Investigations into wall pressure during slug-flow pneumatic conveying

POWDER TECHNOLOGY ELSEVIER

Powder Technology 84 (1995) 91-98

Investigations into wall pressure during slug-flow pneumatic conveying B.Mi, P.W. Wypych Department of Mechanical Engineering~ University of Wollongong, Wollongon& NSW 2522, Australia Received 7 December 1994; revised 24 January 1995

Abstract

During low velocity slug-flow pneumatic conveying, the wall pressure, that is the pressure that is exerted on the pipe wall by a moving slug of particles, causes a resistance to material flow. Most of the conveying pressure energy is consumed to overcome this resistance force. However, very little information is available for the determination of wall pressure. This paper studies the distribution of wall pressure along the length of a moving slug and then measures on a test rig the total pressure (the sum of the static air pressure and wall pressure) during slug-flow pneumatic conveying u n d e r different conveying conditions. Various values of the stress transmission coefficient (such as the ratio of radial stress to axial stress) in particle slugs are obtained from the wall pressure measurements and the calculated axial stress of the slug. Based on the principles of particulate mechanics and the experimental values of the stress transmission coefficient, a semi-empirical expression of stress transmission coefficient is presented for slugs flowing in pipes with rigid and parallel walls.

Keywords: Wall pressure; Slug-flow; Conveying conditions; Stress transmission coefficient

1. Introduction Granular solids can be transported in the form of slugs during low velocity pneumatic conveying. In horizontal flow, there are stationary beds between the slugs, as shown in Fig. 1. While moving forwards, each particle slug is subjected to the air pressure force, wall friction force and resistance force caused by the stationary bed, see Fig. 1. This results in an axial compressive stress trx in the slug. The axial stress trx causes a radial stress trr as the pipe wall prevents the radial deformation of the particle slug. The ratio A= ~

In this paper, the radial stress exerted on the pipe wall is designated as the wall pressure trw. It should be noted that for horizontal slug-flow, the weight of a particle slug also causes a radial stress on the pipe. For convenience and distinction, this pressure is called the gravity pressure trg. The total radial stress exerted on the pipe wall is referred to as the total wall pressure trt~. Fig. 2 show the total wall pressure and its components. As indicated by previous research [1-13], it is important to determine the wall pressure and stress transmission coefficient in low velocity pneumatic conveying. Wall pressure also has a direct effect on the wear of

O) aw

is defined as the stress (force) transmission coefficient. It is analogous to the stress ratio of horizontal to vertical stress for the calculation of stresses in silos and hoppers. Air gap _ "5.L~..~ ...........~... F °J.*,°o ..°. P ~

0

Particle Slug

:.~.~5..~:~..~53:~.2.~~~ ......-..'~....'~*~..'~*.....* ~.°.-°...~...-.~-°...a.°.....°o..

Ra

.52.3:.~.,.~..,55. °~...~°~....°.....°..°...~.°.b .o...°...e°t....=..°o°....~.¢

001! (a)

Rf

Stationary bed Fig. 1. Low velocity pneumatic conveying. 0032-5910/95/$09.50 © 1995 Elsevier Science S.A. All rights reserved

SSD1 0032-5910(95)02974-7

011

(b)

O'tw=O'w+ ~ i i

(c)

Fig. 2. Total wall pressure and components: (a), radial stress at wall due to axial compressive stress, called wall pressure; (b), radial stress at wall due to weight, called gravity pressure; (c) total wall pressure.

B. Mi, P.W. Wypych / Powder Technology 84 (1995) 91-98

92

the conveying pipeline. Although numerous investigations [4-7] have been carried out to determine the wall pressure and stress transmission coefficient in silos and hoppers, very little information is available worldwide for the determination of these parameters during low velocity slug-flow. Consequently, research into low velocity pneumatic conveying has been slow to advance in many areas, particularly pipeline pressure drop prediction. This study undertakes investigations into wall pressure during the particle slug-flow of coarse granular (cohesionless) materials through a horizontal pipe. Also, a method to determine the stress transmission coefficient and its influential parameters (for example static internal friction angle) is presented.

Us

o

Konrad et al. [1] also presented the following equation for the pressure gradient of a single horizontal slug by applying the above boundary conditions to Eq. (2). 13

The interstitial air pressure and stresses acting on an element of a moving particle slug are shown in Fig. 3. Assuming that the axial stress trx and radial stress tr, are functions of x only (that is, these stresses are constant in a cross-sectional area of slug), then A= trw/ trx. Based on an equilibrium of the forces acting on an elemental section of slug, Konrad et al. [1] developed the following equation for the axial stress in a horizontal slug. trx= K e x p ( 4 1 ~ W A X ) + ( - - 2 p b g t t w + - ~ ) D

D

(2)

where K is an integration constant and can be determined by the boundary conditions: tr~=trb at x = 0 and trx= trt at x=/s. Also, note the following assumptions made by Konrad et al. [1] in the development of their theory and Eq. (2). (a) The particles are conveyed only in the single slug and in the regions just in front of and behind it. (b) There is a layer of stationary material in front of and behind the moving slug. (c) The flow pattern resembles that of a gas/liquid system [1]. ~lw

x

Fig. 4. Distribution of wall pressure along slug.

Ap = 4 ~ A 2. Wall pressure distribution

ls

- D

~r,+ 20ug~w

(3)

Rearranging Eq. (3): O'f=

-- 2pbg~l,w"~ ~

Replacing

(

- 2p~g~ + ~

4~v /~

(4)

)°

4tzwA

in Eq. (2) with ef gives trx=Kexp(

4DA ) x+~r,

(5)

Applying the boundary condition, ~x =trb at x = 0, again to Eq. (5), then K = ~ b - o',

(6)

Substituting Eq. (6) into Eq. (5) and replacing ~rx with OVw/~,:

Eq. (7) is a distribution function of wall pressure along the length of a moving slug. According to this equation, a wall pressure distribution curve can be drawn as shown in Fig. 4. It was found [1,8] that the average thickness of stationary bed is approximately equal in front of and behind the slug, so that ¢rf= trb. Therefore, Orw = Ao"t.

o J

XaP x

~ - ~

p+dp~

I 3. Measurement of wall pressure

Fig. 3. Air p r e s s u r e and stresses acting on a particle slug in a horizontal pipe.

Numerous experiments are carried out to measure the wall pressure in the low velocity pneumatic conveying test rig in the Bulk Solids Handling Laboratory. Details on the test facility and major items of equipment have

B. Mi, P.tE. Wypych / Powder Technology 84 (1995) 91-98

93

Table 1 Physical properties of test materials Material

~

d

Pb

(mm)

(kg m -3)

"/b

P, (kg m -3)

~b (°)

~bw (0)

Plastic pellets (white)

0.430

3.12

493

0.49

865

44.70

15.15

Plastic pellets (black)

0.451

3.76

458

0.46

834

43.76

12.95

Wheat Barley Urea

0.440 0.465 0.455

3.47 3.91 1.88

811 722 722

0.81 0.72 0.72

1449 1350 1326

43.73 31.07 35.00

16.01 14.20 13.35

been presented elsewhere [3]. Hence, a brief description only is presented here. The test rig consists of a mild steel pipeline, either 96 or 52 m in length (105 mm i.d.), fitted with flanges, designed to give a continuous uninterrupted bore. Two wall pressure measuring assemblies are installed in a horizontal section of the pipeline. The details of the wall pressure measuring assemblies also have been presented elsewhere [3]. Each assembly includes two types of pressure transducer [3]: type-A (model BHL4400 and -3040) and type-B (model AB/HP-50G), at the same cross-section of pipe. The type-A transducer measures only the static air pressure in the pipe (via a porous plastic membrane), whereas the type-B transducer measures the total wall pressure, which includes the same static air pressure, the gravity pressure (due to material weight) and the radial wall pressure (caused by axial compression along the slug). Hence, the wall pressure is obtained by subtracting the static air pressure and gravity pressure from the total pressure. The crosscorrelation function of the measured wall pressure signals is used also to determine the slug velocity. The distance between the two wall pressure measuring assemblies is 2.49 m. The upstream assembly is far from the end of the vertical lift (about 23 m) so that the slugs can reach a stable size. A sight glass is located between the two wall pressure testing points to confirm slug-flow through this section of pipe. Also, the slugflow pattern can be observed through the sight glass and photographs of slugs and stationary beds can be taken easily. In order to investigate the effect of different materials on wall pressure, four types of material were selected with the relevant physical properties summarized in Table 1. Note the physical properties of another test material, urea, which is considered in a case study at the end of the paper, are included in Table 1.

velocity pneumatic conveying conditions. Fig. 5 shows typical wall pressure and air pressure time records for the black plastic pellets conveyed through the 52 m long pipeline. Each peak in Fig. 5 indicates a particle slug flowing through the pipe. The peaks in Fig. 5(a) basically correspond to the peaks in Fig. 5(b). Fig. 5 also shows that the location of each peak is the same in the two plots, but that the pressure decay after each peak is slower in plot (b) than plot (a). The reason is that the peak of wall pressure only appears when a slug passes through the test point, and the wall pressure 'disappears' as soon as the slug leaves the test point. However, the air pressure still is evident until the slug arrives at the end of the pipe or the slug collapses. Hence the air pressure peak lasts for a longer time. It can also be observed clearly from the wall pressure plots that different slugs cause different 3. . . . .

G (I: it_

'

. . . .

I

. . . . . .

C'H." 0--

2.

(a) to ud

|.

0.

0.

'00.

CYCLE TIME (5ECS]

3~ .....

, ....

i ....

CH. I__

lac

E

2~I.

a: tr~ tAJ

o:

(b) IB.

¢t= ac

4. Experimental results CYCLE TIME ($FC$1

Using the method described in the previous section, wall pressures are measured over a wide range of low

Fig. 5. Plots of wall pressure and air pressure for black plastic pellets. mf=O.064 kg s-~; m , = 0 . 8 5 kg s - L

B. Mi, P.W. Wypych / Powder Technology 84 (1995) 91-98

94

values of wall pressure (that is, the wall pressure peaks have different heights), refer to Fig. 5(a). Hence, an average value of wall pressure is used to represent the wall pressure obtained for a given conveying condition. For horizontal slug-flow, since the type-B transducer is located on the bottom of the pipe of the low-velocity pneumatic conveying test rig, that part of the measured value exceeding the maximum gravity pressure (pbgD) is considered as wall pressure. Otherwise the value is the gravity pressure due to the weight of the slug or stationary bed. Fig. 6 presents the average wall pressure of each test plotted against the mass flow rate of air. It can be seen that the wall pressure for each test material appears to increase linearly as the mass flow rate of air increases during low velocity pneumatic conveying, although the slope of each 'line' may not be constant. The stress transmission coefficient A is a parameter relevant to wall pressure. Since trw~ Ztrf, and trf = o~pbUs2 according to a momentum balance, it is possible to obtain the values of A after the data of wall pressure, stationary bed thickness and slug velocity have been obtained. According to the experimental wall pressure results and the data of slug velocity and stationary bed thickness [8,9], values of stress transmission coefficient are calculated. Table 2 lists the values of the stress transmission coefficient and corresponding operating conditions for wheat. From Table 2, it can be seen that for a given granular material, the stress transmission coefficient is approximately constant. For example, despite the different conveying conditions and pipe lengths an average value 1.6

1.4

4A o

[]

White Plastic Pellets Black Plastic Pellets Wheat Barley

~P[] []

1.2

~ ~l:'+ 4-

&

1.0

9°

+,

&

4-**

0.8 o o

O

O

o

0.6

A 0.4.

A []

0.2 0.04

, 0.05

, 0.06

, 0.07

, 0.08

, 0.09

M a s s F l o w - R a t e o f Air, m f ; (kgs -l) F i g . 6. W a l l p r e s s u r e vs. m a s s f l o w r a t e o f air.

0.10

of 0.572 is calculated for wheat, with all values of A contained in the range - 1 0 to +9%. This indicates that the stress transmission coefficient is a function of the physical properties of the bulk material and pipe. Similar findings are obtained for the other test materials. For example: average A= 0.756 for white plastic pellets ( - 8 to + 9%); 0.806 for black plastic pellets ( - 7 to + 6%); 0.655 for barley ( - 8 to + 6%).

5. Correlation of stress t r a n s m i s s i o n coefficient

After obtaining the experimental values of stress transmission coefficient, it is necessary to correlate A with the relevant influential factors. Some aspects of material strength need to be considered initially for this purpose.

5.I. Strength of particulate medium According to particulate mechanics, the strength of the bulk solid prepared to a specific degree of consolidation can be represented on a shear--compressive stress diagram by a yield locus (YL) together with a wall yield locus (WYL), which represent the limiting stress conditions that can be sustained by a plane wall bounding the bulk material, as shown in Fig. 7. The Mohr circles Co, C1 and C2 in Fig. 7 represent the three different states of an element P within an infinite homogeneous particulate medium. The first is the static state of the particulate medium, that is all the particles are stationary. At this time, the Mohr circle with centre Co lies completely under the wall yield locus. The second state of the particulate medium is represented by the Mohr circle C1 that intersects the wall yield locus, but still lies completely under the yield locus. In this state, shearing occurs at the boundary between the particulate medium and wall, however, due to the Mohr circle C1 being below the yield locus (or called the Mohr failure envelope), the particles are fixed relative to each other. Hence the particulate medium will slip as a rigid body along the wall plane. The points representing the plane of the wall lies at A1 or D~. This state can be referred to as 'en bloc' movement of the particulate medium. The third state of the particulate medium is represented by the Mohr circle C2 that is tangent to the Mohr failure envelope Y U In this state, relative movement exists between the particles. Note that no state of stress (nor Mohr circle) can exist above the Mohr failure envelope since the (flowing) failure occurs in the particulate medium. The failure occurring at the active stress state (try< trz) is called active failure. The failure at the other possible stress state (%> trz) is called passive failure. For the third state of particulate medium, the particles adjacent to the wall plane also slip along the wall and A2 or

B. Mi, P.W. Wypych / Powder Technology 84 (1995) 91--98

95

Table 2 Experimental values of wall pressure and stress transmission coefficient for wheat 52 m pipeline

96 m pipeline

mr-

ms

O"w

O'f

(kg s - ' )

(kg s -1)

(kPa)

(kPa)

0.0495 0.0663 0.0733 0.0828 0.0882 0.0495 0.0665 0.0742 0.0836 0.0885 0.0545 0.0676 0.0747 0.0796 0.0868 0.0657 0.0747 0.0838 0.0885

0.967 0.959 0.957 0.961 0.962 1.450 1.439 1.439 1.439 1.461 1.945 1.986 1.977 1.997 1.996 2.383 2.403 2.388 2.415

0.334 0.671 0.726 0.896 0.978 0.341 0,639 0.785 0.892 0.997 0.442 0.711 0.746 0.814 0.909 0.663 0.780 0.952 1.028

"0.634 1.158 1.346 1.656 1.729 0.622 1.101 1.333 1.604 1.762 0.760 1.234 1.300 1.423 1.597 1.130 1.364 1.688 1.747

~.

0.527 0.579 0.539 0.541 0.566 0.548 0.580 0.589 0.556 0.566 0.582 0.576 0.573 0.572 0.569 0.587 0.572 0.564 0.588

mf

ms

o'w

o'f

(kg s - ' )

(kg s - ' )

(kPa)

(kPa)

0.0557 0.0676 0.0749 0.0838 0.0870 0.0559 0.0678 0.0754 0.0833 0.0869 0.0560 0.0678 0.0739 0.0838 0.0878 0.0677 0.0744 0.0835 0.0873

1.159 1.157 1.161 1.168 1.162 1.494 1.494 1.496 1,497 1.493 1.960 1.957 1.968 1.969 1.964 2.402 2.374 2.373 2.375

0.420 0.702 0.706 0.926 0.969 0.423 0.737 0.796 0.882 1.007 0.423 0.707 0.788 0.948 1.008 0.673 0.781 0.928 0.996

0.768 1.193 1.310 1.688 1.712 0.762 1.223 1.364 1.529 1.623 0.728 1.191 1.346 1.524 1.703 1.190 1.357 1.535 1.709

0.547 0.588 0.539 0.601 0.587 0.555 0.603 0.584 0.577 0.622 0.515 0.594 0.585 0.622 0.592 0.566 0.576 0.605 0.583

5.2. Stress transmission coefficient in a vertical pipe 't r/,t y~

~

YL=EYL

~zy

,~,WYL

O

02

°l

~

O

Fig. 7. Possible state of stress on element P, represented by a series of Mohr circles.

represent the plane of the wall. The state of stress acting at point E in Fig. 7 is the theoretical limiting state of stress on the failure plane at failure. The stress transmission coefficient exhibits the relationship of normal stress on two mutually vertical planes. Its value is affected by the state, strength and some physical properties of bulk material. The following stress analyses attempt to develop a general expression for A.

Fig. 8 shows a particle slug flowing in a vertical pipe with a parallel and rigid wall. The experimental observations found that during slug-flow almost all the particles contained in the slug are fixed relative to each other and the slug moves like a rigid body in the one direction (for example downward). This indicates that failure occurs between the boundary of the material and pipe wall, but there is no failure occurring between the particles. As described in the previous section, such a case is the second state of particulate medium. The Mohr circle representing the stresses of a boundary element P in the second state must be located

z•y

A Single Particle

i

D 2

\i" Fig. 8. Particles flowing in a vertical pipe.

B, Mi, P.W. Wypych / Powder Technology 84 (1995) 91-98

96

between the Mohr failure envelope and wall yield locus of the material in the strength diagram, as shown in Fig. 9. The points A or D where the Mohr circle C intersects the wall yield locus of the material represent the wall plane along which the particles slip. If the normal stress at point A represents the horizontal stress (%) of the element P shown in Fig. 9 (the radial stress of the slug), according to the Mohr circle theory, the normal stress at point B will represent the vertical stress (az) of the element P (the axial stress of the slug). Hence, the stress transmission coefficient in the vertical pipe will be %/trz. As try
(8)

sin to= sin 4w/sin 4s

(9)

For the passive case of stress state, the stress transmission coefficient Zv is Ap= ~ = l + s i n ¢s cos(to+¢w) ¢rz 1 - sin ¢~ cos(to + Cw)

(10)

From Eqs. (8) and (10), it can be found that for a given bulk material, the stress transmission coefficient is determined only by its static internal friction angle and wall friction angle while the material flows in the pipe with a rigid and parallel wall. Obviously, Ap> AA. Note that although Eqs. (8), (9) and (10) are developed for the stress transmission coefficient in a vertical pipe, no special nature of the vertical pipe has YL=EYL

been used during the development of these equations. Hence it is reasonable to assume that the equations are suitable also for the stress transmission coefficient in a horizontal pipe.

6. Static internal friction angle

Equations have been developed in the previous section for the stress transmission coefficient A. However, the value of A is dependent on the static internal friction angle (¢s) as given in Eqs. (8), (9) and (10). Hence, a relationship between ~bs and its influential factors must be developed. As shown in Fig. 10, while a bulk material flows in the form of slugs in a pipe, there are an infinite number of possible Mohr circles representing the stress state of the slug located between the YL and WYL of the material. For example, the stress states represented by the Mohr circles C1 and C2 both satisfy the criterion of the slug flowing in the pipe, that is the particles of the slug slip along the pipe wall, but there is no relative movement between the particles. By drawing a tangent line OF1 to the Mohr circle C1 and tangent line O F 2 to the Mohr circle (22 through the origin of the coordinate system, two different static internal friction angles are obtained as ~bsl and ¢s2. It is difficult to know exactly the actual static internal friction angle. However, there always exists a unique Mohr circle which represents the actual stress state of the slug. Owing to the uncertainty of the actual 'location' of Mohr circle of the slug, it is extremely difficult to predict the internal static friction angle from theory. Hence, an empirical correlation is sought. From the results of A listed in Table 2, static internal friction angles can be calculated by using Eqs. (8) and (9) or (10) for these test materials. It is also noted that all the stress transmission coefficients for the materials are less than 1. This means that the stress

YL=EYL FI

_--- o P,

Fig. 9. Diagram of strength.

"~

ozi

,

~

o

Fig. 10. Possible Mohr circles representing the stress state of a particle slug.

B. Mi, P.W. Wypych I Powder Technology 84 (1995) 91-98 Table 3 Static internal friction angles for test materials

180.

•e~

Material

140 120

Static internal friction angle

Plastic pellets (white)

~b~ (°):

Plastic pellets (black)

15.75

Wheat

13.40

97

Barley

20.08

16.81

7. 4.2

--~ 3

~

I00

..~

80!

.,~

6o .::

.~

40,

-~

20.

~.2

2

'

~

~

.

~

~

1 4.5 ~

8

2.8

m,. (ta%

0 0.01

22

0.02

0.03

0.04

0,05

0.06

0.07

0.08

0.09

0.10

Mass flow-rate of air, m t, (kgsq)

"~

20'

Fig. 12. Low velocity conveying characteristics of urea• L = 9 6 D = 1 0 5 ram.

18

To demonstrate the accuracy of this technique, these parameters were substituted into the pressure drop model presented by Mi and Wypych [3] to generate the conveying characteristics shown in Fig. 12. Note that the curves of constant ms shown in Fig. 12 were predicted by this pressure drop model [3] using only the physical properties of urea listed in this paper. It can be seen that there is excellent agreement between the experimental data and the predicted ms curves. This provides confidence in using the model for scaleup and design purposes (that is for other capacities and pipeline diameters).

16

14.

12

12

, 14

.

, 16

.

, 18

.

, 20

22

Observed ~ s Fig. 11. G o o d n e s s of fit.

state in slugs should be in the active stress case during slug-flow. Hence, Eq. (10) is not used for this calculation. Table 3 lists the calculated static internal friction angle of each test material. Based on the data listed in Table 3, the following expression for static internal friction angle is regressed by applying the method of least squares with the limit of ~bs< ~b. 4 ¢~=- g 4~wYb'"

(11)

where Yb is defined as the bulk specific weight with respect to water at 4 °C. The goodness of fit is shown in Fig. 11. Using Eqs. (8), (9) and (11), the stress transmission coefficient can now be predicted for particle slugs moving through a pipe with rigid and parallel walls, as long as the properties of the material are measured. This procedure is demonstrated by the following case study. 7. Industrial case study

An adhesives company requested assistance to design a pneumatic conveying system for urea. It was required to minimize dust generation and hence, low velocity slug-flow was selected for this application. Firstly, the physical properties of urea were determined. The results are summarized in Table 1. Using Eq. (11), ~bs= 15.97 ° and from Eqs. (8) and (9), A = 0.668.

m;

8. Conclusions

The following conclusions and recommendations are based on the experimental and theoretical results presented in this paper. Wall pressure in a moving slug varies as an exponential function along its length and is determined by the stresses acting on the front and back faces of the slug. For a given material, wall pressure appears to increase linearly as the mass flow rate of air increases. The stress in a horizontal moving slug appears to be in the active state. Values of ~bs and A determined by this technique appear to provide good estimations of pipeline pressure drop for other granular materials (such as urea). However, before a unified theory can be developed and confirmed for low velocity slug-flow, further experimental work is needed with different pipe diameters and types of material, particularly powders.

9. List of symbols

A cross-sectional area of pipe C, Ca Ca, C2 Mohr circle centres

( m 2)

98

d

B. Mi, P.W. Wypych / Powder Technology 84 (1995) 91-98

equivalent volume diameter of particle

%, trf

(m) D g mf~ m s

ap

R Rf R~

internal pipe diameter (m) air pressure force (N) acceleration due to gravity ( m s -2) length of a single slug (m) mass flow rate of air or solids (kg s-1) pressure drop across a single slug (Pa) inner pipe radius (m) wall friction force (N) resistance force due to stationary bed

% trr %~, (r,~ (rx, %, trz (rl tr2, (r2', (r2" ~', ~w -rye, %y

(N) us Upy, Upz

slug velocity (m s - ' ) particle velocity in y direction or z direction (m s -1)

Greek letters E

4, 7b

h hA, hp tzw Pb, P~ o"

bulk voidage internal friction angle (°) static internal friction angle (°) wall friction angle (°) bulk specific weight with respect to water at 4 °C stress transmission coefficient A in active stress case, passive stress case wall friction coefficient bulk density, particle density (kg m -3) normal stress (Pa)

normal stress at back face or front face of slug (Pa) gravity pressure (Pa) radial stress of a particle slug (Pa) total wall pressure or wall pressure (Pa) axial stress, horizontal stress or vertical stress of a particle element (Pa) major principle stress (Pa) minor principle stresses (Pa) shear stress, shear stress at wall (Pa) shear stresses on particle element (Pa) angle defined in Fig. 9 (°)

References [1] K. Konrad, D. Harrison, R.M. Nedderman and J.F. Davidson, Proc. 5th Int. Conf. Pneumatic Transport of Solids in Pipes, London, UK, 16-18 Apr., 1980, BHRA Fluid Engineering, Granfieid, UK, p. 225. [2] D. Legel and J. Schwedes, Bulk Solids Handling, 4 (1984) 399. [3] B. Mi and P.W. Wypych, Powder Technol., 81 (1994) 125. [4] A. Borcz, A. and H.A. Rahim, Powder Handling Process., 1 (1989) 349. [5] A.W. Hancock and R.M. Nedderman, Trans. Inst. Chem. Eng., 52 (1974) 170. [6] A.W. Jenike and J.R. Johanson, J. Structure Div. ASCE, 95 (1968) 1011. [7] J.K. Waiters, Chem. Eng. Sci., 28 (1973) 13. [8] B. Mi, Ph.D. Thesis, University of Wollongong, Australia (1994). [9] B. Mi and P.W. Wypych, Powder Handling Process., 5 (1993) 227.