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Energy dissipation in particle bed comminution
MINPRO-02667; No of Pages 5 International Journal of Mineral Processing xxx (2014) xxx–xxx
Contents lists available at ScienceDirect
International Journal of Mineral Processing journal homepage: www.elsevier.com/locate/ijminpro
Energy dissipation in particle bed comminution Thomas Mütze ⁎ TU Bergakademie Freiberg, Institute of Mechanical Process Engineering and Mineral Processing, Agricolastraße 1, 09599 Freiberg, Germany
a r t i c l e
i n f o
Article history: Received 13 December 2013 Received in revised form 16 July 2014 Accepted 1 October 2014 Available online xxxx Keywords: Comminution Compression Confined particle bed Flow losses
a b s t r a c t In particle bed comminution energy is dissipated by several microprocesses which accompany the flow of particles, their compaction, and breakage. It is unknown how much energy is associated with each microprocess and what the ratios are between the energies dissipated by the different microprocesses. Based on an experimental study, a model has been developed to calculate a critical compaction velocity and the energy dissipated by flow losses. The critical compaction velocity describes the probability of the deaeration of a particle bed. Since the flow losses consume less than 0.1% of the total energy input, this microprocess can be considered negligible compared to others. © 2014 Elsevier B.V. All rights reserved.
1. Introduction The stressing of particle beds aims at forming a compact body or breaking the constituent parts into finer pieces. Both results are achieved by an initial flow of individual particles followed by their deformation and breakage. On this basis the total energy input (energy absorption) has been divided amongst several microprocesses (Schubert, 1967): - energy loss due to friction between the particles as well as between the particles and the confinements of the particle bed (friction losses), - energy absorption till breakage (breakage energy), - energy loss due to irreversible structural changes of particles (plastic deformation work), - energy loss due to friction occurring while displacing the fluid in the pores (flow losses), - energy loss caused by the wear of the confinements of the particle bed, and - energy loss due to thermoplastic effects, sound wave propagation and oscillation of elastically deformed fragments. Unfortunately, it is unknown how much energy is associated with each microprocess. Even the ratio between the energies, which are consumed by the individual microprocesses, is often estimated without any real knowledge. The breakage energy is assumed to make up only a small part of the energy absorption (1.5…12% (Schubert, 1967)). The plastic deformation work is, like the breakage energy, a very material
⁎ Tel.: +49 3731 392918; fax: +49 3731 392947. E-mail address: [email protected].
dependent process reaching the same order of magnitude as the breakage energy for limestone (Heegn, 1986). The plastic deformation work in the case of quartz is important only at high stress intensities due to the higher strength of the material and its lower tendency to agglomerate. Overall the friction losses are estimated to be the most energy intensive microprocess (Müller, 1989). An experimental study was used to derive a model for calculating flow losses and evaluating their relevance in particle bed comminution.
2. Material and methods The flow through a packed particle bed is influenced mainly by the external dimensions of the bed and its internal porosity, the fluid inside the pores, and the fineness of particles. Therefore the porosity and the fineness had to be investigated very carefully since both parameters change during stressing. A range of particle sizes between 1 and 1000 μm was used to vary the granulometry of the feed materials (limestone, quartz, silicon carbide, and glass beads). All together the feeds were manufactured mainly by grinding and size classification as broad fractions (x90/x10 ≈ 40), ten narrow fractions (x90/x10 ≈ 3), and 23 bimodal mixtures (x90/x10 = 10…1000). An example of these particle size distributions is given in Fig. 1. It is obvious that a bimodal mixture of two narrow size fractions does not produce the particle size distribution of a “natural” grinding product, but is considered to give an acceptable approximation of it. Prior to each experiment the materials were dried for more than 10 h at 120 °C to assure an ignorable humidity. Confined particle beds were compressed in a hydraulic press (DP 1600/1, Hegewald & Peschke) at low stress velocity (vs = 0.05 cm/s) and in an energy-controlled spindle press (Müller, 1989) at high stress velocities (10 cm/s). The
http://dx.doi.org/10.1016/j.minpro.2014.10.004 0301-7516/© 2014 Elsevier B.V. All rights reserved.
Please cite this article as: Mütze, T., Energy dissipation in particle bed comminution, Int. J. Miner. Process. (2014), http://dx.doi.org/10.1016/ j.minpro.2014.10.004
2
T. Mütze / International Journal of Mineral Processing xxx (2014) xxx–xxx
100
particle size distribution Q3(x) in %
90
narrow fraction around 10 µm mixture (10 µm - 100 µm = 3:1)
80
mixture (10 µm - 100 µm = 1:1) mixture (10 µm - 100 µm = 1:3)
70
narrow fraction around 100 µm 60
broad fraction (< 100 µm)
50 40 30 20 10 0 0,1
1
10
100
1000
particle size x in µm Fig. 1. Particle size distributions of the feed (limestone): narrow fractions (10 μm, 100 μm), mixtures of narrow fractions (mass ratio 1:3, 1:1, 3:1), a broad fraction b 100 μm.
dimensions of the confined particle beds resulted from the requirements of an ideal particle bed (Mütze et al., 2011): - height 10 mm - diameter 30…70 mm The annular gap between the piston and the die was around 200 μm in order to release excess air which was pressed out of the particle bed by the compaction. Each test was repeated up to five times; the arithmetic mean was used for further calculations. The coefficient of variation was mostly below 2% or is noted otherwise. The particle size distributions of the products were measured by sieving and laser diffraction after sufficient dispersion. The size distributions were used to describe the grinding kinetics and changes in fineness. This focused on the volume-specific surface area SV which is one of the main influencing variables in fluid flow through packed beds. The energy absorption Em was calculated as the area enclosed by the stress and the relief curves and divided with the mass of the stressed particles. The stress and the relief curves were also used to describe the elastic–plastic compaction behaviour dependent on material and process conditions (Mütze, 2012, 2014). 3. Modelling and results Initially the model development focused on an appropriate description of the changes in the porosity of the particle bed as a result of compaction. Furthermore an evaluation was necessary to which extend the fluid is trapped inside the pore volume. Finally the energy consumption due to flow losses was calculated. 3.1. Compaction behaviour As known from literature, the important parameters in uniaxial compaction are the pressure, the material type, the particle size, the particle size distribution, and the stress velocity (Cooper and Eaton, 1962; Heckel, 1961; Kawakita and Lüdde, 1971; Mütze, 2012, 2014; Walker, 1923). Changes in the porosity are closely connected with the deformation of the particle bed. The plastic deformation of the particle bed can be described by the normalized compression Θplast (Eq. (1)). This compression correlates the actual change of the bulk
density ρb,plast due to the compaction pressure p with a maximum possible change indicated by the solid density ρs (Mütze, 2012, 2014): Θplast ðpÞ ¼
ρb;plast ðpÞ−ρb;plast;0 : ρs −ρb;plast;0
ð1Þ
The correlation between plastic deformation and pressure on the particle bed is often approximated by an equation with two parameters (Heckel, 1961; Kawakita and Lüdde, 1971; Walker, 1923). The influences of the mean particle size, the particle size distribution, and the stress velocity on the compaction behaviour have been studied recently by using Eq. (2) (Mütze, 2012, 2014):
Θplast ðpÞ ¼ Θ ln
Θ* p*
p 1þ p
ð2Þ
model parameter 1: reference compression model parameter 2: reference pressure
Due to its mathematical form, Eq. (2) reflects the Θplast(p)-plot of narrow size fractions as well as bimodal mixtures and broad size distributions with adequate accuracy (Fig. 2). The deviation between the measured and the calculated values of Θplast is below ±0.02 for most of the examined size fractions. The two model parameters describe opposite effects of compaction. While the reference pressure describes the resistance against the uniaxial compaction of a powder, the reference compression describes the aptitude of a powder to compaction. These contrary trends allow an exemplified displaying of only one parameter in order to show the effect of for instance the particle size and the stress velocity. The reference pressure and thus the resistance against compaction increases significantly with decreasing mean particle size and increasing stress velocity for narrow size fractions (Fig. 3). This correlates with the predominant compaction mechanism of breakage for coarse-grained materials and the increasing probability of breakage with increasing particle size. This effect is superimposed by the micro-plasticity of fine fractions at which the high strength of particles b10 μm leads to plastic deformation at the contact points when the yield stress is reached (Rumpf, 1965). Due to this
Please cite this article as: Mütze, T., Energy dissipation in particle bed comminution, Int. J. Miner. Process. (2014), http://dx.doi.org/10.1016/ j.minpro.2014.10.004
T. Mütze / International Journal of Mineral Processing xxx (2014) xxx–xxx
3
1
0,04
0,8
plast,model
0,9 0
-0,04
-
-0,08
plast,meas
plast
0,6
difference
0,7
0,5 0,4 0,3 0,2
-0,12
0,1 0
-0,16 0
100
200
300
400
500
600
pressure in MPa limestone 1 µm (measured)
quartz 10 µm (measured)
limestone 1 µm (approx.)
quartz 10 µm (approx.)
glass beads 50 µm (measured) glass beads 50 µm (approx.)
limestone 1 µm (difference)
quartz 10 µm (difference)
glass beads 50 µm (difference)
Fig. 2. Measurement, approximation by model (Eq. (2)), and difference between measurement and model of the normalized compaction as function of pressure (narrow size fractions; stress velocity 0,05 cm/s).
additional mechanism, the resistance against compression increases not proportionally to the decreasing particle size. The mean particle size of a material with a bimodal particle size distribution has a much smaller influence on the reference pressure than that of a material with a narrow unimodal size distribution (Mütze, 2012). Here, the reference pressure corresponds mainly to the behaviour of the finer component meaning that the fine fraction forms a matrix in which individual coarse particles can be embedded. Only if the mass fraction of coarse particles is sufficiently large, approximately 75 wt.%, the reference pressure and thus the compression behaviour of the particle bed changes gradually towards the behaviour of the coarser component.
3.2. Deaeration A critical stress velocity of compaction vs,crit helps to distinguish two cases (Mütze and Husemann, 2008): a) vs b vs,crit: the fluid is able to flow out of the pore volume (slow compaction: deaeration faster than stressing) b) vs N vs,crit: the fluid is trapped inside the pore volume (fast compaction: stressing faster than deaeration) This critical velocity depends on the geometry and the porosity of the particle bed, the pressure drop across the particle bed, the characteristics of the fluid, and the fineness of the particles (Mütze and
100
reference pressure p* in MPa
y = 355.04x-0.81 R² = 1.00
10
y = 176.28x-0.85 R² = 1.00
1
narrow size fractions, 10 cm/s narrow size fractions, 0.05 cm/s bimodal mixtures (1 µm : 10 µm), 0.05 cm/s bimodal mixtures (1 µm : 1000 µm), 0.05 cm/s bimodal mixtures (10 µm : 1000 µm), 0.05 cm/s
0,1 1
10
100
1000
mean particle size xm,3 in µm Fig. 3. Influence of granulometry of the feed material and stress velocity on reference pressure (limestone, narrow size fractions and bimodal mixtures, stress velocities of 0.05 cm/s and 10 cm/s, fitted by power functions).
Please cite this article as: Mütze, T., Energy dissipation in particle bed comminution, Int. J. Miner. Process. (2014), http://dx.doi.org/10.1016/ j.minpro.2014.10.004
4
T. Mütze / International Journal of Mineral Processing xxx (2014) xxx–xxx
Fig. 4. Velocity ratio of deaeration and stress in relation to the mean particle size of the feed (quartz, narrow fractions, parameter: compaction pressure and stress velocity).
Husemann, 2008). The changes of fineness and porosity due to an increasing pressure on the particle bed are described by the compaction model on one hand and the grinding kinetics on the other. The stress velocity itself affects the compression and crushing behaviour of the particle beds so marginally that it has no significant effect on the flow velocity inside the particle bed. Nonetheless a material dependent effect on the flow velocity can be shown: - In a coarse-grained particle bed the flow velocity decreases with increasing compacting pressure because of the high increase of volume-specific surface area and increasing flow resistance. - The flow velocity increases in fine-grained particle beds. Since the size reduction of this material is not as significant as the one of coarser feed fractions, the flow resistance increases only slightly. This small increase is more than counterbalanced by the compaction and a higher pressure gradient of air between the inside and outside of the particle bed.
These findings are illustrated in Fig. 4 using the stress velocity v's which allows the comparison of uniaxial stressing with perpendicular deaeration (Mütze and Husemann, 2008). A material dependent transition point exists between a behaviour more determined by crushing and one more determined by compaction. Hard materials like quartz, which are difficult to be crushed, show this transition between 100 and 1000 μm. Middle-hard materials like limestone show this transition between 20 and 100 μm. This different behaviour of different feed size fractions becomes less important with increasing compaction pressure on the particle bed because of increasing amount of fragments. 3.3. Flow losses Since at least a part of the fluid is trapped in the pore volume at high stress velocities, it will be compacted with the particle bed consuming additional compression energy. This additional energy can be a measure
mass specific adiabatic work Ead,m in mJ/g
10
1
0,1
0,01
limestone (1 µm)
quartz (10 µm)
limestone (100 µm)
quartz (100 µm)
limestone (1000 µm)
quartz (1000 µm)
silicon carbid (10 µm)
glass beads (50 µm)
0,001 1
10
100
1000
bed pressure pmax in MPa Fig. 5. Mass-specific volumetric work as a function of compaction pressure (narrow feed fractions).
Please cite this article as: Mütze, T., Energy dissipation in particle bed comminution, Int. J. Miner. Process. (2014), http://dx.doi.org/10.1016/ j.minpro.2014.10.004
T. Mütze / International Journal of Mineral Processing xxx (2014) xxx–xxx
for the flow losses of a rapidly stressed particle bed. Since the compression is very fast, an adiabatic change of state can be assumed in order to calculate these flow losses. The maximum magnitude of the flow losses was first determined by assuming that the compaction is fast enough to trap all the fluid which is initially in the pore volume. This assumption can be made since the fluid velocity in the pores ranges from 0.1 to 10 cm/s whereas typical stress velocities of roller mills are in the range from 20 to 4800 cm/s at comparable geometric conditions (Mütze and Husemann, 2008). The adiabatic change of state leads to a mass specific adiabatic work Ead,m which has to be performed on the fluid:
Ead ¼ −
pPore;0 V Pore;0 κ−1
"
# V Pore;0 κ−1 1− V Pore
ð3Þ
pPore;0 Ead ¼ Ead;m ¼ − m κ−1
8 3κ−1 9 2 > > = 1−Θplast ðpÞ 1− ρ ρs ρs −ρb;plast;0 < b;plast;0 5 1−4 > ρb;plast;0 ρs > 1−Θplast ðpÞ ; :
m pPore,0 κ Θplast(p)
ð4Þ
mass of stressed particles initial fluid pressure in pore volume isentropic exponent compaction function from Eq. (1)
The adiabatic work depends strongly on the feed material as well as the pressure on and the compaction behaviour of the particles (Fig. 5).
5
4. Conclusions The values of the energy absorption of the examined materials are between 10 and 25 J/g at 300 MPa bed pressure. The flow losses account for less than 0.1% of the total energy input which means that - this microprocess is negligible compared to friction losses, breakage energy, and plastic deformation work, and - the limiting assumptions of the calculation regarding the maximum magnitude of flow losses are justified. References Cooper, A.R., Eaton, L.E., 1962. Compaction behavior of several ceramic powders. J. Am. Ceram. Soc. 45, 97–101. Heckel, R.W., 1961. Density–pressure relationships in powder compaction. Trans. Metall. Soc. AIME 221, 671–675. Heegn, H., 1986. Veränderung der Festkörpereigenschaften bei mechanischer Aktivierung und Feinzerkleinerung. Akademie der Wissenschaften der DDR. Kawakita, K., Lüdde, K.-H., 1971. Some considerations on powder compression equations. Powder Technol. 4, 61–68. Müller, F., 1989. Hochdruckzerkleinerung im Gutbett bei Variation von Feuchte und Beanspruchungsgeschwindigkeit. TU Clausthal. Mütze, T., 2012. Modeling and parameter study of the elastic–plastic deformation. 7th International Conference for Conveying and Handling of Particulate Solids — CHoPS 2012, Friedrichshafen. Mütze, T., 2014. Modellhafte Beschreibung des Beanspruchungsverhaltens geschlossener Gutbetten. Chem. Ing. Tech. 86, 814–820. Mütze, T., Husemann, K., 2008. Compressive stress: effect of stress velocity on confined particle bed comminution. Chem. Eng. Res. Des. 86, 379–383. Mütze, T., Husemann, K., Peuker, U.A., 2011. Das ideale Gutbett. Chem. Ing. Tech. 83, 720–724. Rumpf, H., 1965. Die Einzelkornzerkleinerung als Grundlage einer technischen Zerkleinerungswissenschaft. Chem. Ing. Tech. 37, 187–202. Schubert, H., 1967. Zu einigen Fragen der Kollektivzerkleinerung. Chem. Technol. 19, 595–598. Walker, E.E., 1923. The properties of powders. Part VI. The compressibility of powders. Trans. Faraday Soc. 19, 73–82.
Please cite this article as: Mütze, T., Energy dissipation in particle bed comminution, Int. J. Miner. Process. (2014), http://dx.doi.org/10.1016/ j.minpro.2014.10.004
Contents lists available at ScienceDirect
International Journal of Mineral Processing journal homepage: www.elsevier.com/locate/ijminpro
Energy dissipation in particle bed comminution Thomas Mütze ⁎ TU Bergakademie Freiberg, Institute of Mechanical Process Engineering and Mineral Processing, Agricolastraße 1, 09599 Freiberg, Germany
a r t i c l e
i n f o
Article history: Received 13 December 2013 Received in revised form 16 July 2014 Accepted 1 October 2014 Available online xxxx Keywords: Comminution Compression Confined particle bed Flow losses
a b s t r a c t In particle bed comminution energy is dissipated by several microprocesses which accompany the flow of particles, their compaction, and breakage. It is unknown how much energy is associated with each microprocess and what the ratios are between the energies dissipated by the different microprocesses. Based on an experimental study, a model has been developed to calculate a critical compaction velocity and the energy dissipated by flow losses. The critical compaction velocity describes the probability of the deaeration of a particle bed. Since the flow losses consume less than 0.1% of the total energy input, this microprocess can be considered negligible compared to others. © 2014 Elsevier B.V. All rights reserved.
1. Introduction The stressing of particle beds aims at forming a compact body or breaking the constituent parts into finer pieces. Both results are achieved by an initial flow of individual particles followed by their deformation and breakage. On this basis the total energy input (energy absorption) has been divided amongst several microprocesses (Schubert, 1967): - energy loss due to friction between the particles as well as between the particles and the confinements of the particle bed (friction losses), - energy absorption till breakage (breakage energy), - energy loss due to irreversible structural changes of particles (plastic deformation work), - energy loss due to friction occurring while displacing the fluid in the pores (flow losses), - energy loss caused by the wear of the confinements of the particle bed, and - energy loss due to thermoplastic effects, sound wave propagation and oscillation of elastically deformed fragments. Unfortunately, it is unknown how much energy is associated with each microprocess. Even the ratio between the energies, which are consumed by the individual microprocesses, is often estimated without any real knowledge. The breakage energy is assumed to make up only a small part of the energy absorption (1.5…12% (Schubert, 1967)). The plastic deformation work is, like the breakage energy, a very material
⁎ Tel.: +49 3731 392918; fax: +49 3731 392947. E-mail address: [email protected].
dependent process reaching the same order of magnitude as the breakage energy for limestone (Heegn, 1986). The plastic deformation work in the case of quartz is important only at high stress intensities due to the higher strength of the material and its lower tendency to agglomerate. Overall the friction losses are estimated to be the most energy intensive microprocess (Müller, 1989). An experimental study was used to derive a model for calculating flow losses and evaluating their relevance in particle bed comminution.
2. Material and methods The flow through a packed particle bed is influenced mainly by the external dimensions of the bed and its internal porosity, the fluid inside the pores, and the fineness of particles. Therefore the porosity and the fineness had to be investigated very carefully since both parameters change during stressing. A range of particle sizes between 1 and 1000 μm was used to vary the granulometry of the feed materials (limestone, quartz, silicon carbide, and glass beads). All together the feeds were manufactured mainly by grinding and size classification as broad fractions (x90/x10 ≈ 40), ten narrow fractions (x90/x10 ≈ 3), and 23 bimodal mixtures (x90/x10 = 10…1000). An example of these particle size distributions is given in Fig. 1. It is obvious that a bimodal mixture of two narrow size fractions does not produce the particle size distribution of a “natural” grinding product, but is considered to give an acceptable approximation of it. Prior to each experiment the materials were dried for more than 10 h at 120 °C to assure an ignorable humidity. Confined particle beds were compressed in a hydraulic press (DP 1600/1, Hegewald & Peschke) at low stress velocity (vs = 0.05 cm/s) and in an energy-controlled spindle press (Müller, 1989) at high stress velocities (10 cm/s). The
http://dx.doi.org/10.1016/j.minpro.2014.10.004 0301-7516/© 2014 Elsevier B.V. All rights reserved.
Please cite this article as: Mütze, T., Energy dissipation in particle bed comminution, Int. J. Miner. Process. (2014), http://dx.doi.org/10.1016/ j.minpro.2014.10.004
2
T. Mütze / International Journal of Mineral Processing xxx (2014) xxx–xxx
100
particle size distribution Q3(x) in %
90
narrow fraction around 10 µm mixture (10 µm - 100 µm = 3:1)
80
mixture (10 µm - 100 µm = 1:1) mixture (10 µm - 100 µm = 1:3)
70
narrow fraction around 100 µm 60
broad fraction (< 100 µm)
50 40 30 20 10 0 0,1
1
10
100
1000
particle size x in µm Fig. 1. Particle size distributions of the feed (limestone): narrow fractions (10 μm, 100 μm), mixtures of narrow fractions (mass ratio 1:3, 1:1, 3:1), a broad fraction b 100 μm.
dimensions of the confined particle beds resulted from the requirements of an ideal particle bed (Mütze et al., 2011): - height 10 mm - diameter 30…70 mm The annular gap between the piston and the die was around 200 μm in order to release excess air which was pressed out of the particle bed by the compaction. Each test was repeated up to five times; the arithmetic mean was used for further calculations. The coefficient of variation was mostly below 2% or is noted otherwise. The particle size distributions of the products were measured by sieving and laser diffraction after sufficient dispersion. The size distributions were used to describe the grinding kinetics and changes in fineness. This focused on the volume-specific surface area SV which is one of the main influencing variables in fluid flow through packed beds. The energy absorption Em was calculated as the area enclosed by the stress and the relief curves and divided with the mass of the stressed particles. The stress and the relief curves were also used to describe the elastic–plastic compaction behaviour dependent on material and process conditions (Mütze, 2012, 2014). 3. Modelling and results Initially the model development focused on an appropriate description of the changes in the porosity of the particle bed as a result of compaction. Furthermore an evaluation was necessary to which extend the fluid is trapped inside the pore volume. Finally the energy consumption due to flow losses was calculated. 3.1. Compaction behaviour As known from literature, the important parameters in uniaxial compaction are the pressure, the material type, the particle size, the particle size distribution, and the stress velocity (Cooper and Eaton, 1962; Heckel, 1961; Kawakita and Lüdde, 1971; Mütze, 2012, 2014; Walker, 1923). Changes in the porosity are closely connected with the deformation of the particle bed. The plastic deformation of the particle bed can be described by the normalized compression Θplast (Eq. (1)). This compression correlates the actual change of the bulk
density ρb,plast due to the compaction pressure p with a maximum possible change indicated by the solid density ρs (Mütze, 2012, 2014): Θplast ðpÞ ¼
ρb;plast ðpÞ−ρb;plast;0 : ρs −ρb;plast;0
ð1Þ
The correlation between plastic deformation and pressure on the particle bed is often approximated by an equation with two parameters (Heckel, 1961; Kawakita and Lüdde, 1971; Walker, 1923). The influences of the mean particle size, the particle size distribution, and the stress velocity on the compaction behaviour have been studied recently by using Eq. (2) (Mütze, 2012, 2014):
Θplast ðpÞ ¼ Θ ln
Θ* p*
p 1þ p
ð2Þ
model parameter 1: reference compression model parameter 2: reference pressure
Due to its mathematical form, Eq. (2) reflects the Θplast(p)-plot of narrow size fractions as well as bimodal mixtures and broad size distributions with adequate accuracy (Fig. 2). The deviation between the measured and the calculated values of Θplast is below ±0.02 for most of the examined size fractions. The two model parameters describe opposite effects of compaction. While the reference pressure describes the resistance against the uniaxial compaction of a powder, the reference compression describes the aptitude of a powder to compaction. These contrary trends allow an exemplified displaying of only one parameter in order to show the effect of for instance the particle size and the stress velocity. The reference pressure and thus the resistance against compaction increases significantly with decreasing mean particle size and increasing stress velocity for narrow size fractions (Fig. 3). This correlates with the predominant compaction mechanism of breakage for coarse-grained materials and the increasing probability of breakage with increasing particle size. This effect is superimposed by the micro-plasticity of fine fractions at which the high strength of particles b10 μm leads to plastic deformation at the contact points when the yield stress is reached (Rumpf, 1965). Due to this
Please cite this article as: Mütze, T., Energy dissipation in particle bed comminution, Int. J. Miner. Process. (2014), http://dx.doi.org/10.1016/ j.minpro.2014.10.004
T. Mütze / International Journal of Mineral Processing xxx (2014) xxx–xxx
3
1
0,04
0,8
plast,model
0,9 0
-0,04
-
-0,08
plast,meas
plast
0,6
difference
0,7
0,5 0,4 0,3 0,2
-0,12
0,1 0
-0,16 0
100
200
300
400
500
600
pressure in MPa limestone 1 µm (measured)
quartz 10 µm (measured)
limestone 1 µm (approx.)
quartz 10 µm (approx.)
glass beads 50 µm (measured) glass beads 50 µm (approx.)
limestone 1 µm (difference)
quartz 10 µm (difference)
glass beads 50 µm (difference)
Fig. 2. Measurement, approximation by model (Eq. (2)), and difference between measurement and model of the normalized compaction as function of pressure (narrow size fractions; stress velocity 0,05 cm/s).
additional mechanism, the resistance against compression increases not proportionally to the decreasing particle size. The mean particle size of a material with a bimodal particle size distribution has a much smaller influence on the reference pressure than that of a material with a narrow unimodal size distribution (Mütze, 2012). Here, the reference pressure corresponds mainly to the behaviour of the finer component meaning that the fine fraction forms a matrix in which individual coarse particles can be embedded. Only if the mass fraction of coarse particles is sufficiently large, approximately 75 wt.%, the reference pressure and thus the compression behaviour of the particle bed changes gradually towards the behaviour of the coarser component.
3.2. Deaeration A critical stress velocity of compaction vs,crit helps to distinguish two cases (Mütze and Husemann, 2008): a) vs b vs,crit: the fluid is able to flow out of the pore volume (slow compaction: deaeration faster than stressing) b) vs N vs,crit: the fluid is trapped inside the pore volume (fast compaction: stressing faster than deaeration) This critical velocity depends on the geometry and the porosity of the particle bed, the pressure drop across the particle bed, the characteristics of the fluid, and the fineness of the particles (Mütze and
100
reference pressure p* in MPa
y = 355.04x-0.81 R² = 1.00
10
y = 176.28x-0.85 R² = 1.00
1
narrow size fractions, 10 cm/s narrow size fractions, 0.05 cm/s bimodal mixtures (1 µm : 10 µm), 0.05 cm/s bimodal mixtures (1 µm : 1000 µm), 0.05 cm/s bimodal mixtures (10 µm : 1000 µm), 0.05 cm/s
0,1 1
10
100
1000
mean particle size xm,3 in µm Fig. 3. Influence of granulometry of the feed material and stress velocity on reference pressure (limestone, narrow size fractions and bimodal mixtures, stress velocities of 0.05 cm/s and 10 cm/s, fitted by power functions).
Please cite this article as: Mütze, T., Energy dissipation in particle bed comminution, Int. J. Miner. Process. (2014), http://dx.doi.org/10.1016/ j.minpro.2014.10.004
4
T. Mütze / International Journal of Mineral Processing xxx (2014) xxx–xxx
Fig. 4. Velocity ratio of deaeration and stress in relation to the mean particle size of the feed (quartz, narrow fractions, parameter: compaction pressure and stress velocity).
Husemann, 2008). The changes of fineness and porosity due to an increasing pressure on the particle bed are described by the compaction model on one hand and the grinding kinetics on the other. The stress velocity itself affects the compression and crushing behaviour of the particle beds so marginally that it has no significant effect on the flow velocity inside the particle bed. Nonetheless a material dependent effect on the flow velocity can be shown: - In a coarse-grained particle bed the flow velocity decreases with increasing compacting pressure because of the high increase of volume-specific surface area and increasing flow resistance. - The flow velocity increases in fine-grained particle beds. Since the size reduction of this material is not as significant as the one of coarser feed fractions, the flow resistance increases only slightly. This small increase is more than counterbalanced by the compaction and a higher pressure gradient of air between the inside and outside of the particle bed.
These findings are illustrated in Fig. 4 using the stress velocity v's which allows the comparison of uniaxial stressing with perpendicular deaeration (Mütze and Husemann, 2008). A material dependent transition point exists between a behaviour more determined by crushing and one more determined by compaction. Hard materials like quartz, which are difficult to be crushed, show this transition between 100 and 1000 μm. Middle-hard materials like limestone show this transition between 20 and 100 μm. This different behaviour of different feed size fractions becomes less important with increasing compaction pressure on the particle bed because of increasing amount of fragments. 3.3. Flow losses Since at least a part of the fluid is trapped in the pore volume at high stress velocities, it will be compacted with the particle bed consuming additional compression energy. This additional energy can be a measure
mass specific adiabatic work Ead,m in mJ/g
10
1
0,1
0,01
limestone (1 µm)
quartz (10 µm)
limestone (100 µm)
quartz (100 µm)
limestone (1000 µm)
quartz (1000 µm)
silicon carbid (10 µm)
glass beads (50 µm)
0,001 1
10
100
1000
bed pressure pmax in MPa Fig. 5. Mass-specific volumetric work as a function of compaction pressure (narrow feed fractions).
Please cite this article as: Mütze, T., Energy dissipation in particle bed comminution, Int. J. Miner. Process. (2014), http://dx.doi.org/10.1016/ j.minpro.2014.10.004
T. Mütze / International Journal of Mineral Processing xxx (2014) xxx–xxx
for the flow losses of a rapidly stressed particle bed. Since the compression is very fast, an adiabatic change of state can be assumed in order to calculate these flow losses. The maximum magnitude of the flow losses was first determined by assuming that the compaction is fast enough to trap all the fluid which is initially in the pore volume. This assumption can be made since the fluid velocity in the pores ranges from 0.1 to 10 cm/s whereas typical stress velocities of roller mills are in the range from 20 to 4800 cm/s at comparable geometric conditions (Mütze and Husemann, 2008). The adiabatic change of state leads to a mass specific adiabatic work Ead,m which has to be performed on the fluid:
Ead ¼ −
pPore;0 V Pore;0 κ−1
"
# V Pore;0 κ−1 1− V Pore
ð3Þ
pPore;0 Ead ¼ Ead;m ¼ − m κ−1
8 3κ−1 9 2 > > = 1−Θplast ðpÞ 1− ρ ρs ρs −ρb;plast;0 < b;plast;0 5 1−4 > ρb;plast;0 ρs > 1−Θplast ðpÞ ; :
m pPore,0 κ Θplast(p)
ð4Þ
mass of stressed particles initial fluid pressure in pore volume isentropic exponent compaction function from Eq. (1)
The adiabatic work depends strongly on the feed material as well as the pressure on and the compaction behaviour of the particles (Fig. 5).
5
4. Conclusions The values of the energy absorption of the examined materials are between 10 and 25 J/g at 300 MPa bed pressure. The flow losses account for less than 0.1% of the total energy input which means that - this microprocess is negligible compared to friction losses, breakage energy, and plastic deformation work, and - the limiting assumptions of the calculation regarding the maximum magnitude of flow losses are justified. References Cooper, A.R., Eaton, L.E., 1962. Compaction behavior of several ceramic powders. J. Am. Ceram. Soc. 45, 97–101. Heckel, R.W., 1961. Density–pressure relationships in powder compaction. Trans. Metall. Soc. AIME 221, 671–675. Heegn, H., 1986. Veränderung der Festkörpereigenschaften bei mechanischer Aktivierung und Feinzerkleinerung. Akademie der Wissenschaften der DDR. Kawakita, K., Lüdde, K.-H., 1971. Some considerations on powder compression equations. Powder Technol. 4, 61–68. Müller, F., 1989. Hochdruckzerkleinerung im Gutbett bei Variation von Feuchte und Beanspruchungsgeschwindigkeit. TU Clausthal. Mütze, T., 2012. Modeling and parameter study of the elastic–plastic deformation. 7th International Conference for Conveying and Handling of Particulate Solids — CHoPS 2012, Friedrichshafen. Mütze, T., 2014. Modellhafte Beschreibung des Beanspruchungsverhaltens geschlossener Gutbetten. Chem. Ing. Tech. 86, 814–820. Mütze, T., Husemann, K., 2008. Compressive stress: effect of stress velocity on confined particle bed comminution. Chem. Eng. Res. Des. 86, 379–383. Mütze, T., Husemann, K., Peuker, U.A., 2011. Das ideale Gutbett. Chem. Ing. Tech. 83, 720–724. Rumpf, H., 1965. Die Einzelkornzerkleinerung als Grundlage einer technischen Zerkleinerungswissenschaft. Chem. Ing. Tech. 37, 187–202. Schubert, H., 1967. Zu einigen Fragen der Kollektivzerkleinerung. Chem. Technol. 19, 595–598. Walker, E.E., 1923. The properties of powders. Part VI. The compressibility of powders. Trans. Faraday Soc. 19, 73–82.
Please cite this article as: Mütze, T., Energy dissipation in particle bed comminution, Int. J. Miner. Process. (2014), http://dx.doi.org/10.1016/ j.minpro.2014.10.004