When a fluid is passed upwards through a bed of particles
the pressure loss in the fluid due to frictional resistance
increases with increasing fluid flow. A point is reached when
the upward drag force exerted by the fluid on the particles
is equal to the apparent weight of particles in the bed.
At this point the particles are lifted by the fluid, the separation
of the particles increases, and the bed becomes fluidized.
The force balance across the fluidized bed dictates that
the fluid pressure loss across the bed of particles is equal
to the apparent weight of the particles per unit area of the bed.
Thus:
weight of particles - upthrust on particles
pressure drop = -------------------------------------------
bed cross sectional area
For a bed of particles of density 
p,
fluidized by a fluid of density f
to form a bed of depth H and voidage 
in a vessel of cross sectional area A:
A plot of fluid pressure loss across the bed versus superficial
fluid velocity through the bed would have the appearance of Figure 1.
Figure 1: Pressure drop versus fluid velocity for packed
and fluidized beds
The straight line region OA is the packed bed region.
Here the solid particles do not move relative to one
another and their separation is constant. The pressure loss versus
fluid velocity relationship in this region is described in general
by the Ergun equation, Equation 3. (See Rhodes, 1998, chapter
4 for a detailed analysis of packed bed flow).
The region BC is the fluidized bed region where Equation 1 applies.
At point A it will be noticed that the pressure loss rises above
the value predicted by Equation 1. This rise is more marked in powders
which have been compacted to some extent before the test and is associated
with the extra force required to overcome interparticle attractive forces.
The superficial gas velocity at which the packed bed becomes a fluidized
bed is known as the minimum fluidization velocity, Umf. This is also
sometimes referred to as the velocity at incipient fluidization ("incipient"
means "about to begin"). Umf increases with particle size and particle
density and is affected by fluid properties. It is possible to derive
an expression for Umf by equating the expression for pressure loss in
a fluidized bed (Equation 2) with the expression for pressure loss across
a packed bed. Thus substituting the expression for (-
p)
for a fluidized bed from Equation 2 into the expression for
(-p) for a packed bed from Equation 3:
and so
where Ar is the dimensionless number known as the Archimedes number
and Remf is the Reynolds number at incipient fluidization,
In order to obtain a value of Umf from Equation 7 we need
to know the voidage of the bed at incipient fluidization,
= mf. Taking mf
as the voidage of the packed bed, we can obtain a crude Umf.
However, in practice voidage at the onset of fluidization may be considerably
greater than the packed bed voidage. A typical often used value of
mf is 0.4. Using this value, Equation 7 becomes:
Wen and Yu (1966) produced an empirical correlation for Umf with
a form similar to Equation 8: (Eq. 10 is an alternate expression.)
The Wen and Yu correlation is valid for spheres in the range
0.01 < Remf < 1000.
For gas fluidization the Wen and Yu correlation is often taken as being
most suitable for particles larger than 100 
m, whereas
the correlation of Baeyens (1974) , shown below in Equation 11, is best
for particles less than 100 m.
