Tipos de modulación

The stability criterion of Mutsers and Rietema was:

At the bubble point velocity when e= Emb, the equality would hold.

Foscolo and Gibilaro (1984) followed the approach of Verloop and Heertjes (1970) to reach the criterion for the prediction of transition from particulate to aggregate fluidization using only the hydrodynamics of fluidized bed. Considering gravitational, buoyancy and drag forces acting on a single particle, the elastic wave velocity, Ue, was determined as:

Ue = ^3.2 gdp (1 - e) ( p p - p ) / p p (2.24)

while the voidage propagation velocity, Ug, was obtained as:

U e - n U , ( l - e ) (2.25)

where n and Ut are the Richardson-Zaki indices to be discussed in next section. Therefore on the critical condition proposed by Wallis, the criterion for transition from particulate to aggregate fluidization is:

P-5 , . 0.5

P p - P

Ut

a 0.56 n (1 - 6^)°^ eg”* (2.26)

Pp

where Ey is the voidage at which the transition from particulate to aggregate fluidization occurs. For cases of practical interest, Ey = Emb •

Chapter 2: Transition from Particulate to Aggregate Fluidization 6 ^

In laminar flow conditions, Rct< 0.2, the terminal velocity could be determined by knowing the physical properties of the system as:

U , - W (2.27)

‘ 18n

Substituting (2.27) in (2.26) and taking n = 4.8:

M- 38

((Pp - P) Pp g dp)

g ; ^ a 0 . 1 4 9 ( l - e b r ^ (2.28)

Gibilaro et al. later confirmed (1985) that there were undoubtedly circumstances where interparticle forces could play an important role such as the influence of electrostatic and magnetic forces zind the adhesive effects of moisture in gas fluidized systems. They suggested that interparticle forces could influence the dynamic behaviour of the bed whilst remaining neutral to its equilibrium state. Under equilibrium conditions, all particle layers are equally attracted by the layers immediately below and above: the net force is therefore zero. Thus both the steady-state expansion characteristics and the continuity wave velocity (or voidage propagation velocity, Ue) are unaffected by the inclusion of interparticle forces. Therefore Ug remains as equation (2.25).

Under non-equilibrium conditions the interparticle forces can no longer be neglected; they affect the elastic modulus of the suspension and, as a result, the dynamic or elastic wave velocity Ue which now could be obtained by:

|3 .2 g d p (l-e )(p p -p ) 4mk(l - e)“

u e = J--- + ---"2--- (2.29)

V Pp JtdpPp

where m and k are constants which should be determined for each system. This equation is very general and no method was introduced by them for evaluating the

Chapter 2: Transition from Particulate to Aggregate Fluidization 65

constants explicitly. The first term under the square root sign is identical to equation (2.24) and the second term incorporates the interparticle forces. The role of interparticle attractive forces in stabilizing fluidized beds, according to the improved criterion is illustrated in Fig. 2.6: the continuity wave velocity, Ug, is unaffected, whereas the dynamic wave velocity, Ue, increases with interparticle attraction with the consequence that it makes its first intersection with Ug at progressively higher values of the voidage, e. Voidages below Emf are unattainable in practice, so that an intersection in the range Emf > e> 0 indicates a bed that displays aggregate behaviour from the onset of fluidization; intersections at higher voidages signify an initial range of stable particulate expansion and fully particulate behaviour is achieved where the interparticle attractive forces are sufficient to prevent any intersection occurring.

3 - 3 3 Particulate ___ expansion r a n g e . ^ - ^ FuJJv % r e g a te behaviour ^mf V oidage, e

Fig. 2.6. Fluidized bed stabilization by interparticle forces: the full line represents the continuity wave velocity, Ua and broken lines represent dynamic wave velocities, Ug, for increasing values of inteiparticle forees|(Gibilaro et al., 1985).

Chapter 2; Transition from Particulate to Aggregate Fluidization 6 ^

Reiling (1992) took interparticle forces into account as well as the hydrodynamics to provide a more accurate criterion. The voidage propagation velocity was the same as equation (2.25) however the elastic wave velocity, considering the interparticle forces was found asl (See Appendix A):

\ C p p ( l - i

Ue - . L , ^--- (2.30)

^mb)^mb

where Otjnb is the tensile stress of emulsion phase at the incipient bubbling velocity and C* is the proportionality constant defined as:

interparticle forces across a single layer of particles ^ . “ drag force acting on a single layer of particle ^ ^

* ôot (^tjp - _ Hmb

V

where a tjp is the tensile stress of the loose packed powder and AP*mb is the bed pressure drop at incipient bubbling velocity. AP*mb is a hypothetical quantity that can be thought of as follows: As gas is passed through a bed of particles, the bed pressure increases linearly with gas velocity up to the point of minimum fluidization (Fig. 2.7). As the bed continues to expand above Umf in the homogeneous region, the bed is stabilized by a sort of cavity structure in the region of bed expansion between Umf and Umb- The cohesive interparticle interactions give rise to a restoring force that counteracts the drag force. This restoring force is continuously lowered due to increase in porosity because of a reduction in the number of particle contacts. At a high enough gas velocity, the cross-linked particle networks are finally broken, there is a decrease in the internal stress of the emulsion phase. The bed reaches a maximum voidage and bubbling commences. The pressure drop at Umb is AP^mb- This gives rise to the definition of C* in equation (2.32).

Chapter 2: Transition from Particulate to Aggregate Fluidization 67