BHG - Arching and Ratholes in Silos 2014

5/26/2018 BHG - Arching and Ratholes in Silos 2014

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BHG Doc No. Peschl-002 Rev A

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Bulk solids and powder flow properties testing, silo and hopper design, discharge and conveying technology

Arching and Ratholes in Si los

Paper by:-Dr. Ivan Peschl IPT. Seminar MIT Boston 14

thMay 1999


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Summary

The behaviour of bulk solids in an industrial installation is an interaction between the bulk solidsproperties, silo and the functioning of the installation system process requirements. In anyproduction processes, the proper function of all of the links of a chain of process is responsible for theproper, efficient and reliable system function.

Typically, Silos and hoppers are an import link of the process. Product bridging or ratholing in a silo orhopper can cause discontinuity of any process plant, with high safety, technical and financialconsequences. The influences of the silo geometry are the bulk solids and powder properties, silodiameter, height, hopper angles, pressure relations between the product in storage and of the silodesign, and wall friction.

The influences of the bulk solids handled are the cohesion, angle of internal friction, density andunconfined compressive strength values.

The common interaction is the wall friction and the pressure in the silo.

There have been a number of approaches developed to interface all of the above functions, with themost commonly known by Jenike. However, previous approaches had not considered the size of the

silo or hopper. In this paper, we explain the development of a theoretical model based on soilmechanics, as used in mining and civil engineering for calculation of pits tunnels and shafts.

Introduction

From the basic geometry of the silo, we obtain the main data for calculation of bridges or ratholing asshown in Fig.1A. From bulk solids and powder flow property testing to measure mechanical properties ofthe product, we obtain the necessary requirements to interface the bulk solids characteristics andsilo/hopper design Fig.1B.

There is also an interaction between the discharge equipment and the bulk solids. The silo geometryinfluences the bulk solids via the silo diameter, height and the angle of the hopper. The bulk solids flowproperties influence the silo by the flowability function which may cause bridging and ratholing if notinterfaced correctly.

For calculation of bridges and ratholing in silos, there are two main phenomena to be taken inconsideration:-

The pressure in the silo exerted by the bulk solids is to be taken as the maximum consolidationpressure.

The loading of the bridge has to be calculated on the base of soil mechanics as used for thecalculation of the mining pits and tunnels.

Continued.


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Parameters Influencing Bridging and Ratholing

Fig 1A. Silo Parameters:

D Diameter of the silo.H Silo Height. Angle of the hopper in relation to

the center axis of the silo.

pv Vertical pressure.ph Horizontal pressurepw Wall friction.

Fig 1B. Bulk Solids Flow Characteristics:

Yield Locusc Cohesion.d Unconfined compressivestrength.1 Major principal stress. Angle of internal friction. Wall friction.

Continued.

d 1

c


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Theoretical approach of br idging and ratholing

1. Tangential Forces around the Bridge.

Generally, around a rathole or a mining pit, the missing stresses on the boundary of any bridge orrathole are compensated by tangential pressure. Mohr (lit. 18) developed a model of the stressdistribution in an elastic body with a hole in the middle.

The Mohr equations for radial and tangential stresses for the case of elastic deformation are:

Radial stress: re = x ( 1 r i2/r2) (1)Tangential stresses: te = x ( 1 + r i2/r2 ) (2)Where: ri Radius of bridge.

r Radius

x Stress before the bridge occurs.re Radial stress in elastic body after bridge occurs.te Tangential stress in elastic body after bridge occurs.

The average vertical stress in vertical cross section of silos is: x = p vThe graphical presentation of stresses as function of the radius is shown in Fig.2. However, fromknown mining technology, the material around a pit deforms plastically, and on this basis reduces thetangential and radial stresses near to zero. We can postulate:

Forr = r i is t ~ r ~ 0The relation between the major stresses can not be bigger than relation obtained by shear test. For

internal angle of friction= 30o follows the relation of the principal stresses 1/3


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1. Tangential Forces around the Bridge. Continued.

FIG. 2 Plastic deformation in the area around the funnel or bridge:

ri radius of bridge or funnelrp radius at transition to plastic deformation

Within the possible tangential stresses on the boundary of the bridge according (3), the most stable bridge is when:

t = 0. (6)In mining technology, this allows to secure the pit with a small contra-pressure in form of wooden

supports. This way, relatively weak supports can resist the pressure of hundreds of atmospheres given

by the depth above the pit. In our case, because of cohesion the tangential stress can decrease in theplastic region much faster then for free flowing materials and that means that the loading of the bridgecan decrease to zero.

Because of the described phenomena, we can assume:

1. That the loading of the bridges is only under their own weight.

2. That the material in the bridge is consolidated by a consolidation stress, equal to thevertical pressure (lit 8,14).

A model of a bridge is shown in the following Fig.3.

Continued.

Ri

Vertical pressure

Plastic Elastic

Tangential pressure

Radial pressure

Rp R

pv

ptpr


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1. Tangential Forces around the Bridge. Continued.

Fig 3 Schematic bridge in a hopper.

d Diameter of the bridge.P Load of the bridge.R Radius of the bridge. Angle of the hopper.n Stress in the bridge support.

2. Calculation of the Geometry of a Bridge.

The critical diameter of bridging is dependant on the angle of the cone, coefficient of wall friction and the unconfined compressive strength, corresponding with the silo pressure.

The angle of the hopper cone for mass flow silos is mostly commonly between 15 and 25 degrees tothe vertical. For further calculations we will limit the angle of the cone:

15o < < 25oFor mass flow silos we can assume that the coefficient of friction, especially in the cone, will changeduring use. For a new silo, the coefficient of wall friction can be high, but after a certain time, thepolishing effect will lead to a very small coefficient of wall friction, in which case a value of 5 degree canbe reached. For silos from stainless steel or aluminium, the wall friction can be very low, similar to newsilo too.

The angle of the stress in the bridge support is:0 < < +

A stable arch must fulfil two conditions:

a) The bridge or rathole stresses must be smaller then the unconfined compressive strength.

b) The bridge or rathole support has to be stable.

Independent of all other conditions, a bridge will have a stable support when the angle of the arch in thesupport has an angle smaller than the angle of the hopper. In this case, sliding or rolling of particles inthe support is not possible.

From the measurement of the arch geometry in a model bin (lit 15) it follows, that the preferable angle

in the support of a bridge is between zero and . The observation of bridges in practice confirms thistoo. Therefore, we assume that possible bridges can occur under an angle between zero and .Bridges with an angle bigger than the angle of the hopper, can easily partially slide because of locallysmall coefficient of friction, or break as result local irregularities.

Continued.

D

P

R

n n

2


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2. Calculation of the Geometry of a Bridge. Continued.

For calculation of the critical bridge we will take the limit:

=The critical bridge dimension is when the bridge is at its smallest possible radius. All other bridges with

a bigger radius would have a bigger stress then the critical bridge and would break. All bridges with asmaller radius would slide down along the hopper wall as shown in Fig. 4.

FIG. 4 Stability of bridges depends on the radius of the bridge.

Consequently, the critical bridge will have a radius:

R = d / (2 sin . (7)

Where: d Diameter of the bridge.R Radius of the Bridge. Angle of the hopper.

3. Loading of a Bridge.

If we assume that the critical bridge can just resist the load of the bulk solids own weight, then, allbridges with bigger radius should break under this load. The maximum total force loading the bridge is:

P = * (8)where: A The cross sectional area of the bridge. The density.

Continued.

Bridges slides down

Critical bridge

Bridge breake


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3. Loading of a Bridge. Continued.

The forces in the bridge support are:

P = L*n*sin (9)Where: L The length of the bridge support.n The stress in the bridge support. Angle of the hopper.

From equations (8)and (9)we obtain the relation between the bridge geometry and the loading forcesof the bridge.

A/L =n *sin / (10)

4. The Consoli dation Pressure.

The major pressure in the silo is the vertical pressure. This pressure is responsible for theconsolidation of the powder in the silo. The pressure in the silo is dependent of the silo geometry thedensity of the powder and the wall friction.

The pressure in the silo can be calculated according the theory of Janssen (lit 16). According

Janssen, for the infinite depth of the silo is valid:

ph = D/4 (11)pv = ph/ (12)

After the filling of the silo, the relation of horizontal to the vertical pressure is:

= ph/pv = 0.5. (13)

This corresponds with experimental results on real installations and models (lit 15), especially for

mass flow silos with smooth walls, polished through by the flow of bulk solids.

From equation (11), (12), and (13)we obtain the consolidation pressure for a circular silo:

pcons = pv = 2*D/4 (14)

Where:pcons Consolidation pressure. The density. The wall friction Static Value.

Continued.


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5. The Shear Test.

The stability of a bridge is highly dependent of the unconfined compressive strength of the powder. The relation between the consolidation pressure and the unconfined compressive strength can only

be found by measurement with a bulk solids shear tester (as shown on the front cover page).

By undertaking shear tests of the bulk solids, we obtain the yield loci for various pressures. From theyield loci value, we can determine the unconfined compressive strength for different consolidationpressures, which corresponds with the silo pressure.

The critical unconfined compressive strength d can be expressed as function of consolidationpressure equal to the vertical silo pressure. We call this the flow function as shown in Fig.5.

dcrit = f(pv) = f(cons) (15)FIG. 5 Flow function of a powder.

In many cases the flow function is linear and can be expressed as function of the angle of flow

functionand the cross section through d- axes at the valuedo , as shown in Fig. 6.dcrit = do + tan * pv (16)Where: dcrit Critical unconfined compressive strength.do Unconfined compressive strength. Flow function.

pv Vertical pressure.

Continued.

d d

p= D4

f( , , )D

Pv = 1


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5. The Shear Test.

FIG. 6 Practical approximation of a flow function.

6. Calculation of the Critical Bridge Diameter.

The stresses in the bridge and the bridge support are limited to the value of unconfined compressivestrength, valid for the consolidation pressure in the silo. The equilibrium of the load of the bridge (Fig7) and the stresses in the support depends of the geometry of the silo outlet as can be seen inequation (10).

For further calculations we can postulate:

n = d From the equilibrium of forces we obtain in case of plain symmetry the forces in the support:

For plane symmetry is valid:

A/L = d / 2 (17)

From equations (8), (10) and (17)we obtain the critical diameter of arching for plane symmetry :

dcrp l = 2 * d *sin / (18)For axial symmetry, in case of an circular outlet, it is valid

A/L = d / 4 (19)

Continued.

d

Pv = 1

Flowfun

ction

d0


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n 2

P

n

6. Calculation of the Critical Bridge Diameter. Continued.

From equations 10, and 13 we obtain the critical diameter for arching for the axial symmetry:

dcrax= 4 * d * sin / (20)The din equations 18 and 20 has to be found from the flow-function corresponding with the pressurein the silo.

The critical dfor bridging can be calculated using the data from the flow-function as expressed in theequation (16). Then we obtain:

For Plane Symmetry:

dcrpl = 2sin * (do + tan * pv) / (21)For Axial Symmetry

dcrpl = 4sin * (do + tan * pv) / (22)For dimensioning of the silo outlet, the calculated critical outlet has to be multiplied by a safety factor C:

Doutlet = crit * C (23)For dimensioning of a safe silo-outlet, a number of safety (un-knowledge) coefficients should be used asrecommended below:

For inaccuracy of measurements C1= 1.1For inaccuracy of theory C2= 1.5For fluctuation of material properties C3= 1.2 2For engineering safety C4= 1.5 2

FIG 7. Equilibrium of forces in a bridge.

Continued.


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6. Calculation of the Critical Bridge Diameter. Continued.

REMARKS

The theoretical approach contains a number of conservative assumptions:

1. We assumed that the tangential pressure in the bridge is zero. In reality this is true only for the boundary of the bridge.

2. We calculate the loading area of the bridge as cross section of the outlet which issmaller than the real area.

3. We assumed that the wall friction is zero, but in reality it is higher than zero.

Because of the conservative assumptions, allowances for inaccuracy of measurements, theory, as well as engineering safety may be already covered and we can use the theoretic value for calculation. However, be careful to neglect the safety factors.

7. Example of Calculations

For the calculations, limestone powder has been used for this example.

From the tested and measured yield loci, the flow function can be established - the unconfinedcompressive strength as function of major principal stress:

From the shear test of a limestone powder, we have obtained the following data:

d= 42 g/cm2 by 1 = 123 g/cm2d= 74 g/cm2 by 1 = 254 g/cm2Effective angle of friction e = 40 degrees.Density = 0.8Wall friction = 20 o for calculation of consolidation pressure. = 0ofor calc. of geometry of bridges.

7.1 Silo Data:

For calculation of the outlet diameter we make a choice of silo diameter and hopper angle:

Diameter will be executed for D= 1.5, 2 and 2.5 m

Hopper angle = 15oand 20o

Continued.


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7. Example of Calculations. Continued.

7.1 Silo Data:

The vertical pressure in the silo is calculated according to above powder properties and equation (14).This pressure is the consolidation pressure for the powder in the silo.

D(m) pv = consolidation pressure (g/cm2)

1.5 1642.0 2132.5 274

For the chosen silo diameters and calculated consolidation pressure, we obtain from the flow function the corresponding unconfined compressive strength, as shown in Fig. 8 :

D(m) pv (g/cm2) d(g/cm2)1.5 164 512.0 213 682.5 274 80

FIG. 8. Flow-function comparison of evaluation according Jenike and Peschl

Contiuned.


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7.2 Calculation of Critical Diameter of the Bridge.

Forplane symmetry we use the developed formula (21).

Silo diameter (m) Critical outlet (cm) for = 20 o Critical outlet (cm) for =15 o1.5 44 332.0 58 432.5 68 51

Foraxial symmetry we use the developed formula (22).

Silo diameter (m) critical outlet (cm) for =20 o critical outlet (cm) for = 15 o1.5 88 652.0 116 882.5 136 106

In Fig.9,the critical dimension of the outlet for axial symmetry is shown as the function of the silodiameter.

In Fig. 10, a silo for limestone is shown. The angle of the cone is 15o and the used discharge equipment shown is Siletta with a diameter of 1200 mm. For this limestone, many similar silos with a

diameter of 2.4 m with a circular outlet of a diameter of 1.2 m and with a cone of 15 ohave been built.In all cases, bridging has been eliminated. This means that in the theoretic approach containssufficient conservative assumptions.

FIG. 9 Comparison of Critical Bridge diameter as function of silo diameter and for plane and axialsymmetry, according Peschl and Jenike.

Angle of the outlet = 20oWall friction = 20o

Contiuned.


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7.2 Calculation of Critical Diameter of the Bridge. Contiuned.

Curves according Peschl:

1Axial symmetry, = 20o2Axial symmetry, =15 o3According Jenike for axial symmetry

Contiuned.

65

PESCHL= 20136

116

88

106

88

PESCHL=15

JENIKE

1

2

3

Silo diameter (cm)

100 150 200 250

Archdiameter(cm

)


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FIG. 10 Example of a Mass flow Silo for Limestone, showing a Siletta Discharger feeding a screwconveyor. Outlet Diameter = 1200mm.

Comparison o f Critical Bridge Calculation according to Jenike.

According to Jenike, the construction of a yield locus is based on use of the steady state value of the

consolidation stress as the end of the yield locus. This construction delivers a smaller 1 as comparedto the peak value of the consolidation step during shear testing as shown in Fig.8.

d = 42 g/cm2 by1=100 g/cm2d= 74 g/cm2 by1= 220 g/cm2

The factor ff is found from the Jenike graphically Fig.11valid for

e = 40 degrees = 20 degrees

From Fig.11, follow values can be obtained:

d= 28 g/cm2

Contiuned.


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The critical diameter of the outlet for axial symmetry is:

with = 0.8 we obtain:Dcrax= 2.2* d / = 76 cm

In the optimal area for factor ff , the influence of the wall friction is very small, as shown in Fig 11, andcan be neglected.

In the Jenike calculation procedures, allowance for safety factors are not included. The criticaldiameter according Jenike is plotted in the Fig. 9 as a straight line, as according to Jenike, thediameter of the silo not included in the calculation.

7.3 Calculation for Ratholing.

After a rathole occurs, the radial and tangential stresses has to be calculated according to thehorizontal stress given in equation (16). As discussed in chapter 3 -loading of a bridge, the stressesare very small. In case of the rathole, the average pressure in Fig 2 is px = ph. The loading of arathole is the vertical pressure according Janssen. The vertical pressure is also the consolidationpressure.

The vertical pressure increases from the top of the silo downwards, until the pressure correspondingthe infinite depth of the silo.

With cohesive powders, a rathole can exist until the depth where the unconfined compressive strengthequals the vertical pressure. By growing of the depth of the rathole above this level, the rathole mustcollapse.

A stable rathole over the whole height of the silo can occur when the unconfined compressive strengthis bigger than the vertical pressure at the bottom of the silo.

The diameter of the rathole is practically independent of the pressure because of the reducedtangential and radial stresses. The diameter is predicted only by the size of the outlet. For infinitedepth of the silo, the rathole can be stable when:

d > pv = Dsilo / 4 (24)We can obtain more detailed calculation by calculating the vertical pressure for the upper part of thesilo. There is the pressure dependent from the depth below the surface.

Any cohesive material can form a rathole, and it is only a question of which depth will the pressure behigher than the unconfined compressive strength. In this case the rathole can be stable if:

d >D(1-expK)/4 (25)

where: K = -2H/DH is depth of the rathole

Continued.


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8. Conclusions.

The authors calculation method, based on soil mechanics, is used for similar problems in mining and civil engineering. The methods contain all important data from geometry of the silo as well from the

measured powder characteristics. The results of calculation have been checked many times in actualinstallations and redesigned of existing silos. From experience, with new installations we can concludethat the outlet opening calculations are sufficiently oversized. From the redesigned silos, we found thatin many instances the silo opening was originally (by others) under-dimensioned. Both situations can becalculated using the Peschl formulas and provide good results in practice. In this case we can concludethat the theoretical approach, by the rules of soil mechanic, is sufficiently accurate for application inpractice.

Comments on obtaining true flow property measurements.

In the past, the main critic is, that a number important influences have been ignored: Jenike based hiscalculations of bridges at the assumption of a radial stress distribution in the cone. This however, is onlyvalid for a flow condition while emptying the silo. In reality, the powders has to start from a static fillingcondition - in which case the higher shear forces are acting than for the flow condition. Thus, theconsolidation pressure, corresponding with the static pressure in the cone is higher than proposed. In

this case, Jenikes assumptions can lead to unsafe dimensions for bridging and ratholing.

The height and the diameter of the silo is not considered as a parameter for the calculation of the silo.

The influence of the angle of the cone and the coefficient of wall friction can be found in Jenike graphicsfor obtaining of the factor ff.

The coefficient of wall friction is not or not sufficiently represented in Jenikes calculation method. This

can lead to over or undersized silo outlet. The angle of the wall friction has, in the usual range from 15to 30 degree a very small influence on the ff factor it changes from 1.45 to 1.53. This has practicallyno influence on dimension of outlet. However, theoretically, and from many experiments it is known thatthe coefficient of wall friction has a linear influence on the silo pressure and thus on the consolidationpressure.

The usual angle for mass flow silos of 15-25 degree has a very small influence as well, in the samerange as the influence of wall friction. This has practically no influence for calculation of the criticalunconfined strength.

If the angle of the hopper is equal to zero and the coefficient of friction is low, the Jenikes graphics deliver the highest ff - also the most worse case condition. This is not corresponding with the practice,

where the angle of the hopper equal to zero deliver bridge free solutions. Peschl developed a theory (lit

6) for ideal flow silos, and silos without cone, which are down to very small diameters and free ofbridges. Walker has developed a theoretic model similar to Jenikes, but with the difference that Walkersgraphics for the choice of the coefficient ff deliver optimal solutions for low friction coefficient and angleof the hopper equal to zero degree.

Finally, the construction of the endpoint of the yield locus is not logic and against all laws of known soilmechanics. In practice, there are mixed measured static points with a dynamic endpoint. This approach

is a result of limited possibilities of the Jenike shear tester, which appears to have difficulty in reachinga stationary plastic stress values, and very difficult to achieve reliable measurement of the staticstress in the period of transition from elastic to plastic stage.

The Peschl rotational shear tester is able to measure a static point for the end-Mohr circle too and with thisit is no more necessary to declare points non-valid or argue about possible expansion during the plastic

deformation. Detailed description is given in literature 1-3.

Continued.


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9. Nomenclature.

Hopper angle

Angle of internal friction

e Effective angle of internal friction

Angle of wall friction

n Normal stress in the bridge support

d Unconfined compressive strength

cons Consolidation pressure

Density

Relation between the horizontal to vertical pressureA Surface of the bridge

d Diameter of the bridgedcrit Critical diameter of the bridgedcrax Critical diameter bridge for axial symmetrydcrpl Critical diameter bridge for plane symmetryD Diameter of the siloR Radius of the bridgeL Length of the bridgeP Loading of the bridgeph Horizontal pressure in a silopv Vertical pressure in a silor Radiusri Radius on the boundary of a bridge or a ratholerp Radius for the transition between elastic and plastic region

pr Radial pressurept Tangential pressurepx Average pressure in the infinite distance from a bridge or a rathole.

Contiuned.


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10. References

1. Dr. Ir. I.A.S.Z. PeschlMechanical properties of powdersbulk solids handlingvol. 8, number 5, October 1988

2. Dr. Ir. I.A.S.Z. PeschlEquipment for measurement of mechanical properties of powderspowder handling and processingvolume 1, number 1 march 1989

ERRATA: in original publication the Mohr-diagram of fig 3. and fig.4. is exchanged. In our copy wehave already correct it.

3. Dr. Ir. I.A.S.Z. PeschlMeasurement and evaluation of mechanical properties of powderspowder handling and processingvolume 1, number 2, June 1989

4. Dr. Ir. I.A.S.Z. PeschlQuality control of powders for indust rial applicationpowder handling and processingvolume 1, number 4, November 1989

5. Dr. Ir. I.A.S.Z. PeschlFlowability test of powders

PMI - powder metallurgy internationalpmi 22 (1990) [2}

6. Dr. Ir. I.A.S.Z. PeschlIdeal flow siloAdvances in feed technology, Verlag Moritz Schfernumber 1, 1989

7. Dr. Ir. I.A.S.Z. Peschl Universal blender - a blending method and mixing system for cohesive and free flowingpowders.Bulk Solids Handling volume 6, number 3, June 1986

8. Dr. Ir. I.A.S.Z. PeschlNew silo theory - based on the development of expansion zone.Powder Handling and Processing volume 3, number 1, march 1991

ERRATA: in original publication change:page 25; D - diameter of silo, page 30;K=LAM*MU*Z/D,

9. Dr. Ir. I.A.S.Z. PeschlQualittskontrolle von pulvern in der industriellen PraxisKeramische Zeitschrift47.Jahrgang - Nr 11 - 1995, Seite 7-10

Contiuned.


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10. References. continued.

10. Dr. Ir. I.A.S.Z. PeschlShear Test for Process Control and EngineeringPowder Handling and Processing vol 11 no 1 January/March 1999

11. TerzaghiTheoretical Soil MechanicJohn Wiley & Sons, New York 1966

12. Josselin de Jong -Static and Kinematics in the Failure Zone of Granular Material

Waltman Delft 1959

13. Jenike A.W.Storage and Flow of Solids,University of Utah Engineering Experiment StationBulletin 123, Nov. 1964

14 Peschl/ColijnNon - symmetrical Bin Flow ProblemsBulk Solids Handling, 1981, no 3

15 PeschlTheory of formation of arches in bins

ASME, May 1969, No. 2.

16 JanssenVersuche uber Getreidedrucke in silozellenVDI Verlag Berlin 1895

17 KastnerStatik des Tunnel und Stollenausbaus auf der Grundlage der geomechanischenErkenntnissenSpringer Verlag, Berlin 1971

18 MohrAbhandlungen aus dem Gebiete der technischen Mechanik

2. Auflage , Berlin Verlag Ernst & Sohn

19. Timoschenko und LessellsFestigkeitslehreBerlin 1928 Springer Verlag

20. PeschlSeminar Papers Powder Technology and Quality ControlIPT, FL-9490 Vaduz, PRADAFANT 7

Continued.


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FIG.11. Graphics for flow factor function ff according Jenike

1 0

2 0

3 0

4 0

0 1 0 2 0 3 0 4 0 5 0 0

ff=1 ,

45

ff=4. 0

ff=3,0

ff=2, 5

ff=2, 0

ff=1, 9

ff=1,6

ff= 1, 5

ff=1 , 4

ff=1 ,

53

=

c =

BHG - Arching and Ratholes in Silos 2014

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