Asst. Prof. Dr. Muanmai Apintanapong

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

 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.

 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

all the material in the hopper is in motion at discharge, but not necessarily at the same velocity
Material in motion along the walls
D
Does not imply plug flow with equal velocity
Typically need 0.75 D to 1D to enforce mass flow

If a hopper wall is too flat and/or too rough, funnel flow will appear.
(centrally moving core, dead or nonmoving annular region)
“Dead” or nonflowing region or stagnant zone

mass flow cone with funnel flow above it
Mass Flow bottom section
Funnel Flow upper section

 Ratholing/Piping and Funnel Flow
 Arching/Doming
 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 solid.
• If the bulk solid consolidates increasingly with increasing period of storage at rest, the risk of ratholing increases.
Stable
Annular
Region

-Segregation
-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 arching.
• 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 particles.
• Arching can be prevented by sufficiently large outlets.
Cohesive Arch preventing material from exiting hopper

- 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

 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.

Usually occurs in funnel flow silos where the cone angle is insufficient to allow self draining of the bulk solid.
Remaining bulk solid

 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

Cone Angle Cumulative % of from horizontal hoppers with mass flow
45
60
0
25
70
75
*data from Ter Borg at Bayer
50
70

+ 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

+ 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



Measure
- powder cohesion/interparticle friction
- wall friction
- compressibility/permeability
Calculate
- outlet size
- hopper angle for mass flow
- discharge rates

Conical Pyramidal
Watch for inflowing valleys in these bins!

Wedge/Plane Flow
B
L
L>3B
Chisel


 w
(angle of wall friction)


= effective angle of internal friction

(slope of hopper wall)

 w
(angle of wall friction)



= effective angle of internal friction

(slope of hopper wall)

 c
<

1
: flow
 c
>

1
: arching is stable, no flow

1
= bearing stress,

1
= major principal stress
 c
= unconfined yield strength

 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.



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



1
T = thickness
B = opening dimension




 effective angle of internal friction)
 w
(angle of wall friction)

(slope of hopper wall)

 w funnel flow
Mass flow ff


 Calculate the critical width B for arching of the slot opening of a wedge shaped, mild steel hopper with

= 30

C

55
 For mild steel hopper with wall friction angle = 35

, the maximaum effective angle of friction (

) =
55


ff = 1.25

ff = 1.25
No intersection of ff and this
FF, there is no arching problem

ff = 1.25
 c
= 50
There is an intersection of ff and this
FF, there is arching problem

1
= 65
 From

1
= 65 lb/ft 2 ,
 c
= 50 lb/ft 2 , w = 90 lb/ft 3 and

= 55

, therefore B

50/90

0.6 ft or critical slot with for arching is about 7 inches.

B =
 c,i
H(

)/W
H(

) is a constant which is a function of hopper angle
Bulk density = W


3
2
1
10 20 30 40 50
Cone angle from vertical
60

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
 c = 17º with 3º safety factor
= 25º
& ff = 1.27

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

 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

 Q = 0.58
 b g 0.5
(B - kd p
) 2.5
where Q 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

 Equation is derived from fundamental principles - not empirical
 Q =  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 Q

B 2.5
!

 Slide valve/Slide gate
 Rotary valve
 Vibrating Bin Bottoms
 Vibrating Grates
 others

Quite commonly used to discharge materials from bins.

Dead Region
Better Solution

 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.

 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)
 Q = 0.1196 B 3.1
Horizontal openings, rectangular orifice (12.1% MC db)
 Q = 0.153 W 1.62
L 1.4
Vertical openings, circular orifice (12.7% MC db)
 Q = 0.0351 B 3.3
Vertical openings, rectangular orifice (12.4% MC db)
 Q = 0.0573 W 1.75
L 1.5



 Q = KW n
 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

D 3
Log Q
Slope ~ 2.8-3.2
Log D