Intensification of Precipitation Using Narrow Channel Reactors: Case

Aug 3, 2006 - ... of Chemical Technology (formerly UDCT), Chemical Engineering Department, Matunga, Mumbai 400019, India ... Modeling for agglomeratio...
3 downloads 0 Views 358KB Size
Download PDF
3086

Ind. Eng. Chem. Res. 2007, 46, 3086-3094

Intensification of Precipitation Using Narrow Channel Reactors: Case Study of Hydrotalcite Precipitation Venktesh S. Shirure,† Bhaurao P. Nikhade, and Vishwas G. Pangarkar* Mumbai UniVersity Institute of Chemical Technology (formerly UDCT), Chemical Engineering Department, Matunga, Mumbai 400019, India

Hydrotalcite, which is an anionic clay, has been prepared by precipitation reaction in a conventional stirred semibatch reactor and a novel narrow-channel reactor. For the stirred batch reactor, the effects of various parameters such as feed point location, stirring rate, impeller type, supersaturation ratio, and temperature on particle size (PS) and particle size distribution (PSD) have been studied. For the narrow-channel reactor, the effects of the channel cross-sectional area (two square channels with cross sections of 1 and 2 mm2), the Reynolds number, the type of mixer (Y and cross mixers), and the supersaturation ratio were studied. Modeling for agglomeration was performed to predict the PSD. Comparison of the mean PS of the two reactors, on a power consumption basis, clearly indicates the superiority of the narrow-channel reactor. The results also clearly show that the narrow-channel reactor is composed of much more compact process equipment, which has significantly lower capital and operating costs, as compared to a conventional stirred reactor, yielding, simultaneously, a better product. All these advantages lead to an intensified process. 1. Introduction Precipitation reactions involve fast chemical events, the reaction times of which are of the same order of magnitude as the characteristic mixing steps.1 Indeed, Dankwerts,2 in his pioneering work on characterization of mixing at the molecular level, specifically mentioned precipitation as a process where mixing parameters are very important. Stirred batch/continuous reactors have been extensively used to perform precipitation reactions. However, recent literature indicates that narrow channel reactors give better mixing conditions than the conventional batch reactor.3,4 Trippa and Jachuck3 studied calcium carbonate precipitation in narrow-channel reactors without segmentation by air. This study showed that narrow-channel reactors gave smaller particle sizes than a stirred batch reactor, which reflects the mixing effectiveness of reagents inside the channel. Shirure et al.4 performed a systematic study of the precipitation of Mg(OH)2 in both narrow-channel and stirred batch reactors over a wide range of operating conditions. This study clearly showed that narrow-channel reactors give a far lower particle size (PS), with an order-of-magnitude-lower power consumption per unit mass of the product than stirred batch reactor. Apart from these studies, very limited studies are available on precipitation reactions in narrow-channel reactors. In the present study, hydrotalcite precipitation via the reaction between magnesium and aluminum salts and an alkali was chosen. The solubility of hydrotalcite in aqueous media is extremely low. Hence, the appearance of the precipitate is instantaneous, indicating that micro mixing can have an important role in deciding the particle size distribution (PSD). Hydrotalcite is a compound of commercial importance, and precipitation in narrow-channel reactors is of scientific and industrial interest. Hydrotalcite is a synthetic layered mixed hydroxide that contains exchangeable anions. Because of these exchangeable anions, hydrotalcite has numerous applications * To whom correspondence should be addressed. Tel.: +91-22-2414 5616. Fax: +91-22-2414 5614. E-mail address: [email protected],[email protected]. † Currently with Biocon India, Ltd., Bangalore, India.

in the field of catalysis. Hydrotalcite also finds applications in medicines, in wastewater treatment, as a polyvinyl chloride (PVC) stabilizer, etc.5 Also there is an increasing amount of attention being given to inorganic compounds such as hydrotalcite, because of their ability to undergo endothermic dehydration under fire conditions, for use as environmentally friendly and safe fire retardants.6-8 Moreover, hydrotalcite does not evolve toxic corrosive substances upon combustion and can be used at higher processing temperatures. The key parameter in its use as a fire retardant and catalyst is its PS, agglomeration level, and PSD. Crystals with sizes of at least 0.1 µm are more suitable for fire-retardant compositions. However, although the available hydrotalcites have a small crystal size, the crystal lattice has great strain and strong aggregation occurs, yielding an agglomerated particle size of >70 µm. The objective of this study is to investigate the effect of mixing and reaction parameters on the PS and PSD for precipitation of hydrotalcite in narrow-channel reactors and conventional stirred batch reactors. Furthermore, in view of the importance of agglomeration of the particles, agglomeration modeling of hydrotalcite particles in a narrow-channel reactor has been attempted. In modeling, each segment of the reaction mixture is considered as a batch and the residence time in the reactor is considered as batch time. Using simplified assumptions, population balance equations are solved. The comparison of experimental and predicted PSD curves clearly shows that the assumptions that have been made are pragmatic. 2. Previous Work on Hydrotalcite Although there are many formulations for hydrotalcite, according to the nomenclature used by Cavani et al.,5 hydrotalcite is a compound that has the formula Mg6Al2(OH)16CO3‚ 4H2O. Other compounds that have different molar compositions are categorized as hydrotalcite-like compounds. Carbonate ions are essential for this type of structure. The structure of hydrotalcite can be understood by understanding the structure of brucite. In brucite, when Mg2+ ions are substituted by a trivalent ion that has a radius that is not very different (Al3+ for hydrotalcite), a positive charge is generated in the hydroxyl sheet. This net positive charge is compensated by (CO3)2-

10.1021/ie060371+ CCC: $37.00 © 2007 American Chemical Society Published on Web 08/03/2006

Ind. Eng. Chem. Res., Vol. 46, No. 10, 2007 3087

of the narrow-channel reactor have been given by Shirure et al.4 The effect of channel diameter was studied in two channels with same square configuration but different cross-sectional areas, of 1 and 2 mm2 (henceforth called reactors A and B, respectively), with a Y mixer. The segmentation required was created using air. The reactant streams were injected into the restricted volume, where the dissipation of kinetic energy ensured a thorough mixing. Mixing and segmentation occurred simultaneously in the mixer chamber, out of which the reacting mixture was expelled as a segmented flow. The slurry flow rate was varied in the range of 0.4-1.15 mL/s. 4. Analysis

Figure 1. Schematic diagrams of narrow-channel reactors: (a) Y mixer and (b) T mixer.

anions, which lie in the interlayer region between the two brucite-like sheets. In the free space of this interlayer, the water of crystallization also finds a place to reside. Mg2+ and Al3+ ions can be replaced by other bivalent and trivalent ions. In such cases, the compounds are called hydrotalcite-like compounds, as mentioned previously.5 Parida and Das9 synthesized hydrotalcite via a precipitation method, using nitrates of magnesium and aluminum in stoichiometric proportion. They characterized the particles by powder X-ray diffraction and confirmed the hydrotalcite structure. Several authors have studied the thermal behavior of hydrotalcite. Kanezaki10 studied the effect of the Mg/Al atomic ratio on the thermal stability of hydrotalcite. They synthesized three types of hydrotalcite clays, with Mg/Al atomic ratios of 2, 3, and 4, for which they used reactants in corresponding stoichiometric proportions. Basile et al.11 prepared hydrotalcite via coprecipitation in an aqueous solution at pH 10. They introduced noble metals in hydrotalcite clays. The aforementioned studies pertain mainly to the synthesis of hydrotalcite and its characterization. Very limited literature is available in regard to the effect of mixing parameters on the PS and PSD of the hydrotalcite precipitate. In view of the importance of the PS and PSD, it was thought desirable to perform a systematic study of the precipitation reaction, with special emphasis on the effects of mixing parameters in both stirred batch and narrow-channel reactors. 3. Experimental Section In the present work, magnesium chloride salt (99% purity), anhydrous aluminum chloride (99% purity), sodium carbonate (99% purity), and sodium hydroxide (99% purity) were used to perform the hydrotalcite precipitation. Aqueous solutions were prepared in distilled water. Two reagent feed streams were prepared: the first contained aqueous solutions of salts of magnesium and aluminum, and the second contained NaOH and Na2CO3 solutions in stoichiometric proportions. First, experiments were performed in a stirred cell that was equipped with four baffles and had a capacity of 3 L and internal diameter of 15 cm. Both feed streams were added, maintaining a pH value in the range of 10-11. Two types of 5-cm-diameter impellers were used: a four-bladed pitched-blade downflow impeller (PTD) and a disk turbine impeller (DT). The impeller speed was varied over a range of 5.5-15 rev/s. The flow rate of the reactant was maintained at 8 mL/min. The addition lasted 2 h. Figure 1 shows a schematic diagram of the narrow-channel reactors used for the study. Other details (types and dimensions)

4.1. Composition Analysis. Composition analysis of the slurry was performed by filtering it through Whatman filter paper. The precipitate was dried at a temperature of 115 °C and analyzed for magnesium and aluminum content via complexometric titration analysis.12 4.2. Particle Size Analysis. The outlet slurry was analyzed in a Coulter model LS 230 particle size analyzer, which uses a light scattering technique for analysis. Duplicate particle size analysis was performed for each sample, and the volume average crystal size was determined as

Dp )

∑ Vidp

i

(1)

The reproducibility of the analysis was within (3%. As explained later, the particles in the aqueous product suspension can be present as agglomerates of several particles. To obtain the true intrinsic individual PS, these agglomerates must be broken down with a surfactant. By trial and error, Tween 20 was determined to be the best surfactant for the present system, which is mainly comprised of hydrotalcite particles. 4.3. Scanning Electron Microscopy (SEM) Analysis. A Philips scanning electron microscopy (SEM) system (Model 620) was used to obtain SEM images of the agglomerates. The purpose of the SEM analysis was to obtain the size of the agglomerates as well as the number of the particles in the given agglomerate. Therefore, for this analysis, no surfactant was used. The Coulter particle size analyzer provided the true, intrinsic size distribution, whereas the SEM analysis provided the morphology and extent of agglomeration of the individual particles. 4.4. Supersaturation Ratio (S). The supersaturation ratio (S) was calculated as follows:

supersaturation ratio (S) )

concentration of hydrotalcite solubility (2)

The solubility was determined by adding 2 g of hydrotalcite powder into 200 mL of deionized water. The solution was stirred magnetically, keeping the temperature constant at 35 °C for 3 h. Undissolved hydrotalcite was filtered off via the use of Whatman filter paper. The resulting solution was analyzed for magnesium and aluminum content to determine the solubility. Three experiments were conducted, and an average of the three was obtained and used as the solubility of hydrotalcite; the value obtained was 8.35 × 10-4 mol/L at 35 °C. 5. Results and Discussion 5.1. Stirred Semibatch Reactor Study. The PS that is discussed in regard to the stirred semibatch reactor is the intrinsic PS obtained from the Coulter LS 230 particle size analyzer.

3088

Ind. Eng. Chem. Res., Vol. 46, No. 10, 2007

Figure 2. Effect of the Reynolds number (Re) on particle size (PS) for a stirred batch reactor.

5.1.1. Effect of Feed-Point Location. Initially, some experiments were conducted to study the effect of feed-point location for both reagent streams. Three feed-point arrangements were studied: (i) both feed streams at the impeller discharge, (ii) one in the discharge and the other in the bulk, and (iii) both in the bulk. It was observed that PS is significantly affected by the feedpoint location. When both feed streams were in the discharge of the impeller, smaller particle sizes were obtained than those from the other two feed positions. When the addition was in the more-turbulent region, better mixing conditions can be obtained, which lead to smaller particles. Here, it is necessary to note that the micro-mixing time is very important and has a direct effect on the PS. Therefore, further experiments were conducted with reagent feed points located in the discharge stream of the impeller. 5.1.2. Effect of Temperature. The temperature effect was studied in the 30-90 °C range under the following conditions: a stirring rate of 5.5 rev/s, a supersaturation ratio of S ) 450, and DT impeller. It was observed that, with increases in the temperature from 35 °C to 60 °C, there was no effect on the intrinsic PS (25 µm particle size). However, with further increases in the temperature (from 60 °C to 90 °C), the mean PS decreased by 25%. This behavior can be explained by taking into consideration the relation of temperature with nucleation rate. According to the classical theory of nucleation, the nucleation rate is proportional to the temperatur. The relation can be given as follows:13

[

BN ) A′ exp -

]

C′σ3 T (ln S)2 3

(3)

Hence, with increasing temperature, the nucleation rate increases, which eventually gives a smaller PS. However, at lower temperature, the chances of agglomeration are less; therefore, for further studies, the reactor temperature was maintained at 30 °C. 5.1.3. Effect of Stirring Rate and Impeller Type. The effect of stirring rate on the mean product crystal size is shown in Figure 2, where the stirring rate was converted into a Reynolds number (Re), which was calculated using the following equation:

Re )

nDa2F µ

where F and µ represent the density and viscosity of the water, respectively, and Da is the diameter of the impeller.

Figure 3. Particle size (PS), as a function of power input (P/m), for disk turbine (DT) and pitched-blade downflow (PTD) impellers.

As can be observed, the mean intrinsic PS for both impellers decreases as the Re increases. The mean PS obtained for the DT impeller was less than that for the PTD impeller for all stirring rates. This can be attributed to the higher power number and power input of the DT impeller. The power dissipated is proportional to the power number under otherwise identical conditions. The reported power numbers for the DT impeller is 5.5, and, for the PTD impeller, it is 1.27, under turbulent conditions.14,15 Turbulent flow conditions were ensured in all the experiments, because even at the lowest stirring speed of 5.5 rev/s, turbulent flow conditions were satisfied. The specific power input per unit mass (specific power) was calculated using the following equation, given by Shirure et al.:4

Pm )

5 3 P Npn Da F ) m m

(4)

Figure 3 shows the variation of the mean PS with the power input per unit mass of solid product for the two impellers at a supersaturation ratio of S ) 450. It is evident that the mean PS decreases with the specific power input (P/m) and the dependence can be given as (P/m)-0.21. In the case of sodium perborate tetrahydrate (SPT) crystallization, Mandare and Pangarkar16 reported the PS dependence as (P/m)-0.15 with a solids loading of 33%. However, in the present case, the solids loading is 22% and the exponent is -0.21, which shows stronger PS dependence on the specific power input. This may be due to a greater fragility of hydrotalcite, as compared to SPT. Figure 4 shows the PSD for the PTD impeller at two different stirring speeds: 5.5 and 15 rev/s. It was observed that, as the stirring rate increases, the PSD curve become sharper. This reflects better mixing conditions with increasing stirring rate. 5.1.4. Effect of the Supersaturation Ratio. The variation of the volume-averaged PS with the supersaturation ratio S is shown in Figure 5. It can be observed that, with increasing supersaturation ratio, the mean PS decreases. When the supersaturation ratio is increased from S ) 10 to S ) 150, PS decreased sharply (by ∼40%). On the other hand, at higher supersaturation ratios (S > 150), the extent of decrease in PS is comparatively less (∼10%). Similar behavior was observed by Shirure et al.4 for narrow-channel reactors, and their explanation is applicable in this case also. 5.2. Narrow-Channel Reactors. In the case of narrowchannel reactors, SEM analysis was performed to confirm the agglomeration at high supersaturation ratios, whereas Coulter (LS 230) particle size analysis was used to determine the intrinsic PS.

Ind. Eng. Chem. Res., Vol. 46, No. 10, 2007 3089

Figure 4. Effect of the stirring speed on the particle size distribution (PSD).

Figure 5. Variation of the particle size PS with supersaturation ratio S for the DT impeller.

Figure 6. Effect of segmentation on PS.

5.2.1. Effect of Segmentation on Intrinsic PS. Figure 6 shows the effect of Re on the mean PS with and without segmentation by air in the reactor at two different supersaturation ratios: S ) 50 and S ) 450. The Reynolds number was calculated using the following equation:

Re )

DQF A cµ

The PS values obtained are almost similar in the two cases, under otherwise similar conditions. Hence, it can be concluded that there is no effect of segmentation on the volume-averaged

Figure 7. Effect of Re on PS at various supersaturation ratios.

Figure 8. Effect of feed flow rate on the PSD.

PS of the hydrotalcite particles. However, with the addition of segmenting air, the problem of blockage of the channels was eliminated. Therefore, segmentation is essential for smooth, uninterrupted operation, and, hence, the same was used in all further experiments. 5.2.2. Effect of Reynolds Number (Re) on the Intrinsic Particle Size (PS). Figure 7 shows the effect of Re on the intrinsic PS obtained from the Coulter particle size analyzer for different supersaturation ratios in reactor A. The intrinsic PS decreases sharply with Re at higher supersaturation (S ) 450). The decrease is ∼50% when the Reynolds number is increased from Re ) 1600 to Re ) 4600. Figure 8 shows the variation of the intrinsic PSD for reactor A. The PSD curve is sharper for higher flow rates, which is due to the higher Re, which, in turn, causes better mixing conditions at higher flow rates. A detailed explanation of this behavior has been provided by Shirure et al.4 5.2.3. Effect of the Supersaturation Ratio and Channel Cross Section on the Intrinsic and Apparent Particle Size (PS). As shown in Figure 9 for reactors A and B, as the supersaturation ratio S increases, PS decreases for both reactors. At S ) 450, PS decreases more sharply with Re than at a lower supersaturation ratio (S ) 50) for reactor A. As mentioned in Section 4.0, the very small particles have a tendency to form agglomerates. Agglomeration occurs at a higher supersaturation ratio. To confirm the higher agglomeration tendencies at a higher supersaturation ratio, selected samples were analyzed via SEM. Figure 10 shows the micrograph of the particles at S ) 450

3090

Ind. Eng. Chem. Res., Vol. 46, No. 10, 2007

Figure 9. Effect of Re on PS at various supersaturation ratios in reactors A and B.

and a flow rate of 0.4 mL/s. Agglomerated particles with a size of 200 µm can be clearly observed, and these agglomerates are formed by more than two particles. At a lower supersaturation

ratio (S ) 50), however, there is no agglomeration and only growth controls the PS. A nonagglomerated particle of 10 µm can be observed in Figure 11. This is possible when the initially produced nuclei are allowed to grow. These micrographs support the argument that involves true and apparent particle size. Figure 9 also shows the effect of Re on the mean PS at two different S values in reactors A and B. It can be observed that, as Re increases, PS decreases at all supersaturation ratios that have been investigated. However, the extent of decrease varies for the two reactors. For instance, at S ) 450, when the Reynolds number increased from Re ) 2300 to Re ) 4000 in reactor B, the decrease in PS was 13%, whereas, in reactor A, under otherwise similar conditions, it was 40%. This indicates that the smaller the dimensions of the channel, the better the micro mixing and, hence, the lower the PS. 5.2.4. Effect of the Mixer Type on the Intrinsic Particle Size (PS). The effect of the angle of mixing of the two reactants prior to the reaction was studied with two mixer configurations: a Y mixer and a cross mixer, the details of which can be found elsewhere.4 This effect was studied with a 1-mm2 channel

Figure 10. Micrograph of hydrotalcite particles at S ) 450 and a flow rate of 0.4 mL/s.

Figure 11. Micrograph of a nonagglomerated hydrotalcite particle.

Ind. Eng. Chem. Res., Vol. 46, No. 10, 2007 3091 Table 1. Comparison of Narrow-Channel and Stirred Batch Reactors for the Production of 5000 kg of Hydrotalcite (S ) 450) reactor

power (kW)

volume of reactor (m3)

number of batches

stirred batch narrow channel (1 mm2)

3283 280

0.7 0.205 × 10-6

3

reactor. It was determined that the intrinsic PS obtained with the Y mixer was larger (14.1 µm) than that obtained with the cross mixer (11.2 µm), but the PSD in the Y mixer was sharper than that with the cross mixer. In the cross mixer, the two streams collide with each other, resulting in a high power dissipation in a very small volume (almost a reaction plane with zero thickness). This yields a very high degree of micro mixing, which results in spontaneous nucleation of fine-sized particles. In the Y mixer, the two streams flow parallel to each other before mixing and, hence, the extent of micro mixing is low. The reactants take a longer time to mix than the cross mixer. Thus, the reaction is spread over a certain length of the mixer, rather than a plane, as in the case of the cross mixer. This results in slow nucleation but relatively larger growth rates and, hence, a larger average particle size. 5.2.5. Conversion. It was observed that very high conversions (>95%) could be obtained for all flow rates in both narrowchannel reactors. The highest conversion was 99.5% with a flow rate of 1.15 mL/s in reactor A. This may be attributed to the highly effective mixing in reactor A, which is due to the small channel cross-sectional area and high flow rate. Effectively, the reactor size for a given rate of production is much lower for narrow channel reactors, which, through their use, implies process intensification. 6. Comparison of Intrinsic Particle Size (PS) in Narrow-Channel and Stirred Semibatch Reactors Figure 12 shows a plot of power input versus the average intrinsic PS for the stirred batch reactor and the continuous narrow-channel reactor (reactor A). It can be observed that, with increasing power input, PS decreases in the stirred batch reactor. As the stirring rate increases, the power dissipated increases and attrition due to (i) intraparticle, (ii) particle-impeller, and (iii) particle-solid surfaces collisions increases. Therefore, the average intrinsic PS decreases as the power input increases. The smallest particle size obtained in the stirred batch reactor was 15 µm, versus 7 µm for the narrow-channel reactor. This clearly shows the superiority of narrow-channel reactors over stirred batch reactors. Figure 12 also shows that the specific power requirement in the narrow-channel reactor is an order of magnitude lower than that of the stirred batch reactor. For the

number of narrow channels

time (h)

25

24 24

range of supersaturation ratio covered in this work, the narrowchannel reactor always gave smaller particles than the stirred batch reactor. This is an important advantage of narrow-channel reactors in industrial practice. Table 1 gives a comparison of the reactor volumes for narrow-channel and stirred batch reactors under otherwise similar conditions. It is obvious from Table 1 and Figure 12 that the narrow-channel reactor affords an intensified process, in terms of both capital and operating costs. There is a sharp decrease in PS with increasing P/m for the narrow-channel reactor, whereas, for the stirred batch reactor, a 4-fold increase in P/m is required to decrease the PS by onehalf. This clearly indicates the highly effective utilization of the power input in the narrow-channel reactor. 7. Agglomeration Modeling As described previously, at higher supersaturation ratios, the solid precipitate has a high tendency of agglomeration. In this section, a model of agglomeration for hydrotalcite particles is described. To understand the model, it is essential to know the measurement principle of the Coulter LS 230 particle size analyzer, which was used to obtain the intrinsic PS and PSD of the precipitate. 7.1. Coulter LS (CLS) 230 Particle Size Analyzer and Population Balance. The Coulter CLS 230 instrument is a light scattering particle size analyzer that uses the diffraction of laser light by particles as the main source of information regarding the particle size PS. Also it obtains information via the polarized intensity differential scattering (PIDS) method. The diffraction pattern of each particle is characteristic of its size. The pattern measured by the CLS instrument is the sum of the patterns scattered by each constituent particle in the sample. Detectors measure the composite diffraction pattern. This composite diffraction pattern is decomposed into many diffraction patterns, one for each size classification, and the relative amplitude of each pattern is used to measure the relative volume of the spherical particle of that size. This computation is based on the Fraunhofer model of light scattering by the particles. Size channels in the CLS instrument are spaced logarithmically and are, therefore, progressively wider in span toward larger sizes. The diameter of the particle in a channel is given as

dpk ) antilog

[loglower edge + logupper edge] 2

(5)

Statistical quantities are weighed according to the type of distribution (volume percent, number percent). Xc in the following calculations is the weighed channel center. The weighing factors are calculated based on the spherical particles. For number percent calculations, Xc is simply equal to size of that channel; for volume percent calculations, Xc is given as

(dplc)3π Xc ) 6

Figure 12. Comparison of stirred batch and narrow-channel reactors.

(6)

The smallest particle size that can be measured in this instrument is 0.04 µm, and the largest measurable particle size is 2000 µm; this is measured in 116 channel sizes.

3092

Ind. Eng. Chem. Res., Vol. 46, No. 10, 2007

For segmented flow in the narrow-channel reactor, it was assumed that each segment is a well-stirred batch reactor of very small volume. The time required for the segment to reach outlet is the batch time (i.e., residence time). The generalized population balance equation17,18 can be simplified for the wellstirred tank as follows:

∂f(r,t) ∂(Gf(r,t)) d(ln V) + + f(r,t) ) ∂t ∂r dt Q I f I Q Ef E B+ - A + BR (7) V V where the term f(r,t) dr represents the number of particles at time t per unit volume of suspension in the size range between r and (r + dr); A and BR are, respectively, terms that describe agglomeration and breakage. The population balance equation for size-independent growth, under constant volume and well-mixed conditions, becomes isothermal:

∂f(r,t) ∂f(r,t) +G ) B - A + BR ∂t ∂r

(9)

(10)

where X is the net theoretical number of binary collisions, which involve particles with a radius between r and r + dr; k appears as an efficiency factor, which represents the probability that two colliding particles stick together. This probability factor is dependent on the supersaturation and diameter of the colliding particle. The frequency of collisions between two particles is assumed to be the product of the population densities f(r1,t) and f(r2,t). The rate of agglomeration is then given as the difference between the birth rate (AB) and death rate (AD) of the agglomerate:

A ) AB - AD

∫0r f(r - r′,t)f(r′,t) dr′

(12)

where r′ and r - r′ are the radii of two particles, the agglomeration of which produces a particle of radius r. This equation takes into consideration the fact that the birth of a particle occurs only via the agglomeration of particles smaller than its size. The death term then can be written as

AD ) k′f(r,t)

∫0r f(r′,t) dr′

(13)

This equation takes in account the fact that the death of an agglomerate of size r can happen when it collides with a particle of any size. Consequently, the agglomeration rate is

∫0r f(r - r′,t)f(r′,t) dr′ - k′f(r,t)∫0r f(r′,t) dr′

(14)

(8)

The smallest particles that can be measured by the Coulter instrument (i.e., 0.04 µm in size) were assumed to be the nucleus for all growth/agglomeration. Each particle was assumed to be spherical. 7.2. Agglomeration. Because agglomeration is a function of the particle number density, the model is based on the number of particles, rather than the volume or mass of particles. For modeling purposes, the number of agglomerated particles per 100 nuclei was taken as the basis. The net rate of agglomeration of particles is expressed as

A ) kX

AB ) k

A)k

During the experimentation, it was observed that the reaction is instantaneous and nucleation starts as soon as the two reactants come into contact. Hence, it is a very difficult to determine the nucleation kinetics; however, it is safe to assume that all the nucleation occurs at the beginning and growth follows the nucleation. Assuming very little breakdown in narrow-channel reactors, BR in the aforementioned equation can be neglected. The SEM micrographs reveal that there is a very high degree of agglomeration at S ) 450; hence, it is safe to assume agglomeration control rather than growth control. Thus, for the present model, the growth rate term was neglected. The population balance with these assumptions simplifies to

∂f(r,t) ) -A ∂t

The birth term AB can be expressed as

(11)

The aforementioned model equations require efficiency factors. Assuming that (1) the agglomeration efficiency constants for death and birth are the same (i.e., k ) k′), and (2) all particles in the ith channel are of the same size (the value of A from eq 14 was used in eq 9, after A was obtained via the numerical integration of eq 14) yields the expression

dni dt

m-1

) k{[

m-1

∑1 ∑1 ni(

116

nj)] - nm

∑1 nk}

(15)

The differential term in the aforementioned equation can be discretized, to form linear algebraic equations. Halfon and Kaliaguine19,20 have modeled alumina trihydrate crystallization in a stirred batch reactor, assuming that the probability of all particles sticking together is the same. As an improvement in the present work, it is assumed that the efficiency factor is different for each binary collision. However, in such a case, the system of equations cannot be solved, because the number of unknowns exceeds the number of equations. Hence, it was assumed that the probability of two particles sticking together is solely determined by the efficiency constant of the smaller of the two. To minimize computation time, the following assumptions were made: (1) Particle birth occurs via the agglomeration of smaller particles only. However, all smaller particles do not contribute to the birth of a given particle. For instance, a 10 µm particle cannot form via the binary collision of 5 µm and 1 µm particles, but it can form from the following combination of binary collisions: 5 µm and 5 µm particles, 6 µm and 4 µm particles, 7 µm and 3 µm particles, 8 µm and 2 µm particles, and 9 µm and 1 µm particles. Data regarding the birth of the particles were calculated using Scilab 2.6 software. (2) The particle dies because of the birth of some other particle; hence, the same data as those used for the birth of particles can be used for the death of particles by some rearrangements. For instance, in the aforementioned example, when a particle of 10 µm is forming from the collision of a 7 µm particle and a 3 µm particle, at the same time, particles 7 and 3 µm in size are dying. With the aforementioned assumptions, the system of equations reduces to

Ind. Eng. Chem. Res., Vol. 46, No. 10, 2007 3093

∆n1 ) -k1n1n1 - k1n1n2 - k1n1n3 - k1n1n4 - k1n1n5 ∆t (for i ) 1) (16) l ∆n4 ) k1n1n2 + k2n2n2 - (k1n1 + k2n2 + k3n3 + k4n4 + ∆t k4n5 + k4n6 + k4n7)n4 (for i ) 4) (17) l ∆n5 ) k1n1n4 + k2n2n3 + k2n2n4 + k3n3n3 - (k1n1 + k2n2 + ∆t k3n3 + k4n4 + k5n5 + k5n6 + k5n7 + k5n8 + k5n9)n5 (for i ) 5) (18) l ∆n114 ∆n114 ) k109n109n113 + k110n110n113 + k111n111n112 + ∆t ∆t ) k109n109n113 + k110n110n113 + k111n111n112 + k111n111n113 + k112n112n112 - (k111n111 + k112n112 + k113n113 + k114n114)n114 (for i ) 114) (19) l ∆n116 ) k111n111n115 + k112n112n115 + k113n113n114 + ∆t k114n114n114 (for i ) 116) (20) For the values of i ) 2, 3, 4, ... up to 116, the aforementioned system of simultaneous algebraic equations was solved by the matrix method, using Scilab 2.6. The PSD for an experimental run of 0.80 mL/s was considered as an initial PSD. Values of k were determined for each time interval. Using linear interpolation, the k value for the intermediate flow rate was calculated. Using these k values, the PSD was predicted. Figure 13 shows predicted and experimental PSD curves for S ) 450. Figure 13 shows that, at high supersaturation ratios, the predicted and experimental PSD curves show good agreement. Therefore, it can be concluded that, at higher supersaturation ratios, agglomeration is the dominant phenomenon in hydrotalcite precipitation and particle size PS is solely determined by the level of agglomeration. However, the larger deviation in the PSD curves (especially for low PS) is because of the limitations of the CLS instrument, which can be explained as follows. (1) As explained previously, the channels of the CLS instrument are logarithmically positioned. For instance, when the CLS instrument measures a particle that has a size of 17.18 µm, the very next size that it can measure is 18.86 µm. (2) In this model, only those collisions where particle birth or death occurs were considered. Hence, if particles with sizes of 17.18 µm and 0.412 µm collide and form a new particle, the CLS instrument cannot measure it as a particle of different size. It was observed that, at low S (S ) 50), the predicted and experimental values do not match, which is likely due to the fact that agglomeration does not control the growth of the particles. 8. Conclusion Precipitation of hydrotalcite in narrow-channel reactors shows better results, in terms of lower particle size and power consumption, than a conventional stirred batch reactor. The

Figure 13. Comparison of predicted and experimental PSD curves.

mixing parameters have a significant effect on the particle size distribution (PSD) of hydrotalcite. At higher supersaturation ratios, as the flow rate increases, the particle size decreases, although at lower supersaturation ratios, the flow rate has a negligible effect. The effect of channel diameter was examined, and it was determined that channels with a smaller crosssectional area give smaller particle sizes. Agglomeration was observed to be dominant at high supersaturation. The agglomeration model also supports this claim, because, at high supersaturation, the experimental and predicted PSD curves show good agreement. This, in turn, shows that assumptions made in the model for prediction of the PSD were pragmatic. Comparison of the narrow-channel reactor and the stirred batch reactor clearly indicates that the use of the former results in an intensified process. Nomenclature A ) agglomeration rate Ac ) cross-sectional area of the narrow channel (m2) A′ ) proportionality constant AB ) agglomeration birth rate AD ) agglomeration death rate B ) birth rate (1/s) BN ) nucleation rate (1/s) C′ ) proportionality constant D ) diffusivity (m2/s) Da ) impeller diameter (m) Dp ) average particle diameter (µm) dpi ) diameter of particle in the ith channel (µm) dpk ) diameter of the kth channel (µm) k ) agglomeration efficiency constant L ) length of channel (m) m ) mass (kg) n ) speed of stirring (rev/s) n1, n2, .... ) number of particles per unit volume in 1, 2, ... Np ) power number P ) power (W) P˙ m ) power per unit mass of product (W/kg) ∆P ) pressure drop (N/m2) Q ) flow rate of the reaction mixture (mL/s) QI ) volumetric flow rate at the inlet (m3/s) QE ) volumetric flow rate at the outlet (m3/s) r ) particle diameter (µm) S ) supersaturation ratio t ) time (s) T ) temperature (°C) V ) volume (m3)

3094

Ind. Eng. Chem. Res., Vol. 46, No. 10, 2007

%V ) percentage volume contribution of the particle diameter Dp Vi ) volume fraction in the ith channel X ) net theoretical number of binary collisions Xc ) weighed particle diameter x ) fraction of product in the product slurry (kg/kg) Greek Symbols F ) density (kg/m3) µ ) viscosity (kg/(m s)) σ ) surface tension (N/m) Literature Cited (1) Marcant, B.; David, V. Experimental evidence for and prediction of micromixing effects in precipitation. AIChE J. 1991, 37 (11), 1698. (2) Danckwerts, P. V. The effect of incomplete mixing on homogeneous reactions. Chem. Eng. Sci. 1958, 8, 93. (3) Trippa, G.; Jachuck, R. J. J. Process intensification precipitation of calcium carbonate using narrow channel reactors. Trans. Inst. Chem. Eng. 2003, 81 (A), 766. (4) Shirure, V. S.; Pore, A. S.; Pangarkar, V. G. Intensification of precipitation using narrow channel reactors: Magnesium hydroxide precipitation. Ind. Eng. Chem. Res. 2005, 44, 5500. (5) Cavani, F.; Trifiro, F.; Vaccari, A. Hydrotalcite-type anionic clays: preparation, properties and applications. Catal. Today 1991, 11, 173. (6) Bonin, Y.; Leblanc, J. Fire-resistant polyamide compositions. U.S. Patent No. 4,985,485, January 15, 1991. (7) Miyata, S.; Imahashi, T. Fire-retardant resin composition. U.S. Patent No. 4,729,854, March 8, 1998. (8) Wagner, M.; Peerlings, H. Flame-proof polyester molding compositions comprising hydrotalcite, red phosphorus and melamine cyanurate. U.S. Patent No. 6,649,674, November 18, 2003.

(9) Parida, K.; Das, J. Mg/Al hydrotalcites: preparation, characterization and ketonisation of acetic acid. J. Mol. Catal. A: Chem. 2000, 151, 185. (10) Kanezaki, E. Effect of atomic ratio Mg/Al in layers of Mg and Al layered double hydroxide on thermal stability of hydrotalcite-like layered structure by means of in situ high-temperature powder X-ray diffraction. Mater. Res. Bull. 1998, 33 (5), 773. (11) Basile, F.; Fornasari, G.; Gazzano, M.; Vaccari, A. Synthesis and thermal evolution of hydrotalcite-type compounds containing noble metals. Appl. Clay Sci. 2000, 16, 185. (12) Vogel, A. I. A Text Book of QuantitatiVe Inorganic Analysis, Including Elementary Instrumental Analysis, 3rd Edition; Longmans: London, 1961. (13) Mullin, J. W. Crystallisation; Butterworths: London, 1972. (14) McCabe, W. L.; Smith, J. C.; Harriott, P. Unit Operations of Chemical Engineering, 5th Edition; McGraw-Hill: London, 1993. (15) Genck, W. J. Optimizing crystallizer scale-up. Chem. Eng. Progr. 2003, 99 (6), 36-44. (16) Mandare, P. N.; Pangarkar, V. G. Semi-batch reactive crystallization of sodium perborate tetrahydrate: Effect of mixing parameters on crystal size. Chem. Eng. Sci. 2003, 58, 1125. (17) Randolph, A. A population balance for countable entities. Can. J. Chem. Eng. 1964, 42, 280-281. (18) Randolph, A. D.; Larson, M. A. Theory of Particulate Processes; Analysis and Techniques of Continuous Crystallization. Academic Press: New York, 1971. (19) Halfon, A.; Kaliaguine, S. Alumina trihydrate crystallization; Part 1. Secondary nucleation and growth rate kinetics. Can. J. Chem. Eng. 1976, 54, 160. (20) Halfon, A.; Kaliaguine, S. Alumina trihydrate crystallization; Part 2. A model of agglomeration. Can. J. Chem. Eng. 1976, 54, 167.

ReceiVed for reView March 26, 2006 ReVised manuscript receiVed June 27, 2006 Accepted July 3, 2006 IE060371+