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 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 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( ) 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 tions. Another problem-specific numerical method for solving dC 10 q l q l ; ¼ 3 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 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 1Þ a;3 ð -form crystal nucleation rateÞ (5) application of multidimensional population balance equa- 8 ga 26–30 > kgaðSa 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 1Þ otherwise cal diffusion than the first-order method. l l This article considers a class of numerical algorithms Bb ¼ kbb;1 Sb 1 a;3 þ kbb;2 Sb 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 gb kgb;2 than the previously considered methods for solving PBMs. Gb ¼ kgb;1 Sb 1 exp Sb 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 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). The kinetic parameters kba, kga, and kda corre- spond to the nucleation (#m23 s21), growth (m s21), and disso- 21 @u @p lution (m s ) rates of a-form crystals, respectively, whereas þ ¼ 0; 23 21 @t @L kbb,j and kgb,j correspond to the jth nucleation (#m s ) (13) 2 @ @ and growth (m s 1) for j 5 1 and dimensionless for j 5 2 u p ; @ ¼ @ rates of b-form crystals, respectively, and gi is the growth t L exponential constant of the i-form crystals which may have a value between 1 (for diffusion-limited growth) and 2 (for where u is fseed,i or fnucl,i and p is Gifseed,i or Gifnucl,i. Equa- surface integration-limited growth).36 The Arrhenius equation tion 13 is discretized in the L domain with uniform intervals D 5 D was used to account for the variability of crystal growth rate of size L, Lk k L indicates the crystal size at node k, 5 th with temperature: and Ik [Lk21/2,Lkþ1/2] is the k cell. The conservative approximation to the spatial derivative is used: Ega ; kga ¼ kga;0 exp (9) du ðtÞ 1 8:314ðÞT þ 273 k ^ ^ ; ¼ D pkþ1=2 pk 1=2 (14) dt L Egb k b; ¼ k b; exp ; (10) where u is the value of u at L and the numerical flux pˆ g 1 g 0 8:314ðÞT þ 273 k k kþ1/2 approximates hkþ1/2 5 h(Lkþ1/2) with h(L) implicitly defined 39 21 by where kgi,0 and Egi are the pre-exponential factor (m s ) 21 and activation energy (J mol ) for the growth rate of i-form LþZDL=2 crystals, respectively. The values for densities, volumetric 1 puLðÞðÞ¼ hðÞn dn: (15) shape factors, and parameters for the saturation concentration DL are in Table 1. L DL=2 Secondary nucleation is assumed for both a- and b-form crystals, since it is the dominant nucleation process in seeded For stability, it is important that upwinding is used in con- a crystallization. The growth rate expression for the -form structing the numerical flux pˆkþ1/2. One way is to compute crystals includes both growth (supersaturation) and dissolu- the Roe speed to determine the direction of the wind: tion (undersaturation). Dissolution occurs during the poly- morphic transformation of a-tob-form crystals, where pkþ1 pk ; akþ1=2 ¼ (16) a-form crystals dissolve and b-form crystals nucleate and ukþ1 uk grow. where pk is the value of p at Lk. In the context of process model, the Roe speed is Numerical Methods ; The numerical methods described here differ in terms of akþ1=2 Gi (17) their discretization along the crystal size dimension (L), each of which produces a system of ODEs describing the time and evolution of the crystal size distribution at the chosen discre- if Gi 0 then the wind blows from the left to the right 37 tized points Lk. To provide a fair basis for comparison, the and the numerical fluxes pˆkþ1/2 and pˆk21/2 are approximated implementation of all of the methods integrated the ODEs 2 by p kþ1/2 and pk 1=2, respectively. using a fourth-order orthogonal Runge-Kutta Chebyshev if Gi \ 0 then the wind blows from the right to the left 38 method, which is a class of explicit Runge-Kutta methods and the numerical fluxes pˆkþ1/2 and pˆk21/2 are approximated þ þ with extended stability domains along the negative real axis. by pkþ1/2 and pk21/2, respectively.
124 DOI 10.1002/aic Published on behalf of the AIChE January 2009 Vol. 55, No. 1 AIChE Journal limiter substantially reduces smearing near discontinuities and results in better resolution of corners and local extrema. All WENO methods adopt the following idea. Denote the r candidate stencils by
Sm ¼ ðÞLkþm rþ1; Lkþm rþ2; ...; Lkþm ; m ¼ 0; 1; ...; r 1; (18)
whose corresponding rth order ENO approximation of the flux hkþ1/2 is Figure 1. Computational cells. r p = ¼ q ðpkþm rþ1; ...; pkþmÞ : (19) kþ1 2 m L¼Lkþ1=2 The difference between the values with superscript 6 at Using the smoothest stencil among the r candidates for the the same location Lkþ1/2 is due to the possibility of different 2 approximation of hkþ1/2 is desirable near discontinuities to stencils for cell Ik and for cell Ikþ1, that is, p kþ1/2 is due to the stencil for cell I and pþ is due to the stencil for cell avoid introducing aphysical oscillations. All of the stencils k kþ1/2 are smooth in regions where the solution is smooth, in which Ikþ1 (see Figure 1). In the next sections, five reconstruction þ case it is better to combine the results of multiple stencils to- procedures are described to obtain pk 1=2 and pkþ1/2 only, as þ 5 gether to produce a higher order (higher than rth order, the pkþ1/2 can be readily derived from pk0 1=2+ for cell Ik0 Ikþ1 2 5 order of the original ENO method) approximation to the flux and pk 1=2 can be derived from p k0þ1/2 for cell Ik0 Ik21. 32 hkþ1/2. WENO methods assign a weight xm to each candi- date stencil Sm to obtain the combined approximation of hkþ1/2 as WENO variants All WENO methods discussed here are the derivatives of Xr 1 r p = ¼ x q ðp ; ...; p Þj : (20) the original essentially nonoscillatory (ENO) method devel- kþ1 2 m m kþm rþ1 kþm L¼Lkþ1=2 oped by Harten et al.41 in 1987. This article was the first to m¼0 obtain a self similar (i.e., no mesh size-dependent parameter), uniformly high order accurate, yet essentially nonoscillatory To achieve the essentially nonoscillatory property, the interpolation [i.e., the magnitude of the oscillations decays as weights adapt to the relative smoothness of p on each candi- O(Dxr) where r is the order of accuracy] for piecewise date stencil such that any discontinuous stencil is effectively smooth functions. ENO methods are especially suitable for assigned a zero weight. In smooth regions the weights are problems containing both shocks and complicated smooth adjusted such that the resulting approximation gives an order flow structures, such as those occurring in shock interactions of accuracy higher than r. The differences between WENO with a turbulent flow and shock interaction with vortices. To methods lie on the method for selecting the weights xm and r ... improve the ENO method and further expand its applications, the flux approximations qm(pkþm2rþ1, ,pkþm). The subse- ENO methods based on point values and total diminishing quent WENO methods have r 5 3 with the flux approxima- r ... variation (TVD) Runge-Kutta time discretizations were tions qm(pkþm2rþ1, ,pkþm) constructed based on quadratic developed, which can reduce computational costs signifi- polynomials. cantly for multiple space dimensions.39,40 Then biasing during selection of the stencil was proposed for enhancing stability and accuracy.42,43 Later, WENO methods were Liu et al.’s WENO (LOCWENO) method developed, using a convex combination of all candidate sten- The flux approximations and weights for the fourth-order cils instead of just one as in the original ENO.31–34 accurate LOCWENO method are31 WENO methods improve the accuracy of the original ENO method to the optimal order in smooth regions while 3 ; ; maintaining the essentially nonoscillatory property near dis- qmðpkþm 2 pkþm 1 pkþmÞ continuities. Liu et al.31 converted the rth order ENO method p 2p þ p ¼ kþm kþm 1 kþm 2 ðÞL L 2 into an (r þ 1)th order WENO method with a cell average 2DL kþm 1 approach. Based on the pointwise finite difference ENO pkþm pkþm 2 39,40 þ ðÞL L method and a new smoothness indicator, the WENO 2DL kþm 1 method by Jiang and Shu32 can achieve the optimal (2r 2 p 2p þ p þ p kþm kþm 1 kþm 2 ð21Þ 1)th order accuracy. Jiang and Shu’s WENO version was kþm 1 24 later modified by adding a mapping function for the original nonlinear weight which improves accuracy near smooth and extrema.33 Serna and Marquina34 improved the behavior of Jiang and Shu’s WENO method by introducing the power- k 2 x P m ; eno3 or powermod3 limiter, resulting in an (2r 1)th order m ¼ 2 (22) kj weighted power ENO method. The powereno3 or powermod3 j¼0
AIChE Journal January 2009 Vol. 55, No. 1 Published on behalf of the AIChE DOI 10.1002/aic 125 Table 2. Values of h and dm for LOCWENO, JSHWENO, where km is defined in Eq. 23 and the values of h and dm are and WPower-ENO Methods in Table 2. The smoothness indicators ISm are LOCWENO JSHWENO WPower-ENO 13 1 IS ¼ ðÞp 2p þ p 2 þ ðÞp 4p þ 3p 2; (32) h 32 20 12 k 2 k 1 k 4 k 2 k 1 k d0 1/12 1/10 1/5 d1 1/2 3/5 1/5 13 2 1 2 1/4 3/10 2/5 IS1 ¼ ðÞpk 1 2pk þ pkþ1 þ ðÞpk 1 pkþ1 ; (33) d2 12 4
13 2 1 2 where IS2 ¼ ðÞpk 2pkþ1 þ pkþ2 þ ðÞ3pk 4pkþ1 þ pkþ2 : (34) 8 12 4 > dm > for p 33 < e h kþ1=2 The Henrick mapping k ðISm þ Þ m ¼ > (23) > d2 m þ x 2 x x2 : for p = : ðdm þ dm 3dm þ Þ h k 1 2 gmðxÞ¼ : (35) ðISm þ eÞ 2 x dm þð1 2dmÞ
The values of h and dm in Eq. 23 are in Table 2 and e is a small 2 is used to revise these weights to improve the accuracy near number to avoid division by zero (i.e., e 5 10 4 was used in smooth extrema: this article). The ISm are smoothness indicators given by k 2 2 xHJS P m (36) ðÞp p þ ðÞp p m ¼ 2 IS ¼ kþm 1 kþm 2 kþm kþm 1 k m 2 j¼0 j 2: þ ðÞpkþm 2pkþm 1 þ pkþm 2 ð24Þ with
JS k ¼ gmðx Þ; (37) Jiang and Shu’s WENO method with Henrick mapping m m (JSHWENO) to produce a fifth-order accurate method. The Jiang and Shu’s WENO method used here is based on the quadratic polynomial instead of the original linear approximation. With Weighted power ENO (Wpower-ENO) method Using the definitions in Eqs. 25–27, the flux approxima- d = ¼ p p ; (25) kþ1 2 kþ1 k tions for the Wpower-ENO method are
dkþ1=2 þ dk 1=2 ; Pow = dk ¼ (26) 3 ; ; k 1 2 2 q0ðpk 2 pk 1 pkÞ¼pk 24 D ¼ d = d = ; (27) ðÞL Lk Powk 1=2 Powk 1=2 L Lk k kþ1 2 k 1 2 þ d = þ þ ; ð38Þ DL k 1 2 2 2 DL the flux approximations are 3 ; ; Dk ðL LkÞ Dk L Lk ; q1ðpk 1 pk pkþ1Þ¼pk þ D dk þ D 3 ; ; Dk 1 24 L 2 L q0ðÞ¼pk 2 pk 1 pk pk 24 (39) ðÞL Lk Dk 1 Dk 1 L Lk d = ; 28 þ k 1 2 þ þ ð Þ Pow = DL 2 2 DL 3 ; ; kþ1 2 q2ðpk pkþ1 pkþ2Þ¼pk 24 3 Dk ðÞL Lk Dk L Lk ðÞL Lk Powkþ1=2 Powkþ1=2 L Lk q ðÞ¼p ; p ; p p þ d þ ; þ d = þ ; ð40Þ 1 k 1 k kþ1 k 24 DL k 2 DL DL kþ1 2 2 2 DL (29) where 3 ; ; Dkþ1 q2ðÞ¼pk pkþ1 pkþ2 pk ; ; 24 Powkþ1=2 ¼ powereno3ðDk Dkþ1Þ (41) ðÞL Lk Dkþ1 Dkþ1 L Lk ; þ dkþ1=2 þ ð30Þ 5 DL 2 2 DL is the powereno limiter acting on L Lkþ1/2 where
32 ; ; ; ; and the weights are powereno3ðx yÞ¼minsignðx yÞ power3ðjxj jyjÞ (42) ( k signðxÞ if jxj jyj ; xJS ¼ P m ; (31) minsignðx; yÞ¼ (43) m 2 k ; j¼0 j signðyÞ otherwise
126 DOI 10.1002/aic Published on behalf of the AIChE January 2009 Vol. 55, No. 1 AIChE Journal Table 3. Initial Seed Distribution Parameters for a- and b-forms 6 ^ 6 i ji rseed,i[m] 3 10 Lseed;i[m] 3 10 a 2 3 1010 2.000 30.000 b 2 3 1010 4.000 50.000
x2 þ y2 þ 2½maxðx; yÞ 2 power3ðx; yÞ¼minðx; yÞ : (44) ðx þ yÞ2
The weights xm and parameters km are the same as for the LOCWENO method, except that the smoothness indicators 13 2 1 2 IS ¼ Pow = þ 2p 2p þ Pow = ; (45) 0 12 k 1 2 4 k k 1 k 1 2
13 2 1 2 IS1 ¼ ðÞpk 1 2pk þ pkþ1 þ ðÞpk 1 pkþ1 ; (46) 12 4 Figure 2. Temperature profile used in simulations. 13 2 1 2 IS ¼ Pow = þ 2p 2p Pow = ; (47) 2 12 kþ1 2 4 kþ1 k kþ1 2 or are used. This method is fifth-order accurate. 1 1 pþ ¼ ð3p p Þ¼p ðp p Þ (54) k 1=2 2 k kþ1 k 2 kþ1 k HR method The popular HR method uses second-order discretization Similar inclusion of a flux limiter to the antidiffusion term with a flux limiter to ensure nonoscillatory behavior. For Gi gives 0, a backward second-order discretization is used: þ 1 / 1 : 1 pk 1=2 ¼ pk ðÞpkþ1 pk (55) p p ¼ ð3p 4p þ p Þ 2 wk kþ1=2 k 1=2 2 k k 1 k 2 1 1 This HR method is second-order accurate in smooth regions, ¼ ð3p p Þ ð3p p Þ (48) 2 k k 1 2 k 1 k 2 and first-order accurate near discontinuities. or The second-order finite difference (FD2) method 1 1 ; A second-order finite difference method with correct pkþ1=2 ¼ ð3pk pk 1Þ¼pk þ ðpk pk 1Þ (49) 2 þ 2 2 upwinding uses the fluxes p kþ1/2 and pkþ1/2 given by Eq. 49 and 54, respectively. where the first term is first-order and the second term is com- monly referred to as an ‘‘anti-diffusion term’’ because it reduces numerical diffusion. Applying a flux limiter on the antidiffusion term yields 1 p ¼ p þ /ðw Þðp p Þ ; (50) kþ1=2 k 2 k k k 1 where wk is the upwinding ratio defined by
pkþ1 pk wk ¼ (51) pk pk 1 and /( ) is the flux limiter. In this article, the popular Van Leer flux limiter44 was used:
w þjwj /ðwÞ¼ : (52) 1 þ w
For Gi \ 0, a forward second-order discretization is used:
þ þ 1 p = p = ¼ ð 3pk þ 4pkþ1 pkþ2Þ kþ1 2 k 1 2 2 Figure 3. CSD of nucleated a form at the end of the 1 1 batch for the various numerical methods (DL ¼ ð p þ 3p Þ ð p þ 3p Þð53Þ 2 kþ2 kþ1 2 kþ1 k 5 0.6 lm).
AIChE Journal January 2009 Vol. 55, No. 1 Published on behalf of the AIChE DOI 10.1002/aic 127 Figure 4. CSD of nucleated b form at the end of the Figure 6. CSD of seeded b form at the end of the batch for batch for the various numerical methods (DL the various numerical methods (DL 5 0.6 lm). 5 0.6 lm).
solutions for all CSDs were obtained by using WPower-ENO Simulation Results method with very fine resolution. All the computations were The five numerical methods were applied to the L-glutamic performed using Compaq Fortran 6.6 on a HP workstation acid polymorphic crystallization model. The initial seed dis- XW6400 [Intel Xeon 5150 (2.66 GHz) and 2 GB of RAM]. tributions fseed,i(L,0) for a- and b-forms are described by In an unseeded crystallization the CSD profiles for all Gaussian distributions: methods are nearly coincident with the reference profiles (see ! Figures 3 and 4), indicating that a conventional numerical ^ 2 method such as FD2 might suffice, which is consistent with ji ðL Lseed;iÞ f ; ðL; 0Þ¼pffiffiffiffiffiffi exp ; (56) expectations since no sharp gradients occur in these distribu- seed i pr 2r2 2 seed;i seed;i tions. In the case of seeded crystallization (the usual case in practice), the differences in the CSD profiles between the with the parameters in Table 3 selected so that the distribu- WENO variants and their conventional counterparts are sig- tions would be sharp enough to challenge numerical meth- nificant (see Figures 5 and 6). While the three WENO var- ods. The temperature profile is in Figure 2 where the vertical iants are nearly indistinguishable with the reference profiles, solid line indicates the seeding time (i.e., at t 5 10 min). the HR and FD2 methods exhibit numerical diffusion and do Since an analytical solution is not available, the reference not resolve the peaks accurately. In addition, the FD2 method introduces a spurious oscillation (known as numerical disper-
Figure 5. CSD of seeded a form at the end of the batch Figure 7. Evolution of the error L1 norm with time for for the various numerical methods (DL 5 the various numerical methods (DL 5 0.6 0.6 lm). lm).
128 DOI 10.1002/aic Published on behalf of the AIChE January 2009 Vol. 55, No. 1 AIChE Journal Figure 10. CPU time required for the various numerical Figure 8. Error L norm at the end of the batch vs. DL 1 methods for a given error L norm at the for the various numerical methods. 1 end of the batch. sion) which can occur near sharp gradients with this method. size coordinate of the seeded and nucleated crystal size dis- The HR method does not produce spurious oscillation tributions, respectively. The error L norms from the three because the flux limiter detects the presence of sharp gra- 1 WENO variants are much smaller in magnitude and grow dients and limits the size derivatives. The larger numerical much slower than those from the HR and FD2 methods (see errors in the CSD profiles obtained by HR and FD2 methods Figure 7). In terms of the L norm, the JSHWENO method for the seeded a-form compared with the seeded b-form are 1 gave the smallest numerical errors. Figure 8 indicates that associated with its sharper gradient. the JSHWENO method gave smaller numerical errors for the The prediction errors were quantified in terms of the L 1 full range of DL from 0.1 to 2.0 lm. norm (EL ): 1 The JSHWENO method generally had lower CPU times 1 than the WPower-ENO method, but somewhat higher CPU EL1 ¼ times than the other methods for most values of DL (see Fig- 2ðNgrid;seed þ Ngrid;nuclÞ ()ure 9). To fairly compare the overall efficiency for these ; ; X NXgrid seed NXgrid nucl 3 ref ref ; methods, the CPU time was compared for discretizations that fseed;i;k fseed;i;k þ fnucl;i;k fnucl;i;k produce the same error L1 norm. From Figure 10 it is i¼fga;b k¼1 k¼1 observed that, for any given error L1 norm, the WENO var- (57) iants used less or equal CPU time to the HR and FD2 meth- ref ref where fseed,i,k and fnucl,i,k are the reference solutions for the seeded and nucleated crystals size distributions and Ngrid,seed and Ngrid,nucl are the number of grids used to discretize the
Figure 11. Relative CPU time for the various numerical methods with respect to CPU time from Figure 9. CPU time vs. DL for the various numerical JSHWENO for a given error L1 norm at the methods. end of the batch.
AIChE Journal January 2009 Vol. 55, No. 1 Published on behalf of the AIChE DOI 10.1002/aic 129 L O Table 4. 1 Self-convergence Order ( L1) for the Various ods. These results recommend WENO methods for the simu- Numerical Methods lation of crystallization processes, especially when the distri- DL [lm] LOCWENO JSHWENO WPower-ENO HR FD2 butions are sharp and very high accuracy is desired. These methods combine very high order of accuracy with good 0.1 1.62 1.30 1.53 1.67 1.81 convergence properties even in the presence of sharp varia- 0.2 2.43 1.45 1.97 1.66 1.78 0.3 2.41 2.40 2.12 1.65 1.57 tion in the size distributions. 0.4 2.26 2.84 2.28 1.57 1.37 0.5 2.27 2.87 2.55 1.50 1.20 0.6 2.46 2.98 2.79 1.44 1.07 Notation 0.7 2.41 3.12 2.86 1.34 0.98 ai,1, ai,2, ai,3 5 parameters for the saturation concentration of the 0.8 2.24 3.06 2.68 1.25 0.88 i-form crystals 0.9 2.11 3.06 2.60 1.20 0.82 a 5 Roe speed 1.0 1.94 2.86 2.39 1.09 0.75 kþ1/2 Bi 5 nucleation rate of the i-form crystals Average order 2.22 2.59 2.38 1.44 1.22 C 5 solute concentration Csat,i 5 saturation concentration of the i-form crystals Egi 5 activation energy for the growth rate of i-form crystals ods, and hence the WENO variants were more efficient. The EL 5 prediction errors in terms of the L1 norm 1 5 JSHWENO method was the most efficient for nearly all fi, fseed,i, fnucl,i total, seed, and nucleated crystal size distribution of the desired accuracy levels. Figure 11 shows the relative cost of i-form crystals G 5 growth rate of the i-form crystals the numerical methods with respect to the JSHWENO i Ik 5 the kth cell method. The HR method was more efficient than the FD2 ISm 5 smoothness indicator 5 a method for nearly all desired accuracy levels, and was more kba, kga, kda nucleation, growth, and dissolution rates of -form crys- efficient than the WPower-ENO method for some accuracy tals k b , k b 5 the jth nucleation and growth rates of b-form crystals levels, but was not as efficient as the LOCWENO and b ,j g ,j kgi,0 5 pre-exponential factor for the growth rate of i-form crys- JSHWENO methods. Although the WENO methods are more tals complicated to implement, their efficiency is much better kvi 5 volumetric shape factor of the i-form crystals 5 when sufficiently high accuracy in the size distribution is L, L0 characteristic length of crystals and nuclei 5 desired. Among the WENO variants, the performance of Lk crystals length at the kth discretized point DL 5 discretization size of crystal length JSHWENO is followed by that of the LOCWENO method Lseed,i 5 mean for the seed crystal size distribution of i-form by a small margin, and then followed by that of the crystals 5 self-convergence order WPower-ENO method. OL1 L1 5 2 Another way to assess numerical methods is to compute pˆkþ1/2, pˆk21/2 numerical flux approximation at node k þ 1/2 and k 1/2 the L1 self-convergence order r 5 qm quadratic polynomial flux approximation function Si 5 supersaturation of the i-form crystals EL j D 5 ln 1 2 L Sm candidate stencil EL jD 5 O ¼ 1 L : (58) T crystallizer temperature L1 ln 2 Greek letters This metric provides information on the internal consistency d 5 44 ( ) dirac delta function of the numerical method and its intrinsic convergence. The ji 5 scaling factor for the seed crystal size distribution of L1 self-convergence order for all numerical methods are in i-form crystals l 5 Table 4. For a linear model with a smooth solution, these i,n the nth moment of the i-form crystals x 5 values would correspond to the order of the truncation error m scalar weight to each candidate stencil Sm for the flux approximation for a given numerical method. This is not the case here / 5 flux limiter because of the nonlinearity of the model and the sharp gra- qi 5 density of the i-form crystals q 5 dients in the distribution. On average, the JSHWENO method solv density of the solvent r 5 gives the best L self-convergence order, followed by the seed,i standard deviation for the seed crystal size distribution 1 of i-form crystals WPower-ENO, LOCWENO, HR, and FD2 methods.
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