- Email: [email protected]
An alternative presentation of the design parameters for mass flow hoppers
. .
PowderTechnology,42(1985)79-89
An
Alternative
Presentation
of the
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Design
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Parameters
-
forMass
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Flow
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Hoppers
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:: : .79-=:-
....
13. A. M O O R E and P. C. A R N O L D Department o f Mechanical Engineering, The Universily o f Wollongong, Wollongong, N.S. IV., 2500 (Auslralia)
SUMMARY
T h e p r o c e d u r e f o r d e t e r m i n i n g t h e critical h o p p e r o u t l e t d i m e n s i o n a n d wall s l o p e f o r m a s s f l o w h o p p e r s b y t h e J e n i k e m e t h o d is well established and d o c u m e n t e d . However, e x i s t i n g p r e s e n t a t i o n s r e l a t i n g t h e b u l k solid f l o w p r o p e r t i e s o f e f f e c t i v e angle o f i n t e r n a l f r i c t i o n a n d t h e k i n e m a t i c angle o f wall frict i o n w i t h t h e h o p p e r wall s l o p e a n d f l o w f a c t o r f o r mass f l o w are o f t e n i n c o n v e n i e n t f o r m a n u a l use a n d p r e s e n t d i f f i c u l t i e s w h e n i n c l u d e d in design p r o g r a m s s u i t a b l e f o r microcomputers. T h i s p a p e r details an a l t e r n a t i v e p r e s e n t a t i o n o f t h e original J e n i k e f l o w f a c t o r charts. These alternative charts have been abbreviated to d i s p l a y o n l y t h e critical design valuea in t h e b o r d e r region b e t w e e n mass f l o w a n d f u n n e l f l o w . T h e charts e l i m i n a t e t h e n e e d f o r i m p r e c i s e p a r a m e t e r i n t e r p o l a t i o n s b y disp l a y i n g t h e r e q u i r e d design p a r a m e t e r s in t h e f o r m o f c o n t o u r s o f c o n s t a n t wall s l o p e a n d critical f l o w f a c t o r as a f u n c t i o n o f t h e e f f e c t i v e angle o f i n t e r n a l f r i c t i o n a n d t h e k i n e m a t i c angle o f wall f r i c t i o n . A n illustrative e x a m p l e is p r o v i d e d t o d e m o n s t r a t e t h e i n h e r e n t a d v a n t a g e s o f this presentation for the evaluation of mass flow hopper geometries.
For those installations requiring the operating features of complete emptying of contents, first in-first out discharge sequence, minimum segregation or uniform flow rate the hopper geometry parameters should be chosen to ensure true mass flow discharge. Mass flow hoppers are also employed in single and multiple outlet expanded flow bins. Determination of the mass flow hopper g e o m e t r y p a r a m e t e r s is a t w o - s t a g e p r o c e s s . Firstly, the flow properties of the particular bulk solid are assessed from laboratory tests on a representative sample taking account of such conditions as moisture content, time of storage at rest, and temperature. The calculation of the outlet dimension and the hopper wall slope to ensure that no stable cohesive arches form and that flow occurs along the hopper walls constitutes the second stage. T h i s is a c h i e v e d b y a p p l y i n g t h e t h e o r y o f Jenike [1] to the measured bulk solid flow properties. The scope of this paper does not allow a full discussion of either the laboratory measurement of flow properties, or the general theory of gravity flow of bulk solids in mass flow channels. These are fully documented in R e f s . 1 a n d 2. I t is f i t t i n g , h o w e v e r , t h a t a brief overview of the Jenike design procedure be presented along with various methods of displaying mass flow parameters relationships that have been developed.
INTRODUCTION
The successful design of bulk solid storage facilities both in terms of structural integrity and reliable and predictable operation can be ensured by applying well-established and accepted procedures. These procedures, proven by experience and application in industry, are based on the theory of flow of bulk solids f i r s t e x p o u n d e d b y J e n i k e [ 1 ]. The inherent advantages of using mass flow hoppers in the design of gravity discharge storage facilities is well recognised. 0032-5910/85/$.3.00
D E T E R M I N A T I O N O F T H E MASS F L O W H O P P E R GEOMETRY PARAMETERS The Jenike design procedure [1] takes into account the bulk solid flow properties and the stresses imposed by the hopper, in determining the geometry parameters of critical outlet dimension and wall slope. The required flow properties which describe the variation of strength and frictional characteristics with consolidation pressure are: © Elsevier Sequoia/Printed in The Netherlands
80 % %
b o t h t h e s e p a r a m e t e r s a n d , f u r t h e r , t h e flow factor value i~ influenced b y ~. T h e f l o w - n o f l o w c o n c e p t is i m p l e m e n t e d by approximating the stress conditions occurring in a converging channel during flow by a radial stress field according to the relationship
B
Cond i
~
FF
/ / -
Coh~slvc
-.00
o = 7rS(c~) "
NO-FLOW
Fig. I. Flow-no design.
FLOW
flow concept
The solution of the set of partial differential equations describing the radial stress field under specific boundary conditions a l l o w s t h e v a l u e o f S(ce) t o b e c o m p u t e d and, hence, the flow factor determined by the relation
£z, for mass flow hopper
-- flow functions for both instantaneous and time storage conditions -- effective angle of internal friction 6 --kinematic angle of wall friction ¢ for requir(,d wall lining materials and surface finishes -- bulk density p By utiiising a flow-no flow concept (Fig. 1), t h e s t r e n g t h o f t h e b u l k s o l i d (as r e p r e s e n t e d b y t h e f l o w f u n c t i o n ) is c o m p a r e d with the stresses imposed by the hopper (represented by the flow factor). The flow f a c t o r is d e f i n e d b y : ff -
O I
(i)
Referring to Fig. I, flow will occur for ~i > oc w h e n the cohesive arch across the outlet will fail, since the stress ~, i m p o s e d by the hopper exceeds the unconfined yield s t r e s s o f t h e b u l k s o l i d Oc- T h e c r i t i c a l v a l u e o f a~ o c c u r s a t t h e i n t e r s e c t i o n p o i n t o f t h e flow factor and the flow function. The flow f a c t o r v a l u e is a f u n c t i o n o f t h e w a l l f r i c t i o n angle, the effective angle of internal friction, the hopper wall slope and the form of the hopper outlet (axisymmetric or plane flow). If ¢ o r 5 v a r y w i t h a ~, a n i t e r a t i v e p r o c e d u r e must be carried out until a I converges to the critical value. T h e m i n i m u m o u t l e t d i m e n s i o n B is defined by: B -
e,H(oO pg
-
al H(oO ff
pg
(3)
(2)
The flow factor ff and the hopper wall s l o p e c~ a r e d e s i g n p a r a m e t e r s o f t h e J e n i k e p r o c e d u r e b e c a u s e , as e q n . (2) i n d i c a t e s , t h e c r i t i c a l o u t l e t d i m e n s i o n is d e p e n d e n t o n
ff =
H(cx)(I + sin ~)S(~) 2 sin
(4)
Jenike computed flow factor values and presented the results in the well-known series of charts for axisymmetric and plane flow hoppers [1]. These graphs display c o n t o u r s o f c o n s t a n t f l o w f a c t o r as a f u n c t i o n of the hopper slope and the kinematic angle of wall friction for specific values of effective angle of internal friction (30 ° through 70 ° by 10 ° increments). Examples of the flow factor charts for asymmetric and plane flow hoppers (for an effective angle of int e r n a l f r i c t i o n o f 5 0 °) a r e p r e s e n t e d i n F i g s . 2 and 3 respectively. As depicted in these figures, particularly Fig. 2, the region of the flow factor contours is b o u n d e d b y a l i m i t . T h i s b o r d e r r e p r e s e n t s the limit beyond which the boundary conditions are not compatible with the radial stress field and flow does not develop along t h e h o p p e r w a l l s . T h i s b o u n d is p a r t i c u l a r l y severe for axisymmetric hoppers and, to ensure that mass flow occurs, Jenike recommends that the wall slope be reduced 3 to 5c from the bound to account for any variation of the wall friction value in practice from the laboratory test values. For plane flow h o p p e r s , h o w e v e r , t h e b o u n d a r y is n o t s o stringent, but to prevent the development of excessive non-flowing regions originating f r o m t h e h o p p e r t r a n s i t i o n a d e s i g n l i m i t is suggested (the dashed line, Fig. 3). In determining the mass flow hopper geometry, the values of the hopper wall slope and flow factor are determined from these charts for the combination of effective angle of internal friction, the kinematic angle
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z
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F i g . 2. F l o w
10 °
~_0° 30 ° HOPPER WALL SLOPE o~-OEGREES
z~O°
50 °
factor chart for axi.s.yrnmetric hoppers, ~ = 50 ° (from
Jenike [1]).
s0o
10 °
0 °
0°
10 °
20 °
30 °
40 °
50 °
6~
HOPPER WALL SLOPE c~-OE6REES F i g . 3- F l o w f a c t o r c h a r t f o r p l a n e f l o w h o p p e r s , ~ = 5 0 ° ( f r o m J e n i k e [ 1 ] ) _
o f wall f r i c t i o n a n d t h e h o p p e r o u t l e t s h a p e . A l t h o u g h it is p o s s i b l e t o m i n i m i s e t h e f l o w factor by suitable selection of the hopper s l o p e f o r a g i v e n wall f r i c t i o n v a l u e ( a n d t h u s o p t i m i s e t h e f l o w a b i l i t y o f t h e h o p p e r [ 1 ] ), f r o m a p r a c t i c a l d e s i g n v i e w p o i n t it is p r e f e r a b l e t o m a x i m i s e t h e h o p p e r wall s l o p e . T h i s
is a c h i e v e d b y u s i n g v a l u e s a l o n g t h e s u g g e s t e d d e s i g n limits f o r p l a n e f l o w , o r a l l o w i n g a 3 ° r e d u c t i o n o f wall slope f r o m the mass f l o w - f u n n e l f l o w l i m i t in t h e case o f a x i s y m metric hoppers. Although the Jenike flow factor charts c l e a r l y d i s p l a y t h e l i m i t s o f h o p p e r wall s l o p e
82
between mass flow and funnel flow and the respective flow factor values along the design boundary, they are disadvantaged by the neccssmq¢ p a r a m e t e r i n t e r p o l a t i o n s r e q u i r e d . There are two major problems associated with the pm'ameter interpolations. Firstly, since the charts are presented for specific v a l u e s o f 5, t h o s e d e s i g n s h a v i n g i n t e r m e d i a t e values require the parameters of flow factor and hopper wall slope to be adjusted between the respective charts to ensure accuracy. Secondly, due to the troughed form of the flow factor contours in the region of the desiEn limit, determination of the flow factor value can be difficult. For example, consider determination of the flow factor for ¢ of 15 ° a n d a" o f 4 0 ° f r o m Fig. 3. .-\ c l e a r e r i n d i c a t i o n o f t h e i n f l u e n c e o f on the limit of hopper wall slope for mass flow in c o n i c a l h o p p e r s was p r e s e n t e d b y J o h a n s o n a n d C o l i j n [ 3 ] ( F i g . 4)_ T h i s d i a gram condenses the mass flow limits of the Jenike flow factor charts to one diagram by d i s p l a y i n g c o n t o u r s o f c o n s t a n t 6 as a f u n c t i o n o f ¢ a n d c~_ I n t e r p o l a t i o n f o r i n t e r m e d i a t e v a l u e s o f ~i i n o r d e r t o d e t e r m i n e t h e n ] a x i l n u n ] v a l u e s o f a- is m o r e c o n v e n i e n t than for the denike flow factor charts. .Johanson and Colijn [3] also include an alternative series of flow factor charts to determine the flow factor after a suitable h o p p e r wall slope has b e e n s e l e c t e d f r o m Fig. -t. T h e s e c h a r t s d i s p l a y c o n t o u r s o f c o n s t a n t f l o w f a c t o r as a f u n c t i o n o f 6 a n d .O f o r t h e s p e c i f i c ct" v a l u e s o f 1 0 , 2 0 a n d 3 0 °. The flow factor chart for a hopper wall slope o f 2 0 ° is d e p i c t e d i n F i g . 5. I n t e r p o l a t i o n 50
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}---I'=-~%~--I ---j- WITH
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o-i-!-l-I 0
I0
20
HOPPER
RADIAL-STRESS
I'~"D"¢ 30 ANGLE,
40
~ ~ 50
60
o~
Fig. 4. H o p p e r w a l l s l o p e l i m i t s f o r a x i s y r n m e t r i c hoppers (from Johanson and Colijn [3])-
1 L/
iN',,:,.,
k~ \ 1 ~ !
~h, 'N,"N I ~ J
o1\
(/1111 ! I i
i
I I I t I 1 I I I I
0
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20 WALL
Fig. 5. Critical
flow
30 FRICTION
factors
= 20 ° ( f r o m J o h a n s o n
40
50
ANGLE,
for mass
flow
60
,i~ hoppers,
and Colijn [3]).
b e t w e e n t h e f l o w f a c t o r c o n t o u r s is m o r e convenient in this presentation due to their more uniform nature. However, imprecise f l o w f a c t o r values c a n arise b e c a u s e o f t h e presentation of the charts in 10 ° increments of wall slope. Limits on the selection of hopper wall s l o p e h a v e a l s o b e e n p r e p a r e d f o r auxisymmetric and plane flow hoppers for mass f l o w b y A r n o l d e t el. [ 2 ] a n d are p r e s e n t e d in Fig. 6. This presentation utilises the limits proposed by Jenike [1]. Of more benefit, however, particularly in regard to computer applications, are the equations which are given to represent the design limits. For a-xisymmetric hoppers
0
t180- cos-,p = sin !
\ 2 sin ~} ]
(sinfUl and for plane flow hoppers e x p [ 3 . 7 5 ( 1 . 0 1 ) ( a -30)/I0] __ ¢ c~ =
30
~- ~°l--]'--i-~_~f i 1 A"EANOTCO.~ATI.~E ~o !___~I_ I__!'%.-.1 F,~-D(No ,ASS ,-OW '-
'V I/I/I i,op, I~!\1 - ~°1\1~ \~ i\l\,;.,o ~ ~ ~ I ! I t /i i 1 1
0 . 7 2 5 ( t a n 6 ) °'2
(6)
f o r ¢ < 6 - - 3 ; q~ a n d ~ i n d e g r e e s . Reference to Fig. 6, however, indicates that for the range of common values of ¢ ( a p p r o x i m a t e l y 1 5 t o 3 0 °) f o r m a s s f l o w d e s i g n t h e m a x i m u m b o u n d o f ~ is r e l a t i v e l y insensitive to the variation of/i. The influence of the effective angle of internal friction on the flow factor has been considered by Jenike [4]. In dealing with plane flow hopper design he notes that the e f f e c t o f q~ is m i n i m a l o n t h e f l o w f a c t o r
s~-
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- _
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•
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HOPPER
\ ~ . . ~ .
3"~".
NRLL
SLOPE
40' •
-
50' •
5"IB'.
DEGREES
Fig. 6. Wall slope limits for axisyrnmetric and p l a n e flow h o p p e r s (after A r n o l d et aL [2]). value and presents the variation of the flow factor as a function of ~ (Fig. 7). Although F i g . 7 w a s p r e p a r e d f o r ¢ o f 2 5 ° a n d c~ o f 2 5 ° Jenike states that the flow factor variation will be negligible for different design values, thus having little effect on the outlet dimension B determined from eqn. (2). 2.O
..< ~s
0o
\ 30 °
50 o
70 °
90 o
EFFECTIVEANGLEOFlmErNAtFRICTION Fig. 7. V a r i a t i o n o f flow f a c t o r w i t h effective angle o f i n t e r n a l f r i c t i o n , ~ = 2 5 °, ~ = 25 ° ( f r o m J e n i k e [4]). ALTERNATIVE PRESENTATION OF THE MASS FLOW HOPPER DESIGN PARAMETERS A common factor between the various presentations discussed in the previous section for the presentation of the flow factor a n d t h e h o p p e r w a l l s l o p e is t h e n e e d t o r e f e r to several graphs in determining a satisfactory design. Generally, the selection of the mass flow hopper geometry involves the determination of the maximum hopper wall slope allowable
for the particular bulk solid flow properties and operating conditions. This necessarily involves selecting design values along the mass flow limits (refer to Figs. 2 and 3) and, as such, the remainder of the bounded mass f l o w r e g i o n is o f l i t t l e i n t e r e s t . T h e s e ~-reas are only utilised for such tasks as the verification of existing plant designs or the prediction of flow rate by such methods as that detailed-by Johanson [5]. With these factors in mind, an alternative presentation of the mass flow hopper design p a r a m e t e r s h a s b e e n d e v e l o p e d a n d is p r e sented in Figs. 8 and 9 for axisymmetric and plane flow hoppers respectively. These figures display the variation of the flow factor and hopper wall slope in contour form as a function of the kinematic angle of wall friction and the effective angle of internal friction. The data mesh for the formation of both charts was obtained by utilising the design b o u n d f o r (~ ( a s p r e d i c t e d b y e q n . ( 5 ) a n d ( 6 ) ) a n d t h e p a r t i c u l a r v a l u e s o f (b a n d 6 t o determine the maximum allowable value of o~. W i t h t h e s e t h r e e p a r a m e t e r v a l u e s t h e respective flow factor was determined using differential equation solution techniques and eqn. (4). Thus, only the values of maximum are displayed which will ensure mass flow f o r t h e r e s p e c t i v e v a l u e s o f ¢ a n d ~. T h e
84
45-
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~-
35.045E F F E C T I V E FINGLE OF
SI3556N65INTEFINRL F R I C T I O N - DELTFI
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Fig. S..Alternative presentation of axisymmetrie hopper design parameters. following design constraints have been taken into account in determining the data mesh: - - F o r a x i s y m m e t r i c hoppers, a 3 ° r e d u c t i o n of the hopper wall slope from the radial stress compatibility bound has been effected and the flow factor determined at this margin point. - - F o r plane f l o w hoppers, t h e w a l l s l o p e l i m i t specified in equation (6) and the constraint of ¢ < 5 -- 3 degrees, has been applied. -- The upper limit of the hopper wall slope is 6 0 °. This data mesh lends itself to computer applications for systems that do not have mathematical packages for the solution of differential equations (to determine the flow factor). The values required can be determined by linear interpolation techniques based on the data mesh. A n a d v a n t a g e o f t h i s c h a r t f o r m a t is t h a t a complete overview of the trends of flow factor and hopper wall slope variation with ¢ a n d 5 is p r e s e n t e d . R e f e r e n c e t o F i g s . 8 a n d 9 indicates the direct influence of ¢ on the v a l u e o f c~ a n d t h e m i n o r e f f e c t o f 6 , p a r t i c u larly for plane flow hoppers. The trend of
reducing values of flow factor with increasing is a l s o w e l l d e m o n s t r a t e d . T h i s t r e n d is in agreement with the presentation of Jenike [4], but allows the overall effect of ~ on the flow factor to be appreciated. For a constant value of 50 ° and a range of ¢ of 5 < ¢ < 4 0 ° , t h e f l o w f a c t o r is s e e n t o v a r y f r o m 1 . 2 5 to 1.22, justifying the assumption of Jenike to neglect the ¢ variation in determining an initial measure of flowability. F o r a g i v e n s e t o f v a l u e s f o r ~ a n d ~b, t h e design parameters of ~ and flow factor are determined by referencing the intersection point to the plotted contours. Although the contour increments for Figs. 8 and 9 are necessarily coarse to ensure clarity, large chart formats can utilise smaller contour intervals and reduce prorating of values. Correct interpretation of the data determined from the charts should be stressed. I t is i m p o r t a n t t o b e a r i n m i n d t h a t o n l y t h e maximum value of ~ and the respective flow factor value relevant to the traditional mass flow limits are displayed. Concerning the value of hopper wall slope, the ~ contour specifies the maximum value allowable to
. . . 50
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a= 1 5 . ~= 1 0 . - -
030- -
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45.
RNGLE
50-
OF
55-
INTERNRL
FRICTION
=
6@-
-
65-
o
7@-
OELTR
Fig. 9. Alternative p r e s e n t a t i o n o f plane flow hopper design parameters. ensure mass flow and slip of the bulk solid a l o n g t h e w a l l . V a l u e s o f c~ g r e a t e r t h a n t h a t specified can lead to a funnel flow discharge pattern, while a reduction below this value leads to a more conservative design. Reference to eqn. (2) similarly provides an insight into the variation of the flow factor about the design value. Utilising a flow factor value smaller than the design point will lead to a reduction in oi, a reduced outlet dimension B and, hence, the formation of cohesive arches. Noting that the design point specifies the critical flow factor value, a more conservative design will then result from using an increased value. The advantages of utilising this presentation to determine the hopper geometry based on the bulk solid flow properties have been noted in the preceding discussions. However, care should be exercised in the verification of m a s s f l o w o c c u r r i n g i n existing storage facilities. T h e v a l u e o f h o p p e r w a l l s l o p e c a n be checked knowing 6 and ¢ and ensuring t h a t ~ a c t u ~ < ~chart- D i f f i c u l t i e s a r i s e i n c h e c k i n g t h e h o p p e r o u t l e t d i m e n s i o n (as affected by the flow factor) because values
of ~ not lying on the mass flow limits relate t o f l o w f a c t o r s n o t d i s p l a y e d in Figs. 8 o r 9. Reference to the original Jenike flow factor c h a r t s (Figs. 2 a n d 3) i n d i c a t e t h a t t h e f l o w factor may be above or below that displayed in the alternative presentation, depending on used within the bounded mass flow region. I t is r e c o m m e n d e d t h a t f o r t h i s e x e r c i s e t h e Jenike charts be consulted because the comp l e t e f l o w f a c t o r v a r i a t i o n w i t h ~ is d i s p l a y e d f o r g i v e n v a l u e s o f q~ a n d 8.
I L L U S T R A T I V E EXAMPLE To demonstrate the use of these new design charts, the geometry of a mass flow hopper with a plane flow outlet for an ROM coal storage will be determined. The measured f l o w p r o p e r t i e s o f t h e c o a l a r e p r e s e n t e d as a function of the major consolidation stress a l i n F i g . 1 0 a n d i n e q u a t i o n f o r m m T a b l e 1. Two characteristics, typical of moist cohesive b u l k solids s u c h as c o a l , are d e p i c t e d i n Fig. 10. These are the reduction of ~ and ¢ (for some wall materials) to limiting values with
86 70.
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EFFECTIVE
ANGLE
,
.
,
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OF INTERNAL
FRICTION
-I-
I-----_________
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KINEMflTIC -
A N G L E OF M A L L F R I C T I O N STAINLESS STEEL -2UHHW P O L Y M E R - 3 -
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I
~ I~.
2 • " • "
2
S-
10.
iS.
MAJOR C O N S O L I O R T I O N Fig. I0.
3
t
Flow
prope~ies
of ~ and
¢ ~r
STRESS
the design
-
20-
KPR
example.
T.ABLE 1 Flow property
equations
for the design example
Function
Equation
Instantaneous
flow function
Unit
Oc = 0 . 3 2 a i
+ 1.20
Time flow function (2 days)
Oct = 0.35o" I + 2 . 0 5
Bulk density variation
:'= 850.74
[
H
z
kPa kPa
O1 ~0.0697
~5.----~- )
kg/m 3
f,
1 5~g
"
x
Fig. I1.
55
.
EFFECTIVE
Determination
of the
..
hopper
60 ANGLE
geometry
OF
.
INTERNAL
for the
design
65. FRICTION
example.
-- DEGREES
70.
Iteratio n No.
~ (°)
1
assume
60
ff
al (kPa) 2.07 1.98 1.98
2
64.5
1.111 1.080
3
64.6
1.079
increasing a~. The variation of the wall frict i o n a n g l e .~nay b e d u e t o e i t h e r a c o n v e x upward wall yield locus and/or the tendency of the bulk solid to adhere to the wall lining material at low values of normal stress. A p o r t i o n o f F i g . 9 is p r e s e n t e d i n F i g . 1 1 , w h i c h d i s p l a y s c~ c o n t o u r s a t 1 ° i n c r e m e n t s and the flow factor contours at 0.01 intervals. As previously stated, an iterative procedure must be used to determine the critical hopper geometry if either 8 or ¢ vary with a~. Essentially, the procedure suggested by Jenike [1] will be applied, first determining the critical flow factor value, noting the respective value and then applying the flow factor value to eqn. (2) to yield the hopper outlet d i m e n s i o n . T h e p r o c e d u r e is d e e m e d t o h a v e converged to the critical values when little change results in oi and 8 from successive iterations. Consider the determination of the hopper geometry under instantaneous conditions w i t h t h e U H M W p o l y m e r as t h e w a l l l i n i n g material. To start the procedure initially e s t i m a t e ~ = 6 0 °. W i t h ~b = 2 3 ° c o n s t a n t , from Fig. 11 point A yields a flow factor value of 1.111. Applying the flow-no flow concept (Fig. 1) t o t h e i n s t a n t a n e o u s f l o w f u n c t i o n a n d the respective flow factor produces an initial v a l u e f o r o~ o f 2 . 0 7 k P a . R e f e r e n c e t o t h e v a r i a t i o n d e p i c t e d in F i g . 1 0 f o r t h i s v a l u e o f o~ y i e l d s a n u p d a t e d v a l u e o f 5 o f 6 4 . 5 °. The subsequent values for each iteration in converging to the critical values are presented above.
.
~
P o i n t : :(refer Fig. i i )
"
(o) 23.0 23.0 23.0
:
-
A B C
For the values of 5 (64.6) and ¢ (23.0), t h e f l o w f a c t o r a n d cz c a n b e r e a d f r o m F i g . 1 1 as 1 . 0 7 9 a n d 3 0 . 3 ° ( s a y 3 0 . 5 °) r e s p e c t i v e l y . A s i n d i c a t e d in t h e a b o v e t a b l e , c o n vergence occurs relatively fast. Substitution of the relevant values into eqn. (2) allows the critical outlet dimension to be determined: H(~) = H(~ = 30.5) = 1.15 p = p(al = 1.98) = 788.1 kg/m 3 and hence B = 273 mm (say 275 ram). The critical hopper geometry parameters would then be detailed as ~ = 30.5 ° and B = 275 mm. The procedure becomes further complicated when considering a lining material that has a variable kinematic angle of wall friction. To illustrate the design procedure when both ¢ and $ vary with a l, the critical geometry parameters will be determined for time storage conditions (2 days at rest) and using s t a i n l e s s s t e e l as t h e w a l l l i n i n g m a t e r i a l . In a similar fashion to the previous example, values of 8 and ¢ must be assumed to obtain an initial flow factor value. Assuming 5 = 60 a n d ¢ = 2 0 °, t h e i t e r a t i o n v a l u e s a r e p r e sen.*.ed b e l o w . For the final values of flow factor = 1.097 a n d c~ = 3 5 . 5 °, t h e c r i t i c a l o u t l e t d i m e n s i o n c a n b e d e t e r m i n e d a s 4 8 5 r a m , w i t h H(c~) = 1 . 1 8 a n d p = 8 2 2 . 5 k g / m B. T h e v a r i a t i o n o f ~b w i t h a I c a n b e u t i l i s e d in mass flow hopper design by allowing larger wall slopes for outlet dimensions greater than the critical value determined
Iteration No.
~ (°)
ff
a, (kPa)
¢ (°)
Point (refer Fig. 11)
1 2 3
assume 60 62.3 62.4
1.115 1.098 1.097
3.75 3.66 3.65
assume 20 18.7 18.9
D E T
88 T A B L E o_ Variation o f .¢ and ~5 with (~l and resulting variation o f ~ with B for stainless steel oI (kPa)
5 (¢)
.0 (°)
ff
ot (°)
B (mm)
1.98 2.50 3.00 4.00 5.00 8.50 10.00 15.00
64.6 63.9 63.2 62.0 61.0 58.1 57.1 54.7
21.7 20.4 19.6 18.6 18.0 17.0 16.8 16.4
1.080 1.086 1.091 1.100 1.109 1.134 1.143 1.168
31.8 33.4 34.5 35.8 36.6 38.0 38.3 39.0
275 342 405 528 647 1042 1204 1724
in relation to particular instantaneousor time How functions. The hopper wall slope may be increased due to the reduction in wall friction which occurs from the increased values of a~ resulting from increased outlet dimensions. The variation of o~with B can be determined by incrementingo~ above the critical value, calculating the respective values of ¢ and and, from Fig. 11, reading the respective values of ~ and flow factor. Knowing the flow factor and the value of a , allows the outlet dimensionto be calculated. The variation of 6 and ¢ for a range of o, has been presented as a curve in Fig. 11. This curve is unique to the particular variations of ~b and 6 with a I and, as indicated, the critical design points for instantaneous (point I) and time storage (point T) conditions lie on this locus. Table 2 summarises the variation of ¢ and 5 with a i and the resulting variation of with B for stainless steel. It can be seen that for plane flow hopper outlets in the region of 1 m width, wall slopes of 38 ° could be utilised. This design data can also be applied to increasing the hopper wall slope in a stepwise fashion at intermediate levels above the hopper outlet. As the effective width of the hopper increases due to divergence of the walls, the wall slope can be increased in accordance with the above variation. This technique can prove useful in applications that have headroom and space constraints.
CONCLUSION An alternative method of presenting the mass flow hopper design parameters of flow
factor and hopper wall slope and their variation with the material properties of kinematic angle of wall friction and effective angle of internal friction has been detailed. The advantages resulting from this presentation over existing methods include: (i) T h e a b i l i t y t o o b t a i n a n o v e r a l l a s s e s s ment of the variation of the design parameters of ~ and flow factor along the limits of mass flow chosen for design purposes. (ii) A l l t h e r e l e v a n t d e s i g n d a t a r e q u i r e d for hopper geometry determination are presented on one chart (for either axisymmetric or plane flow hoppers), which reduces the amount of parameter interpolation required. (iii) A c h a r a c t e r i s t i c c u r v e c a n b e p l o t t e d on the chart presentation representing the v a r i a t i o n o f ¢ a n d 6 w i t h i n c r e a s i n g o 1. The critical values of ¢ and 6 for instantaneous and time storage conditions lie on this curve. The respective values of ~ and flow f a c t o r as s p e c i f i e d b y t h e c u r v e c a n b e u t i l i s e d t o p r e s e n t t h e v a r i a t i o n o f c~ f o r v a l u e s o f B greater than the critical. (iv) T h e d a t a m e s h u s e d t o d e t e r m i n e t h e c o n t o u r p o s i t i o n s On t h e c h a r t s c a n b e a p p l i e d to compile a hopper design computer program using linear interpolation techniques (rather than the solution of differential equations) to determine the required design parameters.
ACKNOWLEDGEMENT The authors wish to acknowledge the financial support provided by the National Energy Research, Development and Demonstration Council for the project 'Flow Properties of Coal for Storage Bin Design'.
The work reported here forms part of that project. : . - " . " .... . . T h e a s s i s t a n c e o f Dr: G. J. M 0 n t a g n e r i n facilitating the computer-generated diagrams is g r a t e f u l l y a c k n o w l e d g e d .
LIST OF SYMBOLS
B
ff FF g
H(~) m r
s(~)
6
critical arching dimension for a mass flow hopper flow factor for a converging flow channel flow function for a bulk solid acceleration due to gravity design function of ~ and hopper outlet shape [1] hopper outlet shape, axisymmetric (m = 1) o r p l a n e f l o w ( m = 0) radial dimension from vertex of hopper radial stress function half-angle of hopper or slope of hopper wall measured from the vertical weight bulk density effective angle of internal friction of a bulk solid
:
p :- bulkdensitY0£ a buikS01id -. : -:-::a .. n 0 i T n a l Stress._0r a v e r a g e - o f v e r t i c a l : " : and lateral:stress01 majorc0nsolidating stress 01 majo r stress acting at theabutment of a cohesive-arch ac unconfined yield stress of a bulk solid " . act unconfined yield stress of a-bulk solid under conditions of time storage at rest r shear stress ¢ kinematic angle of wall friction developed between a hopper wall and a bulk solid "
REFERENCES
1 A_ W. J e n i k e , S t o r a g e a n d ]Flow o f S o l i d s , B u l L 123, Eng. E x p t . Sta., Univ. of Utah (1964). 2 P. C. A r n o l d , A . G. M c L e a n a n d A . W. R o b e r t s , Bulk Solids: Storage, Flow and Handling, The University
of
Newcastle
Research
_~_~socla.tes
(TUNRA) Ltd., 2nd edn., 1980. 3 J. R . J o h a n s o n a n d H . C o l i j n , I r o n S t e e l Eng., XLI (1964) 85. 4 A . W . J e n i k e , P o w d e r T e c h n o L , 11 ( 1 9 7 5 ) 8 9 . 5 J . P~ J o h a n s o n , Trans. ~]in. Eng., A I 2 4 E . 2 3 2 (1965)69.
PowderTechnology,42(1985)79-89
An
Alternative
Presentation
of the
-
.
.
--
-'
Design
•
"
Parameters
-
forMass
.
- --
Flow
- :
Hoppers
- r .
:: : .79-=:-
....
13. A. M O O R E and P. C. A R N O L D Department o f Mechanical Engineering, The Universily o f Wollongong, Wollongong, N.S. IV., 2500 (Auslralia)
SUMMARY
T h e p r o c e d u r e f o r d e t e r m i n i n g t h e critical h o p p e r o u t l e t d i m e n s i o n a n d wall s l o p e f o r m a s s f l o w h o p p e r s b y t h e J e n i k e m e t h o d is well established and d o c u m e n t e d . However, e x i s t i n g p r e s e n t a t i o n s r e l a t i n g t h e b u l k solid f l o w p r o p e r t i e s o f e f f e c t i v e angle o f i n t e r n a l f r i c t i o n a n d t h e k i n e m a t i c angle o f wall frict i o n w i t h t h e h o p p e r wall s l o p e a n d f l o w f a c t o r f o r mass f l o w are o f t e n i n c o n v e n i e n t f o r m a n u a l use a n d p r e s e n t d i f f i c u l t i e s w h e n i n c l u d e d in design p r o g r a m s s u i t a b l e f o r microcomputers. T h i s p a p e r details an a l t e r n a t i v e p r e s e n t a t i o n o f t h e original J e n i k e f l o w f a c t o r charts. These alternative charts have been abbreviated to d i s p l a y o n l y t h e critical design valuea in t h e b o r d e r region b e t w e e n mass f l o w a n d f u n n e l f l o w . T h e charts e l i m i n a t e t h e n e e d f o r i m p r e c i s e p a r a m e t e r i n t e r p o l a t i o n s b y disp l a y i n g t h e r e q u i r e d design p a r a m e t e r s in t h e f o r m o f c o n t o u r s o f c o n s t a n t wall s l o p e a n d critical f l o w f a c t o r as a f u n c t i o n o f t h e e f f e c t i v e angle o f i n t e r n a l f r i c t i o n a n d t h e k i n e m a t i c angle o f wall f r i c t i o n . A n illustrative e x a m p l e is p r o v i d e d t o d e m o n s t r a t e t h e i n h e r e n t a d v a n t a g e s o f this presentation for the evaluation of mass flow hopper geometries.
For those installations requiring the operating features of complete emptying of contents, first in-first out discharge sequence, minimum segregation or uniform flow rate the hopper geometry parameters should be chosen to ensure true mass flow discharge. Mass flow hoppers are also employed in single and multiple outlet expanded flow bins. Determination of the mass flow hopper g e o m e t r y p a r a m e t e r s is a t w o - s t a g e p r o c e s s . Firstly, the flow properties of the particular bulk solid are assessed from laboratory tests on a representative sample taking account of such conditions as moisture content, time of storage at rest, and temperature. The calculation of the outlet dimension and the hopper wall slope to ensure that no stable cohesive arches form and that flow occurs along the hopper walls constitutes the second stage. T h i s is a c h i e v e d b y a p p l y i n g t h e t h e o r y o f Jenike [1] to the measured bulk solid flow properties. The scope of this paper does not allow a full discussion of either the laboratory measurement of flow properties, or the general theory of gravity flow of bulk solids in mass flow channels. These are fully documented in R e f s . 1 a n d 2. I t is f i t t i n g , h o w e v e r , t h a t a brief overview of the Jenike design procedure be presented along with various methods of displaying mass flow parameters relationships that have been developed.
INTRODUCTION
The successful design of bulk solid storage facilities both in terms of structural integrity and reliable and predictable operation can be ensured by applying well-established and accepted procedures. These procedures, proven by experience and application in industry, are based on the theory of flow of bulk solids f i r s t e x p o u n d e d b y J e n i k e [ 1 ]. The inherent advantages of using mass flow hoppers in the design of gravity discharge storage facilities is well recognised. 0032-5910/85/$.3.00
D E T E R M I N A T I O N O F T H E MASS F L O W H O P P E R GEOMETRY PARAMETERS The Jenike design procedure [1] takes into account the bulk solid flow properties and the stresses imposed by the hopper, in determining the geometry parameters of critical outlet dimension and wall slope. The required flow properties which describe the variation of strength and frictional characteristics with consolidation pressure are: © Elsevier Sequoia/Printed in The Netherlands
80 % %
b o t h t h e s e p a r a m e t e r s a n d , f u r t h e r , t h e flow factor value i~ influenced b y ~. T h e f l o w - n o f l o w c o n c e p t is i m p l e m e n t e d by approximating the stress conditions occurring in a converging channel during flow by a radial stress field according to the relationship
B
Cond i
~
FF
/ / -
Coh~slvc
-.00
o = 7rS(c~) "
NO-FLOW
Fig. I. Flow-no design.
FLOW
flow concept
The solution of the set of partial differential equations describing the radial stress field under specific boundary conditions a l l o w s t h e v a l u e o f S(ce) t o b e c o m p u t e d and, hence, the flow factor determined by the relation
£z, for mass flow hopper
-- flow functions for both instantaneous and time storage conditions -- effective angle of internal friction 6 --kinematic angle of wall friction ¢ for requir(,d wall lining materials and surface finishes -- bulk density p By utiiising a flow-no flow concept (Fig. 1), t h e s t r e n g t h o f t h e b u l k s o l i d (as r e p r e s e n t e d b y t h e f l o w f u n c t i o n ) is c o m p a r e d with the stresses imposed by the hopper (represented by the flow factor). The flow f a c t o r is d e f i n e d b y : ff -
O I
(i)
Referring to Fig. I, flow will occur for ~i > oc w h e n the cohesive arch across the outlet will fail, since the stress ~, i m p o s e d by the hopper exceeds the unconfined yield s t r e s s o f t h e b u l k s o l i d Oc- T h e c r i t i c a l v a l u e o f a~ o c c u r s a t t h e i n t e r s e c t i o n p o i n t o f t h e flow factor and the flow function. The flow f a c t o r v a l u e is a f u n c t i o n o f t h e w a l l f r i c t i o n angle, the effective angle of internal friction, the hopper wall slope and the form of the hopper outlet (axisymmetric or plane flow). If ¢ o r 5 v a r y w i t h a ~, a n i t e r a t i v e p r o c e d u r e must be carried out until a I converges to the critical value. T h e m i n i m u m o u t l e t d i m e n s i o n B is defined by: B -
e,H(oO pg
-
al H(oO ff
pg
(3)
(2)
The flow factor ff and the hopper wall s l o p e c~ a r e d e s i g n p a r a m e t e r s o f t h e J e n i k e p r o c e d u r e b e c a u s e , as e q n . (2) i n d i c a t e s , t h e c r i t i c a l o u t l e t d i m e n s i o n is d e p e n d e n t o n
ff =
H(cx)(I + sin ~)S(~) 2 sin
(4)
Jenike computed flow factor values and presented the results in the well-known series of charts for axisymmetric and plane flow hoppers [1]. These graphs display c o n t o u r s o f c o n s t a n t f l o w f a c t o r as a f u n c t i o n of the hopper slope and the kinematic angle of wall friction for specific values of effective angle of internal friction (30 ° through 70 ° by 10 ° increments). Examples of the flow factor charts for asymmetric and plane flow hoppers (for an effective angle of int e r n a l f r i c t i o n o f 5 0 °) a r e p r e s e n t e d i n F i g s . 2 and 3 respectively. As depicted in these figures, particularly Fig. 2, the region of the flow factor contours is b o u n d e d b y a l i m i t . T h i s b o r d e r r e p r e s e n t s the limit beyond which the boundary conditions are not compatible with the radial stress field and flow does not develop along t h e h o p p e r w a l l s . T h i s b o u n d is p a r t i c u l a r l y severe for axisymmetric hoppers and, to ensure that mass flow occurs, Jenike recommends that the wall slope be reduced 3 to 5c from the bound to account for any variation of the wall friction value in practice from the laboratory test values. For plane flow h o p p e r s , h o w e v e r , t h e b o u n d a r y is n o t s o stringent, but to prevent the development of excessive non-flowing regions originating f r o m t h e h o p p e r t r a n s i t i o n a d e s i g n l i m i t is suggested (the dashed line, Fig. 3). In determining the mass flow hopper geometry, the values of the hopper wall slope and flow factor are determined from these charts for the combination of effective angle of internal friction, the kinematic angle
"
"
.
50
°
,
.,
•
k
,
-
-
"
.
:
..-
"
"
8 i
.
_
.
!
I I
30°
--I, N
2o °
z
z 10°
0°
F i g . 2. F l o w
10 °
~_0° 30 ° HOPPER WALL SLOPE o~-OEGREES
z~O°
50 °
factor chart for axi.s.yrnmetric hoppers, ~ = 50 ° (from
Jenike [1]).
s0o
10 °
0 °
0°
10 °
20 °
30 °
40 °
50 °
6~
HOPPER WALL SLOPE c~-OE6REES F i g . 3- F l o w f a c t o r c h a r t f o r p l a n e f l o w h o p p e r s , ~ = 5 0 ° ( f r o m J e n i k e [ 1 ] ) _
o f wall f r i c t i o n a n d t h e h o p p e r o u t l e t s h a p e . A l t h o u g h it is p o s s i b l e t o m i n i m i s e t h e f l o w factor by suitable selection of the hopper s l o p e f o r a g i v e n wall f r i c t i o n v a l u e ( a n d t h u s o p t i m i s e t h e f l o w a b i l i t y o f t h e h o p p e r [ 1 ] ), f r o m a p r a c t i c a l d e s i g n v i e w p o i n t it is p r e f e r a b l e t o m a x i m i s e t h e h o p p e r wall s l o p e . T h i s
is a c h i e v e d b y u s i n g v a l u e s a l o n g t h e s u g g e s t e d d e s i g n limits f o r p l a n e f l o w , o r a l l o w i n g a 3 ° r e d u c t i o n o f wall slope f r o m the mass f l o w - f u n n e l f l o w l i m i t in t h e case o f a x i s y m metric hoppers. Although the Jenike flow factor charts c l e a r l y d i s p l a y t h e l i m i t s o f h o p p e r wall s l o p e
82
between mass flow and funnel flow and the respective flow factor values along the design boundary, they are disadvantaged by the neccssmq¢ p a r a m e t e r i n t e r p o l a t i o n s r e q u i r e d . There are two major problems associated with the pm'ameter interpolations. Firstly, since the charts are presented for specific v a l u e s o f 5, t h o s e d e s i g n s h a v i n g i n t e r m e d i a t e values require the parameters of flow factor and hopper wall slope to be adjusted between the respective charts to ensure accuracy. Secondly, due to the troughed form of the flow factor contours in the region of the desiEn limit, determination of the flow factor value can be difficult. For example, consider determination of the flow factor for ¢ of 15 ° a n d a" o f 4 0 ° f r o m Fig. 3. .-\ c l e a r e r i n d i c a t i o n o f t h e i n f l u e n c e o f on the limit of hopper wall slope for mass flow in c o n i c a l h o p p e r s was p r e s e n t e d b y J o h a n s o n a n d C o l i j n [ 3 ] ( F i g . 4)_ T h i s d i a gram condenses the mass flow limits of the Jenike flow factor charts to one diagram by d i s p l a y i n g c o n t o u r s o f c o n s t a n t 6 as a f u n c t i o n o f ¢ a n d c~_ I n t e r p o l a t i o n f o r i n t e r m e d i a t e v a l u e s o f ~i i n o r d e r t o d e t e r m i n e t h e n ] a x i l n u n ] v a l u e s o f a- is m o r e c o n v e n i e n t than for the denike flow factor charts. .Johanson and Colijn [3] also include an alternative series of flow factor charts to determine the flow factor after a suitable h o p p e r wall slope has b e e n s e l e c t e d f r o m Fig. -t. T h e s e c h a r t s d i s p l a y c o n t o u r s o f c o n s t a n t f l o w f a c t o r as a f u n c t i o n o f 6 a n d .O f o r t h e s p e c i f i c ct" v a l u e s o f 1 0 , 2 0 a n d 3 0 °. The flow factor chart for a hopper wall slope o f 2 0 ° is d e p i c t e d i n F i g . 5. I n t e r p o l a t i o n 50
I
i
!
'
i
i
~
'
, BOUNDARY
~
-i ~ --T--i~.60To~o ,o1~1"..~., i--r---~-~o l
....
8=
I I L.--L--_
LINES:
I
I I
I I
-; ....
}---I'=-~%~--I ---j- WITH
F,~,~F-OW~,
o-i-!-l-I 0
I0
20
HOPPER
RADIAL-STRESS
I'~"D"¢ 30 ANGLE,
40
~ ~ 50
60
o~
Fig. 4. H o p p e r w a l l s l o p e l i m i t s f o r a x i s y r n m e t r i c hoppers (from Johanson and Colijn [3])-
1 L/
iN',,:,.,
k~ \ 1 ~ !
~h, 'N,"N I ~ J
o1\
(/1111 ! I i
i
I I I t I 1 I I I I
0
I0
20 WALL
Fig. 5. Critical
flow
30 FRICTION
factors
= 20 ° ( f r o m J o h a n s o n
40
50
ANGLE,
for mass
flow
60
,i~ hoppers,
and Colijn [3]).
b e t w e e n t h e f l o w f a c t o r c o n t o u r s is m o r e convenient in this presentation due to their more uniform nature. However, imprecise f l o w f a c t o r values c a n arise b e c a u s e o f t h e presentation of the charts in 10 ° increments of wall slope. Limits on the selection of hopper wall s l o p e h a v e a l s o b e e n p r e p a r e d f o r auxisymmetric and plane flow hoppers for mass f l o w b y A r n o l d e t el. [ 2 ] a n d are p r e s e n t e d in Fig. 6. This presentation utilises the limits proposed by Jenike [1]. Of more benefit, however, particularly in regard to computer applications, are the equations which are given to represent the design limits. For a-xisymmetric hoppers
0
t180- cos-,p = sin !
\ 2 sin ~} ]
(sinfUl and for plane flow hoppers e x p [ 3 . 7 5 ( 1 . 0 1 ) ( a -30)/I0] __ ¢ c~ =
30
~- ~°l--]'--i-~_~f i 1 A"EANOTCO.~ATI.~E ~o !___~I_ I__!'%.-.1 F,~-D(No ,ASS ,-OW '-
'V I/I/I i,op, I~!\1 - ~°1\1~ \~ i\l\,;.,o ~ ~ ~ I ! I t /i i 1 1
0 . 7 2 5 ( t a n 6 ) °'2
(6)
f o r ¢ < 6 - - 3 ; q~ a n d ~ i n d e g r e e s . Reference to Fig. 6, however, indicates that for the range of common values of ¢ ( a p p r o x i m a t e l y 1 5 t o 3 0 °) f o r m a s s f l o w d e s i g n t h e m a x i m u m b o u n d o f ~ is r e l a t i v e l y insensitive to the variation of/i. The influence of the effective angle of internal friction on the flow factor has been considered by Jenike [4]. In dealing with plane flow hopper design he notes that the e f f e c t o f q~ is m i n i m a l o n t h e f l o w f a c t o r
s~-
~.~.
- , ....
, . . . . . . . . .
t .....
....
~"-'~ ~
~~ /--_',. -".
"
:
:
--
I ....
.
-
-
.
.
' ....
, . . . . . . . .
Pc.she .
.
.
.'l
FLOW
. . . . . . . . .
-
~OPPe~5
BXISYHMETR'
C
HoPPeRS
.
~- - '- --~ . : ~"- ~.~ ~= ~ o
30.
='
%$.-..
: .
""
-
,.,.
B
~ =,~~ ,,--------'-'~<. .~: \ :
"~-,~..~..~
. -
=
-
.
\. ~ . ~ \~. ~ . \ . ~ ~ " , , ~
- _
I ff -
•
;~0. •
HOPPER
\ ~ . . ~ .
3"~".
NRLL
SLOPE
40' •
-
50' •
5"IB'.
DEGREES
Fig. 6. Wall slope limits for axisyrnmetric and p l a n e flow h o p p e r s (after A r n o l d et aL [2]). value and presents the variation of the flow factor as a function of ~ (Fig. 7). Although F i g . 7 w a s p r e p a r e d f o r ¢ o f 2 5 ° a n d c~ o f 2 5 ° Jenike states that the flow factor variation will be negligible for different design values, thus having little effect on the outlet dimension B determined from eqn. (2). 2.O
..< ~s
0o
\ 30 °
50 o
70 °
90 o
EFFECTIVEANGLEOFlmErNAtFRICTION Fig. 7. V a r i a t i o n o f flow f a c t o r w i t h effective angle o f i n t e r n a l f r i c t i o n , ~ = 2 5 °, ~ = 25 ° ( f r o m J e n i k e [4]). ALTERNATIVE PRESENTATION OF THE MASS FLOW HOPPER DESIGN PARAMETERS A common factor between the various presentations discussed in the previous section for the presentation of the flow factor a n d t h e h o p p e r w a l l s l o p e is t h e n e e d t o r e f e r to several graphs in determining a satisfactory design. Generally, the selection of the mass flow hopper geometry involves the determination of the maximum hopper wall slope allowable
for the particular bulk solid flow properties and operating conditions. This necessarily involves selecting design values along the mass flow limits (refer to Figs. 2 and 3) and, as such, the remainder of the bounded mass f l o w r e g i o n is o f l i t t l e i n t e r e s t . T h e s e ~-reas are only utilised for such tasks as the verification of existing plant designs or the prediction of flow rate by such methods as that detailed-by Johanson [5]. With these factors in mind, an alternative presentation of the mass flow hopper design p a r a m e t e r s h a s b e e n d e v e l o p e d a n d is p r e sented in Figs. 8 and 9 for axisymmetric and plane flow hoppers respectively. These figures display the variation of the flow factor and hopper wall slope in contour form as a function of the kinematic angle of wall friction and the effective angle of internal friction. The data mesh for the formation of both charts was obtained by utilising the design b o u n d f o r (~ ( a s p r e d i c t e d b y e q n . ( 5 ) a n d ( 6 ) ) a n d t h e p a r t i c u l a r v a l u e s o f (b a n d 6 t o determine the maximum allowable value of o~. W i t h t h e s e t h r e e p a r a m e t e r v a l u e s t h e respective flow factor was determined using differential equation solution techniques and eqn. (4). Thus, only the values of maximum are displayed which will ensure mass flow f o r t h e r e s p e c t i v e v a l u e s o f ¢ a n d ~. T h e
84
45-
,__ 3 5 .
~-
35.045E F F E C T I V E FINGLE OF
SI3556N65INTEFINRL F R I C T I O N - DELTFI
VO-
Fig. S..Alternative presentation of axisymmetrie hopper design parameters. following design constraints have been taken into account in determining the data mesh: - - F o r a x i s y m m e t r i c hoppers, a 3 ° r e d u c t i o n of the hopper wall slope from the radial stress compatibility bound has been effected and the flow factor determined at this margin point. - - F o r plane f l o w hoppers, t h e w a l l s l o p e l i m i t specified in equation (6) and the constraint of ¢ < 5 -- 3 degrees, has been applied. -- The upper limit of the hopper wall slope is 6 0 °. This data mesh lends itself to computer applications for systems that do not have mathematical packages for the solution of differential equations (to determine the flow factor). The values required can be determined by linear interpolation techniques based on the data mesh. A n a d v a n t a g e o f t h i s c h a r t f o r m a t is t h a t a complete overview of the trends of flow factor and hopper wall slope variation with ¢ a n d 5 is p r e s e n t e d . R e f e r e n c e t o F i g s . 8 a n d 9 indicates the direct influence of ¢ on the v a l u e o f c~ a n d t h e m i n o r e f f e c t o f 6 , p a r t i c u larly for plane flow hoppers. The trend of
reducing values of flow factor with increasing is a l s o w e l l d e m o n s t r a t e d . T h i s t r e n d is in agreement with the presentation of Jenike [4], but allows the overall effect of ~ on the flow factor to be appreciated. For a constant value of 50 ° and a range of ¢ of 5 < ¢ < 4 0 ° , t h e f l o w f a c t o r is s e e n t o v a r y f r o m 1 . 2 5 to 1.22, justifying the assumption of Jenike to neglect the ¢ variation in determining an initial measure of flowability. F o r a g i v e n s e t o f v a l u e s f o r ~ a n d ~b, t h e design parameters of ~ and flow factor are determined by referencing the intersection point to the plotted contours. Although the contour increments for Figs. 8 and 9 are necessarily coarse to ensure clarity, large chart formats can utilise smaller contour intervals and reduce prorating of values. Correct interpretation of the data determined from the charts should be stressed. I t is i m p o r t a n t t o b e a r i n m i n d t h a t o n l y t h e maximum value of ~ and the respective flow factor value relevant to the traditional mass flow limits are displayed. Concerning the value of hopper wall slope, the ~ contour specifies the maximum value allowable to
. . . 50
-
i
I
i
i
a_ z
-
:
.
.
.
.
.
i
.
.
.
.
.
.
.
|
.
/
40-
.
|
•
-..
85
- _.~ r~=l~ °
.
I
~_ 3 S . n'-
u_ 3 0 ~J
~
2S-
tl_
w 20.
J Z
a= 1 5 . ~= 1 0 . - -
030- -
35.
40-
EFFECTIVE
45.
RNGLE
50-
OF
55-
INTERNRL
FRICTION
=
6@-
-
65-
o
7@-
OELTR
Fig. 9. Alternative p r e s e n t a t i o n o f plane flow hopper design parameters. ensure mass flow and slip of the bulk solid a l o n g t h e w a l l . V a l u e s o f c~ g r e a t e r t h a n t h a t specified can lead to a funnel flow discharge pattern, while a reduction below this value leads to a more conservative design. Reference to eqn. (2) similarly provides an insight into the variation of the flow factor about the design value. Utilising a flow factor value smaller than the design point will lead to a reduction in oi, a reduced outlet dimension B and, hence, the formation of cohesive arches. Noting that the design point specifies the critical flow factor value, a more conservative design will then result from using an increased value. The advantages of utilising this presentation to determine the hopper geometry based on the bulk solid flow properties have been noted in the preceding discussions. However, care should be exercised in the verification of m a s s f l o w o c c u r r i n g i n existing storage facilities. T h e v a l u e o f h o p p e r w a l l s l o p e c a n be checked knowing 6 and ¢ and ensuring t h a t ~ a c t u ~ < ~chart- D i f f i c u l t i e s a r i s e i n c h e c k i n g t h e h o p p e r o u t l e t d i m e n s i o n (as affected by the flow factor) because values
of ~ not lying on the mass flow limits relate t o f l o w f a c t o r s n o t d i s p l a y e d in Figs. 8 o r 9. Reference to the original Jenike flow factor c h a r t s (Figs. 2 a n d 3) i n d i c a t e t h a t t h e f l o w factor may be above or below that displayed in the alternative presentation, depending on used within the bounded mass flow region. I t is r e c o m m e n d e d t h a t f o r t h i s e x e r c i s e t h e Jenike charts be consulted because the comp l e t e f l o w f a c t o r v a r i a t i o n w i t h ~ is d i s p l a y e d f o r g i v e n v a l u e s o f q~ a n d 8.
I L L U S T R A T I V E EXAMPLE To demonstrate the use of these new design charts, the geometry of a mass flow hopper with a plane flow outlet for an ROM coal storage will be determined. The measured f l o w p r o p e r t i e s o f t h e c o a l a r e p r e s e n t e d as a function of the major consolidation stress a l i n F i g . 1 0 a n d i n e q u a t i o n f o r m m T a b l e 1. Two characteristics, typical of moist cohesive b u l k solids s u c h as c o a l , are d e p i c t e d i n Fig. 10. These are the reduction of ~ and ¢ (for some wall materials) to limiting values with
86 70.
.
EFFECTIVE
ANGLE
,
.
,
!
OF INTERNAL
FRICTION
-I-
I-----_________
N :
N 4a.
KINEMflTIC -
A N G L E OF M A L L F R I C T I O N STAINLESS STEEL -2UHHW P O L Y M E R - 3 -
~
\
3~u~ J
~
~" m
\
L
3
\
3
I
~ I~.
2 • " • "
2
S-
10.
iS.
MAJOR C O N S O L I O R T I O N Fig. I0.
3
t
Flow
prope~ies
of ~ and
¢ ~r
STRESS
the design
-
20-
KPR
example.
T.ABLE 1 Flow property
equations
for the design example
Function
Equation
Instantaneous
flow function
Unit
Oc = 0 . 3 2 a i
+ 1.20
Time flow function (2 days)
Oct = 0.35o" I + 2 . 0 5
Bulk density variation
:'= 850.74
[
H
z
kPa kPa
O1 ~0.0697
~5.----~- )
kg/m 3
f,
1 5~g
"
x
Fig. I1.
55
.
EFFECTIVE
Determination
of the
..
hopper
60 ANGLE
geometry
OF
.
INTERNAL
for the
design
65. FRICTION
example.
-- DEGREES
70.
Iteratio n No.
~ (°)
1
assume
60
ff
al (kPa) 2.07 1.98 1.98
2
64.5
1.111 1.080
3
64.6
1.079
increasing a~. The variation of the wall frict i o n a n g l e .~nay b e d u e t o e i t h e r a c o n v e x upward wall yield locus and/or the tendency of the bulk solid to adhere to the wall lining material at low values of normal stress. A p o r t i o n o f F i g . 9 is p r e s e n t e d i n F i g . 1 1 , w h i c h d i s p l a y s c~ c o n t o u r s a t 1 ° i n c r e m e n t s and the flow factor contours at 0.01 intervals. As previously stated, an iterative procedure must be used to determine the critical hopper geometry if either 8 or ¢ vary with a~. Essentially, the procedure suggested by Jenike [1] will be applied, first determining the critical flow factor value, noting the respective value and then applying the flow factor value to eqn. (2) to yield the hopper outlet d i m e n s i o n . T h e p r o c e d u r e is d e e m e d t o h a v e converged to the critical values when little change results in oi and 8 from successive iterations. Consider the determination of the hopper geometry under instantaneous conditions w i t h t h e U H M W p o l y m e r as t h e w a l l l i n i n g material. To start the procedure initially e s t i m a t e ~ = 6 0 °. W i t h ~b = 2 3 ° c o n s t a n t , from Fig. 11 point A yields a flow factor value of 1.111. Applying the flow-no flow concept (Fig. 1) t o t h e i n s t a n t a n e o u s f l o w f u n c t i o n a n d the respective flow factor produces an initial v a l u e f o r o~ o f 2 . 0 7 k P a . R e f e r e n c e t o t h e v a r i a t i o n d e p i c t e d in F i g . 1 0 f o r t h i s v a l u e o f o~ y i e l d s a n u p d a t e d v a l u e o f 5 o f 6 4 . 5 °. The subsequent values for each iteration in converging to the critical values are presented above.
.
~
P o i n t : :(refer Fig. i i )
"
(o) 23.0 23.0 23.0
:
-
A B C
For the values of 5 (64.6) and ¢ (23.0), t h e f l o w f a c t o r a n d cz c a n b e r e a d f r o m F i g . 1 1 as 1 . 0 7 9 a n d 3 0 . 3 ° ( s a y 3 0 . 5 °) r e s p e c t i v e l y . A s i n d i c a t e d in t h e a b o v e t a b l e , c o n vergence occurs relatively fast. Substitution of the relevant values into eqn. (2) allows the critical outlet dimension to be determined: H(~) = H(~ = 30.5) = 1.15 p = p(al = 1.98) = 788.1 kg/m 3 and hence B = 273 mm (say 275 ram). The critical hopper geometry parameters would then be detailed as ~ = 30.5 ° and B = 275 mm. The procedure becomes further complicated when considering a lining material that has a variable kinematic angle of wall friction. To illustrate the design procedure when both ¢ and $ vary with a l, the critical geometry parameters will be determined for time storage conditions (2 days at rest) and using s t a i n l e s s s t e e l as t h e w a l l l i n i n g m a t e r i a l . In a similar fashion to the previous example, values of 8 and ¢ must be assumed to obtain an initial flow factor value. Assuming 5 = 60 a n d ¢ = 2 0 °, t h e i t e r a t i o n v a l u e s a r e p r e sen.*.ed b e l o w . For the final values of flow factor = 1.097 a n d c~ = 3 5 . 5 °, t h e c r i t i c a l o u t l e t d i m e n s i o n c a n b e d e t e r m i n e d a s 4 8 5 r a m , w i t h H(c~) = 1 . 1 8 a n d p = 8 2 2 . 5 k g / m B. T h e v a r i a t i o n o f ~b w i t h a I c a n b e u t i l i s e d in mass flow hopper design by allowing larger wall slopes for outlet dimensions greater than the critical value determined
Iteration No.
~ (°)
ff
a, (kPa)
¢ (°)
Point (refer Fig. 11)
1 2 3
assume 60 62.3 62.4
1.115 1.098 1.097
3.75 3.66 3.65
assume 20 18.7 18.9
D E T
88 T A B L E o_ Variation o f .¢ and ~5 with (~l and resulting variation o f ~ with B for stainless steel oI (kPa)
5 (¢)
.0 (°)
ff
ot (°)
B (mm)
1.98 2.50 3.00 4.00 5.00 8.50 10.00 15.00
64.6 63.9 63.2 62.0 61.0 58.1 57.1 54.7
21.7 20.4 19.6 18.6 18.0 17.0 16.8 16.4
1.080 1.086 1.091 1.100 1.109 1.134 1.143 1.168
31.8 33.4 34.5 35.8 36.6 38.0 38.3 39.0
275 342 405 528 647 1042 1204 1724
in relation to particular instantaneousor time How functions. The hopper wall slope may be increased due to the reduction in wall friction which occurs from the increased values of a~ resulting from increased outlet dimensions. The variation of o~with B can be determined by incrementingo~ above the critical value, calculating the respective values of ¢ and and, from Fig. 11, reading the respective values of ~ and flow factor. Knowing the flow factor and the value of a , allows the outlet dimensionto be calculated. The variation of 6 and ¢ for a range of o, has been presented as a curve in Fig. 11. This curve is unique to the particular variations of ~b and 6 with a I and, as indicated, the critical design points for instantaneous (point I) and time storage (point T) conditions lie on this locus. Table 2 summarises the variation of ¢ and 5 with a i and the resulting variation of with B for stainless steel. It can be seen that for plane flow hopper outlets in the region of 1 m width, wall slopes of 38 ° could be utilised. This design data can also be applied to increasing the hopper wall slope in a stepwise fashion at intermediate levels above the hopper outlet. As the effective width of the hopper increases due to divergence of the walls, the wall slope can be increased in accordance with the above variation. This technique can prove useful in applications that have headroom and space constraints.
CONCLUSION An alternative method of presenting the mass flow hopper design parameters of flow
factor and hopper wall slope and their variation with the material properties of kinematic angle of wall friction and effective angle of internal friction has been detailed. The advantages resulting from this presentation over existing methods include: (i) T h e a b i l i t y t o o b t a i n a n o v e r a l l a s s e s s ment of the variation of the design parameters of ~ and flow factor along the limits of mass flow chosen for design purposes. (ii) A l l t h e r e l e v a n t d e s i g n d a t a r e q u i r e d for hopper geometry determination are presented on one chart (for either axisymmetric or plane flow hoppers), which reduces the amount of parameter interpolation required. (iii) A c h a r a c t e r i s t i c c u r v e c a n b e p l o t t e d on the chart presentation representing the v a r i a t i o n o f ¢ a n d 6 w i t h i n c r e a s i n g o 1. The critical values of ¢ and 6 for instantaneous and time storage conditions lie on this curve. The respective values of ~ and flow f a c t o r as s p e c i f i e d b y t h e c u r v e c a n b e u t i l i s e d t o p r e s e n t t h e v a r i a t i o n o f c~ f o r v a l u e s o f B greater than the critical. (iv) T h e d a t a m e s h u s e d t o d e t e r m i n e t h e c o n t o u r p o s i t i o n s On t h e c h a r t s c a n b e a p p l i e d to compile a hopper design computer program using linear interpolation techniques (rather than the solution of differential equations) to determine the required design parameters.
ACKNOWLEDGEMENT The authors wish to acknowledge the financial support provided by the National Energy Research, Development and Demonstration Council for the project 'Flow Properties of Coal for Storage Bin Design'.
The work reported here forms part of that project. : . - " . " .... . . T h e a s s i s t a n c e o f Dr: G. J. M 0 n t a g n e r i n facilitating the computer-generated diagrams is g r a t e f u l l y a c k n o w l e d g e d .
LIST OF SYMBOLS
B
ff FF g
H(~) m r
s(~)
6
critical arching dimension for a mass flow hopper flow factor for a converging flow channel flow function for a bulk solid acceleration due to gravity design function of ~ and hopper outlet shape [1] hopper outlet shape, axisymmetric (m = 1) o r p l a n e f l o w ( m = 0) radial dimension from vertex of hopper radial stress function half-angle of hopper or slope of hopper wall measured from the vertical weight bulk density effective angle of internal friction of a bulk solid
:
p :- bulkdensitY0£ a buikS01id -. : -:-::a .. n 0 i T n a l Stress._0r a v e r a g e - o f v e r t i c a l : " : and lateral:stress01 majorc0nsolidating stress 01 majo r stress acting at theabutment of a cohesive-arch ac unconfined yield stress of a bulk solid " . act unconfined yield stress of a-bulk solid under conditions of time storage at rest r shear stress ¢ kinematic angle of wall friction developed between a hopper wall and a bulk solid "
REFERENCES
1 A_ W. J e n i k e , S t o r a g e a n d ]Flow o f S o l i d s , B u l L 123, Eng. E x p t . Sta., Univ. of Utah (1964). 2 P. C. A r n o l d , A . G. M c L e a n a n d A . W. R o b e r t s , Bulk Solids: Storage, Flow and Handling, The University
of
Newcastle
Research
_~_~socla.tes
(TUNRA) Ltd., 2nd edn., 1980. 3 J. R . J o h a n s o n a n d H . C o l i j n , I r o n S t e e l Eng., XLI (1964) 85. 4 A . W . J e n i k e , P o w d e r T e c h n o L , 11 ( 1 9 7 5 ) 8 9 . 5 J . P~ J o h a n s o n , Trans. ~]in. Eng., A I 2 4 E . 2 3 2 (1965)69.