Processes involving particles, are known to exhibit extremely unpredictable behaviour, mainly due to the mesoscopic nature of granular media. Understanding particulate processes, not only for intellectual satisfaction, but also for process design and operation, basically requires a systems approach in modelling. Because they combine simplicity and flexibility, the stochastic models based on the Markov chain theory are very valuable mathematical tools to this respect. However, they are still largely ignored by the whole core of chemical engineering researchers. This motivates the existence of this review paper, in which we examine the three traditional issues: mixing and transport, separation and transformation.

Applications of Markov Chains in Particulate Process Engineering: A Review

Application of the theory of Markov chains to model different processes in particle technology

This paper is concerned with the investigation of the mixing process in a pilot scale continuous powder mixer, from both experimental and modelling points of view. A Markov chain model is proposed to simulate the residence time distribution (RTD) obtained from tracer experiments and detection by image analysis. The general model consists of 10 continuous stirred tank reactors with identical exchange coefficients and may serve as a basis for other applications. A 2D chain model is also developed and applied to take into account more features of real flow, such as crosswise mixing. Agreement between the model results and experimental data is more than reasonable, and gives a clear physical meaning to the parameters introduced. The characteristics of the process are controlled by the rotational speed of the mixer, which appears to be the key scale-up variable along with stirrer design.

Flow analysis and markov chain modelling to quantify the agitation effect in a continuous powder mixer

A continuous-time Markov chain has been used to establish the residence time distribution (RTD) model in a new type entrained-flow gasifier, which is called entrained-flow gasifier with opposed multi-burner. According to the measurement results of the flow fields in the gasifier, the state transfer diagram of Markov chain formed in the case of the flow fields are simplified. The results show that this method is feasible in modeling the flow system which consists of ideal mixing cells and plug flow regions. The flow pattern of the gasifier is closed to continuous stirred tank reactor (CSTR). The simulation results are in good agreement with the experimental data. The established model has been applied to forecast the RTD in the industrial gasifier.

Markov chain model of residence time distribution in a new type entrained-flow gasifier

The movement of particles in a laboratory fluidized bed has been studied using Positron Emission Tomography (PET). With this non-invasive technique both pulses of various shapes and single tracer particles were followed in 3-D. The equipment and materials used made it possible to label actual bed particles as tracer. This paper briefly describes the data analysis, some of the results and compares them with a stochastic model. The results confirm the basic assumptions of the model qualitatively but not quantitatively. Further analysis of the results indicates that the system exhibits gulf-streaming, a feature which is not yet accounted for in the model, but is common in fluidized beds in practice.

Positron emission tomography applied to fluidization engineering

In this paper we study stochastic models for the transport of particles in a fluidized bed reactor and compute the associated residence time distribution (RTD). Our main model is basically a diffusion process in [0,A] with reflecting/absorbing boundary conditions, modified by allowing jumps to the origin as a result of transport of particles in the wake of rising fluidization bubbles. We study discrete time birth-death Markov chains as approximations to our diffusion model. For these we can compute the particle distribution inside the reactor as well as the RTD by simple and fast matrix calculations. It turns out that discretization of the reactor into a moderate number of segments already gives excellent numerical approximations to the continuous model. From the forward equation for the particle distribution in the discrete model we obtain in the diffusion limit a partial differential equation for the particle density p(t,x) \[ \frac{\partial}{\partial t} p(t,x) =\frac{1}{2} \frac{\partial^2}{\partial x^...

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H. W. Stuut

Papers

Stochastic models for transport in a fluidized bed