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Dimensional Cold-Modeling Criteria for Fluidization Quality K. Gallucci and L. G. Gibilaro* Dipartimento di Chimica, Ingegneria Chimica e Materiali, Universita` di L’Aquila, Monteluca di Roio, 67040 L’Aquila, Italy
Theoretical measures of the extent of instability of a fluidized suspension are shown to provide effective dimensional criteria for cold-modeling applications. The criteria are discussed and their flexibility demonstrated with regard to fluidization-quality predictions in dimensionally matched systems, which nevertheless fail to satisfy the dimensionless scaling rules. Experimental measurements of instantaneous pressure fluctuations in pressurized air-fluidized beds of various powders confirm the applicability of the proposed cold-modeling procedure. Introduction An ever present problem with the adoption of a fluidized bed route to a processing objective concerns the degree of uncertainty regarding the quality of the fluidization which will result in the proposed commercial unitsin particular, if the experimental parameters turn out to differ appreciably from those for which experimental data for fluidization quality are available. Under these circumstances, prudence dictates some degree of experimental evaluation before the costly construction of a final design solution is embarked upon. This calls for a “cold model” of the proposed unit, designed to simulate it with regard to its fluid-dynamic behavior, but constructed at low cost, for operation at ambient temperature and under benign conditions, far perhaps from those envisaged for the full production unit. “Fluidization quality” is a conveniently compact term which refers to the fluid-dynamic conditions brought about by the fluidization process itself. In the present context, it relates to a bubbling system: the strong effect on the fluidized environment of the presence of bubbles. These course upward through the bed faster than the interstitial gas, growing in size as they do so (mainly by coalescence), giving rise to vigorous particle mixing and substantial oscillation of the bed surface, through which they leave. In most applications the mixing action is advantageous; but this is more than offset by the lack of gas-particle contact in the bubbles, which effectively bypass the bulk of the bed, particularly where bubble growth becomes excessive. These and other manifestations of the bubbling fluidized state may critically affect the performance of a fluidized bed reactor and need to be taken into account at the design stage of a proposed application. Failure to do so can have serious economic consequences for the success, or otherwise, of a commercial undertaking. It was in view of these considerations that a number of workers in the early 1980s proposed the application of dimensionless fluid-dynamic scaling relations, which had long established an outstanding record of success in other fields, to cold-modeling studies of fluidized beds (Fitzgerald et al.1 and Glicksman2). This procedure involves no assumptions regarding the specific properties (bubble size, etc.) to be matched by the cold model but merely that the relevant dimensionless parameters * To whom correspondence should be addressed. E-mail:
[email protected].
defining the system dynamics be numerically equal for the model and original system. As these dimensionless numbers are defined in terms of combinations of the physical parameters (particle size and density, fluid viscosity, etc.), different numerical combinations of these can give rise to the same dimensionless number, thus enabling seemingly quite different physical systems to be matched on a dimensionless basis: the problem then becomes that of searching for a suitably realizable cold model on which experiments may be performed, which relates to the prime object of interest. Since the advent of the pioneering studies referred to above, numerous successful applications of the technique have been reported in the literature. The purpose of this paper is to consider alternative cold-modeling criteria (first proposed by Gibilaro3) to those deriving from the formal dimensionless scaling relations and to present results of experiments designed to test their suitability in cold-modeling applications. The criteria are based on the direct matching of parameters which characterize the instability of a fluidized suspension and hence the fluidization quality in a bubbling bed. It will be seen that these provide for a considerable measure of flexibility, which could prove useful in certain circumstances. In addition, the fact that they are less comprehensibly applicable than the dimensionless relations (they are unlikely to provide a match for the minimum fluidization velocity, for example) is of little practical consequence given that it is solely the need to characterize fluidization quality in a proposed industrial application which provides the incentive for cold-modeling studies in the first place. Predictive Parameters for Fluidization Quality Minimum Bubbling Void Fraction Emb. Bubble formation comes about as a result of the condition of instability of the homogeneously fluidized state. For systems which fluidize homogeneously at the minimum fluidization flux Umf and subsequently switch to the bubbling condition at the higher minimum bubbling flux Umb (Geldart group A behavior, characteristic of fine powders of less than about 100 µm diameter), the void fraction mb at the transfer from stable to unstable fluidization has been proposed as a measure of fluidization quality (Kwauk4)sthe higher mb, the better the fluidized environment for most process applications: smaller bubbles and generally smoother operation, approaching the fully homogeneously fluidized state as mb f 1. The fact that mb may be estimated from no
10.1021/ie049407t CCC: $30.25 © 2005 American Chemical Society Published on Web 10/30/2004
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more than a knowledge of the basic system parameterss fluid density and viscosity and particle density and diameter (Gibilaro3)smeans that it can be immediately thought of as a potential cold-modeling parameter, some measure of fluidization-quality equivalence deriving from a match of predicted mb in a proposed system with that in a more simply realizable one which seeks to mimic its fluid dynamic characteristics. The minimum bubbling void fraction mb alone would appear to be an insufficient criterion for the design of a full cold model, directed at uncovering all the relevant fluid-dynamic properties of its productive counterpart; this observation is placed on a quantitative footing below where a further complementary criterion is proposed to make up the perceived shortfall. Before considering that aspect, however, some observations are in order concerning the generality of mb as a measure of relative system instability. It is only for systems displaying Geldart group A behavior that mb has a real physical significance. Formulations of the momentum conservation equation for the particle phase of a fluidized suspension that contain a compressibility term can give rise to predictions of homogeneous fluidization, which may then give way to bubbling fluidization at a higher fluid velocity and a clearly defined value of mb. (Note that it is the compressibility term which, in effect, distinguishes a dispersed phase from a continuous one, “compression” describing the fact that dispersed particles may be brought closer together by the application of a force.) Predictions of mb within the physically realizable range above 0.4 indicate group A behavior; predictions of less than 0.4 are clearly unattainable in practice and simply indicate the system to be unstable and hence to start bubbling at the onset of fluidization: mb ) mf ≈ 0.4 (group B behavior). Although unrealizable in practice, predicted values of mb of less than 0.4 still provide a measure of relative instability, however, and hence of fluidization quality (Gibilaro3). Thus, a system predicted to remain stable up to a void fraction of only 0.3, say, is clearly “more unstable” than one for which stability is predicted up to void fraction 0.4: in this way, when it comes to predicted values for mb, Kwauk’s advocacy of this parameter as a measure of fluidization quality may be extended also to cases of group B and D behavior. To proceed in a quantitative manner, a specific formulation for the defining equations of change for fluidization, the particle bed model (Foscolo and Gibilaro5,6 and Gibilaro3), will henceforth serve as a reference. Wallis7 first presented the general criterion for homogeneous fluidization in terms of two parameters immediately identifiable from the momentum equation for the particle phase of a fluidized suspension: the velocities of the dynamic and kinematic waves, uD and uK, respectively. Stable, homogeneous fluidization occurs when uD > uK, bubbling fluidization occurs when uK > uD, and the transition between these two states occurs at uD ) uK at void fraction mb. The particle bed model quantifies this criterion, the value for mb following from the implicit relation
( )( )[
1.79 gdp n u2 t
0.5
Fp - Ff Fp
0.5
mb1-n
(1 - mb)
0.5
]
-1)0
(1)
where ut is the terminal settling velocity of a single, unhindered particle and n is the exponent in the Richardson-Zaki relation: U/ut ) n.
Growth Rate Gradient Parameter ∆a. It was anticipated above that something in addition to mb would be needed for the cold modeling of a proposed fluidized bed. This can be immediately appreciated from consideration of the different types of dynamic behavior observed in beds possessing the same mb values. For although at void fraction mb the amplitudes of particle concentration waves of all wavelengths remain constant, neither growing nor decaying with time, the situation at void fractions close to mb becomes strongly dependent on the particular system under consideration. This phenomenon is discussed by Gibilaro,3 where comparisons are presented of short perturbation wave amplitude growth rates, a, as functions of void fraction for various systems, all having the same minimum bubbling void fraction, mb ) 0.52, but very different manifestations of instability close to this critical condition. Typical group A behavior is represented by 70 µm alumina powder fluidized by ambient air: at void fractions just below mb the growth rate is very strongly negative and just above mb very strongly positive, indicating a system with a clearly defined minimum bubbling point, switching at mb from unambiguously homogeneous to unambiguously bubbling behavior. The other extreme is illustrated by the fluidization of 5 mm sand particles with ambient water: here the growth rate is very weakly negative for void fractions below mb and very weakly positive above it, indicating the ill-defined transition from stable to unstable behavior found in practice in which “parvoids” rather than bubbles are observed in the vicinity of mb, and whether these grow or decay with time is often difficult to determine. A parameter which could provide a measure of this feature of relative system stability is thus the gradient, ∂a/∂, evaluated at mb. For practical reasons, described by Gibilaro,3 a somewhat more suitable choice is the parameter ∆a, defined in eq 2, and evaluated explicitly from the particle bed model formulation as a function of mb
∆a ) mb
|
∂a ∂
mb
) 0.67
() g dp
0.52(n
- 1) - mb(2n - 1) mb(1 - mb)0.5
(2)
mb and ∆a together provide a predictive measure of the instability of the homogeneously fluidized state. mb is dimensionless and hence quite compatible with the conventional dimensionless scaling relations; ∆a, on the other hand, has the dimensions of reciprocal time. Together they represent a dimensional criterion for fluid-dynamic similarity and may be used to relate fluidization quality of a proposed fluidized bed to one whose dynamic behavior is known. Before turning to experimental results which demonstrate directly their suitability for cold-modeling applications, some consideration will be given to their flexibility vis-a`-vis the established dimensionless scaling relations. Comparison of Dimensional and Dimensionless Equivalence Criteria Some measure of fluidization quality for fluidization by ambient air is provided by the empirical Geldart8 map, which classifies powders (defined in terms of their material density and particle diameter) into groups A-D. We are concerned here mainly with group A and B type behavior, representing, respectively, powders which initially fluidize homogeneously and those which
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Table 1. Systems Matched on the Basis of the Proposed Dimensional Criteria
B1 B2 B3 B4
particle density, kg/m3
particle diameter, µm
air pressure, bar
air density, kg/m3
air viscosity, kg/ms
De
Ga
mb
∆a, s-1
1418 2550 6114 846
495 344 334 727
0.93 7.00 30.40 0.10
1.11 8.33 36.04 0.12
1.80 × 10-5 1.80 × 10-5 1.83 × 10-5 1.80 × 10-5
7.8 × 10-4 3.3 × 10-3 5.9 × 10-3 1.4 × 10-4
4.5 8.5 × 101 1.4 × 103 1.6 × 10-1
0.11 0.12 0.11 0.12
2670 2560 2620 2450
Figure 1. Generalized Geldart map for the fluidization of any powder by any fluid. (Points) Systems of Table 1: matched on the basis of the dimensional criteria.
bubble from the onset of fluidization. The particle bed model provides an immediate prediction of the boundary between group A and B type behavior of the Geldart classification (and an indirect evaluation of the other boundaries) for fluidization by any Newtonian fluid (defined in terms of its density and viscosity), thus enabling a theoretical generalization of the Geldart map to be constructed (Foscolo et al.9,10 and Gibilaro3): particle density of the original Geldart classification is thereby replaced by the particle/fluid density ratio, De-1, and particle diameter by the dimensionless Galileo number, Ga
De ) Ff/Fp, Ga ) gdp3Ff2/µf2
(3)
These two dimensionless groups constitute primary criteria for fluid-dynamic similarity in accordance with the established dimensionless scaling relations. The generalized Geldart map is shown in Figure 1. A point on this map represents a specific system and all its dimensionless cold models. On this diagram the locus of a point having a fixed value of mb is a line running approximately parallel to the group boundaries (the group A/group B boundary is the locus of mb ) 0.4). Also shown in the group B region of Figure 1 is a series of points, all representing systems with mb of approximately 0.12 (i.e., well away from physically realizable values for mb) and ∆a of approximately 2500. The dimensionless parameters, De and Ga, for these points differ widely, so that the four systems provide the potential means for evaluating the proposed dimensional cold-modeling criteria under conditions in which the dimensionless criteria give no reason to expect equivalence and for situations far removed from group A type behavior. The defining parameters (dp, Fp, Ff, and µf) for the four systems are shown in Table 1. They have been arrived at on the basis that the fluid is ambient-temperature air at a pressure which defines its density and viscosity and that the particles represent materials of specified densities. The procedure then involves selecting a key
Figure 2. Pressurized fluidized bed: essentials of layout.
system (B1, say), evaluating its mb and ∆a from eqs 1 and 2, then selecting another particle density and searching for values of particle diameter and air pressure which deliver the same mb and ∆a values as those of the key system. In this way particle densities corresponding to available particle materials may be selected. The results of such a numerical search, achieved using the Mathcad 2000 optimization routine, are shown in Table 1 together with values of the corresponding dimensionless parameters De and Ga. It will be seen that, in common with other routines tested, exact equalities of mb and ∆a are not obtained. The close approximations, however, serve the purpose from a practical point of view. The selected particle densities of systems B1 and B2 of Table 1 correspond to alumina and sand particles, respectively, both available in wide particle size ranges in the fluidization laboratory at L’Aquila. The numerical search yielded the particle size (344 µm) and air pressure (7 bar) for system B2, which lead to mb and ∆a values approximately equal to those of the key, ambient pressure, B1 system. Experimental Procedure and Results Powders for systems B1 and B2 where obtained by sieving the respective wide size distribution stocks to produce bed inventories of the required Sauter mean diameters (495 and 344 µm, respectively). Fluidization experiments were then performed with these systems in a bed of 7.4 cm diameter at various loadings in order to characterize the fluidization quality in each case. For operation at 7 bar, a pressure rig, designed to enable visual observation of the bed through narrow vertical windows, was employed (Foscolo et al.11). Bubbling
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Figure 3. Normalized PSDFs: U - Umf ) 0.047 m/s. Table 2. Systems Matched on the Basis of the Dimensionless Scaling Rulesa
B1 B* a
particle density, kg/m3
particle diameter, µm
air pressure, bar
air density, kg/m3
air viscosity, kg/ms
De
Ga
mb
∆a, s-1
1418 2640
495 303
0.93 2.09
1.11 2.49
1.80 × 10-5 1.80 × 10-5
7.8 × 10-4 9.4 × 10-4
4.5 5.2
0.11 0.12
2670 3250
B1: Umf ) 0.128 m/s, D/dp ) 150, H/dp ) 404. B*: Umf ) 0.087 m/s, D/dp ) 158, H/dp ) 396.
fluidization was observed for all cases at the lower gas fluxes, with a subsequent gradual transition to slugging behavior, particularly for the higher length beds. The experiments consisted simply in monitoring pressure fluctuations in the beds, which are brought about predominantly by bubbles breaking through the bed surface. On this basis, parameters which characterize the pressure fluctuations serve as measures of fluidization quality, thereby permitting comparisons to be made of systems matched in accordance with any chosen criteria. The general layout of the experimental rig is shown in Figure 2. The characterizing parameters adopted were the root mean square (rms) and the dominant frequency of the pressure fluctuations, measured as functions of gas flux over the operating range. The procedure in every case involved fluidizing the bed at a steady gas flux and then sampling the instantaneous pressure signal at 0.01 s intervals over a time period of approximately 80 s. The pressure values were logged in a PC, where they were processed to produce the mean, the rms, and, by means of a fast Fourier transform routine, the power spectrum density function (PSDF), typical examples of which are reported in Figure 3 for the two systems. All experiments were repeated once, both results being reported in the figures that follow, reproducibility being generally very good indeed. System Matching in Accordance with Dimensionless Scaling Criteria. Preliminary fluidization runs were carried out to test the reliability of the equipment and experimental procedure. These consisted of matching system B1 with another system B1* in the conventional manner on the basis of the dimensionless scaling criteria, De and Ga. This procedure requires that the two systems be geometrically similar and that comparisons be made at equal values of the dimensionless gas fluxsconditions which specify the imposition of two further criteria, the length number Le and the flow number Fl
Le ) L/dp, Fl ) U/Umf
(4)
where L is a representative length dimension of the bed and Umf is the measured value of the minimum fluidization flux.
Figure 4. Systems of Table 2, matched on the basis of the dimensionless scaling rules: rms/FpUmf2 vs U/Umf. System B1: open symbols. System B*: solid symbols.
The relevant parameters of the dimensionless matched systems are given in Table 2, and the experimental results, which confirm dimensionless equivalence, are reported in Figure 4. System Matching in Accordance with the New Dimensional Criteria. Whereas for the dimensionless criteria considered above the additional conditions, eq 4, follow more or less directly, the same cannot be said for the dimensional criteria of eqs 1 and 2. Some investigation is required in this respect: what flow rate and loading condition for the cold model may be deemed equivalent to those specified in the system it seeks to mimic? There is no reason to imagine that the dimensionless conditions of eq 4 will be suitable for this purpose. As dimensional similarity relates to bubble size and frequency in the matched systems, it would appear likely that the bed diameters should be the same. This condition is imposed for all experiments reported in Figures 5-11. A similar condition has been imposed with regard to equivalent gas fluxes, the excess gas flux, U - Umf, being chosen as the independent variable for comparative purposes in every reported case. Note that this implies very different mass flows of gas where the matched systems are operated at significantly different pressures, as in the case considered here. The question of comparable bed loading still remains open. Two
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Figure 5. Systems matched on the basis of dimensional criteria. Equal mass loading of 1.3 kg. System B1: open symbols. System B2: solid symbols.
Figure 6. Systems matched on the basis of dimensional criteria. Equal mass loading of 1.9 kg. System B1: open symbols. System B2: solid symbols.
Figure 7. Systems matched on thebasis of dimensional criteria. Equal packed bed heights of 20 cm. System B1: open symbols. System B2: solid symbols.
options have been examined experimentally: equal packed bed heights and equal masses of bed inventory. (i) Equal Masses of Bed Inventory. Figures 5 and 6, showing the systems matched on the basis of equal masses of particles (1.3 and 1.9 kg, respectively) and hence effectively of bed pressure drop, show excellent agreement both with regard to rms and dominant frequency over the full tested range of operation. (ii) Equal Packed Bed Heights. Figures 7 and 8 compare the two matched systems on the basis of equal packed bed heights, 20 and 32 cm, respectively. In both cases there are clear differences in both the rms and the dominant frequency of the pressure fluctuations for the matched systems as functions of excess gas flux. Figures 9 and 10 show these same results for pressure rms of these equal packed bed height systems, this time normalized with respect to bed mass. This has the effect of bringing the two trends very close together for both bed heights, a result compatible with the equal particle mass comparisons of Figures 5 and 6, and also with
numerical simulations of two-dimensional particle bed model simulations for equal height beds reported by Gibilaro3 using the code described by Chen et al.12 Ill-Matched Systems Compared on the Basis of the Additional Dimensional Criteria. As a final check on the validity of mb and ∆a as dimensional equivalence criteria for fluidization quality, runs were performed on a further system, B**, consisting of ambient pressure fluidization of glass particles of diameter 111 µm and density 2570 kg/m3, matched with respect to the additional criteria for dimensional equivalence employed successfully above (equal bed diameter, gas flux, and mass of particles) but with significantly different values for the primary dimensional equivalence criteria, mb and ∆a, 0.29 and 3056 s-1, respectively. The results of fluidization tests are shown in Figure 11. It is clear that this system gives rise to quite different fluidization-quality characteristics than those matched on the basis of equal mb and ∆a, the significantly lower rms values corresponding to smaller bubbles,
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Figure 8. Systems matched on the basis of dimensional criteria. Equal packed bed heights of 32 cm. System B1: open symbols. System B2: solid symbols.
Figure 9. Systems matched on the basis of dimensional criteria. Equal packed bed heights of 20 cm. System B1: open symbols. System B2: solid symbols.
Figure 11. Unmatched systems: equal mass loading of 1.3 kg. System B1: open symbols. System B**: solid symbols.
on the basis of the proposed criteria exhibit the same rms of the pressure fluctuations when their mass loadings are the same; when, on the other hand, initial (packed bed) heights are the same, the rms of pressure per unit mass of bed inventory become closely matched, indicating the same mean bed surface fluctuation and hence the same maximum bubble size. Acknowledgment This work was funded by the Italian Ministry for Education, Universities and Research under their program PRIN 2003. Figure 10. Systems matched on the basis of dimensional criteria. Equal packed bed heights of 32 cm. System B1: open symbols. System B2: solid symbols.
as expected for a system closer to the group A/group B boundary of Figure 1. Conclusions The reported experiments, which confirm the tentative findings of an earlier numerical simulation, illustrate the applicability of novel dimensional criteria (for the relative instability of bubbling fluidized beds) to cold-modeling applications. These criteria can predict an essential fluid-dynamic equivalence for systems differing widely on the basis of the conventional dimensionless scaling relations, thereby providing a degree of flexibility in the search for realizable cold models for proposed commercial fluidized bed applications. At present, the demonstrated equivalence is confined to beds of the same diameter, operated at the same excess gas flux. Under these conditions, beds matched
Nomenclature a ) perturbation amplitude growth rate, 1/s dp ) particle diameter, m D ) bed diameter, m De ) density number g ) gravitational field strength, N/kg Ga ) Galileo number H ) bed height, m L ) representative length, m n ) Richardson-Zaki exponent uD ) dynamic wave velocity, m/s uK ) kinematic wave velocity, m/s ut ) single-particle terminal velocity, m/s U ) gas flux, m/s Umf ) gas flux at minimum fluidization, m/s ) void fraction mb) void fraction at minimum bubbling condition mf ) void fraction at minimum fluidization condition µf ) fluid viscosity, Ns/m2 Ff ) fluid density, kg/m3 Fp ) particle density, kg/m3
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Literature Cited (1) Fitzgerald, T.; Bushnell, D.; Crane, S.; Shieh, Y.-C. Testing of Cold Scaled Modelling for Fluidized Bed Combustors. Powder Technol. 1984, 38, 107. (2) Glicksman, L. R. Scaling Relationships for Fluidized Beds. Chem. Eng. Sci. 1984, 39, 1373. (3) Gibilaro, L. G. Fluidization-Dynamics; Butterworth Heinemann: Oxford, 2001. (4) Kwauk, M. Fluidization: Idealized and Bubbleless, with Applications; Ellis Horwood: New York, 1992. (5) Foscolo, P. U.; Gibilaro, L. G. A Fully Predictive Criterion for the Transition between Particulate and Aggregate Fluidization. Chem. Eng. Sci. 1984, 39, 1667. (6) Foscolo, P. U.; Gibilaro, L. G. Fluid Dynamic Stability of Fluidized Suspensions: the Particle Bed Model. Chem. Eng. Sci. 1987, 42, 1489. (7) Wallis, G. B. One-Dimensional Two-Phase Flow; McGrawHill: New York, 1969. (8) Geldart, D. Types of Fluidization. Powder Technol. 1973, 7, 275.
(9) Foscolo, P. U.; Gibilaro, L. G.; Di Felice, R.; Pistone, L.; Piccolo, V. Scaling Relationships for Fluidization: the Generalised Particle Bed Model. 1990, 45, 1647. (10) Foscolo, P. U.; Gibilaro, L. G.; Di Felice, R. Hydrodynamic Scaling Relationships for Fluidization. Appl. Sci. Res. 1991, 48, 315. (11) Foscolo, P. U.; Germana`, A.; Di Felice, R. An Experimental Study of the Expansion Characteristics of Fine Catalysts under Pressure. In Fluidization IV; Grace, J. R., Shemilt, L. W., Bergougnou, M. A., Eds.; Engineering Foundation: New York, 1989. (12) Chen, Z.; Gibilaro, L. G.; Foscolo, P. U. Two-Dimensional Voidage Waves in Fluidized Beds. Ind. Eng. Chem. Res. 1999, 38, 610.
Received for review July 6, 2004 Revised manuscript received August 26, 2004 Accepted August 30, 2004 IE049407T
