Mechanics of Fluids

242 Laminar flow between solid boundaries
may be reached in which the upward force on the particles equals their
weight. The particles are then supported by the fluid and the bed is said to
be fluidized.
If ρs and ρ represent the density of the solid particles and the fluid respectively
and ε the porosity, then, for a thickness l if the bed, the effective
weight of the particles (i.e. actual weight minus buoyancy) divided by the
cross-sectional area is (ρs −ρ)gl(1−ε). This must be exactly balanced, when
fluidization begins, by the drop in piezometric pressure through the bed as
given by eqn 6.72. Hence the minimum fluidizing velocity is given by
u f =
(ρs − ρ)gε 3
(1 − ε)µk(S/Vs) 2
If the velocity is increased beyond this value, there is little further change
in the pressure drop (although some fluctuation may occur as fluidization
begins because movement of the particles produces non-uniform porosity).
When ρs and ρ are not widely different the bed expands uniformly; this is
known as particulate fluidization. For a large density difference, as with a
gas, bubbles of fluid, mostly devoid of solid particles, may rise through the
bed; the upper surface then looks rather like that of a boiling liquid. This type
of fluidization is called aggregative and the bed is described as a boiling bed.
When the fluid velocity exceeds the settling velocity for a particle the
particle is carried away. Where the particles are not of uniform size, this
process is of course gradual, the smaller particles being removed at lower
velocities.
Fluidization is a valuable method for handling solids such as grain and pulverized
coal: also, because it allows the maximum area of contact between
solid and fluid, it finds wide application in catalytic and heat transfer
processes in the chemical industry.
PROBLEMS
6.1 Glycerin (relative density 1.26, dynamic viscosity 0.9 Pa · s) is
pumped at 20 L · s −1 through a straight, 100 mm diameter pipe,
45 m long, inclined at 15 ◦ to the horizontal. The gauge pressure
at the lower, inlet, end of the pipe is 590 kPa. Verify that the
flow is laminar and, neglecting end effects, calculate the pressure
at the outlet end of the pipe and the average shear stress at
the wall.
6.2 Show that when laminar flow occurs with mean velocity um
between extensive stationary flat plates the mean kinetic energy
divided by mass of the fluid is 1.543 u 2 m /2.
6.3 A Bingham plastic with a yield stress of 120 Pa and apparent
viscosity 1.6 Pa · s is to be pumped through a horizontal pipe,
100 mm diameter and 15 m long at 10 L · s −1 . Neglecting end
effects and assuming that the flow is laminar, determine the
pressure difference required.