A lactose crystallization process can be modeled via mechanistic models, such as the mass balance equations, whereas the dynamic evolution of the crystal size can be modeled by the population balance model (PBM). Vu and others (2003, 2005, 2006) derived dynamic models of an α-lactose monohydrate crystallization process for cooling batch, cooling and evaporative semi-batch, and cooling in continuous mode crystallization. The dynamic model was a mechanistic model derived from mass and population balances that related the processing parameters to the CSD.

#### Process dynamics model

For cooling crystallization process, the model proposed by Vu and others (2003, 2005, 2006) excluded nucleation. However, as evidence in Figure 7 (A), secondary nucleation increases the number of crystals in suspension, and thus, should be included in the model. In 2012, the model (Vu and others 2003, 2005, 2006) was refined by Wong and others (2012) to include the change in the number concentration of lactose crystals generated by nucleation.

Assuming that mass transfer by agglomeration or breakage is minimal, the species conservation of water, dissolved impurity, dissolved lactose, and α-lactose crystals for batch crystallization can be written as follows (Wong and others 2012).

Water concentration (*x*_{1}; g water/mL solution),

- (9)

Lactose concentration (*x*_{2}; g lactose/mL solution),

- (10)

Suspension density of crystals (*x*_{3}) (g lactose crystal/mL solution),

- (11)

Volume equivalent average diameter of crystals (*x*_{4}; m),

- (12)

Number concentration of crystals, *N* (#/m^{3} solution)

- (13)

In this model, the monohydrate crystal (Eq. (11)) is produced from 95% dissolved lactose (Eq. (10)) and 5% water (Eq. (9)). For a seeded batch crystallization process, *N* (Eq. (11)) is the total number density of seed crystals (number/mL solution) introduced at *t* = 0 min. The change in *N* is triggered mainly by nucleation (Eq. (13)) with rate of (*B*^{o}; #/mL-min). G (m/min) is the growth rate (Eq. (12)). The supersaturation (S) is dependent on the temperature for a cooling crystallization process. S can be calculated by Eq. (1)-(4). The cooling profile required to achieve operation along any regions outlined in the lactose “supersolubility” diagram (Figure 8) can be estimated by solving Eq. (9)-(13) and (1)-(4) with the ordinary differential equation (ODE) solver in Matlab R2008a (Wong and others 2012).

#### Population balance model

Population balances are widely used to model dynamic particle or droplet size distributions, such as those seen in aggregation, flocculation, crystallization, bubble or droplets in solvent extraction or flocculation columns. Fundamental to the formulation of PBM is the assumption that there exists a number density of particles at every point in the particle state space (Smart and Smith 1992). The population balance is given as (Randolph and Larson 1988):

- (14)

Here, *n* is the population density,

is the set of internal and external coordinates, where , and

Birth and Death represent empirical birth and death density functions at a point in phase space that occur through aggregation (agg) and breakage (br). This equation, coupled with mass, energy and momentum balances, crystallization kinetics and boundary conditions representing the entry and exit of particle suspension, can be used to completely model the formation of the crystals and the dynamics of the crystallization.

The birth and death function for aggregation of 2 particles (volume *u* and *v* − *u*) into a single particle volume v can be represented by (Hartel and Randolph 1986):

- (15)

- (16)

The aggregation kernel, *K*(*u*,*v*), measures the aggregation frequency between particle (volume *u*) with another one (volume *v*; Bramley and others 1996). The aggregation kernel should be selected based on the aggregation mechanism (Hartel and Randolph 1986; Smit and others 1994, Bramley and others 1996). In a length-based form, (Eq. (15) and (16)) become (Hounslow and others 1988):

- (17)

- (18)

The birth and death terms because of the breakage mechanisms is (Costa and others 2007),

- (19)

- (20)

where γ(*u*) is the number of daughter particles originated from the breakup of a particle of size *u*, *b*(*u*) is the breakup rate of a particle (size *u*), and *p*(*v*/*u*) is the fraction of daughter particles with size between *v* and *v* + d*v*.

#### Solving PBM

PBM can be solved via a stochastic or deterministic framework. The selection is usually based on the importance of the particle population fluctuations (Rawlings and others 1993). Stochastic methods have the advantage of satisfying mass conservation as well as correctly accounting for fluctuations that arise as the system mass accumulates in a small number of large aggregates (Marchisio and others 2003). In lactose crystallization, the CSD data are commonly modeled by deterministic framework. The Method of Moments (MOM) will be reviewed in this section.

In most crystallization systems, some average or total quantities are sufficient to represent the particle distribution. Therefore, the PBM can be transformed into a series of moment equations with regard to the internal coordinate properties (Randolph and Larson 1988). For a crystallization process with the particle size (L) as the only internal coordinate, the PBM (Eq. (14)) can be rewritten as,

- (21)

*B*^{o} and *G* are the nucleation and growth rates.

For size independent aggregation, (Eq. (17) and (18)) and (Eq. (23) and Eq. (24)) can be reduced to (Hounslow and others 1988):

- (25)

- (26)

Here, and is the size-independent aggregation rate.

This simplification greatly reduces the dimension of the internal coordinate of the particle state phase. (Eq. (21)) can be solved together with the mass, momentum and energy conservation equations to yield a complete mathematical description of the crystallization process (Randolph and Larson 1988).

For any total number of moments (I), (Eq. (21)) forms a closed set in terms of the moments m_{i}. Therefore, MOM can be easily implemented with an ODE solver. However, it is a challenge to solve the inverse problem. Full reconstruction of the CSD is limited, since the detailed information of the CSD is lost after the transformation (Giaya and Thompson 2004; John and others 2007). The most common approaches have been reviewed by John and others (2007).

#### Estimating crystallization kinetics using PBM

The power of the PBM in analyzing lactose crystallization is commonly illustrated by its application in mixed suspension mixed product removal (MSMPR) crystallizers (Shi and others 1990; Liang and Hartel 1991; Wong and others 2010). Depending on the assumptions, the PBM was formulated or simplified into forms where crystallization kinetics can be calculated from experimental CSD data.

When GRD is observed in crystallization, the curvature at small sizes (<50 μm) in the semi-logarithmic plot of *n* (number density) versus *L* (particle size) can no longer be accounted for by simple linear extrapolation (Liang and others 1991). Instead, the crystallization kinetics should be calculated by the moment transfer technique, the details of calculation method can be found in Liang and others (1991). The CSD prediction of the GRD models is shown in Figure 9 (B). Compared to the limited range modeled by the linear PBM (Figure 9 A), the GRD model covered the entire range of size distribution. The value of growth and nucleation rates estimated by linear and GRD models are summarized in Table 6. Clearly, the linear model discounted the nucleation rate and overestimates the average growth rate.

Table 6. Crystallization kinetics calculated from 3 methods | Growth rate (G) | Nucleation rate (*B*^{o}, #/mL min) | Aggregation rate (*β*_{o}) |
---|

Linear PBM (Shi and others 1990) | 0.77 μm/min | 3.27E + 03 | – |

Moment transfer technique (Liang and others 1991) | 0.26 μm/min | 3.35E + 04 | – |

Moment transformation method (Wong and others 2010) | 38.1 (μm)^{3}/min | 18E + 6 | 1.34E-12 (m^{3}/#-s) |

As discussed earlier, crystal aggregation might occur during lactose refining. When aggregation occurs in a MSMPR operation, assumption (2) is no longer valid, and should be replaced by alternative terms that describe the aggregation process. To account for aggregation in an MSMPR lactose crystallization process, Wong and others (2010) included size independent aggregation functions (Eq. (25) and (26)) in the PBM. In contrast to the linear and GRD models, Wong and others (2010) solved the PBM using particle volume as the internal coordinate. Therefore, the crystallization kinetics (*G*_{v}, *B*^{o}, *β*_{o}) shown in Table 6 are volume-based kinetics. The kinetic rates of crystallization depend largely on the solution approach to PBM. For lactose crystallization, the linear PBM and GRD approaches have been most commonly used. However, when a large amount of aggregation/breakage is observed, care should be taken to avoid oversimplifying the PBM.

Here, *k*_{b}, *n*, *j*, *k*, *k*_{oN} are considered to be empirical constants, *m*_{i} is the *i*^{th} moment of the CSD, *S* and Δ*c* are the supersaturation, *M*_{T} is the suspension density, *E*_{aN} is the activation energy for nucleation, and *T*_{k} is the temperature in Kelvin (K). These parameters (*k*_{b}, *n*, *j*, *k*, *k*_{oN}, *E*_{aN}) can be regressed from sets of experimental crystallization rate data.

For nucleation rate, empirical power law models comprises of supersaturation (S) and suspension density (*M*_{T}, *m*_{3}) are commonly found. Most models predict *j* close to one, suggesting the dominance of crystals–vessel walls collisions, or, crystal–stirrer rather than crystal–crystal collisions (Garnier and others 2002). Eq. (29) and (30) are based on the standard Arrhenius kinetics model, where the contribution of agitation was not considered. However, the effect of agitation is especially important in contact nucleation, where the values of *k* (Eq. (31)) predicted are in the range of 2 to 4 (Davey and Garside 2000).

The growth rate is correlated in a similar way. Eq. (32) is commonly suggested to represent growth of crystals (Liang and others 1991; Rawlings and others 1993; Arellano and others 2004):

- (32)

where *k*_{oG} and m are empirical constants and *E*_{aG} is the activation energy for crystal growth. The parameters (*k*_{oG}, *m* and *E*_{aG}) are regressed from sets of experimental crystallization rate data. In (Eq. (32)), the temperature dependence was incorporated by an Arrhenius-type expression (Rawlings and others 1993). The value of m is commonly reported in the range of 2 to 3, where a value of m greater than 2 might indicate the effects of the β-lactose impurity on crystal growth (Shi and others 1990). In the literature, the value of *E*_{aG} was between 22 to 24 kcal/mol. The high *E*_{aG} indicated that the growth of lactose crystals is most likely a surface integration mechanism, where the growth rate is determined by the rate that lactose molecules were incorporated into the crystal lattice (Shi and others 1990).

For lactose crystallization, similar expressions for aggregation is lacking. To overcome the complex relationship of aggregation rate and processing parameters, a kinetic model was developed based on artificial neural network (ANN; Wong and others 2010). For an MSMPR crystallizer, the ANN model predicted higher aggregation rate with high amount of seed crystals and low residence time. On the contrary, aggregation reduces at high agitation speed and supersaturation had minimal impact (Wong and others 2010).