Kinematic wave velocity, stability

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. 

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]) ... 

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). 

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... 

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). 

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]. 

Indeed, equating from Equation 37 to C (= C ), again renders the condition derived for neutral stability in Equation 33. Stable modes are obtained for c < c d whereas for unstable modes to exist it is required that c > c > c (since V, >0). Hence, it is the relation between kinematic and dynamic wave velocities which essentially determines the stability, as c > c corresponds to unstable modes, whereas modes with cj > c are attenuated. 

Wallis s criterion for the stability of the state of homogeneous fluidization, md > K, can now be stated explicitly through eqns (5.10) and (6.9) for the kinematic- and dynamic-wave velocities respectively it may be expressed in dimensionless form, ( d — mk)/mk > 0 ... 

The extensive experimental evidenee for the predietive ability of the stability criterion, summarized in the preceding sections, provides indirect support for the constitutive relations employed for the kinematic- and dynamic-wave velocities. We now consider more direct means of measuring these quantities. 

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