In this work, a set of Discrete Element Method (DEM) simulations are performed based on a computer experimental design for the qualitative and quantitative characterization of a periodic section of a powder blender. Mixing of powders is a crucial operation in areas such as pharmaceuticals, cosmetics and food manufacturing since poor blend homogeneity can affect the quality of the final products. It is therefore extremely important to have a good understanding of mixing mechanisms and the effect of parameters such as operating conditions and mixer design on blend uniformity [1-2]. The available modeling approaches for powder blending range from purely first-principle based, (DEM modeling)  to simply data-driven based (Response Surface Methodology or Kriging modeling) , or hybrid-based models which combine first-principle and empirical correlations (Population Balance modeling) . Choosing between the available modeling approaches depends on the desired level of detail as well as the preferred computational cost. DEM is an expensive modeling approach that has been applied to study many different kinds of complex solid-based processes including powder mixing, since it can provide detailed information about every particle involved in the process. However, DEM is very computationally intensive tool which cannot be used for online purposes, optimization and control. The specific aims of this work include (a) the investigation of the effects and significance of different operating variables and design parameters on the performance of a blender and (b) the development of a fast reduced order model (ROM) using the information obtained by the set of DEM simulations which can predict the performance of the blender at unexplored operating regions.
Design of computer experiments for expensive simulations has attracted a lot of attention in the optimization literature [6-8]. Specifically, a lot of work has focused on designing a set of experiments that can lead to an accurate surrogate response surface which is less computationally expensive and more appropriate for process optimization. However, the minimization of sampling cost in the process of attaining the required information about the underlying process is a very important aspect. For this purpose, Latin Hypercube sampling has been considered in this work, due to its ability to produce a sampling set which is shown to be representative of the entire experimental region but smaller than a full factorial design . The six significant input variables considered in this work are the blade speed, blade angle, blade width, fill level, weir height and shaft angle. By applying the periodic section modeling developed in our previous study , effects of these variables on cross-sectional mixing and axial motion are distinguished. As a result, the variance decay rate of the batch-like mixing in the cross-sectional directions (kb), and the mean particle velocity in the axial direction (vx), which are key indices of mixing performance, are considered as output variables.
The first goal of this study is to understand the effect of different operating conditions on the mixing performance of continuous mixing system. To achieve this goal, Projection to Latent Structure (PLS) is applied to the data set of a total of 64 samples. The cross-validation method recommended by Wold et al.  is applied, which is used to derive the confidence intervals of loading scores of these variables in order to assess the sufficiency of the obtained data. The results reveal that increase of blade speed and shaft angle, or decrease of fill level, weir height and blade width, lead to significant increase of vx; increase of blade speed and blade angle, or decrease of fill level, blade width and shaft angle, lead to significant increase of kb. Based on the loading scores of different variables, simultaneously increasing blade speed and decreasing shaft angle is the optimal strategy for the improvement of mixing performance. This conclusion can be used for further design of continuous mixing system. Results from this initial qualitative study agree with experimental observations as well as expected conclusions based on theoretical knowledge of the mixing system. In addition, the bootstrapping approach followed increases the reliability of the database sufficiency for describing the process performance. As a result, the next step of this work is the development of a more quantitative predictive model based on this database.
The second aspect of this work is the development of a Reduced Order Model (ROM) based solely on the data obtained by the performed simulations. ROM modeling based on Principal Component Analysis (PCA) or Proper Orthogonal Decomposition (POD) has attracted a lot of attention in fluid dynamics where processes are modeled by expensive CFD simulations [12-13]. In this work, the collected data is comprised of the steady-state magnitudes of certain state variables inside the periodic section, such as average concentration, average velocity components in x, y and z direction, and the average particle kinetic energy. PCA is employed in order to compress the available data and Kriging models are built for the input-output mappings as well as input-PC loading mappings. Using this approach, the process outputs (kb and vx), and the values of the state variables at different locations of the periodic section can be predicted at unsampled operating conditions and design configurations of the mixer. Results show that the developed model can predict the contours of the desired state variables as well as the outputs at different input variable combinations with good accuracy at a very low computational cost.
Summarizing, this work addresses three challenges associated with expensive DEM simulations for modeling powder blending. Firstly, critical operating conditions and design parameters for the mixing performance of a continuous blender are identified through a PLS model. Secondly, the sufficiency of the data obtained based on a Latin Hypercube sampling design in the case of a high dimensional input variable space is assessed through a cross-validation approach in terms of the accuracy of the PLS loadings values. Finally, a reduced order model is built based on a PCA decomposition of the obtained data and the use of Kriging algorithm. DEM simulations are gaining popularity in modeling powder-based processes in the pharmaceutical industry, however, their high computational requirements prohibits their use for optimization, control and process flowsheet design. To our knowledge, this is the first attempt of developing a ROM model for DEM modeling and thus the proposed methodology will have significant applications in all areas of powder based processing.
1. Portillo, P.M., M.G. Ierapetritou, and F.J. Muzzio, Characterization of continuous convective powder mixing processes. Powder Technology, 2008. 182(3): p. 368-378.
2. Portillo, P.M., M.G. Ierapetritou, and F.J. Muzzio, Effects of rotation rate, mixing angle, and cohesion in two continuous powder mixers--A statistical approach. Powder Technology, 2009. 194(3): p. 217-227.
3. Dubey, A., et al., Computational Approaches for the study of granular dynamics of continuous blending process- I : DEM based methods. Macromolecular Materials and Engineering, 2010(Accepted for Publication).
4. Jia, Z., et al., Predictive Modeling for Pharmaceutical Processes Using Kriging and Response Surface. Journal of Pharmaceutical Innovation, 2009. 4(4): p. 174-186.
5. Ramachandran, R. and P.I. Barton, Effective parameter estimation within a multi-dimensional population balance model framework. Chemical Engineering Science, 2010. 65(16): p. 4884-4893.
6. Kleijnen, J.P.C., et al., State-of-the-Art Review: A User's Guide to the Brave New World of Designing Simulation Experiments. INFORMS JOURNAL ON COMPUTING, 2005. 17(3): p. 263-289.
7. Pistone, G. and G. Vicario, Design for Computer Experiments: Comparing and Generating Designs in Kriging Models, in Statistics for Innovation, P. Erto, Editor. 2009, Springer Milan. p. 91-102.
8. Sacks, J., et al., Design and Analysis of Computer Experiments. Statistical Science, 1989. 4(4): p. 409-423.
9. Simpson, T., D. Lin, and W. Chen, Sampling strategies for computer experiments: design and analysis. International Journal of Reliability and Safety (IJRS), 2001. 2(3): p. 209-240.
10. Gao, Y., M.G. Ierapetritou, and F.J. Muzzio, Periodic section modeling of convective continuous powder mixing processes. AIChE Journal, 2011. In Press: p. DOI: 10.1002/aic.12348.
11. Wold, S., M. Sjöström, and L. Eriksson, PLS-regression: a basic tool of chemometrics. Chemometrics and Intelligent Laboratory Systems, 2001. 58(2): p. 109-130.
12. Alonso, D., A. Velazquez, and J.M. Vega, A method to generate computationally efficient reduced order models. Computer Methods in Applied Mechanics and Engineering, 2009. 198(33-36): p. 2683-2691.
13. Lang, Y.-d., et al., Reduced Order Model Based on Principal Component Analysis for Process Simulation and Optimization†. Energy & Fuels, 2009. 23(3): p. 1695-1706.
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