- Email: [email protected]
New population balance model for predicting particle size evolution in compression grinding
Minerals Engineering xxx (2015) xxx–xxx
Contents lists available at ScienceDirect
Minerals Engineering journal homepage: www.elsevier.com/locate/mineng
New population balance model for predicting particle size evolution in compression grinding V.P.B. Esnault a,⇑, H. Zhou b, D. Heitzmann a a b
Lafarge Centre de Recherche, 95 rue de Montmurier, 38290 Saint Quentin-Fallavier, France Hosokawa Alpine Aktiengesellschaft, Peter-Dörfler-Strasse 13-25, 86199 Augsburg, Germany
a r t i c l e
i n f o
Article history: Received 28 May 2014 Revised 19 December 2014 Accepted 22 December 2014 Available online xxxx Keywords: Grinding Comminution Process modeling HPGR PBM
a b s t r a c t Population Balance Models (PBM) are widely used to predict the evolution of the particle size distribution during various grinding processes, such as ball milling. They represent breakage through the definition of particle destruction and fragments generation rates. Their application to compression grinding (HPGR, vertical mills. . .) has been limited, due to the complexity of interactions between particles of different sizes. In this work, we present a new PBM approach for compression grinding. Complex interactions between size classes are represented in a simplified manner by making particle destruction and fragment generation depend on the bed porosity. Model is tested by confrontation to an extensive collection of experimental results on a piston-die cell, on three different materials (cement clinker, limestone, and quartz). When properly calibrated with preliminary tests, the model is able to predict the evolution of the particle size with accuracy, for any starting grain size distribution and any load. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Compression grinding equipments, such as high pressure grinding rolls (HPGR) or vertical mills, have been now in service for several decades, following the invention of the HPGR by Schönert, 1988. HPGR, vertical mills, but also Horomill (Cordonnier, 1994), despite their differences in terms of geometry, are all based on the same physical principle: the relatively slow compression of a granular bed. Such equipments are now operated worldwide in the mineral and cement industry (Kowatra, 2004). Those mills are reported to save up to 40% of the necessary energy input as compared to ball mill grinding, and thus represent a huge technological improvement, despite remaining process control problems for some applications, most notably with fine powders (Schönert, 1988; Fuerstenau and Abouzeid, 2002; Fuerstenau and Abouzeid, 2007; Wang and Forrsberg, 2003; Aydogan et al., 2006; Musa and Morrison, 2009). We focus here on a well-explored laboratory test to represent the compression grinding process, the piston-die cell. In this test, a small sample of materials (100 g in our case) is put in a cylindrical chamber. A piston is then forced on the materials with a press, to reach levels of pressure in the material of the same magnitude of
⇑ Corresponding author. Tel.: +33 4 74 82 80 97. E-mail address: [email protected] (V.P.B. Esnault).
what is encountered in the industry (up to 100–200 MPa). The test has been extensively used in the literature to model the more complex industrial setting (for example (Aziz and Schönert, 1980; Fuerstenau et al., 1996), or more recently (Oettel et al., 2001; Hosten and Cimilli, 2009 or Dundar et al., 2013)). Most recently, Kalala et al., 2011 address several aspects of HPGR/compression cell equivalence, such as energy draw or mimicking in the cell side effects on HPGR. The object of this communication is to present a new, semiempirical model to represent the evolution of the particle size distribution at the material level during the compression grinding process. More precisely, we exploit the alleged regularity of the stress and strain field in the piston-die cell to observe the behavior of an elementary unit of material, and we want to correlate data such as stress, deformation or relative density with fragmentation level. Such a tool would be a key ingredient for detailed modeling of compression grinding equipments, using techniques such as DEM modeling, for instance. For this, we rely on the use of a Population Balance Model (PBM), a type of model where particle size distributions are divided into size classes. Fragmentation is then represented through the transfer of mass between those size classes. PBM has been extensively used to represent ball milling, since the seminal works of Austin and Luckie, 1972, Whiten, 1974. Reviews on the application of this type of model in this field of application can be found in Benzer, 2000 or Hashim, 2004.
http://dx.doi.org/10.1016/j.mineng.2014.12.036 0892-6875/Ó 2015 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Esnault, V.P.B., et al. New population balance model for predicting particle size evolution in compression grinding. Miner. Eng. (2015), http://dx.doi.org/10.1016/j.mineng.2014.12.036
2
V.P.B. Esnault et al. / Minerals Engineering xxx (2015) xxx–xxx
Given the development of the usage of compression technologies in the industry on one hand, and the wide usage of PBM in the modeling of ball mill circuits on the other, it is surprising that very little attempts have been made to apply PBM to compression grinding. In ball milling, the modeling is greatly facilitated by the possibility to represent the process through linear relations. Breakage rates, for instance, can be considered as constant functions of time, unaffected by the evolution of the grain size distribution (at least in a wide range of fineness). This is no longer possible in the case of compression grinding. First, there is no obvious definition of ‘‘time’’, in compression grinding (should we consider the evolution of the applied load? the increase of material density?. . .). Even more important, the breakage behavior of the particles is deeply affected by the granular environment (contact number, bed density. . .). Any attempt to represent the breakage of the particles in compression through PBM is bound to take into account those non-linearities. Such an attempt to formulate a general non-linear PBM model, taking into account particle interactions, is made by Bilgili et al. in Bilgili et al., 2006. It is applied to compression grinding in Bilgili and Capece, 2012. The general model is very complex, and to apply it to compression grinding, the authors need to apply disputable simplifications (for instance, they neglect secondary breakage, i.e. the breakage of particles that themselves come from an initial breakage event). Even in this simplified form, the model seems to have had limited application in practical cases. Most recently, Dundar et al., in Dundar et al., 2013, used a piston-die test to calibrate a PBM for HPGR crushing. The model relies on important simplifications of the actual process, but is simple to implement and is the first of its type to actually model the real industrial process. Incidentally, it brings elements in favor of the modeling of compression grinding through the piston-die test. Another attempt to represent the evolution of particle size, though not through the PBM approach, has been developed by Schönert and his team in Liu and Schönert, 1996. The model is based on an ‘‘energy split function’’ that divides the total energy input between size classes. The particle size evolution in each class is then considered to be similar as the one observed in a monodisperse test for the same energetical input. Attempts have also been made to apply PBM directly at the process level. Models then do not necessarily represent precisely what actually happen to given sample of material as a function of load, but the comminution effect of the industrial device taken as a whole. Fuerstenau et al., 1991 developed a method to adapt PBM models in the case of HPGR operation. Energy input is used as the ‘‘time’’ variable, and non-linearities are taken into account by introducing a ‘‘retardation effect’’ with increasing pressure, illustrating increased dissipation rate with increasing pressure. The model successfully reproduces the experimental results. More recently, Hinde and Kalala, 2009 proposed a broader correlation, still for HPGR, between starting and final granulometries, using a general law exploiting the energy input to describe the amount of fragmentation provided by the machine. Such models are certainly useful tools in terms of process control, as they allow to represent the behavior of a given equipment (here, mostly HPGR) as a function of operating parameters (power draw, for instance). However, they are not designed to precisely represent the mill heterogeneities and their potential influence on the global process. This difference is most crucial when considering more complicated geometries than the classical HPGR system, such as in vertical mills or Horomills. One remaining difficulties with those existing models, both at the material and at the process level, is that they cannot be considered as fully predictive. They do reproduce well the experimental results, sometimes up to the final industrial process. But some
parameters need to be numerically adjusted for every test, and prediction of results under a new set of conditions (pressure and starting size range) is not possible. The model can then considered at best as a numerical description of what is happening during the test. The present model focuses on the preliminary calibration for a given material, on a given set of experiments. As we see it later in this presentation, once the calibration is done, the model can safely be applied to this material, for any starting granulometry, for any applied pressure. We believe this predictive power is one of the major characteristics for such a model to be one day applied in a broader context. A predictive model is also a condition if we want to provide a full numerical modeling of compression grinding. Material flow in roller compaction or HPGR grinding can for instance already be modeled through Finite Element Modeling (FEM) simulation, originally in 2D (Cunningham, 2005; Michrafy et al., 2011), and now in 3D (Cunningham et al., 2010; Michrafy et al., 2011). PBM model would allow to couple the informations from such FEM modeling with comminution, constituting a powerful tool for the understanding of compression grinding equipments of any geometry. 2. The non-linear PBM model In PBM, the evolution of grain size distribution is modeled through the transfer of material between size classes, through a linear relationship first formulated by Epstein, 1947, and first applied to grinding (of coal) by Broadbent and Calcott, 1956. In a matrix form, the particle size distribution at instant n þ 1; F nþ1 depends of the size distribution at n; F n through relation (1):
F nþ1 ¼ T n F n
ð1Þ
With T n being defined through (2):
Tn ¼ I S þ B S
ð2Þ
I is the identity matrix. S is a diagonal matrix, called selection matrix. Its ii term describe the probability that a particle from class i gets broken between n and n þ 1. B is a triangular matrix, call grinding or repartition matrix. Its term ij describe the proportion of fragments of size j when a particle of class i is broken. So when we consider the terms in Eq. (2), we see that the first term characterizes mass conservation, the second the disappearance of a particle from its size class when broken, the third the repartition of the fragments between the smaller size classes. Baxter et al., in Baxter et al., 2004, formulated a general way to include non-linearities in this relationship (3):
F nþ1 ¼ T n ðF n Þ F n
ð3Þ
Of course making the matrix T n a function of the full size distribution F n would need to a very complex formulation. In this model, our essential assumption is to simplify expression (3) by assuming T n is a function of the granular material porosity, / (or symmetrically, the relative density 1 /). In other words, we assume that the complex effect of the granular environment (the number of contacts, the distribution of the load pattern. . .), can be summarized through the packing density alone. Based on such an assumption, the selection matrix S can be directly determined from experimental tests. Indeed, the disappearance rate of the top size class as a function of / can be calculated from a set of tests on originally monodisperse size fractions. A first estimation for the repartition matrix B can also be obtained experimentally, by considering tests on monodisperse fractions, for a low applied pressure. In this case, the extent of
Please cite this article in press as: Esnault, V.P.B., et al. New population balance model for predicting particle size evolution in compression grinding. Miner. Eng. (2015), http://dx.doi.org/10.1016/j.mineng.2014.12.036
V.P.B. Esnault et al. / Minerals Engineering xxx (2015) xxx–xxx
3
secondary breakage (the re-breakage of fragments) can be neglected. Fragments can then all be assumed to originate from the parent particles. However, it is not possible to monitor directly the evolution of B as a function of /, due to the presence of re-breakage. We then define two variants of the model according to the way we deal with this dependence: In the first one, we simply assume that B is independent from /. The model can then be entirely be calibrated from a given set of tests on monodisperse fractions, without using any parameter fitting. In the second, we adjust B as a function of /. To determine the values of B, we assume that comminution follows Rittinger law, i.e. surface creation is proportional to energy input. More precisely, we define SSðFðxÞÞ the specific surface of size distribution FðxÞ (4):
SSðFðxÞÞ ¼
Z 0
1
6 @FðxÞ dx qx @x
ð4Þ
with q the specific mass of the material. We assume the following relationship (5):
@SSðFðxÞÞ @W SSðF nþ1 ðxÞÞ SSðF n ðxÞÞ ¼ ð/ /n Þ @W @/ nþ1
ð5Þ
The left hand side term is the surface creation between time step n and n þ 1. @SSðFðxÞ is the surface creation as a function of energy input @W W. It is assumed constant and can be measured on calibration tests for a given material. @W can be easily calculated if the pressure P and @/ the piston displacement x are monitored during the test. W is calculated using (6):
W¼
Z
xfin
SPðxÞxdx
ð6Þ
0
with S being the die section. / is also monitored continuously using the variation of the sample thickness. Breakage functions Bi;/ ðxÞ are implemented in the model using a combination of two Rosin– Rammler distributions, following the mathematical form (7):
Bi;/ ðxÞ ¼ 1 pi eðx=k1;i Þ
k1;i
ð1 pi Þeðx=k2;i Þ
k2;i
kinit 1;i ;
ð7Þ
kinit 2;i ;
init k1;i )
Initial values of the parameters (pinit are adjusted to i ; fit the observed distributions. In order to respect (5), we modify Bi;/ ðxÞ by considering parameters pi ; k1;i ; k2;i ; k1;i et k2;i can be established from their initial values using the following transformation (8):
pi ¼ pinit i init
k1;i ; k2;i ¼ ln ð10Þ
SSðBi;/ ðxÞÞ ¼ aSSðBinit i;/ ðxÞÞ
ð9Þ
að/Þ is then representative of the way energy is exploited in fragment generation as compaction progresses. We assume this function does only depend of the granular environment through the porosity, and not of the considered size class. Determining Bi;/ ðxÞ can be done by the following procedure: Determine the value of að/Þ such as (5) is respected. Calculate the corresponding bi and corresponding Bi;/ ðxÞ distributions. Fig. 1 gives an example of a Bi;/ ðxÞ function (for clinker, size class 1–1.25 mm), for a ¼ 1 (i.e. the measured value at low pressure), and a ¼ 5, close to the highest value implemented in the simulations. The a ¼ 5 case corresponds to a situation where the fragment generation is broader, with more fine particles and more surface generation. The model can then be calibrated to a given set of data on monodisperse materials. The selection function Sð/Þ is directly measured from the disparition rate of the top size particles in tests on monodisperse particles. Bð/Þ is established from their initial values measured at low pressure, and modified according the energy-surface relationship (5). The model can also be tested using experiments with the same material, but different starting granulometries or final pressures, to assess its prediction power. In this sense, it is a predictive model, as no further adjustments are made to the model in order to fit those new experimental results.
init
k1;i ; k2;i ¼ bi ð/Þk1;i ; bi ð/Þk2;i 1 1 kinit k1;i 1;i
Fig. 1. Fragment distribution function Bi;/ ðxÞ for clinker 1–1.25 mm, for a ¼ 1 and a ¼ 5.
kinit 1;i ;
ln ð10Þ
1 1 kinit k2;i 2;i
3. Experimental procedure and model calibration
kinit 2;i
ð8Þ
k1;i et k2;i are the two shape parameters of theRosin–Rammler distributions. Multiplying them by a factor bi ð/Þ then allows to widen (if bi ð/Þ < 1) or to narrow (if bi ð/Þ > 1) the particle size distribution of the formed fragments. The operation on k1;i et k2;i aims at conserving the d90 during the transformation (by definition, d90 is the values as that 90% of particles in mass are smaller than d90 ). The transformation is then a mean to modulate the Bi;/ ðxÞ width through its shape parameters, without modifying its top value. It provides a way to adapt empirically the breakage function so the created surface corresponds to the energy input provided by the piston. In order to characterize the modification of the distributions, we define að/Þ, such as (9):
We used for the tests the piston-die cell represented on Fig. 2. Its diameter is 8 cm, and the initial samples thicknesses were around 1.5 mm, a big enough sample size relatively to grain to ignore side effects. Pressure was applied through a 3000 kN capacity press for the highest pressures applied (above 20 MPa), and a 100 kN press for lower pressures. This bigger press would allow for maximum working pressure of 600 MPa, but in practice we limited ourselves to 280 MPa, already far above the pressures regularly encountered in the industry. Applied load was monitored through a force sensor, and bed strain through a LVDT sensor measuring the progression of the piston. Bed final and initial porosity were evaluated by measuring the piston height with a caliper.
Please cite this article in press as: Esnault, V.P.B., et al. New population balance model for predicting particle size evolution in compression grinding. Miner. Eng. (2015), http://dx.doi.org/10.1016/j.mineng.2014.12.036
4
V.P.B. Esnault et al. / Minerals Engineering xxx (2015) xxx–xxx
Fig. 2. Illustration of a test using the piston-die cell.
We considered three different materials in this study: a cement clinker, a quartz, and a limestone sand. For each materials we conducted series of tests on different monodisperse size classes (from 100 lm to 2.5 mm), at different applied pressure (from 1.5 to 280 MPa), in order to map the behavior of the material. Data points collected are summarized in Table 1. The data collection is voluntarily more extensive on cement clinker (55 points), then on quartz and limestone (16 points). We aimed at validating the model in optimal conditions on clinker, then trying to assess its behavior for a cruder, easier to acquire data set on limestone and quartz. Those materials also present very different behavior. Considering Moh’s hardness, for instance, the quartz can be considered as very hard and brittle (with a Moh’s hardness of 7), while limestone is much softer (Moh’s hardness of 3). Clinker presents intermediate properties (Moh’s hardness of 5). Each test was conducted on a 100 g sample. Tests were reproduced three times from each point in order to limit data dispersion. After the compression, the material was dispersed manually (at high pressure, the compression cell produce solid, compact flakes). Particles were sieved at the original size class bottom size to measure the disappearance rate of the initial particles. Fragments particle size distribution was then measured using laser granulometry and specific surface of the grain size distribution is determined according to expression 4. It is to be noted that this expression is very sensitive to small variation of the size distribution in the small size range. Small measurement mistakes on the finest fraction can lead to sizable variation of the calculated surface area. In our case, this problem is limited as we only consider relatively coarse size distributions (negligible amount of particles below 3 lm), and as we consider only materials for which optical properties are well known and for which the technique has been calibrated. The model was calibrated from this data set in accordance with the model described in the preceding chapter. Selection matrix S was calculated form the disappearance rate of the top size particle as a function of density (Fig. 3). Fig. 4 displays the grain size distribution of fragment generated (i.e. excluding the original size class), at different pressures, for clinker (1–1.25 mm). A quick look does suggest that important re-breakage is going on, as those
Fig. 3. Percentage of broken particles as a function of relative density 1 / (clinker, 500–600 lm size class). Selection function as function of density is calculated from the average fitted line.
Fig. 4. Grain size distribution of the fragments generated, i.e. excluding the particles of the starting class size, at different pressures (clinker, 1–1.25 mm size class).
distribution keep going finer, making it unlikely particles have been generated in one generation of breakage only as implemented in Bilgili and Capece, 2012. An initial estimation of the repartition matrix B was made from the fragment size distribution at the lowest pressure tested (1.5 MPa). In between size classes where experimental results were obtained, intermediate properties were established using interpolation. Integration of the size distribution according to formula (4) gives the evolution of the specific surface with pressure in a given
Table 1 Calibration tests realized on monodisperse samples. Material
Size classes
Applied pressure
Clinker
2–2.5 mm, 1–1.25 mm, 500–600 lm 250–300 lm, 100–125 lm
1.5, 3, 5, 8.8, 14, 23, 33.8, 55, 100, 166, 280 MPa 1.5, 3, 5, 9, 14.2, 24, 36.4, 53, 90, 137, 250 MPa
Calcaire
1.25–1.6 mm, 600–800 lm, 315–400 lm, 125–160 lm
3, 15, 60, 180 MPa
Quartz
1.25–1.6 mm, 600–800 lm, 315–400 lm, 125–160 lm
3, 15, 60, 180 MPa
Please cite this article in press as: Esnault, V.P.B., et al. New population balance model for predicting particle size evolution in compression grinding. Miner. Eng. (2015), http://dx.doi.org/10.1016/j.mineng.2014.12.036
5
V.P.B. Esnault et al. / Minerals Engineering xxx (2015) xxx–xxx
Fig. 5. Applied pressure as a function of relative density (1 /).
Fig. 8. Specific surface as a function of energy input, with linear regression, case of quartz, 1.25–1.6 mm.
Fig. 6. Total mechanical energy input as a function of relative density (1 /). Fig. 9. Specific surface as a function of energy input, with linear regression, case of limestone, 1.25–1.6 mm.
Table 2 Calculated value of
for the different calibration tests.
Material
Size fraction
@SS @W
Clinker
2–2.5 mm 1–1.25 mm 500–600 lm 250–300 lm 100–125 lm Average
0.00243 0.00222 0.00171 0.00225 0.00340 0.00240
Quartz
1.25–1.6 mm 600–800 lm 315–400 lm 125–160 lm Average
0.00243 0.00307 0.00225 0.00483 0.00342
Limestone
1.25–1.6 mm 600–800 lm 315–400 lm 125–160 lm Average
0.0067 0.00696 0.00765 Not linear 0.00710
Fig. 7. Specific surface as a function of energy input, with linear regression, case of clinker, 1–1.25 mm.
size class. Energy input is measured as function of density 1 / during the test. Figs. 5 and 6 represent respectively pressure and mechanical energy input as a function of relative density 1 /, in the case of clinker, 1–1.25 mm, 280 MPa. In order to establish the breakage function, it is necessary to @SS , the surface creation to energy ratio. We plot on Figs. 7– evaluate @W 9 specific surface of the sample as a function of energy input, for respectively clinker (1–1.25 mm), quartz (1.25–1.6 mm) and limestone (1.25–1.6 mm). Linearity of the data seems to depend very much of the nature of the material. For clinker and especially @SS quartz, the curves are remarkably linear and a value of @W can be practically established. For limestone, the point at the highest pressure is notably offline. The same phenomenon could be observed for all size classes tested. We attribute this to the ‘‘soft’’ nature of
@SS @W
(in m2 kg
1 1
J
)
the material: fragile breakage of the grains is dominated by phenomenons such as plastic deformation of the grains and reagglomeration at high pressure, leading to parasite energy consumption in the case of limestone. Quartz, by contrast, is very hard and grains continue to be essentially subjected to brittle fracture, even at high pressure. For limestone, the point at the highest pressure as systematically excluded of the simulations.
Please cite this article in press as: Esnault, V.P.B., et al. New population balance model for predicting particle size evolution in compression grinding. Miner. Eng. (2015), http://dx.doi.org/10.1016/j.mineng.2014.12.036
6
V.P.B. Esnault et al. / Minerals Engineering xxx (2015) xxx–xxx
@SS Table 2 summarizes the values obtained for @W for different size classes. Values can vary quite a bit from one size class to another, indicating possible differences in terms of breakage behavior according to the size of the particle. For each material, we use the average value. Those values should be considered with caution as they have nothing in common with the ones that can be expected in the industry. Mütze and Husemann, 2008 defend that loading speed influences the energy draw in a piston-die type of test. This is attributed to the dependance in the loading of phenomenons such as friction between grains. With loading speeds an order of magnitude faster in the industry then in the lab-scale tests, real systems dissipate more leading to very low energy consumption observed at lab-scale. This lower energy consumption in compression cell when compared to real HPGR process has also been observed experimentally and presented in Kalala et al., 2011. Any potential influence of the loading speed was neutralized on the tests by applying the same loading speed to all samples. In the numerical calculations, we considered a total of 40 size classes, from 2.1 to 2.5 mm for the bigger one, to a smaller, unbreakable 0–3 lm size class. Population in each size class is then calculated by finite difference from the initial porosity / to the desired one, using a unitary step of 0.001.
4. Results To assess the prediction power of the model, several tests were performed on mixes of different size classes of the original material. Each mix was pressed at 25 and 100 MPa, and the particle size was analyzed. The mixes tested are listed in Table 3. Like for the monodisperse calibration tests, each point is averaged on three tests to minimize data dispersion.
The result was then compared to the predicted grain size evolution by the model, between the initial and finally observed density. The quality of the prediction was assessed through the calculation of an error function e, define through relation (10):
R1
em ¼ R01 0
jF e ðxÞ F m ðxÞjdðln xÞ jF e ðxÞ F 0 ðxÞjdðln xÞ
ð10Þ
F 0 ðxÞ is the initial size distribution of the mix. F e ðxÞ is the size distribution observed experimentally after the test, and F m ðxÞ is the one predicted by the model. e then gives an estimation of the quality of the model predicted grain size modification compared to the actually observed one. We report in Table 4 the calculated error obtained on the different validation tests. Two values are given, the first one for the variant without dependence of B relatively to /, the second one including the variation of the breakage function considering the energy usage. A first result is that in the majority of the cases, the variable B does improve the quality of the prediction notably. In the case of the quartz, the fixed B even gives totally unacceptable results while the variable B allows for a reasonable prediction. This is most significant as the numerical optimization of the variable B was done on the calibration tests only, while those validation tests were done independently. It has also been concluded that error values are generally higher on small increments of pressure (between 25 and 100 MPa, or 0 and 25 MPa) then on the full range of prediction (0–100 MPa). This is an artifact linked to the definition of the error e: it is normalized by the particle size variation between the initial and the final state. If this variation is small, small imprecisions or errors lead to higher values of e.
Table 3 Prediction evaluation tests realized, on granular mixes. Material
Size mix
n of the mix
Tested pressure
Clinker
20% 1–1.25 mm, 80% 100–500 lm 20% 250–300 lm, 80% <100 lm
1.1 1.2
25 et 100 MPa 25 et 100 MPa
Limestone
20% 1.25–1.6 mm, 80% 315–630 lm 20% 600–800 lm, 80% 125–315 lm
2.1 2.2
25 et 100 MPa 25 et 100 MPa
Quartz
20% 1.25–1.6 lm, 80% 630–800 lm 20% 630–800 lm, 80% 315–630 lm
3.1 3.2
25 et 100 MPa 25 et 100 MPa
Table 4 Evaluation of the prediction error e on the different validation tests. Material
Mix number
Pressure span
Porosity span
Error, fixed B (%)
Error, variable B (%)
Clinker
1.1
0 ! 100 MPa 0 ! 25 MPa 25 ! 100 MPa 0 ! 100 MPa 0 ! 25 MPa 25 ! 100 MPa
0:5790 ! 0:2717 0:5790 ! 0:3859 0:3859 ! 0:2717 0:5465 ! 0:3031 0:5465 ! 0:3960 0:3960 ! 0:3031
18.60 7.81 43.16 22.73 23.24 38.82
7.64 7.19 16.78 20.24 23.74 20.17
0 ! 100 MPa 0 ! 25 MPa 25 ! 100 MPa 0 ! 100 MPa 0 ! 25 MPa 25 ! 100 MPa
0:4349 ! 0:1780 0:4349 ! 0:2542 0:2542 ! 0:1780 0:4353 ! 0:1765 0:4353 ! 0:2690 0:2690 ! 0:1765
34.73 17.00 74.45 47.99 35.45 72.30
11.59 23.01 20.44 25.66 27.88 28.59
0 ! 100 MPa 0 ! 25 MPa 25 ! 100 MPa 0 ! 100 MPa 0 ! 25 MPa 25 ! 100 MPa
0:4570 ! 0:2399 0:4570 ! 0:3352 0:3352 ! 0:2399 0:4624 ! 0:2584 0:4624 ! 0:3618 0:3618 ! 0:2584
65.13 62.13% 70.91 73.76 73.73 77.37
25.96 25.21 39.33 12.86 48.08 6.47
1.2
Limestone
2.1
2.2
Quartz
3.1
3.2
Please cite this article in press as: Esnault, V.P.B., et al. New population balance model for predicting particle size evolution in compression grinding. Miner. Eng. (2015), http://dx.doi.org/10.1016/j.mineng.2014.12.036
V.P.B. Esnault et al. / Minerals Engineering xxx (2015) xxx–xxx
Fig. 10. Starting, observed and predicted grain size distribution, clinker 20% 1– 1.25 mm, 80% 100–500 lm, 0–100 MPa.
Fig. 11. Starting, observed and predicted grain size distribution, clinker 20% 250–300 lm, 80% <100 lm, 0–100 MPa.
7
Fig. 13. Starting, observed and predicted grain size distribution, 20% 630–800 lm, 80% 315–630 lm, 0–100 MPa.
Fig. 14. Starting, observed and predicted grain size distribution, limestone 20% 1.25–1.6 mm, 80% 315–630 lm, 0–100 MPa.
the target. The prediction remains in the good order of magnitude, though, even in those not so favorable cases. Two explanations can be given for those differences in terms of prediction quality:
Fig. 12. Starting, observed and predicted grain size distribution, 20% 1.25–1.6 lm, 80% 630–800 lm, 0–100 MPa.
Figs. 10 (case 1.1 (‘‘coarse’’ clinker)) and 11 (case 1.2 (‘‘fine’’ clinker)) show that the prediction behavior of the model for clinker is very good, with only a slight lack of precision for the finer case. In the most favorable cases, the quality of the prediction is visually excellent, to the point that the prediction error is of the same magnitude of the experimental measurement errors. For limestone and quartz (Figs. 12–15), for which the calibration database is less extended, model prediction can be more off
It seems very important to feed the model with a proper set of data for the calibration. This concerns the number of experimental points (the extent of the data set on clinker was much larger), but also its range. In the case of the ‘‘fine’’ clinker (case 1.2), for instance, most of the particles were actually outside the calibration range where the bottom size is 100 lm. It is then not surprising to see that the model does not predict that well the behavior of those fine particles as it has not been calibrated to do so. Calibration in the fine range would require the constitution of fine, monodisperse samples, which can prove experimentally complicated. It is also likely that the nature of the material affects the quality of the prediction. The model specifically models fragile breakage of the grains, and is not designed to deal with other type of grain transformation, such as plastic deformation or reagglomeration. Limestone is a quite representative case. We could see on the calibration data and sample aspect that the behavior, especially at high pressures, was far from ideal fragile breakage. This could be a reason for the fact that the model exaggerates grinding: the model attributes all the energy to breakage and surface creation, which is no longer the case for high pressures. It also proves difficult to evaluate the particle
Please cite this article in press as: Esnault, V.P.B., et al. New population balance model for predicting particle size evolution in compression grinding. Miner. Eng. (2015), http://dx.doi.org/10.1016/j.mineng.2014.12.036
8
V.P.B. Esnault et al. / Minerals Engineering xxx (2015) xxx–xxx
Fig. 15. Starting, observed and predicted grain size distribution, limestone 20% 600–800 lm, 80% 125–315 lm, 0–100 MPa.
calibrated. Such an effort is likely to be of some interest only for mass produced, standard, and reasonably reproducible materials, such as cement clinker, which was the primary interest in this study. It also remains empirical, in the sense its quality can only be appreciated at the light of its prediction capacities. The physics behind those notions of selection and repartition remains largely unexplored in compression. Most of all, this model would require to be confronted to actual industrial data. The representativeness of the piston-die cell, for instance, remains an open question, especially if we want the model to be really predictive. This would require more experimental data and very good process control, as the evaluation of the bed porosity, in the system, is of primary importance. The current model could be applied directly to HPGR modeling through mixed modeling, with breakage functions being determined on the piston-die cell, and the link between surface creation and energy calculated on the bigger mill. Provided such a link with the actual industrial systems, we believe this model could be of some help for the understanding and the piloting of compression grinding techniques in the industrial world. Finally, the current model can represent the breakage inside an elementary unit of material inside a full numerical simulation of a compression grinding equipment. It seems particularly well suited to be coupled to a FEM model of a mill (where porosity and energy input are constantly evaluated everywhere), allowing to couple the densification of the material as it flows through the mill and the breakage of the grains. Such complete modeling would allow to characterize precisely the behavior of the mills, including for geometries more complex than in the HPGR case, such as in vertical mills and Horomill, and ultimately lead to better design for those equipments.
References Fig. 16. Function að/Þ, for clinker, for case 1.1 (coarse) and 1.2 (fine).
size in the highly agglomerated samples produced by the test, leading to characterization errors, both in the calibration data set and in the demonstration example. Finally, in Fig. 16, we plot alpha as a function of /, for case 1.1 and 1.2 (clinker). One can see that a > 1, a condition we have prescribed for the model to respect the breakage function measured at low pressure. We observe that a follows the same curve in the two cases, a good argument for the physical relevance of the notion. It would tend to indicate that grain breakage generates finer fragments at higher densities and higher pressure. This is coherent with the idea that grains tends to support higher stress values, with more contact points in dense systems, leading to fewer breakage, but more energetic and according to more complex fracture patterns.
5. Conclusion Very few models are available to predict the evolution of particle size distribution in compression grinding, and they cannot be considered as really predictive in the sense they could be calibrated for a given material on a data set, and then applied to any situation with this material. We believe that this aspect makes the present model suitable for application to industrial grinding devices. It also exploits tangible material parameters such as material density and energy input in an explicit manner. The model is still heavy to handle. It should be noted that it requires an extensive set of experimental data to be properly
Schönert, K., 1988. A first survey of grinding with high-compression roller mills. Int. J. Miner. Process. 22 (1-4), 401–412, recent Advances in Comminution. Cordonnier, A., 1994. A New Grinding Process: Horomill, 8th European Symposium on Comminution. Kowatra, S.K., 2004. Adv. Comminut., Soc. Min. Mettal. Explorat. Fuerstenau, D.W., Abouzeid, A.-Z.M., 2002. The energy efficiency of ball milling in comminution. Int. J. Miner. Process. 67 (1-4), 161–185. Fuerstenau, D.W., Abouzeid, A.-Z.M., 2007. The performance of the high pressure roll mill: effect of feed moisture. Int. J. Miner. Process. 82 (4). Wang, Y., Forrsberg, E., 2003. International overview and outlook on comminution technologies, report from Lulea Tekniska Universitet. Consulted online at (accessed 01.29.14). Aydogan, N.A., Ergün, L., Benzer, H., 2006. High pressure grinding rolls (HPGR) applications in the cement industry. Miner. Eng. 19 (2). Musa, F., Morrison, R., 2009. A more sustainable approach to assessing comminution efficiency. Miner. Eng. 22 (7-8), 593–601. Aziz, J., Schönert, K., 1980. Single particle size reduction and bed loading conditions for cement clinker fractions. Zement-Kalk-Gips 43 (2), 65–70. Fuerstenau, D., Gutsche, O., Kapur, P., 1996. Confined particle bed comminution under compressive loads. Int. J. Miner. Process. 44-45, 521–537. Oettel, W., Nguyen, A.-Q., Husemann, K., Bernhardt, C., 2001. Comminution in confined particle beds by single compressive load. Int. J. Miner. Process. 63 (1). Hosten, C., Cimilli, H., 2009. The effects of feed size distribution on confined-bed comminution of quartz and calcite in piston-die press. Int. J. Miner. Process. 91, 3–4. Dundar, H., Benzer, H., Aydogan, N., 2013. Application of population balance model to HPGR crushing. Miner. Eng. 50-51, 114–120. Kalala, J.T., Dong, H., Hinde, A.L. 2011. Using piston-die press to predict the breakage behaviour of HPGR. In: 5th International Conference Autogenous and SemiAutogenous Grinding. 25–28 September, British Columbia, Canada. Austin, L.G., Luckie, P.T., 1972. Methods for determination of breakage distribution parameters. Powder Technol. 5, 215–222. Whiten, W.J., 1974. A matrix theory of comminution machines. Chem. Eng. Sci. 29, 589–599. Benzer, H., 2000. Mathematical modeling of clinker grinding process, PhD thesis, University of Hacettepe. Hashim, S.F.S., 2004. Mathematical modeling of clinker grinding process, PhD thesis, Queensland University. Bilgili, E., Yepes, J., Scarlett, B., 2006. Formulation of a non-linear framework for population balance modeling of batch grinding: beyond first order kinetics. Chem. Eng. Sci. 61, 33–44.
Please cite this article in press as: Esnault, V.P.B., et al. New population balance model for predicting particle size evolution in compression grinding. Miner. Eng. (2015), http://dx.doi.org/10.1016/j.mineng.2014.12.036
V.P.B. Esnault et al. / Minerals Engineering xxx (2015) xxx–xxx Bilgili, E., Capece, M., 2012. A rigorous breakage matrix methodology for characterization of multi-particle interactions in dense-phase particle breakage. Chem. Eng. Res. Des. 90, 1177–1188. Liu, J., Schönert, K., 1996. Modelling of interparticle breakage. Int. J. Miner. Process. 44-45, 101–115. Fuerstenau, D.W., Shuckla, A., Kapur, P.C., 1991. Energy consumption and product size distributions in choke-fed high-compression roll mills. Int. J. Miner. Process. 32, 59–79. Hinde, A.L., Kalala, J.T., 2009. The application of a simplified approach to modelling tumbling mills, stirred media mills and HPGR’s. Miner. Eng. 22, 633–641. Cunningham, J.C., 2005. Experimental studies and modeling of the roller compaction of pharmaceutical powders, PhD thesis, Drexel University. Michrafy, A., Diarra, H., Dodds, J.A., Michrafy, M., Penazzi, L., 2011. Analysis of stress strain rate in roller compaction process. Powder Technol. 208.
9
Cunningham, J.C., Windstead, D., Zavaliangos, A., 2010. Understanding variation in roller compaction through finite element-based process modelling. Comput. Chem. Eng. 34, 1058–1071. Michrafy, A., Diarra, H., Dodds, J.A., Michrafy, M., 2011. Experimental and numerical analyses of homogeneity over strip width in compaction. Powder Technol. 206, 154–160. Epstein, B., 1947. The mathematical description of certain breakage mechanisms leading to the logarithmico-normal distribution. J. Franklin Inst. 244 (6), 471–476. Broadbent, S.R., Calcott, T.G., 1956. A matrix analysis of processes involving particle assemblies. Philos. Trans. Royal Soc. 249, 99–123. Baxter, J., ABu-Nahar, A., Tüzün, U., 2004. The breakage matrix approach to inadvertent particle degradation: dealing with intra-mixture interactions. Powder Technol. 143-144, 174–178. Mütze, T., Husemann, K., 2008. Compressive stress: effect of stress velocity on confined particle bed comminution. Chem. Eng. Res. Des. 86, 379–383.
Please cite this article in press as: Esnault, V.P.B., et al. New population balance model for predicting particle size evolution in compression grinding. Miner. Eng. (2015), http://dx.doi.org/10.1016/j.mineng.2014.12.036
Contents lists available at ScienceDirect
Minerals Engineering journal homepage: www.elsevier.com/locate/mineng
New population balance model for predicting particle size evolution in compression grinding V.P.B. Esnault a,⇑, H. Zhou b, D. Heitzmann a a b
Lafarge Centre de Recherche, 95 rue de Montmurier, 38290 Saint Quentin-Fallavier, France Hosokawa Alpine Aktiengesellschaft, Peter-Dörfler-Strasse 13-25, 86199 Augsburg, Germany
a r t i c l e
i n f o
Article history: Received 28 May 2014 Revised 19 December 2014 Accepted 22 December 2014 Available online xxxx Keywords: Grinding Comminution Process modeling HPGR PBM
a b s t r a c t Population Balance Models (PBM) are widely used to predict the evolution of the particle size distribution during various grinding processes, such as ball milling. They represent breakage through the definition of particle destruction and fragments generation rates. Their application to compression grinding (HPGR, vertical mills. . .) has been limited, due to the complexity of interactions between particles of different sizes. In this work, we present a new PBM approach for compression grinding. Complex interactions between size classes are represented in a simplified manner by making particle destruction and fragment generation depend on the bed porosity. Model is tested by confrontation to an extensive collection of experimental results on a piston-die cell, on three different materials (cement clinker, limestone, and quartz). When properly calibrated with preliminary tests, the model is able to predict the evolution of the particle size with accuracy, for any starting grain size distribution and any load. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Compression grinding equipments, such as high pressure grinding rolls (HPGR) or vertical mills, have been now in service for several decades, following the invention of the HPGR by Schönert, 1988. HPGR, vertical mills, but also Horomill (Cordonnier, 1994), despite their differences in terms of geometry, are all based on the same physical principle: the relatively slow compression of a granular bed. Such equipments are now operated worldwide in the mineral and cement industry (Kowatra, 2004). Those mills are reported to save up to 40% of the necessary energy input as compared to ball mill grinding, and thus represent a huge technological improvement, despite remaining process control problems for some applications, most notably with fine powders (Schönert, 1988; Fuerstenau and Abouzeid, 2002; Fuerstenau and Abouzeid, 2007; Wang and Forrsberg, 2003; Aydogan et al., 2006; Musa and Morrison, 2009). We focus here on a well-explored laboratory test to represent the compression grinding process, the piston-die cell. In this test, a small sample of materials (100 g in our case) is put in a cylindrical chamber. A piston is then forced on the materials with a press, to reach levels of pressure in the material of the same magnitude of
⇑ Corresponding author. Tel.: +33 4 74 82 80 97. E-mail address: [email protected] (V.P.B. Esnault).
what is encountered in the industry (up to 100–200 MPa). The test has been extensively used in the literature to model the more complex industrial setting (for example (Aziz and Schönert, 1980; Fuerstenau et al., 1996), or more recently (Oettel et al., 2001; Hosten and Cimilli, 2009 or Dundar et al., 2013)). Most recently, Kalala et al., 2011 address several aspects of HPGR/compression cell equivalence, such as energy draw or mimicking in the cell side effects on HPGR. The object of this communication is to present a new, semiempirical model to represent the evolution of the particle size distribution at the material level during the compression grinding process. More precisely, we exploit the alleged regularity of the stress and strain field in the piston-die cell to observe the behavior of an elementary unit of material, and we want to correlate data such as stress, deformation or relative density with fragmentation level. Such a tool would be a key ingredient for detailed modeling of compression grinding equipments, using techniques such as DEM modeling, for instance. For this, we rely on the use of a Population Balance Model (PBM), a type of model where particle size distributions are divided into size classes. Fragmentation is then represented through the transfer of mass between those size classes. PBM has been extensively used to represent ball milling, since the seminal works of Austin and Luckie, 1972, Whiten, 1974. Reviews on the application of this type of model in this field of application can be found in Benzer, 2000 or Hashim, 2004.
http://dx.doi.org/10.1016/j.mineng.2014.12.036 0892-6875/Ó 2015 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Esnault, V.P.B., et al. New population balance model for predicting particle size evolution in compression grinding. Miner. Eng. (2015), http://dx.doi.org/10.1016/j.mineng.2014.12.036
2
V.P.B. Esnault et al. / Minerals Engineering xxx (2015) xxx–xxx
Given the development of the usage of compression technologies in the industry on one hand, and the wide usage of PBM in the modeling of ball mill circuits on the other, it is surprising that very little attempts have been made to apply PBM to compression grinding. In ball milling, the modeling is greatly facilitated by the possibility to represent the process through linear relations. Breakage rates, for instance, can be considered as constant functions of time, unaffected by the evolution of the grain size distribution (at least in a wide range of fineness). This is no longer possible in the case of compression grinding. First, there is no obvious definition of ‘‘time’’, in compression grinding (should we consider the evolution of the applied load? the increase of material density?. . .). Even more important, the breakage behavior of the particles is deeply affected by the granular environment (contact number, bed density. . .). Any attempt to represent the breakage of the particles in compression through PBM is bound to take into account those non-linearities. Such an attempt to formulate a general non-linear PBM model, taking into account particle interactions, is made by Bilgili et al. in Bilgili et al., 2006. It is applied to compression grinding in Bilgili and Capece, 2012. The general model is very complex, and to apply it to compression grinding, the authors need to apply disputable simplifications (for instance, they neglect secondary breakage, i.e. the breakage of particles that themselves come from an initial breakage event). Even in this simplified form, the model seems to have had limited application in practical cases. Most recently, Dundar et al., in Dundar et al., 2013, used a piston-die test to calibrate a PBM for HPGR crushing. The model relies on important simplifications of the actual process, but is simple to implement and is the first of its type to actually model the real industrial process. Incidentally, it brings elements in favor of the modeling of compression grinding through the piston-die test. Another attempt to represent the evolution of particle size, though not through the PBM approach, has been developed by Schönert and his team in Liu and Schönert, 1996. The model is based on an ‘‘energy split function’’ that divides the total energy input between size classes. The particle size evolution in each class is then considered to be similar as the one observed in a monodisperse test for the same energetical input. Attempts have also been made to apply PBM directly at the process level. Models then do not necessarily represent precisely what actually happen to given sample of material as a function of load, but the comminution effect of the industrial device taken as a whole. Fuerstenau et al., 1991 developed a method to adapt PBM models in the case of HPGR operation. Energy input is used as the ‘‘time’’ variable, and non-linearities are taken into account by introducing a ‘‘retardation effect’’ with increasing pressure, illustrating increased dissipation rate with increasing pressure. The model successfully reproduces the experimental results. More recently, Hinde and Kalala, 2009 proposed a broader correlation, still for HPGR, between starting and final granulometries, using a general law exploiting the energy input to describe the amount of fragmentation provided by the machine. Such models are certainly useful tools in terms of process control, as they allow to represent the behavior of a given equipment (here, mostly HPGR) as a function of operating parameters (power draw, for instance). However, they are not designed to precisely represent the mill heterogeneities and their potential influence on the global process. This difference is most crucial when considering more complicated geometries than the classical HPGR system, such as in vertical mills or Horomills. One remaining difficulties with those existing models, both at the material and at the process level, is that they cannot be considered as fully predictive. They do reproduce well the experimental results, sometimes up to the final industrial process. But some
parameters need to be numerically adjusted for every test, and prediction of results under a new set of conditions (pressure and starting size range) is not possible. The model can then considered at best as a numerical description of what is happening during the test. The present model focuses on the preliminary calibration for a given material, on a given set of experiments. As we see it later in this presentation, once the calibration is done, the model can safely be applied to this material, for any starting granulometry, for any applied pressure. We believe this predictive power is one of the major characteristics for such a model to be one day applied in a broader context. A predictive model is also a condition if we want to provide a full numerical modeling of compression grinding. Material flow in roller compaction or HPGR grinding can for instance already be modeled through Finite Element Modeling (FEM) simulation, originally in 2D (Cunningham, 2005; Michrafy et al., 2011), and now in 3D (Cunningham et al., 2010; Michrafy et al., 2011). PBM model would allow to couple the informations from such FEM modeling with comminution, constituting a powerful tool for the understanding of compression grinding equipments of any geometry. 2. The non-linear PBM model In PBM, the evolution of grain size distribution is modeled through the transfer of material between size classes, through a linear relationship first formulated by Epstein, 1947, and first applied to grinding (of coal) by Broadbent and Calcott, 1956. In a matrix form, the particle size distribution at instant n þ 1; F nþ1 depends of the size distribution at n; F n through relation (1):
F nþ1 ¼ T n F n
ð1Þ
With T n being defined through (2):
Tn ¼ I S þ B S
ð2Þ
I is the identity matrix. S is a diagonal matrix, called selection matrix. Its ii term describe the probability that a particle from class i gets broken between n and n þ 1. B is a triangular matrix, call grinding or repartition matrix. Its term ij describe the proportion of fragments of size j when a particle of class i is broken. So when we consider the terms in Eq. (2), we see that the first term characterizes mass conservation, the second the disappearance of a particle from its size class when broken, the third the repartition of the fragments between the smaller size classes. Baxter et al., in Baxter et al., 2004, formulated a general way to include non-linearities in this relationship (3):
F nþ1 ¼ T n ðF n Þ F n
ð3Þ
Of course making the matrix T n a function of the full size distribution F n would need to a very complex formulation. In this model, our essential assumption is to simplify expression (3) by assuming T n is a function of the granular material porosity, / (or symmetrically, the relative density 1 /). In other words, we assume that the complex effect of the granular environment (the number of contacts, the distribution of the load pattern. . .), can be summarized through the packing density alone. Based on such an assumption, the selection matrix S can be directly determined from experimental tests. Indeed, the disappearance rate of the top size class as a function of / can be calculated from a set of tests on originally monodisperse size fractions. A first estimation for the repartition matrix B can also be obtained experimentally, by considering tests on monodisperse fractions, for a low applied pressure. In this case, the extent of
Please cite this article in press as: Esnault, V.P.B., et al. New population balance model for predicting particle size evolution in compression grinding. Miner. Eng. (2015), http://dx.doi.org/10.1016/j.mineng.2014.12.036
V.P.B. Esnault et al. / Minerals Engineering xxx (2015) xxx–xxx
3
secondary breakage (the re-breakage of fragments) can be neglected. Fragments can then all be assumed to originate from the parent particles. However, it is not possible to monitor directly the evolution of B as a function of /, due to the presence of re-breakage. We then define two variants of the model according to the way we deal with this dependence: In the first one, we simply assume that B is independent from /. The model can then be entirely be calibrated from a given set of tests on monodisperse fractions, without using any parameter fitting. In the second, we adjust B as a function of /. To determine the values of B, we assume that comminution follows Rittinger law, i.e. surface creation is proportional to energy input. More precisely, we define SSðFðxÞÞ the specific surface of size distribution FðxÞ (4):
SSðFðxÞÞ ¼
Z 0
1
6 @FðxÞ dx qx @x
ð4Þ
with q the specific mass of the material. We assume the following relationship (5):
@SSðFðxÞÞ @W SSðF nþ1 ðxÞÞ SSðF n ðxÞÞ ¼ ð/ /n Þ @W @/ nþ1
ð5Þ
The left hand side term is the surface creation between time step n and n þ 1. @SSðFðxÞ is the surface creation as a function of energy input @W W. It is assumed constant and can be measured on calibration tests for a given material. @W can be easily calculated if the pressure P and @/ the piston displacement x are monitored during the test. W is calculated using (6):
W¼
Z
xfin
SPðxÞxdx
ð6Þ
0
with S being the die section. / is also monitored continuously using the variation of the sample thickness. Breakage functions Bi;/ ðxÞ are implemented in the model using a combination of two Rosin– Rammler distributions, following the mathematical form (7):
Bi;/ ðxÞ ¼ 1 pi eðx=k1;i Þ
k1;i
ð1 pi Þeðx=k2;i Þ
k2;i
kinit 1;i ;
ð7Þ
kinit 2;i ;
init k1;i )
Initial values of the parameters (pinit are adjusted to i ; fit the observed distributions. In order to respect (5), we modify Bi;/ ðxÞ by considering parameters pi ; k1;i ; k2;i ; k1;i et k2;i can be established from their initial values using the following transformation (8):
pi ¼ pinit i init
k1;i ; k2;i ¼ ln ð10Þ
SSðBi;/ ðxÞÞ ¼ aSSðBinit i;/ ðxÞÞ
ð9Þ
að/Þ is then representative of the way energy is exploited in fragment generation as compaction progresses. We assume this function does only depend of the granular environment through the porosity, and not of the considered size class. Determining Bi;/ ðxÞ can be done by the following procedure: Determine the value of að/Þ such as (5) is respected. Calculate the corresponding bi and corresponding Bi;/ ðxÞ distributions. Fig. 1 gives an example of a Bi;/ ðxÞ function (for clinker, size class 1–1.25 mm), for a ¼ 1 (i.e. the measured value at low pressure), and a ¼ 5, close to the highest value implemented in the simulations. The a ¼ 5 case corresponds to a situation where the fragment generation is broader, with more fine particles and more surface generation. The model can then be calibrated to a given set of data on monodisperse materials. The selection function Sð/Þ is directly measured from the disparition rate of the top size particles in tests on monodisperse particles. Bð/Þ is established from their initial values measured at low pressure, and modified according the energy-surface relationship (5). The model can also be tested using experiments with the same material, but different starting granulometries or final pressures, to assess its prediction power. In this sense, it is a predictive model, as no further adjustments are made to the model in order to fit those new experimental results.
init
k1;i ; k2;i ¼ bi ð/Þk1;i ; bi ð/Þk2;i 1 1 kinit k1;i 1;i
Fig. 1. Fragment distribution function Bi;/ ðxÞ for clinker 1–1.25 mm, for a ¼ 1 and a ¼ 5.
kinit 1;i ;
ln ð10Þ
1 1 kinit k2;i 2;i
3. Experimental procedure and model calibration
kinit 2;i
ð8Þ
k1;i et k2;i are the two shape parameters of theRosin–Rammler distributions. Multiplying them by a factor bi ð/Þ then allows to widen (if bi ð/Þ < 1) or to narrow (if bi ð/Þ > 1) the particle size distribution of the formed fragments. The operation on k1;i et k2;i aims at conserving the d90 during the transformation (by definition, d90 is the values as that 90% of particles in mass are smaller than d90 ). The transformation is then a mean to modulate the Bi;/ ðxÞ width through its shape parameters, without modifying its top value. It provides a way to adapt empirically the breakage function so the created surface corresponds to the energy input provided by the piston. In order to characterize the modification of the distributions, we define að/Þ, such as (9):
We used for the tests the piston-die cell represented on Fig. 2. Its diameter is 8 cm, and the initial samples thicknesses were around 1.5 mm, a big enough sample size relatively to grain to ignore side effects. Pressure was applied through a 3000 kN capacity press for the highest pressures applied (above 20 MPa), and a 100 kN press for lower pressures. This bigger press would allow for maximum working pressure of 600 MPa, but in practice we limited ourselves to 280 MPa, already far above the pressures regularly encountered in the industry. Applied load was monitored through a force sensor, and bed strain through a LVDT sensor measuring the progression of the piston. Bed final and initial porosity were evaluated by measuring the piston height with a caliper.
Please cite this article in press as: Esnault, V.P.B., et al. New population balance model for predicting particle size evolution in compression grinding. Miner. Eng. (2015), http://dx.doi.org/10.1016/j.mineng.2014.12.036
4
V.P.B. Esnault et al. / Minerals Engineering xxx (2015) xxx–xxx
Fig. 2. Illustration of a test using the piston-die cell.
We considered three different materials in this study: a cement clinker, a quartz, and a limestone sand. For each materials we conducted series of tests on different monodisperse size classes (from 100 lm to 2.5 mm), at different applied pressure (from 1.5 to 280 MPa), in order to map the behavior of the material. Data points collected are summarized in Table 1. The data collection is voluntarily more extensive on cement clinker (55 points), then on quartz and limestone (16 points). We aimed at validating the model in optimal conditions on clinker, then trying to assess its behavior for a cruder, easier to acquire data set on limestone and quartz. Those materials also present very different behavior. Considering Moh’s hardness, for instance, the quartz can be considered as very hard and brittle (with a Moh’s hardness of 7), while limestone is much softer (Moh’s hardness of 3). Clinker presents intermediate properties (Moh’s hardness of 5). Each test was conducted on a 100 g sample. Tests were reproduced three times from each point in order to limit data dispersion. After the compression, the material was dispersed manually (at high pressure, the compression cell produce solid, compact flakes). Particles were sieved at the original size class bottom size to measure the disappearance rate of the initial particles. Fragments particle size distribution was then measured using laser granulometry and specific surface of the grain size distribution is determined according to expression 4. It is to be noted that this expression is very sensitive to small variation of the size distribution in the small size range. Small measurement mistakes on the finest fraction can lead to sizable variation of the calculated surface area. In our case, this problem is limited as we only consider relatively coarse size distributions (negligible amount of particles below 3 lm), and as we consider only materials for which optical properties are well known and for which the technique has been calibrated. The model was calibrated from this data set in accordance with the model described in the preceding chapter. Selection matrix S was calculated form the disappearance rate of the top size particle as a function of density (Fig. 3). Fig. 4 displays the grain size distribution of fragment generated (i.e. excluding the original size class), at different pressures, for clinker (1–1.25 mm). A quick look does suggest that important re-breakage is going on, as those
Fig. 3. Percentage of broken particles as a function of relative density 1 / (clinker, 500–600 lm size class). Selection function as function of density is calculated from the average fitted line.
Fig. 4. Grain size distribution of the fragments generated, i.e. excluding the particles of the starting class size, at different pressures (clinker, 1–1.25 mm size class).
distribution keep going finer, making it unlikely particles have been generated in one generation of breakage only as implemented in Bilgili and Capece, 2012. An initial estimation of the repartition matrix B was made from the fragment size distribution at the lowest pressure tested (1.5 MPa). In between size classes where experimental results were obtained, intermediate properties were established using interpolation. Integration of the size distribution according to formula (4) gives the evolution of the specific surface with pressure in a given
Table 1 Calibration tests realized on monodisperse samples. Material
Size classes
Applied pressure
Clinker
2–2.5 mm, 1–1.25 mm, 500–600 lm 250–300 lm, 100–125 lm
1.5, 3, 5, 8.8, 14, 23, 33.8, 55, 100, 166, 280 MPa 1.5, 3, 5, 9, 14.2, 24, 36.4, 53, 90, 137, 250 MPa
Calcaire
1.25–1.6 mm, 600–800 lm, 315–400 lm, 125–160 lm
3, 15, 60, 180 MPa
Quartz
1.25–1.6 mm, 600–800 lm, 315–400 lm, 125–160 lm
3, 15, 60, 180 MPa
Please cite this article in press as: Esnault, V.P.B., et al. New population balance model for predicting particle size evolution in compression grinding. Miner. Eng. (2015), http://dx.doi.org/10.1016/j.mineng.2014.12.036
5
V.P.B. Esnault et al. / Minerals Engineering xxx (2015) xxx–xxx
Fig. 5. Applied pressure as a function of relative density (1 /).
Fig. 8. Specific surface as a function of energy input, with linear regression, case of quartz, 1.25–1.6 mm.
Fig. 6. Total mechanical energy input as a function of relative density (1 /). Fig. 9. Specific surface as a function of energy input, with linear regression, case of limestone, 1.25–1.6 mm.
Table 2 Calculated value of
for the different calibration tests.
Material
Size fraction
@SS @W
Clinker
2–2.5 mm 1–1.25 mm 500–600 lm 250–300 lm 100–125 lm Average
0.00243 0.00222 0.00171 0.00225 0.00340 0.00240
Quartz
1.25–1.6 mm 600–800 lm 315–400 lm 125–160 lm Average
0.00243 0.00307 0.00225 0.00483 0.00342
Limestone
1.25–1.6 mm 600–800 lm 315–400 lm 125–160 lm Average
0.0067 0.00696 0.00765 Not linear 0.00710
Fig. 7. Specific surface as a function of energy input, with linear regression, case of clinker, 1–1.25 mm.
size class. Energy input is measured as function of density 1 / during the test. Figs. 5 and 6 represent respectively pressure and mechanical energy input as a function of relative density 1 /, in the case of clinker, 1–1.25 mm, 280 MPa. In order to establish the breakage function, it is necessary to @SS , the surface creation to energy ratio. We plot on Figs. 7– evaluate @W 9 specific surface of the sample as a function of energy input, for respectively clinker (1–1.25 mm), quartz (1.25–1.6 mm) and limestone (1.25–1.6 mm). Linearity of the data seems to depend very much of the nature of the material. For clinker and especially @SS quartz, the curves are remarkably linear and a value of @W can be practically established. For limestone, the point at the highest pressure is notably offline. The same phenomenon could be observed for all size classes tested. We attribute this to the ‘‘soft’’ nature of
@SS @W
(in m2 kg
1 1
J
)
the material: fragile breakage of the grains is dominated by phenomenons such as plastic deformation of the grains and reagglomeration at high pressure, leading to parasite energy consumption in the case of limestone. Quartz, by contrast, is very hard and grains continue to be essentially subjected to brittle fracture, even at high pressure. For limestone, the point at the highest pressure as systematically excluded of the simulations.
Please cite this article in press as: Esnault, V.P.B., et al. New population balance model for predicting particle size evolution in compression grinding. Miner. Eng. (2015), http://dx.doi.org/10.1016/j.mineng.2014.12.036
6
V.P.B. Esnault et al. / Minerals Engineering xxx (2015) xxx–xxx
@SS Table 2 summarizes the values obtained for @W for different size classes. Values can vary quite a bit from one size class to another, indicating possible differences in terms of breakage behavior according to the size of the particle. For each material, we use the average value. Those values should be considered with caution as they have nothing in common with the ones that can be expected in the industry. Mütze and Husemann, 2008 defend that loading speed influences the energy draw in a piston-die type of test. This is attributed to the dependance in the loading of phenomenons such as friction between grains. With loading speeds an order of magnitude faster in the industry then in the lab-scale tests, real systems dissipate more leading to very low energy consumption observed at lab-scale. This lower energy consumption in compression cell when compared to real HPGR process has also been observed experimentally and presented in Kalala et al., 2011. Any potential influence of the loading speed was neutralized on the tests by applying the same loading speed to all samples. In the numerical calculations, we considered a total of 40 size classes, from 2.1 to 2.5 mm for the bigger one, to a smaller, unbreakable 0–3 lm size class. Population in each size class is then calculated by finite difference from the initial porosity / to the desired one, using a unitary step of 0.001.
4. Results To assess the prediction power of the model, several tests were performed on mixes of different size classes of the original material. Each mix was pressed at 25 and 100 MPa, and the particle size was analyzed. The mixes tested are listed in Table 3. Like for the monodisperse calibration tests, each point is averaged on three tests to minimize data dispersion.
The result was then compared to the predicted grain size evolution by the model, between the initial and finally observed density. The quality of the prediction was assessed through the calculation of an error function e, define through relation (10):
R1
em ¼ R01 0
jF e ðxÞ F m ðxÞjdðln xÞ jF e ðxÞ F 0 ðxÞjdðln xÞ
ð10Þ
F 0 ðxÞ is the initial size distribution of the mix. F e ðxÞ is the size distribution observed experimentally after the test, and F m ðxÞ is the one predicted by the model. e then gives an estimation of the quality of the model predicted grain size modification compared to the actually observed one. We report in Table 4 the calculated error obtained on the different validation tests. Two values are given, the first one for the variant without dependence of B relatively to /, the second one including the variation of the breakage function considering the energy usage. A first result is that in the majority of the cases, the variable B does improve the quality of the prediction notably. In the case of the quartz, the fixed B even gives totally unacceptable results while the variable B allows for a reasonable prediction. This is most significant as the numerical optimization of the variable B was done on the calibration tests only, while those validation tests were done independently. It has also been concluded that error values are generally higher on small increments of pressure (between 25 and 100 MPa, or 0 and 25 MPa) then on the full range of prediction (0–100 MPa). This is an artifact linked to the definition of the error e: it is normalized by the particle size variation between the initial and the final state. If this variation is small, small imprecisions or errors lead to higher values of e.
Table 3 Prediction evaluation tests realized, on granular mixes. Material
Size mix
n of the mix
Tested pressure
Clinker
20% 1–1.25 mm, 80% 100–500 lm 20% 250–300 lm, 80% <100 lm
1.1 1.2
25 et 100 MPa 25 et 100 MPa
Limestone
20% 1.25–1.6 mm, 80% 315–630 lm 20% 600–800 lm, 80% 125–315 lm
2.1 2.2
25 et 100 MPa 25 et 100 MPa
Quartz
20% 1.25–1.6 lm, 80% 630–800 lm 20% 630–800 lm, 80% 315–630 lm
3.1 3.2
25 et 100 MPa 25 et 100 MPa
Table 4 Evaluation of the prediction error e on the different validation tests. Material
Mix number
Pressure span
Porosity span
Error, fixed B (%)
Error, variable B (%)
Clinker
1.1
0 ! 100 MPa 0 ! 25 MPa 25 ! 100 MPa 0 ! 100 MPa 0 ! 25 MPa 25 ! 100 MPa
0:5790 ! 0:2717 0:5790 ! 0:3859 0:3859 ! 0:2717 0:5465 ! 0:3031 0:5465 ! 0:3960 0:3960 ! 0:3031
18.60 7.81 43.16 22.73 23.24 38.82
7.64 7.19 16.78 20.24 23.74 20.17
0 ! 100 MPa 0 ! 25 MPa 25 ! 100 MPa 0 ! 100 MPa 0 ! 25 MPa 25 ! 100 MPa
0:4349 ! 0:1780 0:4349 ! 0:2542 0:2542 ! 0:1780 0:4353 ! 0:1765 0:4353 ! 0:2690 0:2690 ! 0:1765
34.73 17.00 74.45 47.99 35.45 72.30
11.59 23.01 20.44 25.66 27.88 28.59
0 ! 100 MPa 0 ! 25 MPa 25 ! 100 MPa 0 ! 100 MPa 0 ! 25 MPa 25 ! 100 MPa
0:4570 ! 0:2399 0:4570 ! 0:3352 0:3352 ! 0:2399 0:4624 ! 0:2584 0:4624 ! 0:3618 0:3618 ! 0:2584
65.13 62.13% 70.91 73.76 73.73 77.37
25.96 25.21 39.33 12.86 48.08 6.47
1.2
Limestone
2.1
2.2
Quartz
3.1
3.2
Please cite this article in press as: Esnault, V.P.B., et al. New population balance model for predicting particle size evolution in compression grinding. Miner. Eng. (2015), http://dx.doi.org/10.1016/j.mineng.2014.12.036
V.P.B. Esnault et al. / Minerals Engineering xxx (2015) xxx–xxx
Fig. 10. Starting, observed and predicted grain size distribution, clinker 20% 1– 1.25 mm, 80% 100–500 lm, 0–100 MPa.
Fig. 11. Starting, observed and predicted grain size distribution, clinker 20% 250–300 lm, 80% <100 lm, 0–100 MPa.
7
Fig. 13. Starting, observed and predicted grain size distribution, 20% 630–800 lm, 80% 315–630 lm, 0–100 MPa.
Fig. 14. Starting, observed and predicted grain size distribution, limestone 20% 1.25–1.6 mm, 80% 315–630 lm, 0–100 MPa.
the target. The prediction remains in the good order of magnitude, though, even in those not so favorable cases. Two explanations can be given for those differences in terms of prediction quality:
Fig. 12. Starting, observed and predicted grain size distribution, 20% 1.25–1.6 lm, 80% 630–800 lm, 0–100 MPa.
Figs. 10 (case 1.1 (‘‘coarse’’ clinker)) and 11 (case 1.2 (‘‘fine’’ clinker)) show that the prediction behavior of the model for clinker is very good, with only a slight lack of precision for the finer case. In the most favorable cases, the quality of the prediction is visually excellent, to the point that the prediction error is of the same magnitude of the experimental measurement errors. For limestone and quartz (Figs. 12–15), for which the calibration database is less extended, model prediction can be more off
It seems very important to feed the model with a proper set of data for the calibration. This concerns the number of experimental points (the extent of the data set on clinker was much larger), but also its range. In the case of the ‘‘fine’’ clinker (case 1.2), for instance, most of the particles were actually outside the calibration range where the bottom size is 100 lm. It is then not surprising to see that the model does not predict that well the behavior of those fine particles as it has not been calibrated to do so. Calibration in the fine range would require the constitution of fine, monodisperse samples, which can prove experimentally complicated. It is also likely that the nature of the material affects the quality of the prediction. The model specifically models fragile breakage of the grains, and is not designed to deal with other type of grain transformation, such as plastic deformation or reagglomeration. Limestone is a quite representative case. We could see on the calibration data and sample aspect that the behavior, especially at high pressures, was far from ideal fragile breakage. This could be a reason for the fact that the model exaggerates grinding: the model attributes all the energy to breakage and surface creation, which is no longer the case for high pressures. It also proves difficult to evaluate the particle
Please cite this article in press as: Esnault, V.P.B., et al. New population balance model for predicting particle size evolution in compression grinding. Miner. Eng. (2015), http://dx.doi.org/10.1016/j.mineng.2014.12.036
8
V.P.B. Esnault et al. / Minerals Engineering xxx (2015) xxx–xxx
Fig. 15. Starting, observed and predicted grain size distribution, limestone 20% 600–800 lm, 80% 125–315 lm, 0–100 MPa.
calibrated. Such an effort is likely to be of some interest only for mass produced, standard, and reasonably reproducible materials, such as cement clinker, which was the primary interest in this study. It also remains empirical, in the sense its quality can only be appreciated at the light of its prediction capacities. The physics behind those notions of selection and repartition remains largely unexplored in compression. Most of all, this model would require to be confronted to actual industrial data. The representativeness of the piston-die cell, for instance, remains an open question, especially if we want the model to be really predictive. This would require more experimental data and very good process control, as the evaluation of the bed porosity, in the system, is of primary importance. The current model could be applied directly to HPGR modeling through mixed modeling, with breakage functions being determined on the piston-die cell, and the link between surface creation and energy calculated on the bigger mill. Provided such a link with the actual industrial systems, we believe this model could be of some help for the understanding and the piloting of compression grinding techniques in the industrial world. Finally, the current model can represent the breakage inside an elementary unit of material inside a full numerical simulation of a compression grinding equipment. It seems particularly well suited to be coupled to a FEM model of a mill (where porosity and energy input are constantly evaluated everywhere), allowing to couple the densification of the material as it flows through the mill and the breakage of the grains. Such complete modeling would allow to characterize precisely the behavior of the mills, including for geometries more complex than in the HPGR case, such as in vertical mills and Horomill, and ultimately lead to better design for those equipments.
References Fig. 16. Function að/Þ, for clinker, for case 1.1 (coarse) and 1.2 (fine).
size in the highly agglomerated samples produced by the test, leading to characterization errors, both in the calibration data set and in the demonstration example. Finally, in Fig. 16, we plot alpha as a function of /, for case 1.1 and 1.2 (clinker). One can see that a > 1, a condition we have prescribed for the model to respect the breakage function measured at low pressure. We observe that a follows the same curve in the two cases, a good argument for the physical relevance of the notion. It would tend to indicate that grain breakage generates finer fragments at higher densities and higher pressure. This is coherent with the idea that grains tends to support higher stress values, with more contact points in dense systems, leading to fewer breakage, but more energetic and according to more complex fracture patterns.
5. Conclusion Very few models are available to predict the evolution of particle size distribution in compression grinding, and they cannot be considered as really predictive in the sense they could be calibrated for a given material on a data set, and then applied to any situation with this material. We believe that this aspect makes the present model suitable for application to industrial grinding devices. It also exploits tangible material parameters such as material density and energy input in an explicit manner. The model is still heavy to handle. It should be noted that it requires an extensive set of experimental data to be properly
Schönert, K., 1988. A first survey of grinding with high-compression roller mills. Int. J. Miner. Process. 22 (1-4), 401–412, recent Advances in Comminution. Cordonnier, A., 1994. A New Grinding Process: Horomill, 8th European Symposium on Comminution. Kowatra, S.K., 2004. Adv. Comminut., Soc. Min. Mettal. Explorat. Fuerstenau, D.W., Abouzeid, A.-Z.M., 2002. The energy efficiency of ball milling in comminution. Int. J. Miner. Process. 67 (1-4), 161–185. Fuerstenau, D.W., Abouzeid, A.-Z.M., 2007. The performance of the high pressure roll mill: effect of feed moisture. Int. J. Miner. Process. 82 (4). Wang, Y., Forrsberg, E., 2003. International overview and outlook on comminution technologies, report from Lulea Tekniska Universitet. Consulted online at
Please cite this article in press as: Esnault, V.P.B., et al. New population balance model for predicting particle size evolution in compression grinding. Miner. Eng. (2015), http://dx.doi.org/10.1016/j.mineng.2014.12.036
V.P.B. Esnault et al. / Minerals Engineering xxx (2015) xxx–xxx Bilgili, E., Capece, M., 2012. A rigorous breakage matrix methodology for characterization of multi-particle interactions in dense-phase particle breakage. Chem. Eng. Res. Des. 90, 1177–1188. Liu, J., Schönert, K., 1996. Modelling of interparticle breakage. Int. J. Miner. Process. 44-45, 101–115. Fuerstenau, D.W., Shuckla, A., Kapur, P.C., 1991. Energy consumption and product size distributions in choke-fed high-compression roll mills. Int. J. Miner. Process. 32, 59–79. Hinde, A.L., Kalala, J.T., 2009. The application of a simplified approach to modelling tumbling mills, stirred media mills and HPGR’s. Miner. Eng. 22, 633–641. Cunningham, J.C., 2005. Experimental studies and modeling of the roller compaction of pharmaceutical powders, PhD thesis, Drexel University. Michrafy, A., Diarra, H., Dodds, J.A., Michrafy, M., Penazzi, L., 2011. Analysis of stress strain rate in roller compaction process. Powder Technol. 208.
9
Cunningham, J.C., Windstead, D., Zavaliangos, A., 2010. Understanding variation in roller compaction through finite element-based process modelling. Comput. Chem. Eng. 34, 1058–1071. Michrafy, A., Diarra, H., Dodds, J.A., Michrafy, M., 2011. Experimental and numerical analyses of homogeneity over strip width in compaction. Powder Technol. 206, 154–160. Epstein, B., 1947. The mathematical description of certain breakage mechanisms leading to the logarithmico-normal distribution. J. Franklin Inst. 244 (6), 471–476. Broadbent, S.R., Calcott, T.G., 1956. A matrix analysis of processes involving particle assemblies. Philos. Trans. Royal Soc. 249, 99–123. Baxter, J., ABu-Nahar, A., Tüzün, U., 2004. The breakage matrix approach to inadvertent particle degradation: dealing with intra-mixture interactions. Powder Technol. 143-144, 174–178. Mütze, T., Husemann, K., 2008. Compressive stress: effect of stress velocity on confined particle bed comminution. Chem. Eng. Res. Des. 86, 379–383.
Please cite this article in press as: Esnault, V.P.B., et al. New population balance model for predicting particle size evolution in compression grinding. Miner. Eng. (2015), http://dx.doi.org/10.1016/j.mineng.2014.12.036