Purely Empirical Models

A number of empirical models have been proposed for cyclone pressure drop. We shall mention two of the most widely used. 

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There are two basic classes of mathematical models (see Fig. 5.3-18) (1) purely empirical models, and (2) models based on physicochemical principles. 

The concept of the reaction-rate model should be considered to be more flexible than any mechanistically oriented view will allow. In particular, for any reacting system an entire spectrum of models is possible, each of which fits certain overlapping ranges of the experimental variables. This spectrum includes the purely empirical models, models accurately describing every detail of the reaction mechanism, and many models between these extremes. In most applications, we should proceed as far toward the theoretical extreme as is permitted by optimum use of our resources of time and money. For certain industrial applications, for example, the closer the model approaches... 

In many cases of practical interest, no theoretically based mathematical equations exist for the relationships between x and y we sometimes know but often only assume that relationships exist. Examples are for instance modeling of the boiling point or the toxicity of chemical compounds by variables derived from the chemical structure (molecular descriptors). Investigation of quantitative structure-property or structure-activity relationships (QSPR/QSAR) by this approach requires multivariate calibration methods. For such purely empirical models—often with many variables—the... 

In general, there is an array of equilibrium-based mathematical models which have been used to describe adsorption on solid surfaces. These include the widely used Freundlich equation, a purely empirical model, and the Langmuir equation as discussed in the following sections. More detailed modeling approaches of sorption mechanisms are discussed in more detail in Chap. 3 of this volume. 

To overcome some of the limitations just mentioned that are associated with purely empirical models, simulations that include various aspects of the inhaled aerosol dynamics have been developed. The simplest of these belong to a class of models we refer to as Lagrangian dynamical models (LDMs), meaning that the model simulates some of the dynamical behavior of the aerosol in a frame of reference that travels with the aerosol (i.e., a Lagrangian viewpoint ). 

Because one-dimensional LDMs include the aerosol dynamics by using solutions of the dynamical equations in simplified versions of parts of the lung geometry and by including empirical data from experiments on inertial impaction, they remain semiempirical in nature. As a result, they share some of the drawbacks of purely empirical models mentioned earlier. In particular, one-dimensional LDMs use the same mouth-throat deposition models used with... 

Because one-dimensional EDMs require the numerical solution of a partial differential equation (as opposed to simple algebraic equations with empirical models and one-dimensional LDMs, or ordinary differential equations with hygroscopic LDMs), EDMs are more difficult to program, require somewhat more computational resources (typically many minutes on a PC), and have only recently been modified to include two-way coupled hygroscopic effects [37], For these reasons, only a few examples exist of one-dimensional EDMs being used with inhaled pharmaceutical aerosols (e.g., Ref. 11), although they have been used to aid in the development of purely empirical models (e.g., the ICRP 1994 [6] model is partly a curve fit to data from the one-dimensional EDM of Ref. 38). 

Of course, one-dimensional EDMs are not without their drawbacks. Indeed, they suffer from several of the same problems that plague empirical and onedimensional LDMs. In particular, their use of empirical mouth-throat deposition models is a serious drawback to modeling of dry powder and metered-dose inhalers, as discussed earlier with purely empirical models. As with onedimensional LDMs, the use of simplified lung geometries and empirical... 

The recommended approach to modeling is to create models based on fundamental balances (of mass, species, energy, population) and basic kinetics and use them to build a complete model of the precipitator, as shown in earlier sections. Such a set of equations is known as a physical or a mechanistic model. Complete physical models are difficult to create and solve because they require identification in advance of all physical and chemical subprocesses, properties, and parameters. That is why the semiempirical models of a form similar to the complete physical models (but usually simpler) and with fewer equations are often used for scaling up. Parameters of such models are often given in lumped form, some of them fitted to available experimental data obtained from the small-scale system. Such a model can be useful for scaling up, but one cannot be sure that the scale-up will be completely correct because there is no guarantee that the model contains the complete mechanism (88). However, scale-up errors should be smaller than in the case of purely empirical models. CFD codes that are based on reasonable simplifications (closures) regarding their accuracy can be placed between the physical and semiempirical models their application was demonstrated earlier. 

An example of the above consideration is the equivalence of the power model proposed by Colombo et al. (1994) and Clausen s model (1993) for emission controlled by internal diffusion in the source. The assumptions made in the latter - more physically based - model lead to a final description of the emission rate at the surface of the source which is equivalent to the description of the former - purely empirical - model. The equivalence of the models is valid when the parameter C of the empirical model takes the value 1. This can readily be seen if we compare the mathematical equations of the two models ... 

This technique can be applied at a range of gate biases and plotted as a function of channel charge density, gate voltage, or applied lateral electric field. Non-linear models have been proposed for the contact resistance as a function of the applied bias (see, for example, [ffS]), which generally assume a Schottky or other diode-like charge injection from the contact into the channel. The transfer line method can also be used to extract a purely empirical model. 

For other metals, such as Cd, Zn, Cu, and Ni, no simple sohd with properties simulating metal solubility in soils exists. Lindsay (1979) previously advocated the concept of a fictitious sohd phase called soil-Cu. There are a number of theoretical and semi-theoretical models that have been used to describe (ad)sorpfion of transition metals onto reactive surfaces (Fe, Mn or Al oxides soil organic matter). While probably more correct in a mechanistic sense than the solubility relations discussed below, these models have not proven to be particularly useful with intact soils because they contain a very complex assemblage of colloidal surfaces. Moreover, they do not seem to adequately predict increases in metal solubility with increases in total soil metal burden. This has led an increasing number of researchers to develop purely empirical models that describe trace-metal solubility as a function of simple soil parameters such as pH, organic matter content, and total metal content (e.g. McBride et al., 1997 Gray et al.,... 

These purely empirical models can often describe metal solubihties with reasonable accuracy. For example, Sauv6 et al. (2000) measured free Cd " activities in 64 soils with diverse Cd levels, pH coefficients, and OM contents. A three-term version of Eq. (13) (the OM term was nonsignificant) could explain 70% of the observed variation in Cd + activities. When Gray et al. (1999) combined their data set with another developed by McBride et al. (1997), they were able to explain 81% of the variation in (Cd +) using a full, four-term version of the model (Eq. (13)). We note that the pH coefficients for these data sets rarely approach the value of -2 suggested by Lindsays (1979), or by limited studies of metal solubility in calcareous soils (e.g. Elfalaky et al., 1991). 

Various models are available for the purpose of corrosion prediction. These models can be broadly classified into four categories mechanistic models, empirical models, semiempirical models, and neural net models. Some models are purely empirical models based on lab experiments and field data, while others are mechanistic models of different physicochemical transport processes involved in corrosion. 

Let us assume that the model of functional reduction is not a purely empirical model that is supposed to capture how we actually carry out reductions. Nor is it a purely normative model that is concerned with how we should carry out reductions. Rather, it should be interpreted as a partial definition. If so, this approach is committed to the claim that at least some of the relata of the reduction relation are concepts (or modes of presentations, or senses of the expressions which refer to the items which are said to reduce), because it is the descriptive content of the terms that makes for diversity. The relevant concepts present us with an item, a. functional property, which is then said to be identical with a type at another level. It is easy to see how this relates to reduction as conceived of here ... 

The pipe flow analogy breaks down beyond the creeping flow range. Thus beyond the creeping flow range, purely empirical models must be employed, the only remaining link with the pipe flow is the structure of the dimensionless groups. 

Solely a purely empirical model, introduced by Kuss [6], applying set points of precision at the boundaries of the operation range, enables adjustment of the calculation and evaluation to the set points of the precision specified but requires the precision at each of the relevant concentrations to be known already. Calibration, which we formerly expected to provide this information, in fact fails. Hence, additional repeated measurements are required, contrary to our original objective, which was to allow less effort in terms of measurements. Unfortunately, the calibration model provides no reliable precision characteristics, but in this case it is no longer required. 

Another purely empirical model was proposed by Zenz. This is a graphical method. The chart of Zenz, which is based on years of practical experience,... 

If we have an industrial reactor operating we can get a purely empirical model that will correlate outlet concentrations as a function of pressure, inlet temperature, inlet concentration and other operating variables For some reactors such a model properly constructed can be rather good, as long as the new X is within a certain space of X If the function Z Mj--(X) is well defined and smooth for a subspace of X, then we can find successive approximations, either linear... 

This purely empirical model assumes that adsorption energy decreases logarithmically with adsorption density, and hence there is no linear relationship between F and C. Adsprpiion density is, in principle, not limited. 

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