Inadequacy of models

The early models yielded approximate concentrations that reflected the understanding of the soil solution at the time. Later models have yielded better predictions of the soil solution s composition, but they are still only approximate. That reflects the complexity of the soil more than the inadequacy of modeling. The models predict ion interactions in the aqueous solution quite well. Reactions at the surface of colloidal particles are more complex, less understood, slower, and hence are more difficult to formulate. In addition, the models are forced to use the solubility products of pure, simple solids. Soil inorganic particles are far from pure compounds, are often poorly crystalline to amorphous, are not at internal equilibrium, and may not be in equilibrium with the aqueous phase. In addition, the reactions of soil organic matter are not known quantitatively aud soils are open systems, meaning that matter is continually being added and removed. 

Theory established unless later observations or experiments show inadequacies of model... 

The most reliable estimates of the parameters are obtained from multiple measurements, usually a series of vapor-liquid equilibrium data (T, P, x and y). Because the number of data points exceeds the number of parameters to be estimated, the equilibrium equations are not exactly satisfied for all experimental measurements. Exact agreement between the model and experiment is not achieved due to random and systematic errors in the data and due to inadequacies of the model. The optimum parameters should, therefore, be found by satisfaction of some selected statistical criterion, as discussed in Chapter 6. However, regardless of statistical sophistication, there is no substitute for reliable experimental data. 

Although the inadequacy of the one-dimensional model [Goldanskii 1959, 1979] is well by now understood, we start with discussing the simplest one-dimensional version of the theory, since it... 

The inadequacy of the worst case model is evident and the statistical nature of the tolerance stack is more realistic, especially when including the effects of shifted distributions. This has also been the conclusion of some of the literature discussing tolerance stack models (Chase and Parkinson, 1991 Harry and Stewart, 1988 Wu et al., 1988). Shifting and drifting of component distributions has been said to be the chief reason for the apparent disenchantment with statistical tolerancing in manufacturing (Evans, 1975). Modern equipment is frequently composed of thousands of components, all of which interact within various tolerances. Failures often arise from a combination of drift conditions rather than the failure of a specific component. These are more difficult to predict and are therefore less likely to be foreseen by the designer (Smith, 1993). 

The inadequacy of the worst case approach to tolerance stack design compared to the statistical approach is evident, although it still appears to be popular with designers. The worst case tolerance stack model is inadequate and wasteful when the capability of each dimensional tolerance is high > 1.33). Some summarizing comments on the two main approaches are given below. 

For more than two decades researchers have attempted to overcome the inadequacies of Flory s treatment in order to establish a model that will provide accurate predictions. Most of these research efforts can be grouped into two categories, i.e., attempts at corrections to the enthalpic or noncombinatorial part, and modifications to the entropic or combinatorial part of the Flory-Huggins theory. The more complex relationships derived by Huggins, Guggenheim, Stavermans, and others [53] required so many additional and poorly determined parameters that these approaches lack practical applications. A review of the more serious deficiencies... 

The various copolymerization models that appear in the literature (terminal, penultimate, complex dissociation, complex participation, etc.) should not be considered as alternative descriptions. They are approximations made through necessity to reduce complexity. They should, at best, be considered as a subset of some overall scheme for copolymerization. Any unified theory, if such is possible, would have to take into account all of the factors mentioned above. The models used to describe copolymerization reaction mechanisms arc normally chosen to be the simplest possible model capable of explaining a given set of experimental data. They do not necessarily provide, nor are they meant to be, a complete description of the mechanism. Much of the impetus for model development and drive for understanding of the mechanism of copolymerization conies from the need to predict composition and rates. Developments in models have followed the development and application of analytical techniques that demonstrate the inadequacy of an earlier model. 

The inadequacy of the two-box model of the ocean led to the box-diffusion model (Oeschger et al, 1975). Instead of simulating the role of the deep sea with a well-mixed reservoir in exchange with the surface layer by first-order exchange processes, the transfer into the deep sea is maintained by vertical eddy diffusion. In... 

In literature, some researchers regarded that the continuum mechanic ceases to be valid to describe the lubrication behavior when clearance decreases down to such a limit. Reasons cited for the inadequacy of continuum methods applied to the lubrication confined between two solid walls in relative motion are that the problem is so complex that any theoretical approach is doomed to failure, and that the film is so thin, being inherently of molecular scale, that modeling the material as a continuum ceases to be valid. Due to the molecular orientation, the lubricant has an underlying microstructure. They turned to molecular dynamic simulation for help, from which macroscopic flow equations are drawn. This is also validated through molecular dynamic simulation by Hu et al. [6,7] and Mark et al. [8]. To date, experimental research had "got a little too far forward on its skis however, theoretical approaches have not had such rosy prospects as the experimental ones have. Theoretical modeling of the lubrication features associated with TFL is then urgently necessary. 

The solution to the learning problem should provide the flexibility to search for the model in increasingly larger spaces, as the inadequacy of the smaller spaces to approximate well the given data are proved. This immediately calls for a hierarchy in the space of functions. Vapnik (1982) has introduced the notion of structure as an infinite ladder of finitedimensional nested subspaces ... 

Values for the heats of hydration of a number of ions that were calculated by the aforementioned methods on the basis of theoretical models and experimental data are reported in Table 7.2. We see that there is a certain general agreement, but in individual cases the discrepancies are large, due to inadequacies of the theoretical concepts used in the calculations. 

The consequence of all these (conscious and unconscious) simplifications and eliminations might be that some information not present in the process will be included in the model. Conversely, some phenomena occurring in reality are not accounted for in the model. The adjustable parameters in such simplified models will compensate for inadequacy of the model and will not be the true physical coefficients. Accordingly, the usefulness of the model will be limited and risk at scale-up will not be completely eliminated. In general, in mathematical modelling of chemical processes two principles should always be kept in mind. The first was formulated by G.E.P. Box of Wisconsin All models are wrong, some of them are useful . As far as the choice of the best of wrong models is concerned, words of S.M. Wheeler of New York are worthwhile to keep in mind The best model is the simplest one that works . This is usually the model that fits the experimental data well in the statistical sense and contains the smallest number of parameters. The problem at scale-up, however, is that we do not know which of the models works in a full-scale unit until a plant is on stream. 

Because earlier experimental results and data analyses (3-10) had led us to anticipate the inadequacy of the simple approach considered above, we also planned and carried out (2) a second order factorial design of experiments and related data analysis. Mathematical analysis (of the results of 11 experiments) based on the second order model showed that all of these results could be represented satisfactorily by an equation of the form... 

Because of the inadequacies of the aforementioned models, a number of papers in the 1950s and 1960s developed alternative mathematical descriptions of fluidized beds that explicitly divided the reactor contents into two phases, a bubble phase and an emulsion or dense phase. The bubble or lean phase is presumed to be essentially free of solids so that little, if any, reaction occurs in this portion of the bed. Reaction takes place within the dense phase, where virtually all of the solid catalyst particles are found. This phase may also be referred to as a particulate phase, an interstitial phase, or an emulsion phase by various authors. Figure 12.19 is a schematic representation of two phase models of fluidized beds. Some models also define a cloud phase as the region of space surrounding the bubble that acts as a source and a sink for gas exchange with the bubble. 

The properties of the two helium isotopes in the liquid state are strongly influenced by quantum effects. In Fig. 2.8, the specific heat of 3He, calculated from the ideal gas Fermi model (Tp = 4.9 K) with the liquid 3He density, is compared with the experimental data. The inadequacy of this model is evident. A better fit, especially at the lower temperatures, is obtained by the Landau theory [25]. 

Parameters subject to calibration within SWAT were selected after a preliminary sensitivity analysis and literature review, to partially compensate for the inadequacy of the initial values assumed for some of them (especially those related to soil type), model structure and other sources of uncertainty. A detailed description about the SWAT parameters can be found in [5, 6], while a brief description of the selected parameters is provided next ... 

The discrepancies between the experimental data and the behavior predicted using the Smoluchowski-Stokes-Einstein model for ksenfluor and ksenphos likely arise from the inadequacies of the simple Smoluchowski-Stokes-Einstein analysis for application to the anthracene/diaryliodonium salt molecular system. For example, the Smoluchowski analysis assumes that the reacting molecules are spherical in... 

A part of these discrepancies is undoubtedly due to the uncertainty in the d and p values and some inadequacy of the combination rules (especially (6)), and is clearly irrelevant to the soundness of the APM itself. The remaining part of the observed discrepancies arises from oversimplifications in the underlying assumptions of the model. We here limit the discussion to two of them ... 

Falkowski PG, Katz ME, Knoll AH, Quigg A, Raven JA, Schofield O, Taylor FJR (2004) The evolution of modem eukaryotic phytoplankton. Science 305 354-360 Flynn KJ (2005) Castles built on sand dysfunctionality in plankton models and the inadequacy of dialogue between biologists and modellers. J Plankton Res 27 205-210 Fontana A (2007) Chemistry of oxylipin pathways in marine diatoms. Pure Appl Chem 79 481 490... 

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