Modeling the Precipitation of l-Glutamic Acid via ... - ACS Publications

CRYSTAL GROWTH & DESIGN 2005 VOL. 5, NO. 3 845-854

Articles Modeling the Precipitation of L-Glutamic Acid via Acidification of Monosodium Glutamate A. Borissova, Y. Jammoal, K. H. Javed, X. Lai, T. Mahmud,* R. Penchev, and K. J. Roberts Institute of Particle Science and Engineering, School of Process, Environmental and Materials Engineering, The University of Leeds, Leeds LS2 9JT, UK

W. Wood Syngenta, Global Specialist Technology (SW6), P.O. Box A38, Leeds Road, Huddersfield, HD2 1FF, UK Received August 9, 2004

ABSTRACT: A new insight for quantitative determination of the complex interaction of kinetics of reactive crystallization is gained by a study of the acidification of monosodium glutamate with hydrochloric acid in a semibatch reactor. The dynamic equilibrium of the L-glutamic acid ions, their simultaneous presence in the solution, and the effect on the crystal size and size distribution are simulated using a mathematical model based on integration of the population balance equation with the crystallization kinetics and thermodynamics of the process. The model includes equations of chemical reactions, pH, nucleation, size-independent growth, and population balance. This allows a precise determination of the driving force for the reactive crystallization and the parameters of the nucleation and crystal growth rate expressions. The model considers the effect of the controlling parameters such as the rate of addition of the acid and the initial concentrations of ionic species and can be used to predict the concentrations in the solution, the pH, and the crystal mass. The effect of process conditions on the crystal size, crystal size distribution, and morphology is studied. Introduction Crystallization and precipitation processes via the acidification of a salt are characterized by complex interactions between thermodynamic, kinetic, and hydrodynamic effects at both micro- and macro-level that make its mathematical modeling a challenging task. The integration of the population balance equation (PBE) for the modeling of particulate formation with the complex chemistry of the precipitating systems is an additional difficulty. This equation represents the balance of the number of particles in the system, and it must be solved together with the classical material balance of species considering the hydrodynamic, thermodynamic, and kinetic relationships. The development of robust mathematical models of precipitation processes is essential for their efficient control and scale-up. The effect of mixing, mainly micromixing, on precipitation processes has been extensively * To whom correspondence should be addressed. Fax: +44 (0)113 3432405. E-mail: [email protected].

studied by Villermaux et al.,1,2 Fitchett & Tarbell,3 Leeuwen et al.,4 Baldyga & Bourne,5 Garside & Tavare,6 and Jones et al.7,8 Two concepts have been adopted in modeling crystallization depending on the mixing models used. The first concept assumes ideal mixing conditions i.e., perfect mixing,1,2,9-13 whereas the models of the second group integrate more complex hydrodynamic models with the crystallization kinetics14-17 [zonal models,6-8 computational fluid dynamics (CFD) integrated,14,16 or a combination of zonal and CFD models8,17]. The PBE is the core part of the overall precipitation model. The concept of the population balance has been applied to modeling crystallization by Randolph and Larson.18 In their landmark textbook,18 Randolph and Larson codified and popularized the population balance modeling approach for the simulation of crystallization processes. Most mathematical models,1-17 developed for precipitation processes, have been directed toward the study and control of inorganic precipitating systems. There

10.1021/cg030087a CCC: $30.25 © 2005 American Chemical Society Published on Web 04/16/2005

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Borissova et al.

Figure 1. Photographs of (a) R and (b) β crystals of LGA. The R-form is metastable and prismatic in shape, whereas the needlelike β-form is stable.

are only a few works on precipitation of organic materials11,12 presumably reflecting the difficulties associated with modeling complex solution chemistry. The precipitation model developed in this study simulates the precipitation of L-glutamic acid as a result of acidification of monosodium glutamate (MSG) with hydrochloric acid (HCl) in a fully automated semi-batch stirred tank reactor. The model represents the semi-batch process as a sequence of batch subprocesses conducted between two consecutive additions of HCl. The full ionic species mass balance is also included in the model. Modeling Precipitation of L-Glutamic Acid via Acidification L-Glutamic acid (LGA) is an amino acid mainly used in medicine and biochemical research and is known to have two polymorphic forms, R and β.19 The R-form is prismatic in shape and is known as the metastable form, whereas the needlelike β-form is thermodynamically stable.20 Photographs of R- and β-form crystals of LGA are presented in Figure 1. Ionic Equilibrium Modeling. LGA has three sites for protonation/deprotonation.21 The amine group has a pKa of about 9.7, i.e., at this pH there are equal proportions of -NH3+ and -NH2 species. The carboxylic acid group at the other end of the molecule has a pKa of 4.57, and the carboxylic acid group next to the amine group has a pKa of 2.17. Therefore, if the pH is lowered (below 2.2), the molecule will be mainly in the fully protonated form with an overall charge of +1 (see Figure 2). At pH between 2.2 and 4.6, it will be mainly the

Figure 2. Mole fractions of different ionic species as a function of pH of the solution. RH+1, RH, RH-1, RH-2 are the mole fractions of the LGA ions in the aqueous solution: RH-2 is the deprotonated form with an overall charge of -2, RH-1 is the deprotonated form with an overall charge of -1, RH is the zwitterion with an overall charge of zero, and RH+1 is the fully protonated form with an overall charge of +1.

zwitterion, i.e., with one negative and one positive charge, with an overall charge of 0. However, close to pH 2.2, there will still be relatively high levels of the +1 form, and closer to pH 4.6 there will be relatively high levels of the -1 form, together with a small proportion of the +1 form.22 This means that if the pH is lowered the LGA molecule becomes protonated i.e., it accepts an H+ ion onto the amine group. If the pH is

Scheme 1

Modeling the Precipitation of L-Glutamic Acid

Crystal Growth & Design, Vol. 5, No. 3, 2005 847

increased the molecule becomes deprotonated, i.e., it loses an H+ ion from one of the acid groups. The isoelectric point, i.e., where the species has an overall zero charge, comprising the zwitterion with equal but smaller amounts of the +1 and -1 forms, is midway between the two pKa values, i.e., at pH 3.37. Thus, with changing pH one or another of the four LGA ions becomes dominant, but one or more are also present in the solution. Chemical Reaction Modeling. LGA has three sites for protonation/deprotonation, and its acid/base chemistry is summarized in Scheme 1. After each drop of HCl is added to the system the following equilibria are assumed to be achieved and the reaction scheme of the precipitation may be written as ka

RH-2 + H+ [\] RH-1(MSG) 3

2

ka

1

+

kAB )

[A][H ] [B]

(2)

where kAB and kBA are the equilibrium constants of the forward and the reverse reaction, respectively. According to the Le Chaˆtelier’s principle, addition or removal of one of the components will cause the equilibrium to reestablish itself. If HCl is added, the reaction will shift to the right to relieve the stress. As a result, the initial concentration of A will decrease by a certain amount, X, the initial concentration of B will increase by the same amount, and the initial concentration of H+ will change by [H+]HCl - X. So, the equilibrium concentrations are H+

B

initial change equilibrium

[A]o -X [A]o - X

[H+]HCl - X [H+]o + [H+]HCl - X

[B]o +X [B]o + X

[H+]

o

Substituting these values into the equilibrium constant expression (eq 2) and solving for X gives X)

(4)

Amino acids are similar to the other polyprotic acids, which ionize stepwise, and an equilibrium constant that can be written for each step. Therefore, the equilibrium concentrations of the individual species can be calculated by means of fractions of dissociating species, Ri (i ) 0, 1, 2, 3), which can be derived for any polyprotic acid.23

ka1[H+]2

(1)

where RH-2 is the deprotonated form with an overall charge of -2, RH-1 is the deprotonated form with an overall charge of -1, RH is the zwitterion with an overall charge of zero, RH+1 is the fully protonated form with an overall charge of +1, and kai (i ) 1, 2, 3) are the respective equilibrium constants. Henceforth, the concentration of the zwitterion in the solution is denoted as [RH]. The reaction of species A with the hydrogen ion to give a species B is represented as follows:

A

[H+] ) [H+]0(solution) + [H+]HCl(added) - X

[RH] ) CRHR1 )

Ksp

RH(GA)solution [\] RHsolid

BA

The hydrogen ion concentration of the solution at equilibrium can be calculated from

[H+]3 (5) CRH + 3 [H ] + ka1[H+]2 + ka1ka2[H+] + ka1ka2ka3

RH(GA)solution + H+ [\] RH+1

kAB

0 < X < [A]0

[RH+] ) CRHR0 )

ka

RH-1(MSG) + H+ [\] RH(GA)solution

xB A + H+ w\ k

where [H+]0,HCl ) [H+]0 + [H+]HCl, and

[A]0 + [H+]0,HCl + kAB ( 2

x{[A]0 + [H+]0,HCl + kAB}2 - 4{[A]0[H+]0,HCl - [B]0kAB} 2

(3)

CRH + 3 [H ] + ka1[H+]2 + ka1ka2[H+] + ka1ka2ka3 ∆[RH] (6) [RH-1] ) CRHR2 ) ka1ka2[H+] (7) CRH + 3 [H ] + ka1[H+]2 + ka1ka2[H+] + ka1ka2ka3 [RH-2] ) CRHR3 ) ka1ka2ka3 (8) CRH + 3 [H ] + ka1[H+]2 + ka1ka2[H+] + ka1ka2ka3 where CRH is the total concentration of the four species. After adding HCl, this concentration can be calculated as follows:

CRH(after adding HCl) ) ([RH-2] + [RH-1] + [RH] + V0 [RH+1]) (9) V0 + Ft where Vo is the initial volume of solution in the reactor, F is the flow rate of HCl, and t is the time. The term ∆[RH] in eq 6 represents the decrease in concentration of the species [RH] as a consequence of its precipitation whenever this concentration is above the supersaturation. The values of the equilibrium constants ka1, ka2, ka3 (eqs 5-8) can be calculated from the values of the corresponding pKa constants pKa1, pKa2, pKa3 using the relation: pKa ) -log(ka). The following values for the constants have been used in this work:24 pKa1 ) 2.2, pKa2 ) 4.25, pKa3 ) 9.67. The corresponding values for the equilibrium constants, calculated using the above formula, are ka1 ) 6.31 × 10-3, ka2 ) 5.62 × 10-5, ka3 ) 2.14 × 10-10.

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Figure 3. Titration of LGA and pH calculations.26 If a strong acid is added to a solution of LGA, two buffer regions and two equivalent points, correspondingly, are established: first, consisting of anion (-2)/anion (-1); second, consisting of anion (-1) and zwitterion.

Buffer Calculations. The Henderson-Hasselbach equation25 was used to calculate pH:

pH ) pKa + log

[base] [acid]

(10)

The numerator [base] in the log function is either the free base or the conjugate base, while the denominator [acid] is either the free acid or the conjugate acid. This is explained in details below. Since at the point of precipitation it is the un-ionized (or undissociated or free) LGA molecule that precipitates out of solution, the denominator [acid] in eq 10 is equal to the solubility of the un-ionized form of the acid (C/RH). Calculated26 values of pH using eq 10 are given in Figure 3. If a strong acid is added to a solution of LGA, e.g., anionic (-2), a buffer region is first established consisting of anion (-2)/anion (-1). The hydrogen ion concentration can be calculated from the relation of equilibrium constant:

[RH-1] [H+] ) ka3 [RH-2]

(11)

Taking the negative logarithm of each side of this equation yields the Henderson-Hasselbach equation:

pH ) pKa3 + log

[RH-2] [RH-1]

[RH-1] [RH]

pH ) pKa1 + log

(12)

(13)

At the point of precipitation, the amount of conjugate base is equal to the concentration of [RH-1] species. The amount converted to the free acid is equal to the

[RHL] [RH+1]

(14)

Modeling Crystallization Kinetics. Particulate matter produced by crystallization has a size distribution over a specific size range. A crystal size distribution (CSD) is most commonly expressed as the distribution of number or mass of crystals over the size range. The two distributions are related and affect many aspects of crystal properties and downstream processing, including appearance, purity, solid-liquid separation, dissolution, drying, milling, etc. Population density n is defined as the number of crystals with specific size L in the unit volume of the system. If ∆N denotes the number of crystals per unit system volume in the size range from L to L + ∆L, then

n ) lim

At the first equivalence point, a solution of anion (-1) exists, and [H+] ) xka3ka2. Beyond this point, an anion/ zwitterion buffer exists; and the pH is given by

pH ) pKa2 + log

/ ) i.e., the precipitation solubility of the free acid (CRH 2 starts when [RH] is equal to C/RH in eq 13. At the second equivalence point, a solution of zwitterions exists, and the concentration of hydrogen ion is given by [H+] ) xka1ka2. After this point, another buffer region is established that consists of zwitterions/ cations; the pH can be calculated by

∆Lf0

∆N dN ) ∆L dL

(15)

The total population density denoted as n j is the number of particles with specific size within the total volume of the system. For a perfectly mixed batch crystallizer, in which it is assumed that crystals are born at size zero, crystal breakage and agglomeration are negligible, and crystal shapes are uniform, the PBE for size-independent growth rate is described by Randolph and Larson:18

∂(nV) ∂(nV) +G )0 ∂t ∂L

(16)



Figure 4. Schematic diagram showing the flowchart associated with the algorithm used to solve the equations of the mathematical model.

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Table 1. Process Conditions and Experimental Results for LGA Precipitation initial MSG concentration wt of MSG/ wt of water [%]

g of MSG/ L of water

mol of MSG/ L of water

amount of HCl added before onset of crystallization [g]a

pH of crystallization

onset of crystalization [min]

time of crystallization [min]

crystal form

total crystal mass [g]

5 10 20 30 40

50 100 200 300 400

0.296 0.591 1.183 1.774 2.365

31.23 34.36 40.57 50.56 52.2

4.36 4.78 5.14 5.37 5.49

62 69 81 101 104

88 80 158 88 75

R R R R R

40 50 95 75 65

a

Addition rate of 6 M HCl ) 0.5 g/min.

Table 2. Comparison between Experimental and Simulated Time of Onset of Crystallization, pH, and Total Crystal Mass for Solutions with Different Initial Concentrations of MSG crystal mass [g/L] wt % MSG 5 10 20 30 40

onset of crystallization [min] experimental predicted 62 69 81 101 104

pH at the onset of crystallization experimental predicted

57 58 60 62 64

4.36 4.78 5.14 5.37 5.49

4.38 4.82 5.18 5.37 5.51

where G is the crystal growth rate and V is the volume of the system. Because the working volume of a semi-batch crystallizer is time-varying, it is convenient to redefine the number density on the basis of the total operating volume of the system such that

n j ) nV

(17)

with boundary conditions:

n j (L,0) ) 0 (19)

where n0 is the population density of nuclei (zero-sized crystals). A finite-difference method27 is applied to discretise eq 18. The two derivates are approximated as: t

t

(20)

t

jL - n j L-1 ∂n j ∆n j n f ) ∂L ∆L ∆L

(21)

Substituting eqs 20 and 21 into eq 18 yields the following algebraic equation for n j tL:

n j tL )

∆t Gn jt ∆L L-1 ∆t G 1+ ∆L


time [min]


150 149 239 189 179

40 50 95 75 65

150 149 239 189 179

48 56 106.8 78.5 73.9

B ) kb∆Cb

(23)

G ) kg∆Cg

(24)

where kb is the nucleation rate constant, and ∆C is given by

n j t-1 L +

(22)

where ∆t and ∆L are the time and size increments, respectively. The constitutive equations required are the nucleation and growth rates. They are conventionally written in

(25)

The exponent b is often referred to as the order of nucleation, and it depends on the physical properties of the system. kg is the growth rate constant, which is a function of temperature, and the exponent, g, expresses the dependence of the crystal growth rate on the supersaturation. It should be noted that values of these kinetic constants are not available for the present chemical system. The nucleation and crystal growth consume the LGA ions, which reduces its concentration in the solution as shown in eq 6. The decrease in concentration can be expressed as

t-1

jL - n jL ∂n j ∆n j n f ) ∂t ∆t ∆t

time [min]

∆C ) [RH] - C/RH (18)

n j (0,t) ) n0

predicted

terms of supersaturation (∆C) as

With this substitution, eq 16 reduces to

∂n j ∂n j +G )0 ∂t ∂L

experimental

-

KvFcryL03 KAFcry ∆[RH] ) -B -G ∆t Mcry 2Mcry

∫0L

max

L2n(L)dL (26)

where ∆[RH]/∆t is the change of LGA concentration due to crystallization, Kv is the volume shape factor, KA is the area shape factor, Lo is the size of nuclei, and Fcry and Mcry are the density and the molecular weight of crystals, respectively. The model is completely defined with the above eqs 3-9 and 22-26, and the algorithm used to solve these equations is given in Figure 4. The parameters kb, kg, b, and g can be determined from the experimental data of concentration and supersaturation, crystal size, and size distribution by minimizing the error between the predicted and experimental data. The number of crystals, the cumulative length, and the mass of the crystals in a finite range from L1 to L2 were obtained18,28,29 from



the zero, first, and third moments of the population density function, n:

N(L1,L2) ) Lcum )

∫LL

∫0L

2

1

2

n(L)dL

(27)

Ln(L)dL

(28)

∫LL

(29)

M(L1,L2) ) FcryKv

2

1

L3n(L)dL

This overall precipitation model is based on the following assumptions: (i) The crystallizer is well mixed, and hence supersaturation is constant throughout the crystallizer. (ii) The crystallizer is under isothermal conditions, and temperature variations during operations are minimal and have a negligible influence on either the mass in the system or the CSD. (iii) A crystal free feed enters the crystallizer at the same temperature as the contents of the crystallizer. (iv) The impeller speed is constant during the process. (v) No agglomeration, breakage, and product classification occur in the crystallizer. (vi) Nuclei are born only at size zero. (vii) All crystals are of uniform shape, and the growth rate G is the same for all crystal sizes. A detailed description of the mathematical model and the solution technique may be found in ref 30. Experimental Work for Model Verification. Experiments were carried out using an HEL Simular reactor system comprising a 2-L double jacketed glass reactor, a Julabo FP50-HD thermostated bath for reactor temperature control, a data interface board, and process control software (WIN-ISO, HEL Ltd.). Reactor stirring was provided using a stainless steel pitched blade stirrer with four blades at 45° rotating at a constant speed of 250 rpm. Solution temperatures were measured using a platinium resistance thermometer (PT100), and transmittance (turbidity) was determined using an in-house built turbidometric fiber optic probe for detection of the onset of crystallization. A diaphragm pump was used for feeding 6 M HCl at a constant rate of 0.5 g/min into MSG solution with five different initial concentrations. The temperature in the reactor was held constant at 25 °C. The quantity of the acid needed to initiate crystallization in the reactor was measured. The experimental facility and procedure have been fully described in ref 31. As a result of addition of HCl to the reactor, the pH of the solution droped, and after a certain quantity of HCl was added (in about 62-104 min), the solution became supersaturated with respect to LGA and crystallization started. During all the experiments only the metastable R-form was obtained. The process conditions and experimental results are presented in Table 1. The total mass of the crystals was measured after each experiment, and pH was recorded throughout the experiment. Results and Discussion A Visual Basic code for the solution of the mathematical model was developed and applied to simulate the precipitation process using different initial concentra-

Figure 5. Experimental and simulated pH vs time for solutions with different initial concentrations of MSG: (a) 5 wt % (b) 10 wt % (c) 20 wt %.

tions of MSG (See Table 1). The code uses the following input data: initial concentration of MSG and HCl, initial pH and volume of the solution, rate of addition of HCl, and temperature of the system. The values of all the parameters of the nucleation, b and kb, and crystal growth, g and kg, models (eqs 23 and 24) were determined based on the experiment carried out using 10 wt % initial MSG concentration so that they minimized the error of prediction of the measured total crystal mass, pH, and the time of the onset of precipitation. The following values of the nucleation and growth rate parameters were obtained: B ) 107∆C3 and G ) 5 × 10-7∆C1.5. These values were used for the simulations of the remaining four experimental cases. The predicted time of the onset of precipitation, pH, and total crystal

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Figure 6. Predicted ionic species of LGA in a solution of 10 wt % MSG. RH+1, RH, RH-1, RH-2 are the mole fractions of the LGA ions in the aqueous solution.

Figure 7. Variation of measured turbidity and predicted crystal mass with time for 20 wt % solution of MSG.

Figure 8. Comparison between the predicted and measured pH of precipitation as a function of the initial concentration of MSG.

mass for all of the five experimental cases are compared with the measured data in Table 2. Prediction of the Onset of Crystallization. It was found that the predicted onset of crystallization was earlier than that observed experimentally. The predicted times of the onset of crystallization given in Table 2 correspond to a predicted mass of crystals 0.5 g/L and number of crystals in the range of 10+10 per liter. The deviation between the measured and predicted time of the onset of crystallization increases with increasing initial concentration of MSG and is in the range 5-40 min. These differences between the experimental and predicted values may be due to

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(i) the above limits of mass and number of crystals per liter set as the criteria for the onset of crystallization in the numerical simulations being too small to be detected by the turbidity probe; (ii) the need to determine the driving force of the process taking into account the metastable zone width (MSZW) of crystallization. Prediction of pH. The pH profiles predicted by the model and obtained from measurements for three experiments with different initial concentrations of MSG solution are shown in Figure 5. As can be seen, the pH value decreases with increasing time due to the protonation of MSG ion to LGA ion from the HCl added to the reactor. At the onset of crystallization the pH increases due to the removal of LGA ions from the solution and their deposition as solids. Both the experimental and predicted results show an increase of the pH at the onset of crystallization. With further addition of HCl the pH starts to decrease again. Prediction of Species Concentration and Crystal Mass. The concentrations of the four ionic species of LGA are predicted by the model, and their variations with pH are presented in Figure 6. Initially, the concentration of LGA in solution increases until saturation is reached when the first crystals start to appear. A sudden drop in the LGA concentration is observed at that moment due to the deposition of LGA mass in crystal form. After this point the concentration of the RH species (LGA) remains constant and later starts to decline slightly. Comparison of Figure 6 with Figure 2, which does not take into account the precipitation and the variation of the volume due to the addition of HCl, shows a significant change of the LGA concentration profile due to the precipitation. The predicted mass of crystals formed and measured variations of turbidity with time for an initial MSG concentration of 20 wt % are presented in Figure 7. The total mass of crystals produced at the end of each experimental run is compared with that obtained from the model in Table 2. The crystal mass is overpredicted for all the experimental runs. The errors of prediction are in the range 4-20%. The difference between the measured and predicted values of the crystal mass may be due to the loss of materials during the experiments. Prediction of pH of Precipitation. The calculation of the pH of precipitation for a given solution of LGA was performed using eq 13. It uses the values for pKa2, the solubilitity of the free or undissociated acid [RH], and the concentration of the -1 ion [RH-1]. The values of pH of precipitation predicted by the model and the experimental data for five experiments with different initial concentrations of MSG solution are presented in Table 2. The predicted and measured variations of pH with initial concentration of MSG solution are also shown in Figure 8. The predicted results show a good agreement with the experimental data. Prediction of Crystal Properties. The results obtained from the solution of the PBE are presented in Figure 9a,b, which shows the population density as a function of time and the size of the crystals. Figure 9a also shows how the peaks of the population density curves shift from a smaller to bigger size, thus representing the crystal growth. At the beginning of the crystallization process, nucleation is dominant, and thus



Figure 9. Change of population density n(L) with size (L) and time: (a) 10 wt % and (b) 20 wt % solution of MSG.

the population density curves decrease. As a result of crystal growth, the peaks of these curves shift toward a larger size. Figure 10 shows the comparison between the measured and predicted CSD for 10 and 20 wt % MSG solutions. From the experimental data, the crystals were found to be within a narrow range of 100300 µm with a mean size of about 150 mm. The predicted CSD for the same solutions of MSG showed a wider range of crystal sizes (0-900 mm) and a mean crystal size of about 250 mm. This discrepancy may reflect the neglect of the crystal breakage in the process model. Conclusions A mathematical model of precipitation of LGA via acidification of MSG with HCl in a semi-batch reactor has been developed in this study. The model includes equations for a detailed chemical reaction mechanism between MSG and HCl, nucleation, size-independent crystal growth, and population balance. An experimental verification of the model results was carried out

Figure 10. CSD functions from experimental data for the runs using 10 and 20 wt % solutions of MSG.

using a computer controlled 2-L semi-batch reactor. The model calibration and validation data sets were used

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to determine the unknown nucleation and growth rate model parameters and to verify the simulated results. The simulation results for pH, crystal mass, and crystal size obtained using the estimated model parameters show good agreement with the experiments. The precipitation process model developed in this work can be extended to include the equations of crystal breakage and agglomeration and also the energy balance equation to account for the heat of reaction, which would give a more accurate representation of the acidification process. An optimization of the values of the nucleation and crystal growth rate model parameters should also be considered in the future work. Acknowledgment. We are grateful to the EPSRC for their specific funding, through Ropa Research Grant (GR/N/20300), of this research in reactive precipitation and for the financial support of one of us (A.B.). In addition, we acknowledge EPSRC and associated industrial sponsors for their support of the in-process batch engineering laboratory at Leeds. The authors are indebted to Dr. R. B. Hammond, University of Leeds, for useful discussions. Notation B b C ∆C F G g KA kai,kAB kb kg KV L M Mcry N n n j n0 pKa T t Vo

nucleation rate order of nucleation concentration concentration driving force flow rate of HCl crystal growth rate order of crystal growth area shape factor equilibrium constants nucleation rate constant crystal growth rate constant volume shape factor characteristic crystal size mass of crystals per unit volume molecular weight number of crystals population density total population density population density of nuclei solubility product temperature time initial volume of solution

Greek letters density of LGA crystals Fcry Subscripts * saturation value

References (1) Marchal, P.; David, R.; Klein, J.; Villermaux, J. Chem. Eng. Sci. 1988, 43, 59-67. (2) Franck, R.; David, R.; Villermaux, J. Chem. Eng. Sci. 1988, 43, 69-77.

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