Reactor-diffusion system

Numerous modeling studies of CVD reactors have been made and are summarized in recent review papers (I, 212). Table 3 in reference 212 lists major examples of CVD models up to mid-1986. Therefore, rather than giving an exhaustive list of previous work, Table V presents a summary of the major modeling approaches and forms the basis for the ensuing discussion, which is most appropriately handled in terms of two groups (1) hot-wall LPCVD systems and (2) cold-wall, near-atmospheric-pressure reactors. In LPCVD reactors, diffusion and surface reaction effects dominate, whereas in cold-wall reactors operated at near-atmospheric pressures, fluid flow and gas-phase reactions are important in predicting performance, as discussed earlier in relation to transport phenomena. 

Part III focuses on spatial instabilities and patterns. We examine the simplest type of spatial pattern in standard reaction-diffusion systems in Chap. 9, namely patterns in a finite domain where the density vanishes at the boundaries. We discuss methods to determine the smallest domain size that supports a nontrivial steady state, known as the critical patch size in ecology. In Chap. 10, we provide first an overview of the Turing instability in standard reaction-diffusion systems. Then we explore how deviations from standard diffusion, namely transport with inertia and anomalous diffusion, affect the Turing instability. Chapter 11 deals with the effects of temporally or spatially varying diffusivities on the Turing instability in reaction-diffusion systems. We present applications of Turing systems to chemical reactions and biological systems in Chap. 12. Chapter 13 deals with spatial instabilities and patterns in spatially discrete systems, such as diffusively and photochemically coupled reactors. 

We consider the TAP reactor as a basic kinetic device for systematic studies of reaction-diffusion systems. In this chapter, we are going to (i) present and analyze models of different TAP configurations with a focus on their possibilities with respect to characterizing active materials and unraveling complex mechanisms, and (ii) demonstrate relationships between TAP models and other basic reactor models, that is, models for the ideal continuous stirred-tank reactor (CSTR), batch reactor (BR) and plug-flow reactor (PER). In some situations, the TZTR can be considered a simple building block for constracting the various models. 

We employ a method of numerical continuation which has been earlier developed into a software tool for analysis of spatiotemporal patterns emerging in systems with simultaneous reaction, diffusion and convection. As an example, we take a catalytic cross-flow tubular reactor with first order exothermic reaction kinetics. The analysis begins with determining stability and bifurcations of steady states and periodic oscillations in the corresponding homogeneous system. This information is then used to infer the existence of travelling waves which occur due to reaction and diffusion. We focus on waves with constant velocity and examine in some detail the effects of convection on the fiiont waves which are associated with bistability in the reaction-diffusion system. A numerical method for accurate location and continuation of front and pulse waves via a boundary value problem for homo/heteroclinic orbits is used to determine variation of the front waves with convection velocity and some other system parameters. We find that two different front waves can coexist and move in opposite directions in the reactor. Also, the waves can be reflected and switched on the boundaries which leads to zig-zag spatiotemporal patterns. 

Below we discuss a particular case of coexisting stable front waves and calculate two-parameter bifurcation diagrams determining wave velocity vs. physical parameter relations. These results are then related to spatiotemporal patterns obtained by directly solving the partial differential equations which describe a bounded system. This system gives rise to an alternating pattern of fronts moving back and forth in the reactor. At the boundaries one type of front is transformed into the other one and is reflected back to the reactor. Such a pattern exists for the reaction-diffusion system (with Neumann boundary conditions) as well as for the reaction-diffusion-convection system (with Danckwerts boundary conditions). Observed zig-zag dynamics is both of theoretical and practical interest for operation of chemical reactors. 

Therefore, in FTS reaction system, there are a lot of factors, crossing several scales, including molecular reaction (element scale), internal and external diffusion (system scale), as well as type of reactor (system scale). And the combined effects of those factors significantly influence the product distribution. It is considered that the mesoscale study of FTS, which cross two or several scales, is the key point of controlling the product distribution in FTS reaction. 

Catalytic partial oxidation of o-xylene and naphthalene is performed mostly in intensively cooled multi-tubular fixed bed reactors, but systems with a fluidized bed were also developed. Typically, V20s/Ti02 catalysts with K2SO4 or A1 phosphates as promoter are used. In fixed bed reactors, the conversion of both feedstocks per pass is around 90%, and the selectivity is in the range 0.86-0.91 mol PA per mol naphthalene and 0.78 mol per mol o-xylene. (Note that the selectivity would be 100%, if only the reactions according to Eqs. (6.13.1) and (6.13.2), respectively, would take place.) The active compounds are distributed on spheres of porcelain, quartz, or silicium carbide (shell catalyst). The thickness of the shell is only around 0.2 mm, and the diffusion paths for the reactants are short. By this means, the influence of pore diffusion is small, and the unwanted oxidation of phthalic acid anhydride to CO2 is suppressed compared to a catalyst with an even distribution of active compounds where the influence of pore diffusion would be much stronger (see Section 4.5.6.3 Influence of Pore Diffusion on the Selectivity of Reactions in Series ). Thus the intrinsic reaction rates are utilized for the modeling of a technical reactor (next Section 6.13.2). 

Although the diffuser system is not strictly part of the primary cooling system, it uses primary water to limit N-16 activity at the pool surface. The diffuser pump and its associated valves and piping is located on the first shield step of the east side of the reactor. The diffuser pump takes its suction from near the top of... 

The basic limitations of these reactor catalyst systems will now be analysed in more detail. In particular, the limitation on product carbon number imposed by fluid-bed technology and the pore diffusion limitation of the larger particles applicable in fixed-bed and ebulliating-bed technology will be discussed. 

The idea of homogenous spatial distribution of the particles is based on the concept of well-stirred reactor. However, even microscopic reactions produce local nonhomogenities, which can not be always eliminated by diffusion. There are many contraversions about the stochastic formulation of reaction - diffusion systems. Three directions in the theory of random fields seem to be able to cope with such complexity the theory of random measures, the theory of stochastic partial differential equations relating to trajectories, and the theory of Hilbert space valued stochastic processes. The details are beyond of our scope. 

In the past few years, however, there has been a rebirth of interest in the formation of dissipative structures in chemically reacting and diffusing systems. This interest has been mainly sparked by the developement of open spatial reactors by groups in Texas [21-28] and in Bordeaux [29-38]. Basically, two types of open reactors are currently operating (i) the two-dimensional continuously fed unstirred reactors where the transport process is essentially natural molecular diffusion and where the feeding is either uniform (continuously fed unstirred reactor [23,24]) or from the lateral boundaries (linear [34, 35], annular [21, 22, 38] or disc [25, 36] gel reactors) and (ii) the Couette... 

As pointed out in the previous section, the spatially extended open Couette flow reactor [27-33] provides a practical implementation of an effectively one-dimensional reaction-diffusion system with an external concentration gradient imposed from the boundaries. With the specific motivation to provide theoretical and numerical support for the recent experimental observations of sustained dissipative structures in the Couette flow reactor, we will consider the standard reaction-diffusion equation ... 

In most experimental runs, the volume and feeding flows of the two CSTRs at both ends of the Couette reactor were large enough for their internal state not to be significantly influenced by the dynamics inside the Couette reactor [33]. This corresponds mathematically to imposing Dirichlet boundary conditions to our model reaction-diffusion system (3). In most of the simulations... 

A similar theoretical analysis can be reproduced for higher dimensional systems [48]. In a prospective paper [63], in collaboration with Pearson and Russo, we have reported preliminary results of a study of two-dimensional reaction-diffusion systems that model sustained front patterns observed in gel reactors. The linear [34, 35] and annular [21, 22, 38] gel reactors are strips of gel that are fed from the lateral boundaries. These reactors have a natural tendency to produce narrow front (linear or circular) structures away from the boundaries [39]. Stationary single-front and multi-front patterns have been observed experimentally [34-38]. According to the Hopf bifurcation mechanism reported in section 5, these front patterns are expected to destabilize into periodically oscillating structures. Since the Hopf mode is likely to be condensed in the active region at the front zones, the reaction-diffusion sys-... 

Mishra, S.K. et al. (2008) Spatiotempo-ral compound wavelet matrix framework for multiscale/multiphysics reactor simulation case study of a heterogeneous reaction/diffusion system. Int.J. Chem. Reactor Eng., 6 (A28), A28-1-A28-42. 

De Kepper has described a fascinating pattern of waves that can be studied in a continuous couette reactor. This is an open spatial reactor that provides a good approximation of a one-dimensional diffusion system, and consists of two concentric cylinders with a narrow gap between them. The inner cylinder can rotate while the outer one is fixed. At each end of the cylinder is a chamber fitted with a stirrer into which reactants can flow in and products flow out. Variation of the rate of rotation causes changes in the... 

The search for Turing patterns led to the introduction of several new types of chemical reactor for studying reaction-diffusion events in feedback systems. Coupled with huge advances in imaging and data analysis capabilities, it is now possible to make detailed quantitative measurements on complex spatiotemporal behaviour. A few of the reactor configurations of interest will be mentioned here. 

If a fluid is placed between two concentric cylinders, and the inner cylinder rotated, a complex fluid dynamical motion known as Taylor-Couette flow is established. Mass transport is then by exchange between eddy vortices which can, under some conditions, be imagmed as a substantially enlranced diflfiisivity (typically with effective diflfiision coefficients several orders of magnitude above molecular difhision coefficients) that can be altered by varying the rotation rate, and with all species having the same diffusivity. Studies of the BZ and CIMA/CDIMA systems in such a Couette reactor [45] have revealed bifiircation tlirough a complex sequence of front patterns, see figure A3.14.16. 

Cyclohexane, produced from the partial hydrogenation of benzene [71-43-2] also can be used as the feedstock for A manufacture. Such a process involves selective hydrogenation of benzene to cyclohexene, separation of the cyclohexene from unreacted benzene and cyclohexane (produced from over-hydrogenation of the benzene), and hydration of the cyclohexane to A. Asahi has obtained numerous patents on such a process and is in the process of commercialization (85,86). Indicated reaction conditions for the partial hydrogenation are 100—200°C and 1—10 kPa (0.1—1.5 psi) with a Ru or zinc-promoted Ru catalyst (87—90). The hydration reaction uses zeotites as catalyst in a two-phase system. Cyclohexene diffuses into an aqueous phase containing the zeotites and there is hydrated to A. The A then is extracted back into the organic phase. Reaction temperature is 90—150°C and reactor residence time is 30 min (91—94). 

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