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236 3. Heterogeneous Processes and Reactor Analysis

and
V V (3.568)
f f p

In eq. (3.564), the gravitational force accounts for the mass of the fluid, which is included
in the volume of the particle. So,

M V ( ) V (3.569)
p p p p f h p
v
Using the aboe equations, the terminal velocity relationships can be deriThe drag ed. v
force is
F ( V g ) (3.570)
d h f p

Finally, a formal definition of terminal velocity is deried from the equations abo v v e:

gV 2  ( ) 0. 5
u  p h f  (3.571)
ter C
  A pr f D  
For a spherical particle,

 d 3 
 p 
V p  6  d 2 p

A  d 2  3 (3.572)
pr p
 
 4 

Note that the “projected area” is the area of the object projected on the perpendicular to
the flow plane; for a sphere, this is equialent to the area of a circle with the same radius. v
Thus, the folloinition is deried (Perry and Green, 1999): wn def wing well-kno v

gd 4  (  0.5
)
u ter  p h f  (3.573)
 3 f C D 

where C D f is the drag coeficient. This is the constant terminal velocity with which a parti-
cle falls when the accelerating effect of gravity balances the drag force.
Within the intermediate Re ter region (0.1 < Re p < 1000), the drag coeficient can be esti- f
mated via the relation

24 0.7
C 1 0.14 Re (3.574)
D p
Re p