Ind. Eng. Chem. Res. 2007, 46, 1333-1337
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GENERAL RESEARCH Effects of Interparticle Forces on the Particle Pressure Wei Wang* Institute of Process Engineering, Chinese Academy of Sciences, P.O. Box 353, Beijing 100080, China
Interparticle forces are important to the particle pressure. Existent models usually adopt the coefficient of restitution to account for the interactions among particles without necessary distinctions in physics. A model is established herein to account for the effect of interparticle forces by using the concept of the Boltzmann distribution with consideration of many-body influences. The predicted particle pressure is compared with reported data in the literature and used to explain stability phenomena in fluidized systems. Introduction
between the KTGF estimation and experimental evidence is suppressed in particle-fluid two-phase flows or fluidized beds.
Granular flows experience two kinds of interactions among particles: collisional/frictional interactions exerted at direct contact and pairwise potentials acting over a certain distance. The energy dissipation through direct contact is taken into account by using the coefficient of restitution in both the kinetic theory of granular flow (KTGF)1-5 and the so-called hard-sphere approach,6 while pairwise potentials beyond direct contact can be incorporated into the hard-sphere model in the form of an effective restitution coefficient.7 The so-called soft-sphere model takes all kinds of interactions as virtual combination of spring, slide, and dash, etc.8,9 All of these interactions, no matter what their physical grounds are, transform the kinetic energy of particles into heat as a criterion for energy dissipation, e.g., inelastic collisions waste the energy through viscous stress, while cohesive forces dissipate the energy through break of energy bonds.10 Interparticle forces originate essentially from intermolecular (or interatom) forces. Derjaguin approximation11 bridges the trans-scale gap between molecules and macroscopic particles with the assumption of additivity, which does not, however, hold for van der Waals forces. For dense particle-fluid twophase flows, the influence of neighboring particles or the manybody effect is important not only to the nonadditive van der Waals force but also to the other additive interactions such as electrostatic forces. As a result, the pairwise potential is dependent on local concentration and structures. A promising solution to the many-body problems is to adopt the mean force, which is pioneered by Gidaspow and Lu12 for application in a gas-solid two-phase flow riser. An experimental correlation was therein proposed to explain the deviation between the KTGF predictions and measured data on the particle pressure. Their approach was further used to account for the concentrated solids adjacent to the wall.13 Moreover, Koch and Hill14 pointed out that the many-body interactions exist even in the dilute limit when studying sedimentations. This article attempts to provide a statistical solution to the effects of interparticle forces on the particle pressure with consideration of many-body influences by which the deviation * To whom correspondence should be addressed. Tel.: +86-1082616050. Fax: +86-10-62558065. E-mail:
[email protected].
Model and Analysis Both fluid-particle interphase forces and interparticle forces influence the spatial distribution and lead to local heterogeneity of particles. Correlating the density distribution with the mean potential gives the well-known Boltzmann distribution as follows
F ) F0 exp(-Emean/Ethermal) where F is the local density of the solid phase, F0 is the bulk density of the solid phase, Emean is the potential of mean force, Ethermal, denoting the thermal or fluctuating part of energy, is KT for molecules (K, Boltzmann’s constant; T, the absolute temperature), and mΘ for macroscopic particles (m, the average mass of a particle; Θ, the granular temperature). The Boltzmann distribution is widely used to study the equilibrium behavior with appropriate selection of the mean potentials, which are, e.g., the chemical potential for molecular or colloidal systems,15 the free energy for crystal nucleation,16 and the gravitational potential for the barometric distribution law.17 Moreover, the interactions resulting from the Boltzmann distribution are related to structure formation and phase separation, etc., for colloids featuring an upper limit of 1 µm to coarse clay and even FCC particles.12,18-23 The key delimitation between macroscopic particles and colloidal particles is distinguished as the random motion (thermal motion) within a comparable time scale.17 It is found that only the fluctuating part of the gravitational energy drives the thermalization and diffusive motion of settling glass beads.24 The particles in a fluidized system, being floated to overcome the gravitational effects and thermalized to diffuse randomly, behave in a manner analogous to colloidal particles in dynamics. Though the effective temperature is hard to be isotropic, even gravity is balanced by drag force, the Boltzmann distribution could still be assumed as a first approximation to the real distribution. As the radial distribution function (RDF) g0 is defined as g0 ) F/F0, the Boltzmann distribution for fluidized particles can be rearranged to the following form
10.1021/ie0514197 CCC: $37.00 © 2007 American Chemical Society Published on Web 01/13/2007
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Ind. Eng. Chem. Res., Vol. 46, No. 4, 2007
E(r) )
{
r < dp ∞, -mΘ ln g0(r), r gdp
(1)
where dp denotes the average diameter of particles. This mean potential E(r) can be related to a hard-sphere potential EHS as follows
E(r) ) EHS(r) + E′(r)
(2)
{
(3)
where
EHS(r) ) E′(r) )
{
∞, r < dp 0, r g dp
r < dp 0, -m Θln g0(r), r gdp
(4)
According to the molecular thermodynamics,25,26 the particle pressure with such a mean potential is defined as follows
2 β p ) 1 - nπβ n s 3
dE(r) dr dr
∫0∞ g0(r)r3
[
ps ) RsFpΘ 1 + 4Rsg0(dp) +
1 2
∫d
p
dg0 4 nπr3 dr 3 dr
[
1 2
∫dad
p
p
1 nπdp3 6
(
3
(4/3)πr
]
(6)
) ]
According to the mean value theorem
[
ps ) RsFpΘ 1 + 4Rsg0(dp) +
4ξ3 dp3
∫dad
p
p
Rs
dg0 dr dr
]
(8)
where dp e ξ e adp. Here the correlation diameter ξ corresponds to the functioning range of interparticle forces. Readjusting this particle pressure to a function of solids volume fraction and integrating it by parts we get a particle pressure that is normalized by its kinetic part as follows
ps* )
ps ) 1 + 4Rsg0 + RsFpΘ kinetic collisional 0 4ξ3 (-Rsg0 - R g0dRs) (9) 3 s dp cohesive
∫
The right-hand side of eq 9 consists of three parts, which correspond to the kinetic, collisional, and cohesive contributions, respectively. This relation of the particle pressure is equivalent to that in the KTGF for elastic particles if only the hard-sphere potential is considered. The energy dissipation, which is described by the coefficient of restitution in the KTGF for
(10)
where Rsm denotes the maximum solid-phase volume fraction at packed state, say 0.636, we obtain the final relation of the particle pressure with consideration of many-body interactions of which the cohesive part reads
()
ps,cohesive ξ 3 [4Rsg0 + 6(RsRsm)1/3(Rs1/3 + 2Rsm1/3) ) RsFpΘ dp 12Rsmlng0] (11)
Figure 1 gives a comparison of this relation, in which ξ is approximated to be the average particle diameter dp, to the experimental correlation of Gidaspow and Lu,12 which reads
-
dg0 dr (7) dr
(1/6)πdp3
g0 ) [1 - (Rs/Rsm)1/3]-1
-
Assuming that the RDF approaches unity a times the particle diameter away, i.e., g0(adp))1
ps ) RsFpΘ 1 + 4Rsg0(dp) +
inelastic collisions, now takes the form of cohesive interactions. Assume Bagnold’s expression27 for the RDF, which reads
(5)
with β ) (mΘ)-1, nm ) RsFp, n being the number density of particles, Rs being the solid-phase volume fraction, and Fp denoting the density of particles. The particle pressure ps can be expressed as a function of RDF after substitution of eqs 1-4 into eq 5 as follows ∞
Figure 1. Cohesive particle pressure ps,cohesive (scaled with RsFpΘ) as a function of solid volume fraction.
ps,cohesive ) 0.73Rs + 8.957Rs2 RsFpΘ
(12)
Good agreement can be found over the measured range of concentration, i.e., 0 < Rs < 0.25, beyond which eq 11 predicts a stronger cohesive effect. Since a many-body effect is expected to be more significant for more concentrated flow, our prediction probably captures the right mechanisms although it needs more direct evidence from measurements. Depending on the relative dominance of the mean potential and fluctuation energy, the many-body effect could work as a repulsive or attractive factor to stabilize particle homogeneity or cause aggregation. Figure 2 describes the radial variation of -ln g0, the mean potential normalized by a particle fluctuating energy mΘ. The RDF value is cited from published data of hard-sphere fluid,28 which is a first approximation to real granular media in fluidized systems. For this hard-sphere approximation the structural barrier, shown as the first peak at 1 < r/dp < 2, is weak compared to their fluctuating energy mΘ since |ln g0| < 1 for all concentrations. Thus, the mean interaction appears cohesive and favoring agglomeration. The depth of the attractive well at contact further determines the agglomeration behavior. For comparative dilute flow, say Rs < 0.314 as shown in Figure 2, the cohesive well is not deep enough to capture particles with fluctuating energy mΘ. As a result, particles getting across the structural barrier will also easily break loose the cohesion well due to their fluctuating nature. For dense flow, say Rs > 0.314, however, the attractive well depth exceeds the particle fluctuating energy, not allowing the
Ind. Eng. Chem. Res., Vol. 46, No. 4, 2007 1335
Figure 2. Variation of the mean potential E(r) normalized by mΘ as a function of reduced radial position (r/dp).
Figure 4. Effect of correlation diameter ξ on the particle pressure.
Figure 5. Effect of coefficient of restitution on the particle pressure. Figure 3. Particle pressure (scaled with RsFpΘ) under different correlation diameter ξ.
trapped particles to flee. This phenomenon coincides with common sense that high-velocity fluidization sees more dynamic evolution of particle clusters. Figure 3 clearly shows different variation tendencies of the dimensionless particle pressure ps* with varying ξ. Cohesive interaction with ξ > dp results in a region with dps*/dRs < 0. The negative region appreciably broadens as ξ increases, while the dilute end varies little. Thus, the many-body effects mainly affect the concentrated region. In the region with negative bulk modulus of elasticity (dps/ dRs < 0), a particulate medium would be eventually unstable in response to concentration perturbations. As a result, an initially uniform state of particles would finally give place to a heterogeneous distribution featuring a sequence of randomly intermittent patterns of various concentrations such as that in bubbling fluidization. In fact, Rietma29 has put forward such a notion on an intuitive ground and ascribed the flow transition to some interparticle cohesion. Buyevich and Kapbasov10 also cited the works of Goldstik that the particulate pressure with a maximum at some value of the concentration reflects potential existence of two phase states of the dispersions, the gas-like and liquid-like ones, which are associated with regions where bulk modulus is positive and negative, respectively. Thus, the broadening negative region as shown in Figure 3 seems to coincide with general knowledge that the solids volume fraction at incipient fluidization for fine particles is lower than that for coarse particles30 as finer particle usually bear stronger interparticle forces or bigger values of ξ. To cover the whole range of volume fraction, a frictional particle relation,31 which is actually correlated from experimental data32 and in the form of a function of solid volume fraction, ps
Figure 6. Comparison of model prediction (line) of the particle pressure in a gas-fluidized bed with Campbell and Wang’s data (symbols, ξ )1.07dp).
) Fpf(Rs), is introduced to account for the state near close packing. Coactions of cohesive interactions and dry frictions result in a behavior resembling that in a vapor-liquid system as depicted in Figure 4, where the interparticle potentials in granular media correspond to the role of the van der Waals forces in fluid. The analogy between these two stability phenomena, however, demands directed measurements, especially focusing on interparticle forces and their influences, which is however beyond the topic of the present paper. For comparison, the prediction of the KTGF for the particle pressure4 is provided in Figure 5. Since it ignores the effects of the interparticle forces, the KTGF misses the region with negative bulk modulus of elasticity for a given value of the granular temperature. Figures 6 and 7 give more verification of the model from its qualitative prediction of the particle pressure in a gas-fluidized bed32 and the collisional particle pressure in a liquid-fluidized
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F ) local density of solid phase, kg/m3 F0 ) bulk density of solid phase, kg/m3 Fp ) particle density, kg/m3 ξ ) correlation diameter, m Θ ) granular temperature, m2/s2 Acknowledgment This work was supported by the National Natural Science Foundation of China under Grant Nos. 20606033, 20490201, and 20221603 and the Chinese Academy of Sciences. Literature Cited Figure 7. Comparison of model prediction (line) of the collisional particle pressure ps,c in a liquid-fluidized bed with Zenit et al.’s data (symbols, ξ )1.07dp).
bed.33 The experimental data are usually provided without distinction of the granular temperature,32-34 of which the anisotropy characterization and measurement remains a challenge to a great extent.35-37 As a result, calculation of the particle pressure is based on estimated magnitudes instead of a model for the granular temperature. Our model for the particle pressure captures the regions with negative bulk modulus of elasticity near the packed state in both data. The data from a gas-fluidized bed cover the range from Θ ) 0.01 to 0.1 m2/s2, since the particles fluctuation are expected to increase with higher gas flow rate and smaller solid fraction. The correlated values of Θ match the area of empirical data38 in gas-fluidized beds. The variation of collisional particle pressure ps,c in a liquid-fluidized bed almost parallels the prediction of our model in the absence of contribution of the kinetic and the frictional parts of the particle pressure. This situation may imply that variation of the particle fluctuation in a liquid-fluidized experiment is not as intensive as that in a gas-fluidized bed. Conclusion A model is established to account for the effect of interparticle forces on the particle pressure with consideration of many-body influences. The predicted cohesive effect is found in coincidence with experimental correlation for FCC particles over measurement range of concentration, beyond which the model predicts a stronger cohesion, leading to a region with negative bulk modulus and instability. This model may allow an alternative by introducing a correlation diameter ξ to take the place of the current coefficient of restitution, although its application needs further verification. Nomenclature dp ) average diameter of particles, m E ) potential of mean force, kg‚m2/s2 Emean ) potential of mean force, kg‚m2/s2 Ethermal ) thermal energy of particles, kg‚m2/s2 g0 ) radial distribution function m ) average mass of a particle, kg n ) number density ps ) particle pressure, Pa ps,cohesive ) cohesive particle pressure, Pa ps,c ) collisional particle pressure, Pa r ) radial distance, m T ) absolute temperature, K Rs ) solid-phase volume fraction Rsm ) maximum solid-phase volume fraction
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ReceiVed for reView December 21, 2005 ReVised manuscript receiVed November 28, 2006 Accepted November 28, 2006 IE0514197
