A Mathematical Model for Conical Hopper Mass Efficiency
Abstract
:1. Introduction
2. Materials and Methods
2.1. Mathematical Model of Granular Material Discharge from the Conical Hopper—Assessments
- —frictional force [N];
- —force normal to the hopper walls [N];
- —coefficient of friction of material sliding along the hopper walls [–];
- —diameter of the hopper outlet [m];
- —height of the considered elementary volume [m];
- —pressure of the material acting on the hopper walls [Pa].
- —weight of the elemental volume [N];
- —density of the granular material [];
- —acceleration of gravity [].
- —packing angle of the material inside the hopper [°];
- —pressure acting on the upper surface of the elemental volume [Pa].
- —downward acceleration of the elemental volume []
- x—distance from the origin to the cross-section [m].
- —height of the elemental volume (EV) of the granular material [m];
- —cross-sectional area, varying with the displacement of the EV [m2].
- —diameter at the upper part of the hopper [m];
- —diameter at the lower part of the hopper [m];
- —hopper height [m];
- —inclination angle of the cone generatrix [°].
- —acceleration of the material in elemental volume [];
- —increase in the cross-sectional area of the elementary volume [m2];
- —weight of the elementary volume of the material [N];
- —angle of repose for the considered material [°];
- —angle of the given material’s external friction (material-to-vessel) [°].
- —the simplifying coefficient [].
- —a solution to the homogenous Equation (21);
- —a partial solution of the non-homogeneous Equation (19).
- B—constant;
- K(z)—a coefficient determined by Formula (15);
- z—the integration variable.
- —the height of the whole cone [m].
- , [m];
- , [m];
- , [Pa];
- —density of the material, [];
- pa—atmospheric pressure, [Pa].
2.2. Mathematical Model of Granular Material Discharge from the Conical Hopper—Simulation
3. Results
4. Discussion
- The results of the experimental studies are consistent with the results of the theoretical modeling.
- The difference between the experimental data and the theoretical modeling results does not exceed 5%.
- For D = 0.05 m and α = 4°, Qk = 12.36 kg/s;
- For D = 0.05 m and α = 12°, Qk = 4.56 kg/s;
- For D = 0.10 m and α = 4°, Qk = 49.33 kg/s;
- For D = 0.10 m and α = 12°, Qk = 21.67 kg/s.
- For D = 0.05 m and α = 4°, Qk = 36.15 kg/s;
- For D = 0.05 m and α = 12°, Qk = 11.59 kg/s;
- For D = 0.10 m and α = 4°, Qk = 144.3 kg/s;
- For D = 0.10 m and α = 12°, Qk = 55.08 kg/s.
5. Conclusions
- the density of the bulk material;
- the angle of particle packing in the overall mass;
- the external friction angle of the material;
- the internal friction angle of the material (angle of repose);
- the coefficient of sliding friction of the bulk material against the hopper walls.
- the generatrix angle of the hopper cone;
- the height of the hopper;
- the diameter of the discharge orifice.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Feature | Buckwheat | Coal |
---|---|---|
Density, ρ [] | 560 | 900 |
Particle packing angle, β [°] | 56 | 35 |
External friction angle, φ [°] | 15 | 25 |
Internal friction angle, ψ [°] | 17 | 34 |
Material-wall friction coefficient [-] | 0.47 | 0.47 |
Cone angle, α [°], step: 2° | 4–12 | 2–12 |
Hopper height, [m] | 0.5 | 0.5 |
Hopper orifice diameter, [m], step: 0.025 m | 0.025–0.125 | 0.025–0.125 |
Factors | Designation | Unit | Levels of Factors | Variation Interval | ||
---|---|---|---|---|---|---|
Upper | Null | Lower | ||||
Code Values | ||||||
+1 | 0 | −1 | ||||
Hopper orifice diameter, D | x1 | m | 1.0 | 0.075 | 0.050 | 0.025 |
Hopper cone generatrix angle, α | x2 | deg. | 10 | 8 | 6 | 2 |
Experiment No. | x1 | x2 | x1·x2 | y1 (QK), Buckwheat, kg/s | y2 (QK), Coal, kg/s | ||
---|---|---|---|---|---|---|---|
1 | +1 | +1 | +1 | 0.3333 | 0.3333 | 25.0 | 65.3 |
2 | 0 | +1 | 0 | −0.6667 | 0.3333 | 13.4 | 35.0 |
3 | −1 | +1 | −1 | 0.3333 | 0.3333 | 5.4 | 14.0 |
4 | +1 | −1 | −1 | 0.3333 | 0.3333 | 37.0 | 103.0 |
5 | 0 | −1 | 0 | −0.6667 | 0.3333 | 20.5 | 57.0 |
6 | −1 | −1 | +1 | 0.3333 | 0.3333 | 8.6 | 24.0 |
7 | +1 | 0 | 0 | 0.3333 | −0.6667 | 29.8 | 80.0 |
8 | 0 | 0 | 0 | −0.6667 | −0.6667 | 16.2 | 43.0 |
9 | −1 | 0 | 0 | 0.3333 | −0.6667 | 6.6 | 18.0 |
∑ | 6 | 6 | 4 | - | - | - | - |
Coefficient of the Regression Equation | Coded Coefficient | Real Coefficient |
---|---|---|
b0 | 43.222 | 3.557 |
b1 | 32.0 | 1039.99 |
b2 | −11.667 | −6.0 |
b12 | −7.0 | −140.0 |
b11 | 5.667 | 9066.72 |
b22 | 2.667 | 0.667 |
Coefficient of the Regression Equation | Coded Coefficient | Real Coefficient |
---|---|---|
b0 | 16.178 | −0.123 |
b1 | 11.867 | 338.676 |
b2 | −3.717 | −1.692 |
b12 | −2.2 | −44.0 |
b11 | 2.033 | 3253.28 |
b22 | 0.783 | 0.196 |
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Bembenek, M.; Dmytriv, V.; Banha, V.; Horodniak, R.; Pawlik, J. A Mathematical Model for Conical Hopper Mass Efficiency. Appl. Sci. 2024, 14, 7373. https://doi.org/10.3390/app14167373
Bembenek M, Dmytriv V, Banha V, Horodniak R, Pawlik J. A Mathematical Model for Conical Hopper Mass Efficiency. Applied Sciences. 2024; 14(16):7373. https://doi.org/10.3390/app14167373
Chicago/Turabian StyleBembenek, Michał, Vasyl Dmytriv, Vasyl Banha, Roman Horodniak, and Jan Pawlik. 2024. "A Mathematical Model for Conical Hopper Mass Efficiency" Applied Sciences 14, no. 16: 7373. https://doi.org/10.3390/app14167373