K-L model

The K-L model, because of its greater simplicity, thus seems to be the model of choice for systems with smaller bubbles. In this paper we shall show how the K-L model can be used to predict the experimental results obtained by Massimilla and Johnstone (1961) on the catalytic oxidation of ammonia. It will be seen that the performance of their system was largely controlled by reaction limitations within the bed s phases. The effects of various parameters on bed performance are examined for such a reaction-limited system, and then the effects of these parameters for a transport-limited system are also discussed. Finally, we consider the effect of using average values of the bubble diameter and transport coefficients on model predictions. 

In order to calculate the expanded bed height, h, for the given catalyst weight of 4 kg, one needs to calculate the fraction of bed occupied by bubbles, 6. From the K-L model... 

In the K-L model, reaction occurs within the bed s phases, and material is continuously transferred between the phases. Two limiting situations thus arise. In one, the interphase transport is relatively fast and transport equilibrium is maintained, causing the system performance to be controlled by the rate of reaction. In the other, the reaction rate is relatively fast and the performance is controlled by interphase transport. It will be shown that the ammonia oxidation example used above is essentially a reaction-limited system. 

The slow-reaction situation has been treated before (Grace, 1974), using a model of bed performance developed well before the K-L model (Orcutt et al., 1962). This earlier work concluded that when the reaction was very slow, the hydrodynamics and the way the hydrodynamics were modeled were unimportant. The analysis given above, using the more sophisticated K-L model, shows that the hydrodynamics can be very important indeed, even when the reaction is slow. In the situation cited, a reduction of 75% in catalyst requirement can be attained by expoitation of the bed hydrodynamics. 

Fryer and Potter (1972), using the model of Davidson and Harrison, reported that a bubble size found at about 0.4h could be used as the single bubble size in that model. Earlier in this paper, the bubble size found at 0.5h was used arbitrarily in calculating the conversion in an ammonia oxida tion system using the K-L model. 

In this paper we have shown that the Kunii-Levenspiel model can be used to accurately predict the results of Massimilla and Johnstone. In addition, we have used the K-L model to predict the changes in conversion with parameter variation under the limiting conditions of reaction control and transport control. Finally, we have shown that significant differences in the averaging techniques occur for height to diameter ratios in the range of 2 to 20. 

Sheiner, L.B. Rosenberg, B. Melmon, K.L. Modeling of individual pharmacokinetics for computer-aided drug dosage. Comp. Biomed. Res. 1972, 5, 441-459. 

The celebrated K-L model did gain its popularity because it is very simple, yet developed for large scale industrially relevant flows and considers most of the pertinent bubbling bed phenomena. 

Over the years the range of uses of the K-L model has been extended to chemical processes that can not be described by first order kinetics. For these problems no anal3dical solution can be obtained so the resulting set of DAE equations are solved numerically. Gascon et al [48], for example, investigated the behavior of a two zone fluidized bed reactor for the propane dehydrogenation and n-butane partial oxidation processes emplo3ung the K-L model framework. 

The main postulates of the K-L model (Kunii and Levenspiel, 1968a, b 1991) are sketched in Figure CSS. lb. The bubble develops a cloud of particles around it as it moves upward at a velocity... 

The assnmptions discnssed above are obvious. Many others are less so and are inherent in almost all flnid-bed models. The most important is with respect to particle size range. Most models do not acconnt for this explicitly, except the K-L model, which gives different equations for fine, intermediate, and coarse particles (Knnii and Levenspiel, 1991). Onr calcnlations for all the five models are for mainly Geldart B class particles and should generally be valid for the so-called fine and intermediate size particles. With the inherent uncertainties of prediction in all the models now available, a finer distinction is not warranted between these two classes of particles. 

Also, the picture is more complicated than depicted in Figure 12.10a. As the bubble rises, it carries with it a small amount of the solids as wake. Thus a rigorous model should really recognize four regions emulsion, bubble, cloud, and wake. In the K-L model, it is assumed that the wake solids are evenly distributed in the cloud phase. This simplifies the computations without seriously affecting the accuracy. 

K-L model is based on two successive mass transfer steps, leading to the coefficients febc for bubble-cloud exchange and for cloud-emulsion exchange. The equations for the K-L model are given in Table 12.5. 

Kunni-Levenspiel (K-L model) developed a matheiiiatical model for fluidised bed catalytic reactors in which aggregate fluidisation occurs, resulting in the formation of gas-solid bubbles (Figure 4.28). 

Mass transfer between bubble and emulsion An important feature of fluidized-bed reactors is mass transfer between bubble and emulsion. Several models have been proposed for this exchange. The Davidson model assumes no cloud, so that only one mass transfer coefficient k, (for direct bubble-emulsion exchange) is involved. On the other hand, the K-L model is based on two successive mass transfer steps, leading to the coefficients k,. (for bubble-cloud exchange) and (for cloud-emulsion exchange). The equations for the K-L model are given in Table 9.2. 

---