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Crystallization and precipitation
Chemical Engineering and Processing 38 (1999) 345 – 353 www.elsevier.com/locate/cep
Crystallization and precipitation A. Mersmann * Department B of Chemical Engineering, Technical Uni6ersity Munich, Boltzmannstr. 15, D-85748 Garching, Germany Received 4 March 1999; accepted 4 March 1999
Abstract This paper deals with the design and operation of cooling and evaporation crystallizers and precipitators for reaction crystallization. It will be shown that the median crystal size, L50, of a crystalline product and the crystal size distribution (CSD) mainly depend on the crystallization kinetics, e.g. the rates of nucleation and crystal growth, and that these kinetics are controlled by supersaturation. Therefore, the main objective in crystallization and precipitation is to choose and to maintain the optimal supersaturation with respect to product quality at all times and all locations in the crystallizer or precipitator. © 1999 Elsevier Science S.A. All rights reserved. Keywords: Crystal growth; Nucleation; Agglomeration; Median crystal size
1. The role of supersaturation for the median crystal size The selection of the apparatus or reactor and its operating condition depend on the chemico-physical properties of the system which can be described by the dimensionless solubility c*/cc (c* is the solubility of the solute and cc the density of the crystalline material) and the slope d(ln c*)/d(ln T) of the solubility curve c*= f(T) with T as the absolute temperature (see Fig. 1). Evaporative crystallization is necessary in the case of flat solubility curves d(ln c*)/d(ln T) B 1. Drowing out crystallization is carried out by the addition of an antisolvent, for instance alcohols or ketones. The maximum possible supersaturation in the absence of any formation of solid matter (e.g. a theoretical supersaturation, not occurring in crystallizers) is given by the ratio
solution flow rate in cooling crystallization or cooling rate
Dedicated to Professor Em. Dr-Ing. Dr h.c. mult. E.-U. Schlu¨nder on the occasion of his 70th birthday. * Tel.: +49-89-289-15652; fax: +49-89-289-15674. E-mail address: [email protected] (A. Mersmann)
the ratio
solution flow rate in evaporative crystallization evaporation rate
or the ratio
solution flow rate in drowning out crystalli antisolvent flow rate
zation. However, we demonstrate that the actual supersaturation Dc (c −c*) or relative supersaturation s=Dc/ c* is the result of crystallization kinetics and the presence of crystals and foreign particles in the apparatus. In reaction crystallization two or more components react chemically, with the result that a new product is formed. If the initial concentration of the reactants is high, and, therefore, the initial concentration cmax of the product is also high, but the solubility c* is very low, the resulting high supersaturation Smax =1+ smax = cmax/c* or smax = Dcmax/c* leads to high rates of primary nucleation. Since foreign particles in the nanometer range are always present in the reactant solutions, heterogeneous nucleation will take place. Homogeneous nucleation can be dominant at a very high supersaturation (Dc/ cc \ 0.1 for non-dissociating products). Primary nucleation is an activated process which means that local or temporary peaks of supersaturation Dc are quickly reduced to the metastable supersaturation Dcmet(Dc Dcmet). The rate, Bprim, of primary nu-
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cleation is a function of the actual local supersaturation s. After this nucleation step particles grow at the moderate supersaturation Dc 5 Dcmet with the growth rate G=f(Dc). If the residence time t of the nuclei in the precipitator or the batch time are small, the size increase DL = Gt is small, too. When the crystal size exceeds approximately 100 mm, attrition fragments are formed which can act as secondary nuclei. The total nucleation rate is the sum B = Bprim +Bsec of the contributions of primary and secondary nucleation. The median crystal size and the crystal size distribution (CSD) depend on the kinetic parameters B and G. In the case of MSMPR (mixed suspension mixed product removal) crystallizers or precipitators, the median crystal size L50 is given by L50 = 3.67
GmT 6aBrc
1/4
,
with mT being the suspension density and rc being the density of the crystals. With reaction crystallization, the suspension density results from a mass balance of the
reactants in the case of complete conversion. According to the last equation, nanoparticles can be obtained by applying very high nucleation rates, Bprim, which require very strong supersaturations Smax. Such high supersaturations can be achieved by: high concentration reactants, products with high concentrations cmax but low solubilities c*, and rapid micromixing at the feed point of the reactants for a fast chemical reaction. If, however, a coarse product is desired, the supersaturation must be kept low by: using low concentration reactants (perhaps by dilution), good macromixing in the entire precipitator, and seeding by high recycling rates of slurry in order to reduce local supersaturation peaks by crystal growth. In systems with high solubilities (c*/cc \ 0.01), the supersaturation is so small (s B 0.1) that activated nucleation does not take place and new nuclei are only formed as attrition fragments. In this case large median sizes L50 can be obtained if: the attrition rate Batt Bsec is low, the crystal growth rate G has the maximum allowable value with respect to crystal purity, and the mean residence time of the slurry is optimum. In the case of batch crystallizers, the median crystal size L50 as well as the CSD are dependent on the actual supersaturation, which varies with time and also with location, if a voluminous apparatus is not ideally mixed. The kinetic crystallization parameters, i.e. the nucleation rate, B, and the growth rate, G, are necessary for the evaluation of the median crystal size.
2. Nucleation rate
Fig. 1. Solubility of various solutes in water.
The rate of primary nucleation depends on the diffusivity DAB, the solubility c*, the density cc of the crystals and the supersaturation S, and can be calculated according to the classical theory of nucleation [1]. However, with respect to a very high particle or nuclei density N (particles m − 3) a very rapid perikinetic agglomeration takes place, with the result that observed nucleation rates, Bprim, of nuclei in the micrometer range are several orders of magnitude smaller than predicted by theory [2]. In addition, foreign particles are present in every solution, leading to heterogeneous nucleation. According to experiments carried out with foreign particles added to the reactants, the nucleation work is reduced to approximately 10% of the value for homogeneous primary nucleation [3,4]. Furthermore, the rate of heterogeneous primary nucleation is proportional to the specific surface area of foreign particles
A. Mersmann / Chemical Engineering and Processing 38 (1999) 345–353
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activated attrition fragments contribute to the rate of the secondary nucleation. Above the curve valid for the rate Bhom = 1015 m − 3 s − 1 of homogeneous nucleation the rate of perikinetic agglomeration becomes higher than Bhom. If agglomeration is not hindered agglomerates and not original nuclei will be obtained.
3. Growth rates Experimental results and theoretical considerations have shown that the mean growth rate, G, of a crystal collective depends mainly on the mass transfer coefficient, kd, the dimensionless solubility, c*/cc, and the relative supersaturation, s=c/c* [5]. In Fig. 4 the ratio 6/kd with G=26 is plotted against s, with c*/cc as a parameter. As can be seen from the shaded area in this figure, the growth rate G is often in the range 10 − 9 m s − 1 B GB10 − 7 m s − 1, and can be calculated for growth controlled by diffusion for s\ 1 from G=
b Dc k , 3a d cc
where a and b are the volume or surface shape factors, respectively. Since the maximum supersaturation is cmet,act after activated nucleation, the supersaturation Dc with Dcmet,het \ Dc \0 is decisive for the growth of nuclei. With batch crystallizers, the growth rate depends on the time averaged supersaturation Fig. 2. Volumetric nucleation rate against the relative supersaturation for activated and attrition controlled nucleation.
present in the solution [3,4]. These results and further considerations allow evaluation of the metastable zone widths for homogeneous and heterogeneous nucleation. Below a certain supersaturation smet,sec or Dcmet,sec/c* the nucleation rate induced by activated nucleation is so small that new nuclei can only originate from attrition fragments [1]. In Fig. 2 the nucleation rate is plotted against the relative supersaturation s for activated (surface) nucleation (theoretical lines) and for attrition induced nucleation (shaded area). Further theoretical considerations and experimental results lead to Fig. 3, where the dimensionless metastable zone width, Dcmet/cc, (valid for B: 1015 m − 3 s − 1) is plotted against the dimensionless solubility, c*/cc, with cc as the density of the solid crystals (valid for nondissociating systems). In such a diagram lines of constant relative supersaturation s can be drawn according to Dc/cc =(Dc/c*)(c*/cc) = s · c*/ cc, and areas show where homogeneous and heterogeneous nucleation are the dominant nucleation mechanisms. In the area ‘attrition controlled’ rates of activated nucleation are so small that only growth
Dc =
1 t
&
t
Dc dt.
0
Here t is the residence time of the crystals in the crystallizer. The curves in Fig. 4 show maximum possible growth rates in the absence of impurities or additives which retard or even block crystal growth. Only agglomeration of submicron particles of high particle density N may enhance the increase of crystal size per unit time. With needle-like crystals or platelets, the definition and prediction of crystal growth rates are difficult. If the crystallizer is not ideally mixed, the local distribution of supersaturation has to be taken into account. This can be done by dividing the total active volume into sections with approximately constant supersaturation.
4. Median crystal size In the case of activated nucleation the rate Bact increases very rapidly with the supersaturation S (Bact S 10 up to Bact S 50 in the absence of agglomeration depending on c*/cc (see Fig. 2)). In reality, however, strong agglomeration takes place for nuclei
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smaller than one micrometer [2]. The higher the initial supersaturation the smaller the median crystal size L50 in the absence of agglomeration. In Fig. 5 the median crystal size is plotted against the mean relative supersaturation s (or the maximum supersaturation smax = Smax −1 for reaction crystallization) for inorganic and organic systems with different solubilities c*. Additionally, the diagram also shows experimental results of the aqueous systems of BaSO4, BaCO3, CaCO3, Mg(OH)2 and MgC2O4 2H2O [15 – 17]. As can be seen from this – figure, the median crystal size drops with increasing relative supersaturation. In reaction crystallization, the relative supersaturation, which is decisive for the median crystal size depends on: the rate constant of the chemical reaction, the number and size of foreign particles present in the reactants, the effectiveness of macro- and micromixing, and the maximum product concentration. Experiments carried out with continuously operated precipitators have shown that the mean crystal size is much larger in comparison to products obtained in batch precipitators, in which large differences of supersaturation occur with respect to time and space. Systems with high solubilities (c*/cc \0.01) lead to median crystal sizes L50 \100 mm. The larger the crystal and the higher the collision velocity wcol of crystals against a resistance (rotor, wall, other crystals), the stronger the attrition effects which lead to attrition
fragments in the size range 1 mmB Latt B 150 mm [6]. In Fig. 6, the relative abraded crystal volume DVc/Vc based on the volume Vc of a single crystal is plotted against the collision velocity wcol for 0.1 and 1 mm KNO3 crystals according to a model of Gahn [6]. The diagram also shows lines according a model developed earlier [7–10]. In all cases, a huge number of attrition fragments are produced in industrial crystallizers with crystals larger than 100 mm. A very simplified equation valid for the effective rate of attrition controlled secondary nucleation Batt, based on the volumetric crystal hold-up, 8v = mT/rc, can be derived from the model of Gahn:
H 5v Kr 3p 2rcoNv Natt,eff 3 Batt = 7 · 10 − 4 3 h whf. · 8v m G 2a 3Ne Natt,tot The hardness Hv, the shear modulus m and the fracture surface energy G are properties of the crystalline matter. The operating conditions of the stirred vessel are characterized by the mean specific power input o, the Newton number Ne and the flow number Nv of the stirrer. The ratio Natt,eff/Natt,tot of the effective attrition fragments (serving as secondary nuclei) based on the total number of such fragments is about 0.01 and depends on supersaturation. The target efficiencies hw and hf can be calculated from equations in the literature [11,6–8]. Since the mean specific power input o depends on the tip speed utip of the stirrer with its diameter D according to
Fig. 3. Metastable zone widths Dcmet/cc against the dimensionless solubility c*/cc for homogeneous, heterogeneous and attrition controlled nucleation.
A. Mersmann / Chemical Engineering and Processing 38 (1999) 345–353
o=
4Ne D p4 T
n
u , H
utip
c− c* exp
G =kg
Fig. 4. Crystal growth rate against the relative supersaturation with the ratio c*/cc as a parameter.
2 3 tip
it is necessary to restrict the circumferential velocity of rotors (stirrer, pumps) to B10 m s − 1 and the mean specific power input in stirred vessels to o B 0.5 W kg − 1 when a coarse product is desired. The growth rate, G, of small attrition fragments is reduced due to crystal deformation and an increased chemical potential [12]. This reduction can be described by
<
349
Gs RTLatt
=
g
D T
3/2
min
=
n
1 Dr 3cw · Lp · g · w 2s 8v 0.088Ne7/18 rL
1/4
with D/T as the diameter ratio (stirrer D and tank T), ws as the settling velocity of the largest crystal with the drag coefficient cw, and with the density difference Dr = rc − rL between the crystal (c) and the liquid (L). 8v is the volumetric crystal hold-up and Ne the Newton number of the stirrer. As can be seen from this equation, crystals with a large settling velocity ws are mostly endangered by attrition.
c*
with kg as the growth coefficient and g as the growth exponent, usually within the range 1B g B 2. The higher the fracture resistance Gs and the smaller the size Latt of the attrition fragment, the more the growth is reduced (see Fig. 7 valid for KNO3 attrition fragments). With respect to individual differences in deformation, the growth rate differs from fragment to fragment (growth rate dispersion, GRD). As a result of these considerations, large median sizes can only be obtained under the following conditions: the supersaturation has to be optimum (see Fig. 8), the residence time (continuously operated crystallizer) or the batch time has to be optimum (see below), the circumferential velocity utip of a rotor should be low, and the specific power input should also be low. However, with respect to the suspension of coarse crystals in a stirred vessel, a minimum value of [utip(D/ T)3/2] of the stirrer is necessary for a given suspension [13].
5. Supersaturation in crystallizers The actual local supersaturation in a crystallizer under isothermal condition is the result of the following parameters and processes: Important for sparingly soluble systems: Concentration of the reactants Rate of the chemical reaction Intensity of macro-and micromixing Dilution of solution Agglomeration Important for highly soluble systems: Crystallization kinetics (nucleation, growth) Generation of crystal surface by attrition Let us assume that precipitation is initiated by a very fast chemical reaction with the result that the progress of this reaction is controlled by macro- and micromixing. This will be explained in more detail.
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A. Mersmann / Chemical Engineering and Processing 38 (1999) 345–353
Fig. 5. Median crystal size against the initial (maximum possible) supersaturation; shaded are results of inorganic and organic systems [11].
5.1. Effect of macro- and micromixing In Fig. 9, the supersaturation S is plotted against a dimensionless time t* which is defined by
t*=
1 V t = tmacro V: A + V: B tmacro
for MSMPR crystallizers, and by t*=t ·
V: A + V: B V
for batch crystallizers. With continuously operated crystallizers, the actual supersaturation depends on the ratio of the mean residence time t= [V/(V: A + V: B)] and the macromixing time tmacro according to
p T 2/3 tmacro,min : 5 :5 1/3 4Ne (o)
1/3
T D
5/3
·
1 . N
Here o is the mean specific power input equivalent to o= (1/V) o dV, T is the tank diameter and N is the
Fig. 6. Relative attrition volume against the collision velocity (crystalresistance) for KNO3 according to different models.
Fig. 7. Growth rate of 356 KNO3 attrition fragments against their initial size.
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increasing t*. The longer the micromixing time tmicro in comparison to the macromixing time tmacro, the smaller is the supersaturation for a given t* and, consequently, the smaller the nucleation rate.
5.2. Effect of dilution (reaction crystallization)
Fig. 8. Optimal supersaturation for cooling crystallization against the dimensionless solubility for different cooling rates or slopes of the solubility curve.
speed of the stirrer. In batch crystallizers the ratio V/(V: A +V: B) is the time of the addition of the two reactants A and B or the solution, respectively. Let us assume that the addition time of a reaction crystallizer is very short compared to the total batch time. If the micromixing time of the liquid mixture with the viscosity n for a segregation degree of 0.1 according to tmicro :5 ln Sc
n 1/2 1/2 o− loc
is very short (tmicro tmacro or tmacro/tmicro ), the maximum possible supersaturation Smax is obtained after a short time (batch crystallizers) or a short residence time (continuously operated crystallizers), see point A in Fig. 3. The supersaturation decreases with
Let us consider a T-mixer and a stirred vessel (see Fig. 10). In the case of a stirred vessel, the flow rates V: A and V: B of the reactants are diluted in the vessel and the degree of dilution increases with the vessel volume V. Therefore, the actual concentration and supersaturation depend on the ratio (V: A + V: B)/V. The actual local supersaturation can be influenced by process of: dilution (s 1/V) (see point B in Fig. 9), macromixing (s 1/N), and 1/2 micromixing (s o − loc ). Therefore, the prediction of the median crystal size requires the prediction of the mixing phenomena and the calculation of local concentrations and supersaturation profiles.
5.3. Effect of attrition (coarse products) In the case of strong attrition (e.g. high settling velocity ws of crystals), the maximum and the median crystal size are reduced with the result that the coefficient of variation, CV, decreases. The CSD can be controlled by attrition (see Fig. 11). In this diagram, the median crystal size is plotted against the residence time t of a continuously operated crystallizer. Since KNO3 (as well as potash alumn and citric acid) are very prone to attrition, the optimum residence time topt is only 1 h. For attrition-resistant materials such as KCl, NaCl, (NH4)2SO4 and tartaric acid, the residence time topt can be 2–3 h. Therefore, it is very important to know the attrition behaviour, which depends mainly on the fracture resistance of the crystalline matter, as has been shown by the equation for Batt/8v [6].
Fig. 9. Supersaturation S against the dimensionless time with tmicro as a parameter.
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A. Mersmann / Chemical Engineering and Processing 38 (1999) 345–353
6.2. Batch crystallizers
Fig. 10. Operation of a T-mixer and a stirred vessel.
6. How to influence the median crystal size The median crystal size L50 depends on the crystallization kinetics via B(S) and G(S) and on the growth time (batch) or t= V/(V: A +V: B) with L50 =3.67 · Gt for MSMPR crystallizers.
6.1. Continuously operated crystallizers Very small particles (nanoparticles) can be produced by: highly concentrated reactants and high Dc/cc (Dc/cc \ 0.1), avoidance of agglomeration (addition of surfactants and or change in pH), rapid quenching or diluting in order to stop growth (combination T-mixer+stirred vessel (see Fig. 10)) fast local micromixing of the reactants but poor macromixing, no dilution (T-mixer) in order to produce many nuclei. The contrary is true for obtaining large particles ( \ 100 mm). The reduction of attrition is the prerequisite for producing coarse crystals. As a rule, particles in the micrometer range are agglomerates formed from nanoparticles [4,2].
In order to produce very fine particles the remarks given in Section 6.1 are valid. The addition tubes of the two reactants should be very close together at a point of a high specific power input (discharge region of the impeller, preferably a Rushton turbine). Another possibility is to premix the reactants in a T-mixer where the fast chemical reaction takes place (see Fig. 10). The suspension is then introduced into the vessel where a very weak (or zero) supersaturation leads to no further activated nucleation (or growth). Large particles can be produced if the supersaturation is optimum (only very few nuclei but sufficient growth), and the residence time of the crystals is long (L50 Gt) [14].
Appendix A. Nomenclature B c cw D DAB f G H Hv Kr kd kg L mT N Ne Nv R S T T t utip V V: ws a, b o
Fig. 11. Median crystal size of KNO3 crystals against the mean residence time.
m n h r
nucleation rate (m−3 s−1) concentration (kmol m−3) drag coefficient stirrer diameter (m) diffusion coefficient (m2 s−1) factor growth rate (m s−1) height (m) Vickers Hardness (J m−3) efficiency mass transfer coefficient (m s−1) growth coefficient (m s−1) crystal size (m) suspension density (kg m−3) stirrer speed (s−1) Newton number Flow number gas constant (J mol−1 K−1) supersaturation tank diameter (m) temperature (K) time (s) stirrer tip speed (ms−1) vessel volume (m3) volumetric flow rate (m3 s−1) settling velocity (m s−1) shape factors (volume, surface) specific power input (W kg−1) shear modulus (J m−3) kinematic viscosity (m2 s−1) target efficiency density (kg m−3)
A. Mersmann / Chemical Engineering and Processing 38 (1999) 345–353
t G Gs 8v Ne Sc n/DAB t*
residence or mixing time (s) fracture resistance (J m−2) fracture resistance(J mol−1 m) volumetric hold-up Newton number of the stirrer Schmidt number dimensionless time
Subscripts and superscripts A, B act att c eff f het hom L max p prim sec tip tot w * 50
component A, B activated attrition crystalline effective flow heterogeneous homogeneous liquid maximum parent primary secondary tip speed total velocity equilibrium 50 percent
.
353
References [1] A. Mersmann, Supersaturation and nucleation, TransIChemE 74A (1996) 812 – 820. [2] H. Schubert, A. Mersmann, How agglomeration processes affect experimentally determined nucleation rates, Proc. Int. Conf. Mixing Cryst., Tioman Island, Malaysia, 1998. [3] M. Angerho¨fer, Untersuchungen zur Kinetik der Fa¨llungskristallisation von Bariumsulfat, Thesis, TU Munich, 1994. [4] H. Schubert, A. Mersmann, Determination of heterogeneous nucleation rates, TransIChemE, 78A (1996). [5] A. Mersmann, General prediction of statistically mean growth rates, J. Cryst. Growth. 147 (1995) 181 – 193. [6] C. Gahn, A. Mersmann, Theoretical prediction and experimental determination of attrition rates, TransIChemE 75A2 (1996) 125. [7] R. Ploss, T. Tengler, A. Mersmann, Scale-up of MSMPR-crystallizers, Ger. Chem. Eng. 9 (1986) 42 – 48. [8] J. Pohlisch, A. Mersmann, The influence of stress attrition on crystal size distribution, Chem. Eng. Technol. 11 (1988) 40–49. [9] T. Tengler, Wachstum und Keimbildung bei der Ku¨hlungskristallisation von Ammoniumsulfat, Thesis, TU Mu¨nchen, 1990. [10] R. Sangl, A. Mersmann, Attrition and secondary nucleation in crystallizers, in: A. Mersmann (Ed.) Proc. 11th Symp. Ind. Cryst., 1990, pp. 331 – 336. [11] A. Mersmann, Crystallization Technology Handbook, Marcel Dekker, New York, 1995. [12] U. Zacher, A. Mersmann, J. Cryst. Growth. 147 (1995) 172. [13] A. Mersmann, F. Werner, S. Maurer, K. Bartosch, Theoretical prediction of the minimum stirrer speed in mechanically agitated suspensions, Chem. Eng. Proc. 37 (1998) 503 – 510. [14] M. Ku¨hberger, A. Mersmann, How to meet product requirements during cooling crystallization by control of supersaturation, Int. Conf. Mixing Cryst., Tioman Island, Malaysia, 1998.
Crystallization and precipitation A. Mersmann * Department B of Chemical Engineering, Technical Uni6ersity Munich, Boltzmannstr. 15, D-85748 Garching, Germany Received 4 March 1999; accepted 4 March 1999
Abstract This paper deals with the design and operation of cooling and evaporation crystallizers and precipitators for reaction crystallization. It will be shown that the median crystal size, L50, of a crystalline product and the crystal size distribution (CSD) mainly depend on the crystallization kinetics, e.g. the rates of nucleation and crystal growth, and that these kinetics are controlled by supersaturation. Therefore, the main objective in crystallization and precipitation is to choose and to maintain the optimal supersaturation with respect to product quality at all times and all locations in the crystallizer or precipitator. © 1999 Elsevier Science S.A. All rights reserved. Keywords: Crystal growth; Nucleation; Agglomeration; Median crystal size
1. The role of supersaturation for the median crystal size The selection of the apparatus or reactor and its operating condition depend on the chemico-physical properties of the system which can be described by the dimensionless solubility c*/cc (c* is the solubility of the solute and cc the density of the crystalline material) and the slope d(ln c*)/d(ln T) of the solubility curve c*= f(T) with T as the absolute temperature (see Fig. 1). Evaporative crystallization is necessary in the case of flat solubility curves d(ln c*)/d(ln T) B 1. Drowing out crystallization is carried out by the addition of an antisolvent, for instance alcohols or ketones. The maximum possible supersaturation in the absence of any formation of solid matter (e.g. a theoretical supersaturation, not occurring in crystallizers) is given by the ratio
solution flow rate in cooling crystallization or cooling rate
Dedicated to Professor Em. Dr-Ing. Dr h.c. mult. E.-U. Schlu¨nder on the occasion of his 70th birthday. * Tel.: +49-89-289-15652; fax: +49-89-289-15674. E-mail address: [email protected] (A. Mersmann)
the ratio
solution flow rate in evaporative crystallization evaporation rate
or the ratio
solution flow rate in drowning out crystalli antisolvent flow rate
zation. However, we demonstrate that the actual supersaturation Dc (c −c*) or relative supersaturation s=Dc/ c* is the result of crystallization kinetics and the presence of crystals and foreign particles in the apparatus. In reaction crystallization two or more components react chemically, with the result that a new product is formed. If the initial concentration of the reactants is high, and, therefore, the initial concentration cmax of the product is also high, but the solubility c* is very low, the resulting high supersaturation Smax =1+ smax = cmax/c* or smax = Dcmax/c* leads to high rates of primary nucleation. Since foreign particles in the nanometer range are always present in the reactant solutions, heterogeneous nucleation will take place. Homogeneous nucleation can be dominant at a very high supersaturation (Dc/ cc \ 0.1 for non-dissociating products). Primary nucleation is an activated process which means that local or temporary peaks of supersaturation Dc are quickly reduced to the metastable supersaturation Dcmet(Dc Dcmet). The rate, Bprim, of primary nu-
0255-2701/99/$ - see front matter © 1999 Elsevier Science S.A. All rights reserved. PII: S 0 2 5 5 - 2 7 0 1 ( 9 9 ) 0 0 0 2 5 - 2
A. Mersmann / Chemical Engineering and Processing 38 (1999) 345–353
346
cleation is a function of the actual local supersaturation s. After this nucleation step particles grow at the moderate supersaturation Dc 5 Dcmet with the growth rate G=f(Dc). If the residence time t of the nuclei in the precipitator or the batch time are small, the size increase DL = Gt is small, too. When the crystal size exceeds approximately 100 mm, attrition fragments are formed which can act as secondary nuclei. The total nucleation rate is the sum B = Bprim +Bsec of the contributions of primary and secondary nucleation. The median crystal size and the crystal size distribution (CSD) depend on the kinetic parameters B and G. In the case of MSMPR (mixed suspension mixed product removal) crystallizers or precipitators, the median crystal size L50 is given by L50 = 3.67
GmT 6aBrc
1/4
,
with mT being the suspension density and rc being the density of the crystals. With reaction crystallization, the suspension density results from a mass balance of the
reactants in the case of complete conversion. According to the last equation, nanoparticles can be obtained by applying very high nucleation rates, Bprim, which require very strong supersaturations Smax. Such high supersaturations can be achieved by: high concentration reactants, products with high concentrations cmax but low solubilities c*, and rapid micromixing at the feed point of the reactants for a fast chemical reaction. If, however, a coarse product is desired, the supersaturation must be kept low by: using low concentration reactants (perhaps by dilution), good macromixing in the entire precipitator, and seeding by high recycling rates of slurry in order to reduce local supersaturation peaks by crystal growth. In systems with high solubilities (c*/cc \ 0.01), the supersaturation is so small (s B 0.1) that activated nucleation does not take place and new nuclei are only formed as attrition fragments. In this case large median sizes L50 can be obtained if: the attrition rate Batt Bsec is low, the crystal growth rate G has the maximum allowable value with respect to crystal purity, and the mean residence time of the slurry is optimum. In the case of batch crystallizers, the median crystal size L50 as well as the CSD are dependent on the actual supersaturation, which varies with time and also with location, if a voluminous apparatus is not ideally mixed. The kinetic crystallization parameters, i.e. the nucleation rate, B, and the growth rate, G, are necessary for the evaluation of the median crystal size.
2. Nucleation rate
Fig. 1. Solubility of various solutes in water.
The rate of primary nucleation depends on the diffusivity DAB, the solubility c*, the density cc of the crystals and the supersaturation S, and can be calculated according to the classical theory of nucleation [1]. However, with respect to a very high particle or nuclei density N (particles m − 3) a very rapid perikinetic agglomeration takes place, with the result that observed nucleation rates, Bprim, of nuclei in the micrometer range are several orders of magnitude smaller than predicted by theory [2]. In addition, foreign particles are present in every solution, leading to heterogeneous nucleation. According to experiments carried out with foreign particles added to the reactants, the nucleation work is reduced to approximately 10% of the value for homogeneous primary nucleation [3,4]. Furthermore, the rate of heterogeneous primary nucleation is proportional to the specific surface area of foreign particles
A. Mersmann / Chemical Engineering and Processing 38 (1999) 345–353
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activated attrition fragments contribute to the rate of the secondary nucleation. Above the curve valid for the rate Bhom = 1015 m − 3 s − 1 of homogeneous nucleation the rate of perikinetic agglomeration becomes higher than Bhom. If agglomeration is not hindered agglomerates and not original nuclei will be obtained.
3. Growth rates Experimental results and theoretical considerations have shown that the mean growth rate, G, of a crystal collective depends mainly on the mass transfer coefficient, kd, the dimensionless solubility, c*/cc, and the relative supersaturation, s=c/c* [5]. In Fig. 4 the ratio 6/kd with G=26 is plotted against s, with c*/cc as a parameter. As can be seen from the shaded area in this figure, the growth rate G is often in the range 10 − 9 m s − 1 B GB10 − 7 m s − 1, and can be calculated for growth controlled by diffusion for s\ 1 from G=
b Dc k , 3a d cc
where a and b are the volume or surface shape factors, respectively. Since the maximum supersaturation is cmet,act after activated nucleation, the supersaturation Dc with Dcmet,het \ Dc \0 is decisive for the growth of nuclei. With batch crystallizers, the growth rate depends on the time averaged supersaturation Fig. 2. Volumetric nucleation rate against the relative supersaturation for activated and attrition controlled nucleation.
present in the solution [3,4]. These results and further considerations allow evaluation of the metastable zone widths for homogeneous and heterogeneous nucleation. Below a certain supersaturation smet,sec or Dcmet,sec/c* the nucleation rate induced by activated nucleation is so small that new nuclei can only originate from attrition fragments [1]. In Fig. 2 the nucleation rate is plotted against the relative supersaturation s for activated (surface) nucleation (theoretical lines) and for attrition induced nucleation (shaded area). Further theoretical considerations and experimental results lead to Fig. 3, where the dimensionless metastable zone width, Dcmet/cc, (valid for B: 1015 m − 3 s − 1) is plotted against the dimensionless solubility, c*/cc, with cc as the density of the solid crystals (valid for nondissociating systems). In such a diagram lines of constant relative supersaturation s can be drawn according to Dc/cc =(Dc/c*)(c*/cc) = s · c*/ cc, and areas show where homogeneous and heterogeneous nucleation are the dominant nucleation mechanisms. In the area ‘attrition controlled’ rates of activated nucleation are so small that only growth
Dc =
1 t
&
t
Dc dt.
0
Here t is the residence time of the crystals in the crystallizer. The curves in Fig. 4 show maximum possible growth rates in the absence of impurities or additives which retard or even block crystal growth. Only agglomeration of submicron particles of high particle density N may enhance the increase of crystal size per unit time. With needle-like crystals or platelets, the definition and prediction of crystal growth rates are difficult. If the crystallizer is not ideally mixed, the local distribution of supersaturation has to be taken into account. This can be done by dividing the total active volume into sections with approximately constant supersaturation.
4. Median crystal size In the case of activated nucleation the rate Bact increases very rapidly with the supersaturation S (Bact S 10 up to Bact S 50 in the absence of agglomeration depending on c*/cc (see Fig. 2)). In reality, however, strong agglomeration takes place for nuclei
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smaller than one micrometer [2]. The higher the initial supersaturation the smaller the median crystal size L50 in the absence of agglomeration. In Fig. 5 the median crystal size is plotted against the mean relative supersaturation s (or the maximum supersaturation smax = Smax −1 for reaction crystallization) for inorganic and organic systems with different solubilities c*. Additionally, the diagram also shows experimental results of the aqueous systems of BaSO4, BaCO3, CaCO3, Mg(OH)2 and MgC2O4 2H2O [15 – 17]. As can be seen from this – figure, the median crystal size drops with increasing relative supersaturation. In reaction crystallization, the relative supersaturation, which is decisive for the median crystal size depends on: the rate constant of the chemical reaction, the number and size of foreign particles present in the reactants, the effectiveness of macro- and micromixing, and the maximum product concentration. Experiments carried out with continuously operated precipitators have shown that the mean crystal size is much larger in comparison to products obtained in batch precipitators, in which large differences of supersaturation occur with respect to time and space. Systems with high solubilities (c*/cc \0.01) lead to median crystal sizes L50 \100 mm. The larger the crystal and the higher the collision velocity wcol of crystals against a resistance (rotor, wall, other crystals), the stronger the attrition effects which lead to attrition
fragments in the size range 1 mmB Latt B 150 mm [6]. In Fig. 6, the relative abraded crystal volume DVc/Vc based on the volume Vc of a single crystal is plotted against the collision velocity wcol for 0.1 and 1 mm KNO3 crystals according to a model of Gahn [6]. The diagram also shows lines according a model developed earlier [7–10]. In all cases, a huge number of attrition fragments are produced in industrial crystallizers with crystals larger than 100 mm. A very simplified equation valid for the effective rate of attrition controlled secondary nucleation Batt, based on the volumetric crystal hold-up, 8v = mT/rc, can be derived from the model of Gahn:
H 5v Kr 3p 2rcoNv Natt,eff 3 Batt = 7 · 10 − 4 3 h whf. · 8v m G 2a 3Ne Natt,tot The hardness Hv, the shear modulus m and the fracture surface energy G are properties of the crystalline matter. The operating conditions of the stirred vessel are characterized by the mean specific power input o, the Newton number Ne and the flow number Nv of the stirrer. The ratio Natt,eff/Natt,tot of the effective attrition fragments (serving as secondary nuclei) based on the total number of such fragments is about 0.01 and depends on supersaturation. The target efficiencies hw and hf can be calculated from equations in the literature [11,6–8]. Since the mean specific power input o depends on the tip speed utip of the stirrer with its diameter D according to
Fig. 3. Metastable zone widths Dcmet/cc against the dimensionless solubility c*/cc for homogeneous, heterogeneous and attrition controlled nucleation.
A. Mersmann / Chemical Engineering and Processing 38 (1999) 345–353
o=
4Ne D p4 T
n
u , H
utip
c− c* exp
G =kg
Fig. 4. Crystal growth rate against the relative supersaturation with the ratio c*/cc as a parameter.
2 3 tip
it is necessary to restrict the circumferential velocity of rotors (stirrer, pumps) to B10 m s − 1 and the mean specific power input in stirred vessels to o B 0.5 W kg − 1 when a coarse product is desired. The growth rate, G, of small attrition fragments is reduced due to crystal deformation and an increased chemical potential [12]. This reduction can be described by
<
349
Gs RTLatt
=
g
D T
3/2
min
=
n
1 Dr 3cw · Lp · g · w 2s 8v 0.088Ne7/18 rL
1/4
with D/T as the diameter ratio (stirrer D and tank T), ws as the settling velocity of the largest crystal with the drag coefficient cw, and with the density difference Dr = rc − rL between the crystal (c) and the liquid (L). 8v is the volumetric crystal hold-up and Ne the Newton number of the stirrer. As can be seen from this equation, crystals with a large settling velocity ws are mostly endangered by attrition.
c*
with kg as the growth coefficient and g as the growth exponent, usually within the range 1B g B 2. The higher the fracture resistance Gs and the smaller the size Latt of the attrition fragment, the more the growth is reduced (see Fig. 7 valid for KNO3 attrition fragments). With respect to individual differences in deformation, the growth rate differs from fragment to fragment (growth rate dispersion, GRD). As a result of these considerations, large median sizes can only be obtained under the following conditions: the supersaturation has to be optimum (see Fig. 8), the residence time (continuously operated crystallizer) or the batch time has to be optimum (see below), the circumferential velocity utip of a rotor should be low, and the specific power input should also be low. However, with respect to the suspension of coarse crystals in a stirred vessel, a minimum value of [utip(D/ T)3/2] of the stirrer is necessary for a given suspension [13].
5. Supersaturation in crystallizers The actual local supersaturation in a crystallizer under isothermal condition is the result of the following parameters and processes: Important for sparingly soluble systems: Concentration of the reactants Rate of the chemical reaction Intensity of macro-and micromixing Dilution of solution Agglomeration Important for highly soluble systems: Crystallization kinetics (nucleation, growth) Generation of crystal surface by attrition Let us assume that precipitation is initiated by a very fast chemical reaction with the result that the progress of this reaction is controlled by macro- and micromixing. This will be explained in more detail.
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Fig. 5. Median crystal size against the initial (maximum possible) supersaturation; shaded are results of inorganic and organic systems [11].
5.1. Effect of macro- and micromixing In Fig. 9, the supersaturation S is plotted against a dimensionless time t* which is defined by
t*=
1 V t = tmacro V: A + V: B tmacro
for MSMPR crystallizers, and by t*=t ·
V: A + V: B V
for batch crystallizers. With continuously operated crystallizers, the actual supersaturation depends on the ratio of the mean residence time t= [V/(V: A + V: B)] and the macromixing time tmacro according to
p T 2/3 tmacro,min : 5 :5 1/3 4Ne (o)
1/3
T D
5/3
·
1 . N
Here o is the mean specific power input equivalent to o= (1/V) o dV, T is the tank diameter and N is the
Fig. 6. Relative attrition volume against the collision velocity (crystalresistance) for KNO3 according to different models.
Fig. 7. Growth rate of 356 KNO3 attrition fragments against their initial size.
A. Mersmann / Chemical Engineering and Processing 38 (1999) 345–353
351
increasing t*. The longer the micromixing time tmicro in comparison to the macromixing time tmacro, the smaller is the supersaturation for a given t* and, consequently, the smaller the nucleation rate.
5.2. Effect of dilution (reaction crystallization)
Fig. 8. Optimal supersaturation for cooling crystallization against the dimensionless solubility for different cooling rates or slopes of the solubility curve.
speed of the stirrer. In batch crystallizers the ratio V/(V: A +V: B) is the time of the addition of the two reactants A and B or the solution, respectively. Let us assume that the addition time of a reaction crystallizer is very short compared to the total batch time. If the micromixing time of the liquid mixture with the viscosity n for a segregation degree of 0.1 according to tmicro :5 ln Sc
n 1/2 1/2 o− loc
is very short (tmicro tmacro or tmacro/tmicro ), the maximum possible supersaturation Smax is obtained after a short time (batch crystallizers) or a short residence time (continuously operated crystallizers), see point A in Fig. 3. The supersaturation decreases with
Let us consider a T-mixer and a stirred vessel (see Fig. 10). In the case of a stirred vessel, the flow rates V: A and V: B of the reactants are diluted in the vessel and the degree of dilution increases with the vessel volume V. Therefore, the actual concentration and supersaturation depend on the ratio (V: A + V: B)/V. The actual local supersaturation can be influenced by process of: dilution (s 1/V) (see point B in Fig. 9), macromixing (s 1/N), and 1/2 micromixing (s o − loc ). Therefore, the prediction of the median crystal size requires the prediction of the mixing phenomena and the calculation of local concentrations and supersaturation profiles.
5.3. Effect of attrition (coarse products) In the case of strong attrition (e.g. high settling velocity ws of crystals), the maximum and the median crystal size are reduced with the result that the coefficient of variation, CV, decreases. The CSD can be controlled by attrition (see Fig. 11). In this diagram, the median crystal size is plotted against the residence time t of a continuously operated crystallizer. Since KNO3 (as well as potash alumn and citric acid) are very prone to attrition, the optimum residence time topt is only 1 h. For attrition-resistant materials such as KCl, NaCl, (NH4)2SO4 and tartaric acid, the residence time topt can be 2–3 h. Therefore, it is very important to know the attrition behaviour, which depends mainly on the fracture resistance of the crystalline matter, as has been shown by the equation for Batt/8v [6].
Fig. 9. Supersaturation S against the dimensionless time with tmicro as a parameter.
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6.2. Batch crystallizers
Fig. 10. Operation of a T-mixer and a stirred vessel.
6. How to influence the median crystal size The median crystal size L50 depends on the crystallization kinetics via B(S) and G(S) and on the growth time (batch) or t= V/(V: A +V: B) with L50 =3.67 · Gt for MSMPR crystallizers.
6.1. Continuously operated crystallizers Very small particles (nanoparticles) can be produced by: highly concentrated reactants and high Dc/cc (Dc/cc \ 0.1), avoidance of agglomeration (addition of surfactants and or change in pH), rapid quenching or diluting in order to stop growth (combination T-mixer+stirred vessel (see Fig. 10)) fast local micromixing of the reactants but poor macromixing, no dilution (T-mixer) in order to produce many nuclei. The contrary is true for obtaining large particles ( \ 100 mm). The reduction of attrition is the prerequisite for producing coarse crystals. As a rule, particles in the micrometer range are agglomerates formed from nanoparticles [4,2].
In order to produce very fine particles the remarks given in Section 6.1 are valid. The addition tubes of the two reactants should be very close together at a point of a high specific power input (discharge region of the impeller, preferably a Rushton turbine). Another possibility is to premix the reactants in a T-mixer where the fast chemical reaction takes place (see Fig. 10). The suspension is then introduced into the vessel where a very weak (or zero) supersaturation leads to no further activated nucleation (or growth). Large particles can be produced if the supersaturation is optimum (only very few nuclei but sufficient growth), and the residence time of the crystals is long (L50 Gt) [14].
Appendix A. Nomenclature B c cw D DAB f G H Hv Kr kd kg L mT N Ne Nv R S T T t utip V V: ws a, b o
Fig. 11. Median crystal size of KNO3 crystals against the mean residence time.
m n h r
nucleation rate (m−3 s−1) concentration (kmol m−3) drag coefficient stirrer diameter (m) diffusion coefficient (m2 s−1) factor growth rate (m s−1) height (m) Vickers Hardness (J m−3) efficiency mass transfer coefficient (m s−1) growth coefficient (m s−1) crystal size (m) suspension density (kg m−3) stirrer speed (s−1) Newton number Flow number gas constant (J mol−1 K−1) supersaturation tank diameter (m) temperature (K) time (s) stirrer tip speed (ms−1) vessel volume (m3) volumetric flow rate (m3 s−1) settling velocity (m s−1) shape factors (volume, surface) specific power input (W kg−1) shear modulus (J m−3) kinematic viscosity (m2 s−1) target efficiency density (kg m−3)
A. Mersmann / Chemical Engineering and Processing 38 (1999) 345–353
t G Gs 8v Ne Sc n/DAB t*
residence or mixing time (s) fracture resistance (J m−2) fracture resistance(J mol−1 m) volumetric hold-up Newton number of the stirrer Schmidt number dimensionless time
Subscripts and superscripts A, B act att c eff f het hom L max p prim sec tip tot w * 50
component A, B activated attrition crystalline effective flow heterogeneous homogeneous liquid maximum parent primary secondary tip speed total velocity equilibrium 50 percent
.
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References [1] A. Mersmann, Supersaturation and nucleation, TransIChemE 74A (1996) 812 – 820. [2] H. Schubert, A. Mersmann, How agglomeration processes affect experimentally determined nucleation rates, Proc. Int. Conf. Mixing Cryst., Tioman Island, Malaysia, 1998. [3] M. Angerho¨fer, Untersuchungen zur Kinetik der Fa¨llungskristallisation von Bariumsulfat, Thesis, TU Munich, 1994. [4] H. Schubert, A. Mersmann, Determination of heterogeneous nucleation rates, TransIChemE, 78A (1996). [5] A. Mersmann, General prediction of statistically mean growth rates, J. Cryst. Growth. 147 (1995) 181 – 193. [6] C. Gahn, A. Mersmann, Theoretical prediction and experimental determination of attrition rates, TransIChemE 75A2 (1996) 125. [7] R. Ploss, T. Tengler, A. Mersmann, Scale-up of MSMPR-crystallizers, Ger. Chem. Eng. 9 (1986) 42 – 48. [8] J. Pohlisch, A. Mersmann, The influence of stress attrition on crystal size distribution, Chem. Eng. Technol. 11 (1988) 40–49. [9] T. Tengler, Wachstum und Keimbildung bei der Ku¨hlungskristallisation von Ammoniumsulfat, Thesis, TU Mu¨nchen, 1990. [10] R. Sangl, A. Mersmann, Attrition and secondary nucleation in crystallizers, in: A. Mersmann (Ed.) Proc. 11th Symp. Ind. Cryst., 1990, pp. 331 – 336. [11] A. Mersmann, Crystallization Technology Handbook, Marcel Dekker, New York, 1995. [12] U. Zacher, A. Mersmann, J. Cryst. Growth. 147 (1995) 172. [13] A. Mersmann, F. Werner, S. Maurer, K. Bartosch, Theoretical prediction of the minimum stirrer speed in mechanically agitated suspensions, Chem. Eng. Proc. 37 (1998) 503 – 510. [14] M. Ku¨hberger, A. Mersmann, How to meet product requirements during cooling crystallization by control of supersaturation, Int. Conf. Mixing Cryst., Tioman Island, Malaysia, 1998.