Paired-interaction closure

The paired-interaction closure (Patterson, 1975, 1985) is one of the simplest closures and depends on a spiked PDF shown in Figure 13-27, which represents the probabilities of zero, maximum, and mean concentrations for each chemical component. The paired-interaction closures for cick for the reaction terms in both equations and for sicj in the segregation equation are as follows ... 

Figure 13-30 Normalized concentration downstream of the feed jet array in the Toor tubular reactor for L = 12 400 L/mol s reactant feed ratio of 1.26 and an average velocity of 0.75 m/s. The experimental values of Vassilatos and Toor (1965) are compared to simulation values using paired-interaction closure. The reaction was a single second-order...

Figure 13-31 Comparison of the Baldyga et al. data for mixed reaction in a static mixer with results of paired-interaction closure for the reaction A - - B p-R - - o-R p-R + o-R -h B -> S AA -I- B Q. (See Baldyga et al., 1997, for details.)...

Figure 13-32 Yield of R in the reaction A- -B R R- -B S from experimental data of Middleton et al. (1986) and from their simulations, which assume no local mixing rate effect. Simulations using paired-interaction closure agree with the Middleton et al. simulations, showing that the controlling mixing rate is not micromixing.

Paired-Interaction Closure for Multiple Chemical Reactions... 

Paired-interaction closure [eqs. (13-29) and (13-30)] may be used in three dimensional simulations of turbulent mixed reactors with multiple simultaneous chemical reactions. The assumptions of the closure are that (1) only one concentration probability need be used to capture the most important aspects of the interaction of mixing and chemical reaction, and (2) that the reactants need only be considered in pairs and that higher-order interactions of the reactants are not important. These assumptions give a closure that is very fast to compute and results very close to experimental results (see Zipp and Patterson, 1998 Randick, 2000). 

In this example eqs. (13-27) to (13-30) (the paired-interaction closure) were incorporated into a subroutine called by Fluent to compute the rates of segregation growth and decay and the rates of the chemical reactions. The subroutine, called Pairin, is available from author Patterson. Pairin may easily be adapted to other fluid dynamics simulators if desired. Pairin may also be used as an example for development of new subroutines using more or less sophisticated closures. In addition, the subroutine developed by Baldyga and co-workers (see Sections 13-5.2 and 13-5.8) for use of the P-PDF may be used instead of the Pairin subroutine. 

In the numerical solution the matrix structure is evaluated from Eqs. (44)-(46). Then Eqs. (47)-(49) with corresponding closure approximations are solved. Details of the solution have been presented in Refs. 32 and 33. Briefly, the numerical algorithm uses an expansion of the two-particle functions into a Fourier-Bessel series. The three-fold integrations are then reduced to sums of one-dimensional integrations. In the case of hard-sphere potentials, the BGY equation contains the delta function due to the derivative of the pair interactions. Therefore, the integrals in Eqs. (48) and (49) are onefold and contain the contact values of the functions... 

It is perhaps surprising that it is possible to solve the SSOZ equation for a number of simple molecules. For diatomic symmetry molecules with hard sphere pair interactions, the SSOZ equation with PY closure has been solved analytically using a Weiner-Hopf technique introduced by Baxter. We consider a diatomic molecule consisting of two fused hard spheres of diameter a, with their centres a distance L apart. For a fluid composed of these molecules, each of the four correlation functions is the same by symmetry, and the SSOZ equation reduces to a scalar equation... 

Spherical clusters and nuclei, as discussed in the above subsections, exist mainly for fermionic shell closures. The strong pairing interaction in nuclei restores spherical... 

V (the potential) is identified with the enthalpy, i.e. the number n of base pairings (contacts), and T corresponds to the entropy. At each stage in the folding process, as many as possible new favourable intramolecular interactions are fonned, while minimizing the loss of confonnational freedom (the principle of sequential minimization of entropy loss, SMEL). The entropy loss associated with loop closure is (and the rate of loop closure exp... 

We should mention here one of the important limitations of the singlet level theory, regardless of the closure applied. This approach may not be used when the interaction potential between a pair of fluid molecules depends on their location with respect to the surface. Several experiments and theoretical studies have pointed out the importance of surface-mediated [1,87] three-body forces between fluid particles for fluid properties at a solid surface. It is known that the depth of the van der Waals potential is significantly lower for a pair of particles located in the first adsorbed layer. In... 

In all the cases considered, stabilization of the cis isomers of semidiones is observed when the cation is present. As a rule, for the nonchelated semidione the trans isomer is more stable than the cis isomer. However, the interaction with cations produces the opposite effect, and the cis isomer appears to be more stable than the trans form. Carbonyl compounds capture an electron and are converted into ketyls, which contain a negatively charged oxygen atom. This atom is a particularly powerful proton acceptor. Therefore, a proton, which steps forward as a cation, can provide the contour closure for ion-pair formation. 

To explain the enantioselectivity obtained with semi-stabilized ylides (e.g., benzyl-substituted ylides), the same factors as for the epoxidation reactions discussed earlier should be considered (see Section 10.2.1.10). The enantioselectivity is controlled in the initial, non-reversible, betaine formation step. As before, controlling which lone pair reacts with the metallocarbene and which conformer of the ylide forms are the first two requirements. The transition state for antibetaine formation arises via a head-on or cisoid approach and, as in epoxidation, face selectivity is well controlled. The syn-betaine is predicted to be formed via a head-to-tail or transoid approach in which Coulombic interactions play no part. Enantioselectivity in cis-aziridine formation was more varied. Formation of the minor enantiomer in both cases is attributed to a lack of complete control of the conformation of the ylide rather than to poor facial control for imine approach. For stabilized ylides (e.g., ester-stabilized ylides), the enantioselectivity is controlled in the ring-closure step and moderate enantioselectivities have been achieved thus far. Due to differences in the stereocontrolling step for different types of ylides, it is likely that different sulfides will need to be designed to achieve high stereocontrol for the different types of ylides. 

Together with an appropriate closure for the pair and triplet distribution functions, one may restrict consideration in this limit to the first two equations in the hierarchy (23). Again, in this approximation, one may follow a course of the Debye-Hiickel approach to obtain the mean field potential, while image forces are accounted for. In this way, the distribution of ions in the system will be known, and interaction forces can be calculated on the basis of this distribution. 

Interactions of lone pairs with empty orbitals can control conformation Ring-closing reactions why five-membered rings form quickly and four-membered rings form slowly Baldwin s rules why some ring closures work well while others don t work at all... 

In copyrolytic reactions of the aminosilylenes with unsaturated ketones or imines (heterodienes) we mainly obtained isomeric mixtures. The chemo- and regioselectivity of main- and byproducts can be explained with multistep-cycloadditions. We assume a primary Lewis acid-base interaction between the lone electron pair of the heteroatom (oxygen or nitrogen) and the electron gap at silylene, which is followed by a [2+l]-cycloaddition and a radical ring-opening ring-closure reaction. 

A-Benzyloxy-)8-lactam 70 in the presence of Rh2(AcO)4 was found to undergo ring closure to provide the carbapenam 71. The initially generated carbenoid may first interact with a nitrogen electron lone pair to give ylide 72. Proton abstraction from the benzylic position by this ylide, followed by N — O bond heterolysis, yields the cyclized product and benzaldehyde (90TL1807 91JOC2688). 

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