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The effect of mixing on the PSD of paracetamol continuous cooling crystallization products using a CFD-PBE simulation Xiaoyan Fu, Dejiang Zhang, Shijie Xu, Bo Yu, Keke Zhang, Sohrab Rohani, and Junbo Gong Cryst. Growth Des., Just Accepted Manuscript • DOI: 10.1021/acs.cgd.7b01671 • Publication Date (Web): 16 Apr 2018 Downloaded from http://pubs.acs.org on April 16, 2018
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Crystal Growth & Design
The effect of mixing on the PSD of paracetamol continuous cooling crystallization products using a CFD-PBE simulation Xiaoyan Fua, Dejiang Zhanga, Shijie Xua, Bo Yu a, Keke Zhanga, Sohrab Rohanid, Junbo Gonga, b, c * a b
School of Chemical Engineering and Technology, Tianjin University, Tianjin, China;
Key Laboratory Modern Drug Delivery and High Efficiency in Tianjin University, Tianjin, China; c
The Co-Innovation Center of Chemistry and Chemical Engineering of Tianjin, Tianjin, China
d
Dept. of Chemical and Biochemical Engineering, University of Western Ontario, London, ON N6A 5B9, Canada *Corresponding author E-mail:
[email protected] Abstract Continuous crystallization is a widely viewed topic as it has many advantages in industrial production. However, it’s very resource intensive and time consuming to determine operation condition through trial and error. Computer simulation appears to an effective method in continuous crystallization design. The continuous cooling crystallization of paracetamol in a 5 L crystallizer was modeled to investigate the effect of mixing on the particle size distribution (PSD). A novel coupled CFD-PBE simulation was developed in which the population balance equation (PBE) was discretized using a high-resolution central scheme and solved simultaneously with CFD equations in ANSYS FLUENT 15.0 utilizing user-defined scalars (UDS). The influences of both internal grid and external (spatial) grid on the final PSD were studied, and an appropriate grid was chosen. The spatial distribution of velocity, temperature, slurry density, and nucleation rate was simulated. The continuous cooling crystallization under different stirrer speed, bath temperature and residence time was investigated. It was found that (a) higher stirrer speeds lead to uniform distribution but smaller PSD , and (b) as the average temperature gets higher, or the residence time gets longer, the influence of mixing on final PSD gets smaller. 1
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Key words: Continuous crystallization; Paracetamol; Particle size distribution; CFD-PBE approach; Mixing
1. Introduction Crystallization is an indispensable separation unit in the chemical, pharmaceutical or food industry. Continuous crystallization offers many advantages including better control of the particle size distribution (PSD) and product morphology. For a long time, researchers believed that continuous production, including continuous crystallization, is only used in large-scale production, and the pharmaceutical industry generally does not reach such a scale. With the vigorous development of online monitoring and control technology, and the increasing demand for drugs in society, the continuous crystallization of drugs has been widely researched in recent decades. Batch crystallization usually demonstrates fluctuations from batch to batch, causing different clinical trial results. And the batch crystallization process is usually rather complex, with crystallization conditions changing significantly with time. Compared to the batch crystallization, continuous crystallization shows great potential in product stability, high product quality, small footprint, high production efficiency, high production rate, and high energy efficiency. The PSD of the product could greatly affect the quality of the product (flowability, bulk density, etc.) and the downstream processes (filtration, granulation, tableting, etc.). In order to get the desired PSD, the process should be properly designed and controlled. The population balance equation (PBE) relating the PSD and the operating conditions is widely employed. The PBE comprises of a hyperbolical integro-partial-differential equation. Analytical solutions could exist only in some special situations 1, 2. Numerical solutions, however, show better adaptability in different situations. One general numerical approach is the method of moments, which could be concluded as the standard method of moments3, the quadrature method of moments4 and its derivatives (direct quadrature method of moments (DQMOM)5, fixed quadrature method of moments (FQMOM)6, automatic differentiation-quadrature method of moments (AD-QMOM)7, differential-algebraic equation-quadrature method of moments 2
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Crystal Growth & Design
(DAE-QMOM)8). Due to its low computational cost, it is widely used in CFD-PBE field. Some stochastic methods such as lattice Boltzmann method9 and Monte Carlo methods10 are also popular in computer simulation. Another type of numerical approach is the multi-class method. Particles are divided into different bins according to the particle size. Every bin undergoes mass exchange with each other. The multi-class method includes the fixed pivot method11, method of weighted residuals (including finite element scheme), high-resolution finite volume method12, etc. The method of moments, through computational efficiency, have some difficulties in reconstructing into PSD. Many reconstruction methods were proposed while no satisfactory unified method can be found in the literature up to now.13, 14 The fixed pivot method and other multi class schemes usually cause numerical diffusion. The high-resolution method can provide high accuracy while avoiding the numerical diffusion and numerical dispersion. Moreover, combined with the finite volume scheme, this method could easily couple with Navier-Stokes equations and solved simultaneously. As it is clear, the crystallization in industry or laboratory is effected/controlled by the operation conditions, which involve both the time dimension and space dimension, especially at industrial scale. The computational fluid dynamics (CFD) is an efficient numerical simulation of the flow field based on the conservation equations. The spatial distribution of operation conditions, including supersaturating, slurry density, temperature, turbulence energy that control nucleation rate, growth rate and particle breakage/aggregation can be simulated. Therefore, coupling of CFD with PBE is necessary in an inhomogeneous system. The CFD-PBE coupled model is investigated widely. The gas-liquid system is the most general field of its application because the PSD of the bubbles has a strong influence on the flow field and mass transfer efficiency. 15 However, the bubbles have a strong tendency for breakage during aggregation, and nearly no nucleation and growth, leading to a difference in numerical approaches with the crystallization system. Similarly, liquid-liquid systems 16, 17 and fluidized bed 5 are also studied. CFD-PBE is usually coupled with a micro mixing model to deal with fast precipitation reactions. 3
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Ulbert et al. (2005) 18 proposed a two-population model to study the imperfect mixing in a MSMPR. Duan Xiaoxia et al. (2016)
19
coupled the CFD-PBE model with the
engulfment model (E-model) to obtain the numerical solution of reaction crystallization in a stirred tank. Metzger et al. (2016)20 modeled fast precipitation processes in the impinging jet using the direct quadrature method of moments–interaction by exchange with
the
mean
(DQMOM-IEM)
approach.
The
computational
fluid
dynamics-population balance equation-probability density function (CFD-PBE-PDF) model was widely used owning to its convenience to incorporate into existing CFD codes. 21-26 The CFD-PBE simulation in a stirred tank has been successfully accomplished before. Wei (1997)27 first integrated CFD and precipitation. Following Wei, Jaworski and Nienow (2003)28 investigated the precipitation reaction of barium chloride and sodium sulphate in a continuous stirred tank, the standard method of moment was used and the influence of agitator speed and residence time were roughly investigated. Vicum et al. (2007)29 modeled the micro-mixing effect in a semi batch stirred tank using a coupled CFD-PBE-PDF method. Different agitation rates, reactant concentrations and feed addition modes were compared with the experimental data. Baldyga et al. (2007)30 simulated a double-feed semi batch precipitation process using a three environmental PDF model. The feed zone was refined and the results showed a satisfactory agreement with experimental data. Wang et al. (2007)31 carried out a finite-mode-probability density function (FM-PDF) model of a continuous stirred tank under cylindrical coordinates. Woo et al. (2006)32 combined a high-resolution scheme with the PDF model. In order to reduce the amount of calculation, a 2D fixed flow field was utilized as background. The effects of agitation speed, addition mode, and scale-up were demonstrated through the CFD model. However, there was no experimental data and the 2D fixed flow field did not represent the 3D stirred tank adequately. Another strategy to reduce computation is called compartmental modeling, during which the flow field is divided into several well-mixed zone proposed by Kramer et al. (1996)33. Cheng Jingcai et al. (2012)34 successfully simulated a 3D stirred tank incorporating not only nucleation and growth, but also aggregation in a continuous precipitation system. 4
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A fixed pivot method with first-order upwind scheme was used to discretize the inner coordinates. Although significant achievements have been made, the CFD-PBE simulation of stirred tank under various conditions using different crystallization technology still needs extensive research. Most of the articles applied the method of moments or its derivatives to solve the population balance equation 4-7, 20, 35-40, though computationally efficient, the real particle size distribution could not be obtained. The PSD curve exhibits a double peak or shows a sharp change, resulting from the deconvolution algorithm. Multi-class method seems necessary to meet the increasing accuracy and demand in particle product simulation. This article tried to utilize the multi-class method with high-resolution scheme to solve the PBE, and combine PBE with CFD model in a continuous cooling stirred tank crystallizer. The high-resolution central discretization scheme was developed for nonlinear conservation laws and convection-diffusions by Kurganov & Tadmor (2000)41 firstly. Benefited by its small numerical viscosity, none nonphysical oscillations and the adaptability to small time steps, high-resolution scheme was adapted to crystallization to solve population balance equations soon, and in variety aspects. For example, Woo et al.(2006, 2009)24,
32
and Pirkle et al.(2015)25 adapted it to solve PBEs in fast
crystallization combined with probability distribution function. Ma et al. (2002a, b)42, 43 and Gunawan et al. (2004)12 adapted it to solve multi-dimensional crystallization process. Researchers like Kumar et al. (2006, 2009)44,
45
deal with aggregation in
population balance using this scheme and did some comparison. Filbet et al. (2004)46 developed the finite volume scheme and Gunawan et al. (2008)47 developed a parallel simulation of PBEs using HR scheme. Recently, Cheng et al. (2017)48 extended this scheme to a general form. To the authors’ best knowledge, there are no article using this high resolution (HR) scheme in continuous crystallization cooling processes. The mathematical models were showed in the second section. In the third section the computation details were listed followed by the simulation process and the algorithm validation. In the last section the results and discussion were presented. 5
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2. Modeling In order to guarantee calculation accuracy and reduce the amount of calculation, the following assumptions were made: (1) The solution was treated as pseudo-homogeneous system, where the particles move with the fluid. (The maximum Stokes number in this study was 0.1296, indicating that the particles have good follow-up ability.) (2) The micro-mixing effect as well as aggregation and breakage were neglected. (3) The size of newly generated crystal was set as 1 µm. (4) The fluctuations of the fluid surface were neglected. The surface was treated as symmetrical. (5) No liquid density change during crystallization was assumed.
2.1 CFD models The commercial CFD package FLUENT 15.0 was used to solve the convection-diffusion equations. There is a great potential in predicting accurately the turbulent flow in the stirred tank using a detached eddy simulation method (DES) proposed by Gimbun (2012)49. The DES method performs large-eddy simulation (LES) away from walls and Reynolds averaged Navier-Stokes (RANS) in boundary layers to deal with the anisotropic flow structures close to the wall. Though there are so many advantages, the DES model is an unsteady model and it would take a long time to simulate flow field. This article still carried out a standard k-epsilon model to simulate the turbulence. The turbulent kinetic energy might be under-predicted near the impeller. 49
2.2 PBE models The population balance model is shown below: 50 [ ]
+ ∇ ∙ − ∇ ∙ Γ ∇ = −
[, ]
+ −
(1)
The number density n (#/m3 solvent/m) is a joint possibility density function which 6
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has a distribution at internal coordinate L and outer coordinate x. The flow velocity at outer and inner coordinate is up and G (L, x), respectively. up (L, x) is the velocity of particles of size L at spatial position x. G (L, x) is the size dependent growth rate and is a function of supersaturation. The supersatuation has a distribution at spatial position x.
Γ is the effective diffusion coefficient, kg/m/s. B and D represents the birth term and death term, respectively, caused by nucleation and dissolution.
To couple PBE with the FLUENT solver, the high-resolution finite-volume scheme12 is used to discretize n at inner coordinate. Particles were divided into several bins according to their size. The growth term was discretized as in equation 2. !" !
− ∆ {'()* [( +
=# −
$
$
"
∆"
+
,
{'()* [()$ − +
( ] − '(-* [(-$ +
∆"
4 5
∆"3* ,
()$
+
(-$ ]}, ∆0 ≥ 0
∆".* ,
] − '(-* [( − +
( ]}, ∆0 < 0
∆" ,
(2)
The derivatives ( were dealt with using a differential approximation, to avoid
the numerical oscillation and negative values, the minmod limiter proposed by Roe 51 was applied.
( = 898:;<
" - ".* ∆"
,
