(70co) Application of the theory of Markov chains to model mixing of granular materials

Every mixing process is an evolution of microstates of a system, in which these states are connected with the components to be mixed. According to its nature mixing of powders is a process, which is discrete in space in advance, and very often its discretization in time can be also be accepted. The most appropriate tool to describe the evolution of such systems is the theory of Markov chains, which was successfully applied by different authors to model particular cases of the process. However, a general methodology of building these models is still not developed to a level, which could allow chemical engineers applying the theory in practical work. The objective of the study is to work out general recommendations on constructing cell models of batch and continuous mixing, on deriving basic operators of the models (such as the state vector, the matrix of transition probabilities, the evolution equation), and calculating technological parameters of mixing and mixtures (such as the mixture homogeneity, the entropic criterion, the residents time distribution and its moments, the variance reduction ratio, etc). The general algorithm for building the matrix of transition probabilities is described for 1D and 2D models, and examples of application of the theory to model the process in a static revolving mixer, and in a continuous blade mixed validated experimentally are presented. In parallel, there arises the problem of the right tracer choice: it is normally supposed that a perfect trace is the one, particles of which have completely the same properties determining a flow pattern as the basic granular material (the tracer particles must have a different property to be distinguished, the flow pattern is completely indifferent to which). However, the tracer really following the basic flow on the one hand brings us minimum information about the flow itself, on the other hand. Restoration of elements of the matrix of transition probabilities is a so called inverse problem, which solution is practically always is not unique. However, this solution can become unique, or localized if some additional constraints are employed in the description. An attempt to combine the RTD analysis with the hold-up – throughput correlation is also described on the basis of 2D array of cells with variable number of rows corresponding to the hold-up. This model allows predicting RTD curves for non-homogeneous flow in a continuous mixer Finally, it is shown theoretically and validated experimentally that the theory of Markov chains is an effective tool to model batch and continuous mixing.