Model Kunii-Levenspiel

As in the fluidized beds analysis (Section 3.8.3), a similar simplification has been made in Kunii-Levenspiel model for the material balances in the emulsion phase, where again the corresponding derivatives have been omitted (eqs. (3.529) and (3.530)). As in the case of liquid flow in trickle beds, the flow of the gas in the emulsion phase is considered too small and so the superficial velocities can be neglected. Thus, in trickle beds, from eq. (3.367),... 

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)... 

Figure 3.64 Levenspiel model (simplified Kunii-Levenspiel model).

The Kunii-Levenspiel model can be simplified by assuming that the derivative terms of eqs. (3.529) and (3.530) are unimportant compared to the rest of the terms. Furthermore, plug flow can be assumed for gas (bubble) phase. Under these assumptions, the set of equations reduces as follows (Carberry, 1976 Fogler, 1999) ... 

Solutions for solids presence in bubble phase This model has been proposed by Chavarie and Grace as a two-phase simplification of the three-phase Kunii-Levenspiel model (Grace, 1984). Here, the solution is given under the following assumptions first-order reaction, gas flow only through the bubble phase (fh 1) and absence of solids in... 

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)... 

Mass transfer For the Kunii-Levenspiel model, the mass transfer coefficient of the gas between bubble and cloud is (eq. 3.544)... 

It is evident that the Kunii-Levenspiel model deviates a lot for low values of superficial velocity and thus low fbuh, which shows that it cannot be used in this region. However, for a high superficial gas velocity, its predictions are slightly better than the other three models. Thus, with the exception of the Kunii-Levenspiel model at low values of /bub, these models have a good behavior for the specified system. 

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. 

The Kunii-Levenspiel Model will be used in conjunction with the correlations of Broadhurst and Becker (1975) and Mori and Wen (1975) to analyze the ammonia oxidation of Massimilla and Johnstone (1961). The reaction... 

The 20% conversion calculated using the Kunii-Levenspiel model compares quite well with the experimental value of 22% measured by Massimilla and Johnstone. 

FIGURE CS5.1 Schematics of several fluidized-bed reactor models (a) Davidson model, (b) Kunii-Levenspiel model, (c) Miyauchi model, (d), (e) Fryer-Potter and Jayaraman-Kulkami-Doraiswamy models. 

The Kunii-Levenspiel model for fluidization is given on the CD-ROM along with a comprehensive example problem. The rale limiting transpon steps are also discussed. See Professional Reference Shelf R12.3. 

It is of interest to compare the efficiency of a fluidized bed in terms of the Kunii-Levenspiel model to that of a corresponding plug-flow reactor. We can do this by comparing the catalyst requirement in a PFR to that in the fluid-bed reactor (FBR) for the same conversion, which in general is the ratio of the effective... 

In this section, we will investigate three kinds of mass balances for a fluidized bed the plug flow, the CSTR, and the hydrodynamical Kunii-Levenspiel models [18]. The ideal models can be utilized only for rather crude, approximate calculations only the design of a fluidized bed should comprise a hydrodynamical model coupled with experiments on a pilot scale. 

The most advanced and realistic description of fluidized beds is the Kunii-Levenspiel model [18]. According to this model, the bubble phase is assumed to move in the reactor following the characteristics of a plug flow, while the gas flow in the emulsion phase is assumed to be negligible. The cloud and wake phases are presumed to possess similar chemical contents. The transport of the reacting gas from the bubble phase to the cloud and wake phases and vice versa prevails. The volume element, AV, therefore consists of three parts, as in Figure 5.34 ... 

FIGURE 5.34 Schematic structure of a fluidized bed according to the Kunii-Levenspiel model. (Data from Levenspiel, O., Chemical Reaction Engineering, 3rd Edition, Wiley, New York, 1999.)... 

As a summary, we can conclude that the Kunii-Levenspiel model for a fluidized bed consists of 3 AT N = number of components) molar balances (Equations 5.251 through 5.253) if all the components are utilized, or, 3 S (S = number of reactions) balances, if the key components are used as in Equations 5.251 through 5.253. The 3 - S balances comprise the model in case the extents of reactions are used in Equations 5.254 through 5.256. In the latter two cases, the concentrations of the components are related through Equations 5.248 through 5.250. 

---