Powder mixing is a crucial process in many industries, such as chemical manufacturing, foods, plastics and drugs. A slight change in the ingredient properties or the operating conditions during the manufacturing of a drug can have great impact on its specifications. The performance of a mixer is also dependent on accurate feeding system. Ingredient agglomeration and intermittent flow are known to depend on feeder design, size, and operating conditions as well as ingredient particle size and cohesion.

However, there is lack of a predictive understanding of the relationships between the operating and design parameters and the performance of the equipments, so that it can be used to find an optimal design. Various approaches have been developed to model powder flow and improve the characterization of powder processes, and can be mainly categorized into the following groups: Monte Carlo simulations, particle-dynamic simulations, heuristic models, and models based on kinetic theory. Discrete-element method (DEM) was initially developed by Cundall and Strack (1979). It simulates the interactions and movements of individual particles based on physical laws. However DEM is computationally very expensive since the Newton's equation of motion for each particle must be solved at each time step.

One of the widely used predictive data-driven models is response surface method (RSM), which was first introduced by Box and Wilson in 1951. It is an input-output mapping by fitting data using a set of basis functions. As the first step of RSM is specification of a sampling set within the local region, design of experiment (DOE) tools are usually required. Hence the prediction given by RSM relies greatly on the sampling points. A set of randomly selected sampling points very likely will lead to a poor predictive function. In this work, a predictive model is proposed that can efficiently characterize the performance of feeders and mixers with respect to operating parameters, and thus can be used to determine optimal process design variables and operating conditions. The Kriging method (Issaks and Srivistava, 1989; Gressie, 1993), which is an optimal prediction method designed for geophysical systems, has been utilized to predict the flow variability of feeders and regional concentration of mixers under different operating conditions. Without building an accurate mathematical model, Kriging method can provide a global predictive model starting with a small number of random sampling points and show where additional samplings should be done so as to iteratively improve the prediction. At each iteration, the first step is to determine semivariance of the each pair of the sampling points. After obtaining the set of semivariance scatterpoints, variogram model coefficients can be obtained using regression. Finally a set of equations are solved to get the kriging weights at each testing point, and the prediction value can be calculated as the weighted sum of the observed value of sampling points.

Two case studies of feeding and mixing processes are provided to illustrate the effectiveness of the proposed approach. The predicted flow variability is compared with the experimental data of loss-in-weight feeders, while for the mixing case the predicted concentrations describing mixture homogeneity are compared with the data obtained using a detailed DEM model.

Reference:

[1] Cundall P.A., Strack O.D.L. A discrete numerical model for granular assemblies. Geotechnique.1979, 29: 47–65.

[2] Box, G. and Wilson, K. On the experimental attainment of optimum conditions. J. Royal Stat. Soc. Series B. 1951, 13:1-45.

[3] Isaaks E. and Srivistava R. Applied geostatistics. New York, NY: Oxford Univ. Press, 1989

[4] Cressie N. Statistics for spatial data. New York, NY: Wiley, 1993.