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),
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),
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).
where S
yz is the area of the batch of material on the yz plane of the conveyor belt [
31].
The volume V
1 (m
3) 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 ≥ h
max, which is inclined at the angle δ.
Table 1.
Lengthwise distance, surface, and volume of a batch of loose material.
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 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 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),
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, (
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 H
1 in the xy plane, see
Figure 5b,
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 H
1 is known, it is possible to calculate the required used loading width b
p of the conveyor belt, according to Relation (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.
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 |
Table 3.
Volume of material Vi(δ–Θ) [cm3] for H1 = 15 mm belt cleat.
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 |
Table 4.
Volume of material Vi(δ–Θ) [cm3] for H1 = 20 mm belt cleat.
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 |
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 H
1 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 H
1 for inclination angle (δ–Θ) (deg).
3.2. Experimentally Determined Distribution of Material and Its Volumes
A batch of material of volume V
1(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 Vi [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 Vi [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 Vi [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 Vi [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 Vi(δ–Θ) [cm3] for H1 = 10 mm belt cleat.
Table 5.
Volume of material Vi(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 |
Table 6.
Volume of material Vi(δ–Θ) [cm3] for H1 = 15 mm belt cleat.
Table 6.
Volume of material Vi(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 |