Reduced models

Various functional forms for / have been proposed either as a result of empirical observation or in terms of specific models. A particularly important example of the latter is that known as the Langmuir adsorption equation [2]. By analogy with the derivation for gas adsorption (see Section XVII-3), the Langmuir model assumes the surface to consist of adsorption sites, each having an area a. All adsorbed species interact only with a site and not with each other, and adsorption is thus limited to a monolayer. Related lattice models reduce to the Langmuir model under these assumptions [3,4]. In the case of adsorption from solution, however, it seems more plausible to consider an alternative phrasing of the model. Adsorption is still limited to a monolayer, but this layer is now regarded as an ideal two-dimensional solution of equal-size solute and solvent molecules of area a. Thus lateral interactions, absent in the site picture, cancel out in the ideal solution however, in the first version is a properly of the solid lattice, while in the second it is a properly of the adsorbed species. Both models attribute differences in adsorption behavior entirely to differences in adsorbate-solid interactions. Both present adsorption as a competition between solute and solvent. 

The integral equation approach has also been explored in detail for electrolyte solutions, with the PY equation proving less usefiil than the HNC equation. This is partly because the latter model reduces cleanly to the MSA model for small h 2) since... 

Finally, we like to mention that the QCMD model reduces to the Born-Oppenheimer approximation in case the ratio of the mass m of the quantum particles to the masses of the classical particles vanishes [6], This implies... 

Binary Mixtures—Low Pressure—Polar Components The Brokaw correlation was based on the Chapman-Enskog equation, but 0 g and were evaluated with a modified Stockmayer potential for polar molecules. Hence, slightly different symbols are used. That potential model reduces to the Lennard-Jones 6-12 potential for interactions between nonpolar molecules. As a result, the method should yield accurate predictions for polar as well as nonpolar gas mixtures. Brokaw presented data for 9 relatively polar pairs along with the prediction. The agreement was good an average absolute error of 6.4 percent, considering the complexity of some of... 

If r]oo <3C ( /. j/o), the Carreau model reduces to a three-parameter model ( 0,k, and p) that is equivalent to a power law model with a low shear limiting viscosity, also known as the Ellis model ... 

Quantitative molecular-property relationship Segregated-flow model Reduced folate carrier protein... 

The reader will recognize these terms as having of the same form as the correction terms in the two-environment model discussed earlier. With N — 1, 6 j = 0 and the model reduces to the laminar-chemistry approximation. With N —2, additional information is obtained concerning the second-order moments of the composition vector. Likewise, by using a larger N, the Mh-order moments are controlled by the DQMOM correction terms found from Eq. (89). 

A simpler and general discrete time scheduling formulation can also be derived by means of the Resource Task Network concept proposed by Pantelides [10], The major advantage of the RTN formulation over the STN counterpart arises in some problems involving many identical pieces of equipment. In these cases, the RTN formulation introduces a single binary variable instead of the multiple variables used by the STN model. The RTN-based model also covers all the features at the column on discrete time in Table 8.1. In order to deal with different types of resources in a uniform way, this approach requires only three different classes of constraints in terms ofthree types of variables defining the task allocation, the batch size, and the resource availability. Briefly, this model reduces the batch scheduling problem to a simple resource balance problem carried out in each predefined time period. 

If the second term in the configurational entropy of mixing, eq. (9.42), is zero, the quasi-chemical model reduces to the regular solution approximation. Here, Aab is given by (eq. (9.21). If in addition yAB =0the ideal solution model results. 

Other synonyms for steady state are time-invariant, static, or stationary. These terms refer to a process in which the values of the dependent variables remain constant with respect to time. Unsteady state processes are also called nonsteady state, transient, or dynamic and represent the situation when the process-dependent variables change with time. A typical example of an unsteady state process is the operation of a batch distillation column, which would exhibit a time-varying product composition. A transient model reduces to a steady state model when d/dt = 0. Most optimization problems treated in this book are based on steady state models. Optimization problems involving dynamic models usually pertain to optimal control or real-time optimization problems (see Chapter 16)... 

In case that the decay of impact kinetic energy due to viscous dissipation is the predominant mechanism in droplet flattening, Madejski s full model reduces to ... 

The semi-empirical Pitzer equation for modeling equilibrium in aqueous electrolyte systems has been extended in a thermodynamically consistent manner to allow for molecular as well as ionic solutes. Under limiting conditions, the extended model reduces to the well-known Setschenow equation for the salting out effect of molecular solutes. To test the validity of the model, correlations of vapor-liquid equilibrium data were carried out for three systems the hydrochloric acid aqueous solution at 298.15°K and concentrations up to 18 molal the NH3-CO2 aqueous solution studied by van Krevelen, et al. 

Table 4.2), provides a convenient hierarchy, each model reducing both the errors and the maximum size of the molecule accessible by about a factor of two. A user-friendly implementation of these models is readily available within the Gaussian 98 suite of programs [25]. 

If the rays possess ideal flexibility to allow application of Gaussian statistics, the resultant structure will resemble a soft sphere. This was the reason why the present author introduced the soft sphere model [38]. This model reduces to dendrimers in the narrow sense when no spacer chains between the branching units are present. 

In this case the extent of conformational change induced in one subunit is the same, in both sign and magnitude, as in the second subunit. The two subunits act concertedly, and this model reduces to the model of Section 4.5. 

The multipole model reduces the crystal electron density to a number of parameters, which can be fitted to experimental structure factors. For CU2O, structure factors for the (531) and higher-order reflections out to (14,4,2) were taken from X-ray measurements. Weak (ooe) (with o for odd and e for even) and very weak (eeo) reflections were also taken from X-ray work. Fig. 6 shows a three-dimensional plot of the difference between the static crystal charge density obtained from the multipole fitting to... 

A good understanding of the detailed chemistry of oxidant formation makes it possible to construct more compact chemical models. These generalized or lumped mechanism models reduce the number of individual chemical reactions by combining similar or sequential reactions and ig-... 

In conclusion, the different shapes of isotherms describing equilibrium distributions of a contaminant, between geosorbents and aqueous or gaseous phases, depend on the sorption mechanism involved and the associated sorption energy. At low contaminant concentration, all models reduce to essentially linear correlation. At higher contaminant concentration, when sorption isotherms deviate from linearity, an appropriate isotherm model should be used to describe the retention process. 

The first two terms represent the Gibbs energy of f.c.c. Al Cr and Fe because the second sublattice consists entirely of unfilled interstitial sites. The next two terms represent the Gibbs energy of the CrC and FeC compounds which are formed when all of the interstitial sites of the f.c.c. Al lattice are filled with C. The second line is the ideal entropy term while the last four lines correspond to the excess term. It can be seen that, when j/ = 0, the model reduces to Eq. (5.21) for the simple substitutional mixing of Cr and Fe. This is particularly useful as thermodynamic assessments of metallic binary systems can be extended to include interstitial elements without the need for changing any of the existing parameters. 

The family of curves represented by eqn. (46) is shown in Fig. 11 and the mean and variance of both the E(f) and E(0) RTDs are as indicated in Table 5. When N assumes the value of 0, the model represents a system with complete bypassing, whilst with N equal to unity, the model reduces to a single CSTR. As N continues to increase, the spread of the E 0) curves reduces and the curve maxima, which occur when 0 = 1 —(1/N), move towards the mean value of unity. When N tends to infinity, E(0) is a dirac delta function at 0 = 1, this being the RTD of an ideal PER. The maximum value of E(0), the time at which it occurs, or any other appropriate curve property, enables the parameter N to be chosen so that the model adequately describes an experimental RTD which has been expressed in terms of dimensionless time see, for example. Sect. 66 of ref. 26 for appropriate relationships. 

Since + V2 add up to 1, there is no dead volume, so at this point our model reduces to Fig. E14.3c. Now relax the plug flow assumption and adopt the tanks-in-series model. From Fig. 14.3... 

A Bayesian hierarchical modeling framework was used to evaluate the effects data for each species and test endpoint (Figure 7.4). Hierarchical models reduce the effect of incomplete data sets, small numbers of tests, inconsistent information on effects among species, and other issues that lend uncertainty to the risk characterization results. 

After algebraic manipulations, the model reduces to the solution of the following four nonlinear equations for x and the monomer and counterion concentrations Cj, C2, and Cf + ... 

While more physically based models provide a picture of the underlying forces that lead to chemical bonding, the bond valence model reduces the rules of chemistry to their simplest mathematical form. In this form it is able to provide insights into the behaviour of the many complex systems found in acid-base chemistry. 

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