Axial Dispersion and Tanks-in-Series Models

The axial dispersion and tanks-in-series models are the two most common of models that have been developed as general semi-empirical correlations of mixing behavior, presumably bearing some relation to the actual flow pattern in the vessel. The model parameters have to be determined from experimental data and are then correlated as functions of fluid and flow properties and reactor configurations for use in design calculations. 

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]. 

For statistically stationary flow, the species continuity equation for the axial dispersion model is (Chapter 7)  

In (12.7.2-1), u is taken to be the mean (plug flow) velocity through the vessel, and Da is an axial dispersion coefficient to be obtained by means of experiments. One important application is the modeling of fixed beds, as discussed in detail in Chapter 11, and then it is usually termed an effective transport model, with Da = Dea- The axial dispersion model has also been used to approximately describe a variety of other reactors. 

One common approach to determine the model parameter Da, is to perform a residence time distribution test on the reactor and to fit the value of Da so that the model solution and the experimental output curve agree. For such a RTD analysis, a transient term, dC/jdt, is to be included in equation (12.7.2-1). Fig. 12.7.2-1 shows the )- curves of the axial dispersion model for an impulse input with closed boundaries (here, ff = 6FIV= 9ulL, with L = length of the reactor) [Carslaw and Jaeger, 1959,1986]. 

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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 fundamentally different basic assumptions, the axial dispersion and tanks-in-series models can be compared on the basis of several criteria such as conversion or parameters and Pe. From the comparison, it is evident that the conditions selected are such that both models obtain similar values of variances o = 1 / for the tanks-in-series model). 

FIGURE 4.35 Comparison of axial dispersion and tanks-in-series models for a consecutive-competitive reaction system (A- -B R- -E,R- -B > S-fE). 

Two main types of models are in common use for describing axial mixing in bubble columns. The most commonly used model is the Dispersion Model. Here, a diffusion-like process is superimposed on piston or plug flow. The stirred tanks-in-series model has also been used to describe flow of liquids in bubble columns. Levenspiel (1 ) presents a number of models incorporating various combinations of mixed tanks to model stagnant regions and backflow. 

In the next section RTD functions are discussed, and in Secs. 6-3 and 6-4 the RTD is evaluated from response data, with illustrations for ideal reactors. Later sections deal with application of the single-parameter axial-dispersion and number-of-tanks in series models. Finally, conversions are evaluated by the various methods and compared with results for ideal reactors to determine the magnitude of the deviations. 

Several parameters have been used to gather information about sample dispersion in flow analysis peak variance [109], time of appearance of the analytical signal, also known as baseline-to-baseline time [110], number of tanks in the tanks-in-series model [111], the Peclet number in the axially dispersed plug flow model [112], the Peclet number and the mean residence time in the diffusive—convective equation [113]. 

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],... 

The following two models are frequently used to account for partial macromixing the dispersion model and the tanks-in-series model. In the dispersion model, deviation from plug flow is expressed in terms of a dispersion or effective axial diffusion coefficient. This model was anticipated in Chapter 12, and the governing equations for mass and heat are listed in Table 12.2 of that chapter. The derivation is similar to that for plug flow except that now a term is included for diffusive flow in addition to that for bulk flow. This term appears as -D ( d[A]/d ), where is the effective axial diffusion coefficient. When the equation is nondimensionalized, the diffusion coefficient appears as part of the Peclet number defined as = itd/D. A number of correlations for predicting the Peclet number for both liquids and gases in fixed and fluidized beds are available and have been reviewed by Wen and Fan (1975). 

It may be noted that both tanks in series model and axial dispersion model predict the same conversion values. 

If we know the number N of the tanks-in-series model, we calculate the conversion by Eq. (4.10.32) for example, for Do = 1 we obtain a conversion of 59.0%, which almost equals the values as determined by the axially dispersed plug flow model (58.6% and 58.3%). 

The dispersion model is an alternative to the tanks-in-series model. This model formally characterizes mass transport in the radial and axial direction as a one-dimensional process in terms of an effective longitudinal dHTusivity Dax that is superimposed on the plug flow. The dimensionless group Dax/(Mf) is called the dispersion number, and the reciprocal value is the Bodenstein number Bo. Calculation of the conversion is possible by equations based on Da and Bo. 

There is no one exact way to compare the tanks-in-series and axial dispersion models, because the responses are never identical. In many ways, the two models are rather similar, although the mathematical details for the tanks-in-series model are much simpler than for the axial dispersion model. No theoretical justification such as Taylor diffusion is possible in general. The model parameter 77, is strictly empirical, but a useful relationship can be obtained from equating the variances for the two models ((12.7.2-4a), (12.7.2-4b), (12.7.2-6)) ... 

The transfer functions and normalized exit age density functions for three plausible one-parameter models are listed in Table 1. They are the axial dispersion model, the N-stirred tanks in series model and the gamma probability density model. 

J is then a parameter which has no direct physical meaning. Both axial dispersion and mixing tanks in series are approximately equivalent, especially at high Pe (or J) values. By setting Pe = 2J, or better Pe = 2(J - 1), as Pe 0 then J -> 1. The mixing tanks in series model is very popular for representing regularly dispersed RTD curves. 

Both the tank in series (TIS) and the dispersion plug flow (DPF) models require traeer tests for their aeeurate determination. However, the TIS model is relatively simple mathematieally and thus ean be used with any kineties. Also, it ean be extended to any eonfiguration of eompartments witli or without reeycle. The DPF axial dispersion model is eomplex and therefore gives signifieantly different results for different ehoiees of boundary eonditions. 

In this section, we develop two simple models, each of which has one adjustable parameter the tanks-in-series (TIS) model and the axial-dispersion or dispersed-plug-flow (DPF) model. We focus on the description of flow in terms of RTD functions and related quantities. In principle, each of the two models is capable of representing flow in a single vessel between the two extremes of BMF and PF. 

The other two methods are subject to both these errors, since both the form ofi the RTD and the extent of micromixing are assumed. Their advantage is that they permit analytical solution for the conversion. In the axial-dispersion model the reactor is represented by allowing for axial diffusion in an otherwise ideal tubular-flow reactor. In this case the RTD for the actual reactor is used to calculate the best axial dififusivity for the model (Sec. 6-5), and this diffusivity is then employed to predict the conversion (Sec. 6-9). This is a good approximation for most tubular reactors with turbulent flow, since the deviations from plug-flow performance are small. In the third model the reactor is represented by a series of ideal stirred tanks of equal volume. Response data from the actual reactor are used to determine the number of tanks in series (Sec. 6-6). Then the conversion can be evaluated by the method for multiple stirred tanks in series (Sec. 6-10). 

Figure 6-lOh is a plot of Eq. (6-33) for various values of n. The similarity between Figs. 6-8 and 6-10 indicates that the axial-dispersion and series-of-stirred-tanks models give the same general shape of response curve. The analogy is exact for = 1, for this curve in Fig. 6-1 Oh agrees exactly with that in Fig. 6-8 for infinite dispersion, DJuL = oo both represent the behavior of an ideal stirred-tank reactor. Agreement is exact also at the other extreme, the plug-flow reactor ( = oo in Fig. 6-1 Oh and DjuL = 0 in Fig. 6-8). The shapes of the curves for the two models are more nearly the same the larger the value of n.

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 order to describe adequately the hydrodynamics of the experimental fixed bed reactor, it is necessary to take into account the axial dispersion in the mathematical model. The time dependent continuity equation including axial dispersion for a fixed bed reactor is given by a partial differential equation (pde) of the parabolic/hyperbolic class. These types of pde s are difficult to solve numerically, resulting in long cpu times. A way to overcome these difficulties is by describing the fixed bed reactor as a cascade of perfectly stirred tank reactors. The axial dispersion is then accounted for by the number of tanks in series. For a low degree of dispersion (Bo < 50) the number of stirred tanks, N, and the Bodenstein number. Bo, are related as N Bo/2 [8].The fixed bed reactor is now described by a system of ordinary differential equations (ode s). No radial gradients are taken into account and a onedimensional model is applied. Mass balances are developed for both the gas phase and the adsorbed phase. The reactor is considered to be isothermal. 

Use tanks-in-series, axial dispersion, and RFR concepts interchangeably in modeling chemical reactors with knowledge of... 

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