Critical simplification

Thus the Rayleigh problem is reduced to the solution of the ODE, (3-126), subject to the two boundary conditions, (3-127) and (3-128). When a similarity transformation works, this reduction from a PDE to an ODE is the typical outcome. Although this is a definite simplification in the present problem, the original PDE was already linear, and the existence of a similarity transformation is not essential to its solution. When similarity transformations exist for more complicated, nonlinear PDEs, however, the reduction to an ODE is often a critical simplification in the solution process. 

Finally, the focus of Chapter 11 is on advanced methods of theoretical analysis, especially computer algebra. Most of the results presented in this chapter, that is, critical simplification, the analytical criteria for distinguishing nonlinear from linear behavior, are original. 

Principle of critical simplification. In accordance with this principle (Yablonsky et al., 2003), the behavior near critical points, for instance ignition or extinction points in catalytic combustion reactions, is governed by the kinetic parameters of only one reaction—adsorption for ignition and desorption for extinction— which is not necessarily the rate-limiting one. 

The main result is that real multiplicity of steady states—the existence of two internal stable steady states and one internal unstable steady state—can be explained using one of the four mechanisms shown in Table 7.3. If the experimentally observed steady-state reaction rate is characterized by two different branches, and multiplicity of steady states is observed, one of these four mechanisms can be used to interpret the data. We prefer to use mechanism B because its steps are characterized by overall reaction orders that are not larger than two. In fact, this mechanism is identical to the mechanism represented by Eq. (7.102), an example of which is the adsorption mechanism for the oxidation of carbon monoxide (Eq. (7.103)). We will return to this mechanism in Chapter 11, in which the problem of critical simplification is discussed. 

The principle of critical simplification was first explained by Yablonsky et al. (Yablonskii and Lazman, 1996 Yablonsky et al., 2003) using the catalytic oxidation reaction as an example. The authors presented a dramatic simplification of the kinetic model for this reaction at critical conditions relating to bifurcation points. In this section, results obtained by GoTdshtein et al. (2015) are also used. 

If Bi = 0 and B2 = 0, the situation is more complex and the standard asymptotic analysis is not applicable at these conditions. Actually, this is a more interesting case because this bifurcation point is the point of maximum bifurcational complexity (MBC). At the same time, it is the point where critical simplification is observed and the following relationship between the kinetic parameters is obtained ... 

This equation is an example of critical simplification. At the bifurcation point, the kinetic parameters k po2 and k pco are interdependent. Knowing, for example, the partial pressures of carbon monoxide and oxygen at the bifurcation point and the adsorption coefficient of carbon monoxide (k ), we can determine the adsorption coefficient of oxygen (k ). 

There are zero, one, or two steady states on this branch, depending on the sign of B2. If B2 < 0, there are no steady states if B2 = 0, there is one steady state and if B2 > 0, there are two steady states with x = 0. Thus, at the bifurcation point B2 = 0, a functional relation exists between the parameters k po2,k2Pco and k. This is another critical simplification. A change of sign of B2 from positive to negative corresponds to ignition. 

The example of critical simplification analyzed in this chapter may serve as a good subject for some generalization. Let us assume that in our system there are three independent variables (chemical concentrations), one mass conservation balance, and one infinitely fast reaction between two surface species. As usual, our system is governed by the mass-action law and there is no question of autocatalysis. In this case, the steady-state value of one variable will be zero (or rather, negligible) and the dynamics of the system will be two-dimensional. Consequently, critical simplification will be observed for this system as well. 

Yablonskh, G.S., Lazman, M.Z., 1996. New correlations to analyze isothermal critical phenomena in heterogeneous catalysis reactions ( Critical simplification , hysteresis thermodynamics ). React. Kinet. Catal. Lett. 

With these simplifications, and with various values of the as and bs, van Laar (1906-1910) calculated a wide variety of phase diagrams, detennining critical lines, some of which passed continuously from liquid-liquid critical points to liquid-gas critical points. Unfortunately, he could only solve the difficult coupled equations by hand and he restricted his calculations to the geometric mean assumption for a to equation (A2.5.10)). For a variety of reasons, partly due to the eclipse of the van der Waals equation, this extensive work was largely ignored for decades. 

The calculations that have been carried out [56] indicate that the approximations discussed above lead to very good thermodynamic functions overall and a remarkably accurate critical point and coexistence curve. The critical density and temperature predicted by the theory agree with the simulation results to about 0.6%. Of course, dealing with the Yukawa potential allows certain analytical simplifications in implementing this approach. However, a similar approach can be applied to other similar potentials that consist of a hard core with an attractive tail. It should also be pointed out that the idea of using the requirement of self-consistency to yield a closed theory is pertinent not only to the realm of simple fluids, but also has proved to be a powerful tool in the study of a system of spins with continuous symmetry [57,58] and of a site-diluted or random-field Ising model [59,60]. 

This procedure constitutes an application of the steady-state approximation [also called the quasi-steady-state approximation, the Bodenstein approximation, or the stationary-state hypothesis]. It is a powerful method for the simplification of complicated rate equations, but because it is an approximation, it is not always valid. Sometimes the inapplicability of the steady-state approximation is easily detected for example, Eq. (3-143) predicts simple first-order behavior, and significant deviation from this behavior is evidence that the approximation cannot be applied. In more complex systems the validity of the steady-state approximation may be difficult to assess. Because it is an approximation in wide use, much critical attention has been directed to the steady-state hypothesis. 

Applying MD to systems of biochemical interest, such as proteins or DNA in solution, one has to deal with several thousands of atoms. Models for systems with long spatial correlations, such as liquid crystals, micelles, or any system near a phase transition or critical point, also must involve a large number of atoms. Some of these systems, including synthetic polymers, obey certain scaling laws that allow the estimation of the behaviour of a large system by extrapolation. Unfortunately, proteins are very precise structures that evade such simplifications. So let us take 10,000 atoms as a reasonable size for a realistic complex system. 

By further simplification, Chang (1961) obtained the critical flux for vertical surfaces,... 

Dynamic simulations are also possible, and these require solving differential equations, sometimes with algebraic constraints. If some parts of the process change extremely quickly when there is a disturbance, that part of the process may be modeled in the steady state for the disturbance at any instant. Such situations are called stiff, and the methods for them are discussed in Numerical Solution of Ordinary Differential Equations as Initial-Value Problems. It must be realized, though, that a dynamic calculation can also be time-consuming and sometimes the allowable units are lumped-parameter models that are simplifications of the equations used for the steady-state analysis. Thus, as always, the assumptions need to be examined critically before accepting the computer results. 

The different dialects of XML (XHTML, KML) are constrained by XML schemas (W3C, 2004). These schemas are critical to the success of XML. They are used to ensure that an XML file adheres to a well-defined structure. Schemas are themselves XML files, which must conform to the XSD specification. Schema designers are free to develop constraints to varying degrees. Forcing an XML file to be compatible with a tightly-constrained schema frees developers from having to write their own data validation procedures. This leads to a great simplification of data manipulation software. 

Critical loads of sulfur and nitrogen, as well as their exceedances are derived with a set of simple steady-state mass balance (SSMB) equations. The first word indicates that the description of the biogeochemical processes involved is simplified, which is necessary when considering the large-scale application (the whole of Europe or even large individual countries like Russia, Poland or Ukraine) and the lack of adequate input data. The second word of the SSMB acronym indicates that only steady-state conditions are taken into account, and this leads to considerable simplification. These models include the following equations. 

We call the centre of the concentration range the critical micelle concentration (CMC). As an over-simplification, we say the solution has no colloidal micelles below the CMC, but effectively all the monomer exists as micelles above the CMC. As no micelles exist below the CMC, a solution of monomer is clear - like the solution of dilute soap in the bath. But above the CMC, micelles form in solution and impart a turbid aspect owing to Tyndall light scattering. This latter situation corresponds to washing the face in a sink. 

The structure and mathematical expressions used in PBPK models significantly simplify the true complexities of biological systems. This simplification, however, is desirable if the uptake and disposition of the chemical substance(s) is adequately described because data are often unavailable for many biological processes and using a simplified scheme reduces the magnitude of cumulative uncertainty. The adequacy of the model is therefore of great importance and thus model validation must be critically considered. 

Process Simplification VIP The process simplification VIP uses the value methodology and is a formal, rigorous process to search for opportunities to eliminate or combine process and utility system steps or equipment, ultimately resulting in the reduction of investment and operating costs. The focus is the reduction of installed costs and critical path schedule while balancing these value improvements with ejq)ected facility operability, flexibility, and over-alllife cycle costs. 

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