Bin and Hopper Design
Karl Jacob
The Dow Chemical Company
Solids Processing Lab
jacobkv@dow.com
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The Four Big Questions

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What is the appropriate flow mode?
What is the hopper angle?
How large is the outlet for reliable flow?
What type of discharger is required and
what is the discharge rate?
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Hopper Flow Modes

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Mass Flow - all the material in the
hopper is in motion, but not necessarily
at the same velocity
Funnel Flow - centrally moving core,
dead or non-moving annular region
Expanded Flow - mass flow cone with
funnel flow above it
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Mass Flow
D
Does not imply plug
flow with equal
velocity
Typically need 0.75 D to 1D to
enforce mass flow
Material in motion
along the walls
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Funnel Flow
“Dead” or nonflowing region
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Expanded Flow
Funnel Flow
upper section
Mass Flow
bottom section
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Problems with Hoppers

Ratholing/Piping
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Ratholing/Piping
Stable Annular
Region
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Problems with Hoppers


Ratholing/Piping
Funnel Flow
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Funnel Flow
-Segregation
-Inadequate Emptying
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Coarse
Fine
Coarse
-Structural Issues
10
Problems with Hoppers



Ratholing/Piping
Funnel Flow
Arching/Doming
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Arching/Doming
Cohesive Arch preventing
material from exiting
hopper
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Problems with Hoppers




Ratholing/Piping
Funnel Flow
Arching/Doming
Insufficient Flow
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Insufficient Flow
- Outlet size too small
- Material not sufficiently
permeable to permit dilation in
conical section -> “plop-plop”
flow
Material under
compression in
the cylinder
section
Material needs
to dilate here
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Problems with Hoppers

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
Ratholing/Piping
Funnel Flow
Arching/Doming
Insufficient Flow
Flushing
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Flushing

Uncontrolled flow from a hopper due to
powder being in an aerated state
- occurs only in fine powders (rough rule
of thumb - Geldart group A and smaller)
- causes --> improper use of aeration
devices, collapse of a rathole
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Problems with Hoppers

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

Ratholing/Piping
Funnel Flow
Arching/Doming
Insufficient Flow
Flushing
Inadequate Emptying
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Inadequate emptying
Usually occurs in funnel flow silos
where the cone angle is insufficient
to allow self draining of the bulk
solid.
Remaining bulk
solid
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Problems with Hoppers







Ratholing/Piping
Funnel Flow
Arching/Doming
Insufficient Flow
Flushing
Inadequate Emptying
Mechanical Arching
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Mechanical Arching


Akin to a “traffic jam” at the outlet of bin
- too many large particle competing for
the small outlet
6 x dp,large is the minimum outlet size to
prevent mechanical arching, 8-12 x is
preferred
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Problems with Hoppers
Ratholing/Piping
 Funnel Flow
 Arching/Doming
 Insufficient Flow
 Flushing
 Inadequate Emptying
 Mechanical Arching
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 Time Consolidation

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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
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Segregation

Mechanisms
- Momentum or velocity
- Fluidization
- Trajectory
- Air current
- Fines
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What the chances for mass flow?
Cone Angle
Cumulative % of
from horizontal hoppers with mass flow
45
0
60
25
70
50
75
70
*data from Ter Borg at Bayer
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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
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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
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How is a hopper designed?


Measure
- powder cohesion/interparticle friction
- wall friction
- compressibility/permeability
Calculate
- outlet size
- hopper angle for mass flow
- discharge rates
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What about angle of repose?
Pile of bulk
solids



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Angle of Repose

Angle of repose is not an adequate
indicator of bin design parameters
“… In fact, it (the angle of repose) is only useful in the
determination of the contour of a pile, and its
popularity among engineers and investigators is due
not to its usefulness but to the ease with which it is
measured.” - Andrew W. Jenike

Do not use angle of repose to design
the angle on a hopper!
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Bulk Solids Testing

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Wall Friction Testing
Powder Shear Testing - measures both
powder internal friction and cohesion
Compressibility
Permeability
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Sources of Cohesion (Binding Mechanisms)


Solids Bridges
-Mineral bridges
-Chemical reaction
-Partial melting
-Binder hardening
-Crystallization
-Sublimation
Interlocking forces
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Attraction Forces
-van der Waal’s
-Electrostatics
-Magnetic
Interfacial forces
-Liquid bridges
-Capillary forces
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Testing Considerations

Must consider the following variables
- time
- temperature
- humidity
- other process conditions
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Wall Friction Testing
Wall friction test is simply Physics 101 - difference for bulk
solids is that the friction coefficient, , is not constant.
P 101
F = N
N
F
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Wall Friction Testing
Jenike Shear Tester
WxA
Bracket
Cover
Ring
SxA
Bulk Solid
Wall Test
Sample
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Wall shear stress, 
Wall Friction Testing Results
Wall Yield Locus (WYL),
variable wall friction
Wall Yield Locus,
constant wall friction
’
Normal stress, 
Powder Technologists usually express  as the
“angle of wall friction”, ’
’ = arctan 
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Jenike Shear Tester
WxA
Bracket
Cover
Ring
SxA
BulkSolid
Solid
Bulk
Shear plane
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Other Shear Testers
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Peschl shear tester
Biaxial shear tester
Uniaxial compaction cell
Annular (ring) shear testers
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Ring Shear Testers
Arm connected to load
cells, S x A
Bulk
solid
WxA
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Bottom cell
rotates slowly
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Shear test data analysis

C
1
fc

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Stresses in Hoppers/Silos

Cylindrical section - Janssen equation
Conical section - radial stress field

Stresses = Pressures

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Stresses in a cylinder
Consider the equilibrium of forces on a
differential element, dh, in a straightsided silo
Pv A
  D dh
h
Pv A = vertical pressure acting from
above
 A g dh = weight of material in element
dh
(Pv + dPv) A
 A g dh
D
(Pv + dPv) A = support of material from
below
  D dh = support from solid friction on
the wall
(Pv + dPv) A +   D dh = Pv A +  A g dh
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Stresses in a cylinder (cont’d)
Two key substitutions
 =  Pw (friction equation)
Janssen’s key assumption: Pw = K Pv This is not strictly true but
is good enough from an engineering view.
Substituting and rearranging,
A dPv =  A g dh -  K Pv  D dh
Substituting A = (/4) D2 and integrating between h=0, Pv = 0
and h=H and Pv = Pv
Pv = ( g D/ 4  K) (1 - exp(-4H K/D))
This is the Janssen equation.
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Stresses in a cylinder (cont’d)
hydrostatic
Bulk solids
Notice that the asymptotic pressure depends
only on D, not on H, hence this is why silos are
tall and skinny, rather than short and squat.
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Stresses - Converging Section

Over 40 years ago, the pioneer in bulk
solids flow, Andrew W. Jenike,
postulated that the magnitude of the
stress in the converging section of a
hopper was proportional to the distance
of the element from the hopper apex.
 =  ( r, )
This is the radial stress field assumption.
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Silo Stresses - Overall
hydrostatic
Bulk solid
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Notice that there is essentially no stress at
the outlet. This is good for discharge
devices!
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Janssen Equation - Example
A large welded steel silo 12 ft in diameter and 60 feet high is to be
built. The silo has a central discharge on a flat bottom. Estimate
the pressure of the wall at the bottom of the silo if the silo is filled
with a) plastic pellets, and b) water. The plastic pellets have the
following characteristics:
 = 35 lb/cu ft
’ = 20º
The Janssen equation is
Pv = ( g D/ 4  K) (1 - exp(-4H K/D))
In this case:
D = 12 ft
 = tan ’ = tan 20º = 0.364
H = 60 ft
g = 32.2 ft/sec2
 = 35 lb/cu ft
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Janssen Equation - Example
K, the Janssen coefficient, is assumed to be 0.4. It can vary
according to the material but it is not often measured.
Substituting we get Pv = 21,958 lbm/ft - sec2.
If we divide by gc, we get Pv = 681.9 lbf/ft2 or 681.9 psf
Remember that Pw = K Pv,, so Pw = 272.8 psf.
For water, P =  g H and this results in P = 3744 psf, a factor of 14
greater!
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Types of Bins
Pyramidal
Conical
Watch for inflowing valleys
in these bins!
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Chisel
Types of Bins
Wedge/Plane Flow
L
B
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L>3B
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A thought experiment
c
1
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c
The Flow Function
Time flow function
Flow function
1
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c
Determination of Outlet Size
Time flow function
c,t
Flow function
c,i
1
Flow factor
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Determination of Outlet Size
B = c,i H()/
H() is a constant which is a function of
hopper angle
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H() Function
H()
3
2
1
10
20
30
40
50
60
Cone angle from vertical
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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) = ’ = 25º
Bulk density =  = 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 ’ = 25º
c = 17º with 3º safety factor
& ff = 1.27
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Example: Calculation of a Hopper
Geometry for Mass Flow
ff = /a
or a = (1/ff) 
a > c
Condition for no arching =>
(1/ff)  = 0.3 1 + 4.3
(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
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Material considerations for hopper design
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Amount of moisture in product?
Is the material typical of what is
expected?
Is it sticky or tacky?
Is there chemical reaction?
Does the material sublime?
Does heat affect the material?
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Material considerations for hopper design



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Is it a fine powder (< 200 microns)?
Is the material abrasive?
Is the material elastic?
Does the material deform under
pressure?
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Process Questions
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How much is to be stored? For how long?
Materials of construction
Is batch integrity important?
Is segregation important?
What type of discharger will be used?
How much room is there for the hopper?
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Discharge Rates



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
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Beverloo equation
W = 0.58 b g0.5 (B - kdp)2.5
where W 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

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Johanson Equation


Equation is derived from fundamental
principles - not empirical
W = 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 W  B2.5!
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Discharge Rate - Example
An engineer wants to know how fast a compartment
on a railcar will fill with polyethylene pellets if the
hopper is designed with a 6” Sch. 10 outlet. The car
has 4 compartments and can carry 180000 lbs. The
bulk solid is being discharged from mass flow silo
and has a 65° angle from horizontal. Polyethylene
has a bulk density of 35 lb/cu ft.
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Discharge Rate Example
One compartment = 180000/4 = 45000 lbs.
Since silo is mass flow, use Johanson equation.
6” Sch. 10 pipe is 6.36” in diameter = B
W = (35 lb/ft3)(/4)(6.36/12)2 (32.2x(6.36/12)/4 tan 25)0.5
W= 23.35 lb/sec
Time required is 45000/23.35 = 1926 secs or ~32 min.
In practice, this is too long - 8” or 10 “ would be a better
choice.
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The Case of Limiting Flow Rates

When bulk solids (even those with little
cohesion) are discharged from a
hopper, the solids must dilate in the
conical section of the hopper. This
dilation forces air to flow from the outlet
against the flow of bulk solids and in the
case of fine materials either slows the
flow or impedes it altogether.
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Limiting Flow Rates
Interstitial gas pressure
Bulk
density
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Note that gas pressure is less than
ambient pressure
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Vertical
stress
66
Limiting Flow Rates

The rigorous calculation of limiting flow
rates requires simultaneous solution of
gas pressure and solids stresses
subject to changing bulk density and
permeability. Fortunately, in many
cases the rate will be limited by some
type of discharge device such as a
rotary valve or screw feeder.
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Limiting Flow Rates - Carleton Equation
4v sin  15  

B
sd
1/ 3
f
2
0
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2/3 4/3
f
0
5/ 3
p
v
g
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Carleton Equation (cont’d)
where
v0 is the velocity of the bulk solid
 is the hopper half angle
s is the absolute particle density
f is the density of the gas
f is the viscosity of the gas
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Silo Discharging Devices





Slide valve/Slide gate
Rotary valve
Vibrating Bin Bottoms
Vibrating Grates
others
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Rotary Valves
Quite commonly used to discharge
materials from bins.
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Screw Feeders
Dead Region
Better Solution
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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.
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