Piping in the flow of granular material from rectangular bins

Int. J Rock Mech. M~n. Scl. VoLg. pp. 1-16. PergamonPress 1971.Printed m Great BrRam

PIPING IN THE FLOW OF G R A N U L A R M A T E R I A L F R O M R E C T A N G U L A R BINS L. W. SAPERSTEINand N. C. OZGEN Pennsylvania State Umversity, University Park, Pennsylvania (Recewed 4 April

1970)

Abstract--An examination of the cond~t,ons that ex,st m bins which are stated to the planestrain idealization indicates that pipe flow is a stable cond,tion in these bros. P~ping is the flow of material in a b,n in a flow field not much larger than the underlying or,rice. Thts flow is counter to the more efficient 'mass', or expanding flow. A model bm was bruit to study th,s type of flow; of interest was a grad technique that was used to determine the velocgies m the failing material. This techmque allows an evaluation of the major principal stress direct,on. Results from the experiment tended to verify the conclusion that ptping is a stable condition. This result implies that inefficient flow can be expected from plane-strata bros. These would be slot b,ns, I,ke the model, bunkers, and certam types of mine stopes. INTRODUCTION THts paper presents a study of the flow of granular material: dry, cohesionless sand, from a model ban with a slot opening. The utility of th,s study hes not only in the conclusions drawn about the mechanics ofthas flow, but also m the experimental procedure that was used. Although not original to thas study, at as felt that thas mostly unknown method gtves an extremely effective way o f studying bulk material that can be modeled by the plane-strain approximataon and should be useful to other experimenters. The intent o f th,s work was to determine the nature of flow from bans o f rectangular geometry and to sce if the inclticzcnt flow pattern termed 'piping' was an inevitable c o m p a n i o n to this geometry. Accordingly, a model study was planned that was consNtent with plane-strata c o u l o m b pla.~t,caty. The aam was to estabhsh generally applicable and theoretacally well-founded c r i t e r i a for the presence of piping m this flow regime. Recently, a large number o f stt, dae.~ have been done to apply the prlncaples of a contmuously fading medium to the problem of gravity-induced flow o f bt, lk material. In 1961, JIZNIKr. [1] tssued a report explaining flow and non-flowcrietrm in terms o f sod plasticity. The report draws on tradational plastic theory [2--4] and on a newer plastic, limtt analysis for granular material [5, 6] to give yield and flow criteria which are based on determanable material and container constants. These criteria are derived from stress and strata characteristics which are, in turn, derived from the M o h r - C o u l o m b yield functaon and the concept o f plastac potential. Much the same approach was taken by PARISEAU[7, 8] in his investigations into orepasses. He relates stresses and velocities through a sliphne net. In a subsequent investagation into mme subsidence conducted with DAHL [9]. he introduced an experimental technique which has evolved into the procedure used in the present study and called the 'grid technique'. PARISEAU [10] has also used the grid technique in an investigation into the utihty o f plasticity theory for analyzing gravity flow o f bulk material. In 1967, SAPERST[tN [I I] related

2

L W. SAPERSTEINAND N. C. OZGEN

flow fields and flow rates in axisymmetric flow to the angle of internal friction of cohesionless material, and developed a piping relationship for this same situation. The present work includes laboratory testing of piping criteria developed from the work of JE.~mE [1], P^RIS~U [7] and SAI'ERSTEIN[11 ]. THEORY The relevant portions of plasticity theory necessary to follow the experimental method of the grid technique are presented here, subsequent sections deal with piping. The planestrain simplification of three-dimensional plasticity considers that deformation occurs in two dimensions only. These are x, the flow or positive gravity direction, and y, normal to x. In the z-direction, stresses and velocities are considered identical. The Mohr-Coulomb yield criterion is selected for use because of its demonstrated applicability in soil plasticity. Under plane-strain conditions, four sets of equations govern plastic behavior [4]. These can be separated into a system of stress equations and a system of velocity equations. The former contain only stress variables and the latter contain both stress and velocity variables. If the material is homogeneous and isotropic such that the same angle of internal friction is observed in both the stress and velocity systems, then the two systems are coupled by 0, the angle between the major principal stress direction and the x-axis. Two families of sliplines exist [1], whose directions are defined by the equation which is common to both systems: dy = tan (0 4- v-) dx

(1)

where p is the angle from the major principal stress direction to the slipline. The geometry of sliplines is shown in Fig. l.

C2

~'4

2

Fzo. !. Characteristicgeometryof sliplines.

THE FLOW OF GRANULAR MATERIAL FROM RECTANGULAR BINS

3

Total strain rates are defined by ~

-

CU 8x

-

(2a)

'~y= Gx= --~

+

.

(2b)

t~t, iv"

Gy --

(2c)

The components of velocity in the x and y-direction are given, respectwe[y, by u and v. Figure 2 shows the sign convention used m thts paper; all quantmes shown are posmve. .-.-.~ v y

+y

yx

~yy,Cyy ~xy'~xy

lxx,xx +x FIG 2 Stgn convention for stresses and total stratn rates (showing poslttve d~rectlons) From the geometry o f a Mohr's circle (Fig. 3) assuming that stresses and strain rates are coupled tan 20 =

a'r

_

1

¢~,r 1

(3)

Combining equation (2) wzth equatton (3) gives

Ou

av

tan 20 = Oy + Ox

Ou

Or"

Ox

Oy

(4)

The grid technique enables the four unknowns of equation (4) to be identified. When typifying flow of bulk material from a bin, piping is defined as flow through a zone whose width is unchanging and not much larger than the width of the underlying orifice. Obviously, pipe flow must be in vertical channels. If the contained material is capable of cohesion or consohdation, it is quite possible for the non-flowing material surrounding the pipe to become cemented into the bin. The opposite of piping is 'mass

4

L.W. SAPERSTEIN AND N. C. OZGEN

\ /

" - - - 2" (°1 + °3)

°=
.(O1=O3)

°

-I

FiG. 3. Mohr diagram for a coulomb material at yield.

flow',[l] and is conceived of as a flow zone whose width increases with the distance from the orifice. This flow zone should ultimately intersect the bin walls. During flow, the stress field is assumed to be symmetric with respect to the central axis in the flowing region. This assumption was subsequently demonstrated to be correct; thus all analyses are performed on a half-bin only. If pipe flow exists, it is useful to know whether it is stable, and what are the boundary conditions at the pipe edge. Therefore the following analysis looks at equilibrium and the variation of principal stress direction in the pipe zone; also at the value of this direction at the vertical boundary. Fulfilling the plane-strain conditions, the stress in the z-direction is either zero or constant; thus it need not appear in the equilibrium equation. Looking at the pipe, flow occurs in the direction of the x-axis and can be considered independent of x. The equations of equilibrium reduce to a simple form G90"z.w

~,

=

--

=

o

a O'py =

~,,

(Sa)

Ob)

where ~, is the bulk density of the tlowing material. To keep numbers dimensionless, n, representing the ratio of pipe width to orifice width, y/D, will be used. Integrating and

THE FLOW OF GRANULAR MATERIAL FROM RECTANGULAR BINS

5

substituting for the component stresses their equivalent expression in terms of the mean normal stress, o, %, = a + cr cos 20 sin ~

(6a)

~ , = a sin 20 sin ¢,

(6b)

gives ya~,

=

-

D (n --

+

2 sin 20 sin ~ dO 1 -- cos 20 s m ¢

(7a)

1)

and de, a

--

=

0

da dn -+ 2 cot 20 dO = - - . a n--I

(7b)

(7c)

Combining equations (7b) and (7c) and eliminating do/a from the result gives a relation for the variation of the major principal stress direction with the flow zone width [11] d0 -dn

=

(sin20) cos 2 0 - - s m ~

(1 -- sin ¢, ~os 20) . n--

(8)

Solutions to equation (8) which result in an increasing value o f n would imply an expanding flow field, that is 'mass flow'; solutions resulting in a finite value of n indicate pipe flow. However, there are no real, that is to say natural, situations which would lead to a zero in the numerator, hence to a solution. From equation (8) it is concluded that for plane-strain conditions there is no possibility for pipe flow to expand into mass flow. Re-examining this last equation, it ,s seen that pipe flow has to exist if there is a zero in the denominator implying --

--~"

OO.

?n This is obviously the case for n = I. Setting cos 2 0 -

s, n4,-=-O

it is found that 0-

4

2

which defines an envelope along which d O / d n - , . oo. This Js the same conclusion found by JENIKE [I ]. G,ven that the boundary between the pipe and rigid zones is vertical, then the slope o f a sl,phne defining the boundary must be [12] dx=tan

0--

--

=0.

(9)

Equation (9) gives that the principal stress direction at the boundary, for a sand with a 36 ° angle of internal friction, should be 27 °. Experiments were devised to show the existence of piping and to determine the angle 0 in the flowing zone.

6

L.W. SAPERSTEIN AND N. C. OZGEN EXPERIMI~--NT

A model bin, 47 in. high, 33 in. wide, 7 in. deep, with a Kin. slot opening centrally located in the base was constructed. The front panel was of Plexiglass, and the entire box was sufficiently rigid to insure the presence of plane-strain conditions. Throughout experimentation, observations were made on the consistency of flow; at no time was the flow nearest the walls slower than that in the center. Hence deformation in any x-y plane was equal to that in any other x - r plane. Sighting, at the top, free surface, along a line parallel to the z-axis, no deviation from parallel, of the surface, was observed. In this manner, the use of the plane-strain idealization was justified. The grid technique, originally devised by DAHL and PARISEAU[9], is sufficiently ingenious to be presented here. The technique has several strengths. Because it is a grid, the intersections are points and the movements of a point can be determined. In addition, the exact location of any discontinuity can be observed by finding the breaks in a grid line. Since the grid is created within the studied material, there is no need to introduce foreign material or to change the physical properties of the sand. The grid lines, themselves, are the shadow image of a wire screen and are the result of impregnating the surface layer o f sand with light=sensitive diazo dye. In subdued light, a quantity of sand is mixed with diazo, coupler, and citric acid and then allowed to dry. In the meantime, the Plexiglass plate is washed and then coated with a thin layer of wetting agent. The treated sand is spread over the plate. If a proper amount of wetting agent has been placed on the plate, a layer of sand will adhere, with the excess falling off, as the plate is raised from the horizontal to the vertical. Working smoothly and quickly, the plate is clamped to the bin, the bin tipped backwards and then flied with clean, dry sand. The danger here is not that the small volume of water introduced by the wetting agent will change the material properties of the sand, but that it will evaporate before the bin is filled, allowing the dyed sand to fall away. Once the bin is safely filled, a wire mesh is clamped to the front of the plate. Any size may be used, although a I-in. grid proved the most convenient. The dye is exposed by illumination with a strong photo-flood lamp. The lamp serves also to drive off any remaining water. Development and fixing of the dye is accomplished by flooding the model with anhydrous ammonia gas. The results of this operation are shown in Fig. 4. The network of principal stress directions in the flow zone was developed through experimental determination of the velocity field. During the flow, controlled by a hinged door, photographs, taken at regular intervals, provided the raw material for data reduction. One such set of pictures is shown in Fig. 5. The following steps outline the data reduction procedure. (!) Within a certain time interval, the motion of each particle was marked as a straight line on a sheet of paper. (2) Conforming to the patterns of particle motion, these lines were smoothly connected to give flow lines throughout the flow zone. (3) The motion of each particle within a unit time interval was measured. Directions of travel were evaluated at the midpoints of increments. (4) Displacements, or velocities as they occur in a unit time interval, were reduced to relative velocities by dividing them with a monotonic quantity. This was the motion of the top, free surface, measured at the center, for the same time interval. (5) The u= and v-components of the relative velocities, in the x- and y-directions respectively, were calculated.

i

l

| II,

4 The model bin and I)plL:ll ~rld hnc,, prt~duLcd b~ d l , l l ( ) ~.lllllpI 1lllltl

Fie.. 5(a)

FI~., $(bJ

Fi(, 5(a-f) l)hotogr,,ph ,., ,.~.h,~h ',,,ere i,tkcn during one c',,pcrlmcnt v, ilh ~-m orili,.¢ wKlth RM

fn

w~

Ir

ii'~l I,

~LI , ~lq|,

i,ll

I,

ii

~, ,i~I~

rA~

t

i

pm lr

THE FLOW OF GRANULAR MATERIAL FROM RECTANGULAR BINS

7

(6) Derivatives of these velocities, those that appear in equation (4), were evaluated graphically to find strain rates in the flow zone. (7) By substitution into equation (4), the angle 0 was calculated throughout the flow zone.

Following the above scheme, flow (strain) lines were constructed, Fig. 6; then lines of

y 0 "25

-30

/ I

I

-25

-20

-15

-10

-5

y (in.)

2

3

4

y (in.)

x (in.) FIG. 6. Experimentally constructed streamline pattern in the flow zone.

x

FIG. 7. Experimentally constructed equal velocity curves (V/Vo) in the flow zone.

equal relative velocity (isovels) were drawn by joining points of equal velocity in the x-y plane. Figure 7 shows the isovels; these figures were plotted as curves of V/Vo and ~b against x and y as running parameters. From these, graphs of the horizontal and vertical components of the velocity vector were constructed; plots of u/V o and v/Vo against x and y

8

L W. S A P E R S T E I N A N D N. C. O Z G E N

2.0

1.8

1'6

1.4 U

Vo

1 •2

1"0 '"l

"8

.6

~ -y=3

"4

"2 0 0

I -5

i -I0

I -15

I -20

I -25

I -30

x (in.) FtG. 8. Vert,cal c o m p o n e n t s o f the relative velocity field O' as parameter).

THE FLOWOF GRANULAR MATERIALFROM RECTANGULARBINS

2.4

2"2

2~0

1"8 u

~o

1"6

~ "

x ""-- ----.

_ ~,.'--~_~,L'~

1.4

~ -12 -10

x,,

-

-

\X

~

x='16 x'-20

"~'~"-'~-Z

1.2

__

x='-18

i'0

"8

"6 x=-22 x=-24 x=-26 x=-28

"4

.2

I

0 0

I

1

I

2

I

3

4

y (tn.)

Fro 9 Verticalcomponentsof the relat,,~evelooty field (x as parameter)

9

!0

L.W. SAPERSTEIN AND N. C. OZGEN

1.0

.9

-8

"7

-6

v__ Vo

,5

• 4

3

y-

2

0 ..I..

t

0

-5

i

i

-10

-IS

t

-20 x (in.)

I

i

-25

-30

Fro. I0. Horizontal components of the relative velocity field (y as parameter).

T H E F L O W O F G R A N U L A R M A T E R I A L F R O M R E C T A N G U L A R BINS

-I-I -i'0

_i -

V

-

°

,)

.

V o

.4

-

-3

J -

.2

-

.l

x:-to

x=-12 x=-[4 x=-16 x=-18

_~

x=-26

x=-27 :<=-28 x=-29

0

.1.

I

19

I

I

I

~

3

4

y (in.)

FIG 1 1. Horizontal components of the relauve velocity ficld (x as parameter)

II

12

L.W. SAPERSTEIN AND N. C. OZGEN

- .25

- .20

- .15

- .05

~

y-4 .--t-----.--4

.00

XX

-5

//1-101

-15

-20

-25

-30

x (in.)

"05

1o[ -15 FIG. 12. Normal strain rates in the x-direction (.y as paramctcr).

.6

.5 .4 .3 .2 Yy ÷

X•

-

!=4

.1

-.1

I

I

l

-5

-10

-15

.I

-20

-, x ( i n . ) Ro. 13. Normal strain rates in the ),-direction (.v as parameter).

J

-25

,

l

THE FLOW OF GRANULAR MATERIAL FROM RECTANGULAR BINS

13

-3

"2

xy ÷

/

,1

~y-2 ~y=4 ,=,3

"0

-,.].

I

0

-5

f

I

f

z

i

-lO

-15

-20

-25

-30

÷X(in.)

FIG. 14. Shear strata rates (y as parameter). are shown in Figs 8-11. From this last group, the total strain rates, under vclocaty components of a unit drawdown, were computed by simply measuring the slopes of the curves at certain intervals. Figures 12-14 show the plots of the strain rates within the flow zone. By inserting these rates into equation (4), the principal stress directions were defined. A net of principal stress directions was constructed by drawing smooth curves tangent to the direcUons defined by 0. Figure 15 shows this net. A slipline field (Fig. 16) was promptly obtained from flus net as the shplines are everywhere inclined to the prinopal stress dlrcctlon by the angles

.= i,i-

i

DISCUSSION

As the sand flowed, it would consistently seek pipe geometry. Directly over the orifice, the flow zone would begin to expand, but instead of continuing into mass flow, it ceased expansion within a few inches and settled into pipe flow. No material entered the pipe through the vertical boundaries; entrance into the pipe occurred only by sliding from the top of the rigid zones. Thus the top of the rigid zone assumed a profile near the material's natural angle of repose. In the portion of the flow zone that is defined by vertical boundaries, the principal stress directions were uniform and mostly unchanging. The value of this angle at the center hne was, of course, 90°; moving toward the boundary, the value abruptly changed to the range 55-65 ° and remained at that throughout the entire pipe zone. Because of severe Rooc 8 / I - - n

14

L.W. SAPERSTEIN AND N. C. OZGEN

-2~

-2~

-22 -20

-18

3oundary

-16

-14

-12 x (J.n.) ÷ -10

-8

-6 -4 -2 0

0

t 1

I

I

I~

3

4

$

y (~.)'+

Fzo. 15. Experimentally determincd dlrcction 6¢id of the r~twork of p r i ~ l

|iz~s.

THE FLOW OF GRANULAR MATERIAL FROM RECTANGULAR BINS

-26

-2~

-22

Characteristic

-2C Characteristic -18

-16 oundary x (in.)

-14

=12 =10

-8

-6

-4

-2

O0

1

2

3

4

Fro. 16. Experimental slipline field.

I

y (i..)

15

16

L.W. SAPERSTFAN AND N. C. OZGEN

distortion at the boundary, the conjectured value o f 27 ° was not observed. As expected, the slipline field was also uniform with the flow zone. The b o u n d a r y was defined by an envelope o f the second family o f characteristics. The unchanging nature o f 0 across the flow zone, plus the lack o f change o f flow characteristics with a change in slot width tend to verify the analysis [equation (8)] which says that pipe flow, in plane-strain conditions, is stable. The efficiency o f this type o f bin, which is a model not only o f slot bins, but o f various bunkers and thin-vein stopes, is low. If this kind o f bin must be used, then a low bin-diameter-to-orifice-opening ratio must be maintained. Acknowledgment--The authors acknowledge the support of the National Science Foundation under Grant

(3K-3056.

I. 2. 3. 4. 5. 6.

7. 8. 9. 10. 1I. 12.

REFERENCES JENIKEA. W. (3ravtty flow of bulk solids. Bull. Utah Eng~ Exp. Sin No. 108, Salt Lake City O961). DRUCKERD. C. Limit analysis of two-and t ~ i m e n s i o n a t soil mechanics problem. ,I. Mech. Phys. Solids 1, 217-226 (1953). HtLL R. The Mathematical Theory of Pla.tticity, p. 130, Clarendon Press, Oxford (19.50). SHtELOR. T. On Coulomb's law of failure in soils, d. Mech. Phys. Solids4, IO-16 0955). SOKOLOVSK!!V. V. Statics o f Granular Media, Pergamon Press, Oxford 0965). Scoa-r P. F. Principles o f Sail Mechanics, Addison-Wesley, Reading, Mar~rachus~tts (1963). PAms~^u W. (3. and F^mHut~r C. The force-penetration characteristics for wedge penetration into rock. Int. J. Rock Mech. Min. Sei. 4, 165-150 (1967). PARI.~tJ W. G. and Pt.LEIOERE. P. Soil plasticity and the movement of material in ore passes. Trans. Am. Min. metulL Engrs 24|, 42 (1968). PARISEAUW. G. and DAllL H. D. Mine subsidence and model analysis. Trans. Am. Min. metall. Engrs 241, 488 0968). PAgtSFAtJW. G. Discontinuous Velocity Fields in Gravity Flow of Granular Materials through Slots, Proceedings of the Symposium on the Mechanics of Granular Materials and Powders, Marianske Lazne, Czechoslovakia, September 15-20 (1969). SAeFt~s-rEiNL. W. The Dynamics of Granular Solids, Unpublished doctoral thesis, Oxford Umversity 0967). JOltANSONJ. R. Stress and velocity fields in the gravity flow of bulk solids, J. appl. Mech., Trans. Am. Sac. mech. Engrs 31, Scr. E, No. 3, 499-506 0964).