6.3 Particulate fluidisation No transition to aggregate fluidisation 54.6 Aggregate behaviour observed Aggregate fluidisation for 0 .5 6 < e <0.93 316.5 Aggregate behaviour observed Aggregate fluidisation for £< 0.97
Table 2.1 The transition from particulate to aggregate fluidisation in liquid beds. Comparison o f the results o f Harrison et al. (1961) with the prediction o f equation( 2.42 ) for various
liquids. System: lead shot; glycerol-water mixtures; pg=11320 kgm"^
( Foscolo et al. , 1984)
dp=0.77mm.
Material Particle Density (k g m Re, Experimental observations Prediction of equation 2.42 Resin 1500 8.7 Particulate fluidisation No transition to aggregate fluidisation Glass 2900 32.5 Particulate fluidisation No transition to aggregate fluidisation Steel 7430 72.1 Aggregate behaviour Aggregate fluidisation for0.5<e<0.95 Lead 11320 100 Aggregate behaviour Aggregate fluidisation for 0.42<e<0.97
T ab le 2.2 The transition from particulate to aggregate fluidisation in liquid beds. Comparison o f the results o f Harrison et al. (1961) with the
prediction o f equation ( 2.42 ) for various particles. System: solid spheres, paraffin;
pf=780 kgm"^ , jLi =0.002 Nsm’^. ( Foscolo et al. , 1984)
Particles
Dynamic wave velocity(mms’^) Ambient water
Mf=0.001 N s PplOOO kgm‘^
Material Pg(kgm'3) dp (pm ) Ug(measured) "el
Acetate 1270 2000 41 89
Acetate 1270 4000 82 126
Deirin 1420 5000 95 167
lead glass 2900 375 55 68
Copper 8640 550 97 96
Table 2.3 Comparison o f experimental dynamic wave
velocities with the predictions o f equation (2 .3 8 ) ( Gibilaro et al. ,1990).
F oscolo et al (1989 ) attributed to the above discrepancy due to the following reasons
a) The fluid pressure acting on the particle in the model was approxim ated by the expression for it's mean value in a homogeneous unperturbed bed . This is given by
where, is the variation o f fluid pressure on a particle, 6 is
the bed voidage, and Pp are the fluid and particle densities.
As a consequence o f this approximation, the particle phase in fluidisation is taken as a non-flow ing fluid, leading to the underestimation o f dynamic wave velocity predicted by equation 2 .3 8 .
b) Added mass is another factor to be taken into considerations when there is a relative acceleration between the particle and fluid phase. This effect is expected to be n egligib le for gas fluidisation, where the particle density is so much greater than that o f the fluid . For liquid fluidisation o f relatively low density material, however, the effect could be significant. This w ill in turn be manifested in an error in the prediction o f the dynamic wave velocity and hence the onset o f instability.
The problem o f fluid pressure field approximation was overcome by W allis(1969) who dealt with the general two-phase problem by using both phase momentum equations to eliminate the common pressure gradient term and hence, arrived at a more complete formulation for dynamic wave velocity given by
(1 + R) 2.44
with
D _ Cp) Pf
(Go Pp) 2.45
where, Ufo is steady-state fluid interstitial velocity, Uei is the dynamic wave velocity derived from the single phase model and Gg, Steady-State bed voidage.
Table 2.4 shows the variation o f dynamic wave velocity as pred icted from equation 2 .4 4 again st the ex p erim en ta lly m easu red v a lu e s for d iffe r e n t d e n sity p a r tic le s . T h e corresponding data obtained using the simple model as described by equation 2.38 are also included for comparison. For the case o f a fluidised bed where the particle density is generally much greater than that o f fluid , the two phase model approaches the single phase approximation, for most cases o f practical interest, which includes all gas fluidised systems and liquid fluidisation o f moderate density particles. W hereas, for lighter particles , the
Dynamic wave velocity (mms'^) Ambient water:
viscosity= 0 . 0 0 1 Nsm"^
D ensity= lOOOkgm"^
Material Pp(kgm"3) dp(jum) u^(measured) ^el ^e2
Acetate 1270 2000 41 89 64
Acetate 1270 4000 82 126 92
Deirin 1420 5000 95 167 127
lead glass 2900 375 55 68 56
Copper 8640 550 97 96 96
Table 2.4 Comparison between experim ental and theoretical values of dynamic wave velocity . u^j: predicted from equation 2.38 and u^ 2 predicted from
equation 2.44.
discrepancy observed in the table 2.4 can be substantially red u ced .
Although the solution to the pressure gradient improves the dynamic velocity values o f low density particles greatly, h ow ever, the d iscrep an cy b etw een the exp erim en tal and th eoretical value o f dynam ic w ave v e lo c ity in d icates the importance o f the added mass which was ignored in the previous two phase analysis.
Gibilaro e al. (1990) improved the theoretical consideration further, by employing the fluid and particle forces derived from the previous analysis (1984, 1987) into à combined momentum equation o f two incompressible phases in a one-dimensional flow derived by Wallis (1989).
Wallis's equation was derived, from the basic assumption o f Guerst (1985) for the kinetic energy density, K .E.j, for two-phase flow . This is given by
K + - ^ P p ( l - e ) v^ + ^ P j C , ( u - v ) 2 46
all the sym bols in the above equation have been described previously. The first and the second terms o f the above equation represent the k in etic energy due to flu id and p article respectively and the last term represents the kinetic energy due to flow field brought about by the relative motion between the
fluid and particle phase. The constant Ci in the last term is defined as the inertial coupling coefficient, and for the case o f a "Maxwellian suspension " adopts the form given by:
C |- —5 ( 1 - e)
2.47
where, C% is the inertial coupling coefficien t and e is the bed v oid age.
Using the appropriate expression for C\, for the case o f a solitary particle travelling through a stagnant in v iscid fluid, delivers the expected result for the total kinetic energy, K.E.j;
K.E^ = -iM v '
2.48
where, M is the sum of the particle mass and one half the mass o f the fluid that the particle displaces. This provided the ground for the work by W allis (1989), who incorporated the added mass effect into the formulation o f combined momentum equations for two incom pressible phases;
■ a v
a v ■
■ a u
a u i
+ v a z j - p , | . a t " " Oz _
+ f , - f p = 02 . 4 9
leading to a similar equation given by
‘2p p + p , ; a v
5v
3p,
L
2 JL a t
~ 0 2.50where, ff and fp represent the forces per unit volume o f fluid and particle phase respectively.
The remarkable conclusion indicates that all analysis based on equation 2.49 can be extended to take into account added mass effects by simply replacing particle and fluid densities everywhere they occur by new values which are increased by factor o f 0.5 as shown in equation 2.50.
The success o f incorporating the added mass into combined momentum equations, initiated the work on the effect o f the added mass on the behaviour o f liquid fluidised beds (Gibilaro et al. 1990). The combined momentum equation for the two-phase particle bed formulation was found to be
■
Pf[lr ^ ^ w] ^ (pp- Pf)g
= 0
2.51 W it h Uo - V 4. 8n 2.52 F„ = ( 1 - e ) ( P p - p , ) g and Ep =3. 2(1 - e ) g d p ( P p - p j 2.53 5 9where, Fj) is the fluid-particle interaction force and Ep is the elastic modulus for the particle phase.
The interaction force Fq and the elastic modulus Ep defined in equation 2.51 are obtained from the previous analysis (Foscolo et al. 1984 andl987).
Equation 2.51 therefore represents an explicit formulation