multi-stage fluidized bed column: hydrodynamic study
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Chemical Engineering and Processing 47 (2008) 957–970
Multi-stage fluidized bed column: Hydrodynamic study
Abhinandan Singh, Rupesh Verma, Kaushal Kishore, Nishith Verma ∗Department of Chemical Engineering, Indian Institute of Technology Kanpur, Kanpur 208016, India
Received 15 March 2006; received in revised form 23 September 2006; accepted 14 March 2007Available online 19 March 2007
bstract
In our recent work we carried out mass transfer study on the multi-stage fluidized bed ion exchanger column with solids and liquid flowingn the counter current directions and demonstrated improved separation efficiency of dissolved anions from waste water in comparison to thatchieved in a fixed bed operation. In this study, we report the results pertaining to hydrodynamic study with a view to ascertaining the type ofuidization prevailing on the column’s stage and the operating range of liquid and solid flow rates for the steady and stable operation of the columnithout loading or flooding with excess solid or water flow rates. Residence time distribution (RTD) study was carried out to investigate the extentf mixing on the stages. In addition, the experimental measurements for pressure-drops were made over a wide range of operating conditionsncluding number of stages, height of the downspout on every stage, and the liquid and solid flow rates. Based on the data, empirical correlations
ere developed using scale-up analysis for predicting pressure-drop, bed porosity and average bed height during cross-flow fluidization apparentlyrevalent on the stage. The results in this study assume significance from the perspective of design and stable operation of staged fluidized bed ionxchangers.2007 Elsevier B.V. All rights reserved.
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eywords: Ion exchange; Fluidization; Dimensionless analysis; Hydrodynamic
. Introduction
Fixed bed ion exchangers are commonly employed in treat-ng waste water for dissolved anionic or cationic impurities.he treatment in fixed bed exchanger is essentially a batch oper-tion, as it requires periodic regeneration following saturationf the ion exchanger resins. The continuous operation of ionxchangers may be achieved in staged column with water flow-ng in the upward direction through the mesh of various stagesnd the fluidized solid particles flowing across the stage andhen onto the next stage through a downspout tube fitted in theolumn. Literature is replete to some extent with studies car-ied out on the multi-staged fluidized bed contactor applied inas–solid applications [1–3]. In these studies the focus is onhe application of the stage-wise operation in continuous dry-ng or cooling of the solid particles using air as the drying or
ooling agent. Some of these studies also focus on the types ofow patterns prevalent in the gas–solid fluidized beds and theevelopment of correlations for estimating gas to particle heat∗ Corresponding author.E-mail address: [email protected] (N. Verma).
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255-2701/$ – see front matter © 2007 Elsevier B.V. All rights reserved.oi:10.1016/j.cep.2007.03.007
d porosity
nd mass transfer coefficients. Commercial reactors for calci-ations based on multi-staging are also in operation [4]. Onhe other hand, the reported literature for the stage-wise opera-ion of the liquid–solid contactors for ion exchange applicationss scant. Although there are commercial units since late 1960stilizing the principles of stage-wise operation in water soft-ning and mineral processing, the technology is patented andhe published data are inadequately informative [5,6]. A listf some of the commercial units based on different types ofiquid–solid contactor may be obtained in Ref. [7]. In some ofhe early works, Dodds et al. [8] have also studied the operationf the stage-wise solid–liquid contactor specifically designedor the ion exchange treatment, albeit in a semi-continuousperation in which resins were periodically transferred to theower stage, except for the top plate on which resins wereontinuously fed. The water to be treated was fed to the bot-om of the contactor. The contactor consisted of a series oftages separated by pairs of especially designed horizontal per-orated plates without downcomer to insure no particle-drainage
ccurred under no-flow condition of water. Gomez-Vaillard andershenbaum [9] later extended the study to investigate theffect of operating conditions on the performance of the reac-or.
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58 A. Singh et al. / Chemical Enginee
In our recent work, we presented the data pertaining to massransfer studies carried out on a five-stage Perspex made col-mn and showed that 90% reduction in the dissolved soluteoncentrations may be achieved relative to the inlet concentra-ion during a typical continuous operation [10]. The experimentsere carried out over varying water and resin flow rates and
nlet solute concentrations. The results showed a gradual level-ng off in the solutes removal efficiency with increasing numberf stages; a mass transfer effect which is a characteristic oftage-wise unit operations. In this work we present the resultsertaining to hydrodynamic study with a view to determin-ng the operating range of water and resin flow rates for themooth and stable operation of the multi-staged column, with-ut flooding of the stages with water or loading of the stagesith resins corresponding to two extreme scenarios of the oper-
tion. The study also includes ascertaining type of fluidizationnd extent of mixing (or channeling) on various stages of theolumn by determining residence time distribution (RTD) ofowing water through voids between the resin solids. The RTDtudy was carried out over a wide range of operating conditionsncluding water and solid flow rates, number of stages and stageeights. Finally, using scale-up analysis various dimensionlessarameters are identified and correlations developed for pre-icting pressure-drop, bed porosity and average bed height asunctions of the operating variables during cross-flow fluidiza-ion of the solids by water. In the following section we havexplained that although the overall operation of the multi-stageduidized bed column may be considered to be counter current,
here is apparently cross-fluidization of the solids on stages,ith water flowing in the upward direction and the fluidized
olid resins flowing horizontally across the stage. We would likeo point out that in recent times computational fluid dynamics
CFD) has been considerably used in addressing a wide rangef flow scenarios including those existing in bubble columns,uidized bed reactors and sieve trays [11–13]. Few experi-ental studies are also reported for re-circulating liquid–solidtfhc
Fig. 1. Schematic diagram of multi
nd Processing 47 (2008) 957–970
ystems [14,15]. However, to the best of the authors’ knowl-dge no study using either CFD or other technique has beeneported concerning hydrodynamics for the staged ion exchangeolumn, with cross-flow fluidization of the solid particles byater.
. Multi-staged fluidized bed column
The configuration of the staged liquid–solid ion exchangeolumn is similar to that of the sieve trays distillation col-mn used for vapour–liquid contacts. Fig. 1 is the schematicf the multi-staged fluidized bed ion exchange column fabri-ated and used in this study. The Perspex made column (500 mm× 100 mm i.d.) essentially consisted of five stages (65 mm
eight per stage) assembled together with flange joints. A brassade mesh with openings smaller than the particle size wastted onto an aluminium ring sandwiched between every pairf adjoining flanges. The fluidized solid resin particles movedcross the stage on to the next stage through a downcomer, asater flowed upward through the mesh openings. Fig. 2 is the
chematic of the single stage configuration. As shown, the down-omer consists of two concentric tubes, assembled with the helpf the rack and pinion arrangement. The arrangement allows thedjustment of the stage height by moving two tubes verticallyp or down independently. Thus the bed heights on two adja-ent stages could be adjusted independently without stopping ornterrupting the operation. In the existing arrangement, the resined heights could be varied between 2 and 20 mm. In principal,he required bed height is determined from the mass transferonsideration and should be equal to mass transfer zone (MTZ).ounter current operation of solid–water flow permits achiev-
ng the required mass transfer rate within a small height [10]. In
he existing configuration, provisions were made on every stageor pressure-drop measurements across the stage height with theelp of a manometer. Fresh resins were fed from the top usingonveyor belt. The solid flow rate was controlled by adjusting-stage fluidized bed column.
A. Singh et al. / Chemical Engineering a
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Fig. 2. Single stage configuration.
he speed of the belt. A centrifugal pump was used to pump
ater through a water distributor underneath the last stage. Theater flow rate was adjusted by a needle valve. The stage hold-ng the water distributor was filled with glass beads for uniformistribution of the water. Necessary arrangements in the fittings
sflmi
Fig. 3. Schematic of cros
nd Processing 47 (2008) 957–970 959
nd fixtures were made to ensure that no air bubbles intrudednto the column during operation.
. Cross-flow fluidization
Fig. 3 is the pictorial representation of flow conditions pro-ressively existing on the stage, beginning with a fixed bed ofesin particle with water flowing upward and then graduallyhanging over to the continuous operation in the column withater and resin flowing counter currently at constant flow rates.o begin with, the resin particles may fill up the bed only to aertain fraction of the downcomer height, the bed height beinglightly smaller than the downcomer height (h < hd). As the flowate of water is gradually increased up to the minimum fluidiza-ion condition, the bed expands till the downcomer height, hd. Ashe resins pour from the adjacent upper stage through the down-pout, difference in the heights of the resin particles developscross the stage from the left (location of the upper downspout)o the right of the stage, resulting in the cross-flow of the flu-dized resin particles on the stage. The fluidized particles flowver the stage height, hd through the downspout to the next lowertage, as shown in Fig. 3. The experiments revealed that for givenowncomer height and water flow rate, there is a certain rangef resin flow rates over which the resin bed completely fluidizesnd the resin particles move smoothly across the stage to the nexttage due to difference in the bed heights across the stage, whileater flows upward through the voids between the solid resins.hus, the state of fluidization on every stage is cross-fluidization,lthough the overall flows of the solid and water phases are in theounter current direction. The flow scenario may be considered
imilar in visualization to that existing during the liquid–vaporows in a distillation column. However, it is important to keep inind that notwithstanding similarity in the physical understand-ng of the flows in multi-staged ion exchanger and in distillation
s-flow fluidization.
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60 A. Singh et al. / Chemical Enginee
olumn, with the liquid flowing upward through the mesh ofhe stages and the fluidized solids flowing across the stage tohe next bottom stage through the downspout, the flow situa-ion for gas–liquid (G–L) systems is often more complex thanhe liquid–solid (L–S) systems due to a number of phenom-na, including froth formation and gas–liquid interfacial surfaceension effects, which are inherently present in the distillationolumn and absent in the solid–liquid ion exchanger. As a conse-uence, the design equations or correlations developed for G–Lystems, for example, to establish flooding and loading criterionn a distillation column cannot be applied in the proposed L–System and should be developed independently as described inhe subsequent section of the paper.
. Characterization of resin particles (determininghysical properties)
The INDION-220 cation exchange resin particles were clas-ified by sieving into various size ranges (0.264–0.707 mm). Theesins are known to be water absorbing (hygroscopic), as theyignificantly swell after soaking into water. The size of the drynd wet resins has been reported in Table 1. As seen in Table 1,pproximate changes in the resin size due to swelling are 70,2, 18, 75 and 17.4% for the average particle size (dry basis)f 0.707, 0.65, 0.55, 0.314 and 0.264 mm, respectively. Threeractions of the resins in different size ranges (0.55, 0.65 and.92 mm) were chosen for hydrodynamic experiments. Due towelling, the bulk density of particles was also altered from500 kg/m3 corresponding to dry state to 1330 kg/m3 corre-ponding to wet resins. The particle size and the bulk densityf resins were also independently verified by comparing thexperimentally obtained settling terminal velocity with the cor-esponding theoretical values.
To measure the terminal velocity, experiments were per-ormed in a vertical glass column of diameter 0.05 m and length.5 m filled with water. The particles were dropped in fromhe top of the column. Time taken by the particles to travel aistance of 0.8 m in vertical direction was noted. To avoid thentrance effects, during which the particle velocity increasesrom zero to settling terminal velocity, the marking on theolumn was made at 0.2 m below from the top of the col-mn. Fig. 4 describes the theoretical and experimental results
or terminal velocity for varying particle sizes. As observedhe average terminal velocities of the particles measured were.0365, 0.0227, and 0.0218 m/s for the particle sizes 0.925,.65, and 0.55 mm, respectively. The corresponding theoreti-able 1esin properties as furnished by the manufacturer
roperty INDION-220 (cationexchange resin)
article size (mm) 0.3–1.3article density (kg/m3) 1330otal exchange capacity (mequiv./ml resin) 1.8oisture content (% kg water/kg wet resin) 54–60
unctional group –SO3−
onic form H+
upbsnttcwtoat
Fig. 4. Determining average particle size via terminal velocity.
al values of the terminal velocities are 0.0368, 0.0276 and.0219 m/s, respectively. The drag coefficient required in thealculation of terminal velocity was calculated from the corre-ation, which is shown to be valid over a wide range of particlee between 0.2 and 1000 [16]. Under the selected operatingonditions, Re was found to be varying in the transition region9.6–27). The effect of the column walls in the aforementionedalculation was neglected due to low particle to column diam-ter ratio (dp/D < 0.0145). The time taken by the particles tochieve terminal velocity was also calculated by solving tran-ient equation for the accelerating region, where the velocityf resin varied with time and was found to be small (less than.1 s), as shown in Fig. 4. The particle size and bulk densityere chosen corresponding to those of the swollen resins in
he theoretical calculation of terminal velocity. From Fig. 4, itay be observed that the experimental data were very close
o the theoretical results with deviations found to be less than%.
. Pressure-drop in the fluidized bed multi-stagedolumn (without solid flow)
Prior to the experiment carried out on the multi-stage col-mn operating under counter current liquid–solid flows, severalressure-drops measurements were taken in fixed and fluidizededs of resins with and without water flow on single and multi-tage for varying amounts of resins and particle sizes. This wasecessitated for delineating the effects of largely resin flowshrough the downspout tubes on the total pressure-drop acrosshe stages. The experimental measurements showed that duringontinuous operation of the column more than 90% of the totalater flow occurred through the mesh. Considering the ratio of
he column to the downspout tube i.d. being approximately 12,
nly a small fraction of water flowed up through the downspouts the solids flowed counter currently in the downward directionhrough the downspout tube.
A. Singh et al. / Chemical Engineering a
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ig. 5. Pressure-drop in fixed and fluidized beds of resins for varying stageeights (i.d. = 100 mm, dp = 0.92 mm, ε = 0.53).
Fig. 5 describes the experimental data for pressure-drop inxed and fluidized beds on the 1st and 5th (last) stage of the col-mn without resin flow. For clarity the data for the intermediatetages are not shown as they lie between those for the 1st andth stage. The data are shown for the solids filled up to 5, 15 and0 mm of the stage heights on each stage, corresponding to 0.02,.06 and 0.08 kg of resins (average particle size = 0.92 mm) pertage, respectively. There are three salient observations that can
e made from the plots. First, the pressure-drop across varioustages increases with increasing stage height or the amount ofhe resins. Under the fluidized bed conditions, pressure-drop ispproximately proportional to the apparent weight of the solidpotfl
able 2easured pressure-drops across the first and last stage for various stage heights or di
. no. uo (m/s) �P (Pa)
Stage # 1
5/0.02a 15/0.06a
1 0.0 4.0 11.52 0.002 4.9 14.13 0.0025 6.2 16.14 0.003 6.9 17.55 0.0035 7.9 22.16 0.004 8.1 24.17 0.0045 8.7 26.28 0.005 8.5 27.59 0.005325 8.1 23.80 0.0055 8.1 23.81 0.006 8.1 23.82 0.007 8.1 23.8
a Values are hd (mm)/W (kg).
nd Processing 47 (2008) 957–970 961
esins (actual weight less buoyancy). The bed fluidization occurst minimum fluidization velocity of particles, which is deter-ined by equating pressure-drop in fixed bed with the effectiveeight of the particles:
Pmf = Lmf(1 − εmf)(ρp − ρ)g = L(1 − ε)(ρp − ρ)g (1)
ere all variables with subscript ‘mf’ correspond to the mini-um fluidization condition. L is the same as the stage height hd.econd, there is no appreciable difference between the pressure-rops across bottom (1st) and top (5th) stages under fixed oruidized conditions. A maximum difference of less than 5%
s observed for the maximum stage height (see Table 2). Theeasured pressure-drops across either fixed or fluidized solid
eds for various stages or stage heights may be comparedo the hydrostatic pressure on the stage (=hρg, where h, thetage spacing = 65 mm). With all five stages in the assembly,he pressure-drop due to hydrostatic head was measured to be.15 kPa. The pressure-drop due to the wall friction at the col-mn’s wall may be calculated to be negligible. Thus, from theoregoing data it may be concluded that most of the pressure-rops in the column is contributed by the hydrostatic pressurever the stages.
As also observed from Fig. 5, the theoretical curves wereound to closely predict the data for fixed bed conditions. Theifferences between the theoretically calculated values and theata observed at low flow rates are attributed due to fluidizationf the small fraction of relatively smaller particles present inhe resin mixture. Fluidization was also observed to begin at aery small flow rate, confined within the small regions aroundhe periphery of the column and fixture and fittings. However,hen the entire region of the bed was fluidized, the measuredressure-drop was found to level at the value equal to the theoret-cal effective weight as calculated from Eq. (1). It is important to
oint out that the continuous operation of the column is carriedut at relatively higher flow rates (past the minimum fluidiza-ion conditions) when the entire bed is fluidized so that the resinsow across the stage through the downcomer to the next stage.fferent amounts of solid resins (without resin flow)
Stage # 5
20/0.08a 5/0.02a 15/0.06a 20/0.08a
16.1 4.5 11.8 17.218.8 5.2 14.9 19.424.2 6.5 16.9 24.326.1 7.1 18.1 26.829.1 8.2 22.9 30.832.8 8.4 24.9 33.534.5 8.7 25.9 35.135.1 8.9 28.1 35.932.1 8.3 24.1 32.932.0 8.3 24.1 32.832.0 8.3 24.1 32.832.0 8.3 24.1 32.8
962 A. Singh et al. / Chemical Engineering a
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ig. 6. Pressure-drop in fixed and fluidized beds of resins for varying particleize (i.d. = 100 mm, hd = 10 mm, ε = 0.53).
ne can also observe in the figure that fluidization started at theelocity approximately equal to Umf calculated by the follow-ng equation and is independent of the total weight of the resinarticles:
mf = d2p (ρp − ρ)gε3
mfφ2s
150μf(1 − εmf), Rep,mf ≤ 20 (2)
n the above equation, Umf is calculated by equating pressure-rop in the fixed bed with the effective weight of the particlesweight less buoyancy force). In principle, as many numbers oforrelations for Umf may be obtained as those of various corre-ations for the fixed bed pressure-drop by equating the effectiveeight of the particles with the respective pressure-drops. Weive above the expression for Umf obtained from the correspond-ng correlation proposed by Ergun [17], since its predictionsere found to be reasonably closer to the experimental data
or resin particles used in the present work. To experimentallyetermine εmf required in the calculation of Umf, the heightsf expanded bed under the fluidized conditions at increasingow rate were measured. The height of the bed at minimumuidization condition was determined by trend line fitting andxtrapolating the expanded bed height up to the experimentallybserved Umf.
The pressure-drop measurements were also made for vary-ng resin particle sizes. Fig. 6 describes the data for the 5thtage (stage height = 10 mm). Similar to the results obtainedn Fig. 5, there was insignificant difference observed between
he pressure-drops for all stages. As observed from the plots,ncrease in the pressure-drop in fixed beds with decreasing par-icle diameter for the same amount of resin (or the bed height)s in accordance with Eq. (2). For the fluidized bed conditionssttm
nd Processing 47 (2008) 957–970
he pressure-drops for three cases are the same and expectedlyqual to the apparent or effective weights of the resin particles.he observed increase in minimum fluidization velocity with
ncreasing particle diameter is also in accordance with the the-ry. There is reasonably good agreement between the data andhe theory predictions.
. Operation of multi-staged column (counter currentater–resin flow)
Prior to determining pressure-drops across various stagesuring continuous operation of the column, with solid resinsnd water flowing in the counter current direction, the operat-ng range of solid and water flow rates was determined underteady-state conditions. In principle the operating velocity ofater should be set between Umf, the minimum fluidizationelocity for the solid particles to fluidize, and Ut, the terminalelocity of the particle to prevent entrainment (carry over) in thepward flowing water. Selecting resin particles of certain sizeets both Umf and Ut. However, there are two additional oper-ting limits (constraints), which must be established a priori forhe smooth and stable operation of the multi-staged column. Ones the minimum solid flow rate for a fixed water flow rate, belowhich the stage may be flooded with water preventing solid par-
icles from flowing down through the downspout tube. The others the maximum solid flow rate above which the stage may beoaded with excess solids. As pictorially described in Fig. 2,uidized resin particles flow on the stage due to the gradient,ifference between the solid-bed heights, hu and hd, across thetage from one side of the column to the other, as water flowspward through the mesh openings. Therefore, it follows that forhe steady-state operation of the column, if the water velocitys set larger than Umf, the solid gradient (driving force for theolids to flow) must be maintained across the stage to meet theequired flow rate of solid particles. At relatively smaller waterow rate, large solid flow rates may result into loading of thetage with solid resin particles, leading to choking of the down-pout tube. In other extreme scenario, at relatively larger waterow rate, flooding may occur on the stages. We describe below
he operating range of water and resin flow rates for the stableperation of the column under the experimental conditions usedn this work.
Fig. 7 describes the resin flow rates as a function of waterow rates corresponding to two limits of operation of the col-mn. The data are shown for the column operated with varyingumbers of stages (1–5). In our previous study we have shownhat the number of stages required in the column is determinedy the mass transfer consideration [10] and in particular, is influ-nced by the solid side mass transfer resistance. In accordanceith the theory, the experimental data in the aforementioned
tudy showed that the extent of separation or exit concentrationf the solute asymptotically leveled off with increase in the num-er of stages. Under the experimental conditions chosen in the
tudy, five stages were sufficient to achieve the desired separa-ion. Re-referring Fig. 7, the dark symbols in the plots representhe maximum solid flow rate at a fixed water flow rate, whichay be used for the steady operation of the column without load-
A. Singh et al. / Chemical Engineering a
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iwTlepocasymptotical convergence. Since the column was typically oper-ated with four to five stages for the maximum efficiency ofseparation, the rather safe and conservative operating limits ofthe water and solid flow rates may be defined from the corre-
ig. 7. Operating range of solid and water flow rates in multi-stage columni.d. = 100 mm, hd = 20 mm, ε = 0.53, dp = 0.92 mm).
ng of the stages with excess solids. Similarly, the blank symbolsepresent the minimum solid flow rate at a fixed water flow rate,hich may be used without flooding of the stage with water.herefore, the difference between two solid flow rates defines
he operating range of fluidization without loading and flood-ng of the stages at given water flow rate. In other words, overhe operating range the solid gradient between hu and hd (referig. 2) is steadily maintained and the column may be operatedontinuously with complete fluidization of the solid particlescross the entire cross-sectional area of the mesh and contin-ous flow of the solids over the stage to the downspout tube.or example, at a water flow rate of 3 liters per min (lpm) theperating range of solid flow rates is between 5 and 15 g/minorresponding to flooding and loading conditions in the columnperated with one stage. Solid flow rate smaller than 5 g/minradually leads to the stage flooded with water, whereas that inxcess of 15 g/min causes choking of the downspout tube dueo excess solid flow rate. In the figure, operating ranges for theolumn operated with one stage are marked with vertical double-eaded arrows for clarity. There are additional information thatay be obtained from Fig. 7. As observed, the minimum waterow rate required for complete fluidization of the solid resins
s relatively larger in the column operated with more number oftages. The resin flow rate is observed to increase with increasingater flow rate for both operating curves, flooding and loading.t fixed water flow rate, the minimum and maximum solid flow
ates also increase with increasing number of stages. However,n either case (increasing water flow rate or increasing numberf stages) the solid flow rates gradually level off. The capacityf the water pump and the maximum speed of the conveyor belt
Fm
nd Processing 47 (2008) 957–970 963
ay be determined by the maximum water and solid flow ratessed in the operation, which are 6 lpm and 45 g/min, respectivelyor the case discussed in Fig. 7. For the particle size of 0.925 mm,
mf and Ut are calculated to be 0.00532 and 0.0295 m/s, whichorrespond to the water flow rate of 2.5 and 14 lpm, respec-ively. Thus the operating water flow rates are within the flowates corresponding to Umf and Ut.
From the data shown in Fig. 7, it may be inferred that formooth operation of the column, both solid and liquid flow ratesust be adjusted to ensure that the operation is within loading
nd flooding limits. Thus, it follows naturally that in the col-mn of relatively larger diameter, flow rate of the upward goingater must also be kept sufficiently high to ensure that there iso settling of the solids. In other words, it is the ratio of theolid to liquid phase velocity (and not the individual velocitiesr flow rates) that sets the criterion for smooth operation in aolumn, whether of large or small diameter. Indeed, it has beenhown in the subsequent section on hydrodynamic modeling thatimensionless group, us/uo, where us and uo are the superficialelocities of the solid and water, is one of the design parametersor the multi-staged column.
Fig. 8 describes the plots reconstructed from the data shownn Fig. 7, for the minimum and maximum ratios of the solid to theater velocity as a function of the superficial velocity of water.he minimum and maximum values of the ratios correspond to
oading and flooding conditions. The results are shown for differ-nt number of stages used in the column. As observed from thelots, the ratio goes through a maximum for one- and two-stageperated column. For more numbers of stages, both operatingurves (loading and flooding) monotonically decrease and have
ig. 8. Operating range of us/uo in multi-stage column (i.d. = 100 mm, hd = 20m, ε = 0.53, dp = 0.92 mm).
964 A. Singh et al. / Chemical Engineering and Processing 47 (2008) 957–970
Table 3(us/uo)loading for varying particle sizes and stage heights (stage no. = 5)
S. no. Qwater (lpm) dp = 0.55 mm dp = 0.92 mm
hd = 5 mm hd = 20 mm hd = 5 mm hd = 20 mm
1 4.0 0.0272 0.0274 0.0274 0.02762 4.3 0.0253 0.0255 0.0256 0.02583 4.6 0.0240 0.0241 0.0241 0.024445
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5.1 0.0197 0.0199 0.0220 0.02225.6 0.0201 0.0204 0.0206 0.0207
ponding plots for the 5th stage of five-stage operated columnnd are shown in the inset of Fig. 8.
The experiments were also carried out for various particleizes and stage heights and the corresponding plots (not shownere) for us/uo were re-constructed from the raw data. For theomparison purpose, Table 3 summarizes the experimental dataus/uo) corresponding to the loading conditions in the columnperated with five stages for the minimum and maximum particleize and bed height used in the present study. Similar behaviourn the pressure-drop variation was also observed for the floodingonditions in the column. The data showed that while both waterow rates corresponding to Umf and Ut increased with increasingarticle size in accordance with the theoretical calculations, theperating limits for loading and flooding varied less than 10%.imilarly, variation in the operating limits with increase in thetage height or the resins hold-up (the amount of solid resinsn the stages) was found to be insignificant (less than 5%). Theegligible influence of the stage height on two operating limitsf the column is consistent with the explanation given in thereceding section that the driving force for the particle to movecross the stage is difference between the upstream resin bednd the downstream stage heights and therefore, under steady-tate the required solid flow rate at the given water flow rate isndependent of the actual hold-up or the amount of the solids onhe stage. It may also be pointed out that the inlet water pressureo the column was increased with increasing number of stageso overcome the pressure-drop in the downspouts due to solidows and in the column due to hydrostatic head.
Fig. 9 describes the pressure-drop measured across the 5thtage (hd = 20 mm) in the five-stage column operated underteady-state conditions, with the resins and water flowingounter currently. The pressure-drops across the remainingtages were measured to be approximately the same (with vari-tion less than 5%). The pressure-drops are shown as a functionf the ratio of the solid to liquid velocities. The correspondingperating flow rates are within the flooding and loading limitss obtained for the results shown in Figs. 7 and 8. Thus the dataor the lowest limit of the ratio, i.e. us/uo = 0, would correspondo the scenarios described previously in Fig. 6, with the columnperated without solid flow. As seen in Fig. 6, the pressure-dropcross the 5th stage (hd = 20 mm) without solid flows was mea-ured to be 32 Pa under fluidized bed conditions. To show the
elative pressure-drop across the stage with and without solidow, a solid horizontal line at 32 Pa has been drawn. The differ-nce between the loading/flooding curves and the horizontal lineay, therefore, be attributed due to the pressure-drop caused byoTst
ig. 9. Pressure-drop in counter current multi-staged column (i.d. = 100 mm,
d = 20 mm, number of stages = 5).
he solid–water cross-flow on the stage and counter current flown the downspout tube. Under the existing conditions a maxi-
um of 43 Pa (=75 − 32 Pa) is observed to occur at the extremease of loading corresponding to the us/uo ratio of 0.0277. Thisrop per stage may be compared to 637 Pa of the hydrostaticressure equivalent to 65 mm of water column height on thetage, i.e. distance between two stages. While we defer dis-ussion on the correlation of pressure-drop per stage with theperating variables in the multi-staged column to the subsequentection in this paper, the comparative pressure-drops in the col-mn due to hydrostatic pressure, apparent weight of the solidesins, cross-flow on the stage and counter current flow in theownspout tube were approximately calculated to be 85, 7, and%, respectively. From the results it is fair to conclude that in thetaged wise operation of the ion exchange column within twoxtreme limits of flooding and loading, pressure-drop across thetages and downspout tubes is inherently smaller in comparisono the hydrostatic pressure in the column. This is primarily dueo the reason that only a small stage height or only small amountf resin per stage is required for ion exchange in the fluidizeded, as the mass transfer zone (MTZ) between solid and waterhase is confined to a very small height. In addition, the flu-dized resin bed as described earlier is in the state of cross-flowuidization, in which case the pressure-drop is expected to bemaller than in the two-phase counter current flow. In the latterase the pressure-drop due to drag is relatively larger. The maxi-um fraction of the water flow in the column during steady-state
peration is through the mesh and less through the downspout.herefore, the solid resins flowing downward through the down-pout due to gravity also experience small frictional loss due tohe solid–liquid flow.
ing and Processing 47 (2008) 957–970 965
7
flocoicebitbwisFuttiecmbcribpm
TOs
S
1
2
3
4
Fh
t4
A. Singh et al. / Chemical Engineer
. RTD measurements (extent of mixing on stages)
After obtaining flooding and loading limits for liquid–solidow rates, the RTD technique was utilized to ascertain the extentf mixing prevalent on the stages (within the solid beds) in theolumn. The measurements were made on the five-stage columnperated over varying solid–liquid flow rates within two operat-ng limits. We have pointed out earlier that during start-up of theolumn there was some fluidization observed around the periph-ry of the downspout tube and other fixture. However, once theed was completely fluidized there was apparently uniform flu-dization observed across the column’s cross-section, at least onhe top surfaces of the bed. The extent of channeling or possi-le bypassing of some of the solid regions by the fluid elementithin the bed could not be visually detected. The RTD study,
ncluding axial dispersion model, was used to gain insight intouch non-ideal flow pattern existing within the fluidized bed.or the RTD study in this work, the spherical glass beads weresed as the solid particles so that the particles were unreactiveo water, when NaOH of known concentration was used as aracer during the step change in the water concentration at thenlet to the column. Water concentrations were measured at thexit of the column and typical RTD data were plotted. Prior toarrying out the RTD experiments several typical pressure-dropeasurements were made in fixed and fluidized beds of glass
eads, similar to those for the solid resins (Figs. 5 and 6) toonfirm the behaviour of fluidized resin beds similar to those ofesins. Similarly, the operating limits for flooding and loading
n the multi-staged column were also established for glass beadsy varying solid and water flow rates independently. For com-arison purpose, Table 4 summarizes the results for the lattereasurements.able 4perating range for the staged column fluidized with glass beads (dp = 0.6 mm,
tage height = 20 mm, number of stages = 5)
. no. Water flow rate (lpm) Solid flow rate (g/s)
Minimum Maximum
st stage1 5 40 1202 6 60 1803 8 60 2404 10 72 300
nd stage1 5 40 1402 7 60 2003 9 72 2604 12 84 320
rd stage1 6 48 1602 8 64 2203 11 72 2604 14 88 340
th stage1 7 60 1802 9 72 2603 12 80 3404 15 88 360
2tctscmiscomotsc(acvmtt4fhsbi
ig. 10. Effect of water flow rate on residence time distribution (dp = 0.5 mm,
d = 10 mm, i.d. = 100 mm, Cinlet = 0.05 M, Qs = 24 g/min).
Fig. 10 describes the RTD data (Cexit/C0) to the step input inhe solute concentration for varying liquid flow rates between000 and 5000 cm3/min. Solid flow rate was held constant at4 g/min. The data are shown as function of dimensionless time/tmean, where tmean is the average residence time of water in theolumn corresponding to the respective water flow rate. To ascer-ain the extent of mixing prevalent between two phases on thetage for varying liquid flow rates, theoretical RTD curves wereonstructed based on the axial dispersion model [18]. As per theodel’s basic postulates the extent of micro-mixing is reflected
n the model’s parameter, dispersion coefficient or the dimen-ionless peclet number, which represents the relative influence ofonvection with respect to axial dispersion. Two extreme valuesf dispersion coefficient (0 and ∞) characterize the flow to beacro or segregated as prevalent in an ideal plug flow and micro
r non-segregated as prevalent in an ideal mixed flow. There arewo salient observations that can be made from Fig. 10. One, atmall water flow rates the model parameter (dispersion coeffi-ient) is required to be adjusted over two concentration rangeslow and high) to fit the data, while at relatively larger flow ratesingle value of dispersion coefficient suffices for the entire
oncentration range. Two, there is a gradual convergence in thealues of dispersion coefficient with increasing flow rates. Asentioned in the figure, dispersion coefficient corresponding
o the fractions of the fluid having relatively smaller residenceime increases from 0.02 to 0.12 as flow rate is increased from000 to 5000 cm3/min, whereas dispersion coefficient decreasesrom 0.25 to 0.12, corresponding to the fractions of the fluid
aving large residence time. The results assume significance,ince at relatively lower water flow rates the fluidized bed maye assumed to be segregated into two zones, one having res-dence time less than 1–1.5τ with small degree of dispersion
9 ring a
aifldfiwsottflotvctrfoSototarbeti
Fh
ui
tTiWttiusapwoofionitmRoti
66 A. Singh et al. / Chemical Enginee
nd the other having residence time larger than the average res-dence time with large dispersion. On the other hand, at largeow rates (still smaller than the flooding operating limit) axialispersion within the fluidized bed may be assumed to be uni-orm across the stage. Similar experimental observations andnferences were made from the RTD data obtained for the caseshen water flow rate in the column was held constant and the
olid resin flow rates were varied from low to high flow rates. Asbserved from Fig. 12, dispersion coefficient was again requiredo be adjusted over two ranges of residence times or exit concen-rations at relatively small flow rates of solids, whereas at largerow rate (smaller than loading operating limit), one single valuef dispersion coefficient could be used for the model fitting tohe data. In addition, similar to the trend observed in the pre-ious scenario there was gradual convergence in the dispersionoefficients corresponding to two fractions of the fluid exitinghe column, one for small residence time and the other for largeesidence time, as the solid flow rates were gradually increasedrom 23 to 45 g/min. From both RTD data (Figs. 10 and 11)ne can also gain insight into the type of mixing on the stages.ince the “tagged” fluid elements (i.e. tracer) are immediatelybserved at the exit of the column and not after one residenceime as expected for an ideal plug flow reactor (PFR), the extentf mixing for the fluidized bed may be considered to be closer tohat prevalent in a mixed flow reactor (MFR), yet not the sames expected under ideal mixed flow conditions since the RTDesponse curves obtained for the staged column are observed toe characteristically different from that of an ideal MFR. Consid-
ring complex flow situations existing on the column’s stage dueo the upward flow of water through the voids between the hor-zontally moving solids, the RTD data and the model results areig. 11. Effect of solid flow rate on residence time distribution (dp = 0.5 mm,
d = 10 mm, i.d. = 100 mm, Cinlet = 0.05 M, Ql = 4.5 lpm).
ntvhR
8c
fleOhalatalqi�
flnoqmf
nd Processing 47 (2008) 957–970
seful in determining the extent of mixing for cross-fluidizationn the staged column.
The RTD experiments were also carried out for varying par-icle sizes (0.5, 0.65 and 0.92 mm) and stage heights (5–20 mm).he effects of these variables on RTD were observed to be
nsignificant and are not described here for the sake of brevity.e would also like to point out that the present study was limited
o the measurement of RTD in the liquid phase only. Althoughhe solids RTD study should be useful in providing additionalnformation on mal-distribution of the solid particles, in partic-lar on the non-uniform voids between the particles, such studyhould be complementary to the liquid phase RTD study forscertaining the extent of mixing between two phases, because inrinciple, non-idealities existing in flow of one of the two phasesill be reflected in the other phase through RTD concentrationsf tracer in that phase. In other words, from the solids RTD datane should be able to corroborate the similar inferences as maderom the liquid RTD study with regard to the extent of channel-ng or non-uniform mixing prevalent on the stage as a functionf solid and liquid velocities. There is another important thing toote from the point of view of measuring tracer concentrationsn the solid phase. It is fair to say that the existing analyticalechniques to measure concentrations within solids are prone to
easurement error and may not be truly relied upon to constructTD response curve. A literature review indeed shows that mostf the studies pertain to injection and measurement of tracer inhe gas or liquid phase in G–S or L–S systems [19–21]. Onlyn recent years, thanks to the development of optical or otheron-intrusive tomography techniques, including X-ray or IR,he visualization of solid voids and the measurement of solid-olume fractions in two-phase flow through packed columnsave been made possible [22–24]. We plan to carry out solidsTD study in a separate future work.
. Correlations for determining �P, εb, and hu duringross-flow fluidization
In general, the experimental variables such as water and resinow rates, downcomer (stage) height, and particle size may beasily measured and controlled during operation of the column.n the contrary, the measurement of bed porosity and upstreameight of the fluidized bed is difficult to measure due to closednd continuous operation of the ion exchanger column. Simi-arly, an empirical correlation for predicting the pressure-dropscross the solid beds under cross-flow fluidization state duringhe continuous operation is required for design and operation oflarge-scale column. The significance of the developed corre-
ations in the present work is, therefore, to predict these threeuantities by the scale-up analysis. The approach to develop-ng correlations follows from the scale-up analysis based on the
-Buckingham theorem. The process variables of cross-flowuidization were mutually correlated by an empirical equation inon-dimensionalized groups. As per the fundamental postulate
f the theorem, every physical relationship between n physicaluantities can be reduced to a relationship between m = n − rutually independent dimensionless groups, whereby r standsor the rank of the dimensional matrix, made up of the physical
A. Singh et al. / Chemical Engineering and Processing 47 (2008) 957–970 967
Table 5Operating variables in the multi-stage column
S. no. W (kg/s) Q (m3/s) dp (m) hd (m) hu (m) m (kg) hCCl4 (m) �P (Pa)
1 0.00022 0.00020 0.00150 0.005 0.0150 0.13312 0.014 218.1482 0.00022 0.00022 0.00150 0.005 0.0160 0.12868 0.014 218.1483 0.00022 0.00023 0.00150 0.005 0.0170 0.11848 0.013 202.5664 0.00022 0.00027 0.00150 0.005 0.0200 0.11417 0.012 186.9845 0.00030 0.00023 0.00100 0.010 0.0300 0.15542 0.014 218.1486 0.00030 0.00025 0.00100 0.010 0.0320 0.15162 0.014 218.1487 0.00037 0.00017 0.00100 0.010 0.0220 0.19660 0.021 327.2228 0.00037 0.00018 0.00100 0.010 0.0240 0.18084 0.020 311.6409 0.00037 0.00022 0.00100 0.010 0.0290 0.15359 0.016 249.312
10 0.00052 0.00018 0.00055 0.015 0.0350 0.17536 0.021 327.22211 0.00052 0.00020 0.00055 0.015 0.0380 0.16877 0.020 311.64012 0.00052 0.00022 0.00055 0.015 0.0420 0.16481 0.019 296.0581 0151 015
qci
p�
phρ
μ
f
A
E
F
R
B
w
vo = Q(12)
TN
S
11111
3 0.00052 0.00023 0.00055 0.4 0.00052 0.00027 0.00055 0.
uantities in question and generally equal to (or in some fewases smaller than) the number of the base quantities containedn their secondary dimensions (Tables 5 and 6).
In the present case, a total of 12 number of independentrocess variables were identified: pressure-drop across a stageP, superficial velocity of water vo, superficial velocity of solid
articles vs, upstream bed height hu, downstream stage heightd, gravity g, bed porosity ε, column diameter D, solid densitys, liquid density ρf, particle diameter dp and fluid viscosityf. Based on these variables the non-dimensional groups were
ormed as follows:
rchimedes no. (Ar) = buyoancy force
inertial force
= (ρs − ρf)gρfd3p
μf(3)
uler no. (Eu) = pressure force
inertial force= �P
ρfv2o
(4)
roude no. (Fr) = inertial force
gravity force= v2
o
gdp(5) w
e
able 6on-dimensionalized groups in correlations
. no. εs Ar Fr Re vs/
1 0.507 49.663 0.036 34.663 0.02 0.542 49.663 0.042 37.552 0.03 0.594 49.663 0.049 40.440 0.04 0.648 49.663 0.064 46.218 0.05 0.712 14.715 0.074 26.960 0.06 0.730 14.715 0.085 28.886 0.07 0.567 14.715 0.037 19.257 0.08 0.620 14.715 0.045 21.183 0.09 0.710 14.715 0.063 25.034 0.00 0.750 2.448 0.083 11.650 0.01 0.770 2.448 0.098 12.709 0.02 0.788 2.448 0.116 13.769 0.03 0.802 2.448 0.134 14.828 0.04 0.834 2.448 0.175 16.946 0.0
0.0455 0.16124 0.019 296.0580.0550 0.15239 0.018 280.476
eynolds no. (Re) = inertial force
viscous force= ρfvodp
μf(6)
ed porosity (ε) = volume of pores
bed volume= V − (m/ρs)
V(7)
superficial velocity of solid particles
superficial velocity of fluid= vs
vo(8)
downspout/stage height
column diameter= hd
D(9)
upstream bed height
column diameter= hu
D(10)
diameter of particle
column diameter= dp
D(11)
The superficial velocities of liquid and solid in the columnere calculated as follows:
A
here A is the effective cross-sectional area of the stage andquals the total cross-sectional area of the column less that of
vo hd/D hu/D dp/D Eu
0729 0.04761 0.14285 0.01428 408.4950673 0.04761 0.15238 0.01428 348.0670625 0.04761 0.16190 0.01428 278.6820547 0.04761 0.19047 0.01428 196.9530421 0.09523 0.28571 0.00952 300.1190393 0.09523 0.30476 0.00952 261.4370731 0.09523 0.20952 0.00952 882.3510665 0.09523 0.22857 0.00952 694.4910563 0.09523 0.27619 0.00952 397.7910621 0.14285 0.33333 0.00523 729.2150569 0.14285 0.36190 0.00523 583.5650525 0.14285 0.40000 0.00523 472.3770488 0.14285 0.43333 0.00523 407.300427 0.14285 0.52381 0.00523 295.430
9 ring a
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v
Itpptcgolsb(Ta
ε
(
E
tbc(
l
l
l
Htec
pgruflsgTd
8
tep1prierEtewtr
l
8.2. Bed height (upstream)
Fig. 13 describes the maximum height or depth of the flu-idized bed as functions of various operating conditions (vs, vo,
68 A. Singh et al. / Chemical Enginee
he downspout, and
s = W
ρshdD(13)
n Eq. (13) the superficial velocity of the solids flowing acrosshe stage was calculated based on the projected area (hd × D)erpendicular to the direction of the flow. This method of com-iling the complete set of dimensionless numbers makes it clearhat the numbers formed in this way cannot contain numeri-al values or any other constant. These appear in dimensionlessroups only when they are established and interpreted as ratiosn the basis of known physical interrelations. Now we can get ateast three independent relationships by compiling a completeet of above dimensionless numbers. These relationships are fored porosity (εs), Euler number (Eu) and (upstream) bed heighthu/D) ratio, which are the dependent dimensionless variables.hree empirical expressions containing dimensional groups ares follows:
= C1
(hd
D
)a1(dp
D
)a2(vs
vo
)a3
(14)
hu
D
)= C2
(hd
D
)b1(dp
D
)b2(vs
vo
)b3
(15)
u = C3Ren1Arn2Frn3εn4s
(hd
D
)n5(hu
D
)n6(dp
D
)n7
×(
vs
vo
)n8
(16)
Dimensional analysis cannot provide any information abouthe exact form of the functions. This information can onlye obtained experimentally or by some additional theoreticalonsiderations. Taking the logarithmic on both sides of Eqs.14)–(16), we obtain the following working expressions:
n εs = ln C1 + a1 ln
(hd
D
)+ a2 ln
(dp
D
)
+a3 ln
(vs
vo
)(17)
n
(hu
D
)= ln C2 + b1 ln
(hd
D
)+ b2 ln
(dp
D
)
+b3 ln
(vs
vo
)(18)
n Eu = ln C3 + n1 ln Re + n2 ln Ar + n3 ln Fr + n4 ln εs
+n5 ln
(hd
D
)+ n6 ln
(hu
D
)+ n7 ln
(dp
D
)
+n8 ln
(vs
vo
)(19)
ere in the above equations C1–C3, n1–n8, a1–a4, and b1–b3 arehe unknowns coefficients which are to be determined from thexperimental measurements. The independent variables, whichan be experimentally controlled or varied are dp, hd, vo and vs.
nd Processing 47 (2008) 957–970
On the experimental side, measurements were made for theressure-drop, bed height, and porosity for different sizes oflass beads, downspout or stage height, solid and water flowates in the column of diameter = 0.10 m. The bed height (depth)pstream of the stage was measured during the counter currentows of glass beads and water. To measure the bed porosity, theolid and water flows were instantly stopped and the weight oflass beads and fixed bed height on the stage were measured.he bulk density and particle sizes of glass beads were a priorietermined.
.1. Bed porosity
Fig. 12 describes the effect of various experimental condi-ions (vs, vo, dp, and hd) on bed porosity. Bed porosity wasxperimentally measured for different water and solid flow rates,article size (0.55, 1.0 and 1.54 mm) and downcomer height (5,0 and 16 mm) by varying one parameter at a time. The bedorosity was found to decrease with increasing particle size andesin flow rate, and on the other hand increase with increas-ng stage height and water flow rate. Trend lines were drawn tostimate the nature of correlation for these cases and the cor-esponding slopes were obtained. These slopes were used inq. (14) to replace the parameters a1, a2 and a3. By substi-
uting these values in Eq. (17) and again comparing with thexperimentally obtained bed porosity, constant ln C1 in Eq. (17)as determined to be −5.849. The final equation obtained for
he determination of bed porosity in cross-flow fluidization isepresented as
n ε = −5.849 + 0.1791 ln
(hd
D
)− 0.3763 ln
(dp
D
)
−0.8676 ln
(vs
vo
)(20)
Fig. 12. Cross-flow fluidization: variation in bed porosity.
A. Singh et al. / Chemical Engineering and Processing 47 (2008) 957–970 969
dduiaAwaifitfa
l
8
iounrcvmf
l
F
sEbaasflarm
9
rospdtfdpflswop
Fig. 13. Cross-flow fluidization: variation in upstream bed height.
p, and hd). The average of the bed heights measured at fiveifferent locations upstream of the resin flow on the stage wassed as the maximum average bed height (hu). The set of exper-mental data for bed heights was collected by varying vs vo, dp,nd hd, each of them one at a time keeping the others constant.s observed from Fig. 13, the height of fluidized bed increasesith increasing water flow rates, particle size, and stage height
nd decreases with increase in the resin flow rates. Substitut-ng the coefficients (slopes of the trend lines) in Eq. (18), andtting the data with the experimentally measured bed height,
he constant ln C2 was determined. The final equation obtainedor the determination of upstream bed height is expresseds
n
(hu
D
)= −6.599 + 0.4216 ln
(hd
D
)− 0.3367 ln
(dp
D
)
−0.9593 ln
(vs
vo
)(21)
.3. Euler number (Eu)
A correlation for the prediction of Eu was developed by min-mizing the residual of Eq. (19). In this method the squaresf the known variables are minimized for various guess val-es of unknown variables. The mathematical regression of largeumber of data was carried out with help of the NAG sub-outine [25]. The subroutine evaluates the coefficients of theorresponding variables from the dimensionless experimentalariables. The final equation obtained for the prediction of Euay be represented in terms of the dimensionless quantities as
ollows:
n Eu = 2702 + 233.6 ln Re + 0.005351 ln Ar + 115.3 ln Fr(hd
) (hu
)
+0.8772 ln εs + 0.1467 lnD+ 0.3559 ln
D
+349.3 ln
(dp
D
)+ 0.02398 ln
(vs
vo
)(22)
tiss
ig. 14. Comparison between data and model predictions for Euler number.
Thus, knowing the operating conditions during the multi-tage cross-flow fluidization, Eu may be determined by usingq. (22), which in turn determines the pressure-drop across theed (refer Eq. (4)). Finally, Fig. 14 compares the experimentalnd model predicted results obtained from Eq. (22). A reason-bly good agreement is observed under the operating conditionselected in this work. In essence, the pressure-drop across theuidized bed, the maximum height of the resins on the stage,nd the bed porosity may be estimated from Eqs. (20)–(22) cor-esponding to the cross-fluidization of the resin particles in theulti-stage fluidized bed column.
. Conclusions
By choosing a proper range of resin vis-a-vis water flowates, multi-stage ion exchanger column was operated continu-usly, with solid resin and water flowing counter currently underteady-state conditions. During steady operation, fluidized resinarticles move across the stage to the next stage through theownspout, as water flows upward through the voids betweenhe solid particles. Loading and flooding limits were obtainedor the stable operation of the column. The maximum pressure-rop occurred in the column due to the hydrostatic head. Theressure-drop across the resin beds during apparently cross-flowuidization on the stage was found to be negligible. The RTDtudy was carried out to ascertain the extent of mixing betweenater and solid particles. The empirical correlations were devel-ped using scale-up analysis to estimate the pressure-drop, bedorosity, and the maximum depth or height of the resin bed on
he stage during the continuous operation of the multi-stage flu-dized bed column. The hydrodynamic data presented in thistudy assume significance from the perspective of design andtable operation of staged fluidized bed ion exchangers.
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A
ACdDghhm�
QQv
v
Wz
Gε
μ
ρ
ρ
φ
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EFR
Smot
R
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[
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70 A. Singh et al. / Chemical Enginee
cknowledgement
The authors acknowledge the partial financial support (DSTo.: SR/S3/CE/09/2002-SERC-Engg dated 31.10.2002) from
he Department of Science and Technology (DST), New Delhio conduct this study.
ppendix A. Nomenclature
cross-sectional area of the column (m2)D drag coefficientp average size of the particle (mm)
diameter of the column (m)gravity (m2/s)
d height of the downcomer (m)u upstream height of the fluidized bed (mm)
mass of the resin (kg)P pressure-drop across the bed (Pa)l flow rate of water (m3/s)s flow rate of resin (kg/min)
o superficial velocity of liquid (m/s)s superficial velocity of resins (m/s)
mass flow rate of resin (kg/s)axial co-ordinates in the bed (m)
reek symbolsbed porosity
f viscosity of fluid (Pa s)density of water (kg/m3)
p bulk density of solid materials (kg/m3)s sphericity of the particles
imensionless numberr Archimedes no., (ρs − ρf)gρfd
3p/μf
u Euler no., �P/ρfv2o
r Froude no., v2o/gdp
e Reynolds no., ρuodp/μf
ubscriptsf minimum fluidization
superficialterminal condition
eferences
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[
[
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