Big Chemical Encyclopedia

Chemical substances, components, reactions, process design ...

Articles Figures Tables About

Kinematic wave velocity

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. [Pg.136]

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. [Pg.139]

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. [Pg.144]

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. [Pg.27]

It is of particular interest to note that the wave velocity at neutral stability is in fact identical to the definition of kinematic wave velocity, (Wallis, [74]) ... [Pg.329]

Equations 40 indicate that the locus for which the kinematic wave velocity is equal to that of the dynamic wave, = C, represents neutral stable wave modes. [Pg.348]

The kinematic-wave velocity mk is the limiting value of dLjjdt as the amplitude of the imposed fluid flux change AU = Ui U ) approaches the infinitesimal limit. Under these conditions eqn (5.8) yields ... [Pg.50]

Figure 6.3 shows dynamic- and kinematic-wave velocities, eqns (6.9) and (5.10) respectively, for these two systems as functions of void fraction. It will be seen that for water fluidization, the dynamic-wave velocity is always well in excess of the kinematic-wave velocity, the reverse being the case for air fluidization - conforming to the well known behaviour of these systems, in which the water fluidization is always homogeneous and the air fluidization is always bubbling. [Pg.58]

Eqn (8.36) is the statement of the general Wallis (1962, 1969) criterion for fluidized bed stability. The specific forms for the dynamic- and kinematic-wave velocities arising from the model formulation, eqns (8.19) and (8.30), yield the closed form of this criterion, which was derived indirectly in Chapter 6, and expressed in eqn (6.10). [Pg.81]

It will be seen that for the larger (150 pm) particles the stability limit (where the dynamic and kinematic wave velocities intersect) occurs at a physically unobtainable void fraction (off scale in Figure 8.3), smaller than the packed bed value of 0.4. This indicates a system predicted to start bubbling ( K > Md) right from the minimum fluidization condition - behaviour typical for normal gas fluidization. For the smaller 70 pm, particles the... [Pg.82]

Figure 8.3 Dynamic- and kinematic-wave velocities as functions of void fraction for the fluidization of alumina particles by air (pp = 1000 kg/m, = 150 and 70 pm). Figure 8.3 Dynamic- and kinematic-wave velocities as functions of void fraction for the fluidization of alumina particles by air (pp = 1000 kg/m, = 150 and 70 pm).
Water-fluidization experiments are discussed in more detail in Chapter 12, after the derivation of a stability eriterion in Chapter 11 that is based on the full, two-phase model whieh is more appropriate for liquid-fluidized systems for which particle and fluid densities are relatively close. It will be seen, however, that the kinematic-wave velocity expression emerging from this more complete description is identical to that of the simplified, one-phase treatment considered here, eqn (9.3). [Pg.97]

Figure 9.12 Kinematic-wave velocities in water-fluidized beds comparison of experimental evaluations (points) with theoretical predictions (continuous curve, eqn (9.3)). Figure 9.12 Kinematic-wave velocities in water-fluidized beds comparison of experimental evaluations (points) with theoretical predictions (continuous curve, eqn (9.3)).
The kinematic-wave velocity relation is well-established, and experimental evaluations of its validity, such as those described above, are easy to conceive and execute. This is by no means the case for the dynamic-wave velocity relation. [Pg.99]

The kinematic-wave velocity mk maintains the same expression as in the single-phase treatment, eqn (11.7). [Pg.131]


See other pages where Kinematic wave velocity is mentioned: [Pg.139]    [Pg.142]    [Pg.15]    [Pg.212]    [Pg.50]    [Pg.80]    [Pg.97]    [Pg.99]    [Pg.255]   
See also in sourсe #XX -- [ Pg.329 ]




SEARCH



Kinematic

Kinematic wave velocity, stability

Kinematic waves

© 2024 chempedia.info