Stirred tank modeling

A stirred-tank model has been proposed, (Daly, 1980), to model the mixing behavior of an air-solid, spouted, fluidised-bed reactor. The central spout is modelled as two tanks in series, the top fountain as a further tank and the down flowing annular region of the bed as 6 equal tanks in series. It is assumed that a constant fraction of the total solids returns from each stage of the annular region into the central two tank region, as depicted below. 

Convection is mass transfer that is driven by a spatial gradient in pressure. This section presents two simple models for convective mass transfer the stirred tank model (Section II.A) and the plug flow model (Section n.B). In these models, the pressure gradient appears implicitly as a spatially invariant fluid velocity or volumetric flow rate. However, in more complex problems, it is sometimes necessary to develop an explicit relationship between fluid velocity and pressure gradients. Section II.C describes the methods that are used to develop these relationships. 

One of the simplest models for convective mass transfer is the stirred tank model, also called the continuously stirred tank reactor (CSTR) or the mixing tank. The model is shown schematically in Figure 2. As shown in the figure, a fluid stream enters a filled vessel that is stirred with an impeller, then exits the vessel through an outlet port. The stirred tank represents an idealization of mixing behavior in convective systems, in which incoming fluid streams are instantly and completely mixed with the system contents. To illustrate this, consider the case in which the inlet stream contains a water-miscible blue dye and the tank is initially filled with pure water. At time zero, the inlet valve is opened, allowing the dye to enter the... 

This simple example illustrates two important features of stirred tanks (1) the concentration of dissolved species is uniform throughout the tank, and (2) the concentration of these species in the exit stream is identical to their concentration in the tank. Note that a consequence of the well-stirred behavior of this model is that there is a step change in solute concentration from the inlet to the tank, as shown in the concentration profile in Figure 2. Such idealized behavior cannot be achieved in real stirred vessels even the most enthusiastically stirred will not display this step change, but rather a smoother transition from inlet to tank concentration. It should also be noted that stirred tank models can be used when chemical reactions occur within the tank, as might occur in a flow-through reaction vessel, although these do not occur in the simple dye dilution example. 

Stirred tank models have been widely used in pharmaceutical research. They form the basis of the compartmental models of traditional and physiological pharmacokinetics and have also been used to describe drug bioconversion in the liver [1,2], drug absorption from the gastrointestinal tract [3], and the production of recombinant proteins in continuous flow fermenters [4], In this book, a more detailed development of stirred tank models can be found in Chapter 3, in which pharmacokinetic models are discussed by Dr. James Gallo. The conceptual and mathematical simplicity of stirred tank models ensures their continued use in pharmacokinetics and in other systems of pharmaceutical interest in which spatially uniform concentrations exist or can be assumed. 

As with the stirred tank model, mass balance equations can be developed to describe mass transfer in plug flow. In this case, it is convenient to define the system as a differential cylindrical section of the tube, with length Az and volume nD2 Az/4, where D is the tube diameter. This system is fixed in space and may... 

Stirred tank model A simple convective flow pattern in tanks, characterized by complete and instantaneous mixing in all directions. Also called the continuously stirred tank reaction or the mixing tank model. See Eqs. (3) and (4) and Figure 2. 

The dispersion and stirred tank models of reactor behavior are in essence single parameter models. The literature contains an abundance of more complex multiparameter models. For an introduction to such models, consult the review article by Levenspiel and Bischoff (3) and the texts by these individuals (2, 4). The texts also contain discussions of the means by which residence time distribution curves may be used to diagnose the presence of flow maldistribution and stagnant region effects in operating equipment. 

Dl/uL -> 0), and so for many practical cases of interest a comparison is possible. Thus, for small deviations from plug flow, either the dispersion model or stirred tanks model may be satisfactorily used depending on one s personal preference. 

Packed, Tray, and Spray Towers Packed and tray towers have been discussed in the subsection Mass Transfer in Sec. 5. Typically, the gas and liquid are countercurrent to each other, with the liquid flowing downward. Each phase may be modeled using a PFR or dispersion (series of stirred tanks) model. The model is solved numerically... 

Series-of-stirred-tanks model The series-of-stirred-tanks model (often referred to as the cell model) is perhaps the simplest type of stagewise model for the backmix reactor. In this model, the reactor is represented by a series of perfectly mixed stages. The degree of backmixing is characterized by the number of stages the... 

INDICATES POINTS FROM STIRRED TANK MODEL... 

Transformation of the process diagram into a corresponding simulation flowsheet is illustrated in the lower part of Fig. 6.5. In principle all plant elements may be represented by a separate model. For practical applications, though, it is sufficient to take into account only a time delay as well as the dispersion of the peak until it enters the column. This can be achieved by a pipe-flow model that includes axial dispersion. The detector (including some connecting pipes) can be represented by a stirred-tank model. 

It is usual in laminar mixing simulations to represent the flow using tracer trajectories. The computation of such flow trajectories in a coaxial mixer is more complex than in traditional stirred tank modelling due to the intrinsic unsteady nature of the problem (evolving topology, flow field known at a discrete number of time steps in a Lagrangian frame of reference). Since the flow solution is periodic, a node-by-node interpolation using a fast Fourier transform of the velocity field has been used, which allowed a time continuous representation of the flow to be obtained. In other words, the velocity at node i was approximated... 

Interpretation of Response Data by the Series-of-stirred-tanks Model... 

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. 

SECTION 6-6 INTERPRETATION OF RESPONSE DATA BY THE SERIES-OF-STIRRED-TANKS MODELS 259... 

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.

The series-of-stirred-tanks model could not represent the RTD for the laminar-flow reactor shown in Fig. 6-7. However, the RTD data given in Example 6-2 can be simulated approximately. The dashed curve in Fig. 6-1 Oh is a plot of this RTD. While no integer value of n coincides with this curve for all 6/6, the curve for = 5 gives approximately the correct shape. Comparison of the fit in Figs. 6-9 and 6-lOh indicates that about the same... 

To use the stirred-tank model, n is found from response data, as illustrated in Sec. 6-6. 

Example 6-7 Using the series-of-stirred-tanks model to simulate the RTD of the reactor described in Example 6-2, predict the conversion for a first-order reaction for which k = 0.1 sec and 0 = 10 sec. 

SECTION 6-10 CONVERSION ACCORDING TO THE SERIES-OF-STIRRED-TANKS MODEL... 

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