Energy-saving potential of aramid-based conveyor belts

Energy-saving

potential

of

Aramid-based conveyor belts

S. Drenkelford

Master

of

Science

Energy-saving potential of

Aramid-based conveyor belts

Master of Science Thesis

For the degree of Master of Science in Mechanical Engineering at Delft

University of Technology

S. Drenkelford

February 23, 2015

Faculty of Mechanical, Maritime and Materials Engineering (3mE) · Delft University of Technology

Delft University of Technology

FACULTY MECHANICAL, MARITIME AND MATERIALS ENGINEERING

Department Marine and Transport Technology

Mekelweg 2 2628 CD Delft the Netherlands Phone +31 (0)15-2782889 Fax +31 (0)15-2781397 www.mtt.tudelft.nl

This report consists of 90 pages and 3 appendices. It may only be reproduced literally and as a whole. For commercial purposes only with written authorization of Delft University of Technology. Requests for consult are only taken into consideration under the condition that the applicant denies all legal rights on liabilities concerning the contents of the advice.

Specialization: Transport Engineering and Logistics Report number: 2014.TEL.7928

Title: Energy-saving potential of aramid-based conveyor belts

Author: S. Drenkelford

Title (in Dutch) Energie besparende potentie van aramide transportbanden

Assignment: Graduation Confidential: Yes

Initiator (university): prof.dr.ir. G. Lodewijks Initiator (company): - H. Van De Ven

- Drs. A.M. Beers Supervisor: MSc M. Zamiralova Date: January 23, 2015

Delft University of Technology

FACULTY OF MECHANICAL, MARITIME AND MATERIALS ENGINEERING

Department of Marine and Transport Technology Mekelweg 2 2628 CD Delft the Netherlands Phone +31 (0)15-2782889 Fax +31 (0)15-2781397 www.mtt.tudelft.nl

Student: Assignment type: Graduation

Mentor: Prof. dr. ir. G. Lodewijks Report number: 2014.TEL.xxxx Specialization: TEL Confidential:

Creditpoints (EC): 35

Subject: Energy consumption of aramid conveyor belts

The bulk material handling industry uses mile after mile of rubber conveyor belts to transport ore and minerals around dry bulk material processing facilities. Research has shown that using aramid

materials in conjunction with an NR/BR rubber drastically reduces the energy consumption of conveyor belts. In particular, the overall maximum energy saving could be as much as 60% depending on the conveyor belt system used.

Currently, data are available to estimate the energy consumption of aramid conveyor belts. However, the estimation is based on specific conveyor belt applications. In order to predict the energy saving more generally, this assignment will improve currently available datasheets and establish the fundamental estimation of energy consumption towards general conveyor belt applications, when aramid conveyor belts are in use. The study and analysis will be based on the norm of DIN 22101. One of the outputs of the assignment will be a GUI application allows users to investigate the energy consumption with respect to general belt conveyor configurations.

The report should comply with the guidelines of the section. Details can be found on blackboard. The professor,

Summary

Belt conveyor systems are globally responsible for the transportation of vast amounts of bulk solid materials. They are the most efficient solution where large quantities of iron ore, coal, limestone or rocks need to be moved from one place to another. Despite of their efficiency, they still consume large amounts of energy to perform their function. This energy consumption can be significantly reduced by applying Aramid products in conveyor belts. The industry however, is not keen on using these products, since they have not yet proven themselves. The Customer Benefit Model was created by Teijin Aramid to quantify the potential cost savings that can be achieved by the application of Aramid products in conveyor belts and by doing so, convincing potential customers of the benefits. For this purpose, there is a need for a prediction of the possible energy savings that can be achieved by Aramid-based conveyor belts. In this research, a method is developed for the determination of this prediction.

A literature survey was done to investigate the current possibilities for the determination of the energy consumption of a belt conveyor. As a primary method of calculation the German belt conveyor standard DIN 22 101 will be used. The energy consuming components of a belt conveyor system are identified and the largest contributor, the indentation rolling resistance, is found to be the most important component that should be determined. Several methods were found to do this, based on viscoelastic half spaces and viscoelastic Winkler foundations. The models that are based on half spaces were found unsuited for this purpose, since a conveyor belt typically consists of a thin layer of rubber material that performs the viscoelastic function. Of the remaining methods, the method of Jonkers and the method of Lodewijks were preferred. To be able to use the method of Lodewijks, Maxwell model parameters need to be determined by approximating the master curves, that were created from measured rubber data from Dynamic Mechanical Analysis, with Prony Series approximations. For the three parameter model that was used and found accurate in literature, the Maxwell model parameters were found to give false results. This was proven by determining the trend of the five parameter model and the trend that was observed for the method of Jonkers, which uses raw data to obtain indentation rolling resistances. It was found that by cutting the master curves at a certain cut-off frequency, the method of Lodewijks can yield quite accurate power requirements for loaded belt conveyors, but underestimates the power requirements for empty conveyors. Due to the complexity of the steps that are required to setup the method of Lodewijks, it is considered not practical and will therefore not be used. The method of Jonkers can be used, although it is known to overestimate the indentation rolling resistance. To reduce this overestimation a modified method of Jonkers is proposed.

The performance of the proposed method and that of the original Jonkers method were assessed in a case study in which the energy consumptions of four belt conveyor systems were evaluated and compared against the results from external research. The modified method of Jonkers gives a better approximation of the power requirements for loaded conveyor belts, but underestimates the power requirements of empty conveyor belts. It is concluded that the modified method of Jonkers should be used to estimate the power requirements for loaded conveyor belts and the original method of Jonkers for the empty conveyor belts. Since the proposed methods and the generated results are all based on theoretical models, it is recommended that a pilot belt is installed, on which actual measurements can be done. These measurements can then be used to improve the proposed modification to Jonkers method.

To make this method usable for employees of Teijin Aramid, a software application has been developed that incorporates both the original and the modified Jonkers method. The software application was tested and is able to perform the analysis that is shown in this study.

Samenvatting

Transportbanden zijn wereldwijd verantwoordelijk voor het transport van grote hoeveelheden stortgoederen. Ze zijn de meest efficiÃńnte oplossing op plaatsen waar grote hoeveelheden ijz-ererts, kolen, kalksteen of steen verplaatst moet worden. Ondanks hun efficiÃńntie verbruiken deze installaties toch grote hoeveelheden energie. Deze energieconsumptie kan significant worden verminderd door het toepassen van Aramide producten in de transportbanden. Deze technolo-gie wordt echter niet met open armen ontvangen door de bestaande industrie, doordat deze nieuwe types transportbanden zichzelf nog niet bewezen hebben. Het Customer Benefit Model is gemaakt door Teijin Aramid om de potentiÃńle kostenbesparing te kwantificeren die mogelijk zijn door de toepassing van deze transportbanden en hiermee klanten te overtuigen van de vo-ordelen. De input van dit model zijn de energieverbruiken van de verschillende types Aramide transportbanden. Dit rapport beschrijft de bepaling en de ontwikkeling van een methode om deze energieverbruiken te berekenen. Deze methode moet precies genoeg zijn om betrouwbare resultaten te leveren die de ordergrootte van de potentiÃńle energiebesparingen tonen die door het toepassen van Aramide producten mogelijk zijn.

De huidige bestaande methodes voor het bepalen van de energieconsumptie van transportban-den zijn onderzocht door middel van een literatuuronderzoek. De Duitse norm DIN 22 101 zal worden gebruikt als basis voor de energieberekeningen. De componenten van het totale energie-verbruik van een transportband zijn geÃŕdentificeerd en de indrukrolweerstand, die het grootste aandeel heeft in het totale energieverbruik, is het belangrijkste component om te berekenen. De literatuur beschrijft verscheidene methoden om de indrukrolweerstand te bepalen, gebaseerd op viscoelastische half-ruimten en viscoelastische Winkler matrasmodellen. De methoden die zijn gebaseerd op een half-ruimtemodel blijken minder bruikbaar, aangezien een transportband gebruikelijk bestaat uit dunne lagen viscoelastisch materiaal. Van de overgebleven methoden zijn de methode van Lodewijks en de methode van Jonkers geselecteerd voor dit onderzoek. Om de methode van Lodewijks te kunnen gebruiken moeten eerst de Maxwell model parameters bepaald worden door met Prony series de master curves te benaderen. Deze master curves zijn gebaseerd op gemeten materiaaleigenschappen die zijn bepaald met Dynamic Mechanical

Analysis. Met deze Maxwell model parameters is de methode van Lodewijks doorgerekend en de

resultaten bleken het tegenovergestelde te vertonen dan was verwacht, namelijk dat de toevoeg-ing van Sulfron een negatieve invloed zou hebben op de roleigenschappen van het rubber. De hypothese dat het drie-parameter model onjuiste resultaten geeft is bevestigd door de berekening van het vijf-parameter model en de berekening van de methode van Jonkers. Het drie-parameter model kan wel gebruikt worden door de master curves af te snijden boven een bepaalde

frequen-tie. De op deze manier behaalde resultaten zijn vrij accuraat voor de beladen transportbanden, maar onderschatten het benodigde vermogen voor de lege transportband. Doordat deze meth-ode erg complex is om op te zetten, wat niet gewenst is voor de industriÃńle toepassing waar dit rapport op doelt, is besloten om de methode van Jonkers verder te gebruiken. Het is echter bekend dat deze methode de vermogensbehoeftes overschat, dus een aangepaste methode van Jonkers is voorgesteld om dit gedrag tegen te gaan.

De prestaties van de aangepaste methode van Jonkers en die van de originele methode van Jonkers zijn getoetst aan de resultaten die zijn gepresenteerd door Lodewijks voor een casus betreffende de Optimum Collieries in Zuid-Afrika. Voor beladen transportbanden geldt dat de aangepaste methode van Jonkers een betere benadering van de vermogensbehoeften geeft dan de originele methode van Jonkers. Voor lege transportbanden is dit niet het geval en zorgt de aanpassing van de methode voor een onderschatting van de vermogensbehoeften. De conclusie is dat de vermogensbehoeften voor de beladen transportbanden dus het beste berekend kunnen worden met de aangepaste methode van Jonkers en de vermogensbehoeften van de lege transportbanden met de originele methode van Jonkers.

Gezien de voorgestelde aangepaste methode van Jonkers en de gegenereerde resultaten enkel zijn gebaseerd op theorie, is het van belang dat er metingen worden gedaan aan een echte transportband, zodat de aangepaste methode van Jonkers indien nodig bijgesteld kan worden. Een software applicatie is ontwikkeld, tegelijkertijd met dit rapport, zodat de medewerkers van Teijin Aramid de originele en de aangepaste methode van Jonkers kunnen gebruiken om vermogensbehoeften en dus energieverbruiken van transportbanden te kunnen bepalen. Deze applicatie is getest en is in staat om de analyse naar behoren uit te voeren.

List of symbols

Capital letters

A First part contact length of idler and belt m

C Secondary resistance factor

-C’ Last part contact length of idler and belt m

CV Correction factor for the Jonkers method

-D Diameter idler roll m

E’ Storage modulus Pa

E” Loss modulus Pa

E* Complex modulus Pa

E1 First Maxwell model parameter Pa

E2 Second Maxwell model parameter Pa

F Total motional resistance N

FAuf Load/belt friction in loading zone N

FGr Belt/belt cleaner resistance N

FGb Belt bending resistance at pulley N

FH Main resistance N

FHi Main resistance of section i N

FN Secondary resistances N

FS Special resistances N

FSchb Load/chute resistance in loading zone N

FSt Gradient resistance N

FT ri Pulley bearing resistance N

Fz Gravitational force of mass N

H Conveyor lifting height m

L Conveying lenght m

Li Conveying length section i m

P Required drive power W

U’ Idler bearing resistance N

U_B00 Belt flexure resistance N

U_E00 Indentation rolling resistance N

Small letters

f Friction factor carry strand and return strand combined

-fi Friction factor indentation rolling resistance

-fb Friction factor idler bearing resistance

-fbe Friction factor belt flexure resistance

-fbu Friction factor bulk flexure resistance

-fij Friction factor Jonkers method

-f_ij0 Friction factor modified Jonkers method

-fo Friction factor carry strand

-fu Friction factor return strand

-g Gravitational acceleration m/s2

h Cover layer thickness m

m0_G Mass belt kg/m

m0_L Mass load kg/m

m0_R Mass rotating parts kg/m

m0_Ro Mass rotating parts (Carry strand) kg/m

m0_Ru Mass rotating parts (Return strand) kg/m

qR Distributed load on an idler roll N/m

v Belt speed m/s

Greek letters

α Wrap angle radians

δ Phase loss angle o or radians

0 Maximum strain

-η Damping coefficient Pa s

θ Inclination angle (total system) o

θi Inclination angle of section i o

λ Trough angle o

µ Friction factor pulley/belt

-σ Applied stress Pa

σ0 Maximum applied stress Pa

List of abbreviations

BEM Boundary Element Model

BR Butadiene Rubber

CEMA Conveyor Equipment Manufacturers Association DIN Deutsche Institut fÃijr Normung

FEM Finite Element Model

MRC Maximum Rolling Coefficient MTPH Metric Tonnes Per Hour

NR Natural Rubber

phr Parts per Hundred Rubber SBR Styrene-Butadiene Rubber SLS Standard Lineair Solid (model) tan-delta Tangent Delta

Contents

List of symbols ix

List of abbreviations xi

1 Introduction 1

1.1 Energy reduction in belt conveyors . . . 2

1.2 Research goal and scope . . . 3

1.3 Report structure . . . 3

2 Aramid conveyor belts 5 2.1 Twaron . . . 5

2.2 Sulfron . . . 8

3 The energy consumption of belt conveyors 9 3.1 Belt conveyor basics . . . 9

3.1.1 Basic setup of a generic belt conveyor . . . 9

3.1.2 Conveyor belt composition . . . 10

3.1.3 Other components of a belt conveyor system . . . 11

3.2 Energy calculations of DIN 22 101 . . . 13

3.2.1 Total motional resistance . . . 13

3.2.2 Resulting power consumption . . . 16

3.3 Energy consumption distribution . . . 16

4.1 General theory . . . 21

4.1.1 Viscoelasticity . . . 21

4.1.2 Dynamic Mechanical Analysis . . . 22

4.1.3 Material models . . . 23

4.1.4 Modelling the belt backing layer . . . 24

4.2 Theoretical methods . . . 26 4.2.1 May et al. . . 26 4.2.2 Hunter . . . 27 4.2.3 Jonkers . . . 28 4.2.4 Spaans . . . 28 4.2.5 Lodewijks . . . 29 4.3 Numerical methods . . . 32 4.3.1 Wheeler . . . 32 4.3.2 Qui . . . 32 4.4 Selected method . . . 33 5 Rubber rheology 35 5.1 Experimental setup . . . 35

5.2 Creating master curves . . . 36

5.3 Maxwell model parameters . . . 38

5.4 Cutting the master curves . . . 44

5.5 Jonkers method . . . 47

5.5.1 Jonkers method with master curve data . . . 47

5.5.2 Jonkers method without master curve data . . . 48

6 Proposed method 49 6.1 Analysis . . . 49 6.2 Set-up . . . 51 7 Verification 57 7.1 Optimum Collieries . . . 57 7.2 Results . . . 59

7.2.1 Verification power requirements for loaded conveyors . . . 59

7.2.3 Discussion results Optimum Collieries verification . . . 65

8 Case study 67 8.1 Case study: stockpile conveyor . . . 67

8.2 Calculation of the energy consumptions with master curve data . . . 69

8.3 Calculation of the energy consumptions without master curve data . . . 74

9 Software application 77 9.1 System of requirements . . . 77

9.2 System architecture . . . 78

9.2.1 Calculation . . . 78

9.2.2 Calculation of a conveyor belt . . . 79

9.3 Recalculation of the case study . . . 81

9.4 Extra options . . . 85

10 Conclusions 87 11 Recommendations 89 Appendix A: research paper 91 Appendix B: draft of research paper for BeltCon 2015 97 Appendix C: results DMA tests 107 11.1 Control sample, 3Hz . . . 107 11.2 Sulfron sample, 3Hz . . . 109 11.3 Control sample, 6Hz . . . 111 11.4 Sulfron sample, 6Hz . . . 114 11.5 Control sample, 10Hz . . . 116 11.6 Sulfron sample, 10Hz . . . 119 11.7 Control sample, 20Hz . . . 122 11.8 Sulfron sample, 20Hz . . . 125 Bibliography 129

Chapter 1

Introduction

Since the industrial revolution, mankind has had an ever-growing need for raw materials, such as iron ore and coal. To fulfil this need, large logistical networks have grown all over the world to move these materials from one place to another. One of the backbones of these networks is the belt conveyor, which can move large quantities of bulk solid material, often with a greater efficiency than for instance trucks or trains. Despite of this great efficiency, they still consume vast amounts of energy. Due to the rapid depletion of fossil fuels around the world and the implied impact on the environment, the urge to reduce this energy consumption keeps increasing.

1.1

Energy reduction in belt conveyors

In the past twenty to thirty years there has been an increasing amount of research done on the subject of reducing the energy consumption of conveyor belts. A German study indicated that the largest part of the energy consumption is due to the indentation of the rubber covers when the belt passes an idler roll [2]. This has resulted in the development of the so-called Low Rolling Resistance (LRR) conveyor belts. These belts were designed to reduce the amount of energy that is lost during the conveying operation by reducing the rubber hysteresis losses of the running cover compounds (see Figure 1.2). Work done by researchers like Nordell [3], Gallagher [4] and Zhang [5] have shown the potential of LLR conveyor belts to reduce the energy consumption of both troughed belt conveyors and pipe conveyors.

Figure 1.2 The energy that is required to indent the belt cover material is not all recovered when the ma-terial is released. Hence, the loss is the grey area.

Teijin Aramid BV, a former branch of AkzoNobel that is now owned by the Teijin Group, produces products that can also substantially reduce the energy consumption of belt conveyor systems, which are called Aramids. Aramid is an acronym for Aromatic Polyamide and it is a high-performance fibre material. These materials can be used in the carcass of a conveyor belt and in its rubber compounds to reduce the hysteresis losses. In Chapter 2 these materials and their application in conveyor belts are explained elaborately. A study performed by Lodewijks suggests that the savings that can be achieved by these Aramid products in belt conveyor systems can be as high as 60% [6]. Despite of these motivating numbers, the industry is very conservative in the adoption of these new products, due to the large financial risks that are involved with the failure of large belt conveyor systems and failures that have happened with early versions of Aramid-based conveyor belts in the 1990’s. Since that time, Aramid conveyor belts have been improved to a level where they can compete with the conventional steel cord belts. To convince potential clients that these conveyor belts can be a good alternative that can save energy and costs, the Customer Benefit Model (CBM) was created by Teijin Aramid. This model uses the potential energy savings and emission data to provide clients with an overview of the reduced environmental impact and the cost reductions that can be achieved by applying Aramids in conveyor belts. The energy savings are currently determined with fixed proportional factors which are used to scale the potential energy consumptions of the different Aramid-based

conveyor belt types in comparison with a conventional steel cord conveyor belt.

1.2

Research goal and scope

The scaling factors for the energy consumption that are used in the CBM are based on a single case study, which can result in varying results when applied to generic belt conveyor systems. In this research, a method is developed to estimate the energy consumption of a generic belt conveyor system under steady state operating conditions that is more accurate than the current solution and therefore yields a more realistic result. With the focus on its usability in a non-technical environment, this study is about finding the balance between the ease-of-use and the technical accuracy. To reach this goal, the following research questions will be answered:

- Which factors are responsible for the energy consumption of a conveyor belt and to what extent?

- What methods to determine the energy consumption of belt conveyors are available in literature and how are they applicable for this topic?

- What is the desired accuracy of this estimation?

- What should be used as comparison to quantify these energy savings?

To make this method usable with the CBM, a user-friendly software application will be developed alongside of this method, that will allow Teijin employees to calculate the first approximation of the energy consumption of a generic belt conveyor system, in order to use it in the Customer Benefit Model to determine the potential cost savings and reductions in emissions. By compar-ison with field test results and calculations by consultancy firms, the results of this method can be verified.

This study is done as a graduation project at the Delft University of Technology for the faculty of Mechanical, Maritime and Materials Engineering (3me) and Teijin Aramid BV. As stated by the assignment (as showed at the start of this report) this research will be based on the German standard DIN 22 101, in which a methodology is stated for the calculation and design of belt conveyor systems.

1.3

Report structure

In Chapter 2 the Aramid products are explained that are used by Teijin to reduce the energy consumption of conveyor belts. Chapter 3 will show how belt conveyors in general consume energy and how this energy consumption is distributed. In Chapter 4 the details of the inden-tation rolling resistance are explained, along with the methods that were found in literature to determine this resistance. The limitations of the rubber data that is used in this report is shown in Chapter 5, after which a proposal is done to modify the only remaining method in Chapter 6.

Chapter 7 shows the verification of this modified method by comparing it with a case study from public literature. Chapter 8 shows a case study of a fictive belt conveyor system that illustrates the function of this method and the performance of the Aramid-based conveyor belts.

In Chapter 9 the software application that has been developed alongside of this method is shown, which will be able to estimate the energy consumption of a generic belt conveyor system. Finally, conclusions and recommendations are stated in Chapters 10 and 11.

Chapter 2

Aramid conveyor belts

Teijin Aramid produces two types of aramid that can reduce the energy consumption of belt conveyors: Twaron and Sulfron. They are engineered to focus on two different aspects of energy losses in belt conveyors, respectively the reduction of belt weight and the reduction of the running resistance.

2.1

Twaron

Twaron is a high-strength fibre material with molecules that are characterized by rigid polymer chains. These molecules are linked by strong hydrogen bonds that transfer mechanical stress very efficiently, making it possible to use chains of a low molecular weight. A picture of their general molecular structure is shown in Figure 2.2. Figure 2.1 shows the various shapes in which this material is produced.

Figure 2.1 The various shapes in which Twaron is produced. The spool of fibre in the middle can be used to create a carcass of a conveyor belt.

Figure 2.2 Molecular structure of an aromatic polyamide.

In general, the carcass of a belt conveyor for heavy duty applications is made out of steel cords that are embedded in the rubber of the conveyor belt. The function of this carcass, which is transmitting the tensional force, can also be fulfilled by a Twaron fabric carcass. Such a carcass with a similar strength class is a lot lighter than a steel cord carcass, since Twaron is five times stronger that steel on a weight-for-weight basis. Twaron fibres can be woven in two carcass shapes (see Figure 2.3). For long overland conveyors the straight warp fabric is most commonly used, since it gives a more sturdy carcass. Figure 2.4 shows a chart in which the strengths of different materials are shown. It is clear that steel and polyester, which are very common materials in carcasses of conveyor belts, do not even come near the strength per mass of Twaron.

Figure 2.4 Strength comparison of various carcass materials. [7]

Due to the lower weight of Twaron in comparison with steel, the weight of the belt decreases when a Twaron carcass is used. The weight reduction is even further enhanced by the fact that less rubber is required to encase the carcass, since there are no significant gaps between the tension members of a Twaron fabric carcass, as is the case with a steel cord carcass. Figure 2.5a shows the composition of a generic steel cord conveyor belt, in which can be seen that there is a lot of rubber between the steel cords. In a Twaron straight warp fabric carcass, as shown in Figure 2.5b, the Twaron cords are very close to each other, forming an approximately flat surface.

(a) Steel cord carcass. [8] (b) Twaron fabric carcass. [7]

2.2

Sulfron

The running resistance that a conveyor belt needs to conquer to maintain its operation is largely due to indentation losses [2]. The second product of Teijin Aramid, Sulfron, is a rubber com-pound ingredient that can reduce the energy that is consumed by the conveyor belt when it passes over each idler roll that support it. During these indentations, the carbon black particles within the material break up. When the rubber is released into its original shape as it passes the idler rolls, these particles bond again, which consumes energy. This is where Sulfron comes in, since it comes in between the formation of the new chemical bonds and thus reducing the consumed energy. Besides this advantage, the addition of Sulfron is also beneficial for the resis-tance to abrasion and the flexibility of the belt. Figure 2.6 shows pellets of Sulfron that can be added to the rubber compounds.

Figure 2.6 Pellets of Sulfron that can be added to a rubber compound to reduce the running resistance of a conveyor belt.

By using Twaron and Sulfron in a conveyor belt, the operational energy consumption can be reduced. Also, if the running resistances are reduced far enough, it could even result in lower requirements for other components in the belt conveyor system, like the drives, the take-up system and the idlers. To determine these possibilities, the energy and tension requirements need to be determined. In the next chapter, the methodology of DIN 22 101 is explained, which will be used as the primal method of calculation.

Chapter 3

The energy consumption of belt

conveyors

in 1982, the Deutsches Institut für Normung (DIN) published a standard that contained a basic guide for the calculation of the various aspects of belt conveyors. It is called DIN 22 101, “Belt

conveyors for bulk materials; bases for calculation and design” [9] and it has been one of the most

important tools that engineers can use to predict, amongst others, the energy requirements of a belt conveyor system. For the estimation of the energy consumption this standard determines the resistance to motion of the combination of the belt, the load and the elevation. In this chapter the basic setup of a belt conveyor system and the accompanying components are explained, after which the procedure of DIN 22 101 to determine the energy consumption of belt conveyors is presented. Finally, the relation of this procedure to experimental results found in literature is shown.

3.1

Belt conveyor basics

It is important to understand the composition of belt conveyor systems in general to be able to perform a good calculation of its energy consumption. In this section this general composition is presented and examples of the components are shown.

3.1.1 Basic setup of a generic belt conveyor

Independent of the exact shape and size, any belt conveyor system contains a number of key components. A schematic picture of a generic belt conveyor system is shown in Figure 3.1, in which these components can be seen. Material is loaded at the tail of the conveyor by either a feeding chute or a feeder belt. It is then transported by the belt to the head of the conveyor, where the material is discharged. For overland belt conveyors the carry side of the conveyor generally has a trough shape, as depicted in Figure 3.2. This configuration allows a much greater capacity than a flat belt. Other configurations are possible, like a two roll trough or a five roll trough, but this study focusses on three roll troughs, since it is the most commonly used configuration.

Figure 3.1 Setup and key components of a generic belt conveyor system.

Two angles are of importance in the cross section of the transported material (see Figure 3.2): the trough angle λ, which is the angle of the side rolls with the horizontal axis and the angle of surcharge β, which is the angle under which the bulk solid material rests on the belt. The angle of surcharge depends on the material and the size of the lumps. In general, an angle of 20 to 25 degrees is used for materials like coal and ore.

Figure 3.2 Cross section area of a conveyor belt in a three roll trough configuration.

3.1.2 Conveyor belt composition

Conveyor belts for long overland belt conveyors can be divided into two general categories. The first are conveyor belts with a steel cord carcass. Long steel cords run in the direction of the conveyor belt to deal with the tensile forces that the conveyor belt experiences along its track. Figure 3.3a shows a schematic representation of such a conveyor belt. The tensile members are encased in the core of the belt, in the rubber of the carcass layer. The top cover layer is generally made of an abrasive-resistant rubber, since it has to deal with the impact and the grinding of

the transported material. The bottom cover is often made of another type of rubber that is designed to reduce the rolling resistance.

Figure 3.3b shows a schematic representation of the second belt category: conveyor belts with a fabric carcass. There are a couple of materials that are frequently used to create these fabric carcasses, like polyester, Nylon and Aramid. For long overland conveyors, only Aramid is a worthy material, since it can withstand far greater tensile forces and its elongation under tension is considerably smaller. Similar as with a steel cord belt, a fabric belt also has an abrasive-resistant top cover and a less-rolling resistant bottom cover. It is however often found that the carcass layer that embeds the fabric is thinner than the carcass layer of a steel cord belt, since the surface pressure is better distributed due to the approximately flat surface of such a fabric layer.

(a) Steel cord carcass. (b) Twaron fabric carcass.

Figure 3.3 Carcass types in conveyor belts.

3.1.3 Other components of a belt conveyor system

As shown in Figure 3.1, a belt conveyor system has at least seven important components besides the conveyor belt itself and the static structures that are required to support the whole system. In this study only the belt and the components that directly influence the belt’s performance are looked into, which are the pulleys, the idlers and the take-up system.

The pulleys are located at the begin and the end of a belt conveyor system and their function is to reverse the direction of the belt. In many belt conveyor systems the head pulley is also the drive pulley that powers the belt conveyor. It transfers the rotational power of a drive motor to the pulling force that moves the conveyor belt. An example of a head pulley is shown in Figure 3.4.

The idlers support the conveyor belt along the entire track. The combination of the idler rolls that support the total width of the belts in one frame is called an idler set and they are located along the entire conveyor path, standing at distances of one to several meters from each other. In Figure 3.5 four of these sets are shown.

Figure 3.4 Example of a head pulley. [10]

Figure 3.5 Example of idler sets. [11]

Figure 3.6 Example of a (gravity) take-up sys-tem. [12]

The take-up system provides the force that is required to keep the belt tension above a certain level. This belt tension is required to be able to transfer the drive power onto the conveyor belt and to prevent belt sag between the idler sets, which would result in an increasing energy consumption. Figure 3.6 shows a take-up system that is powered by gravity. There are also systems that increase the belt tension by tensioning one of the pulleys with the help of winches or threaded axles.

The energy that is consumed by a belt conveyor system depends on these components and the circumstances under which the system is operating. To calculate the magnitude of this energy consumption the methodology of DIN 22 101 can be used, which is explained in the next section.

3.2

Energy calculations of DIN 22 101

DIN 22 101 determines the energy consumption of belt conveyors based on the motional re-sistances of the belt, the material and the surrounding moving equipment. It states that the power consumption of a belt conveyor system during steady state operating conditions can be determined by the following equation:

P = F · v

η (3.1)

in which:

P Required power [W]

F Total motional resistance force [N] v Belt speed [m/s]

η Efficiency of the drive

The power consumption of a belt conveyor that is starting or stopping varies from the power consumption during steady state conditions. Due to the fact that these actions generally con-sume only a fraction of the time that the conveyor runs in a steady state condition, the choice was made to neglect these starting and stopping states. Therefore, the method presented in this report only considers the belt conveyor system during steady state conditions.

3.2.1 Total motional resistance

The total motional resistance force used in Equation 3.1 is the sum of four resistance forces:

F = FH+ FN + FSt+ FS (3.2)

These four resistance forces are explained below:

Main resistance, FH

The main resistance covers the force that is required to move the load an to keep the moving parts of the conveying system in motion. This resistance occurs along the entire length of the conveyor. The main resistance of a belt conveyor can be determined by equation 3.3. One has to note that this equation considers the combination of the carrying strand and the return strand of a belt conveyor by the usage of one combined resistance factor, f .

FH = L · f · g · [m0R+ (2m 0 G+ m 0 L) · cos(θ)] (3.3) in which: FH Main resistance [N] L Conveyor length [m]

f hypothetical friction coefficient for the upper and the lower strand jointly [-] g Gravitational acceleration [m/s2]

m0_R Mass of the rolls per meter of belt length [kg/m]

m0_G Mass of the belt per meter of belt length [kg/m]

m0_L Mass of the load per meter of belt length [kg/m]

In DIN 22 101, the friction coefficient f is chosen from a table for belt filling ratios in the range from 0.7 to 1.1, based on the operating and installation conditions and the experience of the engineer. According to this method, the value of f is between 0.012 and 0.035. A concise version of this table can be seen in Table 3.1. If the different values of the friction factors of the upper and the lower strand are known, the calculation is done in twofold, where the weight of the carried material is only accounted for in the calculation of the upper strand.

Table 3.1 Standard values for the average coefficient of friction f for the combi-nation of the carry strand and the return strand of belt conveyors. [9]

Situation f

Horizontal conveyors, inclined conveyors, gently de-clined conveyors

- Favourable operating conditions 0.017 - Normally constructed and operated installations 0.020

- Unfavourable operating conditions 0.023 to 0.027 - Normally constructed and operated installations up to 0.035 in extremely low temperatures

Declined conveyors (drives operate as dynamos) 0.012 to 0.016

Secondary resistances, FN

The secondary resistances that are experienced by the belt conveyor are those that are caused by friction of components at the head an the tail of a belt conveyor. They consist of the resistance due to belt scrapers, loading of the belt and the resistance of the belt when passing pulleys. Stated otherwise, they are resistances that occur only locally and are therefore not dependent on the length of the conveyor. Equation 3.4 represents these secondary resistances.

FN = FAuf + FSchb+ FGr+ FGb+ FT ri (3.4)

With:

FN Secondary resistances [N]

FAuf Frictional resistance between the load and the belt in the loading zone [N]

FSchb Frictional resistance between the load and the lateral chutes in the loading zone [N]

FGr Frictional resistance caused by belt cleaner [N]

FGb Belt resistance to bending at the pulleys [N]

FT ri Pulley bearing resistance [N]

Since these resistances are independent of the length of the belt, their contribution becomes smaller as the length of the conveyor increases. The secondary resistances are generally ac-counted for by multiplying the main resistance by a factor C, according to the following relation:

C = 1 + FN FH

(3.5)

The 1982 version of DIN 22 101 contains a table in which the magnitude of the factor C can be determined, based on the length of the belt conveyor. A copy of this table is shown in Table 3.2. It can be seen that only conveyors with a length of 80 meters or higher are considered.

Table 3.2 Standard values for the coefficient C for belt conveyors. [9]

L [m] C L [m] C 80 1.92 600 1.17 100 1.78 700 1.14 150 1.58 800 1.12 200 1.45 900 1.10 300 1.31 1000 1.09 400 1.25 1500 1.06 500 1.20 ≥ 2000 1.05

For conveyors that are shorter than 80 meters, or those that have multiple feeding points, the secondary resistances need to be determined more specifically by using Equation 3.4. These special conveyors will not be taken into account in this study, since short conveyors do not benefit enough from the application of Aramids in their conveyor belts to justify the required capital investment and conveyors with multiple feeding points are outside the scope of this research.

Gradient resistance, FSt

For conveying systems of which the head and the tail are situated on different height levels, the gradient resistance FSt describes the required energy to lift the material over this height

difference. This is done by simply using the physical law of potential energy and is shown in the following equation:

FSt= H · g · m0L (3.6)

Where H is the height difference in meters and m0_Lthe mass of the load per meter of belt length, in kilograms per meter.

Special resistances, FS

Special resistances F_Soccur when a conveying system is badly aligned or when special devices are placed to unload the belt at another point than the head of the conveyor. In this research, these circumstances will not be looked into, so the Special resistances will be neglected throughout this report.

3.2.2 Resulting power consumption

The sum of the four resistances can be reduced to a more tidy formula (Equation 3.7), in which the secondary resistances are taken into account by the earlier mentioned factor C and the special resistances are neglected. This formula can then be inserted into Equation 3.1, which yields the equation of the power consumption of a belt conveyor during steady state operating conditions (Equation 3.8). F = C · L · f · g · [m0_R+ (2m0_G+ m0_L)cos(θ)] + H · g · m0_L (3.7) P = v · (C · L · f · g · [m 0 R+ (2m0G+ m0L)cos(θ)] + H · g · m0L) η (3.8)

It is clear that DIN 22 101 does not contain explicit information about the belt’s material properties. These properties can be accounted for by the friction factor f, which is also known as the rolling resistance factor. As stated earlier, this rolling resistance factor is a fictional factor of the upper and the lower strand of the belt conveyor combined and is derived from empirically filled tables. In the next section, the most important components of this rolling resistance factor are identified, which give insight in the possibilities reducing the energy consumption of conveyor belt systems.

3.3

Energy consumption distribution

In 1993 two German researchers, Hager and Hintz, did an extensive study to determine the energy consumption of belt conveyors and the distribution of its components. They identified seven components that together form the total motional resistance of belt conveyor systems [2]:

Indentation rolling resistance, U_E00

The resistance of the conveyor belt rolling over the idler rolls. (see Figure 3.7). When the belt passes over an idler roll, a part of the rubber is indented. A part of the energy that is required for this indentation is transformed into heat and is lost.

Belt flexure resistance, U_B00

The internal friction in the conveyor belt due to the deformation of the belt in between idler sets, which is shown in Figure 3.8. Energy is required to reform the belt into its original shape when it approaches the next idler set.

Bulk flexure resistance, U_F00

The friction in the bulk material caused by the deformation of the bulk solid material in between idler sets (see Figure 3.8). Between these sets the belt deflects downward and the edges deflect to the outside. When the belt is forced into its original shape at the next idler set, the bulk solid material is forced into its original shape too, consuming energy to overcome the internal frictional forces.

Figure 3.8 Bulk flexure resistance

Idler bearing resistance, U0

Frictional resistance in the bearings that support each idler roll.

Secondary resistances, FN

The resistances that are caused by belt cleaners and friction in loading zones.

Extraordinary resistances, FS

The resistances caused by misalignment of components (called the Special resistances in DIN 22 101)

Gradient resistance FSt

The energy that is required to lift the material if the head and the tail of the belt conveyor are located on different altitudes.

After experiments on different configurations of belt conveyor systems Hager and Hintz con-cluded that the largest part of the energy consumption of long horizontal belt conveyors for bulk solid materials is due to the indentation rolling resistance [2]. For a 1000 meter long hor-izontal conveyor, they calculated that the indentation rolling resistance consumed as much as 61% of the total energy consumption of the belt conveying system. A chart of their findings is shown in Figure 3.9. For inclined or declined conveyors, the distribution is different since the gradient resistance is in these cases identified as the largest factor [2] and can be as high as 66% for a belt conveyor with an ascending angle as small as 5%. Of the remaining energy consuming components, the largest one is again the indentation rolling resistance, which contributes 22% to the total rolling resistance. Since there is no way to reduce the amount of energy that is required to lift the material, the indentation rolling resistance is the primary focus to reduce the energy consumption of belt conveyors.

Figure 3.9 The distribution of the consumed energy of a horizontal overland belt conveyor with a length of approximately 1000 meter [2].

Slightly varying results were found by Alspaugh in 2004, who also identified the indentation rolling resistance as the dominant energy consumer for long overland belt conveyors and the lifting of the material as greatest consumer for inclined or declined conveyors [13].

Since the indentation rolling resistance is identified as the largest contributor to the rolling resistance it will be the main focal point of the determination of the energy consumption in this study. Nonetheless, the other main resistances also contribute to the total energy consumption of a belt conveyor system and can therefore not be neglected. To account for these contributions, the results of Hager and Hintz were examined again. These results show that the indentation rolling resistance accounts for 61% of the total resistance in a flat overland conveyor. Within the total main resistance that occurs for this flat conveyor the indentation rolling resistance accounts for 68%. For an ascending conveyor the contribution of the indentation rolling resistance to the total resistance is much lower, but its contribution to the main resistances is found to be 67% (see Figure 3.10). Since these two values are very close to each other, it will be assumed that the indentation rolling resistance always accounts for 68% of the main resistances.

Figure 3.10 The distribution of the consumed energy of a belt conveyor with an inclination angle of 5% [2].

Based on the findings in this chapter, this study will focus on the determination of the indentation rolling resistance. The ratio of this indentation rolling resistance and the main resistances will be used to scale the obtained results and in that way account for the other main resistances, which will not be calculated themselves.

To determine the indentation rolling resistance several methods were found in literature. In the next chapter these methods are explained and the suitable methods will be selected.

Chapter 4

Indentation rolling resistance

Chapter 3 concluded that the largest part of the energy consumption of a belt conveyor system is consumed by the indentation rolling resistance. To be able to make a decent prediction of the energy consumption of a conveyor belt, the indentation rolling resistance factor needs to be determined. In literature, several methods were found that determine the indentation rolling resistance. In this chapter the most influencing methods are explained. The first section of this chapter provides general theory that is required to use the theoretical models that are shown in Section 2. In Section 3 the numerical methods to determine the indentation rolling resistance are explained, after which the methods are selected that will be used in this study.

4.1

General theory

In this section the general theory is presented that will be the basis for the determination of the indentation rolling resistance. It will show the fundamental basis of the material and a number of material models that will be used to describe the material and its behaviour.

4.1.1 Viscoelasticity

The indentation of the rubber bottom cover consumes energy since not all of the energy that is absorbed by the indentation is released back into the system as the rubber is relaxed. This is due to the fact that rubber is a so-called viscoelastic material, by which a material is described that is both elastic and viscous at the same time. The material therefore reacts in an interme-diate way to applied stresses or deformations than normal elastic or viscous materials. Where elastic materials are characterised by Hooke’s law and viscous materials by Newton’s law, the behaviour of a viscoelastic materials is somewhere in between. For a harmonically applied load, the deformation also acts harmonically, but due to the viscous part of the material, its reaction is slower than the applied load: it lags by a certain phase angle that is anywhere between 0 (pure elastic material) and 90 degrees (pure viscous material). The responses of these three materials are shown in Figure 4.1. A viscoelastic material can be described by a complex modulus of elasticity, which is a combination of an elastic modulus (the storage modulus, E0) and a viscous modulus (the loss modulus, E00). The relation between these moduli is graphically shown in Figure 4.2, with E* being the complex modulus and δ the phase lag angle, that relates the time lag in the response of the strain on an applied load.

Figure 4.1 Response to strain of an elastic material, a viscous material and a vis-coelastic material.

From Figure 4.2 it can be seen that the storage modulus and the loss modulus are related by the following equation:

E00

E0 = tan(δ) (4.1)

The tangent of the phase loss angle δ (tan-delta) is a frequently used parameter to describe the performance of a viscoelastic material, since it relates the stored energy to the lost energy in a load cycle. In general: a lower tan-delta represents a material with lower energy losses.

4.1.2 Dynamic Mechanical Analysis

The exact properties of a viscoelastic material are dependent on its composition and the cir-cumstances in which it operates. To determine these properties a technique called Dynamic Mechanical Analysis (DMA) is used. In this technique, a sample of the material is exposed to a

Figure 4.2 Relation between the moduli within a viscoelastic material.

harmonic deformation, while the stress response is measured. This yields the complex modulus, the storage modulus, the loss modulus and the tan-delta of the material. These properties are dependent on the frequency of the load cycle and the temperature, so the output of a DMA is a temperature sweep, in which the loading frequency is constant, or a frequency sweep in which the temperature is constant.

The results of a DMA can be used to create the master curves of a material. These master curves are built up from many small curves, that represent DMA measurements of a single temperature and multiple loading frequencies. These temperature curves can be shifted in the frequency spectrum by using a time-temperature superposition technique called the Williams-Landel-Ferry (WLF) equation, which is given by equation 4.2 [14]. This technique states that for a given temperature and frequency, there is another combination of temperature and frequency for which the same material properties are valid. The frequency shift that is required to reach this new combination is the shift factor a_T. This shift factor can be determined by entering the temperature of the individual temperature curves, a reference temperature and the WLF parameters C₁ and C₂. log(aT) = C1(T − TG) C2+ (T − TG) (4.2) 4.1.3 Material models

To be able to use these material properties to determine the indentation rolling resistance, the bottom cover layer of a conveyor belt needs to be modelled in two ways: the material itself and the layer of material. In literature two models are frequently used to describe the viscoelastic material of conveyor belt materials: the Standard Linear Solid (SLS) model and the generalised Maxwell model. The SLS model is a three parameter model and it is the simplest model that can describe the relaxation behaviour that can be identified in viscoelastic materials. Figure 4.3 shows this model. The spring E1 represents the elastic part of the material and the combination

of spring E2 and the damper η represent the viscous part. The relaxation time of this model,

τ , is defined as:

τ = η

E2

(4.3)

This implicates that this model has a single relaxation time, which is not very accurate for real viscoelastic materials. However, it was found by Lodewijks that a single relaxation time can be sufficient for belt speeds between 0.1 and 10 m/s [15].

Maxwell model. In this model n branches of a spring and a damper can be added, creating a (2n+1)-parameter model (see Figure 4.4). This provides the potential for a more accurate model, but also implies a more complex model.

Figure 4.3 Standard Linear Solid model.

Figure 4.4 Generalised Maxwell model.

4.1.4 Modelling the belt backing layer

The material models presented above need to be modelled as a layer of material before the indentation rolling resistance can be determined. Two methods were found in literature.

The first method is the viscoelastic half space, in which a layer of material is assumed of an infinite thickness, as is displayed in Figure 4.5. This method provides a two-dimensional stress model that can handle the most important stresses within such a material: compression stress and shear stress.

Figure 4.5 Graphic representation of a viscoelastic half space.

The second method that was found is to describe the layer as a Winkler viscoelastic foundation model. In this model, the material is assumed to consist of many separate viscoelastic elements on a rigid base, without interaction between these elements (see Figure 4.6). This model implies that the shear stress and the inertia of the material itself is neglected [15]. The advantage of the Winkler foundation model is that the problem is reduced to a one-dimensional problem, which results in simpler equations.

Figure 4.6 Graphic representation of a Winkler foundation model.

4.2

Theoretical methods

The following researchers have made use of the above material models to derive the indentation rolling resistance forces.

4.2.1 May et al.

In 1959, May et al. laid out the first brick in this field of research with a paper called “Rolling

friction of a hard cylinder over a visco-elastic material" [16]. They defined the viscoelastic layer

as a two-dimensional viscoelastic half space. To describe the behaviour of the material, the SLS model was chosen. The rolling resistance force was determined through the asymmetrical stress pattern that is caused by the stress relaxation within the viscoelastic material. This stress pattern causes a moment around the cylinder’s axis. The rolling resistance force that counteracts this moment was found to be dependent on the belt speed and has a maximum for a certain belt speed. This specific speed corresponded to a peak in the relaxation time distribution, which is material-dependent. The analysis was performed by keeping the indentation depth at a constant level by increasing the load. They concluded that the load to maintain this depth was also dependent on the belt speed, where the load was required to increase for an increasing velocity.

In 1995 Lodewijks rearranged this method so that the vertical load was pre-described, instead of the indentation depth, which is more convenient for evaluating belt conveyor systems. He arrived at the following indentation rolling resistance factor [15]:

f_im∗ = F ∗ i F∗ z (4.4)

In which F_z∗ is the pre-described vertical load and F_i∗ the resulting rolling resistance force, which are defined by:

F_z∗ = E1a 3 0 6Rh " 2 −  _b a0 3 + 3  _b a0 # + 2E2ka 3 0 Rh " 1 −  _b a0 2# (4.5) F_i∗ = E1a 4 0 8R2_h " 1 − 2  _b a0 2 +  _b a0 4# +E2a 4 0k R2_h  k3− k 2 1 +  _b a0 2! +1 3 1 +  _b a0 3! − k(1 + k)  k + b a0  e −1 k _a 0+ b a0    (4.6)

E1, E2 are the Maxwell model parameters that follow from the SLS model. R is the radius of

the roll in meters, h is the thickness of the belt’s bottom cover layer and b is the length of the second part of the contact zone between the belt and the roll. k is the Deborah number, which is defined by:

k = V τ a0

(4.7)

a3₀ = 3FzDh

4E₁ (4.8)

Where D is equal to 2R. Due to the pre-description of the indentation depth, which is generally not the case in the situation of a belt conveyor, this method is not directly applicable in this study.

4.2.2 Hunter

Hunter followed in 1961 with “The rolling contact of a rigid cylinder with a viscoelastic half

space" [17]. Like May et al, he models the belt backing material as a viscoelastic half space and

the material itself conform the SLS model. He uses a more analytical approach than May et al by using integral equations that allow shear stresses. Hunter also relates the rolling resistance force to the asymmetrical stress distribution within the rolling contact. He approaches the problem from another angle: the retardation instead of the relaxation. For the three parameter Maxwell model the following equations can be stated [15]:

f = E2 E1

(4.9)

µD = E1+ E2 (4.10)

Which represent the retardation coefficient and the dynamic shear modulus. The semi-contact length of the indentation at zero velocity, a₀, is defined by Hunter as:

a2₀ = 2(1 − ν)(1 + f )RFz

πµD

(4.11)

The ratio between a and a0, the contact length when the belt has a non-zero speed, can be

determined by: _a 0 a 2 = 1 + 2f∗ h∗ K0(k) K1(k)+ I0(h) I1(h) (4.12)

With k = a/V τ and h = (1 + f )k. K₀, K₁, I₀ and I₁ are modified Bessel functions of the zeroth and first order. Through the entering pressure distribution function Hunter finally reaches the indentation rolling resistance factor:

Γ1 = a − 1 h − 1 2 "_ a0 a 2 − 1 # I0(h) I1(h) ! (4.13) b = V τ − Γ1 (4.14) f_ih∗ = 1 R b − V τ 1 + f − Γ1 _a a0 2! (4.15)

Like May et al, Hunter too reaches the conclusion that the rolling friction coefficient is de-pendent on the belt speed and that it reaches a certain maximum value for a belt speed that corresponds with the relaxation time of the belt material. The usability of the method of Hunter is questionable, since he assumes that the indentation depth is independent of the belt speed [15].

4.2.3 Jonkers

In 1980, Jonkers used a different approach to model the indentation rolling resistance [18]. Instead of using the asymmetrical stress distribution, he uses the rubber hysteresis losses that are caused by the indentation of the viscoelastic material. Jonkers also models the belt’s backing material with the SLS model, but he uses a Winkler model to describe the material layer with a finite thickness [15]. It is assumed that the indentation can be described as half a sinusoidal curve and that the pressure distribution also follows this geometry, with the maximum pressure at the centreline of the roll. Although not correct for typical belt conveying speeds [19], this choice of geometry allowed Jonkers to express the indentation rolling resistance force as a concise equation that is quick and easy in use. The implication is that it overestimates the indentation rolling resistance, which has been shown by, among others, Wheeler [19]. Also, the method of Jonkers does not take the influence of the belt speed on the contact length into account [15]. Despite these downsides, the Jonkers method is still widely used to make a quick comparison of two different belt materials, since it requires very little parameters.

Jonkers defines the indentation rolling resistance force of a single roll as:

Fij = f (δ)  _h E0_D2 1₃ F 4 3 Z (4.16)

In which h and D are respectively the thickness of the bottom cover layer and the diameter of the idler roll and f (δ) is:

fδ= 1 2πtan(δ) " (π + 2δ)cos(δ) 4p 1 + sin(δ) #4₃ (4.17)

If the friction factor is defined as in DIN 22 101 - like a Coulomb friction, the method of Jonkers yields the following equation:

f_ij0 = FW FZ = 1 2πtan(δ) " (π + 2δ)cos(δ) 4p 1 + sin(δ) #4_{3 } FZh E0_D2 1₃ (4.18)

An adaptation of the method of Jonkers was used by Lodewijks to be able to compare it with other methods. This adaptation uses the contact length as determined by Lodewijks’ adaptation of the method of May (which will be shown in Section 4.2.5). In this adaptation, the influence of the belt speed on the contact length is taken into account.

4.2.4 Spaans

Spaans also used the Winkler foundation model for the belt’s backing layer and the SLS model for the material itself. He therefore also neglects the shear forces within the material. Similar

to the approach of Jonkers, he established the indentation rolling resistance by determining the hysteresis losses for the belt travelling over a roll and assumes the maximum pressure at the centre line of the roll. The indentation friction factor was determined by Spaans to be [15]:

fis= 0.5ηi

Fz1/3

(2/3)4/3_E∗1/3_D2/3

0 (1 + (1 − ηi)3/4)4/3

(4.19)

The indentation damping factor ηi and the lateral stiffness E∗ need to be determined from a

harmonic deformation test that is performed on a sample of the total belt [15]. D₀ is a diameter in which both the diameter of the roll and the curvature of the belt at the roll are accounted for.

The method of Spaans is less useful than methods presented earlier, since he uses the lateral stiffness and the indentation damping coefficient of the total belt, which cannot be determined properly without fabricating a piece of the total belt. More drawbacks to use this model are that it neglects the belt speed dependence of the contact length and that the hysteresis loss factor δ is assumed to be a material constant, which is generally not a valid assumption [15]. To still be able to compare this method with those of Hunter, May and Jonkers, an adaptation was provided by Lodewijks through which only the belt’s backing layer is modelled [15]. By using the three parameter Maxwell model, the damping factor ηi of the damping layer is

given as a function of the loss factor tan-delta by [15]:

ηi(δ) =

2πtan(δ)

2 + (π + 2δ)tan(δ) (4.20)

In which the loss factor tan-delta is given by [15]:

tan(δ) = ωηE

2 2

E1E22+ ω2η2(E1+ E2)

(4.21)

If the assumption is made that the lateral stiffness is only caused by the belt’s backing layer, the Winkler model provides the following expression for this stiffness [15]:

E∗ = E1

h (4.22)

The adapted indentation rolling resistance factor of Spaans’ methods can now be expressed:

f_is∗ = 0.5ηi(δ) Fz1/3h1/3 (2/3)4/3_E1/3 1 D 2/3 0 (1 + (1 − ηi(δ))3/4)4/3 (4.23)

A final assumption needs to be made in order to compare this method with those of Hunter, May and Jonkers, which is that under normal circumstances, D₀ will practically be the same as the idler roll diameter D.

4.2.5 Lodewijks

In 1995 Lodewijks created a method for the calculation of the indentation rolling resistance in which he partly follows the method of May et al [19]. The material is modelled by using the

SLS model, but different from May et al., the Winkler foundation model is used to describe the backing layer. An iterative process was established to determine the contact length and the SLS model parameters, by beginning with a simple predictor for the contact length that follows from the Winkler foundation model [15] and is given by equation 4.8. With these SLS model parameters the indentation resistance force is determined and with that, the indentation rolling resistance factor.

The stress distribution for the SLS model can be written as:

σ(x) = a2  _E 1 2Rh _{a − x} a  _{a + x} a  + E2k Rh  (1 + k)  1 − e( −1(a−x) ka )  − _{a − x} a  (4.24)

In which k is defined as in equation 4.7.

a = F 1/3 z  E1 6Rh  2 −_ab3+ 3_ab  +2E2k Rh  1 −b_a2 (1/3) (4.25)

The formula for the rolling resistance force F_i is almost equal to that of May et al., but the contact length is different.

Fi= E1a4 8R2_h " 1 − 2 _b a 2 + _b a 4# +E2a 4_k R2_h " k3−k 2 1 + _b a 2! +1 3 1 + _b a 3! − k(1 + k)  k + b a  e−1k ( a+b a ) # (4.26)

For the use in the procedure of DIN 22 101, the following relation can be used:

fim=

Fi

Fz

(4.27)

To compensate for the neglecting of the shear forces by the Winkler foundation, Lodewijks determined a correction factor fs, based on the method of Hunter, which does incorporate the

shear forces, and the method of May, which also neglects them.

fs=

f_ih∗

f_im∗ (4.28)

Which allows the calculation of the corrected indentation rolling resistance:

fi= fsfim (4.29)

The graphical relation between the method of May et al., Hunter and the modified method of May et al. can be seen in Figure 4.7, which were generated for the parameters from Table 4.1. As can be seen, the modified method of May et al. proposed by Lodewijks yields a lower value for the indentation rolling resistance than its original method and the method of Hunter. The correction factor f_s shifts the results in the direction of those of May et al.

Figure 4.7 Comparison of the methods of Hunter, May et al. and the adjustment of May et al. as pro-poseds by Lodewijks [15].

Table 4.1 Input parameters used by Lodewi-jks to compare the methods of Hunter, May and his modification of May. [15]

R 0.0795 m E1 7 MPa

h 0.008 m E2 250 MPa

Fz 2000 N η 1875 Pa s

V 0.1 - 10 m/s

The accuracy of this method was confirmed by Wheeler in 2003, who compared the results of this method with experimental results [19].

A generalisation of this model was created by Rudolphi and Reicks in 2006, in which they expand the SLS model to the generalised Maxwell material model [20]. This generalised model uses multiple relaxation times, which allows for a more accurate description of rubber behaviour. The methods of Jonkers (both original and the adaptation of Lodewijks) and Spaans show friction factors as shown in Figure 4.8 for input variables of Table 4.1. It is clear that they yield much higher results than the methods of Hunter, May et al. and Lodewijks.

Figure 4.8 Comparison of the methods of Jonkers and Spaans [15].