Flow Behavior of Granular
Materials and Powders
Part III
Asst. Prof. Dr. Muanmai Apintanapong
Bin and Hopper Design
Figure 1: a. pressure in a silo filled with a fluid (imaginary); b. vertical stress
after filling the silo with a bulk solid; c. vertical stress after the discharge of
some bulk solid
Gravity flow through orifices
Law of hydrodynamics do not apply to the flow of
solid granular materials through orifices:
Pressure is not distributed equally in all directions due to
the development of arches and to frictional forces between
the granules.
The rate of flow is not proportional to the head, except at
heads smaller than the container diameter.
No provision is made in hydrodynamics for size and
shape of particles, which greatly influence the flow rate.
Particle history
Hopper Flow Modes
Mass Flow - all the material in the hopper is
in motion, but not necessarily at the same
Funnel Flow - centrally moving core, dead or
non-moving annular region
Expanded Flow - mass flow cone with funnel
flow above it
Mass Flow
all the material in
the hopper is in
motion at
discharge, but
not necessarily at
the same velocity
Material in motion
along the walls
Does not imply plug
flow with equal
Typically need 0.75 D to 1D to
enforce mass flow
Funnel Flow
If a hopper wall is too
flat and/or too rough,
funnel flow will
(centrally moving
core, dead or nonmoving annular
“Dead” or nonflowing region or
stagnant zone
Expanded Flow
mass flow cone with
funnel flow above it
Funnel Flow
upper section
Mass Flow
Problems with Hoppers
Ratholing/Piping and Funnel Flow
Insufficient Flow
Irregular flow
Inadequate Emptying
Time Consolidation - Caking
• Occurs in case of funnel flow.
• The reason for this is the strength
(unconfined yield strength) of the bulk
• If the bulk solid consolidates
increasingly with increasing period of
storage at rest, the risk of ratholing
Funnel Flow
-Inadequate Emptying
-Structural Issues
In case of centric filling, the larger particles
accumulate close to the silo walls, while the smaller
particles collect in the centre.
In case of funnel flow, the finer particles, which are
placed close to the centre, are discharged first while
the coarser particles are discharged at the end. If
such a silo is used, for example, as a buffer for a
packing machine, this behaviour will yield to
different particle size distributions in each packing.
In case of a mass flow, the bulk solid will segregate
at filling in the same manner, but it will become
"remixed" when flowing downwards in the hopper.
Therewith, at mass flow the segregation effect
described above is reduced significantly.
• If a stable arch is formed above the
outlet so that the flow of the bulk solid is
stopped, then this situation is called
• In case of fine grained, cohesive bulk
solid, the reason of arching is the strength
(unconfined yield strength) of the bulk solid
which is caused by the adhesion forces
acting between the particles.
• In case of coarse grained bulk solid,
arching is caused by blocking of single
• Arching can be prevented by sufficiently
Cohesive Arch preventing
large outlets.
material from exiting
Insufficient Flow
- Outlet size too small
- Material not sufficiently
permeable to permit dilation in
conical section -> “plop-plop”
Material under
compression in
the cylinder
Material needs
to dilate here
Irregular flow
Irregular flow occurs if arches and ratholes
are formed and collapse alternately. Thereby
fine grained bulk solids can become fluidized
when falling downwards to the outlet opening,
so that they flow out of the silo like a fluid.
This behaviour is called flooding. Flooding
can cause a lot of dust, a continuous discharge
becomes impossible.
Inadequate emptying
Usually occurs in funnel flow silos
where the cone angle is insufficient
to allow self draining of the bulk
Remaining bulk
Time Consolidation - Caking
Many powders will tend to cake as a function
of time, humidity, pressure, temperature
Particularly a problem for funnel flow silos
which are infrequently emptied completely
What the chances for mass flow?
Cone Angle
Cumulative % of
from horizontal hoppers with mass flow
*data from Ter Borg at Bayer
Mass Flow (+/-)
+ flow is more consistent
+ reduces effects of radial segregation
+ stress field is more predictable
+ full bin capacity is utilized
+ first in/first out
- wall wear is higher (esp. for abrasives)
- higher stresses on walls
- more height is required
Funnel flow (+/-)
+ less height required
- ratholing
- a problem for segregating solids
- first in/last out
- time consolidation effects can be severe
- silo collapse
- flooding
- reduction of effective storage capacity
How is a hopper designed?
- powder cohesion/interparticle friction
- wall friction
- compressibility/permeability
- outlet size
- hopper angle for mass flow
- discharge rates
Types of Bins
Watch for inflowing valleys
in these bins!
Types of Bins
Wedge/Plane Flow
Design diagram for mass flow (wedgeshaped hopper)
(angle of wall friction)
(slope of hopper wall)
= effective angle
of internal friction
Design diagram for mass flow (conical
(angle of wall friction)
(slope of hopper wall)
= effective angle
of internal friction
Stress conditions in the hopper
c < 1 : flow
c > 1 :
arching is stable, no flow
1 = bearing stress, 1 = major principal stress c = unconfined yield strength
Flow function and time flow function
Hopper forms
The design of silos in order to obtain reliable flow is
possible on the basis of measured material properties
and calculation methods. Because badly designed
silos can yield operational problems and a decrease
of the product quality, the geometry of silos should
be determined always on the basis of the material
properties. The expenses for testing and silo design
are small compared to the costs of loss of production,
quality problems and retrofits.
Critical dimensions of hopper openings
To determine critical dimension, failure conditions
must be established for two basic obstructions;
arching (no flow) and piping (flow may be reduced
or limited).
Consider that the strongest possible arch may form,
the critical opening dimension (B) becomes:
B  c/w (for slot opening)
 B  2c/w (for circular opening)
Where w = bulk density
T = thickness
B = opening dimension
Flow factor (ff) depends upon:
effective angle of internal friction)
w (angle of wall friction)
(slope of hopper wall)
Mass flow
Calculate the critical width B for arching of
the slot opening of a wedge shaped, mild steel
hopper with  = 30C
For mild steel hopper
with wall friction
angle = 35, the
maximaum effective
angle of friction () =
ff = 1.25
ff = 1.25
of ff and this
FF, there is
no arching
ff = 1.25
There is an
of ff and this
FF, there is
c = 50
1 = 65
From 1 = 65 lb/ft2, c = 50 lb/ft2, w = 90 lb/ft3 and 
= 55, therefore B  50/90  0.6 ft or critical slot
with for arching is about 7 inches.
Determination of Outlet Size
B = c,i H()/W
H() is a constant which is a function of hopper angle
Bulk density = W
H() Function
Cone angle from vertical
Example: Calculation of a Hopper Geometry for
Mass Flow
An organic solid powder has a bulk density of 22 lb/cu ft. Jenike
shear testing has determined the following characteristics given
below. The hopper to be designed is conical.
Wall friction angle (against SS plate) = w = 25º
Bulk density = W = 22 lb/cu ft
Angle of internal friction =  = 50º
Flow function c = 0.3 1 + 4.3
Using the design chart for conical hoppers, at w = 25º
c = 17º with 3º safety factor
& ff = 1.27
Example: Calculation of a Hopper Geometry for
Mass Flow
ff = /a
or a = (1/ff) 
Condition for no arching =>
(1/ff)  = 0.3 1 + 4.3
a > c
(1/1.27)  = 0.3 1 + 4.3
1 = 8.82 c = 8.82/1.27 = 6.95
B = 2.2 x 6.95/22 = 0.69 ft = 8.33 in
Discharge Rates (Q)
Numerous methods to predict discharge rates
from silos or hopper
For coarse particles (>500 microns)
Beverloo equation - funnel flow
Johanson equation - mass flow
For fine particles - one must consider
influence of air upon discharge rate
Beverloo equation
Q = 0.58 b g0.5 (B - kdp)2.5
where Q is the discharge rate (kg/sec)
b is the bulk density (kg/m3)
g is the gravitational constant
B is the outlet size (m)
k is a constant (typically 1.4)
dp is the particle size (m)
Note: Units must be SI
Johanson Equation
Equation is derived from fundamental
principles - not empirical
Q = b (/4) B2 (gB/4 tan c)0.5
where c is the angle of hopper from vertical
This equation applies to circular outlets
Units can be any dimensionally consistent set
Note that both Beverloo and Johanson show that Q  B2.5!
Silo Discharging Devices
Slide valve/Slide gate
Rotary valve
Vibrating Bin Bottoms
Vibrating Grates
Rotary Valves
Quite commonly used to discharge
materials from bins.
Screw Feeders
Dead Region
Better Solution
Discharge Aids
Air cannons
 Pneumatic Hammers
 Vibrators
These devices should not be used in place of a
properly designed hopper!
They can be used to break up the
effects of time consolidation.
Flow rate equations
From Ewalt and Buelow (1963), measuring flow of
shell corn from straight-sided wooden bins equipped
with test orifices:
Horizontal openings, circular orifice (8.4% MC db)
Horizontal openings, rectangular orifice (12.1% MC db)
Q = 0.153 W1.62 L1.4
Vertical openings, circular orifice (12.7% MC db)
Q = 0.1196 B3.1
Q = 0.0351 B3.3
Vertical openings, rectangular orifice (12.4% MC db)
Q = 0.0573 W1.75 L1.5
Q = KWn
K and n are two constants which can be found either by
substituting experimental data from two sets of tests and
solving the two equations simultaneously or by
determination them directly from the slope and yintercepts of the straight line plot of Q versus one of the
dimensions on log-log graph paper.
Q = f(i, r, d/D, D, bulk density and etc.)
There is no single parameter satisfactory relationship
for estimating Q.
Most important parameter is the opening
diameter (greatly affect on flow rate)
Q  D3
Log Q
Slope ~ 2.8-3.2
Log D

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