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Author’s Accepted Manuscript
Continuous blending of cohesive granular material Avik Sarkar,Carl Wassgren PII:S0009-2509(10)00250-2
DOI:
doi:10.1016/j.ces.2010.04.011Reference:CES 9140
To appear in:Chemical Engineering Science Received date:14August 2009Revised date:7April 2010Accepted date:
13April 2010
Cite this article as:Avik Sarkar and Carl Wassgren,Continuous blending of cohesive granular material,Chemical Engineering Science ,doi:10.1016/j.ces.2010.04.011This is a PDF ?le of an unedited manuscript that has been accepted for publication.As a service to our customers we are providing this early version of the manuscript.The manuscript will undergo copyediting,typesetting,and review of the resulting galley proof before it is published in its ?nal citable form.Please note that during the production process errors may be discovered which could affect the content,and all legal disclaimers that apply to the journal pertain.
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Continuous blending of cohesive granular material
Avik Sarkar a
Carl Wassgren a,b
a School of Mechanical Engineering, Purdue University
b Department of Industrial and Physical Pharmacy (by courtesy), Purdue University
Abstract
Results are presented from discrete element method (DEM) computer simulations of cohesive particles in a periodic slice of a continuous blender. The influence of inter-particle cohesion at various impeller speeds and fill levels is reported. Although increasing cohesion does not significantly change axial flow rates, mixing rates in the transverse plane and axial direction are affected. Mixing is generally enhanced for slightly cohesive materials, but decreases for larger cohesion, similar to trends observed in tumbling batch mixers. Changes in fill level are also shown to affect axial transport rates and mixing. These results suggest that the controllable operating parameters, such as feed rate and impeller speed, may be adjusted for cohesive powder formulations to obtain optimal mixing performance.
Keywords: Cohesion; Discrete element method; Dispersion; Granular materials; Mixing; Particulate processes
1. Introduction
Continuous blending is becoming an increasingly common process in industries that handle particulate materials such as food products, pharmaceuticals, and chemicals (Pernenkil and Cooney, 2006). Continuous blending offers several advantages over more traditional batch blending including larger throughputs, more straightforward scaling, and less operator interaction. Unfortunately, knowledge accumulated from batch blending experience does not always translate to continuous blending. For example, the residence time in a batch blender is independent of impeller speed while in a continuous blender the two are closely connected. Therefore, it is worthwhile to investigate the parameters governing a continuous powder mixing process.
This paper is part of an ongoing effort to study bladed continuous mixers (Portillo et al., 2008; Sarkar and Wassgren, 2009). In particular, this paper is closely related to the work of Sarkar and Wassgren (2009) in which continuous blending of free-flowing granular material was simulated for a range of operating conditions. The difference between this previous work and the current work is that the current work includes the effects of particle cohesion, which has not been systematically studied for continuous blenders. The effects of cohesion on flow rate and mixing over a range of impeller rotation speeds
and fill levels are reported.
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2. Background
Blending of non-cohesive particulate materials in batch blenders has been widely studied by a number of researchers, both experimentally (e.g., Bagster and Bridgwater, 1967, 1970; Bridgwater et al., 1968; Bridgwater, 1976) and computationally (e.g., Moakher et al., 2000; Zhou et al., 2004; Sudah et al., 2005; Bertrand et al.,2005). These studies have provided considerable insight into the physics of blending and segregation processes, such as how the initial loading of materials can result in significantly different blending rates (Sudah et al., 2005), larger numbers of shorter mixing blades are more effective for mixing than fewer, but larger blades (Malhotra and Mujumdar, 1990; Malhotra et al., 1988, 1990; Laurent and Bridgwater, 2002b, 2002c), and that, for certain ranges of speed, fill level has a more significant influence on mixing than rotation speed (Laurent et al.,2000; Laurent and Bridgwater, 2002a).
Studies investigating the influence of cohesion on blending in batch blenders are more recent. McCarthy (2003) and Chaudhuri et al. (2006) both found that mixing rates marginally improve for small values of particle cohesion, but decrease as cohesion increases further. Though neither paper identifies a mechanism for the improved mixing rate, both indicate that their results are consistent with the experimental findings of Shinbrot et al. (1999). Shinbrot et al. (1999) demonstrated the emergence of spontaneous chaotic flow patterns for a fine, cohesive powder, which enhances mixing as compared to free flowing, coarse material.
There have been only a few studies concerning blending in continuous blenders. Reviews concerning continuous mixing can be found in Fan et al. (1990) and Pernenkil and Cooney (2006). Fan et al. (1990) report two continuous mixer types, a rotating drum and a continuous ribbon blender. Harwood et al. (1975) compared the performance of seven continuous mixers blending a dry sand-sugar mixture. Mixers that performed best relied on creating a high shear gradient between two augers or surfaces. Rotating drums with an inlet and an outlet have also been used as continuous mixers. Abouzeid et al. (1974) studied the effects of drum rotation rate, inclination, feed rate, and particle size on residence time mean and variance. The variance of the residence time can be directly correlated to the axial dispersion of the powder particles. Dimensionless variance increased with increasing rotation speed and inclination, and decreasing feed rate. Sherritt et al. (2003) proposed an axial dispersion model which included the drum rotation speed, fill level, and drum and particle diameter. This model was applied to both batch and continuous mixers, since axial dispersion, which is caused by random collisions, is purely diffusive and is unaffected by a bulk axial transport rate.
Laurent and Bridgewater (2000) used positron emission particle tracking (PEPT) to study a bladed mixer with an inlet and outlet, thereby producing a continuous system with superimposed axial transport. They observed that the weirs supporting the blades partitioned the mixer into three compartments, and caused material holdup in each
compartment. Two circulation loops were also identified in each compartment which
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hampered axial mixing. Portillo et al. (2008) experimentally characterized the performance of a horizontal axis, bladed continuous mixer. They reported longer residence times, which also corresponded to better blend homogeneity, for an upward inclination of the mixer axis and for smaller impeller rotation speeds. Portillo et al. (2008) also reported that increasing the number of blades from 29 to 34 slightly improves mixture homogeneity. Sarkar and Wassgren (2009) performed discrete element method (DEM) parametric studies using non-cohesive particles in a periodic slice of a continuous mixer. The fill level and impeller rotation speed were simultaneously varied to cover a wide range of values. Mixing was found to be fastest at small fills and large impeller speeds where diffusive mixing produced during fluidization was the primary mechanism. At larger fills, a smaller impeller speed was found to give better mixing. It was also shown that decreasing the blade spacing, corresponding to increasing the number of blades, improves the mixing rate, consistent with the experimental results of Portillo et al. (2008).
Since the blending dynamics of batch blenders and continuous blenders may be different, due in particular to the fact that there is axial transport and a limited residence time of material in continuous blenders, it is worthwhile to examine how cohesivity influences particle mixing in continuous blenders. Unfortunately there have been no studies to date on this topic. This work uses DEM simulations of a periodic “slice” of a horizontal, bladed, continuous blender to examine the flow rate and axial and transverse mixing as a function of particle cohesion, impeller rotation speed, and fill level.
3. Model
A periodic section of a continuous blender has been modeled using DEM and is shown schematically in Fig. 1. Material exiting the periodic section at one end re-enters the section at the other. The periodic section contains two mutually orthogonal blade stages in order to simulate the geometry found in a GEA Buck Systems experimental blender. Each stage has two blades inclined 45O to the mixer axis. This periodic section simulates an axial region near the middle of the continuous blender. Entrance and exit effects are not captured using this periodic slice. More details regarding the geometry and dimensions of the model can be found in Table 1 and also in Sarkar and Wassgren (2009).
A number of cohesion models have been reported in the literature for use in DEM simulations, including those implementing liquid bridges (Lian et al., 1993; Muguruma et al., 2000), JKR surface cohesion (Johnson et al., 1971; Mishra et al., 2002), constant cohesion (Iordanoff et al., 2005; Chaudhuri et al., 2006), and cohesion as a function of contact area (or contact length in the two-dimensional simulations of Matuttis and Schinner, 2001). In this work, the cohesion model proposed in Luding (2005) is implemented, which is a combination of the hysteretic linear spring normal force model of Walton and Braun (1986) and the analytical elasto-plastic/van der Waals cohesive interaction model of Tomas (2004). The cohesive force acts only after contact is made,
and only during unloading. The loading, unloading, and cohesive forces are continuous,
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linear functions of the contact overlap, with the interaction force equaling zero at
separation (refer to Fig. 2). Three parameters define the model: a loading stiffness, k L , an unloading stiffness, k U , and a cohesive stiffness, k C . When two particles first come into contact, they proceed along the loading path OA until a maximum overlap max δ is reached. The slope of OA is given by the loading stiffness k L . Subsequently, if the particles move apart, they follow the unloading path AB (slope k U ) and BO (slope -k C ). The largest cohesive force experienced during the lifetime of the contact is at point B, with the magnitude given by min C k δ. If a contact is reloaded before completely separating, the force follows the unloading path (along BA in Fig. 2a) till the maximum contact overlap is reached. Subsequent loading beyond the preexisting maximum overlap proceeds along the regular loading path with a linear contact stiffness of k L . The Luding model was chosen for the current studies since it is physically based and straightforward to implement.
A simple sliding friction tangential force model is used in all of the simulations. As with the other cohesive force models mentioned previously, cohesion is assumed to have no influence on tangential force interactions.
Energy is dissipated in the model through non-elastic normal impacts and tangential friction. For non-cohesive particles, the coefficient of normal restitution N ε is related to the particle loading (L k ) and unloading (U k
) stiffness as,
N ε=
. (1) For the present work, a value of 0.75 has been chosen for all normal contact pairs, based on Muller et al. (2008). Muller et al. (2008) report the ranges of normal restitution coefficients of granules for different materials at different impact speeds. The baseline force model parameters used in the simulation, as well as the other simulation parameters are listed in Table 1.
Parametric studies are performed in which the impeller rotational speed, cohesive contact stiffness, i.e. contact strength, and fill volume are varied. Each of these parameters is expressed in dimensionless form. The impeller rotational speed, ω, is expressed in terms of a Froude number, Fr ,
22drum
D Fr g
ω= (2)
where D drum is the containing drum diameter and g is the acceleration due to gravity. The Froude number is a ratio of the characteristic centripetal acceleration acting on a particle due to impeller rotation, to the acceleration due to gravity.
The cohesive contact stiffness, k C , is expressed in dimensionless form as,
*
2C C p k k d g
ρ=
(3)
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where ρ is the particle density and d p is the particle diameter. The dimensionless
cohesive stiffness is a ratio of a characteristic cohesive strength acting on a particle, albeit based on an overlap equivalent to a particle diameter, to a particle’s weight.
A commonly used dimensionless parameter used in cohesive particle studies is the granular Bond number g Bo , which is defined as the ratio of a characteristic cohesive force to a characteristic gravitational force. The dimensionless cohesive stiffness, k C *, defined in Eq. (3) is, in fact, a form of a granular Bond number; however, it is based on a contact overlap equivalent to a particle diameter. Hence, a more easily interpreted Bond number is given in addition to the dimensionless cohesive stiffness. For the Luding
cohesion model a maximum cohesive force cannot be readily defined for an impact since the maximum contact overlap is a function of the impact speed. The cohesive force for the more simple case of a single particle resting on a surface under the action of gravity can be used to determine the characteristic cohesive force. Therefore, a more easily interpreted granular Bond number may be defined as
,min min max N C C U L
g L L U C F k k k k Bo mg k k k k δδ?===?+, (4)
where ,min N F is the magnitude of the largest cohesive force developed during this contact (point B in Fig. 2a) corresponding to a maximum loading force ,max N F mg = (point A in Fig. 2a). Table 2 lists the corresponding g Bo values for the values of the
stiffnesses used in the simulations (reported in Table 1). These values may be used to approximately compare the cohesion level in the current simulations with those in other computational and experimental works. However, a consistent definition of Bond number must be used, where the cohesive force is a result of the particle’s self weight. For example, for the current case, a Bond number of 0.47 corresponds to a surface energy value of 16.4×10-3 J/m 2 in the JKR cohesion model (Johnson et al., 1971) for the particle properties presented in Table 1. At the limit C k →∞, the maximum Bond number
,max g Bo using the current definition is given by ()
,max 2
,1
1U L g L
N pp k k Bo k ε?=
=?,
(5)
where ,N pp ε is the particle-particle coefficient of normal restitution for the non-cohesive case. Near this limit the time-step required to resolve a cohesive impact is infinitesimally small, which cannot be computationally implemented. For the presently used value of ,0.75N pp ε=, the limiting Bond number using the current definition is ,max 0.78g Bo =.
The dimensionless bulk fill fraction has also been varied in this study. Assuming a maximum random packing solid fraction of 0.64 for spheres, the bulk fill fraction ν is
defined as
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()32
2
6
0.644
p p
drum shaft periodic d N D D L π
νπ
=
?
?, (6)
where p N is the number of particles in the simulation, shaft D is the internal shaft
diameter, and periodic L is the mixer periodic length. In a full continuous blender, the bulk fill is dependent on the inlet feed rate, impeller rotation speed, and outlet weir geometry. Currently simulated values of ν listed in Table 1 cover a wide range of fills expected within a typical mixer.
4. Results
4.1. Effect of rotation rate and cohesion
Simulations were performed for seven values of dimensionless cohesive stiffness and four values of Froude number. All measurements were averaged over at least six impeller rotations during steady state operation. The baseline case of 40% bulk fill volume level contained 15,912 particles.
4.1.1. Flow and transit time
Transverse flow patterns for the non-cohesive (*
,0C pp k =) and the largest cohesion case
simulated (*,3000C pp k =) are presented in Fig. 3 for two Froude numbers (Fr = 0.21 and
Fr = 3.35). At the smaller value of Froude number, nearly all of the particles reside at the bottom of the bed, agitated only during intermittent blade passes. Particles roll down the surface of the bed, both over and under the central shaft. Similar flow patterns were observed for non-cohesive particles in Sarkar and Wassgren (2009), and have also been reported by Malhotra et al. (1990) and Laurent and Bridgwater (2002). Videos of the simulation do not reveal any significant agglomeration at the bed surface, but the angle of repose qualitatively appears to be larger (Fig. 3b) than the corresponding non-cohesive case (Fig. 3a). It is difficult to quantify this measurement since the free surface is not well defined due to some fluidization. For the larger Froude number case, the flow
patterns are very similar to the corresponding non-cohesive cases in Sarkar and Wassgren (2009). Particles are fluidized by the fast moving impellers in the free space available above the bed. The differences between the free flowing and cohesive cases are subtle . The number of particles flowing over the shaft slightly increases at larger particle cohesion values. However the overall flow patterns remain nearly unchanged with varying cohesion, particularly for the larger impeller speeds.
The flow rate through the mixer may be expressed in terms of the average axial velocity. The average axial velocity is defined as,
,1
1p
N axial axial i i p v v N ==|,
(7)
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where the average is taken over all particles in the periodic section. Figure 4a plots the average axial velocities scaled by the impeller tip speed (ωD drum /2) for different Froude numbers and dimensionless cohesive stiffnesses. Little variation is seen in the scaled axial velocities for larger impeller speeds. At smaller impeller speeds, the flow rate increases slightly as particles become more cohesive. The formation of tensile force chains results in more coordinated axial particle movement, and thus larger axial flow rates than for uncoordinated, weakly cohesive particles. This effect is not observed at larger Froude numbers since particle momentum differences at larger impeller speeds can easily overcome the cohesive spring forces. This effect is discussed in greater detail in
the following paragraph. Figure 4b re-plots the same data to illustrate trends with
increasing Froude number. Scatter bars have been omitted in this figure for clarity. The scaled axial velocity increases with increasing Froude number, with a nearly linear
dependence for less cohesive contacts and larger Froude numbers. The weakly cohesive linear dependence is consistent with the non-cohesive cases examined in Sarkar and Wassgren (2009), despite differences in particle density and particle-wall friction.
The axial component of the granular temperature has been used to determine if increased cohesion does indeed result in more coordinated axial particle movement. A binning procedure similar to Sarkar and Wassgren (2009) has been used to compute the axial component of granular temperature and the dispersion indices (sections 4.1.2 and 4.2.2). Uniformly spaced, 3D, Cartesian grid points are generated within the simulation domain. A bin is defined as a cubical region of side 3d p with the grid point of interest at the center. Granular temperature or dispersion at a grid point is obtained by averaging over all
particles contained within the corresponding bin. Parameters associated with the binning procedure are provided in Table 3. The equation used to define the axial component of the granular temperature for a group of bin N particles contained within a bin is given by,
(
){}bin
bin
bin 2
2
22
Z,bin ,,,bin 111bin
bin
bin 11
1'N N N Z Z i Z Z i Z i i i i T v v v v v N N N ===§·==?=?¨??1
||| (8) where Z v is the average velocity for the group of bin N particles within the bin. The
overall axial component of the granular temperature of the system is taken as the number weighted average of the axial granular temperature, given by,
2bin bin
2bins Z,overall bin
bins
''Z Z v N T v N ==|| (9)
Dispersion indices quantifying mixing are calculated in an analogous manner (Eqs. (10)-(12)). Figure 5 plots the Z-component of the granular temperature, scaled by the square of blade tip speed. For all impeller speeds, an increased strength in the tensile force chains results in a decrease in the axial component of the granular temperature, which in turn indicates a greater degree of coordinated axial particle movement. The decrease in granular temperature is most noticeable for the smallest Froude number since a small impeller speed is insufficient to break up cohesive contacts formed at larger values of particle cohesion. The reduction in the scaled axial granular temperature is smaller for the larger Froude number cases, likely due to the fact that at larger impeller speeds
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particles have sufficient momentum to overcome the cohesive bonds regardless their strength.
The mean residence time of particles through a blender is directly related to the mixer length and throughput while the width of the residence time distribution may be used to estimate axial dispersion (Levenspiel, 1999). Portillo et al. (2008) measured the residence time of a tracer substance in their full length continuous mixer experiments and found that larger mean residence times correlated with improved mixing. They hypothesize that for larger residence times, the mixing blades make a larger number of passes through the bed and thus improve mixing. Since the current work simulates a periodic section, distributions of particle “transit time” through the periodic section, rather than the residence time in the blender, are examined. The transit time
tr
t for a
particle is defined as the time taken for a particle to travel one periodic length (
periodic
L) in the axial direction (Fig. 6a). Particles may cross a periodic boundary while traversing this length. Figures 6b - e plot the transit time probability distributions in dimensionless form for all of the particles in the periodic section. The abscissa is scaled by the impeller rotation period 2/
πω, thereby presenting the transit time in terms of number of impeller rotations. A set of transit time probability distribution data is obtained by following the trajectories of all particles starting from a given initial configuration of particle positions. The transit time distributions presented in Fig. 6 were obtained by averaging more than a hundred sets of distributions, each corresponding to a different initial configuration of particle positions. The ordinate is scaled by the shaft angular velocity ω, since it is the blade rotation that drives the flow. In these scaled axes, both the mean and width of the distribution are found to decrease with increasing Froude number, consistent with the data in Figs. 4 and 5. A shorter mean transit time distribution corresponds to a higher average axial velocity, which is seen for the larger Froude number cases in Fig. 4. A narrower transit time distribution indicates more coherent axial movement of the particles, reflected in the decrease in the axial component of granular temperature in Fig.
5. Note that most of the particles traverse the periodic length in under two impeller rotations. The transit time distributions show a weak dependence on cohesion, especially for larger Froude numbers. As discussed previously (refer to Figs. 4 and 5), the influence of cohesion is less significant at larger Froude numbers since larger impeller speeds prevent agglomeration and the bed behaves almost like a free flowing material for all values of cohesion. Differences caused by cohesion are prominent only at smaller Froude numbers. The transit time distributions are narrower for the more cohesive particles at smaller Froude numbers, indicating a more coordinated axial particle movement, consistent with the smaller axial granular temperatures in Fig. 5.
4.1.2. Mixing
Following Martin et al. (2007) and Sarkar and Wassgren (2009), dispersion is used to characterize mixing. All results are reported for the particle dispersion produced over one complete rotation of the impellers. Dispersions in the transverse plane and the axial direction are reported separately. Axial dispersion is an important quantity in and of itself since it is directly related to material residence time (or transit time in the present
work). Moreover, axial dispersion helps to remove temporal inhomogeneities in the input
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stream; however, too large of an axial dispersion may not be desirable since the powder leaving the mixer would have widely ranging residence times and be subject to different amounts of work by the blender paddles. Dispersion at a point inside the blender is taken as the dispersion of particles lying within a cubical bin of side 3d p with the grid point of interest as the center of the bin. Mathematically, the dimensionless transverse and axial dispersions for a bin are defined as:
XY,bin M =
Z,bin
M = (11)
where XY,bin M and Z,bin M are the transverse and axial dimensionless dispersion indices for a bin, and bin N is the number of particles in the bin. The parameters i x Δ, i y Δ, and i z Δ are the displacements of the i th particle in the bin, and x Δ, y Δ, and z Δ are the
average displacements of all the particles in the bin. The overall transverse dispersion in the mixer, XY,overall M , is taken to be a number weighted average of the local dispersions: XY,bin
bin
bins
XY,overall
bin
bins
M N M N
=
||.
(12)
The binning procedure is identical to that considered in Sarkar and Wassgren (2009) and bin dimensions are provided in Table 3. The overall axial dispersion Z,overall M is defined an analogous manner. It should be noted that that the dispersion indices depend on the bin size and number of bins, which relate to the scale of scrutiny and number of samples, respectively. For the current study, the dispersion indices have been used as a
comparative measure at a constant bin size and number of bins, with each cubical bin having an edge length 3d p , which is a reasonable bin size to investigate mixing at a particle length scale.
Figure 7 plots the overall transverse dispersion in the mixer for varying cohesion. For all but the 3.35Fr = case, transverse mixing per rotation increases slightly with a small increase in cohesion and reaches a maximum value in the range k *C,pp = 500 – 1000, corresponding to Bo g = 0.16 – 0.26. Thereafter, the mixing rate decreases with a further increase in cohesion. Similar behavior has been reported in batch rotating drums by McCarthy (2003) and Chaudhuri et al. (2006). McCarthy’s (2003) simulations
considered cohesion due to liquid bridges and found that the best mixing occurred at Bo g = 1.0. Although a different cohesive force model (constant cohesive force; Iordanoff et al., 2005) was used, Chaudhuri et al. (2006) found that mixing was best for particle interactions with Bo
g = 0.1. Shinbrot et al . (1999) reported that for fine, cohesive
particles in a rotating drum, cohesion caused stick-slip powder movement which resulted in chaotic mixing patterns. Such behavior was not clearly observed in the current simulations. The cohesive interaction between particles is expected to hinder particle diffusion and promote agglomerate formation, both of which decrease mixing. It is hypothesized that moderate levels of cohesion at small Froude numbers improves
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convective mixing due to more coordinated particle movement, while at the same time not being sufficiently large to adversely affect diffusive mixing. Larger values of
cohesion may significantly hinder diffusive mixing by forming agglomerates and strong cohesive bonds.
Mixing in the transverse plane also generally increases with Froude number, except for dimensionless cohesive stiffnesses less than approximately 1500. For cohesive
stiffnesses smaller than this value, transverse mixing decreases from Fr = 0.21 to Fr = 0.84, but then increases for larger values of Froude number. Similar behavior was observed for the free flowing case in Sarkar and Wassgren (2009). At larger Froude numbers, particle fluidization becomes more apparent with a corresponding increase in mixing. The reason for the dip in transverse mixing at an intermediate value of Froude number for less cohesive material remains unclear.
Figure 8 plots the overall axial dispersion in the continuous mixer. Axial dispersion decreases monotonically with increasing cohesion for all Froude numbers, consistent with the idea that increasing cohesion, especially at smaller Froude numbers, increases the coordination of axial particle movement (Fig. 5). Larger Froude numbers improve axial mixing, consistent with Figs. 5 – 7.
The dispersion trends suggest that for a given set of operating conditions, the best mixing will be achieved for non-cohesive to mildly cohesive material. In addition, larger impeller Froude numbers can significantly enhance mixing as long as care is taken to ensure that the powder experiences a sufficient number of blade passes. Both the
dispersion data and transit time distributions should be considered simultaneously when selecting the operating conditions. Keeping this is mind, the overall dispersions defined by Eqs. (10) - (12) are plotted in Fig. 9 over the average transit time rather than over an impeller rotation. The average transit time for a given cohesion and impeller speed is obtained by taking the arithmetic mean of the corresponding transit time distribution presented in Fig. 6. The dispersion produced over the average transit time measures the mixing achieved as the powder traverses a length periodic L along the mixer. Figures 9a and 9b plot the scaled transverse (,XY transit M ) and axial (,Z transit M ) dispersions,
respectively. The standard deviations of the axial dispersion data, which have been omitted for clarity, are larger than that for the transverse component since the allowable axial particle displacements are not bounded by the drum diameter. For both the
transverse and axial dispersion components, the mixing trends for a given Froude number with respect to cohesion remain essentially the same as those in Figs. 7 and 8 (dispersion per impeller rotation), but the dependence on Froude number is different. The best mixing is obtained at the smallest Froude number for free flowing and small cohesivity material for both components of dispersion. At small speeds, the residence time is large owing to the slow axial flow rate. The material experiences a larger number of blade passes leading to better homogeneity. In contrast, though the mixing achieved per
impeller rotation is greater for the larger Froude numbers, a much smaller residence time leads to the free flowing material passing through the blender without experiencing many blade passes. At the largest cohesion values, an intermediate value of Fr = 0.84 is found
to give the best transverse mixing, suggesting that greater impeller agitation is necessary
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to achieve mixing, but one that is not so large that the number of blade passes that the material experiences is small. The axial dispersion values for the most cohesive case are found to be numerically close to each other, with the Fr = 3.35 case being slightly larger than the others.
4.2. Effect of fill volume fraction
The bulk fill volume fraction in a continuous mixer is a complex function of the inlet feed rate, impeller rotation rate, and the geometry of the outlet weir. In the periodic section simulations performed here, the fill fraction has been artificially prescribed by specifying the number of simulated particles. All results in this section are reported for an intermediate Froude number value of 0.84
Fr= (100 rpm). The number of simulated particles ranged from 9,944 for 25% bulk volume fill to 27,845 for 70% fill.
4.2.1. Flow and transit time
The average scaled axial velocity is plotted in Fig. 10 as a function of the bulk fill volume fraction for two values of cohesion. In both cases, the average axial velocity reaches a maximum near 40% bulk fill volume fraction. This result is similar to that reported for non-cohesive particles in Sarkar and Wassgren (2009), where the peak value is in the range between 45-50% bulk fill fraction. At smaller fill fractions, the blades spend a short fraction of the rotation time period immersed in the bed resulting in a small flow rate. At larger fills, the increased blade thrust (due to a larger time spent in the bed) results in an increased average particle axial velocity. However, increasing the fill fraction also results in an increased wall resistance, which acts to decrease the bed’s axial speed. At the largest fill fraction, the lack of available free volume also hinders particle mobility in both the axial and transverse directions. This competition of effects results in the observed maximum. Increasing particle cohesion increases the average scaled velocity, but only slightly. As was previously discussed Section 4.1.1, increased cohesion results in more coordinated axial particle movement and thus a slightly larger flow rate. The scaled transit time distributions for the varying fill fractions are plotted in Fig. 11. The transit time distributions are broader for smaller fill fractions as compared to those for larger fills. This trend suggests that the axial dispersion is greater at smaller fills (Levenspiel, 1999), and is indeed found to be the case (discussed further in Section
4.2.2). As was demonstrated in Fig. 5 of Sarkar and Wassgren (2009), temporal fluctuations in local axial velocities are larger at smaller bulk fills. With increasing fill fraction, both spatial and temporal variations in axial flow decrease as both blade stages remain immersed longer in the bed, thus continuously propelling the bed forward. A bed that has smaller axial flow rate spatial variations will result in a narrower transit time distribution. The transit time distributions for the larger cohesion cases in Fig. 11b are narrower than the corresponding cases for smaller cohesion (Fig. 11a). Increasing cohesion results in a more coherent axial flow leading to a narrower transit time distribution.
4.2.2. Mixing
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For both cohesion values, transverse mixing per rotation increases with increasing fill up to a fill fraction of 55%, and decreases for larger fill fractions (Fig. 12). A similar trend was observed in Sarkar and Wassgren (2009). The transverse mixing per axial periodic length shows identical behavior. At the 100 rpm impeller speed, increasing fill improves mixing by establishing circulation loops in which particles are moved up by the blades and then cascade down the free surface under the agitator shaft (refer to Fig. 3). As the bulk fill fraction increases, a secondary loop begins to emerge with particles passing over the shaft. For even larger fills where the central shaft is fully immersed in the bed, nearly all particles flow over the shaft. At the largest fills, particle mobility is hampered by the limited free volume available, and a decrease in transverse mixing is observed. Larger cohesion values also hinder particle mobility resulting in a further decrease in transverse mixing.
Figure 13 plots the dimensionless axial dispersion per rotation as a function of bulk fill volume fraction for two values of cohesion. An increase in either fill fraction or cohesion is detrimental to axial mixing (axial dispersion per axial periodic length shows identical behavior). This result is consistent with the transit time distributions (Fig. 11), which are narrower for both increasing fill and increasing cohesion. The same observation has been reported in Sarkar and Wassgren (2009) for free-flowing particles.
5. Conclusions
Results from DEM simulations of a periodic section of a continuous mixer section have been presented. The first set of simulations investigates the influence of varying cohesion and impeller speed on flow and mixing. The second set studies the influence of the mixer bulk fill fraction for varying levels of cohesion.
For varying cohesion and Froude number, changes in flow rates are subtle. Differences in axial flow are more significant at smaller impeller speeds where coordinated particle movement is observed due to cohesion. At larger speeds, larger shear rates overcome particle bonding so that cohesive particles flow in a manner similar to non-cohesive material.
Transverse and axial dispersions are found to be dependent on both cohesion and Froude number. Mixing per rotation improves at larger Froude numbers, aided by an increase in impeller agitation and fluidization. Optimal transverse mixing is observed at a moderate cohesion value, similar to what has been previously observed in batch blenders. Mixing per unit axial length for free flowing powders is found to be best at low impeller speeds, as a larger number of blade passes are encountered during the material’s residence time in the blender. For more cohesive particles, an intermediate value of impeller speed is found to give the best mixing over a fixed axial length, as both the dispersion produced per pass and the number of blade passes felt by the powder need to be sufficiently large.
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Variations in fill fraction produce trends similar to those observed in the non-cohesive work of Sarkar and Wassgren (2009). The flow and mixing trends seen with increasing fill are identical for both the smaller and larger cohesion cases considered presently. A maximum value in the degree of transverse mixing per rotation occurs at a fill fraction of approximately 55%, whereas the largest axial mixing per rotation is observed at the lowest simulated bulk fill volume fraction of 25%.
The average transit time in the blender is a strong function of the Froude number, with larger Froude numbers resulting in smaller average transit times. Although larger Froude numbers produce better mixing, the resulting smaller residence times may lead to the mixture components passing through the blender without being well homogenized.
Operating a continuous mixer at a higher speed would require a longer mixer axial length to ensure sufficient agitation of the material by the blades. Portillo et al. (2008) showed that increasing the backward tilt of the mixer increases residence time and also results in improved mixing. Increasing cohesion narrows the transit time distribution, which in turn decreases axial mixing. Although varying fill fraction does not significantly affect the average transit time, the width of the distribution is affected. Smaller fills have a wider transit time distribution than larger fill fractions. The cause is attributed to the decline in unsteady axial flow behavior as fill fraction increases. Notation g Bo granular Bond number ,max g Bo maximum granular Bond number for the Luding model
drum D mixer drum diameter , [L]
shaft D central shaft diameter, [L] p d
particle diameter, [L]
Fr Froude number given by )22drum
D g ω
,max N F normal contact force in the Luding (2005) force model [MLT -2] ,max N F , ,min N F
maximum and minimum normal force in the Luding force
model, [MLT -2]
()tr f t
transit time probability distribution, [T -1]
g
acceleration due to gravity, [LT -2]
U k , L k , C k
unloading, loading, and cohesive stiffness for the Luding normal force model, [MT -2]
*C k dimensionless cohesive stiffness for the Luding normal force model
*,U pp k , *,C pp k
dimensionless unloading and cohesive particle-particle stiffness for a particle-particle contact, made dimensionless by 2p d g
ρ
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*,U pw k , *,C pw k
dimensionless unloading and cohesive stiffness for a particle-wall contact, made dimensionless by 2p d g ρ blade L impeller blade length, [L]
periodic L mixer periodic length, [L]
,XY bin M , ,Z bin M local radial-plane (XY) and axial (Z) dispersion indices in a (cubical) bin over one shaft rotation
,XY overall M , ,Z overall M overall radial-plane (XY) and axial (Z) dispersion indices for the mixer over one shaft rotation
,XY transit M , ,Z transit M
overall radial-plane (XY) and axial (Z) dispersion indices for the mixer over the average transit time
m
mass of a particle given by 3
16p d πρ, [M] bin N number of particles in a (cubical) bin p N total number of particles in a simulation
Z,bin T axial component of the granular for a group of particles in a bin, [L 2T -2
]
Z,overall T overall axial component of granular temperature taken over all bins, [L 2T -2]
tr t particle transit time, [T] t Δ simulation time step, [T]
,axial i v
axial velocity for particle i , [LT -1]
axial v
average axial velocity averaged over all particles, [LT -1]
'Z v deviation of particle axial (Z) velocity from the mean axial velocity, [LT -1]
,Z i v
axial (Z) velocity of particle i , [LT -1]
Z v
average axial (Z) velocity for a group of particles, [LT -1]
blade w
impeller blade width, [L]
i x Δ, i y Δ, i z Δ
displacement of particle i over one impeller rotation, [L] x Δ, y Δ, z Δ
average displacement of all particles in a bin over one shaft rotation, [L]
δ
contact overlap in the Luding normal force model, [L]
max δ, min δ, res δ
maximum, minimum, and residual overlap in the Luding force model, [L]
,N pp ε particle-particle normal restitution coefficient ,N pw ε particle-wall normal restitution coefficient blade θ impeller blade inclination, degrees pp μ
particle-particle friction coefficient
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pw μ particle-wall friction coefficient
ν bulk fill fraction
ρ particle density, [ML -3]
φ
particle size dispersity; all particle radii fall in the interval
()()1,1p p d d φφao?+?? with uniform probability distribution ω
rotation speed of the impeller blades, [T -1]
Acknowledgements
The authors are grateful to the National Science Foundation Engineering Research Center for Structured Organic Particulate Systems (NSF ERC-SOPS, EEC-0540855) for
financial support. The authors also thank members of the Particulate Systems Laboratory (PSL) at Purdue University for their constructive comments.
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Figure Captions
Figure 1. Schematic showing a periodic section of a continuous mixer.
Figure 2. The inter-particle normal force model. (a) Normal force vs. displacement curve for the Luding (2005) model. (b) Schematic showing the spring stiffness modeled in a contact. Additional detail concerning the force model may be found in Luding (2005).
Figure 3. Instantaneous velocities and solid fractions at different cross sections along the
mixer length for (a) 0.21Fr =, *,0C pp k =, (b) 0.21Fr =, *
,3000C pp k =, (c) 3.35Fr =,
*,0C pp k =, (d) 3.35Fr =, *,3000C pp k =, and (e) locations of cross sectional regions (1)-
(4). Arrows represent the local velocity vectors and the shading represents the solid fraction. Vertical and horizontal lines in regions (2) and (4), respectively, represent the instantaneous orientations of the blades.
Figure 4. Average scaled axial velocity, (a) variation with scaled cohesive stiffness, (b) variation with Froude number. Scatter bars in (a) represent one standard deviation of the average scaled velocity measured over multiple impeller rotations.
Figure 5. Scaled axial (Z) component of granular temperature. Scatter bars represent the scaled standard deviation of the data obtained over multiple impeller rotations.
Figure 6. (a) Transit time tr t is defined as the time it takes for a particle to traverse the length periodic L . Scaled transit time probability distributions plotted against scaled transit time for varying cohesion, (b) Fr = 0.21, (c) Fr = 0.84, (d) Fr = 1.89, (e) Fr = 3.35.
Figure 7. Transverse dispersion in the mixer as a function of dimensionless cohesive strength for varying Froude numbers. Scatter bars represent one standard deviation of the transverse dispersions produced over multiple impeller rotations.
Figure 8. Axial dispersion in the mixer plotted as a function of dimensionless cohesive stiffness for varying Froude number. Scatter bars represent one standard deviation of the axial dispersions produced over multiple impeller rotations.
Figure 9. Scaled overall dispersion over the average transit time versus scaled cohesive stiffness for (a) transverse plane dispersion, and (b) axial dispersion. Scatter bars in (a) represent one standard deviation of the dispersion data over multiple measurement intervals.
Figure 10. Average scaled particle axial velocity scaled by impeller tip speed for varying bulk fill volume fraction. Scatter bars represent one standard deviation of the data obtained over multiple impeller rotations.
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Figure 11. Scaled transit time distributions for varying bulk fill volume fraction, (a) *,500C pp k =, and (b) *,2500C pp k =. The baseline value of 0.84Fr = (100 rpm) has been used in all cases.
Figure 12. Dimensionless overall transverse dispersion over one impeller rotation as a function of bulk fill volume fraction. Scatter bars represent one standard deviation of the data obtained over multiple impeller rotations.
Figure 13. Dimensionless axial dispersion over one impeller rotation as a function of bulk fill volume fraction. Scatter bars represent one standard deviation of the data obtained over multiple impeller rotations.