The prediction of conveyor trajectories

Transcription

1 University of Wollongong Researh Online Faulty of Engineering - Papers (Arhive) Faulty of Engineering and Information Sienes 007 The predition of onveyor trajetories David B. Hastie University of Wollongong, dhastie@uow.edu.au Peter W. Wypyh University of Wollongong, peter_wypyh@uow.edu.au Puliation Details Hastie, DB & Wypyh, PW, The predition of onveyor trajetories, International Materials Handling Conferene: Belton 14, 007, p 1-16, South Afria: International Materials Handling. Researh Online is the open aess institutional repository for the University of Wollongong. For further information ontat the UOW Lirary: researh-pus@uow.edu.au

2 The Predition of Conveyor Trajetories DB Hastie and PW Wypyh Centre for Bulk Solids and Partiulate Tehnologies Faulty of Engineering, University of Wollongong Northfields Avenue, Wollongong, NSW, 5 1 INTRODUCTION From transportation of ulk materials in underground and open-ut mines to appliations in power stations and proessing plants to name ut a few, elt onveyors are widely used as an eonomial ontinuous onveying method. Most of these industries rely strongly on onveyor transfers to divert material from one onveyor to another at some point in their system. The unfortunate reality is that some ompanies will e driven y ost minimisation for the short-term rather than planning for the long-term. This will result in ad deisions eing made and the inorret onveyor transfer eing installed with flow-on effets suh as downtime for modifiations or even replaement. There are many faets to the design of a suessful transfer hute inluding minimising produt impat, degradation, hute wear, noise, dust and spillage while maximising the material veloity to allow the produt to leave the hute at or near the speed of the reeiving onveyor. Fully understanding the ehaviour of a material is paramount to designing a suessful transfer hute. Aurately determining the material disharge and trajetory from the head pulley is the first step in this design proess. This paper will fous on the determination of the material trajetory as it leaves the elt onveyor head pulley. This proess will also determine the point at whih the material leaves the elt, referred to as the disharge angle. There are numerous methods availale in the literature fousing on the modelling of material disharge and trajetory, inluding C.E.M.A. (1966; 1979; 1994; 1997; 005), M.H.E.A. (1977; 1986), Booth (1934), Golka (199; 1993), Korzen (1989), Goodyear (1975) and Dunlop (198). These methods have een evaluated for oth horizontal and inlined elts for a range of elt speeds and pulley diameters to evaluate oth low and high-speed onditions. The result of these evaluations have een ompared to allow omment on their ease of use, ompleteness and potential auray. TRAJECTORY METHODS EXAMINED.1 Disharge Angles Within eah of the methods presented, the disharge angle of material leaving the head pulley is determined analytially or graphially whih then ditates the starting position of the material trajetory. As will e evident in setion 3, it was found that there are numerous approahes to determine this disharge angle with some methods omparale to eah other while some eing vastly different. The resulting trajetories generated will vary signifiantly, also detailed in setion 3. Of ourse there should only e one disharge angle for a given material and set of onditions.. Horizontal and Inlined Conveyor Geometries The Conveyor Equipment Manufaturers Assoiation has released six editions of the C.E.M.A. guide, Belt Conveyors for Bulk Materials sine The first five editions follow the same proedure for determination of material trajetory with only slight adjustments to various values in referene tales. In the 6 th Edition of the C.E.M.A. guide (005), there is a hange to how the time interval is alulated for high-speed elts for determining the trajetory profile, now eing alulated ased on the elt speed rather than the tangential veloity. For all other onditions the time intervals are alulated as per the previous editions. The C.E.M.A. method addresses seven onveyor onditions; - low-speed elts where material wraps around the pulley efore disharge (horizontal, inlined and delined elts), 1

3 - high-speed elts where material disharges at the point of tangeny etween the elt and head pulley (horizontal, inlined and delined elts), and - inlined elts where material disharges at the top most vertial point of ontat etween the elt and the pulley (inlined elts). The equations presented in Tale 1 represent the onditions whih need to e satisfied to determine whih of the seven ases aove is to e used. If low-speed onditions exist, equation (7) is used to determine the disharge angle, α d. Horizontal Inlined gr s s gr Low-speed Medium High-speed < 1 (1) N/A s 1 gr () < osα (3) s 1 gr = (4) s 1 gr > (5) s osα < < 1 (6) gr Tale 1 Belt speed onditions for the C.E.M.A. (1966; 1979; 1994; 1997; 005) and M.H.E.A. (1986) methods s gr = osα (7) d The trajetory produed y the C.E.M.A. method is ased on the entroidal path. To plot the trajetory, a tangent line is drawn from the point at whih the material leaves the pulley. At regular time intervals along this tangent line, vertial lines are projeted down to fall distanes supplied in the C.E.M.A. guide. A urve is then drawn through these points to produe the entroid trajetory. The upper and lower trajetory limits an also e plotted y offsetting from the entroidal urve the distane to the elt and to the load height. It is evident from this proedure that a onstant width trajetory path results. C.E.M.A. notes that for light fluffy materials, a high elt speed will alter the upper and lower limits with oth vertial and lateral spread due to air resistane. The Mehanial Handling Engineer s Assoiation guide, Reommended Pratie for Troughed Belt Conveyors (M.H.E.A. 1977) addresses oth low-speed and high-speed elts via entripetal aeleration. The speed onditions of Tale 1 are used, this time using pulley diameter rather than material entroid, resulting in the determination of the lower trajetory. This method (M.H.E.A. 1977) also provides an approximation for the outer trajetory of the material y first determining the angle, α d, at whih the upper surfae of material starts its trajetory. The M.H.E.A. released a seond edition of the guide (M.H.E.A. 1986) whih etter explains the method of determining the material trajetory. On inspetion, it is idential to the C.E.M.A. method up to and inluding the 5 th Edition. However it uses metri rather than imperial units and the onversion results in minor differenes, ultimately ausing minor variations to the trajetory urves whih are determined exatly the same as for the C.E.M.A. method. Booth (1934) found that while using availale theory, a large disrepany was present etween the theory and that of the atual trajetory. After areful investigation and onfirmation of these errors, Booth onluded that, for one, the effets of the material slip were not eing addressed as material disharged over the head pulley. This led to an analytial analysis to develop a more representative theory.

4 Booth s method egins y determining the angle, α, at whih the partile will leave the head pulley, again ased on the low-speed and high-speed onditions desried y the C.E.M.A. method (1966; 1979; 1994; 1997; 005). For low-speed onditions, an initial estimate of the disharge angle is found using equation (8), followed y the angle at whih material slip first ours on the elt, α r, equation (9). The analytial analysis y Booth produes equation (10) and y setting ( ψ ) = and ψ = αr the onstant of integration, C, an e determined. One C has een defined, equations (8) and (10) are solved simultaneously using ( ψ ) = d and ψ = αd to determine the disharge angle and disharge veloity. osα d = (8) gr gr 1 = osαr sinα r (9) μ ( ψ ) ( ) = ( μ + ) μ 1 osψ 3 μ sinψ gr 4 1 μψ + (10) The trajetory is then plotted alulating a time interval from the tangent line and projeting down the freefall distanes provided y Booth. However there is no mention of how the upper trajetory is determined or how the material height at the disharge point is alulated. Booth aknowledged that this method was tedious and ompliated and as an alternative, developed a hart to minimise the time required to analyse a partiular elt onveyor geometry still with a reasonale auray. Golka s method (199; 1993) for determining material trajetory is ased on the Cartesian oordinate system and is for materials without ohesion or adhesion. For low-speed elts, this method one again follows that of C.E.M.A. in determining the disharge angle of the lower trajetory using equation (). It also alulates a separate disharge angle for the upper trajetory y sustituting the radius of the upper trajetory into equation (). The material height on the elt efore disharge is taken diretly from C.E.M.A. however an adjusted material height, h d, is also alulated for the point at whih the upper trajetory disharges from the pulley, equation (11). Ce h d 0.5 h = R p 1+ 1 (11) R p Two divergent oeffiients have een introdued y Golka (199; 1993), ε 1 for the lower and ε for the upper trajetory, whih takes into aount variales suh as air resistane, size distriution, permeaility and partile segregation. Unfortunately no explanation is given as to how they are determined. A set of urves has een produed using the parameters utilised in setion 3 and varying the divergent oeffiients, as an e seen in Figure 1. Not knowing with any ertainty what divergent oeffiients should e used for a given produt/ondition an result in a sustantial variation of the predited trajetory urves. 3

5 ε1 = 0, ε = 0 ε1 = 0.1, ε = 0.1 ε1 = 0., ε = 0. ε1 = 0.3, ε = 0.3 ε1 = 0.4, ε = DIMENSIONS IN MILLIMETERS Figure 1 ariation in trajetories ased on different divergent oeffiients Golka (199; 1993) uses three ases, Tale, to predit the material trajetory from the pulley depending on predefined onditions. The ritial veloity, r, needs to e determined from equation (1) where the radius represents either the lower or upper trajetory. This determines the veloity at whih the transition from low-speed to high-speed ours. 1 = and is determined from equation (13). Condition α d1 α d CASE 1 1 < r1 and < r Use equation (14) Use equation (15) CASE 1 > r 1 point of tangeny point of tangeny CASE 3 1 < r1 and > r Use equation (14) point of tangeny Tale Disharge angle determination for the Golka method (199; 1993) = grosα (1) r 0.5 h = (13) R p α = (14) os 1 d 1 gr p osα d = g Rp hd ( + ) (15) 4

6 The trajetory is then determined from a series of Cartesian oordinate ased equations for pre-seleted time steps. Golka (199; 1993) does not inlude elt thikness when determining the radii of the lower and upper stream. Material height is inluded, though ased on values taken from the early C.E.M.A. guide (1966; 1979). Of all the methods reviewed, Korzen (1989) is the most omplex in its approah, addressing the issues of adhesive materials, inertia, slip and air drag in its alulations. There is also a distintion etween stati frition, μ s, and kinemati frition, μ k, used in the determination of the disharge veloity, d, and disharge angle, α d. For high-speed elts, α d = α and d =. For low-speed elts the angle at whih material egins to slip on the elt efore disharge, α r, is determined from equation (16). To evaluate the onstant of integration, C, the onditions ( ψ ) = and ψ = α r are used in equation (17). To determine the disharge angle, the relationships ( ψ ) = R g osψ and ψ = α d are used. α μ μ r = tan s ± sin sin ( tan s) R g ( k ) ( 1+ 16μk ) k 4 k ( ψ ) = gr Ce μ ψ σ a γ h (16) 4μ 1 osψ 5μ sinψ + (17) The detailed numerial analysis developed y Korzen is ahieved y a series of suessive approximations whih inorporate orreted air drag oeffiients ased on partile shape and a proportionality fator for air drag. Using the X-displaement for any given position of the trajetory, the Y-displaement, trajetory angle and resultant veloity an e otained. The first approximation is for a free falling partile in a vauum, whih is used as the initial trajetory estimate, where air drag is negleted, for all other approximations air drag is applied. The analysis is ontinued until the differential error etween suessive approximations have deviations no greater than 1% or %. One the analysis has een ompleted for a suitale range of X-displaements, the X and Y oordinates are plotted to produe the trajetory for the entral path. The disharge angle and disharge veloity alulated aove are for the entre height of the material stream. Korzen (1989) also allows for alulation of the upper and lower trajetory limit disharge veloities. For high-speed elts the upper and lower disharge veloities are the same as that for the entre height trajetory ut for a low-speed elt, the disharge veloities are alulated ased on ratios of the radius of the lower and upper trajetories. For partiles over 1g in mass, the effets of air drag an e dismissed (Korzen 1989). This eing the ase, only the first approximation is performed and the trajetory is plotted from those values. This will also result in the trajetory urves having a loser math to those generated y other methods. Although the disharge veloity for the lower and upper trajetories an e determined, there is no method desried for the determination of the orresponding trajetories. The assumption has een made that the same approximation method is used, sustituting the alulated disharge veloity for the appropriate trajetory. Korzen (1989) does not inlude the elt thikness when determining the trajetory urves. Although as a perentage of the head pulley diameter the elt thikness is quite small, this omission will result in a minor vertial offset of the plotted trajetories ompared to methods whih inlude elt thikness. There also appears to e several errors in the worked examples presented y Korzen (1989). As a result of one group of errors, additional approximations were required to ahieve an adequate error level. Also, in several quoted equations there also 5

7 appears to e some typographial errors, whih an raise questions as to whether the method is eing applied orretly. The Handook of Conveyor and Elevator Belting (Goodyear 1975) is quite simplisti in its approah to determining material trajetory. Using the priniples of projetile motion, equation (18) and equation (19) are used to determine the X and Y oordinates of the trajetory. The disharge angle is determined for low-speed elt onditions y satisfying one of the ases in Tale 1, this time using the radius of the entral height of the material. X = t (18) gt Y = (19) Goodyear (1975) provides two ases for horizontal onveyor geometries whih are idential to those of equations (1) and (). For low-speed inlined onveyors equation (3) is again used, however for high-speed inlined onveyors equation (0) is used. The Goodyear method determines the disharge angle for the entral material stream, (i.e. h/), however there is no referene to the determination of the disharge angles or trajetories for the lower or upper trajetories. The assumption has een made that all disharge angles are of equal value, thus resulting in parallel trajetory streams. One the X and Y oordinates and the disharge angle have een determined, produing the trajetory urve is straight forward. gr s > osα (0) Belt thikness is not used in the Goodyear method (1975) and as this method determines only the entre trajetory path, a material height must e known. The worked examples y Goodyear (1975) inorporate half the material height into the radius used to determine the entral trajetory path without impliitly providing details of the origin of the material height. Goodyear states that the atual trajetory may e different to that of the one alulated due to other fores ating on the partile stream whih haven t een used in these alulations. The Dunlop Conveyor Manual (198) uses a graphial method to determine the material trajetory leaving the onveyor for low-speed elts and analytial method for high-speed elts. For high-speed elts, material again leaves the elt at the point where the elt is at a tangent to the pulley. To alulate the X oordinate, the distane travelled tangentially from the elt, equation (18) is used and the Y oordinate, the distane material falls elow the tangent line, is alulated using equation (19). Dunlop states that the mathematis ehind the trajetory for low-speed onveyors is omplex and so developed a graphial method. Knowing the elt speed and pulley diameter, it is a simple proess to determine the disharge angle and X, the inremental distane along the disharge tangent line. At eah of these inremental distanes, Y is projeted vertially down with the same values as for high-speed onveyors. If it is found that the desired elt speed does not interset with the required pulley diameter then the method for high-speed elts should e used. For the low-speed ondition, pulley diameters etween 31mm and 1600mm are presented. Outside of this range there is no way to estimate the disharge angle or X. The resulting trajetory is a predition of the lower oundary. There is referene to material depth in the worked examples in the Dunlop Conveyor Manual ut no explanation of how it is determined and there is an indiation that the flow pattern is onvergent..3 Delined Conveyor Belts Only three methods, C.E.M.A. (1966; 1979; 1994; 1997; 005), M.H.E.A. (1986) and Goodyear (1975) allow for the determination of the disharge angle for delined elt onveyors and while Dunlop (198) makes no speifi mention of delined onveyor elts in its guide, it alludes to a delined onveyor elt having the same disharge angle as a 6

8 horizontal or inlined onveyor via a worked example. For this reason delined onveyors have not een inorporated into the omparisons of setion 3, however will riefly e disussed. Up to and inluding the fifth edition of the C.E.M.A. guide (1966; 1979; 1994; 1997) and M.H.E.A (1986) the low-speed or high-speed onditions are determined from equation (1). High-speed disharge is again at the point of tangeny and for low-speed onditions equation (3) is used, followed y equation (7). gr s osα (1) The sixth edition of C.E.M.A. (005) has a variation to the low-speed ondition, now adding the disharge angle to the elt delination angle. However, in the worked example for this speifi ase, the disharge angle alulated is idential to the disharge angle found in the previous five editions of the C.E.M.A. guide whih have the disharge angle determined from the vertial. So this would in atual fat indiate that the graphial representation of this ase is inorret. The Goodyear (1975) method of determining the disharge angle for delined onveyor elts uses equation (0) for high-speed onditions resulting in disharge from the tangeny point and for low-speed uses equation (3) and equation (7) and is plotted from the vertial as with the earlier C.E.M.A. methods and M.H.E.A..4 Critial Belt Speed The ritial elt speed, touhed on previously, refers to the point of transition from low-speed to high-speed onditions and Tale 3 summarises the five unique equations used. As previously explained, Golka (199; 1993) determines two distint disharge angles and following from this there are also two ritial elt speeds. The method y Korzen (1989) inorporates an adhesive stress omponent, however when the adhesive stress equals zero, the equation is idential to that of Goodyear (1975). For the Dunlop method (198), the ritial elt speed is graphially determined for the lower and upper disharge angles. There are however limitations to it s determination due to the maximum pulley diameter harted eing 1600mm. C.E.M.A. and M.H.E.A. r, gr osα = () R Booth and Golka (low), = gr osα (3) Golka (high), r h = gr 1+ osα (4) R r σ a Korzen r, = gr osα + γ h (5) Goodyear, = gr osα (6) r Tale 3 Critial elt speeds for the trajetory methods 7

9 .5 Effet of Transition Geometry As the onveyor elt moves through the transition zone, stresses are generated at the elt edges and an ause permanent deformation if the elasti limit is surpassed. To minimise or even overome this, the head pulley an e raised to ½ the trough depth thus reduing the stresses on the elt (C.E.M.A. 005). Although this is stated y C.E.M.A. (005), there is no indiation whether this is applied in the determination of onveyor trajetories. Roerts, Wihe et al. (004) state that the transition angle, ε, does effet elt disharge trajetories and as an e seen in Figure, the point of tangeny of the elt to the head pulley eomes α + ε, where ε is determined from equation (7). 1 z ε = tan L (7) t For low-speed onditions, the angle θ 0 is determined, defining the angle that disharge ommenes. For high-speed onditions disharge is at the point of tangeny (i.e. θ 0 = 0º). Figure Transition Geometry (Roerts, Wihe et al. 004) The trajetory oordinates an then e determined using equation (8). X ( α ε θ ) = Y tan os gy ( α + ε θ ) 0 (8) The method presented y Roerts et al. (004) is the only approah whih diretly addresses the transition arrangement when onsidering the determination of the trajetory and even though a half trough transition is reommended y C.E.M.A. (005), there is no indiation it is applied to the trajetory method provided. 3 TRAJECTORY METHODS COMPARED To allow for a diret omparison etween all the methods desried, a onstant set of parameters was seleted as shown in Tale 4. The only variations allowed for are elt speed ( = 1.5, 3.0 and 6.0ms -1 ), pulley diameter (D p = 0.5, 1.0 and 1.5m) and elt inlination angle (0 30 ), all others are fixed. As adhesive stress is only aounted for y the Korzen (1989) method, it has een set to zero to allow omparison of all methods. Although not quantified here, if adhesive stress were to e applied, the result would e an inreased disharge angle whih would result in a trajetory stream falling loser to the head pulley ompared with the other methods. 8

10 PARAMETERS ALUES Belt Width, w 0.76 m Belt Thikness, 0.01 m Belt Speed, 1.5 ms -1 Pulley Diameter, D p 0.5 m Surharge Angle 0 Troughing Angle 0 Belt Inlination Angle, α 0 Divergent Coeffiient, Lower, ε Divergent Coeffiient, Upper, ε Coeffiient of Frition, Stati, μ s Coeffiient of Frition, Kinemati, μ k Adhesion Stress, σ a 0 kpa Equivalent Spherial Partile Diameter m Atmospheri Temperature, T atm 0 C Atmospheri Pressure, P atm 101 kpa Air isosity, η f 1.80E-05 Nsm - Produt Density, ρ 000 kgm -3 Tale 4 Conveyor parameters used for omparisons For the omparisons made, a numer of graphs have een seleted to represent the oserved differenes for oth low-speed and high-speed elt onditions, all for D p = 1.0m. Where trajetory urves for different models vary y no more than % for oth the lower and upper trajetories, at a vertial drop height of 4m, they have een grouped together. As mentioned in setion, some methods do not inlude elt thikness when determining the radius for the lower, entre or upper trajetory radii and as suh minor modifiations to those methods has een implemented y adding the elt thikness, allowing for diret omparisons to e ahieved. Of further note is the radius for the upper trajetory streams varies for some methods regardless of the inlusion of elt thikness due to the method used to determine the height of the material stream. 3.1 Disharge Angles A set of aritrary parameters has een used to ompare the disharge angles for the trajetory methods desried and a seleted group is shown in Figures 3 to 8. Using a onstant pulley diameter and varying the elt veloity only, the following oservations have een made; a) Figure 3 shows two distint groupings of disharge angles with slight variations evident in oth groups ut as the elt veloity inreases, there is a notieale spread, moving away from the initial two groupings, see Figures 3 to 7, ) Figure 7 shows that some methods, C.E.M.A. (1966; 1979; 1994; 1997; 005), M.H.E.A. (1986) and Golka (upper) (199; 1993) have already reahed high-speed onditions, ) Golka (upper) (199; 1993) reahes high-speed onditions ased on the alulated tangential veloity of the upper stream, whereas Golka (lower) (199; 1993) is still under low-speed onditions, refer to Tale 3, d) Figure 8 shows that all methods are now under high-speed onditions, noting that all ritial elt speeds from Tale 4 are elow the given elt speed, ( =1.75ms -1 ), 9

11 CEMA 1,,4,5 BOOTH CEMA 6 GOLKA MHEA86 KORZEN DIMENSIONS IN METRES DUNLOP GOODYEAR Figure 3 Figure 4 Figure 5 D p =0.5m, =0.50ms -1 D p =0.5m, =0.75ms -1 D p =0.5m, =1.00ms Figure 6 Figure 7 Figure 8 D p =0.5m, =1.5ms -1 D p =0.5m, =1.50ms -1 D p =0.5m, =1.75ms -1 e) referring to the two disharge angles for Dunlop (198) in Figure 7, there is an indiation that there is a high onvergene of the lower and upper paths ased on the authors assumption that two distint disharge angles should e determined. There will in all likelihood e a rossing of the streams whih in reality would not our. Low-speed onditions are ahievale over a wider range of elt speeds as the pulley diameter inreases, for example a pulley diameter of m will operate under low-speed onditions up to approximately 3.1ms Low-Speed Trajetories A omparison graph for the low-speed trajetory preditions is presented in Figure 9. Although having a larger pulley diameter than the example of Figure 3, Figure 9 learly shows two main groupings of methods for the produed trajetories. It is also lear that these two groupings have an approximate horizontal variation of 500mm at a drop height of 4000mm 10

12 whih will have a marked effet on the design of a transfer hute dependant on whih trajetory method is applied. Although the figures are not presented in this paper, a further trend found that as pulley diameter inreases for a given elt speed, the trajetories for the models fall loser to the head pulley. The methods of Booth (1934) and Dunlop (198) produe a near idential urve regardless of the fat that one uses a highly analytial approah and the other uses a straightforward graphial approah respetively CEMA1,,4,5,6,MHEA1986 BOOTH,DUNLOP GOODYEAR GOLKA,GOLKA (divergent) KORZEN KORZEN (air drag) DIMENSIONS IN MILLIMETERS Figure 9 Low-speed ondition, horizontal onveyor, lower and upper trajetory path Pulley diameter, D p = 1.0m, elt veloity, = 1.5ms High-Speed Trajetories In order to investigate the hange in trajetory profile due to elt speed, two high-speed elt onditions have een seleted, = 3.0 ms -1 and = 6.0 ms -1. In Figure 10 and Figure 11 the inlusion of air drag y Korzen (1989) results in a trajetory predition learly muh shallower than all other methods. If air drag is negleted, the resulting trajetory predition is loated amongst the other methods. In setion it was explained that Golka (199; 1993) uses divergent oeffiients to otain a etter approximation of the trajetory paths. Without explanation of how these divergent oeffiients have een determined, it is hard to justify using this method with any level of auray. If on the other hand the divergent oeffiients are negleted in the alulations, the resulting trajetories are idential to the Korzen (1989) method when air drag is negleted, as the equations are idential. In oth ases shown, the early C.E.M.A. (1966; 1979; 1994; 1997) and M.H.E.A. (1986) methods generate the highest trajetory urve and as the disharge veloity inreases, the variation from the other urves eomes more defined. As with the low-speed onditions, the Booth (1934) and Dunlop (198) methods again produe near idential trajetory urves. The inorporation of adhesive stress y Korzen (1989) has een previously explained ut it is lear in Figure 1 and Figure 11 that if adhesive 11

13 stress were inluded in the alulations, the resulting trajetory urves would deviate even further from the trajetory urves of the other methods CEMA1,,4,5,MHEA1986 BOOTH,DUNLOP CEMA6,GOODYEAR GOLKA,KORZEN GOLKA(divergent) KORZEN (air drag) DIMENSIONS IN MILLIMETERS Figure 10 High-speed ondition, horizontal onveyor, lower and upper trajetory path Pulley diameter, D p = 1.0m, elt veloity, = 3.0ms DIMENSIONS IN MILLIMETERS CEMA1,,4,5,MHEA1986 CEMA6,BOOTH,DUNLOP,GOODYEAR,GOLKA,KORZEN GOLKA(divergent) KORZEN (air drag) Figure 11 High-speed ondition, horizontal onveyor, lower and upper trajetory path Pulley diameter, D p = 1.0m, elt veloity, = 6.0ms -1 4 ALIDATION 4.1 Experimental A unique onveyor transfer researh faility, Figure 1, will soon e ommissioned at the University of Wollongong whih will allow oth trajetories and variale geometry transfer 1

14 hutes to e experimentally investigated. It is the intention to generate a sustantial dataase of trajetory information from this faility addressing as many of the parameters inorporated into the trajetory methods presented. This, it is hoped, will allow diret validation of the trajetory methods and ultimately determine whih of the methods is most aurate. Figure 1 Experimental onveyor transfer researh faility at University of Wollongong 4. Simulation The distint element method (DEM) is an ideal simulation tool for approahing this type of granular flow prolem. There are a range of a ommerial pakages availale whih will allow trajetory profiles to e simulated. This will produe another method of omparing the trajetory methods presented to the dataase of experimental results. It should e highlighted however, that there is proaly no option for seleting how a trajetory is determined, solely eing ased on the physis of the ommerial DEM odes (whih proaly do not allow for differenes in moisture, ohesion, et). The simulations may or may not orrespond to the trajetory methods presented and will e evaluated as required. 5 CONCLUSION This paper has presented a numer of the more widely used and/or readily availale trajetory predition methods pulished in the literature. They range in omplexity from the asi, Dunlop (198) and Goodyear (1975) to the omplex, Booth (1934), Golka (199; 1993) and Korzen (1989). Some methods inlude a multitude of parameters suh as C.E.M.A. (1966; 1979; 1994; 1997; 005) and M.H.E.A. (1986) while others inorporate additional parameters suh as divergent oeffiients (Golka 199; 1993) and air drag (Korzen 1989). For low-speed elt onditions there appear to e two distint groupings of trajetory preditions, see Figure 8, when the elt speed is relatively low. However as the elt speed inreases, so does the satter of disharge angles as seen in Figure 3 to Figure 8. 13

15 For high-speed elt onditions an oservation made was that some of the asi methods approximated a trajetory whih was also predited y the more omplex methods, see Figure 10 and Figure 11. Inorporating air drag into the trajetory preditions (Korzen 1989) auses a dramati variation from the other trajetory methods. When onsidering air drag, it makes sense that material will drop away muh quiker than the other methods ut whether it is truly representative of an atual trajetory needs to e explored further. It is fair to say that the differing approahes to determining the material trajetory result in some signifiant differenes whih annot all e orret. Based solely on the omparisons made in this paper, it is impossile to say with any ertainty whih of the methods will produe the most aurate predition of an atual trajetory. It may even e the ase that one method predits aurately the low-speed onditions whereas another is etter suited for high-speed onditions. Further researh is definitely needed and with the aid of a unique onveyor transfer researh faility and simulation via DEM at the University of Wollongong, this should provide a means to apply improvements to the availale trajetory methods 6 ACKNOWLEDGEMENTS The authors wish to aknowledge the support of the Australian Researh Counil, Rio Tinto OTX and Rio Tinto Iron Ore Expansion Projets for their finanial and in-kind ontriutions to the Linkage Projet whih allows this researh to e pursued. 7 NOMENCLATURE elt thikness m C onstant of integration - D p pulley diameter m g gravity ms - h material depth m h d material depth at disharge m Lt transition length m P atm atmospheri pressure kpa R aritrary radius m R radius to outer elt surfae m R radius of material entroid/entre m R p pulley radius m t inrement time for trajetory path s T atm atmospheri temperature C 1 disharge veloity of lower oundary ms -1 disharge veloity of upper oundary ms -1 elt veloity ms -1 r ritial veloity ms -1 d veloity of material at disharge point ms -1 s tangential veloity of material at disharge point ms -1 w elt width m z height of the transition m α initial material disharge angle measured from the vertial α elt inlination angle α d material disharge angle measured from the vertial trajetory α d1 material disharge angle measured from the vertial for lower trajetory α d material disharge angle measured from the vertial for upper trajetory α r angle at whih slip egins to our γ speifi gravity of ulk solid knm -3 ε transition angle ε 1 divergent oeffiient - ε divergent oeffiient - η f air visosity Nsm - θ 0 angle at whih disharge ommenes μ oeffiient of frition - 14

16 μ k oeffiient of kinemati frition - μ s oeffiient of stati frition - σ a adhesion stress kpa 8 REFERENCES Booth, E. P. O. (1934). "Trajetories from Conveyors - Method of Calulating Them Correted". Engineering and Mining Journal, ol. 135, No. 1, Deemer, pp C.E.M.A. (1966). Belt Conveyors for Bulk Materials. 1st Ed, Conveyor Equipment Manufaturers Assoiation. C.E.M.A. (1979). Belt Conveyors for Bulk Materials. nd Ed, Conveyor Equipment Manufaturers Assoiation. C.E.M.A. (1994). Belt Conveyors for Bulk Materials. 4th Ed, Conveyor Equipment Manufaturers Assoiation. C.E.M.A. (1997). Belt Conveyors for Bulk Materials. 5th Ed, Conveyor Equipment Manufaturers Assoiation. C.E.M.A. (005). Belt Conveyors for Bulk Materials. 6th Ed, Conveyor Equipment Manufaturers Assoiation. Dunlop (198). Dunlop Industrial Conveyor Manual. Golka, K. (199). "Disharge Trajetories of Bulk Solids". 4th International Conferene on Bulk Materials Storage, Handling and Transportation, Wollongong, NSW, Australia, 6th - 8th July, pp Golka, K. (1993). "Predition of the Disharge Trajetories of Bulk Materials". Bulk Solids Handling, ol. 13, No. 4, Novemer, pp Goodyear (1975). Goodyear Handook of Conveyor & Elevator Belting: Setion 11. Korzen, Z. (1989). "Mehanis of Belt Conveyor Disharge Proess as Affeted y Air Drag". Bulk Solids Handling, ol. 9, No. 3, August, pp M.H.E.A. (1977). Reommended Pratie for Troughed Belt Conveyors, Mehanial Handling Engineer s Assoiation. M.H.E.A. (1986). Reommended Pratie for Troughed Belt Conveyors, Mehanial Handling Engineer s Assoiation. Roerts, A. W., Wihe, S. J., Ili, D. D. and Plint, S. R. (004). "Flow Dynamis and Wear Considerations in Transfer Chute Design". 8th International Conferene on Bulk Materials Storage, Handling and Transportation, Wollongong, NSW, Australia, 5th - 8th July, Institution of Engineers, Australia, pp

17 9 AUTHORS C Mr. David Hastie B.E. (Mehanial), M.E. (Honours) has een employed with the Centre for Bulk Solids and Partiulate Tehnologies at the University of Wollongong, Australia sine He is a Memer of the Institution of Engineers Australia and a memer of the Australian Soiety for Bulk Solids Handling. In July of 005 he took up an Aademi Researh Fellow position within the Faulty of Engineering. Key areas of interest inlude pneumati onveying, rotary valve air leakage, partile segregation and onveyor transfers and trajetories and has extensive experiene in experimental investigations, instrumentation, data aquisition and analysis, omputer programming and digital video imaging and proessing. He was the main researher on the suessful IFPRI, In. funded projet, on the predition of transport oundaries in fluidised dense-phase and low-veloity pneumati onveying and urrently he is Researh Assoiate on a three year projet titled Quantifiation And Modelling Of Partile Flow Mehanisms In Conveyor Transfers. He is also a part-time PhD andidate working on transfer hute quantifiation and modelling. Peter Wypyh B.E. (Mehanial, Honours 1), PhD is the Diretor of the ARC endorsed Key Centre for Bulk Solids and Partiulate Tehnologies at the University of Wollongong. He has een involved with the researh and development of solids handling and proessing tehnology sine Peter Wypyh has pulished over 80 artiles. He is urrently Chair of the Australian Soiety for Bulk Solids Handling. Peter Wypyh is also the General Manager of Bulk Materials Engineering Australia and has ompleted over 450 industrial projets, involving R&D of new tehnologies, feasiility studies, trouleshooting, general/onept design, optimisation, deottleneking, safety/hazard audits and/or rationalisation of plants and proesses for ompanies all around Australia and in the USA, Hong Kong, New Zealand, China, Singapore and Korea. 10 AUTHOR CONTACTS Mr David Hastie Centre for Bulk Solids and Partiulate Tehnologies Faulty of Engineering University of Wollongong Northfields Avenue, Wollongong, NSW, 5, Australia dhastie@uow.edu.au A/Prof Peter Wypyh Centre for Bulk Solids and Partiulate Tehnologies Faulty of Engineering University of Wollongong Northfields Avenue, Wollongong, NSW, 5, Australia wypyh@uow.edu.au 16