Axial dispersion model described

A dense-bed center-fed column (Fig. 22-li) having provision for internal crystal formation and variable reflux was tested by Moyers et al. (op. cit.). In the theoretical development (ibid.) a nonadiabatic, plug-flow axial-dispersion model was employed to describe the performance of the entire column. Terms describing interphase transport of impurity between adhering and free liquid are not considered. 

A well-defined bed of particles does not exist in the fast-fluidization regime. Instead, the particles are distributed more or less uniformly throughout the reactor. The two-phase model does not apply. Typically, the cracking reactor is described with a pseudohomogeneous, axial dispersion model. The maximum contact time in such a reactor is quite limited because of the low catalyst densities and high gas velocities that prevail in a fast-fluidized or transport-line reactor. Thus, the reaction must be fast, or low conversions must be acceptable. Also, the catalyst must be quite robust to minimize particle attrition. 

Liquid-liquid extraction is carried out either (1) in a series of well-mixed vessels or stages (well-mixed tanks or in plate column), or (2) in a continuous process, such as a spray column, packed column, or rotating disk column. If the process model is to be represented with integer variables, as in a staged process, MILNP (Glanz and Stichlmair, 1997) or one of the methods described in Chapters 9 and 10 can be employed. This example focuses on optimization in which the model is composed of two first-order, steady-state differential equations (a plug flow model). A similar treatment can be applied to an axial dispersion model. 

If the model accounts for the effects of axial dispersion, then the system is described by an axial dispersion model in terms of two-point boundary ODEs, i.e., by... 

The mill is described by a discrete mathematical model, developed on the basis of the axial dispersion model... 

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. 

A few reactor models have recently been proposed (30-31) for prediction of integral trickle-bed reactor performance when the gaseous reactant is limiting. Common features or assumptions include i) gas-to-liquid and liquid-to-solid external mass transfer resistances are present, ii) internal particle diffusion resistance is present, iii) catalyst particles are completely externally and internally wetted, iv) gas solubility can be described by Henry s law, v) isothermal operation, vi) the axial-dispersion model can be used to describe deviations from plug-flow, and vii) the intrinsic reaction kinetics exhibit first-order behavior. A few others have used similar assumptions except were developed for nonlinear kinetics (27—28). Only in a couple of instances (7,13, 29) was incomplete external catalyst wetting accounted for. 

Although the above method can give a simple evaluation of Peclet number for the system, the tailing in the RTD curve can cause significant inaccuracy in the evaluation of the Peclet number. Michell and Furzer67 suggested that a better estimation of the axial dispersion coefficient is obtained if the observed RTD is statistically fitted to the exact solution of the axial dispersion model equation with appropriate boundary conditions. For example, a time-domain solution to the partial differential equation describing the dispersion model, i.e.,... 

Flow of liquids or gases through fixed beds is very important in chemical reaction engineering, since many commercially important processes involve reactors that contain beds of catalyst used to promote a desired reaction. The axial dispersion model has been used extensively to model these flows, even though two phases, fluid and solid, are present. Such a pseudo-homogeneous model assumes the same form we have described in the preceding section if the Peclet number is based on particle dimension and the interstitial fluid velocity is used. In this event... 

By and large we can describe the results of the analysis of distributed parameter systems (i.e., flow reactors other than CSTRs) in terms of the gradients or profiles of concentration and temperature they generate. To a large extent, the analysis we shall pursue for the rest of this chapter is based on the one-dimensional axial dispersion model as used to describe both concentration and temperature fields within the nonideal reactor. The mass and energy conservation equations are coupled to each other through their mutual concern about the rate of reaction and, in fact, we can use this to simplify the mathematical formulation somewhat. Consider the adiabatic axial dispersion model in the steady state. 

However, if the impact of the oxygen concentration on the reactions is to be checked, the axial dispersion model is a good approach (Model II). For this, the mass balance in the liquid phase must be extended for the dissolved oxygen. In addition, the mass balance for oxygen in the rising gas bubbles must be established. Both mass balances are coupled by the mass transfer of oxygen from the gas bubbles into the liquid. Finally, the net formation rate of CHP must be described by Eq. (2.18) to include the impact ofthe oxygen concentration. 

In Eq. 12.5a-l, u is taken to be the mean (plug flow) velocity through the vessel, and is a mixing-dispersion coefficient to be found from experiments with the system of interest. One important application is to fixed beds, as discussed in detail in Chapter 11, and then it is usually termed an effective transport model, with = Z> . However, the axial dispersion model can also be used to approximately describe a variety of other reactors. 

The conversion for the tanks-in-series model was presented in Chapter 10 and is similar to the results from the axial dispersion model for n PeJ2 1. In addition, two-dimensional networks of such compartments can be used to describe dispersion—this was briefly discussed in Chapter 11. For a general mathematical treatment, see Wen and Fan [2],... 

Next, let us discuss the various techniques that have been used for the actual parameter estimation, describing their strong and weak points. For the axial dispersion model, this has been done by Boxkcs and Hofmann [96], and illustrates the typical problems involved. 

Roes and van Swaaij [29] used tracer technique to determine the residence time distribution and the extent of axial mixing in both flowing phases. The axially dispersed model was used to describe the degree of axial mixing. 

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