Particulate Stresses in Dense Disperse Flow - Industrial & Engineering

Ind. Eng. Chem. Res. 1999, 38, 731-743

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Particulate Stresses in Dense Disperse Flow Yu. A. Buyevich† Center for Risk Studies and Safety, Department of Chemical & Nuclear Engineering, University of California, 6740 Cortona Drive, Santa Barbara, California 93117

This paper reviews and discusses the progress as achieved to date in the matter of describing particulate stresses of different origins that occur in concentrated suspensions of solid particles. A few physical mechanisms of generating such stresses are explicitly recognized. First, these stresses appear as a result of direct momentum transport over a transient network of particles separated by thin lubricating films of the intervening fluid, so that a part of the apparent suspension viscosity must be attributed to its dispersed phase. Second, normal and tangential particulate stresses originate because of random fluctuations of particles caused by (1) their pair interactions as the particles are brought closer together and pass one another in mean shear flow, (2) the relative fluid flow and an external body force field as they interact with random fluctuations of suspension concentrations, and (3) random macroscopic flow patterns, such as bubbles rising in fluidized beds that produce a system of Reynolds-like stresses. The theoretical predictions pertaining to all these mechanisms are shown to be in good keeping with experimental evidence available for suspension shear flow and fluidized beds. Introduction Apart from conventional Reynolds-like stresses that occur in turbulent flow of relatively dilute dispersions, mean flow of the dispersed phase is conditioned by specific particulate stresses which are known to because of random fluctuations of suspended particles and direct particle-particle contact interactions. These stresses play a major role in the development of a spatial particle distribution in suspension flow, and as such, they exert a decisive influence on averaged flow properties. Suffice it to say that it is the particulate stresses that make possible many shear- and pressure-driven disperse flows observed in practice. A convincing example is presented by steady unidirectional laminar flow in an inclined channel of a suspension whose particles are, say, heavier than the ambient fluid. It is quite evident that the particles will eventually settle down under the action of an effective gravity-buoyancy body force to form a close-packed sediment, either motionless or slowly moving, with the pure fluid flowing above it and filtering through the sediment. Thus, contrary to numerous observations, there will be no steady suspension flow at all, unless the said force is compensated for by a surface force that comes from a system of specific particulate stresses affecting the mean flow of the dispersed phase. Although the particulate stresses are justifiably expected to be of primary importance for highly concentrated suspensions, their impact often proves to be considerable, and accordingly cannot be ignored, even in dilute dispersions. Such situations are well-exemplified by turbulent riser flows of dilute gas-solid mixtures that are extensively used in refineries to effect the catalytic cracking of gas oil. Indeed, no matter how low the mean particle concentration may be, a thin layer inevitably originates near the riser wall in which the local concentration is sufficiently large to render par†

Deceased. Please address all correspondence to Dr. Donald R. Paul, Department of Chemical Engineering, University of Texas at Austin, Austin TX 78712-1062.

ticulate stresses to be comparable to the turbulent Reynolds stresses that act in a dilute core region outside this layer.1-3 The urgent necessity to account for the particulate stresses in continuum field conservation equations that govern disperse flow provided a compelling incentive to formulate appropriate constitutive laws, and also to warrant a comprehensive description of individual flows by means of a suitable, albeit sometimes arbitrary, choice of various adjustable parameters involved in such constitutive laws. Nonetheless, as has been repeatedly expressed and substantiated by Jackson,4,5 there remains a fundamental uncertainty of whether the commonly employed constitutive laws correctly capture the underlying physics of the particle-particle interaction in disperse systems, and whether solutions to the field equations when closed with these laws ensure an adequate description of real flows encountered in practice. Thus, it is really “our understanding of the proper form for these equations” that is likely to be a major limiting factor in assessing basic properties featured by disperse flows.5 There exists a continual tendency in the literature to define the particulate stresses in disperse flow in practically the same way as it has been done in the wellelaborated mechanics of granular flow (there are many representative examples of numerous theoretical works in this field,6-9 whereas a recent review of some constitutive laws proposed for engineering applications is to be found in Gidaspow10). In particular, the modeling of disperse flow on the granular mechanical basis is extensively used even with respect to turbulent riser flow of dilute gas-solid mixtures1-3 and fast or circulating fluidized beds.11,12 Such an approach completely ignores, however, a crucial difference in physical factors that initially produce and maintain the dispersed-phase flow in various assemblages of suspended particles and in granular systems. Unlike granular flows, particles of which move because of externally applied shear stresses that are transmitted directly by means of virtual interparticle

10.1021/ie980370k CCC: $18.00 © 1999 American Chemical Society Published on Web 02/11/1999

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Ind. Eng. Chem. Res., Vol. 38, No. 3, 1999

contacts, the key cause of originating flow of suspended particles consists of their having been brought into motion by the continuous phase through the action of the drag and other constituents of the interphase interaction force. This basic difference in major physical mechanisms that are responsible for producing flows of the two indicated types implies that quite different approaches must be employed when dealing with these flows. In accord with this notion, and contrary to a great many other works on the subject, the very idea of founding a theoretical analysis of the particulate stresses upon considerations characteristic of the granular mechanics is altogether abandoned in the present paper. Instead, the analysis is built up on alternative surmises and concepts specific to the suspension fluid dynamics. For the sake of clarity, it will behoove us to start with a brief enumeration of the primary mechanisms that are in principle capable of originating specific particulate stresses, and that will be reviewed in what follows. The first mechanism to be mentioned is characteristic of any flow of a highly concentrated suspension even if particle fluctuations are completely ignored. It is wellknown that thin lubricating films of intervening fluid obtain sometimes at close contacts between contiguous particles, thus making for an additional viscous energy dissipation within these films.13 In accordance with a recently advanced suggestion,14 such interparticle contacts give rise to a stress tensor associated with a transient network of particles that obtains above a certain percolation threshold. In effect, this tensor describes some nonlocal suspension stresses that are directly transmitted by this network over a correlation length that considerably exceeds the particle size. Incidentally, this line of reasoning results in a significant conclusion that some part of the apparent viscosity of a concentrated suspension must be attributed to mean shearing motion of its dispersed phase, rather than to the mean flow of the continuous phase. Other mechanisms of the particulate stress generation owe their origin to momentum transfer carried out by random particle fluctuations as they arise in suspension flow because of different physical causes. The most evident, and also generally recognized, mechanisms are connected with particle fluctuations in turbulent disperse flow that produce a set of corresponding Reynolds stresses, and also with thermal fluctuations in colloids and Brownian dispersions that produce an effective pressure reminiscent of the osmotic pressure in molecular systems. In the present paper, suspended particles are presumed to be large enough to render their thermal fluctuations totally insignificant. Furthermore, with a mind to pay foremost attention to specific particulate stresses that originate in highly concentrated suspensions and other disperse systems, I would like to completely overlook various ramifications due to an allowance for the turbulent fluctuations. They are of considerable concern in dilute flows, but may be supposed to be effectively suppressed in dense disperse flows. Nevertheless, a few additional physical mechanisms remain that generate intensive particle fluctuations, and so contribute to developing the particulate stresses. First of all, suspended particles experience random displacements over distances of the order of particle size as particulate layers move in shear suspension flow with different velocities parallel to one another. Such displacements come about owing to interparticle interac-

tions, irrespective of whether the contact or long-range hydrodynamic interactions play a dominant role, and they induce a resultant system of specific normal particulate stresses.15 If there is no mean flow characterized by appreciable shear rates, as commonly occurs in fluidized beds, heavier particles are supported in a suspended state by an upward fluid flow. The drag force exerted by the fluid on the particles is a strongly nonlinear function of local concentration, and the effective mixture weight is dependent on the concentration as well. Therefore, random concentrational fluctuations give rise to so-called pseudoturbulent fluctuations, because of the interaction of these fluctuations with both relative fluid flow and gravity field.16 This mechanism plays a role even in cases where the relative flow is not necessarily vertical, and on the whole, it produces another set of particulate normal stresses that is complementary to the shear-induced normal stresses. Moreover, both pseudoturbulent and shear-induced fluctuations of the particles inevitably give rise to corresponding quasi-viscous tangential particulate stresses. These last stresses have much in common with the stresses generated in the flow of molecular gases by the thermal motion of molecules. Hopefully, they can, and will, be tackled in precisely the same way as the gas stresses are treated in the kinetic theory of gases, by introducing an additional effective viscosity of the dispersed phase due to particle fluctuations. At last, random fluctuations of particles can come about as a net result of occasional, but persistent, irregular flow patterns that are ever present in disperse flow. These patterns are macroscopic in the sense that their length scale is admittedly large as compared to the particle size or the mean distance separating the centers of neighboring particles, but this scale may be small relative to the overall flow characteristic dimension. The corresponding particle fluctuations make for the development of additional Reynolds-like particulate stresses that have nothing to do with the turbulent fluctuations arising in consequence of hydrodynamic instability. Familiar examples of such macroscopic flow patterns are given by bubbles that spontaneously originate in fluidized beds and that are practically devoid of particles, and also by relatively short-scale vortices and circulation flow patterns that are frequently observed in disperse flows of various types. A brief description of the particulate stresses due to all these physical mechanisms constitutes a main intended subject of this paper. Earlier works that have a bearing on this subject are indicated and discussed below, in the course of developing necessary models. As a rule, attention is focused on the physical transparency of the considerations advocated, and also on a convincing verification of theoretical inferences by comparing them to experimental data. To obtain a comparatively simple picture of particulate stresses (and to ensure transparency of presentation), a number of simplifying assumptions will be introduced and discussed where and when appropriate. Stresses Independent of Fluctuations. The goal pursued in this section is 2-fold: (1) to formulate a reliable expression for apparent shear viscosity of concentrated suspensions of identical spherical particles and (2) to separate this viscosity into two parts which have to be associated with mean flow of the continuous and dispersed phases. In dilute and moderately concentrated suspensions, the apparent shear viscosity mono-

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tonically increases when the suspension concentration grows as a result of an increase in the viscous dissipation of kinetic energy that is due to the progressive distortion of streamlines of the ambient fluid flow around suspended particles. There are many theoretical and semiempirical formulas known that aim at describing this effect. The simplest formula resulting from a self-consistent model developed as early as 1960 reads17

µ ) µ0M(φ),

M(φ) ) (1 - φ)-5/2

(1)

where M(φ ) is the relative suspension viscosity that depends on the concentration φ of particles by volume. This formula turns to that by Einstein in the dilute limit φ f 0. The apparent viscosity as defined in this equation bears upon the mean flow of the continuous phase of a suspension. Equation 1 fails, however, to account for an additional dissipation of energy in narrow gaps separating some particles in which a lubrication-type creeping flow of the intervening fluid occurs. As the suspension concentration increases, more and more neighboring particles come in close proximity to one another to form such narrow gaps, so that the total number of gaps supposedly tends to zn/2 as a hypothetical state of close packing is approached. Here, z stands for an effective “coordination number” of a sphere in the close-packed state, meaning the mean number of narrow gaps with lubricating films per sphere. An attempt to allow for the said additional dissipation was undertaken by Frenkel and Acrivos13 who decomposed the relative displacement of the contiguous spheres that are separated by a narrow gap into two components associated with the motion along their line of centers and the sliding motion in the direction normal to this line. These authors showed that the contribution to the apparent viscosity that comes from the constituent corresponding to displacements along the line of centers is inversely proportional to 1 - (φ/φ/)1/3 as φ approaches the concentration attributed to the close-packed state, φ/. At the same time, the contribution due to the latter, sliding, relative motion constituent was shown to diverge much slower, according to a -ln[1 - (φ/φ/)1/3] law. This gave a reason to neglect the latter apparent viscosity contribution as compared to the former one, and consequently, to conclude that the relative viscosity must diverge as [1 - (φ/φ/)1/3]-1. The problem of formulating the relative viscosity function for arbitrarily concentrated suspensions has recently been reexamined.18 It has been argued that actually only the sliding constituent of the relative motion of almost touching spheres that is not accompanied by changes in the specific volume of a sphere in the mixture (and accordingly, by ensuing changes in suspension concentration) has a bearing on the first viscosity coefficient, meaning the apparent shear viscosity in suspension flow. At the same time, the other constituent that implies such a change should be relevant, by its very definition, exclusively when determining the second (or bulk) viscosity coefficient, and so it must be overlooked when evaluating the shear viscosity. This assumption seems quite natural for flows at large Peclet numbers in which a stationary flow-induced configuration of particles determine the inner microscopic structure of the flowing suspension. Calculations yield then the following expression for the relative

viscosity for a highly concentrated suspension:18

M(φ) ) -C ln(1 - φ1/3/φ1/3 / ),

C ≈ zφ//2, φ/ - φ , φ/ (2)

Admittedly, this relative viscosity has relevance to the direct momentum transfer that is carried out through interparticle contacts in a transient network of particles, and as such, it has to be associated with the mean flow of the dispersed phase, but not with the mean flow of either the suspension as a whole or its continuous phase. It must be kept in mind that thin lubricating films are by no means originated at each contact of a sphere with its neighbors as the suspension concentration decreases. In dilute and moderately concentrated suspensions, most particle encounters are surely of a transient nature in the sense that they are not accompanied by contact interactions via thin films. This effect can be described, at a semiempirical level, with the help of subtracting a few first terms of a Taylor expansion of function (2) in powers of (φ/φ/)1/3 from this function itself. Allowing for the fact that the probability of the formation of a couple of particles that are separated by a narrow gap filled with a thin film of the intervening fluid should be definitely less than the frequency of sphere encounters that is proportional to φ2, we infer that at least six first Taylor expansion terms have to be subtracted. As a result, we come out with the following approximate expressions for effective shear viscosity coefficients associated with the mean motion of the two suspension phases:

µc )

µ0 (1 - φ)5/2

,

[( ) ()]

µd ) -µ0C ln 1 -

φ1/3

φ1/3 /

6

+

1 φ

∑ j)1 j

j/3

(3)

φ/

whence it follows in particular that both viscosity coefficients scale with µ0. If, and only if, shear rates characteristic of the continuous and phase-dispersed mean flow are approximately the same, so that the relative motion may be ignored as a first approximation, the effective shear viscosity pertaining to the mean flow of the suspension as a whole can be justifiably introduced and defined as µs ) µc + µd. In accordance with the introduction of the effective number z for close contacts of one sphere at which sliding lubrication flows occur, z is looked upon as a quantity that depends merely on the concentration in the close-packed state but does not depend on the shear rate. This appeals to the familiar concept of a flowinduced structure of the suspension (or of a limiting configuration of particles inherent in mean suspension flow) that is entirely independent of the intensity of particle self-diffusion. Such a situation is well-known in obtaining the limit of high Peclet numbers, based on relative viscosity γa and relevant particle self-diffusion coefficient D, where the effect of self-diffusion is negligible. Correspondingly, eq 3 must be compared to the experimental data obtained for high-Peclet-number flows, or under any other conditions favoring the appearance of the flow-induced and diffusion-independent particle configuration. Most reliable and sufficiently accurate experiments on suspension viscosity have been performed for col-

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Figure 1. Dependence on suspension concentration of relative viscosity attributed to the dispersed phase in a suspension without random fluctuations. Curve 1 shows R ) µd/µ0 as calculated from eq 3 at C ) 2; curve 2 gives R as the Carnahan-Starling approximation, χCS(φ ), to the Enskog factor, eq 6, and dots present experimental data on high-frequency colloid viscosity by different authors as collected by Cohen et al.19

loidal systems in which the Brownian self-diffusion of particles is essential. This motion tends to restore the isotropic particle configuration that is characteristic of the thermodynamically equilibrium state of a colloid. The interplay between flow and self-diffusion leads to a substantial increase in energy dissipation, so that the apparent viscosity of the concentrated colloid turns out to be noticeably higher than it would be if there were no thermal fluctuations of the particles.19 (As a matter of fact, the viscosity of colloids and suspensions diverges proportionally to (1 - φ/φ/)-2 at low Peclet numbers, that is, considerably faster than at high Peclet numbers,19,20 in general accord with Krieger’s and some other empirical formulas.21) Understandably, “low-frequency” values of the effective shear viscosity of the colloid as measured under conditions of steady and almost steady flow are in fact similar to those attained in flows with small Peclet numbers in which the actual particle configuration is more or less close to the equilibrium one. Analogously, viscosity values characteristic of flows with large Peclet numbers, in which Brownian motion does not play a significant role in developing the actual particle configuration, are to be conceived as “high-frequency” values of viscosity. Such values pertain to colloid flows induced by externally imposed oscillatory shear fields in the limit of a very high oscillation frequency, where the effect of Brownian self-diffusion is negligible. Thus, we have to compare eq 3 only with experimental evidence on such high-frequency viscosity data. Equation 3 with C ) 2 is compared to the highfrequency viscosity data as collected by Cohen et al.19 in Figure 1. A rather good agreement between theory and experiment is evident, despite the fact that the above simple model has been developed with the aid of somewhat intuitive arguments. It is worth noting in this connection that a similar model has recently been developed also for suspension flow at small Peclet numbers, or colloid low-frequency flow.18 This other model happens to be in very good keeping with data as well, and in particular, complies with the aforementioned (1 - φ/φ/)-2 divergence law for the apparent lowPeclet-number viscosity as opposed to the -ln(1 - φ1/3/ φ/1/3) law for the high-Peclet number viscosity. To conclude, we have to point out that the previously proposed model,13 the development of which was apparently motivated by a natural desire to bring theoretical predictions in conformity with experimental evidence available at that time, greatly overestimates the viscosity of highly concentrated suspensions at large Peclet numbers. At the same time, it falls short in

explaining a sharp increase in the relative viscosity in the case of small Peclet numbers, by greatly underestimating the experimentally observed proportionality to (1 - φ/φ/)-2. As a net result of the above reasoning, we have to conclude that the contributions to stresses in the mean flow of the continuous and dispersed suspension phases have to be described as stresses acting in Newtonian fluids that are characterized by dynamic shear viscosity values of µc and µd. These stresses have relevance to (1) the distortion of flow streamlines by suspended particles and (2) the occurrence of the lubricating flows at particle contacts. Both factors mentioned entail an additional viscous dissipation of energy. This inference is well-supported by experiments, and this mere fact gives sufficient grounds to recommend eq 3 for practical use. Thus, we have the following expression for viscous stresses caused by direct momentum transport through a transient network of particles that are separated by narrow gaps:

1 σˆ ′ ) 2µd E ˆ d - I1(E ˆ d)I , 3

[

]

I1(E ˆ d) ) Ed,ii

(4)

where E ˆ d is the strain rate tensor for the mean flow of ˆ d) stands for its first invariant, the dispersed phase, I1(E and I is the unit tensor. Equation 4 is incomplete in the sense that it describes only viscous stresses that arise in the flow of suspensions under the condition of invariable concentration, and it does not include a stress tensor component proportional to the first invariant of the strain rate tensor that relates to flows in which this concentration varies. Admittedly, the problem of determining this component, and of calculating a corresponding bulk viscosity coefficient, can be addressed with the help of an analysis of dissipation in narrow gaps separating pairs of contiguous spheres that accompany the relative motion along the line of centers of the spheres.13 However, to the best of my knowledge, neither theoretical conclusions nor experimental data are available for the said stress tensor component. Since it would hardly be wise to speculate on this subject when necessary data are seemingly absent, I leave the determination of the second (bulk) viscosity coefficient completely out of the account in the present paper. Nevertheless, it should be noted that the bulk viscosity that supposedly diverges as (1 - φ/φ/)-1 can probably be expected to play a considerable role in the hydrodynamics of fluidized beds just after incipient fluidization, and in particular, in providing for bed stability at superficial fluid velocities that only slightly exceed that of minimum fluidization. Stresses Due to Random Fluctuations. If suspended particles are involved in a more or less chaotic fluctuating motion, and if this motion is characterized by a correlation length scale that is of the order of magnitude of the mean distance separating the centers of neighboring particles, fluctuations of different particles may be approximately regarded as mutually independent. In this case, particulate stresses that appear because of momentum transfer have much in common with the pressure that develops as a result of the thermal motion of gas molecules. This suggests in turn that these stresses may be described, as a first approximation, in the same way as the pressure of molecular gases. In particular, this means that we may use precisely the same expressions for the osmotic pressure function as is characteristic of dense molecular


gases. This approach has been repeatedly described elsewhere.15,16,22 It yields the following equation for the tensor of particulate stresses:

σˆ ′′ ) -FpφG(φ)〈w′ / w′〉

(5)

where G(φ) is the osmotic pressure function, and where the fluctuation velocity variance tensor is introduced, an asterisk signifying the operation of dyadic multiplication. Equation 5 defines an isotropic particulate pressure due to fluctuations only in the case where the particle fluctuations are isotropic as well. Otherwise, stress tensor (5) is anisotropic, and it is not necessarily diagonal in an arbitrary Cartesian coordinate system. Nonetheless, since the velocity variance tensor is symmetric, stress tensor (5) can always be made diagonal by means of a suitable choice of coordinate axes. This is why we have the right to refer to this tensor as that of normal stresses. When dealing with colloids and other disperse systems, the osmotic pressure function is commonly formulated from the well-known approximate statistical model by Carnahan and Starling23 of assemblages composed of identical hard spheres that interact only sterically. In this case, the corresponding expressions for G(φ) and the Enskog factor (that is, the contact value of the pair distribution function) χ(φ) are as follows:

GCS(φ) )

a

1 + φ + φ2 - φ3 , (1 - φ)3 GCS(φ) - 1 1 - 0.5φ ) (6) χCS(φ) ≡ 4φ (1 - φ)3

These expressions are deficient for dispersions whose concentration exceeds approximately 0.50-0.55. This is due to the fact that the Carnahan-Starling model does not reflect the formation of ordered crystalline phases, that is, the phase transition from the liquidlike to the solidlike states with the suspension concentration increasing. To mend this deficiency, a proper singularity must be added to the quantities identified by eq 6 in a manner similar to adding a singularity to eq 1 for relative viscosity in order to get an expression for µp as specified in eq 3. This has been done earlier,18 with an allowance made for the known asymptotic representations of the osmotic pressure function and the Enskog factor.21 The final expressions read

(φ/φ/)3 , G(φ) ) GCS(φ) + 2.9 1 - φ/φ/ (φ/φ/)1/3 (7) χ(φ) ) χCS(φ) + 1.08 1 - φ/φ/ and they are illustrated in Figure 2. Equations 6 and 7 determine, in accordance with eq 5, the normal particulate stresses caused by random fluctuations of particles, granted that the corresponding velocity variance tensor is known. Apart from normal stresses, particle fluctuations also generate tangential stresses. If the particle fluctuating motion were isotropic, these tangential particulate stresses could be described as stresses acting in a Newtonian fluid, with the help of a corresponding effective viscosity which could be introduced by analogy with the kinetic theory of dense gases. Then, these

b

Figure 2. Osmotic pressure function (a) and Enskog factor (b) as functions of concentration; (1) functions resulting from the model by Carnahan and Starling,23 eq 6; (2) asymptotic representations21 G ) 2.9(1 - φ/φ/)-1 and χ ) 1.08(1 - φ/φ/)-1; (3) functions from eq 7.

tangential stresses and viscosity are to be expressed in a familiar manner such as10,22

1 σˆ ′′′ ) 2ηd E ˆ d - I1(E ˆ d)I , 3 5xπ ηd ) H(φ)Fpa〈w′i2〉1/2 ) Kχ(φ)Fpa〈w′2〉1/2 (8) 48

[

]

where 〈w′2〉 ) 3〈w′i2〉, and function H(φ) must be roughly proportional to χ(φ), so that K may be regarded approximately as a numerical coefficient independent of φ. The Quantity 〈w′2〉 is proportional to the so-called “granular temperature” introduced in a number of papers24 that has to be distinguished from the “particulate temperature” that was specified about 20 years earlier,25 and after that extensively used, as the doubled fluctuation kinetic energy per translational degree of freedom of a single particle.15,16,22,26 It is very significant that the new shear viscosity of the dispersed phase, ηd, as caused by fluctuations and identified in eq 8 has nothing in common with the pure fluid viscosity, and it scales in quite a different manner with the root mean square (RMS) fluctuation velocity of particles multiplied by the particle size and density. As we shall see later in this paper, the particle fluctuations are actually not exactly isotropic. For this reason, we have to do this with a tensor of viscosity coefficients, rather than a single scalar viscosity. Unfortunately, a reliable kinetic theory which would enable us to evaluate components of such a viscosity tensor under conditions of anisotropic fluctuating motion has not yet been developed. In such a contingency, the expression for ηd in eq 8 must be looked upon as a reasonable order of magnitude estimate of the viscosity tensor components. It should be noted that a contribution to the dispersed-phase bulk viscosity must arise because of particle fluctuations as well. The problem of

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calculating such a coefficient is left out of the account in this paper. Thus, the particulate stresses that are caused by random particle fluctuations can readily be evaluated with the aid of eqs 5-8, if the velocity variance tensor is known. The form of this tensor depends on the physical mechanism of generating the fluctuations, and we are going to consider two relevant mechanisms in the remainder of the present paper. It is worth emphasizing that all particulate stresses are treated here as functions of the particulate temperature which is regarded as a local quantity. In a more general case, the field conservation equations governing the mean flow of both phases have to be supplemented with a new transfer equation for the particle fluctuation energy, that is, for the particulate temperature, that is analogous to the heat-transfer equation in molecular systems.10,22,24 As a result, this temperature, as well as some of the particulate stresses, become in a sense nonlocal. To simplify the presentation as much as possible, no attention is paid in what follows to particulate temperature transfer phenomena. Shear-Induced Fluctuations. Shear-induced selfdiffusion of particles was intensively studied experimentally27,28 and modeled theoretically14,29-31 in connection with the migration of particles across flow streamlines and the formation of nonuniform particle distributions in suspension flow. Here, the velocity variance tensor for shear-induced fluctuations is presented in accordance with the model recently developed on the basis of a simple rational mechanics consideration.15 Allowing for the obvious fact that the RMS velocity components must scale with γa, γ standing for different mean shear rates and a being the particle radius, we conclude that the velocity variance tensor must be quadratic in the spatial derivatives of the mean velocity of the dispersed phase. Furthermore, this tensor must be dependent on those combinations of the said derivatives which have a bearing on the true deformation in the dispersed phase, but not on its rotation as a whole. Hence, it follows that the tensor under discussion must be quadratic in the components of the dispersedphase strain rate tensor. The most general representation that satisfies the above requirements involves two terms which are proportional to a convolution of the tensor of strain rates with itself and to the second invariant of this tensor. An analysis proves the RMS values of particle fluctuation velocity components in the directions of mean flow and shear, each of which is proportional to the effective collision frequency squared, to coincide between themselves, and the RMS velocity in the direction normal to the shear plane to be π/2 times smaller than any of them.15 This gives an additional relation between coefficients of proportionality in the representation mentioned. As a final result, taking into account that the collision frequency is proportional to φχ(φ), we obtain, accurately to a factor that must be independent of φ,15

〈w′ / w′〉sh ) C′φ2χ2(φ)a2[(π2/4 - 1)E ˆ d‚E ˆ d + I2(E ˆ d)I], 1 I2(E ˆ d) ) Ed,ijEd,ij (9) 2 a subscript “sh” standing for “shear-induced”. This equation closes the expression for the part of normal stresses that owes its origin to shear-induced random particle fluctuations stresses and that immediately results from eq 5.

Figure 3. Theoretical particle concentration distributions across the gap for rotational Couette flow and experimental data by Phillips et al.30 at different values of mean concentration, R being the external cylinder radius.

The developed model of shear-induced stresses has recently been meticulously checked by carefully comparing its predictions pertaining to the particle distributions that establish themselves in steady rotational Couette flow, and also in steady pressure-driven flow in a plane channel.18 In the Couette flow, only the conservation of angular momentum is significant. The particle distribution within the gap of a cylindrical Couette device was obtained by solving the field equation of momentum conservation for the suspension as a whole, in which the apparent suspension viscosity, µs ) µc + µd, as follows from eqs 2 and 3 was used, and also the field equation of momentum conservation of the dispersed phase, into which stresses (5) closed with eqs 7 and 9 were incorporated. This distribution proves to be universal in the sense that the only parameter, the mean concentration averaged over the gap, enters a functional dependence of local concentration on an appropriate nondimensional radial coordinate. In particular, this distribution turns out to be independent of constant coefficient C′. Despite the utmost simplicity of the arguments that have led to the above model, the theoretical predictions excellently agree with experimental data reported by Phillips et al.,30 as shown in Figure 3. The agreement of theoretical profiles of particle concentration, suspension velocity, and fluctuation velocity variance with data for plane channel flow is not so perfect as shown in Figure 3. An example of comparison of such theoretically evaluated profiles18 with the data reported by Lyon and Leal32 is presented in Figure 4. Again, the particle distribution profiles, as well as the velocity profiles related to the maximal velocity as reached at the channel central plane, depend only on the mean bulk concentration as a parameter. This means that unknown coefficient C′ cannot in principle be found from these profiles, in the same way as it could not be determined from experiments conducted with the rotational Couette flow. However, the fluctuation velocity variance distribution depicted in Figure 4c is dependent on C′, so that the variance profiles allow us to approximately estimate this coefficient. It has been shown that C′ ≈ 0.016.18 The fact that the model predictions agree with experimental evidence for pressure-driven channel flow somewhat poorer than that for Couette flow is probably


a

b

c

Figure 4. Profiles of mean suspension velocity, V, as related to its maximal value Vm attained at the flow central plane, particle concentration, and shear-induced fluctuation velocity variance, U2, as related to (a/H)2 Vm2 (a, b, and c, respectively) and experimental data of Lyon and Leal32 for flow of a neutrally-buoyant suspension in a plane channel having the width 2H, at the volume fraction of particles in the total volume flux of the suspension being equal 0.4; the curves in a and b do not depend on C′; the curve in c is drawn at C′ ) 0.016.

caused by the unavoidable appearance of a relative slip velocity in such flows, so that particles lag behind the ambient fluid. The slip velocity (1) inevitably induces pseudoturbulent fluctuations of suspended particles in addition to the shear-induced ones, the former fluctuations naturally contributing to particulate normal stresses, as shown below, and (2) gives rise to a lateral force that makes the particles migrate from regions with higher shear rates to those with lower shear rates, the influence of this force being especially pronounced near the channel walls. Both these factors have not been taken into account, although they are capable of affecting the concentration and velocity distributions to a considerable extent, even if the slip velocity is relatively low. As has been mentioned already, there are other models known aimed at explaining nonuniform particle distributions. These models belong to two main types: either they are founded on a diffusional approach according to which the particle distribution is described in terms of a diffusion fluxes of particles29,30 or they are

developed within the framework of the standard approach based on the field conservation and constitutive laws for a suspension.14,31 In models of the former type, a usual flux down a concentration gradient is complemented by an additional hypothetical flux that is caused by a gradient of the mean shear rate or shear stress. Models of the latter type are essentially dependent on a number of rather strong assumptions pertaining to the definition of stresses. Apparently, these models do not lead to a so convincing agreement with experimental evidence as the model developed here. What is more, most of these models involve certain concepts, and also far-reaching assumptions, about the rheological properties of suspensions which are not wholly clear. Moreover, they include a number of adjustable parameters which essentially influence the resultant concentration and velocity profiles. While a discussion of some such concepts and assumptions has been given elsewhere,15 here I wish only to stress the most important fact. One should be careful in distinguishing between the selfdiffusivity of particles caused by their shear-induced fluctuations and the coefficient of mutual diffusion of the same particles, which is not always done when building up the diffusion-type models. It is not difficult to show that not only the dependence of these coefficients on suspension concentration is by no means the same but also their characteristic scales are quite different. Thus, the self-diffusion coefficient in simple shear flow scales with γa2,28,29 whereas the mutual diffusion coefficient is proven to scale with (γ a)2τ, τ being the relaxation time of a particle in viscous flow.15 Pseudoturbulent Fluctuations. The impact of random particle fluctuations that arise in fluidized systems was first noted by Jackson33 who repeatedly emphasized their significance for both short-scale mixing processes and hydrodynamic properties of fluidized beds, including their stability. He also noted that such fluctuations bear resemblance to the thermal fluctuations of molecules of a gas, and that a particulate pressure that they generate comes about much in the same way as does the pressure in gases. Since a typical uniform fluidized bed is to be visualized as an assemblage of particles supported by an upward flow of a fluid, the mean shear motion of such an assemblage may usually be ignored. This plainly shows that flow patterns with nonzero mean shear rates, if any, must play only a subsidiary role in the uniform fluidized-bed mechanics. For this reason, one has to look first of all for another alternative physical mechanism responsible for the origination of random particle fluctuations, and this mechanism must be different from the mechanism of the generation of the shear-induced fluctuations as considered above. Such an alternative mechanism that gives rise to sonamed pseudoturbulent fluctuations had been invoked as early as in 1966,25 and it was first discussed in the English language literature in 1971.26 After that, this mechanism was thoroughly investigated in a number of papers, and in particular, it has been considered quite recently.16,22,34 The physical essence of this mechanism can be briefly summarized as follows. In an assemblage of particles supported by a flowing fluid, there appear random fluctuations of the local assemblage concentration caused by various chance reasons. On the other hand, the drag force, and also other constituents of the interphase interaction force, are nonlinear functions of the concentration. Therefore, these concentrational

738


fluctuations induce a fluctuating random force that the fluid exerts on any particle, thus violating the balance of gravity, buoyancy, and drag forces that would be specific to the macroscopically uniform assemblage with no concentrational fluctuations. Moreover, the concentrational fluctuations bring about fluctuations in an effective two-phase mixture density, so that an additional fluctuating force also arises because of the action of gravity combined with buoyancy. In a fluidized bed supported by an upward fluid flow, the aforementioned fluctuating forces accelerate particles either upward or downward, depending on a sign of concentrational fluctuations, so that vertical fluctuations of particles occur. Because of collisions between the particles and other interparticle interactions, longrange hydrodynamic interactions of the particles via random fields of the fluid local velocity and pressure being included, horizontal translational, and perhaps also rotational, degrees of freedom of the particles are also excited. Moreover, similar fluctuations originate in the supporting fluid. As a result, the energy of vertical fluctuations is partially transmitted to horizontal fluctuations, and a pseudoturbulent fluctuating motion of both phases develops in the fluidized bed. The pseudoturbulent motion of particles may be anticipated to be highly anisotropic in cases where the exchange of momentum and energy between the particles is carried out primarily by means of the long-range hydrodynamic interactions, as it happens in dilute suspensions of fine particles. However, this same motion must be expected to be almost isotropic (whereas the fluid pseudoturbulence remains nonetheless substantially anisotropic) in cases where collisions and other contact interactions play a dominant role in the said exchange, for exactly the same reasons as those that make isotropic the thermal motion of gaseous molecules. It seems to be quite natural to assume, as a first approximation, that the interparticle exchange as occurs in concentrated disperse systems such as fluidized beds (except for dilute suspensions and beds of very fine particles) is implemented by collisions. This gives a sufficient foundation for regarding the pseudoturbulent motion of particles as approximately isotropic. As a consequence, one becomes able to take advantage of powerful methods of the kinetic theory of gases when developing a corresponding model of this motion.35 This is precisely what has been done previously in a number of works when assessing the properties of the pseudoturbulent fluctuations in real fluidized beds.16,22,34 The model developed in the papers cited above is based on an analysis of a set of stochastic equations stemming from the Langevin equation for a single particle and from field equations of mass and momentum conservation of the continuous phase. Although all necessary analytical calculations, however cumbersome, can be performed in a fairly straightforward way, the resultant formulas to evaluate the RMS particle velocity and other pseudoturbulent quantities are complicated and rather unwieldy. For these reasons, and also to make a representative illustration of typical emerging mathematics, I shall list here, without further comments, merely the major general relations allowing us to calculate the RMS particle pseudoturbulent fluctuation velocity that is sorely needed to close eq 5 for the pseudoturbulent part of particulate stresses in the case of unidirectional vertical suspension flow. Fluidized beds are of course included as a particular case. Since the

pseudoturbulent fluctuations of particles are presumed isotropic, these stresses in fact reduce to a single scalar quantity, the pseudoturbulent particulate pressure. Details of this analysis can be found elsewhere.16,22,34 For fluidized beds and other vertical suspension flows where the pseudoturbulence must be axially symmetric, the RMS particle velocity is to be found from an equation

[

]

1 1 (R + β)Φu 2 2 〈w′i2〉pt ) 〈w′2〉pt ) 〈φ′ 〉 3 2 h

t2(1 - t2)

∫01 (b - t2)2 dt (10)

a subscript “pt” meaning “pseudoturbulent” and the integral being easily expressible in terms of known functions. The following notation is introduced when formulating this equation:

Φ)

(F′1 + F′2u)u + (1 - κ)g 1 + + 1-φ (F1 + F2u)u

(1 +1 F + R +1 β)FNu,

h)β+

1

N)

(F1 + F2u)u + (1 - κ)g, φ

R(R + β)F (1 + F)(2 + F)

2R R+β 1+ , h 2+F 2F2u Ff F) , κ) (11) F1 Fp b)

(

)

Functions F1 and F2 are involved in a familiar twoterm expression for the drag force, fd, that acts on one particle, that is,

κ F1(φ) ) K1(φ), τ C2κ 2a2 F2(φ) ) K2(φ), τ) (12) a 9ν0

fd ) m[F1(φ) + F2(φ)u]u,

and F ′1, F 2′ are the derivatives of F1 and F2 with respect to φ. Functions K1(φ) and K2(φ) tend to unity in the dilute limit φ f 0, and there exist a great many theoretical, empirical, and semiempirical formulas for these functions, as well as for numerical coefficient C2. In particular, the relations were accepted to make the concrete calculations that are discussed later in this paper:

K1(φ) )

1 , (1 - φ)5/2

K2(φ) )

1 , (1 - φ)1.8 C2 ) 0.168 (13)

these relations being well-supported and -confirmed by the experimental evidence, and the first one of them conforming to eq 1 for the relative viscosity of the continuous phase. Vector u designates the mean relative fluid velocity. For conditions of a steady and uniform fluidized bed in the state of mechanical equilibrium, it represents a root of a balance equation for forces experienced by an individual particle in this state, that is,

fd + m(1 - κ)g ) 0

(14)

where fd is proportional to u and also depends on


modulus u in accord with eq 12. This equation degenerates into scalar one for vertical suspension flow. Two additional quantities, R and β, enter eqs 10 and 11. These quantities are to be found from a set of two algebraic transcendental equations that (1) require the pseudoturbulent fluctuations of particles to be isotropic indeed and (2) specify the dissipation of energy at collisions.16,22 These transcendental equations are rather cumbersome, and for this reason I am going to present them in an explicit form merely for fluidized beds of sufficiently small particles in which the drag force is approximately linear relative to fluid velocity, so that functions F2(φ) and F(φ) may be equated to zero in eq 11. After a manipulation, these equations can be formulated in the following form for the case of small particles:34

S2 - 2S

2

∫01bt -dtt2 - ∫01

3R + β ) X(1 - φ)5χ(φ)〈φ′2〉

γt2 - (1 + γ)t4

[

(b - t2)2 t2(1 - t2)

∫01 (b - t2)

dt ) 0

]

1/2

dt

U (15)

where

S)

1 , RU

U)

1 9 1 + 1 + φ (R + β) , β 2 1R+β2 (16) γ) 2 R

[ (

)

]

(

)

The second equation in set (15) is greatly simplified if there is no collisional dissipation of energy, in which case the RHS of the said equation equals zero, and consequently, β ) -3R. Otherwise, roots R and β of the equations in (15) closed with eq 16 depend on an additional parameter, X, that characterizes the dissipation of energy at collisions

X)

16k 1 - κ a3g 2 ν02 9x2π κ

(17)

the coefficient k showing which part of the energy of colliding particles dissipates, on an average, at a collision.22 Note that this parameter is very sensitive to particle radius a, it is proportional to a raised to the 3rd power. All other things being equal, this parameter is approximately 3 orders of magnitude larger for gasfluidized beds than that for liquid-fluidized beds. At last, we have to define variance 〈φ ′2〉 of the concentrational fluctuations as a function of mean concentration φ. It was suggested earlier to make use of the expression for this variance that follows from the thermodynamic theory of fluctuations when supplemented by the model by Carnahan and Starling.16 Until the last year, it was impossible to check this suggestion in a more or less convincing and conclusive way because of the absence of representative reliable experimental data on pseudoturbulent fluctuations. However, such data have been recently reported in a number of works,36-39 and the theoretical inferences as follows from the above model have been carefully compared to these data.34 It has been concluded that these inferences correspond to these data surprisingly well, if another expression is used for the concentrational fluctuation variance, namely, the expression that results from the generalization of the classical Smoluchowski statistics,

originally worked out for dilute colloids, to concentrated colloids.39 This last expression reads

〈φ′2〉 ) φ2(1 - φ/φ/)

(18)

Equations 10-18 are sufficient to evaluate the RMS pseudoturbulent particle velocity, and after that, to find the pseudoturbulent particulate pressure from eq 5, under different flow conditions. The theoretical predictions resulting from the model developed turn out to agree sufficiently well with experimental evidence. By way of example, comparison of these predictions with the data by Cody et al.36,39 obtained for smooth glass spheres fluidized by argon under normal conditions is demonstrated in Figure 5. The theoretical curve that corresponds to X ) 17 seems to be in excellent agreement with the data for beds containing 297 µm spheres. When plotting the theoretical curves for beds of smaller particles, it has been taken into account that parameter X is proportional to a3, so that it must approximately equal 0.75, 2, and 6 for spheres of 105, 149, and 210 µm in diameter, respectively, if coefficient k involved in eq 17 is assumed to be independent of the sphere size. The agreement between the theory and experiments seems to be remarkably good for fluidized beds of comparatively large particles. However, it becomes progressively worse as the particle size further decreases into the Geldart A region. According to a recently advanced suggestion,41 this is probably due to the fact that stable circulation patterns arise in fluidized beds, if very fine particles are used in Cody’s experiments. As a consequence, local mean shear rates markedly deviate from zero, and the resultant shear-induced fluctuations substantially contribute to measured values of the RMS fluctuation velocity. Further evidence to this effect is exemplified by Figure 6 in which the data reported by Menon and Durian37 are shown alongside the data by Cody et al.36,39 Admittedly, circulation patterns were suppressed in Menon and Durian’s experiments, because of their geometric and other peculiarities. The theoretical curve plotted in Figure 6 agrees with their data quite well, even for beds of finer particles. At the same time, Cody’s results for the fluctuation velocity variance in fluidized beds composed of such particles are up to an order of magnitude higher that those obtained by Menon and Durian, presumably because of the contribution caused by the shear-induced fluctuations.41 The findings of the above model have also been compared34 to experimental data obtained in Zenit et al.38 for fluidized beds of large spherical particles (glass, plastic, and steel spheres of a few millimeters in diameter fluidized by water), and also for gravity-driven flow of mixtures containing glass spheres. The needed calculations for fluidized beds of large particles were performed quite similarly to those illustrated above for beds of small particles, but for F1 having been taken zero, instead of F2. Figure 7 proves that the theoretical predictions greatly underestimate measured values of the particulate pressure in a range of low and moderate concentrations. However, they yield acceptable results for higher concentrations, except perhaps for very high concentrations. In accordance with the opinion expressed earlier,34 such a discrepancy at smaller concentrations is caused by the fact that an additional Reynoldslike contribution to the overall particulate pressure has not been taken into account. This contribution is sup-

740


Figure 6. Data on pseudoturbulent fluctuation velocity variance by Cody et al.36,39 and Menon and Durian37 (open and solid circles, respectively) in fluidized beds at a superficial gas velocity twice as high as the minimum fluidization velocity; the solid curve follows from the developed model.

Figure 7. Comparison of data by Zenit et al.38 obtained for beds of large spherical particles fluidized by water with a theoretical dimensionless particulate pressure (R ) P/0.5Fput2, P standing for a sum of the pseudoturbulent pressure defined in accordance with eqs 5 and 10 and the bubble-induced pressure (22); solid lines correspond, in ascending order, to Cp ) 0.025, 0.05, 0.1, and 0.2; dashed line shows the contribution that is due exclusively to pseudoturbulent fluctuations with Cp taken to be identically equal to zero.

Figure 5. Comparison of theoretical predictions for pseudoturbulent fluctuation velocity variance, 〈w′2〉, with experimental data by Cody et al.;36,39 the data are shown by thick curves and the thinner curves depict theoretical dependence at different values of the dissipation parameter X (characters at the curves); (a) data for fluidized beds of 297 µm spheres, the dashed curve corresponding to X ) 0 at the case where 〈φ ′2〉 is taken from the thermodynamic theory of fluctuations combined with the Carnahan-Starling model. (b), (c), and (d) Fluidized beds of spheres of 210, 149, and 105 µm in diameter, respectively.

posedly due to random fluctuations of particles that inevitably accompany the propagation of chaotic voidage waves which were observed in the experiments under discussion, and which actually represents a final product of the evolution of bubbles in a space confined by the walls of a container that holds the fluidized bed. I shall return to this point in the next section where this extra particulate pressure contribution will be evaluated on the basis of an order of magnitude model for a

particulate pressure induced by fully developed bubbles that rise in a fluidized bed. Meanwhile, it should be noted that calculations of the precisely same type as those performed above can readily be carried out for more sophisticated flow situations by using the general approach characteristic of the model developed.16,34 Of special interest is perhaps vertical riser flows of a gas-solid mixture in which the particulate pseudoturbulent stresses always prove to be important within a near-wall layer of increased concentration, even if the mixture is arbitrarily dilute everywhere over a flow core region. In riser flows, the slip velocity is not always dictated by the balance of drag and gravity-buoyancy forces, as it happens in fluidized systems, and it may significantly exceed the particle terminal velocity.42 Therefore, the pseudoturbulent stresses that scale with this velocity squared (and further multiplied by particle material density) are capable of playing an even more considerable role in riser flows than they do in fluidized beds, and for this reason, they certainly must not be overlooked when treating such flows.34


Stresses Induced by Bubbles Rising in a Fluidized Bed. As has been already mentioned, various chaotic, or even more or less deterministic, macroscopic flow patterns that occur in fluidized beds and suspension flows also induce random fluctuations of particles that ultimately result in the appearance of additional Reynolds-like stresses. Such stresses take part in conditioning mean flow of the dispersed phase as well. Because of a wide variety of such patterns, ranging from truly random porosity waves to comparatively ordered objects, such as bubbles originating at fluidization, it is hardly possible to develop a unified model that would be applicable to different types of flow under different conditions. For this reason, we have to confine ourselves to making an order-of-magnitude evaluation of the said Reynolds-like stresses due to bubbles propagating in fluidized beds, in the hope that such an evaluation turns out to be of some help also in cases where particle fluctuations are caused by macroscopic flow patterns of different types. In accord with the two-phase theory of fluidization, which we accept here in the capacity of a reasonable first approximation, all fluid in excess of the amount necessary for initial fluidization goes through a fluidized bed in the form of rising bubbles that are practically devoid of particles. This means that the difference between the actual superficial fluid velocity, Q, and its value Qmf at the incipient fluidization can be evaluated as34

Q - Qmf ≈ (4π/3)R3nbUb ) CbR7/2g1/2nb

(19)

where the well-known proportionality of bubble velocity Ub to a square root of Rg has been taken into account, and Cb is a numerical coefficient. The amplitude and velocity of averaged particle displacements caused by passing bubbles must evidently scale with the bubble radius and rise velocity, that is, with R and (Rg)1/2, respectively. It is obvious that the total number of particles that experience a displacement caused by a single bubble during a unit time can be roughly estimated as a quantity that is proportional to R3n. Hence, what follows is an order of magnitude estimate for a product of the displacement velocity and the total number of particles within a unit volume that experience displacements in a unit time, (R3n)(Rg)1/2nb. Therefore, with an allowance made for eq 19, the RMS particle fluctuation velocity due to bubbles rising in the fluidized bed has to be evaluated as follows:34

σˆ ′′′′ ≈ -PI, P ) (Cp/2)mnFp(Q - Qmf)2 ) (Cp/2)φFp(Q - Qmf)2 (21) where a proper value of constant coefficient Cp does not follow from the analysis, but it must be expected to be approximately an order of magnitude less than unity.34 For fluidized beds of sufficiently large spherical particles where the drag is quadratic in the relative velocity, Q is expressible as (1 - φ )cut,ut being the particle terminal velocity. When taking c ) 2.4 in accord with the known Richardson-Zaki formula and recent experiments,38,43 we come out with the following representation of the Reynolds-like particulate pressure induced by bubbles propagating in a fluidized bed:

P ) Cpφ[(1 - φ)2.4 - (1 - φ/)2.4](Fput2/2)

(22)

It seems instructive now to compare the dimensionless pseudoturbulent pressure for fluidized beds of large particles as shown in Figure 7 to the similar quantity that results from eq 22 and that owes its origin to the bubble-induced fluctuations of particles. As has already been stated and also can be straightforwardly seen from Figure 7, the theoretically evaluated pseudoturbulent pressure turns out to be much smaller than measured values of the particulate pressure at concentrations lower than approximately 0.3.38 On the contrary, a simple calculation proves the bubble-induced particulate pressure to greatly underestimate the measured values at concentrations higher than 0.3, whereas eq 22 gives quite passable results for smaller concentrations. It is easy to see that the mentioned facts offer a natural opportunity to propose a composite semiempirical formula in compliance with which the total particulate pressure is expressed as a sum of mutually independent constituents due to the pseudoturbulent and bubbleinduced fluctuations. The resultant theoretical curves are also plotted in Figure 7, these curves differing by values of Cp used in the calculation. On the whole, these curves satisfactorily agree with experimental evidence on the particulate pressure observed in fluidized beds of large particles. The agreement between theory and experimental data displayed in Figure 7 looks acceptable, especially so in view of the fact that there is no other more or less well-founded model that would permit a concise and comprehensive description of these data in bubbling fluidized beds. Concluding Remarks

〈w′2〉1/2 bi

3

1/2

) Cw(R n)(Rg) nb/n ) (Cw/Cb)(Q - Qmf) (20)

a subscript “bi” standing for “bubble-induced” and Cw being another numerical coefficient. Of course, the above-employed semiempirical method of using a simple dimensionality consideration combined with an order of magnitude evaluation does not allow us to distinguish between bubble-induced normal stresses acting in different directions. All that we may hope to do within the framework of this method is to introduce an isotropic particulate pressure caused by particle fluctuations that accompany the propagation of bubbles in fluidized systems, which in turn permits us to determine the corresponding stresses as follows:

The main conclusion of a general nature that can be drawn from the review presented above suggests that even comparatively simple model concepts turn out to be highly sufficient in the very difficult matter of describing particulate stresses that are caused by different physical mechanisms and factors. Despite the fact that rather strong assumptions have been sometimes accepted in the above reasoning, the agreement with experimental evidence is fairly good. This is something that other, more sophisticated approaches, if any, usually fail to achieve. This agreement is excellent for the relative viscosity of non-Brownian suspensions and particulate stresses caused by shear-induced fluctuations (Figures 1, 3, and 4). It is rather good for the case of particulate pressure induced by the pseudoturbulent fluctuations in fluidized

742


beds of relatively small particles (Figures 5 and 6). However, it is practical only in the order of magnitude when the sum of pseudoturbulent and bubble-induced particulate pressure constituents for fluidized beds of large particles is concerned (Figure 7), which can nevertheless be regarded as quite passable at the given stage of development, in view of a considerable scatter in experimental data available. This proves without any trace of doubt that the simple models as reviewed above provide in fact for a sound springboard for future efforts. Indeed, the success of these models as demonstrated in the present paper is rather conclusive and certainly encouraging. As a result, the perspective of a further development of these models, the development some of which owes much to the brilliant physical intuition of Roy Jackson, as well as the possible application of these models to various technological problems in chemical engineering, looks promising, to say the least. After having said this, I have to point out nonetheless that the models under discussion considerably oversimplify the physical reality, especially in bordering cases where the interplay between different physical mechanisms that result in the origination of particle fluctuations, and also between different types of interparticle interaction, cannot be ignored. In such cases, some key assumptions, such as that of collisional interparticle exchange by momentum and energy with an ensuing inference with respect to the expected isotropy of pseudoturbulent fluctuations of suspended particles, are likely to become invalid, or at least too rough, so that these models have to be either modified or replaced by other, more elaborate models. A good example of the said oversimplification is provided by simple shear flow of dilute gas-solid suspensions that has recently been addressed theoretically, on the basis of the kinetic theory approach and numerical simulations.44 It has been shown in particular that the explicit incorporation into the theory of effects of interparticle collisions driven by both the mean shear and the particle velocity fluctuations, as well as those due to the action of the drag force, predicts the existence of multiple steady states, even under the conditions of simple shear flow. This and other similar examples prove that much remains to be done, both theoretically and experimentally, in properly assessing multifarious properties featured by seemingly simple flows of dispersions. Thus, we have to conclude that there is still very significant room for improvement of the advanced models, and that these models must be regarded not more than first, and somewhat preliminary, attempts in assessing particulate stresses in a practical acceptable way. Nomenclature a ) particle radius b ) parameter in eq 11 E ˆ d ) tensor of rates of strain for mean flow of the dispersed phase fd ) drag force Fj ) functions in eq 12 F ) 2F2u/F1 G ) osmotic pressure function g ) gravity acceleration h ) parameter in eq 11 Kj ) functions in eq 13 k ) fraction of energy dissipated at collisions M ) relative suspension viscosity

m ) particle mass N ) parameter in eq 11 n ) number concentration of particles nb ) number concentration of bubbles Q ) fluid superficial velocity Qmf ) superficial velocity at minimal fluidization P ) particulate pressure R ) bubble radius S ) parameter in eq 16 U ) parameter in eq 16 Ub ) bubble rise velocity u ) mean slip velocity w′ ) particle fluctuation velocity X ) dissipation parameter defined in eq 17 z ) number of contacts containing thin lubricating films per particle Greek Letters R ) intermediate variable introduced in eq 10, also a general notation for variables shown in some of the figures β ) intermediate variable introduced into eq 10 γ ) mean shear rate, also parameter in eq 16 ηd ) dispersed-phase viscosity due to fluctuations of particles κ ) density ratio, eq 11 µ ) dynamic viscosity µ0 ) dynamic viscosity of pure fluid ν0 ) kinematic viscosity of pure fluid F ) density σˆ ) tensor of particulate stresses τ ) particle relaxation time, eq 12 Φ ) parameter in eq 11 φ ) particle concentration by volume φ/ ) particle volume concentration on the state of close packing χ ) Enskog factor (contact value of pair distribution function) Subscripts c ) continuous phase d ) dispersed phase CS ) Carnahan-Starling model bi ) bubble-induced sh ) shear-induced pt ) pseudoturbulent

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Received for review June 1, 1998 Revised manuscript received September 8, 1998 Accepted September 16, 1998 IE980370K

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