Empirical network equations

A second empirical observation related to the short-range forces of the ionic model is the observation that in many crystals the experimental bond valences also obey eqn (3.4) (Brown 9%lb)  

Equations (3.3) and (3.4) have become known respectively as the valence sum rule and the loop, or equal valence, rule, and are known collectively as the network equations. Equation (3.4) represents the condition that each atom distributes its valence equally among its bonds subject to the constraints of eqn (3.3) as shown in the appendix to Brown (1992a). The two network equations provide sufficient constraints to determine all the bond valences, given a knowledge of the bond graph and the valences of the atoms. The solutions of the network equations are called the theoretical bond valences and are designated by the lower case letter 5. Methods for solving the network equations are described in Appendix 3.  

In many compounds, the experimental bond valences, S, and the theoretical bond valences, s, are both found to be equal to the bond fluxes, 1 , within the limits of experimental uncertainty. This is an empirical observation that is not required by any theory. For this reason, and because there are occasions when the differences between them are significant and contain important information about the crystal chemistry, it is convenient to retain a different name for each of these three quantities to indicate the ways in which they have been determined. The bond flux is determined from the calculation of the Madelung field, the theoretical bond valence is calculated from the network equations (3.3) and (3.4), and the experimental bond valence is determined from the observed bond lengths using eqn (3.1) or (3.2). 

The differences in the two observed bond lengths for Ca-F 1 bonds can be attributed to steric effects discussed in Section 12.3.5. 

The network equations (3.3) and (3.4) invite comparison with the Kirchhoff equations (2.7) and (2.11). By choosing the ionic charge, in eqn (2.7) to be the same as the atomic valence, F,-, in eqn (3.3), and recognizing that in unstrained structures is equal to Sy, it follows from eqn (2.11) that loop ijl Cy = 0 which, when compared with eqn (3.4), means that the capacitances in eqn (2.11) must cancel, i.e. they must all be equal. This greatly simplifies the model, since it means that 

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The procedure of arriving at a probable mechanism via an empirical rate equation, as described in the previous section, is mainly useful for elucidation of (linear) pathways. If the reaction has a branched network of any degree of complexity, it becomes difficult or impossible to attribute observed reaction orders unambiguously to their real causes. While the rate equations of a postulated network must eventually be checked against experimental observations, a handier tool in the early stages of network elucidation are the yield-ratio equations (see Section 6.4.3). This approach relies on the fact that the rules for simple pathways also hold for simple linear segments between network nodes and end products. 

Molecula.rMecha.nics. Molecular mechanics (MM), or empirical force field methods (EFF), ate so called because they are a model based on equations from Newtonian mechanics. This model assumes that atoms are hard spheres attached by networks of springs, with discrete force constants. 

Recently, a new approach called artificial neural networks (ANNs) is assisting engineers and scientists in their assessment of fuzzy information, Polymer scientists often face a situation where the rules governing the particular system are unknown or difficult to use. It also frequently becomes an arduous task to develop functional forms/empirical equations to describe a phenomena. Most of these complexities can be overcome with an ANN approach because of its ability to build an internal model based solely on the exposure in a training environment. Fault tolerance of ANNs has been found to be very advantageous in physical property predictions of polymers. This chapter presents a few such cases where the authors have successfully implemented an ANN-based approach for purpose of empirical modeling. These are not exhaustive by any means. 

Even when the above complications are negligible or properly accounted for and when strain-induced crystallization is absent, the stress-strain curves for networks seldom conform to Eq. (7.3). The ratio //(a — 1/a2) generally decreases with elongation. An empirical extension of Eq. (7. IX the Mooney-Rivlin equation, has been used extensively to correlate experimental results ... 

An empirical equation which describes the interchain changes (AU)4V and (AS)4V on strain for the four networks has the form 24,85)... 

Theories based on these concepts all have to take into account the phenomenology of the stress-strain behaviour of networks. In unilateral extension as well as compression one observes, even at moderate extension (1.1 deviations from the Gaussian behaviour, which can be empirically described by the so-called Mooney-Rivlin equation ... 

Molecular Mechanics. Molecular mechanics (MM), or empirical force field methods (EFF), are so called because they are a model based on equations from Newtonian mechanics. This model assumes that atoms are hard spheres attached by networks of springs, with discrete force constants. The force constants in the equations are adjusted empirically to repro duce experimental observations. The net result is a model which relates the "mechanical" forces within a structure to its properties. Force fields are made up of sets of equations each of which represents an element of the decomposition of the total energy of a system (not a quantum mechanical eneigy, but a classical mechanical one). The sum of the components is called the force field eneigy, or steric energy, which also routinely includes the electrostatic eneigy components. Typically, the steric energy is expressed as... 

A linear decrease of KIc with an increase in crosslink density was reported for model PU based on triisocyanate and diols of various molar masses (Bos and Nusselder, 1994), and for epoxy networks (Lemay et al., 1984). It was suggested that the dilational stress field at the crack tip may induce an increase in free volume and a devitrification of the material. A linear relationship between GIc and M XJ2 was verified for these systems, although other empiric equations were found in other cases (Urbaczewski-Espuche et al., 1991). 

Process-scale models represent the behavior of reaction, separation and mass, heat, and momentum transfer at the process flowsheet level, or for a network of process flowsheets. Whether based on first-principles or empirical relations, the model equations for these systems typically consist of conservation laws (based on mass, heat, and momentum), physical and chemical equilibrium among species and phases, and additional constitutive equations that describe the rates of chemical transformation or transport of mass and energy. These process models are often represented by a collection of individual unit models (the so-called unit operations) that usually correspond to major pieces of process equipment, which, in turn, are captured by device-level models. These unit models are assembled within a process flowsheet that describes the interaction of equipment either for steady state or dynamic behavior. As a result, models can be described by algebraic or differential equations. As illustrated in Figure 3 for a PEFC-base power plant, steady-state process flowsheets are usually described by lumped parameter models described by algebraic equations. Similarly, dynamic process flowsheets are described by lumped parameter models comprising differential-algebraic equations. Models that deal with spatially distributed models are frequently considered at the device... 

Interrelations between the simultaneously solved problems of hydro-dynamic (calculation of x, ) and technico-economic (choice of diameters d,) optimization of the network are revealed by taking as initial the empirical Darcy-Weisbach equation... 

The bond flux can also be calculated by recognizing that in the ionic limit each bond is an electric capacitor (represented by flux linking two opposite charges). The network of atoms and bonds is thus a capacitive electrical circuit, and since in most equilibrium structures aU the bond capacitances are empirically found to be equal, the fluxes can be calculated using the Kirchhoff equations (2) and (3) in which the bond capacitances cancel. ... 

One-plus rate equations play a key role in network elucidation. Perhaps the most difficult step in that endeavor is the translation of a mathematical description of experimental results into a correct network of elementary reaction steps. The observed behavior can usually be fitted quite well by a traditional power law with empirical, fractional exponents, at least within a limited range of conditions. This has indeed been standard procedure in times past. However, such equations are highly unlikely to result from a combinations of elementary steps. Their acceptance may be expedient, but as far as network elucidation is concerned they are a dead... 

As discussed in the previous chapter, the network of the inter-connected membrane pores formed during preparation and fabrication may be tortuous or nearly straight depending on the synthesis and subsequent heat treatment methods and conditions. The microstructure of a membrane, particularly the type with tortuous pores, is too complicated to be described by a single parameter or a simple model. Due to the relatively poor knowledge of flow through porous media, an empirical term called " tortuosity" has been introduced and used by many researchers to reflect the relative random orientation of a pore network and is based on the Kozeny-Carmen equation for the membrane flux, J ... 

These authors consider that a system of good kinetic equations for the biomass-char gasification reactions under the true gasifier atmosphere do not exist yet. This conclusion can be discussed and even criticised, of course. The authors are open to a discussion and experimental checking with the person(s) who considers that holds a system of kinetic equations to correctly describe the network of reactions 4a-4d. These authors do not know such system of kinetic equations and have again to look for an empirical and easy-to-use approach. 

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