Flow and Mixing of Cohesive Particles in a Vertical Bladed Mixer

Feb 19, 2014 - John Bridgwater, ... blades of rake angle ϕ fitted to a vertical shaft. ... At a high shaft speed, blades of 90° rake angle can resul...
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Flow and Mixing of Cohesive Particles in a Vertical Bladed Mixer Rohana Chandratilleke,† Aibing Yu,*,† John Bridgwater,‡ and Kunio Shinohara§ †

Laboratory for Simulation and Modeling of Particulate Systems School of Materials Science and Engineering, The University of New South Wales, Sydney, NSW 2052, Australia ‡ Department of Chemical Engineering and Biotechnology, University of Cambridge, Pembroke Street, Cambridge CB2 3RA, U.K. § Particulate Chemical Engineering Laboratory, Sapporo 060-0051, Japan ABSTRACT: Mixing of fine particles is an important operation to obtain products with controlled properties in the pharmaceutical as well as many other industries. Here, the Discrete Element Method (DEM) is used to simulate the mixing behavior of monosized fine particles in a vertically shafted cylindrical bladed mixer. The mixer impeller consists of two rotating blades of rake angle ϕ fitted to a vertical shaft. The particle cohesion is considered to be due to the van der Waals forces and is changed by varying the Hamaker constant Ha for particles of a given size. The aim is to examine the effects of interparticle cohesion, rake angle, and particle-wall cohesion on the flow and mixing behavior of fine particles. The results suggest that all these factors should be carefully selected for obtaining a good mixing performance. In particular, it is shown that the particle bed may be lifted up and remains above the rotating blades without mixing if the interactions between particles and walls are highly cohesive. At a high shaft speed, blades of 90° rake angle can result in a high mixing rate and a small shaft torque, with mixing mechanisms different from those at other rake angles.

1. INTRODUCTION Granular mixing is an important operation to obtain products with controlled properties in many industries. Pharmaceutical products,1 composite material for cutting tools,2 and plastics3 are some examples. However, such mixing operations can be severely affected by the cohesion between particles to be mixed.4 Generally, the cohesion between particles can be caused by capillary force,5 electrostatic force,6 or van der Waals force,7−10 and a combination of all or some of these forces. In fact, interparticle cohesion can be found in a large number of applications involving powders less than 100 μm, such as pharmaceuticals,1 petrochemicals, food, cosmetic powders, and other functional materials made from powder-coated core particles,11 to name a few. Ensuring the uniformity of powder mixtures is a crucial issue in many applications.12,13 For example, in pharmaceutical tableting processes, product uniformity should satisfy strict quality control standards,1,13 failure of which may result in delivery of an overdose of the active ingredients to patients.14 Investigations of powder mixing have been carried out largely by tumblers such as rotating drums,4,5,15 tote blenders,13 Vblenders,16 and sometimes convective mixers such as ribbon mixers.17 Such mixers are also known as batch mixers. In the case of the rotating drums, the effect of particle cohesion has been investigated for both dry4 and wet particles numerically.18 Cohesion of dry particles has been varied artificially by using a square-well potential,4 and that of wet particles by changing the bond number while particle size is fixed.18 In particular, the effect of capillary force on the flow and agglomeration of wet particles in a cylindrical bladed mixer has been investigated.19,20 According to these mixing studies, the homogeneity of a mixture may be improved when there is some cohesion between particles. A too large cohesion, however, causes deterioration in the mixture homogeneity. It is suggested that more work is necessary to fully understand the phenomenon.4,5 © 2014 American Chemical Society

Further, the homogeneity of a mixture of cohesive particles is found to be a function of the rotational speed in the case of a rotating drum and increases with an increase in the rotational speed of the drum.4 Conversely, increasing the rotational speed of a tote-blender from 10 to 15 rpm results in a deterioration in the mixture homogeneity at steady state, which was attributed to the decrease in the number of free-surface avalanches with the speed.13,21,22 Similarly, in the case of a ribbon blender, increasing the blade speed from 20 to 55 rpm results in a deterioration of the mixture homogeneity.17 In the case of convective mixers, the main factors that can affect the mixer performance are known to be the operational variables such as the rake angle23 and shaft speed.17,24 In addition, material properties such as cohesiveness may also affect it, based on the observations from rotating tumblers.4,18 In addition to mixture homogeneity, mixing time is also a factor determining the performance of a mixer. It has been shown that mixing time increases with an increase in particle cohesion, which is due to sluggishness of thinning of lumps of similar particles.4 Higher rotational speeds are suitable for mixing cohesive particles because particles can receive larger shear, which makes it possible to break up bonded particles.4 The choice of a mixer depends on many factors such as the ease of use, power usage, mixture quality achieved, and mixing time.25 Here, we choose for the present investigation a vertically shafted cylindrical bladed mixer, which belongs to the group of high shear mixers.25,26 The mixer has a simple construction and consists of a vertical-axis cylinder and an impeller with two or more flat blades fitted to an axial shaft, and its simplicity has made it the choice for many mixing Received: Revised: Accepted: Published: 4119

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studies23,24,26−32 as well as granulation studies.29,33 The major industrial use of this type of mixers is in the agglomeration of powders for use in the manufacture of chemical, food, pharmaceutical, metallurgical, and household products.26 Other uses of the mixers are considered to be intimate mixing of cohesive powders, distribution of liquid and pastes in powders, and compounding of polymers.26 To date, however, there are limited numerical or experimental studies in the literature dealing with the mixing behavior of cohesive particles in the cylindrical bladed mixers. This paper presents a numerical investigation of the mixing behavior of powders in such a bladed mixer by means of the discrete element method (DEM). The aim is to quantify the effects of three variables: interparticle cohesion, particle-wall cohesion, and the rake angle. The paper is organized as follows. First, a brief introduction is given about the simulation method, cohesive force, the scaling of the variables used in the study, and quantification of mixing, which is followed by the simulation conditions and procedure where facts related to mixer size, material properties, and time-step are discussed. Then the results of the present study are discussed, focusing on the effects of the three variables. Finally, the conclusions are presented.

are then said to be in reduced units, RU. To convert from RU to actual SI units or vice versa, one has to use the appropriate conversion factor, which can be found elsewhere.37 For example, to obtain SI units of velocity and force in RU units, the following conversion factors are used respectively: 1RU = (gLref)1/2 m/s and 1RU = gπLref3ρref/6N. Conversely, their inverse relationships can be used to convert from SI to RU units. Such a scaling strategy is useful to reduce numerical errors when dealing with microsized particles. The cohesive force is implemented in the simulations in terms of the van der Waals force which can be determined for plate−plate, plate-sphere, or sphere−sphere systems.9 Fv = −

dVi = mi g + dt

ki

∑ (Fc,ij + Fd,ij + Fv,ij) j=1

(1)

and Ii

dωi = dt

ki

∑ (Tij + Mij) j=1

(3)

where the Hamaker constant Ha is a material-dependent constant and is considered as the geometric mean of the Hamaker constants, Ha1 and Ha2 of a pair of objects when they come into interaction with each other, i.e., Ha = (Ha1 Ha2)1/2.38 Similarly, R is the effective radius for a pair of spheres of radii r1 and r2 coming into interaction with each other, which is determined by the harmonic mean, r1 r2/(r1 + r2). Thus, for a flat plate, the radius being infinite, R becomes the radius of the particle in contact with the plate. λ (= 10−7 m) is dipole interaction wavelength, and b is a constant equal to 5.32. δ is the separation distance between the two objects and is assumed to have a minimum value of 1 × 10−9 m for the purpose of calculating the cohesive force. That is, if δ decreases below this minimum value, the cohesive force is assumed to remain constant, as performed by other investigators.10 This is a simplified treatment to avoid the singularity when δ approaches zero. Sun et al.39,40 recently proposed a more rigorous treatment for silica nanoparticles, but its application to microparticles is not clear yet. The DEM simulation is conducted by use of an in-house code which was used for a previous study.35 The code is run on a Linux server with 8 Intel Xenon CPU processors (E5420), which takes about 1.5 h to complete a real simulation time of 0.01 s. 2.2. Quantification of Mixing. The mixing behavior of cohesive particles is analyzed by a particle-scale mixing index based on the well-known Lacey index41 using instantaneous positions of particles of a mixture.28 According to this method, samples are taken at each particle to include only its contacting particles (i.e., the immediate neighborhood). Then the concentration (or number fraction pi) of a target type of particles in each sample is found as pi = ni/Ni where ni and Ni are the target-type particles and total number of particles in a sample, respectively. Next, the variance St2 of pi is found at the instant considered with respect to the mean sample concentration p for the mixture using the following weighting method.24

2. THEORETICAL TREATMENTS 2.1. Discrete Element Method. In this work, DEM is used to simulate the flow and mixing of particles. The model used is briefly described below, because the details can be found in the previous studies.34−36 The model takes into account the rolling friction of particles.35,36 The governing equations consist of the following two momentum conservation equations, describing the translational and rotational motions of particle i, respectively. mi

⎛ λ ⎟⎞⎞ HaR ⎛ ⎜1 − 1/⎜1 + ⎟ 2 ⎝ ⎝ bδ ⎠⎠ 6δ

(2)

where mi, Ii, Vi, and ωi are the mass, moment of inertia, translational velocity, and rotational velocity of particle i, respectively, g is the acceleration due to gravity, j represents a particle index, ki is the total particle number in interaction with particle i, Fc represents the contact force, which is the summation of the normal and tangential forces Fcn,ij and Fct,ij, respectively, at the contact point with particle j, Fd represents the damping force, which is the summation of the normal and tangential damping forces, Fdn,ij and Fdt,ij, at the contact point, Fv,ij is the cohesive force between particles i and j which are not necessarily in contact, and Tij and Mij are the torque due to the tangential components of the contact forces and rolling friction torque on particle i from particle j, respectively. The equations for the contact forces and torques in eqs 1 and 2 are given elsewhere.35 The contact model used here is the Hertz nonlinear contact model.36 It should be noted that all the input values for solving the governing equations are scaled by scaling factors formed by the three reference values, length Lref, density ρref, and gravity g as suggested by Asmar et al.37 Here, Lref and ρref are chosen as particle diameter and density, respectively. The output values

N

St 2 =

∑ [(wi/wT)(pi − p)2 ] i=1

(4)

where N is the total number of samples, which is here equal to the total number of particles, and wi/wT is the weighting factor with wi = Ni/N and wT = ∑i N= 1wi. Finally, the mixing index is found by using the Lacey index below: 4120

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Figure 1. (a) Geometry of mixer used in the simulations, and (b) particle bed laid in top−bottom arrangement, with type-2 particles being on top of type-1 ones: blade is rotated in the direction of arrow (counter-clockwise direction viewed from the top).

M=

S0 2 − S t 2 2

S0 − Sr

2

Table 1. Input Variables and Their Values for the Base Case (5)

where S02 = p(1−p) is the variance for the fully segregated state, and Sr2 = p(1−p)/n is the variance for the fully mixed state, where n is the instantaneous average particle-scale sample size, which is equal to ∑Ni=1Ni/N and rounded to the nearest number. The determination of neighborhood particles depends on the minimum interparticle gap used, which may cause an effect on M. However, as discussed previously,28 the variation in M is less than 7% when the interparticle gap, i.e., the separation distance between the surfaces of two particles δ, is changed from zero to 10%d, where d is the particle diameter. The variation may depend on the mixing systems considered. This is an issue that may need a detailed study. In this work, the gap is set to 5%d, as used in our previous study of a similar system.28

input variable

value

vessel diameter blade height blade length blade width shaft diameter rake angle blade gap particle number particle dia., d shaft speed, ω Young’s mod. E Poisson’s ratio, ν μs(P−P) (P−W) μR(P−P) (P−W)

5.0 mm 0.8 mm 2.13 mm 0.2 mm 0.64 mm 135° 0.15 mm 23000 0.1 mm 500 rpm 2.16 × 107 N/m2 0.3 0.3 0.05 d

to turn the impeller will then increase with the blade speed,24,26 the torque also being dependent on the fill level.26 The torque on the impeller can be reduced drastically by decreasing the vertical-projected blade height hv (or blade width) and further by using bevelled edged blades.26 Therefore, blades with a rake angle of 135° having a low hv are first used to find the effect of particle cohesion. Next, the rake angle is varied between 0° and 167° to find its effect on the mixing and torque, using particles of moderate cohesion. Finally, the effect of wall cohesion is investigated at a fixed rake angle, shaft speed, and particle cohesion. 3.2. Material Properties and Time Step. Particle properties are listed in Table 1, where E, ν, μs, and μR are Young’s modulus, Possion’s ratio, sliding friction coefficient, and rolling friction coefficient, respectively, with P−P and P− W, respectively, denoting contacts between particles and between particles and walls. The particles are assumed to be spherical with properties close to those of glass beads: the particle density ρ is 2500 kg/m3 and the damping coefficient is 0.3. However, the Young’s modulus has been reduced from 109 of real glass beads to 2.16 × 107 N/m2 to increase the time step, for which a value of 0.5 μs is used. This strategy has been used previously in a mixing study with DEM and found to produce results comparable to those obtained with PEPT (positron emission particle tracking).27 The Rayleigh time step,42 which is defined as c1πd/2(ρ/G)1/2, where c1 is a constant of about 0.95 and G =E/(1 + ν) is shear modulus, is related to wave propagation speed and is evaluated to be 2.6 μs. The time step

3. SIMULATION CONDITIONS AND PROCEDURE 3.1. Mixer Details. To vary the interparticle cohesion, one can either change particle size and/or the Hamaker constant Ha. Note that according to eq 3, with the reduction of particle size, the cohesive force per unit particle weight increases in proportion to 1/d2. However, if particle size is reduced with the mixer size and fill level unchanged, the particle number will increase, causing a heavy computational demand. Therefore, the second option of changing Ha is adopted here to keep the computational requirements within manageable limits, although the possible effect of particle size representing a given Ha on mixing behavior is not fully understood at this stage. In this study, we focused on three variables only, i.e., interparticle cohesion, particle-wall cohesion, and rake angle, with the particle diameter being set to 0.1 mm. The mixer used is illustrated in Figure 1 and is similar to that used in the PEPT and DEM studies.35 It consists of a vertical cylindrical vessel and a pair of flat blades attached to an axial shaft. To keep the particle number small, the mixer used is 1/ 50th model of that used in an earlier work,35 and the details of the mixer are listed in Table 1. The rake angle ϕ of a blade is defined as the angle that a wider blade surface makes with the horizontal, measured from the forward direction of motion of the blade. Being a high-shear mixer, the mixer is generally operated at high shaft speeds to obtain large shear forces required to break down lumps of bonded particles. However, the torque required 4121

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particles of one type (e.g., type-1) are first deposited sequentially in the mixer and next the other type, with their numbers being equal. The type-2 particles are thus laid on top of type-1 particles in the top−bottom arrangement, as shown in Figure 1b. Here, the two types of particles are different only in their colors. Next, the particles are allowed to settle under the gravity for a sufficiently long time, which is found to be about 0.8 s by observing particle height variation with time. At this time, the impeller is rotated from the stationary state with a constant acceleration until it reaches a predetermined speed, on attainment of which it rotates at that constant speed. It is known that a blade of ϕ = 90° causes compaction in the particles in front of the blades.23 Therefore, ϕ = 135° is used here as the base case, which will make it possible to use the friction of the wider blade surface for the mixing action.26 In the base case, the shaft is rotated at a relatively low speed ω = 500 rpm. This shaft speed is chosen to make the tip speed of the blades roughly equal to that of an earlier work.35 The numerical experiments are performed using three steps. First, the effect of varying the particle cohesion by changing the Hamaker constant Ha is investigated for the base case. In the base case, the Hamaker constants Hap and Haw for particle and wall materials are assumed to be identical, equal to Ha values. A lower Haw value is also tested on the base case to investigate its effect on mixing. Next, the rake angle is varied between 0° and 167° to investigate its effect on mixing at a moderate Hap value and a higher shaft speed ω of 1432 rpm, which is nearly 3 times the shaft speed of the base case. Here, ω = 1432 rpm is chosen to reduce the mixing time.23,24 Finally, the effect of Haw is investigated at a fixed rake angle and shaft speed for highly cohesive materials. The possible effect of air on powder flow and mixing is neglected because particle size is large, and the

used here is reasonably smaller, being about 1/5th of the Rayleigh time step; therefore, particle collisions can be handled before wave propagation. Table 2 shows the values of the Hamaker constant Ha used. It also lists the bond numbers (referred to as Bo hereafter) for Table 2. Bond Number for a Single Contact at Different Hamaker Constants Hamaker constant, Ha 6.50 4.55 2.60 6.50 6.50 6.50

× × × × × ×

10−20 10−20 10−20 10−21 10−22 10−23

bond number (P−P)

bond number (P− W) (Hap = Haw = Ha)

bond number (P−W) (Haw = 3.8 × 10−21 J)

20.02 14.01 8.01 2.0 0.2 0.02

40.05 28.03 16.02 4.0 0.4 0.04

9.86 8.10 6.12 3.06 0.97 0.3

P−P contacts and P−W contacts for the two cases when Hap = Haw = Ha and Haw = 3.8 × 10−21 J with Hap = Ha, where Hap is the Hamaker constant of particles and Haw is that of the wall. Bo is defined here as the ratio of the cohesive force to particle weight, and an interparticle gap of 1 × 10−9 m, the minimum surface gap as discussed above, is assumed in the evaluation of the bond number. It is thus only an approximation. 3.3. Simulation Conditions and Procedure. To start the simulation, a fixed number of particles are generated at a certain height Z within the height limits of Z ± d and then dropped. Such particle groups are continuously generated and dropped until the required total particle number in the particle bed is reached. To investigate the mixing of the cohesive particles,

Figure 2. States of mixing at a transitional and steady state for three cases at ω = 500 rpm, N is the number of shaft revolutions: (a) Ha = 0 J (noncohesive particles), (b) Ha = 4.55 × 10−20 J, and (c) Ha = 6.5 × 10−20 J. 4122

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Figure 3. Effect of Ha on mixing: (a) M as a function of time (or shaft revolutions); (b) steady-state values of M as a function of Ha.

taking place. An increase in the value of Ha can affect both the initial mixing rate and steady-state value of an M curve. Figure 3a shows that when Ha is increased, the M curves shift to the right, which is an indication that the initial mixing is delayed. This effect of Ha on M is not significant until Ha = 6.5 × 10−22 J, but it becomes apparent at Ha = 6.5 × 10−21 J (red curve) and above. When Ha is increased further, even the steady-state value is affected. The variation of the steady-state value of M with Ha is shown in Figure 3b. It shows that the steady-state value initially decreases with an increase in Ha but recovers again after about Ha = 2.6 × 10−20 J. It is of interest to note that the observation that some degree of cohesiveness can improve the mixing has also been observed in other systems such as rotating drums4,18,21,22 and with wet particles in cylindrical mixers.20 After Ha = 4.55 × 10−20 J, mixing drastically deteriorates. Figure 3a shows that M at Ha = 6.5 × 10−20 J does not have the high initial mixing rate as at lower Ha values and cannot reach the fully mixed state, which is supported by the phenomenon seen from Figure 2c2. 4.1.2. Particle-Wall Cohesion and Bed Lifting. Figures 2c2 and 3a show that mixing does not take place for Ha = 6.5 × 10−20 J as a result of the particle bed being lifted by the blades. Knight et al.26 reported a similar phenomenon based on their experimental observations. They explained the phenomenon as follows. Lifted by the passage of each blade, the powder falls to the base of the vessel when the shaft speed is low. However, with an increase in the shaft speed, the powder falls through a distance which is less than the blade height, resulting in a sharp reduction in the torque on the impeller. The rise and fall of the powder bed are governed by a few variables in the present study: blade rake angle ϕ, particle-wall cohesion characterized by the wall Hamaker constant Haw, interparticle cohesion characterized by the particle Hamaker constant Hap, and shaft speed ω. Therefore, these variables should be adjusted to improve the mixing of powders. An inspection of Table 2 shows that Bo number (see section 3.2) for the particle-wall contacts is more than that for particle− particle interaction. The table shows that when Hap = Haw, Bo and thus the cohesive force at the wall is twice as large as that between a pair of particles. This large cohesion is a result of the effective particle radius (see section 2). Therefore, the wall cohesive force should play a significant role in the bed-lifting phenomena observed in Figure 2(c2). Reducing the value of Haw should decrease the resistance to the particle motion at the

blade tip speed is low, being 0.26 m/s at 500 rpm and 0.75 m/s at 1432 rpm.

4. RESULTS AND DISCUSSION The results are presented in three subsections. Section 4.1 focuses on the base case where the rake angle is fixed at 135°, a typical bladed design, to establish some general understanding of the effects of interparticle (P−P) and particle−wall (P−W) cohesions. In particular, significance of P−W cohesion is identified. In section 4.2, the effect of rake angle is investigated at a moderate value of Hap. In section 4.3, the effect of P−P cohesion is examined at different rake angles and a fixed moderate P−W cohesion. The aim is to develop some design rules for bladed mixers. 4.1. General Flow and Mixing Features. 4.1.1. Mixing States and Index. Figure 2 shows the mixing states at a transitional (left) and steady state (right) for the base case of ϕ = 135° at different Ha values. The so-called steady state is obtained when a macroscopic variable, e.g., mixing index, just fluctuates around its mean. Note that Hap = Haw = Ha for the base case, and N represents the number of shaft revolutions from the start of mixing. Figure 2(a2) shows that the mixture has become uniform after N = 23 in the case of noncohesive particles. The particle bed preserves a good shape, with particles going over the blade without much disturbance to the particle bed. Figure 2(b2) shows that the mixture has mixed sufficiently after N = 22 for Ha = 4.55 × 10−20 J. The particle bed gets distorted because of the wall cohesion in transitional stages and finally becomes an expanded one. Figure 2(c2) shows that particles are not mixed even after N = 21 for Ha = 6.5 × 10−20 J. In this case, the particle bed is seen to have been lifted as one block of particles from the very early stage of mixing. The evidence of the occurrence of such a phenomenon comes from the experimental observation that there is a reduction of the impeller torque at a certain critical shaft speed. Figure 3a shows the particle scale mixing index M as a function of time for the above three cases and a few more Ha values. The curve Ha = 0 represents the case of noncohesion, and it approaches a value close to 1 after about N = 20. The Ha = 0 curve serves as a benchmark of mixing in that the closer a curve to the benchmark curve at the steady state, the better the quality of the mixture. All the M curves in Figure 3a show an increase initially at a certain rate and then an asymptotic behavior; the initial increase is an indication that mixing is 4123

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angle on the mixing performance, we consider the mixing rate under different conditions. For simplicity, the mixing rate is measured by the time to reach the mixing index of 0.8 from the start of mixing. The mixing rate is shown in Figure 5b as a function of rake angle. It shows that ϕ = 90° has the fastest mixing rate for moderately cohesive particles.23 The mixing rate for the very commonly used rake angle ϕ = 120° is just about 77% of that for ϕ = 90°. Therefore ϕ = 90° is a suitable rake angle for mixing. 4.2.2. Torque and Coordination Number Characteristics. Figure 6 shows the effect of rake angle on the shaft torque and average coordination number of particles under the conditions described above. The torque shows a dip at ϕ = 90° for this case of cohesive particles. The average coordination number also shows a dip at ϕ = 90°, implying that particle contacts have become sharply reduced. Thus, it is anticipated that the mixing mechanism has changed at ϕ = 90°. It is worthwhile to note that the characteristic of torque versus rake angle for noncohesive particles at lower shaft speeds is such that the torque increases with a reduction in the rake angle in the range of 45° to 135°.23 A similar result can be inferred from the characteristics of the horizontal force on a blade versus rake angle in single blade studies.12,31 Therefore, it further demonstrates that the mixing mechanism has changed at ϕ = 90°, which is investigated below. Figure 7a shows the shaft’s torque and speed variations as a function of shaft revolutions for the case of ϕ = 90° considered above. The speed varies linearly up to about 7.15 revolutions before becoming steady at 150 rad/s. The torque on the shaft shows a gradual increase in the initial 4 to 5 revolutions. But it becomes sharply reduced at 5 revolutions when the shaft speed reaches about 100 rad/s (or 955 rpm). When the shaft rotates steadily, the torque becomes steady with a value of about 37% of the initial peak torque value. Thus, it can be deduced that the particle bed must have been expanded with the increasing shaft speed. Figure 7b shows that the coordination number drops sharply in the initial 5 revolutions in which the shaft accelerates to its steady speed. After this period, the coordination number becomes almost steady at a low value, which shows that the bed has indeed expanded. The step rise in the coordination number that follows is due to the deliberate change in Hap from the value of 2.6 × 10−20 J to 6.5 × 10−20 J. Note that the torque variation becomes less significant when Hap is increased to 6.5 × 10−20 J.

cylindrical walls and thus should improve the up and down motion of particles following the blade motion. To demonstrate the above effect of wall cohesion, Haw is reduced to a low value of 3.8 × 10−21 J; note that PTFE (or Teflon) has a value of about 4.4 × 10−20 J.43 For this purpose, the case of the lifted-particle bed in Figure 3 is used, and the simulations are continued from the final state in Figure 3 with Haw switched to 3.8 × 10−21 J from the value of 6.5 × 10−20 J while keeping Hap = 6.5 × 10−20 J. The variation of particlescale index due to the change in Haw is shown in Figure 4. It

Figure 4. Effect of reducing wall cohesion when ω = 500 rpm, ϕ = 135°, and Hap = 6.5 × 10−20 J.

can be observed that particles are now in fact approaching the well-mixed state. It is clear that the lifted particle bed could not remain above the blades due to the reduction of the wall cohesion. 4.2. Effect of Rake Angle. 4.2.1. Mixing Performance. Here, we investigate the performance of the mixer at different rake angles for cohesive particles of Hap = 2.6 × 10−20 J with Haw = 3.8 × 10−21 J at a shaft speed of ω = 1432 rpm. The value of Hap is chosen because it is moderately cohesive (Bo = 8) as shown in Table 2, and ω=1432 rpm is selected to reduce the mixing time. By setting Haw to the above value, the Bo number for P−W interactions can be reduced considerably (see Table 2). Figure 5a shows the mixing performance at several blade rake angles. The figure shows that the mixing of the moderately cohesive particles can be performed at any rake angle as in the case of noncohesive particles.23 To compare the effect of rake

Figure 5. Effect of rake angle on the mixing performance for cohesive particles of Hap = 2.6 × 10−20 J at 1432 rpm with Haw = 3.8 × 10−21 J: (a) mixing curves, and (b) average mixing rate to obtain 80% of the steady-state mixing index. 4124

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Figure 6. Effect of rake angle on (a), shaft torque, and (b) average coordination number for cohesive particles of Hap = 2.6 × 10−20 J when Haw = 3.8 × 10−21 J at 1432 rpm.

Figure 7. (a) Variations of torques on blades, and (b) coordination number when ϕ = 90° Hap = 2.6 × 10−20 J at 1,432 rpm, where the step increase of coordination number is due to the increase of Hap to 6.5 × 10−20 J.

snapshot at the steady state is shown in Figure 8d, illustrating that the particle bed is considerably expanded. Comparison of coordination numbers at the steady state and the start of mixing confirms that the bed has expanded considerably at the steady state. Curve 5 of Figure 5a shows that the steady-state mixing at ϕ = 90° in the case of Hap = 2.6 × 10−20 J has not reached the fully mixed state, because not all the particles can participate in the mixing process, with some particles being attached to the vessel wall and lid. To confirm this consideration, Hap is raised from 2.6 × 10−20 J to 6.5 × 10−20 J at steady state. Figure 8e shows that the lifted particles begin to coalesce, forming lumps of particles in the space as well as on the walls. The particle lumps on the walls can be seen to slide downward because they are too heavy to be held by the walls. As a result of most particles returning to the particle bed, mixing improves as seen from curve 5 of Figure 5a. Figure 7b also shows that raising Hap

To confirm the state of the particle bed, snapshots of the particle bed at several revolutions are shown in Figure 8, each being captured in a bottom isometric view, with the shaft rotating in the anticlockwise direction viewed from the top. Figure 8a shows that particles are lifted up in front of the blades rather than going over the blades when the shaft has rotated about 2 revolutions at which point the shaft speed is 42.6 rad/s. As the shaft speed further builds, the particles are thrown above the blades as seen from Figures 8b and 8c. Such an expansion in the particle bed happens because the particles in contact with the vertical blade surfaces are set in the vertical upward motion by the rotating blades, the amount of which increases with the shaft speed, especially at the cylinder periphery; note that there is clear evidence that the bed height increases with the blade speed.24 It is also possible to observe that some particles are attached to the top vessel lid at these stages, but they are being replaced by other particles as the shaft rotates. A typical 4125

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Figure 8. Bottom isometric snapshots of the particle bed in the closed vessel at different revolutions when ϕ = 90° at 1432 rpm: (a) to (d) for Hap = 2.6 × 10−20 J, and (e) to (f) for Hap = 6.5 × 10−20 J.

Figure 9. Bottom isometric views of the mixer showing steady-state mixing states at ω = 1432 rpm for the ϕ = 13° blades rotating in direction of the arrow (or anticlockwise direction viewed from the top): (a) noncohesive particles, and (b) cohesive particles of Hap = 6.5 × 10−20 J.

It would be of interest to note that an analytical equation is available for evaluating the torque of an impeller when stirring cohesive wet particles in cylindrical mixers.44 According to the formula, the torque depends on many variables: bed height, vessel diameter, wall friction, wall cohesion per unit area, bed mass, and shaft speed. It consists of two parts, one due to the wall friction and another due to the P−W cohesion. If evaluated for ϕ = 13° and Hap = 2.6 × 10−20 J with Haw = 3.8 × 10−20 J at 1432 rpm, the frictional part of the torque becomes 1.27 × 10−6 Nm and the cohesive part of the torque becomes 1.2 × 10−4

reduces the bed expansion. Figure 8f shows a typical snapshot of the particle bed at steady state after raising Hap to 6.5 × 10−20 J. The particle bed has become lumpier and taller because of the large cohesion and 90° rake angle. Expansion and collapse of the particle bed can be observed in the animations even at this Hap value, thus promoting mixing. Therefore, it can be speculated that ϕ = 90° blades will result in a good mixture even if the mixing is started off from the state of zero shaft speed at this Hap value. 4126

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Figure 10. Variation of overall and partial coordination numbers (CNs) as a function of shaft revolutions for ϕ = 13° and 167° blades; here, T and B represent the initially laid particles at the top and bottom, respectively; (T-T) indicates partial CN for the contacts of particles of T type, (B-T) indicates that for type B and type T particle contacts, and other combinations follow similar meanings.

arrangement as described in section 3.3. The overall and partial coordination numbers (CNs) shown are the average values for the particle beds, and the partial CNs such as (B-T) and (T-B) need not be equal to each other while mixing is not complete as discussed elsewhere.28 The figures show that the four types of partial CNs approach their respective final values in all four cases considered, which indicates that mixing is indeed taking place. However, depending on the cohesion and rake angle, the final values are different from each other. Similarly, the time taken to reach the steady state also varies, as discussed below. For mixing of noncohesive particles, ϕ = 13° blades show an overall CN at the steady state smaller than that of ϕ = 167° blades, indicating that ϕ = 13° blades expand, thus stirring the particle bed more than ϕ = 167° blades (see Figures 10a and 10b and Figures 9a and 9b). A comparison of overall CNs at Hap= 6.5 × 10−20 J and 2.6 × 10−20 J for ϕ = 13° and 167° blades, respectively, shows that ϕ = 13° blades have a slightly lower steady-state CN, thus confirming that the bed expansion and stirring by ϕ = 13° blades is indeed better (see Figures 10c and 10d). It can also be observed that the partial CNs, (B−B) and (T-T) or (T-B) and (B-T), compete with each other until the steady state is reached, the number of revolutions between two successive crossover points being larger for ϕ = 167° than for ϕ = 13° blades, indicating that the ϕ = 13° blades indeed stir the particle mixture better. Further, a comparison of the CNs in Figure 10a,c show that the overall CN for Hap= 6.5 × 10−20 J is larger than that for Hap = 0. Therefore, the bed stirring will deteriorate with the increase of Hap, and the time

Nm. Thus, the cohesive part is dominant, which is 1 order higher than the actual torque of the simulations (see Figure 6). Therefore, the torque relationship cannot be directly applied to the mixing of dry cohesive particles in the cylindrical mixer considered. 4.3. Effect of Interparticle Cohesions. 4.3.1. Mixing Performance. The results from the above subsections show that mixing can take place for moderate cohesion (e.g., for the case when Hap = 2.6 × 10−20 J and Haw = 3.8 × 10−21J) in a wide range of rake angles from 0° to 180°. This section will investigate the effect of particle cohesion at two different rake angles: ϕ = 13° and 167° when wall cohesion is at a moderate level. ϕ = 167° blades are practically used for powder mixing,29,33 which are therefore used here to compare against the ϕ = 13° blades which can produce a higher mixing rate with a lower shaft torque (see Figures 5 and 6). Here, Hap is varied up to 6.5 × 10−20 J. Figures 9a and 9b show the snapshots of the particle beds of the noncohesive (Hap = 0) and cohesive particles (Hap = 6.5 × 10−20 J) when ϕ = 13° at the steady state, respectively. From the visual inspection, there is not much difference between the two mixing states, suggesting that ϕ = 13° blades can mix highly cohesive particles of Hap = 6.5 × 10−20 J corresponding to which Bo = 20 (see Table 2). It can be seen that the noncohesive particles (Figure 9a) appear to be slightly more expanded than the cohesive particles (Figure 9b). To demonstrate the differences in the stirring by the two types of blades, variations of the overall and partial coordination numbers28,45,46 are considered as shown in Figure 10. Note that the two types of particles are laid in the top−bottom 4127

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Figure 11. Effect of interparticle cohesion on mixing: (a) for ϕ = 13°, and (b) for ϕ = 167°; labels 1 to 4 represent Hap = 0, 2.6 × 10−20, 4.55 × 10−20, and 6.5 × 10−20 J, respectively.

Figure 12. Effect of wall cohesion on the mixing performance for blades of ϕ = 13° when mixing cohesive particles of Hap = 6.5 × 10−20 J at 1432 rpm: (a) mixing curves, and (b) average mixing rate to obtain 80% of the steady-state mixing index.

described in section 2.2. It can be envisaged that both the particle and wall cohesions can contribute to good mixing because shear zones are developed near the walls and blades due to the gradients in velocity field, Therefore, we further investigate the effect of wall cohesion for the case of ϕ = 13° as described below. Figure 12a shows the effect of wall cohesion on mixing for Hap = 6.5 × 10−20 J in the case of ϕ = 13° blades. Haw is varied from 1.8 × 10−20 J to 6.5 × 10−20 J. Figure 2 shows that the particle bed would get sliced into two parts by the rotating blades, stopping any further mixing in the bed at ω = 500 rpm when Hap = Haw = 6.5 × 10−20 J. Figure 12a shows that the mixing of particles of Hap = 6.5 × 10−20 J will take place even if Haw is raised to 6.5 × 10−20 J at ω = 1432 rpm. The diagram shows that there is a slight reduction in the mixing rate when Haw is increased, which is quantitatively shown in Figure 12b as a function of Haw. Here, the mixing rate is defined as the overall gradient of the mixing curve of Figure 12a between the points corresponding to M = 0 and M = 0.8. It can be understood that the lift of the bed on each blade pass is very small for ϕ = 13° blades compared to that for ϕ = 135° blades of Figure 2. Therefore, even if the shaft speed is about 3 times that of the case of ϕ = 135°, the results show that the lifted particle bed can still come down before the next blade pass, ensuring mixing. At other rake angles, we have not investigated the effect of Haw on mixing, but some comments can be made regarding

to reach steady-state mixing will be longer as shown in the section below. 4.3.2. Interaction with Other Variables. Figures 11a and 11b show mixing curves for ϕ = 13° and 167° blades, respectively, at different values of Hap as a function of shaft revolutions at a shaft speed of 1432 rpm with Haw = 3.8 × 10−21 J. It can be seen that an increase in the particle cohesiveness delays the mixing for both ϕ tested. The larger the cohesiveness, the more pronounced the delay, which is especially true in the ϕ = 167° case. Such delaying can be expected for ϕ = 167° blades because of their poor stirring as noted above by a larger overall CN. However, despite such delays in mixing with increased Hap, the steady-state mixture quality M is mostly unaffected by Hap for ϕ = 13° and 167° blades as seen from the figures. On the other hand, an improvement in mixing is observed in Figure 5 due to an increase in particle cohesion Hap in the case of ϕ = 90° blades, as explained in section 4.2.2. It should be noted that one key parameter used for evaluating M is the interparticle gap δ, and therefore a test is carried out to evaluate its effect on M for this system. It is found that M is not improved more than 3% even if the interparticle gap is increased from the present value of 5%d to 10%d to account for the slight bed dilation, which can be seen from Figure 9. However, the effect of bed-dilation on M has been accounted for by considering the particle scale sample size n as 4128

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the effect. For blades of ϕ = 90°, the particle bed may become expanded for the case of Haw = 6.5 × 10−20 J as shown in Figures 8e and 8f. Therefore the effect of Haw on mixing may not be so significant. In this case, the expansion of the bed may be reduced due to the increased wall cohesion and mixing time increased as a result. On the other hand, mixing will be affected in the cases of larger rake angles as previously observed from Figure 3. The results therefore highlight the need to consider the interactions among some of the pertinent variables of powder flow and mixing.

impeller, which are not considered here, and these variables may have strong interactions. A systematic study is needed to come up with some rules that can be reliably applied to industrial design and operation of bladed and other mixers.

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AUTHOR INFORMATION

Corresponding Author

*Tel: + 61 2 9385 4429. Fax: + 61 2 9385 5969. E-mail: a.yu@ unsw.edu.au. Notes

5. CONCLUSIONS DEM simulations have been carried out to investigate the mixing of monosized fine particles in a vertically shafted cylindrical bladed mixer, aiming to establish some general understanding of the flow and mixing behavior of cohesive particles. The focus is given to the effects of three variables: cohesion between particles, cohesion between particles and wall, and rake angle. The following conclusions can be drawn from the present study. If particles and walls are made of the same material as considered in the base case, good mixing can be obtained only when the Hamaker constant is lower than a certain value, e.g., Ha = 4.55 × 10−20 J, or more generally, Bo = 14. Effective mixing cannot be achieved when cohesion is too high, e.g., Ha = 6.5 × 10−20 J or Bo = 20, because the particle bed can be lifted, resulting in limited interaction with the rotating blades. When the wall cohesion is reduced, the mixing can be restored, showing the importance of the wall cohesion for ensuring a good mixing performance. Blades of any rake angle (ϕ from 0° to 180°) can mix moderately cohesive particles (i.e., Hap = 2.6 × 10−20 J or Bo = 14) when the wall cohesion is low. The highest mixing rate is achieved for a blade of 90° rake angle, when the bed shows extensive expansion. The bed expansion results in changes to the mixing mechanisms prevalent at other rake angles. Namely, instead of flowing over the blades, particles are thrown vertically upward by the rotating blades, loosening the particle bed, and then letting the particles to fall again, both of which offer opportunity for particles to mix rapidly. This change in the mixing behavior is apparent from the changes in both the average coordination number of particles and torque on the shaft, which decreases drastically in the case of ϕ = 90°. With the increase of the particle cohesion, mixing becomes delayed or the mixing rate decreases. The result has been verified at ϕ = 13° and 167° but is expected to be valid even at other rake angles if the wall cohesion is kept fixed. For moderately cohesive powders, blades of small rake angles (e.g., ϕ = 13°) can stir the particle bed more effectively than blades of large rake angles (e.g., ϕ = 167°) blades. In particular, blades of large rake angle may lead to a considerable delay in reaching a final mixing state with the increase of particle cohesion. Despite the delay in mixing with increased particle cohesion, the steady-state mixture quality is unaffected by particle cohesion for ϕ = 13° and 167° blades if the wall cohesion is fixed. For small rake angles, the effect of wall cohesion is not so significant, offering more stable mixing operations than by blades of large rake angles (e.g., ϕ = 135° or 167°). The findings should be useful for designing and operation of cylindrical bladed mixers when applied to cohesive powders. They also highlight the complication of mixing cohesive powders: there are many variables involved such as the rake angle, shaft speed, wall cohesion, and immersion depth of the

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors gratefully acknowledge the financial support from the Australian Research Council (ARC) for this work.

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NOMENCLATURE b a constant in the van der Waals equation d particle diameter, m E Young’s modulus, N/m2 acceleration due to gravity, m/s2 g Ha Hamaker constant, J Hap Hamaker constant of particle material, J Haw Hamaker constant of wall material, J Lref a reference value for length, m M particle-scale mixing index defined in eq 5 N total number of particles in the mixture Ni the number of particles in a particle-scale sample ni target-type particles in a particle-scale sample n average number of particles in a particle-scale sample overall number fraction of particles to be mixed p R equivalent radius, m S0 standard deviation of sample concentration of one type of particles at fully segregated state Sr standard deviation of the sample concentration of the type of particle considered at fully mixed state for uniform-sized particles of the particle fraction of p St standard deviation of particle concentrations in samples with respect to the mean sample concentration for the mixture at time t wi particle number ratio, Ni/N wT the summation of wi terms over total particle number ratio, N Greek Letters

δ λ

μpp μpw μR ν ρ ρref ϕ ω

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separation of a particle pair, m dipole interaction wavelength in van der Waals equation, m sliding friction coefficient for particle−particle contacts sliding friction coefficient for particle-wall contacts rolling friction coefficient, m Poisson’s ratio material density of particles, kg/m3 a reference density, kg/m3 rake angle, deg impeller or shaft speed, rpm

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