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A preliminary numerical investigation of agglomeration in a rotary drum
Minerals Engineering 15 (2002) 27–33 www.elsevier.com/locate/mine
A preliminary numerical investigation of agglomeration in a rotary drum B.K. Mishra a
a,*
, C. Thornton b, Daksha Bhimji
b
Department of Materials and Metallurgical Engineering, Indian Institute of Technology, Kanpur 208016, India b School of Engineering and Applied Science, Aston University, Birmingham, UK Received 9 March 2001; accepted 3 October 2001
Abstract In this paper, the discrete element method (DEM) is employed to analyse the wet granulation process. It is essentially a computer simulation of various types of interactions of particles in a rotating drum that eventually results in tracking of trajectories of each particle and particle agglomerates. Non-linear spring-dashpot type contact models are used to monitor particle–particle and particle–wall interactions. The contact between the particles in the presence of a binder is modelled by the well-known JKR theory. To establish the potential of this numerical approach, a 10-cm rotating drum is simulated with a polydispersed system of particles of top size 1 mm. A scraper is suitably located within the drum to restrict the motion of the adhered particles along the mill shell. A spray zone is located within the drum such that the particles receive the binder dosage only within this spray zone. A drying time is also specified beyond which the particles no longer adhere together. Implementation of these ideas within the computer program makes the simulation more realistic, which in turn allows prediction of the steady-state size distribution of the agglomerates. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Agglomeration; Simulation
1. Introduction Agglomeration or granulation of particles is achieved in a variety of ways in mechanical equipment where the particles are brought into contact. Once in contact, the admixture of the particulate mass and binder tends to remain intact and subsequently grows to form a granule subject to various competing forces. The process is typically carried out in revolving units that range from tumbling drums and fluidised beds to high-shear mixers and extruders. The objective is to utilise the fines to form spherical agglomerates. This is brought about in stages: formation of green agglomerates followed by drying and induration. While monitoring and controlling the process of agglomeration is known to be quite complex, it is the formation and growth of agglomerates that has been the subject of intense research for an understanding of the mechanisms involved. In the early 1970s, Sastry and Fuerstenau (1973), and subsequently Kapur (1978), detailed the mechanisms for * Corresponding author. Tel.: +91-0512-597263; fax: +91-0512590007. E-mail address: [email protected] (B.K. Mishra).
granule growth that included nucleation; growth; coalescence; layering and crushing. Understanding of these mechanisms helped in the formulation of mathematical models that include population balance models and mechanistic models (Litster and Liu, 1989; Ennis et al., 1991; Adetayo et al., 1995; Adetayo and Ennis, 2000; Manjunath et al., 2001; Sivakumar et al., 2001). These models are used to predict the size distribution of the agglomerates. However, population balance models have several drawbacks such as computational complexity for which exact solutions may not be possible. More importantly, it is quite cumbersome to develop the governing equation that takes into account all the processing parameters that affect the particle size distribution. Nevertheless, as of today population balance models are the only recourse available for quantitative analysis and control. Ennis (1996) gives a summary of recent developments in the area of agglomeration. From his account and further analysis of the literature since then, it appears that much remains to be done particularly with regard to the modelling of the process. The research work of Ennis et al. (1991) and more recently that of Liu et al. (2000) and Talu et al. (2000) indicate the necessary steps
0892-6875/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 8 9 2 - 6 8 7 5 ( 0 1 ) 0 0 1 9 4 - 7
28
B.K. Mishra et al. / Minerals Engineering 15 (2002) 27–33
to further progress. They have shown that a better understanding of the process can be obtained by considering both mechanical bulk properties and the interfacial properties of the agglomerate. However, their models lack the complete framework to predict the size distribution of the agglomerates. This paper considers a mechanistic approach similar to the one considered by Liu et al. (2000) and Talu et al. (2000) to develop a direct simulation tool by using the discrete element method (DEM) to solve agglomeration problems. From a mechanistic standpoint the agglomeration process can be visualised as a two-step process that begins with initiation of shear flow within the particulate mass and subsequent formation of agglomerates in the presence of binders. While the DEM has been successfully used to address the former problem as in the case of tumbling mills (Mishra and Rajamani, 1994; Rajamani et al., 2000) the solution to the latter problem has its basis in the theoretical work of Johnson et al. (1971) known as the JKR theory. DEM coupled with the JKR theory in principle allows a complete description of the agglomeration process. Due to computational restrictions, the model is restricted to two-dimensions (2D). It is quite clear that an adequate amount of information can be gathered from the two-dimension models until high powered computations will enable the use of millions of particles for complete simulation of the process in three-dimensions.
2. Theoretical considerations The granular dynamics model used in this study is the DEM originally proposed by Cundall and Strack (1979). In the simulations of granular dynamics, particles are treated as discrete elements, which are indestructible but interact with each other and any other elements present such as walls that they make physical contact with. The particle interaction rules are based on contact mechanics theories, which relate the contact force to the relative approach of the particle centroids. The computer program, which has been developed to simulate formation and motion of agglomerates (and is now called GR A N U L E ), is capable of modelling elastic, frictional, adhesive or non-adhesive spherical primary particles with or without plastic yield at the interparticle contacts. In this study, the adhesive option with no plastic deformation at the contacts is used. Thornton and Yin (1991) and Thornton and Ning (1998) have reported details of the implementation of the interaction laws. Application of the GR A N U L E code to agglomerate impact breakage has been reported by Thornton et al. (1996), Kafui and Thornton (2000), and Mishra and Thornton (2001). The program has been further modified to demonstrate the effect of adhesion on the agglomeration behaviour of particles in a rotating drum.
2.1. Interaction laws In the DEM particles are treated as discrete entities, which interact with each other at the interface when they are in contact. The particle interaction rules are based on well-established contact mechanics theories, which relate the contact force to the relative approach of particles. The Hertzian theory (see Johnson, 1985) is used to describe the particle interaction where the normal force–displacement relationship for spherical particles 1 and 2 with elastic moduli E1 and E2 , Poisson’s ratios m1 and m2 , and radii R1 and R2 is given as 4 P ¼ E R a3=2 : ð1Þ 3 In the above expression a is the relative approach that is related to the contact radius a by pffiffiffiffiffiffiffiffi ð2Þ a ¼ aR ; where 1 1 m21 1 m22 ¼ þ ; E E1 E2
and
1 1 1 ¼ þ : R R1 R2
The tangential interaction is non-linear and given by the theory developed by Mindlin and Deresiewicz (1953). In the presence of adhesion JKR theory is assumed (Johnson et al., 1971). It provides a relationship between the contact force and the relative approach as follows. The contact area in the presence of adhesion is 0 1=3 3R P a¼ ; ð3Þ 4E where P 0 is the effective Hertzian force, which would produce the same contact area. It is related to the adhesive and applied forces through the following: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð4Þ P 0 ¼ P þ 2Pc 4PPc þ 4Pc2 ; where P is the applied force and Pc is the pull-off force. According to the JKR theory the pull-off force is Pc ¼ 3pcR ;
ð5Þ
where c is the surface energy for each solid. The relative approach of the two spheres is related to the contact area a by rffiffiffiffiffiffiffiffiffiffi a2 4pca : ð6Þ a¼ E R The incremental normal force DF corresponding to an incremental relative approach Da is obtained as pffiffiffiffiffi pffiffiffi 3 P 3 Pc DF ¼ 2E a pffiffiffi pffiffiffiffiffi Da: ð7Þ 3 P Pc The tangential interactions in the presence of adhesion are dealt with by a combination of models (Savkoor and Briggs, 1977; Mindlin and Deresiewicz, 1953).
B.K. Mishra et al. / Minerals Engineering 15 (2002) 27–33
The particle interactions are modelled as a dynamic process, the evolution of which is advanced using an explicit finite difference scheme to obtain the incremental contact forces and then the incremental displacements of the particles. Each cycle of calculations that takes the system from time t to t þ Dt involves the application of incremental force–displacement interaction laws at each contact, resulting in new interparticle forces which are resolved to obtain new out-of-balance forces and moments for each particle. Numerical integration of Newton’s second law of motion yields the linear and rotational velocities of each particle. A second integration yields the incremental particle displacements and using the new particle positions and velocities, both linear and rotational, the calculation cycle is repeated in the next time step. The time step Dt used is a fraction of the critical time step Dtcr determined from the Rayleigh wave speed for the solid particles. During the process of agglomeration, particles come close to each other and form bonds whose strength is determined by the existing adhesive force. In dynamic situations, as it is practised in a rotating drum, the adhesive particles form agglomerates which subsequently grow and break. A steady-state distribution of agglomerate size is eventually attained within several revolutions of the mill. The purpose of this paper is to use DEM to examine the evolution of the agglomerate size spectra and the effects of various factors that affect the agglomeration process.
3. Results of simulation Inside the agglomerating drum particle clusters stabilise and grow. These can be of irregular shape whose trajectories inside the drum are quite difficult to
29
predict. Depending on the shape of the agglomerates the trajectories of particles of same mass may vary. In a series of numerical simulations trajectories of agglomerates of different shapes and sizes were tracked. The results showed that the trajectories were quite sensitive to the orientation of the major axis of the agglomerate with respect to the mill axis. A typical simulation result showing the trajectory of an agglomerate comprising of 50 particles at 60% of the critical speed of the mill is shown in Fig. 1. In this case it is seen that the agglomerate follows a trajectory that is more or less similar to the one that would have been obtained for a single particle. Also, at the end of the trajectory the agglomerate fractured after hitting the mill shell. This can be seen from the difference in the shades of the particles that form part of different daughter agglomerates. In Fig. 1 the insert shows two enlarged views of the agglomerate right before and after impact where the change in shades is more evident. In actual practice, one deals with not just one agglomerate but many agglomerates of varying sizes. To study the evolution of the size spectra of the agglomerates, a 10-cm diameter drum was simulated in two dimensions. A scraper was suitably positioned within the drum to restrict the motion of the adhered particles along the mill shell. A spray zone was located near the toe of the charge such that the particles receive the binder dosage only within this spray zone. Any particle within the spray zone was allowed to acquire a random amount of the peak value of the dosage. The peak value of the dosage was specified by the interfacial energy. A drying time was also specified beyond which the primary particles lose their acquired adhesive property. In other words, once out of the spray zone the particles were only allowed to form agglomerates
Fig. 1. Trajectory of a particle-agglomerate; Insert: agglomerate before and after impact.
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B.K. Mishra et al. / Minerals Engineering 15 (2002) 27–33
for a specified length of time. However, the agglomerates already formed retain their strength; they can only break by the existing breakage mechanisms prevailing within the rotating drum. For example, an agglomerate can break by colliding with other agglomerates or by hitting the mill shell. Once an agglomerate is broken, it cannot reform until it receives a fresh dose of binder. With the above assumptions regarding the agglomeration process, several simulations were carried out with a peak value of the interfacial energy of 1:5 J=m2 and a drying time corresponding to three-quarters of a drum revolution. The initial configuration of the particles at the start of the simulation is shown in Fig. 2(a), where all the particles were assigned the same levels of grey scale. The particle properties are given in Table 1. After all the particles were settled the speed of the mill was increased from rest to about 35% of the critical speed, and to induce the effect of agglomeration the procedure outlined in the previous paragraph was followed. Fig. 2(b) shows a snapshot after one and a quarter revolutions in which the single particles are assigned the same level of grey scale as in Fig. 2(a), but the agglomerates have different shades of grey depending on their size. It was observed that under the simulating conditions, the size distribution of the agglomerates attained steady-state after about just one and half revolutions of the drum. Further simulation beyond this stage led to agglomerate growth and breakage in a repetitive manner.
Visual inspection coupled with quantitative analysis of the agglomerate size distribution provides a better understanding of the evolution of the size spectra. However, the definition of the size of an agglomerate is imprecise. In practice, sieve sizes are used to classify agglomerates into different size ranges, but ambiguities arise since breakage may occur due to the vibratory nature of the sieving process and the fact that elongated agglomerates may or may not pass through the sieve depending on the orientation of their long axis. Similar uncertainties arise in numerical simulations when considering the size of the agglomerates. Here, the size is defined in terms of the mass of an agglomerate m divided by the mass of the total number (5000) of primary particles M. During the process of agglomeration, the particles by virtue of their interfacial energy tend to form agglomerates that grow with time. Depending on the spatial nature and magnitude of the forces acting on an agglomerate, it may either break or retain its size. Eventually, the charge attains a steady-state size distribution of agglomerates. This aspect of evolution in the size spectra is illustrated in Fig. 3. This figure shows the size distribution of the agglomerates at about half (f ¼ 0:5) and one and a third (f ¼ 1:33) revolutions from which it is clear that the size distributions tend to become progressively coarser. The finest size distribution is the feed size distribution at t ¼ 0 (see Table 1) and it becomes coarser at about half a revolution and eventually attains a steady-state distribution at about one and quarter of a
Fig. 2. Agglomeration of particles in a rotating drum: (a) initial state, (b) intermediate state after one and a quarter of revolutions.
Table 1 Primary particle properties Size distribution
Particle properties
Radius (mm)
Weight (%)
Modulus of elasticity (GPa)
70
1.0 0.75 0.5 Total number of particles ¼ 5000
51.4 32.5 16.1
Poisson’s ratio Density (g=cm3 ) Coefficient of friction Interface energy (J=m2 )
0.3 2.65 0.35 1.5
B.K. Mishra et al. / Minerals Engineering 15 (2002) 27–33
Fig. 3. Comparison of the size distribution during the process of agglomeration (Interface energy ¼ 1.5 J=m2 ).
revolution. Beyond this stage, only a marginal change in size distribution was noticed. It was observed that the time taken for the system to attain steady-state was quite sensitive to the magnitude of interfacial energy (strength of the binder) and its distribution over the particles and the drying time. It is quite unlikely that during agglomeration all the particles form agglomerates. In fact, there is finite fraction of the total particles that always remain as singlets (single particles). The dynamic variation of the singlets and the size of the largest agglomerate in terms of number of particles are shown in Fig. 4. It is seen that the process of agglomeration starts with a sharp decrease in the number of singlets, which quickly assumes a steady-state value. The sharp decrease in the number of singlets from the start-up to half a revolution is due to the formation of agglomerates and during this period
Fig. 4. Variation in the singlets and the number of particles associated with the largest agglomerate.
31
very little agglomerate breakage takes place. However, agglomerates do break as the process continues and towards the end of the process, around 1500 particles remain as singlets. However, during the same period, the number of particles constituting the largest agglomerate fluctuates between ca. 2000 and 500. Nevertheless, it is clear that a steady-state size distribution of the agglomerates is attained towards the end of the same period. It has been determined that for a given particulate system, a threshold value of the bond strength must exist between particles for agglomeration to occur. Once the threshold is reached the particulate system is able to sustain itself and reach a dynamic state of equilibrium in terms of the agglomerate size distribution. For the particulate system under study, it was found that the size distribution did not change significantly even by tripling the magnitude of the interfacial energy. This is evident from Fig. 5 in which the size distributions are compared corresponding to peak interfacial energies of 0.5 and 1:5 J=m2 . However, it was observed that beyond a certain maximum value of the interfacial energy the particulate mass tends to lock in position indicating that the shear is not sufficient to induce breakage within the agglomerate mass. While the simulation was able to illustrate the prevailing inter-relationship between the operating parameters and the corresponding size distribution, the understanding of the micro-mechanics of the process was far from clear. In particular it was difficult to explain why the number of particles associated with the largest agglomerate size fluctuated so much. The observed fluctuations in the number of particles were due to the agglomerate growth and breakage. Therefore, it is important from a practical standpoint to determine the
Fig. 5. Variation in size distribution due to change in interfacial energy.
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B.K. Mishra et al. / Minerals Engineering 15 (2002) 27–33
Fig. 6. Variation in mill torque: with and without adhesion.
factors that affect the largest size of the agglomerate. In order to study the variation in the size of the largest agglomerate, a careful analysis of the dynamics of the process was considered by analysing the variation in the torque. The torque required to hold the mass of the charge in motion at any mill speed can be determined from the contact forces acting on the mill shell. At first, the components of the contact force ~ F , and the distance from the mill centre to the contact point~ r, are determined. Then the torque is calculated as X ~ ~ T ¼ r ~ F; ð8Þ n
where n is the number of contacts between particles and the mill shell. The variation in the mill torque is shown in Fig. 6 for a given charge for two different operating conditions: with and without adhesion. The mill torque during agglomeration showed a torque profile that clearly shows a periodic fluctuation in the torque that can be correlated to the charge motion. But in order to do so, it is imperative to establish the same correlation for an identical system where particles do not form agglomerates. These two torque profiles are shown in Fig. 6. The difference in the torque profile lies only in the parameters of the periodicity. Under normal conditions of operations in the absence of adhesion, the sudden fall in the torque is attributed to the collapse of the charge that is sometimes referred to as surging. The same behaviour is observed within the particle system in the presence of adhesion, but this time, since the charge has become stiff the frequency of charge collapse has decreased. It is this collapse in the charge that is considered to be responsible
for the breakage of the larger agglomerates, hence limiting the agglomerate size.
4. Closing remarks The process of agglomeration is quite complex and not amenable to direct simulation. While the population balance technique is invariably used to describe the evolution of size spectra, it has merely served as a black box tool. For the first time an attempt is made here to show the evolution of size spectra by actually observing the formation and breakage of the particles. In this work the agglomeration process was modelled by the DEM that incorporates rigorous contact models for particle–particle interaction in the presence of adhesion. The model was developed to study the agglomeration problem in two-dimensions only. Using this model it was possible to predict the steady-state size distribution of the agglomerate mass for a very hypothetical and simplified system. Nevertheless, the preliminary simulation results clearly illustrate the dynamic nature of the process. In particular it was determined that due to the very nature of the agglomeration process in a rotating drum the largest size of the agglomerate attainable is finite for a given set of operating conditions.
References Adetayo, A.A., Ennis, B.J., 2000. A new approach to modelling granulation processes for simulation and control purposes. Powder Technol. 108 (2–3), 202–209.
B.K. Mishra et al. / Minerals Engineering 15 (2002) 27–33 Adetayo, A.A., Litster, J.D., Pratsinis, S.E., Ennis, B.J., 1995. Population balance modelling of drum granulation of materials with wide size distribution. Powder Technol. 82 (1), 37–49. Cundall, P.A., Strack, O.D.L., 1979. A discrete numerical model for granular assemblies. Geotechnique 29, 47–65. Ennis, B.J., 1996. Agglomeration and size enlargement session summary paper. Powder Technol. 88 (3), 203–225. Ennis, B.J., Tardos, G., Pfeffer, R., 1991. A micro level based characterisation of granular phenomena. Powder Technol. 65, 257– 272. Johnson, K.L., 1985. Contact Mechanics. Cambridge University Press, Cambridge, UK. Johnson, K.L., Kendall, K., Roberts, A.D., 1971. Surface energy and the contact of elastic solids. Proc. R. Soc. Lond., Ser. A 324, 301–313. Kafui, K.D., Thornton, C., 2000. Numerical simulation of impact breakage of a spherical crystalline agglomerate. Powder Technol. 109 (1–3), 113–133. Kapur, P.C., 1978. Kinetics of green pelletisation. Adv. Chem. Eng. 10, 55–61. Litster, J.D., Liu, L.X., 1989. Population balance modelling of fertiliser granulation. In: Proceedings of 5th International Conference on Agglomeration, ICHEME, pp. 611–617. Liu, L.X., Litster, J.D., Iveson, S.M., Ennis, B.J., 2000. Coalescence of deformable granules in wet granulation processes. AICHE J. 3 (46), 529–541. Mindlin, R.D., Deresiewicz, H., 1953. Elastic spheres in contact under varying oblique forces. J. Appl. Mech. 20, 327–344. Mishra, B.K., Rajamani, R.K., 1994. Simulation of charge motion in ball mills: numerical simulations. Int. J. Miner. Process. 40, 187–197.
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Mishra, B.K., Thornton, C., 2001. Impact breakage of particle agglomerates. Int. J. Miner. Process. 61, 225–239. Rajamani, R.K., Mishra, B.K., Venugopal, R., Datta, A., 2000. Discrete element analysis of tumbling mills. Powder Technol. 109 (1–3), 105–112. Sastry, K.V.S., Fuerstenau, D.W., 1973. Ballability index to quantify agglomerate growth by green pelletization. Trans. AIME 252, 254– 258. Savkoor, A.R., Briggs, G.A.D., 1977. The effect of tangential force on the contact of elastic solids in adhesion. Proc. R. Soc. Lond., Ser. A 356, 103–114. Sivakumar, S., Manjunath, S., Sahay, S.S., Ramakrishnan, V., Kapur, P.C., Pradip, Malghan, S.G., 2001. Population balance model for solid state sintering II. Grain growth. Ceram. Int. 27, 63–71. Manjunath, S., Kapur, P.C., Pradip, Malghan, S.G., 2001. Population balance model for solid state sintering I. Pore shrinkage and densification. Ceram. Int. 27, 57–62. Talu, Ilkay, Tardos, Gabriel, I., Khan Irfan, M., 2000. Computer simulation of wet granulation. Powder Technol. 110, 59–75. Thornton, C., Yin, K.K., 1991. Impact of elastic spheres with and without adhesion. Powder Technol. 65, 153–156. Thornton, C., Ning, Z., 1998. A theoretical model for the stick/bounce behaviour of adhesive, elastic-plastic spheres. Powder Technol. 99, 154–162. Thornton, C., Yin, K.K., Adams, M.J., 1996. Numerical simulation of the impact fracture and fragmentation of agglomerates. J. Phys. D: Appl. Phys. 29, 424–435.
A preliminary numerical investigation of agglomeration in a rotary drum B.K. Mishra a
a,*
, C. Thornton b, Daksha Bhimji
b
Department of Materials and Metallurgical Engineering, Indian Institute of Technology, Kanpur 208016, India b School of Engineering and Applied Science, Aston University, Birmingham, UK Received 9 March 2001; accepted 3 October 2001
Abstract In this paper, the discrete element method (DEM) is employed to analyse the wet granulation process. It is essentially a computer simulation of various types of interactions of particles in a rotating drum that eventually results in tracking of trajectories of each particle and particle agglomerates. Non-linear spring-dashpot type contact models are used to monitor particle–particle and particle–wall interactions. The contact between the particles in the presence of a binder is modelled by the well-known JKR theory. To establish the potential of this numerical approach, a 10-cm rotating drum is simulated with a polydispersed system of particles of top size 1 mm. A scraper is suitably located within the drum to restrict the motion of the adhered particles along the mill shell. A spray zone is located within the drum such that the particles receive the binder dosage only within this spray zone. A drying time is also specified beyond which the particles no longer adhere together. Implementation of these ideas within the computer program makes the simulation more realistic, which in turn allows prediction of the steady-state size distribution of the agglomerates. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Agglomeration; Simulation
1. Introduction Agglomeration or granulation of particles is achieved in a variety of ways in mechanical equipment where the particles are brought into contact. Once in contact, the admixture of the particulate mass and binder tends to remain intact and subsequently grows to form a granule subject to various competing forces. The process is typically carried out in revolving units that range from tumbling drums and fluidised beds to high-shear mixers and extruders. The objective is to utilise the fines to form spherical agglomerates. This is brought about in stages: formation of green agglomerates followed by drying and induration. While monitoring and controlling the process of agglomeration is known to be quite complex, it is the formation and growth of agglomerates that has been the subject of intense research for an understanding of the mechanisms involved. In the early 1970s, Sastry and Fuerstenau (1973), and subsequently Kapur (1978), detailed the mechanisms for * Corresponding author. Tel.: +91-0512-597263; fax: +91-0512590007. E-mail address: [email protected] (B.K. Mishra).
granule growth that included nucleation; growth; coalescence; layering and crushing. Understanding of these mechanisms helped in the formulation of mathematical models that include population balance models and mechanistic models (Litster and Liu, 1989; Ennis et al., 1991; Adetayo et al., 1995; Adetayo and Ennis, 2000; Manjunath et al., 2001; Sivakumar et al., 2001). These models are used to predict the size distribution of the agglomerates. However, population balance models have several drawbacks such as computational complexity for which exact solutions may not be possible. More importantly, it is quite cumbersome to develop the governing equation that takes into account all the processing parameters that affect the particle size distribution. Nevertheless, as of today population balance models are the only recourse available for quantitative analysis and control. Ennis (1996) gives a summary of recent developments in the area of agglomeration. From his account and further analysis of the literature since then, it appears that much remains to be done particularly with regard to the modelling of the process. The research work of Ennis et al. (1991) and more recently that of Liu et al. (2000) and Talu et al. (2000) indicate the necessary steps
0892-6875/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 8 9 2 - 6 8 7 5 ( 0 1 ) 0 0 1 9 4 - 7
28
B.K. Mishra et al. / Minerals Engineering 15 (2002) 27–33
to further progress. They have shown that a better understanding of the process can be obtained by considering both mechanical bulk properties and the interfacial properties of the agglomerate. However, their models lack the complete framework to predict the size distribution of the agglomerates. This paper considers a mechanistic approach similar to the one considered by Liu et al. (2000) and Talu et al. (2000) to develop a direct simulation tool by using the discrete element method (DEM) to solve agglomeration problems. From a mechanistic standpoint the agglomeration process can be visualised as a two-step process that begins with initiation of shear flow within the particulate mass and subsequent formation of agglomerates in the presence of binders. While the DEM has been successfully used to address the former problem as in the case of tumbling mills (Mishra and Rajamani, 1994; Rajamani et al., 2000) the solution to the latter problem has its basis in the theoretical work of Johnson et al. (1971) known as the JKR theory. DEM coupled with the JKR theory in principle allows a complete description of the agglomeration process. Due to computational restrictions, the model is restricted to two-dimensions (2D). It is quite clear that an adequate amount of information can be gathered from the two-dimension models until high powered computations will enable the use of millions of particles for complete simulation of the process in three-dimensions.
2. Theoretical considerations The granular dynamics model used in this study is the DEM originally proposed by Cundall and Strack (1979). In the simulations of granular dynamics, particles are treated as discrete elements, which are indestructible but interact with each other and any other elements present such as walls that they make physical contact with. The particle interaction rules are based on contact mechanics theories, which relate the contact force to the relative approach of the particle centroids. The computer program, which has been developed to simulate formation and motion of agglomerates (and is now called GR A N U L E ), is capable of modelling elastic, frictional, adhesive or non-adhesive spherical primary particles with or without plastic yield at the interparticle contacts. In this study, the adhesive option with no plastic deformation at the contacts is used. Thornton and Yin (1991) and Thornton and Ning (1998) have reported details of the implementation of the interaction laws. Application of the GR A N U L E code to agglomerate impact breakage has been reported by Thornton et al. (1996), Kafui and Thornton (2000), and Mishra and Thornton (2001). The program has been further modified to demonstrate the effect of adhesion on the agglomeration behaviour of particles in a rotating drum.
2.1. Interaction laws In the DEM particles are treated as discrete entities, which interact with each other at the interface when they are in contact. The particle interaction rules are based on well-established contact mechanics theories, which relate the contact force to the relative approach of particles. The Hertzian theory (see Johnson, 1985) is used to describe the particle interaction where the normal force–displacement relationship for spherical particles 1 and 2 with elastic moduli E1 and E2 , Poisson’s ratios m1 and m2 , and radii R1 and R2 is given as 4 P ¼ E R a3=2 : ð1Þ 3 In the above expression a is the relative approach that is related to the contact radius a by pffiffiffiffiffiffiffiffi ð2Þ a ¼ aR ; where 1 1 m21 1 m22 ¼ þ ; E E1 E2
and
1 1 1 ¼ þ : R R1 R2
The tangential interaction is non-linear and given by the theory developed by Mindlin and Deresiewicz (1953). In the presence of adhesion JKR theory is assumed (Johnson et al., 1971). It provides a relationship between the contact force and the relative approach as follows. The contact area in the presence of adhesion is 0 1=3 3R P a¼ ; ð3Þ 4E where P 0 is the effective Hertzian force, which would produce the same contact area. It is related to the adhesive and applied forces through the following: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð4Þ P 0 ¼ P þ 2Pc 4PPc þ 4Pc2 ; where P is the applied force and Pc is the pull-off force. According to the JKR theory the pull-off force is Pc ¼ 3pcR ;
ð5Þ
where c is the surface energy for each solid. The relative approach of the two spheres is related to the contact area a by rffiffiffiffiffiffiffiffiffiffi a2 4pca : ð6Þ a¼ E R The incremental normal force DF corresponding to an incremental relative approach Da is obtained as pffiffiffiffiffi pffiffiffi 3 P 3 Pc DF ¼ 2E a pffiffiffi pffiffiffiffiffi Da: ð7Þ 3 P Pc The tangential interactions in the presence of adhesion are dealt with by a combination of models (Savkoor and Briggs, 1977; Mindlin and Deresiewicz, 1953).
B.K. Mishra et al. / Minerals Engineering 15 (2002) 27–33
The particle interactions are modelled as a dynamic process, the evolution of which is advanced using an explicit finite difference scheme to obtain the incremental contact forces and then the incremental displacements of the particles. Each cycle of calculations that takes the system from time t to t þ Dt involves the application of incremental force–displacement interaction laws at each contact, resulting in new interparticle forces which are resolved to obtain new out-of-balance forces and moments for each particle. Numerical integration of Newton’s second law of motion yields the linear and rotational velocities of each particle. A second integration yields the incremental particle displacements and using the new particle positions and velocities, both linear and rotational, the calculation cycle is repeated in the next time step. The time step Dt used is a fraction of the critical time step Dtcr determined from the Rayleigh wave speed for the solid particles. During the process of agglomeration, particles come close to each other and form bonds whose strength is determined by the existing adhesive force. In dynamic situations, as it is practised in a rotating drum, the adhesive particles form agglomerates which subsequently grow and break. A steady-state distribution of agglomerate size is eventually attained within several revolutions of the mill. The purpose of this paper is to use DEM to examine the evolution of the agglomerate size spectra and the effects of various factors that affect the agglomeration process.
3. Results of simulation Inside the agglomerating drum particle clusters stabilise and grow. These can be of irregular shape whose trajectories inside the drum are quite difficult to
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predict. Depending on the shape of the agglomerates the trajectories of particles of same mass may vary. In a series of numerical simulations trajectories of agglomerates of different shapes and sizes were tracked. The results showed that the trajectories were quite sensitive to the orientation of the major axis of the agglomerate with respect to the mill axis. A typical simulation result showing the trajectory of an agglomerate comprising of 50 particles at 60% of the critical speed of the mill is shown in Fig. 1. In this case it is seen that the agglomerate follows a trajectory that is more or less similar to the one that would have been obtained for a single particle. Also, at the end of the trajectory the agglomerate fractured after hitting the mill shell. This can be seen from the difference in the shades of the particles that form part of different daughter agglomerates. In Fig. 1 the insert shows two enlarged views of the agglomerate right before and after impact where the change in shades is more evident. In actual practice, one deals with not just one agglomerate but many agglomerates of varying sizes. To study the evolution of the size spectra of the agglomerates, a 10-cm diameter drum was simulated in two dimensions. A scraper was suitably positioned within the drum to restrict the motion of the adhered particles along the mill shell. A spray zone was located near the toe of the charge such that the particles receive the binder dosage only within this spray zone. Any particle within the spray zone was allowed to acquire a random amount of the peak value of the dosage. The peak value of the dosage was specified by the interfacial energy. A drying time was also specified beyond which the primary particles lose their acquired adhesive property. In other words, once out of the spray zone the particles were only allowed to form agglomerates
Fig. 1. Trajectory of a particle-agglomerate; Insert: agglomerate before and after impact.
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for a specified length of time. However, the agglomerates already formed retain their strength; they can only break by the existing breakage mechanisms prevailing within the rotating drum. For example, an agglomerate can break by colliding with other agglomerates or by hitting the mill shell. Once an agglomerate is broken, it cannot reform until it receives a fresh dose of binder. With the above assumptions regarding the agglomeration process, several simulations were carried out with a peak value of the interfacial energy of 1:5 J=m2 and a drying time corresponding to three-quarters of a drum revolution. The initial configuration of the particles at the start of the simulation is shown in Fig. 2(a), where all the particles were assigned the same levels of grey scale. The particle properties are given in Table 1. After all the particles were settled the speed of the mill was increased from rest to about 35% of the critical speed, and to induce the effect of agglomeration the procedure outlined in the previous paragraph was followed. Fig. 2(b) shows a snapshot after one and a quarter revolutions in which the single particles are assigned the same level of grey scale as in Fig. 2(a), but the agglomerates have different shades of grey depending on their size. It was observed that under the simulating conditions, the size distribution of the agglomerates attained steady-state after about just one and half revolutions of the drum. Further simulation beyond this stage led to agglomerate growth and breakage in a repetitive manner.
Visual inspection coupled with quantitative analysis of the agglomerate size distribution provides a better understanding of the evolution of the size spectra. However, the definition of the size of an agglomerate is imprecise. In practice, sieve sizes are used to classify agglomerates into different size ranges, but ambiguities arise since breakage may occur due to the vibratory nature of the sieving process and the fact that elongated agglomerates may or may not pass through the sieve depending on the orientation of their long axis. Similar uncertainties arise in numerical simulations when considering the size of the agglomerates. Here, the size is defined in terms of the mass of an agglomerate m divided by the mass of the total number (5000) of primary particles M. During the process of agglomeration, the particles by virtue of their interfacial energy tend to form agglomerates that grow with time. Depending on the spatial nature and magnitude of the forces acting on an agglomerate, it may either break or retain its size. Eventually, the charge attains a steady-state size distribution of agglomerates. This aspect of evolution in the size spectra is illustrated in Fig. 3. This figure shows the size distribution of the agglomerates at about half (f ¼ 0:5) and one and a third (f ¼ 1:33) revolutions from which it is clear that the size distributions tend to become progressively coarser. The finest size distribution is the feed size distribution at t ¼ 0 (see Table 1) and it becomes coarser at about half a revolution and eventually attains a steady-state distribution at about one and quarter of a
Fig. 2. Agglomeration of particles in a rotating drum: (a) initial state, (b) intermediate state after one and a quarter of revolutions.
Table 1 Primary particle properties Size distribution
Particle properties
Radius (mm)
Weight (%)
Modulus of elasticity (GPa)
70
1.0 0.75 0.5 Total number of particles ¼ 5000
51.4 32.5 16.1
Poisson’s ratio Density (g=cm3 ) Coefficient of friction Interface energy (J=m2 )
0.3 2.65 0.35 1.5
B.K. Mishra et al. / Minerals Engineering 15 (2002) 27–33
Fig. 3. Comparison of the size distribution during the process of agglomeration (Interface energy ¼ 1.5 J=m2 ).
revolution. Beyond this stage, only a marginal change in size distribution was noticed. It was observed that the time taken for the system to attain steady-state was quite sensitive to the magnitude of interfacial energy (strength of the binder) and its distribution over the particles and the drying time. It is quite unlikely that during agglomeration all the particles form agglomerates. In fact, there is finite fraction of the total particles that always remain as singlets (single particles). The dynamic variation of the singlets and the size of the largest agglomerate in terms of number of particles are shown in Fig. 4. It is seen that the process of agglomeration starts with a sharp decrease in the number of singlets, which quickly assumes a steady-state value. The sharp decrease in the number of singlets from the start-up to half a revolution is due to the formation of agglomerates and during this period
Fig. 4. Variation in the singlets and the number of particles associated with the largest agglomerate.
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very little agglomerate breakage takes place. However, agglomerates do break as the process continues and towards the end of the process, around 1500 particles remain as singlets. However, during the same period, the number of particles constituting the largest agglomerate fluctuates between ca. 2000 and 500. Nevertheless, it is clear that a steady-state size distribution of the agglomerates is attained towards the end of the same period. It has been determined that for a given particulate system, a threshold value of the bond strength must exist between particles for agglomeration to occur. Once the threshold is reached the particulate system is able to sustain itself and reach a dynamic state of equilibrium in terms of the agglomerate size distribution. For the particulate system under study, it was found that the size distribution did not change significantly even by tripling the magnitude of the interfacial energy. This is evident from Fig. 5 in which the size distributions are compared corresponding to peak interfacial energies of 0.5 and 1:5 J=m2 . However, it was observed that beyond a certain maximum value of the interfacial energy the particulate mass tends to lock in position indicating that the shear is not sufficient to induce breakage within the agglomerate mass. While the simulation was able to illustrate the prevailing inter-relationship between the operating parameters and the corresponding size distribution, the understanding of the micro-mechanics of the process was far from clear. In particular it was difficult to explain why the number of particles associated with the largest agglomerate size fluctuated so much. The observed fluctuations in the number of particles were due to the agglomerate growth and breakage. Therefore, it is important from a practical standpoint to determine the
Fig. 5. Variation in size distribution due to change in interfacial energy.
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Fig. 6. Variation in mill torque: with and without adhesion.
factors that affect the largest size of the agglomerate. In order to study the variation in the size of the largest agglomerate, a careful analysis of the dynamics of the process was considered by analysing the variation in the torque. The torque required to hold the mass of the charge in motion at any mill speed can be determined from the contact forces acting on the mill shell. At first, the components of the contact force ~ F , and the distance from the mill centre to the contact point~ r, are determined. Then the torque is calculated as X ~ ~ T ¼ r ~ F; ð8Þ n
where n is the number of contacts between particles and the mill shell. The variation in the mill torque is shown in Fig. 6 for a given charge for two different operating conditions: with and without adhesion. The mill torque during agglomeration showed a torque profile that clearly shows a periodic fluctuation in the torque that can be correlated to the charge motion. But in order to do so, it is imperative to establish the same correlation for an identical system where particles do not form agglomerates. These two torque profiles are shown in Fig. 6. The difference in the torque profile lies only in the parameters of the periodicity. Under normal conditions of operations in the absence of adhesion, the sudden fall in the torque is attributed to the collapse of the charge that is sometimes referred to as surging. The same behaviour is observed within the particle system in the presence of adhesion, but this time, since the charge has become stiff the frequency of charge collapse has decreased. It is this collapse in the charge that is considered to be responsible
for the breakage of the larger agglomerates, hence limiting the agglomerate size.
4. Closing remarks The process of agglomeration is quite complex and not amenable to direct simulation. While the population balance technique is invariably used to describe the evolution of size spectra, it has merely served as a black box tool. For the first time an attempt is made here to show the evolution of size spectra by actually observing the formation and breakage of the particles. In this work the agglomeration process was modelled by the DEM that incorporates rigorous contact models for particle–particle interaction in the presence of adhesion. The model was developed to study the agglomeration problem in two-dimensions only. Using this model it was possible to predict the steady-state size distribution of the agglomerate mass for a very hypothetical and simplified system. Nevertheless, the preliminary simulation results clearly illustrate the dynamic nature of the process. In particular it was determined that due to the very nature of the agglomeration process in a rotating drum the largest size of the agglomerate attainable is finite for a given set of operating conditions.
References Adetayo, A.A., Ennis, B.J., 2000. A new approach to modelling granulation processes for simulation and control purposes. Powder Technol. 108 (2–3), 202–209.
B.K. Mishra et al. / Minerals Engineering 15 (2002) 27–33 Adetayo, A.A., Litster, J.D., Pratsinis, S.E., Ennis, B.J., 1995. Population balance modelling of drum granulation of materials with wide size distribution. Powder Technol. 82 (1), 37–49. Cundall, P.A., Strack, O.D.L., 1979. A discrete numerical model for granular assemblies. Geotechnique 29, 47–65. Ennis, B.J., 1996. Agglomeration and size enlargement session summary paper. Powder Technol. 88 (3), 203–225. Ennis, B.J., Tardos, G., Pfeffer, R., 1991. A micro level based characterisation of granular phenomena. Powder Technol. 65, 257– 272. Johnson, K.L., 1985. Contact Mechanics. Cambridge University Press, Cambridge, UK. Johnson, K.L., Kendall, K., Roberts, A.D., 1971. Surface energy and the contact of elastic solids. Proc. R. Soc. Lond., Ser. A 324, 301–313. Kafui, K.D., Thornton, C., 2000. Numerical simulation of impact breakage of a spherical crystalline agglomerate. Powder Technol. 109 (1–3), 113–133. Kapur, P.C., 1978. Kinetics of green pelletisation. Adv. Chem. Eng. 10, 55–61. Litster, J.D., Liu, L.X., 1989. Population balance modelling of fertiliser granulation. In: Proceedings of 5th International Conference on Agglomeration, ICHEME, pp. 611–617. Liu, L.X., Litster, J.D., Iveson, S.M., Ennis, B.J., 2000. Coalescence of deformable granules in wet granulation processes. AICHE J. 3 (46), 529–541. Mindlin, R.D., Deresiewicz, H., 1953. Elastic spheres in contact under varying oblique forces. J. Appl. Mech. 20, 327–344. Mishra, B.K., Rajamani, R.K., 1994. Simulation of charge motion in ball mills: numerical simulations. Int. J. Miner. Process. 40, 187–197.
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Mishra, B.K., Thornton, C., 2001. Impact breakage of particle agglomerates. Int. J. Miner. Process. 61, 225–239. Rajamani, R.K., Mishra, B.K., Venugopal, R., Datta, A., 2000. Discrete element analysis of tumbling mills. Powder Technol. 109 (1–3), 105–112. Sastry, K.V.S., Fuerstenau, D.W., 1973. Ballability index to quantify agglomerate growth by green pelletization. Trans. AIME 252, 254– 258. Savkoor, A.R., Briggs, G.A.D., 1977. The effect of tangential force on the contact of elastic solids in adhesion. Proc. R. Soc. Lond., Ser. A 356, 103–114. Sivakumar, S., Manjunath, S., Sahay, S.S., Ramakrishnan, V., Kapur, P.C., Pradip, Malghan, S.G., 2001. Population balance model for solid state sintering II. Grain growth. Ceram. Int. 27, 63–71. Manjunath, S., Kapur, P.C., Pradip, Malghan, S.G., 2001. Population balance model for solid state sintering I. Pore shrinkage and densification. Ceram. Int. 27, 57–62. Talu, Ilkay, Tardos, Gabriel, I., Khan Irfan, M., 2000. Computer simulation of wet granulation. Powder Technol. 110, 59–75. Thornton, C., Yin, K.K., 1991. Impact of elastic spheres with and without adhesion. Powder Technol. 65, 153–156. Thornton, C., Ning, Z., 1998. A theoretical model for the stick/bounce behaviour of adhesive, elastic-plastic spheres. Powder Technol. 99, 154–162. Thornton, C., Yin, K.K., Adams, M.J., 1996. Numerical simulation of the impact fracture and fragmentation of agglomerates. J. Phys. D: Appl. Phys. 29, 424–435.