Kinematic waves dispersion

The discussion of dispersion in linear kinematic waves (Chapter 7, pp. 136, [7] = C see also [3b] and [314]) brings out another shape factor, this... 

In Reprint C in Chapter 7, the behavior of a tracer pulse in a stream flowing through a packed bed and exchanging heat or matter with the particles is studied. It is shown that the diffusion in the particles makes a contribution to the apparent dispersion coefficient that is proportional to v2 fi/D. The constant of proportionality has one part that is a function of the kinematic wave speed fi, but otherwise only a factor that depends on the shape of the particle (see p. 145 and in equation (42) ignore all except the last term and even the suffixes of this e, being unsuitable as special notation, will be replaced by A. e is defined in the middle of p. 143 of Chapter 7). In this equation, we should not be surprised to find a term of the same form as the Taylor dispersion coefficient, for it is diffusion across streams of different speeds that causes the dispersion in that case just as it is the diffusion into stationary particles that causes the dispersion in this.7 What is surprising is that the isothermal diffusion and reaction equation should come up, for A is defined by... 

Equation (11) simply means that the total disturbance ffi v(x, t) dr is the same at any point x as it is at x = 0. We could without loss of generality put a0 = fio = 1. Equation (12) shows that the mean time of the disturbance at x differs from that at x = 0 by a quantity strictly proportional to x moreover, if the wave velocity is judged by the progress of this mean time the wave moves with a constant velocity 3vq/2, the kinematic wave velocity. Equation (13) shows that the increase of the variance is also proportional to x and the constant of proportionality is a measure of the rate of dispersion. 

This shows that the mean of the temperature wave moves with the kinematic wave velocity and that an apparent diffusion coefficient may be defined to describe the dispersion. This coefficient is the sum of the diffusion coefficients which would be obtained if each effect were considered independently. Such an additivity has been demonstrated by the author for the molecular and Taylor diffusion coefficients elsewhere (Aris 1956) and is assumed in a paper by Klinkenberg and others (van Deemter, Zuiderweg Klinkenberg 1956) in their analysis of the dispersion of a chromatogram. 

Pauchon and Banerjee (1988), in their analysis of bubbly flows, have shown that the kinematic wave velocity based on a constant interfacial friction is weakly stable. They have also obtained a functional dependence of the interfacial friction factor on the void fraction by assuming the kinetic wave velocity equal to the characteristic velocity (kinetic waves are neutrally stable). They have assumed that turbulence provides the stabilizing mechanism through axial dispersion of the void fraction. 

Consider dispersion of a linear kinematic wave in dimensionless form.[14] The governing equation and boundary/initial conditions are ... 

Soil structure, antecedent soil moisture and input flow rate control rapid flow along preferential pathways in well-structured soils. The amount of preferential flow may be significant for high input rates, mainly in the intermediate to high ranges of moisture. We use a three-dimensional lattice-gas model to simulate infiltration in a cracked porous medium as a function of rainfall intensity. We compute flow velocities and water contents during infiltration. The dispersion mechanisms of the rapid front in the crack are analyzed as a function of rainfall intensity. The numerical lattice-gas solutions for flow are compared with the analytical solution of the kinematic wave approach. The process is better described by the kinematic wave approach for high input flow intensities, but fails to adequately predict the front attenuation showed by the lattice-gas solution. 

Because p is constant along the characteristic curve (the kinematic wave), u is also constant along the curve, and the characteristic must be a straight line. Therefore, on an x-t diagram one such line passes through every point in the diagram below the top of the dispersion, and in a region where the density is continuous the lines (kinematic waves) do not intersect. 

Whether kinematic waves are dispersive or not cannot be ascertained because beyond a moderate frequency (say 3 Hz for mode 3 and 5 Hz for mode 4 in the experimental conditions of Tournaire, 1987a,b) the waves are strongly damped and cannot be studied. For frequencies below these limits, no significant frequency effect on the wave velocity can be detected. 

The general dispersion equation. Equation 16, can be rearranged in terms of the celerity of kinematic wave, C. As indicated by Equation 20, the neutrally stable... 

Numerical solutions to the coupled heat and mass balance equations have been obtained for both isothermal and adiabatic two- and three-transition systems but for more complex systems only equilibrium theory solutions have so far been obtained. In the application of equilibrium theory a considerable simplification becomes possible if axial dispersion is neglected and the plug flow assumption has therefore been widely adopted. Under plug flow conditions the differential mass and heat balance equations assume the hyperbolic form of the kinematic wave equations and solutions may be obtained in a straightforward manner by the method of characteristics. In a numerical simulation the inclusion of axial dispersion causes no real problem. Indeed, since axial dispersion tends to smooth the concentration profiles the numerical solution may become somewhat easier when the axial dispersion terra is included. Nevertheless, the great majority of numerical solutions obtained so far have assumed plug flow. 

The equilibrium theory of binary and multicomponent isothermal adsorption systems appears to have been first developed by Glueckauf. More general and comprehensive treatments have been developed by Rhee, Aris, and Amundson and by Helfferich and Klein. The former treatment exploits the analogy between chromatographic theory and the theory of kinematic waves. The detailed quantitive theory has been developed only for ideal Langmuir systeiris without axial dispersion or mass transfer resistance and in which the initial and boundary conditions represent constant steady states. Subject to these constraints the treatment is sufficiently general to allow the dynamic behavior to be predicted for systems with any number of components, provided only that the separation factors are known. In the... 

Theoretically, Vineyard described GIXD with a distorted-wave approximation in the kinematical theory of x-ray diffraction [4]. In terms of the ordinary dynamical theory of Ewald [5] and Lane [6], Afanas ev and Melkonyan [7] worked out a formulation for the dynamical diffraction of x-rays under specular reflection conditions and Aleksandrov, Afanas ev, and Stepanov [8] extended this formalism to include the diffraction geometry of thin surface layers. Subsequently, the properties of wave Adds constructed during specularly diffracted reflections have been discussed in more detail by Cowan [9] and Sakata and Hashizume [10]. Meanwhile, a geometrical interpretation of GIXD based on a three-dimensional dispersion surface has been proposed by Hoche, Briimmer, and Nieber [11]. 

The determination of the equation for the free surface shape and the eigenvalue relation between the amplification factor (3, the wave number k, and the integer n is carried out just as in the evaluation of the dispersion relation for plane waves. In particular, from the kinematic boundary condition Eq. 

Another approach for analyzing the stability of the flow is based on wave-theory. In deriving the characteristics of kinematic and dynamic waves in two-component flow, Wallis has shown that the relations between the velocities of these two classes of waves govern the stability of the two stratified layers [74]. It has been shown that the condition of equal kinematic and dynamic waves velocities corresponds to marginal stability. Following this approach, Wu et al. determined the stratified/ nonstratified transition in horizontal gas-liquid flows [38]. The relations between the dispersion equation. Equation 16, and stability criteria Equation 33 on one hand, and the characteristics of kinematic and dynamic waves on the other hand, (for = 0), was shown in Brauner and Moalem Maron [45]and Crowley et al. [47]. 

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