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.
chain modelling and experimental investigation of powder-mixing kinetics in static revolving mixers. Keywords: Static mixer Mixing Markov chains Model Mixture quality a b s t r a c tThis study aims to develop a general model that is able to describe powder flow and mixing in static mixers, regardless of the type of mixer or the mixing configurations. The process model is based on a homogeneous Markov chain describing the flow of each component through the mixing zone by a series of interconnected cells. It accounts for the number of mixing elements and their disposition in the mixer, as well as particle segregation via different transition probabilities. Some simulations are given to emphasize this particular aspect. Other outcomes of the model include the number of passages to reach a required mixture quality, as well as the asymptotic distribution of components. A laboratory static mixer of revolving type was designed specially for this study. It comprises up to 10 mixing sections, and its high internal voidage favours free flow of the powder. Segregating and non-segregating mixtures have been used to test the model and adjust unknown parameters. The model gives very satisfying results. In particular, it is able to account for the oscillating character of mixing kinetics due to particle segregation. It is also suggested that these parameters could be linked separately to powder flowability and mixing element characterization.
Nomenclature a Coefficient of the homographic equation (m/s) b Coefficient of the homographic equation (m 2 /s) B Distance between flanks (m) d Diameter of the screw (m) d z Transition probability along the axis z (-) d y Transition probability along the axis yInternal diameter of the barrel (m) eWidth of the screw flank (m)Coefficient of the homographic equation (m) g z The down channel pressure gradient (Pa/m) H Cross channel depth (m) j The transition number (-) L Length of the screw shaft (m) L f Length of the die (m) L s Length of the channel (m) L t Total path length (m) m Row number (-) M Mass of material in the extruder (kg) n Column number (-) P Matrix of transition probabilities (-) P ik Matrix of transition probabilities from kth to ith columns Q Mass flow rate (kg/s) r Throttle ratio (-) S State vector (-) t Time at which the calculated results are obtained (s) T Temperature of the barrel ( • C) v Transition probability describing convection (-) V z Average velocity along the axis z (m/s) V ex Free volume of the extruder (m 3 ) W Channel width (m) Z Zero matrix (-) Greek symbols t Time interval (s) t exp Experimental time of sampling (s) y Height of a cell (m) z Length of a cell (m) ρ Average density of the material (kg/s) µ Viscosity (Pa s)
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