09 chute paper

8/13/2019 09 Chute Paper

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TUNRA Bulk Solids AWR 9.1

9. CHUTE PERFORMANCE AND DESIGN FOR

RAPID FLOW CONDITIONS*

Alan W. Roberts.

Centre for Bulk Solids and Particulate Technologies,

The University of Newcastle, NSW., 2308, Australia

ABSTRACT

Many industrial chute applications are characterised by rapid flow conditions in which the

bulk solid stream thickness or depth is less than the chute width. Under these conditions, it is

possible to describe the stream flow by means of a lumped parameter model which takes into

account the frictional drag around the chute boundaries as well as making allowance for inter-

particle friction. Equations of motion to describe the chute flow are presented and their

application to the determination of chute profiles to achieve optimum flow is illustrated. By

means of design examples, the problems associated with the feeding of bulk solids onto beltconveyors and conveyor transfers are discussed. Criteria for the selection of the most

appropriate chute geometry to minimise chute wear and belt wear at the feed point are

presented. The determination of optimum chute profiles to achieve specified performance

criteria is outlined.

1. Introduction

Chutes used in bulk handling operations perform a variety of operations. For instance,

accelerating chutes are employed to feed bulk materials from slow moving belt or apron

feeders onto conveyor belts as illustrated in Fig. 1. In such installations the feeder speed is

normally limited to about 0.3 m/s, whereas the conveyor speed may be 5 m/s or higher. Inother cases, transfer chutes are employed to direct the flow of bulk material from one

conveyor belt to one or more conveyors, often via a three dimensional path. An example of

such a transfer chute is shown in Fig. 2. In this case the discharge from the delivering

conveyor is split and directed onto either of two receiving conveyors which are at 90oto the

delivering conveyor as shown. The purpose of the dribble chute is to collect and transfer the

cohesive carry-back material from the belt cleaning system.

The majority of industrial chute applications, such as those described above, involve rapid or

accelerated flow conditions. Such conditions are characterised by thin stream flow in which

the thickness or depth of the flowing stream of bulk material remains less than the width of

the chute. Very often the thickness is less than half the stream width. Under these conditions,it is possible to describe the stream flow by means of a lumped parameter model which takes

takes into account the frictional drag around the chute boundaries as well as making

allowance for inter-particle friction [1,2].

The importance of correct chute design to ensure efficient transfer of bulk solids without

spillage and blockages and with minimum chute and belt wear cannot be too strongly

emphasised. The importance is accentuated with the trend towards higher conveying speeds.

The purpose of this paper is to present an overview of the basic principles of chute design

with particular regard to feeding and transfer in belt conveying operations as illustrated in

Figs. 1 and 2.

___________________________________________________*Published in Chem. Eng, Technol. 26 (2003) 2


2/17


Figure 1. Feeding Onto a Belt Conveyor

Figure 2. Example of a Conveyor Belt Transfer

2. Chute Flow Model

Studies performed by Roberts [1] showed that under thin stream accelerated flow through

chutes, approximately 82% of the energy losses are attributed to the bulk material sliding

along the chute bottom, about 9% of the losses being due to sliding against the side walls,

with the remaining 9% due to inter granular friction. Under these conditions the motion can

be described by a lumped parameter model as shown in Fig. 3. The basic theory is nowreviewed.

vb

vb

vex

vey

ve

Rc1

Rc2

DribbleChute

CurvedImpactPlate

Gate

vd

VeVex

Vb

Feeder

Conveyor

Vf

RHT

Vey

o

Vo

V

y

x

h

VeVex

Vb

Feeder

Conveyor

Vf

RHT

Vey

o

Vo

V

y

x

h


3/17


Equivalent Friction for Chutes of Constant Width

The drag force FDdue to Coulomb friction is expressed by

FD= eFN (1)

where FN = normal force and e = equivalent friction factor which takes into account thefriction coefficient between the bulk solid and the chute surface, the stream cross-section and

the internal shear of the bulk solid. For a chute of rectangular cross-section, e isapproximated by

e = [1 + KvH

B ] (2)

where = actual friction coefficient for bulk solid in contact with chute surface

Kv = pressure ratio. Normally Kv= 0.4 to 0.6H = depth of flowing stream at a particular location

B = width of chute

m g

m v.

v.

v

"s"

"n"

"x"

"y"

FD

m

Ho Ao

HAm v

R

R

2

FN

B

B

H

H

v Rectangular Cross Section

Circular Cross Section

VelocityProfile

vo

Figure 3. Chute Flow Model

The equivalent friction for chutes of circular cross-section has been analysed by Roberts and

Scott [2] and by Parbery and Roberts [3]. It has been shown that ecan be expressed by thepower series

e = [1 + a1H

B+ a2(

H

B)

2

+ ] (3)


4/17


where a1 = 0.4, a2= 0.2057 .

For shallow bed conditions, the higher order terms may be neglected so that the remaining

linear relationship becomes identical to equation (2)

Equivalent Friction for Chutes of Varying Width

In some cases, it is necessary to converge the flow, as illustrated in Figure 4. In this case, the

equivalent friction is given by

e= [ 1 + Kv(H

Bo- 2 s tan )(1 +

tan )] (4)

Figure 4. Chute of Varying Width

2.3 Continuity of Flow

For continuity of flow,

A v = Constant (5)

where = bulk densityA = cross-sectional area of flowing stream

2.4 Equations of Motion

Referring to the chute model shown in Fig. 3 and analysing the dynamic equilibrium

conditions, the following differential equations for the stream velocity may be derived:

Kv pn

Bo

pn

p

B

H

B

Ho

H

s

ds

= pn

v

vo

dF


5/17


Moving Coordinates, Tangential s and n Components

dv

ds + e

v

R-

g

v (cos - e sin ) = 0 (6)

Cartesian Components, x and y Components

dx

dx +

x y(x)[y(x) + e] - g [1 - ey(x)]

1 + y(x)2 = 0 (7)

and y = y(x) x (8)

Also v = x2+ y2 (9)

The radius of curvature of the path at any arbitrary location is

R =[1 + y(x)2]1.5

y(x) (10)

2.5 Complete Solution Involving Variation of e

Solutions of the equations of motion need to take account of the variation of the equivalent

friction along the chute. Assuming the bulk density remains approximately constant, the

combination of equations (2) and (5) allows the equivalent friction to be expressed as a

function of stream velocity. That is, for a chute of constant width,

e= ( 1 +C1v

) (11)

where C1=B

HvK oov (12)

vo = initial velocity

Ho = initial stream thickness

For the converging chute of Fig. 4,

e= [1 + C2

v B2] (13)

where C2= KvBo Ho vo(1 +tan

) (14)

B = Bo 2 s tan (15)

Bo= initial chute width

e given, as appropriate, by equations (11) or (14) is substituted into equation (6) or (7)

which may then be solved numerically.


6/17


3. Chutes of Constant Curvature

In order to illustrate the solution of the flow equations, the case of chutes of constant

curvature and rectangular cross-section is considered. The more exact solution involves the

variation of e to be taken into account. However an approximate solution will often givesatisfactory results.

Approximate Solution

If the curved section of the chute is of constant radius R and e is assumed constant at anaverage value for the stream, it may be shown that the solution of equation (6) leads to the

equation below for the velocity at any location .

v =2 g R

4 e2+ 1[(1- 2 e2) sin + 3 ecos ] + K e-2

e (16)

For v = voat = o,

K = {vo2 -2 g R

4 e2+ 1 [(1- 2 e2) sin + 3 ecos ]}e2e (17)

Special Case:

When o= 0 and v = vo, K = vo2 -6 e g R1 + 4 e2

(18)

Equation (11) becomes,

v =2 g R

4 e2+ 1[(1- 2 e2) sin + 3 ecos ] + e-2

e [vo2 -6 eR g4 e2+ 1

] (19)

Chute of Constant Width - Comparison of Complete and Approximate Solutions

The complete and approximate solutions are compared in Fig. 5 for the case of a chute of

radius R = 2.0 m for the initial velocity vo= 4 m/s. The friction angle for the bulk solid incontact with the chute surface is s= 25

o. Two initial bed depths are considered, Ho/B = 0.2

and Ho/B = 0.6.

Fig. 5(a) shows the variation of equivalent friction, while Fig. 5(b) compares the exact and

approximate solutions for the velocity distributions. As shown, there is little variation in efor the smaller Ho/B which explains why the complete and approximate solutions for the

velocity profiles are virtually identical. Even for the thicker stream for which Ho/B = 0.6,

there is close agreement between the exact and approximated velocity distributions. The cut-

off angle shown in Fig. 5(b) is the angle at which the chute profile should be terminated for it

to be self cleaning. This is further discussed in Section 6.3.


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3.3 Converging Chute of Constant Radius

The case of a chute profile of the type shown in Fig. 4 is now examined. As an example, the

chute radius R = 3.0 m, the initial chute width is Bo= 1.0 m, the initial velocity is vo = 4 m/s

at o = 0oand the friction angle s= 25

o. Three profiles are compared, = 0o, that is, constant

width, = 2.5oand = 5o. Fig. 6(a) shows the stream velocities, while Fig. 6(b) shows thevariation in chute width H and stream thickness or depth as a ratio of chute width, that is,

H/B. The angles = 0o to 2.5o give acceptable performance. However, the convergenceangle of = 5o causes the stream thickness to increase significantly beyond the angular

position of = 40o. For = 5o the cut-off angle for the converging section should be notgreater than 40o.

(a) Variation of Equivalent Friction (b) Variation of Velocity

Figure 5. Flow Through Chute of Constant Radius

Vo= 4 m/s; R = 2.0 m; = tan 25o= 0.466

(a) Variation of Stream Velocity (b) Chute Width and Ratio H/B

Figure 6. Flow Through Converging Chute of Constant Radius

Vo= 4 m/s; R =3.0 m; = tan 25o= 0.466

0.4

0.45

0.5

0.55

0.6

0 10 20 30 40 50 60

ue for Ho/B = 0.2

ue for Ho/B = 0.6EQUIVALENTFRIC

TION

ANGULAR POSITION (deg)

Vo = 4 m/s R = 2.0 m

CutOffAngle

3

3.5

4

4.5

5

0 10 20 30 40 50 60

Vel for Variable ue, Ho/B=0.2

Vel for Const. ue = 0.5

Vel for Variable ue, Ho/B =-0.6


VELOCITY

(m/s

)


Vo = 4 m/s R = 2.0 m

Cut-OffAngle

0.4

0.45

0.5

0.55

0.6

0 10 20 30 40 50 60

ue for Ho/B = 0.2

ue for Ho/B = 0.6EQUIVALENTFRIC

TION


Vo = 4 m/s R = 2.0 m

CutOffAngle

3

3.5

4

4.5

5

0 10 20 30 40 50 60

Vel for Variable ue, Ho/B=0.2


Vel for Variable ue, Ho/B =-0.6


VELOCITY

(m/s

)


Vo = 4 m/s R = 2.0 m

Cut-OffAngle

4

4.5

5

5.5

6

0 10 20 30 40 50 60

Velocity for Lambda = 0 deg

Velocity for Lambda = 2.5 deg


ANGULAR POSITION (deg.)

STREAMVEL

OCITY

(m/s)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 10 20 30 40 50 60

B for Lambda = 0 deg

B for Lambda = 2.5 deg


H/B for Lambda = 0 deg

H/B for Lambda = 2.5 deg



STREAMTHICKNESSH(m)andRATIOH/B

4

4.5

5

5.5

6

0 10 20 30 40 50 60


Velocity for Lambda = 2.5 deg



STREAMVEL

OCITY

(m/s)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 10 20 30 40 50 60


B for Lambda = 2.5 deg



H/B for Lambda = 2.5 deg



STREAMTHICKNESSH(m)andRATIOH/B


8/17


4. Inverted Curved Chute Sections

Inverted curved chute sections are often employed in the impact zone of transfer chutes as

illustrated in Fig. 2. The method outlined above for curved chutes may be readily adapted to

inverted curved chute sections as depicted by the model of Fig. 7.

Figure 7. Inverted Curved Chute Model

Noting that FD= eFN, it may be shown that the differential equation in moving coordinatesis given by

-dv

d + ev =g R

v (cos+ e sin ) (20)

where e= equivalent friction factor as previously defined.

For a constant radius and assuming e is constant at an average value for the stream, the

solution of equation (20) is

v =2 g R

4 e2+ 1

[sin (2 e2- 1) + 3ecos ] + K e2e (21)

For v = voat = o, then

K = {vo2-2 g R

1 + 4 e2 [3

ecos o + (2 e2- 1) sin o]}e-2 eo (22)

Special Case: v = voat o=2

K = {vo2-2 g R

1 + 4 e2 [ 2

e2- 1]}e-e (23)

and

H

m g

v

v

m v

m vR

FD

m

"t"

"n"

.

.

2

R

CrossSection

B

FN


9/17


v = (24)

Equations (20) to (24) apply during positive contact, that is, when

v2

R g sin (25)

5. Wear in Chutes

Chute wear is a combination of abrasive and impact wear. Abrasive wear may be analysed

by considering the mechanics of chute flow as will be now described.

5.1 Abrasive Wear Factor for Chutes

In cases where the bulk solid moves as a continuous stream under rapid flow conditions, the

abrasive or rubbing wear is a function of the normal pressure and rubbing velocity. Consider

the general case of a curved chute of rectangular cross-section as illustrated by the chute

flow model of Fig. 3. An abrasive factor Wcfor wear of the chute bottom surface is given by

Wc=Qm Kctan s

B NWR (26)

Wc has units of (N/ms)

NWR is an abrasive wear parameter have units of normal acceleration and is given by

NWR=v2

R + g sin (27)

where Qm = mass flow rate (kg/s)

Kc =vsv

v = average stream velocity at section considered

vs = rubbing velocity for bulk solid on chute bottom surface

s = friction angle for bulk solid on chute surfaceB = chute width

R = radius of curvature

= chute slope angle measured from the vertical

The factor Kc< 1.0. For rapid thin stream flow Kc0.8. As the stream thickness increases

Kc will reduce.

5.2 Special Case Straight Inclined Chute

Equations (26) and (27) apply to any chute profile. For a straight inclined chute, R = Inthis case equation (26) becomes

Wc=

Qm Kctan g sinsB (28)


10/17


For a constant flow rate and assuming that Kcis approximately constant, then, theoretically,

the wear along a straight inclined chute is constant and independent of the velocity variation.

5.3 Abrasive Wear of Chute Side Walls

Equation (26) applies to the chute bottom surface. For the side walls, the wear will be muchless, varying from zero at the stream surface to a maximum at the chute bottom. Assuming

the side wall pressure to increase linearly from zero at the stream surface to a maximum

value at the bottom as depicted in Figure 3, then the average wear on the side walls can be

estimated from

Wcsw =WcKv2 Kc

(29)

Where Kcand Kvare as previously defined. If, for example Kc= 0.8 and Kv= 0.4, then the

average side wall wear is 25% of the chute bottom surface wear.

5.4 Impact Wear in Chutes

Impact wear may occur at points of entry or points of sudden change in direction. For ductile

materials, greatest wear is caused when impingement angles are low, that is in the order of

15o to 300. For hard brittle materials, greatest impact damage occurs at steep impingement

angles, that is angles in the vicinity of 90o.

6. Feeding Onto Belt Conveyor

A common application of gravity flow chutes is in the loading of bulk solids onto belt

conveyors. The application is illustrated in Fig. 1 which shows a gravity feed chute used in

conjunction with a belt or apron feeder. The normal operating speeds of such feeders is quite

low with vf 0.3 m/s. Hence the chute must not only direct the bulk solid onto the belt

without spillage, but it must also allow the bulk solid to be accelerated so that at the point of

discharge onto the belt, the horizontal component, vex, of the discharge velocity matches, as

close as possible, the belt speed.

The bulk solid falls vertically through a height 'h' before making contact with the curved

section of the feed chute. In view of the very low speed of the feeder, the velocity, vo, of first

contact with the curved section of the feed chute will be, essentially, in the vertical direction.

6.1 Free Fall of Bulk Solid

For the free fall section, the velocity vomay be estimated from

vo = vfo2+ 2 g h (30)

Equation (30) neglects air resistance, which in the case of this feeding example, is likely to be

small. If air resistance is taken into account, the relationship between height of drop, h, and

velocity, vo,is,


11/17


h =v2

g loge [

1 -vfov

1 -vo

v

] - (vo- vfo

g) v (31)

where v= terminal velocityvfo= vertical component of velocity of bulk solid discharging from feeder

vo = velocity corresponding to drop height 'h' at point of impact with chute.

6.2 Chute Profile and Belt Wear

For a given overall height of drop, HT, (Fig. 1), various combinations of free fall heights, h,

and chute geometries may be examined. It is essential to select the best geometry to meet the

performance requirements within the practical constraints of the installation. An overriding

consideration is need to minimise chute wear as well as belt wear at the feed point. At the

same time it is necessary to ensure that the chute slope angle at the belt conveyor feedpoint is sufficiently large for the chute to be self cleaning during start-up following each shut-

down with bulk solid still retained in the chute. The latter objective is governed by the

condition

> tan-1(e) + 5o (32)

Since the discharge slope angle of the chute - ), of necessity, there will be an

appreciable vertical component vex of the exit velocity. This component, when combined

with the bulk density gives rise to the impact pressure, vex2, on the belt. When linked with

the relative rubbing velocity (vb- vey), the relative wear parameterWa is obtained [4].

Wa= bvex2(vb- vey) (kPa/m s) (33)

where b= friction coefficient between bulk solid and belt surface = bulk density

7. Chute Design Examples

The system shown in Fig. 1 is considered in which coal is being fed at the rate of 1000 t/h

from a belt feeder moving at 0.3 m/s. The overall height HT = 5 m, the bulk density = 0.8

t/m

3

and the belt speed vb = 6 m/s. The friction angle for the coal on the chute surface is s=25oand the coefficient of friction for the coal in contact with the conveyor belt surface b=0.6. The chute cut-off angle = 35o for which o= 55

o.

7.1 Vertical Drop and Constant Curvature Chutes

Various combinations of drop heights, h, and corresponding radii of curvature, R, to meet the

geometrical constraints of the system are examined. These arrangements are illustrated in

Fig. 8(a). For each of these combinations, the initial velocity vofor the point of contact with

the chute profile following the free fall have been determined and the velocity distribution

around the constant radius chute have been computed together with the belt wear at the feed

point. The results of this exercise are presented in Figs. 8(b) and (c)


12/17


(a) Chute Profiles (b) Stream Velocities

(c) Conditions at Exit End of Chute

Figure 8. Feed Through Drop Height Followed by Chute of Constant Radius R

It is interesting to observe from Fig. 8(b) that the velocities corresponding to = 45o are thesame for all chutes. Despite the wide variation in the initial velocities at the first contact

point with the curved chute sections, the velocities at discharge are almost the same varying

from 6.18 m/s for the drop of h = 4.18 m with the R = 1.0 m radius chute to v e= 6.35 m/s for

the drop h = 1.73 m with the R = 4.0 m radius chute. This is shown in Fig. 8(c) which

presents the discharge velocities, ve, belt wear Waand initial drop height, h, as functions of

chute radius R. It is evident that the smaller the chute radius, the more energy the chute

-0.5 0 0.5 1 1.5 2

0

1

2

3

4

5

R = 1 m

R = 2 m

R = 3 m

R = 4 m

VERTICALDISTANCE

(m)

HORIZONTAL DISTANCE (m)

5.5

6

6.5

7

7.5

8

8.5

9

9.5

0 10 20 30 40 50 60

R=1.0 m; h=4.18 m; Vo=9.06 m/s

R=2.0 m; h=3.36 m; Vo=8.12 m/s

R=3.0 m; h=2.54 m; Vo=7.06 m/s

R=4.0 m; h=1.73 m; Vo = 5.82 m/s


VELOCITY

(m/s)

-0.5 0 0.5 1 1.5 2

0

1

2

3

4

5

R = 1 m

R = 2 m

R = 3 m

R = 4 m

VERTICALDISTANCE

(m)

HORIZONTAL DISTANCE (m)

5.5

6

6.5

7

7.5

8

8.5

9

9.5

0 10 20 30 40 50 60

R=1.0 m; h=4.18 m; Vo=9.06 m/s

R=2.0 m; h=3.36 m; Vo=8.12 m/s

R=3.0 m; h=2.54 m; Vo=7.06 m/s

R=4.0 m; h=1.73 m; Vo = 5.82 m/s


VELOCITY

(m/s)

1

2

3

4

5

6

6.15

6.2

6.25

6.3

6.35

6.4

0.5 1 1.5 2 2.5 3 3.5 4 4.5

Wa (kPa m/s)

h (m)

Ve (m/s)WEARATFEEDPOINTWa(kPam/s)

INITIALDROPHEIG

HTh(m)

EXITVELOCITY

Ve(m/s)

CHUTE RADIUS OF CURVATURE R (m)

1

2

3

4

5

6

6.15

6.2

6.25

6.3

6.35

6.4

0.5 1 1.5 2 2.5 3 3.5 4 4.5

Wa (kPa m/s)

h (m)

Ve (m/s)WEARATFEEDPOINTWa(kPam/s)

INITIALDROPHEIG

HTh(m)

EXITVELOCITY

Ve(m/s)

CHUTE RADIUS OF CURVATURE R (m)


13/17


absorbs and this implies greater wear of the chute surface. The best performance is obtained

with a smaller drop height and larger chute radius.

Constant Curvature and Parabolic Chute Profiles

The example discussed in the previous section is again considered in this case, a comparisonis made between a chute of constant curvature and a chute of parabolic geometry. The coal

is allowed to drop a distance of h = 1.73 m before coming into contact with each chute, the

velocity of initial chute contact being 5.82 m/s. The constant curvature chute has a radius of

R = 4.0 m. The total drop height is HT = 5.0 m and the cut off angle o= 55o. The equation

of the parabolic chute is

y = C x2 where C = 0.218 (34)

This value of C satisfies the cut-off condition of y'(x) = tan(55o) corresponding to x = 3.27 m

which is based on HT= 5.0 m. Substituting y'(x) = 2 C x, y"(x) = 2C and eas defined byequation (11) into equations (7) and (8) and solving to obtain x and y as functions of x. The

velocity v = x2+ y2 is then obtained.

(a) Chute Profiles (b) Velocity Profiles

Figure 9. Comparison of Constant Radius and Parabolic Chutes

The results are presented in Fig. 9. Fig. 9(a) compares the chute profiles, while Fig. 9(b)

presents the velocity profiles. The constant curvature chute shows that the velocity increases

to a maximum and then decreases as the coal is subjected to retardation. On the other hand,

the parabolic chute shows the velocity increasing steadily approaching a maximum value

asymptotic value towards the cut-off point. This indicates that the coal is being accelerated

over the entire chute. The velocity distribution for the parabolic chute is more favourable

than for the constant radius chute. This is because of the increasing radius of curvature of theparabolic chute as given by

0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

3

3.5

Const Radius Chute

Parabolic Chute

COORDINATE y (m)

COORDINATE

x(m)

5.5

6

6.5

7

0 0.5 1 1.5 2 2.5 3 3.5

v (m/s) for Parabolic Chutev (m/s) for Const Radius Chute

VERTICAL COORDINATE POSITION x (m)

STREAMVELOCIT

Y

v(m/s)

0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

3

3.5

Const Radius Chute

Parabolic Chute

COORDINATE y (m)

COORDINATE

x(m)

5.5

6

6.5

7

0 0.5 1 1.5 2 2.5 3 3.5

v (m/s) for Parabolic Chutev (m/s) for Const Radius Chute


STREAMVELOCIT

Y

v(m/s)


14/17


R =1

2C[1 + (2 C x)2 ]1.5 (35)

The energy loss has a more favourable distribution for the parabolic chute profile as is the

chute wear. The non-dimensional chute wear profiles determined in accordance withequation (26) are plotted in Fig. 10. The exit velocity for the parabolic chute is ve= 6.37 m/s

which is almost identical with ve= 6.35 m/s for the constant curvature chute.

Figure 10. Non-Dimensional Wear Profiles for Constant Radius and Parabolic Chutes

8. Discussion

The foregoing examples may be examined in terms of the exit velocity vedetermined from

the energy and frictional work relationship. For the system of Fig. 3, veis given by

ve = {[vfo2 + 2 g HT] - [2 g eye)] [2 e

o

xe

x2y" dx ]}0.5 (36)

where vfo= initial vertical velocity of bulk solid discharging from feederye = horizontal coordinate of curved chute at exit point

The first term in square brackets of equation (36) depends on v foand HT, which are the same

for all chutes considered. The second term in square brackets depends on the horizontal exit

coordinate yewhich, in turn, depends on the chute profile and cut-off angle e. The latter isthe same for all chutes considered. The last term containing the integral depends on the

vertical component of the chute velocity, x, and chute curvature term y". Despite the wide

variation of the chute profiles in the examples considered, there was little variation in the

final exit velocity ve.

0.8

1

1.2

1.4

1.6

1.8

2

0 0.5 1 1.5 2 2.5 3 3.5

NWR for Parabolic Chute

NWR for Const. Radius Chute


ABRASIVEWEARPARAM

ETER

NWR(M/S^2

Initial Velocity = 5.82 m/s

0.8

1

1.2

1.4

1.6

1.8

2

0 0.5 1 1.5 2 2.5 3 3.50.8

1

1.2

1.4

1.6

1.8

2

0 0.5 1 1.5 2 2.5 3 3.5

NWR for Parabolic Chute

NWR for Const. Radius Chute


ABRASIVEWEARPARAM

ETER

NWR(M/S^2

Initial Velocity = 5.82 m/s


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While this paper has concentrated on chutes of specified geometrical form, it is possible to

determine the required chute profile y(x) to maximise either the exit velocity or horizontal

component of the exit velocity or, in other cases, to minimise the transit time. Analytical

methods, such as those based on the calculus of variations, as well as numerical procedures

may be used to obtain solutions in such cases. Examples of research dealing with chute

optimisation are presented in Refs [5- 7]

9. References

1. Roberts A.W. (1969) An Investigation of the Gravity Flow of Non-cohesive GranularMaterials through Discharge Chutes. Transactions of ASME., Jnl. of Eng. in Industry, Vol.

91, Series B, No. 2,pp 373-381

2. Roberts A.W. and Scott O.J. (1981) Flow of Bulk Solids through Transfer Chutes of

Variable Geometry and Profile. Bulk Solids Handling, Vol. 1, No. 4, pp 715.-727

3. Parbery R.D. and Roberts A.W. (1986) On Equivalent Friction for the Accelerated Flowof Granular Materials in Chutes. Powder Technology, Vol 48pp 75-79

4. Roberts AW and Wiche S.J. (1999) Interrelation Between Feed Chute Geometry and

Conveyor Belt Wear. Bulk Solids Handling, Vol. 19 No.1pp 35-39

5. Charlton W.H. and Roberts A.W. (1970) Chute Profile for Maximum Exit Velocity in

Gravity Flow of Granular Materials.Jnl. Agric. Engng. Res., Vol. 15. pp 292-294

6. Charlton W.H., Chiarella C.and Roberts A.W. (1975) Gravity Flow of Granular Materials

in Chutes: Optimising Flow Properties. Jnl. Agric. Engng. Res., Vol. 20,pp. 39-45.

7. Chiarella C., Charlton W.H. and Roberts A.W. (1975) Optimum Chute Profiles in GravityFlow of Granular Materials: A Discrete Segment Solution Method: Transactions of

ASME., Jnl. of Eng. in Industry, Vol. 97, Series B, No. 1,pp 10-13


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TUNRA Bulk Solids AWR 9 17

b Friction coefficient between bulk solid and belt surfacee Equivalent friction factor Chute slope angle measured from the verticalo Initial chute slope angle

09 chute paper

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