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Strategy for Validating a Population Balance Model of a Batch Crystallization Process Using Particle Size Distribution from Image-based Sensor
19th European Symposium on Computer Aided Process Engineering – ESCAPE19 J. JeĪowski and J. Thullie (Editors) © 2009 Elsevier B.V. All rights reserved.
833
Strategy for Validating a Population Balance Model of a Batch Crystallization Process Using Particle Size Distribution from Image-based Sensor Debasis Sarkara, Zhou Yinga, Lakshminarayanan Samavedhamb, Rajagopalan Srinivasana,b a
Institute of Chemical and Engineering Sciences, 1 Pesek Road Jurong Island, Singapore 627833 b Department of Chemical and Biomolecular Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 117576
Abstract The modeling of the transient behavior of the crystal size distribution is essential to predict and effectively control the quality of the end product. The population balance approach provides an appropriate mathematical framework for the modeling of crystal size. For most batch crystallization processes, it is often difficult to obtain on-line relevant information about the crystal size distribution and the dissolved solid concentration in the liquid phase. We have recently developed an automated image analysis strategy that yields real-time measurements of both particle length and width with acceptable accuracy. In this contribution, we present an approach for validation of a population balance model for batch crystallization of monosodium glutamate by using on-line information about the state of the crystallization process provided by our image analysis strategy and other in-situ process analytical tools such ATR-FTIR. Keywords: Crystallization, Population balance modeling, Image analysis, Parameter estimation, Model validation
1. Introduction Crystallization from solution is a widely practiced unit operation in specialty chemical, pharmaceutical, and agrochemical industries for solid-liquid separation, purification, and production of solid crystals with desired shape and size distribution. The crystal size distribution (CSD) and shape or polymorphic form are the most important product quality variables for any crystallization process as these variables strongly influence the effectiveness of the end-use properties of the crystal products (bioavailability, compressibility, stability, dissolution rate) as well as the efficiency of downstream operations (filtration, drying, storage, handling). Therefore, the modeling of the transient behavior of the CSD is essential to predict and effectively control the quality of the end product. The population balance approach provides an appropriate mathematical framework for the modeling of CSD and has been widely studied in the literature [1-3]. However, the traditional one-dimensional population balance modeling where the size of a crystal is represented as volume equivalent diameter of a sphere is inadequate for many organic crystals that present anisotropic morphology. Thus, multidimensional population balance modeling of such systems has recently attracted the growing interest as it provides a means for incorporating crystal shape into the simulation [4-6].
834
D. Sarkar et al.
For most batch crystallization processes, it is often difficult to obtain on-line relevant information about the crystal size distribution and the dissolved solid concentration in the liquid phase. We have recently developed an automated image analysis strategy that combines classical image analysis techniques with multivariate statistics for online analysis of in-situ images from crystallization process [7, 8]. The strategy introduces a novel image segmentation step based on information extracted from multivariate statistical models. Experimental results have shown that the strategy effectively extracts crystal size and shape information from in-situ images and can be used to reliably investigate crystallization kinetics, especially for high-aspect ratio systems. Since the image analysis strategy yields real-time measurements of both particle length and width, it is useful for validating both one-dimensional and multi-dimensional population balance models for crystallization processes. In this contribution, we present an approach for validation of a population balance model for batch crystallization of monosodium glutamate (MSG) by using on-line information about the state of the crystallization process provided by our image analysis strategy and other in-situ process analytical tools such as ATR-FTIR. The real-time measurements provided by these insitu process analytical tools are expected to allow improved understanding and control of crystallization processes.
2. Image analysis based estimation of particle size distribution A major challenge in online automated image analysis is that in situ images from process equipment are very noisy, therefore segmentation, i.e., separating the particles from the background, is nontrivial. Traditionally, image analysis based approaches have relied on edge detection techniques for particle segmentation; however this requires the specification of suitable thresholds. Specification of robust thresholds is difficult given the high noise and other imperfections in the images. We have recently developed a new method to segment images by combining classical image analysis technique with multivariate statistics. The method takes an alternative approach to segmentation by characterizing the image background instead of the particle. A pseudo-image is first created by extracting suitable features from each in situ image, performing Principal Component Analysis, and formulating the Hotelling T2 statistic as an image. This pseudo-image has fewer imperfections compared to the in situ image given the noise elimination obtained by ignoring the smaller Principal Components. It can therefore be segmented robustly using a global threshold. The proposed image analysis strategy uses the background image (without any particle) for estimating this threshold value. The obtained particle outline is further refined through a discrete Fourier transform and signature curve analysis and particle size and shape information extracted. The accuracy, robustness and efficiency of the strategy have been established by comparing its performance with those obtained by manual image segmentation. Experimental results for batch cooling crystallization of MSG show that the method yields reasonably good estimates of the particles length with about 5% median error.
3. Modeling of MSG batch crystallization The crystals of MSG exhibit a rod-like habit. Such high aspect ratio crystals can be modeled as a rectangular body as shown in Fig. 1. Thus, two characteristic sizes, the length (L) and the width (W), are required for the formulation of the population balance model. Let us describe the population of crystal by the number population density
Strategy for Validating a Population Balance Model of a Batch Crystallization Process Using Particle Size Distribution from Image-Based Sensor
835
function ȥ(L,W,t) such that N(L,W,t), the number of crystals with length between L and L+dL and width between W and W+dW is
N ( L, W , t ) = ³
L + dL
L
³
W + dW
W
ψ ( L,W , t )dLdW
(1)
Assuming that the population density does not depend on the spatial coordinate, the two-dimensional population balance for a constant volume (V) crystallizer can be written as
Fig. 1: Two dimensional approximation of a MSG particle
∂ψ ∂ ∂ + ( G Lψ ) + ( GWψ ) = R N + R A − RB ∂t ∂L ∂W
(2)
where RN, RA, and RB indicate the nucleation rate, the agglomeration rate, and the breakage rate, respectively. The growth rate along the length is GL and that along the width is GW. The nucleation rate and the growth rates are functions of supersaturation ı(t) and often expressed as empirical power law models. The supersaturation depends on the concentration of solute in solution C(t) and the equilibrium solubility C*(t). The solute concentration and hence the supersaturation can be measured online by ATRFTIR. The agglomeration rate and the breakage rate are expressed as differences between a birth term (B) and a death term (D). The growth rates (GL and GW) can be directly determined with our image-processing strategy (Fig. 2). As seen from Fig. 2, the width of the crystals remains almost constant after a certain period of time and thus the population density can be described by a single characteristic length, ȥ(L,t). In absence of any significant breakage, the MSG crystallization can thus be described by the following one dimensional population balance equation along with mass balance and other kinetic models.
D. Sarkar et al.
836 300
) m μ( ht g n e L
200
100
0
0
5
10
15
20
15
20
Time (h) 20 15 ) m μ( ht di W
10 5 0
0
5
10 Time (h)
Fig. 2: Time evolution of particle length and width
∂ψ ∂ (GLψ ) + = RN + RA ∂t ∂L
ψ ( L, t = 0) = ψ seed ( L)
RN = kbσ b μ3
GL = k gσ g
∞
μi = ³ Lψ ( L, t )dL i
dC 0 = −3ρ c kv GL μ2 dt
C (t ) − C * (t ) σ (t ) = C * (t )
(3)
RA = Baggl − Daggl
Baggl =
L2 2
β [( L3 − λ 3 )1/3 , λ ] ⋅ψ [( L3 − λ 3 )1/3 , t ] ⋅ψ (λ , t )d λ ³0 ( L3 − λ 3 ) 2/3 L
∞
Daggl = ψ ( L, t ) ³ β ( L, λ ) ⋅ψ (λ , t )d λ 0
kb, b, kg, g are rate constants and ȝi is the i-th moment of the CSD. ȡc is the density of the crystal and kv the volumetric shape factor. The bitth and death terms for agglomeration are expressed in terms of characteristic particle length where the
Strategy for Validating a Population Balance Model of a Batch Crystallization Process Using Particle Size Distribution from Image-Based Sensor
837
agglomeration kernel ȕ(L,Ȝ) is a measure of the frequency at which crystals of size L and Ȝ collide to form an agglomerate [2]. The identification of an appropriate kernel is a major challenge in agglomeration modeling.
4. Validation strategy The population balance model presented in Eq. (3) can be solved numerically by discretizing the length coordinate, which results in a set of ordinary differential equations. The solution of Eq. (3) for a given operating condition will yield the number of particles, N(L,t) and the solute concentration, C(t). The number of particles, N(L,t) can be determined directly form our image analysis strategy. The solute concentration in the crystallizer can be measured online by ATR-FTIR. Thus the time evolution of CSD and solute concentration can be used to estimate the model parameters (ș) through the minimization of an objective function, ij(ș), which is a measure of the deviation between the model and experiment. The weighted least-square form is often a common choice for the objective function [9].
ϕ (θ ) = α1ε [C (θ )] + α 2ε [ψ m (θ )]
(4)
The İ[C(ș)] is the quadratic error between the measured and model-predicted solute concentration. Similarly, İ[ȥm(ș)] is the error between the measured CSD and simulated CSD. Į1, Į2 are scalars to balance the respective weights of the two quadratic criteria.
5. Concluding remarks The modeling of the transient behavior of crystal size distribution is essential to predict and effectively control the quality of the end product. This work proposes a methodology for development and validation of a population balance model for batch crystallization of MSG that exhibits rod-like habit. The strategy involves direct on-line measurement of all key variables that influence CSD, including the on-line measurement of CSD by our recently developed multivariate image analysis technique. Currently, we are applying this validation strategy to the MSG batch cooling crystallization system described above and the results from this study will be reported in future communications.
References [1] [2] [3] [4]
A. D. Randolph and M. A. Larson, Theory of Particulate Processes, second ed. Academic Press, New York, 1988 M. J. Hounslow et al., AIChE Journal 34 (1988) 1821-1832. D. Ramkrishna, Population Balances: Theory and Applications to Particulate Systems in Engineering, Academic Press, New York, 2000. [5] D. L. Ma et al., Industrial Engineering Chemistry and Research 41 (2002) 6217-6223. [6] F. Puel et al., Chemical Engineering Science 58 (2003) 3715-3727. [7] C. Y. Ma et al., Advanced Powder Technology 18 (2007) 707-723. [8] D. Sarkar et al., accepted in Chemical Engineering Science (2008). [9] Z. Ying et al., accpted in Computers & Chemical Engineering (2008). [10] A. Caillet et al., Crystal Growth and Design 7 (2007) 2088-2095.
833
Strategy for Validating a Population Balance Model of a Batch Crystallization Process Using Particle Size Distribution from Image-based Sensor Debasis Sarkara, Zhou Yinga, Lakshminarayanan Samavedhamb, Rajagopalan Srinivasana,b a
Institute of Chemical and Engineering Sciences, 1 Pesek Road Jurong Island, Singapore 627833 b Department of Chemical and Biomolecular Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 117576
Abstract The modeling of the transient behavior of the crystal size distribution is essential to predict and effectively control the quality of the end product. The population balance approach provides an appropriate mathematical framework for the modeling of crystal size. For most batch crystallization processes, it is often difficult to obtain on-line relevant information about the crystal size distribution and the dissolved solid concentration in the liquid phase. We have recently developed an automated image analysis strategy that yields real-time measurements of both particle length and width with acceptable accuracy. In this contribution, we present an approach for validation of a population balance model for batch crystallization of monosodium glutamate by using on-line information about the state of the crystallization process provided by our image analysis strategy and other in-situ process analytical tools such ATR-FTIR. Keywords: Crystallization, Population balance modeling, Image analysis, Parameter estimation, Model validation
1. Introduction Crystallization from solution is a widely practiced unit operation in specialty chemical, pharmaceutical, and agrochemical industries for solid-liquid separation, purification, and production of solid crystals with desired shape and size distribution. The crystal size distribution (CSD) and shape or polymorphic form are the most important product quality variables for any crystallization process as these variables strongly influence the effectiveness of the end-use properties of the crystal products (bioavailability, compressibility, stability, dissolution rate) as well as the efficiency of downstream operations (filtration, drying, storage, handling). Therefore, the modeling of the transient behavior of the CSD is essential to predict and effectively control the quality of the end product. The population balance approach provides an appropriate mathematical framework for the modeling of CSD and has been widely studied in the literature [1-3]. However, the traditional one-dimensional population balance modeling where the size of a crystal is represented as volume equivalent diameter of a sphere is inadequate for many organic crystals that present anisotropic morphology. Thus, multidimensional population balance modeling of such systems has recently attracted the growing interest as it provides a means for incorporating crystal shape into the simulation [4-6].
834
D. Sarkar et al.
For most batch crystallization processes, it is often difficult to obtain on-line relevant information about the crystal size distribution and the dissolved solid concentration in the liquid phase. We have recently developed an automated image analysis strategy that combines classical image analysis techniques with multivariate statistics for online analysis of in-situ images from crystallization process [7, 8]. The strategy introduces a novel image segmentation step based on information extracted from multivariate statistical models. Experimental results have shown that the strategy effectively extracts crystal size and shape information from in-situ images and can be used to reliably investigate crystallization kinetics, especially for high-aspect ratio systems. Since the image analysis strategy yields real-time measurements of both particle length and width, it is useful for validating both one-dimensional and multi-dimensional population balance models for crystallization processes. In this contribution, we present an approach for validation of a population balance model for batch crystallization of monosodium glutamate (MSG) by using on-line information about the state of the crystallization process provided by our image analysis strategy and other in-situ process analytical tools such as ATR-FTIR. The real-time measurements provided by these insitu process analytical tools are expected to allow improved understanding and control of crystallization processes.
2. Image analysis based estimation of particle size distribution A major challenge in online automated image analysis is that in situ images from process equipment are very noisy, therefore segmentation, i.e., separating the particles from the background, is nontrivial. Traditionally, image analysis based approaches have relied on edge detection techniques for particle segmentation; however this requires the specification of suitable thresholds. Specification of robust thresholds is difficult given the high noise and other imperfections in the images. We have recently developed a new method to segment images by combining classical image analysis technique with multivariate statistics. The method takes an alternative approach to segmentation by characterizing the image background instead of the particle. A pseudo-image is first created by extracting suitable features from each in situ image, performing Principal Component Analysis, and formulating the Hotelling T2 statistic as an image. This pseudo-image has fewer imperfections compared to the in situ image given the noise elimination obtained by ignoring the smaller Principal Components. It can therefore be segmented robustly using a global threshold. The proposed image analysis strategy uses the background image (without any particle) for estimating this threshold value. The obtained particle outline is further refined through a discrete Fourier transform and signature curve analysis and particle size and shape information extracted. The accuracy, robustness and efficiency of the strategy have been established by comparing its performance with those obtained by manual image segmentation. Experimental results for batch cooling crystallization of MSG show that the method yields reasonably good estimates of the particles length with about 5% median error.
3. Modeling of MSG batch crystallization The crystals of MSG exhibit a rod-like habit. Such high aspect ratio crystals can be modeled as a rectangular body as shown in Fig. 1. Thus, two characteristic sizes, the length (L) and the width (W), are required for the formulation of the population balance model. Let us describe the population of crystal by the number population density
Strategy for Validating a Population Balance Model of a Batch Crystallization Process Using Particle Size Distribution from Image-Based Sensor
835
function ȥ(L,W,t) such that N(L,W,t), the number of crystals with length between L and L+dL and width between W and W+dW is
N ( L, W , t ) = ³
L + dL
L
³
W + dW
W
ψ ( L,W , t )dLdW
(1)
Assuming that the population density does not depend on the spatial coordinate, the two-dimensional population balance for a constant volume (V) crystallizer can be written as
Fig. 1: Two dimensional approximation of a MSG particle
∂ψ ∂ ∂ + ( G Lψ ) + ( GWψ ) = R N + R A − RB ∂t ∂L ∂W
(2)
where RN, RA, and RB indicate the nucleation rate, the agglomeration rate, and the breakage rate, respectively. The growth rate along the length is GL and that along the width is GW. The nucleation rate and the growth rates are functions of supersaturation ı(t) and often expressed as empirical power law models. The supersaturation depends on the concentration of solute in solution C(t) and the equilibrium solubility C*(t). The solute concentration and hence the supersaturation can be measured online by ATRFTIR. The agglomeration rate and the breakage rate are expressed as differences between a birth term (B) and a death term (D). The growth rates (GL and GW) can be directly determined with our image-processing strategy (Fig. 2). As seen from Fig. 2, the width of the crystals remains almost constant after a certain period of time and thus the population density can be described by a single characteristic length, ȥ(L,t). In absence of any significant breakage, the MSG crystallization can thus be described by the following one dimensional population balance equation along with mass balance and other kinetic models.
D. Sarkar et al.
836 300
) m μ( ht g n e L
200
100
0
0
5
10
15
20
15
20
Time (h) 20 15 ) m μ( ht di W
10 5 0
0
5
10 Time (h)
Fig. 2: Time evolution of particle length and width
∂ψ ∂ (GLψ ) + = RN + RA ∂t ∂L
ψ ( L, t = 0) = ψ seed ( L)
RN = kbσ b μ3
GL = k gσ g
∞
μi = ³ Lψ ( L, t )dL i
dC 0 = −3ρ c kv GL μ2 dt
C (t ) − C * (t ) σ (t ) = C * (t )
(3)
RA = Baggl − Daggl
Baggl =
L2 2
β [( L3 − λ 3 )1/3 , λ ] ⋅ψ [( L3 − λ 3 )1/3 , t ] ⋅ψ (λ , t )d λ ³0 ( L3 − λ 3 ) 2/3 L
∞
Daggl = ψ ( L, t ) ³ β ( L, λ ) ⋅ψ (λ , t )d λ 0
kb, b, kg, g are rate constants and ȝi is the i-th moment of the CSD. ȡc is the density of the crystal and kv the volumetric shape factor. The bitth and death terms for agglomeration are expressed in terms of characteristic particle length where the
Strategy for Validating a Population Balance Model of a Batch Crystallization Process Using Particle Size Distribution from Image-Based Sensor
837
agglomeration kernel ȕ(L,Ȝ) is a measure of the frequency at which crystals of size L and Ȝ collide to form an agglomerate [2]. The identification of an appropriate kernel is a major challenge in agglomeration modeling.
4. Validation strategy The population balance model presented in Eq. (3) can be solved numerically by discretizing the length coordinate, which results in a set of ordinary differential equations. The solution of Eq. (3) for a given operating condition will yield the number of particles, N(L,t) and the solute concentration, C(t). The number of particles, N(L,t) can be determined directly form our image analysis strategy. The solute concentration in the crystallizer can be measured online by ATR-FTIR. Thus the time evolution of CSD and solute concentration can be used to estimate the model parameters (ș) through the minimization of an objective function, ij(ș), which is a measure of the deviation between the model and experiment. The weighted least-square form is often a common choice for the objective function [9].
ϕ (θ ) = α1ε [C (θ )] + α 2ε [ψ m (θ )]
(4)
The İ[C(ș)] is the quadratic error between the measured and model-predicted solute concentration. Similarly, İ[ȥm(ș)] is the error between the measured CSD and simulated CSD. Į1, Į2 are scalars to balance the respective weights of the two quadratic criteria.
5. Concluding remarks The modeling of the transient behavior of crystal size distribution is essential to predict and effectively control the quality of the end product. This work proposes a methodology for development and validation of a population balance model for batch crystallization of MSG that exhibits rod-like habit. The strategy involves direct on-line measurement of all key variables that influence CSD, including the on-line measurement of CSD by our recently developed multivariate image analysis technique. Currently, we are applying this validation strategy to the MSG batch cooling crystallization system described above and the results from this study will be reported in future communications.
References [1] [2] [3] [4]
A. D. Randolph and M. A. Larson, Theory of Particulate Processes, second ed. Academic Press, New York, 1988 M. J. Hounslow et al., AIChE Journal 34 (1988) 1821-1832. D. Ramkrishna, Population Balances: Theory and Applications to Particulate Systems in Engineering, Academic Press, New York, 2000. [5] D. L. Ma et al., Industrial Engineering Chemistry and Research 41 (2002) 6217-6223. [6] F. Puel et al., Chemical Engineering Science 58 (2003) 3715-3727. [7] C. Y. Ma et al., Advanced Powder Technology 18 (2007) 707-723. [8] D. Sarkar et al., accepted in Chemical Engineering Science (2008). [9] Z. Ying et al., accpted in Computers & Chemical Engineering (2008). [10] A. Caillet et al., Crystal Growth and Design 7 (2007) 2088-2095.