Levenspiel conditions

Danckwerts [2] also obtained steady state solutions based on the same boundary conditions and various studies have since been performed by Taylor [19], Arts [20], and Levenspiel and Smith [21],... 

Alternative methods of estimating Q)L are based on the response of the reactor to an ideal pulse input. For example, equation 11.1.39 may be used to calculate the mean residence time and its variance. Levenspiel and Bischoff (9) indicate that for the boundary conditions cited,... 

Figure 23.9 illustrates the model and kinetics scheme for these conditions. We confine our analysis to a single first-order reaction, based on the development of Kunii and Levenspiel (1990 1991, pp. 300-302). However, extension to other reaction orders is straightforward. 

The condition for which the bed is likely to operate at near equilibrium is when the feed rate is low. This is also the condition when longitudinal dispersion may be significant. Equilibrium solutions have been found by Lapidus and Amundson(39) and by Levenspiel and liisnioi i 40 for this case. These take the form ... 

Under some laminar flow circumstances, the dispersion model (see Sect. 5.4) may be a better description of reality than the laminar-flow model discussed above. Ananthakrishnan et al. [48] discuss conditions under which one should opt for a particular model and their data are presented in modified form by Levenspiel [26]. If the tubular reactor aspect ratio, i.e. the length-to-diameter ratio, is less than 100 and the product of the reactor Reynolds, dup/p, and Schmidt, p/p S numbers is greater than approximately 10", then the laminar-flow model should give good predictions of reactor performance. 

If Fig. 12 and Fig. 14 were laid on top of each other, then conditions of equivalence could be determined under which the performance of the tanks-in-series model with specified N would be the same as that of the recycle model, that is the value of R could be found which would result in the same conversion and V/Vpp ratio. Levenspiel [17] gives these values for a variety of conditions for both first- and second-order reactions. His data are reproduced in Table 8. 

Much of Fig. 18 refers to conditions under which the dispersion model should only be used with caution. Levenspiel [26] suggests that, if DluL is greater than unity, then other models may be more appropriate and Dudukovic and Felder [59] comment that the dispersion model should only be used with confidence when the dispersion number is less than 0.05. As is clear from both Figs. 16 and 18, these conditions represent relatively minor deviations from the plug flow ideal. 

We will not discuss the equations and curves for the open-closed or closed-open boundary conditions. These can be found in Levenspiel (1996). 

Bischoff and Levenspiel (1962) have shown that as long as the measurements are taken at least two or three particle diameters into the bed, then the open vessel boundary conditions hold closely. This is the case here because the measurements are made 15 cm into the bed. As a result this experiment corresponds to a one-shot input to an open vessel for which Eq. 12 holds. Thus... 

Inspection of Fig. 8 shows that there is considerable scatter in the data. Part of this may be due to the fact that we are attempting to represent a complex phenomenon with a single parameter, the dispersion coefficient. Errors would also be caused by the common practice of taking measurements at or beyond the exit of the packed section. This neglect of end conditions could lead to large errors in the calculated dispersion coefficients, as pointed out by Bischoff and Levenspiel (B14). Also, all the analyses were based on the assumption of having a perfect pulse, step. 

Bischoff and Levenspiel (B14) considered this problem, and have presented design charts which allow estimation of errors in the calculated dispersion coefficients for various conditions. It was found that when the ratio of injection to tube diameter is less than 20%, or e < 0.2, then the assumption of a point source, or e-> 0, was good to within 5%. Thus for many cases, the neglect of the finite size of injection tube is justified. 

From Levenspiel, O., Chemical Reaction Engineering/ John Wiley, New York, 1962. As given by conditions of incipient fluidization. 

To ensure complete and uniform mixing conditions of the fluid throughout the reactor, in such systems, a special design is required. Such a design has been presented by Levenspiel (1972) and it is shown in Figure 3.22. 

A mixed-flow reactor requires uniform composition of the fluid phase throughout the volume while the fluid is constantly flowing through it. This requires a special design in order to be achieved in the case of gas-solid systems. These reactors are basically experimental devices, which closely approach the ideal flow conditions and have been devised by Carbeny (Levenspiel, 1972). This device is called a basket-type mixed reactor (Figure 3.6). The catalyst is contained in four rapidly spinning wire baskets. 

In the ideal CSTR, the fluid concentration is uniform and the fluid flows in and out of the reactor. Under the steady state condition, the accumulation term in the general material balance, eq. (3.70), is zero. Furthermore, the exit concentration is equal to the concentration in the reactor. For a volume element of fluid (F,), the mass balance for the limiting reactant becomes (Levenspiel, 1972)... 

The time requited to process one reactor volume of feed at specified conditions is called space-time and is defined normally at actual entering conditions (Levenspiel, 1972). 

The number of reactor volumes of feed at specified conditions, which can be treated in a unit time is called space velocity and is (Levenspiel, 1972)... 

Figure 12.9. Particle-to-gas and bed-to-gas heat transfer coefficients under various flow conditions (from Kunii and Levenspiel, 1991).

The fluidised bed is only one of the many reactors employed in industry for gas-solid reactions, as reported by Kunii and Levenspiel.25 Whenever a chemical reaction employing a particulate solid as a reactant or as a catalyst requires reliable temperature control, a fluidised bed reactor is often the choice for ensuring nearly isothermal conditions by a suitable selection of the operating conditions. The use of gas-solid fluidised beds... 

As our first application, we consider the classical Taylor-Aris problem (Aris, 1956 Taylor, 1953) that illustrates dispersion due to transverse velocity gradients and molecular diffusion in laminar flow tubular reactors. In the traditional reaction engineering literature, dispersion effects are described by the axial dispersion model with Danckwerts boundary conditions (Froment and Bischoff, 1990 Levenspiel, 1999 Wen and Fan, 1975). Here, we show that the inconsistencies associated with the traditional parabolic form of the dispersion model can be removed by expressing the averaged model in a hyperbolic form. We also analyze the hyperbolic model and show that it has a much larger range of validity than the standard parabolic model. 

Here x is the conversion of SiH4. combines the effect of the molar expansion in the deposition process as well as the change in the volumetric flow and the dispersion coefficient, D, with temperature. At low pressures and small Re in LPCVD reactors the dispersion occurs mainly by molecular diffusion, therefore, we have used (D/D0) = (T/T0)l 65. e is the expansion coefficient and the stoichiometry implies that e = (xi)q, the entrance mole fraction of SiH4. The expansion coefficient, e is introduced as originally described by Levenspiel (33) The two reaction terms refer to the deposition on the reactor wall and wafer carrier and that on the wafers, respectively. The remaining quantities in these equations and the following ones are defined at the end of the paper. The boundary conditions are equivalent to the well known Danckwerts1 boundary conditions for fixed bed reactor models. 

Woodburn55 showed thai, for Re], 650, the correlations proposed by DeMaria and White, J Sater and Levenspiel,43 and Dunn et al.16 could correlate his data. However, for 650 < ReL < 1,500, the axial dispersion in the gas phase was independent of the liquid rate. Under these liquid flow conditions, the reverse gas flow induced by the counterflowing liquid was measured. Thus, he concluded that an additional dispersive mechanism associated with reverse gas flow becomes operative at ReL 650. 

For most practical conditions, a comparison of k and k from Equations (4) and (5) would suggest that the principal resistance to transfer resides at the outer cloud boundary. However, when (a), (b) and (c) are taken into account, this is no longer the case. In fact, experimental evidence (e.g. 30,31,32) indicates strongly that the principal resistance is at the bubble/ cloud interface. With this in mind, it is probably more sensible to include the cloud with the dense phase (as in the Orcutt (23, 27) models) rather than with the bubbles (as in the Partridge and Rowe (37) model) if a two-phase representation is to be adopted (see Figure 1). If three-phase models are used, then Equations (2) and (5) appear to be a poor basis for prediction. Fortunately the errors go in opposite directions. Equation (2) overpredicting the bubble/cloud transfer coefficient, while Equation (5) underestimates the cloud/emulsion transfer coefficient. This probably accounts for the fact that the Kunii and Levenspiel model (19) can give reasonable predictions in specific instances (e.g.20),... 

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. 

This expression derived by Lin (1979) appears to be correct and needs to be tested with experimental data. There are a number of models for non-ideal flows, that is, flows that fall between the ideal conditions of a perfect mixer and plug flow. Some of the models for non-ideal flow were discussed in Levenspiel (1972) and Rao and Loncin (1974a). 

In simple cases, when the number of byproducts is small, it may be possible to develop a mechanistic model of the reaction kinetics that predicts the rate of formation of the main product and byproducts. If such a model is fitted to experimental data over a suitably wide range of process conditions, then it can be used for process optimization. The development of reaction kinetics models is described in most reaction engineering textbooks. See, for example, Levenspiel (1998), Froment and Bischoff (1990), and Fogler (2005). 

For any industrial reacting system, the relevant parameters appearing in the rate expression (Eq. (5.14)) need to be obtained by carrying out experiments under controlled conditions. It is necessary to ensure that physical processes do not influence the observed rates of chemical reactions. This is especially difficult when chemical reactions are fast. It may sometimes be necessary to employ sophisticated mathematical models to extract the relevant kinetic information from the experimental data. Some references covering the aspects of experimental determination of chemical kinetics are cited in Chapter 1. It must be noted here that in the above development, the intrinsic rate of all chemical reactions is assumed to follow a power law type model. However, in many cases, different types of kinetic model need to be used (for examples of different types of kinetic model, see Levenspiel, 1972 Froment and Bischoff, 1984). It is not possible to represent all the known kinetic forms in a single format. The methods discussed here can be extended to any type of kinetie model. 

---