Kinetic equation closure

In most cases, closure of the terms in the kinetic equation will require prior knowledge of how the mesoscale variables are influenced by the underlying physics. Taking the fluid-drag term as an example, the simplest model has the form... 

Owing to the presence of the pair correlation function, the collision model in Eq. (6.2) is unclosed. Thus, in order to close the kinetic equation (Eq. 6.1), we must provide a closure for written in terms of /. The simplest closure is the Boltzmann Stofizahlansatz (Boltzmann, 1872) ... 

Composition of explosion prbducts changes during the course of their expansion. The chemical equilibrium hypothesis allows closure of the gasdy-namic conservation equations and calculations of the product composition without invoking the chemical kinetics equations. The rate of the change in concentrations of the reaction products depends on instantaneous values of the thermodynamic parameters of a flowing gas, its finiteness implies a certain deviation of the product composition and, consequently, of the heat released in the gas from their equilibrium counterparts. The chemical equilibrium model may be applied only in the case when this deviation is small. This defines the suflS.cient condition for model applicability to explosion products. 

This closure property is also inherent to a set of differential equations for arbitrary sequences Uk in macromolecules of linear copolymers as well as for analogous fragments in branched polymers. Hence, in principle, the kinetic method enables the determination of statistical characteristics of the chemical structure of noncyclic polymers, provided the Flory principle holds for all the chemical reactions involved in their synthesis. It is essential here that the Flory principle is meant not in its original version but in the extended one [2]. Hence under mathematical modeling the employment of the kinetic models of macro-molecular reactions where the violation of ideality is connected only with the short-range effects will not create new fundamental problems as compared with ideal models. 

Furthermore, the closures for the fluid—particle drag and the particle-phase stresses that we discussed were all derived from data or analysis of nearly homogeneous systems. In what follows, we refer to the TFM equations with closures deduced from nearly homogeneous systems as the microscopic TFM equations. The kinetic theory based model equations fall in this category. 

The source terms on the right-hand sides of Eqs. (25)-(29) are defined as follows. In the momentum balance, g represents gravity and p is the modified pressure. The latter is found by forcing the mean velocity field to be solenoidal (V (U) = 0). In the turbulent-kinetic-energy equation (Eq. 26), Pk is the source term due to mean shear and the final term is dissipation. In the dissipation equation (Eq. 27), the source terms are closures developed on the basis of the form of the turbulent energy spectrum (Pope, 2000). Finally, the source terms... 

The kinetic resolution by etherification has also been conducted through the cyclization of epoxy aliphatic alcohols.274 In these reactions catalyzed by monomeric complex 51, the ring closure of acyclic substrates occurred with exclusive / -selectivity (Equation (74)), whereas m -openings were observed in the desymmetrization of... 

A very promising synthesis of /3-lactones has been recently reported, involving the palladium-catalyzed carbonylation reaction of halogeno alcohols. For example, 3-phenyl-2-oxetanone was obtained in 63% yield from 2-phenyl-2-bromoethanol in DMF solution at room temperature under 1 atmosphere pressure of carbon monoxide (equation 115). A proposed mechanism, in which palladium metal inserts into the carbon-halogen bond, followed by insertion of a molecule of carbon monoxide into the carbon-palladium bond and then ring closure, fits kinetics data (80JA4193). 

The attachment to the metal center of the second or third donor atom of the ligand is usually very rapid, so in most cases departure of the first water molecule is rate determining. The rate-determining step is the closure of a chelate ring in complex formation reactions where a steric property of the ligand governs the kinetics of substitution by the second donor atom.241,242 In cases in which Equation 7.21 is applicable, the volume of activation is given by... 

So far, we are able to construct the constitutive equations for qc, Pc, and y. For moderate solids concentrations, we can neglect the kinetic contributions in comparison to the collisional ones. Thus, we can assume P Pc and qk qc. Substituting the constitutive relations into Eqs. (5.274), (5.275), and (5.281), after neglecting the kinetic contributions, yields five equations for the five unknowns ap, Up, and Tc (or ( 2)). Hence, the closure problem is resolved. 

When the statistical moments of the distribution of macromolecules in size and composition (SC distribution) are supposed to be found rather than the distribution itself, the problem is substantially simplified. The fact is that for the processes of synthesis of polymers describable by the ideal kinetic model, the set of the statistical moments is always closed. The same closure property is peculiar to a set of differential equations for the probability of arbitrary sequences t//j in linear copolymers and analogous fragments in branched polymers. Therefore, the kinetic method permits finding any statistical characteristics of loopless polymers, provided the Flory principle works for all chemical reactions of their synthesis. This assertion rests on the fact that linear and branched polymers being formed under the applicability of the ideal kinetic model are Markovian and Gordonian polymers, respectively. 

Closure of such differential equations requires the definitions of both constitutive relations for hydrodynamical functions and also kinetic relations for the chemistry. These functions are specified by recourse both to theoretical considerations and to rheological measurements of fluidization. We introduce the ideal gas approximation to specify the gas phase pressure and a caloric equation-of-state to relate the gas phase internal energy to both the temperature and the gas phase composition. It is assumed that the gas and solid phases are in local thermodynamic equilibrium so that they have the same local temperature. 

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