Stirred tanks in series model

The Stirred Tanks in Series Model Another model that is frequently used to simulate the behavior of actual reactor networks is a cascade of ideal stirred tank reactors operating in series. The actual reactor is replaced by n identical stirred tank reactors whose total volume is the same as that of the actual reactor. 

When DJuL is found to be large and the tracer response curve is skewed, as in Fig. 2.23b, but without a significant delay, a continuous stirred-tanks in series model (Section 2.3.2), may be found to be more appropriate. The tracer response curve will then resemble one of those in Fig. 2.8 or Fig. 2.9. The variance a2 of such a curve with a mean of tc is related to the number of tanks / by the expression a2 = t2/i (which can be shown for example by the Laplace transform method 7 from the equations set out in Section 2.3.2). Calculations of the mean and variance of an experimental curve can be used to determine either a dispersion coefficient Dl or a number of tanks i. Thus each of the models can be described as a one parameter model , the parameter being DL in the one case and i in the other. It should be noted that the value of i calculated in this way will not necessarily be integral but this can be accommodated in the more mathematically general form of the tanks-in-series model as described by Nauman and Buffham 7 . 

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. 

B. A. BufFham and L. G. Gibilaro [AIChE J., 14, 805 (1968)] have shown how the stirred-tanks-in-series model, with noninteger values of n, can be used to fit RTD data. 

For the RTD of Prob. 6-1 determine (a) the value of DJuL which gives the most accurate fit of the dispersion model to the data and (Jb) the most accurate integer value of n if the stirred-tanks-in-series model is used. 

Calculate the conversion for the laminar-flow reactor of Prob. 6-7, using the stirred-tanks-in-series model to represent the RTD. 

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. 

The model consists of i -i- 1 stirred tanks in series, i of which have a common volume, and one of which has a smaller volume... 

Real reactors can have 0 < cr < 1, and a model that reflects this possibility consists of a stirred tank in series with a piston flow reactor as indicated in Figure 15.1(a). Other than the mean residence time itself, the model contains only one adjustable parameter. This parameter is called the fractional tubularity, Xp, and is the fraction of the system volume that is occupied by the piston flow element. Figure 15.1(b) shows the washout function for the fractional tubularity model. Its equation is... 

A reactor is modelled as two stirred tanks in series, of which the first is half the size of the other. Derive equations for the time distributions E(tr), F(tr) and A(tr). 

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. 

A first-order liquid-phase reaction takes place in a baffled stirred vessel of 2 volume under conditions when the flow rate is constant at 605 dm min and the reaction rate coefficient is 2.723 min the conversion of species A is 98%. Verify that this performance lies between that expected from either a PFR or a CSTR. Tracer impulse response tests are conducted on the reactor and the data in Table 6 recorded. Fit the tanks-in-series model to this data by (A) matching the moments, and (B) evaluating N from the time at which the maximum tracer response is observed. Give conversion predictions from the tanks-in-series model in each case. 

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 ... 

Beard and Bassingthwaighte [247] showed that a power function can be represented as the sum of a finite number of scaled basis functions. Any probability density function may serve as a basis function. They considered as basis function a density corresponding to the passage time of a molecule through two identical well-stirred tanks in series. The weighted sum of such to models leads to the power function... 

The Ideal Stirred Tank 533 Multiple Stirred Tanks in Series 536 Applicability of the CSTC Model 536... 

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. ... 

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. 

The tanks-in-series model (Figure 18.4A) assumes a series of N well-stirred compartments with identical residence times (137,138). The equations describing the tanks-in-series model are... 

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). 

In the series-of-stirred-tanks model the actual reactor is simulated by n ideal stirred tanks in series. The total volume of the tanks is the same as the volume of the actual reactor. Thus for a given flow rate the total mean residence time is also the same. The mean residence time per tank is 8Jn. Figure 6-lOu describes the situation. The objective is to find the value of n for which the response curve of the model would best fit the response curve for the actual reactor. To do this the relation between (C/Co)stcp should be developed. 

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... 

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