DROP WEIGHT TEST WORK

Transcription

1 CHAPTER Introduction The purpose of the drop-weight-test work was to establish a consistent quantification of fracture response of the material to the energy spectra predicted by the DEM software. There was a need to develop a multi-dimensional model that took into account the variation in the energy input and the number of events affecting each particle. The initial experiments were concerned with establishing the relationship between energy input and probability of fracture, while the latter experiments addressed the progeny distribution from a fracture event. 4.2 Description of the JKMRC drop weight test machine The work involved breaking single particles by impact using the drop-weight test machine illustrated below. The input energy was determined by the weight of the steel disk (M) that was dropped on the particle from a level (h 1 ). The drop height (h) is the difference between the final (h 2 ) and the initial (h 1 ) position of the steel weight. h 2 varies as it depends on the residue after breakage. Steel weight h 1 Transparent wall Drop height (h) Anvil h 2 (position depends on remaining residue) Figure 4-1 Simple illustration of the drop-weight test machine 40

2 The input energy (J) is calculated as follows: E = Mg( h 1 h2 ) (4.1) where M is the mass of the steel disk in Kg, g is the force due to gravity and h 1 -h 2 is the drop height in metres. 4.3 Establishing a relationship between particle size and energy required for breakage It is known from experience that based on the same energy input per unit mass, larger particles are generally weaker than smaller particles (see section ). The purpose of this experiment was to determine if this apparent weakness can be modelled. The random variation in the strength of particles of otherwise the same size made this exercise somewhat complicated (Bourgeios and King 1993). Hence a large number of particles per size group had to be tested Experimental procedure The particles were closely sized by weighing (relative mass standard deviation in each group never exceeded 6%) and then broken individually using the dropweight test machine. About ten particles were tested in each class though fewer particles were available for the larger rocks (+800g). A particle that was not broken at first attempt was repeatedly impacted until the largest fragment was broken to a size less than 1/3 of the original physical size. The limit for impact attempts was thirty. The procedure involved fixing the drop weight test energy input and breaking singly all the particles of a specific size group. If all the particles were broken as per defined criteria above, a larger size group was introduced until a size group was found in which some of the particles could not be broken. This indicated that the size group was too strong for the set energy level and at this juncture a higher energy level was set on the drop weight tester and the whole procedure was then 41

3 repeated. Thus in this way the minimum energy that is just sufficient to break all the particles in a particular size when the maximum number of impact attempts is thirty, was established Results of the test to find minimum energy required to cause fracture in up to thirty impact attempts The results of the test are summarised in Table 4-1 below and the data meeting the set criteria (highlighted results in the table) are plotted in Figure 4.2 Table 4-1 Relationship between Energy Input and Mass of particles broken Average Std No. of Energy per Ave No of Std Dev % Broken Particle Mass (g) Dev. Particles tested Impact (J) Attempts of No. of attempts * *

4 Though in both cases, 1 particle failed to break it was assumed these data were practically close to the requirement. 100 Energy Input Joules/impact Particle mass (g) Figure 4-2 The relationship between energy required for breakage and particle size Note that the relationship shown above is based purely on the energy input per impact without considering the effect of the number of impact attempts. It is however surprising to see such a good correlation achieved. This relationship is well described by the equation below. Es = 033. X 076. (4.2) where E s is the minimum energy that is just enough to break all the particles of mass X in 30 impact attempts or less per particle Conclusion A clear relationship exists between energy input to achieve fracture in 30 attempts and particle size. However this relationship does not consider the effect 43

5 of the number of impact attempts on breakage and thus further experiments were required to address this problem. 4.4 Probability of fracture with respect to energy input and number of impact attempts Sample preparation Particles of the same size even if they be of the same material tend to vary in strength due to differences in the distribution of flaws. It was thus decided that the relationship between energy input and propensity to breakage be stated in terms of probability. The procedure involved preparing the sample by dividing particles into groups of nearly the same mass. The particle mass groups used were as follows; 3.15g, 5.15g, 8.15g, 12.6g, 28.9g, 38.8g and 67.5g. The mass variation in each group was no more than ±11 %. Each mass group was further divided into four groups of 20 to 30 particles depending on the number of particles available Test-work procedure To each group, a fixed energy input was applied using no more than ten impact attempts per particle. The number of particles broken at each impact attempt and energy level applied were noted. Any particle that lost 10% of its mass was considered broken. This criterion is justifiable if it is noted that breakage is mostly the propagation of cracks rather than initiation. In some instances the crack propagated may not necessarily lie at the centre and thus propagation of a crack anywhere within 80% of the body's volume is included (80% instead of 90 because in almost all cases breakage involving loss of a fragment of about 10% or less occurs only on one end of the body). The 10% mass loss criterion for certifying breakage is also popular amongst German researchers (Dan and Schubert 1990, Klotz and Schubert 1982), albeit they also tend to use a lower ratio between the screens. The 2 1/10 series or higher resolution is often used. 44

6 4.4.3 Results The results for all the different particle sizes are presented in Appendix B1. This is presented in terms of probability versus number of impact attempts and energy input. The corresponding plots of these results are in Appendix B2. It can be seen from the graphs in Appendix B2 the trends are similar. The proposed model discussed later (section 4.4.4) does not fit all the data accurately and it is recommended in future, to user larger samples of particles to ease the problem of particle strength variability. The parameters were established using the excel solver to minimise the least square difference between the model and all the experimental data. For the purpose of discussion the first figure in Appendix B2 is also shown here. Probability of breakage of 3.15g Particles Probabilty of breakage No. of impacts attempted Energy level (J) Figure 4-3 Probability of Breakage versus Input Energy and Impact attempts Discussion of the results It can be seen from the figure above that both the number of impact attempts (n) and the energy level (E) have an effect on the probability of breakage. It is seen 45

7 that when the energy input is low each extra impact steadily increases the probability of breakage while for the higher energy inputs, the change in probability is substantial for the first few attempts before asymptoting. This is in line with the crack extension theory discussed in Chapter 2. In Equation 2.34 it is seen that all else remaining the same, the relationship between energy input and crack growth is non-linear and thus at low energy input, crack growth will be slower and thus more impacts will be required before reaching critical conditions. The overall effect of the energy input and number of impacts is summarised in Figure 4.4 below. High energy Probability of breakage Low energy Number of impact attempts Figure 4-4 The effect of high and low energy input and impact attempts on probability of breakage With this definite trend, a model that produced a good fit to all the data was obtained by modifying an equation developed by Weichert (1990) written as follows: 2 z ( cx W m ) P = 1 e (4.3) where P is the probability of fracture, c and z are material constants. x is the particle size and W m is the specific elastic energy. It can be seen that this equation is a further development of Weibull s Equation

8 From Figure 4.4 it is seen that the probability of breakage is a function of particle size, energy input and number of impact attempts. It was also noted that some minimum energy level had to be exceeded before any breakage could occur. Thus Equation 4.3 was modified accordingly to yield an equation that gave the best fit to the data written as follows: P b e ( X +.5) n (( E E 0 ) / x 0 ) 1 x E = (4.4) P b is the probability of breakage, X is the particle mass (g), n is the number of impact attempts, E is the energy input (J) and E x0 is the barest minimum energy (J) that is required to fracture a particle of mass X. E x0 was an experimental extrapolation as shown in Appendix B3. The following model fitted the E xo data very well E x = 0. X (4.5) 0 19 Some limitations regarding this formula must be mentioned. It does not take into consideration the different force loading configurations that might affect especially the large particles. With larger particles it is possible to achieve breakage with lower energy than the E x0 value if for example the mode of stress application involves using a wedge to open up a crack. An overview of difficulties concerning the determination fracture energy for particles is given by King (2001) Conclusion A model that can be applied to determine the number of particles of known mass that will break if each particle is subjected to a known energy level and a number of impact attempts has been proposed. This model is thus suitable for estimating breakage in the mill from the DEM energy spectra. 47

9 4.5 The breakage distribution function The information about the probability of breakage considered in section 4.4 is of little use without the knowledge of the extent of breakage that is caused by the impact events. Thus some simple way of describing the breakage distribution function solely for the purpose of this work was established. The standard JKMRC drop test used for characterising the particle size distribution function, which apparently ignores the effect of size, is also evaluated using two different size ranges Establishing an event breakage distribution function During the drop-weight test work discussed in section 4.4, the fraction size of the remaining biggest fragment was visually estimated whenever breakage occurred. Four discrete levels of describing breakage were used; A physical appearance showed that the remaining largest fragment was about 90% of the original parent size. B - physical appearance showed that the remaining largest fragment was about 70% of the original parent size. C - physical appearance showed that the remaining largest fragment was about 50% of the original parent size. D or Completely Broken (CB) - entire particle was broken up into small fragments. An illustration of this visual classification of breakage is shown in Figure 4.5 below: 48

10 Original Particle A B C D Figure 4-5 Illustration of the four levels of classifying breakage Event breakage distribution function results The breakage data is shown in the tables in Appendix C. From the data it is seen that, firstly, regardless of the energy input the expected chances of achieving class A, B or C level of breakage was close to the ratio 1:2:2 Secondly, the probability of getting complete breakage (P CB ) i.e. class D, increased with increased energy input (theoretically, excess energy initiates further re-breakage). The relationship between P CB and energy input showed great scatter for the different mass groups as shown in Figure 4.6 which is based on data extracted from the tables in Appendix C. 49

11 Expt data Model Equation (e- 0.38/(E/x) ) Probability of CB Suspected outliers Specific energy input (J/g) Figure 4-6 Relationship between specific energy and probability of complete breakage of a broken particle. The data in Figure 4-6 show a lot of scatter even when the suspected outliers are excluded. This is a common problem for this kind of test due to variability in particle strength. Use of larger samples would be recommended for future tests. In spite of the highlighted problem, a model with desirable features (it asymptotes at 1 and also does not go through the origin, a reflection of the minimum threshold energy required before any breakage can begin to occur) was proposed and is stated as follows: P CB e 0.38 /( E / X ) = (4.6) where E is the energy input and X is the mass of the particle in grams. 50

12 4.6 The JKMRC drop weight test A method developed by JKMRC (Napier-Munn et al 1996) for estimating the Breakage distribution function was used on two different size fractions (16mm and 45mm) Procedure A screen size fraction (-16mm +13.2mm) was divided into four identical samples. The individual particles from the first sample were each subjected to a fixed energy input level. Each particle was only impacted once. At the end of the test, all the fragments including unbroken particles were combined and screen analysed. The procedure was repeated for the other three samples at higher energy input levels. The procedure above was also repeated on the -45mm +38mm fraction Results The tables showing the product size distributions of the two sizes are in Appendix D. Figures 4-7 and 4-8 below show how specific energy input affects the product size distribution. 51

13 100 % Passing J/g 0.46J/g 0.86J/g 1.96J/g Particle size (mm) Figure 4-7 The effect of specific energy input on product size distribution of a mm fraction % Passing J/g 0.24J/g 0.42J/g 0.52J/g Particle Size (mm) Figure 4-8 The effect of specific energy input on product size distribution of a mm fraction 52

14 It can be seen from both figures above that the higher the energy input the finer the product. However to achieve similar levels of size reduction, the finer sample required higher specific energy inputs. The first three results for each size fraction are plotted on the same graph below for comparison. 100 % Passing size Fraction of parent size mm.16J/g mm.46J/g mm.86J/g mm.12J/g mm.24J/g mm.42J/g Figure 4-9 Comparison of effect of Specific energy input on product size distribution of the finer and the coarse fraction Discussion of the product distribution results It is clear from the two sets of results in section that the parent size factor cannot be ignored when estimating the product size distribution from specific energy input, an issue that has been raised constantly against the JKMRC usage of the t 10 model. This anomaly has been addressed in more recent JKMRC publications (Apelt 2002, Banini 2001) Development of a Breakage function distribution model From section the term E x0, a minimum threshold energy that must be absorbed by particle before the remaining balance can be consumed in fracture propagation, was introduced. It was proposed that instead of correlating the breakage distribution function to specific energy, a model that takes into account 53

15 the E x0 values be used. An E x0 based model for scale-up for energy input to any particle of a given mass was proposed as follows: E E E x0 x0 (4.7) The values based on this equation are superimposed on Figure 4.8 to show that this approach gives a better correlation than the specific energies and this is apparent in Figure 4.10 below as the proposed relative scale values apparently give correspondingly similar breakage size distributions for both Big and Small. % Passing size Fraction of parent size Relative Scale based on (E-E x0 )/E x0 factor mm mm mm mm mm mm mm mm Figure 4-10 Comparison of breakage distribution function as a function of Equation 4.7 for both big and small particles To derive a model that will describe the data presented in section 4.6.3, the widely accepted equation for representing the breakage distribution function (Klimpel and Austin 1977) shown below was used. X Φ X j i δ X + (1 Φ) X j 1 1 i β (4.8) The equation is however modified to take into account the relative scale-up factor discussed above and takes the following form when fitted to the two sets of data: 54

16 0.71 E E x Ex E E x E E x E X x i E + X x i e e (4.9) X j 1 X j 1 The results of fitting this equation to experimental data discussed in section are shown in Figures 4.10 and % Passing Expt 0.16J/g Expt 0.46J/g Expt 0.86J/g Expt 1.96J/g Model 0.16J/g Model 0.46J/g Model 0.86J/g Model 1.96J/g Fraction of parent size Figure 4-11 Results of fitting Equation 4.9 to breakage data of small particles ( mm) % Passing Fraction of parent size Expt 0.12J/g Expt 0.24J/g Expt 0.42J/g Expt 0.52J/g Model 0.12J/g Model 0.24J/g Model 0.42J/g Model 0.52J/g Figure 4-12 Results of fitting Equation 4.9 to breakage data of large particles ( mm) 55

17 It can be seen from the fitted data that the proposed model does follow the experimental trends despite failure to successfully match all the data. More dropweight tests involving a wide range of size fractions is recommended for future investigation of this model. This model can be combined with the model that predicts mass loss for a given grinding period if the average energy input to a sample is known. However the energy input levels to particles present in the mill are widely distributed as shown by the DEM outputs shown in section The models proposed by King and Bourgeois (1993) for computing the selection and distribution function using single particle fracture data and DEM spectral energy distribution would be more applicable. 4.7 Conclusion From the drop-weight tests, it has been possible to develop a model that gives an indication of the likelihood of fracture when an event or events occur where a particle is impacted with some specified amount of energy. A simple model for describing the progeny size distribution at each contact event that results in breakage has been proposed. 56