Solid Motion in a Liquid Fluidized Bed

From the analysis of the decomposition of the images, we can observe that the movement ahead is dominant and controls the whole displacement. In the example given in Fig. 4.8, we can observe that the movement ahead (elementary state or process component) presents approximately the same frequency as the backward displacement. A radial movement is also possible but can be neglected if we consider only a very thin band in the centre of the bed. The same has been consid- 

In this case, the elements of the passage matrix of the particularized problem present very clear physical meanings  

If Pj represents the probability for the process that evolved in state I to remains in this state after the interval of time At, then, because all the states are characterized by independent probabilities, we can write  

Here N is the number of the independent states of the process (N = 2 in the analyzed case) and a is the frequency of exchange of an individual state. We can notice that the dimension for a is time 1 (T ). The same consideration as above can be used for the probability of the process evolving in state II. Then P2 is written  

If we consider that the element is the coloured ball, the following relation writes the equations of the probability balance according to the model  

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Chapter 4 is devoted to the description of stochastic mathematical modelling and the methods used to solve these models such as analytical, asymptotic or numerical methods. The evolution of processes is then analyzed by using different concepts, theories and methods. The concept of Markov chains or of complete connected chains, probability balance, the similarity between the Fokker-Plank-Kolmogorov equation and the property transport equation, and the stochastic differential equation systems are presented as the basic elements of stochastic process modelling. Mathematical models of the application of continuous and discrete polystochastic processes to chemical engineering processes are discussed. They include liquid and gas flow in a column with a mobile packed bed, mechanical stirring of a liquid in a tank, solid motion in a liquid fluidized bed, species movement and transfer in a porous media. Deep bed filtration and heat exchanger dynamics are also analyzed. 

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