Properties of the governing equations

FIGURE 6.1. Schematic illustration of the cp, z, e) region in which the solution for the structure of a combustion wave must lie. 

Since the numerator and the denominator on the right-hand side of equation (5-35) or equation (5-36) are bounded in the region of interest, the singularities will be located at the points or lines where both the numerator and the denominator vanish. The denominator for equation (5-35) and equation (5-36) vanishes when 

Since equations (2) and (3) are independent of e, their solutions will define cylindrical surfaces, the generatrices of which are parallel to the e axis. 

From equation (5-34) it is clear that the solution to equation (3) in the region of interest is the plane 

In view of the discussion in Section 5.3.2, we may also want to treat the plane 

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Introducing specific length and velocity scales provides a more intuitive approach to nondi-mensionalization. In this section the thermal-energy equation is also included in the analysis. Assuming constant transport properties and a single-component fluid, a subset of the governing equations is derived from Section 6.2 as... 

Of particular interest in Statistics is that the means of many large samples taken from a particular population are approximately distributed in this Normal fashion, i.e., they are said to be Normally distributed. This is true even when the population data themselves are not Normally distributed. The mathematical properties of a true Normal distribution allow quantitative statements of the area under the curve between any two points on the x-axis. In Section 6.6.1 it was shown that the total area under the Normal curve is 1, or 100%. It is also of interest to know the proportion of the total area under the curve that lies between two points that are equidistant from the mean. These points are typically represented by multiples of the SD. From the properties of the mathematical equation that governs the shape of the Normal curve, it can be shown that ... 

The following discussion is restricted to two-dimensional, steady, incompressible, constant-property flow. For simplicity, the body forces are neglected. The effects of body forces are considered in the chapter on natural convection. To nondimensionalize the appropriately reduced form of the governing equations from Tables 6.1 -6.3, we select a characteristic length L, a reference velocity a reference temperature... 

In this chapter you will learn that proper assessment of mass transport controlled corrosion reactions requires knowledge of the concentration distribution of the reacting species in solution, certain properties of the electrolyte, and the geometry of the system. A rigorous calculation of mass transport controlled reaction rates requires detailed information concerning these parameters. Fortunately, many of the governing equations have been solved for several well-defined geometries. 

In this section an alternative derivation of the governing equations for granular flow is examined. In this alternative method the peculiar velocity C, instead of the microscopic particle velocity c, is used as the independent variable in the particle property and distribution functions. The transformation of these functions and the governing equation follows standard mathematical procedures for changing the reference frame. The translational motion of an individual particle may be specified either by its microscopic velocity c relative to a fixed or Galilean frame of reference, or by its velocity relative to a frame of reference moving with the local velocity of the granular material Yd-... 

The unknown amplitudes a and 3 are determined by solving a system of ordinary differential equations, called the bifurcation equations. In [4,5], it is shown that the symmetries present in the governing equations (2.1) and the eigenfunctions of equations (2.3) are reflected in corresponding symmetry properties of the bifurcation equations. Specifically, since the problem is invariant with respect to translations in T and Y, and with respect to reflections in Y, the analysis leads to bifurcation equations of the form... 

The dimensionless representation of the governing equations shows that the concentration driving force for reaction ij/ and the local extent of reaction of the solid reactant are related to the position rj and time t through a single parameter a. This parameter a, which is analogous to the Thiele modulus in heterogeneous catalysis, incorporates both kinetic and structural properties and is the measure of the relative magnitude of chemical reaction and diffusion rates. 

The solubility of a solid in the liquid phase of a mixture depends on the properties of the two phases for the components that crystallize, the equilibrium is governed by the following equation [ XI... 

The four fundamental equations, which govern the properties of the combined cycle are the equation of state, conservation of mass, momentum and energy equations. 

Kresge et a/.498 have drawn attention to the fact that detritiation of [3H]-2,4,6-trihydroxy- and [3H]-2,4,6-trimethoxy-benzenes by concentrated aqueous perchloric acid gives correlations of log rate coefficient with — H0 with slopes of 0.80 and 1.14 respectively. Protonation to give the carbon conjugate acids is, however, governed by h0lA0 and h0l 9S, respectively, which suggests that the difference in kinetic acidity dependence is a property of the substrate and should not be interpreted as a major difference in mechanism. The kinetic difference can be eliminated by an appropriate comparison of kinetic and equilibrium acidity dependencies. In equation (230)... 

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