# 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. |

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**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