52468_30[1]

445

PART 5

Instrumentation, Modeling, and Simulation

447

Use of Multiphysics Models for the Optimization of Comminution Operations

J.A. Herbst* and J.K. Lichter*

ABSTRACTRecent developments in multiphysics models are providing new opportunities for the optimi-zation of crushing and grinding equipment and their operation. Combinations of discreteelement methods (DEMs), discrete grain breakage (DGB) modeling, and multiphase flow(MPF) modeling are allowing interactions between breakage, fluid flow, equipment designfeatures, and wear processes to be predicted accurately. Therefore, a new era of microscaleoptimization is beginning for comminution. The dream of integrated optimization of com-minution equipment design and process operation should soon be realized.

In this paper, several areas that may benefit from this type of optimization are identi-fied. Examples for crushing, tumbling mill grinding, and stirred milling are presented.

INTRODUCT IONHistorical evidence points to the fact that effective process optimization requires the useof good mathematical models. This is true because accurate interpolation and extrapola-tion (only possible with a good model) are required for all types of optimizationsearches, whether empirical or analytical. Figure 1 shows the progression in comminu-tion models and modeling accuracy that has occurred during the last few decades.

Empirical models at the base of the triangle in Figure 1 are epitomized by the Bondequation (Bond 1952). Phenomenological models get their form from theory, but modelconstants must be determined experimentally. This model type is epitomized by popula-tion balance models (PBMs) (Hulburt and Katz 1964). The highest level of modeling isphysics based, where the model constants are calculated from fundamental materialproperties and actual equipment geometry and mechanical motion.

The generally accepted levels of prediction accuracy for comminution equipmentcapacity are on the order of 10% for the empirical level, 5% for the phenomenologicallevels, and 2% for the current models at the physics-based level.

By the turn of the century, a collection of tools termed high-fidelity simulation(HFS) tools had emerged. The HFS tools consist of DEMs, MPF, and DGB with a strongtie back to the PBM (Herbst and Nordell 2001). Each of these tools is described briefly inthe following paragraphs.

* Metso Minerals Optimization Services, Colorado Springs, Colorado

448 ADVANCES IN COMMINUTION INSTRUMENTATION, MODELING, AND SIMULATION

DEM simulations focus on discrete “particles” by solving Newton’s second law ofmotion applied to a particle of mass mi moving with velocity vi when it is acted upon by acollection of forces fij, including gravitational forces and particle–particle, particle–fluid,and particle–boundary interactive forces:

(EQ 1)

MPF simulations focus on continuous flow behavior of fluids and slurries modeledas pseudofluids by solving the full Navier-Stokes equation with a term for interactionsbetween particles and fluid:

(EQ 2)

where is the fluid density, v is the velocity vector, P is the pressure, g is the gravitationalforce constant, is the local solid fraction, and fi is the particle–fluid interaction force.

DGB simulations focus on discrete particles in the same way that DEMs do, except inthis case, each physical particle is made up of a set of discrete grains into which strainenergy can be stored/released and cracks can propagate along their boundaries, gov-erned by the energy conservation equation that governs the crack extension force, G:

(EQ 3)

where t is the crack width, u is the stored strain energy around the crack, and a is thecrack length.

These physics-based models have been found to be useful in comminution equipmentdesign and most recently have been determined to be extremely valuable for comminutionsystem optimization.

The fundamentals of the multiphysics models are presented briefly in the next sec-tion. The balance of this paper deals with the application of the models to optimizingcomminution operations.

Incr

easi

ng P

hysi

cs C

onte

nt

Increasing Accuracy

Microscale

Macroscale

Circa 2005

Circa 1972

Circa 1952

MPF

DGB

DEM

Physics Based

Population Balance Models

Phenomenological

Single-Parameter Models

Empirical

FIGURE 1 Evolution of modeling tools for comminution system simulation

D mivi

Dt------------------- fij=

DvDt------ P– 2v g 1

1 –----------- fi+ + +=

G 12t----- u

a------=

MULTIPHYSICS MODELS FOR THE OPTIMIZATION OF COMMINUTION OPERATIONS 449

FUNDAMENTALS OF H IGH - F IDEL I TY S IMULAT ION TO OLSDEM is a numerical technique developed for particle-flow simulation. Unlike continuousnumerical approaches, such as the finite element method or finite volume (FV) method,the DEM does not involve the integration of the equation of motion of the continuousmedium. Instead, the progress over time of every particle in the simulated system is fol-lowed by integration of the equations of motion for that particle. In this approach, virtu-ally everything is known about every particle in the system at every moment during thesimulation. The continuous parameters (bulk density, bulk particle velocity, etc.) areobtained by spatial and temporal averaging of the parameters of the motion of DEM par-ticles. In this sense, the DEM is closer to the experiment than to a continuous simulationtechnique; some investigators (Campbell 1997), therefore, prefer to use the term“numerical experiment” instead of “simulation.”

The basic steps of a DEM simulation are presented in Figure 2. After the initializationof the simulation, three basic steps are repeated at every simulation time step: (1) searchfor the particle–particle and particle–boundary contacts; (2) calculation of contactforces; and (3) integration of equations of motion (spatial advance of particles).

At the start of the simulation, information about position and characteristics (veloc-ities, sizes, shapes, etc.) of every particle in the simulation as well as parameters of thesimulation boundaries (geometries, motion parameters, etc.) must be supplied. Prepro-cessing programs usually generate all these values. In order for the basic DEM programto simulate the flow of nonbreakable particles, the boundaries have to be triangulated inthree dimensions or be subdivided into straight-line segments in two dimensions; thesetriangles and straight lines are basic shapes for the calculation of the particle–boundarycontact forces. One can use a variety of particle shapes, depending upon the characteris-tics of material to be simulated.

The next step in the DEM simulation process is to search for particle–particle andparticle–boundary contacts. This step is probably the most important in determining theefficiency and speed of the DEM software because the two remaining steps (force calcu-lation and integration of the equations of motion) are relatively easy to program effi-ciently. One can always simply search for the contacts of every particle against everyother particle and boundary element in the simulated system, but this approach creates

Start

End

Search forContacts

Calculate ContactForces

Integrate Equationsof Motion

FIGURE 2 Flowchart of a typical “soft particle” DEM simulation


an algorithm that is tremendously inefficient for the simulation of large numbers of par-ticles. The authors believe that the DEM programs they use involve a local search routinethat is one of the best available in the world at the present time.

The second cyclic (repeated at every time step) process of the DEM simulation is thecontact force calculation. Two basic approaches are used; the first one is used for non-breakable particle simulations. In this approach, the normal force acting on the particlecontact is the sum of viscous and elastic components. The elastic component is propor-tional to the particle–particle or particle–boundary distance of overlap with a constantcoefficient of proportionality (normal stiffness). The viscous normal force is proportionalto the relative normal velocity of the contacting members, also with constant coefficientof proportionality (contact viscosity). One can demonstrate that for this type of contactforce, the coefficient of restitution of the simulated particles is a velocity-independentvalue that depends upon particle mass and normal stiffness as well as contact viscosity.The tangential force on the contact is proportional to the relative displacement at thecontact point from the origination of the contact with a constant coefficient of propor-tionality (tangential stiffness) up to the limit of equal normal force multiplied by the fric-tion coefficient of the simulated material.

The second basic approach for the contact force calculation is used for breakableparticle simulations with the DGB technique. In this approach, the contacts are subdi-vided into two categories: the so-called “glued” and “collisional” contacts. The gluedcontacts can withstand tensile stresses; these contacts are responsible for holding ele-mentary particles together. One can think about this contact as a set of elastic fibers con-necting together the sides of elementary triangles or tetrahedrons (Potapov, Hopkins,and Campbell 1995; Potapov and Campbell 1996). These fibers have specified normaland tangential stiffness and can be broken (eliminated) once specified stress is reachedin the fiber. The collisional contacts still exist once the glued contacts are broken, as wellas those between different fragments or between fragments and boundaries. These contactscannot withstand tensile forces. The normal force on these contacts is viscous–elastic;the elastic component is proportional to the area or volume of particle overlap, and theviscous component is proportional to the rate of change of this area or volume. The tan-gential contact force is elastic with a frictional limit similar to the nonbreakable DEMapproach. The end result of such a model is that a brittle–elastic material with predict-able elastic and breakage properties is created (Potapov, Hopkins, and Campbell 1995;Potapov and Campbell 1996).

In addition to the contact forces, some noncontact forces are normally added to theparticles in the DEM simulation. These forces can include fluid drag forces in multiphasesimulations, forces of gravity, and so on. The calculation of these forces is usually straight-forward except for the case of two-way, solid–fluid coupling. This two-way couplingrequires simulation of the fluid motion together with DEM simulation of the particles.

To simulate the motion of the fluid, the standard FV technique, or in some instances,the smooth particle hydrodynamics technique, is employed. To achieve geometric flexi-bility, an unstructured grid (triangular in two dimensions and tetrahedral in threedimensions) can be used. It has been found by direct comparison with a structured gridthat the computational time overhead associated with an unstructured grid with respectto the structured grid is minimal. Free surfaces of the fluid in a piece of equipment aretraced by an extension of the volume of fluid (VOF) technique (see description in worksby Ferzinger and Peric [1997]). The authors and their co-workers have developed a versionof the VOF technique that has virtually no numerical diffusion. To achieve greater numeri-cal stability and to be able to employ larger time steps, the implicit pressure-correctionapproach based on the simple algorithm has been utilized.


Several ways of coupling of fluid and solid particle motion are described in the sci-entific and engineering literature. The most exact one is to simulate the flow of the fluidaround every particle. This technique requires several tens of fluid cells per solid parti-cle; it is also necessary to change the fluid mesh at every time step. At the present time,this technique can deal with, at most, tens of solid particles and thus has been found notto be useful for most mineral processing simulations. The authors and their colleagues haveprograms of this type based on meshless, smooth particle hydrodynamics and particle-in-cell techniques that are also used to establish the formulas for fluid–solid interactionterms. However, for the large-scale mill simulation, the approach based on the work ofDi Felice (1994) is used. It has been established by analyzing the experimentallyobtained solid–fluid interaction terms (Ergun 1952; Richardson and Zaki 1954; Foscolo,Gibilaro, and Waldram 1983) that one can describe the effect of the fluid surroundingthe solid particles simply by the summation of drag and pressure forces. The pressureforce term is simply the local pressure gradient multiplied by particle volume, and the dragforce depends only on the local solid fraction and the Reynolds number based on particlesize. Thus, in our program, we simulate the motion of the fluid through the FV tech-nique, taking into account change of the local solid fraction in the pressure-correctionequation and adding solid–fluid interaction force to the momentum equation. We simu-late motion of the solid particles using the DEM technique with additional pressure andsolid–fluid interaction forces applied to the particles. This allows us to implement thetwo-way solid–fluid interaction accurately and inexpensively in terms of computationaltime. This approach is also fully conservative in terms of mass and momentum.

Finally, the last step in the cyclic process (and probably the simplest one) is numeri-cal integration of the equations of motion of the particles. The numerical integration isperformed for both translational and rotational components of motion of every particlein the simulation. There are several possibilities for numerically integrating the equationsof motion. Several integration techniques of different orders of accuracy in time can beapplied. However, we found that a very simple first-order Euler integration is normallysufficient for the simulation purposes. We usually use this technique unless there arespecific requirements for a higher-order integration approach.

A few comments on computational requirements/limitations are in order at thispoint. There are no inherent limitations on particle size, size range, or number of sizefractions. However, irregularity in shape and a wide range of sizes require more compu-tational time than monosize spheres. Current practical limits on the number of particlesmodeled is about 1 million. Simulations presented in this paper were carried out on multi-processor computers with computation times varying between a few hours and a fewweeks.

To take full advantage of the multiphysics models for optimization, it is necessary tocapture the main features of these microscale simulations in another macroscopic simu-lator that can compute at many times real time. This has been accomplished through anenergy-based coupling of HFS results to more traditional PBMs. This coupling hasoccurred within the framework of MinOOcad, Metso’s dynamic flowsheet simulator,which can run entire mine-to-mill simulations at 200–400 times real time. Because ofthis link between HFSs and PBMs, a large number of scenarios can be evaluated quicklyas an optimizing search is being conducted.

RECENT APPL ICAT IONS OF THE HFS TO COMMINUT ION SYSTEM OPT IM IZAT IONDuring the last 3 years, the authors and their co-workers have applied HFS tools to themodeling of a wide variety of comminution operations including crushing, tumbling mill


grinding, and stirred mill grinding (Herbst and Potapov 2003). A few recent examplesare described in the following paragraphs.

Optimizing CrushersThe application of multiphysics modeling to crusher optimization is an emerging fieldwith considerable potential. One primary difference between the application of multi-physics modeling for crushers as compared to more typical applications for mill design isthat the use of breakage modeling, or DGB, is necessitated. Flow of material through acrusher cannot be achieved unless there is size reduction. The combined use of DEM/DGB modeling permits the evaluation of any of the process parameters of a crusher andwill provide information on capacity, crusher power draw, product size gradation, andlocalized forces on the wear components. Parameters that can be evaluated include, butare not limited to

Crusher chamber (wear liner) profile

Crusher speed

Crusher throw

Open side settings (or closed side settings)

HFS allows a search over the parameter space to allow the determination of the optimumoperating parameters.

Applications to date have successfully predicted the effect of crusher chamberdesign on crusher capacity and product gradation for jaw crushers and cone crushers.Figure 3 shows one such data set for a cone crusher. Simulated and predicted size distri-butions are shown.

Breakage modeling is a core element of crusher performance simulations, therefore,the starting point is characterization of the breakage properties of the ore, and the devel-opment of ore-specific DGB particles for the crusher feed. The ore breakage characteriza-tion is a two-stage process. The ore is first tested using a drop-weight tester (Figure 4).The drop-weight test is then repeated as a DGB experiment and the ore-specific crackpropagation energy is determined, which results in the same product size distribution asthat measured in the drop-weight test. This single ore breakage characteristic is typicallyconstant across all particle sizes and energy levels that would be simulated.

Once the ore breakage characteristics have been defined, the crusher can be mod-eled. Given the design of the crusher and the wear liners, the effect of crusher operatingparameters—namely, speed and throw—and the closed side setting can be predicted. Thecurrent challenge lies in the prediction of wear rates of the wear liners and the resultantcrushing chamber profile (e.g., the mantle and the concave liners in a gyratory crusher).The interest in multiphysics modeling for these applications is twofold: to assist in thecorrect selection of wear materials and for the prediction of liner life and the perfor-mance of the crusher as the liners wear.

DEM/DGB is likely to become a vital link in the development of a methodology toselect high-performance wear materials for crusher liners. Current activities include thedevelopment of a test methodology that will evaluate the performance of various wearmaterials based on a series of laboratory tests. Metal samples are subjected to a series oftests that determine a material’s resistance to abrasive wear and cutting or ploughing. Akey component in this testing is a knowledge of the magnitude and frequency of theforces applied to the wear material. A DEM/DGB experiment can provide those data forany configuration of a crusher, and is sensitive to the ore properties. Forces generatedwith soft ores will therefore be less than those generated for hard ores. Figure 5 shows arepresentation of the different wear mechanisms expressed as a function of surface loading


and distance of movement. The insert (top right) shows a laser measurement of a cutresulting from a high-load, long-glide-distance wear event.

Basic measurements plus DGB modeling can soon be used for optimal wear materialproperty selection in crushers.

OPT IM IZ ING SAG AND BALL M ILLSOptimization of tumbling mill performance at the microscale involves controlling impactsand particle transport processes. In practice, this is accomplished by manipulating mill

Time = 3.03 sec

Feed—MeasuredFeed—SimulatedOutput—MeasuredOutput—Simulated

10 10010

20

40

60

80

100

120

Cum

ulat

ive

% P

assi

ng

Particle Size, mm

FIGURE 3 Comparison of simulated (DGB) and measured size distributions for a cone crusher

5

4

3

2

1

0

TranslationalVelocity, m/sec

FIGURE 4 Drop-weight tester (left) and DGB simulation of test (right)


speed, media size and density, mill filling (ore and media), and the character of millinternals (liners and discharge systems).

The first case study presented here is for the optimization of throughput for a34 15.25-ft semiautogenous grinding (SAG) mill at Kinross’s Fort Knox operations (Hol-low and Herbst, in press). An adhoc team of four Metso Minerals Optimization Services(MMOS) specialists and members from the Fort Knox Process Improvement team joinedtogether to execute this project and to evaluate the results. The manipulated variablesexplored were liner design and mill speed. A base case was established for a mediumhardness ore (based on plant sampling and drop-weight tests) for a traditional linerdesign (66 lifter, rail liners) with a fraction of critical speed equal to 81% and a fillingcomposed of 17.4% ore and 12.6% ball by volume in the mill. HFSs were performed tosee the effect of lifter spacing, lifter face angle, lifter height, and plate thickness on wearand throughput using a methodology described in previous publications (Herbst andNordell 2001, 2002; Qiu et al. 2001). The optimum liner profile was selected based onidentifying a balance between high throughput and long wear life. A Metso “NaturalShape” design was chosen, which promised significant benefits from increased lifterspacing (with protection to the plate) and a more open but variable lifter face angle.

The prediction for the Natural Shape liner chosen was 6.1% higher throughput withabout the same wear life for the base case ore. The analysis of breakage and transport(including HFS modeling of the mill discharge system; Figure 6) using the MinOOcadcircuit simulator (Figure 7) suggested that both the new and the old liner designs couldbenefit from a reduced operating speed.

Model predictions and experimental data on capacity versus speed are shown forthe new liner design in Figure 8. The net result of changing liner design and speed wasabout an 8% increase in throughput.

The second tumbling mill case study was designed to determine optimum lifter pro-files, pulp densities, and ball sizes for 13.5 28-ft ball mills at Iron Ore Company of Canada

MicrofatigueMicrocracking

Severe Microcutting(Gouging)

Minimal WearLight Pressure

Abrasion (Grinding)

100

1,000

Sur

face

Loa

d, M

Pa

Gliding Distance

zxy

15˚

5

0mm

–5–0.6 0

mm0.6

18.6

–0.1

–18.8

FIGURE 5 Wear mechanisms as a function of surface loading and distance of movement


10

9

8

7

6

5

4

3

2

1

0


10

9

8

7

6

5

4

3

2

1

0


180 190200

210220

230

240

250

260

270

280

290

300

310

320330

34035001020

30

40

50

60

70

80

90

100

110

120

130

140150

160170

FIGURE 6 Simulation of charge motion (balls and particles) in the mill (left) and particle motion in the SAG discharger (right)

FIGURE 7 MinOOcad simulation SAG circuit performance with new liners


in Labrador City. Microscale breakage parameters were extracted from a combination oflaboratory tests and DEM simulations, as shown in Figure 9 (Herbst 2002).

These parameters were in turn used along with full-scale DEM simulations to pre-dict plant behavior with Metso’s MinOOcad package for different liners and operatingconditions, as shown in Figure 10.

These predictions were verified by plant tests. In Figure 11, a comparison is madebetween predicted and measured size distributions for two liner configurations investi-gated. The liner chosen was found to produce about a 5% increase in the amount of fines<44 m at the same energy consumption.

Optimum operating levels for liner design, mill speed, and percent solids were pro-jected by DEM/PBM simulations. Predicted performance improvements (on the order of25%) were verified after implementation in the plant.

Plant DataModel

Rotation Speed, % of Critical

Rel

ativ

e T

hrou

ghpu

t

0.85

0.90

0.95

1.00

1.05

1.10

1.15

60 65 70 75 80 85

FIGURE 8 SAG throughput for HFS-designed liners relative to base case as a function of percentage of critical speed of mill rotation

0.0008

Impact Power, x250 Watt

0.0006

0.0004

0.0002

0

FIGURE 9 Laboratory mill and associated DEM simulation snapshot


OPT IM IZ ING ST IR RED M ILLMultiphysics modeling has also been applied to the fundamental design evaluation andequipment scaling of stirred mills. One example is the evaluation of the media move-ment in the mill and the resultant forces being brought to bear on the comminution pro-cess. Figure 12 shows a snapshot of a VertiMill DEM simulation depicting the magnitudeof the shear forces developed between media.

The resultant energy spectra (Figure 13) provides a detailed account of the commi-nution environment within the mill. An understanding of the interrelationship betweenscrew design, media selection, and operating parameters can ultimately lead to a meth-odology to optimize the performance of VertiMills for a specific application.

Similarly, a detailed analysis of shear work on the screw can be used to show theareas and mechanisms for screw liner wear and can be used to evaluate the effect ofscrew design and mill operation on the wear of the liners.

CONCLUS IONSPhysics-based models are now available to optimize predictions of crushing and grindingoperations with a high level of accuracy. The HFS tools described herein have beenapplied to crushing, tumbling mill grinding, and stirred milling with confirmed success.Benefits on the order of 5% to 25% increases in throughput have been reported by usersof this technology.

It is expected that these physics-based models will play an increasing role in thecomminution optimization process. Links to PBMs are expected to be strengthened toallow circuit, full plant, and even mine-to-mill optimization in the near future.

FIGURE 10 DEM/PBM simulations of 13×28-ft mills for two liner configurations


ACKNOWLEDGMENTSThe authors thank Iron Ore Company of Canada, Metso Minerals (Mineral Processingand Wear Protection business lines), and Kinross Fort Knox for allowing these results tobe published. The support of several of our colleagues from MMOS is also gratefullyacknowledged.

10 100 1,0000

20

40

60

80

100

% P

assi

ng

Predicted Microscale Mill 7Measured Ball Mill 7FeedPredicted Microscale Mill 8Measured Ball Mill 8

FIGURE 11 Comparison of size distributions measured in plant tests for mills with two different liner configurations and those predicted from DEM/PBM simulations


REFERENCESBond, F.C. 1952. The third theory of comminution. AIME Transactions 193:484–494.

Campbell, C.S. 1997. Computer simulation of powder flows. Pages 777–793 in PowderTechnology Handbook. 2nd edition. Edited by K. Gotoh, H. Masuda, and K. Higashitani.New York: Marcel Dekker.

Di Felice, R. 1994. The voidage function for fluid-particle interaction systems. Interna-tional Journal of Multiphase Flow 20(1):153–159.

Ergun, S. 1952. Fluid flows through packed columns. Chemical Engineering Progress48:89–94.

Ferzinger, J.H., and M. Peric. 1997. Computational Methods for Fluid Dynamics. Berlin:Springer-Verlag.

5.0

4.5

4.0

3.5

3.0

2.5

2.0

1.5

1.0

0.5

0.0

Shear Power,Watts

5.0

4.5

4.0

3.5

3.0

2.5

2.0

1.5

1.0

0.5

0.0

Shear Power,Watts

FIGURE 12 Sectional side view and top view of a DEM simulation of a VertiMill (model VTM 1250)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

0.001 0.01 0.1 1 10 100 1,000

Single Collision Energy, J/kg

Tota

l She

ar E

nerg

y, J

/kg

At the WallAll Balls

FIGURE 13 Simulated shear energy dissipation at the wall and on the surface of balls as a function of the intensity of events


Foscolo, P.U., L.G. Gibilaro, and S.P. Waldram. 1983. A unified model for particulateexpansion of fluidized beds and flow in fixed porous media. Chemical EngineeringScience 38:1251–1260.

Herbst, J., and L. Nordell. 2001. Optimization of the design of SAG mill internals usinghigh fidelity simulation. Pages 1–5 in SAG 2001 Conference, Vancouver, BC.

Herbst, J.A. 2002. A microscale look at tumbling mill scale-up using high fidelity simula-tion. European Comminution Symposium, Heidelberg, Germany.

Herbst, J.A., and L.K. Nordell. 2002. Emergence of HFS as a design tool in mineral pro-cessing. Plant Design Symposium, Vancouver, BC.

Herbst, J.A., and A.K. Potapov. 2003. Radical innovations in mineral processing simulation.Mineral and Metallurgical Processing (October).

Hollow, J., and J. Herbst. In press. Attempting to quantify improvements in SAG linerperformance in a constantly changing ore environment. SAG 2006. Vancouver, BC.

Hulburt, H.M., and S. Katz. 1964. Some problems in particle technology—a statisticalmechanical formulation. Chemical Engineering Science 19:555–574.

Potapov, A.V., and C.S. Campbell. 1996. A three-dimensional simulation of brittle solidfracture. International Journal of Modern Physics C 7:717–729.

Potapov, A.V., M.A. Hopkins, and C.S. Campbell. 1995. A two-dimensional dynamic sim-ulation of solid fracture. Part I: Description of the model. International Journal ofModern Physics C 6:371–398.

Qiu, X., A. Potapov, M. Song, and L. Nordell. 2001. Prediction of wear of mill lifters usingdiscrete element method. SAG 2001 Conference, Vancouver, BC.

Richardson, J.F., and W.N. Zaki. 1954. Sedimentation and fluidization: Part I. Transac-tions of the Institution of Chemical Engineers 42:35–53.