Crystal Growth Kinetics of Calcite in a Dense Fluidized-Bed Crystallizer

Separations

Crystal Growth Kinetics of Calcite

in a Dense Fluidized-Bed Crystallizer

Clifford Y. Tai, W.-C. Chien, and C.-Y. Chen

Dept. of Chemical Engineering, National Taiwan University, Taipei, Taiwan 106

The growth kinetics of calcite crystals are studied in a batch-fluidized-bed crystallizer,

which is maintained at a constant pH. The growth experiments are conducted in the

metastable region explored as part of this research. The crystal growth rates are

e®alu-ated from the consumption rates of calcium ions, using the cured natural calcite or

silica sand as seeding materials. Se®eral operation ®ariables are in®estigated, including

supersaturation, pH, ionic strength, superficial ®elocity, and particle size and type of

seed. The significant factors that affect the crystal growth rate are identified. Then the

crystal growth data of constant pH and ionic strength are analyzed by the two-step

growth model. The mass-transfer coefficients are obtained and compared at ®arious

crystal sizes and superficial ®elocities. Finally, a growth-rate equation of calcite crystal,

which is based on the two-step growth model, is proposed for design purposes.

Introduction

Recently a pellet reactor, which is a reactive, fluidized-bed, growth-type crystallizer, was developed for water softening, fluoride and phosphate removal, and heavy-metal recovery. The water feed and the chemical reagents required to cause deposition are fed to the reactor containing suspended seeds ŽDirken et al., 1990; Seckler et al., 1990; van Dijk and Wilms,

.

1991 . In water softening, the undesired species, calcium ion, reacts to form calcium carbonate and then grows on the seeds, which are later removed from the reactor after they exceed a certain size. In the operation of the pellet reactor the flow pattern in the fluidized bed resembles an ideal plug flow with a limited backmixing, which gives a higher conversion as

com-Ž .

pared to the ideal mix]flow reactor Levenspiel, 1972 . In the design of a pellet reactor for water softening, we need to know the crystal growth kinetics of calcium carbonate and the hydraulics of a fluidized bed. It seems that the former is less understood than the later. The reported growth-rate equation of calcium carbonate is rather empirical, for

exam-Ž .

ple, an overall growth-rate model van Dijk and Wilms, 1991 :

2q

w

x

d Ca _{2y X} 2q

w

x

s_{K A CaT}

_

_CO3 y_{Ks p}

₄

_,

_{Ž .}

₁ dt

Correspondence concerning this article should be addressed to C. Y. Tai.

Ž w 2qx _. w 2q_x

where y d Ca r_{dt is the depletion rate of Ca in the} solution, which is the same as the crystal growth rate of CaCO in moles per unit time; K3 T is a constant; A is the specific surface area of seed crystals; KXs p is the concentra-w 2qxw 2yx X ₄ tion solubility product of calcite; and Ca CO y_K

3 s p

represents the driving force for crystal growth.

Several mechanisms regarding crystal growth have been

Ž .

proposed in the literature Mullin, 1993 , among which the two-step growth model is considered the most useful from the chemical engineering point of view. At steady-state con-ditions, the two steps can be described mathematically by the following equations:

Gs Kd

Ž

s ysi

.

mass transfer

Ž .

2

r

s_K s _{surface reaction.}

_{Ž .}

₃

r i

Although this model is a simplified scheme for crystal growth, it reveals a great deal of useful kinetic information, which

Ž .

was recently summarized by Tai 1997 . The systems studied are mostly soluble salts using seeded techniques. Similar studies of sparingly soluble salts are limited, because large single-seed crystals of the system are difficult to prepare and the application of the technique was limited in the past.

The single-seed crystals of calcite, one of the barely soluble salts, were successfully prepared by using the gel growth

technique, and the crystal growth kinetics of calcite grown in a stirred vessel was investigated using the two-step growth

Ž .

model to analyze the growth-rate data Tai et al., 1993 . It is concluded that the mass-transfer resistance and surface-reac-tion resistance are significant. The mass-transfer resistance is related to the relative velocity between crystal and solution and increases with a decrease in relative velocity. Under the well-suspended conditions, the relative velocity between crys-tal and solution in a fluidized bed is lower than that in a stirred vessel, thus the mass-transfer resistance as well as the surface-reaction resistance for crystal growth of calcite should be considered in a fluidized bed. Therefore, an overall growth-rate equation, such as Eq. 1, is not an adequate ex-pression for scale-up purposes, because different hydrody-namic behaviors exist between scales.

Besides the hydrodynamic effects, the crystal growth of barely soluble salts are affected by the chemical properties of the supersaturated solution. For example, Stubicar et al. Ž1990, 1992 found that the solution properties, including ionic. strength, pH, and lead-to-fluoride ion activity ratio, influ-enced the crystal growth kinetics of lead fluoride;a-PbF and2

b-PbF appeared at different pH values, and crystal growth2

rates were higher at higher ionic strength.

The aim of this study is to investigate the crystal growth kinetics of calcite in a dense fluidized-bed crystallizer using the pH-stat method. Experiments were first conducted in a pH-stat vessel to identify the metastable region, where nucle-ation is suppressed, for the CaCl2]Na CO ]H O system.2 3 2

Then the growth experiments were performed in the metastable region using three crystal sizes, namely 460, 650, and 920mm, which are prepared by sieving the cured natural calcite. The crystal size is the mean size of the apertures of the two closest ASTM standard sieves. For example, crystals of size 460 mm means the fraction of crystals retained be-tween ASTM sieves Nos. 40 and 35, with nominal apertures of 420 and 500mm, respectively. The crystal growth rate was

w 2q_x

evaluated from the depletion rate of Ca in the solution. The growth-rate data were then interpreted in terms of the two-step growth model, thus the mass-transfer coefficient and surface-reaction coefficient were determined. The effects of operating variables, such as ionic strength, pH, superficial ve-locity, crystal size, and seed type, were explored. The ob-tained crystal growth-rate data will serve as the design basis of a pellet reactor for water softening.

Experimental Procedure

Determination of relati©e supersaturation

The expression of the driving force used for crystal growth of barely soluble salts is the relative supersaturation

pro-Ž .

posed by Nielsen and Toft 1984 :

1r2

s s K rK

_Ž

_{i p s p}

_.

y_1,

_{Ž .}

₄ where K_{i p} is the ionic product, defined as K s a_{i p Ca}2q?

a_CO2y; and K_{s p} is the activity solubility product of CaCO .₃ 3

The activity of species, a_Ca2q or a_CO32y, is the product of the

activity coefficient and the concentration of the respective species. The concentrations of ionic species were computed from the measured pH, total calcium concentration, and total

Figure 1. Fraction of total CO present as the respec-2 tive ions at various hydrogen-ion

concentra-( ) 2I _{( )} I _{( ) (}

tion: a CO3 ; b HCO3 ; c H CO2 3 ASTM, )

1994 .

carbonate concentration by successive approximation for the ionic strength, using a computer program that contains mass-action equations, mass balance equations, charge-bal-ance equations, and the modified Debye-Huckel equation for

¨

Ž

calculating the activity coefficient Nancollas, 1996; Tai et al., .

1993 . In the operation of a pellet reactor, supersaturation should be controlled in a suitable range; a low supersatura-tion gives a low growth rate, which means a low removal rate of species, and a high supersaturation causes nucleation, which messes up the operation. The supersaturation is re-lated to the concentration of CO32y, which is a function of pH at a fixed concentration of total carbonate. Therefore, as judged from Figure 1, which is a diagram showing the frac-tion of total carbon dioxide present as the respective ions at

Ž .

various pHs ASTM, 1974 , a suitable operational range of pH is roughly between 8.5 and 10.5, a pH lower than 8.5 giving a low CO32y concentration and a pH higher than 10.5 absorbing too much CO from air. The effect of pH on the2

relative supersaturation, which is obtained by computing, is clearly shown in Figure 2. It means that the control of pH is important in the experiment of studying the crystallization kinetics of CaCO .3

Identification of metastable region of calcite

A key to the successful operation of pellet reactor is good control of supersaturation, which is generated by the chemi-cal reaction. To suppress nucleation of chemi-calcite in growth ex-periments, supersaturation should be kept in the metastable region, which is the area between supersolubility and

solubil-Ž .

ity curves Nielsen and Toft, 1984 . Since the supersolubility curve of the CaCl2]Na CO ]H O system is not available,2 3 2

experiments were first performed to identify the metastable

Ž .

Figure 2. Relative supersaturation of calcium carbonate

[ ]

salt as a function of pH: T s 258C and Ca sT

0.0005 kmolrrrrr_m3_.

3

w_{CO3 T}x s_{0.05 km olrm ;} } ? } COw _{3 T}x s_0.005

3 w x 3

kmolrm ; CO3 Ts0.0005 kmolrm .

Ž8.5, 9.5, and 10.5, respectively , 100 mL of CaCl. 2 and Na CO solutions were mixed and stirred for 2 h. For a spe-2 3

cific concentration of calcium chloride, the solution remained clear at a low concentration of sodium carbonate. When the concentration of Na CO increased, the solution became tur-2 3

bid due to nucleation. The boundaries were recorded for var-ious CaCl2 concentrations, such as those shown in Table 1 for a pH of 8.5. Then the concentrations or activities of Ca2q

and CO32yat the boundaries were calculated using the same computer program as was used for estimating supersatura-tion. Finally, the metastable region was constructed and all the growth experiments were performed in this region.

Crystallization system

The crystallization system, containing a pH-stat fluidized-bed crystallizer, a storage tank, and a pH control system, as shown in Figure 3, was used to measure the growth rate of calcite crystal. The main part of the fluidized-bed crystallizer is a PVC column with a distributor at the bottom. Immedi-ately above the column is an enlarged section to prevent seed crystals from carrying over to the storage tank. A supersatu-rated solution, which was prepared by mixing CaCl2 and Na CO solutions, of 6 L with a desired pH was charged into2 3

the storage tank and another 0.5 L of the same solution was fed into the fluidized bed. The solution in the storage tank was then pumped through a distributor and into the fluidized bed. After that, the solution overflowed to the storage tank. When the flow rate and pH became steady, the pump was stopped and approximately 25 g of seed crystals were intro-duced into the crystallizer. The pump was restarted and the growth experiment began. The pH of the solution was main-tained constant during the operation, using a pH-stat

appara-Table 1. Demarcation of Metastable Region for Calcium

( )

Carbonate CaCO3 at pH s 8.5 and T s 258C 3 wCaCl x Žkmolrm. w_{Na CO} x ₂ 2 3 3 Žkmolrm. 0.05 0.025 0.015 0.005 0.0025 0.0015 0.0005 0.05 T 0.025 T 0.0175 T 0.015 C U 0.005 T T T T C 0.004 T 0.0035 T 0.00325 C 0.003 C 0.0025 T T T C 0.00225 C 0.002 C 0.0015 T C 0.00125 T 0.001 C 0.000875 T 0.0005 T T T C C 0.00045 T 0.0004 T T C 0.000375 C 0.00035 T 0.0003 C 0.00025 T T C 0.000225 T C UU 0.0002 C C 0.00015 C 0.00005 C U_{T means turbid.} UU_{C means clear.}

tus to control the amount of NaOH solution added to the storage tank. A 3-mL sample of the solution was withdrawn from the crystallizer every 30 min by a syringe fitted with a

Figure 3. PH-stat crystallization system.

1. pH and temperature indicator; 2. reagent bottle; 3. pump-ing system of reagent; 4. reagent deliverpump-ing line; 5. burette; 6. thermometer; 7. water bath; 8. storage tank; 9. tempera-ture controller; 10. motor; 11. glass-electrode; 12. reference electrode; 13. thermocompensator; 14. axial-flow impeller; 15. magnetic motor; 16. flowmeter; 17. fluidized-bed crystal-lizer; 18. recycle valve; 19. distributor.

0.22-mm filter and then diluted to 50 mL with deionized wa-ter. The calcium-ion concentration of the diluted solution was then determined by an atomic absorption spectrometer and the growth rate of calcite crystal was estimated from the con-sumption rate of the calcium ion. A typical run lasted 260 min.

Determination of crystal growth rate

The total mass and surface area of crystals are given by

W s nr aL3 ₅

Ž .

p

As nbL2_{, 6}

Ž .

where r is the crystal density; a and b are the volume and_p surface area shape factor, respectively; and n the number of crystals.

The crystal growth rate expressed in kgrm2_{s is}

1 dW

R sg _{A dt}

Ž .

7

or

3ar dLp

R sg _{b dt} .

Ž .

8

Combine Eqs. 7 and 8, the linear growth rate G can be writ-ten as

dL b dW

Gs s _.

_{Ž .}

₉

dt 3ar A dtp

During a growth experiment, the change of crystal mass in a

Figure 4. Typical change in the total calcium concentra-tion over time for run O14.

crystallizer is related to the change of calcium-ion concentra-tion as follows:

w

2q

_x

dW yd Ca

sMV

ž

/

,

Ž

10

.

dt dt

where M is the molecular weight of CaCO , and V is the3

solution volume.

Equation 10 is substituted into Eq. 9 to give

w

2q

_x

bMV d Ca

Gs y _.

_Ž

₁₁

_.

ž

/

3ar Ap dt

Combining Eqs. 5 and 6, we have

Lr

b p

s .

Ž

12

.

a A W

Substituting Eq. 12 into Eq. 11, we obtain the linear growth-rate expression:

w

2q

_x

LMV d Ca Gs y _.

_Ž

₁₃

_.

ž

/

3W dt w 2q_x

Once the concentration profile of Ca , which is deter-mined by an atomic absorption spectrometer, is available, the linear crystal growth rate can be evaluated by assuming that the crystal size and total crystal weight are the same as for the seed crystal. This assumption will not cause much error because the total increase in crystal size or weight is negligi-ble as compared with the seed crystal. For example, the growth rate of calcite is roughly 1=10y10 _{mrs, and thus an}

increase in crystal size for a growth time of 260 min is esti-mated to be 1.5mm, which is negligible when compared with the seed sizes of 460mm or larger. A typical calcium-ion con-centration profile trend is shown in Figure 4.

Results and Discussion

Metastable region of CaCl – Na CO – H O system

_{2 2 3 2} Using the concentration data at the boundary shown in Table 1 for pH at 8.5 and other similar data for pH at

Ž . w x

9.5 and 10.5 not shown here Chen, 1985 , p CaT is plotted

w x

against p CO3 T in Figure 5 to mark the supersolubility

w x w x w x

curves, in which p CaT and p CO3 T represent ylog CaT

w x

and ylog CO3 T, respectively. The solubility curves are also plotted in the figure using the solubility product of CaCO .3

The metastable regions of the three pHs do not agree when w 2q_x

they are projected to the same pH plane. However, if Ca w 2yx

and CO3 at the boundaries, which were calculated from the total concentration data using a computer program for

Ž

estimating supersaturation, are used in the plotting as shown .

in Figure 6 , the metastable region is independent of the pH value, and the solubility and supersolubility curves are bent at both ends. Similar curves for the CaCl2]Na PO ]H O3 4 2

Ž .

system have been reported Furedi-Milhofer et al., 1975 . This

¨

type of curve means that the formation of complex ions is

Ž .

Figure 5. Metastable region of calcium carbonate for the CaCl –Na CO2 2 3 system at pH 8.5, 9.5, and 10.5.

}= } supersolubility curve; } ? } solubility curve.

2q 2y _w 2qx w 2yx

tion of Ca and CO3 , that is, Ca and CO3 , in Figure 6 is replaced by the activity of the respective ion, a_Ca2q

or a_CO2y, the solubility and supersolubility curves become 3

almost straight and parallel lines in the concentration range studied in this experiment, as shown in Figure 7. This simpli-fies the plotting of the metastable region.

[ 2H_{] [} 2I_]

Figure 6. p Ca vs. p CO3 : metastable region of calcium carbonate.

solubility curve:' pH s 8.5, l pH s 9.5, B pH s 10.5; supersolubility curve: ^ pH s 8.5,e pH s 9.5, I pH s10.5.

Figure 7. pa_Ca2H vs. pa_CO2I: metastable region of

cal-3

cium carbonate.

solubility curve; supersolubility curve: ^ pH s 8.5,e pH s 9.5, I pH s10.5.

Factors affecting calcite growth rate in a dense fluidized

bed

In the operation of a fluidized bed, the superficial velocity of the crystals should be lower than the terminal velocity, otherwise, the crystals would be carried over the top of the bed. At low superficial velocity, a boundary between solution and suspension is clearly observed in a dense bed. When the superficial velocity approaches the terminal velocity, particles move freely in the bed and the boundary no longer exists. In this case, it is called a lean fluidized bed. The dense bed is more advantageous as far as seed loading is concerned. Therefore, factors that affect the calcite growth rate in a dense bed are investigated, including supersaturation, pH, ionic strength, superficial velocity, particle size, and type of seeds.

Figure 8 shows the calcite growth rate as a function of su-persaturation at various levels of ionic strength from 0.0025 to 0.0340 kmolrm3. For all levels of ionic strength, which is adjusted by adding NaCl solution, the growth rates increase with an increase in supersaturation. At the same supersatura-tion, the groth rates increase with ionic strength from 0.0025 kmolrm3 _{to 0.0185 kmolrm}3_{; however, a further increase in}

ionic strength is no longer effective. Besides, the effect of ionic strength is less significant at higher supersaturations. One other thing worth noting is that the slope of the line, log G vs. logs , changes for the lowest ionic strength of 0.0025 kmolrm3, implying a change in the strength of various types of resistance to crystal growth. The effects of ionic strength have been reported for the crystal growth of vaterite and lead

Ž .

fluoride. Kralj et al. 1990 found a 10% increase in the growth rate of vaterite crystal when the ionic strength varied be-tween 0.015 kmolrm3_{and 0.315 kmolrm}3_{. As compared with}

Ž 3

the range studied in this experiment 0.0185 kmolrm to 0.034

3

.

kmolrm , the effect of ionic strength was not observed. On

Ž .

Figure 8. log G vs. logs effect of ionic strength on the growth rate of CaCO crystal at Ls 4603 mm, pHs 9.5, T s 258C.

[I s 0.0340 kmolrm3_{;v I s 0.0185 kmolrm}3_;

)

_{I s 0.0105} kmolrm3_{;B I s 0.0025 kmolrm}3_.

rate of PbF is about five times greater at a high ionic strength2

of 0.1 kmolrm3_{as compared with that in water, and a}

differ-ent growth mechanism is speculated. All the findings on the effect of ionic strength are similar to the results of this exper-iment.

Figure 9. Crystal growth rates of calcium carbonate, log G vs. logs , at two pH values: Is0.018 kmolrrrrr_m3_{; Ls 460} mm; Ts258C; us2.36= 10I2 mrrrrr_s.

v pH s 8.5; I pH s 9.5.

The effects of pH on the growth rate of calcite are shown in Figure 9 for two levels of pH, namely 8.5 and 9.5, which is about the range used in a pellet reactor for water softening Žvan Dijk and Wilms, 1991 . Similar to the effects of ionic. strength, different slopes of the log G]logs plot are ob-tained at different pH, and the effects of pH diminish at higher supersaturations. It is possible that the main resis-tance to crystal growth is influenced by ionic strength and pH in the same way. The effects of pH on the growth rate of lead

Ž .

fluoride were studied by Stubicar et al. 1993 , using the con-stant-composition method. The result is consistent with that of calcite crystal studied in this experiment, that is, the crys-tal growth rate increases with pH when the pH is lower than

Ž the isoelectric point, which is 9;10 and 5.6 for calcite Reed,

.

1989 and a-lead fluoride, respectively.

The growth rates of calcite were measured at various su-perficial velocities, that is, 1.42, 2.36, 3.54 and 4.72 mrs, for the seed crystals of size 460-mm. Although the bed voidage and bed expansion changes a great deal, the calcite growth rates are rather constant, as shown in Figure 10. Assuming that the solution velocity does not influence the surface-reac-tion step, the constant growth rate is consistent with Tournie

Ž .

et al.’s 1979 conclusion that the mass-transfer coefficient is practically independent of the liquid velocity, which they ar-rived at by analyzing the extensive dissolution data for both lean and dense fluidized beds reported in the literature. On the other hand, Figure 11 shows the effects of particle size on the calcite growth rate at a superficial velocity of 4.72 mrs. The calcite growth rate increases with an increase in crystal size. The influence of superficial velocity and crystal size on the mass-transfer coefficient in a fluidized bed is discussed later.

Figure 10. Growth rates of calcium carbonate, log G vs. logs , at various superficial velocities: Ls 460 mm; pHs9.5; Ts258C; Is0.0025 kmol rr r rr_m3_. ^ us 1.42 mrs;I us 2.36 mrs; q us 3.54 mrs; v us 4.72 mrs.

Figure 11. log G vs. logs effect of crystal size at pHs 9.5; I s 0.0025 kmolrrrrr_m3_{; u s 4.72 mr}rrrr_{s; T s} 258C.

I Ls 920mm; q Ls 650 mm; v Ls 460 mm.

In the operation of a water-softening pellet reactor, silica Ž sand or quartz sand is used as the seeding material van Dijk

.

and Wilms, 1991 . It is therefore desirable to investigate the growth phenomena of calcite on the surface of silica sand. The concentration profile of the calcium ion is shown in Fig-ure 12 for an experimental run using silica sand as seed. When compared with Figure 4, it is clearly seen that an induction time of calcite growth is required for silica sand. The

induc-Figure 12. Induction time for calcium carbonate growth using silica sand as seed under the follow-ing operation conditions: pHs 9.5; u s 2.36 mrrrrr_s;s s1.3; Ls460 mm.

tion time is 600 min for the specific operating conditions, and is shorter at higher supersaturation. The calcite growth rate on the silica sand seed is lower than that on the calcite seed, as shown in Figure 13, in which curves A and B are the growth rates of sand seed after a growth period of 15 h and 20 h, respectively. The growth rate of sand seed will eventu-ally approach the growth rate of calcite seed when the sur-face of sand seed is fully covered with calcite.

Mass-transfer coefficient of calcite growth in a dense

fluidized bed

According to two different sources summarized by Tai et

Ž .

al. 1993 , the surface reaction order of calcite growth is ap-proximately 2. This is also true for some other systems, such

Ž .

as copper sulfate pentahydrate Tai and Pan, 1985 and

Ž .

potassium alum Tai et al., 1987 . Taking r s 2, Eqs. 2 and 3 are combined to give

s 1 1

'

s _{G q .}

_Ž

₁₄

_.

'

G Kd

_'

Kr

Thus the mass-transfer coefficient, K , can be evaluated fromd

'

'

Ž .

the slope of the plot, sr G vs. G . Figure 14 illustrates the plotting of Eq. 14 for three different crystal sizes at the same pH and ionic strength. The obtained Kd are listed in Table 2. Using the table, we can compare K_d for the differ-ent crystal sizes suspended at a superficial velocity of 3.54 mrs during the first three runs. On the other hand, the K_d values of the 460-mm-sized crystals operated at various su-perficial velocities can be compared using the last three rows in the table. Despite the variation in particle size and superfi-cial velocity, the K are rather constant. The deviations from_d the average value of K , 1.32=10y10

, are less than 10%.

d

Figure 13. Comparison of growth rates of different seed types.

e calcite; v silica sand after 20 h of growth time; w silica sand after 15 h of growth time.

Figure 14. Equation 14 for calcium carbonate system at pHs 9.5 and I s 0.0025 kmolrrrrrm3_.

Ž .

Run No. Crystal Size mm

^Run A-2-2-M14 460

I Run A-4-2-O13 650

`Run A-4-1-O20 920

The mass transfer between fluidized spheres and liquid so-lution has been widely studied in many operations, such as dissolution, crystallization, ion exchange, leaching, and ad-sorption. Numerous correlations to estimate the mass-trans-fer coefficient are proposed in the literature and summarized

Ž .

by Tournie et al. 1997 . However, none of them is derived from or applied to the crystal growth of a barely soluble sys-tem. One of the correlations originally proposed by Fan and

Ž .

modified by Kunii and Levenspiel 1969 for a liquid flu-idized bed with low voidage has the following form when ne-glecting the diffusion term:

Table 2. Mass-Transfer and Surface-Reaction Coefficients of Calcite Crystal Estimated by Using Eq.

14 at pH s 9.5 and I s 0.0025 kmolrrrrr_m3 Superficial Particle K K Velocity d r Size y1 0 y9 Ž . Ž . Ž . Ž . Run No. mm mrs 10 mrs 10 mrs A-4-1-O20 920 3.54 1.45 13.78 A-4-2-O13 650 3.54 1.42 2.25 A-2-2-M14 460 3.54 1.33 0.30 A-1-4-M08,16 460 2.36 1.11 0.31 A-2-4-M12,13 460 1.42 1.29 0.34 K Ld 1r2 _1r3 Sh s_1.5

_Ž

_1ye Re

_.

_{p Sc ,} e F0.84 15

_Ž

_.

ž

D

/

and Lur Re sp _m , 5- Re -120,p

Ž

16

.

where u is the superficial velocity and e the bed voidage. Equation 15 is derived by using the experimental data of b-naphthol and benzoic acid, which are soluble systems. To test the applicability of Eq. 15 to a barely soluble salt, calcite, the diffusivity of calcite in water should be found first. Unfortu-nately, the diffusivity data of calcite are not available in the literature, and the existing correlations for predicting diffu-sivity are only good for dilute solutions, which is quite differ-ent from the supersaturated solution we are dealing with.

Although a correlation to predict the mass-transfer coeffi-cient could not be established for the calcite growth, Eq. 15 is used to test the consistency of the mass-transfer coefficient

wŽ . x1r2

as shown in Table 2. Values of 1ye Rep for crystals of 460, 650 and 920 mm, fluidized at various superficial

veloci-wŽ . x1r2

ties, are tabulated in Table 3. The values of 1ye Rep at various superficial velocities are quite constant for each of the sizes. Therefore an average value is calculated for each size in Table 3. According to Eq. 15, the mass-transfer

coeffi-wŽ . x1r2 _wŽ

cient is proportional to 1ye Rep rL. The ratios of 1y . x1r2

e Re_p r_{L to that of 920} mm are shown in the last column

[( ) ]1rrrrr2

Table 3. Values of 1Ie Rep for Different Sizes of Calcite Fluidized at Various Superficial Velocities 1r2

w Ž . x

Particle Superficial Bed 1ye Re r_L

p

1r2

wŽ_1ye Re. x

Size, L Velocity Voidage p _{w Ž} 1r2

. x 1r2

_

1ye Rep rL

₄

Žmm. Ž_mrs. e _Re wŽ_1ye Re. x _{Average L s 920m m} p p 1.42 0.68 7.14 1.52 2.36 0.77 12.05 1.67 460 _{3.54 0.84 18.08 1.67} 1.57 0.97 4.72 0.91 24.11 1.44 1.42 0.50 10.23 2.26 2.36 0.58 17.06 2.67 650 2.81 0.64 20.46 2.71 2.61 1.13 3.54 0.71 25.55 2.72 4.72 0.79 34.11 2.68 1.42 0.60 24.17 3.11 2.81 0.64 29.00 3.22 920 _{3.54 0.70 36.11 3.29} 3.23 1 4.72 0.77 47.20 3.31

of Table 3. Thus, the ratios of the mass-transfer coefficient for the three sizes are:

K K K

Ž

d

.

Ls 460mm

Ž

d

.

L s 650mm

Ž

d

.

L s 920mm

s s _.

_Ž

₁₇

_.

0.97 1.13 1.0

The ratios are rather constant, meaning that the mass-transfer coefficient of calcite growth in a fluidized bed is in-dependent of crystal size and superficial velocity. This result is consistent with Table 2.

The surface-reaction coefficients at various crystal sizes are also listed in Table 2, which increases with crystal size and are rather constant for the same size. The size-dependent K_r is responsible for the effect of particle size on the calcite growth rate shown in Figure 11. Similar results have been

Ž .

reported for calcite growth in a stirred tank Tai et al., 1993 Ž

and potassium alum in a fluidized bed Budz et al., 1984; Tai .

et al., 1987 . In Table 2 we also note that the values of Kr

are higher than those of K , especially for larger sizes. It is_d interesting to see if there is a controlling step for the calcite growth, which can be judged from the surface-reaction

effec-Ž .

tiveness factorh Garside, 1971 or the ratio of s rs . If h2y

or s rs approaches 1, the crystal growth process is2y

surface-reaction control. On the other hand, if h or s rs2y

approaches 0, the growth process is mass-transfer control. The calculated values of h and s rs range between 0.191 and2y

0.355, and 0.433 and 0.577, respectively, for the experimental run A-2-2-M14, in which the mass-transfer resistance is the lowest among the experimental runs shown in Table 2 as judged from the relative magnitude of Kd to K . Therefore,r

the mass-transfer resistance and surface-reaction resistance are both significant, with the former being higher than the later in a fluidized bed operated at pH 9.5 and an ionic strength 0.0025 kmolrm3. For the largest size of 920mm, the growth is almost mass-transfer controlled.

Growth-rate expression of calcite crystals

Once the surface-reaction order, mass-transfer coefficient, and surface-reaction coefficient are available, the crystal growth rate can be expressed according to the two-step model. For a surface-reaction order r s 2, the expression of crystal

Ž .

growth rate is as follows Mullin, 1993 :

2

K_d K_d

Gs Kd

ž

1q_{2 K s}

/

y

)

½

ž

1q_{2 K s}

/

y1

5

s . 18

Ž

.

r r

The values of Kd and Kr can be determined from the experiment using a lab-scale fluidized bed. Because K_d is almost a constant, which is independent of crystal size and superficial velocity, and K is related to crystal surface prop-_r erties, which should be independent of hydrodynamics, the experimental data of K_d and K obtained in a lab-scale ap-_r paratus can be used for scale-up purposes.

The units of crystal growth rate, G, in Eq. 18 is in mrs. The linear growth rate can be easily converted to mass growth rate or molar growth rate, such as in Eq. 1, by using Eqs. 9 and 10. Equation 18, which is based on the two-step growth model, is more able than Eq. 1, which is an overall growth-rate

equation, to represent the crystal-growth-rate expression of calcite.

Conclusion

The metastable region of the CaCl2]Na CO ]H O sys-2 3 2

tem is explored in this study. The solubility and supersolubil-ity curves, which are the boundaries of the metastable region, are almost parallel straight lines in the concentration range studied, when the precipitation diagram is constructed by plotting pa_Ca2q against pa_CO2y. The nucleation of calcite is

3

suppressed when the growth experiments are conducted in the metastable region, using a fluidized-bed crystallizer.

Factors that affect the calcite growth rate in a dense flu-idized bed are identified, including supersaturation, pH, ionic strength, and particle size and type of seeds. On the other hand, the effects of superficial velocity on growth rate are less significant. An induction period for crystal growth is re-quired for the silica-sand seed, which is one of the seeding materials used in a large-scale water-softening pellet reactor. The growth rate of this type of seed is slower before the seed surface is fully covered with calcite.

When the calcite-growth-rate data of constant pH and ionic strength are analyzed by the two-step growth model, the mass-transfer coefficients so obtained are independent of crystal size and superficial velocity. On the other hand, the surface-reaction coefficient is size dependent. Since the mass-transfer and surface-reaction coefficients are rather in-sensitive to hydrodynamics in a fluidized bed, a growth-rate equation can be given for design purposes according to the two-step growth model, in which the mass-transfer coefficient and surface-reaction coefficient are determined using a lab-scale fluidized bed.

Acknowledgment

The authors gratefully acknowledge the financial support of the National Science Council of Republic of China.

Notation

w_Ca2qx_{scalcium-ion concentration, kmolrm}3

w_{Ca s total calcium concentration, kmolrm}x 3

T

w_CO 2yx_{scarbonate-ion concentration, kmolrm}3 3

w_CO x _{stotal carbonate concentration, kmolrm}3 3 T

Dsdiffusivity, m2_rs

Gs linear crystal growth rate, mrs Is ionic strength, kmolrm3

K ssurface-reaction coefficient, mrsr

Ls crystal size, m

Re s particle Reynolds numberp

ScsSchmidt number ShsSherwood number Ts temperature,8C ts time, s mssolution viscosity, kgrms rssolution density, kgrm3

s sinterfacial relative supersaturation2y

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