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Prediction of Minimum Fluidization Velocity in Three-Phase Fluidized-Bed Reactors Faı1c¸ al Larachi,* Ion Iliuta,† Olivier Rival, and Bernard P. A. Grandjean Department of Chemical Engineering & CERPIC, Laval University, Que´ bec, Canada G1K 7P4
Knowledge of the onset of fluidization is of considerable relevance and the key to three-phase fluidized-bed reactors design and safe operation. Accordingly, using a wide historic ULmf database set up from the open literature, all the quantification methods proposed to predict the minimum fluidization liquid velocity in three-phase fluidized beds have been thoroughly revisited and critically evaluated herein. The database, providing access to diversified information related to over 540 measurements, is dedicated to embracing wide-ranging fluids and bed properties. It covers 30 various particles and 18 liquids and includes data such as aspect ratio, wall effect (or column-to-particle diameter) ratio, and ReLmf ranging from 0.8 to 27, 9 to 127, and 10-2 to 800, respectively. Indeed, the ULmf behavior is largely nonlinear and thus cannot be accurately described using the existing empirical and physical approaches. As a result, multilayer perceptron artificial neural networks have been extensively used to generate two highly accurate, a purely dimensional and a dimensionless, empirical correlations describing the ULmf. Using crosscorrelation analyses, two unsuspected effects, namely, the wall effect ratio and the liquid surface tension, have been unveiled and then incorporated as correlating variables in the neural network correlations. The resulting mean relative error produced by the dimensional correlation is about 16% while the estimated error associated with the dimensionless-based correlation is 30%. The prediction errors from both correlations are found to be insensitive to column-to-particle diameter ratio. Moreover, the neural network approach has been shown to predict with moderate success the minimum fluidization gas velocity, UGmf, in liquid-buoyed gas-activated three-phase fluidized beds containing coarse particles (dv > 1 mm) at high-input gas fractions. 1. Introduction Co-current upflow three-phase fluidized beds are widespread in diverse fields of fuel conversion and purification. For instance, H-Oil and LC-fining hydrogen addition commercial units using fluidized-bed technology are employed for upgrading heavy oil and residual feedstocks to high-quality synthetic crude oils.1 Three-phase fluidized beds are also found to be appropriate for a number of applications in such fields as methanation of coal-derived synthesis gases,2 coal liquefaction, Fischer-Tropsch synthesis, flue gas desulfurization and particulate removal, crystallization, wastewater treatment, fermentation, and so forth.3 Though it has been the focus of intense academic research over the last 30 years, proper design of three-phase fluidized beds, despite their industrial relevance, continues to rely by and large on know-how and rules of thumb. Part of this poor understanding arises from the inability of the current design correlations and models to adequately describe and quantify relevant hydrodynamic phenomena observed in these reactors. The first obvious hydrodynamic parameter of chief importance in the design of a three-phase fluidized bed is the minimum fluidization liquid velocity, ULmf. It represents the smallest superficial liquid velocity, which * To whom correspondence should be addressed. Phone: 1-418-656-3566. Fax: 1-418-656-5993. E-mail: flarachi@ gch.ulaval.ca. † Department of Chemical Engineering, Faculty of Industrial Chemistry, University Politehnica of Bucharest, Polizu 1, 78126 Bucharest, Romania.
at a given superficial gas velocity brings particles in the bed from rest to motion. A great deal of experimental work has been devoted, over the past 3 decades,4-30 to investigating the onset of fluidization in three-phase fluidized beds. Notwithstanding its relevance, the estimation tools aimed at predicting ULmf in three-phase fluidization have not always been successful.25 To predict this critical velocity, Zhang et al.25 derived a semitheoretical phenomenological separated-flow “gas-perturbed liquid” (or GPL) model in the form of an Ergun-like equation. This model is valid at the inception of fluidization when the pressure gradient is offset by the apparent solids weight. The GPL model was tested on a medium-size ULmf database (275 measurements of data)25 spanning almost 3 order of magnitudes for ReLmf. Its predictability was shown to surpass that of the “homogeneous-drift flux” model of Costa et al.,17 the “separated flow” model of Song et al.,21 and the “pseudo-homogeneous fluid” model of Zhang et al.25 Recently, Zhang et al.31 used the liquidperturbed gas model (LPG) to allow prediction of the onset of fluidization by gas activation at tiny superficial liquid velocities and high-input gas fractions. All five models, illustrated in Table 1, require the knowledge of the gas saturation (βGmf) for a given gas superficial velocity and the bed porosity (mf) at conditions of minimum fluidization. Apart from these models, eight fully empirical equations, listed in Table 1, are set based on narrow fluidization conditions. The aim of this contribution is to provide an updated analysis and a systematic comparison between the current ULmf models/correlations and the largest updated minimum liquid-fluidization database elaborated
10.1021/ie990435z CCC: $19.00 © 2000 American Chemical Society Published on Web 12/29/1999
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Table 1. Summary of Empirical Correlations and Models for ULmf and UGmf in Three-Phase Fluidization
upon using most of the ULmf data published since the 1970s (540 measurements). In this context, two alternative correlations are therefore elected on the basis of multilayer perceptron artificial neural network (ANN) modeling techniques and dimensional analysis. Using this approach, a hybrid dimensionless correlation and a dimensional correlation based on the raw operating variables are established.
2. Prevailing Methodologies for Predicting ULmf The eight ULmf empirical correlations are displayed in Table 1. A quick examination of its content reveals that the Begovich and Watson6 dimensionless correlation (eq 3) is asymptotically inconsistent at zero gas superficial velocity. The same observation from the data presented in this table can be similarly recorded further
Ind. Eng. Chem. Res., Vol. 39, No. 2, 2000 565 Table 2. Summary of Experimental Conditions fluid physical properties
operating conditions
780 e FL e 1623 8.9 × 10-4 e µL e 0.0719 0.025 e σL e 0.073 1.145 e FG e 1.159 1.3 × 10-5 e µG e 1.8 10-5
0 0.0102 e ULmf e 0.082 0.0004 e ULmf e 0.082 0 e UG e 0.615 293 e T e 312 P ) 0.1 MPa
packings and columns properties and ranges of dimensionless groups 78 × 10-5 e dv e 0.0101 0.68 e φ e 1 1290 e FS e 7510a 0.068 e dc e 0.1715 0.31 e H0 e 4.0 9 e dc/dv e 127 0.8 e H0/dc e 27 0 e ReG e 148 23.8 e ArL e 1.54 × 107 1.58 × 10-11 e MoL e 6.52 × 10-3 7.9 × 10-3 e φdc/dv e 0.11 0.01 e ReLmf e 828 1.92 e ReGmf e 61.6
Liquids: H2O, H2O + 1.1 M NaCl, H2O + 1% Na2HPO4, H2O + 0.5 N NaOH + 2 mM K4Fe(CN)6 + 2 mM K3Fe(CN)6, H2O + 1% EtOH, H2O + [0.5-1%] n-C5H11OH, H2O + 0.5% t-C5H11OH, H2O + [15-80%] glycerol, C6H12, C2Cl4, kerosene, mineral oil Shell Vitrea 32 Gases: air, N2 Solids: alumina extrudate, Co-Mo alumina extrudate, alumina sphere, alumina-ceria sphere, glass bead, polypropylene extrudate, polypropylene bead Distributors: tapered packed bed (glass bead or intalox saddle) + (1 or 2) perforated plate(s), perforated plate, Sulzer-structured packing, nozzle-type distributor References: 4-7, 9-16, and 18-30 a
For porous particles, the density of the liquid-saturated particles is considered.
for the empirical correlations of Bloxom et al.5 (eq 2), Fortin10 (eq 5), and Costa et al.17 (eq 6). Finally, the Ermakova et al.4 (eq 1), Begovich and Watson6 (eq 4), Nacef24 (eq 8), and Song et al.21 (eq 7) remaining 0 when correlations correctly reduce to ULmf ) ULmf UG f 0. Ranges of coefficient data used to establish a correlation generally affect its implementation by limiting its prediction intervals. To correct the empirical correlations for restriction of range, a number of conceptual semitheoretical models (Table 1) for predicting the minimum fluidization liquid velocity have been developed. Accordingly, Costa et al.17 derived a pseudohomogeneous flow model (eq 9) wherein the fluidparticle interactions are described by combining the Wallis drift flux model32 with the multiparticle drag equation of Wen and Yu.33 The Song et al.21 minimum fluidization model (eq 10) departs from the separate flow approach of Chern et al.34 to derive an equivalent diameter to be used in the Ergun equation expressing the liquid-solids interaction. In the separate flow structure, the gas is viewed to be in the column central core, the solids are adjacent to the column side wall and the liquid is concentrically sandwiched between them. Zhang et al.25 developed two models for predicting the minimum fluidization liquid velocity, namely, the GPL model (eq 11) and the “pseudo-homogeneous fluid” model (eq 12). The latter views the gas-liquid mixture as being a pseudo-homogeneous “slipless” pseudo-fluid. The model is valid only for very small gas-to-liquid velocity ratios and correspondingly for small gas holdups. The GPL model assumes that full support of the solids is provided by the exclusive action of the liquid, the velocity of which is augmented by the volume exclusion due to the gas presence. With this in mind, the LPG model was recently proposed31 (eq 13) to theoretically capture the gas-activated inception of fluidization of liquid-buoyed solids inventories at tiny liquid throughputs.
3. Comprehensive Database Building up the comprehensive database was the first important task of this study; 461 ULmf measurements for three-phase fluidization and 79 for liquid-solid fluidization (UG ) 0) have been collected from 26 references embracing wide-ranging fluids and bed characteristics. The experimental data compiled was obtained using 18 (organic and aqueous) pure solvents and solutions, air or nitrogen at atmospheric pressure, columns of 8 different sizes packed with solids of over 30 various sizes and having 7 different shapes, and over 13 distributors. Bed characteristics also included are the aspect ratios, H0/dc, with respect to bed height at rest and the wall effect ratios, dc/dv, ranging from 0.8 to 27 and 9 to 127, respectively. Key information related to the database is, for clarity purposes, summarized in Table 2 in terms of upper and lower limits of each variable. Though it is quite comprehensive and even advanced, the database does not include all the data appearing in the literature. Indeed, lacking operating parameters or nontractable graphs impeding the conversion of plotted data from original references forbid access to few experimental results. Finally, the database is arranged in a matrix form to ease cross-correlation computations. Each row in the database contains the minimum fluidization liquid velocity, the superficial gas velocity, the liquid and gas densities, the dynamic liquid and gas viscosities, the liquid surface tension, the solids density, the particle equivalent diameter, the particle sphericity, the column diameter, and the bed height at rest. 4. Confrontation of Models/Correlations to Database Listed in Table 1, the available correlations and models designed to predict the minimum fluidization liquid velocity are confronted with the aforementioned
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Table 3. Statistics for the Current Correlations To Predict ULmf from the Compiled Database level 3 test 〈eU〉 (%)
eqs (1) (2) (3) (4) (5) (6) (7) (8)
level 4 test 〈eU〉 (%)
R (%)
Empirical Correlations 95 85 47 2258 0.4 104 87 60 134 63 55 1171 54 131 191 84 51 125 78 144 87 49
48 496 61 81 500 113 68 65
(10) (10)a (11) (12) a
σ (%)
σ (%)
R (%)
60
85
105 96 296 92
88 87 58 89
88
87
Phenomenological Models 46.2 97.0 80.4 96.1 79.8 96.5 35.1 96.1
44.9 51.2 52.8 65.0
Zhang et al.25 correction of equivalent diameter.
database content and then discriminated using the following statistical criteria: (1) The mean relative error 〈eU〉 between the predicted and experimental values of ULmf:
〈eU〉 )
100 N
|
|
ULmf,calc,i
∑1-U N 1
Lmf,exp,i
(14)
(2) The deviation of the relative error around the mean relative error 〈eU〉 computed as a mean quadratic deviation:
σ ) 100 ×
x
1
N
∑1
N-1
(|
1-
| )
ULmf,calc,i ULmf,exp,i
2
- 〈eU〉
(15)
(3) The correlation coefficient to measure the quality of fit by the selected ULmf approach: R ) 100 × N
N
1
1
∑∑(U
x∑
Lmf,exp,i
- 〈ULmf,exp〉)(ULmf,calc,i - 〈ULmf,calc〉)
N
1
x∑ N
(ULmf,exp,i - 〈ULmf,exp〉)2
(ULmf,calc,i - 〈ULmf,calc〉)2
1
(16)
N is the number of data points and the symbol 〈 〉 stands for the ensemble-averaging operator. The statistics were performed at four levels. Level 1 involved a local-scale test of authors’ own data versus authors’ own correlation/model. Level 2 involved a localscale test of a database applicable range to authors’ own correlation/model. In level 3 a global-scale test was conducted over the whole database for each correlation/ model. Finally, in level 4, a global-scale test, the adjustable parameters of each correlation/model were re-optimized over the whole database to check whether the existing tools can be recommended. Only levels 3 and 4 statistics are discussed below. The mean relative error, the quadratic deviation, and the correlation coefficient obtained by comparing the ULmf calculated or predicted results of each correlation or model with the ULmf measured data contained in the database are given in Table 3. As a result, it can be seen that the worst predictions are provided by the Bloxom
et al.5 (eq 2), the Fortin10 (eq 5), and the Costa et al.17 (eq 6) correlations where the ULmf data corresponding to UG ) 0 were excluded. Despite its high standard deviation, the correlation of Ermakova et al.4 (eq 1) leads all the empirical correlations because it exhibits, as shown in Table 3, the less scattered estimations. In general, the results predicted by each correlation are not considered successful either because of unacceptably associated large standard deviations or low correlation coefficients (always 90-100 80-90 70-80 50-70 40-50 30-40 20< 7.9
3.3
12.6
22.9
22.4
〈eU〉 (%) 17.2
ULmf Dimensional Correlation 5.3 10.3 14.1 13.6 13.3
11.0
10.7
9.3
21.8
17.0
〈eU〉 (%) 24.3
ReLmf Dimensionless Correlation 15.9 24.6 22.8 42.7 33.2
47.3
23.7
Table 11. Statistical Tests of the ReGmf Estimation Approachesa model31
Zhang et al. (eq 13) dimensional ANN correlation a
〈eU〉 (%)
σ (%)
R (%)
996.0 64.0
1700.0 92.5
17.1 54.7
Only 78 measurements of data fulfilled UG/(UG + UL) g 0.93.
range of the physical operating variables data used to perform the ANN learning operation. However, the data relative to the input/output neural network variables form sparse and nonuniformly populated clusters within a hypervolume where predominant empty space represents missing information about the minimum fluidization phenomena. As a result, prediction of minimum fluidization velocity outside these clusters is risky and far from being guaranteed. The results obtained illustrate the inherent ability of ANNs to allow the development of effective and powerful correlations with less constraints than physical models. Furthermore, ANN ULmf correlations are more efficient nonlinear approximators and use no simplifying assumptions required by phenomenological models. Using ANN correlations for simulating the minimum fluidization liquid velocity allows trustful ULmf estimation without recourse to sophisticated “computational fluid dynamics” codes. Physical models require in general more efforts to improve their capabilities. A promising approach lies in the use of a hybrid methodology where neural networks can be combined with physical models. Such hybrid models combine the robustness advantage of physical models with the neural networks higher accuracy.38 Acknowledgment The authors gratefully acknowledge the financial support of the Fonds FCAR of Que´bec and the National Research Council of Canada. Nomenclature
Figure 6. Neural network and ReGmf model31 predictions versus experimental minimum fluidization gas Reynolds number for input gas fractions >93%. Dashed lines represent factor 2 above and below envelopes.
correlations proposed predict ULmf with uniform error distribution. Similarly, the estimated relative errors are found to be independent of the measured ULmf data. This observation outlines the suitability of the ANN correlations in predicting ULmf over a wide range of dc/dv ratio. For low liquid velocities and input gas fractions, UG/ (UG + UL), close to unity, the inception of fluidization in coarse-particle bed inventories (dv > 1 mm) is likely to be triggered by gas momentum at high superficial gas velocity flow rather than motion of tiny liquid flow.31 Thus, in the case of liquid-buoyed solids and gasactivated fluidization, UL is known and the superficial gas velocity at minimum fluidization, UGmf, has to be estimated. In this context, 78 measurements of data fulfilling the condition UG/(UG + UL) > 93%31 were isolated from the database. Estimating UGmf through the proposed ANN correlations amounts to solving an inverse problem where the input UGmf is unknown and the output UL is known. Table 11 and Figure 6 illustrate the moderate ability of the dimensional ANN correlation to capture estimates of ReGmf under these conditions. The predictions of ReGmf by the liquid-buoyed solids/ liquid-perturbed gas model (eq 13, Table 1) are also shown in Table 11 and Figure 6 for illustration. It is important to mention that the proposed modeling approach can only be used as a predictive tool over the
Ai ) normalized input variables ArG ) composite gas Archimedes number, ArG ) dv3FG(FS - FL)g/µG2 ArL ) liquid Archimedes number, ArL ) dv3FL(FS - FL)g/ µL2 ArGL ) gas-liquid Archimedes number, ArGL ) ArL((1 - βGmf)/(1 + βGmf)2)((FS - (1 - βGmf)FL)/(FS - FL)) CaL ) capillarity number, CaL ) µLULmf/σL dc ) column diameter (m) dv ) equivalent diameter of sphere having the same volume as the particle (m) 〈eU〉 ) mean absolute relative error Fb ) buoyancy force (Pa) Fc ) capillary force (Pa) Fg ) gravitational force (Pa) FiG ) gas inertial force (Pa) FiL ) liquid inertial force (Pa) Fv ) viscous force (Pa) FrG ) gas Froude number, FrG ) UG2/dφdv FrL ) liquid Froude number, FrL ) UL2/gdv H0 ) static bed height (m) MoL ) liquid Morton number, MoL ) gµL4/FLσL3 P ) pressure (MPa) ReG ) gas Reynolds number, ReG ) FGUGdv/µG R ˜ eLmf ) liquid Reynolds number based on linear velocity, R ˜ eLmf ) (FLULmfdv/(1 - βGmf)mfµL) ReL ) liquid Reynolds number, ReL ) FLULdv/µL S ) network normalized output T ) temperature (K) UG ) gas superficial velocity (m/s) UL ) liquid superficial velocity (m/s) Greek Letters βG ) gas void fraction per unit porous volume ) bed porosity of fixed bed mf ) bed porosity at minimum fluidization
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φ ) sphericity factor µ ) dynamic viscosity (Pa s) F ) density (kg/m3) σ ) standard deviation on 〈eU〉 σL ) liquid surface tension (N/m) ω ) network connectivity weight Subscripts G ) gas L ) liquid mf ) at minimum liquid-solid or three-phase fluidization (m/s) S ) solid Superscripts 0 ) no gas flow
Literature Cited (1) Bridge, A. G. Hydrogen Processing. In Handbook of Petroleum Refining Processes, 2nd ed.; Meyers, R. A., Ed.; McGrawHill: New York, 1996; Chapter 14, pp 14.3-14.68. (2) Blum, D. B.; Toman, J. J. Three-Phase Fluidization in a Liquid-Phase Methanator. AIChE Symp. Ser. 1977, 73, 115. (3) Fan, L. S. Gas-Liquid-Solid Fluidization Engineering; Butterworth Series in Chemical Engineering; Butterworths: Boston, 1989. (4) Ermakova, A.; Ziganshin, G. K.; Slin’ko, M. G. Hydrodynamics of a Gas-Liquid Reactor with a Fluidized Bed of Solid Matter. Theor. Found. Chem. Eng. 1970, 4, 84. (5) Bloxom, V. R.; Costa, J. M.; Herranz, J.; MacWilliam, G. L.; Roth, S. R. Determination and Correlation of Hydrodynamic Variables in a Three-Phase Fluidized Bed; MIT Report N219; Oak Ridge National Laboratory: Oak Ridge, TN, 1975. (6) Begovitch, J. M.; Watson, J. S. Hydrodynamic Characteristics of Three-Phase Fluidized Beds. In Fluidization; Davidson, J. F., Kearins, D. L. , Eds.; Cambridge University Press: Cambridge, 1978; pp 190-195. (7) Lee, J. C.; Al-Dabbagh, N. Three-Phase Fluidized Beds Onset of Fluidization at High Gas Rates. In Fluidization; Davidson, J. F., Kearins, D. L., Eds.; Cambridge University Press: Cambridge, 1978; pp 184-189, (8) Grace, J. R. Fluidized-Bed Hydrodynamics. In Handbook of Multiphase Systems; Hetsroni, G., Ed.; McGraw-Hill: New York, 1982; Chapter 8.1, pp 8.5-8.64. (9) Saberian, M. N.; Wild, G.; Charpentier, J. C. The Minimum Fluidization Velocity in Gas-Liquid-Solid Fluidization. In Frontiers in Chemical Engineering; Doraiswamy, L. K., Mashelkar, R. A., Eds.; Wiley Eastern: New Delhi, 1984; Vol. 1, pp 405-415. (10) Fortin, Y. Re´acteurs a` Lit Fluidise´ Triphasique: Caracte´ristiques Hydrodynamiques et Me´lange des Particules Solides. Ph.D. Thesis, Institut National Polytechnique de Lorraine, Lorraine, France, 1984. (11) Hassanien, S.; Delmas, H.; Riba, J. P. Transfert de Matie`re Liquide-Particules en Fluidisation a` Trois Phases. Entropie 1984, 119, 17. (12) Saberian, M. N.; Wild, G.; Charpentier, J. C. Onset of Fluidisation in Gas-Liquid Fluidised Beds. Hung. J. Ind. Chem. 1984, 12, 75. (13) Saberian, M. N.; Wild, G.; Charpentier, J. C.; Fortin, Y.; Euzen, J. P.; Patoux, R. Contribution a` l’E Ä tude Hydrodynamique des Re´acteurs a` Lit Fluidise´ Gaz-Liquide-Solide. Entropie 1984, 120, 30. (14) Catros, A.; Bernard, J. R.; Briens, C.; Bergougnou, A. “Gas Holdup above the Bed Surface and Grid Gas Jet Hydrodynamics for Three Phase Fluidized Beds. Can. J. Chem. Eng. 1985, 63, 754. (15) Fan, L. S.; Matsuura, A.; Chern, S. H. Hydrodynamic Characteristics of a Gas-Liquid-Solid Fluidized Bed Containing a Binary Mixture of Particles. AIChE J. 1985, 31, 1801. (16) Saberian, M. N. Contribution a` l’Etude de l’Hydrodynamique des Re´acteurs Fluidise´s a` Trois Phases Gaz-LiquideSolide. Ph.D. Thesis, Institut National Polytechnique de Lorraine, Lorraine, France, 1985.
(17) Costa, N.; De Lucas, A.; Garcia, P. Fluid Dynamics of GasLiquid-Solid Fluidized Beds. Ind. Eng. Chem. Process Des. Dev. 1986, 25, 849. (18) Saberian, M. N.; Wild, G.; Charpentier, J. C.; Fortin, Y.; Euzen, J. P.; Patoux, R. Contribution to the Hydrodynamic Study of Gas-Liquid-Solid Fluidized-Bed Reactors. Int. Chem. Eng. 1987, 27, 423. (19) Jean, R. H. Hydrodynamics and Gas-Liquid Mass Transfer in Liquid-Solid and Gas-Liquid-Solid Fluidized Beds. Ph.D. Thesis, The Ohio State University, Columbus, OH, 1988. (20) Nacef, S.; Wild, G.; Laurent, A.; Kim, S. D. Effets d’Echelle en Fluidisation Gaz-Liquide-Solide. Entropie 1988, 143-144, 83. (21) Song, G. H.; Bavarian, F.; Fan, L. S.; Buttke, R. D.; Peck, L. B. Hydrodynamics of Three-Phase Fluidized Bed Containing Cylindrical Hydrotreating Catalysts. Can. J. Chem. Eng. 1989, 67, 265. (22) Tavakoli, T. Hydrodynamique et Transfert Gaz-Liquide Non-Newtonien en Fluidisation Triphasique. Ph.D. Thesis, Institut National Polytechnique de Toulouse, Toulouse, France, 1989. (23) Grandjean, B. P. A. Hydrodynamique et Transfert de Chaleur Liquide Particule en Lit Fluidise´ Triphasique. Ph.D. Thesis, Universite´ de Montre´al, Montreal, Canada, 1989. (24) Nacef, S. Hydrodynamique des Lits Fluidise´s Gaz-Liquide-Solide. Effets du Distributeur et de la Nature du Liquide. Ph.D. Thesis, Institut National Polytechnique de Lorraine, Lorraine, France, 1991. (25) Zhang, J.; Epstein, N.; Grace, J. R.; Zhu, J. Minimum Fluidization Velocity of Gas-Liquid Fluidized Beds. Trans. Inst. Chem. Eng. 1995, 73, 347. (26) Miura, H.; Kawase, Y. Hydrodynamics and Mass Transfer in Three-Phase Fluidized Beds with non-Newtonian Fluids. Chem. Eng. Sci. 1997, 52, 4095. (27) Briens, L. A.; Briens, C. L.; Maragaritis, A.; Hay, J. Minimum Liquid Fluidization Velocity in Gas-Liquid-Solid Fluidized Beds. AIChE J. 1997, 43, 1180. (28) Briens, L. A.; Briens, C. L.; Maragaritis, A.; Hay, J. Minimum Liquid Fluidization Velocity in Gas-Liquid-Solid Fluidized Beds of Low-Density Particles. Chem. Eng. Sci. 1997, 52, 4231. (29) Begovich, J. M. Hydrodynamics of Three-Phase Fluidized Beds. MSc Thesis, Oak Ridge National Laboratory, Oak Ridge, TN, 1978. (30) Zhang, J. Bubble Columns and Three-Phase Fluidized Beds: Flow Regimes and Bubble Characteristics. Ph.D. Thesis, The University of British Columbia, British Columbia, Canada, 1996. (31) Zhang, J.; Epstein, N.; Grace, J. R. Minimum Fluidization Velocities for Gas-Liquid-Solid Three-Phase Systems. Powder Technol. 1998, 100, 113. (32) Wallis, G. B. One-Dimensional Two-Phase Flow; McGrawHill: New York, 1969. (33) Wen, C. Y.; Yu, Y. H. Mechanics of Fluidization. Chem. Eng. Prog. Symp. Ser. 1966, 62, 100. (34) Chern, S. H.; Muroyama, K.; Fan, L. S. Hydrodynamics of Constrained Inverse Fluidization and Semifluidization in a GasLiquid-Solid System. Chem. Eng. Sci. 1983, 38, 1167. (35) Yang, X. L.; Wild, G.; Euzen, J. P. Study of Liquid Retention in Fixed-Bed Reactors with Upward Flow of Gas and Liquid. Int. Chem. Eng. 1993, 33, 72. (36) Cloutier, P.; Tibirna, C.; Grandjean, B. P. A.; Thibault, J. NNFit, Logiciel de Re´gression Utilisant les Re´seaux a` Couches. http://www.gch.ulaval.ca/∼nnfit. (37) Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. Numerical Recipes in Fortran. The Art of Scientific Computing, 2nd ed.; Cambridge University Press: Cambridge, MA, 1992. (38) Iliuta, I.; Larachi, F.; Grandjean, B. P. A. Pressure Drop and Liquid Holdup in Trickle Flow Reactors: Improved Ergun Constants and Slip Correlations for the Slit Model. Ind. Eng. Chem. Res. 1998, 37, 4542.
Received for review June 16, 1999 Revised manuscript received October 22, 1999 Accepted November 2, 1999 IE990435Z
