1. 3/17/00KVJ1 Bin and Hopper Design Karl Jacob The Dow Chemical Company Solids Processing Lab jacobkv@dow.com

2. 3/17/00 KVJ2 The Four Big Questions n What is the appropriate flow mode? n What is the hopper angle? n How large is the outlet for reliable flow? n What type of discharger is required and what is the discharge rate?

3. 3/17/00 KVJ3 Hopper Flow Modes n Mass Flow - all the material in the hopper is in motion, but not necessarily at the same velocity n Funnel Flow - centrally moving core, dead or non-moving annular region n Expanded Flow - mass flow cone with funnel flow above it

4. 3/17/00 KVJ4 Mass Flow Typically need 0.75 D to 1D to enforce mass flow D Material in motion along the walls Does not imply plug flow with equal velocity

5. 3/17/00 KVJ5 Funnel Flow “Dead” or non- flowing region Active Flow Channel

6. 3/17/00 KVJ6 Expanded Flow Funnel Flow upper section Mass Flow bottom section

7. 3/17/00 KVJ7 Problems with Hoppers n Ratholing/Piping

8. 3/17/00 KVJ8 Ratholing/Piping Stable Annular Region Void

9. 3/17/00 KVJ9 Problems with Hoppers n Ratholing/Piping n Funnel Flow

10. 3/17/00 KVJ10 Funnel Flow -Segregation -Inadequate Emptying -Structural Issues Coarse Fine

11. 3/17/00 KVJ11 Problems with Hoppers n Ratholing/Piping n Funnel Flow n Arching/Doming

12. 3/17/00 KVJ12 Arching/Doming Cohesive Arch preventing material from exiting hopper

13. 3/17/00 KVJ13 Problems with Hoppers n Ratholing/Piping n Funnel Flow n Arching/Doming n Insufficient Flow

14. 3/17/00 KVJ14 Insufficient Flow - Outlet size too small - Material not sufficiently permeable to permit dilation in conical section -> “plop-plop” flow Material needs to dilate here Material under compression in the cylinder section

15. 3/17/00 KVJ15 Problems with Hoppers n Ratholing/Piping n Funnel Flow n Arching/Doming n Insufficient Flow n Flushing

16. 3/17/00 KVJ16 Flushing n 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

17. 3/17/00 KVJ17 Problems with Hoppers n Ratholing/Piping n Funnel Flow n Arching/Doming n Insufficient Flow n Flushing n Inadequate Emptying

18. 3/17/00 KVJ18 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

19. 3/17/00 KVJ19 Problems with Hoppers n Ratholing/Piping n Funnel Flow n Arching/Doming n Insufficient Flow n Flushing n Inadequate Emptying n Mechanical Arching

20. 3/17/00 KVJ20 Mechanical Arching n Akin to a “traffic jam” at the outlet of bin - too many large particle competing for the small outlet n 6 x d p,large is the minimum outlet size to prevent mechanical arching, 8-12 x is preferred

21. 3/17/00 KVJ21 Problems with Hoppers n Ratholing/Piping n Funnel Flow n Arching/Doming n Insufficient Flow n Flushing n Inadequate Emptying n Mechanical Arching n Time Consolidation - Caking

22. 3/17/00 KVJ22 Time Consolidation - Caking n Many powders will tend to cake as a function of time, humidity, pressure, temperature n Particularly a problem for funnel flow silos which are infrequently emptied completely

23. 3/17/00 KVJ23 Segregation n Mechanisms - Momentum or velocity - Fluidization - Trajectory - Air current - Fines

24. 3/17/00 KVJ24 What the chances for mass flow? Cone Angle Cumulative % of from horizontal hoppers with mass flow 450 6025 7050 7570 *data from Ter Borg at Bayer

25. 3/17/00 KVJ25 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

26. 3/17/00 KVJ26 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

27. 3/17/00 KVJ27 How is a hopper designed? n Measure - powder cohesion/interparticle friction - wall friction - compressibility/permeability n Calculate - outlet size - hopper angle for mass flow - discharge rates

28. 3/17/00 KVJ28 What about angle of repose?  Pile of bulk solids  

29. 3/17/00 KVJ29 Angle of Repose n 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 n Do not use angle of repose to design the angle on a hopper!

30. 3/17/00 KVJ30 Bulk Solids Testing n Wall Friction Testing n Powder Shear Testing - measures both powder internal friction and cohesion n Compressibility n Permeability

31. 3/17/00 KVJ31 Sources of Cohesion (Binding Mechanisms) n Solids Bridges -Mineral bridges -Chemical reaction -Partial melting -Binder hardening -Crystallization -Sublimation n Interlocking forces n Attraction Forces -van der Waal’s -Electrostatics -Magnetic n Interfacial forces -Liquid bridges -Capillary forces

32. 3/17/00 KVJ32 Testing Considerations n Must consider the following variables - time - temperature - humidity - other process conditions

33. 3/17/00 KVJ33 Wall Friction Testing Wall friction test is simply Physics 101 - difference for bulk solids is that the friction coefficient, , is not constant. P 101 N F F =  N

34. 3/17/00 KVJ34 Wall Friction Testing Jenike Shear Tester Wall Test Sample Ring Cover W x A S x A Bracket Bulk Solid

35. 3/17/00 KVJ35 Wall Friction Testing Results Wall Yield Locus, constant wall friction ’’ Normal stress,  Wall shear stress,  Wall Yield Locus (WYL), variable wall friction Powder Technologists usually express  as the “angle of wall friction”,  ’  ’ = arctan 

36. 3/17/00 KVJ36 Jenike Shear Tester Ring Cover W x A S x A Bracket Bulk Solid Shear plane

37. 3/17/00 KVJ37 Other Shear Testers n Peschl shear tester n Biaxial shear tester n Uniaxial compaction cell n Annular (ring) shear testers

38. 3/17/00 KVJ38 Ring Shear Testers W x A Bottom cell rotates slowly Arm connected to load cells, S x A Bulk solid

39. 3/17/00 KVJ39 Shear test data analysis   C fcfc 11

40. 3/17/00 KVJ40 Stresses in Hoppers/Silos n Cylindrical section - Janssen equation n Conical section - radial stress field n Stresses = Pressures

41. 3/17/00 KVJ41 Stresses in a cylinder h dh P v A D (P v + dP v ) A  A g dh   D dh Consider the equilibrium of forces on a differential element, dh, in a straight- sided silo P v A = vertical pressure acting from above  A g dh = weight of material in element (P v + dP v ) A = support of material from below   D dh = support from solid friction on the wall (P v + dP v ) A +   D dh = P v A +  A g dh

42. 3/17/00 KVJ42 Stresses in a cylinder (cont’d) Two key substitutions  =  P w (friction equation) Janssen’s key assumption: P w = K P v This is not strictly true but is good enough from an engineering view. Substituting and rearranging, A dP v =  A g dh -  K P v  D dh Substituting A = (  /4) D 2 and integrating between h=0, P v = 0 and h=H and P v = P v P v = (  g D/ 4  K) (1 - exp(-4H  K/D)) This is the Janssen equation.

43. 3/17/00 KVJ43 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.

44. 3/17/00 KVJ44 Stresses - Converging Section r  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.

45. 3/17/00 KVJ45 Silo Stresses - Overall hydrostatic Bulk solid Notice that there is essentially no stress at the outlet. This is good for discharge devices!

46. 3/17/00 KVJ46 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 P v = (  g D/ 4  K) (1 - exp(-4H  K/D)) In this case:D = 12 ft  = tan  ’ = tan 20º = 0.364 H = 60 ftg = 32.2 ft/sec 2  = 35 lb/cu ft

47. 3/17/00 KVJ47 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 P v = 21,958 lb m /ft - sec 2. If we divide by gc, we get P v = 681.9 lb f /ft 2 or 681.9 psf Remember that P w = K P v,, so P w = 272.8 psf. For water, P =  g H and this results in P = 3744 psf, a factor of 14 greater!

48. 3/17/00 KVJ48 Types of Bins Conical Pyramidal Watch for in- flowing valleys in these bins!

49. 3/17/00 KVJ49 Types of Bins Wedge/Plane Flow B L L>3B Chisel

50. 3/17/00 KVJ50 A thought experiment 11 cc

51. 3/17/00 KVJ51 The Flow Function  1 cc Flow function Time flow function

52. 3/17/00 KVJ52 Determination of Outlet Size  1 cc Flow function Time flow function Flow factor  c,i  c,t

53. 3/17/00 KVJ53 Determination of Outlet Size B =  c,i H(  )/  H(  ) is a constant which is a function of hopper angle

54. 3/17/00 KVJ54 H(  ) Function Cone angle from vertical 10 20 3040 50 60 1 2 3 H(  ) Rectangular outlets (L > 3B) Square Circular

55. 3/17/00 KVJ55 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

56. 3/17/00 KVJ56 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 + 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

57. 3/17/00 KVJ57 Material considerations for hopper design n Amount of moisture in product? n Is the material typical of what is expected? n Is it sticky or tacky? n Is there chemical reaction? n Does the material sublime? n Does heat affect the material?

58. 3/17/00 KVJ58 Material considerations for hopper design n Is it a fine powder (< 200 microns)? n Is the material abrasive? n Is the material elastic? n Does the material deform under pressure?

59. 3/17/00 KVJ59 Process Questions n How much is to be stored? For how long? n Materials of construction n Is batch integrity important? n Is segregation important? n What type of discharger will be used? n How much room is there for the hopper?

60. 3/17/00 KVJ60 Discharge Rates n Numerous methods to predict discharge rates from silos or hopper n For coarse particles (>500 microns) Beverloo equation - funnel flow Johanson equation - mass flow n For fine particles - one must consider influence of air upon discharge rate

61. 3/17/00 KVJ61 Beverloo equation n W = 0.58  b g 0.5 (B - kd p ) 2.5 where W is the discharge rate (kg/sec)  b is the bulk density (kg/m 3 ) g is the gravitational constant B is the outlet size (m) k is a constant (typically 1.4) d p is the particle size (m) Note: Units must be SI

62. 3/17/00 KVJ62 Johanson Equation n Equation is derived from fundamental principles - not empirical n W =  b (  /4) B 2 (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  B 2.5 !

63. 3/17/00 KVJ63 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.

64. 3/17/00 KVJ64 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/ft 3 )(  /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.

65. 3/17/00 KVJ65 The Case of Limiting Flow Rates n 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.

66. 3/17/00 KVJ66 Limiting Flow Rates Vertical stress Bulk density Interstitial gas pressure Note that gas pressure is less than ambient pressure

67. 3/17/00 KVJ67 Limiting Flow Rates n 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.

68. 3/17/00 KVJ68 Limiting Flow Rates - Carleton Equation

69. 3/17/00 KVJ69 Carleton Equation (cont’d) where v 0 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

70. 3/17/00 KVJ70 Silo Discharging Devices n Slide valve/Slide gate n Rotary valve n Vibrating Bin Bottoms n Vibrating Grates n others

71. 3/17/00 KVJ71 Rotary Valves Quite commonly used to discharge materials from bins.

72. 3/17/00 KVJ72 Screw Feeders Dead Region Better Solution

73. 3/17/00 KVJ73 Discharge Aids n Air cannons n Pneumatic Hammers n 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.