Layer inversion and bed contraction in down-flow binary-solid liquid-fluidized beds.

Escudie, R. ; Epstein, N. ; Grace, J.R. 等

INTRODUCTION

In the last three decades, liquid fluidization technologies have received increasing attention because of the development of new applications, mainly in biochemical processing and waste water treatment (Di Felice, 1995; Epstein, 2003). Liquid-solid fluidization is usually operated with an upward liquid flow, with particles having a density higher than the liquid. In inverse (or down-flow) fluidization, the particle density is lower than the liquid density, and the bed is expanded by downward liquid flow. Inverse fluidization has been mainly applied to waste water treatment (Gonzalez et al., 1992; Meraz et al., 1995 and 1996; Garcia-Calderon et al., 1996 and 1998; Castilla et al., 2000).

A number of researchers have studied the hydrodynamics of inverse fluidized beds in terms of liquid-solid mass transfer (Nikov and Karamanev, 1991), minimum fluidization velocity (Karamanev and Nikolov, 1992b; Renganathan and Krishnaiah, 2003), pressure drop (Ulaganathan and Krishnaiah, 1996), liquid phase mixing (Renganathan and Krishnaiah, 2004) or voidage fluctuations (Renganathan and Krishnaiah, 2005). Other studies have focused on bed expansion, mainly in the prediction of the Richardson-Zaki parameters, viz., the expansion index n (Fan et al., 1982) and [U.sub.e], the value of the liquid superficial velocity U when a linear plot of log U vs. log [epsilon] is extrapolated to [epsilon] = 1 (Karamanev and Nikolov, 1992b; Garcia-Calderon et al., 1998; Renganathan and Krishnaiah, 2004). Karamanev and Nikolov (1992a) and Karamanev et al. (1996) demonstrated the existence of two regimes for the free rise of light particles, depending on the terminal free rise Reynolds number [Re.sub.t] or the related value of Ar/[d.sup.2]. For Ar/[d.sup.2] > 1.18 x [10.sup.4] [mm.sup.-2], where d is the particle diameter and Ar = ([[rho].sub.f] - [[rho].sub.s]) [[rho].sub.f] [d.sup.3]g/[u.sup.2] is the Archimedes number, the sphere follows a spiral rather than straight path, and the drag coefficient does not obey the standard curve. Hydrodynamic characteristics (bed expansion, pressure drop, minimum fluidization velocity, friction factor) have also been investigated in inverse beds fluidized by non-Newtonian fluids, e.g. aqueous solutions of carboxy-methyl cellulose (Femin Bendict et al., 1998; Vijaya Lakshmi et al., 2000).

In a liquid-solid fluidized bed, the properties of the fluidized particles (size, density, shape) are usually non-uniform, with the result that particles may segregate or mix, according to their characteristics, liquid properties and operating conditions. To simplify the understanding of classification phenomena in up-flow fluidization, the majority of studies have been performed using binary-solid mixtures, the segregation of particles being achieved by size (sizing) or density (sorting) (e.g. Pruden and Epstein, 1964; Kennedy and Bretton, 1966; Gibilaro et al., 1985; Epstein, 2005), or by shape ("shaping") (Escudie et al., 2006a). Inverse liquid-solid fluidized beds were recently applied for the separation of waste plastics for recycling: using binary mixtures of different densities (polypropylene, low density polyethylene, high density polyethylene), Hu and Calo (2005) observed the sorting phenomenon, the lower-density particles in this configuration being preferentially close to the distributor, i.e., at the top of the column.

In upward-flow fluidization, "layer inversion" appears when a mixture of two particle species is fluidized, species hS being high-density small particles and species lL being low-density large particles (i.e., where the size ratio and density ratio of the two species are on opposite sides of unity). An idealized description is as follows: at low liquid velocities, the two species form distinctly separate layers, with species hS at the bottom and species lL at the top; at high velocities, two distinct layers are again visible, but the layers are inverted, species lL and species hS being at the bottom and top, respectively. The liquid velocity at which the bed inverts is one in which the two species are homogeneously mixed. Although this over-simplified version of the phenomenon was reported for a mineral dressing process in the first half of the last century (Hancock, 1936), what actually happens was not clarified until the experiments of Moritomi et al. (1982), confirmed in subsequent experimental studies (Jean and Fan, 1986; Gibilaro et al., 1986; Matsuura and Akehata, 1985; Qian et al., 1993). It was found that the lower layer is not for most liquid velocities a mono-component layer, and that its composition depends on the liquid velocity. Only at very low superficial liquid velocities does the lower layer consist of a pure layer of high-density small particles (species hS). Starting from the liquid velocity corresponding to the first entry of a particle of species lL into the bottom layer, the fluidized bed is composed of a mixed layer at the bottom and a pure layer of species lL at the top. The concentration of species lL in the lower layer then increases with velocity until it matches the bulk composition of the overall fluidized bed. Only one layer, totally mixed, is then present: this is the inversion point. With a further increase of liquid velocity, the proportion of species lL continues to increase in the bottom mixed layer until the last particle of species hS from the lower layer enters the upper pure layer to form a clean-cut segregation between the two species, but with the layers reversed at this very high velocity compared to the first clean-cut segregation pattern at very low velocity. Thus, for some binary mixtures, layer inversions can be obtained by varying either the liquid velocity or the bulk bed composition. Many papers based on both experiments and/or modelling have by now been published to describe the inversion phenomenon. Escudie et al. (2006b) compare the capability of the several existing models to predict accurately the inversion velocity for experimental studies of relevant binary mixtures reported in the literature.

The objective of the present work is to initiate study of the layer inversion phenomenon in an inverse (downward-flow) liquid-solid fluidized bed. A simplified model, based on the difference in bulk density of each individual mono-component bed, is first developed to predict the occurrence of the inversion phenomenon, and to serve as a tool for selecting binary-solid systems for the experimental work. The inversion phenomenon is then described and analyzed.

THEORETICAL CONSIDERATIONS

Models developed to predict the layer inversion of conventional (upward-flow) liquid-fluidized beds are based on five different theoretical approaches, as summarized by Escudie et al. (2006b). One of these approaches is based on an over-simplification of the inversion phenomenon by considering only the bulk densities of mono-component beds. It is assumed that the degree of segregation (or mixing) depends only on the difference between the bulk densities of the two particle species when each is fluidized separately in the same column by the same liquid at the same liquid superficial velocity U. The inversion is then assumed to occur at that velocity where the bulk densities of the two mono-component species fluidized separately, become equal, with perfect mixing of the two particle species then occurring without any bed contraction. This theoretical approach was initiated implicitly by Van Duijn and Rietema (1982), elaborated explicitly by Epstein and LeClair (1985), and resurrected by Hu (2002). The main advantage of this approach is its simplicity, but it is unable to predict the dependence on overall bed solids composition of the inversion phenomenon, e.g. of the layer inversion velocity. Nevertheless, if the bulk density of two particle species is plotted against the liquid superficial velocity, an intersection of the two curves in the fluidization region is normally a requirement for layer inversion to occur, even if the intersection velocity is not necessarily equal to the inversion velocity.

Inverse liquid fluidization involves the downward fluidization of solids having a density smaller than that of the liquid. In the case of pure sizing, gravity and drag dictate that, as in conventional upward fluidization, the larger particles remain close to the distributor and therefore, unlike conventional fluidization, at the top of the bed; the opposite is true for the smaller particles, which congregate far from the distributor and therefore at the bottom of the bed. Here "top" and "bottom" have the colloquial meaning of further from and nearer the planet's surface, respectively. In pure sorting, gravity and buoyancy dictate that, unlike in conventional fluidization, the less dense particles remain close to the distributor and therefore, as in conventional fluidization, at the top of the bed, while the opposite is true for the denser particles. This means that layer inversion, if it is to occur, would involve a binary mixture containing relatively high-density (but less dense than the liquid) large particles (species hL) and lower-density smaller particles (species lS).

The bulk density of a mono-component fluidized bed of particle species i is given by:

[[rho].sub.Bi] = [[epsilon].sub.i] [[rho].sub.f] + (1 - [[epsilon].sub.i]) [[rho].sub.i] (1)

At the hypothetical inversion point, the bulk densities of the two mono-component beds should be equal, i.e.:

[[rho].sub.BlS] = [[rho].sub.BhL]

Combining Equations (1) and (2):

[[epsilon].sub.lS][[rho].sub.f] + (1 - [[epsilon].sub.lS]) [[rho].sub.lS] = [[epsilon].sub.hL][[rho].sub.f] + (1 - [[epsilon].sub.hL]) [[rho].sub.hL] (3)

The voidage [[epsilon].sub.i] of a mono-component fluidized bed of species i can be estimated from the correlation of Richardson and Zaki (1954):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

where U is the liquid superficial velocity of a mono-component fluidized bed of species i, [n.sub.i] is the expansion index, and [U.sub.ei] is the value of U where a linear plot of log U vs. log [epsilon] is extrapolated to [epsilon] = 1. Combining Equations (3) and (4) at the hypothetical inversion point leads to:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

Let us assume that the layer inversion phenomenon for the inverse fluidized bed is analogous to that actually observed for the conventional fluidized bed. In that case, the inversion can be depicted as shown in Figure 1. At the inversion velocity, the bed is constituted by one homogeneous mixed layer (Figure 1c). When the liquid superficial velocity is high compared to the inversion velocity ( > [U.sub.b]), the bottom layer (far from the distributor) may be pure species lS and the top layer pure species hL (Figure 1e), and vice versa when U < [U.sub.a] (Figure 1a). For U between [U.sub.a] and [U.sub.b] but not equal to the inversion velocity, the top layer is a mixture of both species, whereas only the bottom layer is one or other of the two pure species. The mono-component bottom layer is then species lS or species hL when the liquid superficial velocity is higher (Figure 1d) or lower (Figure 1b), respectively, than the inversion velocity.

EXPERIMENTAL EQUIPMENT AND MEASUREMENT TECHNIQUE

Experimental Set-Up

The experimental set-up used by Escudie et al. (2006a) for conventional liquid fluidization was adapted to this work. The column is constructed of acrylic, with an inner diameter (D) of 0.127 m and an overall height of 2.58 m. Water at 17[degrees]C is fed from a feed tank and injected at the top of the fluidization column via a centrifugal pump. The tank is equipped with baffles to calm the flow and minimize entrainment of air with the incoming water. The liquid flow rate is measured by two calibrated rotameters.

[FIGURE 1 OMITTED]

The column consists of three sections: at the top, a calming entry section filled with 25 mm plastic spherical Tri-Packs (FABCO) to homogenize the liquid flow before it reaches the liquid distributor, the test section separated from the calming section by the liquid distributor, and a disengagement section at the bottom. The water exits from the bottom of the column and is recirculated back to the storage tank through an overflow weir that maintains a constant water level. The distributor consists of a perforated plate containing 54 holes of 4.8 mm diameter on a square grid, giving a free area ratio of 7.7%.

Along the height of the column, pressure taps are located, starting 64 mm below the distributor and then at 100 mm intervals. A high-accuracy wet/wet differential pressure transducer (Omega model PX938 0.1WBD) measures the dynamic pressure drops over each interval, the pressure difference being recorded for periods of 60 s at 50 Hz. Assuming negligible wall friction and acceleration effects, the dynamic pressure gradient, - [DELTA]P/[DELTA]Z, for fluidized solids of volumetric mean density [[rho].sub.s] between adjacent pressure taps, i.e., the static pressure gradient, corrected for the hydrostatic head, is then given by:

- [DELTA]P/[DELTA]Z = (1 - [epsilon]) ([[rho].sub.s] - [[rho].sub.f]) g

from which [epsilon], the average bed voidage between successive pressure taps, can be determined.

Particle Selection

The selection of the two particle species to form a feasible binary system is based on the bulk densities (Equation (1)) of the two corresponding mono-component fluidized beds. An intersection of the two curves is a requirement for bed inversion to occur, and as the Richardson-Zaki correlation is used to estimate the voidage [epsilon].sub.i], the expansion index n and the velocity [U.sub.e] must be estimated for both species. Although several correlations have been proposed for predicting these parameters for a conventional liquid-solid fluidized bed, less work has been published for inverse liquid-solid fluidization. Karamanev and Nikolov (1992a) and Karamanev et al. (1996) investigated the free rise of a light solid sphere and showed the existence of two regimes:

1. for Ar/[d.sup.2] > 1.18 x [10.sup.4] [mm.sup.-2], the sphere follows a spiral path and the drag coefficient is constant and equal to 0.95 (instead of the free-settling value, which becomes about 0.44 in the Newton regime). The terminal velocity can then be calculated from:

[U.sub.t] = [square root of 4/3 d/0.95 ([[rho].sub.f] - [[rho].sub.s]/[[rho].sub.f]) g] (6)

2. for Ar/[d.sup.2] < 1.18 x [10.sup.4] [mm.sup.-2], the drag coefficient follows the standard curve for isolated free-settling solid spheres and the sphere motion is rectilinear.

In this second regime, the particle terminal velocity [U.sub.t] for isolated spherical particles can be estimated from the correlation of Turton and Clark (1987):

[Re.sub.t] = [Ar.sup.1/3] [[(18/[Ar.sup.2/3]).sup.0.824] + [[(0.321/[Ar.sup.1/3]).sup.0.412]].sup.-1.214] (7)

which applies up to [Re.sub.t] = 260 000 for free settling (Haider and Levenspiel, 1989), but only for regime (2) in the case of free rising, for which [Re.sub.t] = [U.sub.t][[rho].sub.f]d/[mu] is the terminal-velocity Reynolds number. A moderately good correlation proposed by Khan and Richardson (1989) corrects for column wall effects:

[U.sub.e]/[U.sub.t] = 1 - 1.15 [(d/D).sup.06] (8)

where D is the column diameter and [U.sub.t] is the single particle terminal velocity in the absence of significant wall influence, i.e., for a large column with D >> d. Equation (8) is applicable for diameter ratio d/D between 0.001 and 0.2. The assumption in Equation (8) that any deviation between [U.sub.e] and [U.sub.t] is caused entirely by wall effects is, however, at odds with some experimental data (Di Felice, 1995).

In the first regime above, Renganathan and Krishnaiah (2005) proposed a correlation for predicting the free-rising terminal velocity [U.sub.t]:

[Re.sub.t] = [Ar.sup.1/3] [[(18/[Ar.sup.2/3]).sup.0.776] + [(0.183/[Ar.sup.1/3]) .sup.0.388].sup.-1/0.776] (9)

This equation is of the same form as the Turton and Clark (1987) equation, after removing the presumed wall effect in the measured [U.sub.e] using Equation (8). It is based on the available data of Fan et al. (1982), Karamanev and Nikolov (1992b), Garcia-Calderon et al. (1998) and the authors' own experiments. This correlation (Equation (9)), together with Equation (8), was used to estimate the velocity [U.sub.e] for the selection of the particles.

Correlations for predicting the parameter n are rare for down-flow fluidization. Fan et al. (1982) proposed two empirical equations, each for a different range of the terminal particle Reynolds number (Ret), based on a limited gamut of particle characteristics:

n = 15 [Re.sub.t.sup.-0.35] [e.sup.3.9 d/D] (10)

for 350 < [Re.sub.t] < 1250 and

n = 8.6 [Re.sub.t.sup.-0.2] [e.sup.-0.75 d/D] (11)

for [Re.sub.t] > 1250. As a first assumption, the correlation of Khan and Richardson (1989), developed for conventional mono-component particulately fluidized beds of spheres, was used:

4.8 - n/n - 2.4 = 0.043 [Ar.sup.0.57)

In addition, the minimum fluidization velocity [U.sub.mf] of each mono-component bed is required. [U.sub.mf] was estimated from the well established Wen and Yu (1966) correlation recommended by Richardson (1971) for conventional, and by Renganathan and Krishnaiah (2003) for inverse fluidization of spheres:

[Re.sub.mf] = ([C.sub.1.sup.2] + [C.sub.2]Ar).sup.1/2] - [C.sub.1] (13)

where [Re.sub.mf] is the particle Reynolds number at minimum fluidization, and the parameters [C.sub.1] and [C.sub.2] equal 33.7 and 0.0408, respectively.

Three types of spherical particles, corresponding to two possible binary systems, were selected: one was polypropylene (6.35 mm in diameter and 836.5 kg/[m.sup.3] in mean density), and two species of high-density polyethylene (15.0 mm and 895.9 kg/[m.sup.3]; 19.1 mm and 903.0 kg/[m.sup.3]). The mean density of each particle species was measured by weight and volume. Using the dependence of the density of ethanol-water mixtures on their relative concentrations, the density distribution of the spheres was determined. 100 particles of each species were introduced successively in vessels containing ethanol-water mixtures of different densities, and, by their sink-fl oat behaviour, the density distribution could thus be determined. As shown in Figure 2, the distribution is narrow for the two high-density polyethylene (PE) particle species. For the lighter particle species (polypropylene, PP), the density dispersion is greater, 87% of the spheres having densities between 825.0 and 847.5 kg/[m.sup.3]. Based on estimated Richardson-Zaki parameters, [U.sub.e] by Equations (8) and (9) and n by Equation (12), and of [U.sub.mf] by Equation (13) (Table 1), the theoretical bulk densities (Equation (1)) of all three mono-component fluidized beds are plotted in Figure 3 against the liquid superficial velocity U for three particle species. The two intersections in these plots suggest that the layer inversion phenomenon might be expected when the PP particles are mixed with either PE particle species.

RESULTS AND DISCUSSION

Mono-Component Particle Systems

The three different particle species were first fluidized separately to determine their bed expansion characteristics and their Richardson-Zaki parameters. As described by Escudie et al. (2006a), the voidage of the fluidized beds can be determined by three different methods: by visual measurement of the bed height, from the value of the incremental pressure drop if it is constant over the bed height, and from a plot of the dynamic pressure (i.e., static pressure minus hydrostatic head of liquid) vs. vertical position. For a mono-component fluidized bed of spheres (Escudie et al., 2006a), the void fractions based on all three methods are in very good agreement. The method based on the dynamic pressure plot was used in our work.

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

Table 1 reports values of [U.sub.e] and n obtained by fitting our bed expansion measurements with the Richardson-Zaki equation. For the small PP particles (6.35 mm), [U.sub.e] is very close to the value estimated from the correlations developed for conventional fluidization (Equations (7) and (8)). For the large PE particles (15.0 and 19.1 mm), the correlations underestimate [U.sub.e] by about 15%. According to Karamanev et al. (1996), for free rise of a light solid sphere with Ar/[d.sup.2] > 1.18 x [10.sup.4] [mm.sup.-2], the drag coefficient does not follow the standard curve of a free settling sphere and has a constant value of 0.95: Ar/[d.sup.2] is 1.29 x [10.sup.4] [mm.sup.-2] and 1.53 x [10.sup.4] [mm.sup.-2] for the spheres of diameters 15.0 and 19.1 mm, respectively. However, assuming that the two species rise in this regime results in even greater underestimation of Ue because the assumed drag coefficient is higher. The use of the correlation recently proposed by Renganathan and Krishnaiah (2005), Equation (9), does not improve the estimations. The main reason for the poor predictions may be the diameter of the large particles compared to that of the column. The d/D ratio for these particles is about 0.12-0.15, and Equation (8) of Khan and Richardson (1989) probably does not correctly take into account such large wall effects. This explanation accounts for high predicted [U.sub.e], but not so obviously for low predicted [U.sub.e].

The experimental values of the expansion index n are in good agreement with the correlation (Equation (12)) of Khan and Richardson (1989) only for the largest particles. Equations (10) and (11) of Fan et al. (1982) developed for inverse fluidization do not improve the predictions. Therefore, since the particle selection was based on correlations that in most cases gave poor predictions of the Richardson-Zaki parameters, the experimentally measured values of voidage, and hence of bulk density via Equation (1), of the mono-component fluidized beds had to be used in order to predict the segregation-mixing behaviour of the binary systems (Figure 4). As the density distribution of species lS (mean density = 836.5 kg/[m.sup.3]) for the two binary systems is broad, the bulk density of the mono-component bed is estimated in Figure 4 based on the mean and the two extreme values of solid density (825.0 and 847.5 kg/[m.sup.3]) that encompass 87% of the PP particles. The curves corresponding to species lS intersect the curve for the 19.1 mm spheres at liquid superficial velocities between 0.072 and 0.089 m/s, but do not quite intersect the curve for the 15.0 mm particles. Hence, layer inversion (Figure 1c) can be expected in this velocity range for some compositions of the former binary, but probably not for the latter.

[FIGURE 4 OMITTED]

Binary Particle Systems

Experiments were carried out with both binary systems. As the layer inversion depends on the overall bed composition in a conventional liquid-solid fluidized bed, this parameter was also studied in our inverse fluidized beds. The liquid-free volume fraction of particle species lS in the bed is defined as:

[X.sub.lS] = [v.sub.lS]/[v.sub.lS] + [v.sub.hL] (14)

where [v.sub.lS] and [v.sub.hL] are the total volumes of species lS and hL, respectively, in the bed. For the two binary systems, [X.sub.ls] was varied from 20 to 50%. Table 2 summarizes the experimental conditions. The initial total bed mass for each experiment was also varied in order to have a total bed height at the maximum liquid velocity of about two-thirds the column height.

Segregation-mixing patterns

For both species hL diameters (15.0 mm and 19.1 mm) and for every overall bed composition investigated, the segregation/ mixing patterns were qualitatively the same. Figure 5 provides a schematic representation of the observed patterns. For the higher liquid velocities, the regime illustrated by Figure 1d, designated as incomplete segregation by Escudie et al. (2006a), was observed with a mono-component layer of species lS at the bottom of the column (far from the distributor), whereas a mixed layer of species lS and hL was observed at the top of the column, closer to the distributor. The interface between the two layers was clear and stable. When the liquid superficial velocity decreased, this interface oscillated, and some clusters of big particles descended into the pure layer of species lS, before returning to the mixed layer. The incursions intensified when U decreased until the pure layer disappeared: a gradient of solid concentration was observed, the concentration of species hL and of species lS being dominant at the top and bottom of the column, respectively. This regime is called heterogeneous mixing. Table 2 presents the liquid superficial velocity corresponding to the transition between incomplete segregation and incomplete or heterogeneous mixing. The lower (inversion) velocity corresponding to homogeneous mixing was not easy to determine and could not be detected; in fact, for some experiments a concentration gradient of species was clearly observed, even for the lowest fluidization velocities.

The main results of these observations is that the patterns for these experiments and operating conditions can be represented by Figure 1d and by a decreasing velocity approach to Figure 1c, which means that the experiments were mainly performed for liquid velocities higher than the inversion velocity. However, if we refer to the plot of bulk density [[rho].sub.B] versus U (Figure 4) that gives a simplified picture of the inversion phenomenon, we expected to observe a layer inversion for the range of liquid superficial velocities explored, particularly when the 19.1 mm spheres constituted species hL. To understand the divergence between reality and expectation, the characteristics of the mixed layer, when it exists, require analysis.

Mixed layer characteristics

As in the case of the mono-component fluidized beds, the total bed height was estimated from a plot of dynamic pressure vs. vertical position. However, it was impossible to measure accurately the interface between the two layers by this method since the slopes for the two different layers were very similar, and the number of ports available to measure the mixed layer was usually too small. Therefore, the bed height of the mixed layer was determined visually, when the interface between the two layers was stable.

Mixed layer composition

As detailed by Escudie et al. (2006a), it is possible to estimate the liquid-free composition of the mixed layer when incomplete segregation occurs. The quantity of species lS in the bottom pure layer can be estimated based on its height ([h.sub.p]), assuming that this layer has the same voidage as the mono-component fluidized bed at the same liquid velocity. The liquid-free volume fraction of particle species lS within the mixed layer, [x.sub.lS], is then:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

where [m.sub.lS] and [m.sub.hL] are the initial masses of species lS and hL in the bed, respectively, A the column cross-sectional area, and [[rho.sub.lS] and [[rho].sub.hL] the particle densities of species lS and hL, respectively.

[FIGURE 5 OMITTED]

The volume fraction of species lS in the mixed layer is reported in Figure 6 for the two binary systems, and for all initial bed compositions investigated. The value of [x.sub.lS] ranged between 6 and 9% for most of the experiments. Neither the liquid superficial velocity nor the initial bed composition had a significant influence on [x.sub.lS], whereas U had an appreciable effect in the case of conventional fluidization (e.g. Moritomi et al., 1982). However, it is necessary to emphasize that the accuracy of the estimation of [x.sub.lS] is not high, because it is based on visual measurement of the sometimes fuzzy interface between the two layers.

Visual observation of the mixed layer shows that a few spheres of species lS are located within the wake of the larger particles; most of the other small spheres are situated among the larger particles where the concentrations of the latter are lower.

Mixed layer voidage

Starting from the composition and the height ([h.sub.m]) of the mixed layer, its voidage ([[epsilon].sub.m]) was estimated from:

[[epsilon].sub.m] = ([h.sub.m] A - 1/[m.sub.hL]/1 - [x.sub.lS] [[rho].sub.hL])/[h.sub.m] A (16)

Figures 7a and 7b present the voidage of the mixed layer versus the liquid superficial velocity for the two binary systems, in addition to the voidage of each mono-component fluidized bed. For a given liquid superficial velocity, the voidage [[epsilon].sub.m] is consistently lower than the voidages of the two mono-component systems.

In the serial model of Epstein et al. (1981), the overall voidage of a solid mixture at a given liquid superficial velocity is determined by assuming that the bed behaves as if it were a series of entirely segregated mono-component layers, each at its mono-component voidage for the given velocity. In other words, this model is based on the assumption that the volume of a multi-species fluidized bed is the sum of the volumes of each mono-component bed fluidized separately in the same column at the same liquid superficial velocity. In the case of a binary bed, the serial model predicts that:

1/1 - [[epsilon].sub.m] = [x.sub.lS]/1 - [[epsilon].sub.lS] + 1 - [x.sub.lS]/1 - [[epsilon].sub.hL] (17)

where [[epsilon].sub.m] is the overall voidage of the binary-species bed, and [x.sub.lS] the fluid-free volume fraction of species lS in the mixed bed, while [[epsilon].sub.lS] and [[epsilon].sub.hL] are the voidages of the mono-component beds of species lS and hL, respectively. Consequently, if the mixed layers follow the serial model, their voidages necessarily fall between the voidages of the two corresponding mono-component beds. Figure 7 shows that the serial model is inapplicable to the present results and that bed contraction is significant.

[FIGURE 6 OMITTED]

The percentage volume contraction from the additive volume given by the serial model can be estimated by comparing the experimental voidage [[epsilon].sub.m](exp) to the voidage [[epsilon].sub.m] (serial) estimated from the serial model as follows:

Contraction = [absolute value of 1/(1 - [[epsilon].sub.m] (exp)) - 1/(1 - [[epsilon].sub.m] (serial))/1/(1 - [[epsilon].sub.m] (serial))] x 100% (18)

The contraction generally increases with increasing superficial liquid velocity, with its maximum value reaching ~12.5% (Figures 8a and 8b). It is, however, more complicated to characterize the influence of the species hL size and of the composition [x.sub.lS], especially since the precision of [x.sub.ls] manifests itself in both Equations (16) and (17) for [[epsilon].sub.m] (exp) and [[epsilon].sub.m] (serial), respectively .

Several schemes for predicting the contraction effect have been proposed in the literature. The simplest of them, that of Gibilaro et al. (1986), involves treating the expansion of a binary mixture of spheres as if it were a mono-component bed with a Sauter mean diameter and a volumetric mean density of the two particle species. Developed to estimate the composition of the mixed layer, this method of implicitly incorporating a contraction effect was used in their "complete segregation model" of the layer inversion phenomenon in conventional fluidized beds. The same method, but with a different bed expansion equation, that of Di Felice (1994), was later explicated by Epstein (2005). Other layer inversion models based on a similar approach were developed subsequently, e.g. combining the Richardson and Zaki (1954) equation with property-averaged values of the terminal velocity [U.sub.t] and index n (Asif, 1998; Escudie et al., 2006b). However, these models all have the disadvantage of making their predictions of the contraction effect at any superficial liquid velocity from mono-component voidages calculated by various theoretical, empirical or semi-empirical equations, rather than from measured mono-component voidages as in the present case.

[FIGURE 7 OMITTED]

[FIGURE 8 OMITTED]

An entirely different method, initiated by Asif (2001), applies models to liquid-fluidized beds that are based on the packed bed voidage behaviour of binary mixtures of spheres. These packing models utilize the equation of Westman (1936), written with the subscripts lS and hL as:

[(V - V.sub.lS][x.sub.lS]/[V.sub.hL]).sup.2] + 2G (V - [V.sub.lS][x.sub.lS]/[V.sub.hL]) (V - [x.sub.lS] - [V.sub.hL] (1 - [x.sub.lS])/[V.sub.lS] - 1) + [(V - [x.sub.lS] - [V.sub.hL] (1 - [x.sub.lS]/[V.sub.ls] - 1]).sup.2] (19)

where V = [(1 - [epsilon]).sup.-1], [V.sub.lS] = [(1 - [[epsilon].sub.lS]).sup.-1], [V.sub.hL] = [(1 - [[epsilon].sub.hL]).sup.-1]. G incorporates the contraction effect, being unity for the serial model (no contraction). Empirical equations have been proposed in the packed bed literature for estimation of the parameter G. According to Yu et al. (1993), G is only a function of the diameter ratio of the binary spheres:

G = [(1.355r.sup.1.566).sup.-1] (r [less than or equal to] 0.824) (20a)

G = 1 (r [greater than or equal to] 0.824) (20b)

where r = [d.sub.lS]/[d.sub.hL] is the diameter ratio. From Equation (20), G equals 2.84 and 4.12 when [d.sub.hL] is 15.0 and 19.1 mm, respectively. Finkers and Hoffman (1998) included (1 - [[epsilon].sub.lS]) and (1 - [[epsilon].sub.hL]) as additional parameters in their more complicated empirical equation for G applied to non-spherical particles:

G = [r.sup.k.sub.str] + (1 - [[epsilon].sup.-k.sub.hL]) (21)

where

[r.sub.str] = [((1/[[epsilon].sub.hL]) - 1) [r.sup.3]/1 = [[epsilon].sub.lS]] (22)

and k = -0.63.

The mixed layer voidages for our experiments were estimated based on the Westman equation. Two approaches were tested: (a) finding the best-fit constant value of G for a given size ratio; and (b) determining the best-fitting value of k in Equation (21), which depends on both the size ratio and the voidages of the mono-component beds. Figure 9 plots the voidages estimated from the Westman equation and the serial model against the experimental values for all the experiments. The best fits for approach (a) are G = 8.79 and 14.17 for [d.sub.lS] of 15.0 and 19.1 mm, respectively. These values are greater than those given by the correlation of Yu et al. (1993), and thus show more contraction. For approach (b), k = -0.79 shows the best results.

The use of the Westman equation with the best-fit values of G for the prediction of [[epsilon].sub.m] was compared to the serial model (Figure 9). In order to determine quantitatively the accuracy of the predictions, an average absolute % deviation, AAD, was calculated as:

AAD = [N.summation over (i=1)] [[absolute value of [[epsilon].sub.predicted] - [[epsilon].sub.exp]/[[epsilon].sub.exp]].sub.i] x 100/N (23)

where N is the number of runs. For the 48 runs of the present experiments, AAD was 0.69% and 0.55% from the two packing model approaches, (a) and (b), respectively, whereas AAD was 2.11% for the serial model, for which [[epsilon].sub.predicted] almost always exceeded [[epsilon].sub.exp]. The Westman equation thus significantly improved the bed contraction prediction.

Bulk density of the mixed layer

Knowing the composition and voidage of the mixed layer, its bulk density can be estimated from:

[[rho].sub.b] = [[rho].sub.s] (1 - [[epsilon].sub.m]) + [[rho].sub.f] [[epsilon].sub.m] = [[rho].sub.lS] [x.sub.lS] (1 - [[epsilon].sub.m]) + [[rho].sub.hL] [x.sub.hL] (1 - [[epsilon].sub.m]) + [[rho].sub.f] [[epsilon].sub.m] (24)

Figures 10a and 10b plot the bulk density of the mixed layer against the liquid superficial velocity for the two binary systems. For a given U, the bulk density of the mixed layer was always less than those of the two mono-component layers. Therefore, from stability considerations, when a mixed layer co-exists with a pure layer (of species lS in the present experiments), the mixed layer is expected to be located at the top of the column, i.e. near the distributor. This is a direct consequence of the volume contraction of the binary-solid fluidized bed. Indeed, if the voidage of a binary mixture follows the serial model, its bulk density always falls between the bulk densities of the two mono-component beds fluidized at the same liquid velocity. If a contraction occurs within the mixed layer, solid particles displace part of the liquid, and, as the density of the solid is lower than that of the liquid, this contributes to the bulk density of the mixed layer becoming lower than that of the pure layer.

[FIGURE 9 OMITTED]

[FIGURE 10 OMITTED]

CONCLUSIONS

Experiments were carried out to investigate whether the layer inversion phenomenon occurs in a downward-flow liquid-fluidized bed of binary-solid mixtures. Two binary-sphere systems were selected using a simplified theoretical approach. Only one of the five possible steps of the layer inversion progression (Figure 1d) was clearly observed: for the higher liquid velocities, two layers co-existed, a mono-component layer of lower-density smaller (lS) species at the bottom of the column (far from the distributor), and a mixed layer of species lS and the higher-density larger (hL) species at the top, close to the distributor (Figure 1d). In the overall sequence of the layer inversion phenomenon, this pattern occurs "beyond" the inversion, i.e. at velocities exceeding the inversion velocity.

An important characteristic of the mixed layer is that it is subject to bed contraction. The voidage of the mixed layer was well predicted by appropriate regression of the Westman equation with an accuracy of better than 0.7%. The mixed layer, subjected as it was to a volume contraction that gave it a bulk density lower than those of its mono-component constituents, was therefore located at the top of the column.

Additional experiments based on other binary-solid systems would be desirable to achieve all the steps, or at least the middle three, of the bed inversion progression, and to characterize more precisely the volume contraction. The choice of particle species is difficult because of a dearth of spherical particles having a density lower than that of water, and because the correlations for the Richardson-Zaki parameters (and especially the expansion index n) tend to be inaccurate for inverse fluidized beds.

ACKNOWLEDGEMENT

The authors are grateful to the Natural Sciences and Engineering Research Council of Canada for financial support.

NOMENCLATURE

A cross-sectional area of column, [m.sup.2]
Ar Archimedes number = ([[rho].sub.f] - [[rho]
 .sub.S]) [[rho].sub.f] [d.sup.3] g/
 [[mu].sup.2] for inverse fluidization,
 dimensionless
[C.sub.1], [C.sub.2] parameters in Equation (13), dimensionless
d diameter of spheres, mm
[d.sub.i] diameter of particle species i, mm or m
D inner diameter of column, m
g gravitational acceleration, [m.s.sup.-2]
G parameter of Westman equation, dimensionless
[h.sub.m] height of mixed layer, m
[h.sub.p] height of bottom pure layer of species lS, m
k exponent of packing model used in Equation
 (21), dimensionless
[m.sub.i] mass of particle species i in the bed, kg
n Richardson-Zaki expansion index, dimensionless
N number of data points in Equation (23),
 dimensionless
[n.sub.i] Richardson-Zaki expansion index of particle
 species i, dimensionless
r particle diameter ratio (smaller-to-larger),
 dimensionless
[r.sub.str] particle structural ratio of packing model used
 in Equation (22), dimensionless
[Re.sub.t] terminal free-settling (or free-rising) particle
 Reynolds number d[U.sub.t] [[rho].sub.f]/[mu],
 dimensionless
U superficial liquid velocity, [m.s.sup.-1]
[U.sub.e] value of U when a linear plot of log U vs. log
 [epsilon] is extrapolated to [epsilon] = 1,
 [m.s.sup.-1]
[U.sub.inv] superficial liquid velocity at which
 hypothetical
 or actual inversion occurs, [m.s.sup.-1]
[U.sub.mf] minimum fluidization velocity, [m.s.sup.-1]
[U.sub.t] terminal free-settling (or free-rising)
 velocity, [m.s.sup.-1]
[v.sub.i] volume of particle species i in fluidized bed,
 [m.sup.3]
V overall "specific volume" = bed volume/solid
 volume = [(1 - [epsilon]).sup.-1], dimensionless
[V.sub.i] mono-component "specific volume" = [(1 -
 [[epsilon].sub.i]).sup.-1], dimensionless
[x.sub.lS] liquid-free volume fraction of particle species
 lS in mixed layer, dimensionless
[X.sub.lS] liquid-free volume fraction of particle
 species lS in overall bed, dimensionless
Z vertical coordinate, m

Greek Symbols

[DELTA]P dynamic pressure drop, Pa
[DELTA]Z distance between adjacent pressure taps, m
[epsilon] overall voidage, dimensionless
[[epsilon].sub.i] voidage in mono-component fluidized bed of
 particle species i, dimensionless
[[epsilon].sub.m] voidage of mixed layer, dimensionless
[mu] liquid viscosity, [kg.m.sup.-1] [s.sup.-1]
[[rho].sub.b] bulk density of binary-solid fluidized bed,
 [kg.m.sup.-3]
[[rho].sub.Bi] bulk density of mono-component fluidized bed of
 particle species i, [kg.m.sup.-3]
[[rho].sub.f] liquid density, [kg.m.sup.-3]
[[rho].sub.i] density of solid particle species i,
 [kg.m.sup.-3]
[[rho].sub.s] mean density of solid particles, [kg.m.sup.-3]

Subscripts

hL high-density large particle species
i particle species i
lS low-density small particle species

Manuscript received May 25, 2006; revised manuscript received September 27, 2006; accepted for publication October 27, 2006.

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R. Escudie (1,2), N. Epstein (1) *, J.R. Grace (1) and H. T. Bi (1)

(1.) Department of Chemical and Biological Engineering, University of British Columbia, 2360 East Mall, Vancouver, BC, Canada V6T 1Z3

(2.) INRA, UR050, Laboratoire de biotechnologie de l'environnement, avenue des Etangs, Narbonne, F-11100, France

* Author to whom correspondence may be addressed. E-mail address: helsa@chml.ubc.ca

Table 1. Characteristics of the particle species, estimated
and experimental values of the Richardson-Zaki parameters for
the mono-component fluidized beds

 Particle properties d (mm)
 Authors [[rho].sub.s]
 (kg/[m.sup.3])
 Ar

 Renganathan and [U.sub.mf]
 Krishnaiah (2003)

Up-flow Turton and Clark (1987) [Re.sub.t]
fluidization Haider and Levenspiel [U.sub.t]
 (1989)
 Khan and Richardson [U.sub.e]
 (1989)

Down-flow Fan et al. (1982)
fluidization 350<[Re.sub.t]<1250 n
 [Re.sub.t]>1250 n
 Karamanev et al. (1996) Ar/[d.sup.2]
 ([mm.sup.-2])
 Ar/[d.sup.2] < 1.18 x [U.sub.t]
 [10.sup.4] [mm.sup.-2]:
 free settling velocity

 Ar/[d.sup.2] > 1.18 x [U.sub.e]
 [10.sup.4] [mm.sup.-2]
 [C.sub.D] = 0.95

 Renganathan and [Re.sub.t]
 Krishnaiah (2005) [U.sub.t]
 [U.sub.e]

 Experimental values [U.sub.e]
 n

 Particle properties 6.35
 Authors 836.5

 3.48 x
 [10.sup.5]
 Renganathan and 0.0153
 Krishnaiah (2003)

Up-flow Turton and Clark (1987) 939
fluidization Haider and Levenspiel 0.160
 (1989)
 Khan and Richardson 0.130
 (1989)

Down-flow Fan et al. (1982)
fluidization 350<[Re.sub.t]<1250 2.44
 [Re.sub.t]>1250 1.66
 Karamanev et al. (1996) 8.63 x
 [10.sup.3]
 Ar/[d.sup.2] < 1.18 x
 [10.sup.4] [mm.sup.-2]:
 free settling velocity

 Ar/[d.sup.2] > 1.18 x
 [10.sup.4] [mm.sup.-2]
 [C.sub.D] = 0.95

 Renganathan and 1 180
 Krishnaiah (2005) 0.201
 0.163

 Experimental values 0.131
 2.82

 Particle properties 15.0
 Authors 895.9

 2.91 x
 [10.sup.6]
 Renganathan and 0.0217
 Krishnaiah (2003)

Up-flow Turton and Clark (1987) 2 880
fluidization Haider and Levenspiel 0.208
 (1989)
 Khan and Richardson 0.142
 (1989)

Down-flow Fan et al. (1982)
fluidization 350<[Re.sub.t]<1250 2.41
 [Re.sub.t]>1250 1.60
 Karamanev et al. (1996) 1.29 x
 [10.sup.4]
 Ar/[d.sup.2] < 1.18 x 0.146
 [10.sup.4] [mm.sup.-2]:
 free settling velocity

 Ar/[d.sup.2] > 1.18 x 0.0993
 [10.sup.4] [mm.sup.-2]
 [C.sub.D] = 0.95

 Renganathan and 3 710
 Krishnaiah (2005) 0.268
 0.182

 Experimental values 0.166
 3.08

 Particle properties 19.1
 Authors 903.0

 5.56 x Eq.
 [10.sup.6]
 Renganathan and 0.0242 13
 Krishnaiah (2003)

Up-flow Turton and Clark (1987) 4 020 7
fluidization Haider and Levenspiel 0.229 8
 (1989)
 Khan and Richardson 0.144 12
 (1989)

Down-flow Fan et al. (1982)
fluidization 350<[Re.sub.t]<1250 2.41 10
 [Re.sub.t]>1250 1.46 11
 Karamanev et al. (1996) 1.53 x [10.sup.4]

 Ar/[d.sup.2] < 1.18 x 0.159 6
 [10.sup.4] [mm.sup.-2]:
 free settling velocity

 Ar/[d.sup.2] > 1.18 x 0.100 8
 [10.sup.4] [mm.sup.-2]
 [C.sub.D] = 0.95

 Renganathan and 5 210 9
 Krishnaiah (2005) 0.296
 0.187 8

 Experimental values 0.171
 2.43

Table 2. Experimental conditions and transition velocities
from incomplete segregation to heterogeneous mixing

Species hL Total solids Overall bed Transition

diameter volume composition velocity *
 [m.sup.3] [X.sub.IS] m/s (superficial)

15.0 mm 1.58 x [10.sup.-3] 0.2 0.068
 1.80 x [10.sup.-3] 0.3 0.059
 2.10 x [10.sup.-3] 0.4 0.051
 2.52 x [10.sup.-3] 0.5 0.047

19.1 mm 2.70 x [10.sup.-3] 0.2 0.072
 2.44 x [10.sup.-3] 0.3 0.076
 2.75 x [10.sup.-3] 0.5 0.072

* Incomplete segregation above and heterogeneous mixing
below these velocities