Interpretation of calcite growth data using the two-step crystal growth model

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Interpretation of calcite growth data using the two-step crystal growth model

Clifford Y. Tai

∗

, Meng-Chun Chang, Chi-Kao Wu, Yen-Chih Lin

Department of Chemical Engineering, National Taiwan University, Taipei 106, Taiwan Received 15 June 2005; received in revised form 26 January 2006; accepted 25 March 2006

Available online 30 March 2006

Abstract

The crystal growth rates of calcite were evaluated in a dense fluidized-bed crystallizer using a constant-composition method. Several operation variables related to solution properties that affected growth rate, including supersaturation, pH, ionic strength and activity ratio, were systematically investigated. Then the crystal growth-rate data were analyzed by the two-step crystal growth model and thus the mass-transfer and surface-reaction coefficients were obtained. The effects of the solution properties on the two individual coefficients were observed. Good explanations on these effects are needed.

Keywords: Crystallization; Mass transfer; Surface reaction; Fluidization; Unit operations; Two-step growth model

1. Introduction

One of the water softening process is to feed a pellet reac-tor, which is a reactive, ﬂuidized-bed, growth-type crystallizer, with drinking water containing calcium ions and a stream of carbonate aqueous solution (Dirken et al., 1990). Then calcium ions react with carbonate ions to form calcium carbonate and to grow on the seeds suspending in the crystallizer. The seeds are later removed from the reactor after they exceed a certain size. At this stage some fresh seeds are added to the crystal-lizer, and the crystallizer is operated in a semi-batch mode. In the design of a pellet reactor for water softening, the crystal growth kinetics of calcium carbonate is required.

In 1991 van Dijk and Wilms proposed an empirical model to express the calcite growth rate,

d[Ca2+_] dt = KTa Ca2+ [CO2− 3 ] − Ksp , (1)

where (d[Ca2+_{]/dt) is the depletion rate of [Ca}2+_{] in the}

so-lution, which is equivalent to the calcite growth rate in moles

∗_{Corresponding author. Tel./fax: +886 2 23620832.}

E-mail address:cytai@ntu.edu.tw(C.Y. Tai).

per unit time; KT is a constant; a is the speciﬁc surface area

of seed crystals; K_sp is the concentration solubility product of calcite; and{[Ca2+][CO2₃−]−K_sp } represents the driving force for crystal growth. Eq. (1) is considered as an over-all growth rate model, because the driving force used is the difference between the concentration product of bulk solution and the solubility product of calcite.Tai et al. (1999)have demonstrated that the applicability of an over-all growth rate is conﬁned to a limited range of operating variables when the two-step growth model is applied, which can be expressed mathematically by the following equations:

G= Kd( − i) diffusion, (2)

G= Krri surface reaction. (3)

Eq. (2) deals with the diffusion rate of solute between the bulk solution and the solution/crystal interface and Eq. (3) takes into account all the steps occurring on the crystal surface, including adsorption, desolvation, surface diffusion and incorporation, as proposed by Mullin (1993). Although this model is a simpli-ﬁed scheme for crystal growth, it reveals lots of useful kinetic information, which was summarized byTai (1997).

The surface-reaction order of calcite is approximately 2 from different sources (Tai et al., 1993). This is also true for other systems, such as CuSO4· 5H2O (Tai and Pan, 1985) and potassium alum (Tai et al., 1987). Taking r=2, Eqs. (2) and (3)

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are combined to give √ G= √ G Kd + 1 √ Kr . (4)

Thus the mass-transfer coefﬁcient, Kd, and surface-reaction

co-efﬁcient, Kr, can be evaluated from the slope and interception,

respectively, of a plot, (/√G)vs.√G. In a well-suspended crystallizer, the mass-transfer coefﬁcient is a function of system properties only, and can be evaluated by the following equation (Tai et al., 1987), Sh KdL D = 0.306Ga1/3_Mv1/3_Sc1/3 . (5)

Tai et al. (1999) studied the growth kinetics of calcite crystals suspended in a ﬂuidized-bed crystallizer, which was maintained at a constant pH, i.e., a pH-stat technique. They found that supersaturation, pH and ionic strength were sig-niﬁcant variables to affect the growth rate of calcite crys-tals. In the same study a growth rate equation, which is based on the two-step growth model was proposed for design purpose, G= Kd ⎡ ⎣1+ Kd 2Kr − 1+ Kd 2Kr 2 − 1 ⎤ ⎦ . (6)

The disadvantage of the pH-stat technique is that the supersat-uration changes during an experiment and the crystal growth rate is determined from the slope of the calcium concentration proﬁle in the crystallizer. Then the concentrations are con-verted to supersaturations. The crystal growth rate so obtained may be less accurate than that evaluated by the constant-composition method, in which the supersaturation is kept constant.

Tai et al. (2005)studied the effects of solution properties, including supersaturation, pH, activity ratio and ionic strength on the crystal growth rate of calcite in a fluidized-bed crystal-lizer using the constant-composition method. The growth ex-periments were performed in the metastable region, which was identified experimentally byTai et al. (1999)using cured natu-ral calcite in a size range between 390 and 921m. The crystal growth rates were evaluated from the volume depletion rate of titration solution. The results were consistent with those ob-tained by the pH-stat method (Tai et al., 1999). However, the crystal growth-rate data of calcite were not further analyzed to explore the possible effects of solution properties on the dif-ferent steps of crystal-growth process.Tai et al. (1999)applied the two-step crystal growth model to investigate the effects of particle size and superficial velocity on the mass-transfer co-efficient and surface-reaction coco-efficient of calcite growth in a dense fluidized bed. They found that the mass-transfer coef-ficient was independent of crystal size and superficial veloc-ity and the surface-reaction coefficient is size dependent but independent of superficial velocity. These results are similar

to that of readily soluble systems, such as potassium alum (Budz et al., 1984; Tai et al., 1987) and nickel sulfate (Phillips and Epstein, 1974). Besides, the two-step crystal model has been employed to analyze the crystal growth rate data of many readily soluble systems and much useful kinetic information has been revealed (Tai, 1997). It is interesting to see how the solution properties will inﬂuence the mass-transfer and surface-reaction coefﬁcient.

The aim of this study is to investigate the crystal growth kinetics of calcite in a dense fluidized-bed crystallizer using the constant-composition method. Because there are too many factors that affect crystal growth rate of sparingly soluble sys-tem, the variables investigated in this study are confined to solution properties. The growth experiments were performed in the metastable region by keeping the same crystal size, su-perficial velocity and temperature. The calcite crystal growth rates were measured at various supersaturations under speci-fied pH, activity ratio, and ionic strength. The crystal growth rate of calcite was evaluated from the volume depletion rate of titration solution. The crystal growth rate data were fur-ther analyzed by the two-step crystal growth model in order to explore the effect of the solution properties on the mass-transfer coefficient and surface-reaction coefficient, which were evaluated by Eq. (4). Finally, a possible controlling step of calcite growth process was postulated for various operation conditions.

2. Experimental

The experimental apparatus and procedures have been re-ported somewhere else (Tai et al., 2005) and will be described briefly here. A voltage regulator was installed to the experi-mental system in order to reduce noise caused by unstable volt-age. The crystallization system shown inFig. 1contains a flu-idized bed, which is a PVC column with a distributor at the bottom to give an even flow and with an enlarged section on the top to prevent the seed crystals from carrying over to the storage tank, a storage tank for holding solution, and a com-position control system, which consists of two autotitrators for adding reaction solutions and NaOH solution to keep the sys-tem at constant composition and pH. Usually the amount of NaOH added to the system was only a few drops. During an experimental run, the chloride and sodium ion would accumu-late to increase the ionic strength of solution due to a semi-batch operation. However, the change in growth rate was neg-ligible due to an increment of ionic strength within 5%. The volume of titration solution added to the system was recorded automatically on the autotitrator after the seed crystals were charged into the crystallizer and the solution composition and pH became steady. The titration curve was used to calculate the crystal growth rate. After an experimental run, which usually lasted for one hour, the seed crystals were removed, washed and then kept in a saturated solution for next experiment. Dur-ing the experiment, the seed size, superficial velocity, and tem-perature were kept at 774m, 4.7 × 10−2m/s and 298 K, respectively.

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Fig. 1. A constant-composition crystallization system.

3. Determination of crystal growth rate

The crystal growth rate in kg/m2s is expressed as

Rg=

1

A

dW

dt , (7)

where W and A are the total mass and surface area of crystals, which are related to the crystal size by the following forms:

W= n_pL3, (8)

A= nL2, (9)

wherepis the crystal density; and are the volume and the surface area shape factors, respectively; and n is the number of crystals.

Substituting Eqs. (8) and (9) into Eq. (7) gives

Rg= 3p dL dt = 3 L W A dL dt. (10)

Then Eqs. (7) and (10) can be combined to give the linear crystal growth rate in m/s,

G=dL dt = L 3W dW dt . (11)

In a constant-composition operation, the solution volume in-creased but the concentration kept constant. To evaluate dW/dt, the change in volume should be taken into consideration. Thus,

dW dt = M · dVo Ca2+_o+ Va Ca2+_a− (Vo+ Va) Ca2+_o dt = M ·Ca2+ a− Ca2+ o _dV a dt , (12)

where M is the molecular weight of CaCO3, Voand Vaare the

volume of original solution and added solution, respectively, and[Ca2+]oand[Ca2+]a represent the Ca2+ concentration of

the original solution and added solution, respectively. Eq. (12) is substitute into Eq. (11) to yield

G=LM 3W Ca2+ a− Ca2+ o _dV_a dt . (13)

The volume of titration solution was recorded automatically as shown in Fig. 2. Then the titration curve was ﬁtted by a straight line. The slope of the straight line was used to calculate the crystal growth rate according to Eq. (13). It should be noted that the growth rate determined in this study is the average linear growth rate of all crystal faces. Since the crystal growth rate of CaCO3was very slow, in the order of 10−10m/s, it was evaluated by the depletion of titration volume, then converted to linear growth rate. Therefore the two-step growth model described in this paper is based on 1-D analysis.

Once the crystal growth rates are available at different super-saturations, the mass-transfer coefﬁcient, Kd, and the

surface-reaction coefﬁcient, Kr, can be evaluated by Eq. (4), in which

the relative supersaturation,, was deﬁned byNielsen and Toft (1984). = Kip Ksp 1/2 − 1. (14)

A computer program, which contains mass-action equation, mass-balance equation, charge-balance equation, and the modiﬁed Debye–Hückel equation for estimating the activity coefﬁcient, was used to calculate the relative supersaturation (Nancollas, 1966; Tai et al., 1993).

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0 1000 2000 3000 4000 t (s) 0 1 2 3 4 V (ml)

Fig. 2. A titration curve at the following operation conditions:=1, pH=9.5, R= 4.0, and I = 0.018 M. — experimental line; - - - - ﬁtted line.

On the other hand, the crystal growth rate of calcite in a pH-stat operation was estimated by the following equation (Tai et al., 1999): G=LMVo 3W −d[Ca2+] dt . (15)

The concentration profile of calcium ion for the samples with-drawn intermittently from the crystallizer was determined by an atomic absorption spectrometer. Then the concentration profile was fitted by a polynomial, which could be converted to crystal growth rate by taking derivative first and then by using Eq. (15). It is clear that the two techniques for measuring crystal growth rate are quite different by comparing Eqs. (13) and (15).

4. Results and discussion

4.1. Assurance of constant-composition operation

In this experiment the fluidized-bed crystallizer was oper-ated in a dense mode, i.e., a clear boundary existing between solution and suspension for the superficial velocity being oper-ated below the terminal velocity of particles. A dense bed was chosen instead of a lean bed, which is operated at a superfi-cial velocity near the terminal velocity of particles, because it is more advantageous as far as seed loading is concerned. Sup-pose a lean bed was used, the operation time should last longer due to a slow growth rate of calcite.

If the solution composition in the system was controlled by a pH-electrode, a constant composition was difﬁcult to main-tain as shown inFig. 3. A possible reason was that the absorp-tion of CO2from the environment caused a change in pH that triggered the autotitrator. Therefore the calcium ion increased

0 2000 4000 6000 Time (s) 2.6 2.8 3 3.2 3.4 TCA (ppm)

Fig. 3. Total calcium concentration proﬁle using a combined pH-glass elec-trodes to maintain constant composition.

0 2000 4000 6000 8000 10000 Time (s) 2 2.2 2.4 2.6 2.8 3 TCA (ppm)

Fig. 4. Total calcium concentration proﬁle using a calcium ion-selective electrode to maintain constant composition.

gradually. To overcome the problem, two sets of autotitrators, one equipped with a calcium-ion electrode and the other one with a pH electrode, were applied to control the added amount of reaction solutions and NaOH solution, respectively.

To check the constant composition of crystallization solu-tion, solution samples were withdrawn intermittently by using a syringe ﬁtted with a 0.22m ﬁlter, and the calcium ion con-centration was determined by an atomic absorption spectrom-eter. The results shown inFig. 4, for example, indicate that the

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0 0.4 0.8 1.2 1.6 0 2 4 G (10 -10 m/s) σ

Fig. 5. Crystal growth rate of calcite as a function offor three levels of pH: 8.5 (), 9.0 (䊉), and 9.5 (). Other operation variables kept constant are: R= 5.5 and I = 0.018 M.

calcium concentration is nearly constant during a long period of operation up to 8000 s, although the calcium concentration ﬂuctuates a little bit around the initial value.

4.2. Effects of pH on Kd and Kr

The pH range investigated in this study was between 8.5 and 9.5, which is about the range used in a pellet reactor for water softening (Van Dijk and Wilms, 1991). It will encounter difﬁculties beyond this pH range in the operation for kinetic study. If the pH value is higher than 9.5, carbon dioxide will be absorbed from air to mess up the measurement of growth rate. On the other hand, the growth rate will be very slow at a pH below 8 because the concentration of CO2₃−is too low (ASTM, 1974).

The calcite growth rates at various relative supersaturation, from 0.20 to 1.35, were measured for three pH levels, i.e., 8.5, 9.0 and 9.5, with other operation variables kept the same. The growth rate data are plotted inFig. 5. General trends observed are that the growth rate increases with an increase in supersat-uration and pH, and the growth rates become very large when is above 1.2. A large increase in growth rate for above 12 was due to a sudden increase of diffusivity and will be further explained later.

When the growth rate data were analyzed by the two-step model of crystal growth by taking r= 2 (Tai et al., 1993), a plot of Eq. (4) was constructed as shown inFig. 6 for three levels of pH. For each set of data, a straight line for the ﬁrst six data points follows Eq. (4). However, the straight line for pH= 9.5 has an intercept of negative value, which results in a negative Kr with no physical meaning. Thus the growth-rate

data for pH= 9.5 shown inFig. 5were ﬁtted by a straight line

0 5 10 15 20 0 4 8 12 16 20 σ/ G (10 4 s 1/2 /m 1/2 ) √ G (10-6_m1/2_/s1/2₎ √

Fig. 6. Calcite growth-rate data plotted by the two-step growth model with r= 2 at three pH values: 8.5 (), 9.0 (䊉), and 9.5 ().

Table 1

Growth kinetics of calcite at various pH levels

pH value Fitted equation Correlation Kd Kr

coefﬁcient (m/s) (m/s) 9.5 Ya=1.52Xb 0.98 1.52 — 9.0 Yc=0.675Xd 0.83 1.48 50.3 8.5 Yc=0.630Xd+8.627 0.86 1.58 1.34 a_Y_{= G m/s.} b_X₌_. c_Y_{= (}_/√_G)_{× 10}4_s1/2_/m1/2_. d_X₌√_G_{× 10}−6_m1/2_/s1/2_.

with a correlation coefficient of 0.98 and the intercept almost pass the origin. It means that the calcite growth at pH= 9.5 is diffusion-control. The calculated Kd fromFig. 6 is 1.58× 10−10and 1.48×10−10m/s for pH=8.5 and 9.0, respectively, and fromFig. 5for pH= 9.5 is 1.52 × 10−10m/s. The fitted equations and their associated correlation coefficient and the calculated individual coefficients are tabulated inTable 1. The largest difference is less than 10% between Kd’s. It implies

that pH has no inﬂuence on the diffusion step. On the other hand, Kr increases by an order, i.e., from 1.34× 10−10 to

5.03× 10−9m/s, for pH changing from 8.5 to 9.0, and the change in Kr is even higher when pH increases from 9.0 to 9.5.

The results are not surprising because surface reaction takes place at solution/crystal interface and the interface conditions change with pH value. In the region of higher growth rate or higher supersaturation, the growth-rate data do not ﬁt Eq. (4) as shown inFig. 6for pH=8.5 and 9.0, and do not follow straight line shown inFig. 5for pH=9.5 either. A similar phenomenon was observed for the growth of potassium alum (Tai et al.,

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0 0.4 0.8 1.2 1.6 0 2 4 6 G (10 -10 m/s) σ

Fig. 7. Calcite growth rates as a function offor four levels of R: 5.5 (䊉), 2.0 (×), 1.0 (), and 0.4 (). Other operation variables kept constant are: pH= 9 and I = 0.018 M.

1987) and nickel sulfate (Phillips and Epstein, 1974) grown in a ﬂuidized bed. The data points were lower than that predicted by Eq. (4). It is attributed to an increase in Kd, which increases

with diffusivity, at higher supersaturation. For potassium alum systemFukui and Nakajima (1985)measured diffusivity using the interferometric method. They reported that the diffusivity remained quite constant up to a supersaturation (=0.95), then it started to increase appreciably.

The effect of pH on other sparingly soluble salt has been re-ported byStubicar et al. (1993). They found that the growth rate of lead ﬂuoride had a maximum at its isoelectric point, which was at pH= 5.6. The isoelectric point of calcium carbonate is between 9 and 10 (Reed, 1989). The growth rate of calcite, which is one form of calcium carbonate, increases with pH in the range of 8.5 and 9.5. This is consistent qualitatively with the result reported byStubicar et al. (1993)for lead ﬂuoride.

4.3. Effect of activity ratio on Kd and Kr

The activity ratio of calcium carbonate is deﬁned as: R=

a_Ca2+/a_CO2−

3 =[Ca

2+_]·y

Ca2+/[CO23−]·yCO2₃−. In the estimation of activity, yi, for Ca2+and CO23−, the activity coefficient was calculated by the modified Debye–Hückel equation proposed by Davies (Butler, 1964).Tai et al. (2005)studied the effect of activity ratio on the crystal growth rate of CaCO3by keeping other operation variables constant, including pH, ionic strength, supersaturation, seed size and superficial velocity. They found that there existed a maximum growth rate around R= 2.5. Therefore, it is interesting to know how the activity ratio affects Kd and Kr. The growth-rate data of calcite crystal are plotted

inFig. 7for R= 0.4, 1.0, 2.0, and 5.5. Similar to the results

0 10 20 0 4 8 12 16 σ/ G (10 4 s 1/2 /m 1/2 ) √ G (10_√ -6_m1/2_/s1/2₎

Fig. 8. Calcite growth-rate data plotted by the two-step growth model with r= 2 at three ionic-activity ratios: 5.5 (䊉), 2.0 (×), 1.0 (), and 0.4 (). Other variables kept constant are: I= 0.018 M and pH = 9.

reported byTai et al. (2005), a maximum was found for the growth rate, which has the highest value at R=1 instead of 2.5 reported byTai et al. (2005). However, the operation conditions are not identical for the two cases, only ionic strength kept the same. Besides, the growth-rate data for R= 0.4 almost mingle with that for R= 5.5. Then the growth-rate data are rearranged and plotted as/√Gversus√Gshown inFig. 8. The growth-rate data of R= 0.4 and 5.5 were fitted with one straight line in the linear regression analysis because the growth-rate data are difficult to separate. Similar to Fig. 6, straight lines were obtained at lower growth rates and data points at higher growth rate deviated from the straight lines. The intercept of the line for R= 1 and 2 were negative, thus the growth rate data of R= 1 and 2 shown inFig. 7were fitted by straight lines. The fitted equations and their associated correlationcoefficients and the slopes of the fitted straight lines, i.e., Kd, are tabulated

in Table 2, in which the ﬁtted straight line shown in Fig. 8 for the growth rate data of R= 0.4 and 5.5 is also presented. FromTable 2we know that Kr for R= 1 and 2 is too large to

determine and the largest Kd is about two-fold of the smallest Kd. It is concluded that the crystal growth process of calcite

at R = 1 or 2 is diffusion controlled, and the activity ratio has a greater influence on the surface-reaction step. Stubicar et al. (1990) investigated the effect of Pb/F activity ratio on the growth kinetics of lead fluoride. They found that a plot of growth rate versus supersaturation gave different slopes, 1.55 for a_Pb2+/aF−>1 and 4.00 for aPb2+/aF−<1. Although the crystal growth kinetics of CaCO3and PbF2respond differently to different activity ratios, the effects of activity ratio on the sparingly soluble salts are significant.

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Table 2

Growth kinetics of calcite at various values of activity ratio

Activity ratio Fitted equation Correlation Kd Kr

(R) coefﬁcient (m/s) (m/s) 1.0 Ya=2.86Xb+0.360 0.90 2.86× 10−10 — 2.0 Ya=1.98Xb+0.146 0.98 1.98× 10−10 — 0.4 and 5.5 Yc=0.63Xd+1.959 0.83 1.58× 10−10 2.61× 10−9 a_Y_{= G m/s.} b_X₌_. c_Y_{= (}_/√_G)_{× 10}4_s1/2_/m1/2_. d_X₌√_G_{× 10}−6_m1/2_/s1/2_.

4.4. Effect of ionic strength on Kdand Kr

The ionic strength of a solution is calculated by the following equation: I=1 2 n i=1 Z2_iCi, (16) 0 5 10 15 20 0 4 8 12 16 G (10-6m1/2_/s1/2₎ √ σ/ G (10 4 s 1/2 /m 1/2 ) √

Fig. 9. Calcite growth-rate data plotted by the two-step growth model with r=2 at three levels of ionic strength: 0.005 M (), 0.018 M (䊉), and 0.035 M (). Other variables kept constant are: pH= 9.0 and R = 5.5.

Table 3

Growth kinetics of calcite at various levels of ionic strength

Ionic strength Fitted equation Correlation Kd Kr

(M) coefﬁcient (m/s) (m/s) 0.005 Ya=0.8519Xb+1.814 0.98 1.17× 10−10 3.03× 10−9 0.018 Ya=0.6673Xb+0.1251 0.83 1.49× 10−10 6.40× 10−9 0.035 Ya=0.3217Xb+1.495 0.96 3.11× 10−10 4.47× 10−9 a_Y₌_/√_G_{× 10}4_s1/2_/m1/2_. b_X₌√_G_{× 10}−6_m1/2_/s1/2_.

where Ziis the valence of i species and Ciincludes the

concen-tration of H+, OH−, Ca2+, CaOH+, CaHCO+₃, HCO−₃, CO2₃−, Na+ and Cl− (Tai et al., 1993). In this experiment, the ionic strength was adjusted by adding NaCl to the solution because Na+and Cl−were already present in the solution.

Stubicar et al. (1993)reported that the linear growth rate of lead ﬂuoride in a high ionic strength solution (0.1 mol/dm3 KNO3solution), i.e., at conditions of particle charge compen-sation, is almost ﬁve times greater than that in a low ionic strength solution, which is an aqueous solution without adding KNO3. The difference is tremendous so that a further study on the ionic strength to reveal the growth mechanism is worthy.

The crystal growth rates at various supersaturations, from = 0.2 to 1.4, were measured for three levels of ionic strength, i.e., 0.005, 0.018 and 0.035 M, with other operation variables kept the same. The growth rate increases with an increase in ionic strength and supersaturation. Under the same supersatu-ration, a big difference by a factor of 3 in growth rate was found between the highest and the lowest level of ionic strength, a re-sult somewhat similar to that reported byStubicar et al. (1993). According to Eq. (4), the growth-rate data were rearranged to plot /√G versus √G as shown in Fig. 9. It is found that the straight lines tend to converge on the axis of ordinate. Ac-cording to Eq. (4) the calculated Kd’s and Kr’s are tabulated

inTable 3, in which the fitted equations and their associated correlation coefficients are also listed. Judging from the cor-relation coefficients, which are 0.98, 0.83 and 0.96, the fitting of straight lines is acceptable. The value of Kd varies from

1.17×10−10to 3.11×10−10m/s, thus the ionic strength has a great inﬂuence on Kd. On the other hand, Kris less affected by

ionic strength, changing from 3.03× 10−9to 6.40× 10−9m/s. Besides, Kr is greater than Kd by an order, meaning that

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mass-transfer resistance under the speciﬁed operation condi-tions for a second-order surface-reaction rate (Tai et al., 1993). 5. Conclusion

A ﬂuidized-bed crystallizer controlled at constant composi-tion was used to measure the crystal growth rate of calcite. Several operation variables were studied in this experiment, in-cluding supersaturation, pH, activity ratio and ionic strength. These factors are signiﬁcant in a range of relative supersatu-ration () between 0.2 and 1.4, the pH value between 8.5 and 9.5, the ionic strength (I ) between 0.0025 and 0.0034 M, and the activity ratio (R) between 0.4 and 5.5. The crystal growth rate increases with an increase of supersaturation, pH value, and ionic strength, however, the growth rate shows a maximum at an activity ratio around 1.

When the growth-rate data were analyzed by the two-step crystal growth model by assuming r= 2, the calculated mass-transfer coefficient is rather independent of pH and activity ratio, but varies with ionic strength, the surface-reaction coef-ficient increases with pH and activity ratio but rather indepen-dent of ionic strength. As far as the resistance of calcite growth process is concerned, the surface-reaction resistance is less sig-nificant in the range of the operation conditions investigated in this study.

Notation

a speciﬁc surface area of crystals, m2/kg A surface area of crystals, m2

Ci concentration of i species, mol/m3

[Ca2+_] _{calcium-ion concentration, mol/m}3 [Ca2+_]

a calcium-ion concentration of titration solution,

mol/m3 [Ca2+_]

o calcium-ion concentration of original solution,

mol/m3 [CO2−

3 ] carbonate-ion concentration, mol/m3 D diffusivity of solute in solution, m2/s G linear crystal growth rate, m/s Ga Galileo number, dimensionless I ionic strength, M

Kd mass-transfer coefﬁcient, m/s Kip ionic product, (mol)2/m6 Kr surface-reaction coefﬁcient, m/s Ksp solubility product, (mol)2/m6

K_sp concentration solubility product, (mol)2/m6 KT proportional constant in Eq. (1)

L crystal size, m

M molecular weight of CaCO3, kg/kmol Mv density number,[(p− )/], dimensionless n number of crystals, dimensionless

r surface reaction order, dimensionless R activity ratio, dimensionless

Rg crystal growth rate, kg/m2s Sc Schmidt number, dimensionless Sh Sherwood number, dimensionless

t time, s

Va titration volume, m3

Vo volume of original solution, m3 u superﬁcial velocity, m/s

W crystal weight, kg

y_Ca2+ activity coefﬁcient of calcium ion, dimen-sionless

y_CO2−

3 activity coefﬁcient of carbonate ion, dimen-sionless

Zi valence of i species, dimensionless Greek letters

volume shape factor of crystal, dimensionless

surface area shape factor of crystal, dimen-sionless

solution density, kg/m3

p crystal density, kg/m3

relative supersaturation, dimensionless

interfacial supersaturation, dimensionless

Acknowledgment

The authors gratefully acknowledge the ﬁnancial support provided by the National Science Council of Republic of China (Taiwan).

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