- Email: [email protected]
Exact stress and velocity distributions in a cohesionless material discharging from a conical hopper
Pergamon PII:
Chemical Enqineerin O Science, Vol. 51, No. 16, pp. 3931 3942, 1996 Copyright I-~ 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0009 2509/96 $15.00 + 0.04)
S0009-2509(96)00248-5
EXACT STRESS A N D VELOCITY DISTRIBUTIONS IN A COHESIONLESS MATERIAL DISCHARGING FROM A CONICAL HOPPER S. B. M. M O R E E A * and R. M. N E D D E R M A N * Department of Chemical Engineering, Pembroke Street, Cambridge CB2 3RA, U.K.
(First received 4 September 1995; accepted 19 February 1996)
Abstract The method of characteristics has been used to predict the exact stress and velocity fields in an incompressible, cohesionless, Coulomb material discharging from a conical hopper for a great variety of boundary conditions imposed on the top surface. It is found that in all cases the stresses and velocity tend to the radial stress and velocity fields and that convergence is achieved about half way down the hopper. This justifies the use of the radial stress field in Jenike's design method. The velocity fields often show regions of more or less constant velocity separated by velocity characteristics. This justifies the assumptions in Drescher's method of predicting velocity distributions. Copyright ff~ 1996 Elsevier Science Ltd
Key words: hopper, stress, velocity, granular material, characteristics, Coulomb material.
INTRODUCTION One of the major achievements of Jenike's three Bulletins (Jenike, 1961, Johanson and Jenike, 1962; and Jenike 1964) is the development of the radial stress and velocity fields and the presentation of the results in graphical form. These results have been used by numerous workers, though advances in computing now make it as convenient to repeat the calculations for oneself as to interpolate between Jenike's graphs. The radial stress field (commonly denoted by RSF) is a particular solution to the full stress equations for an ideal C o u l o m b material in a conical or wedgeshaped hopper. Apart from the assumption that the material is stationary, incompressible, cohesionless and obeys the C o u l o m b yield criterion, the analysis is rigorous and the accuracy of the solution is limited solely by the step length in the integration and the number of significant figures retained during the calculation. Thus solutions can be obtained to far greater precision than is warranted by the probable accuracy to the which the physical data are likely to be known. The evaluation of the radial stress field appears, therefore, to be a completed topic. However, the radial stress field is only one of many solutions to the full equations and violates the likely boundary conditions at the top of the hopper. There is every reason to believe that the radial stress field is the asymptote towards which the stresses will tend in the vicinity of the apex. Yet the rate of approach to the asymptote has received little attention. Johanson and
*Present address. Silsoe Research Institute, Wrest Park, Silsoe MK45 4HS, U.K. tCorresponding author. Tel.: + 44(0)1223 334777. Fax: + 44(0)1223 334796.
Jenike (1962) presented two sets of calculations showing that the stresses do tend to the asymptote, but these referred to materials with the improbably large angles of friction of 50 and 60 °. Pitman (1988) considered a few cases but his calculations were confined to situations in which the imposed upper boundary conditions were very close to those predicted from the radial stress field. Horne and Nedderman (1978) considered the two-dimensional case, i.e. a wedge-shaped hopper. They found that stress discontinuities were propagated throughout the region, with the stresses varying in a saw-tooth manner about the radial stress field prediction. The amplitude of the stress discontinuities was comparable with the mean stress throughout the hopper and hence that there was no convergence towards the radial stress field. This may to some extent explain Pitman's difficulties. His numerical scheme was not convenient for handling discontinuous solutions and this may be why he was only able to analyse situations in which the magnitude of the discontinuities was kept small. O n the other hand, there may be a distinction between the conical case in which there is convergence to the radial stress field and the wedge-shaped geometry in which there is not. The radial velocity field (RVF) is derived from the radial stress field on the assumption that the material obeys the principle of coaxiality, sometimes known as the principle of isotropy. This states that the principal stress and strain rate directions are coincident and comes simply from arguments of symmetry. It is clearly related to St. Venant's principle commonly used in elasticity. Like the radial stress field it is the asymptote to which the velocities are expected to tend near the orifice and as such, it is therefore an asymptote within an asymptote. As far as the present authors are aware, no-one has investigated the rate of
3931
3932
S.B.M. MOREEAand R. M. NEDDERMAN
approach of the full velocity equations towards the asymptote in a conical hopper, though Baldwin and Hampson (1988) did some calculations for a wedgeshaped geometry. They assumed that the stress distribution was given by the radial stress field and imposed various velocity boundary conditions to the top surface. They found that weak velocity discontinuities were present throughout the hopper, giving velocities which oscillated about the radial stress field solution. The magnitude of the oscillations was comparable with the magnitude of the mean velocity. Thus, we see that calculations in wedge-shaped geometries do not predict a satisfactory approach to the radial stress and velocity fields, and calculations in conical geometries (which do seem to show convergence to the radial stress and velocity fields) have only been successful either for materials of unrealistic angles of friction, or for circumstances in which the stresses at the top surface were close to those predicted by the radial stress field. On the other hand, Cleaver and Nedderman's (1993a, b) experiments showed that the velocities in the lower half of a conical hopper are radial, with magnitudes similar to those predicted from the radial velocity field. The purpose of this paper is to investigate in greater detail the rate of approach of the full stress and velocity equations to the radial stress and velocity fields. The objective is to determine over what proportion of a conical hopper these fields are a close approximation to the full solution. The work will be confined to narrow hoppers in which there is no stagnant zone adjacent to the wall. Roughly speaking, this puts an upper limit of about 4Y to the value of 0,.. THE
STRESS
FIELD
In this work we will confine ourselves to incompressible, cohesionless materials discharging from a conical hopper and will work in terms of a set of spherical coordinates (r, 0, Z) with origin at the virtual apex and the line 0 = 0 directed vertically upwards. In these coordinates the equations of static equilibrium in the r and 0 directions are Oa~ ¢3r
_ _
2%r - or00 - ozz + _1 _0r0r _ r r Or
--
+ z°--5~cot 0 + 7cos0 = 0
(1)
be expressed in the form % = p(1 + sin 05 cos2~)
(3)
ooo = p(l - sin 05cos 2q0
(4)
z0r = p sin 05sin 2~
(5)
where tp is the angle between the major principal stress and the r direction, p is the coordinate of the centre of the Mohr's circle for the r, 0 plane and ~b is the angle of internal friction. Johanson and Jenike (1962) argue that if the material is discharging, 05 should be taken to be the effective angle of internal friction when this differs from the static angle. As can be seen from eqs (1) and (2), we also require an expression for the azimuthal stress ozz. For this we will assume the H a a r - v o n K~irman hypothesis that the azimuthal stress is equal to one of the other principal stresses. In a discharging hopper the material is compressed in the 0 and tp directions but expands in the r direction. Thus, a passive stress state seems inevitable and the azimuthal stress will equal the major principal stress. Hence, ~rzz = p(1 + sin 05).
Substituting eqs (3) (6) into eqs (1) and (2) gives a pair of first-order partial differential equations in Op/Or, ~p/t~O, Otp/0r and 0If/00. These equations are not presented here as they can be found in Jenike (1962) or Nedderman (1992). It should, however, be noted that these equations are highly non-linear containing, for example, trigonometric functions of ~J. A particular solution to these equations can be obtained by making the assumption that p is linear in r and that ¢ is independent of r, i.e. p = 7rq(O), = ¢(0), where q has been made dimensionless by the incorporation of the weight density 7- This leads to a pair of ordinary differential equations in dq/dO and dC,/d0 which can be solved by Runge Kutta integration to give the radial stress field. Johanson and Jenike (1962) present the results graphically, but to avoid the inevitable errors resulting from interpolation, we have repeated the calculation using a fourthorder R u n g e - K u t t a routine with step length 0.01 radians. The boundary conditions for this integration are, on the centre line:
r
OZor
1 0%o
O~- + r ~
3to,
- + -
0 = 0, (%o - axz) cot 0 - 7sin 0 = 0.
(6)
~k = 7z/2
(7)
and at the wall:
F
{2) In a flowing system inertial terms should be included, but with the velocities behaving as r-2, these terms will behave as r -4 and therefore will be of importance only in the immediate vicinity of the orifice. Thus, the equations of static equilibrium are appropriate also for discharging hoppers except they are very close to the orifice. We will assume that the material obeys the Coulomb yield criterion, which can most conveniently
0 = Ow,
~k = ~z/2 + (~o + qSw)/2
(8)
[see for example Nedderman (1992)] where 05w is the angle of wall friction and ~o is defined by sin o9 -
sin 05 sin 05w
(9)
It will be noted that no use has been made of the boundary conditions at the top of the hopper. This is not required, or even possible, since we have made the assumption that the stresses are directly proportional
Stress and velocity fields cohesionless material discharging from a conical hopper to r. The radial stress field cannot therefore be a general solution to the equations and it is usually taken to be the asymptote to which the stresses will tend near the apex. We have chosen to use the method of characteristics (abbreviated as MOC) to investigate the approach of the full solution towards this asymptote. Since discontinuous solutions, at least in their first derivative, are to be expected, the method of characteristics has advantages in that it generates a grid along which the discontinuities propagate and therefore avoids the numerical difficulties associated with modelling discontinuities within a cell. The derivation of the characteristic directions and the equations along the characteristics is standard and leads to the results.
(10)
Equation along the a-characteristics: 1
.
dq-21~qd~ = ~Is,n(O - (o) dO - cos(O - d~)d~f] +/Lqd0(3 + sin~b + cot 0cosqS) +/~q ~ [ c o s ~ b - (1 + sin~b)cot 0 - ~], (11) Equation of the fl-characteristics:
rdO =
dr tan(~, + E).
reduce to dq - 4# d0 = cols~b[ s i n ( 0 - ~b)d0- c o s ( 0 - q ~ ) ~ l +#q(3
+ sincp)dO + #qdr(
-~) (14)
and
'E
dq - 4~d~ = cosq~ sm(O + qS)dO -cos(O + 4))
- ~ q ( 3 + sin4))dO +/~q dcr°(s ~ b r - ~ ) (15)
Equation of the a-characteristics: r dO = dr tan(~b-~:).
3933
(12)
Equation along the fl-characteristics: dq - 2/~qd~, = c-0-~s 1 0 i_sm ~ . (0 - 0) dO - cos (0 - ~b) ~ ] +/tq d 0 ( - 3 - sin ~b + cot 0cos 4)) +/~q ~ [cos ~b- (1 + sin th) cot 0 - ~1, (13) where ~ = ( ~ / 2 - ~b)/2. While the corresponding forms in Cartesian and cylindrical coordinates are well known, we are not aware that the spherical coordinate form has been presented before. We have chosen this coordinate system so that we can use the boundary condition on the wall in the form of eq. (8). The equations are also expressed in terms of q instead of the more usual p, since we are investigating the hypothesis that at small r, q becomes a function of 0 only. The equations are non-linear and iteration was therefore required. Where values of q, ~, r and 0, but not their derivatives, are required in these equations, the mean of the values at the ends of the interval was used. It will be noted that eqs (11) and (13) are indeterminate on the axis of symmetry since here 0 = 0 and ~, = re/2. Using l'H6pital's rule, these equations
on the axis. These forms were used in preference to eqs (11) and (13) whenever 0 < 0.03 radians. The program was tested by imposing as the top boundary condition, values of q and O calculated from the radial stress field on a surface of constant radius. The results are presented in Fig. 1 from which it is seen that this stress field propagates unchanged throughout the hopper. The abscissa of this figure is the dimensionless radius R, being the actual radius divided by the radius a of the surface upon which the boundary conditions were imposed, i.e. R = r/a. The ordinate is the dimensionless stress, P = p/7a. A variety of different boundary conditions was imposed on the top surface as listed in Table 1. In all cases, convergence to the radial stress fields was found by about half way down the hopper, though convergence was rarely monotonic. The stresses were often subject to sudden variations which were propagated across the hopper along the characteristics, so that the fluctuations at the centre line were out of phase with those at the wall. It should be noted that it is not possible to control the spacing of grid points as this is determined by the positions of the characteristics. This accounts for the irregular spacing of the points in the following figures. Often certain multiples of the values of q and O predicted from the radial stress field were used as the top boundary conditions. These latter values are denoted by qRSFand ORsv. Figures 2 and 3 show the results for the upper boundary conditions, O = 0 Rsv and q = 1.5 qRSF and 0.5 qRSF.It is seen that in the former case convergence to the radial stress field prediction is fairly rapid and almost monotonic. By about half way down the hopper the stresses are within a few percent of the radial stress field prediction. The results for the case when the stresses are less than the RSF prediction show more variation, but again approach the RSF prediction by about R = 0.5. In this latter figure, the phenomenon of the sudden changes of stress being propagated across the hopper giving echoes alternately on the axis and on the wall is clearly visible. The effect of imposing a uniform value of q on the top surface, equal to the RSF prediction at the
3934
S.B.M. MOREEA and R. M. NEDDERMAN 0.6
I
o I
-"
'
I
'
I
'
'
I
'
I
'
MOC
0.5 .....
RSF
j
wall
0.4 e~D
E
0.3
0
=
0.2
.~
0.1
,
.."
'
'
'
~
centre line
-
0.0 0.20
0.00
0.40
0.60
0.80
1.00
1.20
dimensionless radial distance, R [-] Fig. 1. Predicted wall and centre-line stresses using the MOC, where the stress field imposed is the solution to the RSF (0w = 30 °, ~b = 30°, qS,~= 10", q = qRSV,~ = ~pRSV).
Table 1. Boundary conditions imposed on the top surface Initial Conditions (i) (ii) (iii) (iv) (V) (vi) (vii) (viii) (ix) (x) (xi) (xii) (xiii) (xiv) (XV) (xvi)
q= q= q= q= q= q= q= q= q= q= q= q _q -_ q= q= q=
1.2q Rsv 1.5 qrSV 0.5qrSV 1.2qrSV 1.2qrsv 1.2qRSV 1.5qrsv 1.5qrsv 1.5qrsv qRSF qrSV q RSF .... q .RSF ... qRSF
qRSF qRSF
4/= ~p = ~, = ~p = ~0 = ¢, = ~p = ~O= tp = 4J = ~0 = ~, = ~P = I~ = ~fi = 6 =
Hopper half-angle 0w C)
lnternal angle of friction q5 (o)
Wall angle of friction q~,, (<)
30 30 30 15 30 20 20 10 20 30 20 20 15 15 20 20
30 30 30 30 30 30 30 25 35 30 30 30 30 30 35 35
10 10 10 10 19 20 20 10 10 10 10 10 5 5 10 10
~RSF ~RSV CRSV ffrsv ~RSV ~pRSF ~ORsv oRSF ~RSV ¢,RSV ~Pii, tp.. @lin
OU, 1.01 ~ORsF 1.05~brsv
Note: 4J,. indicates a linear variation of ~Ofrom n/2 at the centre line to tPw at the wall.
centre, is s h o w n in Fig. 4 which again shows convergence to the R S F in the lower half of the hopper. The greater magnitudes of the fluctuations are inevitable from the imposition of a wall stress which is significantly different from the R S F value. It should be emphasised t h a t these fluctuations in the stresses are not caused by instabilities in the calculations; they are genuine features of the solution of the equations subject to the chosen b o u n d a r y conditions. However, as will be appreciated from a c o m p a r i s o n of these a n d subsequent figures, the m a g n i t u d e a n d position of the fluctuations are very sensitive to the details of the b o u n d a r y condition applied to the t o p surface. We c a n n o t therefore predict the m a g n i t u d e a n d positions of the stress peaks in the realistic case of a conical
h o p p e r attached to the base of a cylindrical b u n k e r until a solution is available for the stress distribution at the base of the bunker. The effect of imposing a stress of m a g n i t u d e predicted by the RSF, but with a different principal stress direction, is s h o w n in Fig. 5. Here a series of s h a r p stress changes is seen to p r o p a g a t e d o w n the h o p p e r reflecting from the wall a n d centre line. This behaviour can be attributed to imposing the a w k w a r d b o u n d a r y condition of ~, = 1.05 × n/2 o n the centre line where ¢ must inevitably equal n/2 a n d a similar discontinuous change in qJ o n the wall. However, a linear variation of ~p between its centre line a n d wall values causes n o p r o b l e m s when the m a g n i t u d e of the stress is the R S F value; see Fig. 6. G r e a t e r fluctuations
Stress and velocity fields cohesionless material discharging from a conical hopper
3935
0.8 --,
0.7
~g
0.6
o
MOC
.....
RSF
..
wall
0.5 I~
s
0.4
r~
-~
0.3
," 'D
0.2
E "~
0.1 0.0 0
0.2
0.4
0.6
0.8
1
1.2
dimensionless r a d i ~ d i s t a n c e , R [-] Fig. 2. Comparison of the wall and centre-line stresses predicted by the MOC with the RSF solutions (0w = 30°, ~b = 30°, ~b,, = 10°, q = 1.5q rsv, ~ = ~RSF).
0.5 p
o
MOC
•
i
~.,
0.4
,.
0.3
.....
RSF
•
.
wall
-"
E
0.2
tre-line
e-
E
0.1
s
.~
J
0.0 0.00
,
,
,
0.20
I
~
,
,
I
,
0.40
L
J
0.60
I
J
,
J
I
,
0.80
,
,
1.00
I
,
,
1.20
dimensionless radial distance, R [-] Fig. 3. Comparison of the wall and centre-line stresses predicted by the MOC with the RSF solutions (0w = 30°, ~b = 30'~, 0w = 10°, q = 0.5 qRSF, t~ = ~RSF).
are found with a linear variation in ~ and a constant value o f q equal to the mean of the R S F prediction; see Fig. 7. Figures giving the stresses for the remaining conditions listed in Table 1 are given by Moreea (1993). These are all qualitatively similar to the results presented here. It can be seen that in all cases, convergence towards the RSF value is rapid with the fluctuations about the mean having decreased to about a quarter of their initial values by about half way down the hopper (R = 0.5). However, when large fluctuations are im-
posed by extreme boundary conditions, their magnitude, though ever-decreasing, can remain appreciable compared to the local RSF prediction. It should be noted that in Figs 4 and 7, the results curve back on themselves. This is due to overlapping characteristics. U n d e r these circumstances, discontinuous solutions would occur but this phenomenon was not investigated. The results presented in these two graphs are therefore approximate. It is, however, believed that the results presented in the remaining figures are accurate.
3936
S.B.M. MOREEA and R. M. NEDDERMAN
0.5
'
'
'
I
o
'
'
I
'
'
'
I
'
'
'
I
'
MOC
'
II
'
'
'
. •
i
t~
'
s
0.4
.....
RSF
- "
wall
"Jl
=
0000
0.3
t~
o
cO
E
0.2
~
f
"~C~3": " ' "
0.1
~
6
~
o
"
i"
~ 4 " ' ' "
centre-line
000
•
0.0 0.00
-"
00 I
,
,
,
,
0.20
I
,
,
0.40
,
I
J
,
0.60
A
I
,
,
0.80
,Ol
,
,
1.00
,
1.20
dimensionless radial distance, R [-] Fig. 4, Comparison of the wall and centre-line stresses predicted by the MOC with the RSF solutions (O,.=30:',qb=3OL~bw= l O , q = q ¢ ,,sF,~k= ~,sF) .
0.40
'
i
0.35
o .....
,g
0.30
eL
'
'
I
'
'
'
I
'
'
1
'
'
I
MOC RSF
'
/
-
I
o
'
'
wall
0.25 co
E
2.°'II ~
°o o,
.J:entre-line
0.20
c~
0.15 e-,
iO
0.10
E
0.05 0.00
""
0.0
0
". - '
,
I
0.2
,
,
,
I
,
0.4
,
,
I
,
,
0.6
~
I
~
,
,
0.8
I
1.2
1.0
dimensionless radial distance, R [-] Fig. 5. Comparison of the wall and centre-line stresses predicted by the MOC with the RSF solutions (0., = 20 c~,~ = 35 '~, ~b., = 10°, q = qKSV,~9 = 1,05~pRsr).
In conclusion, convergence is much more rapid than that found by Horne (1978) for a wedge-shaped hopper, and this can be attributed to the much greater effect of the wall in the axi-symmetric case. In particular, the curvature of the wall means that the wall stresses are immediately aware of the conical nature of the hopper, unlike the wedge-shaped case, in which the wall stresses are totally independent of the existence of the far wall until the first characteristic from that wall arrives. Thus in the conical case, convergence towards the radial stress field can begin im-
mediately, whereas in two dimensions, the wall stresses often diverge from the RSF value in the early stages. THE VELOCITY FIELD
The assumptions of coaxiality and incompressibility lead, respectively, to
(8v, tan~p - ~
18vo ~) + r~-ff +
8vo l~v~ Vo = --~-r + r ~
+ --r (16)
Stress and velocity fields cohesionless material discharging from a conical hopper 0.40
'
'
'
I
o
0.35
..... ~g
0.30
e.,
0.25
'
'
'
I
'
'
'
I
'
'
'
j
I
MOC
wall
RSF
J
i
3937
l
l
~n~-line
o
E
0.20 0.15 0.10 0.05 0.00
JI
I
I
[
,
I
,
,
,
,
0.4
0.2
0.0
1
I
,
,
,
0.6
i
I
[
l
l
t
1.0
0.8
1.2
dimensionlessradialdistance,R[-] Fig. 6. Comparison of the wall and centre-line stresses predicted by the MOC with the RSF solutions (0~ = 15 ~, ~b = 30°, ~bw= 5°, q = qRSF, I]/ = ~/lin).
0.40 "7" t~
;
0
0.35
0.25
E
0.20
'
'
;
I
'
'
1
'
'
'
1
'
MOC
.....
0.30 "~
1
=
1
'
. "
'
'
w~
RSF
"f~ntre-line
0.15
o
'
Oo°o
0.10
~D
s
E 0.05
s
~
O
• s
~ ' , , I , , , I , ,
0.00
0.2
0.0
,
I
0.6
0.4
,
,
,
I
,
0.8
~
~
I
~
~
1.0
1.2
dimensionless radial distance, R [-] Fig. 7. Comparison of the wall and centre-line stresses predicted by the MOC with the RSF solutions (Ow = 20 "~, q9 = 30~, q~,,,= 10°, q = q R. .S F. . . O -- Otin).
(Vo= 0), the radial velocity is given by,
and
•v, ~r
-
1 ~Vo r ~0
2
v,
v0cot 0
r
r
(17)
where qJ is the m a j o r principal stress direction. This p a r a m e t e r is the only interaction between the stress a n d velocity fields. If one m a k e s the a s s u m p t i o n s (a) t h a t ~Ois given by the radial stress field a n d (b) t h a t the flow is radial
v,=-~exp(-3fltan2qJdO ).
(18)
T h e values predicted from eq. (18) are k n o w n as the radial velocity field (RVF) a n d are t h o u g h t to be the a s y m p t o t e to which the velocities will tend for small values of r. As with the radial stress field, the t o p b o u n d a r y condition has not been used.
3938
S. B. M. MOREEAand R. M. NEDDERMAN
The rate of approach to the asymptote can be studied by using the method of characteristics since the equations form a hyperbolic pair. This is the most convenient method of solution since discontinuous solutions are possible. The equations of the characteristics and the ordinary differential equations along the characteristics in spherical coordinates are given below. Equation of the ~-characteristics:
/
dO = t a n - 0 \
(19)
4] r'
Equation along the 7-characteristic: dt,~ + dv0tan ( 0 - 4 )
(
- Vo
+ d013v'tan 20
cot0 l
1 + c°s20/J +
d,
eral set of stress and velocity boundary conditions on the top surface would require interpolation between the stress characteristics in order to provide the values of 0 required for the evaluation of the velocity field. No attempt was made to do so. In the investigations below, we have assumed an analytical form for the variation of 0 with position, often the RSF value, and predicted the resulting velocity fields. Though presented below as functions of the fractional radius, it should be borne in mind that the stress field will not have converged to its RSF values until about half-way down the hopper and the following calculations show the development of the velocity field from conditions imposed at that height. Once again arithmetical difficulties occur on the axis of symmetry, but applying l'H6pital's rule to eqs (20) and (22) yields
2Vr--r = O.
Equation of the fl-characteristic:
+ 3v, tan 20 dO + 2v, --dr= 0
(04+ ) -6'.-
d0=tan
r
(23)
f
(21)
"
dr, + dv0(tan ( 0 + 4 ) - see 20 ) Equation along the/#characteristic:
+ 3Vr tan 20 dO +
dv,+dvotan(O +4)+dO[3vrtan20 -
t,o (
I
c~-s cot0 2~ ~ ] J + 2v,-dr 7 = 0
i
1
f
]
(22)
i
i
'
I
As before these equations were used in preference to eqs (20) and (22) whenever 0 < 0.03 radians. The boundary conditions are vo = 0 on b~th 0 = 0 and 0 = Ow, and some velocity distribution imposed on the top surface, R = 1. The equations were tested by imposing the radial velocity field on R = 1 and the results are presented in Fig. 8 in the form G(O)= R 2 U, where U is the dimensionless velocity v,/vo and
'
'
'
I
'
'
'
I
'
'
'
centre-line 1.0
.... t|
i
CD
0.9
d o
0.8
.,a
0.7 0.6
/
RSF
wall
0.5
. . . . . . . .
^^^OO,.~C
• w l w .
O.4 I 0.00
0.20
(24)
r
It is seen that for these equations, the characteristic direction are 0 -+ ~/4 as opposed to the stress characteristics whose directions are 0 -+ e, where (e = ~/4 qV2). Thus, the characteristics in the two cases are not coincident and any full solution starting from a gen-
1.1
dr 2v~-- = 0.
0.40
0.60
0.80
1.00
dimensionless radial distance, R [-] Fig. 8. MOC predictions of the wall and centre-line velocity functions where the solution to the RVF has been imposed (Ow= 30~, ~b= 30°, 4~w= 10°).
Stress and velocity fields cohesionless material discharging from a conical hopper
v0 is the centre-line velocity at the top of the hopper. With this ordinate the results should be constants and this is seen to be the case. The results for the case of a uniform plug flow imposed on R = 1 are shown in Figs 9 and 10. Figure 10 shows the directions of the velocity vectors; it should, however, be noted that the horizontal scale in this figure has been enlarged by a factor of two to aid clarity. It can be seen that sudden changes in the direction of the velocity occur and that these propagate along the velocity characteristics. The velocity discontinuity starting at the topmost part of the wall, point A, is clearly visible. This reaches the centre line at point X and continues to B. Its mirror image CXD is visible. The discontinuities reflect from the walls and reach the centre line at E.
3.0
'
'
I
I
3939
However, the magnitude of the velocity change across BE and DE is very much less than that across discontinuities AX and CX. The discontinuities continue to reflect backwards and forwards across the hopper below E but with ever decreasing magnitude. On the wall, the velocity function G(O)= U R ~ starts with the value 1 but changes discontinuously to the lower value of cos 0w as shown in Fig. 9. The wall velocity then oscillates with a small amplitude and large wavelength as it approaches the constant value required by the radial velocity field. However, an appreciable disturbance occurs at point D. By contrast, the centre-line velocity decreases continuously from its imposed value down to point X, whereupon a large discontinuity occurs, followed by a smaller
'
'
I
I
'
'
'
0 "7
2.5
¢D
& o ¢J
1.5
centre-line
0 0 (~000000
2.0
~ m ~ ~ m ~ ~
1.0 >
.......
0.5 0.0 0.00
,
I
0.20
,
,
~
.I
,
~
0.40
,
I
~
,
0.60
,
I
,
0.80
~
I
1.00
d i m e n s i o n l e s s radial distance, R [-] Fig. 9. Predicted velocity functions where a uniform plug flow field has been imposed on an initial arc of dimensionless radius 1.0 (0w = 30 °, ~b = 30 °, q~, = 10s).
X\kk~ ~X\{",',~\,
i ' i"_ll,/"~, t i j f t i t t b ,illltl/tl
i/Ill ~1
\'%'~\ \x, ~ t ! I 4 J z i , l
Fig. 10. Predicted velocity corresponding to the velocity field in Fig. 9.
3940
S.B.M.
MOREEA
and R. M. NEDDERMAN
discontinuity at point E. Figure 10 shows remarkable similarity to the velocity fields assumed by Moreea (1993) following arguments developed by Drescher (1991). It was assumed that the velocity field consisted of three regions, a plug flow region in the top part (region ACX of Fig. 10), a region in which the velocity was constant and parallel to the walls (corresponding to region AXD) and a region in which the flow was radial (below line DXB). Examination of Fig. 10 shows that this is not accurate, but none the less is a very good approximation to the theoretical velocity field. This close parallel perhaps explains the success of Drescher's and Moreea's approximate analyses. Figures 11 and 12 show the results for the imposition of a uniform radial velocity as the top boundary
2.0
'
'
I
'
I
condition. The results are similar to those shown in Figs 9 and 10 apart form the absence of the initial discontinuity in the wall velocity at A. Figures 13 and 14 show the results when a uniform velocity parallel to the walls was imposed on R = 1. In this case a velocity discontinuity occurs immediately at O, since an impossible combination of velocities has been imposed at this point, and further velocity discontinuities are generated at A and C. These two sets of discontinuities propagate as effectively parallel discontinuities throughout the hopper, with the discontinuities O P and CD reflecting from the left-hand wall to give PQ and DE. None the less, it can be seen from Fig. 13 that convergence to the radial velocity field has occurred by about R = 0.5, though there are small
'
'
'
I
ooO
°O
'
'
oo
o~
'
I
'
'
1.8 1.6
v
centre-line
~°°Oooo .
1.4 e',
1.2 1.0 0.8 --
0.6 0.00
,
,
.......... ~ ~ l ~ " ~ ,
0.20
I
,
,
,
, : ¢ f f ~ O 0 0 0 I
0.40
,
,
,
I
0 0 ,
0.60
,
,
I
,
0.80
,
,
1.00
dimensionlessradialdistance,R[-] Fig. 11. Predicted velocity functions where a uniform radial field has been imposed on an initail arc of dimensionless radius 1.0 (0., = 30°, 4~ = 30<', ~b.,= 10°).
"~"'~'~ "k\~ ' ~ ," l, t ~t ,t 1,~1'/1 ~z/i/l~~"
Fig. 12. Predicted velocity vectors corresponding to the velocity field in Fig. 11.
Stress and velocity fields cohesionless material discharging from a conical hopper
2.5
'
'
'
I
'
'
'
I
'
'
'
I
i
'
i
I
3941
i
o 0
line
2.0 0 0 o 1.5
8 o
~lIIi~l~~[~l:O~
0
000 O0 0 0 wall
l.O
0 ~
0.00
0
0 0.20
0
0
0
0
o
o
O
000 0 0
0
0.40
o
0.60
0.80
1.00
dimensionless radial distance, R [-] Fig. 13. Predicted velocity functions where a constant velocity parallel to the hopper wall has been imposed on a horizontal surface corresponding to a velocity discontinuity (cylindrical-conical bunker, 0~ = 30 °, 4' = 30°, 4'w = 10°).
A
O
C
( +. ; ?'x X4" l l'Zl i l / / /
"% ",,",,',,
l, 11 d. tll /
/f/"
t¢,/ltlff
Fig. 14. Predicted velocity vectors corresponding to the velocity field in Fig. 13.
disturbances at smaller values of R caused by the reflection of the now much attenuated discontinuities. We have seen that the stress field converges to the radial stress field prediction by about half-way down the hopper and that within the radial stress field region, the velocities converge to the radial velocity field prediction at about the half-depth. It should not therefore be assumed that the velocities do not converge to the radial velocity field until R ~ 0.25. The principal stress direction though still varying in the upper half of the hopper will have values close to the R S F prediction and, therefore, convergence of the velocity field to the R V F value will have started before
the stress field has converged. It is therefore not surprising that Cleaver found agreement with the R V F prediction up to about R = 0.5.
CONCLUSIONS It is generally agreed that the radial stress and velocity field are the asymptotes to which the stresses and velocities tend at great depth in a conical hopper discharging a cohesionless C o u l o m b material. In this paper we have studied the rate of approach to these asymptotes using the method of characteristics. A wide range of boundary conditions have been
3942
S.B.M. MOREEAand R. M. NEDDERMAN
imposed on the top surface. It has been found that in all cases both the stress and velocity fields do tend to the radial field asymptotes and that close convergence is normally achieved by about half-way down the hopper. The approach to the asymptote is however rarely monotonic and the fluctuations about the asymptote can be large especially when conditions significantly different from the radial field predictions are imposed on the top surface. Jenike's design method assumes that the stresses have converged to the radial stress field values in the region immediately above the orifice and these calculations show that this assumption in his method is fully justified. The velocity predictions show that there are regions of substantially constant velocity separated by sudden velocity changes. This gives weight to those approximate analyses based on Drescher's method which assumed constant velocity regions separated by velocity discontinuities. NOTATION A (/
G(O) P P q r R
U D Uo
arbitrary constant, mZ/s height of top surface above the orifice, m parameter defined by G ( 0 ) = UR 2, dimensionless coordinate of the centre of Mohr's stress circle, N / m 2 dimensionless stress ( = p/Ta), dimensionless parameter defined by q = p/Tr, dimensionless radial coordinate, m dimensionless radius ( = r/a), dimensionless dimensionless velocity ( = t'/Vo) velocity component, m/s centre-line velocity at the top of the hopper, m/s
Greek letters
7 e 0 0w it
weight density, N/m 3 angle defined by ~: = x/4 - ,;b/2, rad zenithal coordinate, rad wall angle, rad coefficient of internal friction ( = tan qS), dimensionless
a ~b ~b,. 7,
eJ
normal stress, N/m 2 shear stress, N / m 2 angle of internal friction, rad angle of wall friction rad azimuthal coordinate, rad angle between the major principal stress direction and the r direction, rad angle defined by eq. (9), rad
The superscipt RSF refers to values predicted from the radial stress field. M O C refers to values predicted by the method of characteristics.
REFERENCES
Baldwin, A. J., 1988, velocity distributions in the flow of 9ranular materials. Part II. Tripos Research Project Report. Department of Chemical Engineering, University of Cambridge. Cleaver, J. A. S. and Nedderman, R. M., 1993a, Theoretical prediction of stress and velocity profiles in conical hoppers. Chem. Enynq. Sci. 48, 3696-3702. Cleaver, J. A. S. and Nedderman, R. M., 1993b, Measurement of velocity profiles in conical hoppers. Chem. Engn9 Sci. 48, 3703 3712. Drescher, A., 1991, Analytical Methods in Bin-load Analysis, Vol. 36, Developments in civil engineering, Elsevier, Amsterdam. Hampson, H. A., 1988, The Theoretical Prediction of Velocity Distributions in Granular Materials. Part II. Tripos Research Project Report. Department of Chemical Engineering, University of Cambridge. Horne, R. M. and Nedderman, R. M., 1978, Stress distributions in hoppers. Powder TechnoL 19, 243 254. Jenike, A. W., 1961, Gravity flow of bulk solids. Bull. 108, University of Utah, USA. Jenike, A. W., 1964, Storage and flow of solids. Bull. 123, University of Utah, USA. Johanson, J. R. and Jenike A. W., 1962, Stress and velocity fields in gravity flow of bulk solids. Bull. 116, University of Utah, USA. Moreea, S. B. M., 1993, Gravity flow of bulk solids in axi-symmetric bunkers. Ph. D. Thesis, University of Cambridge. Moreea S. B. M. and Nedderman, R. M., 1993, Discontinuous velocity fields in a plane strain bunker. J. Israel Inst. Chem. Engng 22, 76-80. Nedderman, R. M., 1992, Statics and Kinematics of Granular Materials. Cambridge University Press, Cambridge. Pitman, E. B., 1986, Stress and velocity fields in twoand three-dimensional hoppers. Powder Technol. 47, 219 231.
Chemical Enqineerin O Science, Vol. 51, No. 16, pp. 3931 3942, 1996 Copyright I-~ 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0009 2509/96 $15.00 + 0.04)
S0009-2509(96)00248-5
EXACT STRESS A N D VELOCITY DISTRIBUTIONS IN A COHESIONLESS MATERIAL DISCHARGING FROM A CONICAL HOPPER S. B. M. M O R E E A * and R. M. N E D D E R M A N * Department of Chemical Engineering, Pembroke Street, Cambridge CB2 3RA, U.K.
(First received 4 September 1995; accepted 19 February 1996)
Abstract The method of characteristics has been used to predict the exact stress and velocity fields in an incompressible, cohesionless, Coulomb material discharging from a conical hopper for a great variety of boundary conditions imposed on the top surface. It is found that in all cases the stresses and velocity tend to the radial stress and velocity fields and that convergence is achieved about half way down the hopper. This justifies the use of the radial stress field in Jenike's design method. The velocity fields often show regions of more or less constant velocity separated by velocity characteristics. This justifies the assumptions in Drescher's method of predicting velocity distributions. Copyright ff~ 1996 Elsevier Science Ltd
Key words: hopper, stress, velocity, granular material, characteristics, Coulomb material.
INTRODUCTION One of the major achievements of Jenike's three Bulletins (Jenike, 1961, Johanson and Jenike, 1962; and Jenike 1964) is the development of the radial stress and velocity fields and the presentation of the results in graphical form. These results have been used by numerous workers, though advances in computing now make it as convenient to repeat the calculations for oneself as to interpolate between Jenike's graphs. The radial stress field (commonly denoted by RSF) is a particular solution to the full stress equations for an ideal C o u l o m b material in a conical or wedgeshaped hopper. Apart from the assumption that the material is stationary, incompressible, cohesionless and obeys the C o u l o m b yield criterion, the analysis is rigorous and the accuracy of the solution is limited solely by the step length in the integration and the number of significant figures retained during the calculation. Thus solutions can be obtained to far greater precision than is warranted by the probable accuracy to the which the physical data are likely to be known. The evaluation of the radial stress field appears, therefore, to be a completed topic. However, the radial stress field is only one of many solutions to the full equations and violates the likely boundary conditions at the top of the hopper. There is every reason to believe that the radial stress field is the asymptote towards which the stresses will tend in the vicinity of the apex. Yet the rate of approach to the asymptote has received little attention. Johanson and
*Present address. Silsoe Research Institute, Wrest Park, Silsoe MK45 4HS, U.K. tCorresponding author. Tel.: + 44(0)1223 334777. Fax: + 44(0)1223 334796.
Jenike (1962) presented two sets of calculations showing that the stresses do tend to the asymptote, but these referred to materials with the improbably large angles of friction of 50 and 60 °. Pitman (1988) considered a few cases but his calculations were confined to situations in which the imposed upper boundary conditions were very close to those predicted from the radial stress field. Horne and Nedderman (1978) considered the two-dimensional case, i.e. a wedge-shaped hopper. They found that stress discontinuities were propagated throughout the region, with the stresses varying in a saw-tooth manner about the radial stress field prediction. The amplitude of the stress discontinuities was comparable with the mean stress throughout the hopper and hence that there was no convergence towards the radial stress field. This may to some extent explain Pitman's difficulties. His numerical scheme was not convenient for handling discontinuous solutions and this may be why he was only able to analyse situations in which the magnitude of the discontinuities was kept small. O n the other hand, there may be a distinction between the conical case in which there is convergence to the radial stress field and the wedge-shaped geometry in which there is not. The radial velocity field (RVF) is derived from the radial stress field on the assumption that the material obeys the principle of coaxiality, sometimes known as the principle of isotropy. This states that the principal stress and strain rate directions are coincident and comes simply from arguments of symmetry. It is clearly related to St. Venant's principle commonly used in elasticity. Like the radial stress field it is the asymptote to which the velocities are expected to tend near the orifice and as such, it is therefore an asymptote within an asymptote. As far as the present authors are aware, no-one has investigated the rate of
3931
3932
S.B.M. MOREEAand R. M. NEDDERMAN
approach of the full velocity equations towards the asymptote in a conical hopper, though Baldwin and Hampson (1988) did some calculations for a wedgeshaped geometry. They assumed that the stress distribution was given by the radial stress field and imposed various velocity boundary conditions to the top surface. They found that weak velocity discontinuities were present throughout the hopper, giving velocities which oscillated about the radial stress field solution. The magnitude of the oscillations was comparable with the magnitude of the mean velocity. Thus, we see that calculations in wedge-shaped geometries do not predict a satisfactory approach to the radial stress and velocity fields, and calculations in conical geometries (which do seem to show convergence to the radial stress and velocity fields) have only been successful either for materials of unrealistic angles of friction, or for circumstances in which the stresses at the top surface were close to those predicted by the radial stress field. On the other hand, Cleaver and Nedderman's (1993a, b) experiments showed that the velocities in the lower half of a conical hopper are radial, with magnitudes similar to those predicted from the radial velocity field. The purpose of this paper is to investigate in greater detail the rate of approach of the full stress and velocity equations to the radial stress and velocity fields. The objective is to determine over what proportion of a conical hopper these fields are a close approximation to the full solution. The work will be confined to narrow hoppers in which there is no stagnant zone adjacent to the wall. Roughly speaking, this puts an upper limit of about 4Y to the value of 0,.. THE
STRESS
FIELD
In this work we will confine ourselves to incompressible, cohesionless materials discharging from a conical hopper and will work in terms of a set of spherical coordinates (r, 0, Z) with origin at the virtual apex and the line 0 = 0 directed vertically upwards. In these coordinates the equations of static equilibrium in the r and 0 directions are Oa~ ¢3r
_ _
2%r - or00 - ozz + _1 _0r0r _ r r Or
--
+ z°--5~cot 0 + 7cos0 = 0
(1)
be expressed in the form % = p(1 + sin 05 cos2~)
(3)
ooo = p(l - sin 05cos 2q0
(4)
z0r = p sin 05sin 2~
(5)
where tp is the angle between the major principal stress and the r direction, p is the coordinate of the centre of the Mohr's circle for the r, 0 plane and ~b is the angle of internal friction. Johanson and Jenike (1962) argue that if the material is discharging, 05 should be taken to be the effective angle of internal friction when this differs from the static angle. As can be seen from eqs (1) and (2), we also require an expression for the azimuthal stress ozz. For this we will assume the H a a r - v o n K~irman hypothesis that the azimuthal stress is equal to one of the other principal stresses. In a discharging hopper the material is compressed in the 0 and tp directions but expands in the r direction. Thus, a passive stress state seems inevitable and the azimuthal stress will equal the major principal stress. Hence, ~rzz = p(1 + sin 05).
Substituting eqs (3) (6) into eqs (1) and (2) gives a pair of first-order partial differential equations in Op/Or, ~p/t~O, Otp/0r and 0If/00. These equations are not presented here as they can be found in Jenike (1962) or Nedderman (1992). It should, however, be noted that these equations are highly non-linear containing, for example, trigonometric functions of ~J. A particular solution to these equations can be obtained by making the assumption that p is linear in r and that ¢ is independent of r, i.e. p = 7rq(O), = ¢(0), where q has been made dimensionless by the incorporation of the weight density 7- This leads to a pair of ordinary differential equations in dq/dO and dC,/d0 which can be solved by Runge Kutta integration to give the radial stress field. Johanson and Jenike (1962) present the results graphically, but to avoid the inevitable errors resulting from interpolation, we have repeated the calculation using a fourthorder R u n g e - K u t t a routine with step length 0.01 radians. The boundary conditions for this integration are, on the centre line:
r
OZor
1 0%o
O~- + r ~
3to,
- + -
0 = 0, (%o - axz) cot 0 - 7sin 0 = 0.
(6)
~k = 7z/2
(7)
and at the wall:
F
{2) In a flowing system inertial terms should be included, but with the velocities behaving as r-2, these terms will behave as r -4 and therefore will be of importance only in the immediate vicinity of the orifice. Thus, the equations of static equilibrium are appropriate also for discharging hoppers except they are very close to the orifice. We will assume that the material obeys the Coulomb yield criterion, which can most conveniently
0 = Ow,
~k = ~z/2 + (~o + qSw)/2
(8)
[see for example Nedderman (1992)] where 05w is the angle of wall friction and ~o is defined by sin o9 -
sin 05 sin 05w
(9)
It will be noted that no use has been made of the boundary conditions at the top of the hopper. This is not required, or even possible, since we have made the assumption that the stresses are directly proportional
Stress and velocity fields cohesionless material discharging from a conical hopper to r. The radial stress field cannot therefore be a general solution to the equations and it is usually taken to be the asymptote to which the stresses will tend near the apex. We have chosen to use the method of characteristics (abbreviated as MOC) to investigate the approach of the full solution towards this asymptote. Since discontinuous solutions, at least in their first derivative, are to be expected, the method of characteristics has advantages in that it generates a grid along which the discontinuities propagate and therefore avoids the numerical difficulties associated with modelling discontinuities within a cell. The derivation of the characteristic directions and the equations along the characteristics is standard and leads to the results.
(10)
Equation along the a-characteristics: 1
.
dq-21~qd~ = ~Is,n(O - (o) dO - cos(O - d~)d~f] +/Lqd0(3 + sin~b + cot 0cosqS) +/~q ~ [ c o s ~ b - (1 + sin~b)cot 0 - ~], (11) Equation of the fl-characteristics:
rdO =
dr tan(~, + E).
reduce to dq - 4# d0 = cols~b[ s i n ( 0 - ~b)d0- c o s ( 0 - q ~ ) ~ l +#q(3
+ sincp)dO + #qdr(
-~) (14)
and
'E
dq - 4~d~ = cosq~ sm(O + qS)dO -cos(O + 4))
- ~ q ( 3 + sin4))dO +/~q dcr°(s ~ b r - ~ ) (15)
Equation of the a-characteristics: r dO = dr tan(~b-~:).
3933
(12)
Equation along the fl-characteristics: dq - 2/~qd~, = c-0-~s 1 0 i_sm ~ . (0 - 0) dO - cos (0 - ~b) ~ ] +/tq d 0 ( - 3 - sin ~b + cot 0cos 4)) +/~q ~ [cos ~b- (1 + sin th) cot 0 - ~1, (13) where ~ = ( ~ / 2 - ~b)/2. While the corresponding forms in Cartesian and cylindrical coordinates are well known, we are not aware that the spherical coordinate form has been presented before. We have chosen this coordinate system so that we can use the boundary condition on the wall in the form of eq. (8). The equations are also expressed in terms of q instead of the more usual p, since we are investigating the hypothesis that at small r, q becomes a function of 0 only. The equations are non-linear and iteration was therefore required. Where values of q, ~, r and 0, but not their derivatives, are required in these equations, the mean of the values at the ends of the interval was used. It will be noted that eqs (11) and (13) are indeterminate on the axis of symmetry since here 0 = 0 and ~, = re/2. Using l'H6pital's rule, these equations
on the axis. These forms were used in preference to eqs (11) and (13) whenever 0 < 0.03 radians. The program was tested by imposing as the top boundary condition, values of q and O calculated from the radial stress field on a surface of constant radius. The results are presented in Fig. 1 from which it is seen that this stress field propagates unchanged throughout the hopper. The abscissa of this figure is the dimensionless radius R, being the actual radius divided by the radius a of the surface upon which the boundary conditions were imposed, i.e. R = r/a. The ordinate is the dimensionless stress, P = p/7a. A variety of different boundary conditions was imposed on the top surface as listed in Table 1. In all cases, convergence to the radial stress fields was found by about half way down the hopper, though convergence was rarely monotonic. The stresses were often subject to sudden variations which were propagated across the hopper along the characteristics, so that the fluctuations at the centre line were out of phase with those at the wall. It should be noted that it is not possible to control the spacing of grid points as this is determined by the positions of the characteristics. This accounts for the irregular spacing of the points in the following figures. Often certain multiples of the values of q and O predicted from the radial stress field were used as the top boundary conditions. These latter values are denoted by qRSFand ORsv. Figures 2 and 3 show the results for the upper boundary conditions, O = 0 Rsv and q = 1.5 qRSF and 0.5 qRSF.It is seen that in the former case convergence to the radial stress field prediction is fairly rapid and almost monotonic. By about half way down the hopper the stresses are within a few percent of the radial stress field prediction. The results for the case when the stresses are less than the RSF prediction show more variation, but again approach the RSF prediction by about R = 0.5. In this latter figure, the phenomenon of the sudden changes of stress being propagated across the hopper giving echoes alternately on the axis and on the wall is clearly visible. The effect of imposing a uniform value of q on the top surface, equal to the RSF prediction at the
3934
S.B.M. MOREEA and R. M. NEDDERMAN 0.6
I
o I
-"
'
I
'
I
'
'
I
'
I
'
MOC
0.5 .....
RSF
j
wall
0.4 e~D
E
0.3
0
=
0.2
.~
0.1
,
.."
'
'
'
~
centre line
-
0.0 0.20
0.00
0.40
0.60
0.80
1.00
1.20
dimensionless radial distance, R [-] Fig. 1. Predicted wall and centre-line stresses using the MOC, where the stress field imposed is the solution to the RSF (0w = 30 °, ~b = 30°, qS,~= 10", q = qRSV,~ = ~pRSV).
Table 1. Boundary conditions imposed on the top surface Initial Conditions (i) (ii) (iii) (iv) (V) (vi) (vii) (viii) (ix) (x) (xi) (xii) (xiii) (xiv) (XV) (xvi)
q= q= q= q= q= q= q= q= q= q= q= q _q -_ q= q= q=
1.2q Rsv 1.5 qrSV 0.5qrSV 1.2qrSV 1.2qrsv 1.2qRSV 1.5qrsv 1.5qrsv 1.5qrsv qRSF qrSV q RSF .... q .RSF ... qRSF
qRSF qRSF
4/= ~p = ~, = ~p = ~0 = ¢, = ~p = ~O= tp = 4J = ~0 = ~, = ~P = I~ = ~fi = 6 =
Hopper half-angle 0w C)
lnternal angle of friction q5 (o)
Wall angle of friction q~,, (<)
30 30 30 15 30 20 20 10 20 30 20 20 15 15 20 20
30 30 30 30 30 30 30 25 35 30 30 30 30 30 35 35
10 10 10 10 19 20 20 10 10 10 10 10 5 5 10 10
~RSF ~RSV CRSV ffrsv ~RSV ~pRSF ~ORsv oRSF ~RSV ¢,RSV ~Pii, tp.. @lin
OU, 1.01 ~ORsF 1.05~brsv
Note: 4J,. indicates a linear variation of ~Ofrom n/2 at the centre line to tPw at the wall.
centre, is s h o w n in Fig. 4 which again shows convergence to the R S F in the lower half of the hopper. The greater magnitudes of the fluctuations are inevitable from the imposition of a wall stress which is significantly different from the R S F value. It should be emphasised t h a t these fluctuations in the stresses are not caused by instabilities in the calculations; they are genuine features of the solution of the equations subject to the chosen b o u n d a r y conditions. However, as will be appreciated from a c o m p a r i s o n of these a n d subsequent figures, the m a g n i t u d e a n d position of the fluctuations are very sensitive to the details of the b o u n d a r y condition applied to the t o p surface. We c a n n o t therefore predict the m a g n i t u d e a n d positions of the stress peaks in the realistic case of a conical
h o p p e r attached to the base of a cylindrical b u n k e r until a solution is available for the stress distribution at the base of the bunker. The effect of imposing a stress of m a g n i t u d e predicted by the RSF, but with a different principal stress direction, is s h o w n in Fig. 5. Here a series of s h a r p stress changes is seen to p r o p a g a t e d o w n the h o p p e r reflecting from the wall a n d centre line. This behaviour can be attributed to imposing the a w k w a r d b o u n d a r y condition of ~, = 1.05 × n/2 o n the centre line where ¢ must inevitably equal n/2 a n d a similar discontinuous change in qJ o n the wall. However, a linear variation of ~p between its centre line a n d wall values causes n o p r o b l e m s when the m a g n i t u d e of the stress is the R S F value; see Fig. 6. G r e a t e r fluctuations
Stress and velocity fields cohesionless material discharging from a conical hopper
3935
0.8 --,
0.7
~g
0.6
o
MOC
.....
RSF
..
wall
0.5 I~
s
0.4
r~
-~
0.3
," 'D
0.2
E "~
0.1 0.0 0
0.2
0.4
0.6
0.8
1
1.2
dimensionless r a d i ~ d i s t a n c e , R [-] Fig. 2. Comparison of the wall and centre-line stresses predicted by the MOC with the RSF solutions (0w = 30°, ~b = 30°, ~b,, = 10°, q = 1.5q rsv, ~ = ~RSF).
0.5 p
o
MOC
•
i
~.,
0.4
,.
0.3
.....
RSF
•
.
wall
-"
E
0.2
tre-line
e-
E
0.1
s
.~
J
0.0 0.00
,
,
,
0.20
I
~
,
,
I
,
0.40
L
J
0.60
I
J
,
J
I
,
0.80
,
,
1.00
I
,
,
1.20
dimensionless radial distance, R [-] Fig. 3. Comparison of the wall and centre-line stresses predicted by the MOC with the RSF solutions (0w = 30°, ~b = 30'~, 0w = 10°, q = 0.5 qRSF, t~ = ~RSF).
are found with a linear variation in ~ and a constant value o f q equal to the mean of the R S F prediction; see Fig. 7. Figures giving the stresses for the remaining conditions listed in Table 1 are given by Moreea (1993). These are all qualitatively similar to the results presented here. It can be seen that in all cases, convergence towards the RSF value is rapid with the fluctuations about the mean having decreased to about a quarter of their initial values by about half way down the hopper (R = 0.5). However, when large fluctuations are im-
posed by extreme boundary conditions, their magnitude, though ever-decreasing, can remain appreciable compared to the local RSF prediction. It should be noted that in Figs 4 and 7, the results curve back on themselves. This is due to overlapping characteristics. U n d e r these circumstances, discontinuous solutions would occur but this phenomenon was not investigated. The results presented in these two graphs are therefore approximate. It is, however, believed that the results presented in the remaining figures are accurate.
3936
S.B.M. MOREEA and R. M. NEDDERMAN
0.5
'
'
'
I
o
'
'
I
'
'
'
I
'
'
'
I
'
MOC
'
II
'
'
'
. •
i
t~
'
s
0.4
.....
RSF
- "
wall
"Jl
=
0000
0.3
t~
o
cO
E
0.2
~
f
"~C~3": " ' "
0.1
~
6
~
o
"
i"
~ 4 " ' ' "
centre-line
000
•
0.0 0.00
-"
00 I
,
,
,
,
0.20
I
,
,
0.40
,
I
J
,
0.60
A
I
,
,
0.80
,Ol
,
,
1.00
,
1.20
dimensionless radial distance, R [-] Fig. 4, Comparison of the wall and centre-line stresses predicted by the MOC with the RSF solutions (O,.=30:',qb=3OL~bw= l O , q = q ¢ ,,sF,~k= ~,sF) .
0.40
'
i
0.35
o .....
,g
0.30
eL
'
'
I
'
'
'
I
'
'
1
'
'
I
MOC RSF
'
/
-
I
o
'
'
wall
0.25 co
E
2.°'II ~
°o o,
.J:entre-line
0.20
c~
0.15 e-,
iO
0.10
E
0.05 0.00
""
0.0
0
". - '
,
I
0.2
,
,
,
I
,
0.4
,
,
I
,
,
0.6
~
I
~
,
,
0.8
I
1.2
1.0
dimensionless radial distance, R [-] Fig. 5. Comparison of the wall and centre-line stresses predicted by the MOC with the RSF solutions (0., = 20 c~,~ = 35 '~, ~b., = 10°, q = qKSV,~9 = 1,05~pRsr).
In conclusion, convergence is much more rapid than that found by Horne (1978) for a wedge-shaped hopper, and this can be attributed to the much greater effect of the wall in the axi-symmetric case. In particular, the curvature of the wall means that the wall stresses are immediately aware of the conical nature of the hopper, unlike the wedge-shaped case, in which the wall stresses are totally independent of the existence of the far wall until the first characteristic from that wall arrives. Thus in the conical case, convergence towards the radial stress field can begin im-
mediately, whereas in two dimensions, the wall stresses often diverge from the RSF value in the early stages. THE VELOCITY FIELD
The assumptions of coaxiality and incompressibility lead, respectively, to
(8v, tan~p - ~
18vo ~) + r~-ff +
8vo l~v~ Vo = --~-r + r ~
+ --r (16)
Stress and velocity fields cohesionless material discharging from a conical hopper 0.40
'
'
'
I
o
0.35
..... ~g
0.30
e.,
0.25
'
'
'
I
'
'
'
I
'
'
'
j
I
MOC
wall
RSF
J
i
3937
l
l
~n~-line
o
E
0.20 0.15 0.10 0.05 0.00
JI
I
I
[
,
I
,
,
,
,
0.4
0.2
0.0
1
I
,
,
,
0.6
i
I
[
l
l
t
1.0
0.8
1.2
dimensionlessradialdistance,R[-] Fig. 6. Comparison of the wall and centre-line stresses predicted by the MOC with the RSF solutions (0~ = 15 ~, ~b = 30°, ~bw= 5°, q = qRSF, I]/ = ~/lin).
0.40 "7" t~
;
0
0.35
0.25
E
0.20
'
'
;
I
'
'
1
'
'
'
1
'
MOC
.....
0.30 "~
1
=
1
'
. "
'
'
w~
RSF
"f~ntre-line
0.15
o
'
Oo°o
0.10
~D
s
E 0.05
s
~
O
• s
~ ' , , I , , , I , ,
0.00
0.2
0.0
,
I
0.6
0.4
,
,
,
I
,
0.8
~
~
I
~
~
1.0
1.2
dimensionless radial distance, R [-] Fig. 7. Comparison of the wall and centre-line stresses predicted by the MOC with the RSF solutions (Ow = 20 "~, q9 = 30~, q~,,,= 10°, q = q R. .S F. . . O -- Otin).
(Vo= 0), the radial velocity is given by,
and
•v, ~r
-
1 ~Vo r ~0
2
v,
v0cot 0
r
r
(17)
where qJ is the m a j o r principal stress direction. This p a r a m e t e r is the only interaction between the stress a n d velocity fields. If one m a k e s the a s s u m p t i o n s (a) t h a t ~Ois given by the radial stress field a n d (b) t h a t the flow is radial
v,=-~exp(-3fltan2qJdO ).
(18)
T h e values predicted from eq. (18) are k n o w n as the radial velocity field (RVF) a n d are t h o u g h t to be the a s y m p t o t e to which the velocities will tend for small values of r. As with the radial stress field, the t o p b o u n d a r y condition has not been used.
3938
S. B. M. MOREEAand R. M. NEDDERMAN
The rate of approach to the asymptote can be studied by using the method of characteristics since the equations form a hyperbolic pair. This is the most convenient method of solution since discontinuous solutions are possible. The equations of the characteristics and the ordinary differential equations along the characteristics in spherical coordinates are given below. Equation of the ~-characteristics:
/
dO = t a n - 0 \
(19)
4] r'
Equation along the 7-characteristic: dt,~ + dv0tan ( 0 - 4 )
(
- Vo
+ d013v'tan 20
cot0 l
1 + c°s20/J +
d,
eral set of stress and velocity boundary conditions on the top surface would require interpolation between the stress characteristics in order to provide the values of 0 required for the evaluation of the velocity field. No attempt was made to do so. In the investigations below, we have assumed an analytical form for the variation of 0 with position, often the RSF value, and predicted the resulting velocity fields. Though presented below as functions of the fractional radius, it should be borne in mind that the stress field will not have converged to its RSF values until about half-way down the hopper and the following calculations show the development of the velocity field from conditions imposed at that height. Once again arithmetical difficulties occur on the axis of symmetry, but applying l'H6pital's rule to eqs (20) and (22) yields
2Vr--r = O.
Equation of the fl-characteristic:
+ 3v, tan 20 dO + 2v, --dr= 0
(04+ ) -6'.-
d0=tan
r
(23)
f
(21)
"
dr, + dv0(tan ( 0 + 4 ) - see 20 ) Equation along the/#characteristic:
+ 3Vr tan 20 dO +
dv,+dvotan(O +4)+dO[3vrtan20 -
t,o (
I
c~-s cot0 2~ ~ ] J + 2v,-dr 7 = 0
i
1
f
]
(22)
i
i
'
I
As before these equations were used in preference to eqs (20) and (22) whenever 0 < 0.03 radians. The boundary conditions are vo = 0 on b~th 0 = 0 and 0 = Ow, and some velocity distribution imposed on the top surface, R = 1. The equations were tested by imposing the radial velocity field on R = 1 and the results are presented in Fig. 8 in the form G(O)= R 2 U, where U is the dimensionless velocity v,/vo and
'
'
'
I
'
'
'
I
'
'
'
centre-line 1.0
.... t|
i
CD
0.9
d o
0.8
.,a
0.7 0.6
/
RSF
wall
0.5
. . . . . . . .
^^^OO,.~C
• w l w .
O.4 I 0.00
0.20
(24)
r
It is seen that for these equations, the characteristic direction are 0 -+ ~/4 as opposed to the stress characteristics whose directions are 0 -+ e, where (e = ~/4 qV2). Thus, the characteristics in the two cases are not coincident and any full solution starting from a gen-
1.1
dr 2v~-- = 0.
0.40
0.60
0.80
1.00
dimensionless radial distance, R [-] Fig. 8. MOC predictions of the wall and centre-line velocity functions where the solution to the RVF has been imposed (Ow= 30~, ~b= 30°, 4~w= 10°).
Stress and velocity fields cohesionless material discharging from a conical hopper
v0 is the centre-line velocity at the top of the hopper. With this ordinate the results should be constants and this is seen to be the case. The results for the case of a uniform plug flow imposed on R = 1 are shown in Figs 9 and 10. Figure 10 shows the directions of the velocity vectors; it should, however, be noted that the horizontal scale in this figure has been enlarged by a factor of two to aid clarity. It can be seen that sudden changes in the direction of the velocity occur and that these propagate along the velocity characteristics. The velocity discontinuity starting at the topmost part of the wall, point A, is clearly visible. This reaches the centre line at point X and continues to B. Its mirror image CXD is visible. The discontinuities reflect from the walls and reach the centre line at E.
3.0
'
'
I
I
3939
However, the magnitude of the velocity change across BE and DE is very much less than that across discontinuities AX and CX. The discontinuities continue to reflect backwards and forwards across the hopper below E but with ever decreasing magnitude. On the wall, the velocity function G(O)= U R ~ starts with the value 1 but changes discontinuously to the lower value of cos 0w as shown in Fig. 9. The wall velocity then oscillates with a small amplitude and large wavelength as it approaches the constant value required by the radial velocity field. However, an appreciable disturbance occurs at point D. By contrast, the centre-line velocity decreases continuously from its imposed value down to point X, whereupon a large discontinuity occurs, followed by a smaller
'
'
I
I
'
'
'
0 "7
2.5
¢D
& o ¢J
1.5
centre-line
0 0 (~000000
2.0
~ m ~ ~ m ~ ~
1.0 >
.......
0.5 0.0 0.00
,
I
0.20
,
,
~
.I
,
~
0.40
,
I
~
,
0.60
,
I
,
0.80
~
I
1.00
d i m e n s i o n l e s s radial distance, R [-] Fig. 9. Predicted velocity functions where a uniform plug flow field has been imposed on an initial arc of dimensionless radius 1.0 (0w = 30 °, ~b = 30 °, q~, = 10s).
X\kk~ ~X\{",',~\,
i ' i"_ll,/"~, t i j f t i t t b ,illltl/tl
i/Ill ~1
\'%'~\ \x, ~ t ! I 4 J z i , l
Fig. 10. Predicted velocity corresponding to the velocity field in Fig. 9.
3940
S.B.M.
MOREEA
and R. M. NEDDERMAN
discontinuity at point E. Figure 10 shows remarkable similarity to the velocity fields assumed by Moreea (1993) following arguments developed by Drescher (1991). It was assumed that the velocity field consisted of three regions, a plug flow region in the top part (region ACX of Fig. 10), a region in which the velocity was constant and parallel to the walls (corresponding to region AXD) and a region in which the flow was radial (below line DXB). Examination of Fig. 10 shows that this is not accurate, but none the less is a very good approximation to the theoretical velocity field. This close parallel perhaps explains the success of Drescher's and Moreea's approximate analyses. Figures 11 and 12 show the results for the imposition of a uniform radial velocity as the top boundary
2.0
'
'
I
'
I
condition. The results are similar to those shown in Figs 9 and 10 apart form the absence of the initial discontinuity in the wall velocity at A. Figures 13 and 14 show the results when a uniform velocity parallel to the walls was imposed on R = 1. In this case a velocity discontinuity occurs immediately at O, since an impossible combination of velocities has been imposed at this point, and further velocity discontinuities are generated at A and C. These two sets of discontinuities propagate as effectively parallel discontinuities throughout the hopper, with the discontinuities O P and CD reflecting from the left-hand wall to give PQ and DE. None the less, it can be seen from Fig. 13 that convergence to the radial velocity field has occurred by about R = 0.5, though there are small
'
'
'
I
ooO
°O
'
'
oo
o~
'
I
'
'
1.8 1.6
v
centre-line
~°°Oooo .
1.4 e',
1.2 1.0 0.8 --
0.6 0.00
,
,
.......... ~ ~ l ~ " ~ ,
0.20
I
,
,
,
, : ¢ f f ~ O 0 0 0 I
0.40
,
,
,
I
0 0 ,
0.60
,
,
I
,
0.80
,
,
1.00
dimensionlessradialdistance,R[-] Fig. 11. Predicted velocity functions where a uniform radial field has been imposed on an initail arc of dimensionless radius 1.0 (0., = 30°, 4~ = 30<', ~b.,= 10°).
"~"'~'~ "k\~ ' ~ ," l, t ~t ,t 1,~1'/1 ~z/i/l~~"
Fig. 12. Predicted velocity vectors corresponding to the velocity field in Fig. 11.
Stress and velocity fields cohesionless material discharging from a conical hopper
2.5
'
'
'
I
'
'
'
I
'
'
'
I
i
'
i
I
3941
i
o 0
line
2.0 0 0 o 1.5
8 o
~lIIi~l~~[~l:O~
0
000 O0 0 0 wall
l.O
0 ~
0.00
0
0 0.20
0
0
0
0
o
o
O
000 0 0
0
0.40
o
0.60
0.80
1.00
dimensionless radial distance, R [-] Fig. 13. Predicted velocity functions where a constant velocity parallel to the hopper wall has been imposed on a horizontal surface corresponding to a velocity discontinuity (cylindrical-conical bunker, 0~ = 30 °, 4' = 30°, 4'w = 10°).
A
O
C
( +. ; ?'x X4" l l'Zl i l / / /
"% ",,",,',,
l, 11 d. tll /
/f/"
t¢,/ltlff
Fig. 14. Predicted velocity vectors corresponding to the velocity field in Fig. 13.
disturbances at smaller values of R caused by the reflection of the now much attenuated discontinuities. We have seen that the stress field converges to the radial stress field prediction by about half-way down the hopper and that within the radial stress field region, the velocities converge to the radial velocity field prediction at about the half-depth. It should not therefore be assumed that the velocities do not converge to the radial velocity field until R ~ 0.25. The principal stress direction though still varying in the upper half of the hopper will have values close to the R S F prediction and, therefore, convergence of the velocity field to the R V F value will have started before
the stress field has converged. It is therefore not surprising that Cleaver found agreement with the R V F prediction up to about R = 0.5.
CONCLUSIONS It is generally agreed that the radial stress and velocity field are the asymptotes to which the stresses and velocities tend at great depth in a conical hopper discharging a cohesionless C o u l o m b material. In this paper we have studied the rate of approach to these asymptotes using the method of characteristics. A wide range of boundary conditions have been
3942
S.B.M. MOREEAand R. M. NEDDERMAN
imposed on the top surface. It has been found that in all cases both the stress and velocity fields do tend to the radial field asymptotes and that close convergence is normally achieved by about half-way down the hopper. The approach to the asymptote is however rarely monotonic and the fluctuations about the asymptote can be large especially when conditions significantly different from the radial field predictions are imposed on the top surface. Jenike's design method assumes that the stresses have converged to the radial stress field values in the region immediately above the orifice and these calculations show that this assumption in his method is fully justified. The velocity predictions show that there are regions of substantially constant velocity separated by sudden velocity changes. This gives weight to those approximate analyses based on Drescher's method which assumed constant velocity regions separated by velocity discontinuities. NOTATION A (/
G(O) P P q r R
U D Uo
arbitrary constant, mZ/s height of top surface above the orifice, m parameter defined by G ( 0 ) = UR 2, dimensionless coordinate of the centre of Mohr's stress circle, N / m 2 dimensionless stress ( = p/Ta), dimensionless parameter defined by q = p/Tr, dimensionless radial coordinate, m dimensionless radius ( = r/a), dimensionless dimensionless velocity ( = t'/Vo) velocity component, m/s centre-line velocity at the top of the hopper, m/s
Greek letters
7 e 0 0w it
weight density, N/m 3 angle defined by ~: = x/4 - ,;b/2, rad zenithal coordinate, rad wall angle, rad coefficient of internal friction ( = tan qS), dimensionless
a ~b ~b,. 7,
eJ
normal stress, N/m 2 shear stress, N / m 2 angle of internal friction, rad angle of wall friction rad azimuthal coordinate, rad angle between the major principal stress direction and the r direction, rad angle defined by eq. (9), rad
The superscipt RSF refers to values predicted from the radial stress field. M O C refers to values predicted by the method of characteristics.
REFERENCES
Baldwin, A. J., 1988, velocity distributions in the flow of 9ranular materials. Part II. Tripos Research Project Report. Department of Chemical Engineering, University of Cambridge. Cleaver, J. A. S. and Nedderman, R. M., 1993a, Theoretical prediction of stress and velocity profiles in conical hoppers. Chem. Enynq. Sci. 48, 3696-3702. Cleaver, J. A. S. and Nedderman, R. M., 1993b, Measurement of velocity profiles in conical hoppers. Chem. Engn9 Sci. 48, 3703 3712. Drescher, A., 1991, Analytical Methods in Bin-load Analysis, Vol. 36, Developments in civil engineering, Elsevier, Amsterdam. Hampson, H. A., 1988, The Theoretical Prediction of Velocity Distributions in Granular Materials. Part II. Tripos Research Project Report. Department of Chemical Engineering, University of Cambridge. Horne, R. M. and Nedderman, R. M., 1978, Stress distributions in hoppers. Powder TechnoL 19, 243 254. Jenike, A. W., 1961, Gravity flow of bulk solids. Bull. 108, University of Utah, USA. Jenike, A. W., 1964, Storage and flow of solids. Bull. 123, University of Utah, USA. Johanson, J. R. and Jenike A. W., 1962, Stress and velocity fields in gravity flow of bulk solids. Bull. 116, University of Utah, USA. Moreea, S. B. M., 1993, Gravity flow of bulk solids in axi-symmetric bunkers. Ph. D. Thesis, University of Cambridge. Moreea S. B. M. and Nedderman, R. M., 1993, Discontinuous velocity fields in a plane strain bunker. J. Israel Inst. Chem. Engng 22, 76-80. Nedderman, R. M., 1992, Statics and Kinematics of Granular Materials. Cambridge University Press, Cambridge. Pitman, E. B., 1986, Stress and velocity fields in twoand three-dimensional hoppers. Powder Technol. 47, 219 231.