Modeling Continuous Powder Mixing by Means of the Theory of Markov Chains

Abstract

This article demonstrates the efficiency of the application of the theory of Markov chains as a tool to model and simulate continuous powder mixing to aid in better design of such equipment. Markov chain models allow calculating practically all parameters of the process necessary for its characterization, and in particular those related to particle residence time distribution (RTD). Some numerical examples from the model, which are important for better understanding the process, are also included. It is shown that the main factor defining the efficiency of continuous mixing, through the variance reduction ratio (VRR), is the ratio of the mean residence time and the period of inflows fluctuation, rather than the variance of the RTD. Also, the influence of the dimensions of the mixer outlet on the mean residence time, and in turn on the VRR, is examined as another way of improving the design.

Using Compartment Modeling to Investigate Mixing Behavior of a Continuous Mixer

“…Berthiaux and coworkers [5] stated that the velocity in the radial direction could be neglected because they assumed it is orders of magnitude lower than that of the axial component, suggesting that the dynamic change in concentration occurs due to the rate of dispersion in the radial and axial direction, as well as the axial transport in the axial direction. The mass balance resulting from this equation can be discretized and solved numerically.…”

A review on the continuous blending of powders

“…In a recent publication, Markov chains were applied to model continuous blending of fine powders (Berthiaux et al, 2004). The continuous blender was assumed to be made up of a series of well mixed states.…”

“…The design of the outlet along with residence time and time period of oscillation of the feed were found to be critical parameters. Berthiaux et al (2004) Manual counting of tracer particles - Chen et al (1973) and scale of segregation was shown to be dependent on the scale of scrutiny for powders. Therefore, feeding mechanism and monitoring techniques play a substantial role in design and investigation of continuous blending systems.…”

Predicting bulk powder flow dynamics in a continuous mixer operating in transitory regimes

“…In the past, Markovian models have been applied to various powder mixers. One can refer to the following works: Chen et al [10], Lai and Fan [11], Wang and Fan [12,13], Fan and Shin [14], Aoun-Habbache et al [15], Berthiaux et al [16], Ponomarev et al [17]. As all these papers described the process under steady conditions, only homogeneous Markov chains have ever been considered for mixing.…”