A General Formulation

The derivation of differential mass balances or continuity equations for the components of an element of fluid flowing in a reactor is considered in detail in texts on transport processes, for example, by Bird et al. [I960 2007]. These authors showed that a fairly general form of the continuity equation for a chemical species j reacting in a flowing fluid with varying density, temperature, and composition is 

The terms and symbols used in this equation have the following meaning Cj is the molar concentration of species j (kmol/m fluid), so that dCJ dt s the nonsteady-state term expressing accumulation or depletion V is the nabla , or del , operator. In a rectangular coordinate system, x, y, zwith unit vectors 8 , 8, and 8z, the gradient of a scalar function /is represented by V/ and the divergence of a vector function v by V-v. More explicitly, 

u is the three-dimensional mass-average velocity vector in (m/s) and defined by 

The Jj term is the molar flux vector for speciesy with respect to the mass-average velocity (kmol/m s). When the flow is laminar or perfectly ordered, the term V-Jj results from molecular diffusion only. It can be written more explicitly as an extension, already encountered in Chapter 3, of Pick s law for diffusion in binary systems, as 

Rj is the total rate of change of amount of j because of reaction, as defined in Chapter 1, that is, a/ for a single reaction and 2m for multiple reactions. The ay are negative for reactants and positive for reaction products. The units of Rj depend on the nature of the reaction. If the reaction is homogeneous, the units could be kmol/m s but for a reaction catalyzed by a solid, preference would be given to kmol/kg cat. s, multiplied by the catalyst bulk density in the reactor. 

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Ahn C B and Cho Z H 1989 A generalized formulation of diffusion effects in pm resolution nuclear magnetic-resonance imaging Med. Rhys. 16 22-8... 

Ciccotti G, Ferrario M and Ryckaert J-P 1982 Molecular dynamics of rigid systems in cartesian coordinates. A general formulation Mol. Phys. 47 1253-64... 

Thus in a mixed system, as e.g. in a stirred tank, the rate of agglomeration additionally depends on the shear field and therefore on the energy dissipation e in the vessel. Furthermore, in precipitation systems solution supersaturation plays an important role, as the higher the supersaturation, the stickier the particles and the easier they agglomerate (Mullin, 2001). This leads to a general formulation of the agglomeration rate... 

For a general formulation of mean bubble sizes see Eqs. (261M264). 

In this section, a general formulation will be given for the effect of bubble residence-time and bubble-size distributions on simultaneous and thermodynamically coupled heat- and mass-transfer in a multicomponent gas-liquid dispersion consisting of a large number of spherical bubbles. Here one can... 

Before proposing a general formulation, we illustrate a DIFF with a simple case, in which we consider bone collagen as the body component, and the diet as containingjust two components, protein and non-protein. In the notation, B stands for body 5 values, and D for diet 8 values suffixes distinguish the particular component specified. Thus Bcolla stands for the collagen 8 C, Dp and Dn for the dietary protein and non-protein 6 C values. 

In the same way as described above, we can formulate the multidimensional theory without relying on the complex-valued Lagrange manifold that constitutes one of the main obstacles of the conventional multidimensional WKB theory [62,63,77,78]. Another crucial point is that the theory should not depend on any local coordinates, which gives a cumbersome problem in practical applications. Below, a general formulation is described, which is free from these difficuluties and applicable to vertually any multidimensional systems [30]. 

The stochastic theory of lineshape has been developed by Anderson and Weiss [157], by Kubo [158], and by Kubo and Tomita [159] in order to treat the narrowing of spectral lines by exchange or motion, a generalized formulation having been subsequently presented by Blume [31]. We consider below an application of the theory of Blume to the specific problem of relaxation between LS and HS states in Mossbauer spectra of powder materials which is based on the formulation by Blume and Tjon [32, 33], Accordingly, the probability of emission of a photon of wave vector Ik and frequency m is given as [160] ... 

The optimal operation of mixed production facilities - A general formulation and some approaches for the solution. Comput. Chem. Eng., 20, 897-904. 

Thermal electron transfer from charge-transfer complexes as a general formulation 233... 

THERMAL ELECTRON TRANSFER FROM CHARGE-TRANSFER COMPLEXES AS A GENERAL FORMULATION... 

In the present context, the way to ensure extensivity is to reformulate the CSE so that the RDMCs and not the RDMs are the basic variables. One can always recover the RDMs from the cumulants, but only the cumulants satisfy connected equations that do not admit the possibility of mixing noninteracting subsystems. Connected equations are derived in Section V. Before introducing that material, we first provide a general formulation of the p-RDMC for arbitrary p. 

For a general formulation of the Zintl-Klemm concept, consider an intermetallic AmX phase, where A is the more electropositive element, t3 pically an alkali or an alkaline earth metal. Both A and X, viewed as individual atoms, are assumed to follow the octet rule leading to transfer of electrons from A to X, i.e., A AF, X —> X , so that mp = nq. The anionic unit X arising from this electron transfer is considered to be a pseudoatom, which exhibits a structural chemistry closely related to that of the isoelectronic elements [11]. Since bonding also is possible in the cationic units, the numbers of electrons involved in A-A and X-X bonds of various types (caa and exx> respectively) as well as the number of electrons e not involved in localized bonds can be generated from the numbers of valence electrons on A and X, namely and ex, respectively, by the following equations of balance ... 

These data suggest a general formulation FeliyFe2+[Fe iyFe ,+yCr ]04... 

A general formulation of the equilibrium, involving a monodendate, inner-sphere surface complex, and of the associated constants, is as follows ... 

This method is simple enough to make a general formulation in all orders of a possible.5 0 The reason why I called it a bookkeeping device is that the expansion is formally correct for any value of octc. Yet the condition aic 1 is crucial, because the result has no real meaning as long as it depends on... 

A general formulation of the ATP-dependent carboxylation of an a carbon by 180-enriched bicarbonate is... 

We now write a general formulation for our set of hybrid orbitals. Each hybrid, Xt> is going to be written as a sum of contributions from the s, px, py, pz atomic orbitals, each with a coefficient that tells the extent of its contribution. 

A general formulation of the problem of solid-liquid phase equilibrium in quaternary systems was presented and required the evaluation of two thermodynamic quantities, By and Ty. Four methods for calculating Gy from experimental data were suggested. With these methods, reliable values of Gy for most compound semiconductors could be determined. The term Ty involves the deviation of the liquid solution from ideal behavior relative to that in the solid. This term is less important than the individual activity coefficients because of a partial cancellation of the composition and temperature dependence of the individual activity coefficients. The thermodynamic data base available for liquid mixtures is far more extensive than that for solid solutions. Future work aimed at measurement of solid-mixture properties would be helpful in identifying miscibility limits and their relation to LPE as a problem of constrained equilibrium. 

A third similar question arises in the potential representation of fields E and B. Conventionally, a magnetic scalar potential is not included as part of the solution of Maxwell s equations. However, there is no a priori prohibition for the existence of such solution within a general formulation. Sections III. A—III.E consider previous issues in some detail. 

Nonequilibrium equation of motion (NEOM) method is the simplest approximate approach. In spite of its simplicity, it is very useful in many cases, and is very convenient for numerical implementation. In this section we consider only a general formulation, some particular examples are considered further. 

All types of time evolution are present in dynamical solvation effects. It is difficult, and perhaps not convenient, to define a general formulation of the Hamiltonian which can be used to treat all the possible cases. It is better to treat separately more homogeneous families of phenomena. The usual classification into three main types adiabatic, impulsive and oscillatory, may be of some help. The time dependence of the phenomenon may remain in the solute, and this will be the main case in our survey, but also in the solvent in both cases the coupling will oblige us to consider the dynamic behaviour of the whole system. We shall limit ourselves here to a selection of phenomena which will be considered in the following contributions for which extensions of the basic equilibrium QM approach are used, mainly phenomena related to spectroscopy. Other phenomena will be considered in the next section. 

In general, the selection of a suitable salt10 or co-crystal11 is mainly driven by biophar-maceutical and stability considerations. Where several modifications exist, the selection of the most suitable for formulation purposes is a general formulation issue as well.12... 

A singular point of reactive membrane separation should be denoted as kinetic arheotrope because it is a process phenomenon rather than a thermodynamic phenomenon. The condition for arheotropy can be elegantly expressed in terms of new transformed variables, which are a generalized formulation of the transformed composition variables first introduced to analyze reactive azeotropes. 

A general formulation of the several possible steps in the direct electroreduction of a carbon-halogen bond is shown in Scheme 1. The Scheme differs little from that introduced... 

Sections II-E,4 and 5 contain a general formulation of the isotope effect for any experimental energy distribution function. The distribution may be one typical of a conventional thermal system, or it may be a special function, characteristic of the particular activation technique. The simplest of the latter type, and one which allows the clearest demonstration of the nature of the energy dependence of the kinetic isotope effect, is a monoenergetic excitation function or 5-function. Then the intermolecular isotope effect is described by eq. (10), where the subscripts H and D stand for any two isotopically labeled species, and hydrogen and deuterium are the important cases,... 

A general formulation for the symmetric and antisymmetric coordinate wave function is... 

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