Utility of the Axial Dispersion Model

Chapters 8 and Section 9.1 gave preferred models for laminar flow and packed-bed reactors. The axial dispersion model can also be used for these reactors but is generally less accurate. Proper roles for the axial dispersion model are the following. 

FIGURE 9.10 Relative error in the predicted conversion of a first-order reaction due to assuming piston flow rather than axial dispersion, kt versus Pe. 

Adiabatic Reactors. Like isothermal reactors, adiabatic reactors with a flat velocity profile will have no radial gradients in temperature or composition. There are axial gradients, and the axial dispersion model, including its extension to temperature in Section 9.4, can account for axial mixing. As a practical matter, it is difficult to build a small adiabatic reactor. Wall temperatures must be controlled to simulate the adiabatic temperature profile in the reactor, and guard heaters may be needed at the inlet and outlet to avoid losses by radiation. Even so, it is hkely that uncertainties in the temperature profile will mask the relatively small effects of axial dispersion. 

Laminar Pipeline Flows. The axial dispersion model can be used for laminar flow reactors if the reactor is so long that At/R 0.125. With this high value for the initial radial position of a molecule becomes unimportant. 

The molecule diffuses across the tube and samples many streamlines, some with high velocity and some with low velocity, during its stay in the reactor. It will travel with an average velocity near u and will emerge from the long reactor with a residence time close to F. The axial dispersion model is a reasonable approximation for overall dispersion in a long, laminar flow reactor. The appropriate value for D is known from theory  

As seen in Chapter 8, the stability criterion becomes quite demanding when At/R2 is large. The axial dispersion model may then be a useful alternative to solving Equation (8.12). 

Turbulent pipeline flow is the original application of the axial dispersion model. For most kinetic schemes, piston flow predicts the highest possible conversion and selectivity. The axial dispersion model provides a less optimistic estimate, but the difference between the piston flow and axial dispersion models is usually small. For an open tube in well-developed turbulent flow, the assumption of piston flow is normally quite accurate unless the reaction is driven to near completion. 

A packed reactor has a velocity profile that is nearly flat, and for the usual case of uniform Oia, no concentration gradients will arise unless there is a radial temperature gradient. If there is no reaction exotherm (and if Tin = Twaii), the model of Section 

Pe = sL/(2dp)or Pe 300 for the packed tube of Example 9.1. Again, the assumption of piston flow is quite reasonable unless the reaction goes to near completion. It should be emphasized that the assumption of an isothermal reaction should be based on a small heat of reaction, for example, as in transesterifcation, where the energy of a 

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Some of the numerical problems in nonlinear regression seem to be able to be partially avoidable by utilizing special features of a given transfer function, as proposed by Mixon, Whitaker, and Orcutt [ 104] and extended by 0ster-gaard and Michelsen [ 105]. This is best illustrated by considering the transfer function of the axial dispersion model with semiinfinite boundary conditions ... 

The final goal of the axial dispersion model is its utilization in the modeling of chemical reactors. Below we will consider steady-state models only, which imply that the mass balance of a reacting component i can be written as... 

The dimensionless term (9/u0 L, where 9 is the axial dispersion coefficient, u0 is the superficial fluid velocity, and L is the expanded-bed height) is the column-vessel dispersion number, Tc, and is the inverse of the Peclet number of the system. Two limiting cases can be identified from the axial dispersion model. First, when 9/u0L - 0, no axial dispersion occurs, while when 9/u0 L - 00 an infinite diffusivity is obtained and a stirred tank performance is achieved. The dimensionless term Fc, can thus be utilized as an important indicator of the flow characteristics within a fluidized-bed system.446... 

The macroscopic problem is more intricate. The type of model utilized depends upon the ratio of the diffusion and reaction rates and thus upon the importance of micro- and macro-mixing. In a pipe reactor the values of the axial dispersion coefficients for both phases are required. For modeling, micro-mixing models are used, which describe the mutual interlinking of coalescence and redispersion processes. 

These interesting devices consist of a tube or duct within which static elements are installed to promote cross-channel flow. See Figure 8.5 and Section 8.7.2. Static mixers are quite effective in promoting radial mixing in laminar flow, but their geometry is too complex to allow solution of the convective diffusion equation on a routine basis. A review article by Thakur et al. (2003) provides some empirical correlations. The lack of published data prevents a priori designs that utilize static mixers, but the axial dispersion model is a reasonable way to correlate pilot plant data. Chapter 15 shows how Pe can be measured using inert tracers. 

The "axial dispersion" or "axial dispersed plug flow" model [Levenspiel and Bischoff, 1963] takes the form of a one-dimensional convection-diffusion equation, allowing to utilize all of the clasical mathematical solutions that are available [e.g. Carslaw and Jaeger, 1959, 1986 Crank, 1956]. 

Even though the axial dispersion and tanks-in-series models discussed in this section can be used to handle a wide variety of nonideal flow situations, many reactors contain elements that do not satisfy the fundamental basis of a diffusionlike model and the development of consistent correlations for the dispersion coefficients might not be possible, hi these situations, more fundamental models, as presented in the previous sections, need to be utilized. 

Because of their structural and conformational complexity, polypeptides, proteins, and their feedstock contaminants thus represent an especially challenging case for the development of reliable adsorption models. Iterative simulation approaches, involving the application of several different isothermal representations8,367 369 enable an efficient strategy to be developed in terms of computational time and cost. Utilizing these iterative strategies, more reliable values of the relevant adsorption parameters, such as q, Kd, or the mass transfer coefficients (the latter often lumped into an apparent axial dispersion coefficient), can be derived, enabling the model simulations to more closely approximate the physical reality of the actual adsorption process. 

Model described by Eqs. (6-1) to (6-4) for adsorbents of monodisperse structure or additional Eqs. (6-33) to (6-35) are too complicated for analytical use. As a matter of fact, the model can include four rate parameters in addition to the adsorption equilibrium constant. Those equations finally give the solutions for the first absolute moment, pi, and the second central moment, pj. These two moments are only utilized in actual experiments. Therefore, it might be helpful if comparison is made between this model and the models which include only two parameters including the equilibrium constant. As two-parameter models, typical are 1) a dispersion model that includes axial dispersion coefficient as a sole rate parameter and 2) a two-phase exchange model which has a mass transfer coefficient as an only rate parameter. These two models are considered to be the two extremes of the complicated model used in the earlier section, hence the results of... 

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