Quantitative modeling

B. A. Auld, F. G. Muennemann and M. Riaziat, Quantitative modelling of flaw responses in eddy current testing. In R. S. Sharpe (ed.), Research techniques in nondestructive testing, Vol. 7. Academic, London, 1984. 

In practice, e.g., in nature or in fonnulated products, colloidal suspensions (also denoted sols or dispersions) tend to be complex systems, consisting of many components that are often not very well defined, in tenns of particle size for instance. Much progress has been made in the understanding of colloidal suspensions by studying well defined model systems, which allow for a quantitative modelling of their behaviour. Such systems will be discussed here. 

The first quantitative model, which appeared in 1971, also accounted for possible charge-transfer complex formation (45). Deviation from the terminal model for bulk polymerization was shown to be due to antepenultimate effects (46). Mote recent work with numerical computation and C-nmr spectroscopy data on SAN sequence distributions indicates that the penultimate model is the most appropriate for bulk SAN copolymerization (47,48). A kinetic model for azeotropic SAN copolymerization in toluene has been developed that successfully predicts conversion, rate, and average molecular weight for conversions up to 50% (49). 

Flory-Huggins Theory. The simplest quantitative model foi that iacludes the most essential elements needed foi polymer blends is... 

As a reactant molecule from the fluid phase surrounding the particle enters the pore stmcture, it can either react on the surface or continue diffusing toward the center of the particle. A quantitative model of the process is developed by writing a differential equation for the conservation of mass of the reactant diffusing into the particle. At steady state, the rate of diffusion of the reactant into a shell of infinitesimal thickness minus the rate of diffusion out of the shell is equal to the rate of consumption of the reactant in the shell by chemical reaction. Solving the equation leads to a result that shows how the rate of the catalytic reaction is influenced by the interplay of the transport, which is characterized by the effective diffusion coefficient of the reactant in the pores, and the reaction, which is characterized by the first-order reaction rate constant. 

Temperature, pH, and feed rate are often measured and controlled. Dissolved oxygen (DO) can be controlled using aeration, agitation, pressure, and/or feed rate. Oxygen consumption and carbon dioxide formation can be measured in the outgoing air to provide insight into the metaboHc status of the microorganism. No rehable on-line measurement exists for biomass, substrate, or products. Most optimization is based on empirical methods simulation of quantitative models may provide more efficient optimization of fermentation. 

Although it is probably not possible at present to develop a completely satisfactory quantitative model, a conceptual model that identifies the critical processes can be developed. First, it is apparent that before significant chemical reaction can occur substantial modification must be induced by the shock-compression process. With present knowledge, the problem is one of mechanical deformation, rather than one of chemistry. The materials studies... 

The Emerman quantitative model [40] makes it possible, with sufficient accuracy, to predict the redistribution of filler particles depending on the parameters of processing and to explain the existing (still not extensive) experimental data. Good agreement should be noted between the experimental data and those calculated according to the Emerman model [40],... 

Price (Ref 39) compares the three principal quantitative models of ignition and concludes that there are serious deficiencies in the existing models, so much so that no one theory appears adequate to represent the complexity of ignition of composite solid proplnts. In the gas-phase model (Refs 22 36) the hot oxidizing environ-... 

Here, d is the radius of the cavity around the solute (given in A), the dipole fi is given in A and au, and d is the macroscopic dielectric constant of the solvent. The crucial problem, however, is that the cavity radius is an arbitrary parameter which is not given by the macroscopic model, making the results of eq. (2.18) rather meaningless from a quantitative point of view. A much more quantitative model is provided by the semimicroscopic model described below. 

Models and theories have been developed by scientists that allow a good description of the double layers at each side of the surface either at equilibrium, under steady-state conditions, or under transition conditions. Only the surface has remained out of reach of the science developed, which cannot provide a quantitative model that describes the surface and surface variations during electrochemical reactions. For this reason electrochemistry, in the form of heterogeneous catalysis or heterogeneous catalysis has remained an empirical part of physical chemistry. However, advances in experimental methods during the past decade, which allow the observation... 

We now have enough information to turn our qualitative ideas about a gas into a quantitative model that can be used to make numerical predictions. The kinetic model ( kinetic molecular theory, KMT) of a gas is based on four assumptions (Fig. 4.23) ... 

The topic of molecular motion is an active one in experimental and theoretical polymer physics, and we may expect that in time the simple reptation model will be superseded by more sophisticated models. However, in the form presented here, reptation is likely to remain important as a semi-quantitative model of polymer motion, showing as it does the essential similarity of phenomena which have their origin in the flow of polymer molecules. 

If we can develop accurate quantitative models that simulate how cells respond to various enviromnental changes, we can better utilize the chemical synthesis capabilities of cells. Steps toward this goal are being taken. Models of the common gut bacterium Escherichia coli have been developed from mechanisms of subcellular processes discovered or postulated by molecular biologists. These models have progressed to the point where they can be used with experiments to discriminate among postulated mechanisms for control of subcellular processes. 

From the selection rules of the 6j coefficients (.89), it follows that the biquadratic terms cannot mix the S = I levels with higher spin states. By contrast, the anisotropic symmetric and antisymmetric terms, whose magnitude is related to that of the isotropic component (89), can give rise to a substantial mixing. However, a detailed quantitative model is needed to verify whether the peculiar magnetic properties of [3Fe-4S] + centers can be explained by this mixing. 

Wise, J., K.T. Gillen, and R.L. Clough, Quantitative model for the time development of diffusion-limited oxidation profiles. Polymer, 38(8), 1929-1944, 1997. 

From careful measurements of such processs, it is possible to develop quantitative models to describe the molecular dynamical response to impulsive laser excitation. These enable the fundamental interaction of intense, ultrafast laser light with molecules to be understood from first... 

To develop and apply assumption-free learning frameworks and methodologies, aimed at uncovering and expressing in adequate solution formats performance improvement. . opportunities, extracted from existing data which were acquired from plants that cannot be described effectively through first-principles quantitative models. 

The preceding set of characteristics and properties of the estimators makes our type of mapping procedures, /, particularly appealing for the kinds of systems that we are especially interested to study, i.e., manufacturing systems where considerable amounts of data records are available, with poorly understood behavior, and for which neither accurate first-principles quantitative models exist nor adequate functional form choices for empirical models can be made a priori. In other situations and application contexts that are substantially different from the above, while much can still be gained by adopting the same problem statements, solution formats and performance criteria, other mapping and search procedures (statistical, optimization theory) may be more efficient. 

All of the quantitative models are implicitly dose-dependent. This is of particular importance with respect to interspecies comparisons since it may be possible for the relative susceptibility of species to reverse on going from higher to lower doses. 

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