Conveyor trajectory discharge angles

Transcription

1 University of Wollongong Researh Online Faulty of Engineering - Papers (Arhive) Faulty of Engineering and Information Sienes 007 Conveyor trajetory disharge angles 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, Conveyor trajetory disharge angles, International Conferene for Bulk Materials Storage, Handling and Transportation, 007, p -0, Newastle, Australia: Australian Soiety for Bulk Solids 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 Conveyor Trajetory Disharge Angles DB Hastie and PW Wypyh Centre for Bulk Solids and Partiulate Tehnologies Faulty of Engineering, University of Wollongong Northfields Avenue, Wollongong, NSW, 5 SUMMARY This paper presents a review of the various methods availale to predit the disharge angle of material as it leaves the head pulley of a onveyor elt. This disharge angle predition is a vital omponent in the overall determination of the disharge trajetory of material and as a result will ditate the design of a transfer hute to adequately feed downstream omponents in a system. The disharge angle will also ditate the loation of a stokpile if this is the desired appliation of the onveyor elt. Inluded in this paper are numerial omparisons of the methods presented to quantify similarities and differenes etween individual approahes y using a range of elt speeds and pulley diameters. Oservations made on the various methods will e disussed, inluding advantages, disadvantages and limitations. NOMENCLATURE a height to material entroid, m t elt thikness, m C onstant of integration, - D p head pulley diameter, m g gravity, m s - h material height, m h d height of material stream at disharge, m R aritrary radius, m R elt radius, m R radius to entre/entroid of material stream, m R h radius to outer produt stream, m R p head pulley radius, m veloity of lower stream, m s - veloity of upper stream, m s - r ritial veloity, m s - s tangential veloity, m s - α elt inlination angle, α d disharge angle, α d disharge angle for the lower trajetory, α d disharge angle for the upper trajetory, α r angle of wrap on head pulley, γ speifi gravity of material, kn m -3 μ oeffiient of frition, - μ k oeffiient of kineti frition, - μ s oeffiient of stati frition, - σ a adhesive stress, kn m - ψ angular parameter, INTRODUCTION Whether designing a transfer hute or prediting the loation of a stokpile, determining the disharge angle of a material stream from a onveyor head pulley is essential. Aurately prediting the disharge angle will aid in the design of a transfer hute, leading to flow-on effets for the determination of the material trajetory and interation with transfer hoods or impat plates. The horizontal loation of a stokpile of material from the onveyor head pulley will e influened y the disharge angle. Whether using equipment suh as a spreader for openast mining or a fixed onveyor, inaurate positioning of a stokpile might result in the ontamination of an existing stokpile with another produt, urying equipment or loking aess. There are numerous trajetory models availale in the literature whih determine the disharge angle of the material stream as it leaves the onveyor head pulley. The fous of this paper will e a review of the methods used y C.E.M.A. [,,3,4,5], M.H.E.A. [6], Booth [7], Golka [8,9], Korzen [0], Goodyear [] and Dunlop []. DISCHARGE ANGLE METHODS The seven methods reviewed present a variety of formulae to predit the material disharge angle from the head pulley of a onveyor elt. Some of the methods fous only on the horizontal onveyor geometry [] although inferene is made to oth inlined and delined geometries in the worked example provided. Some fous only on inlined onveyors [7,8,9,0] ut an easily e adapted for horizontal geometries y setting the elt inlination angle to zero, and then there are other methods whih address horizontal, inlined and delined onveyor geometries [,,3,4,5,6,]. Unless otherwise stated in the following setions, for high-speed elt onditions (defined later) material will disharge at the point of tangeny etween the elt and the head pulley.

3 . Horizontal and Inlined Conveyor Belts The methods of C.E.M.A. and M.H.E.A. [,,3,4,5,6] use the same set of equations to determine the disharge angles for the various geometries, with the only quantifiale differenes eing minor variations to the taulated values through susequent editions of the C.E.M.A. guide and the onversion from imperial to metri units for M.H.E.A., see Tale. For horizontal onveyors geometries there are two possile ases, that of lowspeed, determined y equation (), where the material will wrap around the head pulley to a determined disharge angle, equation (7), and that of high-speed, determined y equation (), where the material will disharge at the point of tangeny etween the elt and the head pulley. Inlined onveyor geometries present four possile ases. Low-speed onditions are determined ased on the elt inlination angle, as shown in equation (3), the disharge angle again determined from equation (7). As the elt inlination angle is now eing used, there is also a minor hange in the maximum tangential veloity required for the transition from low- to high-speed onditions and is explained further in setion 3. There is an additional ase where the material will disharge at the vertial entre line of the head pulley, equation (4). For high-speed elts, material will disharge at the point of tangeny of the elt and the head pulley in most irumstanes, equation (5), however there is a speial ase, as shown in equation (6) where the material may leave the elt at the tangeny point ut may again ome in ontat with the urved elt surfae, interfering with the trajetory path to follow. Of note with the C.E.M.A. and M.H.E.A. methods is that the disharge angle alulated is for the entroid (entre of gravity) of the material stream and the methods assume the disharge angle for the lower and upper surfaes of the material trajetory are idential. Tale Belt speed onditions for the C.E.M.A. [,,3,4,5] and M.H.E.A. [6] methods Horizontal Inlined s s Low-speed Medium High-speed < () N/A s () < osα (3) s = (4) s > (5) s osα < < (6) s osα d = (7) The oldest of the methods is provided y Booth [7] and uses an iterative approah to find the disharge angle. Booth s method only determines the disharge angle and susequent trajetory for the lower stream, that is for the material diretly in ontat with the elt. If a disharge angle for the upper trajetory path is to e assigned, an assumption must e made on how it should e determined. Based on C.E.M.A. and M.H.E.A. it appears reasonale to assume that the upper disharge angle is equal to the lower disharge angle. To determine whether low-speed or high-speed onditions apply, either equation (8) or (9) must hold true, see Tale. Tale Belt speed onditions for the Booth [7] method Low-speed s < (8) High-speed s (9) For low-speed onditions, an initial estimate of the disharge angle an e found using equation (0) followed y the angle at whih material slip first ours on the elt, α r, whih is determined y equation (). The analytial analysis y Booth produes equation () and y setting ( ψ ) = and ψ = αr the onstant of

4 integration, C, an e determined. One C has een otained, equations (0) and () an e solved simultaneously using ( ψ ) = d and ψ = αd to determine the disharge angle and also the disharge veloity, if required. osα d = (0) = osαr sin αr () μ ( ψ ) ( ) ( μ + ) μ osψ 3 μ sinψ = + Ce 4 Golka [8,9] determines the ritial veloity where the transition from low-speed to high-speed ours from equation (3), for oth the lower and upper trajetory paths whih are then ompared to the upper and lower stream veloities as shown in Tale 3. Seleting one of the three ases, the final values of the disharge angles are alulated y equations (4) and (5). μψ = osα (3) r () Tale 3 Disharge angle determination for the Golka [8,9] method Condition α d α d CASE and < r < r Use equation (4) Use equation (5) CASE > r Point of tangeny Point of tangeny CASE 3 and < r > r Use equation (4) Point of tangeny α = (4) os d p osα d = g Rp hd ( + ) (5) Where, 0.5 h = + and R p h d 0.5 h = R p + R p The method used y Golka [8,9] for the lower trajetory disharge angle is atually the same disharge angle alulated y Booth s [7] first approximation. The upper trajetory disharge angle is slightly different due to atual alulation of a disharge angle rather than the assumption that it is the same angle as for the lower disharge. Korzen s [0] method of determining the disharge angle is very similar in approah to that of Booth [7], however Korzen ases his analysis on the entral material stream and also uses the determined entral disharge angle for oth the lower and upper trajetory streams. Firstly, the angle at whih slip first ours, α r, is ψ = and ψ = α r as determined, equation (6), in order to evaluate the onstant of integration, C, using ( )

5 the ondition of equation (7). The disharge angle is determined y seleting ψ = α d. ( ψ ) = R g osψ and α μ μ r = tan s ± sin sin ( tan s) R g ( k ) ( + 6μk ) 4μ osψ 5μ sinψ σ a γ h k 4 k ( ψ ) = + Ce μ ψ (6) (7) Goodyear [] provides two ases for horizontal onveyor geometries whih are idential to those of the C.E.M.A. and M.H.E.A. equations () and (). For low-speed inlined onveyors equation (3) is again used, however for high-speed inlined onveyors equation (8) 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 for the lower or upper trajetories. The assumption has een made that all disharge angles are of equal value. s > osα (8) Dunlop [] uses a graphi approah to determine the disharge angles for low-speed onveyor elts. A series of head pulley diameters are plotted on a graph of elt speed versus disharge angle and y projeting the elt speed to the appropriate pulley diameter, the disharge angle is determined. If, however the elt speed does not interept the desired pulley diameter, Dunlop reommends high-speed onditions apply and material disharge is at the point of tangeny. There is one major limitation with this method, the maximum pulley diameter presented is 600mm whih does not allow for the determination of the angle of disharge for head pulleys of larger diameter.. Delined Conveyor Belts Only three methods, C.E.M.A. [,,3,4,5], M.H.E.A. [6] and Goodyear [] allow for the determination of the disharge angle for delined elt onveyors and while Dunlop 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 horizontal or inlined onveyor via a worked example. For this reason delined onveyors have not een inorporated into the omparisons, however will riefly e disussed. Up to the fifth edition of the C.E.M.A. guide [,,3,4] and M.H.E.A [6] the low-speed or high-speed onditions are determined from equation (8). High-speed disharge is again at the point of tangeny whereas for low-speed onditions equation (3) is used, followed y equation (7). The sixth edition of C.E.M.A. [5] 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 [] method of determining the disharge angle for delined onveyor elts uses equation (8) 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 early C.E.M.A. methods and M.H.E.A. 3 CRITICAL BELT SPEEDS The ritial elt speed, touhed on previously, refers to the point of transition from low-speed to highspeed onditions and Tale 4 summarises the five unique equations used. As previously explained, Golka [8,9] determines two distint disharge angles and following from this there are also two ritial elt speeds. The method y Korzen [0] inorporates an adhesive stress omponent, however when the adhesive stress equals zero, the equation is idential to that of Goodyear []. For the Dunlop method [], the ritial elt speed is graphially determined and the inferene provided y the worked examples is that the lower and upper disharge angles are idential. There are however limitations to it s determination due to the maximum pulley diameter depited graphially eing 600mm.

6 Tale 4 Critial elt speeds for the various methods C.E.M.A. and M.H.E.A. r, osα = (9) R Booth and Golka (lower), = osα (0) Golka (upper), r h = + osα () R r σ a Korzen r, = osα + γ h () Goodyear, = osα (3) r 4 COMPARATIE RESULTS An aritrary set of onditions was seleted from whih a range of omparisons were made for the seven disharge methods, see Tale, noting that not all parameters are used in eah method. A ommon elt inlination angle of zero has een used for all omparisons (i.e. horizontal geometry) as there was found to e no differene in the disharge angles for inlined onveyor geometries until the ritial elt speed was reahed as was explained in setion 3. Tale 5 Parameters used for omparisons Belt inlination angle, α 0 elt width, w 0.76 m elt thikness, t 0.0 m surharge angle 0 troughing angle 0 Coeffiient of frition [7], μ 0.5 Stati frition [0], μ s 0.5 Kinemati frition [0], μ k 0.4 Produt density, ρ 000 kg/m 3 Speifi gravity of ulk solids, γ 9.6 kn/m 3 Adhesion, σ a 0 kpa Centroid height [,,3,4], a m Centroid height [5], a m Centroid height [6], a 0.04 m Material height [,,3,4], h m Material height [5], h m Material height [6], h m 4. Effet of Belt Inlination Angle on Critial Belt Speed Following on from setion 3, a quantitative omparison of the ritial elt speeds was undertaken for the disharge angle methods and is presented in Figures to 4. These figures learly show the variation in elt speed as elt inlination angle inreases. Although the atual variation of elt speed for any given method and pulley diameter is quite small (i.e. 0. to 0.5 m s - ), it will have an effet on the speed at whih the transition from low to high-speed onditions ours. Whether the Dunlop [] method is ased on an existing trajetory method is unlear, however the lower stream Dunlop urves do not appear to follow the trend of any one method as is evident in Figures and 3, where the Dunlop urve follows the Golka L [8,9] / Booth [7] urve and then the C.E.M.A. [,,3,4,5] / M.H.E.A. [6] urve respetively. The upper Dunlop [] urve loosely follows the upper Golka [8,9] urve in Figures and ut no upper ritial elt speeds ould e determined for the 500mm diameter pulley due to the atual diameter of the outer surfae of the material eing 76mm whih is outside the range of the graph. No determination of the Dunlop ritial elt speed was possile for the 000mm diameter pulley diameter.

7 CEMA,MHEA GOLKA L,BOOTH GOLKA U KORZEN,GOODYEAR DUNLOP L DUNLOPU Critial Belt Speed (m/s) Critial Belt Speed (m/s) Belt InlinationAngle (deg.) Belt InlinationAngle (deg.) Figure Critial elt speed for D p = 0.5 m Figure Critial elt speed for D p =.0 m Critial Belt Speed (m/s) Critial Belt Speed (m/s) Belt InlinationAngle (deg.) Belt InlinationAngle (deg.) Figure 3 Critial elt speed for D p =.5 m Figure 4 Critial elt speed for D p =.0 m 4. Disharge Angles Applying the parameters of Tale 5 to the disharge angle methods, omparisons were made for four head pulley diameters, 0.5m,.0m,.5m and.0m and for elt veloities ranging from ms - to 3.5ms - in 0.5ms - inrements. Figures 5 to 0 displays a range of elt veloities (0.5ms - to.75ms - ) for a head pulley diameter of D p =0.5m and the following oservations have een made: a) Figure 5 shows two distint groupings of disharge angles with slight variations evident in oth groups; ) As the elt veloity inreases, there is a notieale spread in the disharge angles, moving away from the initial two groupings, see Figures 5 to 9; ) Figure 9 shows that some methods, C.E.M.A. [,,3,4,5], M.H.E.A. [6] and Golka (upper) [8,9] have already reahed high-speed onditions (i.e. disharge at the point of tangeny etween the elt and head pulley); d) Golka (upper) [8,9] reahes high-speed onditions ased on the alulated tangential veloity of the upper stream, whereas Golka (lower) [8,9] is still under low-speed onditions, refer to Tale 3; e) Figure 0 shows that all disharge angle methods are now under high-speed onditions ( =.75ms - ), referring to Figure for the ritial elt veloity where the transition from low-speed to high-speed onditions ours; f) Referring to the two disharge angles for Dunlop [] in Figure 9, there is an indiation that there is a high onvergene of the lower and upper paths ased on the assumption made that two distint disharge angles should e determined (as explained previously). There will in all likelihood e a rossing of the lower and upper trajetory streams whih in reality would not our. Of ourse, this situation only ours under low-speed onditions. For high-speed onditions (Figure 0), the streams are parallel.

8 CEMA,,4,5 BOOTH CEMA 6 GOLKA MHEA86 KORZEN Dimensions in metres DUNLOP GOODYEAR Figure 5 D p =0.5m, =ms - Figure 6 D p =0.5m, =0.75ms - Figure 7 D p =0.5m, =.00ms Figure 8 D p =0.5m, =.5ms - Figure 9 D p =0.5m, =.50ms - Figure 0 D p =0.5m, =.75ms - Further seleted graphial omparisons are presented in Figure to Figure 9 displaying other ominations of pulley diameter and elt speed. Additional oservations have een made from these omparisons: g) For a onstant elt veloity, the disharge angle for a given method inreases as the pulley diameter inreases, see Figures to 3 or Figures 4 to 6; h) As was previously stated, the upper disharge angle for the Dunlop [] method is displaying a highspeed ondition one a pulley diameter of.5m has een reahed due to limitations with the Dunlop graphial method; i) Enforing the statement made in () aove, as elt veloity inreases for a given pulley diameter, there is more spread in the disharge angles determined, see Figures and 5 and also Figures 3, 6 and 9; j) As the elt veloity / pulley diameter ratio dereases for a given elt veloity, the two groupings of disharge angles eome more defined, see Figures to 3 and Figures 4 to 6; k) As pulley diameter inreases there is a wider range of elt veloities availale to produe low-speed onditions.

9 .00 CEMA,,4,5 BOOTH CEMA 6 GOLKA MHEA86.50 KORZEN Dimensions in metres DUNLOP GOODYEAR Figure D p =.0m, =.00ms - Figure D p =.5m, =.00ms - Figure 3 D p =.0m, =.00ms Figure 4 D p =.0m, =.00ms - Figure 5 D p =.5m, =.00ms - Figure 6 D p =.0m, =.00ms Figure 7 D p =.0m, =3.00ms - Figure 8 D p =.5m, =3.00ms - Figure 9 D p =.0m, =3.00ms -

10 Of speial note is the Korzen [0] method. As shown in equation (6) and (3) there is an adhesion omponent present. In the ase of the omparisons presented aove, the adhesion stress has een set to zero to keep omparisons onsistent against other methods, see Tale 5. As an example of the effet of varying the adhesive stress a pulley diameter of D p =.0m was seleted and a range of adhesive stresses from 0 kpa to kpa was applied, see the results in Figure 0. It an learly e seen that as the adhesive stress is inreased for any given elt speed, the resulting disharge angle inreases. Also as the adhesive stress inreases there is also an inrease in the range of elt speeds efore the transition from low-speed to high-speed ondition ours. If the adhesive stress is inreased to.5 kpa for this set of omparisons, a disharge angle of 9.3 results for a elt speed of 0.5 ms - whih is oviously at an angle past the most horizontal point on the head pulley and would in atual fat result in material douling ak on itself. If the adhesive stress ontinues to e inreased no solution is possile ased on equation (6). 90 Pulley Diameter.0m 80 Disharge Angle (deg.) Adhesive stress = 0 kpa Adhesive stress = 0.5 kpa Adhesive stress =.0 kpa Adhesive stress =.5 kpa Adhesive stress =.0 kpa Belt Speed (m s - ) Figure 0 Effet of adhesive stress on disharge angle in the Korzen [0] method 5 CONCLUSION Seven different disharge angle methods have een presented utilising a wide range of formulae and in the ase of Dunlop [], a graphial approah. With suh a range of formulae it is inevitale that there are differenes etween the disharge angles produed y eah method, whih has learly een shown, yet they annot all e orret. It was found that when omparing horizontal and inlined onveyor geometries there is no differene to the disharge angle until nearing the ritial elt speed at whih point, as the elt inlination angle inreases, the elt speed at whih the transition from low-speed to high-speed onditions redues slightly. The methods presented y C.E.M.A. [,,3,4,5] and M.H.E.A. [6] are the same, varying only in the fat that one is ased on imperial units while the other used metri units respetively and also some minor adjustments to the taulated data in different editions of the C.E.M.A. guide. The Korzen [0] method inorporates adhesive stress into its determination of the disharge angle unlike any of the other methods. If a non-adhesive material is eing analysed then the result is the same as that for the Goodyear [] method ut if the material does exhiit adhesive harateristis, the result will e a larger disharge angle as the material stays in ontat with the elt longer. In some ases, some assumptions were required to allow for a diret omparison of all the methods presented. These assumptions were ased on whether the disharge angle determined for the lower trajetory stream ould e used for the upper stream also, suh as Booth [7], or whether a unique upper stream disharge angle needed to e alulated, suh as Golka [8,9]. In the ase of the Dunlop [] method the assumption that two unique disharge angles should e determined from the graphial approah was perhaps an inorret one as disussed and seen in Figure 9. Future work in the area of disharge angles will fous on atual measurement of these angles from the onveyor transfer researh faility eing developed urrently at the University of Wollongong.

11 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 REFERENCES. C.E.M.A. Belt Conveyors for Bulk Materials. st Ed, Conveyor Equipment Manufaturers Assoiation, C.E.M.A. Belt Conveyors for Bulk Materials. nd Ed, Conveyor Equipment Manufaturers Assoiation, C.E.M.A. Belt Conveyors for Bulk Materials. 4th Ed, Conveyor Equipment Manufaturers Assoiation, C.E.M.A. Belt Conveyors for Bulk Materials. 5th Ed, Conveyor Equipment Manufaturers Assoiation, C.E.M.A. Belt Conveyors for Bulk Materials. 6th Ed, Conveyor Equipment Manufaturers Assoiation, M.H.E.A. Reommended Pratie for Troughed Belt Conveyors, Mehanial Handling Engineer s Assoiation, Booth, E. P. O. "Trajetories from Conveyors - Method of Calulating Them Correted". Engineering and Mining Journal, ol. 35, No., Deemer, 934, pp Korzen, Z. "Mehanis of Belt Conveyor Disharge Proess as Affeted y Air Drag". Bulk Solids Handling, ol. 9, No. 3, August, 989, pp Golka, K. "Disharge Trajetories of Bulk Solids". 4th International Conferene on Bulk Materials Storage, Handling and Transportation, Wollongong, NSW, Australia, 6th - 8th July, 99, pp Golka, K. "Predition of the Disharge Trajetories of Bulk Materials". Bulk Solids Handling, ol. 3, No. 4, Novemer, 993, pp Goodyear Handook of Conveyor & Elevator Belting: Setion, Dunlop Industrial Conveyor Manual, 98.