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Ind. Eng. Chem. Res. 2007, 46, 1836-1842
Solid Dispersion Studies in Expanded Beds Srikumar V. Murli, Prakash V. Chavan, and Jyeshtharaj B. Joshi* Department of Chemical Engineering, Institute of Chemical Technology, UniVersity of Mumbai, Matunga, Mumbai, India 400 019
Solid-phase axial dispersion has been measured in 0.15-m-i.d. solid-liquid fluidized beds. The solid phase was ion-exchange resin in the size range of 300-1000 µm. The particle size distribution was measured at 34 axial locations ranging from 0.03 to 1.10 m from the bottom. A model describing the particle classification enabled the estimation of axial dispersion coefficient. Based on the experimental results, a correlation has been developed that has been found to be applicable to all the reported data in the published literature over a wide range of Reynolds number. Introduction Solid-liquid fluidized beds (SLFB) are used in chromatographic separation, ion exchange, crystallization, particle classification, etc. Thus, this equipment finds wide applications in purification from cell suspensions, crude cell lysates, refolded mixtures, or other viscous particulate-loaded streams. In these applications, SLFB is usually operated in the low range of the Reynolds number so that particle motion remains constrained. The particles are also selected with such a size range that the average particle size continuously decreases from bottom to top, which also imparts steadiness to the particle motion. The main advantage of a fluidized bed is that the chromatographic medium is in an expanded form and allows fermented suspensions to pass through the column. Further, fluidization reduces the dead zones. These enable an ease of operation as compared to fixed beds. However, the axial dispersion in solid phase needs to be kept very low to get high throughputs, extent of adsorption, and level of resolution. In the case of the fluidization of monosized particles, Carlos and Richardson1 found the primary mechanism of axial mixing to be turbulent dispersion. This, however, is not the only mechanism when different sized particles are present in the bed. In this case, convection motion of particles also prevails and the flux due to convection motion equals the flux due to axial dispersion. In a multiparticle bed, smaller particles get located toward the top and the larger particles closer to the bottom, leading to a special case of fluidized bed with constrained particle dispersion, called “classified” fluidized bed. This particle classification causes a variation in voidage along the bed height. The superficial velocity is bound on the lower limit by the minimum fluidization velocity of the largest particle and on the upper limit by the entrainment velocity of the smallest particle.2 These limits enable the selection of particle size distribution in such a way that the resulting voidage permits ease of flow, however limiting the particle dispersion to a minimum possible level. The following mathematical model describes the net flux of solid phase as a result of dispersion and classification (Thelen and Ramirez).2
(
)
d∈Si Flux ) -DSi + ∈SiVPi FS dz
(1)
The above equation is written for each size group. In the case of solids in a batch mode, there is no net flux of solids across * To whom correspondence should be addressed. Tel: +91-2224145616. Fax: +91-22-24145614. E-mail:
[email protected].
any horizontal plane. In eq 1, VPi is the classification velocity for particle i and is given by
VPi ) VL/∈L - VSi
(2)
VSi is the hindered settling velocity for particle i. The Richardson-Zaki3 equation is valid for an extreme case of monosized solid particles. Therefore, Joshi4 proposed a modification in the Richardson-Zaki equation for the case of multisized particles by taking a suitable average of individual terminal settling velocities. For these modifications, the following assumptions have been made: (1) The Richardson- Zaki index for a particular particle size (ni) also holds in a multisize particle system. (2) The value of ni remains constant irrespective of any size distribution around the particle i. (3) At a given location, the voidage is uniform across the cross section.
VSi ) VS∞ ∈nLi-1
(3)
m
VS∞ )
VS∞i ∈Si ∑ i)1 m
(4)
∈Si ∑ i)1 Al-Dibouni and Garside5 measured solid-phase dispersion for monosized and multisized particles. For monosized particles, they found that the value of DS reaches a maximum at certain voidage. This is because the value of DS depends upon the intensity of turbulence which, in turn, shows a maximum with respect to voidage. This argument gets supported by the experimental measurements of the turbulence intensity (uz′) by Handey et al.6 and Yutani et al.7 These results have been correlated by Joshi4 in the form of following equation:
uz′ ) 1.5∈SVS
(5)
where VS is the hindered settling velocity and is related to terminal settling velocity by the Richardson-Zaki equation:
VS ) VL/∈L ) VS∞ ∈Ln-1
(6)
substitution of eq 6 in eq 5 gives
uz′ ) 1.5VS∞(1 - ∈L)∈Ln-1
(7)
by taking duz′/d∈L and equating to zero, it can be shown that the maximum turbulence prevails at
∈L,max ) (n - 1)/n
10.1021/ie0612257 CCC: $37.00 © 2007 American Chemical Society Published on Web 02/20/2007
(8)
Ind. Eng. Chem. Res., Vol. 46, No. 6, 2007 1837 Table 1. Summary of Experimental Studies on Particle Mixing in Multisize Particle System
for the turbenlent regime n ) 2.4 and uz′ is maximum at ∈L ) 0.6 which agrees favorably with the observatios of Al-Dibouni and Garside.5 For multisized particles, a maxima in mixing was not observed by Al-Dibouni and Garside,5 Juma and Richardson,8 and Van Der Meer et al.9 This is probably because of the existence of classification velocity. The feature of classification velocity does not exist in the case of monosized particles and hence eq 1 is not applicable. In the experiments conducted by Juma and Richardson8 on binary mixtures of 2-3 and 2-4 mm, they observed that the value of DS for the 2-mm particles is higher in the presence of 4-mm particles as compared to that in the presence of 3-mm particles. This is because of the higher level of turbulence created by 4-mm particles as described by eq 5. Van Der Meer et al.9 performed experiments with binary ionexchange resin particles. They found the effect of column diameter to be negligible. This is because, the solid-liquid fluidized beds operate in a homogeneous regime where the turbulence is generated due to the relative velocity between the particles and the liquid (eq 5). The turbulence created near the wall is negligible as compared with that within the bulk of the bed. In contrast, in the heterogeneous regime, large-scale bulk circulation prevails where the turbulence length scale is of the order of column diameter. Therefore, the bulk turbulence that decides the dispersion coefficient in heterogeneous regime is known to be proportional to D4/3, where D is the column diameter. Asif and Petersen10 based their work on a mass balance approach as against the volume balance approach adopted by the previous authors. In a multisize particle system, the variation in voidage causes a variation in bulk density. However, if the bulk density was taken as a function of height, it would come out to be the same as eq 1. Barghi et al.11 estimated the solid
dispersion coefficient in a binary particle system by a collision method based on an electrochemical technique. Over the range of velocity used in the study, it was observed that the dispersion coefficient increased with an increase in the liquid velocity. It was also observed that the value of DS for a certain equivalent diameter is practically independent of the particle shape. Further, an extent of dispersion was found to be higher near the distributor plate, probably because of the turbulence generated by liquid jets emerging through the distributor holes. Table 1 summarizes the previous work. From Table 1 and the foregoing discussion, it is clear that the case of multiparticle system has not received due attention. Further, as mentioned earlier, operation of chromatographic separation occurs in the low Re∞ range. In the published literature, the low Re∞ range has not been sufficiently investigated. Therefore, it was thought desirable to investigate the solid-phase dispersion in a multiparticle system and also cover the low Re∞ range. Further, it was also thought desirable to make an effort to develop a unified correlation for DS over a wide range of Re∞. Equipment and Methods Column. A 150-mm-i.d. and 1.6-m-long acrylic pipe was used as the column. The schematic diagram of the experiential setup is shown in Figure 1. A positive displacement pump of 125 mL/s capacity was provided with a dampener. A calming section packed with glass beads of 0.3-m height was provided below the distributor. The distributor was a sieve plate with 2-mm holes on a 4-mm square pitch, giving a free area of 20%. A mesh of BSS 200 was attached to the distributors at the top and bottom of the column to restrict the movement of particles across the distributor plate. Pressure tapings were provided at spacing of 0.1 m along the column. The pressure was measured by a manometer, with a liquid mixture of chlorobenzene and chloroform, of 1.25 specific gravity.
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Figure 1. Schematic diagram of experimental setup.
Acrylic ball valves with a bore of 6 mm were designed to provide the bed samples along the bed height. The dimensions of the acrylic ball valve are shown in Figure 2. Acrylic valves were provided at a distance of 0.1 m along the column length, in between syringes were provided, to obtain an intermediate solid sample along the height. Liquid and Particles. Tap water at ∼30 ( 3 °C was used for all the experiments. The solids phase was ion-exchange resin manufactured by Ion Exchange (India) Ltd. and having the commercial name Indion 810. The column was filled with ∼5 kg of resins. The particle size distribution is shown in Figure 3. The swelling of the resin was measured by using a calibrated glass tube. The swelling ratio (i.e., volume of swollen resin to volume of dry resin) was found to be 1.3. The settling properties of the various fractions of particles are given in Table 2. The values of the Galileo number and particle Reynolds number are based on swelled particle diameter and experimental terminal settling velocity. From the table it can be seen that, beyond a superficial liquid velocity of 0.009 m/s, the smallest particles would commence entrainment, and hence, it is the limiting velocity for experimentation.
Figure 2. Schematic diagram of acrylic ball valve.
Sampling and Pressure Measurements. After achieving steady state, pressure measurements were made and the solidphase sampling was performed. The sample volume was ∼10 mL. The samples were dried and sieved to estimate the volume fraction of each particle size group. The total solid sample was equivalent to 0.5% of the entire solids in the bed. The sampling was repeated three times, and the reproducibility of data was found to be within 10%. Estimation of Solid Dispersion Coefficient. The following assumptions have been made for using eq 1: (i) the concentration was assumed to be uniform in the radial direction; (ii) the multiparticle system was divided into seven groups and each group had a representative particle diameter equal to the arithmetic mean; (iii) solid dispersion coefficient of each group
Figure 3. Particle size distribution of Indion 810 sample. Values mentioned are the mean particle size of the different sized particles.
was assumed to be constant irrespective of location in the column; (vi) there is no density difference among the solids.
Ind. Eng. Chem. Res., Vol. 46, No. 6, 2007 1839 Table 2. Settling Characteristics of Different Size Group Particles particle diameter (µm) before swelling
after swelling
912 780 650 550 460 356
995 851 709 600 502 389
experimental particle terminal settling Richardson- Galileo Reynolds velocity, Zaki index, number, number, VS∞ (m s-1) n (-) Ga (-) Re∞ (-) 0.023 0.020 0.017 0.014 0.012 0.009
3.36 3.47 3.60 3.72 3.94 4.25
770.90 482.28 279.09 169.08 98.92 45.85
28.62 21.28 15.07 10.50 7.53 4.37
The pressure gradient at the location z from the bottom is given by the following equation:
-
(dpdz) ) (1 - ∈ z
Lz)(FS
- FL)g
(9)
The values of ∈Lz were estimated from the pressure measurements and was described in the form of the following equation:
∈L ) f1(z)
(10)
Figure 4. Variation of bed voidage along bed length at different superficial liquid velocities. + 0.0021, 4 0.0025, ] 0.003, 0 0.004, and × 0.0053 m s-1.
On the basis of measured particle size distribution, the volume fraction (xi) of individual particles was expressed in the following form:
∈Si ) xi(1 - ∈Lz)
(11)
Volume fraction (xi) of each particle was calculated along the bed height and then used in eqs 3 and 4 to estimate the hindered settling velocity of the particle. The result of hindered settling velocity for each size group was described by the following polynomial equation:
VSi ) f 2(z)
(12)
substitution of eqs 10 and 12 in eqs 1 and 2 gives
DSi
∫∈∈
Si2
Si1
d∈Si ) ∈Si
∫zz 1
2
(
VL
f 1(z)
)
- f 2(z) dz
(13)
the values of DS were found by numerical integration.
Figure 5. Volumetric concentration profile of particles at various heights. Superficial liquid velocity 0.0021 m s-1, fluidized bed height 0.60 m, average voidage 0.58; 0 995, ] 851, 4 709, × 600, - 502, and + 389 µm.
Results and Discussion Solid Concentration Profile along the Bed Height. The voidage variation along the bed at different superficial liquid velocities is shown in Figure 4. The reproducibility was found to be within 4%. Figure 5 shows typical concentration profiles at one superficial liquid velocity (VL). The value of VL was covered over a very wide range in such a way that the average voidage was in the range of 0.45-0.82. Volume fraction for each size group was estimated by the following equation:
∈Si )
1 H
∫0H ∈Si dz
(14)
The value of ∈Si was within 2% of the initial charge of that size. This check was performed to confirm the sampling procedure. It has been observed that the gradient of particle concentration reduces as the superficial liquid velocity increases. A lower concentration gradient is an indication of higher dispersion. After obtaining the concentration profile and hindered settling velocity, the values of DS were calculated and are shown in Figure 6. As explained earlier, larger particles generate greater turbulence intensity, causing greater DS. This is observed in the experiments of the present work. Unlike in monosized particles,
Figure 6. Solid dispersion coefficient variation with average voidage for different size group particles: 0 995, ] 851, 4 709, × 600, - 502, and + 389 µm.
the DS values continuously increase with an increase in the average voidage. Variation of local voidage is such that the voidage of 0.7 is present at some point within the bed but no maxima in mixing was observed. A comparison of our work
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Figure 7. Comparison with literature data. Van Der Meer et al.9([ D ) 21.0 mm; 9 D ) 42.9 mm; 2 D ) 51.1 mm; b D ) 61.0 mm). Present work (0 995, ] 851, 4 709, × 600, - 502, + 389 µm).
with the data of Van Der Meer et al.9 is shown in Figure 7 which shows fairly good agreement between the two sets of data. Correlations for Solid Dispersion. Correlations for solidphase dispersion coefficient available in the literature are limited. Van Der Meer et al.9 proposed a correlation both for their data and for some of the literature data that had a close particle size distribution. Though these correlations suggest a strong dependence on the superficial liquid velocity, they do not take into account the multisize particle system. The correlation, however, predicts identical values of DS for large and small particles irrespective of their size difference. This would be contrary to the observation in the experiments conducted in this work and also by previous authors. Kang et al.12 proposed the dependence of solid dispersion coefficient on liquid superficial velocity and minimum fluidization velocity. The improvement in the above equation was that the particle properties were considered in the definition of Vmf. But the above correlation becomes unsuitable at the point of minimum fluidization; i.e., as the particles are just suspended, the value of solid dispersion coefficient should tend toward zero. Using their own experimental data together with those reported by the others, Asif and Petersen10 proposed a unified correlation when Re∞ (>120). The applicability of this correlation was examined for the present and some additional literature data. In the low Re∞ range the Asif-Petersen correlation was found to overpredict the solid dispersion coefficient (Figure 8). It was thought desirable to examine all the other correlations, and results have been summarized in Table 3. Considering the overall applicability of the previous correlations, it was thought desirable to develop a rational correlation that would fit the data over a wide range of Re∞. Proposed Correlation. The solid dispersion coefficient is expected to be a function of the following parameters:
DS ) f (VL,dP,R,FL,FS - FL,µL,∈L)
Figure 9. Proposed correlation for solid dispersion coefficient. The various cases of investigators are mentioned in Table 1. Table 3. Available Correlation for Solid Dispersion Coefficient investigators
correlations
Van Der Meer et al.9 DS ) 0.04V1.8 [own data] L with 0.5 < ∈L < 0.9 and 0.002 < VL < 0.02 DS ) 0.25V2.2 L with 0.5 < ∈L < 0.9 and 0.002 < VL < 0.3 Kang et al.12 DS ) 2.97 × 10-3(VL + Vmf)0.802
Asif and Petersen10
Fr/PeP ) K1((VL - Vmf)/(VS∞))K2
(15)
where R is the angle of the column inclination. For a vertical column R ) 0. The equivalent diameter is given by
2∈L d De ) 3∈S P
Figure 8. Parity plot of predicted versus experimental solid dispersion coefficient using Asif and Petersen equation. The various cases of investigators are mentioned in Table 1.
(16)
K1 ) 7.9 ( 1.1 and K2 ) 2.141 ( 0.054
limitation did not take into account particle diameter
predicts a definite value of DS at minimum fluidization even when it should tend to zero overpredicts DS by an order of magnitude for low Re∞ systems considered high Re∞ (>120) systems
the effect of parameters such as dP, FL,FS - FL,µL was considered in the estimation of Vmf; the remaining parameters have been used in the definition of the equivalent diameter.
Ind. Eng. Chem. Res., Vol. 46, No. 6, 2007 1841
(2) The following correlation was found to hold over the entire range of Re∞ covered in the published literature: DS ) 2.17De
(VL - Vmf) ∈L
Nomenclature
Figure 10. Parity plot of predicted versus experimental solid dispersion coefficient. The various cases of investigators are mentioned in Table 1.
A plot of DS versus De(VL - Vmf)/∈L is shown in Figure 9. The slope of the line comes out to be 2.17 and thus proposed correlation for the estimation of DS is given by
DS ) 2.17De
(VL - Vmf) ∈L
(17)
The basis of this equation comes from the definition of the liquid dispersion coefficient in pipe flow by Taylor,13 who derived the following equation from first principles:
DL ) 5.05VD x0.5f ′
(18)
where f ′ is the friction factor in the pipe, which is a function of pipe Reynolds number, V is the average velocity in the pipe, and D is diameter of the pipe. For the present case, the effect of particle diameter and the voidage on DS was found to be as follows. (a) Effect of Particle Diameter. For larger particles, De values are higher, which result in higher DS. Also their effect is taken into account in the form of VL, because high values of VL are needed for heavy particles. Hence, values of De(VL Vmf)/∈L also become correspondingly high. (b) Effect of Voidage. It can be seen from the present work that DS values increase with an increase in the average voidage. This observation has been taken into account while calculating the equivalent diameter (De), where an increase in ∈ L causes an increase in De. A parity plot of the experimental versus estimated solid-phase dispersion coefficients (eq 17) is shown in Figure 10. The fluid and solid properties along with the operating parameters covered by various investigators are mentioned in Table 1. The agreement can be seen to be very good. The small deviations could be due to the different configuration of distributor designs that are used by various authors. Also, dispersion effects are more pronounced in the case of low-density particles.14
D ) diameter of column, m De ) equivalent diameter, m DL ) liquid dispersion coefficient, m2 s-1 DS ) solid dispersion coefficient, m2 s-1 dP ) particle diameter, mm dp ) pressure drop across a section under consideration, N m-2 Fr ) Froude number, FLVL2/gdP(FS - FL) f ′ ) friction factor g ) gravitational acceleration, m s-2 Ga ) Galileo number, dp3(FS - FL)FLg/µL2 H ) total height of fluidized bed, m m ) number of particle size groups n ) Richardson-Zaki index PeP ) particle Peclet number based on superficial liquid velocity, dPVL/LDS Re∞ ) particle Reynolds number, FLdPVS∞/µL rH ) hydraulic radius, m uz′ ) bulk turbulence intensity, m s-1 V ) liquid velocity in pipe defined in eq 18, m s-1 VL ) superficial liquid velocity, m s-1 Vmf ) minimum superficial liquid fluidization velocity, m s-1 VP ) particle classification velocity relative to wall, m s-1 VS ) hindered settling velocity of particles, m s-1 VS∞ ) terminal settling velocity of a particle in an infinite medium, m s-1 VS∞ ) average terminal settling velocity in multi size group particle system, m x ) particle volume fraction z ) height from bottom to point of consideration, m Greek Letters R ) angle of column inclination ∈L) average voidage of bed ∈S ) solid fraction ∈S ) overall solid fraction FL ) liquid density, kg m-3 FS ) solid density, kg m-3 µL ) liquid viscosity, Pa‚s ∈Lz ) voidage of bed at height z Subscripts i ) ith particle size in a multisize particle system L ) liquid mf ) at minimum fluidization S ) solid ∞ ) infinite medium Literature Cited
Conclusion (1) The solid-phase axial dispersion coefficient (DS) was measured over a wide range of VL and dP and it was found that DS increases with an increase in both the parameters.
(1) Carlos, C. R.; Richardson, J. F. Solid Movement in Liquid Fluidised Beds. II. Measurement of Axial Mixing Coefficients. Chem. Eng. Sci. 1968, 23, 825-831. (2) Thelen, T. V.; Ramirez, W. F. Bed Height Dynamics of Expanded Beds. Chem. Eng. Sci. 1997, 52 (19), 3333-3344.
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(3) Richardson, J. F.; Zaki, W. N. Sedimentation and Fluidisation: Part I. Trans. Inst. Chem. Eng. 1954, 32, 35-53. (4) Joshi, J. B. Solid Liquid Fluidized Beds: Some Design Aspects. Chem. Eng. Res. Des. 1983, 61, 143-161. (5) Al-Dibouni, M. R.; Garside, J. Particle Mixing and Classification in Liquid Fluidized Beds. Trans. Inst. Chem. Eng. 1979, 57, 94102. (6) Handley, D.; Doraisamy, A.; Butcher, K. L.; Franklin, N. L. A Study of the Fluid and Particle Mechanics in Liquid Fluidised Beds. Trans. Inst. Chem. Eng. 1966, 44, 260-273. (7) Yutani, N.; Ototake, N.; Too, J. R.; Fan, L. T. Estimation of the Particle Diffusivity in a Liquid Solids Fluidized Bed Based on a Stochastic Model. Chem. Eng. Sci. 1982, 37 (7), 1079-1085. (8) Juma, A. K. A.; Richardson, J. F. Segregation and Mixing in Liquid Fluidized Beds. Chem. Eng. Sci. 1983, 38 (6), 955-967. (9) Van Der Meer, A. P.; Blanchard, C. M. R. J. P.; Wesselingh, J. A. Mixing of Particles in Liquid Fluidised Beds. Chem. Eng. Res. Des. 1984, 62, 214-222. (10) Asif, M.; Petersen, J. N. Particle Dispersion in a Binary Solid Liquid Fluidized Bed. AIChE J. 1993, 39 (9), 1465-1471.
(11) Barghi, S.; Briens, C. L.; Bergougnou, M. A. Mixing and Segregation of Binary Particles in Liquid Solid Fluidized Beds. Powder Technol. 2003, 131, 223-233. (12) Kang, Y.; Nah, J. B.; Min, B. T.; Kim, S. D. Dispersion and Fluctuation of Fluidized Particles in a Liquid Solid Fluidized Bed. Chem. Eng. Commun. 1990, 97, 197-208. (13) Taylor, G. I. Diffusion and Mass Transport in Tubes. Proc. Phys. Soc. London, Sect. B 1954, 67, 857-869. (14) Asif, M.; Kalogerakis, N.; Behie, L. A. Hydrodynamics of Liquid Fluidized Bed including Distributor Region. Chem. Eng. Sci. 1992, 47 (15/ 16), 4155-4166. (15) Kennedy, S. C.; Bretton, R. H. Axial Dispersion of Spheres Fluidised with Liquids. AIChE J. 1966, 12 (1), 24-30.
ReceiVed for reView September 18, 2006 ReVised manuscript receiVed December 11, 2006 Accepted January 18, 2007 IE0612257
