- Email: [email protected]
Granular dynamics simulation of segregation phenomena in bubbling gas-fluidised beds
Powder Technology 109 Ž2000. 41–48 www.elsevier.comrlocaterpowtec
Granular dynamics simulation of segregation phenomena in bubbling gas-fluidised beds B.P.B. Hoomans, J.A.M. Kuipers ) , W.P.M. van Swaaij Department of Chemical Engineering, Twente UniÕersity of Technology, P.O. Box 217, 7500 AE Enschede, Netherlands Accepted 20 September 1999
Abstract A hard-sphere discrete particle model of a gas-fluidised bed was used in order to simulate segregation phenomena in systems consisting of particles of different sizes. In the model, the gas-phase hydrodynamics is described by the spatially averaged Navier–Stokes equations for two-phase flow. For each solid particle, the Newtonian equations of motion are solved taking into account the inter-particle and particle–wall collisions. The Ž2D. model was applied to a binary system consisting of particles of equal density, but different sizes where the homogeneous gas inflow velocity was equal to the minimum fluidisation velocity of the bigger particles. Segregation was observed over a time scale of several seconds although it did not become complete due to the continuous back mixing of the bigger particles by the bubbles. An analysis of the dynamics of the segregation in terms of mass fraction distributions is presented. When the particle–particle and particle–wall interactions were assumed to be perfectly elastic and perfectly smooth, segregation occurred very fast and was almost complete due to the absence of bubbles. q 2000 Elsevier Science S.A. All rights reserved. Keywords: Fluidisation; Granular dynamics simulation; Hard-spheres; Segregation; Binary systems
1. Introduction Segregation phenomena play an important role in the fluidisation of systems consisting of particles of different sizes andror densities. Typical examples of such processes are fluidised bed polymerisation and fluidised bed granulation among many others. In order to improve the performance of these processes, detailed knowledge about the distribution of the different solid species throughout the bed in different operating conditions is required. In a system consisting of particles of equal density, but different sizes, the bigger Žheavier. ones tend to reside at the bottom of the bed if the fluidisation velocity does not exceed the minimum fluidisation velocity Ž u mf . of the big particles. The big particles are in this case commonly referred to as jetsam. The smaller Žlighter. ones show the tendency to float and reside at the bed surface. These particles are commonly referred to as flotsam. At gas velocities much higher than the u mf of the big particles, better mixing is normally achieved. Segregation phenom-
) Corresponding author. Tel.: q31-53-489-4798; fax: q31-53-4894774; e-mail: [email protected]
ena in gas-fluidised beds have been the subject of several experimental studies reported in the literature. Nienow et al. w1x, Hoffmann et al. w2x and Wu and Baeyens w3x studied systems consisting of particles of equal density, but different sizes. Nienow and Naimer w4x performed experiments on systems consisting of particles of equal size, but different densities. Results for both types of systems were presented by Rowe et al. w5x. Due to increasing computer power, discrete particle models have become a very useful and versatile research tool in order to study the hydrodynamics of gas-fluidised beds. In these models, the Newtonian equations of motion for each individual particle are solved. Particle–particle and particle–wall interactions are taken into account directly, which is a clear advantage over two-fluid models that require closure relations for the solids-phase stress tensor w6,7x among many others.. When simulating gasfluidised beds with particles of different sizes andror particles of different densities, multi-fluid models can be used w7x, but several difficulties are encountered due to the fact that large sets of continuum equations have to be solved. In addition, and more fundamentally, significant problems arise when closure laws for the mutual interaction of particles belonging to different classes have to be
0032-5910r00r$ - see front matter q 2000 Elsevier Science S.A. All rights reserved. PII: S 0 0 3 2 - 5 9 1 0 Ž 9 9 . 0 0 2 2 5 - 9
42
B.P.B. Hoomans et al.r Powder Technology 109 (2000) 41–48
Fig. 1. The neighbour list principle using two cutoff distances. The shaded particles are stored in the neighbour list of the black one.
formulated. Although the Kinetic Theory of Granular Flow ŽKTGF. offers a theoretical framework to overcome this problem, the mathematics becomes very complicated and has, to the authors’ knowledge, only been applied to binary mixtures w17,18x. A discrete particle approach offers a more natural way to overcome these problems, since each individual particle in the simulation is tracked. Hence, discrete particle models are very useful in order to study in detail segregation phenomena in gas-fluidised beds consisting of particles of different sizes andror densities. Moreover, they can be used to generate data that can subsequently be used to develop closure models for continuum models. However, the number of particles that can be taken into account in a simulation is limited Žtypically10 6 ., which implies that currently, the method can only be applied to rather small systems of rather coarse particles. Tsuji et al. w8x developed a soft-sphere discrete particle model based on the work of Cundall and Strack w9x. In this approach, the particles are allowed to overlap slightly and this overlap is subsequently used to calculate the contact forces. Recently, Kawaguchi et al. w10x presented results for a three-dimensional version of this model. Schwarzer w11x used a model similar to that of Tsuji et al. w8x to simulate liquid-fluidised beds in which lubrication forces were also taken into account. Hoomans et al. w12x used a hard-sphere approach in their discrete particle model meaning that the particles interact through binary, instantaneous, inelastic collisions with friction. Xu and Yu w13x presented a hybrid technique where they used a contact force model in order to determine the inter-particle forces and a collision detection algorithm in order to determine the precise instant at which the particles first come into contact. Mikami et al. w14x recently extended the model of Tsuji et al. w8x in order to include cohesive forces between the particles. Using an extension of the model presented earlier w12x, Hoomans et al. w15x were able to simulate the dynamics of segregation phenomena in gas-fluidised beds for systems consisting of particles of equal density, but
different sizes as well as for systems consisting of particles of equal size, but different densities. Seibert and Burns w16x were able to predict segregation phenomena in liquidfluidised beds using a Monte Carlo simulation technique. However, the Monte Carlo technique is only capable of predicting a certain steady state and is not suitable to simulate the dynamics of segregation. In this paper, we will use a two-dimensional hard-sphere model w15x to study the dynamics of segregation in a gas-fluidised bed consisting of particles of equal density, but different sizes.
2. Granular dynamics Since most details of the model are presented in a previous paper w12x, only the key features will be summarised briefly here. The collision model as originally developed by Wang and Mason w19x is used to describe a binary, instantaneous, inelastic collision with friction. The key parameters of the model are the coefficient of Žnormal. restitution Ž0 F e F 1. and the coefficient of Ždynamic. friction Ž m G 0.. In the case of a perfectly elastic Ž e s 1. and perfectly smooth Ž m s 0. collision, no energy is dissipated. These collisions are referred to as ideal collisions.
Table 1 General parameter settings for the simulations Particles Shape Diameter big, d p, big Diameter small, d p, small Density, r e ew m, m w Total number
Bed Spherical 4.0 mm 1.5 mm 2480 kgrm3 0.96 0.86 0.15 5000
Width Height
150 mm 250 mm
Number x-cells, NX 15 Number y-cells, NY 25 Cell x-dimension Cell y-dimension
10 mm 10 mm
B.P.B. Hoomans et al.r Powder Technology 109 (2000) 41–48
In the case of non-ideal collisions Ž e - 1 andror m ) 0., energy is dissipated during the collision process. 2.1. Sequence of collisions In the hard-sphere approach, a sequence of binary collisions is processed one collision at a time. This requires that a collision list is compiled in which, for each particle, a collision partner and a corresponding collision time are
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stored. A constant time step is used to take the external forces into account and within this time step, the prevailing collisions are processed sequentially. In order to reduce the required CPU time, neighbour lists are used: for each particle, a list of neighbouring particles is stored and only for the particles in this list a check for possible collisions is performed. When simulating a binary system of particles of different sizes, two cutoff distances are used as schematically shown in Fig. 1. The neighbour list consists of all
Fig. 2. Snapshots of the particle configurations for a system of particles of different size with non-ideal particle interactions Ž e s 0.96, m s 0.15. using homogeneous gas inflow conditions at 1.5 u mf of the bigger particles.
B.P.B. Hoomans et al.r Powder Technology 109 (2000) 41–48
44
the small particles whose centres are found within the small square Žcoloured grey. and the big particles whose centres are found within the big square Žcoloured grey.. By using this approach, the number of particles in the neighbour list will never become too high, which significantly reduces both CPU time and memory requirements. 2.2. External forces In this work, we use the external forces analogous to those implemented in the two-fluid model described by Žamong others. Kuipers et al. w6x where, of course, the forces now act on a single particle: mp
dzp dt
s mp g q
Vp b
Ž1y´ .
Ž u y zp . y Vp= p,
Ž 1.
where m p represents the mass of a particle, zp its velocity, u the local gas velocity and Vp the volume of a particle. A similar equation of motion was used by Kawaguchi et al. w10x. In Eq. Ž1., the first term is due to gravity and the third term is the force due to the pressure gradient. The second term is due to the drag force where b represents an interphase momentum exchange coefficient as it usually appears in two-fluid models. For low void fractions Ž ´ 0.80., b is obtained from the well-known Ergun equation:
b s 150
2 Ž 1 y ´ . mg
´
d p2
q 1.75 Ž 1 y ´ .
rg dp
< u y zp < ,
Ž 2.
where d p represents the particle diameter, mg the viscosity of the gas and rg the density of the gas. For high void
fractions Ž ´ G 0.80., the following expression for the interphase momentum transfer coefficient has been used, which is basically the correlation presented by Wen and Yu w20x who extended the work of Richardson and Zaki w21x: 3 ´ Ž1y´ . b s Cd rg < u y zp < ´y2 .65 . 4 dp
Ž 3.
The drag coefficient Cd is a function of the particle Reynolds number:
° 24 Ž1 q 0.15Re C s~ Re ¢0.44 d
0.687 p
.
Re p - 1000
p
Ž 4.
Re p G 1000
where the particle Reynolds number in this case is defined as follows: Re p s
´rg < u y zp < d p mg
.
Ž 5.
The pressure gradient in the third term on the right hand side of Eq. Ž1. is calculated using a first-order approximation. The value at the position of the centre of the particle is then obtained via an area-weighted averaging technique using the values of the pressure gradients at the four surrounding grid nodes. This technique is also used to obtain local gas velocities and local void fractions at the position of the centre of mass of a particle as was described in our previous paper w12x. For the integration of Eq. Ž1., an explicit first-order scheme is used to update the velocities and the positions of the particles.
Fig. 3. Segregation profiles obtained from both single- and multiple-frame analysis for the simulation with non-ideal particle interactions. The height was made dimensionless by dividing by the bed height Ž0.25 m..
B.P.B. Hoomans et al.r Powder Technology 109 (2000) 41–48
3. Gas phase hydrodynamics The motion of the gas-phase is calculated from the following set of equations that can be seen as a generalised form of the spatially averaged Navier–Stokes equations for a two-phase gas–solid mixture w6x. Continuity equation gas phase: E Ž ´rg . q Ž = P ´rg u . s 0. Ž 6. Et Momentum equation gas phase: E Ž ´rg u . q Ž = P ´rg uu . Et s y´= p y S p y Ž = P ´tg . q ´rg g . Ž 7. In this work, transient, two-dimensional, isothermal ŽT s 293 K. flow of air at atmospheric conditions is considered. The constitutive equations can be found in Hoomans et al. w12x. The void fraction Ž ´ . is calculated on the basis of the positions of the particles in the bed as was described in our previous paper w12x. The two-dimensional void fraction obtained in this way is then converted into a volume-based void fraction using the following transformation function: 2 3r2 ´s1y Ž1y´2D . . Ž 8. ' p 3
(
Two-way coupling is achieved through the source term S p in Eq. Ž7.. This source term is expressed in terms of the interphase momentum transfer coefficient b : Npart
Sp s
Ý i
Vp ,i b Ž u y zp ,i .
Ž1y´ .
d Ž x y x p ,i . .
Ž 9.
By using this approach, the influence of each particle on the gas phase is taken into account, which is similar to
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the approach presented by Delnoij et al. w22x. The delta function in Eq. Ž9. has the dimension wmy3 x and it ensures that the force Ždivided by volume. is accounted for at the right position in the system. The reference volume is taken to be equal to: DX DY d p Žwhere DX and DY are the Xand Y-dimensions of a computational cell, respectively, and d p is the particle diameter., which implies that this volume is not the same for particles of different sizes. An interpolation technique mentioned in Section 2 is used to distribute this force per volume to the four nearest nodes in the grid. Note that the dimensions of a computational cell have to be chosen bigger than the diameter of the biggest particles in the simulation.
4. Results 4.1. General parameter settings The system that was simulated consisted of two types of particles of equal density, but different sizes. A homogeneous mixture of 250 glass ballotini particles of 4.0 mm diameter and 4750 particles of 1.5 mm diameter Ž50r50 wt.%. was homogeneously fluidised at the minimum fluidisation velocity of the bigger particles Ž1.7 mrs.. The general parameter settings for the simulations are summarised in Table 1. The coefficients of restitution and friction for particle–particle collisions were assumed to be the same for collisions between both classes of particles. 4.2. Segregation dynamics The first simulation was run for 50 s using the parameter settings described in Section 4.1 and a time step of
Fig. 4. Probability distribution of the jetsam mass fraction at 0.075 m above the distributor plate obtained over 50 s of simulation.
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B.P.B. Hoomans et al.r Powder Technology 109 (2000) 41–48
Fig. 5. Snapshots of the particle configurations for a system of particles of different sizes with ideal particle interactions Ž e s 1.0, m s 0.0. using homogeneous gas inflow conditions at 1.5 u mf of the bigger particles.
10y4 s. Snapshots of particle configurations are shown in Fig. 2. 1 In this figure, it can be observed that segregation does occur with the bigger particles accumulating at the bottom of the bed. However, this is a very dynamic situation as could be observed from animations where it became clear that the bigger particles are continuously transported to the upper regions of the bed by bubble
1 Animations of the simulations presented here are available at http:rrwww.ct.utwente.nlr ; pkranimations.html.
wakes and descend again in the denser regions. An analysis based on a single frame as we presented in a previous paper w15x, therefore, does not provide a complete understanding of the dynamics of segregation. A similar problem is encountered in segregation experiments where segregation profiles are determined after the gas supply is abruptly shut off and the bed is subsequently divided in sections that are separately analysed w2,3x. By repeating the experiments, the reliability of the results can be improved which is, however, rather time consuming. In granular dynamics simulations, the positions of all the particles in
B.P.B. Hoomans et al.r Powder Technology 109 (2000) 41–48
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Fig. 6. Segregation profiles obtained from both single- and multiple-frame analysis for the simulation with ideal particle interactions. The height was made dimensionless by dividing by the bed height Ž0.25 m..
the system are known at each time step. This enables the determination of segregation profiles by averaging over a large number of frames. For the results presented in Fig. 3, an average segregation profile was obtained using the particle configurations at each 0.01 s for a total duration of 50 s. Hence, a reliable average and a corresponding variance could be obtained from 5000 frames. The dimensionless bed height in this figure was obtained by dividing the vertical position by the height of the system Ž0.25 m.. A line indicating the average mass fraction jetsam is included in Fig. 3. In this figure, a segregation profile obtained from a single-frame analysis Žat t s 20.0 s. is included as well. Since this profile differs significantly from the average profile, it clearly shows that one has to be careful with the application of single-frame analysis. It only gives an impression of an instantaneous situation and therefore does not provide a complete understanding of the segregation dynamics. In Fig. 4, the probability distribution of the mass fraction jetsam at a dimensionless bed height of 0.3 Ž h s 0.075 m. obtained over the full duration of the simulation is presented. The wide spread in this distribution again emphasises that a single frame analysis may lead to an incomplete understanding of the segregation dynamics. It is important to note that the segregation profiles obtained from the simulation are based on the actual positions of the particles during fluidisation, while in experiments, the gas supply is always shut off before the collapsed bed is sectioned and analysed. 4.3. Effect of collision parameters Since the collision parameters turned out to have a significant influence on the bed hydrodynamics in previous
work w12x, an additional simulation was performed assuming perfectly elastic Ž e s 1. and perfectly smooth Ž m s 0. collisions. All other parameters were set to the same values as used in the simulation described in the previous paragraph. Snapshots of this simulation are presented in Fig. 5. The result is striking, a segregated state is reached after a few seconds and this situation does not change significantly anymore. Since there is no energy dissipation in this case, the particles do not tend to form regions of lower void fraction, which prevents the formation of bubbles as we reported earlier w12x. In the absence of bubbles, there is no mechanism present to transport the bigger particles to the upper regions of the bed, which stresses the important role of bubbles for solids mixing in fluidised beds. The simulation was continued for 20 s and an average segregation profile over 2000 frames was obtained. This average segregation profile is presented in Fig. 6 where it becomes clear that the segregation is far more pronounced than in the case with non-ideal collision parameters Ž e s 0.96, m s 0.15. and that the variance is much lower as well.
5. Conclusions A gas-fluidised bed consisting of particles of equal density, but different sizes was simulated using a two-dimensional hard-sphere discrete particle model. Using realistic values for the collision parameters Ž e s 0.96, m s 0.15., segregation was observed over a time scale of several seconds although the system never reached a clear steady state. The bigger Žjetsam. particles were continuously transported to the upper regions of the bed by the
B.P.B. Hoomans et al.r Powder Technology 109 (2000) 41–48
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bubbles and then moved down again in the denser regions. An average segregation profile was determined using 5000 frames Ževery 0.01 s for a total duration of 50 s.. The results showed a rather large variance. An analysis based on a single frame was found to lead to an incomplete understanding of the segregation dynamics. When the collisions were assumed to be perfectly elastic and perfectly smooth Ž e s 1.0, m s 0.0., a completely different behaviour was observed. In the absence of bubbles, segregation became almost complete after only a few seconds and this situation did not change significantly anymore during the remainder of the simulation. This stresses the important role of bubbles in the mixing of solids in fluidised beds and leaves us with the challenge to validate these simulation results in experiments where particles will have to be carefully selected on the basis of their collision parameters. We would like to point out the importance of these collision parameters as a means to characterise particles. However, there remains a need for accurate measurements of collision parameters of particles typically used in fluidisation.
6. Nomenclature Cd dp e g mp p Sp T t u zp Vp x
drag coefficient, w – x particle diameter, m coefficient of restitution, w – x gravitational acceleration, mrs 2 particle mass, kg pressure, Pa momentum source term ŽEq. Ž7.., Nrm3 temperature, K time, s gas velocity vector, mrs particle velocity vector, mrs particle volume, m3 position vector, m
Greek symbols b defined in Eqs. Ž2. and Ž3., kgrm3 s ´ void fraction, w – x m coefficient of friction, w – x mg gas viscosity, kgrms t gas-phase stress tensor, kgrms 2 rg gas density, kgrm3
Acknowledgements L.H.C. Heijnen and J.G. Schellekens are gratefully acknowledged for their contribution to the development of the simulation and post-processing software.
References w1x A.W. Nienow, N.S. Naimer, T. Chiba, Studies of segregationrmixing in fluidised beds of different size particles, Chem. Eng. Commun. 62 Ž1987. 52. w2x A.C. Hoffmann, L.P.B.M. Janssen, J. Prins, Particle segregation in fluidised binary mixtures, Chem. Eng. Sci. 48 Ž1993. 1583. w3x S.Y. Wu, J. Baeyens, Segregation by size difference in gas fluidized beds, Powder Technol. 98 Ž1998. 139. w4x A.W. Nienow, N.S. Naimer, Continuous mixing of two particulate species of different density in a gas fluidised bed, Trans. Inst. Chem. Eng. 58 Ž1980. 181. w5x P.N. Rowe, A.W. Nienow, A.J. Agbim, The mechanism by which particles segregate in gas fluidised beds: binary systems of nearspherical particles, Trans. Inst. Chem. Eng. 50 Ž1972. 310. w6x J.A.M. Kuipers, K.J. van Duin, F.P.H. van Beckum, W.P.M. van Swaaij, A numerical model of gas-fluidized beds, Chem. Eng. Sci. 47 Ž1992. 1913. w7x D. Gidaspow, J. Ding, U.K. Jayaswal, Multiphase Navier–Stokes equation solver. Numerical methods for multiphase flows, FED 91 Ž1990. 47. w8x Y. Tsuji, T. Kawaguchi, T. Tanaka, Discrete particle simulation of two dimensional fluidized bed, Powder Technol. 77 Ž1993. 79. w9x P.A. Cundall, O.D.L. Strack, A discrete numerical model for granular assemblies, Geotechnique 29 Ž1979. 47. w10x T. Kawaguchi, T. Tanaka, Y. Tsuji, Numerical simulation of two-dimensional fluidized beds using the discrete element method Žcomparison between the two- and three-dimensional models., Powder Technol. 96 Ž1998. 129. w11x S. Schwarzer, Sedimentation and flow through porous media: simulating dynamically coupled discrete and continuum phase, Phys. Rev. E 52 Ž1995. 6461. w12x B.P.B. Hoomans, J.A.M. Kuipers, W.J. Briels, W.P.M. van Swaaij, Discrete particle simulation of bubble and slug formation in a two-dimensional gas-fluidised bed: a hard-sphere approach, Chem. Eng. Sci. 51 Ž1996. 99. w13x B.H. Xu, A.B. Yu, Numerical simulation of the gas–solid flow in a fluidized bed by combining discrete particle method with computational fluid dynamics, Chem. Eng. Sci. 52 Ž1997. 2785. w14x T. Mikami, H. Kamiya, M. Horio, Numerical simulation of cohesive powder behavior in a fluidized bed, Chem. Eng. Sci. 53 Ž1998. 1927. w15x B.P.B. Hoomans, J.A.M. Kuipers, W.P.M. van Swaaij, Discrete particle simulation of segregation phenomena in dense gas-fluidized beds, in: L.S. Fan, T.M. Knowlton ŽEds.., Fluidization, Vol. IX, 1998, pp. 485–492. w16x K.D. Seibert, M.A. Burns, Simulation of structural phenomena in mixed-particle fluidized beds, AIChE J. 44 Ž1998. 528. w17x V. Kumaran, D.L. Koch, Properties of a bidisperse particle-gas suspension: Part 1. Collision time small compared with viscous relaxation time, J. Fluid Mech. 247 Ž1993. 623. w18x E. Manger, Modelling and simulation of gasrsolids flow in curvilinear coordinates, PhD Thesis, Telemark Institute of Technology, Norway, 1996. w19x Y. Wang, M.T. Mason, Two-dimensional rigid-body collisions with friction, J. Appl. Mech. 59 Ž1992. 635. w20x C.Y. Wen, Y.H. Yu, Mechanics of fluidization, Chem. Eng. Prog., Symp. Ser. 62 Ž62. Ž1966. 100. w21x J.F. Richardson, W.N. Zaki, Sedimentation and fluidization: Part I, Trans. Inst. Chem. Eng. 32 Ž1954. 35. w22x E. Delnoij, F.A. Lammers, J.A.M. Kuipers, W.P.M. van Swaaij, Dynamic simulation of dispersed gas–liquid two-phase flow using a discrete bubble model, Chem. Eng. Sci. 52 Ž1997. 1429.
Granular dynamics simulation of segregation phenomena in bubbling gas-fluidised beds B.P.B. Hoomans, J.A.M. Kuipers ) , W.P.M. van Swaaij Department of Chemical Engineering, Twente UniÕersity of Technology, P.O. Box 217, 7500 AE Enschede, Netherlands Accepted 20 September 1999
Abstract A hard-sphere discrete particle model of a gas-fluidised bed was used in order to simulate segregation phenomena in systems consisting of particles of different sizes. In the model, the gas-phase hydrodynamics is described by the spatially averaged Navier–Stokes equations for two-phase flow. For each solid particle, the Newtonian equations of motion are solved taking into account the inter-particle and particle–wall collisions. The Ž2D. model was applied to a binary system consisting of particles of equal density, but different sizes where the homogeneous gas inflow velocity was equal to the minimum fluidisation velocity of the bigger particles. Segregation was observed over a time scale of several seconds although it did not become complete due to the continuous back mixing of the bigger particles by the bubbles. An analysis of the dynamics of the segregation in terms of mass fraction distributions is presented. When the particle–particle and particle–wall interactions were assumed to be perfectly elastic and perfectly smooth, segregation occurred very fast and was almost complete due to the absence of bubbles. q 2000 Elsevier Science S.A. All rights reserved. Keywords: Fluidisation; Granular dynamics simulation; Hard-spheres; Segregation; Binary systems
1. Introduction Segregation phenomena play an important role in the fluidisation of systems consisting of particles of different sizes andror densities. Typical examples of such processes are fluidised bed polymerisation and fluidised bed granulation among many others. In order to improve the performance of these processes, detailed knowledge about the distribution of the different solid species throughout the bed in different operating conditions is required. In a system consisting of particles of equal density, but different sizes, the bigger Žheavier. ones tend to reside at the bottom of the bed if the fluidisation velocity does not exceed the minimum fluidisation velocity Ž u mf . of the big particles. The big particles are in this case commonly referred to as jetsam. The smaller Žlighter. ones show the tendency to float and reside at the bed surface. These particles are commonly referred to as flotsam. At gas velocities much higher than the u mf of the big particles, better mixing is normally achieved. Segregation phenom-
) Corresponding author. Tel.: q31-53-489-4798; fax: q31-53-4894774; e-mail: [email protected]
ena in gas-fluidised beds have been the subject of several experimental studies reported in the literature. Nienow et al. w1x, Hoffmann et al. w2x and Wu and Baeyens w3x studied systems consisting of particles of equal density, but different sizes. Nienow and Naimer w4x performed experiments on systems consisting of particles of equal size, but different densities. Results for both types of systems were presented by Rowe et al. w5x. Due to increasing computer power, discrete particle models have become a very useful and versatile research tool in order to study the hydrodynamics of gas-fluidised beds. In these models, the Newtonian equations of motion for each individual particle are solved. Particle–particle and particle–wall interactions are taken into account directly, which is a clear advantage over two-fluid models that require closure relations for the solids-phase stress tensor w6,7x among many others.. When simulating gasfluidised beds with particles of different sizes andror particles of different densities, multi-fluid models can be used w7x, but several difficulties are encountered due to the fact that large sets of continuum equations have to be solved. In addition, and more fundamentally, significant problems arise when closure laws for the mutual interaction of particles belonging to different classes have to be
0032-5910r00r$ - see front matter q 2000 Elsevier Science S.A. All rights reserved. PII: S 0 0 3 2 - 5 9 1 0 Ž 9 9 . 0 0 2 2 5 - 9
42
B.P.B. Hoomans et al.r Powder Technology 109 (2000) 41–48
Fig. 1. The neighbour list principle using two cutoff distances. The shaded particles are stored in the neighbour list of the black one.
formulated. Although the Kinetic Theory of Granular Flow ŽKTGF. offers a theoretical framework to overcome this problem, the mathematics becomes very complicated and has, to the authors’ knowledge, only been applied to binary mixtures w17,18x. A discrete particle approach offers a more natural way to overcome these problems, since each individual particle in the simulation is tracked. Hence, discrete particle models are very useful in order to study in detail segregation phenomena in gas-fluidised beds consisting of particles of different sizes andror densities. Moreover, they can be used to generate data that can subsequently be used to develop closure models for continuum models. However, the number of particles that can be taken into account in a simulation is limited Žtypically10 6 ., which implies that currently, the method can only be applied to rather small systems of rather coarse particles. Tsuji et al. w8x developed a soft-sphere discrete particle model based on the work of Cundall and Strack w9x. In this approach, the particles are allowed to overlap slightly and this overlap is subsequently used to calculate the contact forces. Recently, Kawaguchi et al. w10x presented results for a three-dimensional version of this model. Schwarzer w11x used a model similar to that of Tsuji et al. w8x to simulate liquid-fluidised beds in which lubrication forces were also taken into account. Hoomans et al. w12x used a hard-sphere approach in their discrete particle model meaning that the particles interact through binary, instantaneous, inelastic collisions with friction. Xu and Yu w13x presented a hybrid technique where they used a contact force model in order to determine the inter-particle forces and a collision detection algorithm in order to determine the precise instant at which the particles first come into contact. Mikami et al. w14x recently extended the model of Tsuji et al. w8x in order to include cohesive forces between the particles. Using an extension of the model presented earlier w12x, Hoomans et al. w15x were able to simulate the dynamics of segregation phenomena in gas-fluidised beds for systems consisting of particles of equal density, but
different sizes as well as for systems consisting of particles of equal size, but different densities. Seibert and Burns w16x were able to predict segregation phenomena in liquidfluidised beds using a Monte Carlo simulation technique. However, the Monte Carlo technique is only capable of predicting a certain steady state and is not suitable to simulate the dynamics of segregation. In this paper, we will use a two-dimensional hard-sphere model w15x to study the dynamics of segregation in a gas-fluidised bed consisting of particles of equal density, but different sizes.
2. Granular dynamics Since most details of the model are presented in a previous paper w12x, only the key features will be summarised briefly here. The collision model as originally developed by Wang and Mason w19x is used to describe a binary, instantaneous, inelastic collision with friction. The key parameters of the model are the coefficient of Žnormal. restitution Ž0 F e F 1. and the coefficient of Ždynamic. friction Ž m G 0.. In the case of a perfectly elastic Ž e s 1. and perfectly smooth Ž m s 0. collision, no energy is dissipated. These collisions are referred to as ideal collisions.
Table 1 General parameter settings for the simulations Particles Shape Diameter big, d p, big Diameter small, d p, small Density, r e ew m, m w Total number
Bed Spherical 4.0 mm 1.5 mm 2480 kgrm3 0.96 0.86 0.15 5000
Width Height
150 mm 250 mm
Number x-cells, NX 15 Number y-cells, NY 25 Cell x-dimension Cell y-dimension
10 mm 10 mm
B.P.B. Hoomans et al.r Powder Technology 109 (2000) 41–48
In the case of non-ideal collisions Ž e - 1 andror m ) 0., energy is dissipated during the collision process. 2.1. Sequence of collisions In the hard-sphere approach, a sequence of binary collisions is processed one collision at a time. This requires that a collision list is compiled in which, for each particle, a collision partner and a corresponding collision time are
43
stored. A constant time step is used to take the external forces into account and within this time step, the prevailing collisions are processed sequentially. In order to reduce the required CPU time, neighbour lists are used: for each particle, a list of neighbouring particles is stored and only for the particles in this list a check for possible collisions is performed. When simulating a binary system of particles of different sizes, two cutoff distances are used as schematically shown in Fig. 1. The neighbour list consists of all
Fig. 2. Snapshots of the particle configurations for a system of particles of different size with non-ideal particle interactions Ž e s 0.96, m s 0.15. using homogeneous gas inflow conditions at 1.5 u mf of the bigger particles.
B.P.B. Hoomans et al.r Powder Technology 109 (2000) 41–48
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the small particles whose centres are found within the small square Žcoloured grey. and the big particles whose centres are found within the big square Žcoloured grey.. By using this approach, the number of particles in the neighbour list will never become too high, which significantly reduces both CPU time and memory requirements. 2.2. External forces In this work, we use the external forces analogous to those implemented in the two-fluid model described by Žamong others. Kuipers et al. w6x where, of course, the forces now act on a single particle: mp
dzp dt
s mp g q
Vp b
Ž1y´ .
Ž u y zp . y Vp= p,
Ž 1.
where m p represents the mass of a particle, zp its velocity, u the local gas velocity and Vp the volume of a particle. A similar equation of motion was used by Kawaguchi et al. w10x. In Eq. Ž1., the first term is due to gravity and the third term is the force due to the pressure gradient. The second term is due to the drag force where b represents an interphase momentum exchange coefficient as it usually appears in two-fluid models. For low void fractions Ž ´ 0.80., b is obtained from the well-known Ergun equation:
b s 150
2 Ž 1 y ´ . mg
´
d p2
q 1.75 Ž 1 y ´ .
rg dp
< u y zp < ,
Ž 2.
where d p represents the particle diameter, mg the viscosity of the gas and rg the density of the gas. For high void
fractions Ž ´ G 0.80., the following expression for the interphase momentum transfer coefficient has been used, which is basically the correlation presented by Wen and Yu w20x who extended the work of Richardson and Zaki w21x: 3 ´ Ž1y´ . b s Cd rg < u y zp < ´y2 .65 . 4 dp
Ž 3.
The drag coefficient Cd is a function of the particle Reynolds number:
° 24 Ž1 q 0.15Re C s~ Re ¢0.44 d
0.687 p
.
Re p - 1000
p
Ž 4.
Re p G 1000
where the particle Reynolds number in this case is defined as follows: Re p s
´rg < u y zp < d p mg
.
Ž 5.
The pressure gradient in the third term on the right hand side of Eq. Ž1. is calculated using a first-order approximation. The value at the position of the centre of the particle is then obtained via an area-weighted averaging technique using the values of the pressure gradients at the four surrounding grid nodes. This technique is also used to obtain local gas velocities and local void fractions at the position of the centre of mass of a particle as was described in our previous paper w12x. For the integration of Eq. Ž1., an explicit first-order scheme is used to update the velocities and the positions of the particles.
Fig. 3. Segregation profiles obtained from both single- and multiple-frame analysis for the simulation with non-ideal particle interactions. The height was made dimensionless by dividing by the bed height Ž0.25 m..
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3. Gas phase hydrodynamics The motion of the gas-phase is calculated from the following set of equations that can be seen as a generalised form of the spatially averaged Navier–Stokes equations for a two-phase gas–solid mixture w6x. Continuity equation gas phase: E Ž ´rg . q Ž = P ´rg u . s 0. Ž 6. Et Momentum equation gas phase: E Ž ´rg u . q Ž = P ´rg uu . Et s y´= p y S p y Ž = P ´tg . q ´rg g . Ž 7. In this work, transient, two-dimensional, isothermal ŽT s 293 K. flow of air at atmospheric conditions is considered. The constitutive equations can be found in Hoomans et al. w12x. The void fraction Ž ´ . is calculated on the basis of the positions of the particles in the bed as was described in our previous paper w12x. The two-dimensional void fraction obtained in this way is then converted into a volume-based void fraction using the following transformation function: 2 3r2 ´s1y Ž1y´2D . . Ž 8. ' p 3
(
Two-way coupling is achieved through the source term S p in Eq. Ž7.. This source term is expressed in terms of the interphase momentum transfer coefficient b : Npart
Sp s
Ý i
Vp ,i b Ž u y zp ,i .
Ž1y´ .
d Ž x y x p ,i . .
Ž 9.
By using this approach, the influence of each particle on the gas phase is taken into account, which is similar to
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the approach presented by Delnoij et al. w22x. The delta function in Eq. Ž9. has the dimension wmy3 x and it ensures that the force Ždivided by volume. is accounted for at the right position in the system. The reference volume is taken to be equal to: DX DY d p Žwhere DX and DY are the Xand Y-dimensions of a computational cell, respectively, and d p is the particle diameter., which implies that this volume is not the same for particles of different sizes. An interpolation technique mentioned in Section 2 is used to distribute this force per volume to the four nearest nodes in the grid. Note that the dimensions of a computational cell have to be chosen bigger than the diameter of the biggest particles in the simulation.
4. Results 4.1. General parameter settings The system that was simulated consisted of two types of particles of equal density, but different sizes. A homogeneous mixture of 250 glass ballotini particles of 4.0 mm diameter and 4750 particles of 1.5 mm diameter Ž50r50 wt.%. was homogeneously fluidised at the minimum fluidisation velocity of the bigger particles Ž1.7 mrs.. The general parameter settings for the simulations are summarised in Table 1. The coefficients of restitution and friction for particle–particle collisions were assumed to be the same for collisions between both classes of particles. 4.2. Segregation dynamics The first simulation was run for 50 s using the parameter settings described in Section 4.1 and a time step of
Fig. 4. Probability distribution of the jetsam mass fraction at 0.075 m above the distributor plate obtained over 50 s of simulation.
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Fig. 5. Snapshots of the particle configurations for a system of particles of different sizes with ideal particle interactions Ž e s 1.0, m s 0.0. using homogeneous gas inflow conditions at 1.5 u mf of the bigger particles.
10y4 s. Snapshots of particle configurations are shown in Fig. 2. 1 In this figure, it can be observed that segregation does occur with the bigger particles accumulating at the bottom of the bed. However, this is a very dynamic situation as could be observed from animations where it became clear that the bigger particles are continuously transported to the upper regions of the bed by bubble
1 Animations of the simulations presented here are available at http:rrwww.ct.utwente.nlr ; pkranimations.html.
wakes and descend again in the denser regions. An analysis based on a single frame as we presented in a previous paper w15x, therefore, does not provide a complete understanding of the dynamics of segregation. A similar problem is encountered in segregation experiments where segregation profiles are determined after the gas supply is abruptly shut off and the bed is subsequently divided in sections that are separately analysed w2,3x. By repeating the experiments, the reliability of the results can be improved which is, however, rather time consuming. In granular dynamics simulations, the positions of all the particles in
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Fig. 6. Segregation profiles obtained from both single- and multiple-frame analysis for the simulation with ideal particle interactions. The height was made dimensionless by dividing by the bed height Ž0.25 m..
the system are known at each time step. This enables the determination of segregation profiles by averaging over a large number of frames. For the results presented in Fig. 3, an average segregation profile was obtained using the particle configurations at each 0.01 s for a total duration of 50 s. Hence, a reliable average and a corresponding variance could be obtained from 5000 frames. The dimensionless bed height in this figure was obtained by dividing the vertical position by the height of the system Ž0.25 m.. A line indicating the average mass fraction jetsam is included in Fig. 3. In this figure, a segregation profile obtained from a single-frame analysis Žat t s 20.0 s. is included as well. Since this profile differs significantly from the average profile, it clearly shows that one has to be careful with the application of single-frame analysis. It only gives an impression of an instantaneous situation and therefore does not provide a complete understanding of the segregation dynamics. In Fig. 4, the probability distribution of the mass fraction jetsam at a dimensionless bed height of 0.3 Ž h s 0.075 m. obtained over the full duration of the simulation is presented. The wide spread in this distribution again emphasises that a single frame analysis may lead to an incomplete understanding of the segregation dynamics. It is important to note that the segregation profiles obtained from the simulation are based on the actual positions of the particles during fluidisation, while in experiments, the gas supply is always shut off before the collapsed bed is sectioned and analysed. 4.3. Effect of collision parameters Since the collision parameters turned out to have a significant influence on the bed hydrodynamics in previous
work w12x, an additional simulation was performed assuming perfectly elastic Ž e s 1. and perfectly smooth Ž m s 0. collisions. All other parameters were set to the same values as used in the simulation described in the previous paragraph. Snapshots of this simulation are presented in Fig. 5. The result is striking, a segregated state is reached after a few seconds and this situation does not change significantly anymore. Since there is no energy dissipation in this case, the particles do not tend to form regions of lower void fraction, which prevents the formation of bubbles as we reported earlier w12x. In the absence of bubbles, there is no mechanism present to transport the bigger particles to the upper regions of the bed, which stresses the important role of bubbles for solids mixing in fluidised beds. The simulation was continued for 20 s and an average segregation profile over 2000 frames was obtained. This average segregation profile is presented in Fig. 6 where it becomes clear that the segregation is far more pronounced than in the case with non-ideal collision parameters Ž e s 0.96, m s 0.15. and that the variance is much lower as well.
5. Conclusions A gas-fluidised bed consisting of particles of equal density, but different sizes was simulated using a two-dimensional hard-sphere discrete particle model. Using realistic values for the collision parameters Ž e s 0.96, m s 0.15., segregation was observed over a time scale of several seconds although the system never reached a clear steady state. The bigger Žjetsam. particles were continuously transported to the upper regions of the bed by the
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bubbles and then moved down again in the denser regions. An average segregation profile was determined using 5000 frames Ževery 0.01 s for a total duration of 50 s.. The results showed a rather large variance. An analysis based on a single frame was found to lead to an incomplete understanding of the segregation dynamics. When the collisions were assumed to be perfectly elastic and perfectly smooth Ž e s 1.0, m s 0.0., a completely different behaviour was observed. In the absence of bubbles, segregation became almost complete after only a few seconds and this situation did not change significantly anymore during the remainder of the simulation. This stresses the important role of bubbles in the mixing of solids in fluidised beds and leaves us with the challenge to validate these simulation results in experiments where particles will have to be carefully selected on the basis of their collision parameters. We would like to point out the importance of these collision parameters as a means to characterise particles. However, there remains a need for accurate measurements of collision parameters of particles typically used in fluidisation.
6. Nomenclature Cd dp e g mp p Sp T t u zp Vp x
drag coefficient, w – x particle diameter, m coefficient of restitution, w – x gravitational acceleration, mrs 2 particle mass, kg pressure, Pa momentum source term ŽEq. Ž7.., Nrm3 temperature, K time, s gas velocity vector, mrs particle velocity vector, mrs particle volume, m3 position vector, m
Greek symbols b defined in Eqs. Ž2. and Ž3., kgrm3 s ´ void fraction, w – x m coefficient of friction, w – x mg gas viscosity, kgrms t gas-phase stress tensor, kgrms 2 rg gas density, kgrm3
Acknowledgements L.H.C. Heijnen and J.G. Schellekens are gratefully acknowledged for their contribution to the development of the simulation and post-processing software.
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