Crystal growth kinetics of two-step growth process in liquid fluidized-bed crystallizers

Crystal growth kinetics of two-step growth process

in liquid #uidized-bed crystallizers

Cli!ord Y. Tai*

Department of Chemical Engineering, National Taiwan University, Taipei 106, Taiwan, ROC Received 17 March 1999; accepted 27 May 1999

Communicated by M. Schieber

Abstract

The #uidized-bed crystallizers have long been used in the chemical industry and recently for water treatment as the so-called pellet reactors. This report summarizes the experimental results regarding hydrodynamics and crystal growth rates in laboratory-scale #uidized-bed crystallizers. The two-step crystal growth model rather than the over-all model is adopted to explain the observed kinetic behaviors. In using the two-step growth model, the disadvantage is the uncertainty of the surface-reaction order. A reliable method is suggested to explore the surface-reaction order using a lean #uidized-bed crystallizer. The crystal growth process of sparingly soluble salts, which have been studied recently for a few systems, seems more complex than that of soluble salts. Then, a comparison of crystal growth kinetics of soluble salts is made between the dense and lean #uidized-bed crystallizers. As a conclusion, the two-step growth model is suitable for the estimation of crystal growth rates in the design of a liquid #uidized-bed crystallizer.  1999 Elsevier Science B.V. All rights reserved.

Keywords: Crystal growth kinetics; Liquid #uidized bed; Two-step growth model; Sparingly soluble salts; Soluble salts

1. Introduction

Fluidized beds have been adopted as separators for many operations in the chemical industry, espe-cially for the crystallization operation. The most e$cient modern industrial crystallizers are of sus-pended-type bed, in which the crystals are grown in a liquid #uidized zone. A supersaturated solution in the metastable region enters the bottom of the bed and the supersaturation is partially released when the solution passes through the growth zone. For example, the Oslo-Krystal Crystallizers belong to the #uidized-bed type, in which the gentle action

* Fax: #886-2-362-3040.

between crystals or between the crystal and the vessel wall minimizes secondary nucleation and allows crystals to grow to a larger size [1]. There-fore, #uidized-bed crystallizers are suitable for producing many organic and inorganic chemicals. Recently, a #uidized bed has been used in water treatment. A pellet reactor, which is a reactive #uidized-bed growth-type crystallizer, has been de-veloped for water softening of drinking water [2] #uoride and phosphate removal [3,4], and for heavy metal recovery from waste streams [5]. The water feed and the chemicals required to cause deposition are fed to the reactor containing sus-pended seeds of several hundred micrometers. The undesired species react to form insoluble salts, such

0022-0248/99/$ - see front matter  1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 0 2 4 8 ( 9 9 ) 0 0 3 0 0 - 0

Nomenclature

*C concentration di!erence, kg solute/kg solvent

D di!usivity, m/s Da Damkohler number

G linear growth rate, m/s Ga Galileo number

g gravity acceleration m/s

K mass-transfer coe$cient, kg/m s K surface-reaction coe$cient, kg/m s

¸ crystal size, m

Mv density number, (o!o)/o

m exponent of crystal size in Eq. (2)

n exponent of*C in Eq. (2) Re Reynold number

R crystal growth rate, kg/ms r surface reaction order Sc Schmidt number Sh Sherwood number ; super"cial velocity, m/s

; terminal velocity of a single particle, m/s

Greek letters e expanded-bed voidage g e!ectiveness factor k liquid viscosity, kg/ms l kinematic viscosity, m/s o solution density, kg/m o particle density, kg/m p relative supersaturation

pι interfacial relative supersaturation

as calcium ion to form calcite (CaCO) in water softening, and then grown on the seeds, which are later removed from the reactor after exceeding a certain size. When compared with the conven-tional precipitation process, in which tiny precipi-tates form in the bulk solution due to primary nucleation, the major advantages of this newly de-veloped clean technology are the avoidance of sludge formation, the possibility of material recov-ery and the reduction of solid waste.

One of the most important parameters needed in the design of a #uidized bed is the up#owing liquid velocity necessary to keep the crystals in

suspen-sion. Depending on the super"cial velocity applied, two types of beds are classi"ed, i.e., the dense bed and the lean bed. The former is operated at a super-"cial velocity lower than the particle terminal velo-city to give the bed a clear solution-suspension boundary. In the later, the super"cial velocity is approaching the particle terminal velocity and the particles move freely in the bed with no solution-suspension boundary. There are two disadvantages associated with the lean bed, i.e., low seed loading due to a high-bed voidage and carry-over of par-ticles due to a #uctuation of solution velocity. Al-though dense beds are most commonly used in the

chemical industry, the hydrodynamics in the bed is more complicated. The crystals, being present in a large quantity, are subjected to hindered settling and further complication arises if the crystals have irregular shape and size distribution. Besides, the bed voidage changes with the up-#owing liquid velocity, resulting in di!erent liquid #ow patterns. On the other hand, the #ow pattern is simple in a lean bed, which resembles a plug #ow with the relative velocity between the particle and liquid being almost the same as the super"cial velocity. Thus, a lean bed is usually used in the laboratory to study the crystal growth mechanism.

To design a #uidized-bed crystallizers, the crystal growth rates should be available besides knowing the hydraulics of a #uidized bed. The crystal growth is a complex process, which includes many steps such as di!usion, adsorption, dehydration, integration, and others. Thus, there are many fac-tors that a!ect the crystal growth rate. The facfac-tors are summarized by Mullin [1] and Sohnel and Garside [6], including growth rate dispersion, crys-tal size, solution velocity, admixtures, magnetic "eld, temperature, and pH. The in#uence of most of the factors to each of the growth step is still un-known, that a general expression of crystal growth rate is di$cult to establish, especially for the spar-ingly soluble systems. This report summarized the studies related to the crystal growth in a lab-scale #uidized-bed crystallizer, either a lean or a dense one. Then an experimental procedure in search of crystal growth rate, which is used for the design of a #uidized-bed crystallizer, is suggested.

2. Design and operation of 6uidized beds

A typical lab-scale #uidized-bed crystallizer, which provides reliable crystal growth data quick-ly, is designed by Mullin et al. [7] as shown in Fig. 1. The apparatus includes the main crystalliza-tion Seccrystalliza-tion A where crystals grows, the calming Section B to avoid carry-over of crystals, the vari-able speed pump E to deliver di!erent #ow rates of solution, the temperature control system (heat ex-changer J, resistance thermometer D, heating tapes

G, and a temperature controller) to maintain the

desired temperature in the crystallization section,

Fig. 1. A typical laboratory-scale #uidized-bed crystallizer [7].

and the salt catching assembly (three-way cock K, salt box ¸ and solution reservoir N). The important features of this design are adopted later by the researchers in this "eld, perhaps with a slight modi-"cation by inserting a distributor in front of the crystallization section for an even #ow of solution. The major physical operating variables of a #uidized bed are the particle diameter, the super-"cial velocity and the "xed-bed height. These para-meters, together with the temperature and the physical properties of the particle and the solution, completely determine the other properties of the system, such as bed voidage, expanded-bed height, speci"c surface area, head loss, and energy

Table 1

Super"cial velocity-bed voidage relationship for calcite crystals suspended in a #uidized bed Particle size ¸₍lm) Super"cial velocity ; (m/s) Bed voidage e (!) e/(1!e) (!) ;   /¸  (10 m\  s\ ) 460 0.0139 0.68 0.782 6.01 0.0233 0.77 1.479 11.2 0.0353 0.84 2.567 18.4 0.0469 0.91 5.172 25.8 651 0.0139 0.50 0.217 3.21 0.0233 0.58 0.390 5.98 0.0281 0.64 0.593 7.48 0.0353 0.71 0.963 9.84 0.0469 0.79 1.718 13.8 921 0.0236 0.60 0.449 3.25 0.0283 0.64 0.593 4.04 0.0353 0.70 0.898 5.27 0.0469 0.77 1.479 7.41

Fig. 2. A plot of Eq. (2) for calcite crystals suspended in a #uidized bed.

dissipation. Van Dijk and Wilms [5] derived the bed porosity or voidage for the Reynolds number in a range between 5 and 100.

e (1!e) "130 l  g o o!o ;   ¸ , (1)

wheree is the bed voidage, l the kinematic viscos-ity,o the solution density, o the particle density, ; the super"cial velocity and ¸ the particle size. By varying the super"cial velocity between the min-imum #uidization velocity and the particle terminal velocity, they suggest a suitable range of particle size, 0.3 mm(¸(1.5 mm, for the operation of a #uidized bed and conclude that the particles )0.3 mm are too small because of excessive ex-pansion and the particles *1.5 mm are too large because of small surface area for a batch crystal-lizer.

Tai and Wu [8] studied the velocity-voidage relation for the rhombohedron calcite #uidized in a laboratory-scale crystallizer. The agreement be-tween the experimental data and Eq. (1) is fairly good except for a datum point of high voidage (e"0.91) as shown in Fig. 2. The experimental data are tabulated in Table 1.

3. Over-all crystal growth rate model

Before early 1980s, most of the crystal growth rate data measured in a dense #uidized bed were presented by an over-all rate model, which is expressed in terms of the crystal size and the

driving force

G"K¸K(*C)L (2)

where G is the linear crystal growth rate, ¸ the crystal size, *C the concentration di!erence be-tween the supersaturated and saturated solutions,

m and n the constants.

The values of m and n have been determined for several systems, including potassium nitrate [9], sodium chloride [10], potassium alum [11], potas-sium sulfate [12,13], nickel sulfate [14] and citric acid [15].

The over-all growth rate is simple to use for design purpose. However, the applicability of growth rate is limited to the small range of operat-ing variables investigated. It is not reliable to make estimation outside the operating conditions nor in the scale-up of crystallizer.

4. Two-step crystal growth model

Several mechanisms regarding crystal growth have been proposed in Ref. [1]. Among them the two-step growth model is considered most useful from the chemical engineering point of view. The model takes into account the mass-transfer and surface-reaction resistance in series and neglects all other resistance in a crystal growth process. At steady-state conditions, the two steps can be repre-sented by the following equations:

R"K(p!pι) mass transport (3)

and

R"KpPι surface reaction. (4)

Due to the uncertainty of r and unknown pι, the

model was not adopted at the time when it was proposed. Recently, attempts have been made to determine the individual rate constants, K and K, and the surface reaction order r.

Tavare and Chivate [16] and Langer and O!er-mann [17] measured the crystal growth and dis-solution rates of potassium sulfate and potassium alum, respectively, in a dense #uidized-bed crystal-lizer. They assumed that the mass-transfer coe$c-ient of crystal dissolution is identical to that of crystal growth and then determined the kinetics of

the surface-reaction step; i.e., the mass-transfer co-e$cient of crystal dissolution is substituted into Eq. (3) to calculate p_{ι and then K and r were} deter-mined according to Eq. (4).

Assuming that r has a _{`standardizeda value of} 2 for several systems, including copper sulfate, po-tassium alum, magnesium sulfate and sodium thiosulfate, Karpinski and his co-workers [18}22] determined the individual rate constants from the following equation, which is derived by combining Eqs. (3) and (4):

pR\_ "K\

 R #K\

 (5)

The growth kinetics of mass transfer and surface reaction were revealed, for example, the mass-transfer coe$cient of crystal growth is the same as that of crystal growth; the surface-reaction coe$c-ient is a function of crystal size and exhibits a maximum when cation admixture is present; the activation energy of di!usion step changes at 318 K for potassium alum crystal, indicating a change in di!usion mechanism above 318 K.

The use of r"2 should be cautious; however, it may be applied to some systems. For example, the surface-reaction order of potassium alum is proven to be 2 over a wide range of supersaturation in a single crystal experiment [23]. Thus, assuming

r"2, Tai and his co-workers [24,25] analyzed the

crystal growth and dissolution rate data of potassi-um alpotassi-um obtained in a lean #uidized-bed crystal-lizer, which is di!erent from the dense bed used by Karpinski and his co-workers [18] and reached some unexpected results. Suppose the lean #uidized bed is operated near the crystal terminal velocity of a uniform crystal size, the solution velocity is re-lated to the crystal size by the following form: ;"



(. !.)g . k





¸. (6)

When Eq. (6) is substituted into the Froessling equation, which is Sh"0.6Re Sc, the result-ant equation has the following form:

Sh"0.306GaMvSc. (7) The mass-transfer coe$cient of crystal dissolution obtained in the experiment is identical to that of

Fig. 3. Plots of growth rate versus crystal size for potassium alum and potassium sulfate crystals: (䢇) KSO [12]; (䉱) K-alum [25].

crystal growth and follows Eq. (7), which shows that the mass-transfer coe$cient is independent of solution velocity and crystal size, and it is a func-tion of system properties only because the crystal size in Sherwood number and in Galileo number is canceled out. Thus, for the size-dependent growth rate of potassium alum shown in Fig. 3, the surface reaction is responsible for the size-dependent growth and the crystal growth rates tend to be constant for large crystals. Note that a crystal growth rate curve of KSO is included in the "gure, which is reported by Mullin and Gaska [12] and has a similar shape of the K-alum system. Once

r, K and K are known, the controlling step of

crystal growth process can be judged from the e!ectiveness factor, which is de"ned by Eq. (8) for a second-order surface reaction [26]:

g"(1!g Da), (8)

where Da"pK/K.

When Da is large, growth is di!usion controlled. Conversely, when Da is small, growth is surface-reaction controlled. Thus in Fig. 3, the crystal growth process of potassium alum is surface-reac-tion controlled for crystal size below 4;10\ m. As crystal size increases, the mass-transfer resistance becomes signi"cant. After the crystal size exceeds

2;10\ m, both coe$cients are constant, giving a constant growth rate. There is no dominating step for larger sizes of potassium alum crystal as judged from the e!ectiveness factor.

5. Determination of surface-reaction order

One of the disadvantages of using the two-step crystal growth model is the unknown of the sur-face-reaction order. In some reports, the value of

r is simply assumed [18,27] or the mass-transfer

coe$cient of crystal growth is replaced by that of crystal dissolution and then to determine the sur-face-reaction order [16,17]. These two methods are not reliable. Although the mass-transfer process of crystal growth is the reverse process of crystal dis-solution, the corners of the crystal become rounded quickly in a dissolution experiment, causing an error in the determination of K for small crystals [24].

Tai et al. [25] suggested a method to determine the surface-reaction order using a lean #uidized bed. When a #uidized-bed crystallizer is operated near the crystal terminal velocity, the mass-transfer coe$cient is a constant for a given system and can be estimated from Eq. (7). Note that the crystal size in Eq. (7) is canceled out, thus K is a function of system properties only, including di!usivity, densit-ies, and viscosity. Once K is known, pι can be

calculated from Eq. (3) using the R!p data. Then,

r is determined from Eq. (4) by plotting ln R versus

lnp_ι. The surface-reaction orders estimated from the available data of potassium alum from various sources are listed in Table 2, and they are close to 2. One disadvantage of using Eq. (7) is that the di!us-ivity data are available for limited systems, and the determination of di!usivity needs skillful tech-niques [28].

A comparison of surface reaction order, which has been supported by experimental evidence, in di!erent types of crystallizer is presented by Tai [29]. The values of r obtained in a single crystal growth cell and #uidized-bed crystallizer are rather consistent, either 1 or 2, which are the values often predicted by the spiral growth [30]. In a stirred-tank crystallizer, the value of r is usually higher than 2, including potassium sulfate, calcium

Table 2

Surface reaction order of potassium alum, estimated from Eq. (4)

¸;10 (m) ¹(3C) Range ofp r Corr. coef. Reference

530 32.0 0.017}0.087 2.03 0.998 [11] 990 32.0 0.006}0.075 2.13 0.993 [11] 1730 28.5 0.026}0.128 2.06 0.997 [18] 907 28.5 0.026}0.128 1.82 0.999 [18] 1095 23.5 0.059}0.147 2.01 0.993 [24] 194 23.5 0.043}0.142 2.10 0.998 [25]

carbonate, and succinic acid, however; r is close to 2 in the #uidized bed for the two former systems. The high order of surface-reaction kinetics can be explained by the NAN growth model, which is the only growth model that predicts r'2 [30]. An-other thing worth mentioning is that the impurity may a!ect the surface-reaction order and di!erent types of impurity yield di!erent e!ects. For example, the surface-reaction order of magnesium sulfate is 1 for pure system and the additive of dye does not change the order [23]. However, the order changes from 1 to 2 when inorganic species, Cr> or Fe>, is added to the solution [19,31].

6. Crystal growth of sparingly soluble salts The study of crystal growth of sparingly soluble salts in a #uidized bed is scarce because of its limited application. Besides, the growth experiment of a sparingly soluble salt is more di$cult than that of a soluble salt. In the former, the pH should be well controlled in order to prevent nucleation [32]. Recently, the pellet reactor, which is a reactive #uidized-bed crystallizer, has been used to remove calcium ions, #uoride ions, phosphate ions, heavy metal ions and others from water streams. For design purpose, the crystal growth rates of sparing-ly soluble salts, such as carbonates, sul"des, and others are required. So far, the systems of calcium sulfate (gypsum), calcium #uoride (#uorspar) and calcium carbonate (calcite) are reported. However, the sparingly soluble systems will be widely studied because of the urgent need of clean technologies.

For the design of pellet reactors to remove cal-cium ion, which is a major species of hard water, and

#uoride ion, which is a pollutant in the waste stream from a semi-conductor plant, Tai and his co-workers studied the crystal growth of calcium #uoride and calcium carbonate in a batch #uidized bed, using a pH-stat apparatus as shown in Fig. 4 [3,8,34]. The important feature of the apparatusis the mainten-ance of constant pH controlled by an autotitrator. The metastable regions of calcite and #uorspar were found and the growth experiments were suc-cessfully conducted in this region without signi"-cant nucleation. Factors that a!ect the crystal growth rate in a dense #uidized bed, include super-saturation, crystal size, super"cial velocity, pH, ionic strength, and type of seed. The crystal growth process of a sparingly soluble salt seems more com-plicated than a soluble salt. The electrical double layer around a crystal is speculated to play a role in the crystal growth process of a sparingly soluble salt because the crystal growth rate is lower at high-ionic strength, which reduces the thickness of electrical double layer. At the present time a crystal growth model to include all the signi"cant factors is di$cult to establish. However, under similar envi-ronment of pH, ionic strength, and species ratio, the growth rate data in a lean #uidized bed show that the mass-transfer coe$cient is independent of crys-tal size and super"cial velocity and the surface-reaction coe$cient is a function of crystal size, all consistent with the results of soluble systems.

7. Comparison of growth kinetics in dense and lean beds

The hydrodynamics in a dense bed seems quite di!erent from that in a lean bed and the

Fig. 4. A pH-stat crystallization system. (1) pH and temperature indicator, (2) reagent bottle, (3) pumping system of reagent, (4) reagent delivering line, (5) burette, (6) thermometer, (7) water bath, (8) storage tank, (9) temperature controller, (10) motor, (11) glass electrode, (12) reference electrode, (13) thermo-compensator, (14) axial-#ow impeller, (15) magnetic pump, (16) #owmeter, (17) #uidized bed reactor, (18) ball valve, (19) distributor.

mass-transfer process is related to the hydrodyn-amics. Therefore, it is interesting to compare the crystal growth kinetics of the two beds.

Mullin and his co-workers [12,13] measured the crystal growth rates of potassium sulfate in a dense (e"80}85%) and lean (e'0.95)#uidized bed. The results show good agreement for these two types of bed. However, a correlation to predict the mass-transfer coe$cient is not established.

The crystal growth rates of potassium alum in a dense bed and a lean bed are measured by Budz et al. [18] and Tai et al. [24], respectively. The crystal growth rates were calculated by Eq. (3) using the mass-transfer coe$cient in the respective report. The mass-transfer coe$cient of dense bed has been corrected to 296.5 K using the activation energy reported by Budz et al. [20]. Again, a good agree-ment exists between the two sets of data as shown in Fig. 5.

The good agreement for the crystal growth rates between the dense and lean #uidized bed is

under-stood through the following argument. The constant crystal growth rate means that the sur-face-reaction and mass-transfer coe$cients are the same in the two types of #uidized bed. The surface reaction is a surface phenomenon and should be independent of hydrodynamics encountered in the two-types of #uidized bed. As far as the mass-transfer coe$cient is concerned, it is usually con-sidered as a function of crystal size and solution velocity. However, according to Eq. (7), the mass-transfer coe$cient in a lean bed is independent of crystal size and super"cial velocity, which is almost the same as terminal velocity or relative velocity, and is a function of system properties only. In other words, the mass-transfer coe$cient is a constant for a given system. As the super"cial velocity is re-duced, the bed voidage and thus the cross-sectional area for the passage of solution become smaller than that of a lean bed. When the solution passes through the crystal bed, the relative velocity, which is the slip velocity between solution and crystal,

Fig. 5. Comparison of potassium alum growth rates at 296.5 K for dense and lean bed: (*) dense bed [18]; (- - - -) lean bed [24].

becomes higher than the reduced super"cial velo-city. It is possible that the relative velocity, which is the velocity that determines the mass-transfer coef-"cient, in a dense bed is almost equal to the ter-minal velocity of crystals regardless of bed voidage. As a result, the mass-transfer coe$cients in a lean bed and dense bed are identical.

The constant mass-transfer coe$cient is also supported by the evidences in the crystal dissolu-tion experiments. Tournie et al. [33] analyzed ex-tensive dissolution data reported in the literature for both lean and dense #uidized beds, and they concluded that the mass-transfer coe$cient was practically independent of liquid solution and par-ticle size.

8. Conclusions

After analyzing a large amount of crystal growth data, the crystal growth process of soluble salts seems not too complex in a liquid #uidized bed. The two-step crystal growth model is good enough to explain the observed crystal growth kinetics in a lean or dense bed. A reliable method has been proposed to determine the parameters of the two-step growth model, using a lean #uidized bed. For

sparingly soluble salts, the growth process is more complicated, but the two-step model can be applied under a similar environment of pH, ionic strength, and species ratio. Nevertheless, the variation of these operating conditions is quite small in a large-scale operation. Besides, the hydrodynamic behav-iors between large- and laboratory-scale #uidized-bed crystallizers are not much di!erent as com-pared with other types of equipment. Therefore, the estimation of crystal growth rates for design pur-pose is more con"dent after tremendous e!orts have put forth in this "eld.

Acknowledgements

The author gratefully acknowledges the "nancial support of the National Science Council of the Republic of China in the research area of crystal growth through the years.

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