Four Coupled Reactors

As shown in Sect. 13.3.2.3, see Table 13.1, two Turing instabilities can occur in a linear four-reactor array with c = 1, namely 

The behavior of the critical substrate concentration is again qualitatively the same in all four cases and qualitatively the same as for two- and three-reactor arrays. Therefore we show only the two cases most favorable to Turing patterns. Figs. 13.13 and 13.14, which as expected are case 2 and case 4, where the low-substrate reactors are located in the interior of the array. 

As for a linear array of three reactors, we find in case 2 that for a small enough and a large enough, the Turing bifurcation always occurs first, even if the reactor 

3 contains no substrate. We define, as for three coupled reactors, the critical value a by the condition that o 3 c(ct, a ) = 1. The behavior of a as function of a is shown in Fig. 13.15. For a 17.7, two curves of Hopf bifurcation points approach each other, merge and vanish as a is decreased. This results in a jump of c from a value larger than 1 to a value smaller than 1, and the definition of a is no longer applicable. 

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We systematically survey the Turing threshold conditions and the structural mode that becomes unstable, for arrays of two, three, and four coupled reactors with Lengyel-Epstein kinetics. The results illustrate the importance of the value of the coupling strength on the occurrence of a Turing instability. They also provide the... 

The six different network topologies that occur for arrays of four coupled reactors are shown in Fig. 13.2. The Laplacian matrix L of each network can be read off from Fig. 13.2 and is given below. We also list the structural modes of L and the corresponding eigenvalues. 

Fig. 13.2 Topologies for four coupled reactors (i) linear (it) circular (iii) star-shaped (iv) triangle-plus-one (v) circular-plus-diagonal (vi) global or aU-to-all coupling. Reprinted from [207]. Copyright 2004, with permission from Elsevier...

Fig. 13.13 Critical profile, (73 as a function of a for four coupled reactors with Lengyel-Epstein kinetics for case 2 uj = (73 = <74 = <7 u = 50.0, d = 1.07. Reprinted with permission firom [208]. Copyright 2004, American Chemical Society...

Since the cooling jacket has cocurrent flow, the model consists of the set of four coupled initial value differential equations (7.5) to (7.8). Note that the first three DEs (7.5) to (7.7) contain the variable catalyst effectiveness factor rj. Thus there are other equations to be solved at each point along the length 0 < / < Lt of the reactor tube, namely the equations for the catalyst pellet s effectiveness factor rj. 

Based on the kinetic parameters of the coke bum-off and the differential mass and heat balances for the gas and solid phase the regeneration process in an industrial fixed bed reactor was modeled. Thereby the four coupled diffential equations (eq. 6-9) were solved by the mathematical program PDEXPACK, developed at the Institute of Chemical Engineering, in Stuttgart (Germany). 

This novel design shategy to simulate and control thermal runaway in a doublepipe reactor requires the simultaneous solution of four coupled first-order ODEs to describe conversion and temperature profiles within the inner pipe and in the annular region. Mass and thermal energy balances for exothermic reaction within the inner pipe are exactly the same as those discussed above (see equations 4-62 and 4-63). Hence, for one exothermic reaction (i.e., A products) in the inner pipe. 

From a mathematical point of view the SBR is the most complex of the four ideal reactors because of its unsteady operation over the whole reaction period and its changing reactor volume due to the dosage. Therefore it is more convenient to use mole numbers for the description instead of concentrations. In addition it has to be observed that the actual number of moles of the fed component present in the reactor cannot be calculated directly by the stoichiometric coupling introduced but by an extended version only which accounts for conversion and time. The added reactant and the initially charged reactant will be indexed A and B, respectively, in the following text. If the conversion is known the number of moles of B present at any time can be calculated according to ... 

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). 

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. 

The codes in each vessel are read before the first coupling, and the vessels are sorted into four flasks according to the first monomer jxisition (the protected monomer A is coupled in the first flask, protected D in the last, step a). After the first couplings (steps b-e), the vessels are mixed and deprotected in a single reactor (steps f and g) then they are sorted according to the second monomer jxisition (monomers E-H, step h)... 

These basic rate models were Incorporated Into a differential mass balance In a tubular, plug-flow reaction. This gives a set of coupled, non-llnear differential equations which, when Integrated, will provide a simulation model. This model corresponds to the Integral reactor data provided by experimentation. A material balance Is written for each of the four components In our system ... 

Liquid Phase. For liquid-phase reactions in which there is no volume change, concentration is the preferred variable. The mole balances are shown in Table 4-5 in terms of concentration for the four reactor types we have been discussing. We see from Table 4-5 that we have only to specify the parameter values for the system (CAo,Uo,etc.) and for the rate law (i.e., ifcyv. .3) to solve the coupled ordiaaiy differential equations for either PFR, PBR, or batch reactors or to solve the coupled algebraic equations for a CSTR. 

We used the same methodology to evaluate the three other cycles identified as promising. Of the four examined, the Cu-Cl had higher Level 1 and Level 2 efficiencies, 48 and 42.4%, respectively, than the others. In addition, proof-of-concept experiments have been completed for all reactions. It is therefore one of tlie more promising candidates for further development. Both cycles can be coupled with the very high temperature gas reactor. 

Figure 2 Suzuki coupling in a semi-continuous, fixed-bed reactor. Four cycles of reaction with 4-bromobenzotrifluoride at 50 °C. 1) fresh catalyst (initial rate, 0.22 mol/h) 2) water-washed catalyst, reagents replenished (initial rate, 0.017mol/h) 3) water-washed catalyst, fresh reaction solution (initial rate, 0.012 mol/h) 4) water-washed catalyst, hydrogen reduction, fresh reaction solution (initial rate, 0.01 mol/h).

Fig. 13.17 Stationary pattern for the four graphs from Fig. 13.16 with inhibitory coupling and Lengyel-Epstein kinetics. The stationary value of the activator concentration is plotted vs the reactor number. (The inhibitor concentration shows quahtatively the same pattern.) Parameter values a = 10.0, Q = 0.1, a = 1.0, b = 5.0, and k = 23.56824264/ j8jo. Reprinted from [209]. Copyright 2006, with permission from Elsevier...

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