge of fracture. This conclusion immediately leads us to a statistical consideration of the problem. Statistical Description of Primary Fracture Distributions
Limited statistical investigations of breakage have been reported (70. 22, 33, 47). However, two recent treatments (29, 30) have come much closer to a satisfying description of the size-weight distribution of primary fracture. The two treatments are similar in concept, although not in detail. A brief discussion of Gilvarry’s (30) treatment will be given, since it illustrates well the concepts and techniques used. He considers single particle breakage, making the following assumptions. Under applied stress, flaws exist within the solid which are “activated” by the stress and can initiate a fracture surface. The flaws are distributed at random within the volume. Each fracture surface will contain flaws which can originate new fractures; similarly, the edges of the confluence of the fracture surfaces also contain flaws which may originate fresh fracture. The basic hypothesis is made that fracture originates from volume, facial, and edge flaws. If the fracture surfaces are random and if the fracturing stresses are random it is reasonable to assume that volume, facial, and edge fracture flaws each have an independent random distribution. With these assumptions Gil1,arr)- applies the Poisson law to the fracture process. The probability of n active flaws falling in a subdomain of extent t is P(n,t) = e-Y‘(yt)f2/n! where y is the mean densit)- of such flaws. The subdomain may be a volume, fracture surface area, or fracture edge in our case. From the q u a tion, the probability of no flaw being in t is P(0.t) = e-Yt, and the probability of a flaw being in dt is P(7,dt) = ydt. For one subdomain, the probability of no flaws being in t and one flaw being in t dt is obviously P = If to represents the upper limit
+
of t , we must have
lo
