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Building population balance model for fluidized bed melt granulation: lessons from kinetic theory of granular flow
Powder Technology 142 (2004) 103 – 109 www.elsevier.com/locate/powtec
Building population balance model for fluidized bed melt granulation: lessons from kinetic theory of granular flow H.S. Tan a,*, M.J.V. Goldschmidt b, R. Boerefijn c, M.J. Hounslow a, A.D. Salman a, J.A.M. Kuipers d a
Department of Chemical and Process Engineering, University of Sheffield, Mappin Street, Sheffield S1 3JD, UK b Akzo Nobel Chemicals Research, P.O. Box 9300, 6800 SB Arnhem, The Netherlands c Unilever Research, P.O. Box 114, 3130 AC Vlaardingen, The Netherlands d Department of Chemical Engineering, Twente University of Technology, 7500 AE Enschede, The Netherlands Available online
Abstract The purpose of this paper is to develop a theoretically sound basis for the equi-partition of kinetic energy (EKE) kernel recently developed by our group to describe the evolution of granule size distributions in fluidized bed granulation. The approach taken is to show first by distinct element modelling that the statistics of fluctuating velocity and thus frequency of collisions are well described by the kinetic theory of granular flow (KTGF)—that is, Maxwellian in nature. It is then possible to use KTGF to show that the size dependence of the aggregation process should indeed be given by the EKE kernel and that the rate constant depends only on the granular temperature, the particle density, the radial distribution function and the efficiency of collisions. It is shown how the kernel developed can be used to describe the evolution of a granule size distribution when low-density sodium carbonate particles (Light Ash) are sprayed with polyethylene glycol. D 2004 Elsevier B.V. All rights reserved. Keywords: Population balance model; Fluidized bed melt granulation; Kinetic theory of granular flow
1. Introduction Fluidized bed granulation has been an important powder production process in the industry for several decades. Compared to other granulation techniques, its ability to produce granules with higher porosity has made it particularly valuable in the detergent and pharmaceutical industries. While research on fluidized bed granulation dates back from the 1960s [1], population balance techniques have only recently been applied, for example, by Boerefijn et al. [2]. With population balance modelling, it is possible to relate observable properties of the product, for example, the granule size distribution, to the underlying microscopic kinetics of the system by extracting its aggregation rate constant. The main interest of this paper is to demonstrate how a growth kernel can be derived based on the principle of kinetic theory of granular flow (KTGF). This kernel is * Corresponding author. E-mail address: [email protected] (H.S. Tan). 0032-5910/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2004.04.030
analogous to an existing agglomeration kernel assuming equi-partition of kinetic energy (EKE model) derived by Hounslow [2] and applied by Boerefijn et al. [3]. We present our work in two parts: (a) We show first by means of Discrete Particle Modelling (DPM) that the distribution of particle velocities to be found in a fluidized bed are in accord with those expected according to the KTGF and that we can then deduce the rate of collisions to be given by the EKE kernel. (b) We then applied the EKE kernel to predict the evolution of granule size distribution obtained from fluidized bed granulation using Discretized Population Balance (DPB) modelling.
2. Theory Kinetic theory of granular flow is a statistical mechanical theory that describes the mean and fluctuating motion of particles within a continuous granular medium. The actual
104
H.S. Tan et al. / Powder Technology 142 (2004) 103–109
particle velocity (c¯) is decomposed into a local mean solids velocity (u¯) and a random fluctuating velocity (C¯) according to: c¯ ¼ u¯ þ C¯
ð1Þ
Associated with the random motion of the particles, the granular temperature h for an ensemble of particles is defined as: 1 ð2Þ h ¼ < C¯ C¯ > 3 where the brackets denote ensemble averaging. The KTGF accounts for two different transport mechanisms of particle properties: (1) Particles can transport a property by carrying it during free flight between collisions, which also known as kinetic transport. (2) Particle quantities can be transferred during collision. Modelling these transport mechanisms uses the well-known Boltzmann integral – differential equation. Solving the zeroth-order approximation to the particle velocity distribution function will result in a well-known Maxwell distribution function that describes the steady-state equilibrium condition without action of any external forces [4]: f ð0Þ ¼
N ð2phÞ
3=2
e
C 2 2h
ð3Þ
where N is the number of particles per unit volume. Just as the classical kinetic theory for gases the kinetic theory of granular flow assumes ‘molecular’ chaos. This implies that all particles are homogeneously distributed within an ensemble (there is no structure formation), that the particle velocity distribution for all particles is isotropic and that the velocities of two particles involved in a collision are not correlated. With this simple assumption that the particle velocities are isotropically distributed around a local mean velocity, the normalized particle velocity distribution in every direction (x, y, z) can be described using the Gaussian distribution [4]: fx ðCx Þ ¼ fz ðCz Þ ¼
1 ð2phx Þ 1
Cx2
e 2hx ; fy ðCy Þ ¼ 1=2
ð2phz Þ1=2
e
Cz2 2hz
1 ð2phyx Þ
Cy2
e 2hy ; 1=2 ð4Þ
while the normalised overall particle velocity distribution can be described using a Maxwellian distribution: f ðCÞ ¼ 4C 2
1 ð2phÞ
3=2
e
C 2 2h
ð5Þ
3. Simulation conditions The simulations are carried out with the 3D hardsphere discrete particle model that was originally devel-
Table 1 Simulation conditions Bed dimensions Width (x direction) Height ( y direction) Depth (z direction) Number of particles Initial bed height
Particle properties 150 mm
Diameter
2.5 mm
450 mm
Density
2526 kg/m3
15 mm
Volumetric shape factor Minimum fluidization velocity
1
24,750
1.28 m/s
f 150 mm
Particle – particle collision parameters
en l s
Case 1
Case 2
Case 3
Case 4
Case 5
1 0 0
0.97 0 0
0.97 0.10 0.33
0.90 0 0
0.90 0.10 0.33
oped by Hoomans [5]. The model solves the Newtonian equations of motion for each individual particle and the Navier – Stokes equations for interpenetrating continua are applied for the gas-phase hydrodynamics. Energy dissipated due to collisions are described by hard-sphere collision laws, by means of the empirical coefficients of normal restitution (en) and tangential restitution (s) and the coefficient of friction (l). All system dimensions and particle properties are specified in Table 1. Five different sets of collision parameters (Table 1) are used to investigate the sensitivity of the overall gas-fluidized bed dynamics. Some snapshots of the simulations taken at the moment of bubble eruption are shown in Fig. 1. In case I, particle –particle collisions are ideal and a nearly homogeneously expanded bed without bubbles is observed. In case II, the coefficient of normal restitution is slightly nonideal and a smoothly bubbling bed is obtained. In cases I, II and IV, the particles do not rotate since particle – particle contacts are frictionless. The impact parameters applied in case III were measured for 2.5 mm spherical glass beads by Gorham and Kharaz [6]. For rough particles, kinetic energy is transformed into rotation, and, subsequently, more energy is dissipated in collisions. This explains why the bed dynamics observed for cases III and V are most vigorous and the largest bubbles are observed in those cases.
4. Sampling of individual particle velocity To obtain a representative velocity distribution from a discrete particle simulation, the averaging ensemble should contain a large number of particles. In principle, the number of particles in the bed (24,750) might be high enough and the velocity distribution of the whole bed content could be taken as averaging ensemble. However, sampling of the velocity distributions is complicated by the intrinsically unsteady, non-homogeneous behaviour of gas-fluidized
H.S. Tan et al. / Powder Technology 142 (2004) 103–109
105
Fig. 1. Snapshots of the simulations taken at the moment of bubble eruption.
beds, which results in a continuously changing flow pattern within the bed. The particle velocity distribution is also a function of the granular temperature that varies with both time and position in the bed. So variations in flow pattern and granular temperature will interfere with sampling if the instantaneous velocity distribution in the whole bed or a time average of the particles at a fixed position in the bed is taken as averaging ensemble. To remove the influence of the flow pattern and the variation of the granular temperature with position, a grid that splits the bed content into smaller particle ensembles is applied. For these ensembles, the local granular temperature is computed after every time step that is taken to compute the gas phase flow field. To obtain ensembles of a sufficiently large number of particles, as implied in the kinetic theory of granular flow, it is assumed that the particle velocity distribution mainly depends on the granular temperature. Therefore, the observed granular temperature range is split into a discrete number of classes and particle ensembles obtained from tmin to tmax with
granular temperatures within the range of the same granular temperature class are merged. To prevent start-up effects from influencing the sampling results, all analyses are started after 5 s. For this work, an analysis grid of 15 45 1 cells (coarse grid) or 30 90 1 cells (fine grid) is projected on the simulated system. For more details of sampling procedure, readers are referred to Ref. [4].
5. Results and discussions Some sampled particle velocity distributions obtained from sampling on a coarse grid can be found in Fig. 2a and b. For case I, it is obvious that the Gaussian distribution and the velocity distributions in each direction are identical, while the total particle velocity distribution is almost Maxwellian. For particles with realistic collision parameters (case III) the sampled velocity distributions show a slight
Fig. 2. Normalised particle velocity distribution functions fx(Cx), fy(Cy), fz(Cz) and f (C) sampled on coarse grid (h(x,y,z) = h for f (C), h(x,y,z) = hx for fx(Cx), etc.). (a) Case I: Velocity distributions sampled in the range 3.16 10 3 < h(x,y,z) < 5.62 10 3 m2/s2 compared to the Gaussian and Maxwellian distribution for h(x,y,z) = 4.39 10 3 m2/s2. (b) Case III: Velocity distributions sampled in the range 5.62 10 4 < h(x,y,z) < 1.00 10 3 m2/s2 compared to the Gaussian and Maxwellian distribution for h(x,y,z) = 7.8 10 4 m2/s2.
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Fig. 3. Normalized particle velocity distribution functions fx (Cx), fy(Cy), fz(Cz) and f (C) sampled on the fine grid in the range 5.62 10 4 < h(x,y,z) < 1.00 10 3 m2/s2 compared to the Gaussian and Maxwellian distribution for h(x,y,z) = 7.81 10 4 m2/s2.
deviation from the Gaussian and Maxwellian distributions, but the resemblance is still very good. To reduce the influence of solids velocity gradients and volume fraction gradients within the sampling ensembles the fine grid is applied for all further analyses in this work. Fig. 3 shows some particle velocity distributions that are sampled for cases III and V using the fine grid. Comparing Figs. 2(b) and 3(a) makes clear that grid refinement improves the agreement between the sampled velocity distributions and Gaussian and Maxwellian distributions, which indicates that the fluctuating particle motion is not homogeneous and isotropic at the subgrid scale. Fig. 3(b) shows that even in case V, the simulation with the most inelastic particles, the velocity distributions sampled on the fine grid do not deviate significantly from Gaussian and Maxwellian distributions.
6. Population balance modelling The rate of aggregation equation is defined by a kernel, bi, j: ragg
i; j
¼ bi; j Ni Nj
ð6Þ
where Ni and Nj is the number concentration in size classes i and j. An aggregation kernel then assuming equi-partition of kinetic energy (EKE model) was described by Hounslow [2], which assumes that particles collide as a consequence of their random component of velocity and that the random components results in equal distribution of the particles’ kinetic energy: sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 2 bi;j ¼ b0 ðtÞðli þ lj Þ ðEKE modelÞ ð7Þ þ 3 3 li lj where li and lj is the particle size in size classes i and j, and b0 is a rate constant.
6.1. Derivation of an agglomeration kernel based on the principle of kinetic theory of granular flow We have shown that the particle velocity distribution inside a fluidized bed corresponds to the KTGF. Goldschmidt [4] has derived an expression based on a pair distribution function, describing the number of collisions between particles of two different species i and j per unit volume and time, rcoll i, j: " # 1 4 hs mi þ mj 2 2 3 ðj u¯ Þ rcoll i; j ¼ pNi Nj di; j gi; j di; j p 2mi mj 3 ð8Þ where di, j is the inter-particle distance between two particles on collision, gi, j the radial distribution function for mixture, hs the mixture granular temperature, mi and mj the mass of particles i and j and u¯ the ensemble average particulate velocity. In population balance, it is assumed the rate of collision to be described by a second order collision rate constant, Ci, j: rcoll i; j ¼ Ci; j Ni Nj
ð9Þ
Since the rate of aggregation, ragg i, j (as in Eq. (6)), is related to the rate of collision, rcoll i, j, by a success factor for agglomeration upon collision, w: ragg i; j ð10Þ w¼ rcoll i; j the following expression can be deduced bi; j ¼ wCi; j
ð11Þ
Combining with Eq. (8), the following agglomeration kernel can be derived: " # 1 4 hs mi þ mj 2 2 3 bi; j ¼ wpNi Nj di; j gi; j ðj u¯ Þ di; j p 2mi mj 3 ð12Þ
H.S. Tan et al. / Powder Technology 142 (2004) 103–109
By ignoring the divergence in particulate velocity field, and assuming all particles are of equal density (qi = qj = q), the above equation can be transformed into: sffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3hs 1 1 2 ðli þ lj Þ þ 3 bi; j ¼ wgi; j 3 q li lj
ð13Þ
107
sion to the EKE model (Eq. (7)), it shows that the timedependent part b0(t) can be represented by [4]: sffiffiffiffiffiffiffi 3hs b0 ðtÞ ¼ wgi; j q
ð14Þ
6.2. Experiments The success factor for agglomeration is expected to depend on various factors, such as particle wetability, particle velocity, binder type and concentration. By comparing this expres-
.
The experiments reported here are similar to those of Boerefijn et al. [3]. The fluidized bed granulator used has
Fig. 4. Measured ( ) and fitted (—) particle size distribution for 240 g/min spray on rate with sodium carbonate.
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H.S. Tan et al. / Powder Technology 142 (2004) 103–109
Fig. 4 (continued).
dimensions of 0.4 m i.d. and 1 m in height. A wire mesh distributor is used, with a discharge valve centrally located for sampling purposes. The spray nozzle used is a SUE25 two-fluid nozzle that produces an elliptical spray pattern. The powder used is sodium carbonate, with a skeletal density, qs, of 1813kg/m3 and bulk density, qb, of 628 kg/ m3, and the binder used is polyethylene glycol (PEG 4000). The charge material was fluidized at a very turbulent regime, at the superficial velocity, Us, of 23 times the minimum fluidizing velocity of the sodium carbonate. The EKE kernel was fitted to the data by taking bo to be
independent of time and averaged over the whole bed. A value of bo = 5.2946 10 10 kg s 1 m 0.5 provides the least squares best fit seen in Fig. 4. Using the EKE kernel, it can be seen in Fig. 4 that the model predicts the measured size distribution fairly well.
7. Conclusions It has been demonstrated that the discrete particle model is an excellent tool for studying detailed information about
H.S. Tan et al. / Powder Technology 142 (2004) 103–109
the basic particle flow characteristics, which are extremely difficult to obtain from experiments. A novel sampling technique to obtain particle velocity distributions for a dense fluidized bed with different collision coefficients was presented. The result with elastic particles shows excellent result with the kinetic theory of granular flow. Good agreement between the sampling results from DPM and the Maxwellian and Gaussian distribution expected in KTGF supports the applicability of the theory to dense gas-fluidized beds Verification of the velocity distribution function is important since it is subsequently used to derive particle –particle collision frequency as well as the impact velocity distribution. From a kinetic theory expression for collision frequency, a growth model for use in a population balance was derived. It is shown that this model is identical to the existing EKE model for fluidized bed granulation, and indeed gives a theoretical basis for its application. The key defect in the direct application of the model is that in the current model it is assumed that the bed is well mixed. There is a clear need to couple population balance modelling with a kinematic model (such as that provided by KTGF). The work presented here is a step along that path since it shows that a population balance model that is consistent with the KTGF can be deduced and is effective. Despite the well-mixed assumption, the model works fairly well for sodium carbonate granulated with PEG4000. Nomenclature c particle velocity, m s 1 C fluctuating component of particle velocity, m s 1 Ci, j collision rate constant, m5/2 s 1 dp particle diameter, m di, j inter-particle distance between two colliding particles, m en coefficient of normal restitution f0 particle velocity distribution function, s 3 m3 f normalized particle velocity distribution function, s m 1 gi, j radial distribution function for mixture, – l particle size, m m mass of particle, kg N number concentration, m 3 rcoll i, j collision frequency per unit volume, m 3 s 1 ragg i, j aggregation frequency per unit volume, m 3 s 1 t time, s u local average particulate velocity, ms 1 Us superficial velocity, m s 1 w mass density function, m 1
109
Greek letters b0 aggregation rate constant, m5/2 s 1(definition in Eq. (6)) aggregation rate constant, kg s 1 m 0.5 (as in DPB modelling) bi, j aggregation kernel, m3 s 1 l coefficient of frictional restitution h granular temperature for monodisperse system, m 2 s 2 hs mixture granular temperature, kg m2 s 2 (J) q density, kg m 3 qs skeleton density, kg m 3 qb bulk density, kg m 3 s coefficient of tangential restitution w success factor for aggregation Subscripts i particle of size class i j particle of size class j x x direction of particle velocity y y direction of particle velocity z z direction of particle velocity min minimum max maximum Other marks – vector quantity <. . .> ensemble average
Acknowledgements This work is supported by the EPSRC of the United Kingdom and by Unilever Research, Vlaardingen, The Netherlands. References [1] M. Banks, M.E. Aulton, Fluidized bed granulation: a chronology, Drug Development and Industry Pharmacy 17 (11) (1991) 1437 – 1463. [2] M.J. Hounslow, The population balance as a tool for understanding particle rate processes, Kona (1998) 179 – 193. [3] R. Boerefijn, M. Buscan, M.J. Hounslow, Effects of non-ideal powder properties on granulation kinetics, Proc. Fluidization X. United Engineering Foundation Inc., Beijing, 2001, pp. 629 – 636. [4] GoldSchmidt, M.J.V., 2001. Hydrodynamic Modelling of Fluidized Bed Granulation. PhD thesis, Twente University. [5] Hoomans, B.P.B., 2000. Granular Dynamics of Gas – Solid Two-Phase Flow. PhD thesis, Twente University. [6] D.A. Gorham, A.H. Kharaz, Results of the Particle Impact Tests, Impact Research Group IRG13, The Open University, Milton Keynes, UK, 1999.
Building population balance model for fluidized bed melt granulation: lessons from kinetic theory of granular flow H.S. Tan a,*, M.J.V. Goldschmidt b, R. Boerefijn c, M.J. Hounslow a, A.D. Salman a, J.A.M. Kuipers d a
Department of Chemical and Process Engineering, University of Sheffield, Mappin Street, Sheffield S1 3JD, UK b Akzo Nobel Chemicals Research, P.O. Box 9300, 6800 SB Arnhem, The Netherlands c Unilever Research, P.O. Box 114, 3130 AC Vlaardingen, The Netherlands d Department of Chemical Engineering, Twente University of Technology, 7500 AE Enschede, The Netherlands Available online
Abstract The purpose of this paper is to develop a theoretically sound basis for the equi-partition of kinetic energy (EKE) kernel recently developed by our group to describe the evolution of granule size distributions in fluidized bed granulation. The approach taken is to show first by distinct element modelling that the statistics of fluctuating velocity and thus frequency of collisions are well described by the kinetic theory of granular flow (KTGF)—that is, Maxwellian in nature. It is then possible to use KTGF to show that the size dependence of the aggregation process should indeed be given by the EKE kernel and that the rate constant depends only on the granular temperature, the particle density, the radial distribution function and the efficiency of collisions. It is shown how the kernel developed can be used to describe the evolution of a granule size distribution when low-density sodium carbonate particles (Light Ash) are sprayed with polyethylene glycol. D 2004 Elsevier B.V. All rights reserved. Keywords: Population balance model; Fluidized bed melt granulation; Kinetic theory of granular flow
1. Introduction Fluidized bed granulation has been an important powder production process in the industry for several decades. Compared to other granulation techniques, its ability to produce granules with higher porosity has made it particularly valuable in the detergent and pharmaceutical industries. While research on fluidized bed granulation dates back from the 1960s [1], population balance techniques have only recently been applied, for example, by Boerefijn et al. [2]. With population balance modelling, it is possible to relate observable properties of the product, for example, the granule size distribution, to the underlying microscopic kinetics of the system by extracting its aggregation rate constant. The main interest of this paper is to demonstrate how a growth kernel can be derived based on the principle of kinetic theory of granular flow (KTGF). This kernel is * Corresponding author. E-mail address: [email protected] (H.S. Tan). 0032-5910/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2004.04.030
analogous to an existing agglomeration kernel assuming equi-partition of kinetic energy (EKE model) derived by Hounslow [2] and applied by Boerefijn et al. [3]. We present our work in two parts: (a) We show first by means of Discrete Particle Modelling (DPM) that the distribution of particle velocities to be found in a fluidized bed are in accord with those expected according to the KTGF and that we can then deduce the rate of collisions to be given by the EKE kernel. (b) We then applied the EKE kernel to predict the evolution of granule size distribution obtained from fluidized bed granulation using Discretized Population Balance (DPB) modelling.
2. Theory Kinetic theory of granular flow is a statistical mechanical theory that describes the mean and fluctuating motion of particles within a continuous granular medium. The actual
104
H.S. Tan et al. / Powder Technology 142 (2004) 103–109
particle velocity (c¯) is decomposed into a local mean solids velocity (u¯) and a random fluctuating velocity (C¯) according to: c¯ ¼ u¯ þ C¯
ð1Þ
Associated with the random motion of the particles, the granular temperature h for an ensemble of particles is defined as: 1 ð2Þ h ¼ < C¯ C¯ > 3 where the brackets denote ensemble averaging. The KTGF accounts for two different transport mechanisms of particle properties: (1) Particles can transport a property by carrying it during free flight between collisions, which also known as kinetic transport. (2) Particle quantities can be transferred during collision. Modelling these transport mechanisms uses the well-known Boltzmann integral – differential equation. Solving the zeroth-order approximation to the particle velocity distribution function will result in a well-known Maxwell distribution function that describes the steady-state equilibrium condition without action of any external forces [4]: f ð0Þ ¼
N ð2phÞ
3=2
e
C 2 2h
ð3Þ
where N is the number of particles per unit volume. Just as the classical kinetic theory for gases the kinetic theory of granular flow assumes ‘molecular’ chaos. This implies that all particles are homogeneously distributed within an ensemble (there is no structure formation), that the particle velocity distribution for all particles is isotropic and that the velocities of two particles involved in a collision are not correlated. With this simple assumption that the particle velocities are isotropically distributed around a local mean velocity, the normalized particle velocity distribution in every direction (x, y, z) can be described using the Gaussian distribution [4]: fx ðCx Þ ¼ fz ðCz Þ ¼
1 ð2phx Þ 1
Cx2
e 2hx ; fy ðCy Þ ¼ 1=2
ð2phz Þ1=2
e
Cz2 2hz
1 ð2phyx Þ
Cy2
e 2hy ; 1=2 ð4Þ
while the normalised overall particle velocity distribution can be described using a Maxwellian distribution: f ðCÞ ¼ 4C 2
1 ð2phÞ
3=2
e
C 2 2h
ð5Þ
3. Simulation conditions The simulations are carried out with the 3D hardsphere discrete particle model that was originally devel-
Table 1 Simulation conditions Bed dimensions Width (x direction) Height ( y direction) Depth (z direction) Number of particles Initial bed height
Particle properties 150 mm
Diameter
2.5 mm
450 mm
Density
2526 kg/m3
15 mm
Volumetric shape factor Minimum fluidization velocity
1
24,750
1.28 m/s
f 150 mm
Particle – particle collision parameters
en l s
Case 1
Case 2
Case 3
Case 4
Case 5
1 0 0
0.97 0 0
0.97 0.10 0.33
0.90 0 0
0.90 0.10 0.33
oped by Hoomans [5]. The model solves the Newtonian equations of motion for each individual particle and the Navier – Stokes equations for interpenetrating continua are applied for the gas-phase hydrodynamics. Energy dissipated due to collisions are described by hard-sphere collision laws, by means of the empirical coefficients of normal restitution (en) and tangential restitution (s) and the coefficient of friction (l). All system dimensions and particle properties are specified in Table 1. Five different sets of collision parameters (Table 1) are used to investigate the sensitivity of the overall gas-fluidized bed dynamics. Some snapshots of the simulations taken at the moment of bubble eruption are shown in Fig. 1. In case I, particle –particle collisions are ideal and a nearly homogeneously expanded bed without bubbles is observed. In case II, the coefficient of normal restitution is slightly nonideal and a smoothly bubbling bed is obtained. In cases I, II and IV, the particles do not rotate since particle – particle contacts are frictionless. The impact parameters applied in case III were measured for 2.5 mm spherical glass beads by Gorham and Kharaz [6]. For rough particles, kinetic energy is transformed into rotation, and, subsequently, more energy is dissipated in collisions. This explains why the bed dynamics observed for cases III and V are most vigorous and the largest bubbles are observed in those cases.
4. Sampling of individual particle velocity To obtain a representative velocity distribution from a discrete particle simulation, the averaging ensemble should contain a large number of particles. In principle, the number of particles in the bed (24,750) might be high enough and the velocity distribution of the whole bed content could be taken as averaging ensemble. However, sampling of the velocity distributions is complicated by the intrinsically unsteady, non-homogeneous behaviour of gas-fluidized
H.S. Tan et al. / Powder Technology 142 (2004) 103–109
105
Fig. 1. Snapshots of the simulations taken at the moment of bubble eruption.
beds, which results in a continuously changing flow pattern within the bed. The particle velocity distribution is also a function of the granular temperature that varies with both time and position in the bed. So variations in flow pattern and granular temperature will interfere with sampling if the instantaneous velocity distribution in the whole bed or a time average of the particles at a fixed position in the bed is taken as averaging ensemble. To remove the influence of the flow pattern and the variation of the granular temperature with position, a grid that splits the bed content into smaller particle ensembles is applied. For these ensembles, the local granular temperature is computed after every time step that is taken to compute the gas phase flow field. To obtain ensembles of a sufficiently large number of particles, as implied in the kinetic theory of granular flow, it is assumed that the particle velocity distribution mainly depends on the granular temperature. Therefore, the observed granular temperature range is split into a discrete number of classes and particle ensembles obtained from tmin to tmax with
granular temperatures within the range of the same granular temperature class are merged. To prevent start-up effects from influencing the sampling results, all analyses are started after 5 s. For this work, an analysis grid of 15 45 1 cells (coarse grid) or 30 90 1 cells (fine grid) is projected on the simulated system. For more details of sampling procedure, readers are referred to Ref. [4].
5. Results and discussions Some sampled particle velocity distributions obtained from sampling on a coarse grid can be found in Fig. 2a and b. For case I, it is obvious that the Gaussian distribution and the velocity distributions in each direction are identical, while the total particle velocity distribution is almost Maxwellian. For particles with realistic collision parameters (case III) the sampled velocity distributions show a slight
Fig. 2. Normalised particle velocity distribution functions fx(Cx), fy(Cy), fz(Cz) and f (C) sampled on coarse grid (h(x,y,z) = h for f (C), h(x,y,z) = hx for fx(Cx), etc.). (a) Case I: Velocity distributions sampled in the range 3.16 10 3 < h(x,y,z) < 5.62 10 3 m2/s2 compared to the Gaussian and Maxwellian distribution for h(x,y,z) = 4.39 10 3 m2/s2. (b) Case III: Velocity distributions sampled in the range 5.62 10 4 < h(x,y,z) < 1.00 10 3 m2/s2 compared to the Gaussian and Maxwellian distribution for h(x,y,z) = 7.8 10 4 m2/s2.
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H.S. Tan et al. / Powder Technology 142 (2004) 103–109
Fig. 3. Normalized particle velocity distribution functions fx (Cx), fy(Cy), fz(Cz) and f (C) sampled on the fine grid in the range 5.62 10 4 < h(x,y,z) < 1.00 10 3 m2/s2 compared to the Gaussian and Maxwellian distribution for h(x,y,z) = 7.81 10 4 m2/s2.
deviation from the Gaussian and Maxwellian distributions, but the resemblance is still very good. To reduce the influence of solids velocity gradients and volume fraction gradients within the sampling ensembles the fine grid is applied for all further analyses in this work. Fig. 3 shows some particle velocity distributions that are sampled for cases III and V using the fine grid. Comparing Figs. 2(b) and 3(a) makes clear that grid refinement improves the agreement between the sampled velocity distributions and Gaussian and Maxwellian distributions, which indicates that the fluctuating particle motion is not homogeneous and isotropic at the subgrid scale. Fig. 3(b) shows that even in case V, the simulation with the most inelastic particles, the velocity distributions sampled on the fine grid do not deviate significantly from Gaussian and Maxwellian distributions.
6. Population balance modelling The rate of aggregation equation is defined by a kernel, bi, j: ragg
i; j
¼ bi; j Ni Nj
ð6Þ
where Ni and Nj is the number concentration in size classes i and j. An aggregation kernel then assuming equi-partition of kinetic energy (EKE model) was described by Hounslow [2], which assumes that particles collide as a consequence of their random component of velocity and that the random components results in equal distribution of the particles’ kinetic energy: sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 2 bi;j ¼ b0 ðtÞðli þ lj Þ ðEKE modelÞ ð7Þ þ 3 3 li lj where li and lj is the particle size in size classes i and j, and b0 is a rate constant.
6.1. Derivation of an agglomeration kernel based on the principle of kinetic theory of granular flow We have shown that the particle velocity distribution inside a fluidized bed corresponds to the KTGF. Goldschmidt [4] has derived an expression based on a pair distribution function, describing the number of collisions between particles of two different species i and j per unit volume and time, rcoll i, j: " # 1 4 hs mi þ mj 2 2 3 ðj u¯ Þ rcoll i; j ¼ pNi Nj di; j gi; j di; j p 2mi mj 3 ð8Þ where di, j is the inter-particle distance between two particles on collision, gi, j the radial distribution function for mixture, hs the mixture granular temperature, mi and mj the mass of particles i and j and u¯ the ensemble average particulate velocity. In population balance, it is assumed the rate of collision to be described by a second order collision rate constant, Ci, j: rcoll i; j ¼ Ci; j Ni Nj
ð9Þ
Since the rate of aggregation, ragg i, j (as in Eq. (6)), is related to the rate of collision, rcoll i, j, by a success factor for agglomeration upon collision, w: ragg i; j ð10Þ w¼ rcoll i; j the following expression can be deduced bi; j ¼ wCi; j
ð11Þ
Combining with Eq. (8), the following agglomeration kernel can be derived: " # 1 4 hs mi þ mj 2 2 3 bi; j ¼ wpNi Nj di; j gi; j ðj u¯ Þ di; j p 2mi mj 3 ð12Þ
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By ignoring the divergence in particulate velocity field, and assuming all particles are of equal density (qi = qj = q), the above equation can be transformed into: sffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3hs 1 1 2 ðli þ lj Þ þ 3 bi; j ¼ wgi; j 3 q li lj
ð13Þ
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sion to the EKE model (Eq. (7)), it shows that the timedependent part b0(t) can be represented by [4]: sffiffiffiffiffiffiffi 3hs b0 ðtÞ ¼ wgi; j q
ð14Þ
6.2. Experiments The success factor for agglomeration is expected to depend on various factors, such as particle wetability, particle velocity, binder type and concentration. By comparing this expres-
.
The experiments reported here are similar to those of Boerefijn et al. [3]. The fluidized bed granulator used has
Fig. 4. Measured ( ) and fitted (—) particle size distribution for 240 g/min spray on rate with sodium carbonate.
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Fig. 4 (continued).
dimensions of 0.4 m i.d. and 1 m in height. A wire mesh distributor is used, with a discharge valve centrally located for sampling purposes. The spray nozzle used is a SUE25 two-fluid nozzle that produces an elliptical spray pattern. The powder used is sodium carbonate, with a skeletal density, qs, of 1813kg/m3 and bulk density, qb, of 628 kg/ m3, and the binder used is polyethylene glycol (PEG 4000). The charge material was fluidized at a very turbulent regime, at the superficial velocity, Us, of 23 times the minimum fluidizing velocity of the sodium carbonate. The EKE kernel was fitted to the data by taking bo to be
independent of time and averaged over the whole bed. A value of bo = 5.2946 10 10 kg s 1 m 0.5 provides the least squares best fit seen in Fig. 4. Using the EKE kernel, it can be seen in Fig. 4 that the model predicts the measured size distribution fairly well.
7. Conclusions It has been demonstrated that the discrete particle model is an excellent tool for studying detailed information about
H.S. Tan et al. / Powder Technology 142 (2004) 103–109
the basic particle flow characteristics, which are extremely difficult to obtain from experiments. A novel sampling technique to obtain particle velocity distributions for a dense fluidized bed with different collision coefficients was presented. The result with elastic particles shows excellent result with the kinetic theory of granular flow. Good agreement between the sampling results from DPM and the Maxwellian and Gaussian distribution expected in KTGF supports the applicability of the theory to dense gas-fluidized beds Verification of the velocity distribution function is important since it is subsequently used to derive particle –particle collision frequency as well as the impact velocity distribution. From a kinetic theory expression for collision frequency, a growth model for use in a population balance was derived. It is shown that this model is identical to the existing EKE model for fluidized bed granulation, and indeed gives a theoretical basis for its application. The key defect in the direct application of the model is that in the current model it is assumed that the bed is well mixed. There is a clear need to couple population balance modelling with a kinematic model (such as that provided by KTGF). The work presented here is a step along that path since it shows that a population balance model that is consistent with the KTGF can be deduced and is effective. Despite the well-mixed assumption, the model works fairly well for sodium carbonate granulated with PEG4000. Nomenclature c particle velocity, m s 1 C fluctuating component of particle velocity, m s 1 Ci, j collision rate constant, m5/2 s 1 dp particle diameter, m di, j inter-particle distance between two colliding particles, m en coefficient of normal restitution f0 particle velocity distribution function, s 3 m3 f normalized particle velocity distribution function, s m 1 gi, j radial distribution function for mixture, – l particle size, m m mass of particle, kg N number concentration, m 3 rcoll i, j collision frequency per unit volume, m 3 s 1 ragg i, j aggregation frequency per unit volume, m 3 s 1 t time, s u local average particulate velocity, ms 1 Us superficial velocity, m s 1 w mass density function, m 1
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Greek letters b0 aggregation rate constant, m5/2 s 1(definition in Eq. (6)) aggregation rate constant, kg s 1 m 0.5 (as in DPB modelling) bi, j aggregation kernel, m3 s 1 l coefficient of frictional restitution h granular temperature for monodisperse system, m 2 s 2 hs mixture granular temperature, kg m2 s 2 (J) q density, kg m 3 qs skeleton density, kg m 3 qb bulk density, kg m 3 s coefficient of tangential restitution w success factor for aggregation Subscripts i particle of size class i j particle of size class j x x direction of particle velocity y y direction of particle velocity z z direction of particle velocity min minimum max maximum Other marks – vector quantity <. . .> ensemble average
Acknowledgements This work is supported by the EPSRC of the United Kingdom and by Unilever Research, Vlaardingen, The Netherlands. References [1] M. Banks, M.E. Aulton, Fluidized bed granulation: a chronology, Drug Development and Industry Pharmacy 17 (11) (1991) 1437 – 1463. [2] M.J. Hounslow, The population balance as a tool for understanding particle rate processes, Kona (1998) 179 – 193. [3] R. Boerefijn, M. Buscan, M.J. Hounslow, Effects of non-ideal powder properties on granulation kinetics, Proc. Fluidization X. United Engineering Foundation Inc., Beijing, 2001, pp. 629 – 636. [4] GoldSchmidt, M.J.V., 2001. Hydrodynamic Modelling of Fluidized Bed Granulation. PhD thesis, Twente University. [5] Hoomans, B.P.B., 2000. Granular Dynamics of Gas – Solid Two-Phase Flow. PhD thesis, Twente University. [6] D.A. Gorham, A.H. Kharaz, Results of the Particle Impact Tests, Impact Research Group IRG13, The Open University, Milton Keynes, UK, 1999.