Mathematical modeling surface area

The reacting sohd is in granular form. Decrease in the area of the reaction interface occurs as the reaction proceeds. The mathematical modeling is distinguished from that with flat surfaces, which are most often used in experimentation. 

A survey of the mathematical models for typical chemical reactors and reactions shows that several hydrodynamic and transfer coefficients (model parameters) must be known to simulate reactor behaviour. These model parameters are listed in Table 5.4-6 (see also Table 5.4-1 in Section 5.4.1). Regions of interfacial surface area for various gas-liquid reactors are shown in Fig. 5.4-15. Many correlations for transfer coefficients have been published in the literature (see the list of books and review papers at the beginning of this section). The coefficients can be evaluated from those correlations within an average accuracy of about 25%. This is usually sufficient for modelling of chemical reactors. Mathematical models of reactors arc often more sensitive to kinetic parameters. Experimental methods and procedures for parameters estimation are discussed in the subsequent section. 

The previously proposed uptake models were mathematical assumptions and had no physical or chemical basis. Millard and Hedges, on the other hand, considered the chemistry of bone-uranium interactions. With the D-A model, they proposed that U was diffusing into bone as uranyl complexes, and adsorbing to the large surface area presented by the bone mineral hydroxyapatite (Millard and Hedges 1996). Laboratory experiments showed a partition coefficient between uranyl and hydroxyapatite under oxic conditions of 10" -10, demonstrating U uptake in the U state without the need for reduction by protein decay products as proposed by Rae and Ivanovich (1986). 

Note, however, there are two critical limitations to these "predicting" procedures. First, the mathematical models must adequately fit the data. Correlation coefficients (R ), adjusted for degrees of freedom, of 0.8 or better are considered necessary for reliable prediction when using factorial designs. Second, no predictions outside the design space can be made confidently, because no data are available to warn of unexpectedly abrupt changes in direction of the response surface. The areas covered by Figures 8 and 9 officially violate this latter limitation, but because more detailed... 

One must understand the physical mechanisms by which mass transfer takes place in catalyst pores to comprehend the development of mathematical models that can be used in engineering design calculations to estimate what fraction of the catalyst surface is effective in promoting reaction. There are several factors that complicate efforts to analyze mass transfer within such systems. They include the facts that (1) the pore geometry is extremely complex, and not subject to realistic modeling in terms of a small number of parameters, and that (2) different molecular phenomena are responsible for the mass transfer. Consequently, it is often useful to characterize the mass transfer process in terms of an effective diffusivity, i.e., a transport coefficient that pertains to a porous material in which the calculations are based on total area (void plus solid) normal to the direction of transport. For example, in a spherical catalyst pellet, the appropriate area to use in characterizing diffusion in the radial direction is 47ir2. 

Modern N2 sorption porosimeters are very sophisticated and generally reliable. Typically they come supplied with customized user-friendly software which enables the experimental data to be readily computed using the above models and mathematical expressions. Usually the raw isotherm data is displayed graphically along with various forms of the derived pore size distribution curve and tabulated data for surface area, pore volume and average pore diameter. 

The porous structure of either a catalyst or a solid reactant may have a considerable influence on the measured reaction rate, especially if a large proportion of the available surface area is only accessible through narrow pores. The problem of chemical reaction within porous solids was first considered quantitatively by Thiele [1] who developed mathematical models describing chemical reaction and intraparticle diffusion. Wheeler [2] later extended Thiele s work and identified model parameters which could be measured experimentally and used to predict reaction rates in... 

Surface and Interfacial Tension. Some properties of liquid surfaces are suggestive of a skin that exercises a contracting force or tension parallel to the surface. Mathematical models based on this effect have been used in explanation of surface phenomena, such as capillary rise. The terms surface tension (gas—liquid or gas—solid interface) and interfacial tension (liquid—liquid or liquid—solid) relate to these models which do not reflect the actual behavior of molecules and ions at interfaces. Surface tension is the force per unit length required to create a new unit area of gas—liquid surface (mN/m (= dyn/cm)). It is numerically equal to the free-surface energy. Similady, interfacial tension is the force per unit length required to create a new unit area of liquid—liquid interface and is numerically equal to the interfacial free energy. 

There are several properties of a chemical that are related to exposure potential or overall reactivity for which structure-based predictive models are available. The relevant properties discussed here are bioaccumulation, oral, dermal, and inhalation bioavailability and reactivity. These prediction methods are based on a combination of in vitro assays and quantitative structure-activity relationships (QSARs) [3]. QSARs are simple, usually linear, mathematical models that use chemical structure descriptors to predict first-order physicochemical properties, such as water solubility. Other, similar models can then be constructed that use the first-order physicochemical properties to predict more complex properties, including those of interest here. Chemical descriptors are properties that can be calculated directly from a chemical structure graph and can include abstract quantities, such as connectivity indices, or more intuitive properties, such as dipole moment or total surface area. QSAR models are parameterized using training data from sets of chemicals for which both structure and chemical properties are known, and are validated against other (independent) sets of chemicals. 

Since atmospheric aerosols comprise particles with a wide range of sizes, it is often convenient to use mathematical models to describe the atmospheric aerosol distribution (Seinfeld and Pandis, 1998). A series of mathematical models have been proposed, of which the lognormal distribution has been the most used in atmospheric applications (Seinfeld and Pandis, 1998 Horvath, 2000). Useful discussions of the various aerosol size distribution models are provided by Seinfeld and Pandis (1998) and Jaenicke (1998). In general, atmospheric aerosols size distributions are shown graphically in terms of the volume (or mass) distributions, surface area distributions, or number distributions as a function of particle size (Jaenicke, 1998). 

First of all, we used this mathematical model to correlate the in vitro and in situ permeabilities of grepafloxacin and ciprofloxacin [39], and the area correction factor Sf obtained was around 4, in accord with results obtained by other authors [52]. This difference is explained by the differences in absorptive surface in the in situ versus the in vitro model, as the latter presents microvilli but not villi or folds. Now we have expanded the number of element of the correlation to all the quinolones included in Table 4.2, and the area correction factor does not suffer any variation (see Fig. 4.8, p. 104). Even if this model has been constructed using very simplistic assumptions, the results are promising and demonstrate that a good modeling approach helps to identify the system critical parameters and how the system behavior changes from the in vitro to the in situ level [39]. It is important to notice that with this linear correlation we make the assumption that the main difference between both systems is the actual effective area for transport. Nevertheless, since the plot is far from being perfect, it is probable that there are more differences in both experimental systems, such as different paracellular resistance or different expression levels of the transporter, that account for the deviation. 

Although the pH-partition hypothesis relies on a quasi-equilibrium transport model of oral drug absorption and provides only qualitative aspects of absorption, the mathematics of passive transport assuming steady diffusion of the un-ionized species across the membrane allows quantitative permeability comparisons among solutes. As discussed in Chapter 2, (2.19) describes the rate of transport under sink conditions as a function of the permeability P, the surface area A of the membrane, and the drug concentration c (t) bathing the membrane ... 

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