Equivalence principle model

Tanaka, Chatani, and Tadokoro improved this model by refining the crystal structure of polyisobutene (182). The resulting structure is a 2/1 helix in which the structural unit contains four nonequivalent monomer units. In the crystal cell there are always eight monomer units arranged in three turns but the 8/3 helical symmetry is no longer retained. This example represents one of the most notable exceptions to the equivalence principle. Displacement from the exact helical conformation is small, however, and all the pairs of torsion angles fall inside the same energy well. 

First, and as already noted, the model universe is populated exclusively by primitive particles that possess solely the property of enumeration, and hence quantification. Consequently, all motions in the model universe are effectively gravitational, and we model this circumstance by constraining all such motions to satisfy the weak equivalence principle, by which we mean that the trajectory of a body is independent of its internal constitution. This constraint can be expressed as follows ... 

Transport phenomena modeling. This type of modeling is applicable when the process is well understood and quantification is possible using physical laws such as the heat, momentum, or diffusion transport equations or others. These cases can be analyzed with principles of transport phenomena and the laws governing the physicochemical changes of matter. Transport phenomena models apply to many cases of heat conduction or mass diffusion or to the flow of fluids under laminar flow conditions. Equivalent principles can be used for other problems, such as the mathematical theory of elasticity for the analysis of mechanical, thermal, or pressure stress and strain in beams, plates, or solids. 

The time-temperature equivalence principle makes it possible to predict the viscoelastic properties of an amorphous polymer at one temperature from measurements made at other temperatures. The major effect of a temperature increase is to increase the rates of the various modes of retarded conformational elastic response, that is, to reduce the retarding viscosity values in the spring-dashpot model. This appears as a shift of the creep function along the log t scale to shorter times. A secondary effect of increasing temperature is to increase the elastic moduli slightly because an equilibrium conformational modulus tends to be proportional to the absolute temperature (13). 

It is of a great significance to understand how the mechanical behaviours and properties of rock masses change with temperature, such as for nuclear waste repositories and deep mining at certain temperatures. The key to this problem is how to make predictions to long-term response of rocks based on mechanical models and test results within a short time of experiments. It is put forward in this paper that the problem can be resolved by means of time-temperature equivalent principle for rocks. 

The prediction of conductivity, a (electrical, ionic or proton), based on pure component values for miscible and phase separated systems can employ the same models as used for modulus and permeability, where a (S/cm) can be substituted for modulus (E) or permeability (P) in the log additivity relationship for miscible systems and the parallel model, series model or equivalent box model for phase separated systems. While these expressions have not been generally employed for conductivity modeling, the principles on which they are based are analogous to modulus and permeability values. 

Reaction kinetic models can be simulated not only by solving the kinetic system of differential equations but also via simulating the equivalent stochastic models. Computer codes are available that solve the stochastic kinetic equations. One of these is the Chemical Kinetics Simulator (CKS) program that was developed at IBM s Almaden Research Centre. It provides a rapid, interactive method for the accurate simulation of chemical reactions. CKS is a good tool for teaching the principles of stochastic reaction kinetics to students and trainees. 

In principle, such propositions resemble the bipolaron model, which presents the physicist s view of the electronic properties of doped conducting polymers 53-159) The model was originally constructed to characterize defects in solids. In chemical terminology, bipolarons are equivalent to diionic spinfree states of a system (S = 0)... 

Figure 8.42. ID structural models with inherent loss of long-range order, (a) Paracrystalline lattice after HOSEMANN. The lattice constants (white rods) are decorated by centered placement of crystalline domains (black rods), (b) Lattice model with left-justified decoration, (c) Stacking model with formal equivalence of both phases (no decoration principle)... 

This diffusive flow must be taken into account in the derivation of the material-balance or continuity equation in terms of A. The result is the axial dispersion or dispersed plug flow (DPF) model for nonideal flow. It is a single-parameter model, the parameter being DL or its equivalent as a dimensionless parameter. It was originally developed to describe relatively small departures from PF in pipes and packed beds, that is, for relatively small amounts of backmixing, but, in principle, can be used for any degree of backmixing. 

The examples of polymer crystal structures shown in the previous sections are ideal structures, which can be described with the traditional concepts of the principles of equivalence and close packing or the new concepts of symmetry breaking146 and frustration.154 The models of perfect crystals are characterized by a long-range positional order for all the atoms (disregarding thermal motion). The X-ray diffraction patterns of such crystals, oriented with the chain axes along one direction (as in oriented fibers), present sharp reflections organized in layer lines. 

In principle, the same rules hold true when zeolitic alkylation catalysts are used. A detailed study of the influence of PO and OSV on the performance of zeolite H-BEA in a backmix reactor was reported by de Jong et al. (80). The authors developed a simple model of the kinetics, which predicted catalyst lifetimes as a function of P/O and OSV. Catalyst lifetime (which is equivalent to the catalyst productivity, the reciprocal of acid consumption) increased with increasing P/O ratio and decreasing OSV. Furthermore, the authors persuasively demonstrated the superiority of a backmix reactor over a plug flow reactor. Qualitatively similar results were obtained by Taylor and Sherwood (222) employing a USY zeolite catalyst in a backmix reactor. The authors stressed the detrimental effect of unreacted alkene on the catalyst lifetime and product quality. Feller et al. (89) tested LaX zeolites in a backmix reactor and found the catalyst productivity to be nearly independent of the OSV within the examined OSV range. At higher values of OSV, the catalyst life was shorter, but in this shorter time the same total amount of product was produced. The P/O ratio had only a moderate influence on the catalyst performance. 

Although, as explained in Chapter 9, many optimization problems can be naturally formulated as mixed-integer programming problems, in this chapter we will consider only steady-state nonlinear programming problems in which the variables are continuous. In some cases it may be feasible to use binary variables (on-off) to include or exclude specific stream flows, alternative flowsheet topography, or different parameters. In the economic evaluation of processes, in design, or in control, usually only a few (5-50) variables are decision, or independent, variables amid a multitude of dependent variables (hundreds or thousands). The number of dependent variables in principle (but not necessarily in practice) is equivalent to the number of independent equality constraints plus the active inequality constraints in a process. The number of independent (decision) variables comprises the remaining set of variables whose values are unknown. Introduction into the model of a specification of the value of a variable, such as T = 400°C, is equivalent to the solution of an independent equation and reduces the total number of variables whose values are unknown by one. 

In principle, reactions which are subject to electrophilic catalysis by protons can be catalysed by metal ions also (e.g. Tee and Iyengar, 1988 Suh, 1992). However, metal ions may function in other ways, such as to deliver a hydroxide ion nucleophile to the reaction centre (e.g. Dugas, 1989 Chin, 1991), and it is often difficult to decide between kinetically equivalent mechanisms without resorting to extensive (and intensive) model studies. Use of the Kurz approach may help to resolve such ambiguities, as shown below. 

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