Ind. Eng. Chem. Res. 1996, 35, 1985-1994
1985
Formation and Aggregation of Polymorphs in Continuous Precipitation. 1. Mathematical Modeling Debasish Chakraborty and Suresh K. Bhatia* Department of Chemical Engineering, Indian Institute of Technology, Powai, Bombay 400 076, India
In many cases of precipitation the product comprises of two or more polymorphs, either due to their simultaneous nucleation or due to transformation of a metastable polymorph. These polymorphs may have different growth rates as well as aggregation tendencies. In the present work the population balance equation is applied to predict the size distribution of the precipitate particles allowing for these features. The Hounslow method for discretization of the population balance equation is modified for improved stability and accuracy, and used to solve the model for polymorph precipitation in a continuous system. The effect of the aggregation tendencies of the polymorphs, as well as of the ratios of their nucleation rates and of their grow rates, on the particle size distribution and product composition is studied. Introduction The estimation of particle size distribution and precipitation rates from supersaturated solutions is a problem of great importance for researchers engaged in studies of crystallization and precipitation. While this is a straightforward matter when only the processes of nucleation and growth are involved (Randolph and Larson, 1985), numerous cases are rendered more complex due to an attendant problem of aggregation. Among the early studies reporting the latter phenomenon, Maruscak et al. (1971) observed agglomeration of calcium carbonate crystals, while Halfon and Kaliaguine (1976) found the effect to be significant in alumina trihydrate crystallization. More recently the importance of agglomeration for calcium carbonate precipitation has also been confirmed by other workers (Hostomsky and Jones, 1991; Chakraborty et al., 1994; Tai and Chen, 1995). Various other studies have shown agglomeration to be important in the precipitation of other compounds such as nickel ammonium sulfate (Tavare et al., 1985), titania (Lamey and Ring, 1986), and copper sulfate pentahydrate (Zumstein and Rousseau, 1989) as well. The effect appears to be important in the precipitation of organic salts also. Thus, Franck et al. (1988) found agglomeration in salicylic acid precipitation, while Hartel et al. (1986) and Hounslow et al. (1988) report this for calcium oxalate. In fact, in the latter case the studies indicate agglomeration of calcium oxalate, rather than particle growth, to be the dominant factor responsible for formation of urinary stones. The mathematical representation of the precipitation process is generally approached via the population balance equation (Hulburt and Katz, 1964), whose solution in the presence of crystal agglomeration is a difficult numerical problem (Marchal et al., 1988; Hounslow, 1990; Bhatia and Chakraborty, 1992). Even more complex situations arise when the precipitate is present as a mixture of polymorphs, each with a different nucleation and growth rate as well as aggregation tendency. In such a case the conventional population balance formulation must be extended to allow for these complications, but this has hitherto not been attempted. An important example where such a more elaborate analysis is essential is that of calcium carbonate pre* Author to whom correspondence should be addressed. Now with Department of Chemical Engineering, The University of Queensland, St. Lucia, Brisbane, QLD 4072, Australia.
S0888-5885(95)00401-5 CCC: $12.00
cipitation. It has long been known (Brooks, 1950; Wray and Daniels, 1957) that calcium carbonate precipitates as a mixture of calcite, vaterite, and aragonite with the relative proportions depending on process conditions. Nevertheless even recent studies (Hostomsky and Jones, 1991; Tai and Chen, 1995), while confirming the polymorphism, have used the conventional single-species formulation. The need for a more elaborate multispecies analysis is most clearly evident from the recent study of the authors and co-workers (Chakraborty et al., 1994) which demonstrated the simultaneous nucleation of calcite and vaterite from the solution under conditions of heterogeneous nucleation and only vaterite under conditions of homogeneous nucleation. The results, obtained using cryo-electron microscopy, are not readily possible with conventional scanning electron microscopy or X-ray diffraction because of instability of the vaterite. Indeed, it was demonstrated that the homogeneously nucleated vaterite, being morphologically imperfect, began to transform to calcite within the precipitator itself which was operated in the continuous mode. The gradual transformation of vaterite into calcite in the supersaturated solution has also previously been reported (Sohnel and Mullin, 1982). Further complications arise from the different aggregation tendencies of calcite and vaterite, for it was noted (Chakraborty et al., 1994) that calcite agglomerated much more strongly than vaterite. It may be noted that the transformation of different phases is a well known phenomenon. The formation of calcite following the nucleation of vaterite particles in the presence of cholesterol is speculated as due to the transformation of vaterite (Dalas and Koutsoukos, 1988). The effect of citrate ions in hydrolysis of ferric chloride is well reported in the literature (Kandori et al., 1992). A faster transformation of β-FeOOH into cubic and rhomboidal hematite particles is achieved in the presence of 0.05 mol % of citrate ions which reduces the transformation period by 50%. As reported by Kandori et al. transformation into the most stable state takes a litter longer time without any other agent. It is evident from the above discussion that situations involving polymorphic precipitation, such as that of calcium carbonate, will require extension of the conventional population balance model and of its solution methods. Among the numerical methods reported the most accurate is that of Bhatia and Chakraborty (1992) which avoids the finite domain error inherent to the © 1996 American Chemical Society
1986
Ind. Eng. Chem. Res., Vol. 35, No. 6, 1996
Table 1. Aggregation Model type of particle species 1 species 2 species 1 species 1 species 2 aggregate
type of particle + + + + + +
species 1 species 2 species 2 aggregate aggregate aggregate
f f f f f f
product
constant aggregation kernel
aggregate aggregate aggregate aggregate aggregate aggregate
β1 β2 β1 β1 β1 β1
other techniques (Gelbard and Seinfeld, 1978; Chang and Wang, 1984a,b; Marchal et al., 1988; Hounslow, 1990). However, the method is computationally intensive and unlikely to be practical for the more complex situation of polymorphic precipitation. For such cases the method of Hounslow may be more attractive and can readily be modified for enhanced accuracy and applied. In the present work we solve this problem for two possible situations. In the first case we consider the simultaneous nucleation, growth, and agglomeration of two different polymorphs. A constant aggregation kennel is used, while assuming each polymorph to have a different agglomeration tendency. In the second case nucleation of a single polymorph is considered, but which subsequently transforms to a second one. The latter continues to grow and aggregate along with the first. As seen above these situations are representative of CaCO3 precipitation. While we present the models and their solution here, in the sequel article we apply these to our experimental data on this system. Theory Simultaneous Nucleation of Two Polymorphs. Population Balance Formulation. As indicated above the situation of simultaneous nucleation of two polymorphs is found in the case of calcium carbonate where, in the presence of heterogeneous nucleation and high supersaturation, both calcite and vaterite are formed (Chakraborty et al., 1994). While not easily evident by conventional methods this feature could be confirmed by on-line video scans and by cryo-electron microscopy. Further support is available from the curvature of the particle size distribution (Bhatia and Chakraborty, 1992; Chakraborty and Bhatia, 1995). In light of these findings we consider here the case of simultaneous nucleation with different growth rates and aggregation tendencies. The two polymorphs are assigned nucleation rates B1 and B2 and growth rates G1 and G2, respectively. The growth rate of aggregates is taken to be Ga, and a constant aggregation kernel is assumed for each species. The constant aggregation kernel is taken as β1 and β2 for the primary particles. Further, we consider species 1 to be much more aggregating in nature than species 2. Thus, species 1 and species 2 aggregate with a constant kernel β1, and aggregates form mainly due to the aggregating nature of species 1. Once the aggregate is formed, it attains a different growth rate, Ga. Aggregates are considered to further aggregate with constant kernel β1, which is the same as that of the more aggregating primary particles. The various aggregation processes are depicted in simplified form in Table 1. The assignment of kernel β1 to events concerning all aggregates is clearly approximate, as some of these would comprise only of species 2. However, since this species is much less agglomerating such aggregates would be relatively few and are neglected, thus avoiding further complications of subdividing aggregates.
The population balance equation (PBE) for species 1 in a well mixed continuous precipitator at steady state is given by
G1
dn1 ) -n1(l)β1 dl
∫0∞n dl -
n1 τ
(1)
where n1(l) is the population density of species 1 and l is particle size. The PBE of species 2 can be expressed as
G2
dn2 ) dl -n2(l)β1
∫0∞n dl - n2(l)(β2 - β1)∫0∞n2 dl -
n2 (2) τ
where n2(l) represents the population density of species 2. The PBE of aggregates can be obtained from the formation by aggregation of smaller particles and the loss of aggregates of a given size due to agglomeration. Thus, PBE of aggregates can be written as
Ga
dna β1 2 ) l dl 2
3 1n(λ)n((l
∫0
- λ3)1/3)
(l3 - λ3)2/3
(β2 - β1) 2 l 2
∫01
dλ +
n2(λ)n2((l3 - λ3)1/3) (l3 - λ3)2/3
dλ -
∫0∞n dl -
na(l)β1
na (3) τ
where na(l) is the population density of aggregates. Now the total number density is given by
n(l) ) n1(l) + n2(l) + na(l)
(4)
while boundary conditions are
n1(0) ) B1/G1
(5)
n2(0) ) B2/G2
(6)
na(0) ) 0
(7)
Dimensionless Formulation. Equations 1 and 2 may be readily integrated to yield in dimensionless form
q1(y) ) exp[-R(1 + φ1m0)y]
(8)
q2(y) ) exp[-(1 + φ1m0 + b(φ2 - φ1)µ02)y]
(9)
and
Here
m0 )
∫0∞(Rq1 + bq2 + bδqa) dy
(10)
∫0∞q2(y) dy
(11)
and
µ02 )
Equations 9 and 11 readily combine to yield a relation between m0 and µ02 2 b(φ2 - φ1)µ02 + (1 + φ1m0)µ02 - 1 ) 0
(12)
Further, we define the dimensionless total population
Ind. Eng. Chem. Res., Vol. 35, No. 6, 1996 1987
density
q(y) )
G2n(l) 1 ) (Rq1 + bq2 + bδqa) B2 b
(13)
which combines with the dimensionless form of eq 3 to yield
φ1 dq - δq + bδ y2 dy 2
3
yq(λ)q((y
∫0
(φ2 - φ1) 2 y bδ 2 bδφ1q
∫0y
the constant aggregation kernel β1 is assigned to the aggregates formed from the primary particles. These features of the aggregation are considered the same as those in the simultaneous nucleation model (Table 1). Population balance of species 1 in a well-mixed continuous precipitator at steady state then yields
G1
3 1/3
-λ ) )
(y3 - λ3)2/3 3
dλ +
(y3 - λ3)2/3
dλ -
∫0∞q(λ) dλ + q1(1 + φ1m0)Rb(δ - R) +
G1
µ0 )
∫0 q(y) dy ) m0/b
(18)
dn2 ) dl -n2(l)β1
n2 - kn2 τ (19)
(15)
Equation 14 represents the general PBE of a simultaneous nucleation system which includes the different growth rates and aggregation frequencies of two different species, following the model of Table 1. The dimensionless total number of particles (µ0) can be obtained analytically from integration of eq 13 as ∞
n1 τ
∫0∞n dl - n2(l)(β2 - β1)∫0∞n2 dl -
(1 + φ1m0)(δ - 1)q2 - µ02b(φ2 - φ1)q2 ) 0 (14) R R q(0) ) q1(0) + q2(0) ) + 1 b b
∫0∞n dl -
where n1(l) is the population density of species 1 and l is particle size. The PBE of species 2 can be expressed as
3 1/3
q2(λ)q2((y - λ ) )
dn1 ) kn2 - n1(l)β1 dl
where n2(l) is the population density of species 2. PBE for aggregates can be obtained from the formation by aggregation of primary particles and the destruction of the aggregates due to the aggregation. Thus, PBE of aggregates can be written as
Ga
dna β1 2 ) l dl 2
∫01
n(λ)n((l3 - λ3)1/3)
(16) (β2 - β1) 2 l 2
Integration of eq 14 over the domain [0, ∞] leads to a second nonlinear algebraic equation relating m0 and µ02
φ1 2 1 m + (m - 1) - (1 + φ1m0)µ02 2b 0 b 0 b 2 (φ - φ1)µ02 ) 0 (17) 2 2 Equations 12 and 17 can be solved simultaneously to obtain m0 and µ02. Substitution of m0 in eq 16 gives the dimensionless total number of particles in the system. This directly obtained zeroth moment can be compared with the moment obtained from numerical solution of the PBE of the present system represented by eq 14 to check the adequacy of the latter. Nucleation Followed by Transformation. Population Balance Formulation. Here we consider the nucleation of a single species and subsequent transformation into another polymorph. It is assumed that nucleation of species 2 occurs at rate B2, and this is subsequently transformed into species 1. This transformation is assumed irreversible and governed by a first-order rate constant k, i.e. k
species 2 98 species 1 Transformed particles grow with a different growth rate G1 while the mother particles are considered to have a growth rate of G2. These two species are considered prone to aggregation with constant aggregation kernel β1 and β2, respectively. Once the aggregates are formed, they are assumed to have a different growth rate Ga. As in the earlier section, species 1 is considered to be more aggregating in nature. Thus, species 1 and species 2 aggregate with a constant of aggregation β1. Further,
(l3 - τ3)2/3 3 ln2(λ)n2((l
∫0
dλ +
- λ3)1/3)
(l3 - λ3)2/3 na(l)β1
dλ -
∫0∞n dl -
na (20) τ
where na(l) represents the population density of aggregates. The total number density is as before given by eq 4, while boundary conditions follow
n1(0) ) 0
(21)
and eqs 6 and 7. Dimensionless Formulation. Equations 18 and 19 readily integrate to the dimensionless forms
q1(y) ) K × {R(1 + φ1µ0) - [1 + K + φ1µ0 + µ02(φ2 - φ1)]} (q2 - exp[-R(1 + φ1µ0)y]) (22) and
q2(y) ) exp(-[1 + K + φ1µ0 + µ02(φ2 - φ1)]y) (23) Here µ0 ) ∫∞0 q dy, as before, and the dimensionless density q1 is now defined with respect to the nucleation rate of species 2 (B2). The dimensionless total population density is now given, following eq 4, as
q(y) ) Rq1 + q2 + δqa
(24)
which combines with the dimensionless form of eq 20
1988
Ind. Eng. Chem. Res., Vol. 35, No. 6, 1996
to yield
-
φ1 dq - δq + δ y2 dy 2
∫0y
(φ2 - φ1) 2 y δ 2
3
q(λ)q((y - λ ) ) (y3 - λ3)2/3
3 yq2(λ)q2((y
∫0
δφ1q
3 1/3
dλ +
- λ3)1/3)
(y3 - λ3)2/3
dλ -
∫0∞q(λ) dλ + q1R(1 + φ1µ0)(δ - R) +
q2[(1 + φ1µ0)(δ - 1) + K(R - 1) - µ02(φ2 - φ1)] ) 0 (25) with q(0) ) 1. Here the aggregation kernels are now made dimensionless by means of the nucleation rate B2. Integration of eq 23 now yields a second-order equation for µ02 as 2 (φ2 - φ1)µ02 + (1 + Kφ1µ0)µ02 - 1 ) 0
(26)
while integration of eq 25 over the domain [0,∞] leads to a second nonlinear algebraic equation relating µ0 and µ02
δφ1 2 δ(φ2 - φ1) 2 µ + µ02 + 1 - δµ0 2 0 2 K(δ - R) × {R(1 + φ1µ0) - [1 + K + φ1µ0 + µ02(φ2 - φ1)]}
ties. A departure from the original presentation of Hounslow is that we use the discretized form only for i g 2, while approximating Q1 analytically after neglecting aggregation at the small sizes in the first interval (i.e. for i ) 1). Additionally, we also consider the particles in the region O e y e y1, having dimensionless number Q0, in formulating the discretized equation, with Q0 being evaluated in the same way as Q1. With these modifications the method was found to have improved stability as well as accuracy for a given value of N, in tests for the single-species case (Chakraborty, 1994). In the subsequent subsections we present the discretized forms for the PBE for the two cases of polymorphic precipitation dealt with here. Simultaneous Nucleation. For this case discretization of the PBE in eq 14, following the procedure of Hounslow (1990), and considering also the particles in [0,y1], leads to
[
i-1
i-1
Qi
N
1
[
i-2
(2) δb(φ2 - φ1) Qi-1
Q(2) i
[(1 + φ1µ0)(δ - 1) + K(R - 1) - µ02(φ2 - φ1)]µ02 ) 0
1
N
(2) (2) 2j-iQ(2) ∑ j - Qi ∑Qj + j)1 j)i
]
R b
(2) (δ - R)(1 + φ1m0)Q(1) i + (1 + φ1m0)(δ - 1)Qi +
µ02b(φ2 - φ1)(δ - 1)Q(2) i +
Numerical Solution
yi+1 ) ryi
(28)
where the ratio r is taken as 21/3. In each interval we define dimensionless numbers of particles by
Qi )
∫yy
q(y) dy
∫y
q1(y) dy
i+1
i
Q(2) i )
i
∫yy
i+1
i
2
[
r
Qi-1 + Qi -
(1 + r)yi r2 - 1
r
]
Qi+1 r2 - 1 δQi ) 0, i ) 2, 3, ..., N (32)
where superscripts (1) and (2) represent species 1 and 2, respectively. Now N - 1 nonlinear algebraic equations are solved simultaneously to obtain Qi, i g 2, while Q0 and Q1 are obtained analytically by neglecting the aggregate part in eq 13. With qa neglected, eqs 8 and 9 combine to yield, upon integration,
Q0 )
Q1 )
[1 - exp(-Rv1y1)] [1 - exp(-v2y1)] + (33) bv1 v2
[exp(-Rv1y1) - exp(-Rv1y2)] + bv1 [exp(-v2y1) - exp(-v2y2)] (34) v2
where v1 and v2 are defined as
yi+1
)
(29)
+
1 2-i (2) (2) 1 1-i (2) (2) 2 Qi-1Q0 - 2 Qi Q0 + 3 3
(27)
The above models of polymorphic precipitation require a numerical solution due to the nonlinear nature of the overall PBE. In an earlier work (Bhatia and Chakraborty, 1992) we have presented an approach based on an adaptation of the method of weighted residuals which is more accurate than other procedures since it avoids the finite domain error. However, the method is computationally intensive and was not considered practical for the present case of more than one species, particularly in view of the need to fit experimental data, as dealt with in the sequel article (Chakraborty and Bhatia, 1996). Consequently the Hounslow method (Hounslow, 1990) is used here with some modification to improve stability and accuracy. In this method we consider N subintervals in the domain (y1, yN+1) with boundary points satisfying
]
(2) 2 2j-i+1Q(2) ∑ j + (Qi-1) 2 j)1
i-1
Equations 26 and 27 can be solved simultaneously to yield µ0 and µ02. The directly obtained zeroth moment µ0 can be subsequently compared with the numerically obtained value from solution of the PBE in eq 25.
1
2j-iQj - Qi∑Qj + 22-iQi-1Q0 - 21-iQiQ0 ∑ 3 3 j)1 j)i
[R(1 + φ1µ0)µ02 - 1] +
Q(1) i
1
2 2j-i+1Qj + Qi-1 ∑ 2 j)1
δbφ1 Qi-1
q2(y) dy
(30)
v1 ) 1 + φ1m0
(35)
(31)
v2 ) 1 + φ1m0 + b(φ2 - φ1)µ02
(36)
and discretize the overall PBE in terms of these quanti-
The jth moment may be obtained accurately by modify-
Ind. Eng. Chem. Res., Vol. 35, No. 6, 1996 1989
ing Hounslow’s method and considering also the particles below the first interval and is given by N
µj )
∑ i)1
(
)
(1 + r) j yi Qi + 2
[
j!
R
b (Rv )j+1 1
j
1-
j! (v2)j+1
[
∑ (Rv1y1)k k)0
[1 - exp(-u2y1)] (41) u2
exp(-Rv1y1) k!
∑ (v2y1)k k)0
]
[exp(-Ru1y1) - exp(-Ru1y2)] + Q1 ) -C1 Ru1 [exp(-u2y1) - exp(-u2y2)] (42) (C1 + 1) u2
+
k!
j ) 0, 1, 2, 3 (37) in which the last two terms represent the contribution from the particles in [0,y1]. The present method also gives the individual zeroth moment of the different species. The zeroth moment of species 1 is obtained analytically from integration of eq 8 and is expressed as
µ01 )
1 R(1 + φ1m0)
(38)
while the zeroth moment for species 2, µ02 follows eq 12. µ0a is obtained from substitution of µ01 and µ02 into the integrated form of eq 13.
(39)
where µ0 is the overall zeroth moment defined in eq 16. Nucleation Followed by Transformation. For this case discretization of the PBE in eq 25, considering also the particles in [0,y1], leads to
[
i-1 1 2 δφ1 Qi-1 2j-i+1Qj + Qi-1 - Qi 2j-iQj 2 j)1 j)1
∑
∑
N
1
]
1
Qj + 22-iQi-1Q0 - 21-iQiQ0 ∑ 3 3 j)i
Qi
[
i-2
(2) δ(φ2 - φ1) Qi-1
Q(2) i
(43)
u2 ) 1 + K + φ1µ0 + µ02(φ2 - φ1)
(44)
and
C1 )
RK {R(1 + φ1µ0) -[1 + K + φ1µ0 + µ02(φ2 - φ1)]} (45)
Now the jth dimensionless moment may be obtained accurately by modifying Hounslow’s method where all the particles below the first interval are considered, and is given by N
( ) ( [
∑ i)j C1
(1 + r) j yi Qi 2 j!
(Ru1)j+1
1-
j
∑
(Ru1y1)k
k)0
[
+
1
[
µ02 -
(2) (1 + φ1µ0)(δ - 1)Q(2) i + µ02(φ2 - φ1)(δ - 1)Qi +
[
Qi-1 + Qi -
(1 + r)yi r2 - 1
∑
]
K × {R(1 + φ1µ0) - [1 + K + φ1µ0 + µ02(φ2 - φ1)]}
]
r
+
j exp(-u2y1) j! 1(u2y1)k (C1 + 1) k! k)0 (u2)j+1
1 1-i (2) (2) 2 Qi Q0 + R(δ - R)(1 + φ1µ0)Q(1) i + 3
2
k!
µ01 )
(2) (2) 2-i (2) 2j-iQ(2) Qi-1Q(2) ∑ j - Qi ∑Qj + 2 0 3 j)1 j)i
K(R - 1) +
])
exp(-Ru1y1)
j ) 0, 1, 2, 3 (46)
1
N
u1 ) 1 + φ1µ0
in which the last two terms represent the contribution from the particles in [0,y1]. The zeroth moment estimated by the numerical method may be compared with the value of µ0 obtained from the solution of eqs 26 and 27. We also obtain the analytical zeroth moment of species 1 (i.e. µ01) by integration of eq 22 as
(2) 2 2j-i+1Q(2) ∑ j + (Qi-1) 2 j)1
i-1
where u1 and u2 are defined as
µj )
1 µ0 ) (Rµ01 + bµ02 + b δµ0a) b
i-2
[1 - exp(-Ru1y1)] Q0 ) -C1 + Ru1 (C1 + 1)
exp(-v2y1)
j
1-
]
Consequently, upon integrating eqs 22 and 23
r
]
Qi+1 -
r2 - 1
]
1 (47) R(1 + φ1µ0)
while the zeroth moment of the aggregates µ0a is obtained analytically from substitution of µ01 and µ02 into the integrated form of eq 24
µ0 ) Rµ0I + µ02 + δµ0a
(48)
Results and Discussion
δQi ) 0
i ) 2, 3, ..., N (40)
(2) with Qi, Q(1) i , and Qi following eqs 29-31. As before N - 1 nonlinear algebraic equations are solved simultaneously to obtain Qi, i g 2, while Q0 and Q1 are obtained analytically by neglecting the aggregates in eq 24.
The set of N - 1 nonlinear algebraic equations provided by eqs 32 or 40 was solved by a Marquardt Levenberg method implemented in subroutine NEQNF of the IMSL library. The mainframe CYBER 180 series computer was used for the computations, and convergence of the solution was obtained in 10-20 cpu
1990
Ind. Eng. Chem. Res., Vol. 35, No. 6, 1996
Figure 1. Effect of ratio of nucleation rates, b, on (a) fractional oversize distribution and (b) dimensionless moments, for R ) δ ) 1 and φ1 ) φ2 ) 10.
seconds. With the modified method the solution was obtained with N ) 30 and y1 ) 0.02, achieving faster convergence than the original work of Hounslow (1988, 1990) where N ) 50 and y1 ) 0.002 were recommended. The solution obtained in the form of the population density is transformed into fractional oversize, since the latter form is often that in which experimental data are available. The fractional oversize of the individual species as well as that of the total population is estimated as
∫y∞q1 dy θ1(y) ) ∞ ∫0 q dy
(49)
∫y∞q2 dy θ2(y) ) ∞ ∫0 q dy
(50)
∫y∞qa dy θa(y) ) ∞ ∫0 q dy
(51)
∫y∞q dy θ(y) ) ∞ ∫0 q dy
(52)
Simultaneous Nucleation. Initially computations were done for this case through the solution of eq 32, with φ1 ) φ2 ) 10. For these values β1 ) β2, and the assumption that all aggregates have aggregation kernel β1 remains valid. Figure 1a depicts the results for the fractional oversize distribution, with R ) δ ) 1, for various values of b. The curvature is consistent with that reported for aggregating systems (Bhatia and Chakraborty, 1992). With increase in b the aggregation terms become more important, as is readily seen from eq 14, and the particle size distribution becomes finer. Figure 1b shows the variation of the various moments with b. At low values of b the rate of nucleation of species 1 is relatively higher, leading to a larger value of Rµ01/b as compared to that of µ02. Also the number of aggregates is determined by species 1 so that µ0a is closer to Rµ01/b, as is the total number represented by Rµ0 (curve 4). At high values of b, however, species 1 has a relatively low nucleation rate so that Rµ02 exceeds
Rµ0/b. Similarly, the number of aggregates and the total number of particles are now determined by species 2 so that µ0a as well as µ0 is closer to µ02. Curve 5 shows that the dimensionless total precipitation rate, represented by µ3, decreases with a increase in b, consistent with the increased importance of aggregation mentioned above. Figure 2a illustrates the effect of R on the fractional oversize distribution, with δ ) R, b ) 1, and φ1 ) φ2 ) 10. For this situation an increase in R implies a relatively lower growth rate of species 1 and the aggregates, leading to a finer distribution. This also leads to a drop in dimensionless precipitation rate, given by the third moment µ3, as shown in Figure 2b. Figure 3a depicts the effect of aggregation parameter φ1 on the oversize distribution, with φ2 ) 0 and R ) δ ) b ) 1. For these parameter values the contributions of all the species may be expected to be significant. As expected, with an increase in φ1 the increasing significance of aggregation leads to a finer particle size distribution. Figure 3b depicts the variation of the various moments with φ1. At low values of φ1 the number of aggregates is smaller than that of species 1 or 2; however, with an increase in φ1 the proportion of aggregates subsequently becomes higher than that of these species. Further, since species 2 is nonaggregating (i.e. φ2 ) 0) its amount is always greater than that of species 1. For the case of φ1 ) 0 with R ) δ ) b ) 1, it may be anticipated that the oversize distribution results match those for the single-species case, with µ0 ) 2 and µ3 ) 12 (since each species has a dimensionless zeroth moment of unity and third moment of six). This was indeed borne out by a test computation which yielded µ0 ) 2.024 and µ3 ) 12.357 following eqs 32 and 37. Nucleation Followed by Transformation. In this case it has been considered that only species 2 is nucleated, which subsequently grows and aggregates while also gradually undergoing transformation to species 1. Nucleation of vaterite followed by transformation to calcite is an example of such a system, besides others, as discussed earlier. The overall population density distribution follows eq 25 which has been discretized in eq 40. Figure 4 depicts the fractional oversize distribution obtained from the solution of the latter equation, for parameter values φ1 ) 5, φ2 ) 0.1, R ) 1, δ ) 0.8, and K ) 0.1. For this low value of the dimensionless transformation rate constant K, species
Ind. Eng. Chem. Res., Vol. 35, No. 6, 1996 1991
Figure 2. Effect of ratio of growth rates, R, on (a) fractional oversize distribution and (b) dimensionless third moment, µ3, with δ ) R, b ) 1, and φ1 ) φ2 ) 10.
Figure 3. Effect of aggregation parameter φ1 on (a) fractional oversize distribution and (b) dimensionless moments, for R ) δ ) b ) 1 and φ2 ) 0.
Figure 4. Fractional oversize distributions for various constituents and the overall product, for φ1 ) 5, φ2 ) 0.1, R ) 1, δ ) 0.8, and K ) 0.1.
2 is much more numerous than 1, as seen from curves 2 and 1, and at small sizes the former dominates. At large size, however, aggregates dominate, following the relatively large value of φ1, as seen from curves 3 and 4. Figure 5a,b depicts the effect of K on the oversize distributions for species 1 and 2, respectively, with the
other parameters having values φ1 ) 10, φ2 ) 0, R ) 1, and δ ) 0.8. With an increase in the value of K the transformation of species 2 to 1 is more rapid, so that the proportion of the latter increases at small size. However, because of the large aggregating tendency of species 1, reflected in the value of φ1 ) 10, this leads to a decrease in its proportion at large size as is evident in Figure 5a. For the nonaggregating species 2 (φ2 ) 0), however, the effect of an increase in K consistently leads to a drop in its proportion at all sizes due to the transformation. Figure 6 shows the effect of K on the overall fractional oversize distribution, indicating a significant shift toward finer sizes with an increase in K. This is due to the higher aggregation tendency of species 1 resulting from the transformation. Such behavior is qualitatively consistent with that observed by us in CaCO3 precipitation, where calcite aggregates much more strongly than vaterite, as discussed earlier by Chakraborty et al. (1994) and in the sequel article (Chakraborty and Bhatia, 1996). Figure 7 depicts the effect of growth rate ratio R on the fractional oversize distribution, with δ ) R and other parameters having values φ1 ) 10, φ2 ) 0, and K ) 1. With an increase in R the relative growth rate of species 1 and of the aggregates (since δ ) R) is reduced, and the distribution is finer. As a result the overall dimensionless precipitation rate, represented by the third
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Figure 5. Effect of dimensionless transformation rate constant, K, on fractional oversize distribution of (a) species 1 and (b) species 2, for φ1 ) 10, φ2 ) 0, R ) 1, and δ ) 0.8.
Figure 6. Effect of dimensionless transformation rate constant, K, on overall fractional oversize distribution for φ1 ) 10, φ2 ) 0, R ) 1, and δ ) 0.8.
Figure 7. Effect of ratio of growth rates, R, on fractional oversize distribution with δ ) R, φ1 ) 10, φ2 ) 0, and K ) 1.
moment µ3, also reduces with increase in R as seen in Figure 8. The effect of the dimensionless transformation rate constant and the aggregation parameter φ1 on the various moments is depicted in parts a and b of Figure 9, respectively. With an increase in K the total number density of species 1 increases while that of
Figure 8. Dependence of dimensionless third moment, µ3, on ratio of growth rates, R, with δ ) R, φ1 ) 10, φ2 ) 0, K ) 1.
species 2 reduces, as expected, as shown by curves 2 and 3, respectively, in Figure 9a. Simultaneously, because of the greater aggregation tendency of species 1 the number density of the aggregates also increases while the overall population density reduces (cf. curves 4 and 1, respectively). The higher aggregation tendency of species 1 also leads to a reduction in the dimensionless precipitation rate, µ3, with an increase in K was shown by curve 5. This behavior is also consistent with that depicted in Figure 9b for the effect of φ1 for K ) 1. At constant K, an increase in φ1 causes a reduction in the total number density as well as that of species 1 and 2, as seen from curves 1, 2, and 3. However, the number concentration of aggregates initially increases due to formation from agglomeration events involving species 1, but then reduces due to loss from events involving agglomerates themselves. This also leads to a dominance of species 2 at large values of φ1, while at low values both species 1 and 2 are comparable in number. Summary The case of polymorphic precipitation is often encountered in industrial practice, with more than one polymorph simultaneously nucleating. In some cases this is due to the transformation of one polymorph to another within the precipitator itself. The formation of vaterite
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Figure 9. Variation of dimensionless moments with (a) dimensionless transformation rate constant, K, for φ1 ) 10 and (b) dimensionless aggregation parameter, φ1, for K ) 1 and R ) δ ) 1 and φ2 ) 0.
and calcite in CaCO3 precipitation is among the most well-known examples of this situation, and depending upon the operating conditions either of the above mechanisms is followed. In the present work a population balance model for the simultaneous nucleation as well as transformation mechanism in a continuous precipitation is developed and solved using an extension of Hounslow’s method. The results show the product composition and particle size distributions to be a strong function of the ratio of growth rates and the ratio of nucleation rates, as well as the aggregation tendencies of the polymorphs. Acknowledgment This work was supported by a grant (No. DST/III4(17)/86-ET) from the Department of Science and Technology of the Ministry of Science and Technology, Government of India. Notation b ) ratio of nucleation rates, B2/B1 B1 ) nucleation rate of species 1 B2 ) nucleation rate of species 2 G1 ) growth rate of species 1 G2 ) growth rate of species 2 Ga ) growth rate of aggregates k ) transformation rate constant K ) dimensionless transformation rate constant, kτ l ) particle size m0 ) eq 10 n(l) ) population density at size l n1(l) ) population density of species 1 n2(l) ) population density of species 2 na(l) ) population density of aggregates q(y) ) dimensionless population density, n(l)G2/B2 q1(y) ) dimensionless population density of species 1, n1(l)G1/B1 in case of simultaneous nucleation and n1(l)G1/B2 in case of nucleation of species 2 followed by transformation to species 1 q2(y) ) dimensionless population density of species 2, n2(l)G2/B2 qa(y) ) dimensionless population density of aggregates, na(l)Ga/B2 Qi ) eq 29 Q(1) i ) eq 30 Q(2) i ) eq 31 r ) yi+1/yi ) (2)1/3 y ) dimensionless particle size, l/G2τ
y1 ) lower size of first interval, dimensionless Greek Symbols R ) ratio of growth rates, G2/G1 β1,β2 ) constant aggregation kernels δ ) ratio of growth rates, G2/Ga φ1 ) agglomeration parameter of species 1, β1B1τ2 in case of simultaneous nucleation and β1B2τ2 in case of nucleation of species 2 followed by transformation to species 1 φ2 ) agglomeration parameter of species 2, β2B1τ2 in case of simultaneous nucleation and β2B2τ2 in case of nucleation of species 2 followed by transformation to species 2 µ0 ) dimensionless zeroth moment, ∫∞0 q(y) dy µ01 ) dimensionless zeroth moment of species 1, ∫∞0 q1(y) dy µ02 ) dimensionless zeroth moment of species 2, ∫∞0 q2(y) dy µ0a ) dimensionless zeroth moment of aggregates, ∫∞0 qa (y) dy µj ) jth moment of q(y) θ(y) ) fractional oversize, eq 52 θ1(y) ) eq 49 θ2(y) ) eq 50 θa(y) ) eq 51
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Received for review July 3, 1995 Accepted March 19, 1996X IE9504011
X Abstract published in Advance ACS Abstracts, May 1, 1996.
