Transport performance of a steeply situated belt conveyor

energies Article Transport Performance of a Steeply Situated Belt Conveyor Leopold Hrabovský1,* and JiˇríFries 2   Citation: Hrabovský, L.; Fries, J. Transport Performance of a Steeply Situated Belt Conveyor. Energies 2021, 14, 7984. https://doi.org/10.3390/ en14237984 Academic Editor: Andrea Mariscotti Received: 21 October 2021 Accepted: 25 November 2021 Published: 29 November 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1Faculty of Mechanical Engineering, Institute of Transport, VSB-Technical University of Ostrava, Poruba, 708 00 Ostrava, Czech Republic 2Department of Machine and Industrial Design, Faculty of Mechanical Engineering, VSB-Technical University of Ostrava, Poruba, 708 00 Ostrava, Czech Republic; [email protected] *Correspondence: [email protected]; Tel.: +420-597-323-185 Abstract: The paper presents a methodology for determining the volume of a batch of conveyed material located before a transverse partition of a certain height and the distance over which the batch of material extends on the working surface of the conveyor belt along its longitudinal axis. Knowing the geometric dimensions of the transported batch of material makes it possible to appropriately set the spacing of the belt cleats and thereby to optimally determine the conveying performance of the inclined belt conveyor. When the angle of inclination of a conveyor with a straight idler frame is equal to the angle of surcharge of the conveyed material, then no layer of material is carried on the surface of the belt. If the conveyor belt is guided along a trough idler frame, only the lower cross-section of the filling of material is used. An increase in the cross-section of the belt load of a conveyor inclined at an angle, which exceeds the angle of repose of the conveyed material, can be achieved by installing regularly spaced belt cleats around the circumference of the working surface of the endless loop of the conveyor belt. The volume of the batch of material retained by the belt cleat depends on the height and width of the cleat and whether or not the conveyor belt is provided with corrugated side edges. The paper presents theoretically determined relationships that can be used to determine the size of the transverse and longitudinal area and the volume of the batch of material spread on the surface of the conveyor belt in front of the cleat. The experiments performed provide the distances of the material distribution on the surface of the conveyor belt depending on the height of the cleat and the angle of inclination of the conveyor belt. Keywords: batch volume of conveyed material; belt cleats; straight loading profile of the conveyor belt; flat roller support of the belt; angle of inclination of the conveyor 1. Introduction Classic belt conveyors [ 1 , 2 ] are continuously operating machines widely used in industry [ 3 ], designed for moving bulk and piece materials up to distances of thousands of meters [ 4 – 6 ]. The endless loop of the conveyor belt [ 7 ] is composed of sections mutually interconnected by vulcanization or mechanical means [8]. Current standards relating to calculations for a belt conveyor of classic construction, define how to determine the amount of transported material per unit of time. The amount of material transported per unit of time [ 9 , 10 ] by continuously operating conveyor equipment, which includes the belt conveyor, is the so-called transport capacity by mass Q m (t · h −1 ) or volume Q v (m 3· h −1 ) and can be regulated with the conveying speed v (m · s −1 ), the width of the conveyor belt B [m], and the type of idler frame (roller stand). The size of the cross-section of the conveyor belt load S (m 2 ) [ 9 ] depends on the shape of the conveyor belt loading profile (flat or trough idler frame) [ 11 ], the belt loading width b [m], and the angle of repose Θ(deg) of the transported materials. On a flat loading profile of a horizontally placed conveyor belt (i.e., the angle of inclination of the conveyor δ = 0 deg) of the belt conveyor, the total cross-sectional area S (m 2 ) of the load of the transported material is determined according to [ 9 , 12 , 13 ]. If the Energies 2021,14, 7984. https://doi.org/10.3390/en14237984 https://www.mdpi.com/journal/energies Energies 2021,14, 7984 2 of 15 conveyed mass is fed onto the inclined part of the belt ( δ > 0), the reduction of the part of the cross-section S 1 is determined by the inclination factor k (-) (1) [ 9 , 10 ], where k 1 (-) is the correction factor for the belt filling canopy. k=1−S1 S·(1−k1) (-), k1=scos2δ−cos2Θ 1−cos2Θ(-), (1) In practice, transport of finely lumpy or fine-grained materials on the surface of the conveyor belt is required at the highest possible angle of inclination of the conveyor; for example, to reduce the built-up area of the conveyor and to shorten the length of the endless loop of the conveyor belt. The article [ 12 ] discusses the advantages of belt conveyors with a high angle of inclination in the application of the mines. One of the numerous possible ways [ 13 , 14 ] to achieve efficient transport of grains of a conveyed mass transported at an angle of inclination of the conveyor which exceeds the limiting angle of inclination of a smooth conveyor belt δm (deg) of a belt conveyor of conventional construction, is to use a conveyor belt with belt cleats [15,16], Figure 1. Energies 2021, 14, x FOR PEER REVIEW 2 of 15 conveyed mass is fed onto the inclined part of the belt (δ > 0), the reduction of the part of the cross-section S1 is determined by the inclination factor k (-) (1) [9,10], where k1 (-) is the correction factor for the belt filling canopy. ( ) 22 S cos δ - cos Θ 1 k = 1 - 1 - k (-), k = (-) 112 S1 - cos Θ  , (1) In practice, transport of finely lumpy or fine-grained materials on the surface of the conveyor belt is required at the highest possible angle of inclination of the conveyor; for example, to reduce the built-up area of the conveyor and to shorten the length of the endless loop of the conveyor belt. The article [12] discusses the advantages of belt conveyors with a high angle of inclination in the application of the mines. One of the numerous possible ways [13,14] to achieve efficient transport of grains of a conveyed mass transported at an angle of inclination of the conveyor which exceeds the limiting angle of inclination of a smooth conveyor belt δm (deg) of a belt conveyor of conventional construction, is to use a conveyor belt with belt cleats [15,16], Figure 1. Figure 1. Conveyor with conveyor belt with belt cleats. 1—conveyor belt, 2—batch of transported material, 3—belt cleat In order to ensure ecological transport without failure requirements for the transport of bulk materials by continuous conveyor systems [17], careful attention should be given to the proper shape of the conveyor belt [18]. McTurk, J. R. discussed in [19] the development of specialized belt conveyors. The most common failures encountered during their operation are mainly those directly related to the conveyor belt itself [20]. In order to achieve the high standards of safety and operational reliability required for belt conveyors, there is a requirement for systematic testing of conveyor belts and their joints [21,22]. Krol, R. et al. state in [23] that the key elements of a belt conveyor, the energy-efficient conveyor belt, and optimized carrying idlers have been developed for the new generation of underground conveyors. Sandwich belt high angle conveyor technology [24] provides an economic solution for high volume steep and vertical conveying. Reliable operation of the pipe conveyors requires continuous monitoring and evaluation of their selected indicators. Asymmetrical tensioning of the conveyor belt is the serious negative and undesirable operational situations [25]. Figure 1. Conveyor with conveyor belt with belt cleats. 1—conveyor belt, 2—batch of transported material, 3—belt cleat. In order to ensure ecological transport without failure requirements for the transport of bulk materials by continuous conveyor systems [ 17 ], careful attention should be given to the proper shape of the conveyor belt [18]. McTurk, J.R. discussed in [ 19 ] the development of specialized belt conveyors. The most common failures encountered during their operation are mainly those directly related to the conveyor belt itself [20]. In order to achieve the high standards of safety and operational reliability required for belt conveyors, there is a requirement for systematic testing of conveyor belts and their joints [21,22]. Krol, R. et al. state in [ 23 ] that the key elements of a belt conveyor, the energy-efficient conveyor belt, and optimized carrying idlers have been developed for the new generation of underground conveyors. Sandwich belt high angle conveyor technology [ 24 ] provides an economic solution for high volume steep and vertical conveying. Reliable operation of the pipe conveyors requires continuous monitoring and evaluation of their selected indicators. Asymmetrical tensioning of the conveyor belt is the serious negative and undesirable operational situations [25]. Energies 2021,14, 7984 3 of 15 In [ 26 ], Fedorko G. et al. deal with the implementation of the Finite Element Method (FEM) model in the concept of digital twins designed to measure key properties and characteristics of the rubber-textile conveyor belts of tube conveyors. Wozniak D. et al. [ 9 , 27 ] present that in steel cord conveyor belts, rubber penetrates the steel ropes during vulcanization. In [ 11 , 28 ], Marasova D. et al. state that rubber-textile conveyor belts, in particular, are an essential element of a belt conveyor. Authors Gierz Ł. et al. describe [ 29 ] the problems of transporting bulk material at different speeds and angles of inclination of the conveyor belt. Results of tests into the energy-efficiency of belt conveyor transportation systems indicate that the energy consumption of their drive mechanisms can be limited by lowering the main resistances in the conveyor described in [30] Bajda M. et al. It is now known that the maximal cross-section surface of the lose material filling at the conveyor with a smooth belt can be (for conveyor belts modified based on the shape of the trough) expressed as the sum of the cap cross-section surface and the surface of the bottom part of the belt filling. Various industrial fields require an increased inclination angle of the belt conveyor transport [ 31 ], or the use of belt conveyors of special designs [ 13 , 14 ]. The permitted inclination angle of the transport by belt conveyors of a standard design [ 9 ] is particularly determined by the friction coefficient between the contact surface of the material grains (incapable of rotation) and the surface of the given conveyor belt. Material grains that are capable of rotation (grains of a spherical or cylindrical shape) must be prevented from rolling along the surface under any angle of the given inclined conveyor belt. For horizontally carried conveyor belts, we also need to prevent rotation of the spherical grains, which can occur due to the effects of the forces along the contact surface of the grains with the conveyor belt and the inertia forces. In order to increase the permitted angle of the belt conveyor transport, we need to: (a) Increase the friction coefficient along the work surface of the conveyor belt [15,32], (b) Structurally modify the surface of the conveyor belt [ 33 , 34 ], making sure that movement of the transported material grains along the belt is prevented (against the transport direction for the uphill transport and in the transport direction for downhill transport) [13,35], (c) Increase the pressure of the transported material to the conveyor belt [36]. The paper presents a methodology for determining the volume of a batch of conveyed material located before a transverse partition of a certain height and the distance over which the batch of material extends on the working surface of the conveyor belt along its longitudinal axis. The experiments performed provide the distances of the material distribution on the surface of the conveyor belt depending on the height of the cleat and the angle of inclination of the conveyor belt. The paper presents a mathematical expression of the sizes of the transverse and longitudinal cross-section surfaces and of the volume of the loose material spread on the surface of the conveyor belt before the transverse partition of the belt conveyor, inclination angle of which exceeds the limit transport angle permitted for the belt conveyor of a classic design. 2. Materials and Methods One of the series of conveying machines used for conveying bulk materials at an angle of inclination exceeding the critical angle of inclination of a conveyor belt of conventional construction is a belt conveyor with a specifically modified working surface of the conveyor belt (Figure 1). A specific construction modification of the conveyor belt consists in the installation (mechanical or by vulcanization) of belt cleats (Figure 2b), which are spaced at regular intervals along the entire length of the closed loop of the working surface of the conveyor belt. Energies 2021,14, 7984 4 of 15 Energies 2021, 14, x FOR PEER REVIEW 4 of 15 Figure 2. Conveyor belt (a) with belt cleats, (b) types of belt cleats, (c) with belt cleats and corrugated side edges. Belt cleats of suitable height create supports for grains of material, thereby preventing their slippage or rotation along the surface of the belt, counter to the transport direction during their conveyance by the conveyor belt of a steeply inclined conveyor. 2.1. Experimental Testing Equipment and Measured Parameters The test equipment (Figure 3) consists of two basic parts, which are the trough I (simulating a section of the conveyor belt) and a tilting mechanism II. The front part of the trough I is fitted with a sliding plate slide 1, which represents a belt cleat of a given height. To accurately determine the angle of inclination of the trough, additional parts are installed on the test device: a digital inclinometer 2 [6] and a digital protractor 3 [7]. To determine the distance z(x) of the material spread along the length of the trough I, a steel ruler 4 is attached to the upper surface of the trough I. Figure 3. Testing device to determine shape, volume, and length of spread material batches (I) - trough, (II) - tilting mechanism, (1) - sliding plate, (2) - digital inclinometer, (3) - digital protractor, (4) - ruler. The measured parameters behind the test equipment are the length of the distribution z(x) and the width bp of the transported material in the plane yz and xy. The surface of the trough I is inclined at the beginning of the measurement with respect to the horizontal plane by an angle δ-Θ = 10 deg using the screw of the tilting mechanism II. For experimental tests, the upper edge of the plate slide 1 is gradually extended to a height of 10, 15, and 20 mm above the upper surface of the trough I. A batch of conveyed material is Figure 2. Conveyor belt ( a ) with belt cleats, ( b ) types of belt cleats, ( c ) with belt cleats and corrugated side edges. Belt cleats of suitable height create supports for grains of material, thereby preventing their slippage or rotation along the surface of the belt, counter to the transport direction during their conveyance by the conveyor belt of a steeply inclined conveyor. 2.1. Experimental Testing Equipment and Measured Parameters The test equipment (Figure 3) consists of two basic parts, which are the trough I (simulating a section of the conveyor belt) and a tilting mechanism II. The front part of the trough I is fitted with a sliding plate slide 1, which represents a belt cleat of a given height. To accurately determine the angle of inclination of the trough, additional parts are installed on the test device: a digital inclinometer 2 [ 6 ] and a digital protractor 3 [ 7 ]. To determine the distance z (x) of the material spread along the length of the trough I, a steel ruler 4 is attached to the upper surface of the trough I. Energies 2021, 14, x FOR PEER REVIEW 4 of 15 Figure 2. Conveyor belt (a) with belt cleats, (b) types of belt cleats, (c) with belt cleats and corrugated side edges. Belt cleats of suitable height create supports for grains of material, thereby preventing their slippage or rotation along the surface of the belt, counter to the transport direction during their conveyance by the conveyor belt of a steeply inclined conveyor. 2.1. Experimental Testing Equipment and Measured Parameters The test equipment (Figure 3) consists of two basic parts, which are the trough I (simulating a section of the conveyor belt) and a tilting mechanism II. The front part of the trough I is fitted with a sliding plate slide 1, which represents a belt cleat of a given height. To accurately determine the angle of inclination of the trough, additional parts are installed on the test device: a digital inclinometer 2 [6] and a digital protractor 3 [7]. To determine the distance z(x) of the material spread along the length of the trough I, a steel ruler 4 is attached to the upper surface of the trough I. Figure 3. Testing device to determine shape, volume, and length of spread material batches (I) - trough, (II) - tilting mechanism, (1) - sliding plate, (2) - digital inclinometer, (3) - digital protractor, (4) - ruler. The measured parameters behind the test equipment are the length of the distribution z(x) and the width bp of the transported material in the plane yz and xy. The surface of the trough I is inclined at the beginning of the measurement with respect to the horizontal plane by an angle δ-Θ = 10 deg using the screw of the tilting mechanism II. For experimental tests, the upper edge of the plate slide 1 is gradually extended to a height of 10, 15, and 20 mm above the upper surface of the trough I. A batch of conveyed material is Figure 3. Testing device to determine shape, volume, and length of spread material batches ( I)—trough , ( II )—tilting mechanism, ( 1 )—sliding plate, ( 2 )—digital inclinometer, ( 3 )—digital protractor, (4)—ruler. The measured parameters behind the test equipment are the length of the distribution z (x) and the width b p of the transported material in the plane yz and xy. The surface of the trough I is inclined at the beginning of the measurement with respect to the horizontal plane by an angle δ – Θ = 10 deg using the screw of the tilting mechanism II. For experimental tests, the upper edge of the plate slide 1 is gradually extended to a height of 10, 15, and 20 mm above the upper surface of the trough I. A batch of conveyed material is poured Energies 2021,14, 7984 5 of 15 onto the surface of the trough I in such a way that the material is distributed on the trough. For these conditions, the distance z (x) from the distribution of the batch of material on the trough at the level xy corresponds to the value b p Equation (7). For these conditions, the distance z (x) from the distribution of the batch of material on the trough surface is measured. The volume V 1(δ–Θ) of the batch of material distributed on the surface of the trough in front of the belt cleat is then transferred into a measuring cylinder, in which its size is determined. The theoretically determined volume V 1 of conveyed material for a conveyor where the inclination angle δ (deg), the dynamic angle of repose Θ (deg) and the conveyor belt width B (m) is given in Table 1. The volume V 1(δ–Θ) of the material batch (Tables 2–4) was redistributed on the upper surface of the trough I and subsequently the angle of inclination of the trough δ was gradually increased by 5 deg. The volume V 2(δ–Θ) of the poured batch of material over the upper edge of the transverse partition was transferred to a measuring cylinder, in which its size was determined. The distribution of the batch of bulk material in the zy plane was read on the ruler 4 and noted in tables (see Section 4. Discussion). 2.2. Properties of the Conveyed Material Experimental tests performed on the testing device to measure the volumes V 1(δ–Θ) to V 3(δ–Θ) of the respective batches of material spread on the conveyor belt were performed at known values of the height H (or H 1 ) of the belt cleat and at a given angle of inclination of the conveyor belt δ . The material used in the laboratory tests was buckwheat with a natural angle of repose ψ = 27 deg (Figure 4a) and a dynamic angle of repose Θ = 16 deg and barley groats with a natural angle of repose ψ = 29 deg (Figure 4b) and a dynamic angle of repose Θ= 19 deg. Energies 2021, 14, x FOR PEER REVIEW 5 of 15 poured onto the surface of the trough I in such a way that the material is distributed on the trough. For these conditions, the distance z(x) from the distribution of the batch of material on the trough at the level xy corresponds to the value bp Equation (7). For these conditions, the distance z(x) from the distribution of the batch of material on the trough surface is measured. The volume V1(δ–Θ) of the batch of material distributed on the surface of the trough in front of the belt cleat is then transferred into a measuring cylinder, in which its size is determined. The theoretically determined volume V1 of conveyed material for a conveyor where the inclination angle δ (deg), the dynamic angle of repose Θ (deg) and the conveyor belt width B (m) is given in Table 1. The volume V1(δ–Θ) of the material batch (Tables 2–4) was redistributed on the upper surface of the trough I and subsequently the angle of inclination of the trough δ was gradually increased by 5 deg. The volume V2(δ–Θ) of the poured batch of material over the upper edge of the transverse partition was transferred to a measuring cylinder, in which its size was determined. The distribution of the batch of bulk material in the zy plane was read on the ruler 4 and noted in tables (see 4. Discussion). 2.2. Properties of the Conveyed Material Experimental tests performed on the testing device to measure the volumes V1(δ–Θ) to V3(δ–Θ) of the respective batches of material spread on the conveyor belt were performed at known values of the height H (or H1) of the belt cleat and at a given angle of inclination of the conveyor belt δ. The material used in the laboratory tests was buckwheat with a natural angle of repose ψ = 27 deg (Figure 4a) and a dynamic angle of repose Θ = 16 deg and barley groats with a natural angle of repose ψ = 29 deg (Figure 4b) and a dynamic angle of repose Θ = 19 deg. Figure 4. Angles of repose of the transported material (a) chickpeas, (b) barley groats. The results obtained from the performed experimental measurements (Tables 5,6) on a special testing device (Figure 3) allowed us to express the true distribution of bulk material on the surface of the upwardly inclined trough with a front slide that simulates a steeply inclined conveyor belt with a belt cleat. The placement of the batch V1 of bulk material on the surface of the trough at the start of the measurement was performed in a static state of the trough, i.e., when the conveyor belt was at rest. After pouring a predetermined volume V1 of bulk material onto the surface of the trough, which at this moment was inclined at an angle δi, the surface of the trough was tilted by an angle δi+j and then subjected to vibrations. The vibrations were induced by repeatedly tapping the bottom surface of the trough five times. The tangent of the slope of the initial delivered batch of bulk material [30], which rested against the belt cleat with its front surface, to the longitudinal axis of the conveyor belt in the horizontal plane is determined according to theoretical relations, Equations (2), (4) and (5), by the difference of angles δ-Θ (deg). 3. Result The paper [34,30] presents an expression that defines the relation for the analytical Figure 4. Angles of repose of the transported material (a) chickpeas, (b) barley groats. The results obtained from the performed experimental measurements (Tables 5and 6) on a special testing device (Figure 3) allowed us to express the true distribution of bulk material on the surface of the upwardly inclined trough with a front slide that simulates a steeply inclined conveyor belt with a belt cleat. The placement of the batch V 1 of bulk material on the surface of the trough at the start of the measurement was performed in a static state of the trough, i.e., when the conveyor belt was at rest. After pouring a predetermined volume V 1 of bulk material onto the surface of the trough, which at this moment was inclined at an angle δi , the surface of the trough was tilted by an angle δi+j and then subjected to vibrations. The vibrations were induced by repeatedly tapping the bottom surface of the trough five times. The tangent of the slope of the initial delivered batch of bulk material [ 30 ], which rested against the belt cleat with its front surface, to the longitudinal axis of the conveyor belt in the horizontal plane is determined according to theoretical relations, Equations (2), (4) and (5), by the difference of angles δ–Θ(deg). 3. Result The paper [ 30 , 34 ] presents an expression that defines the relation for the analytical calculation of the size of the canopy cross-sectional area of the conveyed mass S 1 in the Energies 2021,14, 7984 6 of 15 xy plane. Conveyor belts with belt cleats (Figure 2a) and a straight or trough-shaped loading profile [ 2 , 3 , 13 , 14 ] are used for the transport of bulk and piece materials at angles of inclination of up to 60 deg. For the vertical transport of bulk materials, conveyor belts with a flat loading profile with belt cleats and corrugated side edges [ 4 , 5 , 36 , 37 ] are used (Figure 2c). 3.1. Transverse, Longitudinal Cross-Section and Batch Volume of Transported Material The transverse, longitudinal cross-section of the transported material S yz (m 2 ), see Figure 5a, between two neighboring cleats of sufficient height H ≥ h max and mutual spacing of L ≥z(x), is shown by the relationship Equation (2), Syz =1 2·y(x)·z(x)=y2 (x) 2·tan(δ−Θ)[m2], (2) where y(x) [m] is the vertical distance of the point of the curve (parabola) from the x-axis. Energies 2021, 14, x FOR PEER REVIEW 6 of 15 calculation of the size of the canopy cross-sectional area of the conveyed mass S1 in the xy plane. Conveyor belts with belt cleats (Figure 2a) and a straight or trough-shaped loading profile [2,3,13,14] are used for the transport of bulk and piece materials at angles of inclination of up to 60 deg. For the vertical transport of bulk materials, conveyor belts with a flat loading profile with belt cleats and corrugated side edges [4,5,36,37] are used (Figure 2c). 3.1. Transverse, Longitudinal Cross-Section and Batch Volume of Transported Material The transverse, longitudinal cross-section of the transported material Syz (m2), see Figure 5a, between two neighboring cleats of sufficient height H ≥ hmax and mutual spacing of L ≥ z(x), is shown by the relationship Equation (2), ( ) 2 y(x) 12 S = y z = [m ] yz (x) (x) 2 2 tan δ -   , (2) where y(x) [m] is the vertical distance of the point of the curve (parabola) from the x-axis. From the equation of a parabola (see [31,35]), the vertex of which is located at a distance hmax (m) from the x-axis, written in canonical form, the expression Equation (3) follows for y(x). According to [38] it is possible to express the relation for the calculation of the parabola parameter p [m]. If the expression for the parameter of the parabola p is substituted into the relation Equation (3), the vertical distance y(x) of the point of the parabola with respect to the position on the x-axis can be expressed according to Equation (3), 22 x x tanΘ y = h - = h - [m] (x) max max 2 p b   , (3) z(x) [m] is the distance over which the conveyed material extends on the surface of the conveyor belt in the axial direction z [31]. Figure 5. Transverse and longitudinal cross-section through a batch of transported material (a) H ≥ hmax, (b) H1  hmax. The volume of transported material V1 (m3) supplied to the conveyor belt of a belt conveyor inclined at an angle δ, spread on the surface of the conveyor belt between two belt cleats (according to Figure 5a), can be determined according to relation Equation (4). ( ) 2 y b/2 b/2 b/2 (x) 3 V = S dx = 2 S dx = dx [m ] 1 yz yz tan δ - -b/2 0 0         , (4) where Syz is the area of the batch of material on the yz plane of the conveyor belt [31]. The volume V1 (m3) of the batch of conveyed material located between two adjacent cleats is shown in Figure 5a. The theoretically determined volume V1 of conveyed material for a conveyor where the inclination angle δ = 30 ÷ 60 deg, the dynamic angle of repose Θ = 10 deg and the conveyor belt width B = 0.4, 0.5 per 0.65 m is given in Table 1. The volume V1 is calculated for the operating condition of the belt conveyor, when the conveyed Figure 5. Transverse and longitudinal cross-section through a batch of transported material (a) H ≥hmax, (b) H1≤hmax. From the equation of a parabola (see [ 31 , 35 ]), the vertex of which is located at a distance h max (m) from the x-axis, written in canonical form, the expression Equation (3) follows for y (x) . According to [ 38 ] it is possible to express the relation for the calculation of the parabola parameter p [m]. If the expression for the parameter of the parabola p is substituted into the relation Equation (3), the vertical distance y (x) of the point of the parabola with respect to the position on the x-axis can be expressed according to Equation (3), y(x)=hmax −x2 2·p=hmax −x2·tan Θ b[m], (3) z (x) [m] is the distance over which the conveyed material extends on the surface of the conveyor belt in the axial direction z [31]. The volume of transported material V 1 (m 3 ) supplied to the conveyor belt of a belt conveyor inclined at an angle δ , spread on the surface of the conveyor belt between two belt cleats (according to Figure 5a), can be determined according to relation Equation (4). V1= b/2 Z −b/2 Syz ·dx=2· b/2 Z 0 Syz ·dx = b/2 Z 0 y2 (x) tan(δ−Θ)·dx [m3], (4) where Syz is the area of the batch of material on the yz plane of the conveyor belt [31]. The volume V1(m3) of the batch of conveyed material located between two adjacent cleats is shown in Figure 5a. The theoretically determined volume V 1 of conveyed material for a conveyor where the inclination angle δ = 30 ÷ 60 deg, the dynamic angle of repose Θ= 10 deg and the conveyor belt width B = 0.4, 0.5 per 0.65 m is given in Table 1. The volume V 1 is calculated for the operating condition of the belt conveyor, when the conveyed material is delivered to the working surface of the conveyor belt with belt cleats of a height H≥hmax, which is inclined at the angle δ. Energies 2021,14, 7984 7 of 15 Table 1. Lengthwise distance, surface, and volume of a batch of loose material. δ(deg) 30 35 40 45 50 55 60 δ–Θ(deg) 20 25 30 35 40 45 50 B/b/y(x) (mm) 400/310/13.7 z(x) (mm) 37.6 29.3 23.7 19.5 16.3 13.7 11.5 Syz (mm2)256.5 200.2 161.7 133.4 111.3 93.4 78.4 V1(mm3)42.4 33.1 26.7 22.1 18.4 15.4 13.0 B/b/y(x) (mm) 500/400/17.6 z(x) (mm) 48.5 37.8 30.5 25.2 21.0 17.6 14.8 Syz (mm2)427.1 333.4 269.3 222.0 185.3 155.5 130.4 V1(mm3)91.1 71.1 57.4 47.4 39.5 33.2 27.8 B/b/y(x) (mm) 650/535/23.6 z(x) (mm) 64.8 50.6 40.9 33.7 28.1 23.6 19.8 Syz (mm2)764.1 596.4 481.7 397.2 331.4 278.1 233.4 V1(cm3)218.0 170.2 137.4 113.3 94.6 79.4 66.6 If the material is fed to the working surface of the conveyor belt, which is inclined at a known angle ( δmin < δ ) at the filling point and the inclination angle of the conveyor belt (already carrying material) gradually increases, the original volume V 1(δ–Θ) of the material batch gradually decreases as the conveying angle increases (see Tables 2–4and Figures 6–8) in front of the belt cleat (volume V2(δ–Θ)spills over the transverse partition). Energies 2021, 14, x FOR PEER REVIEW 7 of 15 material is delivered to the working surface of the conveyor belt with belt cleats of a height H ≥ hmax, which is inclined at the angle δ. Table 1. Lengthwise distance, surface, and volume of a batch of loose material. δ (deg) 30 35 40 45 50 55 60 δ–Θ (deg) 20 25 30 35 40 45 50 B/b/y(x) (mm) 400/310/13.7 z(x) (mm) 37.6 29.3 23.7 19.5 16.3 13.7 11.5 Syz (mm2) 256.5 200.2 161.7 133.4 111.3 93.4 78.4 V1 (mm3) 42.4 33.1 26.7 22.1 18.4 15.4 13.0 B/b/y(x) (mm) 500/400/17.6 z(x) (mm) 48.5 37.8 30.5 25.2 21.0 17.6 14.8 Syz (mm2) 427.1 333.4 269.3 222.0 185.3 155.5 130.4 V1 (mm3) 91.1 71.1 57.4 47.4 39.5 33.2 27.8 B/b/y(x) (mm)] 650/535/23.6 z(x) (mm) 64.8 50.6 40.9 33.7 28.1 23.6 19.8 Syz (mm2) 764.1 596.4 481.7 397.2 331.4 278.1 233.4 V1 (cm3) 218.0 170.2 137.4 113.3 94.6 79.4 66.6 If the material is fed to the working surface of the conveyor belt, which is inclined at a known angle (δmin  δ) at the filling point and the inclination angle of the conveyor belt (already carrying material) gradually increases, the original volume V1(δ–Θ) of the material batch gradually decreases as the conveying angle increases (see Tables 2–4 and Figures 6 and 7–8) in front of the belt cleat (volume V2(δ–Θ) spills over the transverse partition). Figure 6. Batch volume of the transported material (see Table 2) (a) V1(10)*1 (cm3), (b) V2(15)*1 (cm3), (c) V3(15)*1 (cm3), (d) V2(20)*1 (cm3). The volume of the batch of material V2(δ–Θ), which overflows over the upper edge of the belt cleat at a given angle of inclination of the conveyor belt, is given in Tables 2–4 for belt width B, dynamic angle of repose and belt cleat height y(x). The volumes V2(δ–Θ) and V3(δ–Θ) (Tables 2–4) of batches (remaining in front of the belt cleat and overflowing across the belt cleat) of the transported mass were determined from the models created in the 3D CAD environment of SolidWorks, see Figure 6 (and also Figures 7–9). If the height of the transverse belt cleat H1 is less than the maximum height hmax of the canopy of the cross-sectional area of the conveyed mass S1 on the xy plane, the volume of conveyed material V3(δ–Θ) [m3] spread on the surface of the conveyor belt surface between two belt cleats (see Figure 5b) can be determined according to the relationship Equation (5), c/2 V = V - V = V - 2 S dx = 1yz 2( - ) 1( - ) 3( - ) 1( - ) 0           ( ) ( ) ( ) 2 2 2h c - x tanΘ y c/2 c/2 1 1(x) 3 = V - dx = V - dx [m ] 1( - ) 1( - ) 2 tan δ - c tan δ - 00       , (5) Figure 6. Batch volume of the transported material (see Table 2) ( a ) V 1(10)*1 (cm 3 ), ( b ) V 2(15)*1 (cm 3 ), ( c ) V 3(15)*1 (cm 3 ), (d) V2(20)*1 (cm3). Energies 2021, 14, x FOR PEER REVIEW 8 of 15 where S1yz [m2] is the longitudinal cross-section of a batch of conveyed material of volume V3(δ–Θ), which has spilled over the belt cleat of height H1, (Figures 5b and 6), c/2 [m] Equation (6) is the horizontal distance of the intersection of the curve (= parabola) of the layer of conveyed material with the upper edge of the belt cleat of height H1 in the xy plane, see Figure 5b, ( ) 82 b h - H 2 b h max 1 1 h = h - H [m], c = p h = = [m] 1 max 1 1 tan tanΘ     , (6) The volume of the batch of conveyed material V3(δ–Θ), which is located between two adjacent cleats spaced apart by the value L  z(x) (Figure 5b) is shown in Figures 7–9. The experimentally obtained values on the test equipment were created in the Laboratory Research and Testing of Institute of Transport, Faculty mechanical Engineering, VSB - Technical University Ostrava. If the height of the belt cleat H1 is known, it is possible to calculate the required used loading width bp of the conveyor belt, according to Relation (7), 4·H1 b = [m] ptan , (7) The volume V1 = V1(δ–Θ) of the bulk material spread on the surface of the trough in front of the belt cleat can be expressed by the relation Equation (4) for the angle of inclination of the belt conveyor, provided that the belt cleat reaches a height of at least H1. Table 2. Volume of material Vi(δ–Θ) [cm3] for H1 = 10 mm belt cleat. H1 (mm)/bp (mm) 10/149.3 δ–Θ (deg) 10 15 20 25 30 z(x)(δ–Θ) (mm) 56.7 37.3 27.5 21.5 17.3 V1(δ–Θ) (cm3) 22.6 1,2 14.9 10.9 8.5 6.9 - V3(δ–Θ) (cm3) 18.1 1,2 12.8 9.8 7.8 - V2(δ–Θ) (cm3) 4.5 1,2 2.1 1.1 0.7 - - - 14.3 2 10.7 9.1 - - - 3.8 1,2 2.1 5.8 - - - - 11.5 2 9.0 - - - - 2.7 2 1.7 - - - - - 9.5 2 - - - - - 2.0 2 1 see Figure 6, 2 see Figure 7. Figure 7. Volumes of material batches remaining on the conveyor belt and spilling over the transverse belt cleat of a height H1 = 10 mm. Figure 7. Volumes of material batches remaining on the conveyor belt and spilling over the transverse belt cleat of a height H1= 10 mm. Energies 2021,14, 7984 8 of 15 Energies 2021, 14, x FOR PEER REVIEW 9 of 15 Table 3. Volume of material Vi(δ–Θ) [cm3] for H1 = 15 mm belt cleat. H1 (mm)/bp (mm) 15/224.0 δ–Θ (deg) 10 15 20 25 30 z(x)(δ–Θ) (mm) 85.1 56.0 41.2 32.2 26.0 V1(δ–Θ) (cm3) 76.2 50.1 3 36.9 28.8 23.3 - V2(δ–Θ) (cm3) 61.0 43.4 3 33.1 26.4 - V3(δ–Θ) (cm3) 15.2 6.8 3 3.8 2.4 - - - 48.0 36.3 3 28.9 - - - 13.0 7.1 3 4.2 - - - - 38.8 30.9 3 - - - - 9.2 5.4 3 - - - - - 32.5 -- - - - - 6.3 3 see Figure 8. Figure 8. Volume of material remaining on the conveyor belt and spilling over the belt cleat of a height of H1 = 15 mm. Table 4. Volume of material Vi(δ–Θ) [cm3] for H1 = 20 mm belt cleat. H1 (mm)/bp (mm) 20/298.6 δ–Θ (deg) 10 15 20 25 30 z(x)(δ–Θ) (mm) 113.4 74.6 55.0 42.9 34.6 V1(δ–Θ) (cm3) 180.6 118.9 87.5 4 68.3 55.2 - V3(δ–Θ) (cm3) 144.5 102.8 78.5 4 62.5 - V2(δ–Θ) (cm3) 36.1 16.1 9.0 4 5.8 - - - 113.8 85.9 67.9 4 - - - 30.7 16.9 10.6 4 - - - - 92.1 72.3 - - - - 21.7 13.6 - - - - - 76.1 - - - - - 16.0 4 see Figure 9. The theoretically calculated volumes Vi(δ–Θ) of the respective batches of transported material (according to relation Equation (4) and relation Equation (5), depending on the inclination of the conveyor belt δ (resp. (δ–Θ)) are given in Tables 2–4) are determined in Figure 8. Volume of material remaining on the conveyor belt and spilling over the belt cleat of a height of H1= 15 mm. The volume of the batch of material V 2(δ–Θ) , which overflows over the upper edge of the belt cleat at a given angle of inclination of the conveyor belt, is given in Tables 2–4for belt width B, dynamic angle of repose and belt cleat height y (x) . The volumes V 2(δ–Θ) and V 3(δ–Θ) (Tables 2–4) of batches (remaining in front of the belt cleat and overflowing across the belt cleat) of the transported mass were determined from the models created in the 3D CAD environment of SolidWorks, see Figure 6(and also Figures 7–9). Energies 2021, 14, x FOR PEER REVIEW 10 of 15 Figures 7–9 (to check the accuracy of the calculation) according to the models created in the 3D CAD environment of the SolidWorks system. Figure 10 shows the volumes V3(δ–Θ) of partial batches of the transported material captured by a belt cleat with a height of H1 for inclination angle (δ–Θ). Figure 9. Batch volume V3(δ–Θ) material, spread on the conveyor belt, retained by a belt cleat with a height of H1 = 20 mm. Figure 10 shows the volumes V2(δ–Θ) of the transported material, which spilled over the upper edge of the belt cleat of a height H1 for inclination angle (δ-Θ) (deg). Figure 10. Batch volume V2(δ–Θ) of material spilling over a belt cleat of a height H1 = 20 mm. 3.2. Experimentally Determined Distribution of Material and Its Volumes A batch of material of volume V1(10) was applied to the upper surface of the trough I of the test equipment (Figure 3) inclined by an angle (see Tables 2–4). The slope of the trough was then increased, always by increments of 5 deg, and the values (for three successive measurements) of the distance of material distribution on the trough surface and the volume of material that spilled over the belt cleat were recorded in Tables 5–6, as determined by subtraction in the measuring cylinder. Figure 9. Batch volume V 3(δ–Θ) material, spread on the conveyor belt, retained by a belt cleat with a height of H1= 20 mm. If the height of the transverse belt cleat H 1 is less than the maximum height h max of the canopy of the cross-sectional area of the conveyed mass S 1 on the xy plane, the volume of Energies 2021,14, 7984 9 of 15 conveyed material V 3(δ–Θ) [m 3 ] spread on the surface of the conveyor belt surface between two belt cleats (see Figure 5b) can be determined according to the relationship Equation (5), V2(δ−Θ)=V1(δ−Θ)−V3(δ−Θ)=V1(δ−Θ)−2· c/2 R0 S1yz ·dx = =V1(δ−Θ)− c/2 R0 y2 1(x) tan(δ−Θ)·dx=V1(δ−Θ)− c/2 R0 (h1·c−x2·tan Θ)2 c2·tan(δ−Θ)·dx [m3], (5) where S 1yz [m 2 ] is the longitudinal cross-section of a batch of conveyed material of volume V 3(δ–Θ) , which has spilled over the belt cleat of height H 1 , (Figures 5b and 6), c/2 [m] Equation (6) is the horizontal distance of the intersection of the curve (= parabola) of the layer of conveyed material with the upper edge of the belt cleat of height H 1 in the xy plane, see Figure 5b, h1=hmax −H1[m], c =p8·p·h1=r2·b·h1 tan Θ=r2·b·(hmax −H1) tan Θ[m], (6) The volume of the batch of conveyed material V 3(δ–Θ) , which is located between two adjacent cleats spaced apart by the value L ≥z(x) (Figure 5b) is shown in Figures 7–9. The experimentally obtained values on the test equipment were created in the Laboratory Research and Testing of Institute of Transport, Faculty mechanical Engineering, VSB - Technical University Ostrava. If the height of the belt cleat H 1 is known, it is possible to calculate the required used loading width bpof the conveyor belt, according to Relation (7), bp=4·H1 tan Θ[m], (7) The volume V 1 = V 1(δ–Θ) of the bulk material spread on the surface of the trough in front of the belt cleat can be expressed by the relation Equation (4) for the angle of inclination of the belt conveyor, provided that the belt cleat reaches a height of at least H 1 . The theoretically calculated volumes V i(δ–Θ) of the respective batches of transported material (according to relation Equation (4) and relation Equation (5), depending on the inclination of the conveyor belt δ (resp. ( δ – Θ )) are given in Tables 2–4) are determined in Figures 7–9(to check the accuracy of the calculation) according to the models created in the 3D CAD environment of the SolidWorks system. Table 2. Volume of material Vi(δ–Θ)[cm3]for H1= 10 mm belt cleat. H1(mm)/bp(mm) 10/149.3 δ–Θ(deg) 10 15 20 25 30 z(x)(δ–Θ)(mm) 56.7 37.3 27.5 21.5 17.3 V1(δ–Θ)(cm3) 22.6 1,2 14.9 10.9 8.5 6.9 -V3(δ–Θ)(cm3) 18.1 1,2 12.8 9.8 7.8 -V2(δ–Θ)(cm3) 4.5 1,2 2.1 1.1 0.7 -- - 14.3 210.7 9.1 -- - 3.8 1,2 2.1 5.8 -- - - 11.5 29.0 -- - - 2.7 21.7 -- - - - 9.5 2 -- - - - 2.0 2 1see Figure 6,2see Figure 7.