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                  3.9 P article Analysis                                 233


                                                  Table 3.15

                                              The density def initions
                  Density                   Volume included in definition
                                 Symbol  Solid  Open pore  Closed pore  Interparticle
                                          olume material v  volume  volume  void volume

                  Bulk             b       √             √           √          √
                  Particle         p       √             √           √
                  Skeletal         s       √             √
                  True             t       √




                  where     is the fluid densityis introduced . In the present book,  the term “hydraulic density”
                        f
                  in order to highlight the use of this kind of density in hydrodynamic calculations in olv- v
                  ing   suspension   of particles.
                    Despite the fact that this hydraulic density is essential to many calculations in olving v
                  fluidization and the suspension of particles, it is characteristic that in the related literature,
                  authors use the terms “particle density” or “solid density” without specifying if the fluid
                  in the pores has been taken into account. Ho the subject of hydraulic density has
                   v
                   er
                   we
                   ,
                  been analyzed in studies of the behavior of impermeable aggreates in fluids. As these g
                  aggregates could be seen as porous particles, the releant analysis is interesting and will v
                  be presented here.
                  The case of hydraulic density
                    g
                  The case of impermeable spherical aggreates has been analyzed by Johnson   et al  .
                               ork,
                  (Johnson   et al  ., 1996). In this w the settling velocity of aggreates in liquids using g
                  Stoke’s law has been studied and a modified Stokw has been introduced. The dif s la fer- e’
                  ence from the classical Stokw is that the density difference is expressed in terms of e’
                  s la
                  g the aggreate density.
                                                         (1     )                   (3.559)
                                              a   a  f  s  a

                  where:
                             a    the aggreate porosity g
                               the aggregate density
                            a
                               the density of the particles composing the aggre g ate.
                            s
                  Thus, the aggreate–fluid density dif ference becomes g

                                                  (1
                                                    )
                                   a  f  a  f  s   a   f                            (3.560)
                                             (1         (1  f  )  a    (1    )(       )  s
                                               )
                                                                     f
                                                               a
                                               a
                                           s
                  To highlight the difference from the classical Stoks equation, the density difference in
                   e’
                  the latter is (       ), where     is the skeletal density of the rigid particle. No it is easy to w ,
                             s  f        s
                  illustrate the analogy to a porous particle. A porous particle can be viewed as an aggregate,
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