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Chute Design Considerations For Feeding and Transfer

01/07/2014

CHUTE DESIGN CONSIDERATIONS FOR FEEDING AND TRANSFER A .W. Roberts Emeritus Professor and Director. Centre for Bulk Solids and Particulate Technologies, University of Newcastle, NSW, A ustralia. SUMMA RY Chutes used in bulk handling operations are called upon to perform a variety of operations. For instance, accelerating chutes are em ployed to feed bulk m aterials from slow m oving belt or apron feeders onto conveyor belts. In other cases, transfer chutes are em ployed to direct the flow of bulk m aterial from one conveyor belt to another, often via a three dim ensional path. The im portance of correct chute design to ensure efficient transfer of bulk solids without spillage and block ages and with m inim um chute and belt wear cannot be too strongly em phasised. The im portance is accentuated with the trend towards higher conveying speeds. The paper describes how the relevant flow properties of bulk solids are m easured and applied to chute design. Chute flow patterns are described and the application of chute flow dynam ics to the determ ination of the m ost appropriate chute profiles to achieve optim um flow is illustrated. The influence of the flow properties and chute flow dynam ics in selecting the required geom etry to m inim ise chute and belt wear at the feed point will be highlighted. 1. INTRODUCTION Undoubtedly the m ost com m on application of chutes occurs in the feeding and transfer of bulk solids in belt conveying operations. The im portance of correct chute design to ensure efficient transfer of bulk solids without spillage and block ages and with m inim um chute and belt wear cannot be too strongly em phasised. These objectives are accentuated with the trend towards higher conveying speeds. W hile the basic objectives of chute design are fairly obvious, the following points need to be noted: chute should be sym m etrical in cross-section and located central to the belt in a m anner which directs the solids onto the belt in the direction of belt travel in-line com ponent of the solids velocity at the ex it end of the chute should be m atched, as far as possible, to the belt velocity. This is necessary in order to m inim ise the power required to accelerate the solids to the belt velocity, but m ore im portantly to m inim ise abrasive wear of the belt norm al com ponent of the solids velocity at the ex it end of the chute should be as low as possible in order to m inim ise im pact dam age of the belt as well as m inim ise spillage due to particle re-bounding slope of the chute m ust be sufficient to guarantee flow at the specified rate under all conditions and to prevent flow block ages due to m aterial holding-up on the chute bottom or side walls. It is im plicit in this objective that the chute m ust have a sufficient slope at ex it to ensure flow which m eans that there is a norm al velocity com ponent which m ust be tolerated adequate precautions m ust be tak en in the acceleration zone where solids feed onto the belt in order to m inim ise spillage. O ften this will require the use of sk irtplates in the case of fine powders or bulk solids containing a high percentage of fines attention needs to be given to design details which ensure that during feeding aeration which leads to flooding problem s, is m inim ised. For this to be achieved, free- fall zones or zones of high acceleration in the chute configuration should be k ept to a m inim um . Chute design has been the subject of considerable research, a selection of references being included at the end of this paper [1-29]. However, it is often the case that the influence of the flow properties of the bulk solid and the dynam ics of the m aterial flow are given too little attention. The purpose of this paper is to focus on these aspects, indicating the basic principles of chute design with particular regard to feeding and transfer in belt conveying operations. 2. BOUNDA RY FRICTION, COHESION A ND A DHESION 2.1 Boundary or Wall Yield Locus For chute design, wall or boundary surface friction has the m ajor influence. It has been shown that friction depends on the interaction between the relevant properties of the bulk solid and lining surface, with ex ternal factors such as loading condition and environm ental param eters such as tem perature and m oisture having a significant influence. The determ ination of wall or boundary friction is usually perform ed using the Jenik e direct shear test as illustrated in Figure 1(a). The cell diam eter is 95m m . The shear force S is m easured under varying norm al force V and the wall or boundary yield locus, S versus V, or m ore usually shear stress τ versus norm al stress σ is plotted.

Figure 1. Boundary or W all Friction Measurem ent The Jenik e test was originally established for hopper design for which the norm al stresses or pressures are always com pressive. In the case of chute design, the pressures are norm ally m uch lower than in hoppers, and often tensile, particularly where adhesion occurs due to the cohesive nature of the bulk solid. The Jenik e test of Figure 1(a) does not allow low com pressive pressures to be applied since there is always the weight of the bulk solid in the shear cell, the shear ring and lid which form s part of the norm al load. To overcom e this shortcom ing, the inverted shear tester of Figure 1(b) was developed at the University of Newcastle. The shear cylinder is retracted so as to m aintain contact between the bulk solid and the sam ple of the lining m aterial. In this way, it is possible to m easure the shear stress under low com pressive and even tensile stresses. The inverted shear cell has been m anufactured with a diam eter of 300m m in order to allow m ore representative size distributions of bulk solids to be tested. The boundary or wall yield loci (W YL) for m ost bulk solids and lining m aterials tend to be slightly convex upward in shape and, as usually is the case, each W YL intersects the wall shear stress ax is indicating cohesion and adhesion characteristics. This characteristic is reproduced in Figure 2. The wall or boundary friction angle is defined by:

= tan-1 [

τw ] σw

(1)

where τw = shear stress at the wall; σ w = pressure acting norm al to the wall

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Figure 2. W all or Boundary Friction and Adhesion Characteristics The W YL for cohesive bulk solids is often convex upward in shape and, when ex trapolated, intersects the shear stress ax is at τo The Wall Friction Angle, , will then decrease with increase in norm al pressure. This is illustrated in Figure 3 which shows the wall friction angles for a representative cohesive coal in contact with a dull and polished m ild steel surfaces. It is to be noted that the wall friction angle cannot be larger than the effective angle of internal friction δ which is an upper bound lim it for . Thus, for very low norm al pressures where the friction angle 4 can be quite large, the bulk solid will fail by internal shear rather than by boundary shear, leaving a layer of m aterial on the surface.

Figure 3. Friction Angles for a Particular Coal on Mild Steel Surfaces The nature of adhesion and cohesion is quite com plex ; a study of this subject would require a detailed understanding of the physics and chem istry of bulk solid and surface contact. It is k nown, for ex am ple, that cohesion and adhesion generally increase as the wall surface becom es sm oother relative to the m ean particle size of the adjacent bulk solid. Also adhesion and cohesion generally increase as m oisture content of the bulk solid increase, particularly in the case of very sm ooth surfaces. No doubt, in such cases, surface tension has a significant influence. Cohesion and adhesion can cause serious flow block age problem s when corrosive bonding occurs, such as when m oist coal is in contact with carbon steel surfaces. The bonding action can occur after relatively short contact tim es. Im purities such as clay can also seriously aggravate the behavior due to adhesion and cohesion. 2.2 Types of A dhesion Problems In order that build-up and hence block ages can be avoided, it is necessary for the body forces generated in the bulk m ass to be sufficient to overcom e the forces due to adhesion and shear. Figure 4 illustrates the types of build-up that can occur.

Figure 4. Build-Up on Surfaces S = Shear Force; B = Body Force; F o = Adhesive Force The body forces are norm ally those due to the weight com ponent of the bulk solid but m ay also include inertia forces in dynam ic system s such as in the case of belt conveyor discharge or, in other cases, when vibrations are applied as a flow prom otion aid. 2.3 Mechanisms of Failure W hen the body forces are sufficient to cause failure and, hence, flow, the m ode of failure will depend on the relative strength versus shear conditions ex isting at the boundary surface and internally within the bulk solid. As discussed by Scott [29], the following failure conditions are considered: (a) Failure Envelopes - General Case In this case the shear stress versus norm al stress failure envelope for a cohesive bulk solid is always greater than the failure envelope at the boundary. This is illustrated in Figure 5. For such cases, it is ex pected that failure will occur at the boundary surface rather than internally within the bulk solid.

Figure 5 Failure Envelopes - General Case (b) Failure Envelopes - Special Case In cases of high m oisture content cohesive bulk solids it is possible for the failure envelope of the bulk solid at lower consolidation stresses or pressures to give lower internal strength than the corresponding strength conditions at the boundary. This is depicted in Figure 6.5. The body forces m ay then cause failure by internal shear leaving a layer of build solid adhering to the chute surface. This layer m ay then build up progressively over a period of tim e.

Figure 6. Failure Envelopes - Special Case O ften such problem s arise in cases where the bulk solid is transported on belt conveyors leading to segregation with the fines and m oisture m igrating to the belt surface as the belt m oves across the idlers. The segregation condition m ay then be transferred to chute surfaces. O ther cases occur when the very cohesive carry-back m aterial from conveyor belts is transferred to chute surfaces. (c) Failure Envelope - Free Flowing Bulk Solids. For free flowing, dry bulk solids with no cohesion, the boundary surface failure envelope is higher than the bulk solid failure envelope. In this case, adhesion of the bulk solid to a chute surface will not occur. Figure 6.6 illustrates this condition.

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Figure 7. Failure Envelopes - Free Flowing Bulk Solids The foregoing cases indicate that for failure and, hence, flow to occur, the shear stress versus norm al stress state within the bulk solid near the boundary m ust lie above the failure envelope. 2.4 Example Consider the case of a cohesive coal of bulk density ρ = 1 t/m which has a m easured adhesive stress of σ o = 1 k Pa for contact with m ild steel, a typical value. The coal is attached to the underside of m ild steel surface as illustrated in Figure 8. A vibrator is proposed as a m eans of rem oving the coal. Assum ing a failure condition as depicted by Figure 5, the stable build up, denoted by the h b , is given by

Figure 8. Adhesion Problem σo

hb =

(2)

a ) g

ρ g (1+

where a = (2 π f) Xf = am plitude of applied acceleration W ithout vibration (that is a = 0), h b = 0.1 m = 100 m m , a substantial am ount. To reduce h b to say 10m m , an acceleration a = 9.2 g is required. There are m any com binations of frequency and am plitude to achieve this. For instance, a frequency of f = 151 Hz and am plitude of X = 0.1m m would suffice. This ex am ple indicates the difficulty of overcom ing adhesion problem s. 3. FEEDING OR LOA DING CONVEYOR BELTS Figure 9 illustrates the application of a gravity feed chute to direct the discharge from a belt or apron feeder to a conveyor belt. The bulk solid is assum ed to fall vertically through a height 'h' before m ak ing contact with the curved section of the feed chute. Since, norm ally, the belt or apron speed vf ≤ 0.5 m /s, the velocity of im pact vi with the curved section of the feed chute will be, essentially, in the vertical direction. As a com m ent, the alternative to the use of an accelerating chute is to em ploy a short accelerating conveyor. These are high m aintenance devices and still require head room . Feed chutes m ay be regarded as the better proposition. 3.1 Free Fall of Bulk Solid For the free fall section, the velocity vi m ay be estim ated from __________ vi =

√ vfo + 2 g h

(3)

Equation (3) neglects air resistance, which in the case of a chute, is lik ely to be sm all. If air resistance is tak en into account, the relationship between height of drop and velocity Vi (Figure 9) is, v∞

1-

vfo

vi - vo

v∞

h = ----- log e [ -------- ] - ( ------- ) v∞ vi g 1g v∞

(4)

where v∞ = term inal velocity vfo = vertical com ponent of velocity of bulk solid discharging from feeder Vi = velocity corresponding to drop height 'h' at point of im pact with chute.

Figure 9. Feed Chute Configuration 3.2 Flow of Bulk Solid around Curved Chute of Constant Radius The case of 'fast' flow around curved chutes is depicted by the chute flow m odel of Figure 10. The relevant details are The drag force F D is due to Coulom b friction, that is FD = E N

(5)

where E = equivalent friction which tak es into account the friction coefficient between the bulk solid and the chute surface, the stream cross-section and the internal shear of the bulk solid. E is approx im ated by E

= [ 1 + Kv H/B ]

(6)

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= actual friction coefficient for bulk solid in contact with chute surface Kv = pressure ratio. Norm ally K~ = 0.4 to 0.8. H = depth of flowing stream at a particular location B = width of chute

For continuity of flow, ρ A v = Constant

(7)

where ρ = bulk density A = cross-sectional area of flowing stream It follows, therefore, that equation (6) can also be written as

E

=[1+

C1

]

(8)

v

Figure 10. Chute Flow Model For a chute of rectangular cross-section C1 =

Kv vo H o B

(9)

where vo = initial velocity at entry to chute H o = initial stream thick ness Analysing the dynam ic equilibrium conditions of Figure 10 leads to the following differential equation: dv g R + Ev = (cos θ - E sin θ) dθ v

(10)

If the curved section of the chute is of constant radius R and ~E is assum ed constant at an average value for the stream , it m ay be shown that the solution of equation (10) leads to the equation below for the velocity at any location θ.

v=

√

__________________________________________ 2g R [( 1 - 2 E) sin θ + 3 E cos θ ] + K e -2Eθ 4E+1

(11)

For v = vo at θ = θ o, K = { vo -

2g R 4E+1

[( 1 - 2 E) sin θ o + 3 E cos θ o ]} e -2Eθo

(12)

Special Case: W hen θ o = 0 and v = vo, K=vo -

6Eg R 1+4E

(13)

Equation (11) becom es,

v=

√

_____________________________________________________ 6ER g 2g R [( 1 - 2 E) sin θ + 3 E cos θ ] + K e -2Eθ [vi 4E+1 4E+1

(14)

4 TRA NSFER CHUTES The foregoing discussion has focused on curved chutes of concave upward form in which contact between the bulk solid and the chute surface is always assured by gravity plus centrifugal inertia forces. In the case of conveyor transfers, it is com m on to em ploy chutes of m ultiple geom etrical sections in which the zone of first contact and flow is an inverted curve. This is illustrated in Figure 11 in which the use of curved im pact plates is em ployed in a conveyor transfer. The lining is divided into two zones, one for the im pact region under low im pact angles, and the other for the stream lined flow. The concept of rem ovable im pact plates, used in conjunction with spares allows ready m aintenance of the liners to be carried out without interrupting the production.

Figure 11. Transfer Chute Showing Im pact Plates 4.1 Inverted Curved Chute Sections The m ethod outlined in Section 3.2 for curved chutes m ay be readily adapted to inverted curved chute section as illustrated in Figure 12. Noting that F D = E N , it m ay be shown that the differential equation is given by -

dv + v = g R (cos θ + sin θ) E E dθ v

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For a constant radius and assum ing E is constant at an average value for the stream , the solution of equation (15) is

v=

√

_____________________________________________ 2g R [ sin θ (2 E - 1) + 3 E cos θ ] + K e 2Eθ 4E+1

(14)

For v = vo at θ = θ o, then 2g R [ 3 cos θ o + (2 E - 1) sin θ o ]} e -2Eθo 1+4E

K = {vo -

(17)

Figure 12. Inverted Curved Chute Model Special Case: v = vo at θ o = π/2 2g R [ 2 E - 1]} e -Eπ 1+4E

K = {vo -

(18)

and

v=

√

__________________________________________________________________ 2 R g (2 E + 1) 2g R [ sinθ (2 E - 1) + 3 E cosθ ] + e -E(π - 2θ) [vo ] 4E+1 4E+1

(19)

Equations (16) to (19) apply during positive contact, that is, when v ≥ sinθ R g

(20)

Figure 13. Minim um Velocities for Im pact Chute Contact The m inim um bulk solid velocities for chute contact as a function of contact angle for three curve radii are presented in Figure 13. 4.2 Convex Chute Sections O n som e occasions, it m ay be desirable to incorporate a convex curve as illustrated in Figure 14 in order reduce the adhesion effects and assist the discharge process.

Figure 14. Convex Curved Chute Section For F D = E N ,it m ay be shown that the differential equation is given by dv - v = g R (cosθ - sinθ) E E dθ v

(21)

This holds for v sinθ ≥ R g ·

(22)

It is noted that Figure 13 also applies in this case with the vertical ax is now representing the m ax im um value of the velocity for chute contact. For a constant radius and assum ing E is constant at an average value for the stream , the solution of equation (21) is

v=

√

_____________________________________________ 2g R [ (1 + 2 E)sin θ - E cos θ ] + K e 2Eθ 4E+1

(23)

For v = vo at θ = θ o, then K = {vo -

2g R [( 1 + 2 E) sin θ o - E cos θ o ]} e -2Eθo 1+4E

(24)

Special Case: v = vo at θ o = π/2 K = {vo -

2g R 1+4E

[1 + 2 E]} e -Eπ

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and

v=

√

________________________________________________________ 2g R 2R g [(1 + 2 E) sinθ + E cosθ ] + e E(2θ - π) [vo 4E+1 4E+1

]

(26)

5. WEA R IN CHUTES Chute wear is a com bination of abrasive and im pact wear. Abrasive wear m ay be analysed by considering the m echanics of chute flow as will be now described. 5.1 A brasive Wear Factor of Chutes In cases where the bulk solid m oves as a continuous stream under 'fast' flow conditions the abrasive or rubbing wear m ay be determ ined as follows:

Figure 15. Chute Flow Model (a) Wear of Chute Bottom Consider the general case of a curved chute as shown in Figure 15, the chute being of rectangular cross-section. An abrasive wear factor W c ex pressing the rate of rubbing against the chute bottom has been derived as follows: Wc =

Q m g Kc tan NWR B

(27)

W c has units of N/m s NWR is the non-dim ensional abrasive wear num ber and is given by, NWR =

v + sin θ R g

(28)

The various param eters are = chute friction angle B = chute width (m ) Kc = ratio vs/v vs = velocity of sliding against chute surface Q m = throughput k g/s R = radius of curvature of the chute (m ) v = average velocity at section considered (m /s) θ = chute slope angle m easured from the vertical The factor Kc < 1. For 'fast' or accelerated thin stream flow, Kc ~/= 0.6. As the stream thick ness increases Kc, will reduce. Two particular chute geom etries are of practical interest, straight inclined chutes and constant radius curved chutes. (i) Straight Inclined chutes In this case R = ∞ and equation (27) reduces to Wc =

Q m Kc tan g sinθ B

(29)

O n the assum ption that Kc is nom inally constant, then the wear is constant along the chute and independent of the velocity variation. (ii) Constant Radius Curved Chutes In this case R is constant and the wear W c is given by equations (27) and (28).

(a) Velocity Variation

(b) Abrasive W ear Figure 16. Velocities and W ear in Chutes of Constant Curvature Q =30 t/h; vo = 0.2m /s; ρ = 1 t/m ; b=0.5m ; E = 0.6;. = 30 The velocity variation around a constant radius curved chute is given by equations (11-14). By way of ex am ple, Figure 16(a) shows the variation of velocity, and Figure 16(b) the corresponding abrasive wear num ber as functions of angular position for constant curvature chutes of radii 1m , 2 m , 3 m and 4 m . It is interesting to observe that as R increases, the increase in NWR becom es progressively sm aller. In Figure 16(a), the lim iting cut-off

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angle for the chutes to be self cleaning is indicated. (b) Chute Side Walls It is to be noted that the wear plotted in Figure 16 applies to the chute bottom surface. For the side walls, the wear will be m uch less, varying from zero at the stream surface to a m ax im um at the chute bottom . Assum ing the side wall pressure to increase linearly from zero at the stream surface to a m ax im um value at the bottom , then the average wear on the side walls can be estim ated from W csw =

W c Kv

(30)

2 Kc

Kc and Kv are as previously defined. If, for ex am ple, Kc = 0.8 and Kv = 0.4, then the average side wall wear is 25% of the chute bottom surface wear. 5.2 Impact Wear in Chutes Im pact wear m ay occur at points of entry or points of sudden change in direction. For ductile m aterials, greatest wear is caused when im pingem ent angles are low, that is in the order of 15 to 30. For hard brittle m aterials, greatest im pact dam age occurs at steep im pingem ent angles, that is angles in the vicinity of 90. 6. WEA R OF BELT A T FEED POINT An im portant application of feed and transfer chutes is to direct the flow of bulk solids onto belt conveyors. The problem is illustrated in Figure 17.

Figure 17 Feeding a Conveyor Belt The prim ary objectives are to m atch the horizontal com ponent of the ex it velocity vex as close as possible to the belt speed reduce the vertical com ponent of the ex it velocity vey so that abrasive wear due to im pact m ay be k ept within acceptable lim its load the belt centrally so that the load is evenly distributed in order to avoid belt m istrack ing ensure stream lined flow without spillage or block ages 6.1 A brasive Wear Parameter An abrasive wear param eter ex pressing the rate of wear for the belt m ay be established as follows: Im pact pressure p vi = ρ vey

(k Pa)

(31)

where ρ = bulk density, t/m ; vey = vertical com ponent of the ex it velocity, m /s Abrasive wear param eter W a = b ρ vey (vb - vex)

(k Pa m /s)

(32)

W here b = friction coefficient between the bulk solid and conveyor belt; vb = belt speed The wear will be distributed over the acceleration length L a. Equation (32) m ay be also ex pressed as W a = b ρ ve Kb

(33)

where Kb = cosθ e (vb/ve - sinθ e)

(34)

θ e = chute slope angle with respect to vertical at ex it Kb is a non-dim ensional wear param eter. It is plotted in Figure 18 for a range of ve/vb values. As indicated, the wear is quite severe at low chute angles but reduces significantly as the angle θ e increases.

Figure 18. Non-Dim ensional W ear Param eter versus Slope Angle θ For the chute to be self cleaning, the slope angle of the chute at ex it m ust be greater than the angle of repose of the bulk solid on the chute surface. It is recom m end that α ≥ tan -1 ( E) + 5 o

(35)

6.2 A cceleration Length The acceleration length L a over which slip occurs is given by La =

vb - vey 2 g mb

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7. WEA R MEA SUREMENT The abrasive wear of chute lining and conveyor belt sam ples m ay be determ ined using the wear test apparatus illustrated in Figure 19.

Figure 19. W ear Test Apparatus As illustrated, the rig incorporates a surge bin to contain the bulk m aterial, which feeds onto a belt conveyor. The belt delivers a continuous supply of the bulk m aterial at a required velocity to the sam ple of m aterial to be tested, which is held in position by a retaining brack et secured to load cells that m onitor the shear load. The bulk m aterial is drawn under the sam ple to a depth of several m illim etres by the wedge action of the inclined belt. The required norm al load is applied by weights on top of the sam ple holding brack et. The bulk m aterial is cycled back to the surge bin via a buck et elevator and chute. The apparatus is left to run for ex tended periods interrupted at intervals to allow m easurem ent of the test sam ple's weight and surface roughness if required. The m easured weight loss is then converted to loss in thick ness using the relationship given in equation (37). Thick ness Loss =

M.10 mm Aρ

(37)

where M = Mass loss (g) A = Contact Surface Area (m ) ρ = Test Sam ple Density (k g/m )

Figure 20. W ear Test Results Tests have been conducted on sam ples of solid woven PVC conveyor belt using black coal as the abrading agent. A typical test result for a norm al pressure of 2 k Pa and a velocity of 0.285 m /s is given in Figure 20. The graph indicates a wear rate of approx im ately 1.3 m /hour. This inform ation m ay be used to estim ate the wear ex pected to tak e place due to loading of coal on this type of conveyor belt. 8. CONVEYOR BELT DISCHA RGE CHA RA CTERISTICS 8.1 General Discussion Figure 21 shows the transition of a conveyor belt which m ay cause som e initial lift of the bulk solid prior to discharge. The bulk solid will also have the tendency to spread laterally as the belt troughing angle decreases through the transition. The am ount of spreading is m ore pronounced for free flowing bulk solids than for cohesive bulk solids. The spreading is also m ore pronounced at lower belt speeds. O nce the bulk solid on the belt reaches the drum , a velocity profile m ay develop as illustrated in Figure 22. As a result of the velocity profile there will be a spread in the discharge trajectories

(a) Conveyor Discharge

(b) Section X at Idler Set

(c) Section Y at Discharge Drum

Figure 21. Conveyor Belt Transition Geom etry and Load Profiles

Figure 22 Belt Conveyor Transition There will also be a variation in the adhesive stress across the depth of the stream . In m ost cases, however, the segregation that occurs during the conveying process will result in the m oisture and fines m igrating to the belt surface to form a thin boundary layer at the surface. This layer will ex hibit higher adhesive stresses than will occur for the rem ainder of the discharging bulk solids stream . The m agnitude of the adhesive stresses at the interface of the boundary layer and the belt surface will determ ine the ex tent of the carry-back on the belt 8.2 Profile of Bulk Solid on Belt The conveyor throughput is given by Q m = ρ A vb

(38)

Referring to Figure 21(b), the cross-sectional are at Section X is given by A=U b

(39)

where U = non-dim ensional cross-sectional area factor b = contact perim eter Assum ing a parabolic surcharge profile, for a three-roll idler set, the cross-sectional area factor is given by U =

1 r tanλ {r sinβ + sin2β + [1 + 4 r cosβ + 2 r (1 + cos2β)]} (1 + 2 r) 2 6

where r = C/B

(40)

β = troughing angle λ = surcharge angle

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8.3 Height of Bulk Solid on Belt at Idlers

a. O verall Height H (Figure 21(b)) H = C sin β + (B + 2 C cos β)

tan λ 4

(41)

b. Mean height h a h a = C sin β + (B + 2 C cos β) tan λ 6

(42)

8.4 Cross-Sectional Profile at Drive Drum The profile shape at the drive drum , Figure 21(c), is difficult to determ ine precisely. It depends on the conveyor speed, cohesive strength of the bulk solid and the troughing configuration. The m ean height h m ay be assum ed to be h =

A (B + C)

(43)

where A = cross-sectional area determ ined from equation (39). 8.5 A ngle at which Discharge Commences In order that the conditions governing discharge m ay be considered, the m odel of Figure 23 is considered. In general, slip m ay occur before lift-off tak es place. Hence, the acceleration ·/v and inertia force Δm ·/v are included in the m odel, v being the relative velocity. However, it 15 unlik ely that slip will be significant so it m ay be neglected.

Figure 23 Conveyor Discharge Model For an arbitrary radius r, the condition for discharge will com m ence when the norm al force N becom es zero. F v = g cos θ + A r Δm

(44)

where Δ m = ρ ΔA (h - Δr) = m ass of elem ent σ o = adhesive stress

F A = σ oΔA = adhesive force ρ = bulk density

8.6 Minimum Belt Speed for Discharge at First Point of Drum Contact In m ost cases, the speed of the conveyor is such that discharge will com m ence as soon as the belt m ak es contact with the discharge drum . In this case θ = - α, where α = slope of the belt at contact point with the drum . The critical case will be for the belt surface, that is, when Δr = 0. The m inim um belt speed for discharge at the first point of drum contact is

vb =

√

________________ σo R g (cos α + ) ρ g h

(45)

Figure 24 Belt Speed for Discharge at First Point of Drum Contact R=0.5m , ρ=1 t/m , α = 0, σ o = 1k Pa. Figure 24 illustrates the application of equation (45). The m inim um belt velocity for discharge to occur at the first point of belt contact is plotted against bulk solids layer thick ness 'h'. The graph applies to the case when α = 0 and σ o = 1 k Pa. The need for higher belt speeds to achieve lift-off as the layer thick ness decreases is highlighted. This indicates the difficulty of rem oving the thin layer of cohesive bulk solid that becom es the carryback that is required to be rem oved by belt cleaners. 8.7 Discharge Trajectories In m ost cases the influence of air drag is negligible. Hence the equations of m otion sim plify. The equation of the path is y = x tan θ +

1 g 2

x v cos θ

(46)

The bounds for the trajectories m ay be determ ined for the two radii (R + h) and R for which the angle θ is obtained from equation (45). 9. CURVED IMPA CT PLA TES Section 4.1 presented an analysis of flow around curved im pact plates. Referring to Figure 22, the radius of curvature of the discharge trajectory is given by [1 + ( Rc =

g x vb cosθ

) ] 1.5

---------------------g

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(47)

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vb cosθ For contact to be m ade with a curved im pact plate of constant radius, the radius of curvature of the trajectory at the point of contact m ust be such that, Rc ≥ R

(48)

where R c = chute radius Example Consider the case of a conveyor discharge in which vb = 4 m /s, θ = 0 The radius of curvature R c as a function of horizontal distance 'x ' is shown in Figure 25.

Figure 25. Radius of Curvature of Path Vb = 4 m /s; θ = 0 The curved im pact plate m ay be positioned so that the chute radius m atches the radius of curvature at the point of contact. This is illustrated in Figure 26.

Figure 26. Im pact Plate and Trajectory Geom etry 10. CHUTES OF THREE DIMENSIONA L GEOMETRY Although this paper has concentrated on chutes of two dim ensional geom etry, the concepts presented m ay be readily ex tended to the three dim ensional case. Using the lum ped param eter m odel approach, the equations of m otion m at be ex pressed in the m ost convenient co-ordinate system relevant to the particular chute profiles. The ex am ple of a transfer chute for the receiving conveyor at 90 to the delivering conveyor is sum m arised. Problem Specification: Bulk Material - Baux ite Bulk density as loaded r = 1.3 t/m throughput Q m = 2500 t/h Belt speed, delivery belt vb = 5 m /s Belt speed, receiving belt, vb = 5 m /s; Surcharge angle of baux ite on belt λ = 25 Conveyor inclination α = 10 Effective drive drum diam eter = 1.2 m Idler inclination angle β = 35 Receiving belt at right angle to delivery belt

Figure 27. Transfer Chute Ex am ple Referring to Figure 26 and 27, the design param eters are: Impact Chute: R c1 = 2.8m ; Contact angle θ c = 74.2; vc = 5.12 m /s; x c = 0.238m ; yc = 1.96m ; vd = 6.75m /s Feed Chute: R c2 = 3.0m ; ve = 5.57 m /s at cut off angle θ = 55; vex = 4.57 m /s; vey = 3.12m /s; W ear W a = 2.26 k Pa m /s 11. CONCLUDING REMA RKS An overview of chute design with special reference to belt conveying operations has been presented. Particular attention has been directed at the need to m easure the relevant flow properties of the bulk solid and to integrate these properties into the chute design process. Chute flow patterns have been described and the application of chute flow dynam ics to the determ ination of the m ost appropriate chute profiles to achieve optim um flow has been illustrated. 12. REFERENCES

1. CHARLTO N, W. H., CHIARELLA, C. and RO BERTS, A. W., "Gravity Flow of Granular Materials in Chutes: O ptim ising Flow Properties". Jnl. Agric. Engng. Res., Vol. 20, 1975, pp. 39-45.

2. CHARLTO N, W. H and RO BERTS, A. W., "Chute Profile for Max im um Ex it Velocity in Gravity Flow of Granular Materials". Jnl. Agric. Engng. Res., Vol. 15, 1970.

3. CHARLTO N, W. H. and RO BERTS, A. W., "Gravity Flow of Granular Materials: Analysis of Particle Transit Tim e". Paper No. 72, MH-33, A.S.M.E., (presented at 2nd Sym posium of Storage and Flow of Solids, Chicago, U.S.A., Septem ber 1972)

4. CHIARELLA, C. and CHARLTO N, W. H., "Chute Profile for Minim um Transit Tim e in the Gravity Flow of Granular Materials". Jnl. Agric. Engng. Res., Bol. 17, 1972.

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5. CHIARELLA, C., CHARLTO N, W. H. and RO BERTS, A. W., "O ptim um Chute Profiles in Gravity Flow of Granular Materials: A Discrete Segm ent Solution Method": Paper No. 73, MH-A, A.S.M.E., 1972.

6. CHIARELLA, C., CHARLTO N, W. H. and RO BERTS, A. W., "Gravity Flow of Granular Materials: Chute Profiles for Minim um Transit Tim e". Paper Presented at Sym posium on 'Solids and Slurry Flow of and Handling in Chem ical Process Industries', AIChE, 77th National Meeting, June 1974, Pittsburg, U.S.A.

7. Chute Design, First International Conference, Bionic Research Institute, Johannesburg, South Africa, 1991. 8. "Chute Design Problem s and Causes", Sem inar, Bionic Research Institute, Johannesburg, South Africa, February 1992. 9. LO NIE, K.W., "The Design of Conveyor Transfer Chutes". Paper Reprints, 3rd Int Conf on Bulk Materials Storage, Handling and Transportation, Newcastle, NSW , Australia, June 1989, pp 240-244.

10. MARCUS, R.D., BALLER, W.J. and BARTHEL, P. "The Design and O peration of the Weber Chute". Bulk Solids Handling, Vol. 16, No.3, July/Septem ber 1996. (pp.405-409).

11. RO BERTS, A. W., "The Dynam ics of Granular Materials Flow through Curved Chutes". Mechanical and Chem ical Engineering Transactions, Institution of Engineers, Australia, Vol. MC3, No. 2, Novem ber 1967.

12. RO BERTS, A. W., "An Investigation of the Gravity Flow of Non-cohesive Granular Materials through Discharge Chutes". Transactions of A.S.M.E., Jnl. of Eng. in Industry, Vol. 91, Series B, No. 2, May 1969

13. RO BERTS, A.W ., "Bulk Solids Handling : Recent Developm ents and Future Directions". Bulk Solids Handling, 11(1), 1991, pp 17-35. 14. RO BERTS, A.W., O O MS, M. and W ICHE, S.J., "Concepts of Boundary Friction, Adhesion and Wear in Bulk Solids Handling O perations". Bulk Solids Handling, 10(2), 1990, pp 189-198.

15. RO BERTS, A.W., SCO TT, O .J. and PARBERY, R.D., "Gravity Flow of Bulk Solids through Transfer Chutes of Variable Profile and Cross-Sectional Geom etry". Proc of Powder Technology Conference, Publ by Hem isphere Publ Corp, W ashington, DC, 1984, pp 241-248.

16. RO BERTS, A.W. and CHARLTO N, W.H., "Applications of Pseudo-Random Test Signals and Cross-Correlation to the Identification of Bulk Handling Plant Dynam ic Characteristics". Transactions of A.S.M.E., Jnl. of Engng. for Industry, Vol. 95, Series B, No. 1, February 1973.

17. RO BERTS, A. W., CHIARELLA, C. and CHARLTO N, W. H., "O ptim isation and Identification of Flow of Bulk Granular Solids". Proceedings IFAC Sym posium on Autom atic Control in Mining, Mineral and Metal Processing, Inst. of Engrs., Aust., Sydney, 1973.

18. RO BERTS, A. W. and ARNO LD, P. C., "Discharge Chute Design for Free Flowing Granular Materials". Transactions of A.S.A.E., Vol. 14, No. 2, 1971.

19. RO BERTS, A. W. and SCO TT, 0. J., "Flow of Bulk Solids through Transfer Chutes of Variable Geom etry and Profile". Proceedings of Powder Europa 80 Conference, W iesbaden, W est Germ any, January 1980.

20. RO BERTS, A. W. and MO NTAGNER, G. J., "Identification of Transient Flow Characteristics of Granular Solids in a Hopper Discharge Chute System ". Paper presented at Sym posium on Solids and Slurry Flow and Handling in Chem ical Process Industries, AIChE, 77th National Meeting, June 2-5, 1974, Pittsburg, Pa., U.S.A.

21. RO BERTS, A. W . and MO NTAGNER, G. J., "Flow in a Hopper Discharge Chute System ". Chem . Eng. Prog., Vol. 71, No.2, February 1975. 22. RO BERTS, A. W., SCO TT, O . ,J. and PARBERY, R. D., "Gravity Flow of Bulk Solids through Transfer Chutes of Variable Profile and CrossSectional Geom etry". Proceedings of International Sym posium on Powder Technology, Kyoto, Japan, Septem ber 1981.

23. RO BERTS, A. W. and SCO TT, O . J., "Flow of Bulk Solids through Transfer Chutes of Variable Geom etry and Profile". Bulk Solids Handling, Vol. 1, No. 4, Decem ber 1981, pp. 715.

24. RO BERTS, A.W . "Basic Principles of Bulk Solids Storage, Flow and Handling", TUNRA Bulk Solids, The University of Newcastle, 1998. 25. RO BERTS, A.W. and W ICHE, S.J. "Interrelation Between Feed Chute Geom etry and Conveyor Belt Wear". Bulk Solids Handling, Vol. 19, NO .1 January, March 1999

26. RO BERTS, A.W. "Mechanics of Buck et Elevator Discharge During the Final Run-O ut Phase". Lnl. of Powder and Bulk Solids Technology, Vol. 12, No.2, 1988. (pp.19-26)

27. SAVAGE, S. B., "Gravity Flow of Cohesionless Granular Materials in Chutes and Channels". J. Fluid Mech. (1979), Vol. 92, Part 1, pp. 53-96. 28. SAVAGE, S.B., 'The Mechanics of Rapid Granular Flows". Adv Appl Mech, 24, 1984, pp 289. 29. SCO TT, O .J., "Conveyor Transfer Chute Design, in Modern Concepts in Belt Conveying and Handling Bulk Solids". 1992 Edition. The Institute for Bulk Materials Handling Research, University of Newcastle, 1992, pp 11.11-11.13.

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