Dispersion model, nonideal reactors

ILLUSTRATION 11.6 USE OF THE DISPERSION MODEL TO DETERMINE THE CONVERSION LEVEL OBTAINED IN A NONIDEAL FLOW REACTOR... 

For any more complex flow pattern we must solve the fluid mechanics to describe the fluid flow in each phase, along with the mass balances. The cases where we can still attempt to find descriptions are the nonideal reactor models considered previously in Chapter 8, where laminar flow, a series of CSTRs, a recycle TR, and dispersion in a TR allow us to modify the ideal mass-balance equations. 

Here we use a single parameter to account for the nonideality of our reactor. This parameter is most always evaluated by analyzing the RTD determined from a tracer test. Examples of one-parameter models for a nonideal CSTR include the reactor dead volume V, where no reaction takes place, or the fraction / of fluid bypassing the reactor, thereby exiting unreacted. Examples of one-parameter models for tubular reactors include the tanks-in-series model and the dispersion model. For the tanks-in-series model, the parameter is the number of tanks, n, and for the dispersion model, it is the dispersion coefficient D,. Knowing the parameter values, we then proceed to determine the conversion and/or effluent concentrations for the reactor. 

The dispersion model is also used to describe nonideal tubular reactors. In this model, there is an axial dispersion of the material, which is governed by an analogy to Pick s law of diffusion, superimposed on the flow. So in addition to transport by bulk flow, UAqC, every component in the mixture is transported through any cross section of the reactor at a rate equal to [—DaAddCldz)] resulting from molecular and convective diffusion. By convective diffusion we mean either Aris-Taylor dispersion in laminar flow reactors or turbulent diffusion resulting from turbulent eddies. 

Correlations exist for the amoimt of dispersion that might be expected in common packed-bed reactors, so these systems can be designed using the dispersion model without obtaining or estimating the RTD. This situation is perhaps the only one where an RTD is not necessary for designing a nonideal reactor. 

Computational fluid dynamics (CFD) has emerged as a very valuable tool in modeling the real flow patterns in chemical reactors. It represents a quantum leap from the idealized reactor models or their modifications, such as the tanks-in-series or axial-dispersion models to account for nonidealities. It has the potential to account for flow and reactions inside a reactor in their entirety. CFD has been used successfully to predict the flow patterns and reactor performance in the case of reactions involving macro-mixing effects. 

In an ideal fixed-bed reactor, plug flow of gas is assumed. This is, however, not a good assumption for reactive solids, because the bed properties vary with position, mainly due to changing pellet properties (and dimensions in most cases), and hence the use of nonideal models is often necessary. The dispersion model, with all its limitations, is still the most practical one. The equations involved are cumbersome, but their asymptotic solutions are simple, particularly for systems... 

A nonideal reactor model based on the axial dispersion (one-dimensional) equation is readily written down following the procedure used in deriving the PFR relationship. Thus, equation (5-20) becomes, for steady-state conditions. 

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. 

We probably don t have to, at this point, say that the overall model here, or even considerable simplifications of it, are best left to numerical solution. The axial dispersion model seems to work pretty well, at least for cases where the holdup/ wetting does not vary much with position in the reactor. For large changes in this factor, or for nonideal flows involving stagnant zones or liquid/gas bypassing, some version of one of the combined models will be required. It will be understood that these will be very specific to the particular design under consideration. 

If the deviations are small then they can be described by the dispersion model (additional dispersive flow is is superimposed on the plug flow) or cell model (cascade of ideal stirred tanks). For larger deviations the calculation of nonideal reactors is generally difficult. A more simply treated special case occurs when the volume elements flowing through the reactor are macroscopically but not microscopically mixed (segregated flow). This case can be solved by the Hofmann-Schoenemann method (see below). 

ILLUSTRATION 11.6 Use of the Dispersion Model to Determine the Conversion Level obtained in a Nonideal Reactor... 

The Dispersion model has been widely used, especially for describing relatively small deviations from plug flow in packed beds and empty tubes. The availability of correlations that can be used to estimate D/uL for common reactor configurations makes this model especially convenient. Nevertheless, there are many situations, primarily high values of D/uL, for which the Dispersion model is not appropriate. Two alternative approaches to describing nonideal reactors are considered in the final sections of this chapter. 

The Dispersion model can be used to predict the performance of a nonideal reactor in the absence of a measured RTD. However, the geometric parameters and the flow conditions of the nonideal reactor must fall within the range of existing correlations for the intensity of dispersion. ... 

In this section, we apply the axial dispersion flow model (or DPF model) of Section 19.4.2 to design or assess the performance of a reactor with nonideal flow. We consider, for example, the effect of axial dispersion on the concentration profile of a species, or its fractional conversion at the reactor outlet. For simplicity, we assume steady-state, isothermal operation for a simple system of constant density reacting according to A - products. 

In the quantitative development in Section 24.4 below, we assume the flow to be ideal, but more elaborate models are available for nonideal flow (Chapter 19 see also Kastanek et al., 1993, Chapter 5). Examples of types of tower reactors are illustrated schematically in Figure 24.1, and are discussed more fully below. An important consideration for the efficiency of gas-liquid contact is whether one phase (gas or liquid) is dispersed in the other as a continuous phase, or whether both phases are continuous. This is related to, and may be determined by, features of the overall reaction kinetics, such as rate-determining characteristics of mass transfer and intrinsic reaction. 

In a bubble-column reactor for a gas-liquid reaction, Figure 24.1(e), gas enters the bottom of the vessel, is dispersed as bubbles, and flows upward, countercurrent to the flow of liquid. We assume the gas bubbles are in PF and the liquid is in BMF, although nonideal flow models (Chapter 19) may be used as required. The fluids are not mechanically agitated. The design of the reactor for a specified performance requires, among other things, determination of the height and diameter. 

Nonideal tubular reactor models, inclusion of interpellet axial dispersion in,... 

Interpretation of tracer data by means of residence time theory, in the extremes of complete and minimum segregation, has been reviewed and extended to treat transient response under reacting conditions. While residence time theory was initially developed for industrial application to nonideal steady state reactors, its transient extension seems especially well suited for describing segments of natural flows which are too complex to interpret using simpler models, such as dispersion. 

Yet another approach is based on the following simple notion. The characteristic C i) curve of Figure 4.4(b) for responses intermediate to the ideal limits is broader and more diffuse than that of the pulse input response for the plug-flow limit. This suggests that some type of diffusion or dispersion term might be incorporated into the basic plug-flow model to represent the effects of nonideal flows on reactor performance. 

Experimental data of stearic acid decarboxylation in a laboratory-scale fixed bed reactor for formation of heptadecane were evaluated studied with the aid of mathematical modeling. Reaction kinetics, catalyst deactivation, and axial dispersion were the central elements of the model. The effect of internal mass transfer resistance in catalyst pores was found negligible due to the slow reaction rates. The model was used for an extensive sensitivity study and parameter estimation. With optimized parameters, the model was able to describe the experimentally observed trends adequately. A reactor scale-up study was made by selecting the reactor geometry (diameter and length of the reactor, size and the shape of the catalyst particles) and operating conditions (superficial liquid velocity, temperature, and pressure) in such a way that nonideal flow and mass and heat transfer phenomena in pilot scale were avoided. 

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. 

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