A numerical investigation into the solids phase chromatography using a combined continuous and discrete approach

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Particuology 6 (2008) 557–571

A numerical investigation into the solids phase chromatography using a combined continuous and discrete approach Wei Yang a , Fang Yang b , Yulong Ding a,b,∗ a

Institute of Particle Science & Engineering, University of Leeds, Leeds LS2 9JT, UK On visiting from University of Birmingham, Edgbaston, Birmingham B15 2TT, UK

b

Received 8 May 2008; accepted 15 July 2008

Abstract Solids phase chromatography for particle classification is based on different retention times of particles with different properties when they are elutriated through a confined geometry. This work aims at a fundamental understanding of such a technology by using the combined continuous and discrete method. A packed bed is employed as the model confined geometry. The numerical method is compared first with experimental observations, followed by a parametric analysis of the effects on the flow hydrodynamics and solids behaviour of various parameters including the number of injected particles, the superficial gas velocity, the contact stiffness and the diameter ratio of the packed column to the packed particles. The results show that the modelling captures some important features of the flow of an injected pulse of fine particles in a packed bed. An increase in the number of injected particles or the superficial gas velocity reduces the retention time, whereas the contact stiffness does not show much effect over the range of 5 × 102 to 5 × 104 N/m. It is also found that the effect on the retention time of the diameter ratio of the packed column to the packed particles seems complex showing a non-monotonous dependence. © 2008 Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy of Sciences. Published by Elsevier B.V. All rights reserved. Keywords: Solids phase chromatography; Gas–solid two-phase flows; Packed bed; Combined continuous and discrete method; Modelling

1. Background Solids phase chromatography (SPC) is a technology proposed for particle classification in the dry form under the process conditions (Ding, 2006). SPC uses a confined geometry, called separation channel, through which particles to be fractionated are pneumatically conveyed by a carrying gas under controlled humidity and temperature. The separation channel can be empty or structurally packed and the surface of the channel and packing may be coated with anti-electrostatic materials. Various detectors can be used to measure evolution of pressure along the channel, particle velocity and retention time. This work considers packed beds as the separation channels. Fundamentally, classification using such separation channels relies on different lengths of retention time of particles with different properties due to interactions among the carrying gas, particles to be classified, separation channel wall and packing. Our recent experiments,

∗

Corresponding author. Tel.: +44 113 343 2747; fax: +44 113 343 2405. E-mail address: [email protected] (Y. Ding).

using packed beds as the separation channels, showed that, given a packed column and a superficial gas velocity, the retention time of particles depended on the size, density, and shape of particles, with larger, spherical or lighter particles passing more quickly through the channel (Yang & Ding, in press). We cannot explain the experimental observations, particularly why large particles elutriate faster than smaller particles. This forms one of the motivations of this work. There are various industrial processes involving a gas–particle mixture flowing through packed beds. Examples include deep bed filtration of particle laden flue gases, pulverised coal injection into blast furnaces, and sorption enhanced reaction processes. As a results, a number studies have been carried out over the past few decades on the hydrodynamics of a steady-state flow of gas–solid mixtures through packed beds in a continuous manner (Chen, Akiyama, Nogami, & Yagi, 1994; Hidaka, Matsumoto, Kusakabe, & Morooka, 1999; Shibata, Shimizu, Inaba, Takahashi, & Yagi, 1991; Song, Wang, Jin, & Tanaka, 1995; Wang, Ding, & Ghadiri, 2004). These studies are different from the SPC where a pulse of gas–solid mixture is injected into the packed bed. The

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doi:10.1016/j.partic.2008.07.007

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Nomenclature a CD dp dp,i Dc e E fn fp,ij ft Fgp g G Ii k ki kn L mi nˆ pg P R Rep Rc r Sn St ˆt t Tp,ij Ug vg vi vp vr <. . .>

radius of the circular area of contact drag coefficient diameter of packed particle diameter of particle i diameter of packed column coefficient of restitution elastic modulus normal contact force interaction forces between particle i and j tangential contact force gas–particle interaction force gravitational acceleration shear modulus moment of inertia of particle i contact stiffness number of particles in contact with particle i normal contact stiffness length of packed column mass of the particle i unit vector in the normal direction gas pressure normal force radius of sphere particle Reynolds number radius of packed column radial position particle displacement in normal direction particle displacement in tangential direction unit vector in tangential direction time intraparticle torque superficial gas velocity velocity vector of gas velocity of particle i velocity vector of particle relative velocity calculation of an average value

Greek letters α relative approach of the centres of the two contiguous spheres ε porosity η coefficient of damping μfric coefficient of friction μg viscosity of gas ν Poisson’s ratio ρg density of gas viscosity stress tensor of gas τg ωi rotational velocity of particle i

second motivation of this work is therefore to understand the unsteady-state solids motion and hydrodynamics of gas–solid mixtures flowing through the packed column.

Packed beds have a complex interior structure. Several modelling approaches have been employed in the past to predict the flow field in the pores of the packing with different degrees of success. The most sophisticated models are based on direct numerical simulation where the packed particles are treated as the solid boundaries (Manz, Gladden, & Warren, 1999). This approach uses very fine meshes to characterise the flow in the pores and hence is capable of addressing some important issues such as predicting non-parallel flows. Direct numerical simulation is advantageous in revealing details of the flow fields but requires very expensive computational powers—a major limitation for applications in simulating real industrial processes. Another shortcoming is probably the difficulties in handling the collisions between particles. As a consequence, direct numerical simulation is rarely used in simulating gas–solid flows in packed beds. A more conventional (and most popular) way of simulating gas–solid flows in packed beds is the so-called network modelling (Martins, Laranjeira, Lopes, & Dias, 2007), where the local packing structure is represented by a network of regular geometries, usually spheres and cylinders or constricted tubes. For example, Kozeny theory treats the pores in the packed bed as an ensemble of channels of various crosssections (Fan & Zhu, 1998), whereas the constricted tube model was used by Pendse, Chiang, and Tien (1983) who found success in improving the prediction of axial dispersion. Imdakm and Sahimi (1991) combined the Monte Carlo method with the network model to study particle transportation in porous media and obtained quantitative agreement with experimental data. An obvious disadvantage of the network modelling is its inability of providing detailed information at the single particle level. Another popular approach of modelling gas–solid flows in the packed beds is the two-fluid model where both gas and the suspended particle phases are treated as interpenetrating continua. Such an approach uses separate mass and momentum conservation equations for each phase and hydrodynamic interactions between the two phases are through the drag coefficient; see for example Dong, Zhang, Pinson, Yu, and Zulli, 2004 and Li, Ding, Wen, and He (2006). This approach requires the use of empirical relationships for both the drag coefficient and the porosity distribution. This work employs the combined continuous and discrete method (CCDM)—a technique that is capable of providing detailed single particle level information hard to obtain by using other approaches (Xu & Yu, 1997). This also forms the third motivation of this work—to interpret the experimental observations by using the CCDM modelling results. Additionally, the CCDM could also provide information for improving the SPC. 2. Model formulation and numerical solution The combined continuous and discrete particle method considers the fluid phase as a continuum with the locally averaged Navier–Stokes equations governing the momentum transport (Anderson & Jackson, 1967), and the particle phase as discrete with the motion of individual particles governed by Newton’s second law of motion. The two phases are coupled through the Newton’s third law of motion. The concept of the dis-

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crete element method was originally developed by Cundall and Strack (1979) and has been extensively explored and used for modelling various particle systems, particularly gas–solid and granular flow systems over the past two decades; see for example Xu and Yu (1997), Mikami, Kamiya, and Horio (1998) and Hoomans, Kuipers, and van Swaaij (2000) on gas fluidisation, Tsuji, Tanaka, and Ishida (1992), Di Maio and Di Renzo (2004) on pneumatic conveying, Foo, Sheng, and Briscoe (2004) on agglomeration, Moreno, Ghadiri, and Antony (2003), Morrison and Cleary (2004) on particle breakage, Hsiau and Yang (2005) on shearing granular materials, and Zou et al. (2003) and Sobolev and Amirjanov (2004) on particle packing. In the following, governing equations for both the fluid and particle phases and the coupling equation between the two phases are given first in Section 2.1, whereas the numerical schemes and simulation conditions are detailed in Section 2.2.

where Rep is the particle Reynolds number defined by   ρg ε vp  − vg  dp  Rep = . μg

2.1. Governing equations

Ii

(8)

The velocity of individual particles is determined by Newton’s second law of motion: i  dvi fp,ij , = mi g + ff,i + dt

k

mi

(9)

j=1

where mi and vi are, respectively the mass and velocity of particle i, fp,ij is the interaction forces between particles i and j, and ki is the number of particles in contact with particle i. Interaction between particles may cause particle to rotate and the rotational motion is calculated by i  d␻i = Tp,ij , dt

k

(10)

j=1

The governing equations for the gas phase include the mass and momentum balances given, respectively by ∂ρg ε = ∇ · (ρg εvg ), ∂t

(1)

with Ii the moment of inertia, ωi the rotational velocity of particle i and Tp,ij the torque. ff,i in Eq. (9) is the force exerted by the fluid on the particle and can be expressed by ff,i =

∂ρg εvg + ∇ · (ρg εvg vg ) = −ε∇pg − ε∇ · ␶g + ρg εg + fgp , ∂t (2) where ρg , vg , pg , and τ g are the density, velocity vector, pressure and viscosity stress tensor, respectively, ε is the porosity, and g is the gravitational acceleration. fgp in Eq. (2) is the gas–particle interaction force expressed by:   fgp = β · vp  − vg , (3) where vp  is particle velocity vector averaged in a computational cell, and β is the interaction term given by either the Ergun equation (Ergun, 1952), Eq. (4), or the Wen and Yu correlation (Wen & Yu, 1966), Eq. (5): β= 150

 (1 − ε)ρg  (1 − ε)2 μg + 1.75 vp  − vg  εdp  dp 

(ε ≤ 0.8),

3 πdp,i



6

 β (vg − vi ) − ∇Pg , 1−ε

(11)

where dp,i is the diameter of particle i. When the long-range interparticle forces (e.g. van der Waals, electrostatic, liquid bridge) are negligible, the most important interparticle force is the direct contact force due to collisions (Di Renzo & Di Maio, 2004). There are several approaches to deal with the contact mechanics; see for example Johnson, Kendall, and Roberts (1971), Cundall and Strack (1979) and Thornton and Yin (1991). The linear spring-dashpot model proposed by Cundall and Strack (1979) was employed in this work. Fig. 1 shows a schematic diagram of the approach. Here the spring represents the elastic contribution to the contact while the dashpot accounts for the energy dissipation by the plastic deformation. By using the model proposed by Cundall and Strack (1979), the contact force is expressed by fp = fn + ft ,

(12)

(4)

β=

 3 (1 − ε)ε−1.7  ρg CD vp  − vg  4 dp 

(ε > 0.8),

(5)

with dp  the average diameter of particles in the cell, μg the gas viscosity, and CD the drag coefficient for single particles given by CD =

24 (1 + 0.15Re0.687 ) p Rep

CD = 0.44

(Rep > 1000),

(Rep ≤ 1000),

(6) (7)

Fig. 1. Models of contact forces: (a) normal force; (b) tangential force.

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Table 1 Simulation conditions. Length of the column Diameter of the column Diameter of packed particles Friction coefficient of walls Friction coefficient of particles Density of particles No. of particles Particle sizes Restitution coefficient Contact stiffness Density of gas Viscosity of gas Superficial gas velocity

20 mm 9.5, 14.3, 19 mm 2 mm spheres 0.3 0.2, 0.5, 0.8 2500 kg/m3 5250–52,500 75–90, 106–125 ␮m 0.9 5 × 102 , 5 × 103 , 5 × 104 , 5 × 106 1.205 kg/m3 1.8 × 10−5 Pa s 1.5, 1.7, 1.9 m/s

where fn and ft are, respectively the normal and tangential contact forces given by fn = −kSn − ηvn , ft = −kSt − ηvt , vn = (vr · n)n, vt = vr − vn ,

(13)

where Sn and St are displacements in the normal and the tangential directions, respectively, k is the contact stiffness, η is the damping coefficient, and n is the unit vector in the normal direction. If sliding takes place, Eq. (13) for ft should be replaced by ft = −μfric |fn | t,

Fig. 2. Scalar cell and staggered cell.

(14)

where μfric is the coefficient of friction, and t is the unit vector in the tangential direction. 2.2. Numerical method and simulation conditions A 20-mm long vertically oriented column was considered in this work. The majority of simulations were on a 9.5-mm diameter column, the one used in our experiments. Two additional column diameters, 14.3 and 19 mm, were also simulated to examine the effect of diameter ratio of the packed column to packed particles. The separation channels were packed with 2 mm spheres, which gave an average packing porosity of 0.45. Fine particles of two sieve cuts, 75–90 and 106–125 ␮m (with normal size distributions in each sieve cut), were carried upward by an airflow from the bottom of the packed column. These fine particles were generated just below the entrance of the packed column. Table 1 shows the simulation conditions. The inlet velocity and outlet pressure of the carrying gas were given and the non-slip conditions were assumed at the column wall. The simulations were performed in the Itasca PFC3D environment. In such an environment, the fluid scheme is invoked at each fluid timestep between the fluid motion and particle dynamics calculations. The fluid scheme is executed when the accumulative mechanical time equals or exceeds the predicted time for the next fluid step. The pressure and velocity of fluid in each cell are calculated by applying the semi-implicit method for pressure linked equations (SIMPLE) algorithm (Patankar, 1980). The fluid–particle interaction forces are calculated during the iteration. Fig. 2 shows some cells used in the fluid scheme. The

porosity and pressure are defined at the centre of scalar cells. Staggered cells, which are shifted a half size relative to the scalar cells in each direction, are used for discretising the momentum equations, so that the fluid velocity in each direction is defined at the boundary of the scalar cells. The parameters (diameter and velocity) for particles used for calculating interaction forces are averaged over each staggered cell. It is computationally prohibitive to check all possible pairs of particles within each timestep. Contact detection in this work is done by mapping particles into rectangular cells (Cundall, 1988). Only particles contained within the same cell are recognised as neighbours and detected for contact. As individual particles move during the course of a simulation, they are remapped and tested for contact with new neighbours. Contact is said to occur when the centre-to-centre distance between two particles is equal to or less than the summation of the radii of the two particles. Timestep for particles is determined by the trade-off between computation time and numerical stability. Cundall and Strack (1979) proposed the following relation for the mechanical timestep determination.  m Timestep < 2 , (15) k where m is the mass of the particle and k is the stiffness. Xu and Yu (1997) used a drop test to evaluate timestep in simulations of fluidised beds and found the smallest timestep occurred when a free falling particle bounds back to the same height as the original height. Timestep is also affected by the smallest particle size (dpmin ). Kafui, Thornton, and Adams (2002) calculated

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the critical timestep based on the Rayleigh wave speed of force transmission on the surface of elastic bodies and proposed the following relations for critical timestep determination. πdpmin Critical Timstep = 2λ



ρp , G

(16)

with λ = 0.1631␯ + 0.8766, ν the Poisson ratio, G the Young’s modulus and ρp the particle density. In this work, the method proposed by Cundall and Strack (1979) was used. The fluid timestep is generally greater than the mechanical timestep for particles. PFC3D chooses the minimum value from the following three values for the fluid timestep: (a) 100 times the mechanical timestep; (b) half of the minimum value over all fluid cells defined by the size of the fluid cell divided by the magnitude of the fluid velocity in each direction; and (c) half of the minimum value over all fluid cells defined by the size of the fluid cell divided by the magnitude of the averaged particle velocity in each direction. If the fluid timestep calculated by this procedure is smaller than the mechanical timestep, the mechanical timestep is chosen for the fluid timestep. Plastic deformation occurs when the relative impact velocity between two colliding spheres is higher than the critical yield velocity (Fan & Zhu, 1998). In the model of Cundall and Strack (1979), a damping coefficient was introduced to take into account kinetic energy loss in an inelastic collision. However, it is difficult to determine this coefficient theoretically due to involvement of multi-collision dynamics (Xu & Yu, 1997). Tsuji, Kawaguchi, and Tanaka (1993) proposed a simplified relation between the damping coefficient and the coefficient of restitution (e): η=2

mk . (π/ln(e))2 + 1

(17)

In this work, the drop test proposed by Xu and Yu (1997) was used.

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3. Results and discussion The first step of the work was to compare the modelling results with experimental observations on a qualitative basis. The work was then focused on understanding the mechanisms of particle classification in packed columns. This was done by examining the effects of various factors on the hydrodynamics of gas–solid flow and particle dynamics. These factors included the number of fine particles, the superficial gas velocity, the material property (contact stiffness), and the size ratio of the packed column to the packed particles. The results are presented and discussed in the following. In this section, except for otherwise indicated, the friction coefficient is taken as 0.5 for particles and 0.3 for the wall, the stiff coefficient as 5 × 103 , the number of particles as 5250 and particle size as 106–125 ␮m sieve cut. 3.1. Preparation of packed columns The column was packed by the free-fall method, as shown in Fig. 3. First, packed particles of 2 mm spheres were generated in a hopper above the column. The initial velocities of these particles were set to zero. Driven by the gravity, these particles fell downward into the column. The packing process was terminated when the ratio of maximum unbalanced forces to contact forces was equal to or below 0.001. Then, these particles were permanently bonded to one another so that there was no relative motion among the packed particles and between the packed particles and the column during the rest of the simulation. 3.2. Comparison between experiments and simulations Experiments were carried out using a column of the same diameter and packed with the same sized particles as used in the modelling. However, the column was 2 m long, much longer than that used in the modelling. The amount of fine particles for the classification experiments was about 2 g, which are far more than we could handle using the discrete method within a reasonable period of time. As a consequence, the comparison will be on a qualitative basis. Despite the difference in the length

Fig. 3. Schematic illustration of the procedure of making a packed column: (a) generation of particles in a hopper above the column; (b–d) free fall of particles by gravity; (e) removal of extra particles and placing a mesh at the outlet to prevent particles from escaping.

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of the column and the amount of fine particles, pressure distribution was measured along the column in experiments, which could provide a means for comparison as will be discussed in the following. Fig. 4(a) and (b) shows the measured and calculated pressure responses after fine particles are injected at the inlet and a position some distance from the inlet in the packed column, respectively. Several observations can be made: • Both experimental measurements and modelling results show a rapid increase in the pressure at the inlet when fine particles are injected. Such rapid increase in the pressure is due to the local decrease in the bed porosity as a result of injection of fine particles, and energy consumed for transporting the fine particles and for overcoming the energy loss due to collisions between fine particles, fine particles and packed particles and fine particles and the wall; see more discussion on this later. The pressure quickly reaches a maximum and then declines slowly to approach the pressure level before the injection; see the upper curves in Fig. 4(a) and (b). A similar pressure response is observed at a downstream position with a time delay (lower curves in Fig. 4(a) and (b)). Note that the time delay is of the order of the distance divided by the superficial gas velocity. • Both the modelling and experiments show that the pressure response peaks at the downstream positions are lower than that at the inlet, which is likely due to the damping of the packed bed on the pressure signal propagation and also dispersion of fine particles so that the decrease in the local porosity is lower than that at the inlet; see later for more discussion. • Both the experiments and modelling show that the smaller particles (75–90 ␮m) give a higher pressure peak and narrower pressure distribution than the larger particles (106–125 ␮m), though the predicted pressure distributions are sharper and narrower than the measured ones. • There are small-scale fluctuations in the predicted outlet pressure, which does not occur in the measured pressure because the data acquisition rate in the experiments is much lower than in the modelling. The fluctuation disappears in predicted pressure drop between the inlet and outlet, similar to the experimental observation. This is shown in Fig. 4(c) where the pressure drop is normalised by the total particle weight per unit cross-sectional area (wp /A) of the packed column. It is interesting to note that the elutriation time ratio of the experiment to that of prediction is about 20, which compares well with the length ratio of the column used in the experiments to that used in the simulation. The above observations suggest that modelling captures some main features of the flow hydrodynamics of a pulse of fine particles through a packed bed. 3.3. Effect of the injected number of fine particles Numerical simulations were performed on the injection of 5250, 26,250 and 52,500 particles of sieve cut 106–125 ␮m at a superficial gas velocity 1.9 m/s. Fig. 5 shows the time evolu-

Fig. 4. Pressure responses: qualitative comparison between experiments and simulations at superficial gas velocity of 1.5 m/s (2 g of fine particles in experiments, 5250 of 106–125 ␮m and 13,718 of 75–90 ␮m particles in simulations). (a) Measured pressure response at the inlet and 0.4 m from inlet (P1 pressure sensor); (b) calculated pressure responses at the inlet and outlet (20 mm from the inlet); (c) normalised pressure drop for the injection of 106–125 ␮m particles.

tions of the porosity and pressure drop where the porosity is the average of the packed column. It can be seen that injection of fine particles results in a reduction in the average porosity of the bed as expected and the extent of the reduction increases with

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Fig. 6. Effect of number of injected fine particles on the radial occupancy of the fine particles: column with 20 mm length and 9.5 mm diameter at a superficial gas velocity of 1.9 m/s, injected particles of a sieve cut of 106–125 ␮m.

Fig. 5. Effect of the number of injected fine particles: column with 20 mm length and 9.5 mm diameter, superficial gas velocity at 1.9 m/s and injected particles with a sieve cut of 106–125 ␮m. (a) Average porosity of the packed column; (b) pressure drop across the packed column.

increasing number of injected particles; an injection of 52,500 particles causes a decrease in the porosity by nearly eight times in comparison to the injection of 5250 particles; see Fig. 5(a). The porosity decrease corresponds to the increase in the pressure drop across column as shown in Fig. 5(b); an injection of 52,500 particles leads to a peak pressure drop of around 20 kPa, which is about 2.5 times the peak pressure drop caused by injecting 5250 particles. It is interesting to note that the time at which the peak pressure drop occurs (Fig. 5b) differs from that the peak porosity occurs (Fig. 5a)—the peak time for the pressure drop decreases with increasing number of injected particles, whereas that for the porosity is little dependent on the number of injected particles. Such a difference is likely due to the porosity data being averaged across the whole volume of the packed column and hence they do not reflect the true dynamics of the system. The pressure drop data, on the other hand, reflect the dynamics of the system. An increase in the number of injected particles causes a decrease in the local porosity, hence an increase in the local gas velocity and a fast pressure response.

Fig. 6 shows the radial occupancy distribution of the fine particles in the packed column, where the occupancy is defined as the time ratio of fine particles spending in an annular voxel to the total time that the particles spend in the column. In Fig. 6, Rc is the column radius, r is the radial co-ordinate, and the horizontal scale is normalised by the packed particle diameter. One can see that the fine particles have the highest occupancy at the wall (Rc − r)/dp = 0, where the porosity is the highest. The occupancy falls quickly to a minimum at (Rc − r)/dp ∼0.5, i.e. half particle diameter away from the wall. With increasing value of (Rc − r)/dp , a periodical change of the occupancy with the radial position is observed, similar to the radial porosity distribution observed experimentally; see for example de Klerk (2003) and Ding et al. (2005). Fig. 6 also shows that the number of injected particles has a small effect on the radial occupancy distribution of fine particles except for the wall region where a greater number of injected fine particles give a slightly lower occupancy. Fig. 7 illustrates the effect of injected fine particle number on the mean velocity and the kinetic energy of the particles where the particle velocity refers to the translational velocity and the kinetic energy accounts for both translational and rotational motions of the particles. One can see that both the mean velocity and kinetic energy of the fine particles increase with increasing number of injected fine particles. This is consistent with the pressure drop increase and porosity decrease as discussed above—a larger number of fine particles leads to a reduction in the local porosity, which gives a higher local gas velocity, hence the higher particle velocity and kinetic energy. The effect of injected fine particle number on their breakthrough behaviour is shown in Fig. 8, which reveals that a larger number of injected particles give a shorter retention time and hence a faster breakthrough. Fig. 8 also suggests that the decrease in the retention time is significant when the number of injected particles increases from 5250 to 26,250, but further increase in the number of particles gives a very small effect. The implication of this observation is that the SPC for particle classification should be operated with a moderate amount of fine

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Fig. 9. Normalised contacts for a superficial gas velocity of 1.9 m/s, 5250 injected particles of a sieve cut of 106–125 ␮m.

Fig. 7. Effect of injected fine particle number on the mean velocity and kinetic energy of particles for a superficial gas velocity of 1.9 m/s and injected particles of a sieve cut of 106–125 ␮m. (a) Mean translational particle velocity; (b) kinetic energy.

Fig. 8. Effect of the number of injected particles on the breakthrough behaviour of fine particles at a superficial gas velocity of 1.9 m/s and injected particles of a sieve cut of 106–125 ␮m.

particles. As the span between 5250 and 26,250 is large, the exact range of particle number needs further investigation. Illustrated in Fig. 9 are normalised contact numbers with respect to the total number of particles for an injection of 5250 particles at superficial gas velocity 1.9 m/s. It can be seen that, at the very beginning of the injection, collisions among the injected particles dominate. Once the particles enter the packed bed, contacts between the injected and the packed particles arise quickly and soon overtake the collisions among the injected particles. In contrast, the number of contacts between the injected particles and the wall are much lower than that among the injected particles in the initial stage and that between the injected and the packed particles after the initial stage. It is therefore concluded that the interactions among the injected particles and interactions between the injected and packed particles mainly determine the retention time. Fig. 10 compares the normalised particle contacts with different numbers of injected fine particles. One can see that an increase in the injected particle number increases the number of contacts among the injected particles, whereas the number of contacts between the injected and packed particles increases very little when the number of injected particles increases from 5250 to 26,250 and little change to the contact number occurs when the number of injected particles increases from 26,250 to 52,500. This seems to indicate some sort of saturation in terms of contact between the injected particles and the packed particles when particle concentration exceeds a certain limit—similar to the adsorption process. A particle may slide upon contact with another particle or the wall. This can cause kinetic energy loss usually in the form of heat. Fig. 11 shows the mean accumulative frictional work for three different numbers of injected particles, calculated by normalising the accumulative frictional work with respect to the total number of injected particles. The mean accumulative frictional work is seen to vary significantly for the different numbers of injected particles. A larger number of injected particles give a greater amount of mean frictional work. As the number of contacts between the injected and packed particles is less sensitive to the injected particle number (Fig. 10), the difference in the

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Fig. 10. Effect of injected fine particle number on particle collisions for a superficial gas velocity of 1.9 m/s and injected particles of a sieve cut of 106–125 ␮m. (a) Collisions among injected particles; (b) collisions between injected and packed particles.

565

Fig. 12. Effect of superficial gas velocity on the time evolutions of mean particle velocity and kinetic energy for 5250 particles of 106–125 ␮m sieve cut. (a) Mean particle velocity; (b) kinetic energy.

mean frictional work as shown in Fig. 11 is most likely to be due to the increased interactions among the injected particles. The associated energy loss is reflected in the pressure drop increase as discussed above (Figs. 4 and 5). 3.4. Effect of superficial gas velocity

Fig. 11. Effect of injected particle number on the mean accumulative frictional work for a superficial gas velocity of 1.9 m/s and particles of the sieve cut of 106–125 ␮m.

Fig. 12 plots the time evolutions of the mean velocity and the total kinetic energy of the injected particles for three superficial gas velocities. One can see that an increase in the superficial gas velocity from 1.5 to 1.9 m/s (∼1.27 times) leads to an increase in the peak mean particle velocity from ∼0.58 to ∼0.75 m/s (∼1.29 times) and the corresponding peak kinetic energy increase is about 1.58 times. Note that the slightly higher extent of the increase in the mean particle velocity than that in the superficial gas velocity increase does not violate the first law of thermodynamics as the peak values are used. If there were no kinetic energy loss, the increase in the mean particle velocity would have implied a kinetic energy increase of ∼1.66 times. The actual lower value of 1.58 times is due to kinetic energy loss to other forms of energy loss such as heat due to friction.

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Fig. 14. Effect of superficial gas velocity on the frictional energy loss for 5250 particles of 106–125 ␮m sieve cut.

Fig. 13. Effect of superficial gas velocity on the breakthrough behaviour for 5250 particles of 106–125 ␮m sieve cut. (a) Normal breakthrough curves; (b) normalised breakthrough curves with respect to (L/Ug ).

The increased mean particle velocity also implies a fast particle elutriation and hence a shorter retention time. This is shown in Fig. 13(a) where particle breakthrough curves are seen to shift towards the left when the superficial gas velocity is increased. It is interesting to note that all the breakthrough curves collapse into a single curve (Fig. 13b) if the elutriation time is scaled with respect to L/Ug where L is the column length. Increasing the superficial gas velocity increases the accumulative frictional loss; see Fig. 14. However, the normalised number of contacts seems to be insensitive to the change of the superficial gas velocity in the range of 1.5–1.9 m/s as illustrated in Fig. 15. This is likely to be due to a shorter retention time of fine particles at a higher superficial gas velocity. It is also found that the superficial gas velocity imposes little effect on the radial distribution of particle occupancy (Fig. 16), which is in agreement with the experimental observations by Ding et al. (2005) using the positron emission particle tracking technique.

Fig. 15. Effect of superficial gas velocity on particle collisions for 5250 particles of 106–125 ␮m sieve cut. (a) Contacts between injected and packed particles; (b) contacts among injected particles.

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Fig. 16. Effect of superficial gas velocity on radial particle occupancy distribution for 5250 particles of 106–125 ␮m sieve cut.

3.5. Effect of contact stiffness As discussed above, collisions among injected fine particles and that between injected and packed particles dominate at different stages of fine particle motion through the packed bed. The contact stiffness is expected to impose a significant effect on the retention time of fine particles. This was investigated by using a range of stiffness values between 5 × 102 and 5 × 106 N/m in the simulation with 5250 particles of 106–125 ␮m sieve cut at a superficial gas velocity of 1.9 m/s. Fig. 17 shows the simulated breakthrough curves for the different values of the contact stiffness. It can be seen that the breakthrough curves for the contact stiffness ranging from 5 × 102 to 5 × 104 N/m are almost overlapping and are in qualitative agreement with the experimental observations as shown earlier, whereas the breakthrough curve for the contact stiffness of 5 × 106 N/m is very much different from the experimental observations. By looking at the detailed particle motion, it was found that the use of contact stiffness of 5 × 106 N/m leads to rebounding of the injected particles at the inlet of the packed column upon injection.

Fig. 17. Effect of contact stiffness on fine particle breakthrough behaviour for 5250 particles at a superficial gas velocity of 1.9 m/s.

Fig. 18. Effect of the contact stiffness on the mean gas velocities and pressure drop for 5250 particles of 106–125 ␮m sieve cut at a superficial gas velocity of 1.9 m/s. (a) Mean gas velocity; (b) pressure drop.

Fig. 18 illustrates the effect of the contact stiffness on the mean gas velocity and pressure drop. One can see that both the mean gas velocity and the pressure drop depend little on the value of the contact stiffness over the range of 5 × 102 and 5 × 104 N/m. This is also true for the mean velocity and kinetic energy of the fine particles as shown in Fig. 19. However, the change in the contact stiffness changes the number of contacts between the injected and packed particles as shown in Fig. 20, which indicates that the normalised accumulative number of contacts increases with increasing contact stiffness. This suggests that particles of higher contact stiffness are more heavily intercepted by the packed particles. However, as shown in Fig. 17, the breakthrough behaviour is little dependent on the stiffness value over the range considered in Fig. 20, other factors must have played a role to compensate the interception of particles. The frictional energy loss could be a main factor as shown in Fig. 21, where increasing the contact stiffness decreases the accumulative frictional work. This implies that higher contact stiffness results in less kinetic energy loss of particles, because the frictional force is directly

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Fig. 21. Effect of contact stiffness on the accumulative frictional work for 5250 particles of 106–125 ␮m sieve cut at a superficial gas velocity of 1.9 m/s.

related to the contact force, which is proportional to the contact stiffness. 3.6. Effect of the ratio of packed column diameter to packed particle diameter

Fig. 19. Effect of contact stiffness on the mean particle velocity and the total kinetic energy for 5250 particles of 106–125 ␮m sieve cut at a superficial gas velocity of 1.9 m/s. (a) Mean particle velocity; (b) total kinetic energy.

Fig. 20. Effect of contact stiffness on the accumulative number of contacts between the injected and the packed particles for 5250 particles of 106–125 ␮m sieve cut at a superficial gas velocity of 1.9 m/s.

The diameter ratio of the packed column to the packed particles has a strong impact on the packing structure (de Klerk, 2003). To investigate such an effect, packed column diameter was changed to give diameter ratios of 4.75, 7.13 and 9.50 and the corresponding injected fine particle numbers were 5250, 11,813 and 21,000, respectively to keep the same volume fraction. In these simulations, fine particles of 106–125 ␮m sieve cut were used; the superficial gas velocity was 1.9 m/s and the contact stiffness was taken as 5 × 103 N/m. Fig. 22 shows the breakthrough curves for the different diameter ratios. One can see that the diameter ratio of 7.13 gives the fastest breakthrough suggesting that the effect of the diameter ratio is complex. On one hand, an increase in the diameter ratio

Fig. 22. Effect of diameter ratio of the packed column to the packed particles on the breakthrough curves for 5250 particles of 106–125 ␮m sieve cut at a superficial gas velocity of 1.9 m/s.

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Fig. 24. Effect of diameter ratio of the packed column to the packed particles on the mean particle velocity for particles of 106–125 ␮m sieve cut at a superficial gas velocity of 1.9 m/s.

Fig. 23. Effect of diameter ratio of the packed column to the packed particles on the pressure drop and mean gas velocity for 5250 particles of 106–125 ␮m sieve cut at a superficial gas velocity of 1.9 m/s. (a) Pressure drop; (b) mean gas velocity.

from 4.75 to 9.5 leads to a reduction in the peak porosity from 0.428 to 0.396 (not shown in figures). This should lead to an increase in the breakthrough time. However, the increase in the diameter ratio also leads to an increase in the mean gas velocity (and hence an increase in the pressure drop) as shown in Fig. 23. The consequence of this is that the injected particles would have a greater velocity (Fig. 24). The combination of the above two factors seems to provide an explanation to the fastest breakthrough for the diameter ratio of 7.13. The above analyses also suggest existence of an optimal size ratio for a shortest breakthrough time. The diameter ratio of the packed column to the packed particles has an effect on the radial distribution of particle occupancy. Fig. 25 shows the results. Increasing the diameter ratio leads to a decreased peak height of the oscillatory dependence of radial occupancy distribution. This is because an increase in the diameter ratio provides more paths for the injected particles and hence particles are more widely and evenly distributed in the radial direction. The results shown in Fig. 25 also reveal that a greater

Fig. 25. Effect of diameter ratio of the packed column to the packed particles on the radial distribution of particle occupancy for 5250 particles of 106–125 ␮m sieve cut at a superficial gas velocity of 1.9 m/s.

diameter ratio gives a reduced particle occupancy at the wall and hence an increased occupancy in the centre region of the column. 4. Concluding remarks This work aims at a fundamental understanding of the SPC for particle classification by using the CCDM. The following conclusions are obtained: • The numerical simulations capture some important features of the flow hydrodynamics and solids behaviour of injected fine particles flowing through a packed bed. The simulated time evolutions of the pressure and pressure drop agree at least qualitatively with the experimental observations. • Injection of fine particles into a packed bed reduces the average bed porosity, leading to an increased local gas velocity

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and hence an increased mean particle velocity and a reduced particle retention time. • The radial particle occupancy changes periodically with radial position with the highest occupancy occurring at the wall where the porosity is the highest. Such an observation is little affected by the number of injected particles and the superficial gas velocity. However, the effect of diameter ratio of the packed column to the packed particles is considerable; an increase in the diameter ratio leads to a decreased peak height of occupancy distribution, indicating a more evenly particle distribution across the column cross-section. • Collisions among the injected particles and that between the injected and packed particles are dominant in determining the retention time of the injected particles, with the former dominates the initial stage of the injection and the latter dominates the rest of the elutriation process. An increase in the number of injected particles increases the number of contacts among the injected particles, but this does not affect much the number of contacts between the injected and packed particles, indicating ‘saturation’ of the contact similar to the adsorption process. • The contact stiffness over the range of 5 × 102 to 5 × 104 N/m imposes some effects on the number of contacts between particles. However, such a parameter has little influence on particle retention time, mean gas velocity, pressure drop and mean particle velocity. The results reported in this paper are for one sieve cut, focusing on the effects of the number of fine particles, the superficial gas velocity, the material properties (contact stiffness), and the size ratio of the packed column to the packed particles. Further work is underway investigating numerically the effects of particle size and surface energy, which is expected to provide clues as for why large particles elutriate faster than small ones. References Anderson, T. B., & Jackson, R. (1967). A fluid mechanical description of fluidized beds. Industrial & Engineering Chemistry Fundamentals, 6(4), 527–539. Chen, J., Akiyama, T., Nogami, H., & Yagi, J. I. (1994). Behavior of powders in a packed bed with lateral inlets. ISIJ International, 34(2), 133–139. Cundall, P. A. (1988). Formulation of a three-dimensional distinct element model I: A scheme to detect and represent contacts in a system composed of many polyhedral blocks. International Journal of Rock Mechanics and Mining Sciences, 25(3), 107–116. Cundall, P. A., & Strack, O. D. L. (1979). Discrete numerical model for granular assemblies. Geotechnique, 29(1), 47–65. de Klerk, A. (2003). Voidage variation in packed beds at small column to particle diameter ratio. AIChE Journal, 49(8), 2022–2029. Di Maio, F. P., & Di Renzo, A. (2004). Analytical solution for the problem of frictional-elastic collisions of spherical particles using the linear model. Chemical Engineering Science, 59(16), 3461–3475. Di Renzo, A., & Di Maio, F. P. (2004). Comparison of contact-force models for the simulation of collisions in DEM-based granular flow codes. Chemical Engineering Science, 59(3), 525–541. Ding, Y. (2006). Method and apparatus for solids phase chromatography. WO 2006/082431 A1. Ding, Y., Wang, Z., Wen, D., Ghadiri, M., Fan, X., & Parker, D. (2005). Solids behaviour in a gas–solid two-phase mixture flowing through a packed particle bed. Chemical Engineering Science, 60(19), 5231–5239.

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