Homogeneous fluidization stability

The fact that the total number of particles must be conserved during the development of occasional disturbances in a uniform vertical flow or in a homogeneous fluidized bed in itself results in the formation of kinematic waves of constant amplitude, as was first demonstrated by Kynch [48]. Both particle inertia and the nonlinear dependence of the interphase interaction force on the suspension concentration cause an increase in this amplitude. This amounts to the appearance of a resultant flow instability with respect to infinitesimal concentration disturbances and with respect to other mean flow variable disturbances. Various dissipative effects can slow the rate at which instability develops, but cannot actually prevent its development. Therefore, investigating the linear stability of a flow without allowing for interparticle interaction leads inevitably to the conclusion that the flow always is unstable irrespective of its concentration and the physical parameters of its phases. This conclusion contradicts experimental evidence that proves suspension flows of sufficiently small particles in liquids to be hydrodynamically stable in wide concentration intervals [57-59]. Moreover, even flows of large particles in gases may be stable if the concentration is either very low or very high. 

Equations 8.4 hold true for unperturbed states of a homogeneously fluidized bed the stability of which is under question. When the bed consists of small particles, these equations yield... 

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 ... 

This stability criterion features prominently in later chapters, where it will be shown to provide reliable predictions of the stability of the homogeneously fluidized state for a vast range of experimentally tested systems. For now we simply illustrate its ability to differentiate between typical gas- and liquid-fluidized beds by means of a simple example sand particles of diameter 200 pm and density 2500 kg/m fluidized first by water (density 1000 kg/m, viscosity 10 Ns/m ) and then by air (density 1.3 kg/m, viscosity 1.7 x lO Ns/m ). 

The contemporaneous, unpublished work of Wallis (1962), referred to earlier, contained an additional term in the momentum equation that resulted in a rather different conclusion concerning the stability of the homogeneously fluidized state this will be considered in some detail in the following chapters. 

A trivial solution to eqns (7.9) and (7.10) is simply the steady-state condition, e = eo (a constant) and Up = 0, which reduces the momentum equation, eqn (7.10), to = 0. Given a constitutive expression for F, this relation delivers the constant, steady-state solution eo for void fraction throughout the bed, a function solely of the fluid flux C/q. Such a solution represents the condition of homogeneous fluidization, and always satisfies the above particle-phase equations. The question now to be posed concerns the stability of this steady-state condition is it sustainable in the face of small fluctuations in void fraction or particle velocity Such... 

Eqns (7.21) and (7.22) provide a clear answer to the stability question for the rather general formulation of the problem considered above, k, the wave number, is a real, positive quantity, so must be positive. Equation (7.22) thus yields the condition that C > Bv and hence, from eqn (7.21), that a must always be positive. The simple conclusion arising from the analysis is that the homogeneously fluidized state is intrinsically unstable. 

The stability of the homogeneously fluidized state is analysed in terms of the full set of system equations, eqns (8.21)-(8.24), in Chapter 11. For the case of gas fluidization, however, where pp pf, terms in the fluid momentum equation that are proportional to fluid density will be negligible compared to the drag and pressure gradient terms, which are involved in supporting the fluidized particles. Equation (8.24) then reduces to ... 

In Chapters 9 and 12, the predietions of eqn (8.36) will be compared with the copious body of observations reported for the stability condition of a wide variety of experimentally investigated fluidized beds. By way of introduction to these comparisons, an example of the particle bed model predictions of the effect of particle diameter on the stability of gas fluidized beds reflecting the observations reported at the start of this chapter, which confirmed the existence of the stable, homogeneously-fluidized state is illustrated in Figure 8.3. 

Rietema, K. and Mutsers, S.M.P. (1978). The effect of gravity upon the stability of a homogeneously fluidized bed, investigated in a centrifugal field. Fluidization. Cambridge University Press. 

This relation is precisely Wallis s criterion for the linear stability limit for homogeneous fluidization (the minimum bubbling point), the left- and right-hand sides comprising the squares of, respectively, the familiar forms for the dynamic-wave speed mq, and the kinematic-wave speed un ... 

The ultimate cause of bubble formation is the universal tendency of gas-solid flows to segregate. Many studies on the theory of stability [3, 4] have shown that disturbances induced in an initially homogeneous gas-solid suspension do not decay but always lead to the formation of voids. The bubbles formed in this way exhibit a characteristic flow pattern whose basic properties can be calculated with the model of Davidson and Harrison [30], Figure 5 shows the streamlines of the gas flow relative to a bubble rising in a fluidized bed at minimum fluidization conditions (e = rmf). The characteristic parameter is the ratio a of the bubble s upward velocity u, to the interstitial velocity of the gas in the suspension surrounding the bubble ... 

Anderson, T.B. Jackson, R.A. Fluid mechanical description of fluidized beds stability of the state of uniform fluidization. I EC Fundam. 1968, 7,12. Verloop, J. Heertjes, P.M. Shock waves as a criterion for the transition from homogeneous to heterogeneous fluidization. Chem. Eng. Sci. 1970, 25, 825. 

Koch DL, Sangani AS (1999) Particle pressure and marginal stability limits for a homogeneous monodisperse gas-fluidized bed Kinetic theory and numerical simulations. J Fluid Mech 400 229-263... 

Hydrodynamic stability of uniform vertical suspension flow has been theoretically treated for more than 30 years (see reference [15,20,29,32-34,43,47], and also reference [48-36]). Much of this work has been undertaken when analyzing the important problem of reasons causing the transition from homogeneous (particulate) fluidization to nonhomogeneous (aggregative) fluidization, and subsequently, providing for the spontaneous origination in a fluidized bed of cavities (bubbles) almost devoid of particles. 

---