1. The Dow Chemical Company
Bin and Hopper Design
Karl Jacob
The Dow Chemical Company
Solids Processing Lab
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2. The Four Big Questions 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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3. Hopper Flow Modes
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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4. 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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5. Funnel Flow Active Flow Channel “Dead” or non-flowing region 3/17/00
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6. Expanded Flow Funnel Flow upper section Mass Flow bottom section
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7. Problems with Hoppers
Ratholing/Piping
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8. Ratholing/Piping
Void
Stable Annular Region
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9. Problems with Hoppers
Ratholing/Piping
Funnel Flow
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10. Funnel Flow Coarse Coarse -Segregation -Inadequate Emptying
-Structural Issues
Coarse
Coarse
Fine
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11. Problems with Hoppers Ratholing/Piping Funnel Flow Arching/Doming
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12. Arching/Doming Cohesive Arch preventing material from exiting hopper
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13. Problems with Hoppers Ratholing/Piping Funnel Flow Arching/Doming
Insufficient Flow
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14. Material under compression in the cylinder section
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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15. Problems with Hoppers Ratholing/Piping Funnel Flow Arching/Doming
Insufficient Flow
Flushing
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16. 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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17. Problems with Hoppers Ratholing/Piping Funnel Flow Arching/Doming
Insufficient Flow
Flushing
Inadequate Emptying
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18. 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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19. Problems with Hoppers Ratholing/Piping Funnel Flow Arching/Doming
Insufficient Flow
Flushing
Inadequate Emptying
Mechanical Arching
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20. 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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21. Problems with Hoppers Ratholing/Piping Funnel Flow Arching/Doming
Insufficient Flow
Flushing
Inadequate Emptying
Mechanical Arching
Time Consolidation - Caking
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22. 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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23. Segregation Mechanisms - Momentum or velocity - Fluidization
- Trajectory
- Air current
- Fines
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24. What the chances for mass flow?
Cone Angle Cumulative % of
from horizontal hoppers with mass flow
*data from Ter Borg at Bayer
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25. 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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26. 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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27. 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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28. What about angle of repose?
Pile of bulk solids   
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29. 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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30. Bulk Solids Testing Wall Friction Testing
Powder Shear Testing - measures both powder internal friction and cohesion
Compressibility
Permeability
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31. Sources of Cohesion (Binding Mechanisms)
Solids Bridges
-Mineral bridges
-Chemical reaction
-Partial melting
-Binder hardening
-Crystallization
-Sublimation
Interlocking forces
Attraction Forces
-van der Waal’s
-Electrostatics
-Magnetic
Interfacial forces
-Liquid bridges
-Capillary forces
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32. Testing Considerations
Must consider the following variables
- time
- temperature
- humidity
- other process conditions
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33. Wall Friction Testing P 101 F
Wall friction test is simply Physics difference for bulk solids is that the friction coefficient, , is not constant.
P 101 N F
F = N
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34. Wall Friction Testing Jenike Shear Tester W x A Bracket Cover Ring
Wall Test Sample
Ring
Cover
W x A
S x A
Bracket
Bulk Solid
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35. Wall Friction Testing Results
Wall Yield Locus (WYL), variable wall friction
Wall shear stress, 
Wall Yield Locus, constant wall friction ’
Normal stress, 
Powder Technologists usually express  as the “angle of wall friction”, ’
’ = arctan 
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36. Jenike Shear Tester W x A Bracket Cover Ring S x A Bulk Solid
Shear plane
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37. Other Shear Testers Peschl shear tester Biaxial shear tester
Uniaxial compaction cell
Annular (ring) shear testers
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38. Ring Shear Testers W x A Arm connected to load cells, S x A Bulk solid
Bottom cell rotates slowly
W x A
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39. Shear test data analysis C fc 1 
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40. Stresses in Hoppers/Silos
Cylindrical section - Janssen equation
Conical section - radial stress field
Stresses = Pressures
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41. (Pv + dPv) A +   D dh = Pv A +  A g dh
Stresses in a cylinder
Consider the equilibrium of forces on a differential element, dh, in a straight-sided silo
Pv A = vertical pressure acting from above
 A g dh = weight of material in element
(Pv + dPv) A = support of material from below
  D dh = support from solid friction on the wall
Pv A h
  D dh dh
(Pv + dPv) A
 A g dh D
(Pv + dPv) A +   D dh = Pv A +  A g dh
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42. 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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43. 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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44. 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.  r
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45. Silo Stresses - Overall
hydrostatic
Bulk solid
Notice that there is essentially no stress at the outlet. This is good for discharge devices!
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46. 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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47. Janssen Equation - Example
K, the Janssen coefficient, is assumed to be 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 = lbf/ft2 or psf
Remember that Pw = K Pv,, so Pw = psf.
For water, P =  g H and this results in P = 3744 psf, a factor of 14 greater!
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48. Types of Bins Conical Pyramidal
Watch for in-flowing valleys in these bins!
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49. Types of Bins
Chisel
Wedge/Plane Flow L B
L>3B
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50. A thought experiment 1 c
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51. The Flow Function
1 c
Flow function
Time flow function
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52. Determination of Outlet Size
1 c
Flow function
Time flow function
c,t
c,i
Flow factor
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53. Determination of Outlet Size
B = c,i H()/
H() is a constant which is a function of hopper angle
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54. Rectangular outlets (L > 3B)
H() Function 3
Circular
H() 2
Square
Rectangular outlets (L > 3B) 1 10 20 30 40 50 60
Cone angle from vertical
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55. 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 
Using the design chart for conical hoppers, at ’ = 25º
c = 17º with 3º safety factor
& ff = 1.27
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56. Example: Calculation of a Hopper Geometry for Mass Flow
ff = /a or a = (1/ff) 
Condition for no arching => a > c
(1/ff)  = 0.3  (1/1.27)  = 0.3 
1 = c = 8.82/1.27 = 6.95
B = 2.2 x 6.95/22 = 0.69 ft = 8.33 in
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57. Material considerations for hopper design
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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58. Material considerations for hopper design
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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59. Process Questions 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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60. 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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61. 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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62. 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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63. 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 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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64. Discharge Rate Example
One compartment = /4 = 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= 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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65. 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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66. Limiting Flow Rates Vertical stress Interstitial gas pressure Bulk
density
Vertical stress
Note that gas pressure is less than ambient pressure
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67. 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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68. Limiting Flow Rates - Carleton Equation
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69. 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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70. Silo Discharging Devices
Slide valve/Slide gate
Rotary valve
Vibrating Bin Bottoms
Vibrating Grates
others
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71. Rotary Valves Quite commonly used to discharge materials from bins.
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72. Screw Feeders
Dead Region
Better Solution
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73. 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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