Magnetically assisted gas--solid fluidization in a tapered vessel: first report with obeservations and dimensional analysis.

Hristov, Jordan

INTRODUCTION

Fluidization in tapered vessels is a useful fluid-solid contacting technique with a variety of application in drying (Becker and Salans, 1960; Mathur and Epstein, 1974), combustion (Khoshnoodi and Weinberg, 1978), gasification (Salam and Bhattacharya, 2006), waste water cleaning (Scott and Hancher, 1976), bioreactors (Scott and Hancher, 1976), food processing (Depypere et al., 2005), plasma (Flamant, 1994) and microwave (Feng and Tang, 1998) assisted processing, etc.

Originally conceived (Mathur and Epstein, 1974) for gas-fluidization of Geldart's D particles (Geldart, 1973) this fluidization technique spreads towards other applications such as fluidization of cohesive particles (Deiva et al., 1996; Erbil, 1998), gas-liquid fluidization of highly viscous liquids (Anabtawi, 1995), drying of agriculture by-products (Wachiraphansakul and Devahastin, 2007), sea foods (Tapaneyasin et al., 2005), pasty materials (Passos et al., 1997), and pharmaceutical coating processes (Jono et al., 2000) due to its specific feature to assure cyclic gas-solid contact during the particle rising through the spout. In most of the cases of practical applications these devices work with sticky solids due to the high liquid content. The latter causes particle aggregation and affects the bed hydrodynamics.

Recently, Bacelos et al. (2007) have published an extensive study on spouted bed hydrodynamics with controlled effect of interparticle forces by adding glycerol to glass spheres with large size distribution. Nowadays, there are a few articles on the effect of interparticle forces on spouted bed behaviour (Passos et al., 1997; Passos and Mujumdar, 2000; Charbel et al., 2004; Trindade et al., 2004; Bacelos et al., 2007). In all of them the interparticle forces are mainly generated by capillary forces and cannot be controlled remotely. The present article addresses spouted bed behaviour in the case of gas-fluidized magnetic particles and interparticle forces induced by an external magnetic field that creates a new branch in the magnetically assisted fluidization (Hristov, 2002, 2003a,b, 2004, 2006, 2007b).

Magnetically assisted fluidization deals with magnetic solids fluidized by liquid (Hristov, 2006), gas (Hristov, 2002, 2003a), or gas-liquid flow (Hristov, 2007b) with two basic magnetization modes: Magnetization FIRST (magnetization of a fixed bed undergoing fluidization) and Magnetization LAST (magnetization of preliminarily fluidized beds). With Magnetization FIRST mode the induced interparticle forces yield particle aggregation (i.e., magnetic flocculation) that alters the bed behaviour and creates a Meta-regime commonly known as a magnetically stabilized bed (MSB). MSB is a fixed bed with induced interparticle forces of magnetic nature that, to some extent, give it a mechanical strength enabling bed expansion without fluidization at velocities beyond the minimum fluidization point in absence of field (Rosensweig, 1979; Penchev and Hristov, 1990a,b; Hristov, 2002). Macroscopically, it resembles the homogeneously expanded powders of the Geldart's Group A (Geldart, 1973). The hydrodynamic behaviour of such beds is basically reviewed (Hristov, 2002, 2003a,b, 2004, 2006) and we will avoid more reference information here.

Addressing magnetically assisted tapered bed (MATB), we have to note that this is the first attempt to perform a magnetically assisted fluidization in a vessel different from cylindrical columns used in this field over 40 years (Filippov, 1961; Hristov, 2002, 2006). However, in order to be exact, we should mention that some attempts to use a tapered 2-D vessel with an axial magnetic system (Helmholtz coils) were performed by the group of Jovanovic et al. (2004) as a part of low-gravity experiments with L-S magnetically assisted beds. Additionally, Jones et al. (1982) have conceived a magnetic field coupled sponted bed system in view of a magnetic control of the particle flow by means of a short coil placed at the top of the draft tube of a classical draft tube-spouted bed device. These inventions and the experiments thereof are by far away from the subject of the present work, the common issue is either the vessel shape or the name; an analysis of them is available elsewhere (Hristov, 2006).

Paper Outline

The article addresses several basic features pertinent to the magnetic field effects on fluidized solids and the vessel shape effect with magnetization FIRST mode, among them:

* Bed behaviour, pressure drop histories, bed expansion (porosity) variations and the physics behind them in presence of an external transverse magnetic field.

* Effects of solid mass charged to the vessel, particle size, etc.

* Critical velocities, pressure drops at minimum fluidization and minimum spouting points.

* Pressure drop hysteresis cycles and graphical determination of critical velocities.

* Dimensional analysis of both the conical vessel and magnetic field assistance; Data correlations.

Some preliminary thoughts and assumptions elucidating the background of the experiments reported here are developed next.

Paper Aim and Some Preliminary Thoughts

The introduction generally defines that the present articles addresses magnetically assisted gas fluidization in a tapered vessel. Due to the large varieties in both the spouted bed technology and the magnetic field assisted fluidization some preliminary comments are needed for better understanding of the explanations further developed in the article. First of all, we use a single vessel and do not vary its cone angle unlike the common practice in articles dealing with spouted beds. We especially avoid this additional process parameter, in this first article on MATB, and address the field effects and the fluidization regimes. This approach allows to draw easily a parallelism between the new results and those known from fluidization in cylindrical vessels (Rosensweig, 1979; Hristov, 2002). To this end, the cone angle is an important process parameter and its effect has to be studied but this draws experiments beyond the scope of the present work.

Further, the field is oriented normally to the fluid flow and the vessel axis. There are several reasons for that, among them: (a) Applying a transverse field we avoid the field imposed axial channelling in the bed annulus (Hristov, 2002). (b) The saddle coils system (Penchev and Hristov, 1990b; Hristov, 2002, 2005) allows a simple extension of the vessel volume in both axial (increase in height) and lateral (increase in bed diameter) direction since it ensures a homogeneous magnetic field over approximately 90 of its internal volume. (c) Additionally, the bed behaviour could be naturally observed through the coils "window" that is practically impossible when closely located short coils are used to create axial fields (Hristov, 2002). To this end, however, irrespective of these preliminary assumptions, the axial field assisted fluidization in tapered vessels has not been investigated yet but this challenging problem is beyond the scope of this article. At the end, in this text "tapered bed" and "spout-bed" will be used as equivalent terms albeit under some circumstances the fluidization cannot reach the spouted bed regime.

EXPERIMENTAL

Experiments with magnetically assisted beds were performed in a conical vessel (15[degrees], opening angle, 30 mm ID-bottom diameter and 190 mm ID-top diameter). The field was generated by a saddle coils magnetic system (Penchev and Hristov, 1990b; Hristov, 2002) with 200 mm ID and 400 mm in height. The field lines were oriented transversely to the cone axis of symmetry and the fluid flow (see the inset). The field was steady and the maximum field intensity attained in these experiments was about 27 kA/m. All magnetic materials used in the experiments are listed in Table 1. Air was used as a fluidizing agent and the flow (controlled by a mechanical valve) was measured by a calibrated rotameter. The pressure drop was measured by a U-tube water manometer connected between the gas inlet and a fine tube (pressure probe) placed above the bed top surface. Figure 1 presents schematically this experimental setup.

Due to the impossibility to measure the bed height directly from the top of the column (the magnetic system hinders the access) a scale placed at the vessel wall was used. The bed height is calculated by [h.sub.b] = [h.sub.L] cos a (see the inset in Figure 1), that is in the present case we have [h.sub.b] = 0.991[h.sub.L].

[FIGURE 1 OMITTED]

RESULTS

Observations 1: Phase Diagrams

Commonly the bed behaviour in conical vessels is illustrated by pressure drop curves (Jing et al., 2000; Devahastin et al., 2006; Zhong et al., 2006) and photos or schematic pictures. The magnetically assisted fluidization provides an additional illustrative tool, that is, a phase diagram in U-H coordinates (Rosensweig, 1979; Hristov, 2002), relating two macroscopic process variables controlling the fluidization. Following the basic approach in the spouted bed mechanics (Mathur and Epstein, 1974) the gas flow rate is represented here by its volumetric flow rate Q ([m.sup.3]/s) rather than the superficial fluid velocity U (m/s) in the gas entrance orifice (Devahastin et al., 2006; Zhong et al., 2006; Bacelos et al., 2007). To some extent, however, the superficial gas velocity either at the gas inlet or at the bed top surface will be used further in this article addressing mainly data correlations by equations similar to those developed for non-magnetic beds.

The phase diagrams in Q-H coordinates (Figures 2a-c) represent the bed behaviour with variations in the amount of the solids as a process parameter. In general, the solids behave similarly to those in cylindrical vessels (Hristov, 2002) but different parts of the bed situated along its axis of symmetry (from the bottom towards the top) are under different flow conditions, in spite of the fact that the magnetizations are identical (the field is homogeneous along the radius of the vessel). The schematic pictures in Figure 3a illustrate fluidization regimes existing under conditions imposed by the gas flow and the field applied.

At a given gas flow rate, the bottom part may pass into a fluidized state, while the top remains completely magnetically stabilized (i.e., with a fixed bed structure). The flow range corresponding to MSB is relatively narrow and bounded by two critical flow rates: (1) [Q.sub.e] denoting the onset of bed expansion that indicates the pass into a magnetically stabilized bed-MSB) and (2) [Q.sub.mf-1] at which the bottom part near the flow entrance becomes fluidized; while the upper bed layer remains fixed--see B2 in Figure 3a. Increase in the gas flow rate yields an extension of the fluidized section in height and the bed exhibits a classical behaviour with bubbling and no fountain formation--see B3 in Figure 3a. The process of transient bed expansion continues to its upper bound denoted as [Q.sub.mf-2] when different structures, dependent on the combination of field intensity and the gas flow rates, can exist.

The transient bed expansion (or developing fluidization) is complex in nature and will be discussed separately in the next section. In brief, the complete bed fluidization beyond the upper bound [Q.sub.mf-2] depends on the field intensity applied. At low fields (branch B in Figure 3a) the bed passes into a fluidized state with large bubbles originating from the cone bottom (B4) and travelling almost along the vessel axis. With higher field intensities the bed attains a wave-like structure due to gas slits (B5) travelling from the bottom to the top. In fact the slits are bubbles deformed in shape by the field. The latter implies that the field imposes extensional forces to any non-magnetic voids along the field lines and a compression in direction normal to them (Hristov, 2002, 2003a) that finally transform any voids into horizontal slits. The boundary between the bubbling bed and the "moving slit" regime is not easily detectable either through pressure drop records or bed height thus only visual observations may distinguish them: for this reason a dashed line marks the transition points.

With higher field intensities (branch C in Figure 3a) the bubbling bed and the travelling slits regimes are replaced by an unstable fountain (B6 or C4) within a range bounded by [Q.sub.ms-U] and [Q.sub.ms-S]. The term "unstable fountains" means that the gas flow rate ([Q.sub.ms-U] < Q < [Q.sub.ms-S]) is not enough to create a stable central channel; the fountain is practically internal with respect to the entire bed body and does not reach the bed top which remains magnetically stabilized (a stabilized Hat). Beyond [Q.sub.ms-S] the stabilized "Hat" is completely destroyed (C5) and we have a magnetically controlled spouted bed.

Before further comments on the complex behaviour of the transient bed expansion (from [Q.sub.e] to [Q.sub.ms-S]) we have to address the effect of the initial bed height (represented here by the solids inventory) on the flow transition. With low solids charged in the vessel the bed is relatively short (shallow) with [h.sub.b0] = 170 mm and a top diameter [D.sub.L] = 98 mm that imposes it to high gas superficial velocities defined by the narrow cross-section of the vessel. These conditions make the transition region (bounded by dotted lines in Figure 2a) almost unclear. Increase in the bed depth (solids inventory) makes the transitions more easily detectable since we get a relatively deep bed with sections exhibiting different fluidization behaviour. For such deep beds, for instance, a short section near the bottom behaves like the shallow one mentioned above, while the upper parts exhibit gradual transition from stabilized into fluidized (or spouted bed) that are easily detectable both by visual observations and records (pressure drop, bed height and gas flow rate).

[FIGURE 2 OMITTED]

Observations 2: Transient Bed Expansion

In the regime of transient bed expansion, that is, from the initial fixed bed into the completely fluidized state, the bed height is almost constant due to specific stages in the bed behaviour, undergoing fluidization, namely:

1. The top is stabilized by the field and since the gas flow rate is not enough either to expand or to fluidize it, this stabilized "hat" stays at an almost fixed position that provides a constant bed height (bed porosity [[epsilon].sub.a]) even though the gas flow rate increases. See C3 and C4 in Figure 3a.

2. At the same time the bottom part is completely fluidized and increase in the gas flow rate yields a growing fluidized section. The top particles of the fluidized section reach the bottom of the "frozen hat" and with increase in the flow rate detach particles from its bottom, thus decreasing its depth. In other words, the bottom fluidized section expands with almost constant pressure drop across it (from C3 to C4). When the "frozen hat" reaches a certain critical thickness it breaks down and the bed becomes completely fluidized.

3. At moderate field intensities the fluidization manifests itself by an approximately bubbleless (see the Kwauk's terminology, Kwauk, 1992) fluidization. The regime resembles an upward pseudo-wave flow (B5) due to almost horizontal thin slits propagating from the bottom to the top and the absence of a visible solids recirculation typical of tapered fluidized beds (Mathur and Epstein, 1974; Asenjo et al., 1977; Zhong et al., 2006; San et al., 2006).

4. Beyond a certain critical velocity this wave-like fluidized structure (B5) becomes unstable and large pseudo-spherical bubbles grow upward in the bed--see B4. No central channel exists and a typical spout-bed structure cannot be reached with increase in the gas velocity.

5. With increase in the field intensity the bed behaviour remains partially unchanged (upper sections) but beyond a certain critical gas flow rate a central (axially oriented) channel starts to propagate upwards from the cone. The channel grows in height with increase in the gas velocity (see C4) but it cannot reach the bed top that remains magnetically stabilized (a "frozen hat"). With increase in the gas flow rate, the developing channel reaches the upper stabilized section, destroys it and a fountain, typical of spouted beds bursts the bed top surfaces at [Q.sub.ms-U]. This fountain becomes instable and can easily collapse. The stable fountain at [Q.sub.ms-S] > [Q.sub.ms-U] and the solids circulations (from the centre towards the wall and then down to the cone bottom) completely corresponds to a stable bed spouting regime.

6. Reduction in gas flow rate ([Q.sub.ms-U] < Q < [Q.sub.ms-S]) or increase in field intensity yields an unstable central channel and suppression of the bed spouting.

Figure 3b presents schematically observations addressing the height of parts (sections) of the bed undergoing fluidization. The branch C in Figure 3a with strong fields is selected to demonstrate the evolution of the bed internal structure since under these conditions the top section is fixed that results in an unchanged bed depth even though the gas flow increases. Besides, in accordance with the phase diagrams in Figure 3 the corresponding field intensities allow to achieve spout formation and consequent jet spouting regime.

[FIGURE 3 OMITTED]

Pressure Drop

The pressure drop across the bed undergoing fluidization is a complex response of the particulate system to gas flow and to the additional conditions imposed by the vessel shape and the external field applied (Figures 4a-d). Before further comments concerning pressure drop curves obtained in this study we refer to a parallelism between the bed behaviour described above and that observed in a non-magnetic spouted bed represented schematically by the pressure drop curve of San Jose et al. (1993) (see Figure 3c).

As a first step of the experimental program, pressure drop curves of non-magnetized beds were measured with variations in the bed weight as a process parameter. The plots in Figure 4a are typical of spouted beds but the only new feature is that the shallow bed of 1 kg exhibits the highest pressure drop. This could be simply explained by the fact that with a cone of 15[degrees] a bed of 1 kg particles practically is packed close to the flow entrance. This section of the vessel allows high superficial gas velocities and higher flow dissipation rates. Besides, almost the entire bed cross section is subjected to this "high velocity" flow in contrast to the deeper beds (of 2 and 3 kg) where with increase of the bed depth the greater part of the annulus is subjected to lower superficial velocities than those in the central bed section. As a result, high pressure drops represent the bed reaction to the gas flow.

In the context of the previous comments, we would mention that these results agree with the non-magnetic experiments of Bacelos et al. (2007). More precisely, similar changes in bed pressure drop with variations in bed height and interparticle forces strongly indicate a redistribution of the gas flow between the central part and the annulus. That is, the greater part of the gas flow passes through the axial zone of the bed while the lateral sections (the annulus) are less penetrated by the flow.

[FIGURE 4 OMITTED]

The above-mentioned behaviour is easily detectable from the pressure drop curves when different field intensities are applied (Figure 4d). High field intensities result in stable interparticle contacts, low bed mobility and low gas flow through the annulus (i.e., high gas flow through the central part and the channel). In fact, this situation resembles the experiments of Bacelos et al. (2007) where the interparticle contacts of glass spheres are stabilized by addition of glycerol to the bed.

Referring to the present situation, at low (H < 5 kA/m) and high field intensities (H > 15 kA/m), the pressure drop curve indirectly reveals that the flow path is preferentially through the bed central zone. At moderate fields (7-15 kA/m) the larger particles exhibit lower pressure drops than the finer ones (Figure 4b). However, the increase in the field intensity reduces the differences (Figures 4c and d) even though the material's properties are almost similar (KM-1 is an artificial magnetite with promoters addressing ammonia synthesis that yields a reduction in its magnetization--see the last column of Table 1).

With high field intensities, beyond the maximum, corresponding to [Q.sub.e] and the sharp decrease in the pressure drop curve, all the beds exhibit almost a flow-independent behaviour. Explicitly, in spite of the oscillations of the pressure drop it varies around an almost constant value that is a result of the creation of a stabilized bed section (MSB) or a stabilized "hat" above the fluidized bottom section. This effect seems strange, but has a simple explanation, namely:

(i) The stabilized bed section (denoted as MSB in Figure 3a), or the "hat," is a fixed bed structure connected in a series with the bottom fluidized bed. The fixed "hat" structure, for instance, should exhibit increasing pressure drop if its depth remains unchanged with increase in the gas flow rate.

(ii) However, the "hat" depth decreases (the MSB depth too) with increase in the gas flow since the flow swirls and fluidized particles detach particles from its bottom. The pressure loss in the "hat" is almost constant due to its decreasing depth with increasing flow rate.

(iii) The fluidized bed at the bottom exhibits an almost constant pressure drop too, that finally yields an almost fluid flow-independent pressure drop across the entire bed.

This simple mechanistic model explains the almost horizontal section of the pressure drop curves since the pressure drop across the bed is strongly related to the variations in the bed porosity (the annular bed porosity) as it is discussed next.

Bed Expansion (Annular Bed Porosity)

The bed porosity [[epsilon].sub.a], known as annular porosity was calculated through a modification (1b) of the Bacelos et al. (2007) Equation (1a), that is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1a)

where [D.sub.L] = [D.sub.b] + 2[h.sub.b] tan([alpha]/2) with interparticle forces created by liquid ([v.sub.L]) supplied permanently to the particle bed.

In absence of a liquid ([v.sub.L] = 0) and with [h.sub.b] = [h.sub.L] cos [alpha] Equation (1a) reads:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1b)

with [D.sub.L] = [D.sub.b] + 2[h.sub.L] sin([alpha]/2).

Obviously, this is a formalistic approach trying to represent the present data by exiting models of non-magnetic spouted beds and to demonstrate how the magnetic field modifies the bed expansion (porosity variation) profiles.

The plots in Figure 5 reveal a raise in the bed height with increase in the field intensity irrespective of the solids inventory in the vessel and the particle size. The bed height (bed porosities) attained with increase in the gas flow rate is strongly attributed to the magnetic interparticle forces and the fled orientations. In general, the experimental results indicate increase in the maximum attainable bed porosity from an average 0.65 in absence of a field up to 0.7-0.8 with field assistance. The bed expansion described in the previous section is not homogeneous. Hence, the evolution of the bed internal structure, that is, bottom fluidization, formation of a stabilized "hat," internal fountain, etc., affects the porosity curves. In general, the plateaux in the [epsilon] - f (Q) curves (see inset in Figure 5) correspond to the formation of stabilized "hats" and developing fluidization with internal fountain, as it is commented next.

A comparison of a pressure drop curve and its corresponding porosity evolution with increase in gas flow rate (Figure 6) points to the link between the bed structure and its response to the fluid flow, that is, the pressure drop. It is obvious that the pressure drops attain maximum within the flow range corresponding to the first plateaux of the porosity curve. The second plateaux match the region of completely developed spouted beds at high gas flow rates with almost independent pressure drop--see, for example, Figures 4d and 5d. In this context, the common articles on spouted beds (Asenjo et al., 1977; Deiva et al., 1996; Jing et al., 2000) following the ideas of Kwauk (1992), try to calculate the pressure drop by means of modifications of the Ergun's equation with assumption of homogeneous flow distribution across the bed cross-sections at all levels: from the bottom to the top surface. However, we suggest that such equations are inadequate to the physical situation in MATB since the bed structure is generally heterogeneous. Nevertheless, this calls for creation of adequate models that is beyond the scope of the present article.

Critical Values at Transition Points

Onset of stabilized bed or fluidization at the vessel bottom

The phase diagrams are useful tools enabling the presentation of complex phenomena by macroscopic variables such as Q and H. However, from purely fluidization standpoint we are interested in critical values of both the gas flow and the pressure drop that mark the transitions between the regimes. Following the basic rules in the area of spouted beds (Mathur and Epstein, 1974) we are interested in the maximum pressure attainable as a function of the operating conditions. Figure 7 presents critical gas flow rates [Q.sub.e] corresponding to the maximum pressure drop [DELTA][P.sub.e]. In summary, the increase in bed weight and particle size yields higher values of [Q.sub.e] that are almost independent of the field intensity.

The pressure drop [DELTA][[P.sub.e] corresponding to [Q.sub.e] is affected by the field intensity and the solids inventory (Figures 8a and b). At low field intensities the field enables a decrease in [DELTA][[P.sub.e], while at moderate fields the data do not indicate a clear tendency. With increase in the field strength and reaching condition allowing formation of a fountain the value of [DELTA][[P.sub.e] increases but some data obtained with KM-1 (500-613) and Magnetite (200-315) show a field-independent behaviour.

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

The critical gas flow [Q.sub.e] and the drop [DELTA][[P.sub.e] may give more adequate information through the gas power dissipation rate [N.sub.e] = [Q.sub.e] [DELTA][[P.sub.e] (see Figures 9a and b). The energy required to start the bed deformation decreases as the field intensity is increased that contrasts with the data obtained with cylindrical vessels (Hristoy, 2002)--see the inset in Figure 9. This could be attributed to the fact that field action orients the particles along the field lines and makes the central zone of the bed (along its axis of symmetry) more fragile (weaker) that enables easily fluidization and channel formation. This tendency corresponds to field intensities below 15 kA/m, while beyond the range of 12-15 kA/m the power dissipation required to deform the bed increases parallel to field. These tendencies correspond to different bed behaviours and regimes beyond [Q.sub.e]. The declining branches of the [N.sub.e] = f (H) plots, for instance, correspond to easily developing wave-like bubbling regimes and to the absence of stable fountains. The rising branches correspond to strong bed stabilization, formation of "hats" and development of internal fountain. The internal fountain breaks the stabilized structures located above it and certainly needs more power to be dissipated in the bed.

[FIGURE 7 OMITTED]

[FIGURE 8 OMITTED]

[FIGURE 9 OMITTED]

Minimum spouting conditions

The onset of the spouting was commented with the analysis of the bed behaviour and some information is provided by the phase diagrams. Now, we address the velocity range bonded by [Q.sub.ms-U] and [Q.sub.ms-S] and the conditions affecting it. With low solids in the vessel and a shallow bed of l kg the range [Q.sub.ms-U] < [Q.sub.ms] < [Q.sub.ms-S] is relatively wider (Figure l0a) than those exhibited by the thicker beds of 2 and 3 kg (Figure 10b).

Irrespective of the amount of the solids in the beds the critical flows ([Q.sub.ms-U] and [Q.sub.ms-S]) bounding the fountain development increase parallel to the field intensity that is a logical effect of the stabilizing mechanism of the induced interparticle forces. The increase in [DELTA][Q.sub.ms] = [Q.sub.ms-U] - [Q.sub.ms-S] with increase in the amount of solids in the vessel could be attributed to redistribution of the gas flow between the central channel and the annulus with increase in the bed depth; that is, with increase in the solids inventory (depth) the gas passes preferentially through the central bed zone that predetermines the spout formation as in the non-magnetic beds (Mathur and Epstein, 1974). With a short bed (1 kg solids) the transition from [Q.sub.ms-U] to [Q.sub.ms-s] is accompanied by large bubbles, plugging and difficult to define flow structures.

[FIGURE 10 OMITTED]

[FIGURE 11 OMITTED]

The above comments address the behaviour of the magnetite particles that are almost spherical due to their nature (derived from natural magnetite sands). The KM-1 catalyst particles were obtained by crushing and by a consequent attrition to get almost spherical shapes. The KM-1 catalyst particles exhibit behaviour similar to the shallow (1 kg) magnetite bed. The previous logical explanation does not work here since the initial bed depth does not vary. Nevertheless, we refer to the fact that KM-1 has lower magnetic properties than the magnetite. In this context, the creation of a flow structure corresponding to stable spouting requires stronger fields and higher flow rates to be applied. No further information could be extracted from the present results and this definitively calls more precise future experiments on the minimum spouting conditions in MATB.

Further, addressing the stable spouting flow rate [Q.sub.ms-S] it is practically equal for beds of 2 and 3 kg, while the shallow bed (1 kg) exhibits low values (Figure 11). The difference practically disappears at high field intensities (H > 171kA) when it might suggest that the magnetic interparticle forces dominate those generated by the gravity.

In the context of the boundaries of the unstable fountain formation, the data representing the pressure drop at the [Q.sub.ms-U] and [Q.sub.ms-S] are quite illustrative of the simultaneous effect of both the field intensity and the solids inventory. In general, the higher fields, the lower pressure drops at the minimum spouting points. The differences in the pressure drop across the central channel (it might suggest that the almost entire flow passes through it) that characterize the low field range (H < 10 kA/m) practically disappear with increase in the field intensity irrespective of the solids inventory (Figure 12a). The KM-1 particles (Figure 12b) exhibit sharper decrease in the pressure drop at [Q.sub.ms-U] and [Q.sub.ms-S] at H < 5 kA/m. The almost smooth plots at H > 10 kA/m, in contrast to the plots in Figure 12a, have no logical explanations at this moment that also calls for future experiments focused on the minimum spouting conditions.

[FIGURE 12 OMITTED]

Pressure Drop Hysteresis and Results Thereof

Normal and abnormal cycles

The fluidization experiments provide not only visual observations of the bed behaviour but also records of pressure drop curves that are commonly used to analyze the bed behaviour as it has been done earlier in this work. Further, the common pressure drop curve of a spouted bed, irrespective of the bed geometry has several characteristic points that distinguish the regimes and the critical gas velocities. The pressure drop is a response of the bed to the fluid flow and every physical phenomenon occurring in the bed will affect the shape of the [DELTA]P-U curve. Two types of pressure drop hysteresis cycles were obtained: (1) "Normal" cycles with de-fluidization curves located below the branches corresponding to the increasing gas flows (see Figure 13). This is the common type of pressure drop curves known from the nonmagnetic spouted beds. The intersection of the branches defines the points denoted as [U.sub.H] (the subscript H means "hysteresis") that might be the minimum fluidization point or the minimum spouting point depending on the field intensity applied. (2) "abnormal" cycles with de-fluidization branches located above than those corresponding to increasing gas flow (see Figure 14). These cycles were observed with Magnetite (200-315) only. The braches of the abnormal cycles have common points (intersections) close to the minima of the pressure drop curves corresponding to the increasing gas flow.

Characteristic points of the hysteresis cycles

Commonly the minimum spouting velocity is detected graphically from the pressure drop curves (Wang et al., 2004) either from the minimum in the branch corresponding to increasing gas flow or from the hysteresis cycle, that is, the intersection of the lines approximating the upward and downward fluidization curves. Hence, it is of primary interest to test this approach with pressure drop hysteresis cycles of magnetically assisted tapered beds.

The common pressure drop curve (increasing flow) of a spouted bed exhibits a minimum that in absence of a magnetic field or other interparticle forces defines the minimum spouting point ([Q.sub.ms]). In magnetically assisted beds such minima define sudden changes of the bed hydraulic resistance that occur at the onset of fluidization (low field intensities) or MSB at the top and fluidization at the vessel bottom. Due to the delayed fluidization caused by magnetic field assistance the system attains complete fluidization conditions at greater than velocities than those defined by the minima. With the "normal cycles" the intersections of the fluidization and de-fluidization branches define UH close to the minimum fluidization or the minimum spouting points. As a rule the values of [U.sub.H] are greater than the critical velocities determined visually. Only two cycles in Figure 13 shows points of [U.sub.H] close to [Q.sub.mf] (Figure 13b) and [Q.sub.ms] (Figure 13g).

The "abnormal" cycles have more than one intersection points of the fluidization and de-fluidization branches. As mentioned above the first one is close to the minimum of the fluidization branch and as rule defines [U.sub.H] close to the onset of MSB with fluidized bottom section (Figures 1d-f). The second intersection point is close to the onset of fluidization (Figure 14b and f). The attempt to approximate the sections of the hysteresis cycle by straight lines (see the dashed lines in Figure 14) provides points denoted as [U.sub.H]-? since there is no unique characteristic point defined by the straight lines. As a rule the lines approximating the descending sections of the fluidization curves and the fixed bed-sections of the de-fluidization branches define [U.sub.H]-? points close to the minima. The plots in Figure 14a, c, e, and f reveal that these are the minimum fluidization points. In two other cases (Figure 14b and d) the intersections define [U.sub.H]-? close to the onset of MSB.

The second idea, applied to the abnormal cycles, is to approximate by straight lines the fluidization branch beyond the minimum and corresponding to the de-fluidization curve (within the same velocity range). The intersections of these straight lines are close to the minimum fluidization point (Figures 14a and b) or the minimum spouting point (Figure 14f). Only in the case of the cycle shown in Figure 14e the both approaches define unique point [U.sub.H]. Further, this method yields some strange results (Figures 14c and d) with intersections located in the sections corresponding to the initial fixed beds that is unrealistic. However, bearing in mind the inherent inexactness of the graphical method it might suggest that these intersections are close to the minima that, in fact, correspond to the results of the first method.

[FIGURE 13 OMITTED]

[FIGURE 14 OMITTED]

The comments in the previous sections concerning the results of the graphical methods tested reveal that the magnetic field assistance alternate the bed behaviour with respect to the nonmagnetic counterparts and the approaches based on the treatment of pressure drop hysteresis cycle should be re-evaluated. To this end, this conclusion envisages more detailed studies within broad range magnetic particulate materials, cones of different angles and field intensities that should provide enough experimental data enabling application of graphical methods to pressure drop hysteresis cycles. The graphical methods demonstrated above only mark what happens but the thorough analysis is beyond the scope of this work.

Dimensional Analysis and Data Correlation

Dimensional analysis--preliminarily thoughts

Non-magnetic background. Tapered fluidization involves more geometrical characteristics of the bed in the group of variables that certainly increase the number of dimensionless ratios, since all these geometric characteristics have dimension of length L (m). Such ratios exist practically in every article on spouted bed, and good examples could be found elsewhere (Kmiec, 1980; Olazar et al., 1993; Jing et al., 2000; Zhong et al., 2006). Most of them, with minor variations, repeat the Mathur and Gishler (1955):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2a)

The common approach is to correlate the bed geometry through dimensionless ratios such as ([D.sub.B]/[D.sub.c] and ([h.sub.b]/[D.sub.c]) (Shirvanian and Calo, 2004) while the phase properties are commonly expressed by the density ratio either as ([[rho].sub.s] - [[rho].sub.f])/[[rho].sub.f] or [[rho].sub.s]/[[rho].sub.f] (Shirvanian and Calo, 2004; San et al., 2006). This yields relationships as the following, for example (Wu et al., 1987):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2a)

Commonly the critical gas velocities ([U.sub.ms]) are normalized (scaled) by [square root of (2g[h.sub.b])] (Wu et al., 1987; San et al., 2006) as in (1b) or through the particle diameter based Reynolds number [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] = [U.sub.ms][d.sub.p][[rho].sub.f]/[[eta].sub.f] (Aravinth and Murugesan, 1997; Mgalhaes and Pinho, 2006) with [U.sub.ms] defined by the cross-sectional area of the gas inlet orifice. In other cases dimensional correlations are used such as (Costa and Taranto, 2003):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1c)

In this context the pressure drop correlations are either in dimensional [DELTA][P.sub.max] (San et al., 2006) or dimensionless form [DELTA][P.sub.max]/[[rho].sub.b]g[h.sub.b] (Costa and Taranto, 2003) with bed height used as a length scale.

The common dimensionless groups (see (1a)-(1c)) are the particle-diameter-based Reynolds number [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] = [U.sub.ms][d.sub.p][[rho].sub.f]/[[eta].sub.f] (Aravinth and Murugesan, 1997; Mgalhaes and Pinho, 2006) and the Archimedes number (Costa and Taranto, 2003; San et al., 2006). In some correlations (He et al., 1997), additional Froude number g[d.sub.p]/[U.sup.2.sub.0] is used.

This brief presentation is by far of completeness but it focuses the attention on the common manners to correlate the spouted bed characteristics irrespective to the vessel geometry. This figures the existing non-magnetic background (correlations derived for non-magnetic beds) that would enable easily to understand the dimensionless analysis developed further in this work. Only two works (Hristov, 2006, 2007a) have been devoted so far to dimensionless analysis of magnetic field assisted fluidization. They deal mainly with process physics and development of dimensionless groups rather then with vessel geometry effects that inherently relate them to cylindrical fluidized beds. To this end, in the case of non-magnetic conical beds, we especially refer to the work of Bi (2004) where comprehensive collections of dimensionless scaling relationships pertinent to [U.sub.ms] through [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] = [U.sub.ms][d.sub.p][[rho].sub.f]/[[eta].sub.f] are thoroughly analyzed.

Problems with the basic units of the mechanical and magnetic systems of units and the origin of the "pressure transform". In the course of the development of our analysis we will avoid the formalism of the classical dimensional analysis and will address a simple approach referring to the physical basis (Hristov, 2006, 2007a). This approach employs a preliminary transformation of the variables provoked mainly by the fact that the magnetic system is based on four units (M, L, T, I), while the mechanical counterpart has only three units, namely (M, L, T). Besides, the dimension of the field intensity H is (A/m) that does not match the dimensions of mechanical variables. This problem could be easily avoided by formation of a new variable representing the magnetic field action on the granular media, namely the magnetic pressure, [P.sub.m] = [[micro].sub.0] MH (see Hristov, 2006, 2007a). In accordance with this approach, termed "pressure transform" all the variables involved in the process of fluidization can be grouped into a new set of variables with a unique dimension of pressure (Pa). The text developed in Appendix demonstrates the approach and gives the basic rules of building-up dimensionless correlations addressing the physical basis of the phenomena.

Power-law: data correlations

Two basic characteristics of fluidized system under consideration were correlated to experimental data: pressure drop and fluid velocity at the minimum spouting point. In accordance with the general expressions developed in the Appendix, the following relationships were used:

Pressure drop:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3a)

or in a dimensional form as:

[DELTA] [equivalent to] [[rho].sub.s]g[d.sub.p][f.sub.1](Ga;[Bo.sub.g-c];[DELTA][D.sub.bL]) (3b)

Fluid velocity (superficial velocity at the gas inlet orifice):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4a)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4b)

where [U.sub.f] = [U.sub.ms-S] = [Q.sub.ms-S]/[S.sub.b].

Examples of correlations developed are summarized in Tables 2, 3a and 3b. For seek of clarity some additional comments addressing the correlation equations will be developed next.

Data correlations--brief comments on the equations chosen

(1) Simple multiplication of two terms representing the non-magnetic (Ga [DELTA][D.sub.bL]) and magnetic conditions ([A.sub.dp]-[B.sub.dp] [Bo.sub.g-m]) imposed to the bed is chosen to create the data correlations, namely:

[DELTA][P.sub.ms]/[[rho].sub.s]g[d.sub.p] = (Ga[DELTA][D.sub.bL])([A.sub.dp]-[B.sub.dp][Bo.sub.g-m] (5a)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (5b)

The form (5b) is developed in Example 3 of Appendix and tries to convert (5a) into an expression more familiar to the people dealing with non-magnetic spouted beds (see Equations (1a)-(1c)).

(2) The form of Equations (5a) and (5b) chosen in this work do not satisfy exactly the rules of the dimensionless analysis (Kline, 1965; Barrenblatt, 1996) teaching that, for example, the correct form is a product of power-law terms, namely:

[DELTA][P.sub.ms]/[[rho].sub.s]g[d.sub.p] = m[(Ga).sup.[alpha]][([DELTA][D.sub.bL]).sup.[beta]][([Bo.sub.g-m]).sup.y] (6)

with m, [alpha], [beta], [gamma] determined through scaling to experimental data.

Misinterpretations, however, might appear with the extremes pertinent to the Bond number rather mathematically than physically, namely: (1) with [Bo.sub.g-m] [right arrow] 0 (cohesionless particles) that gives ([DELTA][P.sub.ms]/[[rho].sub.s]g[d.sub.p]) [right arrow] 0 as well as with (2) with [Bo.sub.g-m] [right arro] [infinity] (too sticky particles that is impossible to fluidize) yielding ([DELTA][P.sub.ms]/[[rho].sub.s]g[d.sub.p]) [right arrow] [infinity]. In order to avoid such misreading of (6) the form of Equations (5a) and (5b) was chosen. Some reasonable standpoints supporting this approach are:

* First of all, the Bond number [Bo.sub.g-m] never goes to zero since even though in absence of magnetic fields there is a residual particle magnetization. The extreme [Bo.sub.g-m] [right arrow] 0, in fact, means that the induced magnetic cohesion is negligible with respect to the gravity effects on the interparticle contacts.

* Further, the extreme [Bo.sub.g-m] [right arrow] [infinity] is a mathematical boundary than a physical one because the upper field intensity limit is imposed by the magnetization at saturation [M.sub.s] of the particle material.

* Last, for real materials employed by magnetically assisted fluidization (Hristov, 2002) the Bond number (Hristov, 2006; Valverde and Castellanos, 2007) gets finite values. In this context, the classic form of data correlation (6) will never attain these extremes and the data can easily be correlated through it providing numerous values of exponents. However, for seek of clarity of the physical explanations in this first appearing article on MFATB we present the data in the forms ((5a) and (5b)) which clearly separate the magnetic and the nonmagnetic effects on the bed behaviour. Besides, we especially accept the exponents [alpha], [beta] and the pre-factor m equal to 1 that enable to concentrates all magnetic effects in the term [([Bo.sub.g-m]).sup.y] represented as linear approximation, namely [([Bo.sub.g-m]).sup.y] [approximately equal to] ([A.sub.dp] - [B.sub.dp][Bo.sub.g-m]).

* An alternative scaling was performed (see Table 3b) in the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (7a)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (7b)

This form avoids misinterpretations occurring with the linear approximation at [Bo.sub.g-m] [right arrow] 0 where unreasonable negative terms appear as result of the regression analysis performed.

Data processing performed with a part of the experimental results yields equations summarized in Tables 2, 3a and 3b. The exponential approximation, for instance, of the minimum spouting velocity [U.sub.ms-s] is more realistic then the linear equation with negative terms that enables only to fit the numerical data but is illogical. Further analysis of dimensional analysis and data approximation problems are available elsewhere (Hristov, in preparation).

DISCUSSIONS

This work draws a new idea to perform magnetically assisted fluidization in conical vessels unlike the common practice (over more than 45 years after the first work of Filippov (1961)) to employ cylindrical vessels (Hristov, 2002, 2003a,b, 2004, 2006). Analysis was drawn through the entire text but in this brief discussion we address some key points, among them:

(1) Parallelism with non-magnetic beds.

(2) Dimensional analysis applied.

(3) Trends and new problems.

(4) Reasonable application of magnetically assisted fluidization in tapered vessels.

Parallelism in Bed Behaviour with Non-Magnetic Counterparts

Addressing the stage in the fluidization development, we have to mention that similar flow structures have been observed with scrap-wood particles of different size and humidity (Leslous et al., 2004), for example. First, the particle humidity (causing interparticle forces) has an effect similar to that one imposed by the increasing field intensity and leads to increasing pressure drop. Besides, the slugging observed fine particles with low humidity (low degree of interparticle forces) exactly corresponds to the wave-like regime observed with low field intensities creating unstable particle aggregates. In this context, the increase in particle size of rather dry particles (Leslous et al., 2004) results in a spouted bed tapped by a fixed bed. That is, the increase in the interparticle dry friction and particle size delays the development of the spout towards the bed surface. These effects are similar to those observed with increased interparticle magnetic forces and formation of aggregates.

In the context of interparticle forces effects on the bed behaviour we have to mention too that progressive development of internal spouts and top located "caps" (the term "hats" is used here) have been observed in liquid-fluidized (Peng and Fan, 1997) and gas-fluidized (Wang et al., 2005) non-magnetic beds due to strong particle interlocking (term used by Wang et al. (2005)) that in general is equivalent to performance of interparticle forces.

These short comments on the parallelism draw only some basic lines of similarity but further experimental work has to be done in that direction. This is a problem of the future and depends mainly on the way how the idea of MFATB will be accepted by the people working on fluidization in interdisciplinary areas. Some ideas are drawn next but the main development of MFATB fluidization needs a lot of intuition, imagination and a real physical analysis of problems that might be solved through this technique.

Dimensional Analysis Applied

The dimensional analysis applied to the MFATB data does not follow the classical rules and originates in two recent works (Hristov, 2006, 2007a) but has a starting point in the scaling of differential equations similar to those used by He et al. (1997), for example. Scaling terms of a certain differential equation and consequent non-dimensionalization procedures generate dimensionless groups as pre-factors of dimensional terms of order of unity that is the mathematical side of the coin. But physically, in fact, these operations simply mean the comparison of effects of physical fluxes entering an elementary volume of the medium. In cases where no developed mathematical models exist, but with a clear standpoint about the main factors affecting the system of interest, the definition of the physical fields contributing the process is the primary step. Then, by simple definition of the surface forces acting on it, in fact, we define the scales of these fluxes that exactly correspond to the initial step of the scaling procedure of differential equations. This idea is sketched in Example 4 of Appendix. The idea of the "pressure transform" is simple and addresses the useful fact that applying the classical dimensional analysis we might forget some variables or to use some inadequate ones. However, we can never forget the physical fields (gravity, fluid flow, cohesion, magnetic, or electric) and their fluxes expressed by the surface forces caused by their action on the system if the preliminary analysis is correct. The approach is quite fruitful in interdisciplinary areas of research with complicated cross-field effects and undeveloped models, as it was demonstrated by Hristov (2006, 2007a) and the present work.

Graphical Methods Applied

The results on magnetic field assisted tapered fluidization raised too many questions (Bi, 2008) concerning the application of the well-known methods from non-magnetic beds to the new system. In this context, the present report defines that with "normal" hysteresis cycles the classical graphical method to determine the minimum spouting velocity (Wang et al., 2004) provides results that match the onset of MSB, the minimum fluidization point and, to some extent, the minimum spouting point. These deviations are mainly attributed to the altered bed behaviour imposed by the field induced anisotropy in particle arrangement and the strong particle aggregation with increase in the field intensity. The graph ical method applied to the "abnormal" hysteresis cycles provides points close to the minima of the fluidization branches of the fluidization curves.

The test to apply the classical graphical method to the hysteresis cycles of MFATB is only a "shoot" to the new system with an old weapon that clearly indicates that this could be done with caution since the results might be strange or inspected. However, these challenging problems call new experiments and further deep analyses.

Trends and Unsolved Problems

1. Fluidization with magnetization LAST mode.

2. Axial field application.

3. Fluidization of admixture both for stabilization and segregation studies with different magnetization modes (FIRST, LAST, or ON-OFF) (Hristov, 2002, 2003b).

4. Drying of wet non-magnetic materials since the magnetic materials may absorb electromagnetic energy supplied by external electromagnetic fields superimposed to the stabilizing DC field.

5. Liquid-solid and three-phase fluidization as a counterpart version of the existing practice to amply cylindrical columns only (Hristov, 2002, 2003b, 2006).

6. Dimensional analysis, modelling and scale-up.

Reasonable Applications of MFATBs--Some Suggestions

The basic data concerning the bed behaviour with magnetization FIRST mode allow envisaging some applications of MFATB, among them:

1. Separation of magnetic and non-magnetic particles in both batch and continuous modes utilizing the high velocity range with well formed central channel and stable particle circulation: the magnetic grains remain in the vessel while the non-magnetic ones are entrained by the air flow.

2. Deep bed filters for particle capture from dusty gases employing various regimes emerging with increase in gas velocity, among them:

* Homogeneous bed expansion with MSB regime and gas velocities greater that those existing in cylindrical beds under the same condition imposed by the field.

* Regime with a magnetically stabilized HAT and a bottom fluidized section as a promising combination of two filter sections connected in a series.

3. Heat transfer devices for low temperature gases operating below the Currie point of the magnetic material employed. In this context, the heat transfer might be combined with deep bed filter application mentioned above.

4. Bioreactors (L-S or G-L-S) with cells or enzymes immobilized on magnetic supports since both the variable vessel cross-section and the induced magnetic cohesion might be a suitable combination avoiding many problems in such devices (Hristov and Ivanova, 1999; Hristov, 2006).

These are only ideas based on the current status of the magnetic field assisted fluidization (Hristov, 2002, 2003a,b, 2004, 2006, 2007b) and might be expected that articles with new results will appear soon.

CONCLUSIONS

The article presents first experimental results on magnetically assisted gas-solid fluidization in a tapered bed in presence of an external transverse magnetic field. This is a novel branch in the magnetically assisted fluidization and some principle results will be outlined, among them:

* The bed behaviour is controlled by the intensity of the external magnetic field. The principle process variables such as bed depth, field intensity, particle size, cone angle were detected. Phase diagrams similar to those used to describe cylindrical beds were created with definition of new critical velocities separating the regimes.

* The field intensity increase yields increase in all critical velocities such as: minimum velocity of bed expansion, minimum spouting velocity known from the non-magnetic fluidization.

* The pressure drop hysteresis cycles of magnetically assisted tapered beds might "normal" as the classic one observed with non-magnetic beds or "abnormal" with inverted location of the fluidization and de-fluidization branches.

* The graphical methods commonly applied to pressure drop hysteresis cycles of non-magnetic beds are valid for "normal" cycles of MATB too, but defines either the onset of MSB or points that do not match the visual detected minimum fluidization or minimum spouting points. Additional graphical approach applied to "abnormal" pressure drop cycles provides points close to the minima of the fluidization branches of the fluidization curves.

* A detailed analysis and parallelism to behaviour exhibited by non-magnetic spouted beds of cohesive particles were performed. Similarly to the non-magnetic counterparts, the increases in bed weight and particle size yields increase in the maximum pressure drop. The external magnetic field augments the maximum pressure drop exhibited by the bed before the fluidization onset but the tendency with respect to the minimum spouting pressure drop is just the opposite.

* Dimensional analysis utilizing a "pressure transform" of the initial set of process variables is applied to develop the principle scaling equations. Samples providing scaling equations concerning both the pressure drop and gas velocity at the minimum spouting point are developed.

NOMENCLATURE

[A.sub.dp], [A.sub.hb] dimensionless coefficients
Ar = [d.sup.3.sub.p]
 [[rho].sub.f]
 ([[rho].sub.s]-
 [[rho].sub.f])g/
 [[eta].sup.2.sub.f] Archimedes number
[B.sub.dp], [B.sub.hb] dimensionless coefficients
[Bo.sub.g-c] =
 [P.sub.c]/[P.sub.g]
 = [P.sub.c]/
 [[rho].sub.s] Bond number of granular materials with a
 g[d.sub.p] natural cohesion
[Bo.sub.g-m] =
 ([[micro].sub.0]MH)/
 [[rho].sub.s]
 g[d.sub.p] magnetic Bond number for granular materials
[D.sub.b] diameter of the flow entrance (denoted also
 as [D.sub.i] in the equations summarized in
 Table 2) (m)
[D.sub.c] column diameter (diameter of the cylindrical
 section above the cone) (m)
[D.sub.t] tube diameter (diameter of the cylindrical
 section above the cone)--a symbol used in
 the equations summarized in Table 2 (m)
[DELTA][D.sub.bL] dimensionless ratio of bed geometric
 characteristics defined by Equation (A-2c)
[d.sub.p] particle diameter (m)
[MATHEMATICAL
 EXPRESSION NOT
 REPRORUCIBLE IN dimensionless pre-factor in exponential
 ASCCII.] data correlation (Equations (7a) and (7b))

Fr = [U.sup.2.sub.f]/
 g[d.sub.p] particle diameter defined Froude number
Ga = ([d.sup.3.sub.p]
 g[[rho].sub.f])/
 [[eta].sup.2.sub.f] Galileo number
[G.sub.SCR] solids circulation rate (kg/[m.sup.2]s)
g gravity acceleration (9.81 [m.sup.2]/s)
H magnetic field intensity (A/m)
[g.sub.b] bed height (m)
[h.sub.b0] initial bed height (m)
[h.sub.L] bed length at the wall (see Figure 1) (m)
K dimensionless coefficient (see the equation
 of Wu et al. (1987) in Table 2)
[K.sub.sf] exchange coefficient defined by Gidaspow
 (1994) and related to the Reynolds number
[K.sub.dp] dimensionless exponent in data correlations
 (Equations (7a) and (7b))
M magnetization (A/m)
[M.sub.s] magnetization at saturation (A/m)
[M.sub.p] mass of particles charged into the vessel(kg)
N = [DELTA]PQ gas flow power dissipated in the bed (W)
[N.sub.e] = [DELTA] gas flow power dissipation in the bed
 [P.sub.e][Q.sub.e] required to deform the bed body (W)
[P.sub.c] cohesion (Pa)
[P.sub.g] =
 [[rho].sub.s] gravity pressure peer unit surface of
 g[d.sub.p] interparticle contacts (Pa)
[DELTA]P pressure drop (Pa)
[DELTA][P.sub.e] pressure drop at the onset of bed expansion,
 that is, the maximum pressure drop attainable
 before the bed expansion onset (Pa)
Q volumetric gas flow rate ([m.sup.3]/s)
[Q.sub.e] volumetric gas flow rate at the onset of
 initial bed expansion ([m.sup.3]/s)
[Q.sub.mf-1] volumetric gas flow rate at the fluidization
 onset in the bottom part of the bed
 ([m.sup.3]/s)
[R.sub.ep] =
 [[rho].sub.f]
 [d.sub.p][U.sub.f]
 /[[eta].sub.f] particle Reynolds number
[S.sub.b] = [pi] cross-section area of the gas inlet
 [D.sup.2.sub.b]/4 orifice ([m.sup.2])
U superficial gas velocity (denoted also as
 [U.sub.f] or [U.sub.o]--see the text) (m/s)
[U.sub.mf] minimum fluidization velocity (depends on
 the cross-section of the bed specified) (m/s)
[U.sub.ms] minimum spouting velocity (depends on the
 cross-section of the bed specified) (m/s)
[Q.sub.mf-2] volumetric gas flow rate at the fluidization
 onset over the entire bed ([m.sup.3]/s)
[Q.sub.ms-U] volumetric gas flow rate at the minimum
 spouting (unstable) point ([m.sup.3]/s)
[Q.sub.ms-S] volumetric gas flow rate at the minimum
 spouting (stable) points ([m.sup.3]/s)
[v.sub.L] liquid volume added to the bed in the
 experiments of Bacelos et al. (2007)
 ([m.sup.3])
[v.sub.p] =
 [M.sub.p]/
 [[rho].sub.s] volume occupied by the solids ([m.sup.3])
[V.sub.bed] = ([pi]
 [h.sub.bo]/12)
 ([D.sup.2.sub.L]
 + [D.sub.L][D.sub.b]
 + [D.sup.2.sub.b])
 -([m.sup.3]) particle bed volume

Greek Symbols

[alpha] cone angle ([degrees])
[epsilon] porosity
[[epsilon].sub.0] initial bed porosity
[[epsilon].sub.a] annular bed porosity
[[theta].sub.s] particle sphericity
[micro] magnetic permeability (Wb/A m) or (H/m)
[[micro].sub.0] magnetic permeability of the space (Wb/Am)
[phi] internal friction angle of particle phase (Equation
 (A-6)) ([degrees])
[[eta].sub.f] fluid dynamic viscosity (denoted also as
 [eta] for simplicity of the expressions)
 (Pa s)
[upsilon] fluid kinematic viscosity ([m.sup.2]/s)
[[rho].sub.f] fluid density (kg/[m.sup.3])
[[rho].sub.g] gas density (kg/[m.sup.3])
[[rho].sub.s] solid particle density (kg/[m.sup.3])
[tau] shear-stress in a liquid (Pa)

Subscripts

f fluid
G gas
max maximum
ms minimum spouting
p particle
p-ms a particle diameter related value at
 the minimum spouting conditions
s solids

Abbreviations

MFATB magnetic field assisted spouted bed
MASB magnetically assisted spouted bed

APPENDIX A: DIMENSIONAL ANALYSIS--A RATIONALE OF THE PRESSURE TRANSFORM APPROACH

Tapered fluidization involves more geometrical characteristics of the bed in the group of variables than cylindrical bed. This certainly increases the number of dimensionless ratios, since all geometric characteristics of both the vessel and the bed have dimension of length L (m). Such ratios exist practically in every article on spouted bed, and good examples could be found elsewhere (Kmiec, 1980; Olazar et al., 1993; Jing et al., 2000, Zhong et al., 2006). Most of them, with minor variations, repeat the Mathur-Gishler (MG) correlation (see (la) and (lc)) for the minimum spouting velocity and others concerning the maximum pressure drop.

In the course of the development of our analysis we will avoid the formalism of the classical dimensional analysis and will treat the physical basis (Hristov, 2006, 2007a) of the phenomena. This approach employs a preliminary transform of the variables involved in the dimensional analysis provoked mainly by the fact that the magnetic systems is based on four units (M, L, T, I), while the mechanical counterpart has only three units, namely (M, L, T). Besides, the dimension of the field intensity H is (A/m) that does not match the dimensions of mechanical variables. This problem could easily be avoided by the formation of a new variable representing the magnetic field action on the granular media, namely the magnetic pressure, [P.sub.m] = [[micro].sub.0] MH (see Hristov, 2006, 2007a). In accordance with this approach, termed "pressure transform," all the variables involved in the process of fluidization can be grouped into a set of new variables with a unique dimension of pressure (Pa) (Hristov, 2006, 2007a).

Let us see how dimensionless groups emerge through a transformation of the initial set of variables into a new set having homogeneous dimensions of pressure, that is, N/[m.sup.2] = Pa.

The initial variables pertinent to the magnetically assisted tapered fluidization and provided by the principle sub-systems forming the fluidized bed are as it follows:

The fluid (gas) participates with: [[rho].sub.g], [[eta].sub.f], [U.sub.f], g, [DELTA]P (A-1a)

The solids participate with: [[rho].sub.s], [d.sub.p], g, [h.sub.b0], [P.sub.c] (cohesion) (A-1b)

The vessel geometry provides: [D.sub.B], [D.sub.L], tg ([alpha]/2) (A-1c)

The "pressure transform" means that we can read the set of variables as:

Fluid: [[rho].sub.f] [U.sup.2.sub.f] = [P.sub.U], [[eta].sub.f],g, [DELTA]P (A-2a)

Solids: [P.sub.g] = [[rho].sub.s]g[d.sub.p] (gravity pressure), [P.sub.c] (cohesion) (A-2b)

Vessel: [DELTA][D.sub.bL] = [D.sub.L] - [D.sub.b]/2[h.sub.b0] = tg ([alpha]/2) (A-2c)

Besides, since the tapered bed hydrodynamics has neither a prescribed velocity scale nor a length scale due to its specific geometry, the use of [U.sub.f] in a dimensionless group is a matter of discussion. Using the particle diameter [d.sub.p] as a length scale the fluid flow group provides [Re.sub.p] = ([[rho].sub.f][d.sub.p]/[[eta].sub.f]) [U.sub.f] and Fr = [U.sup.2.sub.f]/g[d.sub.p] that are the Reynolds and Froude number, respectively--pertinent to particle dynamics in the fluid flow. However, both dimensionless groups employ the velocity [U.sub.f] that is hard to be defined properly (see bellow) and we cannot use correctly these dimensionless groups. To be precise, in cylindrical fluidized beds, for instance, the fluid superficial velocity based on the tube diameter which does not vary along the bed axis and can be used as a reliable velocity scale; that allows to use both Fr and Re numbers based on it. In a conical bed (and in all types of tapered vessels too), there is no unique superficial velocity that might be used as a velocity scale since the vessel cross-section varies along the bed axis. In most of the works on spouted beds the superficial gas velocity at the gas inlet orifice is used for correlations. However, we have to be aware that in this way we refer to the fluid dynamics of the flow entrance orifice but not to fluid-particle dynamics. The velocity in Fr and Re numbers is the fluid-particle relative velocity, while the superficial velocity based on the inlet vessel diameter (and Fr and Re based on it) is relevant to the hydrodynamics of the entering fluid jet but not to fluid-particle dynamics. The present work does not address this crucial point in the modelling of spouted beds and the discussion is beyond the scope of its topic. The comments just done only refer to the fact that Fr and Re numbers are missing in the set of developed dimensionless groups and try to explain what the physical reasoning leading to this standpoint is. If physics is ignored and the superficial velocity is based mechanistically on the gas entrance diameter, the result is a huge amount of equations pertinent to particular devices. In fact, this is a result of formalistic creations (or of tradition, inertia or conventionalism, or something similar) of correlations not based on the process physics. The next paragraph explains how to avoid the problem through merging Fr and Re numbers into one dimensionless group without [U.sub.f].

The problem with the unknown velocity scale, just commented above, could be avoided in a classical manner by the definition of the Galileo number [(Re).sup.2]/Fr = Ga = [d.sup.3.sub.p]g[[rho].sub.f]/[[eta].sup.2.sub.f] immobilizing all variables provided by the fluid flow field. This classical approach, for instance, exists in many textbooks dealing with sedimentation or with fluidization.

Therefore, now the new set of variables becomes:

Fluid: Ga = [d.sup.3.sub.p]g[[rho].sub.f]/[[eta].sup.2.sub.f] and [DELTA]P (A-3a)

Solids: [P.sub.g] = [[rho].sub.s]g[d.sub.p] (gravity pressure), [P.sub.c] (cohesion) (A-3b)

Vessel: [DELTA][D.sub.bL] = [D.sub.L] - [D.sub.b]/2[h.sub.b0] = tg ([alpha]/2) (A-3c)

Using [P.sub.g] = [[rho].sub.s]g[d.sub.p] as a natural pressure scale at the particle level scale (Hristov, 2007a) we get two dimensionless ratios, namely [Bo.sub.g-c] = [P.sub.c]/[P.sub.g] that is the granular Bond number (Hristov, 2006, 2007a) as a measure of the stability of the interparticle contacts and the ratio [DELTA]P/[P.sub.g] commonly used in fluidization to make the pressure drop dimensionless. Therefore, the final set of variables becomes:

Fluid: Ga = [d.sup.3.sub.p]g[[rho].sub.f]/[[eta].sup.2.sub.f] and [DELTA]P/[[rho].sub.s]g[d.sub.p] (A-4a)

Solids: [Bo.sub.g-c] = [P.sub.c]/[[rho].sub.s]g[d.sub.p] (A-4b)

Vessel: [DELTA][D.sub.bL] = [D.sub.L]-[D.sub.b]/2[h.sub.b0] = tg ([alpha]/2]) (A-4c)

The solids and the vessel "provide" the independent variables while the fluid flow "generates," depended variables, namely:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (A-5)

This is the basic dimensionless relationship in the case of gas-fluidized beds with negligible buoyancy and significant interparticle forces whatever is their nature--cohesion, capillarity, electric, or magnetic. If the buoyancy takes place (liquid-solid fluidization), the Archimedes force has to be accounted for by a simple multiplication, that is:

Ga ([[rho].sub.s] - [[rho].sub.f]/[[rho].sub.f] = [d.sup.3.sub.p][[rho].sub.f]([[rho].sub.s]-[[rho].sub.f])g/[[eta].sup.2.sub.s] = Ar

providing the Archimedes number. To this end, we have to recall that although Ga and Ar are similar in their mathematical derivations they have different physical meanings. Besides, in order to avoid misunderstanding, we have to mention that the use of the Archimedes number in the case of gas-fluidized beds is physically incorrect since the buoyancy is negligible, albeit there is an astonishing plethora of correlations doing that mechanistically. These comments address problems beyond the scope of the present work but clarify why Ar does not appear in the group of independent variables used in the correlations developed in this analysis. To those of the readers who take care about the effect of the density difference effect on the bed behaviour let see the correlations ((A-8a)-(A8c)-Example 2) bellow where the ratio [[rho].sub.s][[rho].sub.f] appears automatically in the left-side part of the relationships. Brief comments complementing the above remarks are available in Shirvanian and Calo (2004), where at [[rho].sub.s][[rho].sub.f] the Archimedes number is presented as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

To this end, the ratios such as [[rho].sub.s]/[[rho].sub.F] and [h.sub.b]/[d.sub.p] or [D.sub.c]/[d.sub.p] frequently appearing in literature (see Table 2 for examples) as results of dimensional analyses performed clearly indicate that initial choices of scales were not well designed. The use of the particle diameter as a length scale, for example, is valid at the particle level only and cannot be used to make dimensionless either the bed height [h.sub.b] or the column diameter [D.sub.c], since at this macroscopic level it is insignificant.

The effect of the interparticle forces expressed by the Bond number has to disappear from the group of independent variables if their origins are cohesion, capillarity or electrostatic forces and the bed is fluidized by liquid. It persists as an independent variable in the case of magnetic interparticle forces only (in liquid-solid magnetically assisted beds) through the magnetic bond number, [Bo.sub.g-m] = [[micro].sub.0] MH/[[rho].sub.s]g[d.sub.p] (Hristov, 2006, 2007a; Valverde and Castellanos, 2007) since the liquids do not affect the magnetic interaction of the particles.

With dominating buoyancy the basic dimensionless relationship pertinent to MFATB is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (A-6)

with magnetization at saturation Ms is used in the nominator of [Bo.sub.g-m], (Hristov, 2006, 2007a).

This general expression provides well-known equations concerning spouted bed characteristics as it is exemplified next.

Example 1

Pressure drop at a critical point (minimum spouting or maximum attainable pressure drop) in a dimensionless form is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (A-7)

of coarse particles and negligible interparticle forces

Example 2

Since the bed is an obstacle to the fluid flow, the pressure drop across it is proportional to [[rho].sub.f][U.sup.2.sub.f], that is [DELTA]P [equivalent to] [[rho].sub.f] [U.sup.2.sub.f], we obtain from (A7)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (A-8a)

This relationship can be read as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (A-8b)

That with some algebraic manipulations provides the famous Mathur-Gishler formula, namely:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (A-8c)

The second term in (A8c) can be expressed in various ways as a function of the process parameters and bed geometry as it is illustrated briefly by Equations (1a)-(1c).

Example 3 (the bed weight effects)

The analysis performed above omits the bed weight contribution to the set of initial variables pertinent to the effect of the gravity (see (A-1b)). This is a macroscopic effect in contrast to the microscopic gravity pressure [[rho].sub.s]g[d.sub.P]. The data reported in this work clearly reveal that the pressure drop depends on the solids inventory in the bed similar to non-magnetic spouted bed. If the bed weight G will be included in the set of gravity-related variables, then the question is: what is the surface S that might provide the macroscopic gravity pressure, that is, how to define [P.sub.G] = (G/S)? The bed volume is (Finlayson et al., 1997) [V.sub.bed] = ([pi] [h.sub.b0]/12) ([D.sup.2.sub.L] + [D.sub.L][D.sub.b] + [D.sup.2.sub.b]) that gives G = [[rho].sub.s] (1-[[epsilon].sub.0])g[V.sub.bed] = [[rho].sub.s] (1-[[epsilon].sub.0])g[h.sub.b0] ([pi]/12) ([D.sup.2.sub.L] + [D.sub.L][D.sub.B] + [D.sup.2.sub.B]). Defining an effective surface as S = ([pi]/12) ([D.sup.2.sub.L] + [D.sub.L][D.sub.B] + [D.sup.2.sub.B]) the bed weight per unit area is [P.sub.G] = G/S = [[rho].sub.s] (1-[[epsilon].sub.0])g[h.sub.b0]. Then, the ratio of both gravity pressure scales, that is, [P.sub.G] and [P.sub.g] yields:

[P.sub.G]/[P.sub.g] = [[rho].sub.s] (1-[[epsilon].sub.0])g[h.sub.b0]/[[rho].sub.s]g[d.sub.p] = (1-[[epsilon].sub.0]) ([h.sub.b0]/[d.sub.p]

This operation means that the gravity pressure at the microscopic level is the basic pressure scale used for non-dimensionalization as it was done earlier in this text. Finally, the scaling of the pressure drop across the bed becomes (see (A5a)):

[DELTA]P/[P.sub.g] = [DELTA]P/[[rho].sub.s]G[d.sub.p]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Therefore, the scaling [DELTA]P/[[rho].sub.s]g[d.sub.p] is determined by Ga, [DELTA][D.sub.BL], [h.sub.b0]/[d.sub.p] defining the initial bed geometry and conditions, while [B.sub.og-m] is the unique independent variable controlled by the external magnetic field. This example demonstrates how the ratio [h.sub.b0]/[d.sub.p] emerges in the group of dimensionless process variable through the accepted "pressure transform approach." To this end, the ratio ([DELTA]P/[P.sub.g]) [([P.sub.G]/[P.sub.g]).sup.-1] simply gives [DELTA]P/[[rho].sub.s]g[h.sub.b0] and (A-9a) takes the form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (A-9b)

as in most of the non-magnetic correlations (see Equations (1a)-(1c)) since (1-[[epsilon].sub.0]) is almost constant. Moreover, the initial bed conditions expressed bed (1-[[epsilon].sub.0]) has to appear in the prefactor of the power law in accordance with the rules of the dimensional analysis (Kline, 1965; Barrenblatt, 1996), so the correct form of the RHS of the second expression in (A-9b) is (1-[[epsilon].sub.0])f(Ga;[Bo.sub.g-c]; [DELTA][D.sub.bL]).

Example 4: the physical meaning of the "pressure transform"

Last but not least, the principle advantage of the pressure transform approach (Hristov, 2006, 2007a) is its physical adequacy and avoidance of formal algebraic calculations. By expression of the final set of variables as surface forces (dimension of pressure Pa) we in fact compare the significance of the fluxes of the physical fields acting on an elementary volume of the system. Looking deeply at the physics, in the case of fluid flow for example, the surface forces are proportional to the fluxes of the convection and the diffusion momentum transfer. That is, the fluid momentum flux transferred by convection is U ([[rho].sub.f]U) = [[rho].sub.f] [U.sup.2] with a pre-factor U of the fluid dynamic pressure [[rho].sub.f] [U.sup.2], while its diffusion counterpart is [tau] = - ([upsilon]/[d.sub.p]) ([[rho].sub.f]U) (Newton's law), with a pre-factor [upsilon]/[d.sub.p] and [d.sub.p] as a length scale at the particle level. The ratio of these fluxes, that is the comparison of their significance, is the Reynolds number [[rho].sub.f][U.sup.2]/[tau] = Re. Similarly the ratio of the convection momentum flux U([[rho].sub.f]U) = [[rho].sub.f][U.sup.2] to that generated by gravity related body forces [[rho].sub.f]g[d.sub.p] yields the Froude number Fr = [U.sup.2]/g[d.sub.p].

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Manuscript received December 17, 2007; revised manuscript received February 14, 2008; accepted for publication February 22, 2008.

Jordan Hristov *

Department of Chemical Engineering, University of Chemical Technology and Metallurgy, 1756 Sofia, 8 Kl. Ohridsky Blvd., Bulgaria

* Author to whom correspondence may be addressed. E-mail address: jyhoa actm.eda; jordan.hristovaa mail.bg; http://hristov.com/jordan

Can. J. Chem. Eng. 86:470-492, 2008

[c] 2008 Canadian Society for Chemical Engineering DOI 10.1002/cjce.20046

Table 1. Materials used in the experiments.

Material Fraction Density Magnetization
 ([micro]m) (kg/[m.sup.3]) of saturation
 Ms(kA/m) *

Magnetite ([Fe.sub.3] 200-315 5140 477.36
[O.sub.4]) sand 315-400

Ammonia catalyst 500-613 5100 236.34
KM-1, H. Topsoe 613-800

* See Penchev and Hristov (1990a) and Hristov (2002).

Table 2. Examples of data correlations through the scaling rules
developed in the present work and addressing the magnetic field
effect through the magnetic bond number.

Materials and Equation (Minimum Comments
bed conditions spouting pressure drop,
 [DELTA][P.sub.ms])

[Fe.sub.3] [O.sub.4] P-a [DELTA][P.sub.ms]/ Particle
(315-400 [micro]m) [[rho].sub.s]g[d.sub.p] = diameter as
 (Ga.[DELTA][D.sub.bL]) a length
 ([A.sub.dp] - [B.sub.dp] scale
 [Bo.sub.g-m])

 P-b [DELTA][P.sub.ms]/ Bed depth
 [[rho].sub.s]g[h.sub.b0] = as a length
 [(Ga.[DELTA][D.sub.bL]) scale
 ([A.sub.dp] - [B.sub.dp]
 [Bo.sub.g-m])] x
 ([d.sub.p]/[h.sub.b0])
 (1 - [[epsilon].sub.0])

G = 1 kg P 1-a [DELTA][P.sub.ms]/
 [[rho].sub.s]g[d.sub.p] =
 (Ga.[DELTA][D.sub.bL])
 (18.176 - 0.0105
 [Bo.sub.g-m])
 R = 0.975; N = 7 data
 points; SD = 1.584;
 P = 2.347 x [10.sup.-4]

 P 1-b [DELTA][P.sub.ms]/
 [[rho].sub.s]g[h.sub.b0] =
 [(Ga.[DELTA][D.sub.bL])
 (18.176 - 0.0105
 [Bo.sub.g-m])] x
 (1.255.[10.sup.-3])

G = 2 kg P 2-a [DELTA][P.sub.ms]/
 [[rho].sub.s]g[d.sub.p] =
 (Ga.[DELTA][D.sub.bL])
 (7.140 - 0.00381
 [Bo.sub.g-m])
 R = 0.7454; N = 6 data
 points; SD = 1.881;
 P = 0.089

 P 2-b [DELTA][P.sub.ms]/
 [[rho].sub.s]g[h.sub.b0] =
 [(Ga.[DELTA][D.sub.bL])
 (7.140 - 0.00381
 [Bo.sub.g-m])] x
 (0.865.[10.sup.-3])

G = 3 kg P 3-a [DELTA][P.sub.ms]/
 [[rho].sub.s]g[d.sub.p] =
 (Ga.[DELTA][D.sub.bL])
 (3.611 - 0.00309
 [Bo.sub.g-m])
 R = 0.915; N = 5 data
 points; SD = 0.4624;
 P = 0.01128

 P 3-b [DELTA][P.sub.ms]/
 [[rho].sub.s]g[h.sub.b0] =
 [(Ga.[DELTA][D.sub.bL])
 (3.611 - 0.00309
 [Bo.sub.g-m])] x
 (0.721.[10.sup.-3])

Notes: (1) For sake of clarity of presentation the numerical values
of the term ([d.sub.p]/[h.sub.b0]) (1 - [[epsilon].sub.0]) in the
equations denoted as "b" are especially located at the end of the
numerical expressions, thus repeating their analytical forms. (2) The
data correlations were performed by Origin 6.0. Information (common
nomenclature is used) pertinent to accuracy of approximation is
available close to each equation.

Table 3a. Examples of data correlations through the scaling rules
developed in the present work and addressing the magnetic field
effect through the magnetic bond number. Linear fits.

Bed Equation Equation: Linear fits Comments
weight Code (Minimum Spouting Velocity,
 [U.sub.ms-S] = [Q.sub.ms-S/
 [S.sub.b]), [Fe.sub.3] [O.sub.4]
 (315 - 400 [micro]m)

 UL-a [[rho].sub.f][U.sup.2.sub.ms-S]/ Particle
 [[rho].sub.s]g[d.sub.p] = diameter as
 (Ga.[DELTA][D.sub.bL]) a length
 ([A.sub.Udp] - [B.sub.Udp] scale
 [Bo.sub.g-m]) [right arrow]
 [U.sup.2.sub.ms-s]/g[d.sub.p] =
 [(Ga.[DELTA][D.sub.bL])
 ([A.sub.Udp] - [B.sub.Udp]
 [Bo.sub.g-m])] ([[rho].sub.s]/
 [[rho].sub.f])

 UL-b [U.sup.2.sub.ms-S]/g[h.sub.b0] = Bed depth
 [(Ga.[DELTA][D.sub.bL]) as a length
 ([A.sub.Udp] - [B.sub.Udp] scale
 [Bo.sub.g-m])] x
 [([[rho].sub.s]/[[rho].sub.f]) x
 ([d.sub.p]/[h.sub.b0])
 (1 - [[epsilon].sub.0])]

G = 1 kg UL-1-a [U.sup.2.sub.ms-S]/g[d.sub.p] =
 (Ga.[DELTA][D.sub.bL])
 (-1.386 + 0.0628[Bo.sub.g-m]) x
 5745.2;
 R = 0.9607; N = 7 data points;
 SD = 1.13; P = 5.732 x [10.sup.-4]

 UL-1-b [U.sup.2.sub.ms-S]/g[h.sub.b0] =
 (Ga.[DELTA][D.sub.bL])
 (-1.386 + 0.0628 [Bo.sub.g-m]) x
 5.24

G = 2 kg UL-2-a [U.sup.2.sub.ms-S]/g[d.sub.p] =
 (Ga.[DELTA][D.sub.bL])
 (-12.29 + 1.01 [Bo.sub.g-m]) x
 5745.2;
 R = 0.9274; N = 6 data points;
 SD = 22.38; P = 0.00771

 UL-2-b [U.sup.2.sub.ms-S]/g[h.sub.b0] =
 (Ga.[DELTA][D.sub.bL])
 (-12.29 + 1.01 [Bo.sub.g-m]) x
 3.358

G = 3 kg UL-3-a [U.sup.2.sub.ms-S]/g[d.sub.p] =
 (Ga.[DELTA][D.sub.bL])
 (-6.021 + 0.7506 [Bo.sub.g-m]) x
 5745.2;
 R = 0.9827; N = 5 data points;
 SD = 0.784; P = 4.463.[10.sup.-4]

 UL-3-b [U.sup.2.sub.ms-S]/g[h.sub.b0] =
 (Ga.[DELTA][D.sub.bL])
 (-6.021 + 0.7506 [Bo.sub.g-m]) x
 2.966

Notes: For sake of clarity and coherence with the dimensional analysis
developed the correlations are expressed through the ratios
[U.sup.2.sub.ms-U]/g[d.sub.p] and [U.sup.2.sub.ms-U]/g[h.sub.b0].

Table 3b. Examples of data correlations through the scaling rules
developed in the present work and addressing the magnetic field effect
through the magnetic Bond number. Exponential fits.

Bed Equation Equation: Exponential fits Comments
weight Code (Minimum Spouting Velocity,
 [U.sub.ms-S] = [Q.sub.ms-S]/
 [S.sub.b]), [Fe.sub.3][O.sub.4]
 (315 - 400 [micro]m)

 UE-a [[rho].sub.f][U.sup.2.sub.ms-S] = Particle
 (Ga.[DELTA][D.sub.bL]) [E.sub.Udp] diameter as
 exp ([k.sub.dp] [Bo.sub.g-m]) a length
 [right arrow] [U.sup.2.sub.ms-s]/ scale
 g[d.sub.p] =
 [(Ga.[DELTA][D.sub.bL]) [E.sub.Udp]
 exp ([k.sub.dp] [Bo.sub.g-m])]
 ([[rho].sub.s]/[[rho].sub.f])

 UE-b [U.sup.2.sub.ms-S]/g[h.sub.b0] = Bed depth
 [(Ga.[DELTA][D.sub.bL]) [E.sub.Udp] as a length
 exp ([k.sub.dp] [Bo.sub.g-m])] x scale
 [([[rho].sub.s]/[[rho].sub.f]) x
 ([d.sub.p]/[h.sub.b0])
 (1 - [[epsilon].sub.0])]

G = 1 kg UE-1-a [U.sup.2.sub.ms-S]/g[d.sub.p] =
 [(Ga.[DELTA][D.sub.bL]) 0.578 exp
 (0.0181 [Bo.sub.g-m])] x 5745.2

 UE 1-b [U.sup.2.sub.ms-S]/g[h.sub.b0] =
 [(Ga.[DELTA][D.sub.bL]) 0.5 78 exp
 (0.0181 [Bo.sub.g-m])] x 5.24

G = 2 kg UE-2-a [U.sup.2.sub.ms-S]/g[d.sub.p] =
 [(Ga.[DELTA][D.sub.bL]) 7.89 exp
 (0.022[Bo.sub.gm])] x 5745.2

 UE-2-b [U.sup.2.sub.ms-S]/g[h.sub.b0] =
 [(Ga.[DELTA][D.sub.bL]) 7.89 exp
 (0.022 [Bo.sub.gm])] x 3.358

G = 3 kg UE-3-a [U.sup.2.sub.ms-S]/g[d.sub.p] =
 [(Ga.[DELTA][D.sub.bL]) 11.238 exp
 (0.0168 [Bo.sub.g-m])] x 5745.2

 UE-3-b [U.sup.2.sub.ms-S]/g[h.sub.b0] =
 [(Ga.[DELTA][D.sub.bL]) 11.238 exp
 (0.0168 [Bo.sub.g-m])] x 2.966

Notes: For sake of clarity and coherence with the dimensional
analysis developed the correlations are expressed through the ratios
[U.sup.2.sub.ms-U]/g[d.sub.p] and [U.sup.2.sub.ms-U]/g[h.sub.b0].