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234 3. Heterogeneous Processes and Reactor Analysis

where is the density of the solid part (skeletal density) and is the porosity of the par-
s a
ticle. Then, it is clear that Johnson’ate density is the hydraulic density defined in s aggre g
eq. (3.558), since the term (1 – ) is equal to the particle density .
s a p
Several forms of the modified Stoke’s law have been used in a series of studies concern-
ing aggregates in the liquid phase (Li and Yuan, 2002). In these studies, the particle–liquid
density difference has been further modified and adopted for the cases of impermeable bio-
logical/microbial aggregates, permeable aggre and fractal aggre ates, g g ates.
The introduction and use of a hydraulic density, termed in a different way, in liquid–porous
solid fluidization has been done by Nesbitt and Petersen (1998). They point out that for
resins, which are porous in nature, it might be more correct to use an “apparent density” of
fluidization ( ), a property relevant only when the resin is in a suspension, with the fluid
ap
phase intruding into the pores. However, the authors did not use eq. (3.558), but an e xperi-
mental technique, measuring the terminal velocity of the resin particles and e v aluating the
“apparent density” using the Shiller and Naumann terminal velocity model:

Ga Re 18 Re 2.7 1.687 (3.561)
ter ter

where:
dg p 3 f ( ap )
f
Ga (3.562)

2
f
du
Re fp te r (3.563)
ter

for 3.6 10 Ga

ference is also involved in the Shiller and As in Stoke’s equation, the solid–fluid density dif
v v Naumann model and so the ealuated density is equialent in nature to the hydraulic den-
sity defined aboe by Johnson, which is different from skeletal and particle density . The
v
benefit in this case is that the experimental determination of particle voidage and particle
density, which could be quite complicated in some cases, is a v oided and the hydraulic den-
v sity is directly ealuated without using eq. (3.558). The same approach has been follo wed
by Menoud et al . (1998) in a study of a fluidized bed using a chelating resin. et density W
has been used by Feng et al . (2003) for a fluidized-bed ion-exchange system. Finally ,
Griffith et al . (1997) used a gravimetric method for the evaluation of the so-called “effective
particle density during fluidization.”
The particle and bulk densities are commonly used in mass balance equations, since the
mass and the external volume of the particles are involved. On the other hand, the hydraulic
density should be preferably used in hydrodynamic calculations, because b y forces yanc uo
are involved, and so the total mass of the particle should be taken into account, including
the fluid in the open pores. It is obvious that the particle density is equal to the skeletal and
hydrodynamic density in the case of nonporous particles. Moreover, in the case of a porous
solid in a gas–solid system, the gas density is much lower than the particle density, and thus