Ind. Eng. Chem. Res. 2007, 46, 1125-1137
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Reactive Precipitation in Gas-Liquid Systems Sajan J. Kakaraniya and Anurag Mehra* Department of Chemical Engineering, Indian Institute of Technology, Bombay, Powai, Mumbai-400076, India
The phenomena of precipitation, driven by a reaction, in a gas-liquid system is studied, theoretically as well as experimentally, using the carbon dioxide-calcium hydroxide-calcium carbonate system. Experimental data was generated by contacting carbon dioxide gas with aqueous calcium hydroxide solution in model gas-liquid, stirred contactor, at different operating parameters. The particle-size distribution of the precipitated calcium carbonate particles was found using microscopy and optical image analysis techniques. A modeling approach, based on Higbie’s penetration theory, for the diffusion-reaction-precipitation system, combined with population balances for the particles, has been proposed. The model computations and the experimental results agree reasonably well. Introduction The precipitation phenomena in multiphase systems are much more complex than what prevails in single-phase situations, on account of the intrinsic complexity of the precipitation process and its interaction with the interphase mass transfer effects. A significant feature of gas-liquid precipitators is the possibility of crystal formation occurring simultaneously in the bulk (aqueous) phase as well as in a thin interfacial region of the liquid surrounding the gas bubbles, provided the reaction is fast enough to occur in this zone. The product is generated at both locations, under very different conditions. The product particles, which are produced near the interface, enter the bulk phase when the fluid in which these are contained moves from the interface to the bulk, where these may grow further via bulk reaction and precipitation. The reverse exchange may also take place in which particles from the bulk may grow in the interfacial region when these are swept within the penetration elements to the gas-liquid interface. The precise extent of growth and nucleation that occurs at each of these locales will depend upon and, in turn, influence the prevailing levels of supersaturation as well as the existing particle-size distributions in the respective regions. This process of exchange of material between the interfacial elements and the bulk liquid via convective mixing and the simultaneous occurrence of nucleation and growth at both locales thus leads to a complex situation so that the overall, measured size distribution at any time will be a “summation” of all the particles produced “everywhere”. Therefore, an important step toward understanding these multiphase systems lies in developing the ability to quantitatively resolve the total precipitated material into that being formed near the gas-liquid interface and in the bulk, respectively. This is necessary in order to estimate the nature and extent of control that may be exercised over the precipitate characteristics, such as particle-size distribution (PSD) and the various crystal properties, by changing the operating conditions, i.e., agitation intensity, gas-phase partial pressure, and liquid-phase reactant concentration. The present work focuses on a study of these factors using the CO2-Ca(OH)2-CaCO3 system. Literature Review The available literature in the area of multiphase precipitation describes some of the chemical systems that have been studied. * To whom all correspondence should be addressed. E-mail: mehra@ iitb.ac.in.
Yagi et al.1 carried out experiments to obtain crystals of calcium carbonate by absorbing pure carbon dioxide or from its mixture with sulfur dioxide into aqueous solutions of calcium hydroxide, in a stirred vessel with a flat gas-liquid interface. These experiments were run in the continuous mode, and the gross, overall nucleation and growth rates were determined by applying the calculation of the conventional mixed-slurry-mixed-productremoval (MSMPR) crystallizer operating at steady state and assuming a size-independent growth rate. Kotaki and Tsuge2 also studied the precipitation of calcium carbonate in gas-liquid and liquid-liquid systems in the continuous MSMPR mode by using a power-law model to relate the precipitation kinetics to the operating conditions. Yagi et al.3 used a sparged, stirred vessel that was operated in batch and semibatch modes, and the effect of suspension density and agitation was observed on the flocks formed by the precipitated particles. The conventional approach used in the above studies implicitly assumes that all precipitationsnucleation and growthsoccurs uniformly everywhere in the liquid phase. The multiphase nature of the reactor and the possibility of precipitation near the gas-liquid interface were not considered here. Therefore, such models cannot account for the micromechanisms that actually prevail at the point of precipitation. The first theory of mass transfer with chemical reaction and precipitation was pioneered by Wachi and Jones.4 These authors used the film-penetration theory to model the precipitation processes taking place in the interfacial zone close to the gasliquid interface and in the bulk liquid. Some experimental observations on the effect of gas-liquid mass transfer on the crystal-size distribution of calcium carbonate during batch precipitation were also reported by Wachi and Jones5 along with qualitative comparisons with the theory suggested in their earlier paper. It was found that larger primary particles were formed at higher stirring speeds and that agglomeration of the carbonate particles also occurred. The same authors subsequently presented a dynamic model for computing the particle-size distributions and degree of agglomeration during the precipitation phase.6 In the theoretical formulations given in the above papers, there are some conceptual problems. (1) For poorly diffusing (heavy) solutes or for nondiffusing species (such as particles), the film-penetration theory cannot account for the convective mixing that occurs between the interfacial liquid and the bulk. The only species that are able to reach the bulk phase are those that can diffuse across the diffusion film. Since particle diffusivities, as computed from
10.1021/ie0605128 CCC: $37.00 © 2007 American Chemical Society Published on Web 01/17/2007
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Ind. Eng. Chem. Res., Vol. 46, No. 4, 2007 Table 2. Experimental Conditions temperature liquid volume, VL stirring speed initial Ca(OH)2 concentration, Cb,in B CO2 partial pressure
25 °C 5 × 10-4 m3 2.16 rev/s 1.16 rev/s 13.5 × 10-3 kmol/m3 9.0 × 10-3 kmol/m3 1 atm (Ptotal ) 1 atm) 0.5 atm (Ptotal ) 1 atm)
Table 3. Values of Mass Transfer Coefficients at Different Stirring Speeds in the Stirred Contactor; VL ) 5 × 10-4 m3, aGL ) 6.36 × 10-3 m2
Figure 1. Schematic sketch of stirred contactor apparatus.
Table 1. Reactor Dimensions reactor diameter reactor height gas-liquid contact area, aGL liquid impeller type gas impeller type
90 mm 260 mm 6.46 × 10-3 m2 flat-blade turbine (4-bladed, 0.014 m diam) flat-blade turbine (6-bladed, 0.014 m diam)
the Stokes-Einstein relationship, even for particles ∼0.01 µm are at least 2 orders of magnitude less than those for typical molecular moieties (such as ions or the dissolved gas), the precipitating particles (or at least the larger ones among these) will simply build up in the interfacial liquid and “never” reach the bulk liquid. Particles above a micrometer in size, for instance, may be considered to be stationary for all practical purposes and these will never be able to reach the bulk phase. (2) Further, these model formulations make no distinction between the reactor batch time and the time for which liquid elements will typically remain near the gas-liquid interface, because the latter time does not appear in the film-penetration models. Because the process of accumulation and growth of these precipitating particles in the liquid film region will continue over the entire batch time (a few minutes or more), the application of such models to the emergent, large crystal sizes is untenable. Therefore, these models ought to be limited to systems with particle sizes for which the particle diffusivities and the molecular diffusion coefficients of other accompanying species are of the same order of magnitude. More recently, Jones and co-workers7,8 have also reported studies that use computational fluid dynamics (CFD) to generate the hydrodynamic characteristics in multiphase precipitators. This hydrodynamics still has to be coupled with a transportreaction engineering model. These models are not discussed further because their focus is on predicting the fluid flow characteristics in the reactor and they are specific to given types of reactors. The appropriate generic models to use for the case of particles in the micrometer size range thus have to be built on the basis of those mass transfer theories that explicitly postulate the convective exchange of liquid between the interfacial (film) zone and the bulk liquid, such as the penetration model of Higbie or the surface renewal model due to Danckwerts.9 In such a
stirring speed (rev/s)
kL (m/s)
1.16 2.16
3.72 × 10-5 5.66 × 10-5
scenario, the particlesslarge or smallsmove into the bulk largely because of the convective motion of particle-laden, interfacial liquid elements or eddies. Hostomsky and Jones10 reported a penetration-theory model based on Higbie’s mass transfer approach. However, this model uses only the moments of the particle-size distribution and, therefore, provides no information about the particle-size distribution. The effect of particle diffusion has not been assessed. The boundary conditions used at x f ∞ are inconsistent with the initial conditions. Also, the specific material balances that connect the interfacial concentrations with those in the bulk liquid have not been reported, nor has the model been compared with any experimental data. Therefore, it was thought desirable to comprehensively develop and apply the penetration model for mass transfer accompanied by reactive precipitation to gas-liquid, batch precipitators, using the common but industrially important CO2Ca(OH)2-CaCO3 system. Experimental Section In the view of the limited amount of experimental data available in the literature, it was thought desirable to obtain the product particle-size distribution (PSD), the average particle size, and the carbon dioxide (specific) absorption rate data over a range of operating conditions involving mass transfer coefficients, carbon dioxide partial pressures, and initial calcium hydroxide concentrations. All the experiments were carried out in a stirred contactor at room temperature (25 °C) and pressure. The setup includes an acrylic stirred contactor of internal diameter 90 mm and height 260 mm mounted between two steel discs with the help of four rods (Figure 1). The specifications of the reactor are listed in Table 1. The upper plate and the lower plate carried the shafts of impellers of the gas and liquid phases, respectively, which in turn were connected to a twostepped pulley driven by geared motors. Mechanical seals were provided for both stirrers to prevent any leakage. The fourbladed steel impellers were of 7 mm length, each. The twostepped pulley of the liquid-phase stirrer can provide speeds of 1.16 and 2.16 rev/s only. Baffles were placed in the liquid phase to prevent vortex formation. The top and bottom plates of the reactor were provided with two ports each, for the entry and exit of gas and liquid phases, respectively. Two sample collection ports were also provided on the acrylic reactor, 30 and 40 mm above the reactor bottom. These ports were packed with latex septum. A polythene balloon was filled with pure carbon dioxide and connected to a soap bulb film meter, which was used to measure the rate of absorption of carbon dioxide. This balloon was used to perform experiments with pure carbon
Ind. Eng. Chem. Res., Vol. 46, No. 4, 2007 1127
Figure 2. Image of sample taken from stirred contactor at a stirring speed of 2.16 rev/s, at (a) 60 s, (b) 180 s, and (c) 360 s; other conditions: Cb,in B ) 13.5 × 10-3 kmol/m3, C/A ) 2 × 10-2 kmol/m3.
dioxide dead-end mode. Experiments with 50% partial pressure of carbon dioxide were conducted, in the continuous flow mode, by passing nitrogen gas and carbon dioxide gas together through the reactor at equal flowrates, which were measured by rotameters connected to the respective gas cylinders. A constant flowrate of 1 × 10-4 m3/s was maintained, and the carbon dioxide absorption rates were not measured in the partial pressure experiments. The various possible values of the operating parameters in the experiments conducted here are listed in Table 2. First, the mass transfer coefficients, kL, at different stirring speeds were determined by carrying out the physical absorption of carbon dioxide into pure distilled water and measuring the absorption rates using the soap film meter. The values are listed in Table 3, and the procedure followed is described elsewhere.11 For a normal run, ∼5 × 10-4 m3 to 6 × 10-4 m3 of distilled water was stirred with an excess amount of analytical-grade calcium hydroxide powder for ∼1/2 h so that the water was saturated with calcium hydroxide. Then the solution was filtered with a 42-grade Whatman filter paper under vacuum. The solution (0.1 × 10-4 m3) was pipetted out, and its concentration was determined by titrating with standardized oxalic acid solution. The solution was then diluted to get the desired concentration. Before commencement of the experimental run, the reactor, all the slides, the cover slips, and the sampling bottles were thoroughly washed with a dilute acid and then with fresh water and were then dried. The reactor was then loaded with 5 × 10-4 m3 of calcium hydroxide solution and then purged with pure carbon dioxide at a flowrate of 1.33 × 10-4 m3/s for 30 s to remove the air present in the contactor. After purging, all the ports were closed and the pure carbon dioxide port, via the soap film meter, was opened. The stirrers were started and samples were drawn every minute, up to 6 min, with the help of a syringe. Samples of 2 × 10-6 m3 (2 cm3) slurry were collected at different batch times through the septum. Also, the absorption rates were measured through the soap film meter. For the partial pressure experiments, no purging was done. Both
of the gases, nitrogen and carbon dioxide, were passed through the contactor, the flowrates were adjusted to 1 × 10-4 m3/s, and then the stirrers were started. Two, and in some cases three, runs for each set of operating conditions were taken. A few drops of the crystal slurry were put on a slide from the sampling bottles with the help of a syringe and immediately covered by a cover slip. The slide was placed on the microscope table and magnified, and multiple monochrome images were grabbed and stored on a PC. About 50 images of each slide at different positions were taken and analyzed. The image analysis system consisted of a transmission microscope (Olympus) connected to a digital camera and a grabber card. ImagePro software was used to grab and analyze the images stored on the PC. The image analysis works by detecting the difference in the optical properties of the continuous phase to that of the particles. For one batch time, multiple slides were prepared. Also, an empty slide image was taken to provide a background reference. This background image was subtracted from the image of the sample in order to remove the background objects and particles. A watershed filter was used to break the loosely agglomerated particles into individual ones for each slide. Around 10 000 particles were grabbed for each batch time, for a set of operating conditions. A calibrated scale was loaded, and the diameter of all the objects in the image were measured and stored. This data was used to generate PSD and number average size data. Some sample images are shown in Figure 2 at different batch times. We can see that the concentration and size of the particles increases with batch time. Model Development The following assumptions are made in developing the theory of mass transfer with chemical reaction accompanied by precipitation: (1) Higbie’s penetration model of mass transfer is used as the basic framework for the theoretical development. (2) Isothermal conditions are assumed everywhere and may
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Figure 3. Schematic sketch of physical scenario for reaction and precipitation and concentration profiles for the four dissolved species in a penetration element.
be justified because all the species under consideration are present in dilute form. (3) In writing the diffusion-reaction species balances, even for ionic species, the diffusion coefficients are taken to be noninteracting with respect to each other and essentially concentration independent. Thus, every species is assumed to behave like a dilute, nonionic entity. This usual simplifying assumption further sets all the diffusivity values to be equal in order to account for electroneutrality. (4) Agglomerate formation or breakage has not been incorporated since the focus is exclusively on the evolution of the primary crystal sizes. Therefore, the proposed theory may be applied only for “small” or “early” (batch) times when the system is dominated by the presence of primary, single crystals rather than clusters or agglomerates. (5) Nucleation is taken to occur only at a fixed size so that there is no dispersion of the nucleation rate across the particlesize coordinate. (6) The proposed model is essentially developed for assessing the sizes of the particles without regard for the variety of morphologies and phases that may be present. (7) Redissolution of the precipitated product has not been considered. Such a process can occur only under acidic conditions12 and is likely, for instance, when carbonation is continued even after all the lime has been consumed. The CO2-Ca(OH)2-CaCO3 system can be described by a variety of chemical reaction schemes involving the different intermediate species.13 Here, we have used the simplest kinetic scheme possible, because it reduces the computation intensity drastically while still leading to results that are broadly similar to those that may be obtained from a more complex kinetic scheme. Selected computations were actually carried out using a more complex scheme containing the bicarbonate (HCO3-) ion and compared with results from the simpler scheme mentioned below. The scheme is as follows:14-16
CO2 (A) + 2OH- (B) f CO32- (D) + H2O [k21] Ca2 + (E) + CO32- (D) f CaCO3 [N,G] This scheme implies that the carbonate ion forms directly and irreversibly, by the reaction of dissolved carbon dioxide and the hydroxyl ion; the presence of the bicarbonate species is ignored. Step 2, in the above scheme, represents the precipitation step via nucleation and growth. The use of the simplest kinetic scheme, and the drastic reduction in computational efficiency, facilitates the application of the models to actually compute
PSDs. Use of more complex schemes can make PSD calculations prohibitive. Higbie’s penetration model of mass transfer essentially states that the gas-liquid interface is constituted by a mosaic of liquid elements which move, to and fro, between the interfacial region and the well-mixed bulk liquid. Every element that arrives at the interface from the bulk carries within it the conditions prevailing in the bulk; this element then spends a fixed time, called the contact time (tc), which can be related to the stirring intensity or the mass transfer coefficient, at the interface during which the gaseous reactant penetrates into it. At the end of the contact period, the element returns to the bulk phase where it mixes with the rest of the fluid and loses its identity. Thus, at any given batch time, the gas-liquid interface is made up of liquid elements having different ages (0 < t < tc) and the fraction of elements of all ages are equal. More details about the penetration model of mass transfer may be found elsewhere.9,17 A schematic sketch of a penetration element is given in Figure 3. The unsteady-state material balances, within the penetration elements present at the gas-liquid interface, are given by
DA
DB
DD
∂2CD ∂x2 DE
∂2CA
∂CA + k21CACB ∂t
(1)
∂CB + 2k21CACB ∂t
(2)
∂CD - k21CACB + FN + FG ∂t
(3)
2
)
∂x
∂ 2C B ∂x
)
2
)
∂2CE 2
∂x
)
∂CE + FN + FG ∂t
(4)
The terms FN and FG denote the rate of consumption of the given species due to nucleation and growth of the precipitated product, respectively, and are described below. The initial conditions in the penetration element, just when it arrives from the bulk liquid at the gas-liquid interface, are written as (for all x)
t)0
CA ) CbA
(5)
CB ) CbB
(6)
CD ) CbD
(7)
CE ) CbE
(8)
whereas the boundary conditions are given by (at t > 0)
x)0
CA ) C/A
(9)
∂CB )0 ∂x
(10)
∂CD )0 ∂x
(11)
∂CE )0 ∂x
(12)
Ind. Eng. Chem. Res., Vol. 46, No. 4, 2007 1129
xf∞
CA ) fA(CbA,t)
(13)
CB ) fB(CbB,t)
(14)
CD ) fD(CbD,t)
(15)
CE ) fE(CbE,t)
(16)
(17)
where η is the particle-size distribution function and denotes the volumetric number concentration of the particles of size (diameter) between a and a + da. The first term in this equation is a measure of the net change in the number concentration of particles of size a, at some x and t, due to the diffusive (Brownian) motion of the particles. The diffusivity of particles of diameter a is given by the Stokes-Einstein relation, namely,
Dp )
kT 3πµa
η ) ηb(a)
value 2 × 10-2 1 × 10-2 9 × 10-3 13.5 × 10-3 3.72 × 10-5 5.66 × 10-5 12.72 2.2 × 10-9 100 2710 1.24 × 104 2.35 × 102 6.0 2.0 3.47 × 10-9
C/A Cb,in B
aGL/VL DAa Mw Fp k21 kN n g Ksp a
xf∞
G)
(20)
η(a) ) fn(ηb(a),t)
(21)
∫a∞ ajη da
m/s
measured
m2/m3 m2/s kg/kmol kg/m3 m3/(kmol s) m3n/(kmoln s)
measured Jones et al.21 Carr and Frederick22 Juvekar and Sharma14 Packter19 Packter19 House18 Hostomsky and Jones10
da ) kG(xCDCE - xKsp)gU(CDCE - Ksp) dt
(24)
whereas the volumetric nucleation rate is
N ) kN(xCDCE - xKsp)nU(CDCE - Ksp)
(25)
which implies that the molar rate of nucleating product becomes
FN )
FG )
Fp N Mw
(26)
Fp Mw
∫a∞ dtd (π6 a3)η da
(27)
o
which reduces to
FG )
Fp π G Mw 2
∫a∞ a2η da
(28)
o
By substituting the definition of the second moment of the particle-size distribution, this becomes
FG )
Fp π GN Mw 2 2
(29)
(22)
which states that the flux of fresh particles into the distribution, i.e., Gη|a)ao, is equal to the number nucleation rate, at the boundary a ) ao, and νo is the volume of a fresh nucleate (νo ) (π/6)ao3). Further, the jth moment of the PSD is defined as
Nj )
measured
and nucleation processes still have to be specified. On the basis of the suggestions of House18 and Packter,19 we have chosen the nucleation and growth equations as described below. The nucleation and growth rate constants have been taken to be independent of crystal size as well as agitation intensity. The growth rate for a single particle is taken to be
(19)
which are analogous to those stated above for nonvolatile, dissolved species. Because particles are assumed to nucleate only at a fixed size ao, the particle-size distribution (PSD) is affected by nucleation only through the boundary condition
t>0
kmol/m3
and the molar rate of precipitating product contributing to growth, assuming spherical particles, is given by
∂η(a) )0 ∂x
N a ) a oη ) Gνo
source/remarks Perry and Chilton20
Diffusivity values of all other species taken to be equal to this value.
whereas the boundary condition on x can be written as
x)0
units kmol/m3
kmol2/m6
(18)
The initial condition on the population balance is
t)0
parameter
kL
The concentration gradients of all the nonvolatile species (B, D, and E) are set to zero at the gas-liquid interface, and carbon dioxide (A) is taken to be present at a concentration equal to its solubility at the interface. The conditions in eqs 13-16 arise from the concept of a penetration element “isolated” from the bulk liquid17 and essentially state that effects due to diffusion of the given solute have not been able to penetrate so “far away“. This implies that, far away from the interface, the variation in the concentration of the diffusant with t is simply given by the solution of eqs 1-4 without considering the diffusion terms and results in the solution denoted by eqs 13-16. To account for the (Brownian) diffusive motion of the particles, which may be important for very small particles, a population balance may be written.
∂ ∂η ∂2η + (Gη) ) Dp(a) 2 ∂t ∂a ∂x
Table 4. Values of Parameters for Calcium Carbonate Precipitation at 298 K
(23)
o
The model formulation for a penetration element is now complete. However, the rate equations to represent the growth
Here, U is the Heaviside function (step function) and its use above indicates that precipitation will occur only when the supersaturation levels exceeds the solubility product of the solid, precipitating species. Supersaturation is defined here as the product of the concentrations of the carbonate and the calcium ions. The values of the various parameters are given in Table 4. These data constitute the input values to the proposed theory and essentially pertain to the carbonation of lime solutions. The output obtained from the solution of the above set of equations consists of the concentration profiles of the dissolved species,
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Figure 4. Particle-size distribution at gas-liquid interface (x ) 0) and at a remote location (x ) (DAtc)0.5) for t ) tc, showing the effect of particle diffusivity. Data used: kL ) 3.33 × 10-5 m/s, C/A ) 7.23 × 10-3 kmol/m3, -3 kmol/m3, Cb ) Cb /2, Cb ) Cb ) 0, ηb,in ) 0, k ) Cb,in G B ) 10.4 × 10 E B A D 2.2 × 10-3 m7/(kmol2 s), aGL/VL ) 8.33 m2/m3, ao ) 1 × 10-8 m; other parameter values from Table 4.
namely, CA(x,t), CB(x,t), CD(x,t), and CE(x,t), and the particlesize distribution function η(a,x,t) [and its moments, Nj(x,t) (j ) 0, 1, 2, 3, and 4)]. The above system of partial differential equations was solved by finite-difference techniques. The spatial concentration profiles of the four dissolved species in the penetration element, at the end of the contact period of the element with the gasliquid interface, resemble Figure 3. The contact time is related to the empirically determined, liquid-side mass transfer coefficient by
tc )
4DA πkL2
(30)
It can be seen from Figure 3 that the reaction between dissolved carbon dioxide and the hydroxyl ion is nearly instantaneous and occurs within a very narrow reaction zone. The profile of the product carbonate ion shows a maximum with respect to distance from the gas-liquid interface, and this arises because the generation rate of this ion is largest near the reaction zone and tapers off as one moves toward the gas-liquid interface or toward the bulk end of the element. Also, model predictions suggest that (Figure 4), even for a fresh nucleate as small as 0.01 µm, there is negligible difference in the computed distributions when the particles are taken to be stationary (Dp ) 0) and when the particle diffusivity is defined by the Stokes-Einstein equation. This difference starts becoming somewhat significant if the particle diffusion coefficients are taken to be 10 times higher than what is suggested by the Stokes-Einstein relationship (eq 18); however, this is merely a hypothetical value and the computed PSD functions are also shown in Figure 4. Corresponding spatial profiles for the first four moments of the PSD are given in Figure 5. There is hardly any effect of the particle diffusion phenomena, and these profiles are nearly the same as those obtained by assuming stationary particles. It may be noted that the entire population of the particles ranges in size from 0.01 µm to ∼0.2 µm (for the values of the nucleation and growth constants used), which is on the lower side compared to the primary crystal sizes that have been observed experimentally in the literature as well as in this study. Large particles will have even more negligible values for the diffusion coefficient. It may also be seen that significant diffusive tendencies (such as that embodied in the hypothetical case of Dp(ao) ) 4.36 × 10-10 m2/s) lead to a
Figure 5. Spatial profiles of the first four moments of the PSD at t ) tc. Data used: kL ) 3.33 × 10-5 m/s, C/A ) 7.23 × 10-3 kmol/m3, Cb,in B ) 10.4 × 10-3 kmol/m3, CbE ) CbB/2, CbA ) CbD ) 0, ηb,in ) 0, kG ) 2.2 × 10-3 m7/(kmol2 s), aGL/VL ) 8.33 m2/m3, ao ) 1 × 10-8 m; other parameter values from Table 4.
flattening out of the spatial profiles of the PSD moments on account of the diffusion-driven migration of the particles from near the gas-liquid interface to more remote locations away from the interface. Therefore, the particle concentrations as well as the volume of solid present near the interface are pushed down, whereas further away in the element, these are raised. Thus, the contributions arising from the particle-diffusion term may usually be neglected, and then setting Dp ) 0 in eq 17 implies that no particles are added or removed from the distribution in the size domain ao < a < ∞ (other than those added by nucleation at size ao). So the population-balance equation becomes
∂η ∂ + (Gη) ) 0 ∂t ∂a
(31)
The boundary conditions stated by eqs 20 and 21 now become redundant. In such a situation, it is possible to deal directly with moments of the distribution rather than with the detailed PSD. To get the moments, we integrate eq 31 with respect to a for the zeroth moment,
∂N0 N ) Gη(ao) ) ∂t νo
(32)
The transient evolution of higher (jth) moments of the population balance is obtained by multiplying the population balance by aj, and integrating by parts, over the domain ao to ∞, to yield
∂Nj N ) jGNj-1 + ajo ∂t νo
(33)
In carrying out these integrations, the closure condition given by the above equation is helpful and implies that Gη(∞) ) 0. The initial conditions for the above moment equations are
t)0
N0 ) Nb0
(34)
t)0
Nj ) Nbj
(35)
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The output quantities of interest, which may now be examined, are defined below: the number average particle diameter, d10,
N1 d10 ) N0
The initial conditions for the above balances are
θ)0
(36)
and local supersaturation, Ssat,
Ssat )
x
CDCE Ksp
(37)
The variations in the crystal number average size, d10, with time t, as a function of the distance from the gas-liquid interface, x, are plotted in Figure 6a. The average size increases with time, reaching a few hundred times the size of the fresh nucleate at the end of the contact period of the element, and declines along increasing x. The behavior of the local supersaturation is given in Figure 6b. It may be seen that the local supersaturation peaks at x ) 0 soon after the element arrives at the gas-liquid interface; this peak then moves forward, away from the interface, as time proceeds, and its magnitude falls as the supersaturation is “consumed’’ by the precipitation process. Calculations show that the variation in nucleation rate, FN, also follows the same trend as produced by the local supersaturation. The growth rate follows a more complex trend in that it builds up with time and then decreases (Figure 6c). The initial buildup phase is due to the production of crystal area on which growth takes place and the later fall because of exhaustion of supersaturation. The model for a single penetration element now has to be incorporated into material balances on the bulk liquid for a given reactor. For the case of a batch (uniformly mixed) precipitator, where the liquid is held in batch mode under a constant, unvarying gaseous environment, the material balances are described below. These have not been reported in any of the earlier studies on multiphase precipitation. Material balances of this type were first described by Mehra.23 The following balances may be written as follows:
dCbA dθ
∫0
∞
)
dCbB ) dθ
(CA(x,tc) -
CbA)
aGL 1 dx - k21CbA CbB (38) V L tc
(42)
CbB ) Cb,in B
(43)
CbD ) Cb,in D
(44)
CbE ) Cb,in E
(45)
The population balance for the precipitating solids in the bulk liquid phase of the batch reactor is given by
∂ηb ∂ + (Gbηb) ) ∂θ ∂a
a
∫a∞ (η(x,tc) - ηb) dx VGLL t1c
(46)
o
where the initial condition on this bulk population balance is
ηb ) ηb,in(a)
θ)0
(47)
and the boundary condition, for particles nucleating at a fixed size ao, is given by
a ) a oη b )
θ>0
Nb Gbνo
(48)
Further, the jth moment of the bulk PSD is defined as
Nbj )
∫a∞ ajηb da
(49)
o
Similarly, the bulk growth rate for a single particle is given by
Gb )
da ) kG(xCbDCbE - xKsp)gU(CbDCbE - Ksp) (50) dθ
whereas the volumetric nucleation rate is n b b b b Fν,b N ) kN(xCDCE - xKsp) U(CD CE - Ksp)
(51)
which implies that the molar rate of nucleating product in the bulk becomes
FbN )
Fp b N Mw
(52)
and the molar rate of product contributing to growth in the bulk, assuming spherical particles, is given by
a
∫0∞ (CB(x,tc) - CbB) dx VGLL t1c - 2k21CbA CbB
CbA ) Cb,in A
FbG )
(39)
Fp Mw
∫a∞ dθd (π6 a3)ηb da
(53)
o
which reduces to
dCbD ) dθ
aGL 1 ∞ b (C (x,t ) C ) dx + k21CbA CbB - FbN D c D 0 VL tc
∫
FbG dCbE dθ
)
a
∫0∞ (CE(x,tc) - CbE) dx VGLL t1c - FbN - FbG
The first term on represents the net liquid phase due elements between
(40)
FbG )
Fp π b G Mw 2
∫a∞ a2ηb da
(54)
o
By substituting the definition of the second moment of the particle-size distribution, this becomes
(41)
the right-hand side of the above equations rate of exchange of a species into the bulk to convective motion of the penetration the interfacial region and the bulk phase.
FbG )
Fp π b b G N2 Mw 2
(55)
The bulk population balance in eq 46 may be integrated with respect to a to directly yield the rate of evolution of the jth moment of the bulk PSD, as a function of the batch time θ.
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Figure 6. Spatial profiles of (a) number average size, (b) supersaturation, and (c) growth rates of precipitate, at different values of t. Data used: kL ) 1.14 -3 kmol/m3, Cb ) Cb /2, Cb ) Cb ) 0, ηb,in ) 0, k ) 2.2 × 10-2 m7/(kmol2 s), a /V ) 8.33 × 10-5 m/s, C/A ) 7.23 × 10-3 kmol/m3, Cb,in G GL L B ) 10.4 × 10 E B A D m2/m3, ao ) 1 × 10-8 m; other parameter values from Table 4.
Ind. Eng. Chem. Res., Vol. 46, No. 4, 2007 1133
Alternatively, for the zeroth moment, we have
∂Nb0 ) ∂θ
∫0
∞
(N0(x,tc) -
Nb0)
aGL 1 Nb dx + VL tc νo
the fraction of precipitate formed near the gas-liquid interface
(56)
and for the higher jth moments, obtained by multiplying the population balance by aj, and integrating by parts, over the domain ao to ∞, we get
∂Nbj ) ∂θ
∫0
∞
F ) i
aGL 1 Nb b (Nj(x,tc) - Nbj ) dx + jGbNj-1 + ajo V L tc νo (57)
Nb,i 3
(65)
b,b Nb,i 3 + N3
where
∂Nb,i 3 ) ∂θ
a
∫0∞ (N3(x,tc) - Nb3) dx VGLL t1c
∂Nb,b 3 Nb ) 3GbNb2 + ao3 ∂θ νo
(66)
(67)
The initial conditions for the above moment equations are
θ)0
Nb0 ) Nb,in 0
(58)
θ)0
Nbj ) Nb,in j
(59)
The output obtained from the solution of the above set of equations essentially consists of the bulk concentration of the dissolved species, namely, CbA(θ), CbB(θ), CbD(θ), and CbE(θ), and the bulk particle-size distribution function ηb(a,θ) and its moments, Nbj (θ) for j ) 0, 1, 2, 3, and 4. The solution of eqs 56 and 57 is optional because this information can also be obtained from ηb(a,θ) by the use of eq 49. The above equations were solved by the use of an implicit finite difference partial differential equation (PDE) solver. The output quantities of interest in the bulk liquid, which may now be examined, are defined as follows: the number average particle diameter, db10,
Nb1
db10 )
(60)
Nb0
the mass/volume averaged particle diameter, db43,
Nb4
db43 )
(61)
Nb3
and the bulk supersaturation, Sbsat,
Sbsat )
x
CbD CbE Ksp
(62)
Further, according to the penetration theory, the gas-liquid interface at any time θ is composed of elements which have spent times (at the interface) between t ) 0 and t ) tc. The fraction of elements of a given age are equal to that of any other age, so that the rate of absorption of the gaseous solute, averaged over the entire surface, amounts to taking a time (t) average of the instantaneous rate, RiA, which is given by
RiA ) -DA
( ) ∂CA ∂x
x)0
(63)
from which the average rate of absorption may be obtained from
RA )
1 tc
∫0t RiA dt c
(64)
Equation 66 accounts for the change in bulk crystal volume by material brought in from the interface, and eq 67 accounts for the contribution of material precipitated in the bulk phase itself, to the net change in crystal volume. These equations are obtained from eq 57. Results and Discussions Model Assessment and Analysis of Experimental Data. The model fitting to the experimental data was done by using the mean particle size versus time data. This allows us to use a simpler version of the model that uses only moments and runs much faster than the code that also solves for the full PSD. More discussion on the particle distributions is given later. The experimentally obtained average particle size versus time and the corresponding computed values are shown in Figure 7. The agreement between experiments and theory is very reasonable, and the procedure used for fitting is discussed a little later. Figure 7 also shows the effect of mass transfer coefficients (stirring speeds), kL. It can be clearly seen that, at short batch times, smaller mean sizes are obtained at the higher agitation intensity, whereas at larger times, this trend is reversed. This is because, at the beginning of a batch run, precipitation occurs in the penetration element and not in the bulk region, and hence, short times are available for growth in the element, at high kL values, i.e., low tc values, which results in smaller mean sizes. However, the rate of circulation of the penetration elements between the interfacial zone and the bulk liquid is higher at greater stirring speeds, which causes the supersaturation to be transferred from the interfacial region into the bulk liquid faster and the precipitation locale to tend to shift to the bulk liquid. Thus, we can see that the plot shows a gradual increase in mean size and the net change is not too large at the lower agitation intensity. This kind of trend in the average crystal size thus arises from the specific nature of the interaction between the precipitation occurring at the two different locales. In order to carry out the regression and estimate the best values of the fitted parameters, we examine the effect of fitting parameters, namely, kN, ao, and kG, on the (mass/volume averaged) mean size versus batch time profile; the results are summarized in Figure 8. The black, filled circles represent typical experimental data, and the computed values have been obtained from the model developed here, incorporating the simple kinetic scheme. Changing the value of ao essentially modifies the upper portion of these transient profiles while making very little difference to the early part of the profile. At the larger value of ao, the proportion of larger particles increases and the mean sizes become large, at later times. A change in the value of kN has a similar effect on the computed mean sizes. At a higher value of kN, a larger number
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Figure 7. Variation of number average size with batch time, at different values of mass transfer coefficient. Data used: C/A ) 2 × 10-2 kmol/m3, -3 kmol/m3, Cb,in ) Cb,in/2, Cb,in ) Cb,in ) 0, ηb,in ) 0; other Cb,in B ) 9 × 10 E B A D parameter values from Tables 4 and 5.
Figure 9. Variation in bulk supersaturation with batch time, at different values of mass transfer coefficient. Data used: C/A ) 7.23 × 10-3 kmol/ -3 kmol/m3, Cb ) Cb /2, Cb ) Cb ) 0, ηb,in ) 0, k m3, Cb,in G B ) 10.4 × 10 E B A D -2 7 ) 2.2 × 10 m /(kmol2 s), aGL/VL ) 8.33 m2/m3, ao ) 1 × 10-8 m; other parameter values from Table 4.
Figure 10. Variation of fraction of precipitate formed near the interface with batch time, at different values of mass transfer coefficient. Data used: -3 kmol/m3, Cb ) Cb /2, Cb C/A ) 7.23 × 10-3 kmol/m3, Cb,in B ) 10.4 × 10 E B A b b,in -2 7 ) CD ) 0, η ) 0, kG ) 2.2 × 10 m /(kmol2 s), aGL/VL ) 8.33 m2/m3, -8 ao ) 1 × 10 m; other parameter values from Table 4. Figure 8. Effect of model parameters, that can be regressed, on the volume average size versus batch time profile. Frame at bottom right shows a “best -3 fit’’ line; Data used: C/A ) 7.23 × 10-3 kmol/m3, Cb,in B ) 10.3 × 10 b,in b,in b,in b,in ) 0, a /V ) 8.33 m2/m3; kmol/m3, Cb,in GL L E ) CB /2, CA ) CD ) 0, η other parameter values from Table 4.
of small crystals are produced, creating a situation analogous to that obtained for the case of smaller ao; similarly, smaller values of kN are akin to using larger values of ao. In contrast, an increase in the growth rate constant, kG, shifts the entire mean size versus batch time profile upward, i.e., toward greater sizes. This is because growth rates are increased everywhere. Thus, for the purpose of estimating values of model parameters by using the experimental data, ao and kN cannot be treated as being “independent’’. Therefore, for a fixed, arbitrary value of ao, kN and kG may be estimated by fitting (or for a fixed kN, ao and kG may be estimated). The bottom right frame in Figure 8 shows a regressed line that passes through the experimental data points. It can be seen that the agreement produced between the theory and the experiment is excellent. The theoretical plots in Figure 7, 12, and 13 (db10 vs θ) have been produced by finding the “best” values of kG and ao. The results are shown in Table 5. In order to develop some more insights into the problem at hand, we examine the typical variation of some other important bulk quantities with time. Figure 9 shows the shift of supersaturation, from near the gasliquid interface into the bulk liquid with increasing θ, at different
Figure 11. Variation of average rate of absorption with mass transfer coefficient, at different values of batch times. Data used: C/A ) 2 × 10-2 -3 kmol/m3, Cb,in ) Cb,in/2, Cb,in ) Cb,in ) 0, ηb,in kmol/m3, Cb,in B ) 9 × 10 E B A D ) 0; other parameter values from Tables 4 and 5.
values of mass transfer coefficients, kL. The Sbsat values first build up in the bulk and then decrease because of consumption of the carbonate and calcium ions by conversion to the solid phase. The largest bulk supersaturation is achieved at the highest agitation intensity. Figure 10 shows the fraction of the precipitated solid that has formed near the gas-liquid interface along θ. This fraction declines rapidly with time as more of the
Ind. Eng. Chem. Res., Vol. 46, No. 4, 2007 1135
Figure 12. Variation of number average size with batch time, at different -3 kmol/m3, values of gaseous solubility. Data used: Cb,in B ) 13.5 × 10 b,in b,in b,in b,in ) 0, k ) 5.66 × 10-5 m/s; other Cb,in ) C /2, C ) C ) 0, η L E B A D parameter values from Tables 4 and 5.
Figure 14. Evolution of bulk particle-size distribution with batch time. -3 kmol/m3, Cb ) Data used: C/A ) 2 × 10-2 kmol/m3, Cb,in B ) 9 × 10 E CbB/2, CbA ) CbD ) 0, ηb,in ) 0, kL ) 5.66 × 10-5 m/s; other parameter values from Tables 4 and 5.
Figure 13. Variation of number average size with batch time, at different values of initial liquid-phase reactant concentration. Data used: C/A ) 2 × b,in b,in b,in b,in ) 0, k ) 3.72 × 10-5 10-2 kmol/m3, Cb,in L E ) CB /2, CA ) CD ) 0, η m/s; other parameter values from Tables 4 and 5.
Table 5. Values of Fitting Parameters for Calcium Carbonate Precipitation at 298 K kG ao
6 × 10-2 1 × 10-8 0.6 × 10-8
m3g + 1/(kmol s) m m
for all the computations -3 kmol/m3 for Cb,in B ) 13.5 × 10 -3 kmol/m3 ) 9 × 10 for Cb,in B
material precipitation shifts to the bulk. The shift is most rapid for the highest kL value, and for all the cases, Fi flattens out with θ. Some more experimental data is shown in the next few figures. Figure 11 shows the variation of the average rate of absorption, RA, versus batch time, at different values of kL. As expected, the specific rate of absorption falls with batch time and mass transfer coefficients, and the experiments and computations are in reasonable agreement. Figure 12 shows the number average size along with batch time, at two different values of carbon dioxide partial pressures. Larger mean sizes occur at the higher value of C/A and smaller sizes occur at the lower partial pressure. Basically, the effect of decreasing the gas-phase partial pressure of carbon dioxide is essentially to push the reaction and precipitation processes toward the interfacial region, relative to the bulk liquid, resulting in lower sizes, as expected (if we recall the earlier discussion that the bulk is a more growth-friendly region in contrast to the interfacial zone, which is more nucleation-friendly). The effect of change in the liquid reactant concentration, Cb,in B , on the average size at different batch times, is in contrast to the effect of C/A seen above; see Figure 13. At low batch
Figure 15. Variation of (A) particle number concentration, (B) fraction of precipitate formed near the interface and conversion of hydroxyl ion, (C) volumetric holdup, and (D) bulk supersaturation with batch time. Data -3 kmol/m3, Cb ) Cb /2, Cb used: C/A ) 2 × 10-2 kmol/m3, Cb,in B ) 9 × 10 E B A ) CbD ) 0, ηb,in ) 0, kL ) 5.66 × 10-5 m/s; other parameter values from Tables 4 and 5.
times, the size increases with liquid reactant concentration, but at later times, the size decreases with the concentration. The reason for such a trend is that, with an increase in Cb,in B , the amount of reaction and precipitation in the penetration elements is increased and this increase is much more than that which occurs in the bulk. If the Cb,in B value is large, then for a fixed contact time, tc, growth rates in the element are greater. Also, the starting mean sizes in the slurry are greater. Conversely, if the Cb,in B value is small, the starting sizes are lower but the bulk contribution to crystal volume production is greater and the mean sizes undergo the larger change so that, toward larger batch times, this concentration produces the bigger crystals. Figure 14 shows experimentally the nature and transient evolution of the bulk particle-size distribution (as dimensionless number density function versus size) at different batch times (θ), for a specific and typical set of operating conditions. The bulk PSDs decrease mildly in height and become broader with increasing time. At later times, the curves simply shift rightwards (without any change in the total numbers of particles) on account of growth, indicating negligible amounts of fresh nucleation.
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Nomenclature
B ) OHC ) liquid-phase concentration of dissolved species, kmol/m3 CA ) liquid-phase concentration of dissolved gas, A, kmol/m3 C/A ) solubility of A in liquid phase, at gas-liquid interface, kmol/m3 CB ) liquid-phase concentration of B, kmol/m3 CsB ) solubility of B in liquid phase, kmol/m3 Cb,in B ) initial liquid-phase concentration of B in bulk liquid, kmol/m3 d10 ) number average particle size, m D ) CO32DA ) diffusivity of A in liquid phase, m2/s DB ) diffusivity of B in liquid phase, m2/s Dp ) diffusivity of product particles in liquid phase, m2/s E ) Ca+2 f ) function defined in boundary conditions of species/particles, kmol/m3 i F ) fraction of precipitate formed near gas-liquid interface FG ) rate of conversion to solid phase via growth, kmol/(m3 s) FN ) rate of conversion to solid phase via nucleation, kmol/ (m3 s) g ) power index of supersaturation driving force in growth rate G ) growth rate for single particle, m/s k ) Boltzmann constant, 1.38 × 10-23 J/K k21 ) second-order rate constant for reaction between A and B, m3/kmol kG ) growth rate constant in expression for G, m3g+1/kmolgs kL ) liquid-side mass transfer coefficient for gas-liquid contact, m/s kN ) nucleation rate constant in expression for N, m3n/(kmoln s) Ksp ) solubility product for precipitated species, kmol2/m6 Mw ) molecular weight of solid product, kg/kmol n ) power index of supersaturation driving force in nucleation rate N ) volumetric nucleation rate of crystals, 1/s N0 ) zeroth moment of the particle-size distribution (also, number concentration), 1/m3 N1 ) first moment of the particle-size distribution, 1/m2 N2 ) second moment of the particle-size distribution (also proportional to solid area per unit volume of liquid), 1/m N3 ) third moment of the particle-size distribution (also proportional to solid volume per unit volume of liquid) N4 ) fourth moment of the particle-size distribution, m Ptotal ) total pressure, atm RA ) specific rate of absorption of gaseous solute A, kmol/(m2 s) RN ) number rate of generation of crystals, 1/(m3 s) RG ) volumetric rate of generation of solid phase, 1/s Ssat ) supersaturation ShR ) Sherwood number for particle dissolution (based on radius) t ) time on penetration element contact time scale, s tc ) contact time of a penetration element (Higbie’s model), s T ) temperature, K VL ) volume of liquid phase, m3 x ) distance from gas-liquid interface in penetration element, m XB ) conversion of hydroxyl ion
English
Greek Letters
a ) size of product particles (coordinate), m ao ) size of fresh nucleate, m aGL ) gas-liquid interfacial area, m2
νo ) volume of fresh nucleate, m3 η ) same as η(R,x,t); particle-size distribution function, 1/m4 µ ) viscosity of aqueous medium, kg/(m s)
The theoretical PSDs, also shown in Figure 14, have been computed from the proposed theory using the “best’’ parameter values, as determined by the procedure discussed earlier and the results shown in Table 5. For such a complex system, the proximity of the experiment and the theory are remarkable, especially given that the regressed values of kG and ao have been obtained by fitting, not the PSDs, but the mean particlesize data (see Figure 7). Figure 15 shows some relevant and related computed quantities. Graph (A) shows the bulk particle number density, Nb0, versus batch time, θ. It is apparent that the fresh particle production rate ceases to be significant after ∼100 s, though of course the volume of solid precipitate formed continues to increase with time (Graph C). The bulk supersaturation peaks at ∼50 s and declines to low values at ∼300 s (Graph D). It is interesting to note that the hydroxyl ion exhausts at ∼300 s, when the (cumulative) fraction of precipitate formed in the bulk has almost reached 99%; only initially does significant precipitation occurs in the film region (Graph B). Conclusions The experimental data for the precipitation of calcium carbonate formed by the carbonation of lime solutions in a flat interface reactor shows that the number concentration and size of the particles increases with batch time. At later times, agglomerates start appearing. Given the sizes of particles that are observed to form in actual experiments, the particle diffusivity has been shown to have an insignificant effect on the particle-size distributions of the precipitating product. This implies that the particles formed near the gas-liquid interface can move out of this region into the bulk liquid only by convective motion of the penetration elements that constitute the gas-liquid boundary, and therefore, such phenomena can be modeled only by using models that incorporate this feature. The significant observations, with respect to the effect of different operating parameters, are that (1) As the agitation intensity is raised (and so is the mass transfer coefficient), smaller mean particle sizes are observed at short batch times, but this trend reverses at large batch times. (2) With increasing partial pressure of the gaseous solute, larger mean sizes appear, and this is valid at all batch times. (3) A decrease in the liquid-phase reactant concentration leads to larger mean sizes at large batch times, but the reverse trend is observed at earlier times. It can be generalized that usually the nucleation process dominates the interfacial region, while crystal growth is the dominant feature in the bulk phase. Therefore, operating parameter changes that shift the locale of precipitation to the bulk will ultimately promote growth (larger particles), whereas a shift into the interfacial region will produce the opposite effect. The experimental data on mean size of the product particle versus time can be fitted well to the computational results of the proposed theory. The best-fit values of the model parameters thus produced, when used to compute the detailed PSDs, result in computed values that are in remarkable proximity with the experimentally determined distribution.
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Fp ) density of solid product, kg/m3 θ ) batch time (coordinate), s Superscripts b ) bulk i ) pertaining to interface in ) initial Subscripts A ) pertaining to species A B ) pertaining to species B D ) pertaining to species D E ) pertaining to species E Literature Cited (1) Yagi, H.; Iwazawa, A.; Sonobe, R.; Matsubara, T.; Hikita, H. Crystallization of Calcium Carbonate Accompanying Chemical Absorption. Ind. Eng. Chem. Fundam. 1984, 23, 153-158. (2) Kotagi, Y.; Tsuge, H. Reactive Crystallization of Calcium Carbonate by Gas-Liquid and Liquid-Liquid Reactions. Can. J. Chem. Eng. 1990, 68, 435-442. (3) Yagi, H.; Nagashima, S.; Hikita, H. Semibatch Precipitation Accompanying Gas-Liquid Reaction. Chem. Eng. Commun. 1988, 65, 109119. (4) Wachi, S.; Jones, A. G. Mass Transfer with Chemical Reaction and Precipitation. Chem. Eng. Sci. 1991, 46, 1027-1033. (5) Wachi, S.; Jones, A. G. Effect of Gas-Liquid Mass Transfer on Crystal Size Distribution During Batch Precipitation of Calcium Carbonate. Chem. Eng. Sci. 1991, 46, 3289-3293. (6) Wachi, S.; Jones, A. G. Dynamic Modeling of Particle Size Distribution and Degree of Agglomeration During Precipitation. Chem. Eng. Sci. 1992, 47, 3145-3147. (7) Al-Rashed, M. H.; Jones, A. G. CFD Modeling of Gas-Liquid Reactive Precipitation. Chem. Eng. Sci. 1999, 54, 4779-4784. (8) Rigopoulos, S.; Jones, A. G. Dynamic Modeling of a Bubble Column for Particle Formation via a Gas-Liquid Reaction. Chem. Eng. Sci. 2001, 56, 6177-6184. (9) Doraiswamy, P. V.; Sharma, M. M. Heterogeneous Reactions; Wiley: New York, 1984.
(10) Hostomsky, J.; Jones, A. G. A Penetration Model of the GasLiquid Reactive Precipitation of Calcium Carbonate Crystals. Trans. Inst. Chem. Eng. 1995, 73, 241-245. (11) Venugopal, B. V.; Mehra, A. Gas Absorption Accompanied by Fast Chemical Reaction in Water-in-Oil Emulsions. Chem. Eng. Sci. 1994, 49, 3331-3336. (12) Wallin, M.; Bjerle, I. The Use of the Penetration Model for the Dissolution of Limestone in Carbon Dioxide-Water System. Chem. Eng. Commun. 1990, 91, 91-111. (13) Koutsoukos, P. G.; Kontoyannis, C. G. Precipitation of Calcium Carbonate in Aqueous Solutions. J. Chem. Soc., Faraday Trans. 1984, 80, 1181-1192. (14) Juvekar, V. A.; Sharma, M. M. Absorption of Carbon Dioxide in a Suspension of Lime. Chem. Eng. Sci. 1973, 28, 825-837. (15) Sada, E.; Kumazawa, H.; Lee, C. H. Chemical Absorption into Concentrated Slurry. Chem. Eng. Sci. 1984, 39, 117-120. (16) Mehra, A. Gas Absorption in Reactive Slurries: Particle Dissolution Near Gas-Liquid Interface. Chem. Eng. Sci. 1996, 51, 461-477. (17) Danckwerts, P. V. Gas-Liquid Reactions; McGraw Hill: London, 1970. (18) House, W. A. Kinetics of Crystallization of Calcite from Calcium Bicarbonate Solutions. J. Chem. Soc., Faraday Trans. 1981, 77, 341-359. (19) Packter, A. The Precipitation of Sparingly Soluble Alkline-Earth Metal and Lead Salts: Nucleation and Growth Orders During the Induction Period. J. Chem. Soc. 1968, 859-862. (20) Perry, H. P.; Chilton, C. H. Chemical Engineers’ Handbook; McGraw Hill: Tokyo, 1973. (21) Jones, A. G.; Hostomsky, J.; Zhou, L. On the Effect of Liquid Mixing Rate on Primary Crystal Size During the Gas-Liquid Precipitation of Calcium Carbonate. Chem. Eng. Sci. 1992, 47, 3817-3824. (22) Carr, F. P.; Frederick, D. K. Calcium carbonate. In Encyclopedia of Chemical Technology, Vol. 4; Wiley-Interscience: New York, 1996. (23) Mehra, A. Intensification of Multiphase Reactions through the Use of MicrophasesI. Chem. Eng. Sci. 1988, 43, 899-912.
IE0605128 ReceiVed for reView April 24, 2006 ReVised manuscript receiVed October 30, 2006 Accepted November 20, 2006 IE0605128
