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Analysis of binary and ternary granular mixtures behavior using the kinetic theory approach
Powder Technology 151 (2005) 117 – 125 www.elsevier.com/locate/powtec
Analysis of binary and ternary granular mixtures behavior using the kinetic theory approach Hadjira Iddira, Hamid Arastoopoura,*, Christine M. Hrenyab a
Department of Chemical and Environmental Engineering, llinois Institute of Technology, 10 W. 33rd Street, Illinois, Chicago 60616, United States b Department of Chemical and Biological Engineering, University of Colorado, Boulder, Colorado 80309, United States Received 1 November 2004 Available online 28 January 2005
Abstract The kinetic theory was extended to granular mixtures of different mechanical properties (size, density and/or restitution coefficient) where each particle group was considered as a separate phase with different average velocity and granular energy. This model was applied to simple shear flow of binary and ternary mixture of particles. In the parameter range studied, the rheological behavior of a binary mixture showed a good quantitative agreement with the molecular dynamics simulation and predicted qualitatively well the experiment of Savage and Sayed [S. Savage, M. Sayed, J. Fluid Mech., 142 (1984)] and Feitosa and Menon [K. Feitosa, N. Menon, Phys. Rev. Lett., 88 (7) (2002)]. Furthermore, the investigation of a ternary granular mixture showed interesting results: the fluctuations were strongly damped and the granular mixture showed less resistance to the flow when the number of particulate phases with smaller particle size increased in the mixture, while the total particle concentration remained unchanged. D 2004 Elsevier B.V. All rights reserved. Keywords: Kinetic theory; Binary particle mixture; Ternary particle mixture; Granular shear flow; Energy non-equipartition
1. Introduction Fluid particle flow exhibits complex behavior due to the coexistence of different regimes within the flow. Collisions between particles are responsible for the energy transfer and dissipation that lead to phenomena such as agglomeration, attrition and breakage of particles. Furthermore, collisions of particles of different mechanical properties and corresponding fluctuations that occur during the flow make the system even more complicated and challenging. The process of fluid particle flow is encountered in numerous applications. In nature or industry, we deal with either highly concentrated flow, where the solid phase interactions are dominant (such as the flow of avalanches or flow in solids mixers), or highly dilute flow where the interaction between the solid and fluid is important (such as in pneumatic conveyors or fluid catalytic cracking). * Corresponding author. Tel.: +1 312 567 3040; fax: +1 312 567 8874. E-mail address: [email protected] (H. Arastoopour). 0032-5910/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2004.11.033
To understand the behavior of fluid particle flow, one may need first to understand granular flow where the interstitial fluid effects may be neglected [1,2]. In granular medium, the particle–particle interaction is dissipative, thus modeling the flow behavior is non-trivial. Flow of a monodisperse mixture has been extensively studied. Jenkins and Savage [3] used the kinetic theory approach to develop a model that predicted the behavior of inelastic particles undergoing a rapid shear flow. Lun et al. [4] developed a theory and studied simple shear flow behavior for wide range of the restitution coefficients. Monodisperse granular flow is still attracting researchers’ interest from several disciplines [5–8]. In recent years, granular flow of a binary mixture has become the subject of intense studies, in which the kinetic equation was solved at the near equilibrium steady state assuming energy equipartition to obtain the hydrodynamic properties for nearly elastic particles [9–11]. Zamankhan [12] solved the kinetic equation using the Grad method and assumed that the particles fluctuate with the same energy; he concluded that
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energy non-equipartition must be included in mixtures with different particle properties. Garzo´ and Dufty [13] showed theoretically that the energy equipartition is violated for inelastic particles of a binary mixture. Later, the energy nonequipartition in a binary mixture was experimentally confirmed [14,15]. In a recent publication, Garzo´ and Dufty [16] solved the kinetic equation for systems away from equilibrium; hence, their model could capture not only the energy nonequipartition, but also the flow behavior for a broader range of restitution coefficients. However, their model is restricted to dilute systems, where the radial distribution function was close to unity. Iddir and Arastoopour [17] developed a model that predicts the flow of particles of different mechanical properties, based on the kinetic theory of dense gases. The model predicted well the energy non-equipartition and the stresses of a binary granular mixture. In this study, we investigate the behavior of a binary and a ternary mixture of particles undergoing a simple shear flow using the kinetic theory approach. We also investigate how a third particulate phase may affect the flow behavior. A brief summary of our model is provided in the Appendix.
where the thermal equilibrium may be reached when the generation due to continuous shearing is balanced by the dissipation due to collisions. In the case of simple shear flow the dimensionless energy equations for particles of phase i interacting with particles of any other phase p reduce to the following non-linear algebraic equations (note that the bar on the quantities below represents their dimensionless form): ! a¯ip 1 1 dv¯ 2 þ ¼ N¯ii þ N¯ip ð1Þ l¯ ii þ dy¯ 2 h¯i h¯p where: "
# 2 5 4 4d¯i ð1 þ eÞe2i q¯i gii l¯ii ¼ 1 þ ð1 þ eÞgii ei þ 5 5m¯ i 16d¯i2 gii sffiffiffiffiffiffiffiffiffi m¯ i h¯i ð2Þ p pffiffiffi m¯ i m¯ p m¯ i m¯ p 3=2 p ¯4 2 d gip 2 a¯ip ¼ ð1 þ eÞq¯ i q¯ p ei ep R¯ 1 ¯hi h¯ p 48 ip 5 m¯ 0 ð3Þ
2. Model application to simple shear flow Our model (see Appendix) is based on the extension of the kinetic theory to hard, smooth, non-cohesive (typically larger than 50 Am) and spherical particles of different type (size and/or density). Each group of particles with similar properties is considered as a separate phase, which is characterized by its own average velocity and fluctuating energy or granular temperature. The interaction between the phases occurred at the interface, namely the collision between particles of different size/or density. As a consequence of this assumption, we solved the Boltzmann equation for each particulate phase using the Chapman– Enskog procedure by adding a perturbation to the Maxwellian velocity distribution function [18]. However, we assumed at the interface that the velocity distribution function remains Maxwellian. To investigate the effect of the system parameters (diameter ratios, total solid fraction, solid fraction ratios and restitution coefficients) on the hydrodynamics of a binary and ternary particle mixtures, a very simple flow system, such as simple shear flow, was chosen. In such a system, the velocity gradient is constant; hence, segregation by size or density does not occur. The flow field consists of two infinite parallel boundaries set to move with velocities FV 0/2. The mean motion of the granular mixture was only in the x-direction and was considered fully developed so that all the flow parameters are a function of y. In this study, x and y are the axes parallel and perpendicular to the moving boundaries, respectively. In this example, we are interested in the steady state regime,
pffiffiffi m¯ i m¯ p 3=2 p 3 N¯ ip ¼ d¯ ip2 ðe þ 1Þgip q¯ i q¯ p ei ep m¯ 0 4 h¯ i h¯ p m¯ p 1 R¯ B¯ ip R¯ 5 þ ðe 1Þ m¯ 0 6 1
ð4Þ
and R¯ 1, R¯ 5 and B¯ ip are defined in the Appendix. The granular mixture normal and shear stresses are: s¯ T ¼
N X N dv¯ X l¯ ip þ l¯ ii dy¯ i¼1 p¼1
ð5Þ
s¯ N ¼
N X N X ei q¯ d¯ ip þ i h¯ i m¯ i i¼1 p¼1
ð6Þ
with l¯ ip ¼
a¯ ip 2
1 1 þ ¯hi ¯hp
! ð7Þ
and d¯ ip ¼
d¯ ip 3p
1 gip m¯ 0 48
m¯ i m¯ p h¯ i h¯ p
3=2
ei ep q¯ p q¯ i 1 þ eip R¯ 0
ð8Þ
Where R¯ 0 is defined in the Appendix. The above quantities were made dimensionless by the following characteristic parameters: m* for the mass, d* for the diameter, v* for the velocity and y* for y (in the case of binary and ternary mixtures, the characteristic mass, diameter, velocity and length are: m*=m, d*=d, v*=v and y*=H). The subscript 1 represents the largest particle.
H. Iddir et al. / Powder Technology 151 (2005) 117–125
The above system of equations representing a binary and ternary mixture of particles was solved numerically for h¯ i (i=1, 2, 3), particles of same density (q¯i =q¯p ipp), total solid volume fraction e=0.3 and the same flow parameters used in Ref. [25]. The model described above is compared with results obtained using molecular dynamics simulations. Three-dimensional simulations of binary mixtures are applied to frictionless inelastic spheres that engage in instantaneous collisions, thereby mimicking the corresponding assumptions used in model development. Simple-shear flow in an H H H box is achieved via Lees–Edwards boundary conditions [26]. Because no body forces are present, particles travel in straight-line trajectories between collisions. Collisions are resolved according to a hardsphere algorithm, and the simulation proceeds in time using an event-driven technique [27]. The inputs to the simulation include the box length H, the particle diameters d 1 and d 2, the particle masses m 1 and m 2, the total solids volume fraction. e the solids volume fraction of species 1 e1, the restitution coefficient e and the shear rate c. Hence, the dimensionless parameters that characterize the system are identical to those of the theory (d 1/d 2, m 1/m 2, e, e 1/e 2, e, c) with the addition of H/d 1 (since a domain size is not needed for application of theory to infinite simple shear flow but is needed for the moleculardynamics simulation). Simulation outputs include species energy, mixture energy, and stress components. Further details on the simulation, including a description of initialization and determination of a statistical steady state, are reported elsewhere [21,25].
3. Results and discussion For comparison purposes, we first studied the behavior of a single-phase flow using our model over a range of solid volume fractions. Fig. 1 shows the variation of the dimensionless normal and shear stresses of a monodisperse particulate system
Fig. 1. Variation of the dimensionless normal, and shear stresses for monosize particles with solid volume fraction and comparison of our theory with the Monte Carlo simulation results of Hopkins et al. [22] for e=0.9.
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Fig. 2. Variation of the granular energy ratio with the diameter ratio. Comparison of our theory with the MD simulation, for a binary mixture of particles with equal density, composition of e 1/e 2=0.5 and total solid volume fraction of e=0.3.
with the solid volume fraction, at a restitution coefficient of e=0.9. The comparison of the Monte Carlo (MC) simulation of Hopkins and Shen [22] agreed well with both shear and normal stresses values calculated from our model. This comparison shows also that our model reduces to a single-phase mixture, when all the particle properties are the same. We studied the effect of the flow parameters on the granular energies and the stresses for binary and ternary particle mixtures (different particle size and same density). The diameter of particle 1 (large particle) was kept constant and the diameter of particles 2 and 3 (small particles) was varied. Our model predicts well the non-equipartition of energy of particles in two particulate phases with two different particle sizes and the same density (see Fig. 2). As observed by several investigations [16,21,23], Fig. 2 shows that the fluctuating granular energy of the large particles increases relative to that of the small particles with an increase in the large to small diameter ratio. Smaller particles dissipate more energy when they collide with larger particles. Fig. 2 also shows a very good agreement between our calculated fluctuating energy with our molecular dynamics simulation (MD) simulation. Our model was applied to a mixture of equal mass and two different size particles (size ratio varying from 1 to 4), for e=0.8; it showed that the granular energy ratio, h 1/h 2, of the two phases varied by less than 2% between d 1/d 2=1 and d 1/d 2=4. However for a mixture of equal size and mass ratio of 3, our theory predicted that the h 1/h 2=1.15. This result is in good qualitative agreement with the experimental results of Feitosa and Menon [14]. They measured the granular energy of an equal-mass mixture in a vibrating twodimensional bed; they found that the granular energies ratio was close to unity. Furthermore they found that energy nonequipartition prevailed when the mass ratio is significantly greater than 1; this behavior is the same as our model prediction.
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Fig. 3. Variation of the dimensionless normal and shear stresses with the diameter ratio. Comparison of our theory with the MD simulation for a binary mixture of particles with equal density and composition, for a total solid volume fraction of e=0.3 and restitution coefficient of e=0.95.
Figs. 3 and 4 show a comparison between our model prediction and our MD simulation of the normal and shear stresses for a mixture of particles of equal density and composition with different diameter ratios. Our results clearly showed that the shear and normal stresses decrease with an increase in the particle size ratio in a binary mixture; in other words, our model predicts the maximum normal and shear stresses for monosize larger particles. This behavior was also confirmed by the experiment of Savage and Sayed [2] which indicated that the stresses in a binary mixture (same density, different sizes) were about five times lower than those for a monosize mixture. Normal and shear stresses predicted by our model were also compared with the MD simulation results (Fig. 3) and showed a good agreement at e=0.95. However at lower restitution coefficient values (see Fig. 4), our theory slightly over-predicted our MD simulation results for the normal stress; this may be due to the fact that our theory does not include the velocity distribution function
Fig. 4. Variation of the dimensionless normal and shear stresses with the diameter ratio. Comparison of our theory with the MD simulation for a binary mixture of particles with equal density and composition, for a total solid volume fraction of e=0.3 and restitution coefficient of e=0.8.
Fig. 5. Variation of the dimensionless mixture granular energy, with the system size for a monosize mixture of particles, for solid volume fraction of e=0.3 and restitution coefficient of e=0.95.
anisotropy. Earlier investigations [8,19,23,24] showed that anisotropy of the stress tensor for an ideal granular shear flow becomes important for more dissipative and dilute systems. These studies showed that the anisotropy of the stress tensor is related to the anisotropy of the velocity distribution function. In fact, as pointed out in [19], at high dissipation, there is a formation of particles concentrations below or above the bulk average concentration. This leads to different values for stresses and other properties predicted by theories that assume spatial homogeneity. Fig. 5 shows the variation of the dimensionless granular energy of a monodisperse mixture with the system size (H/ d 1), H being the box length and d 1 the particle diameter. For a fixed solid volume fraction, decreasing the size of the particles is equivalent to increasing their number in the system and thus lowering their granular energy. This is due to the increase in the frequency of the collisions with the increase in the number of particles in the system, and, hence, damping the fluctuations that may lead to a spatial nonhomogeneity in the system [19,20]. Furthermore, in the range of parameters investigated, the effect of increased mean free path in dilute systems is also responsible for the behavior shown in Fig. 6. For equal density particles, this
Fig. 6. Variation of the dimensionless mixture granular energy, with the solid volume fraction for monosize, binary and ternary granular mixtures of equal density and composition for e=0.95.
H. Iddir et al. / Powder Technology 151 (2005) 117–125
τ N /(γ 2 ρd 12 )
10
121
Ternary,d1/d2=2,d3/d1=1.5 Ternary,d1/d2=2,d1/d3=3 Ternary,d1/d2=2,d1/d3=4 Binary,d1/d2=2
1
0.1 0.1
0.2
0.3
0.4
0.5
ε Fig. 7. Variation of the dimensionless normal stress with the solid volume fraction for monosize, binary and ternary granular mixtures of equal density and composition for e=0.95.
figure shows that the fluctuations of the particles decrease with an increase in the solid volume fraction. It also shows that the granular energy is higher for a monodisperse system, and decreases with an increase in the number of the solid phases with smaller size particles and constant composition. For instance, at a solid volume fraction of e=0.3, compared to the monosize mixture of particles, the granular fluctuating energy decreases in binary and ternary mixtures by about 90% and 98%, respectively. Figs. 7 and 8 show a comparison of the dimensionless normal and shear stresses with the total solid volume fraction for monosize (d 1=H/9), binary (d 1/d 2=2) and ternary (d 1/d 2=2, d 1/d 3=3) particle mixtures. In this range of total solid volume fraction, the normal and shear stresses are higher for monodisperse systems, and decrease with an increase in the number of particulate phases in the system. For example, at e=0.3, our theoretical predictions for the shear stresses for binary and ternary mixtures are, respectively, 64% and 80% lower than those for monosize mixture of larger particles (see Fig. 8). Because the total volumetric concentration of all particles remains constant,
Fig. 8. Variation of the dimensionless shear stress with the solid volume fraction for monosize, binary and ternary granular mixtures of equal density and composition for e=0.95.
Fig. 9. Variation of the dimensionless normal stress with the solid volume fraction and the diameter ratio d 1/d 3, for a ternary granular mixture of equal density and composition (e=0.95).
addition of smaller particles causes reduction in the normal and shear stresses. We also observe from these two (Figs. 7 and 8) that the normal and shear stresses decrease with an increase in the solid volume fraction, reach a minimum and then increase. This minimum shifts to a higher value of solid volume fraction when the number of solid phases with smaller size particles increases. For instance, the minimum is at ec0.15, 0.2 and 0.25 for monosize, binary and ternary particle mixtures, respectively. The shift in the minimum confirms our previous observations, where the granular mixture becomes less resistant to the flow motion when particles of smaller sizes are added. Figs. 7 and 8, also, show that two regions exist for all three cases: a fluctuation dominant region in which the particle fluctuation decrease controls the dynamics of the flow and results in an overall decrease in the normal and shear stresses. The second region, to the right of the minimum, is a collision dominant region; the particle fluctuation is damped and solid volume fraction increase is dominant, this in turn, results in an overall increase in the normal and shear stresses.
Fig. 10. Variation of the dimensionless shear stress with the solid volume fraction and the diameter ratio d 1/d 3, for a ternary granular mixture of equal density and composition (e=0.95).
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Figs. 9 and 10 show the variation of the normal and shear stresses as a function of the total solid volume fraction e, for a d 1/d 2=2 (recall that d 1 is also fixed), with the same particle density and composition (e 1=e 2=e 3=e/3). These results show the effect of the particle size of the third phase on the normal and shear stresses. This clearly shows that the stresses have considerably lower magnitudes compared to its counterpart binary mixture cases when the particle size of the third phase is smaller than the one of the second phase. However, an increase in the stresses was obtained, if the particles of larger size than those of the second phase were added to the mixture. This suggests that in the range of parameters investigated a granular mixture shows less resistance to the flow when the number of solid phases with smaller particle size increases in the mixture.
4. Conclusion Based on the kinetic theory of a mixture of hard and smooth spherical particles, we developed a model to predict the behavior of multitype (different size and density) particle flow systems. The fluctuating energy ratio and the stresses of the two particulate phases as a function of the diameter ratio agreed well with the molecular dynamics simulation for a simple shear flow of a binary mixture. The theoretical results of the stresses and the granular energy compared qualitatively well with the experiment Savage and Sayed [2] and Feitosa and Menon [14], respectively. For a fixed total solid volume fraction, the fluctuations were strongly damped when the number of particulate phases with smaller particle size increases in the granular mixture. For particles of the same density, a ternary mixture showed less resistance to the flow when the number of smaller particles is added into the system, compared to its equivalent binary mixture. In general, we have shown that the size and composition of different particulate phases have great effect on both granular energy and stresses. Furthermore, the increase in the number of particulate phases in the mixture considerably changes the flow behavior.
Notation cYs Instantaneous velocity1 Y Cs Peculiar velocity V0 dv ¼ Shear rate dy H ds Diameter of the particle d ip =(d i +d p )/2 Average diameter of particles i and p e ip Restitution coefficient g Gravitational acceleration
Radial distribution function at contact particles i and p g ss Radial distribution function at contact particles of the same size H Gap between the two plates I¯ Identity tensor Y Y k ¼ k ip Unit vector connecting the centers of particles ms Mass of the particle of phase s m 0=m i +m p Total mass of two colliding particles ns ¼ ems qss Number density of phase s Y v Center of mass mean velocity Y vs Mean velocity of phase s VY0 Velocity of the plates es Solid volume fraction of phase s e Total solid volume fraction c=V 0 /H Shear rate qs Solid density of phase s Y Y hs ¼ 13 ms hC s d C s i Fluctuating granular energy or temperature of the phase s Subscript 1 Large Subscript 2,3 Small g ip
In the following, the subscript s can be either i or p.
between
the two
granular
Acknowledgement C.M. Hrenya is grateful for the funding support provided by the American Chemical Society Petroleum Research Fund (ACS PRF 38065-AC9).
Appendix A In the following, we give the results of the model developed; the details are given elsewhere [17]. We consider a mixture of N solid phases (N is the number of solid phases). The number density n 1 of any phase i is defined as ni ðrY ; t Þ ¼
Z
fi1 cYi ; rY ; t dcYi
ðA1Þ
fi1 ðcY ; rY ; t Þ is the single velocity distribution function, defined as the probability of finding a particle of phase i at any time t, having a velocity between cYi and cYi þ dcYi , in Y unit of volume, Ydr .Hence, the mean value of any property of phase i; wi ; ci is defined as: 1 hwi i ¼ ni
Z
wi ðcYi Þfi1 ðcYi ; rY ; tÞdcYi
ðA2Þ
The instantaneous velocity cYi is defined as the sum of the Y average velocity, vY i and peculiar velocity C i : Y
1
between
Y
cYi ¼ vYi þ C i with hC i i ¼ 0 and vYi ¼ hcYi i
H. Iddir et al. / Powder Technology 151 (2005) 117–125
The rate of change of the mean value of any property w i of phase i in a fixed element of volume drY may be expressed as: N X ni Y Bwi B hðwci Þjp i¼ ðni hwi iÞþjd ni hwi cYi i F iext d Y Bt mi Bci p¼1 ðA3Þ PN
p¼1 ðwci Þjp is defined as the difference between the postcollisional and the precollisional properties of particle i due to all possible collisions P with all the particles in the mixture. The average of Np¼1 ðwci Þjp is defined as: N X
hðwci Þjp i ¼
p¼1
N X
¼
N ZZZ X
ðwiV wi Þfip2 ðrYi ; cYi ; rYp ; cYp Þ
p¼1 Y
Y
dip2 ðcYip d k ip Þdk ip dcYi dcYp
ðA4Þ
For impending collisions, the integrals in Eq. (A4) must be carried out only for values of cYip kYip N0. fip2 ¼ fip2 cYi ; Y ri ; Y cp ; Y rp is the complete pair distribution function defined as the probability of finding, at time t, two particles i and p such that they are centered on rYi and rYp and having velocities within the range cYi ; cYi þ dcYi and 2 cYp ; cYp þ dcYp . Following the assumption of chaos [3], f ip can be written as the product of the single velocity distributions, f i1, f p1 and the pair-distribution function at contact, g ip (e i , e p ). dip Y Y Y dip k ;cp ;r þ fip2 cYi ;rY 2 2 d dip ip Y k fp1 cYp ;rY þ ¼ gip ei ;ep Þfi1 cYi ;rY ðA5Þ 2 2 Y Y where cY ip ¼ ci cp is the relative instantaneous velocity. di þ dp dip ¼ 2 Y k ip is the unit vector connecting the centers of the two particles, located at rYi and rYp respectively and is directed from i to p. Y Y In the remaining text, we consider k ip ¼ k gip ðei ; ep Þ ¼ ½dp gii ðei ; ep Þ þ di gpp ðei ; ep Þ =2dip ðA6Þ
where gii ðei ; ep Þ ¼ 1=ð1 ðei þ ep Þ=emax Þ þ 3di =2
N X
The expression of g pp (e i , e p ) is obtained by interchanging the indices i and p. collisional rate of production per unit of volume, PThe N ðhw iV wi iÞjp , was evaluated in [17] p¼1
p¼1
N X p¼1
ð jd vcip þ ccip Þ
ðA9Þ
Y
ðcYip d k Þdk dcYi dcYp ðA10Þ Here v cip and c cip are the collisional fluxes and sources, respectively. Substituting the above expressions (A9) and (A10) into the equation of change (A3), the continuity, momentum and fluctuating energy equations were obtained for w i equal to m i , mi cYi and 12 mi ci2 , respectively. Continuity equation The continuity equation for the solid phase i may be written as: Bei qi þ jd ðei qi Y vi Þ ¼ 0 ðA11Þ Bt e i q i =n i m i is the phase i density, e i is its solid volume fraction and q i is its material density. Momentum equation The momentum equation for phase i may be expressed as: X N N X D Y ei q Y Y ¯ ¯ ei qi ðvi Þ þ Pcip þ Pki i F iext ¼ F Dip Dt m i p¼1 p¼1 ðA12Þ D ð Þ is the material derivative with Dt Y Y P¯ ki ¼ qi ei hC i C i i is the kinetic pressure tensor ¯
Pcip ¼
vcip ðmi cYi Þ
is the collisional pressure tensor
ðA13Þ ðA14Þ ðA15Þ
Y
F Dip ¼ ccip ðmi cYi Þ is the collisional momentum source
Fluctuating energy equation ðA7Þ
hwiV wi ijp ¼
ZZZ dip3 ðwiV wi Þfip2 vcip ¼ 2 dip Y Y Y dip Y cYi ; rY k ; cp ; r þ k 2 2 Y Y cYip k Þdk dcYi dcYp ZZZ 2 ðwiV wi Þfip2 ccip ¼ dip dip Y Y Y dip k ; cp ; r þ cYi ; rY 2 2
ðA16Þ ep =dp
p¼1
N X
where
Y
hwiV wi ijp
p¼1
123
ðA8Þ
The fluctuating energy equation for solid phase i may be expressed as: N N X X 3 ei qi Dhi Y þ jd qYki þ qcip P¯ ki þ P¯ cip : jvYi 2 mi Dt p¼1 p¼1 ¼
N X p¼1
Y
ðNip Y vi F Dip Þ
ðA17Þ
124
H. Iddir et al. / Powder Technology 151 (2005) 117–125
where 1 Y Y hi ¼ mi hC i C i i ðA18Þ 3 h i is the fluctuating granular energy of the solid phase i Y qki
Y
¼ qi ei hC i Ci2 i is the kinetic heat flux
qYcip ¼ vcip ðmi c2i Þ is the collisional energy flux
Momentum interaction mi mp mi mp 3=2 gip 1 þ eip np ni 4 m0 hi hp pffiffiffi Y Y dip p ni 3 hi pR2 d vp vi þ R0 jln jln 12 np 2 hp dip p mp jhp 5mi mp mi jhi þ R10 þ 16 192m20 h2p h2i mijhp mpjhi 5mi mp jhp jhi Bip 2 þ 2 R4 R3 þ 96m0 h2p h2i hp hi
dip2 Y FDip ¼
ðA19Þ ðA20Þ
Nip ¼ ccip ðmi c2i Þ is the collisional energy dissipation ðA21Þ In the above governing equations, the relevant variables describing the flow field are the average velocities, the solid volume fractions and the granular temperatures evaluated at location Y r of the center of the particle at time t. The kinetic equations that characterize the flow of multiphase system are: B FY þY c þ iext d j > Y c fi 1 Bt mi N d2 Z Z X ip Y ½gip ðrY ; Y ¼ r þ k d ip =2Þfi1 ðcYi1 ; Y r ; tÞfi1 ðA22Þ 4 p¼1
Y
Y
Y Y Y 1 ðcY i2 ; r þ k dip =2; tÞ gip ðr ; r k d ip =2Þfp Y Y Y ðcYp1 ; Y r ; tÞfp1 ðcYp2 ; Y r þ k d ip =2; tÞ Y cip d k ÞdcYp dk
where gip ðY r Þ is the spatial pair-distribution function when the particles i and p are in contact. A solution of Eq. (A22) near the equilibrium was obtained using Chapman–Enskog method [18]. fi1 ¼ fi 0 ð1 þ /i Þ where f i 0 is the Maxwellian velocity distribution 3=2 mi mi Ci2 f i 0 ¼ ni exp 2phi 2hi
ðA23Þ
Energy dissipation
Nip ¼
mi mp 3=2 mi mp 3 2 dip eip þ 1 gip ni np 4 hi hp m0 Y m jd v pffiffiffi 3dip p mi jd Y vi p p Bip pd R5 þ d R10 40 hp hi mp vp dip p mi mp jd Y jd Y vi þ Bip þ R4 þ eip 1 4 m0 hp hi m0 pffiffiffi Y Y dip p mp jd vp p mi jd vi d R1 þ Bip d R4 6 16 hp hi vp dip p mi mp jd Y jd Y vi þ R3 ðA27Þ 48 m0 hp hi
Collisional heat flux ðA24Þ
/ i is a perturbation to the Maxwellian velocity distribution. It is a linear function of the first derivative of n i , h i and vY i . Note that / i is a function of the phase mean velocity, vY i , and not the total flow velocity, because as mentioned in the introduction each kind of particles is treated as a separate phase and the interaction is at the interface. After solving Eq. (A22) [17] for each phase, we obtain the constitutive equations: Particulate phase pressure tensor P¯ ip ¼
ðA26Þ
dip3 mi mp mi mp 3=2 gip ni np ð1 þ eip Þ 48 m0 hi hp s vp pffiffiffi mi mp 1 6 j˙ s Y j˙ Y vi ¯ pR0 I pdip þ m0 3 5 hp hi Y 2 jd vp jd Y vi ¯ 5 4 þ þ I R1 2 1þ ð1þeÞgii ei 5 hp hi 8di gii rffiffiffiffiffiffiffiffiffi mi hi ˙ Y j svi þ ni hi I¯ ðA25Þ p
mp mi 3=2 mp mi 1 þ eip ni np gip qYcip0 ¼ dip3 2m0 hi hp pffiffiffi dip p np Bip 3 hi þ j ln Bip j ln R5 þ 2 16 ni hp 8 mpjhp mijhi mp mi mijhp mpjhi 2 R6 þBip 2 m0 h2p hi h2p h2i mp mi jhp mp jhi 1 eip R7 þ R9 þ 2 2 2m0 8m0 hp hi dip p np 3 hi 3p Y Y v vp ÞR0 R1 þ jln þ jln 6 2 10 i ni hp pffiffiffi pffiffiffi dip p mp jhp dip p m2i mi jhi þ þ R 9 8 2 m20 h2p h2i pffiffiffi mp mi mi jhp mp jhi R8 þ dip pBip 2 2 m0 hp hi 2 jhp jhi 75 3 2 þ 2 R7 1þ ð1þeÞ gii ei 5 64di2 gii h2p hi sffiffiffiffiffiffiffiffi hi ðA28Þ jhi mi p
H. Iddir et al. / Powder Technology 151 (2005) 117–125
Where e ip is the normal restitution coefficient between particles i and p mi mp hp hi m i hp þ m p hi Aip ¼ ; Bip ¼ ; 2hi hp 2m0 hi hp m i m p m p hp þ m i hi ðA29Þ Dip ¼ 2m20 hi hp R0 ¼
R1 ¼
R2 ¼
R3 ¼
R4 ¼
R5 ¼
R6 ¼
R7 ¼
R8 ¼
R9 ¼
1 3=2
5=2
Aip Dip 1
3=2 Aip Dip3
þ
1 3=2 2Aip Dip2
1 3=2
5=2
7=2
1 5=2
Aip Dip3 1 7=2 Aip Dip3
þ
3 5=2
2Aip Dip4 1 3=2
2Aip Dip4 1 5=2
Aip Dip3
R10 ¼
1 5=2
7=2
5=2
35Bip2 7=2
9=2
2Aip Dip 5Bip2 7=2
Aip Dip4 7Bip2 9=2 Aip Dip4
þ
þ
10Bip2 7=2
Aip Dip5 6Bip2 0
þ
5=2
Aip Dip5 15Bip2 7=2
Aip Dip4
þ
25Bip2 7=2
9=2
11=2
8Aip Dip
9=2
25Bip4 Aip Dip6 70Bip4 9=2
Aip Dip5
ðA33Þ
ðA34Þ
[10] [11] [12] [13] [14] [15] [16] [17]
ðA35Þ [18]
Aip Dip6 7=2
þ: : :
þ: : :
35Bip4
þ 7=2
2Aip Dip
[8]
þ: : : 11=2
441Bip4
126Bip4 11=2 5Aip Dip5
þ
[6]
[9]
7=2
9=2
[1] [2] [3] [4]
ðA32Þ
8Aip Dip
Aip Dip5
þ
ðA31Þ
315Bip4
14Bip4
References
[5]
þ: : :
7=2 2Aip Dip4
þ
ðA30Þ
We have showed in [17] that the series in the above expressions converge after the second term.
[7]
15Bip4
þ 9=2
þ: : :
þ: : :
7=2 2Aip Dip5
þ
9=2
8Aip Dip
30Bip4
2Aip Dip
þ 5=2
Aip Dip
175Bip4
þ
21Bip2
þ
þ
þ
5=2 Aip Dip3
þ
þ
7=2
3Bip2
þ
Aip Dip
Aip Dip
5=2
2Aip Dip
9Bip2 5=2 4 Aip Dip
þ 7=2
3
15Bip2
þ
þ: : :
ðA36Þ
[19] [20] [21]
þ: : :
ðA37Þ
þ: : :
ðA38Þ
[22] [23] [24] [25]
[26]
þ: : :
1225Bip4 9=2
9=2
24Aip Dip
ðA39Þ
þ: : :
ðA40Þ
125
[27]
R.A. Bagnold, Proc. R. Soc. Lond., A 225 (49) (1954) 49 – 63. S. Savage, M. Sayed, J. Fluid Mech. 142 (1984) 391 – 430. J.T. Jenkins, S.B. Savage, J. Fluid Mech. 130 (1983) 187 – 201. C.K.K. Lun, S.B.D. Savage, J. Jeffrey, N. Chepurniy, J. Fluid Mech. 140 (1984) 223 – 256. J.W. Dufty, A. Santos, J.J. Brey, Phys. Rev. Lett. 77 (7) (1996) 1270 – 1273. J.M. Montanero, A. Santos, M. Lee, J.W. Dufty, J.F. Lutsko, Phys. Rev., E 57 (1) (1998) 546 – 556. L. Bocquet, W. Losert, D. Schalk, T.C. Lubensky, J.P. Gollub, Phys. Rev., E 65 (2001) (011307)1 – (011307)19. M. Strumendo, P. Canu, Phys. Rev., E 66 (2002) (041304)1 – (041304)20. J.T. Jenkins, F. Mancini, Phys. Fluids, A Fluid Dyn. 1 (12) (1989) 2050 – 2057. B.O. Arnarson, J.T. Willits, Phys. Fluids 10 (6) (1998) 1324 – 1328. J.T. Willits, B.O. Arnarson, Phys. Fluids 11 (10) (1999) 3116 – 3122. P. Zamankhan, Phys. Rev., E 52 (5) (1995) 4877 – 4891. V. Garzo´, J.W. Dufty, Phys. Rev., E 60 (5) (1999) 5706 – 5713. K. Feitosa, N. Menon, Phys. Rev. Lett. 88 (7) (2002) (031301)1 – (031301)4. R.D. Wildman, D.J. Parker, Phys. Rev. Lett. 88 (6) (2002) (064301)1 – (064301)4. V. Garzo´, J.W. Dufty, Phys. Fluids 14 (4) (2002) 1476 – 1490. H. Iddir, H. Arastoopour, bModeling of Multi-Type Particle Flow Using the Kinetic Theory ApproachQ submitted to AiChE Journal (2004). S. Chapman and T.G. Cowling, Cambridge at the University press (1970). M.A. Hopkins, M.Y. Louge, Phys. Fluids, A Fluid Dyn. 3 (1) (1991) 47 – 57. E.D. Liss, B.J. Glasser, Powder Technol. 116 (2001) 116 – 132. R. Clelland, C.M. Hrenya, Phys. Rev., E 65 (2002) (031301)1 – (031301)9. M.A. Hopkins, H.H. Shen, J. Fluid Mech. 244 (1992) 477 – 491. M. Alam, S. Luding, J. Fluid Mech. 476 (2003) 69 – 103. J.T. Jenkins, M.W. Richman, J. Fluid Mech. 192 (1988) 313 – 328. J.E. Galvin, S.R. Dahl, C.M. Hrenya, bOn the role of nonequipartition in the dynamics of rapidly-flowing, granular mixturesQ, submitted to J. Fluid Mech., (2004). A.W. Lees, S.F. Edwards, J. Phys. C. Solid State Phys. 5 (1972) 1921 – 1929. M.P. Allen, D.J. Tildesley, Computer Simulation of Liquids, Oxford University Press, New York, 1989.
Analysis of binary and ternary granular mixtures behavior using the kinetic theory approach Hadjira Iddira, Hamid Arastoopoura,*, Christine M. Hrenyab a
Department of Chemical and Environmental Engineering, llinois Institute of Technology, 10 W. 33rd Street, Illinois, Chicago 60616, United States b Department of Chemical and Biological Engineering, University of Colorado, Boulder, Colorado 80309, United States Received 1 November 2004 Available online 28 January 2005
Abstract The kinetic theory was extended to granular mixtures of different mechanical properties (size, density and/or restitution coefficient) where each particle group was considered as a separate phase with different average velocity and granular energy. This model was applied to simple shear flow of binary and ternary mixture of particles. In the parameter range studied, the rheological behavior of a binary mixture showed a good quantitative agreement with the molecular dynamics simulation and predicted qualitatively well the experiment of Savage and Sayed [S. Savage, M. Sayed, J. Fluid Mech., 142 (1984)] and Feitosa and Menon [K. Feitosa, N. Menon, Phys. Rev. Lett., 88 (7) (2002)]. Furthermore, the investigation of a ternary granular mixture showed interesting results: the fluctuations were strongly damped and the granular mixture showed less resistance to the flow when the number of particulate phases with smaller particle size increased in the mixture, while the total particle concentration remained unchanged. D 2004 Elsevier B.V. All rights reserved. Keywords: Kinetic theory; Binary particle mixture; Ternary particle mixture; Granular shear flow; Energy non-equipartition
1. Introduction Fluid particle flow exhibits complex behavior due to the coexistence of different regimes within the flow. Collisions between particles are responsible for the energy transfer and dissipation that lead to phenomena such as agglomeration, attrition and breakage of particles. Furthermore, collisions of particles of different mechanical properties and corresponding fluctuations that occur during the flow make the system even more complicated and challenging. The process of fluid particle flow is encountered in numerous applications. In nature or industry, we deal with either highly concentrated flow, where the solid phase interactions are dominant (such as the flow of avalanches or flow in solids mixers), or highly dilute flow where the interaction between the solid and fluid is important (such as in pneumatic conveyors or fluid catalytic cracking). * Corresponding author. Tel.: +1 312 567 3040; fax: +1 312 567 8874. E-mail address: [email protected] (H. Arastoopour). 0032-5910/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2004.11.033
To understand the behavior of fluid particle flow, one may need first to understand granular flow where the interstitial fluid effects may be neglected [1,2]. In granular medium, the particle–particle interaction is dissipative, thus modeling the flow behavior is non-trivial. Flow of a monodisperse mixture has been extensively studied. Jenkins and Savage [3] used the kinetic theory approach to develop a model that predicted the behavior of inelastic particles undergoing a rapid shear flow. Lun et al. [4] developed a theory and studied simple shear flow behavior for wide range of the restitution coefficients. Monodisperse granular flow is still attracting researchers’ interest from several disciplines [5–8]. In recent years, granular flow of a binary mixture has become the subject of intense studies, in which the kinetic equation was solved at the near equilibrium steady state assuming energy equipartition to obtain the hydrodynamic properties for nearly elastic particles [9–11]. Zamankhan [12] solved the kinetic equation using the Grad method and assumed that the particles fluctuate with the same energy; he concluded that
118
H. Iddir et al. / Powder Technology 151 (2005) 117–125
energy non-equipartition must be included in mixtures with different particle properties. Garzo´ and Dufty [13] showed theoretically that the energy equipartition is violated for inelastic particles of a binary mixture. Later, the energy nonequipartition in a binary mixture was experimentally confirmed [14,15]. In a recent publication, Garzo´ and Dufty [16] solved the kinetic equation for systems away from equilibrium; hence, their model could capture not only the energy nonequipartition, but also the flow behavior for a broader range of restitution coefficients. However, their model is restricted to dilute systems, where the radial distribution function was close to unity. Iddir and Arastoopour [17] developed a model that predicts the flow of particles of different mechanical properties, based on the kinetic theory of dense gases. The model predicted well the energy non-equipartition and the stresses of a binary granular mixture. In this study, we investigate the behavior of a binary and a ternary mixture of particles undergoing a simple shear flow using the kinetic theory approach. We also investigate how a third particulate phase may affect the flow behavior. A brief summary of our model is provided in the Appendix.
where the thermal equilibrium may be reached when the generation due to continuous shearing is balanced by the dissipation due to collisions. In the case of simple shear flow the dimensionless energy equations for particles of phase i interacting with particles of any other phase p reduce to the following non-linear algebraic equations (note that the bar on the quantities below represents their dimensionless form): ! a¯ip 1 1 dv¯ 2 þ ¼ N¯ii þ N¯ip ð1Þ l¯ ii þ dy¯ 2 h¯i h¯p where: "
# 2 5 4 4d¯i ð1 þ eÞe2i q¯i gii l¯ii ¼ 1 þ ð1 þ eÞgii ei þ 5 5m¯ i 16d¯i2 gii sffiffiffiffiffiffiffiffiffi m¯ i h¯i ð2Þ p pffiffiffi m¯ i m¯ p m¯ i m¯ p 3=2 p ¯4 2 d gip 2 a¯ip ¼ ð1 þ eÞq¯ i q¯ p ei ep R¯ 1 ¯hi h¯ p 48 ip 5 m¯ 0 ð3Þ
2. Model application to simple shear flow Our model (see Appendix) is based on the extension of the kinetic theory to hard, smooth, non-cohesive (typically larger than 50 Am) and spherical particles of different type (size and/or density). Each group of particles with similar properties is considered as a separate phase, which is characterized by its own average velocity and fluctuating energy or granular temperature. The interaction between the phases occurred at the interface, namely the collision between particles of different size/or density. As a consequence of this assumption, we solved the Boltzmann equation for each particulate phase using the Chapman– Enskog procedure by adding a perturbation to the Maxwellian velocity distribution function [18]. However, we assumed at the interface that the velocity distribution function remains Maxwellian. To investigate the effect of the system parameters (diameter ratios, total solid fraction, solid fraction ratios and restitution coefficients) on the hydrodynamics of a binary and ternary particle mixtures, a very simple flow system, such as simple shear flow, was chosen. In such a system, the velocity gradient is constant; hence, segregation by size or density does not occur. The flow field consists of two infinite parallel boundaries set to move with velocities FV 0/2. The mean motion of the granular mixture was only in the x-direction and was considered fully developed so that all the flow parameters are a function of y. In this study, x and y are the axes parallel and perpendicular to the moving boundaries, respectively. In this example, we are interested in the steady state regime,
pffiffiffi m¯ i m¯ p 3=2 p 3 N¯ ip ¼ d¯ ip2 ðe þ 1Þgip q¯ i q¯ p ei ep m¯ 0 4 h¯ i h¯ p m¯ p 1 R¯ B¯ ip R¯ 5 þ ðe 1Þ m¯ 0 6 1
ð4Þ
and R¯ 1, R¯ 5 and B¯ ip are defined in the Appendix. The granular mixture normal and shear stresses are: s¯ T ¼
N X N dv¯ X l¯ ip þ l¯ ii dy¯ i¼1 p¼1
ð5Þ
s¯ N ¼
N X N X ei q¯ d¯ ip þ i h¯ i m¯ i i¼1 p¼1
ð6Þ
with l¯ ip ¼
a¯ ip 2
1 1 þ ¯hi ¯hp
! ð7Þ
and d¯ ip ¼
d¯ ip 3p
1 gip m¯ 0 48
m¯ i m¯ p h¯ i h¯ p
3=2
ei ep q¯ p q¯ i 1 þ eip R¯ 0
ð8Þ
Where R¯ 0 is defined in the Appendix. The above quantities were made dimensionless by the following characteristic parameters: m* for the mass, d* for the diameter, v* for the velocity and y* for y (in the case of binary and ternary mixtures, the characteristic mass, diameter, velocity and length are: m*=m, d*=d, v*=v and y*=H). The subscript 1 represents the largest particle.
H. Iddir et al. / Powder Technology 151 (2005) 117–125
The above system of equations representing a binary and ternary mixture of particles was solved numerically for h¯ i (i=1, 2, 3), particles of same density (q¯i =q¯p ipp), total solid volume fraction e=0.3 and the same flow parameters used in Ref. [25]. The model described above is compared with results obtained using molecular dynamics simulations. Three-dimensional simulations of binary mixtures are applied to frictionless inelastic spheres that engage in instantaneous collisions, thereby mimicking the corresponding assumptions used in model development. Simple-shear flow in an H H H box is achieved via Lees–Edwards boundary conditions [26]. Because no body forces are present, particles travel in straight-line trajectories between collisions. Collisions are resolved according to a hardsphere algorithm, and the simulation proceeds in time using an event-driven technique [27]. The inputs to the simulation include the box length H, the particle diameters d 1 and d 2, the particle masses m 1 and m 2, the total solids volume fraction. e the solids volume fraction of species 1 e1, the restitution coefficient e and the shear rate c. Hence, the dimensionless parameters that characterize the system are identical to those of the theory (d 1/d 2, m 1/m 2, e, e 1/e 2, e, c) with the addition of H/d 1 (since a domain size is not needed for application of theory to infinite simple shear flow but is needed for the moleculardynamics simulation). Simulation outputs include species energy, mixture energy, and stress components. Further details on the simulation, including a description of initialization and determination of a statistical steady state, are reported elsewhere [21,25].
3. Results and discussion For comparison purposes, we first studied the behavior of a single-phase flow using our model over a range of solid volume fractions. Fig. 1 shows the variation of the dimensionless normal and shear stresses of a monodisperse particulate system
Fig. 1. Variation of the dimensionless normal, and shear stresses for monosize particles with solid volume fraction and comparison of our theory with the Monte Carlo simulation results of Hopkins et al. [22] for e=0.9.
119
Fig. 2. Variation of the granular energy ratio with the diameter ratio. Comparison of our theory with the MD simulation, for a binary mixture of particles with equal density, composition of e 1/e 2=0.5 and total solid volume fraction of e=0.3.
with the solid volume fraction, at a restitution coefficient of e=0.9. The comparison of the Monte Carlo (MC) simulation of Hopkins and Shen [22] agreed well with both shear and normal stresses values calculated from our model. This comparison shows also that our model reduces to a single-phase mixture, when all the particle properties are the same. We studied the effect of the flow parameters on the granular energies and the stresses for binary and ternary particle mixtures (different particle size and same density). The diameter of particle 1 (large particle) was kept constant and the diameter of particles 2 and 3 (small particles) was varied. Our model predicts well the non-equipartition of energy of particles in two particulate phases with two different particle sizes and the same density (see Fig. 2). As observed by several investigations [16,21,23], Fig. 2 shows that the fluctuating granular energy of the large particles increases relative to that of the small particles with an increase in the large to small diameter ratio. Smaller particles dissipate more energy when they collide with larger particles. Fig. 2 also shows a very good agreement between our calculated fluctuating energy with our molecular dynamics simulation (MD) simulation. Our model was applied to a mixture of equal mass and two different size particles (size ratio varying from 1 to 4), for e=0.8; it showed that the granular energy ratio, h 1/h 2, of the two phases varied by less than 2% between d 1/d 2=1 and d 1/d 2=4. However for a mixture of equal size and mass ratio of 3, our theory predicted that the h 1/h 2=1.15. This result is in good qualitative agreement with the experimental results of Feitosa and Menon [14]. They measured the granular energy of an equal-mass mixture in a vibrating twodimensional bed; they found that the granular energies ratio was close to unity. Furthermore they found that energy nonequipartition prevailed when the mass ratio is significantly greater than 1; this behavior is the same as our model prediction.
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Fig. 3. Variation of the dimensionless normal and shear stresses with the diameter ratio. Comparison of our theory with the MD simulation for a binary mixture of particles with equal density and composition, for a total solid volume fraction of e=0.3 and restitution coefficient of e=0.95.
Figs. 3 and 4 show a comparison between our model prediction and our MD simulation of the normal and shear stresses for a mixture of particles of equal density and composition with different diameter ratios. Our results clearly showed that the shear and normal stresses decrease with an increase in the particle size ratio in a binary mixture; in other words, our model predicts the maximum normal and shear stresses for monosize larger particles. This behavior was also confirmed by the experiment of Savage and Sayed [2] which indicated that the stresses in a binary mixture (same density, different sizes) were about five times lower than those for a monosize mixture. Normal and shear stresses predicted by our model were also compared with the MD simulation results (Fig. 3) and showed a good agreement at e=0.95. However at lower restitution coefficient values (see Fig. 4), our theory slightly over-predicted our MD simulation results for the normal stress; this may be due to the fact that our theory does not include the velocity distribution function
Fig. 4. Variation of the dimensionless normal and shear stresses with the diameter ratio. Comparison of our theory with the MD simulation for a binary mixture of particles with equal density and composition, for a total solid volume fraction of e=0.3 and restitution coefficient of e=0.8.
Fig. 5. Variation of the dimensionless mixture granular energy, with the system size for a monosize mixture of particles, for solid volume fraction of e=0.3 and restitution coefficient of e=0.95.
anisotropy. Earlier investigations [8,19,23,24] showed that anisotropy of the stress tensor for an ideal granular shear flow becomes important for more dissipative and dilute systems. These studies showed that the anisotropy of the stress tensor is related to the anisotropy of the velocity distribution function. In fact, as pointed out in [19], at high dissipation, there is a formation of particles concentrations below or above the bulk average concentration. This leads to different values for stresses and other properties predicted by theories that assume spatial homogeneity. Fig. 5 shows the variation of the dimensionless granular energy of a monodisperse mixture with the system size (H/ d 1), H being the box length and d 1 the particle diameter. For a fixed solid volume fraction, decreasing the size of the particles is equivalent to increasing their number in the system and thus lowering their granular energy. This is due to the increase in the frequency of the collisions with the increase in the number of particles in the system, and, hence, damping the fluctuations that may lead to a spatial nonhomogeneity in the system [19,20]. Furthermore, in the range of parameters investigated, the effect of increased mean free path in dilute systems is also responsible for the behavior shown in Fig. 6. For equal density particles, this
Fig. 6. Variation of the dimensionless mixture granular energy, with the solid volume fraction for monosize, binary and ternary granular mixtures of equal density and composition for e=0.95.
H. Iddir et al. / Powder Technology 151 (2005) 117–125
τ N /(γ 2 ρd 12 )
10
121
Ternary,d1/d2=2,d3/d1=1.5 Ternary,d1/d2=2,d1/d3=3 Ternary,d1/d2=2,d1/d3=4 Binary,d1/d2=2
1
0.1 0.1
0.2
0.3
0.4
0.5
ε Fig. 7. Variation of the dimensionless normal stress with the solid volume fraction for monosize, binary and ternary granular mixtures of equal density and composition for e=0.95.
figure shows that the fluctuations of the particles decrease with an increase in the solid volume fraction. It also shows that the granular energy is higher for a monodisperse system, and decreases with an increase in the number of the solid phases with smaller size particles and constant composition. For instance, at a solid volume fraction of e=0.3, compared to the monosize mixture of particles, the granular fluctuating energy decreases in binary and ternary mixtures by about 90% and 98%, respectively. Figs. 7 and 8 show a comparison of the dimensionless normal and shear stresses with the total solid volume fraction for monosize (d 1=H/9), binary (d 1/d 2=2) and ternary (d 1/d 2=2, d 1/d 3=3) particle mixtures. In this range of total solid volume fraction, the normal and shear stresses are higher for monodisperse systems, and decrease with an increase in the number of particulate phases in the system. For example, at e=0.3, our theoretical predictions for the shear stresses for binary and ternary mixtures are, respectively, 64% and 80% lower than those for monosize mixture of larger particles (see Fig. 8). Because the total volumetric concentration of all particles remains constant,
Fig. 8. Variation of the dimensionless shear stress with the solid volume fraction for monosize, binary and ternary granular mixtures of equal density and composition for e=0.95.
Fig. 9. Variation of the dimensionless normal stress with the solid volume fraction and the diameter ratio d 1/d 3, for a ternary granular mixture of equal density and composition (e=0.95).
addition of smaller particles causes reduction in the normal and shear stresses. We also observe from these two (Figs. 7 and 8) that the normal and shear stresses decrease with an increase in the solid volume fraction, reach a minimum and then increase. This minimum shifts to a higher value of solid volume fraction when the number of solid phases with smaller size particles increases. For instance, the minimum is at ec0.15, 0.2 and 0.25 for monosize, binary and ternary particle mixtures, respectively. The shift in the minimum confirms our previous observations, where the granular mixture becomes less resistant to the flow motion when particles of smaller sizes are added. Figs. 7 and 8, also, show that two regions exist for all three cases: a fluctuation dominant region in which the particle fluctuation decrease controls the dynamics of the flow and results in an overall decrease in the normal and shear stresses. The second region, to the right of the minimum, is a collision dominant region; the particle fluctuation is damped and solid volume fraction increase is dominant, this in turn, results in an overall increase in the normal and shear stresses.
Fig. 10. Variation of the dimensionless shear stress with the solid volume fraction and the diameter ratio d 1/d 3, for a ternary granular mixture of equal density and composition (e=0.95).
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H. Iddir et al. / Powder Technology 151 (2005) 117–125
Figs. 9 and 10 show the variation of the normal and shear stresses as a function of the total solid volume fraction e, for a d 1/d 2=2 (recall that d 1 is also fixed), with the same particle density and composition (e 1=e 2=e 3=e/3). These results show the effect of the particle size of the third phase on the normal and shear stresses. This clearly shows that the stresses have considerably lower magnitudes compared to its counterpart binary mixture cases when the particle size of the third phase is smaller than the one of the second phase. However, an increase in the stresses was obtained, if the particles of larger size than those of the second phase were added to the mixture. This suggests that in the range of parameters investigated a granular mixture shows less resistance to the flow when the number of solid phases with smaller particle size increases in the mixture.
4. Conclusion Based on the kinetic theory of a mixture of hard and smooth spherical particles, we developed a model to predict the behavior of multitype (different size and density) particle flow systems. The fluctuating energy ratio and the stresses of the two particulate phases as a function of the diameter ratio agreed well with the molecular dynamics simulation for a simple shear flow of a binary mixture. The theoretical results of the stresses and the granular energy compared qualitatively well with the experiment Savage and Sayed [2] and Feitosa and Menon [14], respectively. For a fixed total solid volume fraction, the fluctuations were strongly damped when the number of particulate phases with smaller particle size increases in the granular mixture. For particles of the same density, a ternary mixture showed less resistance to the flow when the number of smaller particles is added into the system, compared to its equivalent binary mixture. In general, we have shown that the size and composition of different particulate phases have great effect on both granular energy and stresses. Furthermore, the increase in the number of particulate phases in the mixture considerably changes the flow behavior.
Notation cYs Instantaneous velocity1 Y Cs Peculiar velocity V0 dv ¼ Shear rate dy H ds Diameter of the particle d ip =(d i +d p )/2 Average diameter of particles i and p e ip Restitution coefficient g Gravitational acceleration
Radial distribution function at contact particles i and p g ss Radial distribution function at contact particles of the same size H Gap between the two plates I¯ Identity tensor Y Y k ¼ k ip Unit vector connecting the centers of particles ms Mass of the particle of phase s m 0=m i +m p Total mass of two colliding particles ns ¼ ems qss Number density of phase s Y v Center of mass mean velocity Y vs Mean velocity of phase s VY0 Velocity of the plates es Solid volume fraction of phase s e Total solid volume fraction c=V 0 /H Shear rate qs Solid density of phase s Y Y hs ¼ 13 ms hC s d C s i Fluctuating granular energy or temperature of the phase s Subscript 1 Large Subscript 2,3 Small g ip
In the following, the subscript s can be either i or p.
between
the two
granular
Acknowledgement C.M. Hrenya is grateful for the funding support provided by the American Chemical Society Petroleum Research Fund (ACS PRF 38065-AC9).
Appendix A In the following, we give the results of the model developed; the details are given elsewhere [17]. We consider a mixture of N solid phases (N is the number of solid phases). The number density n 1 of any phase i is defined as ni ðrY ; t Þ ¼
Z
fi1 cYi ; rY ; t dcYi
ðA1Þ
fi1 ðcY ; rY ; t Þ is the single velocity distribution function, defined as the probability of finding a particle of phase i at any time t, having a velocity between cYi and cYi þ dcYi , in Y unit of volume, Ydr .Hence, the mean value of any property of phase i; wi ; ci is defined as: 1 hwi i ¼ ni
Z
wi ðcYi Þfi1 ðcYi ; rY ; tÞdcYi
ðA2Þ
The instantaneous velocity cYi is defined as the sum of the Y average velocity, vY i and peculiar velocity C i : Y
1
between
Y
cYi ¼ vYi þ C i with hC i i ¼ 0 and vYi ¼ hcYi i
H. Iddir et al. / Powder Technology 151 (2005) 117–125
The rate of change of the mean value of any property w i of phase i in a fixed element of volume drY may be expressed as: N X ni Y Bwi B hðwci Þjp i¼ ðni hwi iÞþjd ni hwi cYi i F iext d Y Bt mi Bci p¼1 ðA3Þ PN
p¼1 ðwci Þjp is defined as the difference between the postcollisional and the precollisional properties of particle i due to all possible collisions P with all the particles in the mixture. The average of Np¼1 ðwci Þjp is defined as: N X
hðwci Þjp i ¼
p¼1
N X
¼
N ZZZ X
ðwiV wi Þfip2 ðrYi ; cYi ; rYp ; cYp Þ
p¼1 Y
Y
dip2 ðcYip d k ip Þdk ip dcYi dcYp
ðA4Þ
For impending collisions, the integrals in Eq. (A4) must be carried out only for values of cYip kYip N0. fip2 ¼ fip2 cYi ; Y ri ; Y cp ; Y rp is the complete pair distribution function defined as the probability of finding, at time t, two particles i and p such that they are centered on rYi and rYp and having velocities within the range cYi ; cYi þ dcYi and 2 cYp ; cYp þ dcYp . Following the assumption of chaos [3], f ip can be written as the product of the single velocity distributions, f i1, f p1 and the pair-distribution function at contact, g ip (e i , e p ). dip Y Y Y dip k ;cp ;r þ fip2 cYi ;rY 2 2 d dip ip Y k fp1 cYp ;rY þ ¼ gip ei ;ep Þfi1 cYi ;rY ðA5Þ 2 2 Y Y where cY ip ¼ ci cp is the relative instantaneous velocity. di þ dp dip ¼ 2 Y k ip is the unit vector connecting the centers of the two particles, located at rYi and rYp respectively and is directed from i to p. Y Y In the remaining text, we consider k ip ¼ k gip ðei ; ep Þ ¼ ½dp gii ðei ; ep Þ þ di gpp ðei ; ep Þ =2dip ðA6Þ
where gii ðei ; ep Þ ¼ 1=ð1 ðei þ ep Þ=emax Þ þ 3di =2
N X
The expression of g pp (e i , e p ) is obtained by interchanging the indices i and p. collisional rate of production per unit of volume, PThe N ðhw iV wi iÞjp , was evaluated in [17] p¼1
p¼1
N X p¼1
ð jd vcip þ ccip Þ
ðA9Þ
Y
ðcYip d k Þdk dcYi dcYp ðA10Þ Here v cip and c cip are the collisional fluxes and sources, respectively. Substituting the above expressions (A9) and (A10) into the equation of change (A3), the continuity, momentum and fluctuating energy equations were obtained for w i equal to m i , mi cYi and 12 mi ci2 , respectively. Continuity equation The continuity equation for the solid phase i may be written as: Bei qi þ jd ðei qi Y vi Þ ¼ 0 ðA11Þ Bt e i q i =n i m i is the phase i density, e i is its solid volume fraction and q i is its material density. Momentum equation The momentum equation for phase i may be expressed as: X N N X D Y ei q Y Y ¯ ¯ ei qi ðvi Þ þ Pcip þ Pki i F iext ¼ F Dip Dt m i p¼1 p¼1 ðA12Þ D ð Þ is the material derivative with Dt Y Y P¯ ki ¼ qi ei hC i C i i is the kinetic pressure tensor ¯
Pcip ¼
vcip ðmi cYi Þ
is the collisional pressure tensor
ðA13Þ ðA14Þ ðA15Þ
Y
F Dip ¼ ccip ðmi cYi Þ is the collisional momentum source
Fluctuating energy equation ðA7Þ
hwiV wi ijp ¼
ZZZ dip3 ðwiV wi Þfip2 vcip ¼ 2 dip Y Y Y dip Y cYi ; rY k ; cp ; r þ k 2 2 Y Y cYip k Þdk dcYi dcYp ZZZ 2 ðwiV wi Þfip2 ccip ¼ dip dip Y Y Y dip k ; cp ; r þ cYi ; rY 2 2
ðA16Þ ep =dp
p¼1
N X
where
Y
hwiV wi ijp
p¼1
123
ðA8Þ
The fluctuating energy equation for solid phase i may be expressed as: N N X X 3 ei qi Dhi Y þ jd qYki þ qcip P¯ ki þ P¯ cip : jvYi 2 mi Dt p¼1 p¼1 ¼
N X p¼1
Y
ðNip Y vi F Dip Þ
ðA17Þ
124
H. Iddir et al. / Powder Technology 151 (2005) 117–125
where 1 Y Y hi ¼ mi hC i C i i ðA18Þ 3 h i is the fluctuating granular energy of the solid phase i Y qki
Y
¼ qi ei hC i Ci2 i is the kinetic heat flux
qYcip ¼ vcip ðmi c2i Þ is the collisional energy flux
Momentum interaction mi mp mi mp 3=2 gip 1 þ eip np ni 4 m0 hi hp pffiffiffi Y Y dip p ni 3 hi pR2 d vp vi þ R0 jln jln 12 np 2 hp dip p mp jhp 5mi mp mi jhi þ R10 þ 16 192m20 h2p h2i mijhp mpjhi 5mi mp jhp jhi Bip 2 þ 2 R4 R3 þ 96m0 h2p h2i hp hi
dip2 Y FDip ¼
ðA19Þ ðA20Þ
Nip ¼ ccip ðmi c2i Þ is the collisional energy dissipation ðA21Þ In the above governing equations, the relevant variables describing the flow field are the average velocities, the solid volume fractions and the granular temperatures evaluated at location Y r of the center of the particle at time t. The kinetic equations that characterize the flow of multiphase system are: B FY þY c þ iext d j > Y c fi 1 Bt mi N d2 Z Z X ip Y ½gip ðrY ; Y ¼ r þ k d ip =2Þfi1 ðcYi1 ; Y r ; tÞfi1 ðA22Þ 4 p¼1
Y
Y
Y Y Y 1 ðcY i2 ; r þ k dip =2; tÞ gip ðr ; r k d ip =2Þfp Y Y Y ðcYp1 ; Y r ; tÞfp1 ðcYp2 ; Y r þ k d ip =2; tÞ Y cip d k ÞdcYp dk
where gip ðY r Þ is the spatial pair-distribution function when the particles i and p are in contact. A solution of Eq. (A22) near the equilibrium was obtained using Chapman–Enskog method [18]. fi1 ¼ fi 0 ð1 þ /i Þ where f i 0 is the Maxwellian velocity distribution 3=2 mi mi Ci2 f i 0 ¼ ni exp 2phi 2hi
ðA23Þ
Energy dissipation
Nip ¼
mi mp 3=2 mi mp 3 2 dip eip þ 1 gip ni np 4 hi hp m0 Y m jd v pffiffiffi 3dip p mi jd Y vi p p Bip pd R5 þ d R10 40 hp hi mp vp dip p mi mp jd Y jd Y vi þ Bip þ R4 þ eip 1 4 m0 hp hi m0 pffiffiffi Y Y dip p mp jd vp p mi jd vi d R1 þ Bip d R4 6 16 hp hi vp dip p mi mp jd Y jd Y vi þ R3 ðA27Þ 48 m0 hp hi
Collisional heat flux ðA24Þ
/ i is a perturbation to the Maxwellian velocity distribution. It is a linear function of the first derivative of n i , h i and vY i . Note that / i is a function of the phase mean velocity, vY i , and not the total flow velocity, because as mentioned in the introduction each kind of particles is treated as a separate phase and the interaction is at the interface. After solving Eq. (A22) [17] for each phase, we obtain the constitutive equations: Particulate phase pressure tensor P¯ ip ¼
ðA26Þ
dip3 mi mp mi mp 3=2 gip ni np ð1 þ eip Þ 48 m0 hi hp s vp pffiffiffi mi mp 1 6 j˙ s Y j˙ Y vi ¯ pR0 I pdip þ m0 3 5 hp hi Y 2 jd vp jd Y vi ¯ 5 4 þ þ I R1 2 1þ ð1þeÞgii ei 5 hp hi 8di gii rffiffiffiffiffiffiffiffiffi mi hi ˙ Y j svi þ ni hi I¯ ðA25Þ p
mp mi 3=2 mp mi 1 þ eip ni np gip qYcip0 ¼ dip3 2m0 hi hp pffiffiffi dip p np Bip 3 hi þ j ln Bip j ln R5 þ 2 16 ni hp 8 mpjhp mijhi mp mi mijhp mpjhi 2 R6 þBip 2 m0 h2p hi h2p h2i mp mi jhp mp jhi 1 eip R7 þ R9 þ 2 2 2m0 8m0 hp hi dip p np 3 hi 3p Y Y v vp ÞR0 R1 þ jln þ jln 6 2 10 i ni hp pffiffiffi pffiffiffi dip p mp jhp dip p m2i mi jhi þ þ R 9 8 2 m20 h2p h2i pffiffiffi mp mi mi jhp mp jhi R8 þ dip pBip 2 2 m0 hp hi 2 jhp jhi 75 3 2 þ 2 R7 1þ ð1þeÞ gii ei 5 64di2 gii h2p hi sffiffiffiffiffiffiffiffi hi ðA28Þ jhi mi p
H. Iddir et al. / Powder Technology 151 (2005) 117–125
Where e ip is the normal restitution coefficient between particles i and p mi mp hp hi m i hp þ m p hi Aip ¼ ; Bip ¼ ; 2hi hp 2m0 hi hp m i m p m p hp þ m i hi ðA29Þ Dip ¼ 2m20 hi hp R0 ¼
R1 ¼
R2 ¼
R3 ¼
R4 ¼
R5 ¼
R6 ¼
R7 ¼
R8 ¼
R9 ¼
1 3=2
5=2
Aip Dip 1
3=2 Aip Dip3
þ
1 3=2 2Aip Dip2
1 3=2
5=2
7=2
1 5=2
Aip Dip3 1 7=2 Aip Dip3
þ
3 5=2
2Aip Dip4 1 3=2
2Aip Dip4 1 5=2
Aip Dip3
R10 ¼
1 5=2
7=2
5=2
35Bip2 7=2
9=2
2Aip Dip 5Bip2 7=2
Aip Dip4 7Bip2 9=2 Aip Dip4
þ
þ
10Bip2 7=2
Aip Dip5 6Bip2 0
þ
5=2
Aip Dip5 15Bip2 7=2
Aip Dip4
þ
25Bip2 7=2
9=2
11=2
8Aip Dip
9=2
25Bip4 Aip Dip6 70Bip4 9=2
Aip Dip5
ðA33Þ
ðA34Þ
[10] [11] [12] [13] [14] [15] [16] [17]
ðA35Þ [18]
Aip Dip6 7=2
þ: : :
þ: : :
35Bip4
þ 7=2
2Aip Dip
[8]
þ: : : 11=2
441Bip4
126Bip4 11=2 5Aip Dip5
þ
[6]
[9]
7=2
9=2
[1] [2] [3] [4]
ðA32Þ
8Aip Dip
Aip Dip5
þ
ðA31Þ
315Bip4
14Bip4
References
[5]
þ: : :
7=2 2Aip Dip4
þ
ðA30Þ
We have showed in [17] that the series in the above expressions converge after the second term.
[7]
15Bip4
þ 9=2
þ: : :
þ: : :
7=2 2Aip Dip5
þ
9=2
8Aip Dip
30Bip4
2Aip Dip
þ 5=2
Aip Dip
175Bip4
þ
21Bip2
þ
þ
þ
5=2 Aip Dip3
þ
þ
7=2
3Bip2
þ
Aip Dip
Aip Dip
5=2
2Aip Dip
9Bip2 5=2 4 Aip Dip
þ 7=2
3
15Bip2
þ
þ: : :
ðA36Þ
[19] [20] [21]
þ: : :
ðA37Þ
þ: : :
ðA38Þ
[22] [23] [24] [25]
[26]
þ: : :
1225Bip4 9=2
9=2
24Aip Dip
ðA39Þ
þ: : :
ðA40Þ
125
[27]
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