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High-Order Simulation of Polymorphic Crystallization Using Weighted Essentially Nonoscillatory Methods

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High-Order Simulation of Polymorphic Crystallization Using Weighted Essentially Nonoscillatory Methods High-Order Simulation of Polymorphic Crystallization Using Weighted Essentially Nonoscillatory Methods Martin Wijaya Hermanto Dept. of Chemical and Biomolecular Engineering, National University of Singapore, Singapore 117576 Richard D. Braatz Dept. of Chemical and Biomolecular Engineering, University of Illinois at Urbana-Champaign, IL 61801 Min-Sen Chiu Dept. of Chemical and Biomolecular Engineering, National University of Singapore, Singapore 117576 DOI 10.1002/aic.11644 Published online November 21, 2008 in Wiley InterScience (www.interscience.wiley.com). Most pharmaceutical manufacturing processes include a series of crystallization processes to increase purity with the last crystallization used to produce crystals of desired size, shape, and crystal form. The fact that different crystal forms (known as polymorphs) can have vastly different characteristics has motivated efforts to under- stand, simulate, and control polymorphic crystallization processes. This article pro- poses the use of weighted essentially nonoscillatory (WENO) methods for the numeri- cal simulation of population balance models (PBMs) for crystallization processes, which provide much higher order accuracy than previously considered methods for simulating PBMs, and also excellent accuracy for sharp or discontinuous distributions. Three different WENO methods are shown to provide substantial reductions in numeri- cal diffusion or dispersion compared with the other finite difference and finite volume methods described in the literature for solving PBMs, in an application to the polymorphic crystallization of L-glutamic acid. Ó 2008 American Institute of Chemical Engineers AIChE J, 55: 122–131, 2009 Keywords: pharmaceutical crystallization, polymorphism, hyperbolic partial differential equation, weighted essentially nonoscillatory, high resolution, finite difference Introduction motivated by patent, operability, profitability, regulatory, and 1–3 Most manufacturing processes include a series of crystalli- scientific considerations. The different polymorphs can zation processes in which product quality is associated with have orders-of-magnitude differences in properties such as the crystal final form (such as crystal size and shape distribu- solubility, chemical reactivity, and dissolution rate, which can have an adverse effect on downstream operability and per- tion). Recently, there is a rapid growth of interest in poly- 2 morphism (when a substance has multiple crystal forms) formance of the crystal product. As a result, controlling polymorphism to ensure consistent production of the desired polymorph is very critical in those industries, including in Correspondence concerning this article should be addressed to M.-S. Chiu at drug manufacturing where safety is of paramount importance. [email protected]. Numerical simulations for polymorphic crystallizations Ó 2008 American Institute of Chemical Engineers enable the investigation of the effects of various operating 122 January 2009 Vol. 55, No. 1 AIChE Journal conditions and can be used for optimal design and control.4–6 with Henrick mapping (JSHWENO),32,33 and the weighted Generally, the most widely accepted approach to modelling power ENO method (Wpower-ENO).34 These WENO meth- particulate processes is based on population balance equa- ods are compared with the HR finite volume method and a tions.7 Solving population balance equations is particularly second-order finite difference (FD2) method, for polymorphic challenging when the partial differential equations (PDEs) crystallization of L-glutamic acid under conditions in which are hyperbolic with sharp gradients or discontinuities in the the distribution contains sharp gradients. distribution.8 Standard first-order methods require a very This article is organized as follows. First, the PBM for the small grid size in order to reduce the numerical diffusion polymorphic crystallization of L-glutamic acid is summar- (i.e., smearing), whereas standard higher order methods intro- ized. Then the five numerical methods are discussed and duce numerical dispersion (i.e., spurious oscillations), which compared. Finally, conclusions are provided. usually results in a crystal size distribution with negative val- ues. Efficient and sufficiently accurate computational meth- ods for simulating the population balance equations are required to ensure the behavior of the numerical solution is Process Model determined by the assumed physical principles and not by This section presents a kinetic model for the polymorphic the chosen numerical method. crystallization of metastable a-form and stable b-form crys- 35 There have been many papers on the numerical solution of tals of L-glutamic acid (Hermanto et al. ). The population population balance models. The method of moments approxi- balance equations are mates the distribution by its moments,9 which under certain @f ; @ðÞG f ; i conditions, converts the hyperbolic PDEs into a small num- seed i þ i seed ¼ 0; (1) @t @L ber of ordinary differential equations (ODEs) that describe ÀÁ characteristics of the distribution. The method of moments @ @fnucl;i Gifnucl;i does not apply to PBEs which do not satisfy moment closure þ ¼ BidðÞL À L0 ; (2) @t @L conditions. The method of weighted residuals approximates the size distribution by a linear combination of basis func- where fseed,i and fnucl,i are the crystal size distributions of the tions,10 which results in a system of ODEs. For most practi- i-form crystals (i.e., a-orb-form crystals) obtained from 24 cal crystallizations, a large number of basis functions is seed crystals and nucleated crystals (#m ), Bi and Gi are 2 2 needed to approximate the distribution, which results in high the nucleation (#m3 s 1) and growth rate (m s 1) of the i- computational cost. The Monte Carlo method tracks individ- form crystals, L and L0 are the characteristic size of crystals ual particles, each of which exhibits stochastical behavior (m) and nuclei (m), and d(Á) is a Dirac delta function. These according to a probabilistic model.11–13 This approach is too equations are augmented by the solute mass balance: computationally expensive for most industrial crystalliza- 3 ÀÁ tions. Another problem-specific numerical method for solving dC 10 q l q l ; ¼3 q akvaGa a;2 þ bkvbGb b;2 (3) population balance equations is the method of characteris- dt solv tics.14,15 This method solves each population balance equa- 2 tion by finding curves in the characteristic size-time plane where the nth moment of the i-form crystals (#mn 3)is that reduce the equation to an ODE. Although the method is Z1 ÀÁ highly efficient when the kinetics are simple, the approach n l ; ¼ L f ; þ f ; dL; (4) does not generalize to more complex kinetics. Most publica- i n nucl i seed i tions on numerical methods for solving PBEs involve various 0 21 types of discretizations and go by a variety of names includ- C is the solute concentration (g kg ), qsolv is the density of 23 ing ‘‘method of classes’’ and ‘‘discretized population balance the solvent (kg m ), qi is the density of the i-form crystals 16–20 23 equations.’’ In recent years there have been several (kg m ), kvi is the volumetric shape factor of the i-form 3 efforts to reduce the numerical diffusion and numerical dis- crystals (dimensionless) as defined by vi 5 kviL , where vi is persion for distributions which contain sharp gradients or dis- the volume of the i-form crystal (m3), and 103 is a constant (g continuities, which is common in batch crystallizations. kg21) to ensure unit consistency. The kinetic expressions are High-resolution (HR) finite volume methods (FVMs) popular 21–25 l a ; in astrophysics and gas dynamics were extended to the Ba ¼ kbaðSa À 1Þ a;3 ð -form crystal nucleation rateÞ (5) application of multidimensional population balance equa- 8 ga 26–30 > kgaðSa À 1Þ if Sa 1 tions. A typical implementation applies a first-order <> method near discontinuities or sharp gradients and a second- ða-form crystal growth= Ga (6) order method everywhere else, which results in less numeri- ¼ > ; > dissolution rateÞ cal dispersion than the second-order method and less numeri- : 26 kdaðSa À 1Þ otherwise cal diffusion than the first-order method. ÀÁ ÀÁ l l This article considers a class of numerical algorithms Bb ¼ kbb;1 Sb À 1 a;3 þ kbb;2 Sb À 1 b;3 known as weighted essentially nonoscillatory (WENO) meth- ðb-form crystal nucleation rateÞ; ð7Þ ods which were developed for especially accurate simulation of shock waves and provide much higher order accuracy ÀÁ gb kgb;2 than the previously considered methods for solving PBMs. Gb ¼ kgb;1 Sb À 1 exp À Sb À 1 Three WENO methods are considered: Liu et al’s version of b ; WENO (LOCWENO),31 Jiang and Shu’s version of WENO ð -form crystal growth rateÞ ð8Þ AIChE Journal January 2009 Vol. 55, No. 1 Published on behalf of the AIChE DOI 10.1002/aic 123 Table 1. Model Parameters for L-glutamic Acid The stability properties of this method make it suitable for Polymorphic Crystallization stiff problems. Parameters Values Parameters Values It is advantageous for a numerical method to be conserva- tive, that is, to ensure that a quantity remains conserved by q ln (kba) 17.233 solv 990 calculating a single flux which describes the flow of that ln (k a ) 1.878 qa 5 qb 1540 g ,0 quantity between neighbouring cells.22,23 Although flux con- ga 1.859 kva 0.480 ln (Ega) 10.671 kvb 0.031 servative schemes are normally formulated using finite vol- 23 ln (kda) 210.260 aa,1 8.437 3 10 umes, a finite difference scheme is utilized here based on the 3 22 39 ln (kbb,1) 15.801 aa,2 3.032 10 approach described in Shu. ln (kbb,2) 20.000 aa,3 4.564 23 ÀÁ ln (kgb,0) 52.002 ab,1 7.644 310 21 @ @ ln (k b ) 20.251 ab 21.165 3 10 fnucl;i Gifnucl;i g ,2 ,2 þ ¼ 0; (11) gb 1.047 ab,3 6.622 @t @L ln (Egb) 12.078 Bi fnucl;iðÞ¼L0; t : (12) 2 Gi where Si 5 C/Csat,i and Csat,i 5 ai,1T þ ai,2T þ ai,3 are the supersaturation and the saturation concentration (g kg21)of the i-form crystals, respectively, and T is the solution temper- To simplify notation, (1) and (11) are written in the same form ature (8C).
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