A qualitative comparison between population balances and stochastic models for non-isothermal antisolvent crystallization processes
Introduction
Antisolvent aided crystallization is an advantageous separation technique when the solute is highly soluble or heat sensitive. The driving force in crystal formation is the supersaturation that establishes the thermodynamic equilibrium for the solid–liquid separation. In crystallization operations in general, and in antisolvent crystallization in particular, control of the crystal size and the crystal size distribution is a crucial issue and several factors can affect the size and the widening of the size distribution. Modeling of antisolvent crystallization processes has been tackled, for example, in the systems paracetamol (Trifkovic et al., 2008, Zhou et al., 2006) and sodium chloride (Nowee, Abbas, & Romagnoli, 2008a, 2008b). Recently, cooling has been combined with antisolvent crystallization, and the joint process has been modeled for lovastatin (Nagy, Fujiwara, & Braatz, 2008) and for acetylsalicylic acid (Lindenberg, Krattli, Cornel, Mazzotti, & Brozio, 2009). Crystallization theory states that the temperature influence is extremely important for the growth rate because it is correlated with the kinetic of the growth rate and nucleation rate as well as the solvent composition that is affected by the antisolvent feed rate (Mullin, 2001).
The development of effective mathematical models describing the crystal growth dynamics is a crucial issue toward finding the optimal process performance and to control the crystal size and distribution. Typically, mathematical models based on population balance equations (PBE) are taken into account to describe the time evolution of crystal particles size distributions. Randolph and Larson (1988) first implemented the population balance approach. Since then there have been a number of applications. Examples are the crystallization of: citric acid (Caillet, Sheibat-Othman, & Fevotte, 2007), lovastatin (Nagy, Fujiwara, et al., 2008), paracetamol (Nagy et al., 2008a, Trifkovic et al., 2008, Worlitschek and Mazzotti, 2004), potassium aluminium (Nowee, Abbas, & Romagnoli, 2007), potassium chloride (Mohameed, Abu-Jdayil, & Al Khateeb, 2002), ammonia sulphate (Abbas et al., 2006, Abbas and Romagnoli, 2006, Abbas and Romagnoli, 2007) and sodium chloride (Nowee et al., 2008a, Nowee et al., 2008b). In general, PBEs are widely used as a modeling framework for the study of different applications as: adsorption of impurities (Févotte & Févotte, 2010), jet crystallizers (Woo, Nagy, Tan, & Braatz, 2009) and phase transitions (Févotte & Alexandre, 2007). Multidimensional population balance models also have been proposed for crystals whose growth is not homogeneous across each size axis (Ma et al., 2002, Puel et al., 2003). Population balance models have been used to optimize crystallization operating conditions for various systems (Nagy et al., 2007, Nowee et al., 2008a, Trifkovic et al., 2008). Computational fluid dynamics (CFD) models have been combined with population balance models for modeling mixing effects in crystallization systems (Woo et al., 2006, Woo et al., 2009, Zauner and Jones, 2000).
At the core of the structured population dynamics, the number of crystals in a semi-batch crystallizer is increased by nucleation and decreased by dissolution or breakage. In structured population balances, the crystals are classified by their size. Therefore, population balance-based approaches provide more detailed information regarding the crystal size distribution in the crystallization unit. However, such a detailed description demands a great deal of knowledge on the thermodynamic properties associated with the solute and solvent to be incorporated into the structured population balances.
An alternative and novel approach to deal with particulate systems characterized by mean crystal size (MCS) and crystal size distribution (CSD) was formulated and implemented recently and it is based on modeling the growth process in terms of a Fokker–Planck Equation (FPE) (Galan et al., 2009, Grosso et al., 2011, Grosso et al., 2009). In this approach, the fluctuations of the particle state due to different uncertainty sources (e.g., turbulence at micro-scale mixing, temperature fluctuations, etc.) during the crystallization can be modeled as a random process. Thus, in an effort to explain the observed macroscopic behavior of crystal growth in antisolvent aided crystallization, the Fokker–Planck equation (FPE) is incorporated as the centerpiece of the approach. Models based on Fokker–Planck equations have been used in atmospheric sciences (Egger, 1981, Vallis, 1988), financial market dynamics (Michael & Johnson, 2003), polymerization (Matsoukas and Lin, 2006, Hosseini et al., 2012, Hosseini et al., 2013), among others. Pertaining the latter topic, the FPE approach has been recently used in the framework of the emulsion polymerization (Hosseini et al., 2013), and it was shown that the addition of a stochastic term in the growth term representing the evolution of the particles leads to a better description of the experimental data. Within this context, the use of FPE represents a new direction in developing a population balance model, taking into account the natural fluctuations present in the crystallization process, and allowing a novel description, in a compact form, of the PSD in time.
In this work the comparative analysis between the different approaches to model the time evolution of the Particle Size Distribution (PSD), is presented. The study is solely focused on the growth process. The mathematical modeling for both approaches includes the influence of the two operating parameters, antisolvent feedrate and temperature. These operating variables are used in a direct manner for the population balance approach, where the model is based on first principle assumptions. On the other hand, the phenomenological parameters appearing in the FPE models are formulated considering a polynomial relationship in the model parameters dependencies with the input manipulated variables (antisolvent flowrate and temperature) toward a global model to be used within all possible operating regimes. In this formulation, the input-parameter models should have simple linear or quadratic dependences. Validations against experimental data are presented for the NaCl–water–ethanol crystallization system for both models.
Section snippets
Experimental set-up
The experimental rig is comprised of a 1 l jacketed reactor connected to a Thermo Scientific® cooling/heating bath circulator that provides to keep constant the temperature inside the reactor by an embedded PID controller and a thermocouple wired inside the reactor. The antisolvent is added using a Masterflex® peristaltic pump calibrated for each experiment. The crystal size distribution is determined from the pictures of the samples taken from the reactor using a digital camera mounted in a
Population balance approach
The developments of rigorous mathematical models describing the crystal growth dynamics in crystallization processes are based-on population balances (Randolph & Larson, 1988). The idea of population balances has been widely used the modeling of particulate systems in chemical engineering, with applications including crystallization (Lindenberg et al., 2009, Nagy et al., 2008b, Nowee et al., 2007, Nowee et al., 2008a, Nowee et al., 2008b, Randolph and Larson, 1988, Trifkovic et al., 2008, Zhou
Stochastic modeling approach
Population balance modeling by first principle assumption represents a rigorous way to describe the crystal growth dynamics and assess its definite relationships with the operating conditions. However, PBE modeling requires a clear connection of the nucleation and growth phenomena from the driving force of the process that is the system supersaturation. Thus, a great deal of knowledge on the complex thermodynamics associated with the solute and solvent properties are needed.
An alternative,
Numerical comparisons
Fig. 1 reports, in linear scale, the comparison of the computed CSD with the two models and the available experimental data at different time for three of the nine temperature-antisolvent flow rate combinations: high temperature-and high flow rate; medium temperature–medium flow rate and low temperature-low flow rate. Similar behaviors were observed also for the other operating conditions. For sake of comparison the CSD evaluated through the PBE, Eq. (2), has been normalized (ψ(L, t) = n(L, t)/M0
Conclusions
In this work the comparative analysis between two different approaches, population balance (PBE) and Fokker–Plank (FPE) equations, to model the time evolution of the Particle Size Distribution (PSD) for antisolvent and temperature mediated crystal growth processes, is presented. Both modeling approaches are able to quantitatively capture the shape of the PSD, although the Fokker–Plank approach may seem to perform generally slightly better.
The two approaches show their pros and cons, and there
Acknowledgments
J. Romagnoli kindly acknowledges Regione Sardegna for the support, through the program “Visiting Professor 2011” and the financial support by NSF Award #1132324.
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