Transport Performance of a Steeply Situated Belt Conveyor

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.

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 [4,5,6]. 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 (m³·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 (m²) [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 (m²) of the load of the transported material is determined according to [9,12,13]. If the 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.

$$\mathrm{k}=1-\frac{{\mathrm{S}}_1}{\mathrm{S}}\cdot \left({1-\mathrm{k}}_1\right){(-),\mathrm{k}}_1=\sqrt{\frac{{\cos }^2{\mathsf{\delta }-\cos }^2\mathsf{\Theta }}{{1-\cos }^2\mathsf{\Theta }}}(-),$$

(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 δ_m (deg) of a belt conveyor of conventional construction, is to use a conveyor belt with belt cleats [15,16], Figure 1.

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].

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.

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.

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 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 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 V_1(δ–Θ) of the material batch (Table 2, Table 3 and Table 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 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.

The results obtained from the performed experimental measurements (Table 5 and Table 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 δ_i, the surface of the trough was tilted by an angle δ_i+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 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 S_yz (m²), 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),

$${\mathrm{S}}_{\mathrm{yz}}=\frac{1}{2}\cdot {\mathrm{y}}_{\left(\mathrm{x}\right)}\cdot {\mathrm{z}}_{\left(\mathrm{x}\right)}=\frac{{\mathrm{y}}_{\left(\mathrm{x}\right)}^2}{2\cdot \tan \left(\mathsf{\delta }-\mathsf{\Theta }\right)}{[\mathrm{m}}^2],$$

(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 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),

$${\mathrm{y}}_{\left(\mathrm{x}\right)}{=\mathrm{h}}_{\max }-\frac{{\mathrm{x}}^2}{2\cdot \mathrm{p}}{=\mathrm{h}}_{\max }-\frac{{\mathrm{x}}^2\cdot \tan \mathsf{\Theta }}{\mathrm{b}}\left[\mathrm{m}\right],$$

(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 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).

$${\mathrm{V}}_1={\displaystyle \underset{-\mathrm{b}/2}{\overset{\mathrm{b}/2}{\int }}{\mathrm{S}}_{\mathrm{yz}}\cdot \mathrm{dx}}=2\cdot {\displaystyle \underset{0}{\overset{\mathrm{b}/2}{\int }}{\mathrm{S}}_{\mathrm{yz}}\cdot \mathrm{dx}}={\displaystyle \underset{0}{\overset{\mathrm{b}/2}{\int }}\frac{{\mathrm{y}}_{\left(\mathrm{x}\right)}^2}{\tan \left(\mathsf{\delta }-\mathsf{\Theta }\right)}\cdot \mathrm{dx}}{[\mathrm{m}}^3],$$

(4)

where S_yz 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 material is delivered to the working surface of the conveyor belt with belt cleats of a height H ≥ h_max, 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

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 (δ_min < 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 Table 2, Table 3 and Table 4 and Figure 6, Figure 7 and Figure 8) in front of the belt cleat (volume V_2(δ–Θ) spills over the transverse partition).

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 Table 2, Table 3 and Table 4 for belt width B, dynamic angle of repose and belt cleat height y_(x). The volumes V_2(δ–Θ) and V_3(δ–Θ) (Table 2, Table 3 and Table 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 Figure 7, Figure 8 and Figure 9).

If the height of the transverse belt cleat H₁ is less than the maximum height h_max of the canopy of the cross-sectional area of the conveyed mass S₁ on the xy plane, the volume of conveyed material V_3(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),

$$\begin{gathered} {\mathrm{V}}_{2(\mathsf{\delta }-\mathsf{\Theta })}{=\mathrm{V}}_{1(\mathsf{\delta }-\mathsf{\Theta })}{-\mathrm{V}}_{3(\mathsf{\delta }-\mathsf{\Theta })}{=\mathrm{V}}_{1(\mathsf{\delta }-\mathsf{\Theta })}-2\cdot {\displaystyle \underset{0}{\overset{\mathrm{c}/2}{\int }}{\mathrm{S}}_{1\mathrm{yz}}\cdot \mathrm{dx}=} \\ {=\mathrm{V}}_{1(\mathsf{\delta }-\mathsf{\Theta })}-{\displaystyle \underset{0}{\overset{\mathrm{c}/2}{\int }}\frac{{\mathrm{y}}_{1\left(\mathrm{x}\right)}^2}{\tan \left(\mathsf{\delta }-\mathsf{\Theta }\right)}\cdot \mathrm{dx}}{=\mathrm{V}}_{1(\mathsf{\delta }-\mathsf{\Theta })}-{\displaystyle \underset{0}{\overset{\mathrm{c}/2}{\int }}\frac{{\left({\mathrm{h}}_1\cdot {\mathrm{c}-\mathrm{x}}^2\cdot \tan \mathsf{\Theta }\right)}^2}{{\mathrm{c}}^2\cdot \tan \left(\mathsf{\delta }-\mathsf{\Theta }\right)}\cdot \mathrm{dx}}{[\mathrm{m}}^3], \end{gathered}$$

(5)

where S_1yz [m²] 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 H₁, (Figure 5b and Figure 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,

$${\mathrm{h}}_1{=\mathrm{h}}_{\max }{-\mathrm{H}}_1\left[\mathrm{m}\right],\mathrm{c}=\sqrt{8\cdot \mathrm{p}\cdot {\mathrm{h}}_1}=\sqrt{\frac{2\cdot \mathrm{b}\cdot {\mathrm{h}}_1}{\tan \mathsf{\Theta }}}=\sqrt{\frac{2\cdot \mathrm{b}\cdot \left({\mathrm{h}}_{\max }{-\mathrm{H}}_1\right)}{\tan \mathsf{\Theta }}}\left[\mathrm{m}\right],$$

(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 Figure 7, Figure 8 and Figure 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 b_p of the conveyor belt, according to Relation (7),

$${\mathrm{b}}_{\mathrm{p}}=\frac{{4\text{·}\mathrm{H}}_1}{\tan \mathsf{\Theta }}\left[\mathrm{m}\right],$$

(7)

The volume V₁ = 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 H₁.

Table 2. Volume of material V_i(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.

Table 2. Volume of material V_i(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.

Table 3. Volume of material V_i(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.

Table 3. Volume of material V_i(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.

Table 4. Volume of material V_i(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.

Table 4. Volume of material V_i(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 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 Table 2, Table 3 and Table 4) are determined in Figure 7, Figure 8 and Figure 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 V_3(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 10 shows the volumes V_2(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).

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 Table 2, Table 3 and Table 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 Table 5 and Table 6, as determined by subtraction in the measuring cylinder.

Figure 11. Batch volume V_i [cm³] buckwheat spread on the surface of the conveyor belt for H₁ = 10 mm. Length of the distribution z_(x) for δ–Θ (deg) (a) 15 deg, (b) 20 deg and (c) 25 deg.

Figure 11. Batch volume V_i [cm³] buckwheat spread on the surface of the conveyor belt for H₁ = 10 mm. Length of the distribution z_(x) for δ–Θ (deg) (a) 15 deg, (b) 20 deg and (c) 25 deg.

Figure 12. Batch volume V_i [cm³] of barley groats spread on the surface of the conveyor belt at H₁ = 15 mm. Length of the distribution z_(x) for (δ–Θ) (deg) (a) 10 deg, (b) 15 deg, (c) 20 deg and (d) 25 deg.

Figure 12. Batch volume V_i [cm³] of barley groats spread on the surface of the conveyor belt at H₁ = 15 mm. Length of the distribution z_(x) for (δ–Θ) (deg) (a) 10 deg, (b) 15 deg, (c) 20 deg and (d) 25 deg.

Table 5. Volume of material V_i(δ–Θ) [cm³] for H₁ = 10 mm belt cleat.

Material	Buckwheat					Barley Greats
H1/bp (mm)	10/149.3
δ–Θ (deg)	10	15	20	25	30	10	15	20	25	30
z(x)(δ–Θ) (mm)	60	44	32 5	28 5	21	59	42	31	24	19
	63	42 5	34	26	22	60	45	33	22	21
	61	41	33	29	21	59	43	32	21	21
Σ z(x)(δ–Θ)	184	127	99	83	64	178	130	96	67	61
Σ z(x)(δ–Θ)/n	61.3	42.3	33.0	27.6	21.3	59.3	43.3	32.0	22.3	20.3
V3(δ–Θ) (cm3)	23	18	14	11	10	23	19	15	12	10
V2(δ–Θ) (cm3)	-	5	4	3	2	-	4	4	3	2

⁵ see Figure 11.

Table 5. Volume of material V_i(δ–Θ) [cm³] for H₁ = 10 mm belt cleat.

Material	Buckwheat					Barley Greats
H1/bp (mm)	10/149.3
δ–Θ (deg)	10	15	20	25	30	10	15	20	25	30
z(x)(δ–Θ) (mm)	60	44	32 5	28 5	21	59	42	31	24	19
	63	42 5	34	26	22	60	45	33	22	21
	61	41	33	29	21	59	43	32	21	21
Σ z(x)(δ–Θ)	184	127	99	83	64	178	130	96	67	61
Σ z(x)(δ–Θ)/n	61.3	42.3	33.0	27.6	21.3	59.3	43.3	32.0	22.3	20.3
V3(δ–Θ) (cm3)	23	18	14	11	10	23	19	15	12	10
V2(δ–Θ) (cm3)	-	5	4	3	2	-	4	4	3	2

⁵ see Figure 11.

Table 6. Volume of material V_i(δ–Θ) [cm³] for H₁ = 15 mm belt cleat.

Material	Buckwheat					Barley Greats
H1/bp (mm	15/224
δ–Θ (deg)	10	15	20	25	30	10	15	20	25	30
z(x)(δ–Θ) (mm)	100	89	49	39	31	94	86	52	39	28
	96	85	53	40	30	98 6	84	47 6	40	26
	98	86	51	36	28	97	83 6	49	36 6	29
Σ z(x)(δ–Θ)	294	260	153	115	89	289	253	148	115	83
Σ z(x)(δ–Θ)/n	98.0	86.6	51.0	38.3	29.7	96.3	84.3	49.3	38.3	27.6
V3(δ–Θ) [cm3]	77	60	49	39	33	77	61	48	39	32
V2(δ–Θ) [cm3]	-	17	11	10	6	-	16	13	9	7

⁶ see Figure 12.

Table 6. Volume of material V_i(δ–Θ) [cm³] for H₁ = 15 mm belt cleat.

Material	Buckwheat					Barley Greats
H1/bp (mm	15/224
δ–Θ (deg)	10	15	20	25	30	10	15	20	25	30
z(x)(δ–Θ) (mm)	100	89	49	39	31	94	86	52	39	28
	96	85	53	40	30	98 6	84	47 6	40	26
	98	86	51	36	28	97	83 6	49	36 6	29
Σ z(x)(δ–Θ)	294	260	153	115	89	289	253	148	115	83
Σ z(x)(δ–Θ)/n	98.0	86.6	51.0	38.3	29.7	96.3	84.3	49.3	38.3	27.6
V3(δ–Θ) [cm3]	77	60	49	39	33	77	61	48	39	32
V2(δ–Θ) [cm3]	-	17	11	10	6	-	16	13	9	7

⁶ see Figure 12.

4. Discussion

The experimentally obtained values of the distribution distance of the bulk material (see Table 5 and Table 6) on the surface of the trough differ from the values of z_(x), theoretically calculated according to [30] and also according to Table 2, Table 3 and Table 4. This fundamental difference of the two values can be explained by the impossibility of determining the exact value of the dynamic angle of repose Θ of the material batch, which is reported by the performed partial experimental measurements on the test equipment (Figure 3). The static angle of repose (see Figure 4) of the bulk material is generally defined as the angle that the tangent of the slope of the bulk material makes to the plane of the base surface on which the bulk material is poured. The dynamic angle of repose Θ arises, if the base oscillates, as a result of the grain rearrangement in the layered cone of bulk material with the tangent slope of the material to the plane of the base surface at an angle ψ at the moment of the beginning of shaking. The vibration of the base surface can, when bulk material is carried away on the surface of the conveyor belt of the belt conveyor, be expressed by the repeated deflection of the conveyor belt (so-called rolling resistance) between two adjacent roller supports during its movement along the circumference of the entire working length of the belt conveyor. The amount of deflection of the conveyor belt is influenced, among other things, by the centrifugal forces [2,5] acting on the conveyor belt and the conveyed material as it moves along the deflection curve [38]. The need to increase the tension in the belt with a view to reducing the drag is indicated in connection with higher belt speeds [39].

The results of theoretical assumptions confirmed by experimental measurements show that the distribution of loose material in the longitudinal direction on the surface of the trough (conveyor belt) decreases with an increasing angle of inclination δ of the conveyor and with increasing value of dynamic angle of repose Θ. The value of the dynamic angle of repose of the bulk material decreases from its maximum Θmax = ψ [deg] to its minimum Θmin = Θ [deg]. The actual value of the dynamic angle of repose of the conveyed material (and thus also the volume of each sub-batch) between the two adjacent belt cleats installed on the conveyor belt depends on the degree (i.e., intensity and length of time) of shaking the bulk material batch.

The size of the batch V_3(δ–Θ) of material, which is prevented from moving on the surface of the conveyor belt due to the installed belt cleat, depends on the height H and the length b_p of the cleat. The maximum length of the cleat b_p is generally chosen so as to prevent the conveyed grains of material from falling over the edges of the conveyor belt. The length b_p of the cleat is optimally equal to the used loading width of the conveyor belt b [10,11]. A lower selected cross-sectional length causes a reduction in the cross-section of the bulk material S_xy [31], since the calculated cross-sectional area of the belt filling S₁ [11] is proportional to the square of the used loading width of the belt b.

The minimum height of the belt cleat H should be chosen with a size corresponding to the height h_max (Figure 5a) of the parabolic cross-sectional area S₁ of the transported material. The lower height of the belt cleat (see H₁ in Figure 5b) causes the volume V_2(δ–Θ) of the batch of conveyed material to spill over the belt cleat at the moment the material is fed to the conveyor belt on a steeply inclined belt conveyor.

To prevent the material grains from overflowing over the upper edge of the belt cleat of height H conveyed at an angle of inclination δ (which is the maximum of all inclinations of the angled belt conveyor sections) it is necessary to feed a precisely measured volume of material V1(δ–Θ), the magnitude of which can be determined according to Equation (4), onto the conveyor belt, into the space between the two neighboring belt cleats, if the angle ψ is input after the angle of repose Θ.

5. Conclusions

Continuous transport of loose fine and coarse-grained bulk materials by belt conveyors, the angle of inclination of which exceeds the limit angle of inclination of transport of belt conveyors of classical construction, can be performed by conveyor belts with belt cleats. The transverse belt cleats, of the respective heights and construction, are mounted on the working surface of the endless loop of the conveyor belt at regular intervals, the size of which must be suitably designed. The choice of the mutual spacing of belt cleats is influenced in particular by the longitudinal distribution of the batch of transported material retained by the belt cleat of a given height and length.

The maximum distance over which a batch of conveyed bulk material extends in the longitudinal axis of the conveyor belt can be expressed according to the relationship given in the paper. The size of this distance as well as the volume of the batch of transported material significantly affects the height of the transverse partition and the value of the difference between the angle of inclination of the transport and the angle of repose of the material. The paper also presents the equations for calculating the volume of a batch of conveyed material, according to which it is possible to analytically determine the theoretical volume of bulk material, which is retained by a belt cleat installed on the working surface of the conveyor belt, and which is angled in relation to horizontal plane at an angle of δ.

Capacity increase of transport capability, i.e., the amount of transported material per time unit, of a steeply inclined belt conveyor can be influenced by the speed of movement of the conveyor belt, and by the volume and the spacing of the batches of material carried by the conveyor belt.

The volumetric size of the batch of material which extends along the working surface of the conveyor belt in front of the belt cleat is proportional to the width and height of the belt cleat and is greatly affected by the angle expressed as the difference between the transport angle and the material angle of repose.

The cross-section of the filling of the straight and trough loading profile of the conveyor belt is expressed in the professional literature and technical standards using the dynamic angle of repose Θ. When grains of loose bulk material are carried by a conveyor belt, the individual grains of the material are exposed to dynamic effects, e.g., due to belt deflection between adjacent roller supports, which cause the material grains to slide down a slope (whose tangent to the horizontal plane is at an angle ψ) of bulk material spread on the conveyor belt and spreading them over a larger area.

It is difficult and practically impossible to express in general what actual values the angle of repose (minimum Θ and maximum ψ) of the transported material acquires at a given moment of the total time when the material is transported by the conveyor belt.

The test device described in the paper was created in the laboratory. Experimentally obtained values of the distance over which the batch of conveyed bulk material extends on the longitudinal axis of the conveyor belt, on the test device created, can be used for validation with mathematically determined values (see Table 2, Table 3 and Table 4) and also for verification of practically measured data (Table 5 and Table 6) with theoretical data, which are listed in Table 1.