Mixing tanks in series model

The cell model is a generalization of a class of models such as the completely mixed tanks-in-series model and the back-flow mixed tanks-in-series model. The common characteristic of this model is that the basic mixing unit is a completely mixed or stirred tank. This model has been employed extensively from the early days of chemical engineering to the present (1. ... 

The above expressions are inserted into the appropriate balance equations, for example, for tanks-in-series, segregated tanks-in-series, and maximum-mixed tanks-in-series models. The models are solved numerically [3], and the results are illustrated in the diagrams presented in Figure 4.29, which displays the differences between the above models for second-order reactions. The figure shows that the differences between the models are the most prominent in moderate Damkohler numbers (Figure 4.29). For very rapid and very slow reactions, it does not matter in practice which tanks-in-series model is used. For the extreme cases, it is natural to use the simplest one, that is, the ordinary tanks-in-series 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. 

As explained in Section 4.4.4, there exists an equivalency between tubular dispersion models and stagewise or tank in series models. The stagewise model, used in CHROMPLATE considers the chromatographic column to consist of a large number of well-mixed stirred tanks, arranged in series and thus represents an alternative modelling approach to that of the tubular dispersion model CHROMDIFF. The same two-component separation process is modelled and simulated in both cases. 

As our next fairly simple system let us consider a process in which two energy balances are needed to model the system. The flow rate f of oil passing through two perfectly mixed tanks in series is constant at 90 ft min. The density p of the oil is constant at 40 and its heat capacity Cp is 0.6 Btu/lb °F. The volume... 

One might intuitively expect that infinite recycle rates associated with a system as described by eqn. (61) would produce a completely well-mixed volume with concentration independent of location. This is indeed so and under these conditions, the performance tends to that of an equal sized CSTR. At the other extreme, when R is zero, PFR performance pertains. Fractional conversions at intermediate values of R may be determined from Fig. 14. The specific form of recycle model considered is thus seen to be continuously flexible in describing flow mixing between the PFR and CSTR extremes just as was the tanks-in-series model. The mean and variance of this model are given by eqns. (62) and (63) and these may be used for moments matching purposes of the type illustrated in Example 6. 

Fit the tanks-in-series model to the following mixing cup output data to a pulse input. 

Dispersion and Tanks in Series Model. The first attempts at modeling naturally tried the simple one-parameter models however, observed conversion well below mixed flow cannot be accounted for by these models so this approach has been dropped by most workers. 

The use of the tanks-in-series model for packed beds can be more strongly justified. The fluid can be visualized as moving from one void space to another through the restrictions between particles. If the fluid in each void space were perfectly mixed, the mixing could be represented by a series of stirred tanks each with a size the order of magnitude of the particle. This has been discussed in detail by Aris and Amundson (A14). The fluid in the void spaces is not perfectly mixed, and so an efficiency of mixing in the void spaces has to be introduced (C6). This means that the analogy is somewhat spoiled and the model loses some of its attractiveness. In laminar flow the tanks-in-series model may be still less applicable. 

We have discussed the tanks-in-series model in the sense that the composition in the system was constant over a cross-section. Recently Deans and Lapidus (D12) devised a three-dimensional array of stirred tanks, called a finite-stage model, that was able to take radial as well as axial mixing into account. Because of the symmetry, only a two-dimensional array is needed if the stirred tanks are chosen of different sizes across the radius and are properly weighted. By a geometrical argument. Deans and Lapidus arrived at the following equation for the (i, j) tank ... 

Plug Flow with Dispersion - Plug flow with dispersion is a concept that is often used to describe one-dimensional flow systems. It is somewhat more flexible in computational models because the mixing within the system is not dependent on reactor size, as with complete mix tanks in series. Plug flow with dispersion will be described in the second half of this chapter because special techniques are needed for the analysis. 

EXAMPLE 6.10 Air-Stripping tower with first-order degradation, modeled as plugfiow, plug flow with dispersion, and mixed tanks-in-series reactors... 

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. 

Stratco unit with the single mixer on one end is approximated by a single mixed tank, as shown in the upper part of the figure. However, the Kellogg cascade unit has a series of compartments with mixers and olefin is sparged into each compartment to keep the concentration low so that it reacts with the isobutane rather than polymerizing. The tank-in-series model may be used to model this type of unit and this is shown in the lower part of the figure. A mass balance can be made for a stirred tank reactor readily because the composition is the same everywhere in the vessel. 

There are situations where the entire system can be regarded as one well-stirred tank (N = 1), e.g., when sample dispersion is large [2], This also holds when a device with a large inner volume, such as a mixing chamber, is placed in the analytical path. The tanks-in-series model with (N = 1) is then a suitable tool for describing sample dispersion. 

Inside a mixing chamber (Fig. 3.7), the inlet solutions are thoroughly mixed by the action of centripetal forces, usually assisted by a stirring device, e.g., a small magnetic stirring bar [71]. The mathematical function describing the analyte concentration as a function of time is that of the tanks-in-series model with N = 1, i.e., the concentration at the chamber outlet exhibits an exponential response to a stepwise change in the... 

Additional examples of the use of mixing chamber to FIA are the first version of FIA titrations [10], where a mixing chamber was used to increase the width of the injected zone and to conform with the one-tank-in series model (see Section 4.10) the work of Tyson [1062, 1245], who used the mixing chamber for automated calibration in atomic absorption and Stewart [427], who applied a mixing chamber to automated dilution. 

Reactors packed with inert spheres can be visualized as consisting of a series of uniform mixing stages, and can be therefore described by the tanks-in-series model. By rewriting Eq. (3.11) the mixing height H can be obtained from the peak width at the base (W) and from the mean residence time T, since... 

The chemical kinetics in a packed reactor have been studied in a detail by Reijn, Poppe, and van der Linden [554] with the aid of the tanks-in-series model. They based their theory on three assumptions that the number of tanks N in the reactor is constant that the chemical reactions are (pseudo) first order and that adequate mixing is ensured. 

Finer details of the curves can also be matched through use of higher moments, but, especially with experimental data, these higher moments are almost impossible to obtain. For the axial dispersion or tanks-in-series models that have only one (macro) mixing parameter, the higher moments give no further information (beyond model consistency) but more complex flow models can have more parameters. 

---