Combined models, mixing

Identify the flow pattern of the prototype system by subjecting it to an impulse, step, or sinusoidal disturbance by injection of a tracer material as reviewed in Chapter 8. The result is classified as either complete mixing, plug flow, and an option between a dispersion, cascade, or combined model. 

Reaction kinetics model combined with mixing model... 

The hydrodynamic data will depend on the mixing characteristics of the reactor. In terms of the modeling of the reactor, the hydrodynamic data will depend on the mixing characteristics of each phase and the combinations of mixing characteristics for the phases. Table 7.3 gives typical data for various combinations of mixing characteristics for gas-liquid reactors16. 

A perfectly mixed tank can also be used, of course, to represent a real stirred tank. Since the patterns of flow in many real stirred tanks are rather complicated, more complex models are often required. This whole topic will be discussed in Section IV on combined models. Thus we will only discuss here the use of a series of perfectly mixed tanks to... 

The following brief discussion shows how combined models are being used to characterize flow in two broad classes of process equipment, stirred tanks and fluidized beds. Other types of mixed models have also been devised for various purposes by Bartok et al. (B3), Cholette and Cloutier (C16), Handles et al. (H3), Pansing (P3), and Singer et al. (S17). Eguchi (E2) presents and discusses some of the models used to date. 

Equations can be formulated for many complex patterns, combinations of mixed and plug flow, with decanting of supernatant liquor that contains the smaller crystals and so on. A modification to the CSTC model by Jandc and Garside (1976) recognizes that linear crystal growth rate may be size-dependent in one instance... 

The construction of a combined model starts with one image (created, supposed or seeded) where it is accepted that the flow into the device is composed of distinct zones which are coupled in series or parallel and where we have various patterns of flow flow zones with perfect mixing, flow zones with plug flow, zones with stagnant fluid (dead flow). We can complete this flow image by showing that we can have some by-pass connections, some recycled flow and some slip flow situations in the device. 

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. 

Appelo C. A. J. and Willemsen A. (1987) Geochemical calculations and observations on salt water intrusions, a combine geochemical/mixing cell model. J. Hydrol. 94,... 

A notable adaptation that combines complex mixing rules, the Stryjek-Vera a correlation, and the Peng-Robinson equation is the PRWS model of Wong... 

In [16] a model was used, which combines micro-mixing and reaction with the disintegration of eddies by inertial forces. The structure of the model enables the disintegration of larger eddies by convective inertial forces to be quantitatively... 

Markov models are used to describe disease as a series of probable transitions between health states. The methodology has considerable appeal for use in phar-macometrics since it offers a method to evaluate patient compliance with prescribed medication regimen, multiple health states simultaneously, and transitions between different sleep stages. An overview of the Markov model is provided together with the Markovian assumption. The most commonly used form of the Markov model, the discrete-time Markov model, is described as well as its application in the mixed effects modeling setting. The chapter concludes with a discussion of a hybrid Markov mixed effects and proportional odds model used to characterize an adverse effect that lends itself to this combination modeling approach. 

A class of what is essentially combined models can also be used to represent micromixing effects in reaction systems. We will not try to present them here as a separate class of mixing models, for in essence they duplicate much of the formalism given in the present development, although their interpretation is rather different. This is more clearly illustrated when applications to chemically reacting systems are involved, as will be discussed later. 

Mixing-Cell, Axial Dispersion, and Combined Models The Unsteady State... 

The latter model type describes the experimentally determined relations between dependent and independent variables with the help of statistical methods and neglects the known physicochemical relations. Such models are primarily used on reactor types difficult to describe deterministically. The cell models are composed of specific networks of mixing cells (e.g. stirred reactor cascades) or of combinations of mixing cells and transport cells (ideal tube reactors). The so-called continuum models, however, handle each phase as a continuum. The continuum models are further distinguished as homogeneous and heterogeneous reactor models. In the heterogeneous reactor model, the fluid phases and the solid phase (catalyst) are considered and mathematically described as individual items. 

Figure 6.56 gives an impression of the work related to this type of structured modeling, showing the balances and auxiliary correlations for Sq, P/V, Fp, and qQ, which are used to calculate the total oxygen transfer capacity of the reactor (OTR) as combined from mixed zone and bubble zone (cf. Fig. 4.6). 

The volumetric mass transfer coefficient may be considered as the reciprocal value of a micro-mixing time. This coefficient can be used in combined models describing micro-mixing and chemical reaction, which will be treated in section 5.2.3. 

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