A Novel Process for the Study of Breakage Energy versus Particle Size

Transcription

1 Geomaterials, 2013, 3, Published Online July 2013 ( A Novel Proess for the Study of Breakage Energy versus Partile Size Elias Stamboliadis Tehnial University of Crete, Chania, Greee elistah@mred.tu.gr Reeived April 18, 2013; revised May 20, 2013; aepted May 28, 2013 Copyright 2013 Elias Stamboliadis. This is an open aess artile distributed under the Creative Commons Attribution Liense, whih permits unrestrited use, distribution, and reprodution in any medium, provided the original work is properly ited. ABSTRACT The energy size relationship is examined, either as the speifi energy required breaking a partiulate material from an initial size d 1 to a final size d 2, where d is usually the d 80 size, or as the speifi energy required to break a single partile. The present work uses the results obtained using a ontrolled frequeny entrifugal rusher to rush partiles of a predetermined size lass under different rotation frequenies related to the kineti energy of the partiles at the moment of rushing. The paper alulates the relationship between the rotation frequeny and the kineti energy of the partiles before rushing and examines the size distribution of the produts. The study results allow presenting the relationship between the kineti energy of the partiles and the mass of partiles produed below the initial size lass. The work also produes the optimum mathematial model that desribes this relationship among three proposed ones. Aording to this model one an alulate the energy required breaking half of the initial mass below the initial size lass and the orresponding speifi energy is appointed to the average size of the lass. The parameters of the mathematial model an be used to ompare the grindability of the different materials. The proess an be used as an alternative to the drop weight tehnique used so far for the study of the breakage energy of minerals and roks. Keywords: Centrifugal Crusher; Kineti Energy; Breakage Energy; Size-Energy Relationship 1. Introdution Comminution of minerals is an energy onsuming operation and is responsible for the main energy ost in mineral proessing plants. Modern mineral uses demand finely ground materials for building and hemial Industries and environmental appliations. The most modern appliation is the use of magnesium bearing minerals for the apture of arbon dioxide, P. Renforth et al. [1]. It is well known that the speifi energy, energy per unit mass, required to break a mineral partile inreases rapidly as the partile size is redued. Based on grinding data several theories of the energy-size relationship have been proposed and the most ommon ones are those of Rittinger [2], Bond [3], Kik [4], Charles [5], Stamboliadis [6] and Stamboliadis et al. [7], to name some of them. Aording to these theories the energy size relationship refers to the speifi energy e 1,2 required to grind a material from an initial size x 1 to a final size x 2 and is given by Equation (1), where x 1 and x 2 are not the sizes of a speifi partiles but the sreen sizes at whih a predetermined fration of the material passes. Usually, x is the sreen size at whih 80% of the material will pass. 1 1 (1) e 1,2 C x n x n 2 1 In Equation (1), C is a onstant and the differene between these theories is the value of the exponent n and the way it an be measured. This work refers mainly in the study of the breakage energy required to break single partiles and this onsists a different approah than the one mentioned above. The first experiments to determine the energy required to break a single partile were made using the drop weight tehnique. Aording to it a weight of mass M is allowed to fall on a mineral partile of mass m, from a height h. The initial potential energy E p of the falling weight is E p = M g h, where g = 9.81 m/s 2 is the aeleration of gravity. At the moment the weight strikes on the partile its potential energy has been transformed in to kineti E k = MV 2 /2, where V is the veloity obtained by the falling weight, obviously, E p = E k. Initially the energy was determined by measuring the height but in our days one an also measure the veloity at the moment of impat, thanks to tehnologial developments in high speed

2 E. STAMBOLIADIS 103 ameras [8]. The present work uses a new method to provide the energy required to break a partile. This is done using a entrifugal rusher that aelerates partiles on a rotating dis that esape from the dis with a kineti energy that depends on the rotation frequeny [9]. The partiles strike vertially on a speially designed wall and break. The size distribution of the daughter partiles depends on the strength of the material and the kineti energy of the initial partile at the moment of impat. Suh data have been presented reently by E. Stamboliadis et al. [10], but here the analysis is going a step further to provide a mathematial model to determine the energy-size relationship for breakage. The model one should look for is one that gives the mass fration of the feed material that breaks below the size lass of the feed as a funtion of the kineti energy that the feed partiles have obtained. The maximum fration that an break is unit (100%) and the model one should look for is unit model that varies from zero to one. The proposed model an be used to ompare the grindability of different types of roks and minerals using the parameters of the model. This work also provides the relationship between the rotation frequeny of the dis and the kineti energy that the partiles aquire when leaving the dis. As it will be shown below the speifi energy, energy per unit mass, of any partile leaving the dis is independent of its size and depends only on the rotation frequeny and the dis diameter that is standard for the partiular equipment used. The rotation frequeny is used as the parameter that influenes the kineti energy. Three different models have been tested to find the one that fits the results obtained. They all have a parameter ΔH x = kj/kg that indiates the speifi energy required to break a partile of size x. The inverse of this parameter k x = 1/ΔΗ x gives the breakage rate k x = kg/kj that shows the mass of the partiles of size x that break per unit of speifi energy provided. Eah model gives a urve that differs from the data obtained. The sum of the squares of the differenes between the measured and the alulated value gives the auray of the model. For eah ase one an vary the value of the parameter ΔΗ x, and hoose the one that gives the least value to the sum of squares. This sum is the best auray that a ertain model an provide. Comparing the sums of least squares for the three models tested one an selet the best one that gives the minimum sum of least squares. As will be explained later, the model hosen an help to answer the question, when does a partile break, and obviously the answer is a statisti one beause for pratial reasons the partiles tested are not all of equal mass or size but they have been hosen to belong to the same size lass that is as narrow as possible. Consequently when one speaks for the size of a partile he atually denotes the average size of the size lass and only when one refers to the size of a sreen below whih ertain partiles will pass the size is absolute. 2. The Crusher Used 2.1. Desription of the Crusher The equipment used is a loally made entrifugal rusher desribed in detail by D. Stamboliadis [9]. It onsists of a horizontal rotating dis, 500 mm in diameter, surrounded by a homoentri, ylindrial ell 900 mm in diameter. The dis rotation axis is vertial and is linearly and diretly onneted to the axis of an eletri motor through a obbler. The rotation frequeny of the motor and onsequently of the dis is ontrolled by an inverter in the range of 700 to 2500 rpm. Radialy on the dis there are two symmetri, vertial blades that oblige any partile on the dis to rotate. The partiles are introdued at the enter of the dis, through a vertial shaft, and are obliged to rotation by the radial blades. As a result of the rotation, a entrifugal fore ats on the partiles and drives them to the periphery of the dis along the blades. As the partiles move from the enter to the periphery of the dis their rotation veloity, whih is vertial to the radius, inreases ontinuously and so does the entrifugal fore that gives them a veloity on the diretion of the radius. At the moment the partiles reah the periphery of the dis they esape with the two veloity omponents that are vertial to eah other and equal in magnitude, as alulated below. Their resultant is the vetor sum of the two veloities and its diretion is at 45 degrees to the radius of the dis at the moment of esape. This means that the resultant veloity vetor is not vertial to the homoentri ell surrounding the rotating dis and the partiles will not rush on it at an angle of 90 degrees. In order to ensure that the partiles leaving the dis will rush on a surfae vertial to the diretion of their veloity the inner side of the surrounding ell is lined by blades of hard steel at an angle of 45 degrees to the radius. Figures 1 and 2 give an outside and an inside view of the rusher. Figure 1. External view of the rusher.

3 104 E. STAMBOLIADIS Figure 2. Internal view of the rusher Calulation of the Kineti Energy The alulation of the kineti energy of the partiles at the moment of impat has been desribed by D. Stamboliadis [9] and is as follows. Let R be the radius of the disk and N the rotation frequeny. Assume a partile of mass m been at a distane r from the enter of rotation. The peripheral veloity V p at this point it given by Equation (2): Vp 2π rn (2) A entrifugal fore F ats on the partile that is related to its peripheral veloity aording to Equation (3): 2 mv p F (3) r The entrifugal fore moves the partile to the perimeter with an aeleration alulated by Newton s law given by Equation (4) F m (4) Substituting (2) and (3) into (4) one obtains Equation (5) 2 π N 2 r (5) From the laws of motion one has the relationship between the entrifugal veloity V, the time t and the entrifugal aeleration given by Equation (6), as well as, the relationship between the entrifugal veloity, the time and the radius given by Equation (7). dv (6) dt dr V dt (7) Equating and deleting dt from (6) and (7) one has Equation (8) dv dr or V dv dr (8) V Substituting (5) into (8) one has the differential Equa- tion (9) that relates the entrifugal veloity to the distane of the partile from the enter of the rotation. 2 V dv 2π N r dr (9) The integration of (9) gives Equation (10) V 2π Nr C (10) For r = 0 then V = 0 and onsequently C = 0. At the moment when the partile esapes from the dis r = R and the entrifugal radial veloity is given by (11). V 2π R N (11) At the same moment the peripheral veloity is given by Equation (12) as is equal but vertial to the entrifugal veloity. Vp 2π R N (12) The vetor sum of these two veloities is the atual esaping veloity V that is alulated from Equation (13) V V V (13) p Taking into onsideration (11) and (12) the final veloity is given by Equation (14) and has a diretion of 45 relative to the radius of the dis at the moment of esape. V 2 2π R N or V 2 π DN (14) where D is the dis diameter D = 2R. The kineti energy E of a partile with veloity V is given by Equation (15) 1 2 E m V (15) 2 Substituting (14) into (15) the kineti energy of the partile at the esape point from the dis is given by Equation (6) E m2π R N 2 2 or E mπ DN (16) The speifi energy e = E/m is then given by Equation (16) and it is independent of the partile mass. N 2 2 e 2π R or eπ DN (17) Applying the above formulas to the present ase one alulates that for the partiular rusher, with a dis 500 mm in diameter, the frequeny required to ahieve a speifi energy e = 3600 (J/kg) or the same 1 (kwh/ton), whih a usual speifi energy required, is 2293 rpm and is independent of the size of the partile. This frequeny is within the apaity of the mahine manufatured. 3. Experimental 3.1. Materials Used Two different roks are studied, namely 1) mirorystal-

4 E. STAMBOLIADIS 105 line limestone from the operating quarry of Hordaki near Chania, in the island of Crete, Greee and 2) serpentine from the area of Mantoudi in the Island of Euboea, Greee. There are two reasons for the seletion of these mate- rials. The first is that limestone is more or less homogenous ompared to serpentine that is weathered and these roks are expeted to have different behavior regarding the energy size relationship. The seond reason is that limestone is used widely as a building material, while serpentine is a soure of MgO that is studied for the apture of arbon dioxide in environmental appliations [1]. In both ases the energy ost for size redution is very important. The mirosopi struture of the samples appears in polished setions presented in Figures 3 and 4 respe- tively. One an see that limestone is mirorystalline and the rystals are not distinguished giving a homogeneous appearane at the sale of the partiles tested. On the other hand serpentine rystals an be distinguished but they appear to be weathered and the spae between them onsists of the weathering produt that is expeted to be weaker than the healthy rystals Experimental Proedure The experimental proedure followed during the test work Figure 3. Limestone. is desribed by E. Stamboliadis et al. [4]. A quantity about 30 kg of eah material tested was rushed to minus 30 mm using a laboratory jaw rusher. The material is then lassified into the following size frations of very narrow size range ( mm), ( mm), (4-5.6 mm), (2-2.8 mm) and (1-1.4 mm). The geometri average size of eah size fration is alulated to be (18.93 mm), (9.47 mm), (4.73 mm), (2.37 mm) and (1.18 mm) respetively. Eah size fration is rushed in the entrifugal rusher at different rotation frequenies using 1 kg of the partiular feed size fration at a time. The frequenies used and the orresponding speifi energies are presented in Table 1. The rushed produt of eah test is olleted and lassified in size frations using the sreens 16, 8, 4, 2, 1, 0.5, 0.25, and mm. The mass distribution of eah produt is the umulative mass % finer than the orresponding sreen and is plotted versus the sreen size in the same figure for all the speifi energies used for the same feed fration. From suh a figure one an see the effet of the speifi energy to the size analysis of the produts. The results are also presented in a way that for the same feed size, it gives the mass fration of the material broken below the feed size as a funtion of the speifi energy. The results obtained from this kind of presentation are used to derive the mathematial model that fits them. This model gives the energy required to break eah feed fration size and ontains parameters that allow omparing the different materials tested The Results Obtained for Limestone The results obtained for eah feed fration of limestone tested at different rotation frequenies are tabulated in tables. Table 2 presents the data obtained for the fration mm of limestone. It is reminded that eah rotation frequeny orresponds to a ertain speifi energy as shown in Table 1. The top row of Table 2 shows the sreen size (mm), while the left olumn gives the speifi energy of the feed partiles (J/kg). The values presented in the table give the measured mass fration of the produt that passes through the orresponding sreen for the indiated speifi energy. The horizontal plot of the values of Table 2 gives Figure 5 that shows the mass fration of partiles finer than the sreen size for the different speifi energies. The Table 1. Experimental frequenies and orresponding speifi energies. rpm e = J/kg Figure 4. Serpentine. e = kwh/ton

5 106 E. STAMBOLIADIS T able 2. Limestone f eed mm, mass fration passing. Speifi energy Sreen size mm J/kg Figure 5. Mass finer versus sreen size. h igher the speifi energy the urves move to finer sizes. The vertial plot of Table 2 gives Figure 6 that presents the mass fration of the produt that passes the indiated sreen size, for all speifi energies applied. All urves tend to 1, whih is the maximum mass fration that an be produed below any size. Obviously the oarse partiles are produed at a higher rate than the finer ones and as shown in Figure 6 only a small fration of the fines is produed at the maximum speifi energy tested (4500 J/kg),whih atually is the limit of the mahine used. The fat that the urves of Figure 6 have a maximum indiates the type of mathematial equations that an be applied to desribe the phenomenon. The same type of urves, are obtained for all the feed frations tested for eah type of rok tested. Table 3 presents the results obtained for limestone from all size frations tested at different speifi energy inputs. The data give the mass fration of the produt that passes below the initial feed lass. As an example, for the feed fration (1-1.4) mm, the table shows the mass of the feed that passes through the (1) mm sreen for all the speifi energies applied. Figure 7 presents the mass fration of limestone broken below the feed lass tested, denoted by its average size, as a funtion of the speifi energy. Figure 8 presents the same results for all energy levels applied as a funtion of the average size of the feed lass. Both figures show, eah one in a different way, that for any feed Figure 6. Mass produed versus. Figure 7. Mass fration broken below the size lass versus speifi energy. Figure 8. Mass fration broken below the size lass versus lower size. size the mass fration of broken partiles inreases with

6 E. STAMBOLIADIS 107 Table 3. Limestone, mass fration produed below the size lass. Speifi energy Size lass mm J/kg speifi energy but it dereases with feed size. It is more diffiult to break the small partiles than oarser ones. 4. Derivation of the Model 4.1. Data Proessing The data obtained so far are further proessed mathematially in order to reveal a model that ould desribe them. As already mentioned, the mass fration of the produt at any test that is broken below its initial lass size is expeted to be a number between zero and unity, that is the ase when all the partiles produed pass below this size lass. Consequently the type of the funtion one should look for is a unit funtion. Suh a funtion should give the mass fration of the material of a given feed size that breaks as a funtion of the energy provided and should also vary from zero to one that is the maximum fration of the material that an be broken. It should also inorporate a parameter that depends on the oherene of the material and will show how it relates to the size of the material tested. Three types of suh funtions have been hosen that inorporate a parameter ΔΗ (J/kg) related to the energy required for breakage, as follows. Type M-B B x x exp Type EXP Bx 1exp H x (18) (19) Type LAN B x (20) x Equation type M-B (18) is atually the Maxwell- Boltzman distribution law and an be found in textbooks of physial hemistry, Glasstone [11]. The same equation has also been used to format a omminution theory [12]. Equation EXP (18) is the exponential equation derived when assuming that the rate of depletion of a mass is proportional to the remaining mass. The mathematial derivation an be found in textbooks of differential equation [13]. This kind of model is found to explain grinding data [14]. Finally, equation LAN is the one used by Langmuir to desribe the volume V of a gases hemisorbed as a monolayer on a solid surfae, as referred by Shaw [15]. This equation is also referred homographi and is used to desribe phenomena that tend to a maximum. The form of eah equation is presented in Figures 9 and 10 as a funtion of the speifi energy ε, for ΔΗ values 10 and 100 J/kg respetively. For low ΔΗ, Figure 9, whih means that the partiles break easily, all models give a quik breakage rate. The EXP is faster and the M-B is the slowest. For high ΔΗ, Figure 10, whih means that the partiles are diffiult to break, all models give a slow breakage rate. Again the EXP is faster and the M-B is the slowest showing a delay at low values of the speifi energy ε provided. Eah model is ompare d with the atual results obtained for all feed sizes at the atual values of speifi energy applied. For eah test an estimated value of ΔΗ is seleted and the alulated mass fration predited f al is ompared to the mass fration f meas atually measured at the energy level i provided. One alulates the square of the differene between them f f 2 and finds the sum of squares f 2 al fmeas i n 1 al meas i for all n energy levels applied. Changing the values of ΔΗ one an find the value that gives the least sum of squares and assign it to the size lass of the feed tested. A plot of the sums of least squares versus th e average size of the lass tested is presented in Figure 11 for limestone. The results vary aording to the size and one an take the average sums of least squares for all sizes are plot them in Figure 12 where one an see that the LAN model gives the least deviation from the experimental data. Similar results are obtained for serpentine presented in

7 108 E. STAMBOLIADIS Figures 13 and 14. Here again the LAN model gives the least deviation from the experimental data. In Figure 15 it is easy to ompare the alulated values of the mass fration produed for all sizes to the obtained ones versus the energy for limestone. The orresponding results for serpentine are plotted in Figure 16. These figures show that the agreement of the model seleted to the measured data is satisfatory and there for will used to study the relationship of the speifi breakage energy to the size of the partiles broken. Figure 9. Mass fration broken for ΔΗ value 10 J/kg The Relationship of Energy versus Size Having established the mathematial model that desribes the experimental data one an use it to onlude the speifi energy ε required to break a partile. It is Figure 10. Mass fration broken for ΔΗ value 100 J/kg. Figure 13. Sums of least squares. Figure 11. Sums of least squares. Figure 14. Average sums of least squares. Figure 12. Average sums of least squares. Figure 15. Mass fration broken.

8 E. STAMBOLIADIS 109 Table 4. Energy-size data. lass size mm ΔΗ x J/kg Limestone Serpentine Figure 16. Mass fration broken. understood that the feed partiles are not individual partiles of a partiular size but rather partiles that belong to the same size lass as explained above in the experimental proedure. Consequently the size x of the lass is the geometri average of the x min and x max that is x xmin xmax. The next step is to define the speifi energy ε x required to break the average partile of size x. The data obtained shows the mass fration B x of the size lass that breaks below the minimum size as a funtion of ε presented in Figures 15 and 16. This problem is solved by defining the parameter ΔΗ x, alulated for eah feed size, as the speifi energy ex required breaking the average partile x of the lass. In other words the speifi energy to break a partile of size x is e x = ΔΗ x. From the same Equation (19) it is obvious that the units of ΔΗ x are (J/kg) and at the time e x = ΔΗ x it is alulated that B x = 0.5 meaning that any size lass breaks when half of its mass breaks below this lass. The alulated values of ΔΗ x orresponding to eah average size lass both for limestone and serpentine are given in Table 4 and presented in log-log sale on Figure 17. The relationship of the speifi energy for breakage versus average partile size for limestone is presented by the equation below Hx 1248 x (21) while for serpentine it is presented by the following one 5. Disussion and Conlusions Hx 6188 x (22) This paper has presented the design harateristis of a entrifugal rusher that aelerates partiles and gives them a kineti energy that an be ontrolled by the rotation frequeny of its dis. Its is alulated that the speifi energy of the partiles, at the moment they esape from the dis, hit the opposite wall and break, depends on the dis diameter that is kept onstant during the test work as well as on the rotation frequeny that an be adjusted on will. When partiles of a ertain size lass break they reate small daughter partiles and their size Figure 17. Speifi energy for breakage. distribution depends on the speifi energy of the parent partile. For the purpose of the test work the feed partiles to be broken are lassified in size lasses having a x max /x min ratio equal to 2. This work onentrates on the rate that the partiles belonging to the same size lass are broken to finer partiles and pass the x min size. Three different mathematial models are tested to desribe the rate of breakage for eah one of the different size lasses tested for limestone and serpentine. The model that gives the best fit is the one given by Equation (19) that inorporates a parameter ΔΗx whih an be related to the speifi energy (J/kg) required to break a partile of size x. The opposite of ΔΗ x is a parameter k x that gives the rate of breakage k x = 1/ΔΗ x (kg/j). Equation (19) gives the mass fration of the size lass with average size x xmin xmax that breaks below the lass and it is a pure number with no units. Aording to Equation (19) when the speifi energy ε of the partiles to be broken equals ε = ΔΗ x half of their mass will pass below the partiular size lass. This speifi energy ΔΗ x is defined as the energy required to break the partile with the average size of the lass. The plot of the speifi energies required for breakage versus the partile size gives the relationship of speifi energy versus size that is a power funtion of the form a x b. As expeted the speifi energy required to break a partile inreases as its size dereases. The oeffiient a measured in J/kg as well as the exponent b, whih is a

9 110 E. STAMBOLIADIS number, vary with the rok type tested. For limestone that is shown to be mirorystalline and more homogeneous, at the size ranges tested, the exponent is found to be ( 0.57), while for serpentine that seems to be marorystalline and also weathered the exponent equals ( 1.47). Coeffiient a for limestone is 1250 J/kg, while for serpentine it is 6190 J/kg. These parameters an be used to haraterize the minerals and roks and lassify them aording to their ohesion that is expressed by oeffiient a = J/kg as well as by their homogeneity expressed by the exponent b, whih is a number. As a result of this onsideration one ould say that the small healthy rystals of serpentine are harder to break than the equivalent ones of limestone. At larger sizes serpentine partiles, onsisting of more than one rystal onneted with weathered material, they break easier than the homogeneous partiles of limestone. REFERENCES [1] P. Renforth and T. Kruger, Coupling Mineral Carbonation and Oean Liming, Energy & Fuels, [2] P. R. Rittinger, Learning Book of Mineral Proessing, Tehnial University of Berlin, Berlin, [3] F. C. Bond, The Third Theory of Comminution, Transations of the Amerian Institute of Mining, Metallurgial and Petroleum Engineers, Vol. 193, 1952, pp [4] F. Kik, The Low of Proportional Resistanes and Its Appliations, Arthur Felix, Leipzig, [5] R. J. Charles, Energy-Size Redution Relationships in Comminution, Transations of the Amerian Institute of Mining, Meta llurgial and Petroleum Engineers, Vol. 208, 1957, pp [6] E. T. Stamboliadis, Impat Crushing Approah to the Relationship of Energy and Partile Size in Comminution, European Journal of Mineral Proessing & Environmental Protetio, Vol. 3, No. 2, 2003, pp [7] E. Stamboliadis, S. Emmanouilidis and E. Petrakis, A New Approah to the Calulation of Work Index and the Potential Energy of a Partiulate Material, Geo Materials, Vol. 1, No. 2, 2011, Artile ID: [8] L. M. Tavares and R. P. King, Single Partile Frature under Impat Load, International Journal of Mineral Proessing, Vol. 54, No. 1, 1998, pp [9] D. Stamboliadis, Design of a Centrifugal Crusher for Minerals, Based on the Design of Strutures, Undergraduate Thesis, Tehnologial Eduation Institute of Piraeus, Piraeus, (in Greek) [10] E. Stambloiadis, D. Stamboliadis, K. Kiskira and A. C. Emejulu, Crushing of Mineral Partiles by Control of Their Kineti Energy, Sientifi Journal of Riga Tehnial University, Series 1, Material Siene and Applied Chemistry, RTU Press, Riga, 2013 [11] S. Glasstone, Textbook of Physial Chemistry, 2nd Edition, Mamillan, London, [12] E. Stamboliadis, The Theory of Surfae Potential Energy, Minimum Partile Size and Non-Fratal Nature of Fragmentation, Canadian Metallurgial Quarterly, Vol. 49, No. 2, 2012, pp [13] N. Piskounov, Differential and Integral Calulus, 2nd Edition, MIR, Mosou, [14] E. T. Stamboliadis and V. J. Gaganis, Appliation of Computer Assisted Mathematial Analysis for the Interpretation of Mineral Grinding Data, 1st International Conferene From Sientifi Computing to Computational Engineering Athens, 8-10 September [15] D. J. Shaw, Introdution to Colloid and Surfae Chemistry Surfae Chemistry, 2nd Edition, Butterworths, London, 1970.