Stirred reactors, modeling

Mathematical models of the electron-capture process are based on the stirred reactor model of Lovelock [145] and the kinetic model of Wentworth [117,142,146] as further modified by others [129,134-136,147-150]. The ionization ch2uaber is considered... 

From these examples it is apparent that one needs to be cautious when using steady-state methods and continuation procedures near turning points. While the solutions may converge rapidly and even appear to be physically reasonable, there can be significant errors. Fortunately, a relatively simple time-stepping procedure can be used to identify the nonphysical solutions. Beginning from any of the solutions that are shown in Fig. 15.9 as shaded diamonds, a transient stirred-reactor model can be solved. If the initial solution (i.e., initial condition for the transient problem) is nonphysical, the transient procedure will march toward the physical solution. If the initial condition is the physical solution, the transient computational will remain stationary at the correct solution. 

Simulate the process using a perfectly stirred reactor model. For the nominal processing conditions, plot the O-atom number density as a function of pressure for... 

Yield-Based Reactor Fractional Conversion Reactor Combined Specification Model Well-Stirred Reactor Model Plug Flow Reactor Model Two Phase Chemical Equilibrium General Phase and Chemical Equilibrium... 

K. K. Boon, "A Flexible Mathematical Model for Analy2ing Industrial P. F. Furnaces," M.S. thesis. University of Newcasde, AustraUa, Sept. 1978. R. H. Essenhigh, "A New AppHcation of Perfectly Stirred Reactor (P.S.R.) Theory to Design of Combustion Chambers," TechnicalEeport FS67-1 (u), Peimsylvania State University, Dept, of Euel Science, University Park, Pa., Mar. 1967. 

Concluding, it is essential to represent complex, real-life flow situations by computationally tractable models that retain adequate details. As an example, a computational snapshot approach that simulates the flow in stirred reactors or other vessels for any arbitrary impeller has been developed [5]. This approach lets the engineer simulate the detailed fluid dynamics around the impeller blades with much less computations that would otherwise be required. Improvements in CFD technique are likely to encourage further work along these lines. 

Non-ideal reactors are described by RTD functions between these two extremes and can be approximated by a network of ideal plug flow and continuously stirred reactors. In order to determine the RTD of a non-ideal reactor experimentally, a tracer is introduced into the feed stream. The tracer signal at the output then gives information about the RTD of the reactor. It is thus possible to develop a mathematical model of the system that gives information about flow patterns and mixing. 

Mixing Models. The assumption of perfect or micro-mixing is frequently made for continuous stirred tank reactors and the ensuing reactor model used for design and optimization studies. For well-agitated reactors with moderate reaction rates and for reaction media which are not too viscous, this model is often justified. Micro-mixed reactors are characterized by uniform concentrations throughout the reactor and an exponential residence time distribution function. 

Secondly, these quotations emphasize the fact that the same river input that fuels longitudinal heterogeneity in reservoirs also forms a strong link between the reservoir and its watershed (e.g., [6]). This link has been conceptualized mostly in the form of load-response empirical models [7, 8], or mass-balance approaches [9]. Curiously, empirical modelers usually consider reservoirs as stirred reactors, ignoring the longitudinal spatial heterogeneity present in most situations and processes. 

The completely segregated stirred tank can be modeled as a set of piston flow reactors in parallel, with the lengths of the individual piston flow elements being distributed exponentially. Any residence time distribution can be modeled as piston flow elements in parallel. Simply divide the flow evenly between the elements and then cut the tubes so that they match the shape of the washout function. See Figure 15.12. A reactor modeled in this way is said to be completely segregated. Its outlet concentration is found by averaging the concentrations of the individual PFRs ... 

A process for the depolymerisation of Nylon 6 carpet fibre in the presenee of steam under medium pressure (800 to 1500 KpA, 100 to 200 psig) is described. The feasibility of the seheme was demonstrated using a small laboratory apparatus and the best run produced a 95% yield of crude eaprolaetam. The data obtained were used to construct a computer model of the process for both batch and continuous flow stirred reactors. 6 refs. 

Fig. 2. Properties of model and biological particle systems Micro scale related particle diameter dp/riL versus maximum energy dissipation e , in stirred reactors explanations see Table 3 and Table 4...

For reactors with free turbulent flow without dominant boundary layer flows or gas/hquid interfaces (due to rising gas bubbles) such as stirred reactors with bafQes, all used model particle systems and also many biological systems produce similar results, and it may therefore be assumed that these results are also applicable to other particle systems. For stirred tanks in particular, the stress produced by impellers of various types can be predicted with the aid of a geometrical function (Eq. (20)) derived from the results of the measurements. Impellers with a large blade area in relation to the tank dimensions produce less shear, because of their uniform power input, in contrast to small and especially axial-flow impellers, such as propellers, and all kinds of inclined-blade impellers. 

GP 9] [R 16] By finite-element reactor modeling, it was shown that for conversions as large as 34%, concentration differences within the mini wide fixed-bed reactor of only less than 10% are found [78], Thus, the reactor approximates a continuous-stirred tank reactor (CSTR). This means that the mini wide fixed-bed reactor yields differential kinetics even at large conversions, larger than for reactors used so far (< 10% conversion). 

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. 

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... 

We can characterize the mixed systems most easily in terms of the longitudinal dispersion model or in terms of the cascade of stirred tank reactors model. The maximum amount of mixing occurs for the cases where Q)L = oo or n = 1. In general, for reaction orders greater than unity, these models place a lower limit on the conversion that will be obtained in an actual reactor. The applications of these models are treated in Sections 11.2.2 and 11.2.3. 

Determination of Conversion Levels Based on the Cascade of Stirred Tank Reactors Model... 

ILLUSTRATION 11.7 USE OF THE CASCADE OF STIRRED TANK REACTORS MODEL TO PREDICT REACTOR PERFORMANCE... 

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. 

Particular attention is to be paid to closure models exploiting various types of PDFs such as beta, presumed, or full PDFs (e.g., Baldyga, 1994 Fox, 1996, 2003 Ranade, 2002). While PDFs have successfully been exploited for describing chemical reactions in turbulent flames, tubular reactors (Baldyga and Henczka, 1997), and a Taylor-Couette reactor (Marchisio and Barresi, 2003), they have never been used successfully in stirred reactors so far. 

A reactor model based on solid particles in BMF may be used for situations in which there is deliberate mixing of the reacting system. An example is that of a fluid-solid system in a well-stirred tank (i.e., a CSTR)-usually referred to as a slurry reactor, since the fluid is normally a liquid (but may also include a gas phase) the system may be semibatch with respect to the solid phase, or may be continuous with respect to all phases (as considered here). Another example involves mixing of solid particles by virtue of the flow of fluid through them an important case is that of a fluidized bed, in which upward flow of fluid through the particles brings about a particular type of behavior. The treatment here is a crude approximation to this case the actual flow pattern and resulting performance in a fluidized bed are more complicated, and are dealt with further in Chapter 23. 

Fermentation systems obey the same fundamental mass and energy balance relationships as do chemical reaction systems, but special difficulties arise in biological reactor modelling, owing to uncertainties in the kinetic rate expression and the reaction stoichiometry. In what follows, material balance equations are derived for the total mass, the mass of substrate and the cell mass for the case of the stirred tank bioreactor system (Dunn et ah, 2003). 

ASCSTR - Continuous Stirred Tank Reactor Model of Activated Sludge System... 

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