Linear-parabolic process

Mathematical modeling of mass or heat transfer in solids involves Pick s law of mass transfer or Fourier s law of heat conduction. Engineers are interested in the distribution of heat or concentration across the slab or the material in which the experiment is performed. This process is usually time varying and eventually reaches a steady state. This process is represented by parabolic partial differential equations with known initial conditions and boundary conditions at two ends. Both linear and nonlinear parabolic partial differential equations will be discussed in this chapter. We will present semianalytical solutions for linear parabolic partial differential equations and numerical solutions for nonlinear parabolic partial differential equations based on the numerical method of lines. 

The second step involves the apphcation of a non-linear parabolic equation [72] to the two selected quickbird scenes. This equation allows selective enhancement and smoothing in addition to simultaneously preventing the blurring of the edges. The processing is quite effective for image classification in urban areas. The equation is stated as ... 

Dependences of thermal oxidation rates are of great importance for silicon carbide device technology. It is not necessary to elaborate special technology for the oxidation of SiC monocrystals. It can be carried out using silicon technology but at higher temperatures. In general, the oxidation process can be described by a linear-parabolic equation. 

The oxidation rate of (OOOl)C is 5-10 times higher than of (OOOl)Si in the temperature range 1000-1300°C in dry and wet hydrogen flow at 15 liters per minute (Fig. 5). The oxidation rate of other facets, for instance, (1120), has an intermediate value compared with 0001 facets. The oxidation process of the (0001)C facet is limited by diffusion of the reaction components through the growing oxide film. Hence the oxidation rate of this facet does not depend on the crystal surface treatment, type and level of doping, or polytype structure. The constants of the linear-parabolic equation for (0001)C oxidation are shown in Table 4. 

When a sample is injected into the carrier stream it has the rectangular flow profile (of width w) shown in Figure 13.17a. As the sample is carried through the mixing and reaction zone, the width of the flow profile increases as the sample disperses into the carrier stream. Dispersion results from two processes convection due to the flow of the carrier stream and diffusion due to a concentration gradient between the sample and the carrier stream. Convection of the sample occurs by laminar flow, in which the linear velocity of the sample at the tube s walls is zero, while the sample at the center of the tube moves with a linear velocity twice that of the carrier stream. The result is the parabolic flow profile shown in Figure 13.7b. Convection is the primary means of dispersion in the first 100 ms following the sample s injection. 

The aforementioned inconsistencies between the paralinear model and actual observations point to the possibility that there is a different mechanism altogether. The common feature of these metals, and their distinction from cerium, is their facility for dissolving oxygen. The relationship between this process and an oxidation rate which changes from parabolic to a linear value was first established by Wallwork and Jenkins from work on the oxidation of titanium. These authors were able to determine the oxygen distribution in the metal phase by microhardness traverses across metallographic sections comparison of the results with the oxidation kinetics showed that the rate became linear when the metal surface reached oxygen... 

The volume ratio (see Section 1.9) for cuprous oxide on copper is 1 7, so that an initially protective film is to be expected. Such a film must grow by a diffusion process and should obey a parabolic law. This has been found to apply for copper in many conditions, but other relationships have been noted. Thus in the very early stages of oxidation a linear growth law has been observed (e.g. at 1 000°C) . 

If the PBR is less than unity, the oxide will be non-protective and oxidation will follow a linear rate law, governed by surface reaction kinetics. However, if the PBR is greater than unity, then a protective oxide scale may form and oxidation will follow a reaction rate law governed by the speed of transport of metal or environmental species through the scale. Then the degree of conversion of metal to oxide will be dependent upon the time for which the reaction is allowed to proceed. For a diffusion-controlled process, integration of Pick s First Law of Diffusion with respect to time yields the classic Tammann relationship commonly referred to as the Parabolic Rate Law ... 

Kinetic expressions for appropriate models of nucleation and diffusion-controlled growth processes can be developed by the methods described in Sect. 3.1, with the necessary modification that, here, interface advance obeys the parabolic law [i.e. is proportional to (Dt),/2]. (This contrasts with the linear rate of interface advance characteristic of decomposition reactions.) Such an analysis has been provided by Hulbert [77], who considers the possibilities that nucleation is (i) instantaneous (0 = 0), (ii) constant (0 = 1) and (iii) deceleratory (0 < 0 < 1), for nuclei which grow in one, two or three dimensions (X = 1, 2 or 3, respectively). All expressions found are of the general form... 

FIGURE 4.4 Schematic of threshold behavior of ionization processes. Under ideal conditions, one expects a step function for photoionization, a linear variation with energy under electron impact, and a parabolic dependence for double ionization by electron impact. 

The most predominant effect of H20 in the oxidizing ambient is to increase the parabolic rate constant (90). As a result, the effect of the interface reaction as the rate-controlling process increases with increasing H20 content. A relatively small HzO concentration (25 ppm) in 02 is already sufficient to increase the parabolic rate constant by factors of 1.3 and 1.6 for <111>-and <100>-oriented silicon wafers, respectively (99). The linear rate constant increases more gradually over the range of added H20 (0-2000 ppm) (90). 

This section presents a fundamental development of Sections V and VI. Here a linear dielectric response of liquid H20 is investigated in terms of two processes characterized by two correlation times. One process involves reorientation of a single polar molecule, and the second one involves a cooperative process, namely, damped vibrations of H-bonded molecules. For the studies of the reorientation process the hat-curved model is employed, which was considered in detail in Section V. In this model a hat-like intermolecular potential comprises a flat bottom and parabolic walls followed by a constant potential. For the studies of vibration process two variants are employed. 

The linear correlation between BCF and Kow apparently breaks down for chemicals with a log Kow greater than approximately 6 (Figure 9.4), resulting in a "parabolic" or "bilinear" type relationship between the BCF and Kow (Bintein 1993, Meylan et al., 1999). For these superhydrophobic chemicals, the BCF appears to be much lower than expected from the chemical s octanol-water partition coefficient. A loss of linear correlation between the BCF and Kow can be caused by a number of experimental artifacts (described in section 9.4.3) and physiological processes, including metabolic transformation, fecal egestion, and growth. 

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