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The Design of High Speed Belt Conveyors
G. Lodewijks, The Netherlands.
SUMMARY
This paper discusses aspects of high-speed belt conveyor design. The capacity of a belt conveyor is determined by the belt speed given a belt width and troughing angle. Belt speed selection however is limited by practical considerations, which are discussed in this paper. The belt speed also affects the performance of the conveyor belt, as for example its energy consumption and the stability of it's running behavior.
ENERGY CONSUMPTION
Clients may request a specification of the energy consumption of a conveyor system, for example quantified in terms of maximum kW-hr/ton/km, to transport the bulk solid material at the design specifications over the projected route. For long overland systems, the energy consumption is mainly determined by the work done to overcome the indentation rolling resistance . This is the resistance that the belt experiences due to the visco-elastic (time delayed) response of the rubber belt cover to the indentation of the idler roll. For in-plant belt conveyors, work done to overcome side resistances that occur mainly in the loading area also affects the energy consumption. Side resistances include the resistance due to friction on the side walls of the chute and resistance that occurs due to acceleration of the material at the loading point.
The required drive power of a belt conveyor is determined by the sum of the total frictional resistances and the total material lift. The frictional resistances include hysteresis losses, which can be considered as viscous (velocity dependent) friction components. It does not suffice to look just at the maximum required drive power to evaluate whether or not the energy consumption of a conveyor system is reasonable. The best method to compare the energy consumption of different transport systems is to compare their transport efficiencies.
1 TRANSPORT EFFICIENCY
There are a number of methods to compare transport efficiencies. The first and most widely applied method is to compare equivalent friction factors such as the DIN f factor. An advantage of using an equivalent friction factor is that it can also be determined for an empty belt. A drawback of using an equivalent friction factor is that it is not a 'pure' efficiency number. It takes into account the mass of the belt, reduced mass of the rollers and the mass of the transported material. In a pure efficiency number, only the mass of the transported material is taken into account.
The second method is to compare transportation cost, either in kW-hr/ton/km or in $/ton/km. The advantage of using the transportation cost is that this number is widely used for management purposes. The disadvantage of using the transportation cost is that it does not directly reflect the efficiency of a system.
The third and most "pure" method is to compare the loss factor of transport . The loss factor of transport is the ratio between the drive power required to overcome frictional losses (neglecting drive efficiency and power loss/gain required to raise/lower the bulk material) and the transport work. The transport work is defined as the multiplication of the total transported quantity of bulk material and the average transport velocity. The advantage of using loss factors of transport is that they can be compared to loss factors of transport of other means of transport, like trucks and trains. The disadvantage is that the loss factor of transport depends on the transported quantity of material, which implies that it can not be determined for an empty belt conveyor.
2 INDENTATION ROLLING RESISTANCE
For long overland systems, the energy consumption is mainly determined by the work done to overcome the indentation rolling resistance. Idler rolls are made of a relatively hard material like steel or aluminum whereas conveyor belt covers are made of much softer materials like rubber or PVC. The rolls therefore indent the belt's bottom-cover when the belt moves over the idler rolls, due to the weight of the belt and bulk material on the belt. The recovery of the compressed parts of the belt's bottom cover will take some time due to its visco-elastic (time dependent) properties. The time delay in the recovery of the belt's bottom cover results in an asymmetrical stress distribution between the belt and the rolls, see Figure 1. This yields a resultant resistance force called the indentation rolling resistance force. The magnitude of this force depends on the visco-elastic properties of the cover material, the radius of the idler roll, the vertical force due to the weight of the belt and the bulk solid material, and the radius of curvature of the belt in curves in the vertical plane.
Figure 1: Asymmetric stress distribution between belt and roll
It is important to know how the indentation rolling resistance depends on the belt velocity to enable selection of a proper belt velocity.
Figure 2: Loss factor (tanb) of typical cover rubber
Firstly, the indentation rolling resistance depends on the vertical load on the belt, which is the sum of the belt and the bulk material weight. If the vertical load on the belt decreases with a factor 2 then the indentation rolling resistance decreases with a factor 2.52 (2 ^4/3). The bulk load decreases with increasing belt speed assuming a constant capacity. Therefore, the indentation rolling resistance decreases more than proportionally with increasing belt speed.
Secondly, the indentation rolling resistance depends on the size of the idler rolls. If the roll diameter increases with a factor 2 then the indentation rolling resistance decreases with a factor 1.58 (2 ^2/3). In general the idler roll diameter increases with increasing belt speed to limit the bearing rpm's to maintain acceptable idler life. In that case the indentation rolling resistance decreases with increasing belt speed.
Thirdly, the indentation rolling resistance depends on the visco-elastic properties of the belt's cover material. These properties depend on the deformation rate, see Figure 2. The deformation rate in its turn depends on the size of the deformation area in the belt's bottom cover (depending on belt and bulk load) and on the belt speed. In general the indentation rolling resistance increases with increasing deformation rate (and thus belt speed), but only to a relatively small account.
Fourthly, the indentation rolling resistance depends on the belt's bottom cover thickness. If the bottom cover thickness increases with a factor 2 then the indentation rolling resistance increases with a factor 1.26 (2 ^1/3). if a bottom cover is increased to account for an increase in belt wear with increasing belt speed, then the indentation rolling resistance increases as well.
It should be realized that the indentation rolling resistance, although important, is not the only velocity dependent resistance. The rolling resistance of the idler rolls for example depends on the vertical load as well as on their rotational speed. The effect of the vertical load, which directly depends on the belt speed, is large. The effect of the rotational speed is much smaller. Another resistance occurs due to acceleration of the bulk solid material at the loading point. This resistance increases quadratically with an increase in belt speed assuming that the bulk material falls straight onto the belt. This will affect smaller, in plant belt conveyors in particular.
3 RUBBER COMPOUNDS
The indentation rolling resistance depends on the visco-elastic properties of the belt's bottom cover as discussed in the preceding section. This implies that the rolling resistance can be decreased by selecting a special low indentation rolling resistance (rubber) compound that is available on the market today. A small premium has to be paid for this special compound, but costs can be limited by applying it for the bottom cover only and using a normal wear-resistant compound for the carrying top cover. In that case turnovers are required to fully use the energy saving function of the bottom compound.
A Quantitative indication of the level of indentation rolling resistance is the indentation rolling resistance indicator tan/E ^1/3, where tan is the loss angle and E' the storage modulus of the compound. Compounds with a reasonable indentation rolling resistance performance have indicators below 0.1. Figure 3 shows these indicators for typical medium to good performing rubbers. As can be seen in that figure, the choice for a specific rubber compound affects the energy consumption of the belt conveyor, in particular as a function of the ambient temperature.
One comment (warning) must be made. A special belt with low indentation rolling resistance compound should never be selected if only one conveyor belt manufacturer offers it. In that case the conveyor system can only perform in accordance with its design specifications when that specific belt is used. It is much better, also cost wise, to specify the upper limit of the resistance indicator as given above that can be met by more than one conveyor belt manufacturer.
Figure 3: Indentation rolling resistance indicators for four?different rubbers as a function of temperature.
CONCLUSION
It is not easy to determine the relationship between the belt speed and the belt's energy consumption. This is partly because the calculation of the indentation rolling, which forms the largest part of the rolling resistance, requests detailed knowledge of the visco-elastic properties of the used rubber compound. In addition the (unknown) velocity dependent components of the coulomb friction and seal and viscous drag of the roller bearings play an important role. Also the resistances that occur at transfer stations, in particular due to the acceleration of the bulk solid, play a role especially at high belt speeds.
高速帶式輸送機(jī)的設(shè)計(jì)
G. Lodewijks, The Netherlands. G. Lodewijks 荷蘭
SUMMARY摘要：This paper discusses aspects of high-speed belt conveyor design.本文論述的是高速帶式輸送機(jī)的設(shè)計(jì)。The capacity of a belt conveyor is determined by the belt speed given a belt width and troughing angle.帶式輸送機(jī)的輸送能力是由帶的寬度和槽角給定的帶的速度決定的。然而皮Belt speed selection however is limited by practical considerations, which are discussed in this paper.帶速度的選擇受限于實(shí)際的考慮因素，這是本文將討論的。皮帶速度也影響輸送帶的工作表現(xiàn)，如其能量的消耗和運(yùn)行的穩(wěn)定性。The belt speed also affects the performance of the conveyor belt, as for example its energy consumption and the stability of it's running behavior.
3 ENERGY CONSUMPTION能源消耗
Clients may request a specification of the energy consumption of a conveyor system, for example quantified in terms of maximum kW-hr/ton/km, to transport the bulk solid material at the design specifications over the projected route.客戶可以要求一個(gè)輸送系統(tǒng)能源消耗的詳細(xì)說明，例如設(shè)計(jì)規(guī)格超過計(jì)劃規(guī)格，運(yùn)輸散裝固體材料就要最大限度的量化條件kw-hr/ton/km。長(zhǎng)久陸地系統(tǒng)For long overland systems, the energy consumption is mainly determined by the work done to overcome the indentation rolling resistance [9].，能源消耗主要取決于所做的工作，以克服壓痕滾動(dòng)阻力。皮帶所受的阻力取決于橡膠帶粘彈性（時(shí)間延遲），粘彈性的效應(yīng)掩飾了托輥的壓痕。作For in-plant belt conveyors, work done to overcome side resistances that occur mainly in the loading area also affects the energy consumption.為在廠的帶式輸送機(jī)所做的工作是克服沿邊阻力，沿邊阻力主要發(fā)生在負(fù)重區(qū)，也影響能源的消耗。Side resistances include the resistance due to friction on the side walls of the chute and resistance that occurs due to acceleration of the material at the loading point.沿邊阻力包括溜槽壁的摩擦產(chǎn)生的阻力和在裝料點(diǎn)物料的加速度產(chǎn)生的阻力。
皮帶輸送機(jī)The required drive power of a belt conveyor is determined by the sum of the total frictional resistances and the total material lift.所需的驅(qū)動(dòng)功率，是由摩擦阻力的總數(shù)和所要提升物料的總和決定的。The frictional resistances include hysteresis losses, which can be considered as viscous (velocity dependent) friction components.這個(gè)摩擦阻力包括滯后損耗，它可以被視為粘性（速度而定）摩擦組件。 It does not suffice to look just at the maximum required drive power to evaluate whether or not the energy consumption of a conveyor system is reasonable.它不足以僅從最大所需驅(qū)動(dòng)功率來評(píng)價(jià)一個(gè)輸送系統(tǒng)的能源消耗是否是合理的。 The best method to compare the energy consumption of different transport systems is to compare their transport efficiencies.比較不同運(yùn)輸系統(tǒng)的能源消耗，最好的方法是比較它們的運(yùn)輸效率等。
3.1 TRANSPORT EFFICIENCY1 運(yùn)輸效率
There are a number of methods to compare transport efficiencies.有許多方法來比較運(yùn)輸效率。The first and most widely applied method is to compare equivalent friction factors such as the DIN f factor.首先并最廣泛應(yīng)用的方法是比較當(dāng)量摩擦因素，如按DIN F系數(shù)。利用當(dāng)量摩擦系數(shù)An advantage of using an equivalent friction factor is that it can also be determined for an empty belt.的優(yōu)點(diǎn)是，它也可以由一個(gè)空皮帶確定。采用當(dāng)量摩擦系數(shù)的缺點(diǎn)是，它不是一個(gè)‘純'效率數(shù)字。It takes into account the mass of the belt, reduced mass of the rollers and the mass of the transported material.考慮到皮帶的大量化，就要減少了托輥的數(shù)量和運(yùn)送物資的數(shù)量。In a pure efficiency number, only the mass of the transported material is taken into account.純粹的效率高低，僅有材料的運(yùn)送量，是在考慮之列。
The second method is to compare transportation cost, either in kW-hr/ton/km or in $/ton/km.第二種方法是比較運(yùn)輸成本，要么kw-hr/ton/km要么$/ton/km。The advantage of using the transportation cost is that this number is widely used for management purposes.比較運(yùn)輸成本的優(yōu)點(diǎn)在于輸送成本廣泛用于管理策劃。The disadvantage of using the transportation cost is that it does not directly reflect the efficiency of a system.采用運(yùn)輸成本的不利之處是，它沒有直接地反映著一個(gè)體系的效率。
The third and most "pure" method is to compare the loss factor of transport [10].第三種且最“純粹的”方法是比較運(yùn)輸損耗因子。運(yùn)輸The loss factor of transport is the ratio between the drive power required to overcome frictional losses (neglecting drive efficiency and power loss/gain required to raise/lower the bulk material) and the transport work.損耗因子是驅(qū)動(dòng)電源必須克服的摩擦損失（忽略傳動(dòng)效率和功率損耗/增益須提高/降低大宗材料）和運(yùn)輸工作兩者的比例。The transport work is defined as the multiplication of the total transported quantity of bulk material and the average transport velocity.運(yùn)輸工作是被定義為原料的總運(yùn)輸量和平均運(yùn)輸速度的乘積。采用運(yùn)輸損耗因子的The advantage of using loss factors of transport is that they can be compared to loss factors of transport of other means of transport, like trucks and trains.好處是它們能與其他運(yùn)輸方式的損失因子作比較，如卡車和火車。The disadvantage is that the loss factor of transport depends on the transported quantity of material, which implies that it can not be determined for an empty belt conveyor.缺點(diǎn)是運(yùn)輸損耗因子依賴于材料的運(yùn)輸量，這意味著它不能被一個(gè)空載的帶式輸送機(jī)決定。
3.2 INDENTATION ROLLING RESISTANCE2 壓痕滾動(dòng)阻力
For long overland systems, the energy consumption is mainly determined by the work done to overcome the indentation rolling resistance.長(zhǎng)期陸路系統(tǒng)，能源消耗主要取決于所做的工作，以克服壓痕滾動(dòng)阻力。 Idler rolls are made of a relatively hard material like steel or aluminum whereas conveyor belt covers are made of much softer materials like rubber or PVC.托輥是由比較硬質(zhì)的材料制成，如鋼或鋁，而輸送帶由許多柔軟材料做成，如橡膠或PVC 。The rolls therefore indent the belt's bottom-cover when the belt moves over the idler rolls, due to the weight of the belt and bulk material on the belt.因此當(dāng)皮帶在托輥上運(yùn)動(dòng)時(shí)，輥?zhàn)泳涂s進(jìn)在皮帶的下面，這取決于在皮帶上的皮帶和散裝物料的重量。皮帶下層的壓縮部分的The recovery of the compressed parts of the belt's bottom cover will take some time due to its visco-elastic (time dependent) properties.回彈，將需要一些時(shí)間，由于其粘彈性（時(shí)間依賴性）的性能。皮帶下層彈復(fù)The time delay in the recovery of the belt's bottom cover results in an asymmetrical stress distribution between the belt and the rolls, see Figure 2.時(shí)間的延遲導(dǎo)致了帶和托輥間應(yīng)力的不對(duì)稱分布，見圖1 。This yields a resultant resistance force called the indentation rolling resistance force.這個(gè)產(chǎn)生抵抗的力量稱為壓痕滾動(dòng)阻力。The magnitude of this force depends on the visco-elastic properties of the cover material, the radius of the idler roll, the vertical force due to the weight of the belt and the bulk solid material, and the radius of curvature of the belt in curves in the vertical plane.這股力量的大小，依賴于皮帶下層材料的粘彈性性能、托輥的半徑和壓力，壓力取決于皮帶和散裝固體物質(zhì)的重量，以及在垂直面內(nèi)變形部分的曲率半徑。
Figure 2: Asymmetric stress distribution between belt and roll [7].圖1 皮帶和輥之間的不對(duì)稱應(yīng)力分布
It is important to know how the indentation rolling resistance depends on the belt velocity to enable selection of a proper belt vel重要的是要知道壓痕滾動(dòng)阻力是如何取決于皮帶速度，以便選擇一個(gè)適當(dāng)?shù)膸佟?
Firstly, the indentation rolling resistance depends on the vertical load on the belt, which is the sum of the belt and the bulk material weight.首先，壓痕滾動(dòng)阻力取決于皮帶上的垂直載荷，這個(gè)載荷是由帶和散裝材料的重量的總和產(chǎn)生的。 If the vertical load on the belt decreases with a factor 2 then the indentation rolling resistance decreases with a factor 2.52 (2 ^4/3).如果垂直載荷對(duì)帶減少一個(gè)因子2 ，壓痕滾動(dòng)阻力就減小一個(gè)因子2.52（ 2 ^ 4 / 3 ）。假設(shè)恒容量，The bulk load decreases with increasing belt speed assuming a constant capacity.部分負(fù)荷減少于帶速的增加。 Therefore, the indentation rolling resistance decreases more than proportionally with increasing belt speed.因此，壓痕滾動(dòng)阻力減少的比例比帶速的增加要多。
Secondly, the indentation rolling resistance depends on the size of the idler rolls.其次，壓痕滾動(dòng)阻力取決于托輥的尺寸。 If the roll diameter increases with a factor 2 then the indentation rolling resistance decreases with a factor 1.58 (2 ^2/3).如果托輥直徑增大的一個(gè)因子2，壓痕滾動(dòng)阻力就減小的一個(gè)因子1.58 （ 2 ^ 2 / 3 ）。In general the idler roll diameter increases with increasing belt speed to limit the bearing rpm's to maintain acceptable idler life.一般托輥直徑隨帶速的增大而增大，以限制軸承的轉(zhuǎn)速，保持可接受的托輥壽命。 In that case the indentation rolling resistance decreases with increasing belt speed.在此情況下，壓痕滾動(dòng)阻力隨帶速的增加而減小。
Thirdly, the indentation rolling resistance depends on the visco-elastic properties of the belt's cover material.第三，壓痕滾動(dòng)阻力取決于皮帶覆蓋材料的粘彈性性能。 These properties depend on the deformation rate, see Figure 3.這些性能在很大程度上依賴于變形速度，見圖2 。皮帶往復(fù)運(yùn)行的The deformation rate in its turn depends on the size of the deformation area in the belt's bottom cover (depending on belt and bulk load) and on the belt speed.變形速度又取決于在皮帶下層（取決于皮帶及散裝負(fù)載）變形區(qū)的規(guī)模和皮帶速度。In general the indentation rolling resistance increases with increasing deformation rate (and thus belt speed), but only to a relatively small account.一般壓痕滾動(dòng)阻力隨變形速度（因而帶速）的增加而增加， 但相對(duì)來說僅僅是一小部分。
Figure 3: Loss factor (tanb) of typical cover rubber [7]圖2 典型橡膠層的損耗因子
Fourthly, the indentation rolling resistance depends on the belt's bottom cover thickness.第四，壓痕滾動(dòng)阻力取決于皮帶下層的厚度。 If the bottom cover thickness increases with a factor 2 then the indentation rolling resistance increases with a factor 1.26 (2 ^1/3).如果下層厚度增大一個(gè)因子2，則壓痕滾動(dòng)阻力就增大一個(gè)因子1.26（ 2 ^ 1 / 3 ）。if a bottom cover is increased to account for an increase in belt wear with increasing belt speed, then the indentation rolling resistance increases as well.如果皮帶下層寬度的增加并提高輸送速度，則壓痕滾動(dòng)阻力也會(huì)有所增加。
It should be realized that the indentation rolling resistance, although important, is not the only velocity dependent resistance.應(yīng)當(dāng)看到，壓痕滾動(dòng)阻力雖然重要但不是速度唯一取決于的阻力。 就托輥的The rolling resistance of the idler rolls for example depends on the vertical load as well as on their rotational speed.滾動(dòng)阻力舉例來說，滾動(dòng)阻力取決于豎向荷載以及其轉(zhuǎn)速。The effect of the vertical load, which directly depends on the belt speed, is large.垂直載荷的影響直接取決于皮帶速度，影響是大的。 相比之下，The effect of the rotational speed is much smaller.轉(zhuǎn)速的影響是小多了。 Another resistance occurs due to acceleration of the bulk solid material at the loading point.另一阻力是在裝載處由于散裝固體物質(zhì)下落產(chǎn)生的加速度產(chǎn)生的。假定該散裝物料直線落在皮帶上，This resistance increases quadratically with an increase in belt speed assuming that the bulk material falls straight onto the belt.這種阻力的增加是帶速增加的二次方。This will affect smaller, in plant belt conveyors in particular.這影響也是較小的，尤其是輸送木板類的帶式輸送機(jī)。
3.3 RUBBER COMPOUNDS3 橡膠化合物
作為之前部分的討論，The indentation rolling resistance depends on the visco-elastic properties of the belt's bottom cover as discussed in the preceding section.壓痕滾動(dòng)阻力取決于皮帶下層的粘彈性性能。 This implies that the rolling resistance can be decreased by selecting a special low indentation rolling resistance (rubber) compound that is available on the market today.這意味著通過選擇一個(gè)特殊壓痕低滾動(dòng)阻力（橡膠）化合物，滾動(dòng)阻力可減小，這種橡膠化合物在當(dāng)今市場(chǎng)上是可用的。A small premium has to be paid for this special compound, but costs can be limited by applying it for the bottom cover only and using a normal wear-resistant compound for the carrying top cover.一個(gè)小的補(bǔ)價(jià)必須支付這一個(gè)特殊化合物，但成本可能被限制，如僅僅申請(qǐng)它為皮帶下層或用一個(gè)正常耐磨材料復(fù)合成運(yùn)載皮帶的上層。 In that case turnovers are required to fully use the energy saving function of the bottom compound.在這種情況下，要求充分地使用底部化合物的節(jié)能作用。
壓痕滾動(dòng)阻力A Quantitative indication of the level of indentation rolling resistance is the indentation rolling resistance indicator tan/E ^1/3, where tan is
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