Ind. Eng. Chem. Res. 2004, 43, 4405-4412
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Stochastic Simulation of Hydrodynamics of a Liquid-Solid Inverse Fluidized Bed T. Renganathan and K. Krishnaiah* Department of Chemical Engineering, Indian Institute of Technology Madras, Chennai 600 036, India
The stochastic method of Monte Carlo simulation is used to predict the hydrodynamic characteristics of a liquid-solid inverse fluidized bed for various particle sizes and densities. Pressure drop, bed voidage, and minimum fluidization velocity under steady-state conditions are predicted. Unsteady-state bed expansion/contraction for a step change in liquid flow rate is also simulated. To validate the simulation, experiments are conducted to measure the different hydrodynamic variables such as pressure drop, bed voidage, and minimum fluidization velocity under steady-state conditions and unsteady-state bed expansion/contraction. The predicted values of the hydrodynamic variables compare satisfactorily with the experimental data under both steady- and unsteady-state conditions. 1. Introduction An inverse fluidized bed (IFB) is an apparatus in which solid particles lighter than the continuous liquid medium are fluidized downward in the direction of gravity by downflowing liquid. IFBs can be operated as two- (liquid-solid) or three- (gas-liquid-solid) phase systems. The advantages of this new contacting pattern are large gas holdups, high rates of heat and mass transfer, low attrition of solid particles, and easy refluidization. Because of these advantages, IFBs find major applications in biochemical and environmental engineering.1 Information on hydrodynamics, heat- and masstransfer characteristics, and kinetics and contacting patterns is essential for the successful design of an IFB. Existing studies on two-phase IFBs have focused on the measurement of different hydrodynamic aspects such as pressure drop, bed voidage, and minimum fluidization velocity under steady-state conditions and their prediction using empirical approaches. For predicting pressure drop, empirical equations similar to the Ergun equation have been proposed by Ulaganathan and Krishnaiah2 and Vijayalakshmi et al.3 Similarly, for predicting bed voidage, Fan et al.,4 Karamanev and Nikolov,5 Ulaganathan and Krishnaiah,2 and Bendict et al.6 used different category of models,5 including the Richardson-Zaki equation,7 a multiparticle drag function, and a direct empirical relation. Likewise, correlations similar to the Wen and Yu equation8 have been proposed by Ulaganathan and Krishnaiah2 and Vijayalakshmi et al.3 for estimating the minimum fluidization velocity. It can be clearly seen that the methods used to predict the different hydrodynamic variables are empirical in nature. Therefore, in this paper, a stochastic simulation approach using the Monte Carlo (MC) technique is adopted to predict the above steady-state hydrodynamic characteristics. The study has also been extended to predict unsteady-state bed expansion/ contraction. The term Monte Carlo derives from the casinos in Monte Carlo, Monaco. Monte Carlo simulation is a * To whom correspondence should be addressed. Tel.: 91044-2257 8211. Fax: 91-044-2257 0509. E-mail: krishnak@ iitm.ac.in.
Figure 1. Simulation space.
stochastic method of simulation that utilizes sequence of random numbers to perform the simulation. Monte Carlo simulation has had a great impact in different fields of computational science. It is widely applied in computational physics and chemistry for predicting molecular arrangements and thermodynamic properties. The application of Monte Carlo simulation to predict characteristics of fluid particle systems, however, is quite limited.9 Recently, Seibert and Burns10 used Monte Carlo simulation to predict bed voidage in classical liquid-solid (L-S) fluidized beds. To the best of our knowledge, this is the only work available in the literature on the use of MC simulation for L-S fluidized beds, and no such work is available for IFBs. In this paper, pressure drop, bed voidage, and minimum fluidization velocity under steady-state conditions and unsteady-state bed expansion/contraction are simulated using the Monte Carlo (MC) technique for different particle characteristics and fluid velocities. Experimental data are collected on a two-phase IFB system and the simulated values are compared with the experimental data. 2. Simulation Methodology 2.1. Simulation Space. The fluidized bed is modeled using a small representative number of particles. The simulation space (Figure 1) consists of a solid boundary
10.1021/ie0304513 CCC: $27.50 © 2004 American Chemical Society Published on Web 06/23/2004
4406 Ind. Eng. Chem. Res., Vol. 43, No. 15, 2004
at the top, with infinite depth and fully periodic boundaries, and with dimensions 10 times the particle diameter in the X and Z directions. By providing 10 times the particle diameter in the X and Z directions, the effects of periodic boundary conditions on the simulation results is avoided. 2.2. Particle Placement and Movement. A typical simulation consists of two stages, viz., (1) random placement of particles to generate an initial configuration and (2) random movement of particles until a final configuration is reached. In the first stage, a pseudorandom number generator computes random particle coordinates, and particles are sequentially placed in the simulation space in a nonoverlapping configuration. If an attempted particle placement results in an overlap with an already-placed particle, the particle being placed is removed, and another set of center coordinates is generated until all of the particles are successfully placed in the simulation space. Even though the simulation space is of infinite depth in the Y direction, the particles are initially placed up to a height that is 3 or 4 times the length in the X and Z directions. The initial height of the bed does not have any effect on the simulation of the bed voidage. In the second stage, particle movement begins with the random selection of a single particle in the representative system. The selected particle is moved a distance s in a random direction, i.e., within a shell of radius s in three dimensions. The distance s is taken as the average distance available for the particles to move at the existing void fraction and is given by
s)
[(
)
1 - p 1-
1/3
]
- 1 dp
(1)
Here, is the average void fraction of the system, given by
)1-
NVp AH
(-∆E KE )
FB ) (Fl - Fp)Vpg
(4)
Several models, both theoretical and empirical, exist for the drag force acting on a particle within a flow field. For loc (the local void fraction) > p (the packed-bed voidage), we have chosen to use the equation of Khan and Richardson,11 given by
FD ) CD-2n loc
Flul2 πdp2 2 4
(5)
where CD is the drag coefficient, given by
4 Ar 3 CD ) Ret2
(6)
Ret ) Ret∞fu
(7)
with
()
fu ) 1 - 1.15
dp D
0.6
(8)
Ret∞ is determined using the standard drag curve equations given in Clift et al.12 The exponent n in eq 5 is given by the following relation11
n) (2)
4.8 + 0.1032Ar0.57fn
(9)
1 + 0.043Ar0.57fn
where
where H is the current expanded bed height of the system. If the movement of the particle results in an overlap with an adjacent particle, the selected move is rejected, and the selected particle is returned to the original coordinates. Once a nonoverlapping move has been made, the move is accepted or rejected in accordance with the acceptance criteria given below. 2.3. Acceptance Criteria. Nonoverlapping moves are accepted or rejected depending on the change in the energy of the particle resulting from its movement. Energetically favorable moves (moves that result in a lowering of the energy) are accepted with 100% probability. An energetically unfavorable move might still be accepted, but only with finite probability. As in the MC method of molecular simulation, the acceptance probability of an energetically unfavorable move is given by the equation
pa ) exp
of a failure of the acceptance criteria resets the selected particle’s coordinates to their previous values. 2.4. Energy Calculations. In the simulation, the change in the potential energy of the particle resulting from its movement is calculated using buoyancy, gravity, and fluid dynamic forces. The net of buoyancy and gravity forces affecting a submerged body is given by
(3)
where ∆E is the change in the energy of the particle resulting from the move and KE is the average kinetic energy of the particle. Once the change in the energy is calculated, a uniform random probability, pr, between 0 and 1 is generated, and the particle move is accepted if pa is greater than this value. A move declined because
()
fn ) 1 - 1.24
dp D
1.27
(10)
When loc ) p, the drag force is determined by the Ergun equation, given by
[
FD ) Vp
150µlul(1 - loc) dp23loc
+
]
1.75Flul2 dp3loc
(11)
This concept of using different drag relations depending on the void fraction has also been suggested by Gidaspow.13 In this work, p is taken as 0.45, corresponding to the typical void fraction of a buoyant packed bed immersed in water. The drag force given by eq 5 or 11 is a function of loc, the local void fraction. The local void fraction at the plane of the particle is calculated by passing a horizontal plane through the selected particle and dividing the sum of intersected particle areas by the total cross-sectional area. The intersected area of each particle is given by
(
Ai ) π
)
dp2 - ∆l2 4
(12)
where ∆l is the distance between the selected particle’s
Ind. Eng. Chem. Res., Vol. 43, No. 15, 2004 4407
center and the plane of intersection. The local void fraction is then defined as
loc ) 1 -
∑Ai A
(13)
Using eqs 4-13, the net change in the potential energy of the particle resulting from its movement is given by
∆E ) (FB - FD)(ynew - yold)
(14)
where yold and ynew are the y coordinates of the particle before and after movement, respectively. The particles undergo semi-random fluctuations because of the fluid motion. Although the fluid motion is not wholly random, the effect of fluctuations in the fluid motion on the voidage characteristics of the fluidized bed is approximated using the interstitial velocity of the fluid in the definition of the kinetic energy. Therefore, the average kinetic energy of the particle is calculated using the relation
( )
ul 1 KE ) mp 2 loc
(15)
NFD A
(16)
where FD is given by eq 5 or 11. For the purpose of comparison with the experimental data, the pressure drop is scaled by the ratio of static bed heights used in the experiment and in the simulation. 2.6. Prediction of Bed Height Dynamics. Monte Carlo simulation is directly applicable only for steadystate simulations. Hence, to use Monte Carlo simulation to predict the dynamics of bed expansion/contraction, the MC steps have to be converted to equivalent real time steps. In steady-state conditions, the average particle velocity in a fluidized bed is zero. Under unsteady-state conditions, the magnitude of the particle velocity is given by14
up ) |u2 - utn|
(17)
where u2 is the final velocity after the step change. The modulus symbol is used to take care of both step increase and decrease in velocity. The real time corresponding to M Monte Carlo steps is given by
t)
u jp ) )
|
∫up d
1 - 1
| [
1
|
(19)
ut 1 (n+1 - 1n+1) u2( - 1) - 1 n+1
]|
(20)
where is the average void fraction after M Monte Carlo steps. The time found from eq 18 is scaled by the ratio of the static bed height used in the experiment to that used in the simulation. For a particular particle, the simulation is run so that the bed reaches a steady-state configuration at the chosen initial velocity. Then, the velocity is set to the final value (similar to applying a step change in a real experiment), and once again, the simulation is run until the bed reaches the steady state at the final velocity. The void fraction at each MC step is converted to bed height using the experimental static bed height and the static bed void fraction using the equation
2
The change in the potential energy of the particle and the kinetic energy are then substituted into eq 3, and the result is compared to pr to determine whether the specific move is accepted or declined. Once such a singleparticle move has been completed, the process of selection, movement, overlap check, and acceptance check is repeated until the average void fraction in the system reaches a constant value that does not change with further moves. 2.5. Pressure Drop Calculation. The drag force experienced by all of the particles in the system is manifested as the pressure drop of the liquid. Thus the pressure drop is calculated using the relation
∆P )
where u j p is the average particle velocity at a void fraction of , given by
total distance moved by N particles in M steps Nu jp (18)
H)
(1 - 0)H0 (1 - )
(21)
2.7. Simulation Details. A flowchart showing the entire simulation procedure for calculating the void fraction, pressure drop, and bed height dynamics is presented in Figure 2. To simulate bed height dynamics, u1 and u2 are included in the input box along with computations specified in dotted box in the flowchart. For steady-state simulations, u1 and u2 and the step specified in dotted box are not required. The code for the simulation was written in C and run using the Microsoft Visual C++ 6.0 compiler. The simulations were performed on Pentium III 933 MHz processor. The computational time required to run a simulation was not significant. For example, a typical simulation using 1000 particles required a CPU time of the order of 5 min. 3. Experimental Details To validate the simulation, experiments were conducted in a two-phase liquid-solid inverse fluidized bed. A schematic diagram of the experimental setup of the IFB is shown in Figure 3. The column was made of acrylic with an i.d. of 89 mm and an o.d. of 97 mm using multiple sections. The total height of the column was 2.8 m, and that of the test section was 1.86 m. The column was marked with graph paper along its length for measurement of the height of the bed. The liquid was distributed by a liquid distributor from the top of the column. A packed bed of berl saddles was used to provide a uniform distribution of liquid. Water was pumped from a storage tank through calibrated rotameters and admitted at the top of the column through a distributor. Different rotameters were used to cover a wide range of liquid flows. To conduct unsteady-state experiments, a solenoid valve and two control valves were fixed on the downstream side of the rotameters to give step changes in the liquid flow rate. The water exited from the bottom of the column and recirculated back to the storage tank through an overflow weir. The overflow weir helps to maintain a constant water level in the column. A 1-in. hole was provided in the middle of the test section for the loading
4408 Ind. Eng. Chem. Res., Vol. 43, No. 15, 2004
Figure 2. Flowchart for Monte Carlo simulations.
and unloading of particles. A wire mesh at the top of the column and another above the bottom distributor help to retain the particles within the test section. Provisions were made for online measurement of local (cross-sectional-average) void fractions by the conductivity method15,16 to measure the overall bed void fraction accurately. The conductivity was measured by SS 314 electrodes (10 × 10 mm2 cross section and 20 mm length) that were fixed to the wall of the column with the inside surface flush. This arrangement does not hinder the free movement of fluidized particles in the column. A pair of electrodes was placed diametrically opposite with the longer side perpendicular to the axis of the column. Eighteen sets of such electrodes were placed at equal distances of 10 cm along the length of
the column. To record the online conductivity measurements, an electronic circuit consisting of a switching circuit, a conductivity meter, and an A/D card (16 bit, 16 channel, loaded in a computer) was fabricated. The switching circuit used to switch between the 18 electrodes can be operated either manually or automatically using a software trigger from the computer. In a typical steady-state experiment, for a chosen particle and static bed height, liquid at very low velocity was admitted into the column. The system was allowed to reach steady state as indicated by constant manometer and conductivity meter readings. After steady state had been reached, the pressure drop across the bed and the bed height were noted. The conductance at each of the 18 electrodes was recorded (in terms of proportional
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Figure 4. Effect of number of particles on simulated void fraction. Figure 3. Schematic diagram of experimental setup. Table 1. Characteristics of Particles dp (mm)
Fp (kg/m3)
Ar
0.180 0.328 2.89 6.1 5.0 6.1 8.0 12.2 12.9 12.6
693 688 911 917 849 860 846 835 610 250
17.6 108.0 21 074 184 815 185 164 311 736 773 499 2 939 222 8 213 017 14 717 766
voltage) at a sampling frequency of 10 Hz for 30 s each using the online data acquisition system. The liquid flow rate was varied in small intervals to cover the packedand fluidized-bed regimes with the above measurements being repeated every time until either the bed expanded to fill the entire test section or the maximum possible flow rate was reached. This procedure was repeated for various static bed heights (15, 30, and 45 cm) and particles (Table 1). In a typical unsteady-state experiment, for a chosen particle (dp ) 6.1 mm and Fp ) 917 kg/m3) and static bed height (H0 ) 45 cm), the system was allowed to reach steady state at a chosen initial flow rate. A predetermined step change in flow rate was then given using the solenoid valve. The variation of the conductance (in terms of voltage) with respect to time at the first electrode (below the top mesh) was recorded online at a frequency of 25 Hz using the conductivity meter and data acquisition system until the entire bed had reached steady state at the final flow rate. The simultaneous measurement of variations of voidage with time at different heights (electrodes) required as many conductivity meters as the number of electrodes. Therefore, the experiment with the same step change was repeated every time to record the voltage variation with respect to time at each of the remaining electrodes covered by the bed. The dynamic experiments were performed for different step increase (from 306 to 407, 509, 607, 708, 812, 918, 1016, and 1116 lph) and decrease (from 1116 to 760, 509 and 306 lph) in flow rates, where the units lph denote liter per hour. To estimate the holdup of solids using the conductivity method, single-phase (water) conductance measurements are required. These were obtained for each set of experiments by measuring the conductance of water
at every electrode. This conductance (Γ0loc) measured at each electrode was used to normalize the corresponding two-phase value (Γloc). The normalized conductance ratio (γloc) was converted to void fraction (loc) using the calibration equation
loc ) γm loc
(22)
where
γloc )
Γloc Γ0loc
(23)
and the exponent is obtained by calibration. The measured local void fractions are averaged using the equation
)
1 H
∫0Hloc dy
(24)
to get the average bed void fraction. 4. Results and Discussion For chosen number of spherical particles of given characteristics (Table 1), a Monte Carlo simulation was run corresponding to the different experimental conditions. 4.1. Effect of Number of Particles. For a chosen particle (dp ) 6.1 mm and Fp ) 917 kg/m3), the simulation was carried out using different numbers of particles in the simulation space. Figure 4 shows the variation of the void fraction with the number of particles used for simulation for different liquid velocities. It is evident that the predicted void fraction is independent of the number of particles (for sufficiently large numbers of particles) for all liquid velocities examined. Therefore, all results that follow were obtained using 1000 particles. 4.2. Prediction of Pressure Drop. The pressure drops predicted by the simulations are plotted as a function of liquid velocity for typical particle characteristics in Figure 5 along with the experimental data. As experimentally observed, the simulation also shows a fixed-bed region at low liquid velocities in which the pressure drop increases with liquid velocity and a fluidized-bed region in which the pressure drop is constant. The intersection of these two regions gives the minimum fluidization velocity.
4410 Ind. Eng. Chem. Res., Vol. 43, No. 15, 2004
Figure 5. Comparison of experimental and simulated pressure drops.
Figure 6. Comparison of experimental and simulated void fractions.
It can be seen from Figure 5 that the simulation predicts the experimental pressure drop quite satisfactorily for 5 mm particles. However, for micron-sized particles, the pressure drop obtained from experiment is higher than that predicted by the model in the packed-bed region. The reason for this is that the actual particles have a size distribution whereas the simulation assumes monosized particles. This makes the simulated pressure drop lower than the experimental values because the size distribution in the actual particles gives a greater experimental pressure drop because of the lower average voidage. 4.3. Prediction of Bed Voidage. The variation of void fraction predicted by the simulation is shown as a function of liquid velocity for typical particle characteristics in Figure 6 along with the experimentally measured void fractions. As with the simulated pressure drop, the simulated bed voidage also shows a fixed-bed region of constant void fraction at low liquid velocities and a regime of increasing void fraction beyond the minimum fluidization velocity. It can be observed from Figure 6 that the simulated and experimental void fractions agree satisfactorily for 5 mm particles. The reason for the discrepancy observed in Figure 6 for the 180 µm particles might again be that, whereas the actual particles used in the experiments have a size distribution, the simulation assumes monosized particles. The particles with an average diameter of 180 µm were obtained from sieves of size -200 + 160 µm. At any particular velocity, the particles around 160
Figure 7. Comparison of experimental and simulated minimum fluidization velocities.
µm in diameter would have expanded more than the particles of 200 µm diameter. This might be the reason for the higher bed expansion observed experimentally than in the simulation carried out for 180 µm particles. 4.4. Prediction of Minimum Fluidization Velocity. Using the simulated pressure drop curve for a particular particle, the minimum fluidization velocity is determined as the velocity at which the straight lines representing the packed-bed and fluidized-bed regimes intersect. This procedure of finding the minimum fluidization velocity is used for particles of different diameters and densities (Table 1). The same information (minimum fluidization velocity) can be obtained from simulated voidage vs velocity plots. The Reynolds number calculated from the simulated minimum fluidization velocities for different Archimedes numbers are plotted in Figure 7. The experimental data of the present work and the literature are compared with the simulated curve in the same figure. It is interesting to see that the Wen and Yu correlation8 found to be valid for both classical and inverse twophase fluidized beds also satisfactorily represents the simulated and experimental data obtained in this work. 4.5. Prediction of Unsteady-State Bed Expansion/Contraction. The bed heights obtained at different MC steps are plotted against the real time (obtained by converting the MC steps) in Figures 8 and 9 for various step increase and decrease in velocity, respectively. The time at which the bed surface crosses any
Ind. Eng. Chem. Res., Vol. 43, No. 15, 2004 4411
Figure 10 compares the experimental data on the variation of bed height with time for a step increase and decrease in flow rate of the same magnitude with those predicted by MC simulation. For the same magnitude of step change in flow rate, the time taken to reach steady state is more during bed expansion than during bed contraction because of instabilities arising during bed expansion.14 This hysteresis phenomenon occurring during dynamic bed response experiments is effectively brought out by the simulation, as can be seen in Figure 10. 5. Conclusions
Figure 8. Comparison of experimental and simulated temporal variation of bed height for step increase in velocity.
It can be concluded that Monte Carlo simulations are a powerful tool contributing to numerical experiments providing information on almost all hydrodynamic variables. The experimental data on hydrodynamics under steady-state conditions, viz., pressure drop, bed voidage, and minimum fluidization velocity, as well as unsteady-state bed expansion/contraction have been satisfactorily predicted using Monte Carlo simulation. Nomenclature
Figure 9. Comparison of experimental and simulated temporal variation of bed height for step decrease in velocity.
Figure 10. Hysteresis of bed height dynamics: simulation vs experiment.
electrode (at a particular height) is obtained from the measured spatiotemporal variations of the void fraction. These experimental data of bed height with respect to time are also plotted in Figures 8 and 9 for comparison. In general, a satisfactory agreement is seen between the simulated and measured temporal profiles. The reason for some disagreement between the predictions and experiments might be the nonconsideration of dispersion phenomena in the simulation.17 Further, the scaling of time steps using the ratio of the static bed heights might not be strictly valid in the presence of dispersion.
A ) cross-sectional area of the simulation space, m2 Ar ) Archimedes number ) dp3(Fl -Fp)Flg/µl2 Ai ) intersected area of each particle, m2 CD ) drag coefficient D ) column diameter, m dp ) particle diameter, m ∆E ) change in potential energy of a particle, N‚m fn ) wall correction factor for n fu ) wall correction factor for ut FB ) net buoyant force, N FD ) drag force acting on a particle, N g ) acceleration due to gravity, m/s2 H ) expanded bed height, m H0 ) static bed height, m KE ) average kinetic energy of a particle, N‚m ∆l ) distance between a selected particle’s center and the plane of intersection, m M ) number of MC steps m ) exponent in the voidage-conductance calibration equation mp ) mass of a particle ) πdp3Fp/6, kg n ) Richardson-Zaki exponent N ) number of particles ∆P ) pressure drop, N/m2 pa ) probability of acceptance pr ) uniform random probability Remf ) Reynolds number at minimum fluidization ) umfdpFl/µl Ret ) Reynolds number based on the terminal velocity ) utdpFl/µl Ret∞ ) Ret in infinite medium ) ut∞dpFl/µl s ) distance moved by a particle in one move, m t ) time, s u1 ) initial liquid velocity before a step change, m/s u2 ) final liquid velocity after a step change, m/s ul ) superficial liquid velocity, m/s umf ) minimum fluidization velocity, m/s up ) particle velocity, m/s u j p ) average particle velocity, m/s ut ) terminal velocity of particle, m/s ut∞ ) ut in an infinite medium, m/s Vp ) volume of a particle ) πdp3/6, m3 y ) distance along the Y axis, m ynew ) y coordinate of selected particle after movement, m yold ) y coordinate of selected particle before movement, m
4412 Ind. Eng. Chem. Res., Vol. 43, No. 15, 2004 Greek Symbols 0 ) void fraction of the static bed 1 ) void fraction before a step change in velocity loc ) local void fraction p ) void fraction of the packed bed ) average void fraction of the bed γloc ) local normalized conductance at any electrode in contact with the L-S mixture Γloc ) arithmetic mean of instantaneous local conductances at any electrode in contact with the L-S mixture, Ω-1 Γ0loc ) arithmetic mean of instantaneous local conductances at any electrode in contact with the liquid only, Ω-1 Fl ) liquid density, kg/m3 Fp ) particle density, kg/m3 µl ) liquid viscosity, Pa‚s
Literature Cited (1) Fan, L.-S. Gas-Liquid-Solid Fluidization Engineering; Butterworths: Stoneham, MA, 1989. (2) Ulaganathan, K.; Krishnaiah K. Hydrodynamic Characteristics of Two-Phase Inverse Fluidized Bed. Bioprocess Eng. 1996, 15, 159. (3) Vijayalakshmi, A. C. V.; Balamurugan, M.; Sivakumar, M.; Samuel, T. N.; Velan, M. Minimum Fluidization Velocity and Friction Factor in a Liquid-Solid Inverse Fluidized Bed Reactor. Bioprocess Eng. 2000, 22, 461. (4) Fan, L.-S.; Muroyama, K.; Chern, S.-H. Hydrodynamic Characteristics of Inverse Fluidization in Liquid-Solid and GasLiquid-Solid Systems. Chem. Eng. J. 1982, 24, 143. (5) Karamanev, D. G.; Nikolov, L. N. Bed Expansion of LiquidSolid Inverse Fluidization. AIChE J. 1992, 38 (12), 1916. (6) Bendict, R. J. F.; Kumaresan, G.; Velan, M. Bed Expansion and Pressure Drop Studies in a Liquid-Solid Inverse Fluidised Bed Reactor. Bioprocess Eng. 1998, 19, 137. (7) Richardson, J. F.; Zaki, W. N. Sedimentation and Fluidization: Part 1. Trans. Inst. Chem. Eng. 1954, 32, 35. (8) Wen, C. Y.; Yu, Y. H. Mechanics of Fluidization. Chem. Eng. Prog. Symp. Ser. 1966, 66, 101. (9) Dickinson, E.; Euston, S. Monte Carlo Simulation of Colloidal Systems. Adv. Colloid Interface Sci. 1992, 42, 89. (10) Seibert, K. D.; Burns, M. A. Simulation of Fluidized Beds and Other Fluid Particle Systems Using Statistical Mechanics. AIChE J. 1996, 42 (3), 660. (11) Khan, A. R.; Richardson, J. F. Pressure Gradient and Friction Factor for Sedimentation and Fluidization of Uniform Spheres in Liquids. Chem. Eng. Sci. 1990, 45, 255. (12) Clift, R.; Grace, J. R.; Weber, M. E. Bubbles, Drops and Particles; Academic Press: New York, 1978.
(13) Gidaspow, D. Multiphase Flow and Fluidization: Continuum and Kinetic Theory Descriptions; Academic Press: San Diego, CA, 1994. (14) Slis P. L.; Willemse, TH. W.; Kramers, H. The Response of the Level of a Liquid Fluidized Bed to a Sudden Change in the Fluidizing Velocity. Appl. Sci. Res. 1959, A-8, 209. (15) Begovich, J. M.; Watson, J. S. An Electroconductivity Technique for the Measurement of Axial Variation of Holdups in Three-Phase Fluidized Beds. AIChE J. 1978, 24 (2), 351. (16) Ibrahim, Y. A. A.; Briens, C. L.; Margaritis, A.; Bergongnou, M. A. Hydrodynamic Characteristics of a Three-Phase Inverse Fluidized-Bed Column. AIChE J. 1996, 42 (7), 1889. (17) Thelen, T. V. and Ramirez, W. F. Modeling of Solid-Liquid Fluidization in the Stokes Flow Regime Using Two-Phase Flow Theory. AIChE J. 1999, 45 (4), 708. (18) Fan, L.-S.; Muroyama, K.; Chern, S. H. Some Remarks on Hydrodynamics of Inverse Gas-Liquid-Solid Fluidization. Chem. Eng. Sci. 1982, 37, 1570. (19) Shimodaira, C.; Yushina, Y. Biological Waste Water Treatment with Downflow Fluidized Bed Reactor. In Proceedings of the 3rd Pacific Chemical Engineering Congress; Kim, S., Ihm, S. K., Eds.; Korean Institute of Chemical Engineers: Seoul, South Korea 1983; Vol. IV, p 237-242. (20) Legile, P.; Menard, G.; Laurent, C.; Thomas, D.; Bernis, A. Contribution to the Study of an Inverse Three-Phase Fluidized Bed Operating Countercurrently. Int. Chem. Eng. 1992, 32, 41. (21) Krishnaiah, K.; Guru, S.; Sekar, V. Hydrodynamic Studies on Inverse Gas-Liquid-Solid Fluidization. Chem. Eng. J. 1993, 51, 109. (22) Biswas, S. K.; Ganguly, U. P. A Preliminary Study on the Voidage-Velocity Relationship in Reverse Fluidization. Indian Chem. Eng. 1997, 39 (4), 303. (23) Calderon, D. G.; Buffiere, P.; Moletta, R.; Elmaleh, S. Anaerobic Digestion of Wine Distillery Wastewater in Down-Flow Fluidized Bed. Water Res. 1998, 32 (12), 3593. (24) Banerjee, J.; Basu, J. K.; Ganguly, U. P. Some Studies on the Hydrodynamics of Reverse Fluidization Velocities. Indian Chem. Eng. 1999, 41 (1), 35. (25) Buffiere, P.; Moletta, R. Some Hydrodynamic Characteristics of Inverse Three Phase Fluidized-Bed Reactors. Chem. Eng. Sci. 1999, 54, 1233. (26) Cho, Y. J., Park, H. Y.; Kim, S. W.; Kang, Y.; Kim, S. D. Heat Transfer and Hydrodynamics in Two- and Three-Phase Inverse Fluidized Beds. Ind. Eng. Chem. Res. 2002, 41, 2058.
Received for review May 27, 2003 Revised manuscript received May 6, 2004 Accepted May 10, 2004 IE0304513
