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On the numerical calibration of discrete element models for the simulation of bulk solids
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
On the Numerical Calibration of Discrete Element Models for the Simulation of Bulk Solids Torsten Gr6ger
a Andr6
Katterfeld b
aITASCA Consultants GmbH, Leithestrafle 111, 45886 Gelsenkirchen, Germany b IFSL, OvG-University of Magdeburg, PF4120, 39106 Magdeburg, Germany Abstract Due to the rapid increase of the computational power direct particle simulations, such as simulations on the basis of the Discrete Element Method (DEM), become increasingly popular in the field of bulk solids handling and processing. In order to obtain realistic simulations these projects require an accurate characterisation of the bulk solid properties in the Discrete Element Model. Therefore, the so called calibration of bulks solids deserves particular attention. The purpose of the numerical calibration is the adjustment of microscopic parameters, such as particle stiffness and inter-particle friction, in order to fit the macroscopic numerical behaviour, e.g. stress-strainbehaviour, measured on real experiments. The paper discusses the influence and effects of the microscopic parameters and explains the need for the development of new calibration methods.
Keywords: Materials Handling, Process Engineering, Discrete Element Simulation, Particle Flow Code, Calibration 1. Introduction In the past years the interest of materials handling industries and materials processing industries in Discrete Element Simulations has risen noticeable. The main reason for this is the enormous increase in the computational power available on the PC market. Today, ITASCA performs large scale simulations with more than 300,000 particles by means of the Particle Flow Code (PFC3d) for regular consulting jobs. Transition chutes as shown in Fig. 1 are representative examples for this.
Fig 1.: The depicted conveyors and transition chutes are examples for large scale simulations for material handling industries. Discrete Element Simulations can be considered as numerical experiments, which enable the contact-less measurement of microscopic quantities. These data cannot only be used to visualize the simulated process in a very illustrative manner but also to
533
T. GrOger and A. Katterfeld
534
compute macroscopic quantities, such as stresses and mean velocities, which are of particular interest for the design and the optimisation of equipment. In contrast to continuum mechanical methods the Discrete Element Method enables modelling of both fast flowing and resting zones of particulate materials with the same constitutive equations. This requires that all important microscopic quantities can be determined and mathematically modelled. However, experiences from consulting projects at ITASCA as well as research projects running at the IFSL, University of Magdeburg show that a lot of open questions exist regarding the calibration of the microscopic models. Issues arising from that are not restricted to mere numerical questions but rather concern the fundamental understanding of the characterization of flow properties by microscopically parameters.
2. The Principle of the Discrete Element Method The Discrete Element Method was developed by Cundall (1979) and a lot of detailed descriptions have been published ever since. Therefore only a brief survey will be given here. For algebraic modelling, the particles of bulk solids need to be represented by well defined geometrical objects. For performance reasons, spheres or sphere conglomerates are preferred. The particles themselves are assumed to be rigid however they are allowed to overlap. These overlaps are regarded as contact deformation from which an elastic contact force arises. Dependent on the applied contact model (Fig. 2) other types of contact forces can contribute to the total contact force. Accumulating all contact forces on a particle delivers the resulting force and moment for this particle. With the mass and the momentum of inertia the Newtonian equation can be integrated for a very short time step. This places a particle onto its new position and hence a new contact detection has to be performed as existing contacts may have vanished or new contacts may have formed. The described cycle needs to be executed in a loop until the desired process time is reached.
L Fig. 2: Example of a contact model for spherical particles. Spring - elastic force-displacementlaw, dashpot - viscous damping law, frictional element - Coulomb friction, meniscus liquid bridge (attractive force)
3. Contact properties 3.1. Elastic contact properties In the simplest case the elastic contact deformations can be modelled by a linear spring law. However, for spherical particles a Hertzian law is more appropriate. Only in very rare cases where the real particles exhibit a spherical shape Young's modulus and Poisson's ratio of the solid material can be used directly. If more complex particles are modelled by spheres this simplification needs to be compensated by a calibration of the contact law. For geo-mechanical applications the particle stiffness is adjusted by means of numerical triaxial tests with the goal to fit a measured macroscopic stress-straincurve. With models of very coarse geo-materials numerically stable simulations can be achieved with the realistic stiffness and the realistic masses. For quasi-static processes it
On the Numerical Calibration o f Discrete Element Models
535
is often applicable to up-scale volumes and/or masses in order to achieve numerical stability. Unfortunately, the majority of processes from the field of materials handling and process engineering exhibits both fast flow regimes and comparatively small particles, which do not allow a mass or volume scaling. In order to obtain numerical stable time steps that enable a reasonable computing time (less than a month for most consulting jobs), though, the particle stiffness needs to be reduced. For instance, large scale simulations on high-end PC's require the stiffness of minerals to be decreased by a factor of 100 or higher. Therefore, it is currently not possible to calibrate the particle stiffness for the majority of applications from process engineering and materials handling. It is recommended to choose the particle stiffness as high as the overall computational time allows it.
3.2. Damping Very often the size of the simulated particles is large enough that global damping effects of the surrounding medium can be neglected. For fine particles or surrounding fluids an appropriate damping law can be applied if needed. However, it is essential for most cases of handling and processing of bulk solids to consider the contact damping. Usually, contact damping is modelled in dependency on the relative velocity of the contact partners and occasionally dependent on the contact deformation. Except for nearly spherical particles that enable the measurement of the rebound height of a dropped particle no experiments are known that could be used for a calibration procedure. Practically relatively high contact damping coefficients are required. It is noted that higher damping forces can be achieved for a larger contact stiffness. 3.3. Friction In process engineering and materials handling the macroscopic friction angle of bulk solids is of particular importance. Besides cohesion, friction determines the flow properties of a particulate material significantly. Simultaneously, it is one of the most complex parameters since macroscopic friction is the result off particle friction and rolling friction on the microscopic level as well as the particle shape, the standard deviation of the particle size distribution, the packing structure and the packing density. In general, shear tests are performed numerically and experimentally in order to compare the inclination of the yield loci, which is a measure of the macroscopic friction. Fig. 3 shows examples of simulated yield loci. It is evident that the particle shape has a considerable influence on the macroscopic friction angle
kPa ~ ~ i~'~i ~
,
15
-20
~ e~
--
-10
~k
-
0
10 normal
20
kPa
40
stress
Fig. 3: Yield loci obtained from simulated shear tests. The inclination is a measure for the macroscopic friction. Cohesion (intersection with the ordinate) was caused by liquid bridges.
T. GrOger and A. Katterfeld
536
Unforttmately, the depicted particles composed of a number of spheres demonstrate two disadvantages. Firstly, with an increasing number of primary spheres the computational effort increases, too, and secondly in sections the particles can roll without any resistance. Therefore, it can be of advantage to introduce a rolling resistance (moment) that arises from an offset of the contact force from the centre of mass as depicted in Fig. 4.
Fig. 4: Examples for an offset of the contact force from the centre of mass. There are a number of factors that can be responsible for the force offset, such as the deformation due to rolling (Fig. 4 left), the particle shape (Fig. 4 middle) and asperities on the surface of the particles (Fig. 4 right), as well. These effects can all be covered with the coefficient of rolling friction, which is multiplied with the particle radius to obtain the amount of the offset (lever of the force). Fig. 5 shows the influence of the particle friction coefficient and the rolling friction coefficient on the macroscopic friction of a particulate system that is subjected to direct shearing in a Jenike shear cell. Obviously, the same macroscopic friction can be obtained from different combinations of rolling friction and particle friction (e.g. along the lines between two hatched areas). Since it is desirable to find the pair of coefficients that is valid for all flow conditions, regardless if it is slow shearing or fast flowing material, a single type of experiment seems to be insufficient for the determination of the two unknowns. Therefore ITASCA and the Institute of Materials Handling (IFSL) investigate further methods of measuring the macroscopic friction. Currently, the angle of repose formed in a rotating drum as well as formed by a vertical cylinder is investigated (Fig. 6).
macroscopic friction [°]
pr 0.3 pr( rolling friction p u.z p O. 1
Coulomb friction
Fig. 5: Simulated macroscopic friction angle [o] dependent on the particle (Coulomb) friction ~t[-] and rolling friction ~tr[-] for spheres d=2.3 to 2.6 mm in a shear tester.
537
On the Numerical Calibration of Discrete Element Models
Although no results can be presented, yet, it is reasonable to assume that in the process of forming the angel of repose the coefficients of friction and rolling friction have a differently weighted influence compared to shearing a consolidated system. This will lead to diagrams of the same type shown in Fig. 5. However different gradients are expected to be apparent. Hence, overlaying two of these diagrams should deliver an intersection at the desired macroscopic friction coefficient that delivers the pair of frictional coefficients that is representative for the majority of flow conditions. The described procedure is numerically expensive and further research is needed to find short cuts for the calibration process.
Fig. 6: Different experimental methods for the investigation of the angle of repose and their numerical representation.
4. Cohesion Macroscopic cohesion may arise from a number of microscopic causes, such as Vander-Waals-Forces and liquid bridges. The attractive forces on the microscopic level are comparatively well investigated and several mathematical models exist, which can be embedded in the contact model. Apart from sintering processes attractive forces become relevant for particle sizes smaller than lmm (Fig. 7). 10 2
10 4
~
10 0
~ 10-4 ._
10-2
10 .3
10-4 10-2
10-1
10 0
10 4
10 2
10 3
particle size (~tm)
Fig. 7: Influence of microscopic forces on the macroscopic tensile strength in dependency on the particle size.
538
T. GrOger and A. Katterfeld
Since smaller particle sizes are usually associated with a higher number of particles, large scale DEM-simulations are often restricted to relatively coarse particle systems. Therefore, only the relatively large forces arising from liquid bridges are currently of particular interest for the simulation of industrial applications. Fig. 8 shows two yield loci used for the calibration of cohesion of a wet particulate system consisting of glass spheres with a mean diameter of d=684~tm. The calibration process was straight forward for this particular system as the surface tension could be taken from a table and the volume of the bridge could be calculated from the water content (Gr5ger et. al. 2003). For bulk solids used in industrial applications this will not be possible in most cases. However, the procedure of calibration by means of direct shear tests is comparatively simple if the surface tension is known. The yield locus can then be shifted along the ordinate by varying of the volume of the liquid bridge.
1.6 1.4 1.2
/
"6' a_ 1.0 ~• .~. gg 0.8
-
~-~ l-
~0.4i
!
~
-- ring shear tester
1
0.2 ~ -
. . . . -- ![. . . . .... ~i]
•
DEM simulationr
i
..~
0.0 -0.5
Fig. 8"
0.0
0.5
1.0 1.5 normal stress [kPa]
2.0
2.5
3.0
Comparison of yield loci obtained from shear experiments and simulated shear tests on wet particle systems (d=6841am).
5. Summary Several microscopic parameters used for the direct simulation of particulate systems, such as powders and bulk solids have been discussed and their influence of the flow behaviour was explained. Currently, not all parameters can be calibrated to represent the properties of particulate systems realistically. In case of the elastic properties this is caused by the limitations of the available computational power. In other cases, such as contact damping and friction, fundamental experimental methods for the determination of these properties are still to be developed. The methods from geo-mechanics and soil mechanics are not sufficient to calibrate the more complex flow behaviour of materials from the fields of materials handling and process engineering.
References P.A. Cundall, 1979, Cundall, P.A.; Strack, O. D. L.: A discrete numerical model for granular assemblies. Geotechnique 29, No. 1, 47-65 T. Gr6ger, U. Ttiztin, D. Heyes, 2003, Modelling and Measuring of Cohesion in Wet Granular Materials, Powder Tech, 133,203-215
On the Numerical Calibration of Discrete Element Models for the Simulation of Bulk Solids Torsten Gr6ger
a Andr6
Katterfeld b
aITASCA Consultants GmbH, Leithestrafle 111, 45886 Gelsenkirchen, Germany b IFSL, OvG-University of Magdeburg, PF4120, 39106 Magdeburg, Germany Abstract Due to the rapid increase of the computational power direct particle simulations, such as simulations on the basis of the Discrete Element Method (DEM), become increasingly popular in the field of bulk solids handling and processing. In order to obtain realistic simulations these projects require an accurate characterisation of the bulk solid properties in the Discrete Element Model. Therefore, the so called calibration of bulks solids deserves particular attention. The purpose of the numerical calibration is the adjustment of microscopic parameters, such as particle stiffness and inter-particle friction, in order to fit the macroscopic numerical behaviour, e.g. stress-strainbehaviour, measured on real experiments. The paper discusses the influence and effects of the microscopic parameters and explains the need for the development of new calibration methods.
Keywords: Materials Handling, Process Engineering, Discrete Element Simulation, Particle Flow Code, Calibration 1. Introduction In the past years the interest of materials handling industries and materials processing industries in Discrete Element Simulations has risen noticeable. The main reason for this is the enormous increase in the computational power available on the PC market. Today, ITASCA performs large scale simulations with more than 300,000 particles by means of the Particle Flow Code (PFC3d) for regular consulting jobs. Transition chutes as shown in Fig. 1 are representative examples for this.
Fig 1.: The depicted conveyors and transition chutes are examples for large scale simulations for material handling industries. Discrete Element Simulations can be considered as numerical experiments, which enable the contact-less measurement of microscopic quantities. These data cannot only be used to visualize the simulated process in a very illustrative manner but also to
533
T. GrOger and A. Katterfeld
534
compute macroscopic quantities, such as stresses and mean velocities, which are of particular interest for the design and the optimisation of equipment. In contrast to continuum mechanical methods the Discrete Element Method enables modelling of both fast flowing and resting zones of particulate materials with the same constitutive equations. This requires that all important microscopic quantities can be determined and mathematically modelled. However, experiences from consulting projects at ITASCA as well as research projects running at the IFSL, University of Magdeburg show that a lot of open questions exist regarding the calibration of the microscopic models. Issues arising from that are not restricted to mere numerical questions but rather concern the fundamental understanding of the characterization of flow properties by microscopically parameters.
2. The Principle of the Discrete Element Method The Discrete Element Method was developed by Cundall (1979) and a lot of detailed descriptions have been published ever since. Therefore only a brief survey will be given here. For algebraic modelling, the particles of bulk solids need to be represented by well defined geometrical objects. For performance reasons, spheres or sphere conglomerates are preferred. The particles themselves are assumed to be rigid however they are allowed to overlap. These overlaps are regarded as contact deformation from which an elastic contact force arises. Dependent on the applied contact model (Fig. 2) other types of contact forces can contribute to the total contact force. Accumulating all contact forces on a particle delivers the resulting force and moment for this particle. With the mass and the momentum of inertia the Newtonian equation can be integrated for a very short time step. This places a particle onto its new position and hence a new contact detection has to be performed as existing contacts may have vanished or new contacts may have formed. The described cycle needs to be executed in a loop until the desired process time is reached.
L Fig. 2: Example of a contact model for spherical particles. Spring - elastic force-displacementlaw, dashpot - viscous damping law, frictional element - Coulomb friction, meniscus liquid bridge (attractive force)
3. Contact properties 3.1. Elastic contact properties In the simplest case the elastic contact deformations can be modelled by a linear spring law. However, for spherical particles a Hertzian law is more appropriate. Only in very rare cases where the real particles exhibit a spherical shape Young's modulus and Poisson's ratio of the solid material can be used directly. If more complex particles are modelled by spheres this simplification needs to be compensated by a calibration of the contact law. For geo-mechanical applications the particle stiffness is adjusted by means of numerical triaxial tests with the goal to fit a measured macroscopic stress-straincurve. With models of very coarse geo-materials numerically stable simulations can be achieved with the realistic stiffness and the realistic masses. For quasi-static processes it
On the Numerical Calibration o f Discrete Element Models
535
is often applicable to up-scale volumes and/or masses in order to achieve numerical stability. Unfortunately, the majority of processes from the field of materials handling and process engineering exhibits both fast flow regimes and comparatively small particles, which do not allow a mass or volume scaling. In order to obtain numerical stable time steps that enable a reasonable computing time (less than a month for most consulting jobs), though, the particle stiffness needs to be reduced. For instance, large scale simulations on high-end PC's require the stiffness of minerals to be decreased by a factor of 100 or higher. Therefore, it is currently not possible to calibrate the particle stiffness for the majority of applications from process engineering and materials handling. It is recommended to choose the particle stiffness as high as the overall computational time allows it.
3.2. Damping Very often the size of the simulated particles is large enough that global damping effects of the surrounding medium can be neglected. For fine particles or surrounding fluids an appropriate damping law can be applied if needed. However, it is essential for most cases of handling and processing of bulk solids to consider the contact damping. Usually, contact damping is modelled in dependency on the relative velocity of the contact partners and occasionally dependent on the contact deformation. Except for nearly spherical particles that enable the measurement of the rebound height of a dropped particle no experiments are known that could be used for a calibration procedure. Practically relatively high contact damping coefficients are required. It is noted that higher damping forces can be achieved for a larger contact stiffness. 3.3. Friction In process engineering and materials handling the macroscopic friction angle of bulk solids is of particular importance. Besides cohesion, friction determines the flow properties of a particulate material significantly. Simultaneously, it is one of the most complex parameters since macroscopic friction is the result off particle friction and rolling friction on the microscopic level as well as the particle shape, the standard deviation of the particle size distribution, the packing structure and the packing density. In general, shear tests are performed numerically and experimentally in order to compare the inclination of the yield loci, which is a measure of the macroscopic friction. Fig. 3 shows examples of simulated yield loci. It is evident that the particle shape has a considerable influence on the macroscopic friction angle
kPa ~ ~ i~'~i ~
,
15
-20
~ e~
--
-10
~k
-
0
10 normal
20
kPa
40
stress
Fig. 3: Yield loci obtained from simulated shear tests. The inclination is a measure for the macroscopic friction. Cohesion (intersection with the ordinate) was caused by liquid bridges.
T. GrOger and A. Katterfeld
536
Unforttmately, the depicted particles composed of a number of spheres demonstrate two disadvantages. Firstly, with an increasing number of primary spheres the computational effort increases, too, and secondly in sections the particles can roll without any resistance. Therefore, it can be of advantage to introduce a rolling resistance (moment) that arises from an offset of the contact force from the centre of mass as depicted in Fig. 4.
Fig. 4: Examples for an offset of the contact force from the centre of mass. There are a number of factors that can be responsible for the force offset, such as the deformation due to rolling (Fig. 4 left), the particle shape (Fig. 4 middle) and asperities on the surface of the particles (Fig. 4 right), as well. These effects can all be covered with the coefficient of rolling friction, which is multiplied with the particle radius to obtain the amount of the offset (lever of the force). Fig. 5 shows the influence of the particle friction coefficient and the rolling friction coefficient on the macroscopic friction of a particulate system that is subjected to direct shearing in a Jenike shear cell. Obviously, the same macroscopic friction can be obtained from different combinations of rolling friction and particle friction (e.g. along the lines between two hatched areas). Since it is desirable to find the pair of coefficients that is valid for all flow conditions, regardless if it is slow shearing or fast flowing material, a single type of experiment seems to be insufficient for the determination of the two unknowns. Therefore ITASCA and the Institute of Materials Handling (IFSL) investigate further methods of measuring the macroscopic friction. Currently, the angle of repose formed in a rotating drum as well as formed by a vertical cylinder is investigated (Fig. 6).
macroscopic friction [°]
pr 0.3 pr( rolling friction p u.z p O. 1
Coulomb friction
Fig. 5: Simulated macroscopic friction angle [o] dependent on the particle (Coulomb) friction ~t[-] and rolling friction ~tr[-] for spheres d=2.3 to 2.6 mm in a shear tester.
537
On the Numerical Calibration of Discrete Element Models
Although no results can be presented, yet, it is reasonable to assume that in the process of forming the angel of repose the coefficients of friction and rolling friction have a differently weighted influence compared to shearing a consolidated system. This will lead to diagrams of the same type shown in Fig. 5. However different gradients are expected to be apparent. Hence, overlaying two of these diagrams should deliver an intersection at the desired macroscopic friction coefficient that delivers the pair of frictional coefficients that is representative for the majority of flow conditions. The described procedure is numerically expensive and further research is needed to find short cuts for the calibration process.
Fig. 6: Different experimental methods for the investigation of the angle of repose and their numerical representation.
4. Cohesion Macroscopic cohesion may arise from a number of microscopic causes, such as Vander-Waals-Forces and liquid bridges. The attractive forces on the microscopic level are comparatively well investigated and several mathematical models exist, which can be embedded in the contact model. Apart from sintering processes attractive forces become relevant for particle sizes smaller than lmm (Fig. 7). 10 2
10 4
~
10 0
~ 10-4 ._
10-2
10 .3
10-4 10-2
10-1
10 0
10 4
10 2
10 3
particle size (~tm)
Fig. 7: Influence of microscopic forces on the macroscopic tensile strength in dependency on the particle size.
538
T. GrOger and A. Katterfeld
Since smaller particle sizes are usually associated with a higher number of particles, large scale DEM-simulations are often restricted to relatively coarse particle systems. Therefore, only the relatively large forces arising from liquid bridges are currently of particular interest for the simulation of industrial applications. Fig. 8 shows two yield loci used for the calibration of cohesion of a wet particulate system consisting of glass spheres with a mean diameter of d=684~tm. The calibration process was straight forward for this particular system as the surface tension could be taken from a table and the volume of the bridge could be calculated from the water content (Gr5ger et. al. 2003). For bulk solids used in industrial applications this will not be possible in most cases. However, the procedure of calibration by means of direct shear tests is comparatively simple if the surface tension is known. The yield locus can then be shifted along the ordinate by varying of the volume of the liquid bridge.
1.6 1.4 1.2
/
"6' a_ 1.0 ~• .~. gg 0.8
-
~-~ l-
~0.4i
!
~
-- ring shear tester
1
0.2 ~ -
. . . . -- ![. . . . .... ~i]
•
DEM simulationr
i
..~
0.0 -0.5
Fig. 8"
0.0
0.5
1.0 1.5 normal stress [kPa]
2.0
2.5
3.0
Comparison of yield loci obtained from shear experiments and simulated shear tests on wet particle systems (d=6841am).
5. Summary Several microscopic parameters used for the direct simulation of particulate systems, such as powders and bulk solids have been discussed and their influence of the flow behaviour was explained. Currently, not all parameters can be calibrated to represent the properties of particulate systems realistically. In case of the elastic properties this is caused by the limitations of the available computational power. In other cases, such as contact damping and friction, fundamental experimental methods for the determination of these properties are still to be developed. The methods from geo-mechanics and soil mechanics are not sufficient to calibrate the more complex flow behaviour of materials from the fields of materials handling and process engineering.
References P.A. Cundall, 1979, Cundall, P.A.; Strack, O. D. L.: A discrete numerical model for granular assemblies. Geotechnique 29, No. 1, 47-65 T. Gr6ger, U. Ttiztin, D. Heyes, 2003, Modelling and Measuring of Cohesion in Wet Granular Materials, Powder Tech, 133,203-215