Solid-liquid fluidized beds equations

Using these assumptions, the fundamental transport equations for solid-liquid fluidized beds are given next. 

Equation (24) gives the stability criterion for the solid-liquid fluidized beds. Equation (24) can be rewritten as... 

Equation (61) is the transition criterion provided the conditions given by equation (52) are satisfied. From Table I it can be seen that these conditions are satisfied only in the case of gas-solid fluidized beds and in some cases of solid-liquid fluidized beds where ps Pl- Therefore, for other multiphase dispersions [such as gas-liquid (bubble columns) and solid-liquid fluidized beds (where pl is not negligible)] the comparison of dynamic wave velocity with continuity wave velocity is not valid for deciding the bed stability. Further, the above analysis holds for transition from region I to II (point P in Fig. 1) and not for III to II (point Q). Therefore, the criterion does not hold for bubble columns and dilute dispersions. 

Equation (36) gives the voidage propagation velocity (sharp front or continuity wave velocity) for gas-solid and solid-liquid fluidized beds. However, for the other multiphase dispersions, the procedure given by Eqs. (32) to (37) should be used. Thus, for gas-liquid dispersions, the sharp front velocity is given by Eq. (37). 

This criterion involves the assumption that the gas phase stress terms are negligible. This assumption may not be valid in case of solid-liquid fluidized beds or liquid-liquid dispersions. In this case, the criterion is of the same form as Eq. (172), with different definitions of the parameters Ml, M2, and M3, which are given in Table VII. Table VII also gives the parameters of the criterion when the dispersion terms are not included in the continuity equations of both the phases. 

In the case of solid-liquid fluidized beds, the setthng velocity of the particle is less than the terminal settling velocity. The hindered settling velocity is given by the well-known Richardson-Zaki equation,... 

In a solid-liquid fluidized bed, the bed expansion is determined by the Richardson-Zaki (1954) equation ... 

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. 

This is vahd for beds operated with a gas or a hquid. The solid curves (= const.) for expanding homogeneous (liquid) fluidized beds are based on equations presented by Anderson (Anderson 1961) compared witli Richardson and Zaki (Richardson and Zaki 1954). In the case of heterogeneous (low pressure gas) fluidized beds only the boundaries between the minimum fluidizing velocity ( 0.4) and the pneumatic transport (e 1) are well known. The curve for... 

This equation has been experimentally verified in liquids, and Figure 2 shows that it applies equally well for fluidized solids, provided that G is taken as the flow rate in excess of minimum fluidization requirements. In most practical fluidized beds, bubbles coalesce or break up after formation, but this equation nevertheless gives a useful starting point estimate of bubble size. 

Particulate fluidization, where the fluidizing medium is usually a liquid, is characterised by a smooth expansion of fhe bed. Liquid-solid fluidized beds are used in continuous crystallisers, as bioreactors in which immobilised enzyme beads are fluidized by the reactant solution and in physical operations such as the washing and preparation of vegetables. The empirical Richardson-Zaki equation (Richardson... 

Transfer of mass from liquid into solid particles of an expanded fountain-fluidized bed is developed in the annulus and fountain according to the following criterion equation... 

The rise velocity of a single spherical cap bubble in an infinite liquid medium can be described by the Davies and Taylor equation [Davies and Taylor, 1950] (Problem 9.6). Experimental results indicate that the Davies and Taylor equation is valid for large bubbles (4oo > 0.02 m, in general) with bubble Reynolds numbers greater than 40, while for bubbles in fluidized beds, the bubble Reynolds numbers are typically on the order of 10 or less [Clift, 1986]. By analogy, the rise velocity of an isolated single spherical cap bubble in an infinite gas-solid medium can be expressed in terms of the volume bubble diameter by [Davidson and Harrison, 1963]... 

Narayanan et al.88 correlated gas holdup to the superficial gas velocity empirically and found hG oc U0G for U0G < 6.7 cm s 1 and hG oc UoG 8 for 6.7 cm s-1 < Uoc < 21.34 cm s"1. Hovmand and Davidson43 proposed a slug-flow model to correlate the gas holdup in gas-liquid-solid fluidized beds at superficial gas velocities in excess of that required for the incipient fluidization of solids. They correlated the gas holdup and superficial gas velocity by the equation... 

Recommendations At present, the best available correlation for the liquid-solid mass-transfer coefficient in a three-phase fluidized-bed column is given by Eq. (9-61), and its use is recommended. The equation, however, needs to be checked against the experimental data with hydrocarbon systems. 

In the previous section, stability criteria were obtained for gas-hquid bubble columns, gas-solid fluidized beds, liquid-sohd fluidized beds, and three-phase fluidized beds. Before we begin the review of previous work, let us summarize the parameters that are important for the fluid mechanical description of multiphase systems. The first and foremost is the dispersion coefficient. During the derivation of equations of continuity and motion for multiphase turbulent dispersions, correlation terms such as esv appeared [Eqs. (3) and (10)]. These terms were modeled according to the Boussinesq hypothesis [Eq. (4)], and thus the dispersion coefficients for the sohd phase and hquid phase appear in the final forms of equation of continuity and motion [Eqs. (5), (6), (14), and (15)]. However, for the creeping flow regime, the dispersion term is obviously not important. 

It has been mentioned earlier that the voidage propagation velocity is the same as the sharp front velocity. Thus, the equation for is the same in gas-solid fluidized beds and liquid-solid fluidized beds. However, for bubble columns, the sharp front velocity is given by the following equation ... 

Begovich and Watson (63) found experimentally that the mixture conductivity in a liquid-solid fluidized bed is proportional to the liquid holdup. Their equation can be written as... 

Particulate and Aggregative Fluidization. When the fluid phase is liquid, the difference in the densities of fluid and solid is not very large, and the particle size is small, the bed is fluidized homogeneously with an apparent uniform bed structure as the fluid velocity exceeds the minimum fluidization velocity. The fluid passes through the interstitial spaces between the fluidizing particles without forming solids-free bubbles or voids. This state of fluidization characterizes particulate fluidization. In particulate fluidization, the bed voidage can be related to the superficial fluid velocity by the Richardson-Zaki equation. The particulate fluidization occurs when the Froude number at the minimum fluidization is less than 0.13 [9],... 

The simulation also provides some information about the bubble particle interaction as shown in Fig. 25. As the bubble rises in the liquid-solid fluidized bed, an interaction between the bubble and particles takes place. In the simulation model, the bubble-particle interaction is accounted for by adding a surface-tension-induced force to the particle motion equation. This force is also added to the source term of the liquid momentum equation for the liquid elements in the interfacial area to account for the particle effect on the interface. The particle movement is determined from the resulting total force acting on the particle. From the simulation results, it is seen that most particles contacting the bubble do not penetrate the bubble only one or two particles penetrate. Instead, they pass around the bubble surface. When the particles penetrate the bubble, they fall through quickly to the bubble base because of the low viscosity and density of the gas phase. 

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