Determination of rolling resistance of belt conveyors using rubber data01/07/2014 Determination of rolling resistance of belt conveyors using rubber data: fact or fiction? Prof.dr.ir. Gabriel Lodewijks Delft University of Technology SUMMA RY This paper discusses the nature and im portance of the indentation rolling resistance for m odern belt conveyors. It provides a m athem atical m odel to calculate it and describes rheological tests that can be done to m easure the viscoelastic properties of the conveyor belt's cover m aterial. Finally, it provides an answer to the question: determ ination of rolling resistance of belt conveyors using rubber data: fact or fiction? 1 INTRODUCTION A belt conveyor is a m echanical conveyor frequently and worldwide used to continuously transport a certain m aterial or people from a place A to a place B at a capacity C. W hen ordering a belt conveyor, a client norm ally is concerned about issues lik e perform ance (can we m ove C from A to B?), reliability, m ax im um wear rates, total cost of ownership, com plex ity of the system etc. During large projects the client norm ally provides specifications but does not specify specific types or sizes of com ponents, although m ost m ajor clients have a preferred supplier list. Assum ing that the perform ance, reliability, m ax im um wear rates etc. are guaranteed by the belt conveyor supplier, they can select the actual com ponent types and sizes. To reduce the investm ent and operating costs of a belt-conveyor system it is im portant to determ ine and analyse the influences of the plant param eters and the operating param eters on the energy consum ption. In term s of the indentation rolling resistance this im plies that the dependence of this resistance on the roll radius, idler spacing, belt speed and radius of curvature should be k nown. It is also im portant to k now the influence of the belt m aterial and belt structure on the indentation rolling resistance and therefore on the energy consum ption of the belt. O ne of the m ost im portant com ponents of a belt conveyor is the conveyor belt itself. The conveyor belt can m ak e up till about 70% of the costs of a conveyor and the rolling resistance associated with the rubber (the indentation rolling resistance) can account for about 50% of the total rolling resistance [1]. The selection procedure of the conveyor belt should therefore be tak en seriously. It is well k nown that using standardised design m ethods lik e DIN 22101 or CEMA to calculate the power consum ption of a belt conveyor generally leads to an overestim ation of the power consum ption and thus of the belt tensions. O ne reason is that these design m ethods fail to tak e the viscoelastic or m echanic/dynam ic rubber com pound properties into account. They can therefore not distinguish between the power characteristics of a belt m ade off one rubber com pound or the other. Since the late fifties of the last century quite a few researchers work ed on m odels that can be used to predict that part of the rolling resistance that stem s from the rubber com pound: the indentation rolling resistance. The use of these m odels provided insight into the nature of this resistance [2]. W ith this insight m odels have been developed that enable a link between the m echanic/dynam ic properties of rubber com pound and the later system s power consum ption [3]. Although unk nown power consum ption m ay seem only a m atter of costs, it also seriously affects the conveyors perform ance. Knowledge of rubber com pound properties is therefore im portant because it partly determ ines the size and settings of com ponents lik e m otors and brak e system s. For ex am ple, the application of a low loss rubber com pound on a belt of a long overland system is a good way to reduce the overall operating costs. In case of an incline belt conveyor however, the ex tra costs of belting are not worth the effort since m ost of the power is used to raise the m aterial. The total power consum ption is therefore not noticeably decreased by the use of a low loss rubber. O n a decline belt conveyor the application of a low loss rubber m ay be a bad idea since it m ay increase the size and com plex ity of the brak e system . [ 1] Gabriel Lodewijks is full professor of Transport Engineering and Logistics in the faculty of Mechanical Engineering and Marine Technology of Delft University of Technology, the Netherlands, he is the chairman of the department of Marine and Transport Engineering in the same faculty, and president of Conveyor Experts B.V. 2 RECENT SOUTH A FRICA N PROJECTS In the last three years, three South African projects involving long overland belt conveyors have been realised: 1. CRU-II, Middelburg for Ingwe, 2. O ptim um , Hendrina for Ingwe, and 3. Savm ore, Piet Relief for the Kanga Group. During all three projects the quality, in particular of the rubber covers, and the supplier of the conveyor belting were serious issues for discussion. The nex t three paragraphs ex plain the specific m atters. 2.1 CRU-II. After adjudicating tenders from several top-rank ing world contenders, Middelburg Mine Services awarded the contract for a 14,5 k m overland conveyor system to BATEMAN. The project was ex ecuted by Batem an Engineered Technologies. The conveyor system is part of phase II of the R480M Coal Resources Utilisation Project (CRU II) initiated by Ingwe Coal Corporation Lim ited and was com m issioned in May 2000. In the tendering stage of the project, the belt conveyor system was presented, and later sold, as a high tech system utilising low indentation loss com pound for the belt's covers. The biggest advantage of using a low loss rubber for the belt was a serious decrease in ex pected power consum ption of the total system . The designs of the individual belt conveyors then were based on using belts with low loss covers. The anticipated supplier for the CRU-II belting was Bridgestone. During the course of the project however, the client requested that they could use alternative (read non low-loss rubber) belts as a replacem ent belt. The m ain reason for this was that Ingwe wanted to have a better position to negotiate for replacem ent belting. The design was therefore slightly altered, in particular the settings of m ajor com ponents as the drives and the brak e system s, to enable the application of alternative belting. After com pletion of the system O ptim um used a Dunlop SA belt to replace part of the original Bridgestone belting without any serious problem . 2.2 OPTIMUM. Ingwe has awarded BATEMAN a turnk ey contract for a 21 k m overland-conveyor system to be supplied to O ptim um Colliery. It includes all design, supply and erection, inclusive of civil work s. The system will com prise five belt conveyors ranging in length from 2.7 k m up to 61 k m . The O ptim um project, see Figure 1, k new a short-track design phase. The design of the system was in principle a further developm ent of the system designed for CRU-II, including the application of conveyor belting with low loss rubber com pounds. However, because of the short-track developm ent there was not enough steel cord belting available on the world-m ark et at the tim e. As a result, the client had to buy a m ix ture of Dunlop and Bridgestone belting. O ne of the m ain design principles of the O ptim um overland belt conveyor system was standardisation of com ponents. Therefore, all belt conveyors in principle should allow for the use of either Dunlop or Bridgestone belting. Because the problem with shortage of belt supply was k now at a relatively early stage of the project, m ost com ponents of the belt conveyors could still standardised but were tuned for the specific belting used on individual conveyors. The author has been involved in all three projects as main design engineer for the overall design of the involved belt conveyors. The CRU-II project was done in 1998 through Conveyor Dynamics, Inc. (Bellingham, WA, USA), the Optimum and Savmore projects in 2000 and 2001 through the author's current company Conveyor Experts B.V. (Emmer-Compascuum, the Netherlands). Figure 1: Belt Conveyor KW -05 of the O ptim um overland system . http://saimh.co.za/beltcon/beltcon12/paper1205.htm 1/7 htm (6) 2/7 .(1/3trσ)I is the deviatoric stress tensor and γ d = γ . there was only one long overland belt conveyor in the Savm ore project and Savm ore decided to buy the belt directly from Goodyear and provided it to Batem an as a free issue.) τj j=l (3) This section is based on Chapter 3 of [3]. It can be written as: ∞ t ψ(t) = ψ ∞ + ∫ g(τ) ex p(. were unk nown during the com m issioning stage. 3 VISCOELA STICIT In this section a m odel will be presented that can be used to represent the viscoelastic behaviour of the m aterial of a conveyor belt's cover. As a result the perform ance of that specific conveyor belt. The m ost im portant environm ental param eters that affect the dynam ic response of visco-elastic m aterials are tem perature. has awarded BATEMAN a contract for a 6. The conveyor will link Savm ore's new Maquassa West shaft with the ex isting plant at Maquassa East and will carry 1 000 t/h of run-of-m ine coal. The Savm ore project was developed at the sam e tim e as O ptim um and therefore the Kanga Group had the sam e belt shortage problem as Ingwe.5 k m overland conveyor. or a so called standard linear solid m odel. and thus the conveyor system . This m aterial m odel is k nown as the generalised Max well m odel. Figure 3: Generalised uniax ial Max well m odel. The fourth order tensor function ψ(t) is called the relax ation function and specifies the stress response to a unit strain increm ent. which fits for that range. [2]. Although the design of the Savm ore belt conveyor was based on the assum ption that it should be able to utilise basically any m odern conveyor belt. In such a case a three param eter m odel. If this range is relatively sm all for a specific application then it is sufficient to use one relax ation tim e. see Figure 4.) dτ τ 0 (2) where g(τ) is the relax ation spectrum which can be discrete or continuous and τ the relax ation tim e. It is also im portant to k now the ex act com pound of the m aterial. In this m odel a num ber of dam ping coefficients η j is used which are related to specific relax ation tim es τj.τj) is used then the relax ation function is equal to: N t ψ(t) = ψ ∞ + Σ g j ex p(. The relax ation function of the three param eter m odel is: ψ(t) = E1 + E2 ex p (- t ) τ (4) where the relax ation tim e τ = η/E2. Most belt covers are m ade of rubber or polyester m aterial. Figure 3 shows this uniax ial case.(1/3trγ )I the deviatoric strain tensor. For a three param eter Max well m odel the storage m odulus is: E' (ω) = E1E2 + ωη (E1 + E2) E2 + ωη (5) where ω is the circular frequency of deform ation. The loss factor tanδ is in this case defined by: E" tanδ = E' = ωηE2 E1E2 + ωη (E1 + E2) http://saimh. Figure 4: Three param eter Max well m odel (standard linear solid m odel).3 SA VMORE Kangra Group (Pty) Ltd's Savm ore Colliery. If in the uni-ax ial case a pulse-spectrum g(τ) = N/Σ/j=l gjδ(τ . near Piet Relief in the Mpum ulanga Province of South Africa.co.za/beltcon/beltcon12/paper1205. 2. In rubber for ex am ple the am ount of carbon black influences the m aterial properties considerably [5]. The constitutive behaviour of these m aterials is visco-elastic as can be learned from the tim e-dependency of the stress-strain relations.t') ∂t' -∞ (1) in which σ d = σ . in order to be able to represent the constitutive behaviour of a m aterial for a wide range of loading frequencies. results that is the sim plest m odel that can describe the relax ation of a m aterial and situations of constant stress or high strain rates. The constitutive equation for a isotropic linear visco-elastic m aterial can be written in general tensor form [6]: t ∂yd(t') dt' σ d (t) = ∫ ψ(t . frequency and the am plitude of an im posed load [4]. The dynam ic/m echanic properties of the Goodyear belt however were not k nown and Goodyear was not able or unwilling to supply either rubber m echanic/dynam ic properties or a sam ple of the specific rubber used. the dynam ic/m echanic properties of the specific conveyor belt were still required to optim ise the system by tuning the com ponents. However.Determination of rolling resistance of belt conveyors using rubber data 01/07/2014 Figure 2: The Savm ore overland belt conveyor. Dynam ic tests can be m ade using free oscillations at the resonance frequency of the test m aterial (for ex am ple. an oscillatory strain is applied to a sam ple. 4. and thus the loss m odulus through a m ax im um . A typical test holds two of these constant while system atically varying the third. Figure 7: The loss-factor as a function of tem perature and deform ation rate [2]. In a dynam ic test. and viscoelastic m aterials. and the resulting stress is m easured. an oscillating strain (sinusoidal or other waveform ) is applied to a sam ple and the resulting stress developed in the sam ple is m easured. The choice for a specific test m ode is determ ined by the required inform ation and the nature and geom etry of the sam ple. The elastic stress is a m easure of the degree to which the m aterial behaves as an elastic solid. and tim e/cure are the basic test m odes. G' and loss m oduli. and 7 as a function of tem perature and deform ation rate. A steady test uses continuous rotation to apply the strain and provide a constant shear rate The resultant stress is then m easured when the sam ple reaches a steady state. By em ploying predict m aterial behavior outside the range of conventional 4. others slow.za/beltcon/beltcon12/paper1205. The ratio of the elastic stress to strain is the storage (or elastic) m odulus E'. 6. (Newton's law). to m aterial properties through The ratio of the viscous stress and G" designate the storage 4. These changes are not instantaneous. or stress. of the three param eter Max well m odel can then be written in term s of the loss factor: η i (δ) = 2 π tanδ 2 + (π + 2δ)tanδ (7) From ex perim ents it can be learned that a real rubber cannot be m odelled with one relax ation tim e. 4 This section is based on [7]. tests m ust be perform ed in the sam e tim e scale as the phenom enon under study. and test tem perature in a dynam ic test. Figure 5: The storage-m odulus as a function of tem perature and deform ation rate [2]. the stress signal is out of phase with the strain signal. segm ental. The viscous stress is a m easure of the degree to which the m aterial behaves as an ideal fluid. Figure 8 By separating the stress into these com ponents. Forced frequency rheom eters control oscillation frequency. then the stress is proportional to the strain rate. This relax ation tim e m ust be chosen in agreem ent with the tim e it tak es for a m aterial point of the belt cover to pass the contact zone between belt and roll. to strain is the loss (or viscous) m odulus E". som e are quick . As can be seen in Figure 7. Strain sweeps. respectively. the torsion pendulum ). frequency sweeps. be m easured sim ultaneously. and transient. If the sam ple is a fluid and it behaves ideally. The storage m odulus. viscous. http://saimh. a sweep being a continuous variation of the param eter in operator. and a viscous stress T" that is in phase with the strain rate (90 out of phase with the strain). No single test m ode can span the total tim e range. The seconds is the the Boltzm ann rheom eters. In this case. The elastic and viscous stresses are related the ratio of stress to strain. are described that can be used to gather inform ation on the viscoelastic properties of rubber com pounds.1 m /s to 10 m /s. the resulting stress is proportional to the strain am plitude. and the stress and strain signals are in phase. data can be obtained to m easurem ent in the desired logarithm ic tim e scale can be the m aterial tim e and laboratory tim e (the tim e scale t in to steady data through the Cox -Merz relation. tem perature sweeps.htm 3/7 .1 RHEOLOGICA L TEST MODES A ND METHODS The response of a viscoelastic m aterial to m echanical deform ation involves a series of m olecular.co.selected steps.Determination of rolling resistance of belt conveyors using rubber data 01/07/2014 The dam ping factor η j. In a transient test. W hen testing is done in shear rather than in tension or com pression. tim e sweeps. 4 RHEOLOGICA L TESTING4 In this section rheological tests. To obtain accurate and useful data. the m aterial's dependence on strain am plitude and strain rate can Figure 8 shows the behavior of elastic. dynam ic. and dynam ic data can be directly related principle and tim e-tem perature superposition. strain rate. then it is sufficient to choose one relax ation tim e. The stress signal generated by a viscoelastic m aterial can be separated into two com ponents: an elastic stress T that is in phase with the strain. the m odulus. leading the strain signal by 90. if the differences in belt speed are not too large. the resilience of the SBR rubber of the belt cover passes through a m inim um . in fact the belt speed of conveyor belts varies from 0. The results are depicted in the Figures 5. From the data obtained. obtained from the rheological tests equivalent data for the other type k ey elem ent is that the dynam ic frequency of oscillation directly link s reciprocal of the frequency ou). or with a sinusoidal (or other waveform ) oscillation at a forced frequency chosen from a wide available range. The net effect is that the response of a viscoelastic m aterial to m echanical deform ation can spread over a wide and continuous tim e spectrum ranging from years to m icroseconds.2 DYNA MIC MECHA NICA L TEST In a dynam ic m echanical test.3 DYNA MIC TEST MODES. the response of a m aterial as a function of tim e is m easured after subjecting the m aterial to an instantaneous change in strain. oscillation am plitude. Figure 6: The loss-m odulus as a function of tem perature and deform ation rate [2]. However. and conform ational changes. the loss m odulus and the loss factor have been obtained from ex perim ents for a SBR rubber[2]. There are three conventional test m odes that can be used to obtain data: steady. in particular dynam ical m echanical tests. For solids that behave ideally and follow Hook e's law. m inim um viscosity. transitions occur at higher tem peratures. som e transitions shift different am ounts. If the m echanic/dynam ic properties of vulcanized rubber are to be determ ined then norm ally a linear test geom etry is used. In this paper only the indentation rolling resistance is considered since only that resistance is determ ined by the rubber com pound of the belt's covers 5. the tem perature of the secondary transition shifts m ore than does that of the glass transition. is used. perhaps. and the idler pitch and trough angle. sam ples are ex posed in an oven to an elevated tem perature. Also. For solids.za/beltcon/beltcon12/paper1205. is used to m easure the initial viscosity. For ex am ple. besides being used in studies of therm al transitions in solids.1 INTRODUCTION The rolling resistance accounts for the m ajor part of the resistances. This fact helps locate som e transitions in m ulticom ponent system s. the vertical force due to the weight of the belt and the bulk solid m aterial. Beyond this critical strain level. If a rotational test geom etry is used then the sam ple is sheared and G' and G" are determ ined. tension. tim e sweeps can be used for studying chem ical and therm al degradation of m aterials. frequency. tem perature. This results in an asym m etric stress distribution between the belt and the roll which yields a resultant resistance force. The three point bending geom etry is not often used because som e slippage can occur during testing at the k nife edges. the rheological properties of a viscoelastic m aterial are independent of strain up to a critical strain level. or therm al stability can be sensitively assessed in a tim e sweep by sim ply m easuring the m odulus or viscosity at a constant tem perature. the rotation inertia of the rolls of the idlers and the friction of the bearings of the idler rolls.4 TEST EQUIPMENT For perform ing m echanic/dynam ic tests on elastom ers as rubber rotational and linear test geom etries are available. 5 THE INDENTA TION ROLLING RESISTA NCE 5.co. and strain in a selected atm osphere over an ex tended period of tim e. 4. the indentation rolling resistance force. The glass transition is detected as a sudden and considerable decrease in the storage m odulus E' and an attendant peak in the tan δ curve. since k ey transitions are k nown to shift with oscillation frequency. Time sweep A m aterial's chem ical. the m ost sensitive m eans for m easuring the glass transition and other secondary transitions. k nowledge of which can identify softening points and useful tem perature ranges in solid m aterials. m aterial. and the radius of curvature of the belt. In addition. The best choice of test geom etry is the subject of an industry funded research project between the Transport Engineering and Logistics group of Delft University of Technology and the Rubber Technology Group of Twente University. Due to the visco-elastic properties of the cover m aterial the recovery of the com pressed part will tak e som e tim e. The two m ost frequently used geom etries are the dual cantilever and tension m ode as both enable pretension of the sam ple without the danger of slippage of the sam ple in the brack et. In general. Time/Cure The tim e/cure m ode. and optim um heating-rate of therm osets during curing. Frequency sweep In a frequency sweep. m easuring the strain am plitude dependence of the storage and loss m oduli is usually the first step tak en in characterizing viscoelastic behavior: A strain sweep will establish the ex tent of the m aterial's linearity. The strength of this resistance force depends on the constitutive behaviour of the cover m aterial. So. If a linear test geom etry is used then the sam ple is tensioned (or com pressed) and E' and E" are determ ined. as frequency changes. m echanical. Idler rolls are m ade of a relatively hard m aterial lik e steel or alum inium whereas the belt covers are m ade of m uch softer m aterial lik e rubber or PVC. Therefore the belt cover is indented by the roll due to the weight of the belt and the bulk m aterial when the belt m oves over a roll. approx im ate gel point. Temperature sweep Tem perature sweeps characterize the tem perature dependence of the m aterial's rheological param eters. In this paper the test geom etries are restricted to the linear geom etries. If a linear test geom etry is used then one of the following four geom etries can be chosen (also see Figure 9): three point bending dual cantilever tension com pression Figure 9: Linear test geom etries.Determination of rolling resistance of belt conveyors using rubber data 01/07/2014 Strain sweep Usually.htm 4/7 . in conventional therm al degradation studies. the radius of the idler roll. Norm ally conveyor belt m anufactures supply vulcanized sheets of rubber as sam ple m aterial for testing the m echanic/dynam ic properties. if one com ponent is m ore frequency-dependent than another. on the other hand. These transitions are characterized by m easuring the dynam ic m oduli and tan δ at a selected frequency in a tem perature sweep. Figure 10: Rheom etric RSA II Rheom eter. This test m ode is especially im portant in testing of solid sam ples. m easurem ents are m ade at different oscillation frequencies at a constant oscillation am plitude and tem perature. as the frequency is increased. the m echanic/dynam ic properties of unvulcanized rubber are to be determ ined then a circular test geom etry is used. For m any m aterials. Y c. http://saimh. In general the rolling resistance consists of the indentation rolling resistance. the behavior is non-linear and the m oduli decline. For the determ ination of the m echanic/dynam ic properties of rubber com pound of conveyor belts norm ally the tem perature sweep test m ode. lateral load. both located in the Netherlands. or som etim es the frequency sweep test m ode. the diam eter and m aterial of the idler rolls. inform ation vital in processing liquid m aterials. This test m ode provides. the belt param eters such as width. Any of the test geom etries shown in Figure 9 can be used in a rheom eter as for ex am ple the in Figure 10 shown Rheom etrics RSA-II that is widely used. The tem perature at which this transition occurs is called the glass transition tem perature T g. Com pression is ideal geom etry for elastom ers but its use is restricted by possible overload of the rheom eter. depending on their degree of frequency-dependence. the degree of crystallinity and other m orphological features can be ex am ined in this way. Param eters that determ ine the rolling resistance of the belt are the belt speed. If . The visco-elastic foundation of depth h. Using the sim ple W ink ler visco-elastic foundation m odel and the three param eter solid Max well m odel yields the stress distribution between roll and belt cover: E σ(x ) = a { 2Rh1 ( a -x a )( a -x a E k 1 (a. and thus the accuracy of the W ink ler approach. The indentation rolling resistance according to May is : http://saimh. see Figure 12. A visco-elastic m aterial relax es m ore slowly than it is com pressed so that the belt and the roll separate at a point (x =-b. the length a can be calculated from equation (9). The various sources of energy dissipation in rolling m ay be classified into those that arise through m icro-slip and friction. The ratio b/a can be calculated with equation (8) since σ(-b) = 0. In order to calculate the rolling friction. The inertia of the foundation m aterial is also neglected.1 +b f* a . those that are due to inelastic properties of the m aterial and those due to roughness of the (rolling) surfaces. A m ore convenient approach to determ ine the pressure distribution at any point of the contact area is to assum e that the belt covers can be m odelled by a sim ple W ink ler visco-elastic foundation m odel rather than by a visco-elastic layer. m om ents have to be tak en about the centre of the roll: a M = ∫ σ(x )x dx -b (10) The total distributed frictional force then follows from : Fi = 1 3 M = R E1a 4 b b [ 1 -2 ( a ) + ( a )4] + 8R2h E2a 4k Rh k b [ k .co. defined as used in DIN 22101. E2) (12) in which D is the diam eter of the roll. The indentation rolling resistance according to Hunter is : f ih* = 1 R V τ ( b .htm 5/7 . If the pressure distribution at any point of the contact area has to be calculated analytically then the solution of an integral equation for the pressure is required.za/beltcon/beltcon12/paper1205. rests on a rigid base and is com pressed by the rigid roller. the indentation rolling resistance factor.x ) ) a ) + Rh2 [ (1 + k )(1 . Figure 11: Idler rolling over incom pressible half space. h the effective belt cover thick ness and k =(VbT)/a. can be estim ated by com paring the results obtained by Hunter [8] and May [9]. x '=a').2 ROLLING CONTA CT OF LINEA R VISCO-ELA STIC BODIES The constitutive behaviour of the rubber belt cover m aterial can be m odelled by a three param eter Max well m odel as described in section 3. x '=-a') closer to the centre line (x = 0) than the point where they first m ak e contact (x =a.k )-( a -x a ) ]} (8) where E1 and E2 are constants from the three param eter Mawell m odel. In this section the rolling friction due to the inelastic properties of the belt cover m aterial is considered that form s the largest contribution.2 (1 + ( a )) + ( 1 + ( ba ) ) .k (a1 + k )( k + ba )e -1/k(a+b/a)] (11) Finally. The effect of the interaction of the springs. If the belt m oves at a constant speed then the distributed vertical force can be calculated by integrating equation (9): a F z = ∫ σ(x )dx -b (9) Since F z is constant for a stationary m oving belt and the ratio b/a is k nown from equation (8). There is no interaction between the springs of the foundation which im plies that shear between adjacent elem ents of the m odel is ignored. follows from : f im = Fi Fz = F 1/3zh 1/3 D 2/3 F RM (k . R the radius of the idler rolls. During rolling the m aterial lying in front of the contact zone between belt and roll is being com pressed whilst that at the rear is being relax ed.Γ1 ( a0 ) ) (13) where a and a 0 can be calculated from the boundary conditions. If the indentation depth is sm all com pared to the thick ness of the belt cover and it is assum ed that the carcass m aterial is undeform able then the visco-elastic W ink ler m odel can be applied to approx im ate the deform ation of the belt covers due to the indentation of the roll. In the figures the belt and the roll are depicted upside down which is done for sim plicity only.Determination of rolling resistance of belt conveyors using rubber data 01/07/2014 5. see Figure 11. 5 The analysis set forth in this chapter is based on Chapter 5 of [3].ex p(. The asym m etric contact-phenom enon and the resulting asym m etric stress distribution result in a resistance force. a b . Figure 12: W ink ler foundation m odel. E1. The solution that evolves from this approach is relatively com plicated and cannot be used directly when calculating the rolling resistance of belt conveyors. Determination of rolling resistance of belt conveyors using rubber data f ih* = f i* 01/07/2014 (14) f z* where the vertical (norm al) force and the indentation resistance force are: f z* = f i* = b a0 E1a 0 6Rh E1a 04 8Rh b b ( 2 . which enables com parison in term s of indentation rolling resistance force and/or factor. 7 CONCLUSIONS The analysis set forth in this paper can be sum m arised as follows. The specific equipm ent and the applicability of one test m ethod versus another however are not.za/beltcon/beltcon12/paper1205. Chapter 3 then introduced the concept of visco-elasticity that could be determ ined by perform ing m echanic/dynam ic tests described in Chapter 4. The repeatability.co. of the results of m echanic/dynam ic tests of rubber com pound is still under research. Today. As an ex am ple. As a result. The m ost im portant conclusion in that paper is that the deviation between theory and practice is around 5% (which is ex cellent). the deviation between the power consum ption of belt conveyors predicted by theoretical m odels and m easured in practice can be less than 5%. there are scientifically accepted m ethods to m easure the m echanic/dynam ic properties of rubber. It is therefore not yet possible to ex change calculation results obtained with one design m ethod and the other using the sam e set of m echanic/dynam ic param eters. are standardised. as described in Chapter 4. It is fiction that: the application of m echanic/dynam ic properties of rubber m easured at a specific rheom eter can be used in any design m odel yielding the sam e accurate prediction of the power consum ption of the system . The k ey question now is: how accurate is the prediction of the indentation rolling resistance of conveyor belting using the theory given in Chapter 5 (or another theory) and the test procedures given in Chapter 4? The only way to accurately m easure the power consum ption of a belt conveyor is to m easure torque in the shaft of the drive pulley. Statem ents that the deviation between theory and practice can be less than 5% are not based on scientific evidence. then the results of the m echanic/dynam ic tests can still differ. this m odel has been adopted by a num ber of institutes and com panies around the world. The procedures them selves. The total indentation resistance factor then is equal to: f i = f sf im (18) 6 DISCUSSION In Chapter 1 of this paper the im portance of the indentation rolling resistance of rubber conveyor belts was highlighted and its effect on three projects was illustrated in Chapter 2. the m easurem ent of power consum ption of a new belt conveyor should be done as soon as possible after installation to enable com parison between theory and practice The rubber properties change rapidly during the first half year after installation and therefore power m easurem ents before half a year of running do not give a representative im age of the power consum ption of the system . In other system s. m odels for the determ ination of the indentation rolling resistance. need to be tuned for a specific rheom eter. the perform ance of two rubber com pounds can be com pared to each other This com parison can only be done when it is based on tests perform ed on one specific rheom eter. and therefore design m ethods.htm 6/7 . the power requirem ents of a belt conveyor can be estim ated using com putational design tools (see Chapter 5) provided that they are tuned for the results of the m echanic/dynam ic rubber com pound tests perform ed on a specific test facility. a high loss com pound m ay be beneficial. the use of low loss rubber is irrelevant. and as stated before subject for further study in the Netherlands. This m eans that using the results of m echanic/dynam ic tests of a certain m achine as input param eters of a specific m odel for the indentation rolling resistance m ay yield substantial errors. the indentation rolling resistance is always the driving design param eter for long overland system s. Since 2000 an ex tensive research project has been initiated by the author to ex tend the use of the theory given in Chapter 5 to application in pouch conveyors and pipe conveyors [11] From com parison between the results of field m easurem ents and theoretical predictions it could be concluded that the deviation between theory and practice is between 5% for conventional belt conveyors and 15% for pipe conveyors if the effect of the repeatability of m echanic/dynam ic tests is k nown [12]. Chapter 5 presented a m odel to predict the indentation rolling resistance using m easured m echanic/dynam ic properties of the belt. lik e m ajor incline conveyors.k (1 + k )( k + Rh b a0 b (1 . lik e down hill system s. The results of the m echanic/dynam ic test then can be used as input param eters for the m odel presented in Chapter 5. O ne m ajor problem today is that there is still no standardised way to m easure the m echanic/dynam ic properties of a rubber com pound for application in conveyor belting. In addition it was found that the error m ade during the field tests is at least 5%.( a 0 ) + 3 ( a 0 )) + b b [ 1 . This procedure is described by Lodewijk s and Kruse in [10]. Even if at two independent laboratories the ex act sam e rubber is tested with two identical rheom eters.( a 0 ) ) (15) k [ k . Figure 13 shows the results of tests done at the laboratory of Transport Engineering and Logistics of Delft University of Technology and at another ex tern laboratory.2 (1 + ( )e -1/k(a +b/a )] 0 0 (16) The correction factor to tak e shear in the rubber into account then is: fs = f ih* (17) f im* which indicates the accuracy of the W ink ler m odel. This effect should http://saimh. It is a fact that: there are theoretical m odels to describe the visco-elastic behaviour of rubbers and that predict the power consum ption of belt conveyors. and equal im portant the ex changeability. As can be seen in that figure the deviation between the test results can be a factor two for low tem peratures! Figure 13: Com parison of the results of m echanic/dynam ic tests (scaled to 1 MPa). or 15% (theoretical overestim ation) in case of design calculations. In som e conveyor system s.2 ( a 0 ) + ( a 0 )4] + 1 b 2E2k a 0 Rh E2a 04k )) + 3 (1 + ( a 0 )) . TT. Conference on transportation of bulk solids m aterials.co. L. and Atack . reportnr. 1215-1232. Ph. 10.D. The Power Consum ption of Belt Conveyors.htm 7/7 . (1978). pp. Kruse. Com parison of theory and practice in power predictions of belt conveyors. Dusseldorf.L. D. Kunststoffen. Bulk Solids Handling 18. 2. 415-427..0145-9. May. brochure. (1961). Science and Technology of Rubber. Morris. van den (1989). Lodewijk s. O ctober 2000. The power of field m easurem ents . "The Rolling Contact of a Rigid Cylinder with a Viscoelastic Half Space". 8.A. S. "Rolling Friction of a Hard Cilinder over a Viscoelastic Material". London. 4.D. van (1969). Lodewijk s. G. Hunter. pp. 8 REFERENCES 1. Lodewijk s. (2002).za/beltcon/beltcon12/paper1205. pp. Eirich. G.C. 2002. Bulk Solids Handling 15. (1959). The rolling resistance of conveyor belts. W . (1997).F. Rubber Chem istry and Technology 21. R. Understanding Rheological Testing. pp. (1995). 1713-1724. Lodewijk s. ISBN 90-370. 3. Thesis TU Delft. (1998). 611-617. G. "Dynam ic-Mechanical Characteristics of Rubber Com pounds".5212. Lodewijk s.Determination of rolling resistance of belt conveyors using rubber data 01/07/2014 also be tak en into account when m easuring the m echanic/dynam ic properties of the rubber com pound. 66-74. S. E. http://saimh. J. 6. and Berg. Lecture book Twente University. Lodewijk s. 12. Struik . G. 7. G. G. G. Rheom etric Scientific (1997). Enschede. Dynam ics of Belt System s. 15-22. 9. (1996). pp.part I. Report Delft University of Technology. 11. Journal of Applied Mechanics 28. pp.C.J.J. de and Am erongen. Academ ic Press.Modern Closed Belt Conveyor System s. BULK 5 (2). 5. D.. Journal of Applied Physics 30.E. (2000). Mey.W .
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