Kunii-Levenspiel Parameters

For the mass transfer coefficients, K, i and Kcej, as well as for the volume fractions, Vc/Vh and Ve/Vb, empirical correlations exist. The average residence time for the bubbles is defined as 

For the estimation of the average bubble velocity, Wb, the minimum fluidization velocity Wfn( is required. If the bed resides in a condition of minimum fluidization, the particles float freely. The gravitation force is thus compensated for by the pressure drop in the bed. Let us consider a bed cross-section (A), where there are n pieces of particles. The gravitation force, AF, is then given by 

This pressure drop can be regarded as equal to the pressure drop in Equation 5.216. Let us consider the following expression  

Moreover, a = 150 and b = 1.75. Inserting Equations 5.262 and 5.263 into Equation 5.261, followed by a setting of Equation 5.260 equal to Equation 5.261, gives the following expression for the calculation of the minimum fluidization velocity (Wmf)  

For spherical particles (cp = 1), Equation 5.265 yields emf = 0.383. The average velocity of the bubble velocity, wt, can now be obtained from the equation [9] 

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A one-parameter model, termed the bubbling-bed model, is described by Kunii and Levenspiel (1991, pp. 144-149,156-159). The one parameter is the size of bubbles. This model endeavors to account for different bubble velocities and the different flow patterns of fluid and solid that result. Compared with the two-region model, the Kunii-Levenspiel (KL) model introduces two additional regions. The model establishes expressions for the distribution of the fluidized bed and of the solid particles in the various regions. These, together with expressions for coefficients for the exchange of gas between pairs of regions, form the hydrodynamic + mass transfer basis for a reactor model. 

Applying the appropriate material balances for the solids and the gas, the fraction of the bed occupied by the bubbles and wakes can be estimated using the Kunii-Levenspiel model. The fraction of the bed occupied by that part of the bubbles which does not include the wake, is represented by the parameter d, whereas the volume of the wake per volume of the bubble is represented by a. Consequently, the bed fraction in the wakes is a and the bed fraction in the emulsion phase (which includes the clouds) is 1 — <5 — ot<5. Then (Fogler, 1999)... 

Tliis model is simpler that the Kunii-Levenspiel model and eliminates the unsubstantiated expression for cloud-to-emulsion transfer employed by Kunii and Levenspiel (Grace, 1984). Furthermore, compared to the previous models, the introduction of the parameter yb in the model leads to better results as the assumption that there is no solids in the bubble phase may lead to the underestimation of conversion in fast reactions. For slow reactions, the value of yb is of minor importance. However, for fast reactions the model may become sensitive to this parameter and the actual conversion should be bounded between the predicted ones using the upper and lower limits of yh, i.e. 0.01 and 0.001, respectively (Grace, 1984). 

For the Kunii-Levenspiel model, we need some additional hydraulic parameters. The fraction of the bed occupied by bubbles is given by (eq. 3.494)... 

Some aspects of fluidized-bed reactor performance are examined using the Kunii-Levenspiel model of fluidized-bed reactor behavior. An ammonia-oxidation system is modeled, and the conversion predicted is shown to approximate that observed experimentally. The model is used to predict the changes in conversion with parameter variation under the limiting conditions of reaction control and transport control, and the ammonia-oxidation system is seen to be an example of reaction control. Finally, it is shown that significant differences in the averaging techniques occur for height to diameter ratios in the range of 2 to 20. 

In a general case of nonlinear kinetics, the fluidized bed model is solved numerically with an algorithm suitable for differential algebraic systems. The calculation procedure for fluidized beds with the Kunii-Levenspiel model involves numerous steps, as evidenced by the treatment. As a summary, the path from the minimum fluidization quantities (emf, Wfn() to the reactor model is presented in Table 5.6. The superficial velocity (wo) and the physical parameters are assumed to be constant. 

A catalytic oxidation process is going to be carried out in a fluidized bed with spherical catalyst particles. Calculate all the parameters of oxygen needed for the Kunii-Levenspiel model, starting from the physical data given below ... 

The main model parameter, the mean bubble diameter, db, can be estimated using various correlations. It depends on the type of particle and the nature of the inlet distributor. For small, sand-like particles that are easily fluidized, an expression is given for db as a function of bed height x by Werther (Kunii and Levenspiel, 1991, p. 146) ... 

Several models have been proposed to account for reaction in the freeboard. Yates and Rowe (1977) developed a simple model based upon complete mixing of particles in the freeboard, coupled with either BMF or PF of the freeboard gas. Two model parameters are the rate of particle ejection from the bed, and the fraction of wake particles ejected. Kunii and Levenspiel (1990 1991, pp. 305-307) proposed a model of freeboard reaction which accounts for the contact efficiency of the gas with the solid, and the fraction of solid in the freeboard. A comprehensive freeboard entrainment model is... 

Here we have sketched the three regimes of CFB and their general behavior however, we have not presented their performance equations. The reason is that the parameters for their reasonable models are uncertain today hence, the predictions based on these models will likewise be uncertain. Still the general material balances and the form of the performance equations are available, see Kunii and Levenspiel (1991, 1997). 

The complete mixing of solids in the emulsion phase is necessary for considering the various parameters, involving the mass or volume of solids constant, throughout the reactor. That is exactly the case in the two-phase model and the Levenspiel-Kunii three-phase model. This is achieved by circulation of the solids through their entrainment by bubbles, as shown in Figure 3.61. As solids fall from the upper portions of the bed, they follow... 

The set of Equations (CS 12.18) through (CS 12.20) generally cannot be solved analytically. However, they represent a total generalization with respect to feed size distribution, reaction kinetics, and presence of an elutriation stream. The use of these equations for calculating the various reactor and operating parameters such as solids feed rate, exit bed size rate, product size distribution, and so on is outlined below (see Kunii and Levenspiel, 1969, for details) for the simpler case of singlesize feed. 

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