# A Parametric Energy Model for Energy Management of Long Belt Conveyors

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Conveyor Model

#### 2.1. Conveyor Resistances

#### 2.1.1. Primary and Slope Resistances

#### 2.1.2. Special Resistance

#### 2.1.3. Secondary Resistance

#### 2.2. Modeling Energy Consumption

#### 2.3. Modeling Bulk Material Flow

**Figure 2.**Wave-like property of material flow on the belt conveyor (BC). (

**a**) time = t (15 min); (

**b**) time = t + $\delta t$ (39 min).

## 3. Model Verification

#### 3.1. Steady-State Power Calculations

Unit | Component | |||
---|---|---|---|---|

Total | ${\phi}_{1}$ | ${\phi}_{2}\xb7\overline{q}$ | ${C}_{5}\xb7{\overline{q}}^{2}$ | |

L = 500 m | ||||

kW | 191.3 | 41.7 | 145.7 | 3.9 |

% | 100.0 | 21.8 | 76.1 | 2.1 |

L = 2 km | ||||

kW | 508.1 | 123.7 | 380.4 | 3.9 |

% | 100.0 | 74.9 | 24.3 | 0.8 |

**Figure 4.**Percentage differences in power consumption calculations for the ZX and non-linear models relative to the proposed model.

#### 3.2. Variable Loading Calculations

**Figure 9.**Calculated (Algorithm 1) and modeled amounts of material delivered by the conveyor after a given amount of time.

## 4. Parameter Identification

#### Parameter Estimation

**Figure 12.**Convergence of the no-load parameter, ${\phi}_{1}$, for different measurement noise levels.

**Figure 13.**Convergence of the density parameter, ${\phi}_{2}$, for different measurement noise levels.

## 5. Application Case-Study

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations/Nomenclature

Abbreviations: | |

BC | Belt conveyor. |

CBS | Conveyor belt system. |

FDM | Finite difference method. |

PDE | Partial differential equation. |

SS | steady-state. |

Symbols: | |

η | Conveyor drive system efficiency. |

π | Hourly electricity price R/kWh. |

${\phi}_{1}$ | Energy model no-load parameter, N. |

${\phi}_{2}$ | Energy model density parameter, ${\text{m}}^{2}/{\text{s}}^{2}$. |

A | Cross-sectional area of material on the belt. |

C | A belt resistance factor. |

E | Energy consumption of the conveyor, kWh. |

F | Force on the conveyor belt, N. |

H | Conveyor elevation height, m. |

I | Input material feed rate, kg/s. |

${M}_{\text{in}}$ | Mass of material entering the conveyor, kg. |

${M}_{\text{out}}$ | Mass of material discharged by the conveyor, kg. |

${G}_{n},{\mathbf{b}}_{n}$ | Matrix and vector values of the discrete model. |

${S}_{{\phi}_{1}}^{P},{S}_{{\phi}_{2}}^{P}$ | Model parameter sensitivity to output power. |

$S{T}_{n}$ | Storage level at sample time n. |

$S{T}_{L},S{T}_{U}$ | Lower and upper storage bound. |

${N}_{x},{N}_{t}$ | Number of space and time samples in discrete domain. |

L | Total length of the conveyor belt. |

$\tilde{L}$ | Length of CB covered by material with a uniform $q(x,t)$. |

P | Power consumed by the conveyor, kW. |

$q(x,t)$ | Material mass per unit length at position x at time t, kg/m. |

${q}_{\text{max}}$ | Maximum mass per unit length. |

$\overline{q}$ | Average $q(x,t)$ for the whole belt at a given time. |

${\mathbf{q}}_{n}$ | Vector of $q(i,n)$’s at a given time n. |

t | Time. |

v | Conveyor belt speed, m/s. |

x | Position; distance from conveyor’s tail, m. |

Subscripts and superscripts: | |

i | Position on the belt-in discrete domain. |

n | Time-in discreet domain. |

${\{\xb7\}}_{\text{est}}$ | An estimate of a quantity/variable. |

## References

- Mathaba, T.; Xia, X.; Zhang, J. Analysing the economic benefit of electricity price forecast in industrial load scheduling. Electr. Pow. Syst. Res.
**2014**, 116, 158–165. [Google Scholar] [CrossRef] - Granell, R.; Axon, C.J.; Wallom, D.C.H. Predicting winning and losing businesses when changing electricity tariffs. Appl. Energy
**2014**, 133, 298–307. [Google Scholar] [CrossRef] - Mathaba, T.; Xia, X.; Zhang, J. Optimal scheduling of conveyor belt systems under critical peak pricing. In Proceedings of the International Power and Energy Conference (IPEC) 2012, Ho Chi Minh City, Vietnam, 12–14 December 2012; pp. 315–320.
- Fedorko, G.; Molnar, V.; Marasova, D.; Grincova, A.; Dovica, M.; Zivcak, J.; Toth, T.; Husakova, N. Failure analysis of belt conveyor damage caused by the falling material. Part I: Experimental measurements and regression models. Eng. Fail Anal.
**2014**, 36, 30–38. [Google Scholar] [CrossRef] - Wheeler, C.A.; Roberts, A.W.; Jones, M.G. Calculating the Flexure Resistance of Bulk Solids Transported on Belt Conveyors. Part. Part. Syst. Charact.
**2004**, 21, 340–347. [Google Scholar] [CrossRef] - Zamorano, S. Main Article Belt Conveyor Technology Long Distance Conveying—Choosing the Right Option; Bulk Solids Handling: Houston, TX, USA, 2011. [Google Scholar]
- Ristić, L.B.; Jeftenić, B.I. Implementation of fuzzy control to improve energy efficiency of variable speed buk materail transportation. IEEE Trans. Ind. Electron.
**2012**, 59, 2959–2968. [Google Scholar] [CrossRef] - Lodewijks, G. The design of high speed belt conveyors. In Proceedings of the Conference on Belt Conveying-BELTCON 10, Midrand, South Africa, 19–21 October 1999.
- Zhang, S.; Xia, X. Modeling and energy efficiency optimization of belt conveyors. Appl. Energy
**2011**, 88, 3061–3071. [Google Scholar] [CrossRef] - Hiltermann, J.; Lodewijks, G.; Schott, D.L.; Rijsenbrij, J.C.; Dekkers, J.A.J.M.; Pang, Y. A methodology to predict power saving of troughed belt conveyors by speed control. Particul. Sci. Technol.
**2011**, 29, 14–27. [Google Scholar] [CrossRef] - Jeftenić, B.; Ristić, L.; Bebić, M.; Štatkić, S.; Mihailović, I.; Jevtić, D. Optimal utilization of the bulk material transport system based on speed concontrol drives. In Proceedings of the International Conference on Electrical Machines—ICEM 2010, Rome, Italy, 6–8 September 2010.
- Hou, Y.; Meng, Q. Dynamic characteristics of conveyor belts. J. China Univ. Min. Technol.
**2008**, 18, 629–633. [Google Scholar] [CrossRef] - Zhang, S.; Xia, X. Optimal control of operation efficiency of belt conveyor systems. Appl. Energy
**2010**, 87, 1929–1937. [Google Scholar] [CrossRef] - Middelberg, A.; Zhang, J.; Xia, X. An optimal control model for load shifting—With application in the energy management of a colliery. Appl. Energy
**2009**, 86, 1266–1273. [Google Scholar] [CrossRef] - ISO. ISO 5048:1989 Continous Mechanical Handling Equipment-Belt Conveyor with Carrying Idlers—Calculation of Operating Power and Tensile Forces; Technical Report; International Organization for Standardization: Geneva, Switzerland, 1989. [Google Scholar]
- CEMA. Belt Conveyor for Bulk Material, 6th ed.; Conveyor Equipment Manufacturers Association: Naples, FL, USA, 2005. [Google Scholar]
- Phoenix Conveyor Belts Design Fundamentals-New DIN 22101. Hannoversche Strasse 88, D-21079 Hamburg, Germany. Available online: http://www.phoenix-ag.com (accessed on 16 April 2014).
- Mulani, I.G. Calculation of Artificial Friction Conveying Coefficient f, and a Comparision between ISO and CEMA; Bulk Material Handling by Conveyor Belt 5; Society for Mining, Metallurgy and Exploration: Littleton, CO, USA, 2004; pp. 55–63. [Google Scholar]
- Strauss, W.A. Partial Differential Equations-an Introduction, 2nd ed.; John Wiley & Sons, Inc.: Hoboken, NJ, USA, 2008; pp. 10–11. [Google Scholar]
- Van Delft, T.J. Modeling and Model Predictive Control of a Conveyor-Belt Dryer-Applied to the Drying of Fish Feed. Master’s Thesis, Norwegian University of Science and Technology, Trondheim, Norway, 2010. [Google Scholar]
- Tannehill, J.C.; Anderson, D.A.; Pletcher, R.H. Computational Fluid Mechnics and Heat Transfer, 2nd ed.; Taylor & Francis: Washington, DC, USA, 1997. [Google Scholar]
- Trefethen, L.N. Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations; Cornell University: Ithaca, NY, USA, 1996. [Google Scholar]
- Zhang, S.; Xia, X. A new energy calculation model of belt conveyor. In Proceedings of the IEEE AFRICON 2009, Nairobi, Kenya, 23–25 September 2009.
- Tessier, J.; Duchesne, C.; Bartolacci, G. A machine vision approach to on-line estimation of run-of-mine ore composition on conveyor belts. Miner. Eng.
**2007**, 20, 1129–1144. [Google Scholar] [CrossRef] - Ananthan, T.; Vaidyan, M.V. An FPGA-based parallel architecture for on-line parameter estimation using the RLS identification algorithm. Microprocess. Microsyst.
**2014**, 38, 496–508. [Google Scholar] [CrossRef] - Olivier, L.; Huang, B.; Craig, I. Dual particle filters of state and parameter estimation with application to a run-of-mine ore mill. J. Process Contr.
**2012**, 22, 710–717. [Google Scholar] [CrossRef] - Tariff and Charges Booklet 2013/14. Available online: www.eskom.co.za/tariffs (accessed on 24 January 2014).

© 2015 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons by Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Mathaba, T.; Xia, X. A Parametric Energy Model for Energy Management of Long Belt Conveyors. *Energies* **2015**, *8*, 13590-13608.
https://doi.org/10.3390/en81212375

**AMA Style**

Mathaba T, Xia X. A Parametric Energy Model for Energy Management of Long Belt Conveyors. *Energies*. 2015; 8(12):13590-13608.
https://doi.org/10.3390/en81212375

**Chicago/Turabian Style**

Mathaba, Tebello, and Xiaohua Xia. 2015. "A Parametric Energy Model for Energy Management of Long Belt Conveyors" *Energies* 8, no. 12: 13590-13608.
https://doi.org/10.3390/en81212375