Kinematic wave

Equation 34 has the form of the kinematic wave equation and represents a transition traveling with the wave velocity given by... 

Ketonization of acids, catalysts for, 19 89 Kinematic waves, 37 258-259 Kinetics... 

The signals were recorded as electrical potential in millivolts. The well known Maxwell s formula and an adjustable empirical coefficients were used to obtain the equivalent volume fraction of liquid [43]. Since it is known that kinematic waves exist only in the frequency of some few Hertz, hardware low pass filter with 20 Hz cutting frequency was included for each channel in the electronic unit. The filter was tuned at 82.66 Db. Data were acquired by a computer at 100 Hz. The following comments regarding the above apparatus description are in order ... 

A. Soria. Kinematics Waves and Governing Equations in Bubble Columns and Three Phase Fluidized Beds. PhD thesis. University Western Ontario, Canada, 1991. 

The speed of the adsorption wave can be readily derived by introducing the linear isotherm assumption and the chain mle derivative of q with respect to t. The wave speed results because the assumptions turn Eq. (9.10) into a kinematic wave equation and the wave speed W is instantly recognized as ... 

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... 

The theory of kinematic waves, initiated by Lighthill Whitham, is taken up for the case when the concentration k and flow q are related by a series of linear equations. If the initial disturbance is hump-like it is shown that the resulting kinematic wave can be usually described by the growth of its mean and variance, the former moving with the kinematic wave velocity and the latter increasing proportionally to the distance travelled. Conditions for these moments to be calculated from the Laplace transform of the solution, without the need of inversion, are obtained and it is shown that for a large class of waves, the ultimate wave form is Gaussian. The power of the method is shown in the analysis of a kinematic temperature wave, where the Laplace transform of the solution cannot be inverted. 

A kinematic wave may be called linear if the relationship between the flow and the concentration can be expressed by one or more linear equations, algebraic or differential. The term linear may also be applied when a diffusion term is included in the continuity equation as is done in 3 of Lighthill ... 

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. 

That this is essentially a kinematic wave is seen by dropping the conductivity term in (24) and writing k = Hf + Hs, the concentration of heat and q = vhfT, the flow of heat. We then recover the kinematic wave equation given by Lighthill Whitham. If thermal equilibrium were instantaneously attained so that... 

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 ... 

Pauchon C, Banerjee S (1988) Interphase momentum interaction effects in the averaged multifield model, P art II Kinematic waves and interfacial drag in bubbly flows. Int J Multiphase Flow 14(3) 253-264... 

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