Reaction crystallization in strained fluid films

Chemical Engineering Science 56 (2001) 3257}3273

Reaction crystallization in strained #uid "lms Magnus Lindberg, As ke C. Rasmuson* Department of Chemical Engineering and Technology, Royal Institute of Technology, Teknikringen 28, SE-100 44 Stockholm, Sweden Received 9 December 1999; received in revised form 4 August 2000; accepted 7 November 2000

Abstract The detailed conditions during the ultimate stage of micromixing of the reactants in a reaction crystallization process are analysed. A mathematical model is developed to describe mass transfer, chemical reaction, and crystallization of a molecular compound in strained lamellar structures of reactant solutions inside the smallest vortices. The numerical calculations show that the supersaturation varies signi"cantly in space and time, and suggest that signi"cant crystallization may occur inside these vortices in the case of low-soluble and sparingly soluble compounds. At the end of the vortex lifetime, the crystal size distribution is quite dependent on the properties of the system and on the processing conditions. The number of crystals generated correlates strongly to the maximum supersaturation occurring during the vortex lifetime, and this maximum supersaturation is as a "rst approximation well described by simpli"ed mass transfer models where crystallization is neglected. Often a signi"cant supersaturation remains at the end of the vortex lifetime and the size of the crystals leaving the vortex is determined by the growth rate rather than by nucleation and mass constraint. The mean size is usually larger than the limiting size for Ostvald ripening in the bulk and the size distribution is quite narrow. The results show that neglect of the detailed conditions in reaction crystallization of a molecular compound may not be justi"ed.  2001 Elsevier Science Ltd. All rights reserved. Keywords: Reaction crystallization; Precipitation; Modelling; Micromixing; Mass transfer; Population balance

1. Introduction Crystallization is widely used in the chemical and pharmaceutical industries for separation and puri"cation. In reaction crystallization (precipitation) supersaturation is generated by the production of the compound of interest through a chemical reaction. Precipitation is used in the production of certain bulk chemicals, but is especially widely used in the production of pharmaceuticals, biochemicals, agrochemicals, pigments, catalysts, photographic materials, and advanced ceramic precursors. Reaction crystallization involves mixing, chemical reaction and crystallization processes such as nucleation and crystal growth. Often the chemical reaction and the crystallization processes are rapid compared to the rate of mixing of the reactants. Thus, crystallization often starts already at the feed point under conditions of segregation, and the local mixing conditions will in#uence the

* Corresponding author. Tel.: #46-8-790-8227; fax: #46-8-1052-28. E-mail address: [email protected] (As . C. Rasmuson).

properties of the crystalline product. Product crystal mean size and size distribution determine "ltration properties and in#uence on further downstream processing such as drying, granulation, and packaging. The product crystal mean size is determined by the number of particles that shares the total mass that is crystallized and is hence determined by the rate of nucleation during the crystallization process. Primary nucleation usually dominates in reaction crystallization because of the high levels of supersaturation and the very non-linear dependence of primary nucleation on supersaturation. Hence, this explains why the feed point conditions often signi"cantly in#uence the product properties (e.g. Tovstiga & Wirges, 1990; Marcant & David, 1991, 1993; As slund & Rasmuson, 1992; Franke & Mersmann, 1993, 1995; David & Marcant, 1994; Philips, Rohani, & Baldyga, 1999), although the published experimental results are sometimes contradictory concerning the e!ects of di!erent process parameters. In the mixing process, inertial forces divide the solution into smaller volume elements that are distributed and deformed. The size of these elements is continuously reduced. Tongues of reactant #uid engulf the surrounding #uid through the action of vorticity. Viscous forces

0009-2509/01/$ - see front matter  2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 0 0 ) 0 0 5 2 3 - 6

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further reduce the scale of segregation by straining, and molecular di!usion becomes important. The "nal stage of mixing, often referred to as micromixing, can be described as straining of and di!usion in a lamellar structure of "nite reactant "lms (Baldyga & Bourne, 1986). This lamellar structure has a lifetime often referred to as the vortex lifetime, which can be estimated by (Baldyga & Bourne, 1986)





 t "12.7 . + 

(1)

t varies in water (
"10\ m/s) from 0.1 to 0.0004 s + when the speci"c energy dissipation rate, , varies from approximately 0.01 to 1000 W/kg. In comparison, induction time measurements (Mohanty, Bhandarkar, Zuromski, Brown, & Estrin, 1988; As slund & Rasmuson, 1992) show that the formation of detectable crystals can occur within fractions of milliseconds. Hence, it is quite possible that nuclei are generated already in the smallest primary vortices at the feed point where reactants make contact, and accordingly the conditions in these vortices can be of importance for the mean size of the product crystals. Marcant and David (1991) pointed out that especially the early moments of a semibatch reaction crystallization process are very important for the "nal product. Reaction crystallization has been modelled taking micromixing into consideration (Pohorecki & Baldyga, 1983, 1985, 1988; Baldyga, Pohorecki, Podgorska, & Marcant, 1990; Marcant & David, 1991; David & Marcant, 1994; Baldyga, Podgorska, & Pohorecki, 1995; Chen, Zheng, & Chen, 1996; Baldyga & Orciuch, 1997; Philips et al., 1999). The micromixing is described by the Engulfment model (E-model) (Baldyga & Bourne, 1989) or by the Interaction by Exchange with the Mean model (IEM-model) (Harada, Arima, Eguchi, & Nagata, 1962; Costa & Trevissoi, 1972; Villermaux & Devillon, 1972) or by models that are similar. However, in these models the micromixing is described as an exchange of mass between a #uid element and the surroundings, and the detailed concentration pro"les that develop in the lamellar structures of alternating "lms of reactant solutions, are not considered. We have shown previously (Lindberg & Rasmuson, 1999b), for a molecular compound, that although the gradients may level out within the vortex lifetime, before that the supersaturation varies signi"cantly in space as well as in time. The evolution of the supersaturation (both in space and time) depends sensitively and in a complex way on reactant concentrations and di!usivities, and cannot be described by a simple mean value. In addition, in some cases the gradients do not level out during the vortex lifetime. In computational #uid dynamics (CFD) modelling of precipitation, Leeuwen (1998) concluded that if homogeneous nucleation dominates,

neglect of small-scale #uctuations can introduce large errors. In the present work, a model over mass transfer, chemical reaction and crystallization during the "nal stage of mixing is developed. The conditions within the lamellar structures of alternating "lms of reactant solutions, and how these depend on mass transfer properties and crystallization kinetics as well as on processing conditions like concentration of reactant solutions and mixing intensity, are analysed. It is examined how far the crystallization may proceed before the initial vortex ceases to exist, and it is discussed to what extent the crystals born in these vortices may in#uence the overall size distribution of the process as a whole. In particular, the work addresses the case when a molecule is formed by the chemical reaction, even though some comments with respect to ionic systems are given in the discussion.

2. Modelling 2.1. The model The model considers the one-dimensional, unsteady molecular di!usion of reactants A and B, an instantaneous bimolecular chemical reaction to form product C: A#BPC product molecular di!usion, crystallization and particle di!usion in "nite strained "lms in contact in a lamellar geometry; see Fig. 1. Primary nucleation of crystals and crystal growth are accounted for. It is assumed that (i) concentrations are low so that bulk di!usion can be neglected, (ii) all di!usivities are constant with respect to space and time, and components behave as non-ionic molecules, (iii) the viscosity and density are constant in time and space, (iv) the crystals are in"nitely small compared to the supersaturation gradients, and (v) the crystal shape is constant. Precipitation of a product may occur either (i) via the formation of a sparingly soluble solute molecule that crystallizes into a solid product (A#BPCPP) or (ii) by direct integration of reactants into the crystalline lattice (A#BPP) (Villermaux, 1990; Klein & David, 1995). The former situation is usually the case in reaction crystallization of organic compounds in the "ne chemicals and pharmaceuticals industry and is the case focused upon in this work. The supersaturation is approximated by c /c . ! ! The governing equations describing reactant concentrations, product concentration, and the particle population density, expressed in a "xed length coordinate, x,

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Fig. 1. The mass transfer situation.

and real time, t, are

IC2: t"0, 0)x(d ,

c c c G #r , G # x G "D B x G x G t

(2)





  ! ! k ¸ B ;(t!t )#3k Gn¸ d¸ , 4  .  4 M !  n (Gn)  n n # x # " D B x ¸ x . x t

(3)

c c  "0, "0, x x

#B (¸!¸ );(t!t ) (4) .   with the su$x i" reactants A and B. The second term in each of these equations describes the convection transport due to straining of the "lms, where  x is the correB sponding convection velocity. The rate of strain,  , is B assumed to be spatially uniform. The relation between rate of strain and half-slab thickness, d, is expressed as  ln d  " . B t

(5)

The last term in Eq. (4) includes a Heaviside step function, and describes the fact that nucleation does not start instantaneously at the establishment of supersaturation, but that there is a nucleation time lag. For reasons of symmetry, only two adjacent half-reactant "lms are considered, i.e. 0)x)d #d . Initially,  reactant A is homogeneously distributed in the "nite "lm to the right and reactant B in the "lm to the left. Initially, there are no crystals present. The interface between the two reactant solutions is initially at x"d , where d is the half-slab thickness of reactant B's solution. The following initial and boundary conditions apply: c "c ,  

c n ! "0, "0, x x c c  "0, "0, x x

BC2: t'0, x"0,




IC1: t"0, d (x)d #d ,  c "0, c "c , ! !

c "c , ! ! BC1: t'0, x"d #d , 

c c c ! # x ! "D ! #r B ! ! x x t




c "0, c "c , 

n c ! "0, "0, x x BC3: 0(t(t

 




c  D  x

, x"x , Q


 c x

, VV>Q VV\Q BC4: 0(t(t , x )x)d #d , c "0,   Q  BC5: 0(t(t , 0)x)x , c "0,   Q  BC6: t*0, ¸"0, n"0, "!D

BC7: t*0, ¸PR,

n"0,

where x denotes the x-value of the location of the reacQ tion plane at every moment of time. BC3}BC5 describe the fact that the reaction plane has to be supplied by stoichiometric amounts of reactants and that the reactants cannot coexist due to the instantaneous nature of the chemical reaction. The chemical reaction behaves as a moving continuous source for product generation




c  r "(x!x )D ! Q  x

> Q

VV

"!(x!x )D Q


 c x

VV\Q (6)

and proceeds until one or both reactants are totally consumed (t"t ).  

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surface integration resistance and may be expressed as

2.2. Transformation Two steps of variable transformation are introduced. Firstly, the time and the length scales are transformed by the introduction of the dimensionless time  and the dimensionless length coordinate z (Ou & Ranz, 1983)



(7)

x z" d

(8)

"

RD  dt, d 

by which the convection term due to straining, i.e. the "lm thickness thinning, is fully integrated into the accumulation and production terms. Hence, besides the reaction term the problem is mathematically transformed into a simple di!usion problem with a moving boundary (the reaction plane at z "x /d) between two "xed Q Q boundaries, z"0 and 2. These "xed boundaries correspond to the centre lines of each of the two contacting "nite reactant solution "lms. Below the results are presented in these dimensionless variables. However, in order to proceed with the numerical solutions, two further transformations are introduced, one for each reactant "lm (Crank, 1984): z Z " , (9) z Q z!2 for z (z)2, Z " . (10) Q  z !2 Q By these transformations, the length scales on either side of the reaction plane are adjusted such that the centre lines of the two reactant "lms are located at Z "0 and Z "0, respectively, and the reaction plane becomes  "xed at Z "Z "1. Details of the transformed equa tions and numerical solution methods are given in the appendix. for 0)z(z , Q

3. Kinetics The kinetics of crystallization describes the rates of nucleation and crystal growth. The nucleation rate is described by the Gibbs-Volmer theory (Mullin, 1993)






K . B "K exp ! . (11) . . ln(c /c ) ! ! At least for certain parameter values and in certain regions of supersaturation, this equation describes an extremely non-linear dependence of nucleation rate on supersaturation (see further below). The crystal growth is the result of mass transfer over two resistances acting in serie: the boundary layer di!usion resistance and the






G E G"k (c !c )! (12) E ! ! k B if no consideration is given to the detailed mechanism of surface integration (i.e. power law expression). This equation becomes explicit in G when the order of the surface integration, g, equals 1 or 2. When particles are very small, the relative velocity between the particle and the #uid is negligible even in a turbulent "eld. In this case and for spherical geometry the Sherwood number approaches the value of 2 (Armenante & Kirwan, 1989), i.e. D < k " ! K. B r

(13)

The di!usivity of the particles, D , is a function of the . particle size, and is estimated by the Stoke}Einstein equation: k ¹ D " . . 3
¸

(14)

The in#uence of the mixing intensity is included in the initial "lm thickness, the rate of strain, and the vortex lifetime. The initial mean "lm thickness is assumed to be equal to the Kolmogoro! length scale





  d #d " "  ) 

(15)

and the rate of strain is described by (Baldyga & Bourne, 1984) (/v  "! . B (4#(/v)t

(16)

The vortex lifetime, t , is taken as the maximum time of + the system, and depends on the speci"c energy dissipation rate; see Eq. (1). The corresponding dimensionless vortex lifetime is evaluated by integration of Eq. (7) from t"0 to t . d is obtained by inserting Eq. (16) into Eq. (5) + using Eq. (15) for the initial slab thickness. The dimensionless vortex lifetime equals 2800D /
, i.e. it is indepen dent of the speci"c local energy dissipation rate. When
/D "10, i.e. for water, the dimensionless vortex life time is equal to 2.8. 3.1. Parameter values In this work we study the in#uence of crystallization kinetics, speci"c local energy dissipation rate, di!usivities and initial "lm reactant concentrations. In all cases there is no product present initially. The model allows for unequal "lm thicknesses, but the "lms are assumed to be equally thick in this study, i.e. d "d . The crystal den sity and molar mass are also kept constant. The crystal size is assumed to be the size of a sphere having the same

M. Lindberg, A.s C. Rasmuson / Chemical Engineering Science 56 (2001) 3257}3273 Table 1 Parameters held constant Parameter

Explanation

Value

Unit


D   ! M ! k ? k T c !

Kinematic viscosity Di!usivity of reactant A Solid density Molar mass Area factor Volume factor Solubility

10\ 10\ 1000 122

/6 0.03

m/s m/s kg/m kg/kmol

kmol/m

volume as the actual crystal. Table 1 speci"es the parameter values that are unchanged throughout all the simulations, and Table 2 presents the values of the parameters that are varied. Two sets of kinetic parameters are used for the rate of primary nucleation. One set corresponds to the kinetics of benzoic acid presented by As slund (1994) (K "17 . and K "10 no./m s). In the other set, K is given . . its theoretical value of 10 no./m s (Mullin, 1993, p. 179) and K "200 which corresponds to the value . that is obtained for benzoic acid if the surface energy is calculated by the expression given by Mersmann (1990). Concerning growth rate parameters: g is either unity or 2, and k is varied over 6 orders of magnitude. E The nucleation time lag, t , is strongly dependent on  supersaturation (e.g. As slund, 1994) and the relation can be treated almost as a step function. In the present work, the nucleation time lag is assumed to be constant and very short compared to other time constants. The solubility is taken as 0.03 kmol/m. This is the solubility of benzoic acid in water at 203C * the system studied by As slund and Rasmuson (1992). The size at birth, ¸ , is  assumed to be constant "10\ m. Three levels of speci"c local energy dissipation rate are examined: 1, 6.25, and 100 W/kg, the values can be regarded as typical for regions of higher mixing intensity in a stirred tank. The corresponding vortex lifetimes are 13, 5, and 1.3 ms, and the Kolmogoro! length scales are 32, 20, and 10 m, in pure water at room temperature.

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The initial "lm reactant concentrations have been varied while retaining constant local productivity or constant global productivity. The local productivity is de"ned as the amount of product per unit volume formed at 100% conversion inside the lamellar structures. When the "lm thicknesses of the two reactants are equal, the constraint of constant local productivity leads to min(c , c )  c  " . ! 2

(17)

At constant global productivity, we restrict the combination of initial process reactant concentrations so that the requirement of constant productivity of a semi-batch crystallization process is ful"lled. When a certain production requirement is imposed and an overall stoichiometry is retained, the concentrations of the initial process reactant solutions, c and c , must ful"l the equation  



stoichiometry: yield < c c 
" "   " ! < < #< < c "< c     



c c "   c #c  

(18)

when the reaction is irreversible and complete, and there are no volume changes upon mixing. In the present simulations the initial "lm reactant concentrations are set equal to these initial process reactant concentrations, when the constraint of constant global productivity is adopted. Five di!erent sets of di!usivities are investigated, and are selected as previously (Lindberg & Rasmuson, 2000): (a) all di!usivities are equal, (b) the product di!usivity is lower than both reactant di!usivities, (c) and (d) the product di!usivity is less than one of the reactant diffusivities but higher than the other (in addition, in (c) the product of D /D and D /D is less than unity, and in (d) !  ! the product is greater than unity), and (e) the product di!usivity is higher than both reactant di!usivities.

Table 2 The values of the parameters varied Parameter

Explanation

Default value

Additional values

c /c  c 
! c  ! D /D  D /D !  k E g  K . K .

Ratio of initial reactant "lm concentrations Global productivity, see Eq. (18) Local productivity, see Eq. (17) Relative reactant di!usivity Relative product di!usivity Coe$cient of surface integration Exponent of surface integration Local energy dissipation rate Nucleation rate parameter Nucleation rate parameter

5 * 0.5 2/3 2 10\ 1 6.25 10 200

1/5, 1/2, 1, 2 0.125, 0.25, 0.5 0.75 See Table 3 See Table 3 10\, 10\, 1, 10 2 1, 100 10 17

Unit

kmol/m kmol/m

m/s(kmol/m)E W/kg no./m s

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4. Results and evaluation The raw data generated by the model simulations are reactant and product concentrations, and full population density distributions in every discretized point in time and space. Fig. 2 shows an example of concentration pro"les plotted versus the dimensionless length scale, z, at di!erent dimensionless times, , for a case using the default values speci"ed in Table 2.  "0.36 corresponds  to the moment when the reaction terminates,  , and    "2.8 is the vortex lifetime. The upper diagram in  Fig. 2 shows the development of the reactant concentration pro"les. During the course of the process the reaction plane moves to the right into the region originally occupied by reactant A. At  , the reaction terminates   and after that the remaining concentration gradients of B continue to level out. The lower diagram shows the product concentration pro"les. As seen, the product concentration varies considerably in space and time. The product is distributed in space by the moving reaction plane, and by mass transfer. For ( the maximum   product concentration occurs at the reaction plane, and for ' the product concentration gradients con  tinue to level out like those of reactant B. In addition, the product concentration is partly consumed by the crystallization. Crystals nucleate and grow, and are transported by di!usion and straining. Due to the existence of supersaturation gradients, the nucleation and growth rates

Fig. 2. Development of reactant and product concentration pro"les ("0.005, 0.05, 0.1, 0.15, 0.25, 0.36, 0.7, 1.4, and 2.8).

vary within the "lm. Fig. 3 shows how the population density distribution develops in space and time. In the "rst six diagrams, the population density increases, especially near the reaction plane, due to massive nucleation. In addition, the crystals increase in size due to growth. In the three "nal diagrams, there is no nucleation, only continued crystal growth. Results describing the overall conditions with respect to the whole "lm are given in Fig. 4. The overall

Fig. 3. Development of population density distribution.

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Fig. 4. Development of overall conditions in the vortex (ND "  7.7;10, ¸D "4.8;10\). 

maximum product concentration (c ), the overall !  fraction of the molecular product formed that has crystallized (M /M ), the overall number mean size !Q ! (¸ ), and the total crystal number concentration (N )   are all plotted versus the dimensionless time. The overall number mean size and total crystal number concentration are normalized with respect to the corresponding maximum values during the vortex lifetime. The maximum product concentration, c , increases until the !  reaction terminates ( "0.36), and then c starts to  !  decrease as a result of mass transfer and crystallization. The total number of particles increases strongly during the period of high maximum product concentration, but nucleation is negligible when c is low. As long as there !  are crystals present in a supersaturated solution they grow. However, the overall mean size exhibits a minimum that corresponds to the massive generation of nuclei at the maximum value of c . In this particular simulation, the !  supersaturation is only partly consumed when the vortex lifetime is reached. Particle di!usion generally has a minor in#uence, because the crystals grow rapidly into sizes for which the particle di!usivity is low. However, when the growth rate is very low, particle di!usion contributes to a spreading out of the crystals. Below, we concentrate the presentation of results to the "nal values (i.e. at " ) of the total crystal number + concentration, the overall number mean size, and the overall fraction of product generated that has crystallized in each simulation. 4.1. Inyuence of specixc local energy dissipation rate In Fig. 5, the "nal total crystal number concentration (ND ), "nal overall number mean size (¸D ), and "nal   overall fraction of product formed that has crystallized (MD /MD ) are plotted versus the local speci"c energy !Q ! dissipation rate () at di!erent sets of nucleation kinetics. The "nal crystal mean size and fraction of product formed that has crystallized decrease with increasing energy

Fig. 5. In#uence of the local speci"c energy dissipation rate and the nucleation kinetics on the "nal overall conditions.

dissipation rate. Hence, the remaining supersaturation increases with increasing energy dissipation rate. In general, the total crystal number concentration decreases with increasing . When the mixing intensity, i.e. the speci"c local energy dissipation rate, , is increased, the same volume of feed solution is distributed over an increased number of smaller vortices. However, the total volume of vortex solution is assumed to be constant. Hence, the simulated total crystal number concentration is a quantity that can be directly compared in the evaluation of the in#uence of the mixing intensity. The speci"c local energy dissipation rate a!ects the initial "lm thickness, the rate of straining, and the vortex lifetime. However, the in#uence of  is incorporated into the dimensionless length and time, leading to that reactant concentration pro"les and the rate of production at equal dimensionless time are independent of energy dissipation rate (Lindberg & Rasmuson, 1999a). At negligible consumption even the product concentration pro"le at equal dimensionless time is independent of . Hence, the main in#uence of  is that the mixing process evolves faster at higher values of , leading to that nucleation and crystal growth are given less time, and this will signi"cantly in#uence the conditions in the vortex at the end of its lifetime. At a high degree of supersaturation consumption, the product solute concentration pro"le at equal dimensionless time will depend on . There are cases when the total crystal number concentration may go through a maximum with increasing , as seen in Fig. 5. Fig. 6 shows the development of c !  versus real time, t, for the cases employing the second set of kinetics in Fig. 5. As seen in Fig. 6, there is

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commonly used power law equation of nucleation rate versus supersaturation: B "k (c !c ). (19) N L ! ! Assume that this power law equation is de"ned such that its function value and "rst derivative value equal the corresponding values of the classical nucleation expression (11). Then the exponent value, np, for each set of kinetics and supersaturation can be determined as Fig. 6. Development of product maximum concentration in real time for three cases corresponding to the points for the second set of nucleation kinetics in Fig. 5.

a supersaturation peak at "6.25 and 100 W/kg, that is much lower at "1 W/kg. Thus, although t decreases + with increasing , ND may increase due to higher max imum values of supersaturation. This explains why ND passes through a maximum with increasing , for the  second set of nucleation kinetics. 4.2. Inyuence of crystallization kinetics As seen in Fig. 5, the total crystal number concentration, ND , is always higher for cases employing the second  set of nucleation kinetics. This is because, this set of kinetics leads to higher nucleation rates in the present range of maximum supersaturations (S"19}23). The maximum in ND versus  occurs only for cases employ ing the second set of nucleation kinetics. This is because this set of kinetics leads to a much stronger dependence of the rate of nucleation on the supersaturation. This dependence can be expressed as an exponent value in the

2K (S!1) np" . . S(ln S)

(20)

For the present interval of maximum supersaturation, the exponent value varies between 1.1 and 1.3 for the "rst set of nucleation kinetics, while it varies between 12 and 15 for the second set of kinetics. Thus, the non-linearity of the dependence of the nucleation rate on the supersaturation is very pronounced in the case of the second set of kinetics and actually very weak for the "rst set. Within the present range of maximum supersaturation, changes in c  will have a dramatic in#uence on ND in the case !   of the second set of kinetics, while for the "rst set the in#uence is quite weak. Fig. 7 shows how the "nal overall total crystal number concentration, "nal overall number mean size, and "nal overall degree of consumption depend on the surface integration rate constant, k , for di!erent values of g and E nucleation kinetics. At high values of k , the resistance to E boundary layer di!usion controls and the particles obtain a mean size in the order of 0.1}1 m at the end of the vortex lifetime. At lower k , the degree of supersaturation E consumption is lower and the crystals become smaller.

Fig. 7. In#uence of the growth and the nucleation kinetics on the "nal overall conditions.

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Fig. 8. In#uence of the ratio of the initial "lm concentrations, the di!usivities and the nucleation kinetics at constant local and at global productivity, respectively, on the "nal overall crystal number concentration.

This is also the case when g increases since the numerical value of the driving force has a value below unity in the expression used. In general, the "nal overall crystal mean size does not depend strongly on the nucleation kinetics because the size is often determined by the crystal growth rate and not by mass constraint. However, when there is a high degree of consumption, the mean size depends on the total number of crystals sharing the crystallized mass and is hence governed by the nucleation. In this case the mean size is lower for cases employing the second set of nucleation kinetics, due to that a larger number of crystals is formed.

especially for cases employing the second set of nucleation kinetics, and the in#uence depends on whether the comparison is made at constant local or at global productivity. At constant local productivity, ND increases  with increasing deviation from unity of the ratio of initial reactant concentrations. However, at constant global productivity ND passes through a maximum instead,  that is not always located at unity.

5. Discussion 5.1. The model

4.3. Inyuence of initial reactant concentrations and diwusivities Fig. 8 shows the "nal total number concentration, ND ,  versus the ratio of the initial reactant concentrations for "ve di!erent sets of di!usivities given in Table 3. In the left-hand column of diagrams, the local productivity is kept constant, and in the right-hand column of diagrams, the global productivity is kept constant, in both cases at 0.5 kmol/m. As seen, ND varies considerably with the  initial reactant concentration ratio and di!usivities, Table 3 Di!erent sets of di!usivities Case

D /D ! 

D /D !

D /D 

a b c d e

1 1/2 2 1/2 2

1 1/3 1/3 3 3

1 3/2 6 1/6 2/3

The model does not describe the total process of mixing reactants in a crystallization process, but only the "nal stages of micromixing. Hence, when the results are used in a discussion of the overall process it is assumed that the feeding rate is slow or moderate so that the in#uence of mesomixing can be neglected (Baldyga, Bourne, & Yang Yang, 1993), and that the crystallization steps are rapid. The lamellar structure may be an oversimpli"cation. However, in the mixing process the reactants eventually contact by molecular di!usion and at this moment we expect to "nd the highest possible supersaturation. In addition, the model only accounts for mixing of fresh reactant solutions of various initial concentrations, and hence only re#ects the situation in the very beginning of a semi-batch process. However, again this is the moment when we expect to "nd the highest supersaturations. Since nucleation strongly depends on the level of supersaturation, the maximum supersaturation is of key importance and hence even though the model only describes a part of the entire process and in

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3266

an idealized manner, the results may give "rst a very rough indication of the in#uence of di!erent variables on the outcome of the overall process. It is assumed in the modelling that the crystals are small compared to spatial concentration gradients in the "lms. This is obviously not always ful"lled. However, during the early nucleation, the crystals are the smallest and the "lms have the maximum thickness. Thus, the total number generation may still be reasonably well predicted by the model. At the end of the vortex lifetime, the "lm thickness in water equals 2.5, 1.6, and 0.8 m at  equal to 1, 6.25, and 100 W/kg. The crystal size is of the same order of magnitude. However, at this stage the supersaturation pro"le has often levelled out signi"cantly and the crystals will experience an environment with less strong supersaturation gradients.

Fig. 9. Critical size for benzoic acid.

5.2. The importance for the overall process Usually, the crystal mean size of the product from a crystallization process is governed by the number of crystals sharing the total mass. Hence, the rate of nucleation determines the "nal size if the supersaturation is essentially completely consumed at the end of the process. The simulations of the present work suggest that it is quite likely that crystals may form in the lamellar structures of the smallest vortices. The values on the "nal overall total crystal number concentration of the present simulations are of the same order of magnitude as those reported in the literature for the product of reaction crystallization processes. Although all the solution in a semibatch crystallizer does not necessarily engage in contacts of the type described in the present work, this still indicates that it is possible that a signi"cant fraction of the crystals in the product are actually formed locally in the smallest vortices containing the initial contact of reactant solutions. This is also supported by the normally very strong dependence of the nucleation rate on the level of supersaturation. In the simulations presented in this work, the overall crystal mean size at the end of the lifetime of the smallest vortices where fresh reactant solutions make contact is typically in the range of 0.1}1 m. In addition, the results suggest that signi"cant supersaturation may remain when the vortex has lost its energy and is further mixed with additional solution. The solubility of a crystal increases with decreasing size for small crystals according to the Gibbs}Thomson relationship (Mullin, 1993, p. 103):






c (¸) 2k  M ? Q ! . ln ! " c (R) 3k
 R¹¸ ! T !

(21)

This relation also speci"es for each supersaturation the critical size for a crystal in order to be stable, i.e. the size

Fig. 10. Final total cumulative number concentration distribution.

at which the concentration equals the solubility. Hence, small crystals dispersed from the feed zone into the bulk solution may dissolve depending on the particle size and on the supersaturation in the bulk. Only crystals larger than the critical crystal size will survive and appear in the "nal product. Fig. 9 shows the critical size for survival of benzoic acid crystals according to the Gibbs}Thomson equation. Two di!erent values of surface energy,  , are Q adopted that correspond to the values for K in the . expression of the nucleation rate. Obviously, already at rather low supersaturation, the critical size for survival is below 0.1 m. Hence, the crystals that may form in the smallest vortices where fresh reactant solutions make contact are likely to survive in the bulk and accordingly may signi"cantly in#uence the "nal size distribution of the entire process. According to our simulations, the crystal size distribution generated in the vortex is in general quite narrow. A few examples are given in Fig. 10 as total cumulative oversize number concentration at the end of the vortex lifetime. This leads us to believe that the problem can be somewhat simpli"ed by stating that either all the crystals from a particular vortex survive in the bulk or that essentially all dissolve in the bulk by ripening. 5.3. The importance of local maximum supersaturation The "nal crystal number concentration in the vortex depends on the rate of nucleation and the vortex lifetime. The dependence of the rate of nucleation on the supersaturation is strongly non-linear. The more

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Fig. 11. In#uence of the ratio of the initial "lm concentrations, the ratio of di!usivities and the nucleation kinetics at constant local and at global productivity, on c  . ! 

non-linear the dependence is, the more important is the actual distribution in time and space of the supersaturation, and especially the maximum value of supersaturation. Hence, the nucleation is concentrated to volumes and moments in time where particularly strong supersaturation prevails, e.g. until +0.5 in the example shown in Fig. 2. Here, we "nd the explanation for the results for ND versus relative initial "lm concentration  and di!usivities shown in Fig. 8. The corresponding values of the maximum value of product concentration during the contact, c  , for the di!erent cases are !  shown in Fig. 11. At constant local productivity, the behaviour of ND in Fig. 8 very well re#ects the behaviour  of c  in Fig. 11. At constant global productivity, the !  behaviour of ND rather well correlates to the behaviour  of c  for the second set of nucleation kinetics but not !  that well for the "rst set. Since, the "rst set of nucleation kinetics exhibits a weaker dependence on supersaturation, we do expect to "nd a weaker correlation between ND and c  .  !  Fig. 12 presents the correlation between the "nal total number concentration of crystals at the end of the vortex lifetime and c  during that lifetime at "6.25 W/kg, !  for the two di!erent sets of nucleation kinetics. All the cases in Fig. 8 are included. In addition, all the cases in Fig. 8 are recalculated for two additional levels (0.25 and 0.125 kmol/m) of the local productivity and one additional level (0.75 kmol/m) of global productivity, and all these results are also included in both diagrams. As shown, there is a very good correlation between ND and  c  in the case of the second set of nucleation kinetics, !  and a fairly good correlation for the "rst set. This "rmly supports the hypothesis that c  is a very !  important property characterizing the di!erent cases.

Fig. 12. Correlation of the "nal overall number concentration versus c  . ! 

The correlation between ND and c  can be described  !  by a function similar to Eq. (11) into which the maximum supersaturation is inserted. The pre-exponential factor becomes a function of K , K , and , while the . . exponential term remains the same. In previous contributions we have analysed the conditions in the "lms when crystallization is neglected. The maximum product concentration for such cases can be found analytically for semi-in"nite "lms, c (Lindberg !  & Rasmuson, 2000) and numerically for "nite "lms, c  (Lindberg & Rasmuson, 1999b). These results were !  used for analysis of how to design a process in order to maximize the product crystal size. The underlying assumption was that the product mean size is essentially

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Fig. 13. Correlation of c  versus the initial maximum product concentration to the left and the maximum c in the absence of consumption to !  !  the right.

governed by the maximum product concentration that is generated. In Fig. 13, c  calculated by the present !  model is plotted versus c and c  from the pre!  !  vious models. In both cases the correlation is quite good. In particular, the correlation is good with c  at lower !  values and with c at higher values. The fact that !  c  and c correlate better at lower and at higher !  !  values of supersaturation respectively is related to the degree of consumption. At negligible consumption c  equals c  , and at a high degree of consumption !  !  the maximum will occur early, i.e. c  Pc " !  !  c . Obviously, prediction of the maximum con!  centration in the non-crystallizing case is quite useful as a "rst estimate of how the crystallizing case will behave. 5.4. Local maximum supersaturation in ionic systems In an ionic system, i.e. a type (ii) precipitation, there is no solute molecule formed and the supersaturation is described by (c c /K ) if activity coe$cients are ne QN glected. Consider mass transfer in a lamellar structure of "nite strained reactant "lms where crystallization is neglected. Then, the reactant concentrations are described by (Crank, 1975, p. 16)




     

c  d /d#4n!z D  c "  erf 2 D 2( L\ d /d!4n#z D  , #erf D 2(







c  !d /d#4n#z c "   erf  2 2( L\ !d /d!4n!z #erf . 2(




(22)

(23)

Eqs. (22) and (23) can be used to evaluate the maximum value of (c c ) occurring during the vortex lifetime, 

Fig. 14. The maximum value of the square root of the product of reactant concentrations versus the initial reactant concentrations.

(c c ) . The upper diagram of Fig. 14 shows (c c )     at constant global productivity versus the ratio of initial "lm reactant concentrations at di!erent values of reactant di!usivities. Reactant "lms are assumed equally thick. The in#uence of the ratio of initial reactant "lm concentrations is quite strong. However, also the e!ect of unequal di!usivities is signi"cant which deviates from the "ndings of Nielsen (1964). Nielsen concluded for semiin"nite "lms that the reactant concentrations have a great in#uence on the maximum supersaturation while the e!ect of D /D is weak. The in#uence of di!usivities  is more clearly shown in the lower diagram of Fig. 15, in which (c c ) has been normalized with respect to the   square root of the product of the reactant concentrations at complete mixing




c c   . 2 2

(24)

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5.5. Implementation into an overall process model

Fig. 15. The di!usion correction factor versus the reactant di!usivities at di!erent "lm thicknesses.

The lower diagram of Fig. 15 shows that the relative in#uence of unequal di!usivities is independent of the ratio of initial "lm reactant concentrations. That is, for a certain value of reactant di!usivities, the ratio of the maximum supersaturation and the square root of the product of initial "lm concentrations is constant. In addition, at equal di!usivities the maximum supersaturation coincides with the supersaturation at perfect mixing. The correction factor by which (c c ) is increased when   the di!usivities are unequal is shown in Fig. 15, versus the ratio of di!usivities for di!erent ratios of initial thickness of reactant "lms. The behaviour of ionic systems is not as case dependent as that of molecular systems. The E-model (Baldyga & Bourne, 1989) is commonly used to describe micromixing e!ects in precipitation. Most work has been devoted to ionic cases. The product of the local reactant concentrations according to the E-model is < < !< #  c c c "  c , (25)
  
  <  < # # where < corresponds to the initial eddy volume of  feed solution, < equals the increasing eddy volume, and # c stands for the mean concentration in the sur
  rounding liquid. The maximum of c c can be evalu
   ated by di!erentiating Eq. (25) with respect to < /< .  # The maximum appears when 2< "< :  # c c (26) (c c ) " 
 
    4 which is equal to the reactant concentration product at complete mixing, Eq. (24). Thus according to this simple analysis, the E-model may predict the in#uence of reactant concentrations on the maximum supersaturation in an ionic system and the actual value is correct when di!usivities are equal. For unequal di!usivities the E-model systematically underestimates the supersaturation level. The underestimation increases with increasing deviation from unity of the ratio of reactant di!usivities. Thus, for every set of di!usivities the E-model seems to capture the general behaviour of maximum supersaturation for ionic systems.

In modelling reaction crystallization of an ionic compound in a tubular reactor, Baldyga and Orciuch (1997) used probability density functions (PDF) to capture likely contacts between volumes of di!erent compositions in a Computational Fluid Dynamics model. For reaction crystallization of a molecular compound, a PDF must be formulated also for c , apart from for c and c . Then in !  principle the local population density in every contact can be calculated by the present model. However, in practice this becomes a too heavy work load. The strong correlation between N and c  suggests a short-cut.  !  The speci"c values of reactant concentrations may be used to estimate c  by cD and c , which in turn !  !  !  gives a value of the number of nuclei born from a correlation as discussed above. Di!erent compositions of reactant solutions can be accounted for in the boundary conditions, and initial product concentration can be superimposed on the concentration generated by the chemical reaction. Local values of high product concentration are most likely to occur especially at primary contact of fresh reactant solutions as well as at "rst order of secondary contacts, and these contacts will dominate the generation of new crystals.

6. Conclusions A reaction crystallization model over the ultimate stage of micromixing of the reactant solutions is developed. The model describes mass transfer, instantaneous reaction, and subsequent crystallization of a molecular compound in a lamellar structure of strained "nite #uid reactant "lms. Reasonable values over initial reactant concentrations, di!usivities, crystallization kinetics, and mixing intensity are inserted for numerical solution. For such values it is shown that for low-soluble or sparingly soluble compounds crystallization may start already in these "lms. At the end of the lifetime of these structures where fresh reactant solutions make ultimate contact, the crystal size distribution is quite dependent on the properties of the system and on the processing conditions. Often a signi"cant supersaturation remains and in this case the crystal mean size is governed by the growth rate not by the nucleation rate. Simulations show that the mean size and number concentration in general increase with decreasing speci"c local energy dissipation rate. At the end of the vortex lifetime the crystals have reached in many cases a mean size in the range of 0.1}1 m and the crystal size distribution is quite narrow. The crystals born in these primary vortices are likely to survive in the low supersaturation environment of the bulk. The total crystal number concentration in the vortex is of the same order of magnitude as the product number concentration of reaction

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crystallization processes reported in the literature. Thus, the simulations indicate that the contacts in the local "lms may signi"cantly contribute to the "nal crystal size distribution of the overall process. The neglect of the detailed conditions in reaction crystallization of a molecular compound may not be justi"ed. The number of crystals generated is strongly correlated to the maximum supersaturation during the lifetime of the vortex. It is shown that this maximum supersaturation depends in a complex way on the initial "lm reactant concentrations, di!usivities, relative "lm thicknesses, and degree of consumption, but can as a "rst approximation be well described by simpli"ed models where the crystallization is neglected.

Notation B . c
  c ! c !G c !  c !  cDGL !  c  !  cQCKG !  c 
! c  ! c G c G c G D G D D . d d G F g G J k ? k k B k E

rate of nucleation, no./m s concentration of the reactant in the bulk, kmol/m solubility of substance C, kmol C/m initial product concentration in the "lm of substance i, kmol C/m maximum concentration of substance C in the "lms, kmol C/m initial maximum product concentration, kmol C/m maximum c according to model in Lindberg !  and Rasmuson (1999b), kmol C/m maximum c during the local contact, !  kmol C/m maximum c according to model in Lindberg !  and Rasmuson (2000), kmol C/m global productivity, kmol C/m local productivity, kmol C/m concentration of substance i, kmol i/m initial "lm concentration of substance i, kmol i/m initial process concentration of substance i, kmol i/m di!usivity of substance i, m/s di!usivity of the limiting substance, m/s particle di!usivity, m/s mean half-slab thickness, m reactant i's half-slab thickness, m system of non-linear equations growth rate parameter growth rate, m/s the Jacobian of the system of non-linear equations, F area factor Bolzmann constant mass transfer rate constant surface integration rate constant

k L k T K . K . K QN ¸ ¸  ¸  ¸D  M ! M !Q MD !Q MD ! M ! n np N  N  ND  r r G R S t t  t + t   ¹ < # < G < K x x Q y y P z z Q Z G

nucleation rate parameter Volume factor nucleation rate parameter, no./m s nucleation rate parameter solubility product, kmol/m crystal size, m nucleus size, m overall number mean crystal size, m "nal overall number mean crystal size, m molecular weight of the product, kg/kmol amount of solid product, kmol "nal amount of solid product, kmol "nal amount of formed product, kmol amount of formed product, kmol population density, no./m m nucleation rate parameter overall cumulative number concentration, no./m overall crystal number concentration, no./m "nal overall crystal number concentration, no./m particle radius, m reaction rate of substance i, kmol i/m s gas constant, kJ/kmol K supersaturation ("c /c ) ! ! time, s nucleation time lag, s vortex lifetime, s reaction time, s temperature, K eddy volume, m volume of reactant i, m molecular volume, m space coordinate, m position of the reaction plane, m variable vector rth iterative value of the variable vector y dimensionless space coordinate de"ned by Eq. (8) dimensionless position of the reaction plane dimensionless space coordinate de"ned by Eqs. (9) and (10)

Greek letters  B   1

)
 !      

rate of strain, s\ speci"c local energy dissipation rate, W/kg surface tension, kJ/m Kolmogoro! microscale, m dynamic viscosity, N s/m kinematic viscosity, m/s density of the product, kg/m dimensionless time dimensionless nucleation time lag dimensionless reaction time

M. Lindberg, A.s C. Rasmuson / Chemical Engineering Science 56 (2001) 3257}3273

Acknowledgements The authors gratefully acknowledge the "nancial support of the Industrial Association for Crystallization Research and Technology (IKF), the Swedish Research Council for Engineering Science (TFR), and the School of Chemistry and Chemical Engineering at the Royal Institute of Technology (KTH).

The transformed governing equations and initial and boundary conditions are as follows: In the "lm of reactant B (0(Z (1)





Z z c 1 D c Q " # , z  Z z D Z 1 Q  8

c ! 

Z z c 1 D c Q !# ! ! " z  Z z D Z 1 Q  8





1 D c ! z D Z Q 

8 

(A.1)

4



 

(Z !1)



Gn¸ d¸ ,

(A.2)

d (Gn) Z z n Q "! # D ¸ z  Z 8  Q D 1 n dB . (¸!¸ );(! ). # . #   D z Z D  Q  (A.3)

In the "lm of reactant A (0(Z (1) 



 c  

Z z c 1 c Q # , "  z !2 (z !2)  Z Z  8 Q Q  

c ! 

D c Z z c 1 Q !# ! ! "   Z D Z z !2 (z !2)     8 Q Q



 

(A.4)



Gn¸ d¸ ,

c "0, 

c "c , c "c , ! ! c  "0, BC1: '0, Z "0,  Z  c c n ! "0, "0, "0, Z Z Z    c  "0, BC2: '0, Z "0, Z n "0, Z

BC3: 0(( , Z "Z "1,    1 D 1 c c  "! , D z Z z !2 Z Q  Q  8 8  (c )  "(c )  , (n)  "(n)  , !8 !8 8 8 BC4: 0(( , 0)Z )1, c "0,    BC5: 0(( , 0)Z )1, c "0,    BC6: *0, ¸"0, n"0, BC7: *0, ¸PR,

(A.5)




n"0.

When z "0, i.e. when reactant B is totally consumed, Q BC2}BC5 are replaced by BC2: '

, Z "1,  c c n  "0, ! "0, "0. Z Z Z    When z "2, i.e. when reactant A is totally consumed, Q BC1, BC3}BC5 are replaced by  

 

, Z "1,

c c ! "0, "0, Z Z

d  ! k ¸ B ;(! ) ! 4  .  D M  ! 

The initial and boundary conditions transform into

BC1: '

1 c  ! (Z !1)  (z !2) Z  Q  8

#3k 4

d (Gn) Z z n Q "! #  ¸ D z !2  Z    Q 8 D 1 n dB . (¸!¸ );(! ). # . #   D (z !2) Z D   Q  (A.6)





 n 

n 

c c ! "0, "0, Z Z

d  ! k ¸ B ;(! ) ! 4  .  D M  ! #3k




IC1: "0, 0)Z (1,  c "c , c "0, c "c ,   ! ! IC2: "0, 0)Z (1,

Appendix. Transformed governing equations and numerical solution

c 

3271

n "0. Z

The governing equations are discretised according to the Crank}Nicholson approximation (Lapidus & Pinder, 1982). Firstly, the governing equations of the reactant concentrations are solved separately since they are independent of the governing equations of the product concentration and the population density, because the

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chemical reaction is irreversible. Although the system of di!erential equations for the reactants is linear, the position of the reaction plane has to be iteratively calculated in every new time step. Solution of these equations yields the position of the reaction plane and the production rate at this plane, and this provides the necessary input to the governing equations of the product concentration. The solution of the governing equations of the product solute concentration is divided into two parts * before crystallization starts and after crystallization has started. Before crystallization starts, the governing equations of the product solute concentration result in a system of linear equations. Initially the "nite "lms can be approximated as semi-in"nite. Hence, to initiate the numerical calculations the analytical solution for the reactant concentrations, and the closed form expression over the product concentration in the case of no crystallization (Lindberg & Rasmuson, 1999a), are used. When crystallization has started, the discretized governing equations of the product solute concentration and of the population density become a system of non-linear equations, F(y). The system of equations is solved by applying Newton's method, i.e., the product concentration and population density for each time step are iteratively calculated. The Jacobian, J(y), is expressed in closed form. For each iterative step in Newton's method, the following implicit system has to be solved: J(y )(y !y )"F(y ) (A.7) P P> P P which is solved by the conjugate gradients squared method (cgs.m in Matlab). The time step is varied along the course of a calculation. The convergence of a calculation is checked by comparing the mass balances of the reactants and the product (both as solute and as solid). In addition, the calculations have been validated by comparison with analytical and numerical calculations for limiting cases. References Armenante, P. M., & Kirwan, D. J. (1989). Mass transfer to microparticles in agitated systems. Chemical Engineering Science, 44, 2781}2796. As slund, B. L. (1994). Kinetics and processing in reaction crystallization of benzoic acid. Ph.D. treatise, Royal Institute of Technology, Stockholm, Sweden, TRITA-KET R6. As slund, B. L., & Rasmuson, As . C. (1992). Semibatch reaction crystalization of benzoic acid. A.I.Ch.E. Journal, 38, 328}342. Baldyga, J., & Bourne, J. R. (1984). Mixing and fast chemical reaction * VIII. Initial deformation of material elements in isotropic, homogeneous turbulence. Chemical Engineering Science, 39, 329}334. Baldyga, J., & Bourne, J. R. (1986). Principles of micromixing. In N. P. Cheremisino! (Ed.), Encyclopedia of yuid mechanics, vol. 1 (pp. 147}201). Houston, TX: Gulf Publishing Company. Baldyga, J., & Bourne, J. R. (1989). Simpli"cation of micromixing calculations I. Derivation and application of new model. Chemical Engineering Journal, 42, 83}92.

Baldyga, J., Bourne, J. R., & Yang Yang, X. (1993). In#uence of feed pipe diameter on mesomixing in stirred tank reactors. Chemical Engineering Science, 48, 3383}3390. Baldyga, J., & Orciuch, W. (1997). Closure problem for precipitation. Transactions of the Institution of Chemical Engineers, Part A, 75, 160}170. Baldyga, J., Poldgorska, W., & Pohorecki, R. (1995). Mixing-precipitation model with application to double feed semibatch precipitation. Chemical Engineering Science, 50, 1281}1300. Baldyga, J., Pohorecki, R., Podgorska, W., & Marcant, B. (1990). Micromixing e!ects in semibatch precipitation. In A. Mersmann (Ed.), Proceedings of the 11th symposium on industrial crystallization, Garmish-Partenkirchen, Federal Republic of Germany (pp. 175}180). Chen, J. F., Zheng, C., & Chen, G. T. (1996). Interaction of macro- and micromixing on particle size distribution in reactive precipitation. Chemical Engineering Science, 51, 1957}1966. Costa, P., & Trevissoi, C. (1972). Reactions with non-linear kinetics in partially segregated #uids. Chemical Engineering Science, 27, 2041}2054. Crank, J. (1975). The mathematics of diwusion (2nd ed.). Oxford, UK: Clarendon Press. Crank, J. (1984). Free and moving boundary problems. Oxford, UK: Clarendon Press. David, R., & Marcant, B. (1994). Prediction of micromixing e!ects in precipitation: Case of double-jet precipitators. A.I.Ch.E. Journal, 40, 424}432. Franke, J., & Mersmann, A. (1995). The in#uence of the operational conditions on the precipitation process. Chemical Engineering Science, 50, 1737}1753. Franke, J., & Mersmann, A. (1993). Precipitation of CaCO and  CaSO *2H O. Proceedings of the 12th international symposium on   industrial crystallization, Warzaw, Polen (pp. 2-067}2-072). Harada, H., Arima, K., Eguchi, W., & Nagata, S. (1962). Micro-mixing in a continuous #ow reactor (coalescence and redispersion models). The Memoirs of the Faculty of Engineering, Kyoto University, 24, 431}446. Klein, J. P., & David, R. (1995). Reaction crystallization. In A. Mersmann (Ed.), Crystallization technology handbook. New York: Marcel Dekker Inc. (Chapter 5). Lapidus, L., & Pinder, G. F. (1982). Numerical solution of partial diwerential equations in science and engineering. New York: Wiley (Chapter 4). Leeuwen, M. L. J. van. (1998). Precipitation and mixing. Ph.D. dissertation, Delft University, Delft, Netherlands. Lindberg, M., & Rasmuson, As . C. (1999a). Product concentration pro"le in strained reacting #uid "lms. Chemical Engineering Science, 54, 483}494. Lindberg, M., & Rasmuson, As . C. (1999b). Maximum supersaturation in reaction crystallization of molecular compounds. Proceedings of the 14th international symposium on industrial crystallization, Cambridge, UK. Lindberg, M., & Rasmuson, As . C. (2000). Supersaturation generation at the feed point in reaction crystallization of a molecular compound, Chemical Engineering Science, 55, 1735}1746. Marcant, B., & David, R. (1991). Experimental evidence for and prediction of micromixing e!ects in precipitation. A.I.Ch.E. Journal, 37, 1698}1710. Marcant, B., & David, R. (1993). In#uence of micromixing on precipitation in several crystallizer con"gurations. Proceedings of the 12th international symposium on industrial crystallization, Warzaw, Polen (pp. 2-021}2-026). Mersmann, A. (1990). Calculation of Interfacial tension. Journal of Crystal Growth, 102, 841}847. Mohanty, R., Bhandarkar, S., Zuromski, B., Brown, R., & Estrin, J. (1988). Characterizing the product crystals from a mixing tee process. A.I.Ch.E. Journal, 34, 2063}2068.

M. Lindberg, A.s C. Rasmuson / Chemical Engineering Science 56 (2001) 3257}3273 Mullin, J. W. (1993). Crystallization (3rd ed.). Manchester, UK: Butterworth Heinemann. Nielsen, A. E. (1964). Kinetics of precipitation. Oxford: Pergamon Press (Chapter 2). Ou, J. J., & Ranz, W. E. (1983). Mixing and chemical reactions. Chemical Engineering Science, 38, 1005}1013. Philips, R., Rohani, S., & Baldyga, J. (1999). Micromixing in a semifeed semi-batch precipitation process. A.I.Ch.E. Journal, 45, 82}92. Pohorecki, R., & Baldyga, J. (1983). The use of a new model of micromixing for determination of crystal size in precipitation. Chemical Engineering Science, 38, 79}83. Pohorecki, R., & Baldyga, J. (1985). The e!ect of micromixing on the course of precipitation in an unpremixed feed continuous tank crystallizer. Proceedings of xfth European conference on mixing, Wurzburg, West Germany (pp. 105}114).

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Pohorecki, R., & Baldyga, J. (1988). The e!ects of micromixing and the manner of reactor feeding on precipitation in stirred tank reactors. Chemical Engineering Science, 43, 1949}1954. Tovstiga, G., & Wirges, H.-P. (1990). The e!ect of mixing intensity on precipitation in a stirred tank reactor. In A. Mersmann (Ed.), Proceedings of the 11th symposium on industrial crystallization, Garmish-Partenkirchen, Federal Republic of Germany (pp. 169}174). Villermaux, J. (1990). Precipitation reaction engineering. In A. Mersmann (Ed.), Proceedings of the 11th symposium on industrial crystallization, Garmish-Partenkirchen, Federal Repbulic of Germany (pp. 157}162). Villermaux, J., & Devillon, J. C. (1972). Representation de la Coalescence et de la Redispersion des Domaines de Segregation Dans un Fluid par un Modele D'Interaction Phenomenologique. Proceedings of the second international symposium on chemical reaction engineering, Amsterdam (pp. B1-13}B1-24).