Chemical Engineering January 2016 - 61

50
50
40
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30
30
20
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10
10
10
20
30
Hopper angle from vertical, deg
FIGURE 9a. The flow factor (ff) is the ratio of the arch stress to the major
consolidation stress and depends on the effective angle of internal friction,
the hopper geometry, the hopper angle, and the wall friction angle. The
above diagram gives flow factors for conical hoppers,  = 50 deg
arch is always greater than the
material's cohesive strength, and
there is no minimum outlet dimension
requirement to prevent cohesive
arching
* The Flow Function lies above the
flow factor and the curves do not
intersect. The bulk solid will not
flow due to gravity alone, and another
means of discharging the
powder must be employed
* The Flow Function and flow factor
intersect, as shown in Figure 10.
At the point where the two lines
intersect, the arch stress and the
cohesive strength of the bulk solid
are the same and equal to the critical
stress (crit)
The minimum outlet diameter or
width to prevent a cohesive arch
from developing, Bmin, can then be
calculated from Equation (2):
crit
(2)
The function H(') is shown in Figure
11. For funnel flow hoppers,
the outlet must be large enough to
prevent a cohesive arch and stable
rathole from developing. To prevent
the formation of a stable rathole, the
hopper outlet diagonal should equal
or exceed the critical rathole diameter,
DF. The critical rathole diameter
is calculated by first determining the
40
50
10
20
30
40
50
Hopper angle from vertical, deg
FIGURE 9b. The diagram shown here gives flow factors for hoppers with flat
walls and slotted outlets,  = 50 deg
major consolidating pressure, 1, on
the powder. In some cases, the consolidating
load can be estimated by
the Janssen equation (Equation (3)):
(3)
The critical rathole dimension is then
calculated using Equation (4):
(4)
In Equation (4), fC is the cohesive
strength of the powder at the calculated
consolidation pressure.
For wedge-shaped and pyramidal
hoppers, stable ratholes will not form
if the diagonal of the outlet is equal
to DF or greater. The diameter of the
outlet of a conical funnel-flow hopper
should not be less than the critical
rathole dimension.
If a hopper with a circular outlet
is designed with an opening large
enough to prevent the development
of a stable rathole, cohesive
arching will not occur. For wedgeshaped
hoppers, the width of the
slotted outlet must be large enough
to prevent a cohesive arch from developing.
The same procedure that
is used to determine the minimum
outlet width to prevent arching in a
ChemiCal engineering www.Chemengonline.Com january 2016
wedge-shaped mass--flow hopper
is followed, except that a flow factor
of 1.7 is used.
Note that these analyses assume
continuous handling of the powder. If
the powder is to be stored at rest for
a period of time, time tests should be
conducted. Time tests are described
in ASTM D-6128 and D-6773 [2, 3].
Feeder considerations
Feeders can also be a source of
hopper flow problems if the incorrect
devices are used or if they are improperly
designed. This is especially
true for hoppers with slotted outlets,
where feeders should be designed
to draw uniformly along the entire
cross-section of the outlet in order
for mass flow to occur. However,
even hoppers with round outlets can
have uneven flow if a proper interface
is not utilized.
Rotary valves are often used beneath
hoppers with round outlets. They
are particularly useful for applications
where a seal must be provided to prevent
air from flowing out of or into the
hopper outlet. If a rotary valve is used,
a short vertical spool section should
be installed between the hopper outlet
and valve inlet. Otherwise, material
may flow preferentially from the upside
of the valve and affect the flow pattern
inside the vessel. This is shown in Figure
12.
Screw feeders are primarily used
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Angle of wall friction, deg
Angle of wall friction, deg
ff = 1.1
ff = 1.2
ff = 1.2
ff = 1.4
ff = 1.6
ff = 4
ff = 1.8
ff = 2.5
ff = 3
ff = 4
ff = 1.3
ff = 3
ff = 2.5
ff = 4
ff = 3
ff = 2.5
ff = 1.8
ff = 2
ff = 1.4
ff = 1.6
ff = 2.5
ff = 1.3
ff = 1.6
ff = 1.4
ff = 2
ff = 1.8
ff = 2
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