Two Coupled Reactors

The expressions for the coefficients c of the characteristic polynomial and the Hurwitz determinants A in terms of b, a, d, ay, and 02 are obtained using computational algebra software, such as Mathematica (Wolfram Research, Inc., Champaign, IL, 2(X)2) and Maple (Waterloo Maple Inc., Waterloo, Ontario, 2002). They are very lengthy already for only two coupled reactors and will not be displayed here. The condition C4 = 0 yields the Turing threshold 

As expected, this expression is identical with the result (13.63) in Sect. 13.3.2.1 for K =, since the parameter a does not affect stationary bifurcations as discussed above. The Turing bifurcation exists if a Omm-p = n/175 = 13.228756. For a homogeneous array, i.e., ay = 02 = a, the Turing bifurcation occurs before the Hopf bifurcation if ct alia-, 2) (the argument after the semicolon denotes the number of coupled reactors), where 

For the inhomogeneous two-reactor array, let reactor 1 be the high-substrate reactor, ay alia 2), and reactor 2 the low-substrate reactor, ct2 2). We vary ay 

These results lead to the conclusion that the USS of the array is more susceptible to the Hopf instability than to the Turing instability. If one reactor, i.e., half the system, contains no substrate, 02 = 1, then the Hopf bifurcation occurs first, no matter how large the substrate concentration oy in the other reactor. In fact, for a = 50.0 the USS of the inhomogeneous two-reactor array will undergo a transition to oscillations first as b is decreased as long as 02 5.41038. Further, the 

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The earliest studies of coupled chemical cells were theoretical investigations and focused on the diffusion-induced instability to nonuniform steady states [354, 443] in two coupled reactors. More recently, Epstein and Golubitsky investigated the Turing instability in linear and circular arrays of Brusselators [109]. Their approach makes use of underlying symmetry groups. 

For two coupled reactors, the Laplacian matrix of the graph Q reads... 

No Turing instability occurs in two coupled reactors, if k > 3/2. We illustrate our results for the specific case k = 1, where contributions from reaction and diffusion terms are evenly balanced. For a and d, we choose a = 50.0 and d = 1.07. The latter value is the ratio of the diffusion coefficients of the activator and inhibitor in aqueous solution. We use these values in all examples in this section. For these... 

Our results for two coupled reactors, the circular three-reactor array, and the four-reactor array with all-to-all coupling suggest that /Cjup for arrays with global coupling is given by /Cjup = 3/n, which turns out to be inde the case, see Theorem 13.13 and (13.59). 

Fig. 13.9 Critical profile, <72 vs aj, for two coupled reactors with Lengyel-Epstein kinetics for a = 50.0, d = 1.07. Reprinted with permission from [208]. Copyright 2004, American Chemical Society...

The behavior of the critical profile is qualitatively the same in both cases and qualitatively the same as for two coupled reactors. A comparison of Figs. 13.10 and... 

CO in the synthesis gas mixture for the methanol synthesis does not seem to take part directly in the reaction, but it does influence the process through two effects First the water-gas shift reaction and, secondly, through its effect on the surface morphology (and possibly also composition). For thermodynamic reasons, however, it would be desirable if CO could be hydrogenated directly via Eq (18) instead of going through two coupled equations (3) and (19), since it would yield a higher equilibrium concentration of methanol at the reactor exit. 

K.Y. Chang and W.L. Luyben. Design and control of coupled reactor/column systems- Part 3. A reactor/stripper with two colnmns and recycle. Comput. Chem. Eng., 21(l) 69-86, 1997. 

In FCC there are therefore four separate residence times, and the riser and regenerator are each described by two coupled mass-balance equations. The mass flow rates of catalyst between the reactors are the same since catalyst is recycled. 

In a first reactor, where benzoylformate decarboxylase (BFD) is retained, benz-aldehyde and acetaldehyde are coupled to yield (S)-hydroxy-l-phenylpropanone. This hydroxy ketone is then reduced to the corresponding diol in a second reactor by an alcohol dehydrogenase (ADH). Regeneration of the necessary cofactor is achieved by formate dehydrogenase (FDH) or by other methods. To avoid additional consumption of redox equivalents by unselective reduction of residual starting material from the first reactor, the volatile aldehydes are removed via an inline stripping module between the two membrane reactors. In this setup the diol was produced with high optical purity (ee, de > 90%) at an overall space-time yield of 32 g L d . ... 

Now, from eqs. (5.249) and (5.267), or (5.250) and (5.269), it is clear that the reactor model becomes a system of two coupled differential equations, which should be solved simultaneously. 

Reactor models At fust, we write the general model for the reactor consisting of two coupled differential equations. 

The control of the coupled reactor-regenerator is challenging because of the interaction between the two vessels and the neat operation in terms of energy. The temperatures in both vessels must be controlled at levels that are just below the metallurgical limits of the equipment materials. 

This paper describes previously developed models of the S-I/HyS cycle and a PBMR-268. A general coupling methodology via the IHX is developed, and applied to these models. Finally, two nuclear reactor driven transient scenarios are considered. 

Two coupled synthesis reactors turned out to be an efficient means to increase the yield of target products and the conversion degree of alkyl-and arylchlorides. 

Nalitham, R. V., Lee, J. M., Lamb, C. W., and Johnson, T. W., Two-Stage Coal Liquefaction Process Performance with Close-Coupled Reactors, Fuel Proc. Technol., 17, 13-27 (1987). 

We may compare results presented here with those obtained in two types of inductively coupled reactors [, 3]. One is the reactor we have used for many years [4], in which the portion of the reactor inserted into the r.f. coil is smaller than the main portion of the reactor, in which plasma polymer is collected. Monomer flux is directed into the main portion of the reactor, not through the r.f. coil. Electron bombardment of plasma polymer and substrate is reduced in this way [ ]. Active species are formed mainly under the r.f. coll and are transported by diffusion to the entire volume of the reactor. Interaction of these non-polymerizable energy carrying species (e.g. electrons, excited atoms) with the monomer entering the reactor leads to plasma polymerization [ ]. 

For the same reason, the synthesis of acrylic acid from propylene must be carried out in two separate reactors, one for the oxidation of propylene to acrolein and one for the oxidation of the aldehyde to acrylic acid. This is due to the fact that the requirements needed for the two steps make the two reactions incompatible. Acidity is needed in the second step, to favour the desorption of acrylic acid and save it from unselective consecutive reaction, while on the other hand, acidity is detrimental for the first reaction, because it favours the transformation of propylene to undesired products. Therefore, the development of a process for the one-step transformation of propane to acrylic acid will be possible when a catalyst is developed which possesses active sites able to perform quickly the complete transformation of adsorbed propane to the acrylic acid, the latter being the only product which finally desorbs into the gas phase. Accordingly, best performances in the oxidation of propane to acrylic acid have been reported to be obtained on heteropolyoxomolybdates (26), which are known to couple tuneable acid and redox properties. In this case, acid properties may facilitate the desorption of acrylic acid. 

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