Performance Enhancement of Abrasive Waterjet Cutting

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Proefschrift

ter verkrijging van de graad van doctor

aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus prof. dr. ir. J.T. Fokkema,

voorzitter van het College voor Promoties,

in het openbaar te verdedigen

op maandag 19 mei 2008 om 10.00 uur

door

Vu Ngoc PI

Master of Engineering

Hanoi University of Technology, Vietnam

Geboren te Thai Binh, Vietnam

Dit proefschrift is goedgekeurd door de promotoren:

Prof. Dr.-Ing. habil B. Karpuschewski toegevoegd promotor: dr. ir. A.M. Hoogstrate

Samenstelling promotie commissie:

Rector Magnificus voorzitter Prof. Dr.-Ing. habil B. Karpuschewski Otto-von-Guericke-Universität Magdeburg, promotor Dr. ir. A.M. Hoogstrate TNO Science and Industry, toegevoegd promotor Prof. dr. ir. J.R. Duflou Katholieke Universiteit Leuven Prof. Dr.-Ing. H. Louis Leibniz Universität Hannover Prof. dr. ir. A.J. Huis in ‘t Veld Universiteit Twente Prof. dr. U. Staufer Technische Universiteit Delft Prof. dr. M.A. Guitierrez De La Merced Technische Universiteit Delft, reservelid

ISBN: 978-90-9023096-2

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All rights reserved. No part of this publication may be reproduced, utilized or stored in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without permission from the copyright holder.

Dedicated to my wife Hoang Thi Tham

Acknowledgement

First of all, I would like to express my deep and sincere gratitude to Prof. Dr.-Ing. habil. Bernhard Karpuschewski, my promoter, for his guidance, consideration, and critical review of the present thesis.

I would like to give sincere appreciation to Dr. ir. André Hoogstrate, my adjunct supervisor, for his useful discussion, for his detailed and constructive comments, and for his important support throughout this work.

Then I would like to express my special thanks to Dr. Mohamed Hashish (Flow International Cooperation), Dr. Jey Zeng (OMAX Corporation), Dr. Eric Chalmers (AccuStream Inc.), Dr. Greg Mort (KMT Waterjet Systems Inc.), Dr. Andreas Höfner (GMA Garnet (Europe) GmbH), and Prof. Deng Jianxin (Shandong University), for their discussions, documents, and encouragement.

Furthermore I would like to thank Paolo Golfiotti, my Italian MSc student, for his helping in the abrasive recycling experiments.

Not to forget I would like to record my gratitude to the staff members of the department of Precision and Microsystems Engineering, especially Associate Prof. Marcel Tichem, Dr. Sebastiaan Berendse, Harry Jansen and Marli Guffens, for their supports of my works.

My work is a cooperation between the Delft University of Technology and the Vietnamese Government. The work is supported by Training Scientific and Technical Cadres in Institutions Overseas with the State Budget (Project 322) and the Management Centre for International Cooperation (CICAT). I would like to give my appreciation to all members of CICAT and 322, especially Dr. Paul Althuis, Veronique van der Varst, Willemijn van der Toorn, and Ngoc Lien, for their helps and encouragement.

I would like to thank all my colleagues and friends in and outside TU Delft for their encouraging and helping. Very special thanks to Tolga Susuzlu for his help in my experimental work and to Dr. Thieu Quang Tuan and Jeroen Derkx for their review of my thesis. Also, thanks to Nguyen Thanh Hoan for his help by taking pictures of my experimental setup.

Also I would like to give my appreciation to Prof. Nguyen Dang Binh, Associate Prof. Phan Quang The, Associate Prof. Nguyen Dang Hoe and Associate Prof. Vu Quy Dac from the Thai Nguyen University of Technology, for their support and encouragement.

i Last but not least, I would like to thank my mother, my mother-in-law, my sister and brothers, for their love and their encouragement. Thanks to my nephew Vu Quang Dien for his help by designing the cover of my book. I would like to thank my wife, Hong Tham, for her love, patience, enormous support, review of my thesis, and finally for taking care of our children. I also would like to thank my daughters, Thu Trang and My Hanh, for their love and back up.

Delft, May 2008,

Vu Ngoc Pi

ii Summary

Abrasive Waterjet (AWJ) Machining is a recent non-traditional machining process. Major part of this technology is a very high-pressure beam of water and abrasives, which is used for machining. The typical water pressure ranges from 300 to 380 MPa and the typical beam diameter varies between 0.6 and 1.2 mm. This technology is widely used in industry for cutting difficult-to-machine- materials, milling slots, polishing hard materials, cleaning contaminated surfaces, etc. AWJ machining has many advantages, e.g. it can cut net-shape parts, no heat is generated during the cutting process, it is particularly environmentally friendly as it is clean and it does not create dust.

Although AWJ machining has many advantages, a big disadvantage of this technology is its relatively high cutting cost. Consequently, the reduction of the machining cost and the increase of the profit rate are big challenges in AWJ technology.

To reduce the total cutting cost as well as to increase the profit rate, this research focuses on performance enhancement of AWJ cutting with two possible solutions including optimization in the cutting process and abrasive recycling.

The first solution to enhance the AWJ cutting performance is the optimization of the AWJ cutting process. As a precondition, it is necessary to have a cutting process model for optimization. In order to use that model for this purpose, several important requirements are given. The most important requirement for such a model is that it can describe the “optimum relation” between the optimum abrasive mass flow rate and the maximum depth of cut.

To develop a cutting process model which can be used for the AWJ optimization, many available models have been analyzed. Since the most important requirement for a process model (see above) can be obtained from Hoogstrate’s model, an extension of this model is carried out. The extension model consists of three sub-models including pure waterjet model, abrasive waterjet model and abrasive-work material interaction model. The pure waterjet model enables to determine the energy transfer from pressurized water to the pure waterjet. The abrasive waterjet model is used to calculate the energy transfer from pure waterjet to the abrasive particles. The abrasive– work material interaction model is used to identify the relation between the work material characteristics, the abrasive characteristics and the cutting efficiency in the process of removing work material chips by using the kinetic energy of abrasive particles.

The extension cutting process model is more accurate than the original one and it is capable to

iii optimize AWJ systems. The influence of many process parameters such as the water pressure, the abrasive mass flow rate, the nozzle diameter, the abrasive particle diameter etc. have been taken into account. By modeling the work material coefficient, the extension model can be used for various work materials. Also, by giving a model for the abrasive material coefficient, the model can be applied for several most common abrasive types.

Up to now, there has not been a model for the prediction of AWJ nozzle wear. Therefore, modeling the nozzle wear rate has been carried out and a model for the wear rate of nozzles made from composite carbide has been proposed. The model can be used in the optimization problems as well as in the calculation of the AWJ cutting regime.

Based on the extension cutting process model, two types of optimization applications have been carried out. They are related to technical problems and economical problems. The optimization problems have been solved in order to determine the optimum exchange nozzle diameter and the optimum abrasive mass flow rate for getting different objectives including the maximum depth of cut (for technical problems), the minimum total cutting cost and the maximum profit rate (for the economical problems). From the results of these considerations, regression models for determining the optimum nozzle exchange diameter and the optimum abrasive mass flow rate for various objectives have been proposed.

In AWJ machining, there are many cutting process parameters. Therefore, the ways to select other process parameters optimumly have also been investigated. The procedure for the determination of an optimum cutting regime then is given.

The other solution to enhance the cutting performance is abrasive recycling. In the present study, GMA garnet, the most popular abrasives for blast cleaning and waterjet cutting, has been chosen for the investigation. The recycling of GMA abrasives has been investigated on both technical side and economical side. On the technical side, the reusability and the cutting performance of the recycled and recharged abrasives have been analysed. The influence of the recycled and recharged abrasives on the cutting quality was studied. Also, the optimum particle size of recycled and recharged abrasives for the maximum cutting performance has been detected. On the economical side, first, the prediction of the cost of recycled and recharged abrasives was done. Then, the economic comparisons for selecting abrasives have been carried out. In addition, the economics of cutting with recycled and recharged abrasives have been studied. Several suggestions for an abrasive recycling process which promises a more effective use of the grains have been proposed.

By optimization in the cutting process and by abrasive recycling, the cutting performance can be increased, the total cutting cost can be reduced, and the profit rate can be enlarged considerably. Consequently, the performance of AWJ cutting can be enhanced significantly.

iv Samenvatting

Abrasief waterstraal snijden (AWS) is een recent, niet conventioneel verspaningsproces. Het is een technologie waarbij een waterstraal onder hoge druk, gemengd met abrasief, gebruikt wordt voor de verspaning van diverse materialen. De waterdruk ligt tussen de 300 en 380 MPa; de waterstraaldiameter ligt tussen de 0.6 en 1.2 mm. Waterstraaltechnologie wordt veel gebruikt in de industrie voor het snijden van moeilijk bewerkbare materialen, het boren van gaten, het polijsten van harde materialen, het reinigen van vervuilde oppervlakken etc. AWS bewerken heeft vele voordelen waaronder: het maken van “near-net-shape” onderdelen, geen warmte ontwikkeling tijdens het verspaningsproces en het is bijzonder milieuvriendelijk omdat het schoon is en er geen fijnstof of gevaarlijke stoffen vrij komen.

Naast de vele voordelen die AWS snijden biedt zijn de hoge kosten een belangrijk nadeel. Daarom zijn de reductie van de bewerkingskosten en het verhogen van de winstmarge belangrijke uitdagingen in de AWS technologie.

Om zowel de totale bewerkingskosten te reduceren alsook de winstmarge te verhogen, concentreert dit onderzoek zich op de prestatieverbetering van AWS snijden. Twee mogelijke oplossingen worden bekeken: optimalisatie van het bewerkingsproces en hergebruik van abrasief.

De eerste oplossing om de prestatie van AWS bewerken te verbeteren is de optimalisatie van het AWS proces. Voorwaarde hiervoor is de beschikbaarheid van een procesmodel van de verspaning voor de optimalisatie. Om een model te kunnen gebruiken voor dit doel moet het aan enkele belangrijke voorwaarden voldoen. De belangrijkst daarvan is dat het model de relatie tussen de abrasief massa stroom en de maximale snedediepte beschrijft.

Voor de ontwikkeling van een procesmodel dat gebruikt kan worden voor de AWS optimalisatie zijn vele beschikbare modellen geanalyseerd. Omdat aan de belangrijkste voorwaarde voor een proces model (zie boven) kan worden voldaan door het model van Hoogstrate, wordt een uitbreiding van dit model uitgevoerd.

Het uitgebreide model bestaat uit 3 deelmodellen: het pure waterstraalmodel, het abrasieve waterstraalmodel en het abrasief-werkstukmateriaal interactie model. Het pure waterstraal model maakt het mogelijk de energie overdracht te bepalen van het samengeperste water naar de pure waterstraal. Het abrasieve waterstraalmodel wordt gebruikt om de energie overdracht te berekenen van de pure waterstraal naar de abrasieve deeltjes. Het abrasief-werkstukmateriaal interactie model

v wordt gebruikt om de relatie te bepalen tussen de werkstuk materiaal eigenschappen, het abrasief en de verspaningsefficiëntie tijdens het verwijderen van spanen van het werkstuk door gebruik te maken van de kinetische energie van de abrasieve deeltjes.

Het uitgebreide procesmodel is nauwkeuriger dan het originele model en kan gebruikt worden voor het optimaliseren van AWS systemen. De invloed van diverse procesparameters zoals de waterdruk, de abrasief massastroom, de orifice diameter, de deeltjesgrootte etc. zijn in het model meegenomen. De introductie van een werkstukmateriaal-coëfficiënt maakt het model bruikbaar voor diverse werkstuk materialen. Tevens kan het model gebruikt worden voor verschillende abrasief materialen door de introductie van een abrasiefmateriaal-coëfficiënt.

Tot dusver was er geen model beschikbaar voor de voorspelling van de slijtage van de AWS focusbuis. Daarom is de slijtagesnelheid van de focusbuis gemodelleerd en een model voor de slijtagesnelheid van gesinterde wolfraamcarbide focusbuizen voorgesteld. Dit model kan zowel worden gebruikt voor de optimalisatie van het AWS proces.

Twee types van optimalisaties zijn uitgevoerd, gebaseerd op het uitgebreide procesmodel. Deze zijn gerelateerd aan technische en economische optimalisatie. De optimalisatie functie is zodanig opgesteld, dat de optimale focusbuis wissel diameter en de optimale abrasief massastroom konden worden bepaald. Dit is gedaan voor verschillende doelstellingen waaronder de maximale snedediepte (de technische doelstelling) en de minimale bewerkingskosten en maximale winstmarge (de economische doelstellingen). Gebaseerd op de resultaten van deze overwegingen zijn regressie modellen voorgesteld voor het bepalen van de optimale focusbuis wissel-diameter en de optimale abrasief massastroom voor de verschillende doelstellingen.

Er zijn vele procesparameters in AWS bewerken. Daarom zijn de diverse methodes om de optimale procesparameters te bepalen ook onderzocht. De procedure voor het bepalen van een optimaal verspaningsregiem wordt vervolgens gegeven.

De andere oplossing om de verspaningsprestatie te verbeteren is het hergebruik van het abrasief. In dit onderzoek is gebruik gemaakt van het meest populaire abrasief voor waterstraal snijden en reinigen: GMA garnet. Zowel de technische als de economische kant van het hergebruik van GMA garnet zijn onderzocht. Op het technische vlak zijn de herbuikbaarheid en de verspaningsprestatie van het hergebruikte abrasief geanalyseerd. Hierbij is zowel het batch-gewijze hergebruik van abrasief, alsook het gradueel opmengen van gebruikt en nieuw abrasief geanalyseerd. De invloed van beide recycle-methodes op de verspaningskwaliteit is onderzocht. Tevens is de optimale deeltjesgrootte voor hergebruik bij beide methodes, gerelateerd aan de maximale verspaningsprestatie bepaald. Op het economische vlak is allereerst een voorspelling gedaan van de kosten van beide recycle-methodes; vervolgens is een economische vergelijking voor de selectie van abrasieven gedaan. Daarbij zijn ook de kosten bestudeerd van het bewerken met hergebruikt

vi en met toegevoegd abrasief. Er zijn verschillende suggesties gedaan voor een hergebruik proces dat een effectiever gebruik van het abrasief materiaal belooft.

Door optimalisatie van het verspaningsproces en hergebruik van het abrasief kan de verspaningsprestatie worden verhoogd, de totale verspaningskosten worden gereduceerd en de winstmarge aanmerkelijk worden vergroot. Daardoor kan de prestatie van AWS bewerken significant worden verbeterd.

vii

viii

Contents

Acknowledgement ...... i Summary ...... iii Samenvatting...... v Nomenclature...... xiii 1 Introduction ...... 1 1.1 Historical review...... 1 1.2 Introduction to AWJ Technology...... 2 1.2.1 Introduction to an AWJ cutting system ...... 2 1.2.2 Parameters of an AWJ machining process ...... 5 1.2.3 Advantages and disadvantages of AWJ Technology ...... 6 1.3 Challenges in AWJ Technology ...... 6 2 State of the art in optimization of AWJ machining...... 9 2.1 State of the art in AWJ technical optimization...... 10 2.1.1 Optimum combination of focusing tube and orifice diameter ...... 10 2.1.2 Optimum focusing tube length ...... 11 2.1.3 Optimum abrasive mass flow rate...... 12 2.1.4 Optimum abrasive particle size...... 15 2.1.5 Optimum standoff distance ...... 16 2.2 State of the art in AWJ cost calculation and cost optimization...... 16 2.2.1 State of the art in AWJ cost calculation ...... 16 2.2.2 State of the art in AWJ cost optimization...... 18 2.3 State of the art in AWJ abrasive recycling ...... 21 2.4 Conclusions...... 25 3 Project definition ...... 27 3.1 Aim of the investigations...... 27 3.2 Outline of the thesis ...... 28 4 Used experimental and measuring equipment ...... 29 4.1 AWJ machining setup ...... 29 4.2 Abrasive particles ...... 30

ix 4.2.1 Abrasive properties...... 30 4.2.2 Abrasive size distribution and abrasive particle diameter ...... 33 4.3 Work materials...... 35 4.4 Experimental setup for measuring the water flow rate ...... 36 4.5 Experimental setup for measuring the reaction force ...... 37 4.6 Experimental setup for determining the maximum depth of cut...... 38 4.7 Experimental setup for collecting abrasives ...... 38 4.8 Experimental setup for determining surface roughness ...... 40 4.9 Other measuring equipment...... 41 4.9.1 Microscope...... 41 4.9.2 Surface roughness measurement device ...... 41 5 Frame work of modeling and AWJ optimization approach...... 43 5.1 Frame work of modeling ...... 43 5.2 AWJ optimization approach ...... 45 5.2.1 Introduction to optimization ...... 45 5.2.2 Statement of an AWJ optimization problem...... 51 5.2.3 Solutions for AWJ optimization problems...... 51 6 Modeling the cutting process for AWJ optimization ...... 53 6.1 Requirements for an AWJ cutting process model ...... 53 6.2 State of the art in AWJ cutting process modeling...... 54 6.2.1 Studies of Hashish...... 54 6.2.2 Studies of Zeng and Kim...... 56 6.2.3 Other studies ...... 57 6.3 Introduction to Hoogstrate’s model...... 58 6.3.1 Model description ...... 58 6.3.2 Discussion...... 61 6.4 Extension of Hoogstrate’s model ...... 62 6.4.1 Pure waterjet modeling...... 62 6.4.2 Abrasive waterjet modeling...... 71 6.4.3 Abrasive - work material interaction modeling ...... 80 6.5 Modeling the AWJ cutting process ...... 91 6.6 Conclusions...... 94 7 Optimization in AWJ cutting process ...... 95 7.1 Cost and profit analysis...... 95 7.1.1 Cost analysis...... 95 7.1.2 Profit analysis...... 99 7.2 Optimization for determining optimum nozzle lifetime...... 100

x 7.2.1 Nozzle lifetime and nozzle wear in AWJ machining...... 101 7.2.2 Relation between the nozzle lifetime and the feed speed ...... 106 7.2.3 Optimization for determining optimum nozzle lifetime for minimum cutting cost..... 109 7.2.4 Optimization for finding optimum nozzle lifetime for maximum profit rate ...... 115 7.2.5 Benefits of cutting with optimum nozzle lifetime...... 117 7.2.6 Conclusions...... 120 7.3 Optimization for determining the optimum abrasive mass flow rate ...... 121 7.3.1 Optimization for determining the optimum abrasive mass flow rate for maximum cutting performance...... 121 7.3.2 Optimization for determining the optimum abrasive mass flow rate for minimum cutting cost ...... 124 7.3.3 Optimization for determining optimum abrasive mass flow rate for maximum profit rate 126 7.3.4 Benefits of cutting with the optimum abrasive mass flow rate ...... 130 7.3.5 Conclusions...... 132 7.4 Selection of process parameters for the optimum cutting regime...... 133 7.4.1 Optimum selection of the number of jet formers, the orifice diameter, and the nozzle diameter 134 7.4.2 Optimum selection of abrasive type and size ...... 136 7.4.3 Procedure for determination of the optimum AWJ cutting regime ...... 137 7.5 Conclusions...... 138 8 Recycling and recharging of abrasives ...... 139 8.1 Reusability of abrasives ...... 139 8.1.1 Experimental setup...... 139 8.1.2 Results and discussions...... 140 8.2 Cutting performance and cutting quality of recycled abrasives...... 141 8.2.1 Experimental setup...... 141 8.2.2 Results and discussions...... 142 8.3 Cutting performance and cutting quality of recharged abrasives ...... 147 8.3.1 Experimental setup...... 147 8.3.2 Results and discussions...... 148 8.3.3 Multi-recharging of abrasive...... 150 8.4 Conclusions...... 151 9 Economics of abrasive recycling ...... 153 9.1 Cost calculation for recycled and recharged abrasives...... 153 9.1.1 Cost analysis ...... 153 9.1.2 Results and discussions...... 154

xi 9.2 Economic comparisons for selecting abrasives...... 157 9.3 Economics of cutting with recycled and recharged abrasives ...... 159 9.3.1 Economics of cutting with recycled abrasives ...... 159 9.3.2 Economics of cutting with recharged abrasives...... 166 9.3.3 Comparisons among cutting with new, recycled and recharged abrasives ...... 170 9.4 Suggestions for abrasive recycling process...... 170 9.5 Conclusions...... 171 10 Conclusions and recommendations for further research ...... 173 10.1 Conclusions...... 173 10.2 Recommendations...... 176 References...... 177 Appendix: Recycling system ...... 183 A.1 WARD 1 ...... 183 A.2 WARD 2 ...... 184 About the author...... 187

xii Nomenclature

Symbols

Symbol Unit Definition

A m2 cross section area C € cost c - coefficient d m diameter E MPa elasticity

3 ec J/m specific cutting energy F N force

Grecy kg/h recycling capacity g h m depth k - coefficient l m length

Nm - machinability number n - number m kg/s mass flow rate

P w power Pr € profit p Pa pressure Q - quality number r % reusability R - abrasive load ratio

Re - Reynolds number T s time v m/s velocity x - number

xiii η - momentum transfer efficiency coefficient κ - power transfer efficiency δ m/s wear ξ - cutting efficiency coefficient ρ Kg/m3 density

Subscripts

Subscript Definition a related to average abr related to abrasive actual related to actual awj related to abrasive waterjet c related to cutting com compressible d related to discharge de related to depreciation e related to electrical en related to energy f focusing tube / nozzle h related to hour inc related to incompressible int related to interest l related to length la related to labor m related to mass ma related to maintenance max maximum min minimum msh related to manned shifts mt related to machine tool inc incompressible

xiv op related to optimum ori orifice ov related to overhead p related to particle q related to quality rech related to abrasive recharging recy related to abrasive recycling ro related to occupied room rpl related to replacement sal related to sale sqm related to squared meter sh related to shift th theoretical use related to time of use ut related to utilization w related to water wa related to wages wj related to waterjet wor related to working y related to year

xvi 1 Introduction

Abrasive Waterjet (AWJ) Machining is a recent non-conventional machining process. In this technology, a very high-pressure beam of water and abrasives is used for machining. This technology is widely used in industry as it has many advantages.

In this chapter an introduction to Abrasive Waterjet (AWJ) Technology is provided. A review of the AWJ history is first carried out to draw a picture of the progress in this technology. Brief descriptions of the schema and the main components of an AWJ system are also given. Advantages and drawbacks of the AWJ technology are then evaluated. Challenges of the technology are discussed in the end.

1.1 Historical review

AWJ machining has been developed from Waterjet machining. The earliest use of the water beam in coal mining was in the former Soviet Union and New Zealand [Summ95]. This mining technique was also used for removing blasted rocks from working areas into collection tunnels.

From 1853 to 1886, pressurized water was used for excavating soft gold rocks. The pressurized water for coal mining was also used in Prussia in the early 1900s and then in Russia in the 1930s [Summ95].

In 1936, Peter Tupitsyn, who was working for the Donetsk Coal Basin in Ukraine, proposed the idea of using a waterjet beam to cut boreholes in the coal bed [Chri03].

In the 1950s, Dr. Norman Franz, a forestry engineer, was the first who studied the use of a waterjet beam as a cutting tool for wood processing [Flow08]. However, the first patent of a waterjet cutting system was granted for the staff of McCartney Manufacturing Company, a division of the Ingersoll-Rand Corp. [Tikh92]. In 1971, the first commercial waterjet machine was introduced into the market by this company [Tikh92].

In 1979, Dr. Mohamed Hashish, who has worked for Flow International Cooperation, invented the abrasive waterjet cutting method by adding abrasives into the pure waterjet [Flow08]. Soon after this, in 1980, abrasive waterjet was first used to cut glass, steel, and concrete [Flow08]. The invention of AWJ led to a huge expansion of applications of cutting with high-pressure water. Since then, AWJ has been widely used in various industries such as cutting of a wide variety of sheet

1 materials, cleaning of contaminated surfaces, polishing of hard-to-machine materials, etc.

1.2 Introduction to AWJ Technology

1.2.1 Introduction to an AWJ cutting system

There are two types of waterjets: pure (or plain) waterjet and abrasive waterjet. In pure waterjet cutting, only a pressurized stream of water is used to cut through materials. This type of cutting is used to cut soft materials such as cardboard, leather, textiles, fibre plastics, food or thin plates of aluminium. In AWJ cutting, an abrasive waterjet entrainment system mixes abrasives with the waterjet in a mixing chamber following an orifice (Figure 1.1). The abrasive particles are accelerated by the water stream and then leave the focusing tube (or the nozzle) with the stream. AWJ cutting is used for cutting harder materials such as stainless steel, glass, ceramics, titanium alloys, composite materials, and so forth.

Intensifier Attenuator High-pressure water

Mixing chamber

Orifice

Abrasive particles

Electric Hydraulic Directional Focusing motor pump control Inlet tube valve water Presure Water generation preparation Jet system former system

Figure 1.1: AWJ entrainment system schema

A typical AWJ entrainment system (as shown in Figure 1.1) consists of four main parts: the water preparation system, the pressure generation system, the jet former, and the abrasive supply system. A brief description of these parts is given below:

• The water preparation system:

2 The water preparation system is used for supplying purified water for the pressure generation system. Generally, particles larger than 1 μm have to be filtered out to prevent unacceptable wear of the critical parts of the pressure generation system [Hoog00].

• The pressure generation system:

This system is equipped with a pump to ensure a continuous and stable flow of high pressure. Three types of pumps, namely intensifier, crankshaft and direct pumps can be distinguished.

Figure 1.2: Direct pump (Courtesy of Flow International Cooperation)

Direct pumps are used for applications with low pressure such as cleaning, or washing a desk or a work place etc. In a direct pump, the movement of three plungers is transmitted directly from the electric motor (see Figure 1.2).

Figure 1.3: Double-acting intensifier

Intensifier pumps (Figure 1.3) are used for applications with water pressure up to 600 MPa. In an intensifier pump, a double-acting cylinder in which the movement of the piston is driven by a hydraulic system is used. Two small diameter cylinders at each end of the hydraulic cylinder help to

3 pressurize the water alternately as the hydraulic piston moves back and forth. By connecting two intensifier pumps in series, the output water pressure can be up to 800 MPa [Susu04].

The third type is the crankshaft pump, which can provide the pressure up to 345 MPa [Chri03]. An example of this pump is shown in Figure 1.4. It is known that the efficiency of crankshaft pumps is higher than that of intensifier pumps because crankshaft pumps do not require a power-robbing hydraulic system.

Figure 1.4: Crankshaft pump (Courtesy of OMAX Corp. Kent, WA)

• The jet former:

The jet former is used to transfer part of the hydraulic water energy into kinetic energy of water, and then into kinetic energy of abrasive particles. Figure 1.5 shows a typical jet former for AWJ cutting [Hoog04]. To form the abrasive waterjet, first, the high pressure water is forced through an orifice to create a high speed waterjet. Then the high speed waterjet passes through a mixing chamber, which is installed downstream of the orifice. Because of the Venturi effect, a vacuum is created in the mixing chamber. As a result, the abrasive particles and some air are sucked into the mixing chamber through a feed line. After entering the mixing chamber, the particles are accelerated by the high-speed waterjet (velocity about 600 to 900 m/s) and then passing through a focusing tube (or nozzle).

As mentioned above, the orifice, the mixing chamber and the focusing tube are the main parts of a jet former. Orifices can be made of sapphire, ruby or diamond with a diameter ranging from 0.08 to 0.8 mm [Hoog00]. The lifetime of a diamond orifice is about 1000 to 2000 hours while it is only 40 to 70 hours for sapphire [Koel02]. However, sapphire orifices are most commonly used because they are much cheaper than diamond orifices (the price of a diamond orifice can be $435 while it is

4 only $14.5 for a sapphire one [Bart08]).

Most of AWJ nozzles are made from composite carbide materials. They are available on the market under specific product names such as ROCTEC 100 and ROCTEC 500 from Kennametal Inc. ROCTEC composite carbide is a very dense, sintered, tungsten carbide based hardmetal. The common inner diameter of the focusing tube is from 0.5 to 1.5 mm, and the common length is from 70 to 100 mm.

• The abrasive supply system:

The abrasive supply system is used for accurate supply of abrasives with a pre-required mass flow rate. In practice, there are many types of abrasives which are used in AWJ machining. They can be garnet (for example Barton garnet (a trade mark of Barton Mines Company) and GMA garnet (a trade mark of GMA garnet Pty Ltd) – two most common garnets), olivine, aluminum oxide, silica- sand etc. Generally, in AWJ machining, the abrasive mass flow rate is about 0.08 to 0.5 kg/min (15 to 30 kg/h [Trum97]) and the abrasive size varies between 0.1 and 0.3 mm.

high pressure water orifice abrasive supply

mixing chamber focusing tube

Figure 1.5: A typical jet former for AWJ cutting [Hoog04]

1.2.2 Parameters of an AWJ machining process

There are many parameters involved in an AWJ machining process. In general, these parameters can be divided into two groups: process parameters and target parameters [Momb98]:

• Process parameters:

The process parameters include parameters relating to the forming of the AWJ beam. These parameters can be sorted into four following sub-groups [Momb98]:

-Hydraulic parameters including water pressure and orifice diameter.

5 -Mixing parameters including focusing tube (or nozzle) diameter and focusing tube length.

-Abrasive parameters including abrasive material, abrasive particle size, abrasive shape, and abrasive mass flow rate.

-Cutting parameters including standoff distance, impact angle, traverse rate and number of passes.

• Target parameters:

The target parameters consist of parameters related to the target of the machining. These parameters are the work material, the depth of cut and the cutting quality.

1.2.3 Advantages and disadvantages of AWJ Technology

AWJ cutting has various advantages over other non-conventional techniques such as laser and Electrical Discharge Machining (EDM). The advantages can be presented as follows:

-AWJ can machine a wide range of materials including titanium, stainless steel, aerospace alloys, glass, plastics, ceramics, and so on.

-AWJ can cut net-shape parts and near net-shape parts.

-No heat is generated in the cutting process. Therefore, there is no heat-affected area and thus no structural changes in work materials occur.

-AWJ cutting is particularly environmentally friendly as it does not generate any cutting dust or chemical air pollutants.

-The abrasives after cutting can be reused which allows for possible reduction of the AWJ cutting cost.

-Only one nozzle can be used to machine various types of work materials and workpiece shapes.

-AWJ machining can be easily automated and therefore can be run with unmanned shifts.

Although AWJ cutting is a truly useful machining process and can be used for various applications, the technology still has two major disadvantages:

-The total cutting cost is relatively high;

-The cutting quality is not always satisfying and unstable.

1.3 Challenges in AWJ Technology

As mentioned above, although AWJ cutting has many advantages, its high cutting cost is the most

6 prominent disadvantage. In AWJ cutting, the total cutting cost depends on many cost components such as machine tool cost, abrasive cost, nozzle wear cost, wages including overhead cost and so on. High AWJ cutting cost, for example, in Europe, the cutting cost per hour is about 150…200 (€/h), makes the AWJ business less competitive. As a result, the reduction of the total cutting cost and cutting time as well as the increase of the profit rate (or profit per hour) in AWJ machining are big challenges for this technology.

8 2 State of the art in optimization of AWJ machining

As addressed in Chapter 1, one of the biggest disadvantages of AWJ cutting is its high cost. The AWJ cutting cost per hour, for example, can be 150…200 (€/h) in Europe. Therefore, finding solutions to reduce the total cutting cost to increase the profitability for AWJ users is an important task of AWJ technology.

In the AWJ cutting cost, the abrasive cost (including disposal cost) is usually the largest component (Figure 2.1). This can amount to 20% up to 70% of the total cutting cost, depending on parameters such as the abrasive mass flow rate, the number of cutting heads, the abrasive price, the AWJ system’s cost and so on. However, the abrasives after cutting can be reused, which can reduce the abrasive cost and the disposal cost.

Machine tool cost (23.94%)

Abrasive cost (53.98%) Wages including overhead cost (16.89%)

Orifice cost (0.92%)

Nozzle cost (3.05%) Water cost (1.31%)

Figure 2.1: A typical AWJ cost breakdown [Hoog06]

In practice, the AWJ optimization and abrasive recycling are two main ways to increase the profitability for AWJ users. Especially, optimization can reduce both the cutting time (or increase the cutting performance) and the cutting cost and can increase the profit rate. Therefore, AWJ optimization and abrasive recycling have been the objectives of many studies.

The optimization problems in AWJ machining can be divided into two categories including AWJ technical optimization and AWJ economical optimization. The technical optimization, based on the physical relationships between process parameters, aims to determine optimum values of the

9 process parameters in order to fulfill the maximum cutting performance or the minimum cutting time. The economical optimization, based on the economical relations as well as the physical relations between the process parameters, aims at the optimum values of the process parameters for getting the minimum total cutting cost per product (or per unit length of cutting) or the maximum profit rate.

Up to now, there have been many studies on both AWJ optimization and abrasive recycling. To have a clear picture on this, a literature review is carried out. The review is split into three parts: AWJ technical optimization in Section 2.1, AWJ cost calculation and cost optimization in Section 2.2, and abrasive recycling in Section 2.3.

2.1 State of the art in AWJ technical optimization

In the AWJ cutting process, there are various factors affecting the material removal process or the cutting performance. These factors include the jet-parameters (the water pressure, the orifice diameter, the focusing tube diameter, the focusing tube length, the abrasive mass flow rate, the abrasive size, the abrasive shape and type) and the cutting parameters (e.g. the standoff distance, the workmaterial, the feed speed).

2.1.1 Optimum combination of focusing tube and orifice diameter

90 35 /s)

3 80 (mm) max 70 30

50 25 40 p =240 MPa; d =0.25 mm p =240 MPa; v=1.67 mm/s w ori w v =1.67 mm/s; l =50 mm d =0.25; d =1.2 mm 30 f f ori f

R=0.3; AlMgSi0.5 Maximum depth of cut h R=0.3; AlMiSi0.5 Material removal rate Q (mm 20 20 0.5 1 1.5 2 2.5 3 0 20 40 60 80 100 Focusing tube diameter d (mm) Focusing tube length l (mm) f f Figure 2.2: Focusing tube diameter versus Figure 2.3: Focusing tube length versus material removal rate [Blic90] maximum depth of cut [Blic90]

H. Blickwedel [Blic90] investigated the relationship between the focusing tube diameter and the volume removal rate. The author notes that the final abrasive particle velocity depends on the density of the abrasive-water-air mixture: a denser mixture creates a higher particle velocity. Also,

10 as the focusing tube diameter increases, the density of the mixture decreases and therefore the particle velocity decreases. However, a small focusing tube diameter leads to more interactions between particles and nozzle wall, and particles with each other and thus reduces the particle velocity. Therefore, an optimum value of the focusing tube diameter exists for the material removal rate (see Figure 2.2).

H. Blickwedel [Blic90] proposed an optimum ratio between the focusing tube diameter and the orifice diameter as follows:

d f = 34… (2.1) d ori U. Himmelreich and W. Riess [Himm91] confirmed that the above ratio is a good value for AWJ formation. E.J. Chalmers [Chal91] observed that the maximum depth of cut will occur for the ratio of nozzle to orifice diameter of 3. Zeng and Munoz [Zeng94] also reported that the highest cutting performance is achieved when using the following optimum combination of focusing tube/orifice: 3.3 (0.023”/0.007”), 3.2 (0.032”/0.01”), and 3.14 (0.044”/0.014”).

2.1.2 Optimum focusing tube length

Figure 2.3 shows the relation between the focusing tube length and the maximum depth of cut [Blic90]. The depth of cut, at first, increases linearly with the increase of the nozzle length. This is because a certain acceleration distance is necessary to accelerate the injected abrasive particles [Momb98]. Beyond this critical acceleration distance, the friction due to the spreading water jet increases. This leads to a reduction of the particle velocity and therefore a decrease of the depth of cut [Momb98]. The optimum acceleration distance, as noted by M. Heβling [Heβl88], depends strongly on the abrasive material density. Figure 2.4 shows the relation between the focusing tube length and the maximum depth of cut for different abrasive materials [Heβl88]. It is observed that round steel cast abrasive material is most influenced by the nozzle length while broken abrasive material and quartz sand are only lightly affected (Figure 2.4).

H. Blickwedel [Blic90] suggested the optimum focusing tube length lf,op based on his experimental results:

l fop, = 25… 50 (2.2) d f

M. Hashish [Hash91] indicated that the depth of cut and the kerf width both depend on the length of the focusing tube. The depth of cut and the kerf width reduce as the focusing length increases up to a length of about 50 to 70 times of the focusing tube diameter. Also, it is noted that no change in the depth of cut and the kerf width occurs when the focusing tube length increases further beyond 50 to 70 times of the tube diameter [Hash91].

11 Generally, the wear of the focusing tube is affected by the tube length. M. Hashish [Hash94] addressed that the nozzle exit bore wear rate reduces as the nozzle length increases. This conclusion was also confirmed later by K.A. Schwetz et al [Schw95] and M. Nanduri et al. [Nand00]. Figure 2.5 illustrates the relation between the tube length and the exit bore diameter wear rate [Nand00]. It follows that when the tube length is smaller than a certain value (in this case around 75 mm), a decrease of the tube length will lead to a significant increase of the exit bore wear rate. Beyond this value, the exit bore wear rate is almost unaffected by the tube length.

In practice, the length of the focusing tubes is determined for both a high cutting performance and a long nozzle lifetime. The nozzle lengths are standardized in some common sizes of 76 mm (3”), 89 mm (3.5”) and 101.6 mm (4”). It is known that the most commonly used nozzle length is 76 mm, offering the best cost-to-wear-life ratio [Chal06].

80 30 3 p =310 MPa; m =3.8 g/s Steel cast, angular (7400 kg/m ) w a 70 3 d =0.38; d =1.14 mm Steel cast, round (7400 kg/m ) 25 ori f0

(mm) Nozzle material: WC/C Quartz sand, round (2650 kg/m3) o

max 60 Abrasive: aluminum oxide #80 20 50

40 15

30 10 20 p =200 MPa; v =20 mm/s w f d =0.6; d =600 µm ori p m =30 g/s 5 10 a Maximum depth of cut h Exit diameter increase rate (%) 0 0 0 50 100 150 20 40 60 80 100 120 Focusing tube length l (mm) f Focusing tube length l (mm) f Figure 2.4: Focusing tube length versus Figure 2.5: Nozzle length versus nozzle exit bore maximum cutting depth [Heβl88] increase rate [Nand00]

2.1.3 Optimum abrasive mass flow rate

Typical relations between the abrasive mass flow rate and the maximum depth of cut are shown in Figure 2.6. It follows that the depth of cut, at first, increases as the abrasive mass flow rate increases. However, when the abrasive mass flow rate exceeds a certain value, the depth of cut will drop (Figure 2.6). This relation can be explained by the following equation [Hash89]:

v v =⋅η wj (2.3) awj 1/+ mmaw

In which, vwj is the velocity of water leaving the orifice, vawj is the velocity of abrasive particles

12 leaving the nozzle, η is momentum transfer efficiency, m a is the abrasive mass flow rate, and m w is the water mass flow rate.

Previous studies ([Mill91], [Clau98] and [Susu06]) indicate that the momentum transfer efficiency η decreases as the abrasive mass flow rate increases. Hence, it can be deduced from Equation (2.3) that an increase of the abrasive mass flow rate can lead to a decrease of the abrasive particle velocity vawj . Moreover, an increase of the abrasive mass flow rate also results in an increase of the impact frequency of particles. Therefore, a critical value of the abrasive mass flow rate exists at which the benefit of the impact frequency balances the loss in particle velocity [Zeng94]. This critical value is the optimum abrasive mass flow rate for the maximum depth of cut.

(mm) 30

max 28

26 d =0.8 mm 24 f d =1.2 mm f d =1.6 mm 22 f 20 p =240 MPa; v =1.67 mm/s 18 w f d =0.25 mm; l =50 mm ori f

Maximal depth of cut h 16 AlMgSi0.5 14 0 5 10 15 20 25 Abrasive mass flow rate (g/s)

Figure 2.6: Abrasive mass flow rate versus maximum depth of cut [Owei89]

The optimum abrasive mass flow rate for the maximum cutting performance (or for the maximum depth of cut) depends on many parameters. These are the water pressure [Chal91], [Guo94a], [Guo94b], orifice diameter [Chal91], [Guo94b], the focusing tube diameter [Chal91], [Guo94a], [Hoog05] and the focusing tube length [Guo94a].

Table 2.1: Optimum abrasive to water mass flow rate [Chal91]

Nozzle/orifice combination mmaw/ for hmax mmaw/ for 0.85⋅ hmax 0.76 mm / 0.25 mm 0.3 0.17 1.14 mm / 0.38 mm 0.19 0.12 1.65 mm / 0.53 mm 0.19 0.1

Figure 2.7 shows the effects of jet-parameters on the optimum abrasive mass flow rate according to experimental data of Guo [Guo94a]. It follows that the optimum abrasive mass flow rate increases with the increase of the water pressure (Figure 2.7a), of the water mass flow rate (Figure 2.7b), and of the focusing tube diameter (Figure 2.7c). The relation between the optimum abrasive

13 mass flow rate and the focusing tube length is shown in Figure 2.7d.

To determine the optimum abrasive mass flow rate, E.J. Chalmers [Chal91] found that cutting with the ratio of nozzle to orifice of 3:1 results in the maximum depth of cut at a specific value of

mmaw/ for a given size of the nozzle. In addition, to avoid excessive use of abrasives, Chalmers

[Chal91] assumed the optimum depth of cut is defined as occurring at 0.85hmax. The optimum abrasive to water flow rate is shown in Table 1 [Chal91].

9 9

8.5 8.5

8 8

7.5 7.5

7 7

6.5 6.5

6 d =0.25; d =0.95 mm 6 ori f v =1.67 mm/s; d =355 µm p =240 MPa; v =1.67 mm/s f p w f 5.5 d =0.95 mm; d =355 µm minersiv 5.5 f p minersiv 5 5 100 150 200 250 300 350 15 20 25 30 Optimal abrasive mass flow rate (g/s) Water pressure p (MPa) Water mass flow rate (g/s) w Optimal abrasive mass flow rate (g/s)

a) b) 12 10 p =240 MPa; v=1.67 mm/s w 9.5 d =0.25 mm; d =0.95 mm 11 ori f d =355 µm; minersiv 9 p

10 8.5

8 9 7.5 8 7

7 6.5 p =240 MPa; v=1.67 mm/s 6 w 6 d =0.25 mm; d =0.95 mm ori f d =355 µm; minersiv 5.5 p 5 5 0.5 0.7 0.9 1.1 1.3 1.5 20 30 40 50 60 70 80

Optimal abrasive mass flow rate (g/s) Focusing tube diameter d (mm) Focusing tube length l (mm) f f Optimal abrasive mass flow rate (g/s)

c) d) Figure 2.7: Effect of factors on the optimum abrasive mass flow rate [Guo94a]

14 In spite of recent efforts, the optimum abrasive mass flow rates are predicted for specific combinations of the focusing tube and the orifice diameter only. No model has been developed for determination of the optimum abrasive mass flow rate for more general combinations.

2.1.4 Optimum abrasive particle size

The effect of abrasive particle sizes on the depth of cut was investigated by J. Ohlsen (Figure 2.8a) [Ohls97]. This effect can be classified into two cases: brittle behaving materials (e.g. glass in Figure 2.8a) and ductile behaving materials (e.g. AlMgSi0.5 in Figure 2.8a). In the former case, the workmaterial seems less sensitive to the impact frequency [Momb98]. Therefore, the maximum depth of cut increases with the increase of the particle diameter. In the latter case, for small particles (smaller than 100 μm), a larger particle diameter causes a higher depth of cut. This is

3 because a larger particle means a higher kinetic energy, i.e. Edp ∝ p [Momb98]. In contrast, for larger particles (larger than 100 μm), an increase in size of abrasive particles can lead to a reduction of the maximum depth of cut (Figure 2.8a). Momber et al. [Momb98] noted that this phenomenon was also observed by Nakamura et al. [Naka89], Guo et al. [Guo92] and Momber et al. [Momb96].

H. Oweinah [Owei89] investigated the effect of the abrasive particle diameter on the depth of cut for various abrasive mass flow rates (Figure 2.8b). It is concluded that large particles have a significant influence on the depth of cut when the abrasive mass flow rate varies, while smaller particles are not sensitive against the changes of the abrasive mass flow rate (Figure 2.8b).

45 80

p =400 MPa; v =0.83 mm/s AlMgSi0.5 70 w f 40 d =0.25; d =1.08 mm Glass ori f AlMgSi1; corundum 60 (mm) (mm)

35 max

max 50

30 40

30 25 p =300 MPa (AlMgSi0.5) d =0.25 mm w p p =100 MPa (glass) w 20 d =0.029 mm d =0.25; d =0.9 mm p ori f 20 v=1.67 mm/s; m =5 g/s a 10 garnet Maximal depth of cut h

Maximal depth of cut h 15 0 0 100 200 300 400 500 600 0 5 10 15 20 Abrasive particle diameter d (g/s) p Abrasive mass flow rate (g/s) a) [Ohls97] b) [Owei89]

Figure 2.8: Abrasive particle diameter versus maximum depth of cut

2.1.5 Optimum standoff distance

The effect of the standoff distance on the maximum depth of cut was first investigated by R.E. Barton [Bart82]. It was found that the depth of cut decreases almost linearly with the increase of the standoff distance. Figure 2.9 shows the relationship between the standoff distance and the maximum depth of cut [Blic90]. R.A Tikhomirov et al. [Tikh92] reported the same result for the relation between the standoff distance and the maximum feed speed. The authors noted that at a small increase of the standoff distance, the maximum feed speed first remained constant and then decreased according to an almost linear relation [Tikh92]. The effect of the standoff distance on the depth of cut was also confirmed by Blickwedel [Blic90], Kovacevic [Kova92] and Guo et al. [Guo94b]. In addition, Guo et al. suggested that the optimum standoff distance is about 2 mm [Guo94b].

45 d =0.25; d =1.2 mm ori f 40 l =50 mm; v =1.67 mm/s f f m =8 g/s (mm) 35 a max

15 p =300 MPa 10 w p =200 MPa

Maximal depth of cut h w 5 0 10 20 30 40 50 Standoff distance (mm)

Figure 2.9: Standoff distance versus maximum depth of cut [Blic90]

2.2 State of the art in AWJ cost calculation and cost optimization

2.2.1 State of the art in AWJ cost calculation

• Study of J. Zeng and T. J. Kim:

To calculate the cutting cost per length, J. Zeng and T. J. Kim [Zeng93] first introduced a model for prediction of the feed speed:

1.15 ⎛⎞Np⋅⋅1.25 m 0.687 ⋅ m 0.343 v = mw w a (2.4) f ⎜⎟0.618 ⎝⎠Cqhd⋅⋅⋅f

16 The cutting cost per length was then determined by the following equation:

C h C l = (2.5) v f

Where, Ch is the total hourly cost ($/h), which is calculated as follows:

CChmhlhthphdh=++++ CCC C (2.6)

In which Cmth is the machine hourly cost, Clh is the labor hourly cost, Cth is the material hourly cost which considers the abrasive cost, water cost, focusing tube cost and orifice cost, Cph is the power hourly cost, and Cdh is the cost of maintenance and disposal.

In this study, many cost components were taken into account. Additionally, the effects of many jet- parameters on the cutting cost were also investigated through a model for prediction of the feed speed. After all, the effect of the number of jet formers as well as the effect of the nozzle wear on the cutting cost was still not well-understood.

• Study of D.A. summers et al.:

To compare the AWJ cutting cost per part in both cases with and without abrasive recycling, D.A. Summers et al. [Summ01] carried out a study in which the influence of many cost parameters on the total cutting cost, e.g. the abrasive cost, the disposal cost, the power cost, the water cost and the nozzle wear cost were investigated. Also, the optimum cutting performance was predicted by a tabulated method. The authors concluded that by cutting with the recycled abrasives using particles larger than 100 µm, the cutting cost can be reduced significantly. Nevertheless, the effects of several cost components such as the machine cost, the labor cost and the maintenance cost were not considered. Besides, although the nozzle wear cost was taken into account empirically, there is still no model for calculation of the nozzle wear.

• Study of M. Hashish:

M. Hashish [Hash04] compared the cutting cost in two cases: with water pressure of 400 MPa and 600 MPa. The effects of the water pressure on different cost elements such as abrasives, pump and machine maintenance, water, power and the nozzle wear were studied.

Hashish noted that the feed speeds when cutting at 600 MPa should be at least equal to those at 400 MPa while using 33% less abrasives and water. The author also found that cutting at a pressure of 600 MPa can save the total cutting cost 10 % to 25 % over that at 400 MPa [Hash04].

Although the study compared the cutting cost when cutting with high and low pressure, the effect of water pressure on the AWJ system’s utilization and the nozzle wear were not investigated. In practice, cutting with high pressure can increase not only the pump maintenance cost but also the downtime due to the pump’s maintenance. As a result, the total available cutting time when cutting with high pressure will be reduced and thus the cutting cost will increase accordingly. In addition,

17 due to the high pressure, the nozzle wear also increases (in the study the nozzle wear was constant) and leads to an increase of both the nozzle wear cost and the downtime because of replacement of the nozzle. These effects of high pressure on the utilization and on the nozzle wear cost should therefore be taken into account.

2.2.2 State of the art in AWJ cost optimization

• Study of P. J. Singh and J. Munoz:

P.J. Singh and J. Munoz [Sing93] noted that the AWJ cost optimization problem is very complicated to solve because there are a lot of parameters affecting the total cutting cost. However, local suboptimization can be used as a solution for the problem. For cost analysis, the authors divided the cost elements into three main components: the operating costs, the labor costs Cl, and the capital investment costs Ce [Sing93]. The operating costs consist of the abrasive cost Ca, the power cost

Cp, the water cost Cw, the focusing tube cost Cf, orifice cost Cori, and maintenance cost Cmai. The total cutting cost per centimeter is then determined as follows [Sing93]:

10 ⋅C h C lc, = (2.7) 60 ⋅⋅vkfcf

Where, vf is the feed speed (mm/s) calculated by the model by Zeng and Kim (Equation 2.4); kcf is a contour factor which considers the necessary slow down of the system during turns; Ch is the total cutting cost per hour which is calculated by the following equation [Sing93]:

CCCCCCChapwforimaile=++++ + ++ CC (2.8)

A sub-optimization problem is performed by considering the orifice diameter as an independent variable. Other parameters are then chosen based on this variable [Sing93]. From the results of the optimization problem, the authors concluded that use of smaller orifices is more cutting efficient, i.e. the cutting length per unit of power is higher. However, larger orifices are more cost efficient as the feed speed can be increased so that the labor cost and the capital cost are reduced [Sing93]. In addition, the authors found that cutting with multiple-heads reduces the total cutting cost since the combination of the higher efficiency of smaller orifices with higher throughput of multiple-heads [Sing93]. It is noted that careful cost analysis and cost optimization can save 10 to 30% of the total AWJ cutting cost [Sing93].

As the above sub-optimization study was carried out by considering only one variable, the applications are therefore limited. The effect of process parameters as well as the effect of cost elements on the total cutting cost should be taken into the sub-optimization problem.

• Study of J. Zeng and J. Munoz:

18 J. Zeng and J. Munoz [Zeng94] presented a study on optimum selection of the abrasive mass flow rate in order to fulfill the minimum cutting cost. In this study, the total cutting cost per length

Cl ($/m) is calculated according to the approach of Zeng and Kim [Zeng93]:

CmChaam+⋅60 ⋅, C l = (2.9) 60 ⋅v f

Where, Ch is the total hourly cost ($/h) excluding abrasive cost; vf is the feed speed (m/min); m a is the abrasive mass flow rate (kg/min) and Ca is the abrasive cost per kilogram ($/kg).

The total cutting cost per unit length Cl1 when cutting with m a1 and vf1 can be compared to Cl2, when cutting with m a2 , vf2 by the following equation [Zeng94];

CvCmC+⋅60 ⋅ lf11=⋅111aa (2.10) CClaaf22+⋅60 mCv 222 ⋅

The ratio Cl1/Cl2 was determined for three different combinations of the orifice and focusing tube diameter with Ch1=Ch2=$62 and Ca1=Ca2=$0.59/kg ($0.32/lb) (see Table 2.2) [Zeng94].

Table 2.2: Relative cost using different abrasive mass flow rates [Zeng94]:

Orifice/tube 0.113 0.227 0.34 0.454 0.567 0.68 combination kg/min kg/min kg/min kg/min kg/min kg/min

0.177/0.584 1.04 1.00 0.254/0.81 1.23 1.00 1.00 0.356/1.12 1.31 1.11 1.00 1.02 1.01

The optimum abrasive mass flow rates found for various combinations of the orifice and focusing tube diameter are 0.227 kg/min for 0.177/0.584, 0.34 kg/min for 0.254/0.81, and 0.454 kg/min for 0.356/1.12 [Zeng94].

A main advantage of this method is that the optimum abrasive mass flow rate can be determined rather easily. However, the results are valid for pre-set combinations of orifice and nozzle diameter only. Also, the effects of the abrasive mass flow rate on the nozzle wear and on the abrasive disposal cost were not investigated. Moreover, the effects of cost elements on the optimum values of abrasive mass flow rate should also be taken into account.

• Study of M. Mono:

For solving the AWJ cost optimization problem, M. Mono [Mono97] first introduced graphical relations between the water pressure, the abrasive mass flow rate, the feed speed and the surface roughness. In his cost model, various cost elements were taken into account such as the abrasive cost, the water cost, the nozzle cost, the orifice cost, etc. In order to get a fixed value of the

19 surface roughness, graphical relations between the depth of cut, the ratio between the feed speed and the abrasive mass flow ratevmfa/ , the depth of cut and the minimum cutting cost were constructed.

After all, the use of the graphical relations for selecting AWJ parameters is not straightforward. More important, the approach is only valid for cutting with aluminum [Mono97] and at fixed values of water pressure and surface roughness.

• Study of A. Henning and E. Westkämper:

To find the optimum values of the abrasive load ratio R for getting the maximum cutting performance and the minimum cutting cost per meter, A. Henning and E. Westkämper [Henn04] gave a cost study in which many cost components were taken into account. These cost elements consist of the electricity cost, the water cost, the abrasive cost, the labor cost, the occupancy cost, the nozzle wear cost, the orifice cost, the revenue etc. It was found that the optimum values of the abrasive load ratio for the maximum cutting performance and for minimum cutting cost are different (see Figure 2.10). The authors concluded that, in many cases, an abrasive load that is between these optimum points can be chosen as the optimum abrasive load ratio.

16 12 14

10 8

6 performance 4 4 cost Cutting cost ( /m) Cutting performance (m/h) 2

0 0 0 5 10 15 20 25 30 35 40 Abrasive load ratio R (%)

Figure 2.10: Cutting performance and cutting cost versus abrasive load ratio [Henn04]

The effect of the hydraulic power on the cutting performance as well as on the cutting cost per meter was also investigated in the study. It was found that the optimum cost to abrasive load ratio decreased when cutting with higher hydraulic power [Henn04]. Also, it was noted that the optimum abrasive mass flow rate never exceeded 1.3 kg/min (within experiments in the study). Moreover, in the study, the cutting cost calculation when cutting with multiple cutting heads was carried out. The authors found that it is possible to gain more benefits from high power by cutting with multiple

20 cutting heads [Henn04].

• Study of U. Andersson and G. Holmqvist:

Recently, U. Andersson and G. Holmqvist [Ande05] have carried out a study on strategies for cost and time effective AWJ cutting. In their cost structure, the cost elements are classified into two groups: fixed costs and running costs. The fixed costs include the AWJ system cost (including software) and the labor cost. The running costs consist of the abrasive cost, the water cost, the electricity cost, the cutting head cost (including nozzle, orifice, valve etc.), and the maintenance cost. In addition, the influence of other factors such as the utilization, the economic life of the AWJ system and the interest are also considered.

Andersson and Holmqvist found that the fixed cost is usually half or up to two thirds of the total cutting cost per unit length. They also noted that cutting with two cutting heads instead of one can reduce the cutting cost significantly.

It is noted that three factors have to be considered to reduce the total cutting cost. These are the optimized abrasive mass flow rate, the optimized lifetime of cutting head consumables (focusing tube, orifice, valve etc.), and optimized water pressure [Ande05]. However, in their cost structure, the effect of these factors on the cutting cost was still neglected.

2.3 State of the art in AWJ abrasive recycling

In the AWJ cutting process, the breaking (or the fragmentation) of abrasive particles occurs in two stages: first, during the mixing process (due to interactions between particles and the walls of the mixing chamber and the focusing tube and between particles with each other), and second, during the cutting process (because of the interactions between particles with the workmaterial and particles and each other). Therefore, understanding of the fragmentation of abrasive particles is highly relevant to a study on abrasive recycling.

The fragmentation of abrasive particles has been studied intensively. G. Galecki and M. Mazurkiewicz [Galec87] were the first who studied the fragmentation during the mixing process. The authors found that a large number, i.e. 70 to 80%, of initial particles are disintegrated during the mixing process [Galec87]. They also noted that this number depends on the initial abrasive size, the water pressure, the abrasive mass flow rate and the focusing tube diameter.

T.J. Labus et al. [Labu91] carried out a fundamental research in which the influence of the process parameters on the particle size distribution after the mixing process and after the cutting process was investigated. It was found that low water pressure levels (from 0 to 205 MPa) can have more affect on the main mass fraction change than those by high pressure levels (from 274 to 342 MPa). The mixing tube length does not affect the particle size distribution after the mixing process but the

21 mixing chamber geometry does. The authors noted that for a garnet #80, the main particle breakdown process is a shift from 180 micron particles into 63 micron particles or less. Particles which have the size from 75 to 150 microns do not seem to be affected during the cutting process. The workmaterial thickness also influences the particle disintegration. It was concluded that the recycling is more applicable for thin workmaterials than for thick ones, since more of the main abrasive mass fraction remains intact [Labu91].

H. Louis et al. [Loui95] investigated the effect of cutting parameters on the particle size distribution after the cutting process. The average particle size after cutting is found to be a bit smaller than that after the focusing tube. The influence of workmaterial types on the fragmentation of the abrasive particles was also investigated in this study. The authors noted that cutting stainless steel can reduce the average particle size more than when cutting aluminum. Also, the effect of the abrasive material on the fragmentation of particles after the cutting process was investigated with two types of abrasives (garnet and olivine). It is observed that olivine produces a bit smaller average particle size than garnet [Loui95]. Finally, the effect of the cutting quality was discussed. The authors found that high quality cutting causes a bit smaller average particle size than rough cutting [Loui95].

J. Ohlsen [Ohls97] carried out a systematic study on the recycling of Barton garnet. To evaluate the fragmentation of the abrasive particles, Ohlsen introduced a “disintegration number” which is defined as follows:

d ap, out φD =−1 (2.11) d ap, in

In which, dap,in and dap,out are the average diameter of input and output particles, respectively.

There are many process parameters that affect the magnitude of the disintegration number, for example, the water pressure, the abrasive mass flow rate, the abrasive particle diameter, the focusing tube diameter and the focusing tube length. The effects of these parameters are discussed below (see also Figure 2.11a through 2.11f).

It was observed that the disintegration number increases linearly with the water pressure (see Figure 2.11a). The abrasive mass flow rate affects the particle disintegration significantly only when this rate is smaller than a certain value (4 g/s). Above this value the influence on the particle fragmentation is negligibly small (Figure 2.11b). Figure 2.11c shows an almost linear relation between the initial particle diameters and the disintegration number. Figure 2.11d describes a monotonous decrease of the disintegration number with the increase of the focusing tube diameter.

22 0.8 0.5 d =0.045−0.063 mm p d =018−0.25 mm p 0.45 0.7 d =0.5−0.71 mm p 0.4 0.6

0.35 0.5 0.3 0.4 0.25 0.3 d =0.25; d =0.9 mm ori f 0.2 p =300 MPa; garnet w Disintegration number (−)

0.2 Disintegration number (−) d =0.25; d =0.9 mm ori f 0.15 m =5 g/s; garnet a 0.1 0.1 100 150 200 250 300 350 400 0 2 4 6 8 10 12 Water pressure p (MPa) w Abrasive mass flow rate (g/s)

a) b)

0.5

d =0.25; d =0.9 mm ori f m =5g/s; p =300 MPa 0.6 0.45 a w d =0.18−0.25 mm p

0.5 0.4

0.4 0.35

0.3 0.3

d =0.25; d =0.9 mm ori f 0.2 0.25 Disintegration number (−) Disintegration number (−) p =300 MPa; m =5 g/s w a

0.1 0.2 0 0.2 0.4 0.6 0.8 0.5 1 1.5 2 Abrasive particle diameter d (mm) Focusing tube diameter d (mm) p f

c) d)

180 Conventional chamber design 170 Optimized chamber design 0.6 d =0.25; d =0.9 mm 160 ori f m =5 g/s; d =0.18−0.25 mm a p 0.5 150 0.4 140

0.3 130 d =0.25; d =0.9 mm ori f m =5 g/s; p =300 MPa 0.2 a w 120 Disintegration number (−) d =0.5−0.71 mm p

110 0.1 100 150 200 250 300 350 400 20 40 60 80 100 Water pressure p (MPa) Focusing tube length l (mm) Outlet abrasive particle diameter (µm) w f e) f) Figure 2.11: Effect of parameters on the abrasive particle disintegration [Ohls97]

23 The effects of the focusing tube geometry (Figure 2.11e) and the focusing tube length (Figure 2.11f) on the fragmentation are small. Although the focusing tube length increases 5 times, the disintegration number increases only about 10%.

J. Ohlsen [Ohls97] reported that particles smaller than 60 µm lead to a very small depth of cut, poor cutting quality and can cause abrasive clogging in the mixing head. Moreover, the author found that the cutting performance and the cutting quality of the recharged abrasives are slightly better than those of the new abrasives. The reason is that the particle size distribution of recycled abrasives lies in the range from 125 to 180µm. This range of the particle size can lead to the maximum depth of cut and a lower surface roughness.

M. Kantha Babu and O.V. Krishnaiah Chetty [Babu03] introduced a study on the recycling of a local garnet (origin: Southern India). The authors found that the reusability (or the recycling capability which is determined by the percentage of abrasives that can be reused) with the particles larger than 90 µm is 81, 49, 26 and 15% after the first, second, third and fourth recycling, respectively [Babu03]. The effect of recycled abrasives of three cycles on the depth of cut, on the surface roughness and on the kerf width was investigated. It was observed that the maximum depth of cut of the first and second recycled abrasives is approximately 82 and 79% of the new abrasives. Also, cutting with the first and the second recycled abrasives can reduce both the surface roughness and the kerf taper [Babu03].

In practice, after recycling, the abrasives (recycled abrasives) can be used as a new abrasive or used as addition to new abrasives. The process in which new abrasives are added to recycled abrasives is called abrasive recharging. The recharging aims at maintaining the amount of input abrasives, so as to increase the cutting performance or to maintain the maximum cutting performance at all times.

M. Kantha Babu and O.V. Krishnaiah Chetty [Babu02] carried out a study on abrasive recharging. In their study, the recycled abrasives (with the size more than 90 µm) were recharged with new of abrasives at 20, 40, 60, 80 and 100% of the recycled abrasive mass. The influence of the recharging on the depth of cut, on the surface roughness, and on the kerf width for cutting with aluminum was investigated. It was noted that an increase of the added new abrasives up to 40% led to a significant increase of the depth of cut and a slight increase thereafter [Babu03]. Consequently, for getting maximum depth of cut, the recharging at 40% of the recycled abrasive mass is recommended [Babu03]. It is found that the surface roughness is minimum at 60% recharging of recycled abrasives with the size larger than 90 µm. Also, the top and bottom kerf widths increase marginally when the amount of added new abrasives increases [Babu03].

24 2.4 Conclusions

In this chapter a literature review has been carried out on the AWJ optimization and the abrasive recycling. The review consists of three parts including the state of the art in the AWJ technical optimization, in the AWJ cost calculation and cost optimization and in the abrasive recycling.

So far, various attempts have been carried out to determine the optimum values of the process parameters for the maximum cutting performance, e.g. the optimum combinations of the nozzle and the orifice diameter, the optimum abrasive mass flow rate, the optimum abrasive particle size, etc. However, the optimum abrasive mass flow rate can only be predicted for several combinations of the nozzle and orifice diameter. There is still lack of a model for the determination of the optimum abrasive mass flow rate. In addition, only the optimum combinations of the nozzle and the orifice diameter have been found. The optimum values of the orifice diameter, of the nozzle diameter as well as of the optimum number of the jet formers are still not well understood.

Although there have been many researches on the AWJ cost calculation and cost optimization, there is still room to improve the existing models. In particular, the nozzle wear and its influence on the cutting performance have not been investigated in detail. Also, the effects of various cost elements on the total cutting cost should be considered as variables so that the results of the cost optimization problem are more reliable and more applicable for many users in different places. Moreover, the effect of the number of the jet formers as well as the effect of the nozzle lifetime on the total cutting cost should be taken into account. The cost problem for optimum selecting the abrasive size and the abrasive type should also be investigated. Finally, the AWJ optimization problem for getting the maximum profit rate, which is a very important objective, has not been studied.

Until now, the fragmentation of abrasive particles and the abrasive recycling have been investigated in many studies. However, the optimum abrasive size for recycled abrasives and for recharged abrasives has not been mentioned. Also, the economics of the abrasive recycling have not been evaluated. Moreover, the recycling of GMA garnet, the most popular abrasives for blast cleaning and waterjet cutting, has not been understood. Finally, it is reported that only a small amount of garnet is recycled [Surv07]. Until now, abrasive recycling seems impracticable, which can be explained by the fact that, on one hand the price of new abrasives is low (in Europe, the price of GMA garnet is 0.2 to 0.3 €/kg [GMA07a]) and on the other hand an effective solution for recycling and recharging is lacking [Pi07a].

26 3 Project definition

3.1 Aim of the investigations

The aim of this work is to combine a cost model and a profit model in order to optimize the AWJ machining process for getting the maximum cutting performance (or the minimum cutting time), the minimum total cutting cost and the maximum profit rate.

As mentioned in Chapter 2, although there have been many studies in this area so far, there is still room to improve the existing models. From the restrictions of previous researches, it has been found that there are possible solutions for the optimization of AWJ machining.

• Optimization of the AWJ cutting process

The AWJ optimization consists of two optimization problems, namely technical optimization and economical optimization. The technical part aims to determine the optimum values of the process parameters, e.g. the abrasive mass flow rate, the orifice diameter, the nozzle diameter, the abrasive size and type, etc, in order to maximize the cutting performance. The economical part aims to get the optimum process parameters for the minimum total cutting cost and for the maximum profit rate.

• Recycling of abrasives

Since the abrasive cost, as mentioned in Chapter 2, is usually the largest component in the AWJ total cutting cost, the recycling of abrasives can be a good way to reduce the total cutting cost as well as to increase the profit rate. To find an effective way for the abrasive recycling, the optimum particle size of recycled and recharged abrasives for the maximum cutting performance as well as for the minimum cutting cost and for the maximum profit rate should be determined. In addition, the economics of cutting with recycled and recharged abrasives must be investigated.

• Unmanned machining

One of the benefits of AWJ cutting is that it can run safely for a long time without manual supervision. Consequently, it is possible to reduce the cost of wages strongly by running unmanned shifts [Hoog06].

This thesis will focus on the optimization of the AWJ machining process and on the abrasive

27 recycling. Besides, the effect of unmanned machining on the total cutting cost and the profit rate will be taken into account.

3.2 Outline of the thesis

In Chapter 4, the experiment equipment including the used AWJ system and measuring devices are described. The properties of abrasives as well as of work materials are summarized. Also, set-ups for experiments in the present study are given.

The frame work of modeling of the AWJ optimization is presented in Chapter 5. Also in this chapter, the introduction to optimization, the statements of an AWJ optimization problem and solutions for AWJ optimization problems are described.

In Chapter 6, modeling of the cutting process for AWJ optimization is carried out. This part of the thesis can be seen as an extension of Hoogstrate’s model [Hoog00]. For the modeling, firstly, the requirements for a cutting process model which can be used in the AWJ optimization are given. Next, a review on the AWJ cutting process modeling is presented. Finally, to extend Hoogstrate’s model, three sub-models consisting of discharge coefficient model, momentum transfer efficiency model, and cutting efficiency model are built by combining physical-mathematical models and experimental analyses.

The optimization of the AWJ cutting process is conducted in Chapter 7. In this chapter, the cost analysis in which the effects of various cost elements are taken into account is addressed. The optimization problems are then performed to determine the optimum nozzle lifetime (for the minimum cutting cost and for the maximum profit rate) and the optimum abrasive mass flow rate (for the maximum cutting performance, for the minimum cutting cost and for the maximum profit rate).

Next, in Chapter 8, the recycling and recharging of abrasives are described. The investigation consists of the reusability of abrasives, the optimum particle size for the recycling and recharging for the maximum cutting performance. In addition, the cutting performance and the cutting quality of recycled abrasives and recharged abrasives are presented.

The economics of abrasive recycling and recharging is investigated in Chapter 9. To do this, first, a cost analysis for the recycled and recharged abrasives is conducted. Next, two ways for economical comparison including the comparison of minimum total cutting cost per unit length and the comparison of maximum profit rate are proposed. Based on that, in the final step, the economics of cutting with recycled and recharged abrasives are pointed out.

In the final, Chapter 10, conclusions and recommendations for further researches are discussed.

28 4 Used experimental and measuring equipment

This chapter describes the experiment and measuring equipment that is used. The setup of the AWJ system, the setups of experiments to measure the water flow rate, the reaction force of the pure and the abrasive waterjet, to determine the maximum depth of cut and the surface roughness, and to collect the abrasive for the abrasive recycling investigation are summarized. In addition, the properties and the particle size distributions of abrasives as well as the properties of work materials used in the present research are outlined. Finally, specifications of several used measuring devices are presented.

4.1 AWJ machining setup

Robot Abrasive feeder Pendant

Jet former Pump Abrasive feeder controller

Robot controller

Figure 4.1: AWJ setup

29 Figure 4.1 shows the AWJ setup that is used for experiments in this research. In the setup, two intensifier pumps (see Section 1.2) from the Resato Company, Noordenveld, The Netherlands, are used. They are type PJE-3-3800 with a maximum pressure of 380 MPa which is used for cutting with pressure less than 380 MPa and type PJE-2-8000 with a maximum water pressure of 800 MPa which is used for tests with pressure above 380 MPa.

The jet former (see Section 1.2) is hold by a SCARA robot (Model SR8438-F00, Sankyo Seiki Mfg. Co., Ltd.) (see Figure 4.8). The robot is controlled by a robot controller with Sankyo Buzz program. Therefore, the jet former is able to move in the horizontal plane. The z-axis has to be manually adjusted. The robot can also be controlled manually by an optional hand held device called a Pendant.

The abrasive feeder was developed in the Laboratory of Precision Manufacturing and Assembly [Hoog00]. In the feeder system, the abrasives are fed by a timing belt driven by a stepper motor. By controlling the frequency of the steps, the velocity of the belt and therefore the abrasive mass flow rate are controlled.

4.2 Abrasive particles

In practice, various types of abrasives are used in AWJ machining. G. Mort [Mort95] noted that most of the AWJ shops use garnet (90% of the shops), followed by olivine (15% of the shops), slag (15% of the shops), silica sand (11% of the shops), and aluminum oxide (11% of the shops). Among garnet types, Barton and GMA garnet are the most common abrasives. Therefore, Barton garnet has been chosen as the main abrasive for experiments to extend the cutting process model in this study. Also, GMA garnet and olivine (origin: Norway) have been used to determine their effect on the cutting process. In addition, as mentioned in Chapter 2, there have been several studies on the recycling of Barton garnet [Ohls97] and a local garnet in India [Babu02 and Babu03]. However, the recycling of GMA garnet, the most popular abrasive for blast cleaning and waterjet cutting, has not been investigated. Consequently, GMA garnet has been chosen as the objective of the abrasive recycling investigation.

It is important to understand the characterization of the abrasives. Therefore, this section deals with the properties and the particle size distribution of abrasives used in the present research.

4.2.1 Abrasive properties

As mentioned above, Barton garnet, GMA garnet and olivine are chosen as abrasives using in the experiments in this study. The properties of these abrasives including the chemical composition and

30 the physical characteristics are shown in Table 4.1, Table 4.2 and Table 4.3 for Barton garnet, GMA garnet, and olivine, respectively.

Table 4. 1: Properties of Barton garnet [Bart07]

Feature Comments

General description Combination of Almandite and Pyrope Garnet, a homogeneous mineral, contains no free chemicals. Iron and aluminum ions are partially replaceable by calcium, magnesium and manganese

Oxides and dioxides are combined chemically as follows: FAl32() SiO 43

Chemical analysis Silicon Dioxide (SiO2) 41.34 % Ferrous Oxide (FeO) 9.72 %

Ferric Oxide (Fe2O3) 12.55 %

Aluminum Oxide (Al2O3) 20.36 % Calcium Oxide (CaO) 2.97 % Magnesium Oxide (MgO) 12.35 % Manganese Oxide (MnO) 0.85 % Physical characteristics Hardness (Mohs) 8-9 Melting point 1315ºC Specific gravity 3.9-4.1 g/cm3 Magnetism Slightly magnetic (volume susceptibility =9.999375). Particle shape Sharp, angular, irregular Colour Red to pink Strength Friable to tough Cleavage Pronounced laminations, irregular cleavage planes. Crystallization Cubic (isometric) system as rhombic dodecahedrons or tetragonal trisoctahedrons (trapezohedrons) or in combinations of the two. Quartz None

Electrostatic properties -Mineral conductivity: 18000 volts -Non-reversible Moisture absorption Non-hygroscopic, inert Pathological effects None Harmful free silica content None (silicosis free).

31 Table 4. 2: Properties of GMA garnet [GMA07b]

Feature Comments

Mineral composition (typical) Garnet (Almandite) 97-98 % Ilmenite 1-2 % Zircon 0.2 % Quartz (free silica) <0.5 % Others 0.25 %

Average chemical composition (typical) SiO2 36 %

Al2O3 20 % FeO 30 %

Fe2O3 2 %

TiO2 1 % MnO 1 % CaO 2 % MgO 6 % Physical characteristics (typical) Bulk density 2.3 T/m3 Specific gravity 4.1 g/cm3 Hardness (Mohs) 7.5-8.0 Melting point 1250 0C Shape of natural grains sub-angular

Table 4. 3: Properties of olivine [Oliv92]

Feature Comments

Average chemical composition (typical) MgO 49-50 %

SiO2 41.5-42.5 %

Fe2O3 6.8-7.3 %

Cr2O3 0.2-0.3 %

Al2O3 0.4-0.5 % NiO 0.3-0.35 % MnO 0.05-0.1 % CaO 0.05-0.1 % Physical characteristics (typical) Bulk density 1.7-1.9 T/m3 Specific gravity 3.3 g/cm3 Hardness (Mohs) 6.5-7.0 Melting point 1760 0C Colour Pale green Grain surface Smooth Grain shape Sub-angular to angular

32 4.2.2 Abrasive size distribution and abrasive particle diameter

In practice, the size of abrasive particles can be measured by several methods such as laser diffraction analysis, microscope-based analysis, image analysis, sedimentation analysis, and sieve analysis. Among these methods, sieve analysis is most commonly applied in industries as it is cheap and readily usable.

It is impracticable to estimate each particle individually. Therefore, size analysis is carried out by dividing abrasive particles into a number of suitably narrow size ranges [Momb98]. The results of sieve analysis for Barton garnet, GMA garnet and Olivine for various sizes are shown in Table 4.4, Table 4.5 and Table 4.6 respectively.

From sieve analysis results, the particle size distributions for Barton garnet, GMA garnet, and Olivine are shown in Figure 4.2a, Figure 4.2b, and Figure 4.2c, respectively. The olivine analysis was done during this study.

In is known that, in AWJ machining, the particle size is usually given in mesh designation according to the Tyler standard screen sieve series that barely mentions the related particle size distribution of the used abrasive particles [Momb98]. Also, there is a relation between the average particle

2 diameter d p (mm) and the mesh of the abrasive (with R =0.998) [Momb98]:

−1.0315 dmeshp =⋅17.479 (4.1)

Table 4. 4: Typical sieve analysis for Barton garnet [Bart07]

Nominal aperture Retained mass (%) sizes (μm) #150 HPX #120 HPX #80 HPX #50 HPX 53 1 63 4 75 14 6 90 27 7 106 38 16 125 14 28 3 150 1 30 7 180 11 17 212 2 24 2 250 31 15 300 15 29 355 3 37 425 16 500 1

33 Table 4. 5: Typical sieve analysis for GMA garnet [GMA07B]

Nominal aperture Retained mass (%) sizes (μm) #120 #80 #50

106 0.9 125 6 150 23 4 180 45 10 0.9 212 24.99 35 3 250 0.01 30 11 300 18 40 355 2 31 425 12 500 2

Table 4. 6: Typical sieve analysis for Olivine

Nominal aperture Retained mass (%) sizes (μm) #90 #60

63 0.69 75 2.50 0.44 90 3.06 0.63 106 6.25 1.31 125 19.69 1.94 150 28.38 4.06 180 21.44 6.00 212 11.56 13.44 250 5.19 29.13 300 1.25 24.88 355 17.81 425 0.38

In practice, there are many grades of abrasive mesh depending on the type of abrasive. For example, with Barton garnet, we can have #50, #65, #80, #85, #120, #150 and #220; with GMA garnet, there are four grades including #50, #60, #80 and #120. It is noted that most of the AWJ shops (86%) use the mesh #50, #60, #80 and #100 [Mort95], followed by mesh #120 and #150 (8%). Therefore, Barton garnet with four grades of mesh #50, #80, #120 and #150, GMA garnet with three grades of #50, #800, and #120, and Olivine with the mesh of #60 and #90 have been chosen for experiments in this research.

34 50 50 #150 HPX #120 HPX #120 #80 HPX #80 40 #50 HPX 40 #50

30 30

20 20 Retained mass (%) Retained mass (%) 10 10

0 0 0 100 200 300 400 500 100 200 300 400 500 Nominal aperture size (µm) Nominal aperture size (µm)

a) Barton garnet [Bart07] b) GMA garnet [GMA07b]

#90 25 #60

c) Olivine 15

10 Retained mass (%) 5

0 0 100 200 300 400 500 Nominal aperture size (µm)

Figure 4.2: Abrasive particle size distribution

4.3 Work materials

In the present study, three types of work materials including Al6061T6, stainless steel SS304 and mild steel AISI 1018 were used in experiments for determining the maximum depth of cut, the surface roughness as well as for using in the experiment for abrasive recycling. The chemical composition of the work materials are given in Table 4.7, Table 4.8 and Table 4.9 for Al6061T6, SS304 and AISI 1018, respectively.

35 Table 4. 7: Chemical composition of work material Al6061T6 [Matw07]

Element Al Cr Cu Fe Mg Mn Si Ti Zn % 95.8- 0.04- 0.15- ≤ 0.7 0.8- ≤ 0.15 0.4- ≤ 0.15 ≤ 0.25 98.6 0.35 0.4 1.2 0.8

Table 4. 8: Chemical composition of work material SS304 [Matw07]

Element C Cr Fe Mn Ni P S Si % ≤ 0.08 18-20 66.345- ≤ 2 8-10.5 ≤ 0.045 ≤ 0.03 ≤ 1 74

Table 4. 9: Chemical composition of work material Mild steel AISI 1018 [Matw07]

Element C Fe Mn P S % 0.14-0.2 98.81-99.26 0.6-0.9 ≤ 0.04 ≤ 0.05

4.4 Experimental setup for measuring the water flow rate

Jet former

Collecting pipe

Water collector

Figure 4.3: Water flow rate measurement setup

The experimental setup for measuring the water flow rate is shown in Figure 4.3. The AWJ system used in this experiment is shown in Figure 4.1. A jet former without mixing chamber and focusing tube is used. The water passing through the orifice will be collected by a collecting pipe and a water collector (see Figure 4.3). A digital scale with 20 kg of capacity and 0.01 kg of accuracy is used to weight the collected water in order to determine the water mass flow rate.

36 4.5 Experimental setup for measuring the reaction force

Strain gauge

Jet former

a) b)

Amplifier and A/D converter

c) Figure 4.4: Force measurement setup

Figure 4.4 shows the experimental setup for the waterjet (both the pure and abrasive waterjet) reaction force measurement. To measure the reaction forces of the jet, a full-bridge straingauge sensor (or four-train gauge sensor) is used. The sensor is fixed on one side to the robot frame and mounted to the cutting head on the other side. The working principle of this sensor is based on the measuring strain in the beam sensor, which is caused by a force applied to it. As a force is applied,

37 the beam sensor is bent and therefore, a strain in the material of the beam can be detected. This strain can be measured by measuring the change in resistance of the strain gauges.

Two types of jet formers, with and without the mixing chamber and focusing tube, were used to measure the reaction forces of the abrasive and pure waterjet, respectively (Figure 4.4b and Figure 4.4a, respectively).

The output signal is processed by a computer with LabVIEW program. Since the measured output voltage is very low, an amplifier is used for increasing the power of the signal. Also, an A/D converter is used to convert the signal from analog to digital signal (see Figure 4.4c).

Before running the test, the sensor signal is calibrated in order to compensate the dead weight of the jet former and the pre-strain in the gauges.

4.6 Experimental setup for determining the maximum depth of cut

Jet former

Workpiece

a) b) Figure 4.5: Experimental setup for determining maximum depth of cut

Figure 4.5 shows the experimental setup for determination of the AWJ maximum depth of cut. A jet former with mixing chamber and nozzle is used for cutting. To determine the maximum depth of cut easily, trapezoidal workpieces are used for this experiment (see Figure 4.5b). Two types of work materials including Al6061T6 and SS304 (see Table 4.7 and Table 4.8) will be tested.

4.7 Experimental setup for collecting abrasives

Figure 4.6 shows the experimental setup for collecting the abrasives for the recycling investigation of GMA garnet. The work material is mild steel AISI 1018 (25x80x200 mm3) (see Table 4.9). To

38 collect the abrasives, a big tank is used as a special catcher. To slow down the abrasive particles without breaking them any further, the catcher is filled with water. After collecting, the abrasives are washed and then dried by gas. Since mild steel is chosen as the work material, the chips can be separated by using magnetic separation. For abrasive sieving, a sieve shaker and thirteen sieves (International standard -ISO3310-1) are used (see Figure 4.7). The nominal aperture sizes of the sieves are 45, 63, 75, 90, 106, 125, 150, 180, 215, 250, 300, 355 and 425 micrometer.

Jet former

Catcher-cover Workpiece

Jet former

Workpiece

Catcher-cover

Water Abrasives Catcher

a) b) Figure 4.6: Experimental setup for collecting abrasives

a) The sieve shaker with sieves b) A sieve Figure 4.7: Sieves and sieve shaker for sieving abrasives

39 4.8 Experimental setup for determining surface roughness

Robot

Jet former

Workpiece

Figure 4.8: Experimental setup for determining surface roughness

Figure 4.9: Keyence VHX-100 microscope Figure 4.10: Talysurf Series 120L - Surface roughness measurement device

The aim of the experiment is to investigate the effect of recycled abrasives on the surface roughness. In this setup (see Figure 4.8), the AWJ system and the jet former are the same as used in the experiment for determining the maximum depth of cut (see Section 4.5). The work material is Al6061T6 (40x50x160 mm3) (see Table 4.7).

40 4.9 Other measuring equipment

4.9.1 Microscope

To investigate the effect of the recycling on the abrasive particle shape, a Keyence VHX-100 digital microscope with a VH-Z450 high range zoom lens is used. The magnification of the microscope is up to 3000X. The incorporated CCD camera allows a maximum resolution of 18 million pixels. Also, the high quality depth composition function of the microscope allows making sharp 3D images. The setup of the microscope with the computer for image acquisition is shown in Figure 4.9.

4.9.2 Surface roughness measurement device

To measure the surface roughness of AWJ cutting samples, a tactile stylus tip device (Taylor Hobson Form Talysurf-120L) is used (see Figure 4.10). The resolution of the inductive gauge is 16 nm. A wide range pick-up tool is used to record the profile shapes of the surfaces.

42 5 Frame work of modeling and AWJ optimization approach

In this chapter, a frame work of modeling and the optimization approach in AWJ cutting are introduced. To give the frame work of modeling, in Section 5.1, three subsequent steps for increasing the cutting performance, reducing the total cutting cost, and increasing the profit rate are determined. The frame work of modeling is then illustrated by a schema. To describe AWJ optimization approach, in Section 5.2, firstly, an introduction to optimization including the statement of an optimization problem and methods for solving optimization problems are presented. Finally, solutions for solving AWJ optimization problems are suggested.

5.1 Frame work of modeling

For AWJ optimization (including technical and economical optimization), a predictive structure of a frame work of modeling is conducted. The structure is shown in Figure 5.1. The process for increasing the AWJ cutting performance (or reducing the cutting time), reducing the total cutting cost and increasing the profit rate includes three subsequent steps:

• First step:

In this step, input parameters will be selected. The input parameters include two main categories: technical and economical parameters. The technical parameters include the water pressure, the orifice diameter, the focusing tube diameter and the abrasive mass flow rate, the depth of cut and the quality number (see Subsection 6.3.1). The economical parameters include the fixed cost (viz. the machine tool cost, the wages including overhead cost) and the variable cost (viz. the nozzle wear cost, the orifice cost, the water cost and the abrasive cost). Also, the input parameters consist of the AWJ machine parameters (e.g. the total power of system, the accuracy of machine motions, etc.) and the abrasive parameters (the abrasive type and size, new or recycled abrasives).

• Second step:

In the second step, models which are necessary for both the technical optimization and the economical optimization will be built. They include the following models:

-Technical models for the AWJ cutting process: These models describe the relation between the jet parameters (the water pressure, the orifice diameter and the focusing tube diameter), the abrasive parameters (the abrasive type, the abrasive size and the abrasive mass flow rate), the cutting

43 parameters (the feed speed, the standoff distance, the work material), and the target parameters (the depth of cut). The technical models will be used not only for the technical optimization but also for the economical optimization as well as for the recycling process. Moreover, these models can also be used for the calculation of the AWJ cutting regime in industry.

Input

Abrasive cled Machine Jet y New

former Rec (max, min of feed of (max, min speed/ movements)

Model Output Technical parameters -Cutting process -Jet-parameters (pw; df; dori; ) m a models Increase of cutting -Cutting parameters (h) -Recycling -Required quality number (Q). performance

models Technical optimization

Economical parameters

Cost and -Fixed cost (machine tool cost; -Cost profit wages); reduction models for: -Variable cost (nozzle cost; orifice -AWJ cutting -Profit rate

cost; water cost; abrasive cost) Economical -Recycling optimization increase

Figure 5.1: Structure of proposal models for AWJ optimization

-Technical models for abrasive recycling: To increase the cutting performance when cutting with recycled abrasives, recycling models should be built. In order to do it, firstly, the reusability of the abrasives, the optimum abrasive particle size for recycling and recharging should be determined. Next, the effects of recycled and recharged abrasives on the cutting performance have to be investigated. Finally, the effect of the recycled abrasives on the cutting quality should also be understood.

-Economical models for the AWJ cutting process: To deal with the economical optimization problems economical models need to be found. The economical models describe the relation between cost elements, the total cutting cost, and the profit rate. Besides, the effects of technical parameters, e.g. the water pressure, the abrasive mass flow rate, the orifice diameter, the focusing tube diameter, etc., on the total cutting cost as well as on the profit rate should also be taken into

44 account.

-Economical models for abrasive recycling: The economics of abrasive recycling will be found by solving the cost and profit comparison problems between new and recycled abrasives. For these problems, the cost of recycled and recharged abrasives should be determined, and therefore, cost models for recycled and recharged abrasives must be built. In these models, the effects of cost elements as well as other parameters such as the reusability, the cutting performance of recycled and recharged abrasives, etc., have to be taken into account.

• Third step:

After having the above models, the optimization problems must be solved in order to get the optimum values of the process parameters for different objectives: for minimum cutting time (or maximum cutting performance), for minimum cutting cost and for maximum profit rate. The optimum problems including the technical and the economical optimization problems will be carried out for both the AWJ cutting process and the abrasive recycling process.

5.2 AWJ optimization approach

5.2.1 Introduction to optimization

Optimization is an important tool to obtain the best result under given conditions. It is used to solve any engineering problem, for example, to minimize the cost of a building, to minimize the weight of airplanes, or to maximize the profit or minimize the cost in a manufacturing process.

To give a brief introduction to optimization, in this subsection, the statement of an optimization problem is described. Also, the classification and optimization methods are introduced.

5.2.1.1 Statement of an optimization problem

Mathematically, an optimization problem can be expressed as follows:

Find an n-vector Xxxx= ()12,,,… n of design variables to minimize an objective (or cost) function

fX()= fxx (12,,,… xn ) (5.1) subject to the constraints

gXj ()≤ 0 ; j = 1 to m (5.2)

lXj ()= 0 ; l = 1 to p (5.3)

Where X is an n-dimensional vector called the design variable vector, fX( ) is the objective

45 function or the criterion, g j ()X and lXj ( ) are called inequality and equality constraints, respectively.

• Objective functions of optimization problems

To design an optimization problem, the identification of an objective function is very important. Depending on the requirements of the optimization problem, the criterion or the objective function needs to be minimized or maximized. For example, in gearbox design, the total transmission ratio can be split to obtain the minimum weight of the gearbox. In this case, the objective of the optimization problem is the minimum weight.

In some cases, there may be more than one objective to be minimized or maximized. In these cases, the optimization problem is called a multi-objective problem. For instance, the total transmission ratio of a gearbox is split for two objective functions: maximum power efficiency and minimum gearbox weight.

• Constraints of optimization problems

Mathematically, a constraint is a restriction or a condition that a solution to an optimization problem must satisfy. An optimization problem can contain a constraint or a set of constraints. Identifying constraints and expressing them into mathematical expressions is an important step for designing an optimization problem. Constraints of a problem can be equality (Equation 5.2) or inequality (Equation 5.3). The number of independent equality constraints must be less than or equal to the number of design variables, i.e., p ≤ n [Aror04].

When p=n, the solutions of the equality constraints are the only candidates for the optimum design. Therefore, no optimization of the system is necessary. When p>n, either the constraints are inconsistent or there are several constraints linearly depending on other constraints. In that case, the system will become over-determined [Aror04].

5.2.1.2 Classification of optimization problems

Optimization problems can be classified into the following types [Rao96]:

-Based on the existence of constraints: optimization problems are classified into constrained and unconstrained problems.

-Based on the nature of design variables: optimization problems are classified into static and dynamic optimization problems. If the design variables of an optimization problem are the function of one or more parameters we have the dynamic problem. If not, we have the static optimization problem.

-Based on the physical structure of the problem: optimization problems are classified into optimum

46 control and non-optimum control problems.

-Based on the nature of the equations involved: optimization problems are classified into linear, nonlinear, geometric, and quadratic programming problems.

-Based on the permissible values of the design: optimization problems are classified into integer- valued, real-valued programming problems.

-Based on the deterministic nature of the variables: optimization problems are classified into deterministic and stochastic programming problems.

-Based on the separability of the functions: optimization problems are classified into separable and non-separable programming programs.

-Based on the number of objective functions: optimization problems are classified into single- and multi-objective programming problems.

More details about the above types of optimization problems can be found in [Rao96], [Noce99], and [Aror04].

5.2.1.3 Optimization methods

There are various optimization methods for the solution of different types of optimization problems. The main optimization methods can be summarized as follows:

-Graphical method: This method is used to solve the problems containing two optimization variables. Using this method, the result of the problem can be obtained by drawing contours of constraint functions and the objective functions [Bhat00].

-Simplex method: The Simplex method, created by George B. Dantzig in 1947, is the most efficient and popular method for solving linear programming problems [Rao96]. The method can be used to solve problems with thousands of variables and constraints.

-Classical methods: These methods can be used for the problems which include several variables. To use these methods, the function of the problem is differentiated twice with respect to the design variables, and the derivatives are continuous [Rao96]. If the problem has equality constraints, the Lagrange multiplier method can be used to find the optimum point. If not, the Kuhn-Tucker conditions can be used. The classical methods have limited scope in practical applications because the objective functions of several problems are not continuous and/or differentiable [Rao96].

-Methods for unconstrained problems: These methods are classified into two broad categories: direct search methods and descent methods (see Table 5.1) [Rao96]. The direct search methods do not require the partial derivatives of the function. Therefore, these methods are usually called the non-gradient methods. Also, these methods are known as zeroth-order methods because they do

47 not use the derivatives of the function. The direct search methods are most suitable for simple problems which have a small number of variables.

Table 5.1: Methods for unconstraint problems

Direct search methods Descent methods Random search method Steepest descent (Cauchy) method Grid search method Fletcher-Reeves method Univariate method Newton’s method Pattern search method Marquardt method - Powell’s method Quasi-Newton methods - Hooke-Jeeves method - Davidon-Fletcher-Powell method Rosenbrok’s method - Broyden-Fletcher-Goldfarb-Shanno method Simplex method

The descent methods are also known as gradient methods [Rao96]. As the descent (or gradient) methods require the first and, in some cases, the second derivatives, they are classified into first- order methods and second-order methods. The first-order methods require only first derivatives while the second-order methods need both first and second derivatives of the functions. Generally, the descent methods are more efficient than the direct search methods. This is because the descent methods use more information on the function being minimized (through the use of derivatives) [Rao96].

Table 5.2: Methods for constraint problems

Direct search methods Indirect search methods Random search method Transformation of variables technique Heuristic search method Sequential unconstrained minimization techniques Complex method Interior penalty function method Objective and constraint approximation Exterior penalty function method method Augmented Lagrange multiplier method Sequential linear programming method Sequential quadratic programming method Methods of feasible directions - Zoutendijk’s method - Rosen’s gradient projection method Generalized reduced gradient method

-Methods for constrained problems: Optimization methods for constrained problems can be classified into two direct search methods and indirect search methods (see Table 5.2) [Rao96]. In the direct search methods, the constraints are in explicit manner, while in most of the indirect

48 search methods, the constrained problem is solved as a sequence of unconstrained minimization problems [Rao96].

-Global optimization methods: These methods consist of two major categories: deterministic and stochastic methods [Aror04]. Deterministic methods are used to find the global minimum by an exhaustive search over a set of feasible points. Deterministic methods for global optimization are classified into finite exact and heuristic methods. Using finite exact methods the global minimum can be found in a finite number of steps. Heuristic methods offer an empirical guarantee of finding the global optimum. Stochastic or probabilistic programming is used in the situations where some or all of the parameters of the problem are described by stochastic (or probabilistic) variables. Depending on the equations involved in the problem, a stochastic optimization problem can be a stochastic linear, geometric, dynamic or non-linear programming problem [Rao96].

The above optimization methods can be learned in [Rao96], [Noce99], [Bhat00] and [Aror04]. Next, two optimization methods including suboptimization method and Golden search method are described in detail since they will be used in the AWJ optimization problems in this study.

-Golden ratio search method: This is one of the most efficient methods for finding the minimum of a function of one variable. The method can be described as follows [Math87]:

To find the minimum of a function fx( ) in a given interval, the function is evaluated many times and searched for a local minimum. For reducing the number of function evaluations, a good strategy to determine where fx( ) is to be evaluated has been found and a ratio called the Golden

Ratio has been given (the Golden Ratio is r =−(51/21/2 ) [Math87]). To use this method, the function must have a proper minimum in the given interval.

If the function fx() is unimodal on [a,b], it is possible to replace the interval by a subinterval on which fx() takes on its minimum value. For the Golden search, two interior points including ca=+−()()1 r ⋅ ba − and darba= +⋅( −) are required. Consequently, we have acdb<<<.

As the function fx()is unimodal, the function values fc( ) and fd( ) are less than max{fa() , fb ()} . From this, there are two cases to consider (see Figure 5.2):

If fc()≤ fd (), there must be the minimum in the subinterval [a,d] and we replace b with d and continue the search in the new subinterval. If fd( ) < fc( ) , the minimum must occur in the subinterval [c,b]. In this case, a will be replaced by c and the search will then be continued.

49 y=f(x) y=f(x)

If fc()≤ fd ()then squeeze from If fd( ) < fc( ) then squeeze from the right and use [a, d] the left and use [c, b]

Figure 5.2: Process of suboptimization [Math87]

Component Component Component

Original system

Component Component Component

Suboptimize component i

Component Component Component

Suboptimize component j and i

Component Component Component

Suboptimize component k, j and i (complete system)

Figure 5.3: Process of suboptimization [Rao96]

-Suboptimization method: can be used to solve dynamic optimization problems. Figure 5.3 illustrates the process of suboptimization which can be explained as follows [Rao96]:

Instead of trying to optimize a complete system as a single unit, it would be desirable to split the system into components which could be optimized more or less individually. To avoid a poor solution, a logical procedure needs to be used for splitting the system and for component sub-

50 optimization. In the schema in Figure 5.3, we assume that the system can be split into three components: i, j, and k. Also, the last component (component i) influences no other components. As other components are not affected by the last component, it can be sub-optimized independently. After that, the last two components (components i and j) can be considered together as a single component and can be suboptimized without adversely affecting any of the downstream components. This process can be continued to a larger group of end components (i.e. components i, j, and k) as a single component and sub-optimize it (see Figure 5.3).

5.2.2 Statement of an AWJ optimization problem

The statement of an AWJ optimization problem, like other optimization problems, can be expressed by Equations 5.1, 5.2 and 5.3 with the objective functions and the constraints are defined as follows:

• Objective functions of AWJ optimization problems

To reduce the cost and time as well as to increase the profit in AWJ cutting, three requirements are given for the optimization problems. They are the maximum cutting performance (for getting the minimum cutting time), the minimum total cutting cost, and the maximum profit rate. As a result, two optimization problems called technical and economical optimization problems need to be done. The objective functions of the optimization problems are therefore identified:

-For technical optimization problems: The time of AWJ cutting will be minimum when the cutting performance is maximum. In this case, the depth of cut reaches the maximum. As a result, the objective function of the technical problem is the maximum depth of cut.

-For economical optimization problems: these problems consist of two objective functions including the minimum total cutting cost and the maximum profit rate.

• Constraints of AWJ optimization problems

For AWJ optimization problems, the constraints are the restrictions of the process parameters such as the water pressure, the orifice diameter, the nozzle diameter, the abrasive mass flow rate and so on (see Subsection 1.2.2). In practice, the process parameters are usually ranged from their minimum to their maximum. Therefore, the constraints are unequality. For example, the constraints of the abrasive mass flow rate and the orifice diameter can be expressed as follows:

mmmaaa,min≤≤ ,max (5.4)

dddori,min≤≤ ori ori ,max (5.5)

5.2.3 Solutions for AWJ optimization problems

As mentioned in Chapter 2, although there have been several studies on AWJ optimization, there is

51 still room to improve existing results. For instance, the optimum abrasive mass flow rate, the optimum water pressure, the optimum orifice diameter, etc. have not been well understood. Also, the maximum profit rate, a very important objective, has not been investigated. It seems that AWJ optimization problems are complicated to solve. The reason for that can be explained as follows:

-For a complete AWJ optimization problem, there are many process parameters which need to be optimized. These parameters are, for example, the water pressure, the abrasive mass flow rate, the nozzle lifetime, etc. Therefore, the complete optimization problem is dynamic.

-For an individual AWJ objective function, for example, the minimum total cutting cost per unit length, it is very complicated to minimize by classical methods (see Subsection 5.2.1).

-In practice, AWJ cost elements, such as the abrasive cost per kilogram, the hourly machine tool cost, the hourly wages including overhead cost, etc. are varied depending on the market, the waterjet system, and the policy of waterjet companies. Also, the cost elements can be changed with time. Consequently, to have good and flexible optimum results, not only the process parameters but also the cost elements should be taken into account, which makes the optimization problems more difficult.

Based on the above remarks as well as the analyses of previous studies (see Chapter 2), to solve AWJ optimization problems, the following solutions are suggested:

-Using optimum results of previous studies in order to reduce the variables in the complete optimization problem; for instance, using the optimum ratio of the nozzle diameter to the orifice diameter, the optimum standoff distance, etc. (see Section 2.1).

-Using the suboptimization method for solving the complete optimization problem - the dynamic problem (see Subsection 5.2.1);

-Using the Golden Ratio search method to find the minimum of the objective function in suboptimization problem;

-Considering the process parameters and the cost elements as variables in AWJ optimization problems.

52 6 Modeling the cutting process for AWJ optimization

In this chapter an extension of Hoogstrate’s model [Hoog00] is presented to arrive at a more accurate cutting process model which can be used for AWJ optimization. First, the requirements for an AWJ cutting process model for an optimization problem are discussed. Then, a literature review of existing models on the modeling of AWJ cutting process is carried out. Finally, the formulations of the extension model consisting of three sub-models, viz. pure waterjet model, abrasive waterjet model, and particle – work material interaction model are addressed.

6.1 Requirements for an AWJ cutting process model

Generally, AWJ cutting process models determine the maximum depth of cut. Also, there are several process models which predict the material removal [Momb98]. Figure 6.1 describes the structure of an AWJ cutting process model [Hoog02a]. The process model consists of three models including jet model, kinematics model and material model. The jet model is used to calculate the power of the abrasive particles. The kinematics model describes the effect of the feed speed on the cutting process. The material model is taking into account the effect of the work material on the cutting process.

From the above structure of the AWJ cutting process model, the following requirements for an AWJ cutting process model for use in the optimization problem are given:

-The effects of the process parameters including the water pressure, the abrasive mass flow rate, the orifice diameter, the focusing tube diameter, and the feed speed on the cutting process should be taken into account.

-Various types of work materials should be taken into consideration. As one of the advantages of AWJ cutting is the capability of cutting a variety of types of work materials, the model should be used for calculation with a wide range of materials such as stainless steel, mild steel, ceramics, titanium, and so on.

-As mentioned in Chapter 2, with a certain setup of the process parameter, there is an optimum value of the abrasive mass flow rate for the maximum depth of cut. When the abrasive mass flow rate is less than this value, the maximum depth of cut will increase if it increases. Beyond the value, the opposite is true. Therefore, for optimization in AWJ machining, we need this type of the relation

53 between the abrasive mass flow rate and the maximum depth of cut in the cutting process model.

-The effects of various types and sizes of abrasives should also be investigated. This is because there are several abrasive types which have been used in AWJ cutting process such as garnet (for example Barton garnet, GMA garnet etc.), olivine, and so on. In addition, each type of abrasives has many different sizes, for example, Barton garnet has #50, #80, #120, #150 etc.

-For use in industries, the model accuracy must be sufficiently high, i.e. in 95% of the cases the required workpiece accuracy and quality is obtained in the first run [Hoog00].

-The models can be easily used for AWJ programming as well as for using in a workshop environment.

material model kinematics model jet model pure materials machine motion pure waterjet

alloyed materials machine accuracy abrasive waterjet

laminated materials energy density

required cutting intensity feed speed available power density

process model material removal model max. depth of cut

process quantity model output quantity model quality

taper angle

Figure 6.1: Structure of AWJ process model [Hoog02a]

6.2 State of the art in AWJ cutting process modeling

6.2.1 Studies of Hashish

Hashish [Hash89a] introduced a model for the calculation of the maximum depth of cut. To develop the model, Hashish divided the AWJ cutting process into two modes: the cutting wear mode and the deformation wear mode. In the cutting wear mode, the material is removed by particle impact at low impact angles. In the deformation wear mode, the excessive plastic deformation causes the material removal at high impact angles.

Based on the micro cutting analysis of Finnie [Hash89a] an improved model of erosion by an impacting abrasive particle can be written as:

54 2.5 7 ⋅ mV⎛⎞ δv =⋅⎜⎟sin2αα ⋅ sin (6.1) πρ⋅ pK⎝⎠C where, CK is a characteristic abrasive velocity:

3/5 3 ⋅⋅σ Rf C K = (6.2) ρp

The depth of cut in the cutting wear mode was found as

2/5 Cd⋅ ⎛⎞14 ⋅ mv h = j ⎜⎟abr abr (6.3) c 2.5 ⎜⎟πρ⋅⋅⋅vd C ⎝⎠fjabrK

To calculate the depth of cut in the deformation wear mode, Hashish used the following model which was introduced by Bitter [Bitt63]:

2 mvpabre⋅−() v δV = 2 ⋅σ f

The depth of cut in deformation wear mode then was given by

1 h = (6.4) d πσ⋅⋅⋅dv C v jff + f abr 2 dv− v 21⋅−()cm ⋅abr ⋅ ( v abr − v e ) j abr e

The maximum possible depth of cut equals the sum of the depth of cut by the cutting wear mode and the deformation wear mode:

hhhmax =+cd (6.5)

To determine the maximum depth of cut, Hashish assumes that the flow strength of work material

σ fM≈ E /14 with E M being the modulus of the elasticity of the work material. The correlation coefficient of Hashish’s model when calculated with this parameter and the experimental data was over 0.9 for most materials (the experiments were carried out with 23 materials). The effects of many process parameters such as the abrasive mass flow rate, the focusing tube diameter, and the transverse speed, were investigated. In addition, the effect of the abrasive size was taken into account by using the particle roundness factor Rf . However, there are still some limitations:

-The calculated depth of cut is too high for shallow cuts (<15 mm) and too low for deep cuts (>30 mm) [Momb98].

-The model contains the abrasive particle velocity which is difficult to measure and is calculated by approximation methods [Momb98].

-The optimum trend of the relation between the abrasive mass flow rate and the maximum depth of

55 cut (see Subsection 6.1) is not taken into account. Therefore, the model is not realistic and it can not be used for optimization problems.

-The calculation procedure consists of 8 steps [Hash89a], which is impractical.

Particles velocity vo

Cutting wear Particles zone trajectories x

Deformation h wear zone

Step removal by deformation wear h

Figure 6.2: Two wear zones in the AWJ cutting process [Hash89a]

6.2.2 Studies of Zeng and Kim

Zeng and Kim [Zeng92 and Zeng93] derived an empirical model for the abrasive cutting process. To determine the maximum possible depth of cut, the authors proposed the concept of a “Machinability Number”, which was determined by the following equation:

Ch⋅⋅ d0.618 ⋅ v 0.866 N = max ff (6.6) m 1.25 0.687 0.343 pqmwwa⋅

The machinability numbers for 27 types of engineering materials were found based on experiments (Figure 6.2). The maximum possible depth of cut then was defined by their well-known formula:

1.25 0.687 0.343 Npmw⋅⋅ q w m a hmax = 0.618 0.866 (6.7) Cd⋅⋅ff v

The model (6.7) fit quite well with their experimental data (with the determination coefficient R2=0.911). In practice, Zeng and Kim’s model has been used widely in the waterjet industry and for research purposes. The model and the “machinability number” are still used in programs for calculation of the AWJ cutting regime as well as for CNC machines for AWJ machining. After all, the model still contains several drawbacks:

Figure 6.3: Machinability numbers of various engineering materials [Zeng92]

-Because the model was built based on low values of the water pressure (pw from 138 to 276 MPa), it may not be valid for higher pressure use.

-Like the Hashish model [Hash89a], the trend of the effect of the abrasive mass flow rate on the maximum depth of cut is not taken into account.

-The effects of the abrasive types and abrasive sizes were not considered in the model.

6.2.3 Other studies

Besides the above studies, there have been many other studies on the modeling of AWJ cutting process so far. In order to reference them easily, the basic models for the depth of cut of these studies are summarized in Table 6.1.

57 Table 6.1: Models for maximum depth of cut in AWJ cutting

Author Reference Equation Notes

H. Oweinah [Owei89] η ⋅⋅vm2 -Energy conservation model; habra ηh is efficiency parameter. 2 ⋅vbfM⋅⋅ε

H. Blickweden [Blic90] -Energy conservation and Cpp0 ⋅−( wth) 0.86 2.09 /v regression model; C0 and pth v + f f are regression coefficients.

S. Matsui et al. [Mats91] 104.74 -Regression model; H is the

0.67 material hardness; ε is the vH⋅⋅ε s fs() strain.

R. Kovacevic [Kova92] dmp0.756⋅⋅ 0.221 1.47 -Regression model; the model 0.00139 faw ⋅ 0.74 0.139 is used for cutting mild steel. vsfd⋅

k 2 Y. Chung [Chun92] -Regression model; k1 , k2 , k3 mppawth⋅−( ) kk13⋅ + and p are regression vW⋅ th ft coefficients.

D.G. Taggart [Tagg97] 6.28 -Regression model; the model et al. pddm0.071⋅⋅⋅ 0.44 1.61 0.00474 ⋅ v 0.697 is used for very low pressure worifa f (<150 MPa).

J. Wang [Wang07] 1.186 0.156 -Regression model; the model 6 mpaa⋅ d p 1.974×⋅ 10 is used for cutting alumina ρ ⋅dv⋅⋅ s wffd ceramics.

6.3 Introduction to Hoogstrate’s model

6.3.1 Model description

Hoogstrate, in his PhD. thesis [Hoog00], introduced a model for calculation of the maximum possible depth of cut, based on the energy-flow per time unit through the cutting system. In his theory, the energy-flow through the cutting system from the hydraulic power to the cutting action can be divided into three subsequent steps:

-The first step: transformation of the potential energy of an amount of high pressure water into kinetic energy of a high speed plain waterjet.

-The second step: transformation of a part of the kinetic energy of high speed water to the kinetic energy of the abrasive particles by accelerating the particles.

-The third step: using the kinetic energy of the abrasive particles for removing small chips of the work material.

The basic assumption of the model is that only the abrasive particles contribute to the cutting process. This assumption is not valid for very soft work materials such as paper and plastics which

58 can also be cut by plain waterjet. This is also the case for brittle work materials (for example granite or marble) in which the propagation of cracks significantly contributes to the cutting process. The model therefore is used for all ductile work materials.

Based on the above principles, to find the model for AWJ cutting, first, the pure waterjet model which describes the transformation of the energy from the high pressure water into the plain waterjet is determined:

π 2 3/2 2 Pcwj=⋅⋅ d ⋅ pd w ⋅ ori (6.8) 4 ρw

Next, the abrasive waterjet model which describes the transformation energy from the pure waterjet into the abrasive particles is determined as follows:

PkPabr=⋅ wj (6.9)

R π 2 Pcpd=⋅η 23/22 ⋅⋅⋅ ⋅ ⋅ (6.10) abr2 d4 ρ w ori ()1 + R w

Where κ – power transfer efficiency; κ is determined by:

2 R κη=⋅ 2 (6.11) ()1 + R

The relation between the abrasive load ratio R and the power transfer efficiency κ is shown in Figure 6.4. It is observed that there is an optimum value of the abrasive load ratio for the maximum power transfer efficiency. That means the model can describe the optimum trend of the relation between the abrasive mass flow rate and the maximum depth of cut. Consequently, it can be used for the AWJ optimization problem to determine the optimum abrasive mass flow rate.

cd -coefficient of discharge which is determined experimentally; from typical cutting process conditions, it was chosen cd = 0.71.

η -momentum transfer efficiency coefficient; η is calculated by the following equation:

η =−⋅ccR12 (6.12)

In which, c1 and c2 are empirical constants; R=ṁa/ṁw is the abrasive load ratio.

In the study, c1=1 and c2=1.6 were determined for nozzle diameter df of 0.8 mm for cutting with Barton garnet #150. Consequently, Equation 6.12 is valid for only this case. The model for the cutting process is based on the following relation:

Qmat∝ P abr

59 0.07

0.06

0.05

0.04

0.03

0.02

0.01 Power transfer efficiency k (−) 0 0 0.1 0.2 0.3 0.4 0.5 0.6 Abrasive load ratio R (−) Figure 6.4: Power transfer efficiency κ [Hoog00]

Using the concepts of the specific cutting energy of a work material ec and of the cutting energy efficiency coefficient ξ, the material removal rate of work material is determined as follows [Hoog00]:

ξ ⋅Pabr Qmat = (6.13) ec

On the assumption that the cutting width is uniform over the depth of cut and equal to the focusing tube diameter df , the maximum depth of cut is predicted as:

Pabr hvmax =⋅ξ ()f (6.14) edvcff⋅⋅

Table 6.2: Typical specific cutting energy of materials [Hoog00]

9 3 Materials Specific cutting energy ec (10 J/m ) SS 304 7.5 Aluminum 2.7 Glass 1.5

In the above equation, the cutting efficiency coefficient is determined empirically as:

0.134 ξ ()vvff=⋅0.9113 (6.15)

The coefficient ec is also determined experimentally. Typical values of the coefficient are shown in

Table 6.2. Hoogstrate also found the relation between the machinability number Nm and the specific cutting energy ec as shown:

60 6.11× 1011 ec = (6.16) N m

To quantify the quality of the cut, Hoogstrate gives a term called the quality number of the cut Q which is determined as follows [Hoog00]:

1 Q = (6.17) h With

h h = actual (6.18) hmax Where, h is the dimensionless depth of cut.

Figure 6.5 shows the definition of the quality number Q with its value from 1 to 10 [Hoog00]. The quality number describes not only the qualitative level of the surface roughness but also the tape angle and the specific cut surface characteristics like fiber condition. In addition, it can be used to define an arbitrary criterion for users (see Figure 6.5).

Quality number

Surface rough smooth smooth roughness

Tape strongly small square angle tapered tapered

Fiber broken cut fiber condition fiber

User defined criterium

Figure 6.5: Definition of the quality number [Hoog00]

6.3.2 Discussion

• Advantages of Hoogstrate’s model

-By modeling the AWJ cutting process based on three subsequent steps of the energy flow through the cutting system, the physics of the model is very clear and easy to understand.

-The effects of the process parameters such as the water pressure, the abrasive mass flow rate, the

61 focusing tube diameter, the orifice diameter, the traverse speed are taken into account.

-By modeling the power transfer efficiency, the optimum trend of the effect of the abrasive load ratio as well as of the abrasive mass flow rate on the power transfer efficiency (Figure 6.4) and therefore the maximum depth of cut are taken into account. This character of the model is very important for the AWJ optimization problem in order to determine the optimum abrasive mass flow rate.

-By modeling the specific cutting energy coefficient, the model can be used for the calculation of the maximum depth of cut for various work materials.

• Implications for further improvements

-In Hoogstrate’s model, the discharge coefficient is taken as a constant (see Subsection 6.3.1). In practice, it depends on several parameters including the water pressure, the diameter and the shape of the orifice [Hash02]. Therefore, a more thorough study on the discharge coefficient is needed to improve the model accuracy [Hash02].

-As mentioned above, Equation 6.13 which is used to predict the momentum transfer efficiency is valid only for the nozzle diameter of 0.8 (mm). In general, however, Susuzlu [Susu06] claimed that c1 is smaller than 1 and c2 is a variable. Consequently, the model for momentum transfer efficiency should be improved in order to take into account the influence of the process parameters.

-It is noted that the cutting efficiency coefficient ξ depends on the feed speed, the types and the sharpness of the abrasives. Also, it depends on the abrasive and the work material. However, since the cutting efficiency was not a main objective of Hoogstrate’s study, only the feed speed has been taken into account for the prediction of the coefficient ξ. As a result, the modeling of the cutting efficiency coefficient should be investigated in more detail in order to increase the model accuracy.

6.4 Extension of Hoogstrate’s model

In this section the extension of Hoogstrate’s model is carried out. The proposed structure of the extension model is shown in Figure 6.6. The extension of the model is based on the extensions of three following sub-models: The pure waterjet model in Sub-section 6.41, the abrasive waterjet model in Sub-section 6.42, and the abrasive-work material interaction model in Sub-section 6.4.3.

6.4.1 Pure waterjet modeling

As mentioned in Section 6.3, the pure waterjet model enables to determine the energy transfer from pressurized water to the pure waterjet. From the model of the output power of the pure waterjet (see Equation 6.8), it is clear that the more accurate the discharge coefficient can be determined the more accurate the power of the pure waterjet is deduced. Therefore, to have a

62 more accurate pure waterjet model, in this subsection, the extension of the coefficient of discharge model is carried out. In addition, the reaction force of the pure waterjet, which is used for calculating the momentum transfer efficiency, is investigated.

Abrasive-work material interaction model Jet model

Work materials Pure waterjet Abrasive Abrasive

Required feed speed Available power density

Process model

Maximum depth of cut

Figure 6.6: Proposed structure for extension of Hoogstrate’s model

6.4.1.1 Reaction force of pure waterjet

• Experimental design

The setup of the experiment for measurement of the reaction forces of the abrasive waterjet was shown in Figure 4.4a. The jet parameters used in the experiment are presented in Table 6.3.

Table 6.3: Experimental parameters for the pure waterjet reaction force

Parameter Unit Level number Values

Orifice diameter mm 5 0.125; 0.15; 0.175; 0.2 and 0.25 Water pressure MPa 8 150; 210; 260; 320; 360; 430; 510 and 610

A full factorial experimental design at all levels (5x8=40 measurements) would have been required, which is too time-consuming. In addition, the water pressure and the orifice diameter should be combined properly with each other such as a small diameter of orifice should be used for tests with high water pressures. Therefore, a selected subset of the full factorial has been performed, resulting in 24 individual measurements with two replications.

• Results and discussions:

Figure 6.7 plots the results of the force measurements of the pure waterjet, which shows most likely a linear relation with the water pressure as discussed in [Susu06] and [Clau98]. It is observed that the reaction force increases with the increase of the water pressure as well as with the increase of the orifice diameter. This can be explained that either the increase in the water pressure

63 or the increase of the orifice diameter will lead to the increase in the water volume flow rate and therefore an increase in the reaction force.

The data of the pure waterjet reaction force measurements will be used to determine the momentum transfer efficiency in Subsection 6.4.2.

(N) d =0.25 mm ori d =0.175 mm wj,0 ori

5 200 250 300 350 400

Pure waterjet reaction force F Water pressure p (MPa) w

Figure 6.7: Reaction force of the pure waterjet

6.4.1.2 Extension of the discharge coefficient model

dori

dwj

Vena contracta

Figure 6.8: Schema of the discharge from an orifice

In waterjet cutting, for making a high-pressure beam of water, the water is forced to pass through a small hole called an orifice. At the downstream, the diameter of the water beam is not equal to the orifice diameter (Figure 6.8). As the convergence of the streamline, the cross section of the water beam slightly decreases until a point called vena contracta. The cross section area Awj at this

64 point is minimum. Beyond the vena contracta, the velocity of the waterjet decreases because of the increase of the friction force between the water beam and the orifice wall and the air. The actual volume flow rate of water qwa, at the downstream is less than the theoretical water volume flow rate qwth, . This is because of the reduction in the jet velocity and the decrease of the water volume flow rate due to the sudden changes of the fluid-mechanic condition on the orifice outlet [Momb98]. The actual water volume flow rate can be expressed as

qw, a=⋅Av wj wj (6.19)

This can be rewritten as:

Av wj wj qAvccqcqw,0,,, a=⋅ ⋅⋅=⋅⋅=⋅ wj inc c vc wj th d w th (6.20) Av0,wj inc

With

q ccc=⋅ =wa, (6.21) dcvc qwth,

In the above equations, vwj,inc is the velocity of the incompressible waterjet; for the case that the water is considered incompressible and frictionless the velocity is calculated from the Bernoulli equation:

2 ⋅ pw v wj, inc = (6.22) ρ0

The theoretical water volume flow rate qwth, then can be calculated by:

π ⋅d 2 2 ⋅ p ori w qAvwth,0,=⋅ wjinc = (6.23) 4 ρ0

Where, cc, cvc, and cd are contraction coefficient, velocity compressible coefficient and coefficient of discharge, respectively [Hash02], [Susu06]. The contraction coefficient is defined by the following equation [Susu06]:

cAAcwj= / 0 (6.24)

The velocity compressible coefficient cvc was described by Hashish [Hash02] and Susuzlu [Susu06].

As water is compressible, the relationship between the water pressure pw and the water density ρ can be expressed as [Susu06]:

1/n ⎛⎞np⋅ ρ p =+1 w ⋅ρ (6.25) ()w ⎜⎟0 ⎝⎠E w 0

65 3 In which, ρ0 is the ambient water density (ρ0=1000 kg/m ); Ew0=2135 MPa and n=7.15 are experimental coefficients which were found from the data of Bridgman [Brid70]. By solving the Bernoulli equation and the Equation 6.25, Susuzlu [Susu06] found the following equation for prediction of the waterjet velocity:

11/− n 2 ⋅E ⎡⎤⎛⎞np⋅ v =+−w 0 ⎢⎥⎜⎟11w (6.26) wj, com ρ ⋅−n 1 ⎢⎥⎜⎟E 0 ()⎣⎦⎝⎠w 0

The velocity compressible coefficient cvc was then calculated by the following equation [Susu06]:

11/− n v E ⎡⎛⎞np⋅ ⎤ c ==wj, com w 0 ⋅+⎢⎜⎟11w −⎥ (6.27) vc vEpn⋅−1 ⎢⎜⎟⎥ wj,0 incw ()⎣⎝⎠ w ⎦

In practice, the compressible waterjet velocity vwj,com can also be reduced by the friction in the orifice. This effect is considered through an additional coefficient cvf and the formulation for the coefficient of discharge becomes [Susu06]:

ccccdcvcvf=⋅⋅ (6.28)

In practice, the typical values of the coefficient of discharge are between 0.6 and 0.8 (for sharp- edged sapphire orifices) [Momb98]. The coefficient of discharge plays an important role in the calculation of the actual power of the pure waterjet and consequently in waterjet cutting process models. As a result, there have been many studies on the discharge coefficient.

0.76 (−) d

0.74

d =0.152 mm ori 0.72 d =0.254 mm ori d =0.356 mm ori d =0.457 mm 0.7 ori d =0.508 mm ori d =0.584 mm ori 0.68

0.66

0.64 100 150 200 250 Coefficient of discharge c Water pressure p (MPa) w Figure 6.9: Effect of pressure on the coefficient of discharge at low pressure [Hash89b]

Hashish [Hash89b, Hash97a, and Hash02] carried out several studies on the coefficient of discharge. The relation between the water pressure and the coefficient of discharge was shown in [Hash89b] and [Hash97a] for various values of the orifice diameters. For the water pressure from 105 to 240 MPa and the orifice diameter from 0.152 to 0.584 mm, Hashish found that the coefficient of discharge decreases with the increase of the water pressure as well as the increase of

66 the orifice diameter (Figure 6.9) [Hash89b]. In addition, Hashish [Hash02] introduced a linear equation, which is accurate within 8% for his experimental data for sharp edge sapphire orifices:

cpddwori=−0.785 0.00014 ⋅−⋅ 0.197 (6.29)

X. Claude et al. [Clau98] confirmed the relationship between the water pressure and the discharge coefficient. The authors noted that the coefficient of discharge reduces from 0.74 to 0.64 as the pressure increases from 50 to 350 MPa (Figure 6.10).

The coefficient of discharge in the cases of high-pressure (up to 700 MPa) was investigated by Hashish et al. [Hash97a]. It was noted that the coefficient values spread around 0.6 (Figure 6.11).

0.74 0.64 (−) d (−)

0.72 d 0.63

0.62 0.7

0.61 d =0.137 mm 0.68 ori d =0.182 mm 0.6 ori d =0.226 mm ori 0.66 0.59

0.64 0.58

0.62 Coefficient of discharge c 0.57

Coefficient of discharge c 50 100 150 200 250 300 350 400 300 400 500 600 700 Water pressure p (MPa) Water pressure p (MPa) w w

Figure 6.10: Effect of pressure on the coefficient Figure 6.11 Effect of high-pressure on the of discharge [Clau98] coefficient of discharge [Hash97a] In the preceding studies, the effects of the water pressure as well as the orifice diameter on the coefficient of discharge have been investigated. However, there has not been a good model for the prediction of the coefficient. Since Hashish’s model (Equation 6.29) was developed at low water pressure (from 105 to 240 MPa), its application for high pressure may not be suitable. As a result, more studies on this area are needed to be done in order to determine a more accurate model.

In this subsection, based on the above theory of modelling the discharge coefficient [Susu06] and by combining a physical-mathematical model and an experimental method, a model for prediction of the coefficient of discharge is derived.

• Proposed model for the discharge coefficient

As aforementioned, the coefficient of discharge can be determined according to the following equation [Susu06]:

ccccdcvfvc=⋅⋅ (6.28 repeated)

67 The coefficient of discharge also depends on the quality of the orifice such as the orifice edge condition and the inlet radius [Hash02]. In practice, it is very difficult to calculate the effect of the friction loss as well as the effect of the orifice quality. For simplification, the friction losses in the orifice, the orifice quality effect and the contraction of the jet are combined in a single quality coefficient cq. Therefore, the coefficient of discharge then can be calculated by:

cccdqvc=⋅ (6.30)

From Equations 6.21 and 6.30 we have

q ccc=⋅ =wa, (6.31) dqvc qwth,

Where, cvc is determined by Equation 6.27.

From Equation 6.31 the coefficient of discharge can be determined directly from the measurements of the actual water volume flow rate and the theoretical water volume flow rate. As a result, the coefficients cq can be found by regression method based on the data of the measurements.

The coefficient cq depends mainly on the water pressure pw as well as the orifice diameter. Hence, the coefficient cq can be expressed via the following function:

cfpdq = ()wori, (6.32)

The effect of the water pressure on cq can be expressed through the effect of the velocity of the incompressible waterjet vwj,inc (see Equation 6.22) on it. This results in:

cfvdq = ()wj, inc, ori (6.33)

Based on the Buckingham Pi theorem, cq can be expressed as a function of the above parameters in a dimensionless form:

k 2 ⎛⋅ρ vd ⋅ ⎞ ck=⋅0,wj incom ori (6.34) q 1 ⎜⎟ ⎝⎠μ0

In the above equation

ρ0,⋅⋅vdwj incom ori = Re 0 (6.35) μ0

0 Where, µ0 is the dynamic viscosity of water at room temperature (i.e. at 21 C this value is

-1 -1 approximate µ0 ≈ 0.001002 kg.s .m ); Re0 is the Reynolds number of the incompressible water flow at room temperature. In fluid mechanics, the Reynolds number is used to identify different flow regimes, such as turbulent or laminar flow. This number will be discussed in more detail in the following part.

68 • Experiment for determining the coefficient of discharge

For calculation of the coefficient of discharge (by Equation 6.31), an experiment was carried out in order to measure the water volume flow rate. The water through the orifice was collected by a water collector (see Figure 4.3) and then weighted to determine the actual water volume flow rate. The experiment was conducted with a full factorial design with two replications for three levels of the sapphire orifice diameter (0.150, 0.170 and 0.255 mm) and seven levels of the water pressure (60, 100, 150, 200, 250, 300 and 350 MPa).

14000 1.6 d =0.25 mm ori 12000 d =0.2 mm ori 1.4 d =0.15 mm ori 10000 1.2 d =0.255 mm 8000 ori 1 d =0.17 mm ori d =0.15 mm ori 6000 Turbulent region 0.8

4000 0.6 Transient region

Reynolds number Re (−) 2000 0.4 Laminar region

0 Water volume flow rate (l/min) 0.2 0 100 200 300 400 500 50 100 150 200 250 300 350 Water pressure p (MPa) Water pressure p (MPa) w w Figure 6.12: Reynolds number of the water flow Figure 6.13: Water pressure versus water in the upstream tube volume flow rate

It is known that the upstream tube conditions determine the coherency of the waterjet. Hashish [Hash02] found that the upstream tube length has the highest influence. He suggested that the ratio of the length to the diameter must not be less than 20. In the present system, this ratio is 57.5 (the tube length is 230 mm, the inside diameter is 4 mm), which satisfies well the above requirement. Hashish [Hash02] also noted that the jet coherency is affected by the upstream Reynolds number. A laminar flow regime in the upstream tube promotes a coherent waterjet [Susu04]. Figure 6.12 shows the relation between the upstream Reynolds number and the water pressure with different orifice diameters. It is observed that in most cases the upstream flow is not laminar. Therefore, it is not practical to keep the flow laminar [Susu04]. In addition, as noted by Hashish [Hash89b], an increase of the water pressure leads to an increase of the upstream Reynolds number and therefore deteriorates the jet coherency.

• Results and discussions

Figure 6.13 shows the relationship between the measured water volume flow rates and the water pressure. Based on the measured flow rates, the coefficient of discharge was calculated by Equation 6.31 and presented in Figure 6.14. It follows that the coefficient of discharge decreases

69 as the orifice diameter increases. As noted in [Hash89b] and [Clau98], the coefficient of discharge decreases when the water pressure increases (Figure 6.14).

0.95

(−) Data − d =0.255 mm

d ori 0.9 Data − d =0.17 mm ori Data − d =0.15 mm ori 0.85 Model − d =0.255 mm ori Model − d =0.17 mm 0.8 ori Model − d =0.15 mm ori 0.75

0.7

0.65

0.6 50 100 150 200 250 300 350 Water pressure p (MPa) Coefficient of discharge c w

Figure 6.14: Water pressure versus coefficient of discharge

By conducting a regression analysis (consisting of 42 data points), the quality coefficient cq and then the discharge coefficient (Equation 6.30) are determined as follows:

−0.2343 ⎛⋅ρ vd ⋅ ⎞ c =⋅10.9638 0,wj inc ori (6.36) q ⎜⎟ ⎝⎠μ0

−−0.2343 1 1 / n ⎛⋅ρ vd ⋅ ⎞E ⎡ ⎛np⋅ ⎞ ⎤ c =⋅10.9638⎜⎟0,wj inc ori ⋅w 0 ⋅+−⎢ ⎜⎟ 1w 1⎥ (6.37) d ⎜⎟μ pn−1 ⎢ ⎜⎟E ⎥ ⎝⎠00w ()⎣ ⎝⎠w ⎦

0.9

0.85

(−) 0.8 d,cal 0.75

0.7

0.65 of discharge c

Calculated coefficient 0.6

0.55 0.6 0.7 0.8 Experimental coefficient of discharge c (−) d,ex

Figure 6.15: Correlation between experimental and calculated coefficient of discharge

The correlation between the experimental and the calculated coefficient of discharge is shown in Figure 6.15. The correlation between the experimental data and the model is very good (with the

70 coefficient of determination R2=0.97). The model is accurate to within 3% for all the data. Figure 6.14 shows the relation between the water pressure and the coefficient of discharge with different values of orifice diameter.

6.4.1.3 Conclusions

In this subsection, a model for calculation of the coefficient of discharge has been proposed. The model has been built by combining a physical-mathematical model and an experimental analysis. By modeling the coefficient of discharge the pure waterjet model has been extended and it becomes more accurate.

The effects of water pressure and the orifice diameter on the discharge coefficient are taken into account in the coefficient of discharge model.

The extended pure waterjet model allows developing an accurate AWJ cutting model, which can be found by incorporation of the pure waterjet model, the abrasive waterjet model, and the abrasive- work material interaction model.

6.4.2 Abrasive waterjet modeling

The abrasive waterjet model is used to determine the energy transfer from the pure waterjet to the abrasive particles. From the abrasive waterjet model (Equation 6.10), it is obvious that the accuracy of the model depends on the accuracy of the momentum transfer efficiency model and the pure waterjet model. However, as mentioned in Section 6.3, the model for the momentum transfer (Equation 6.12) is built only for the nozzle diameter of 0.8 mm and for Barton garnet #150. Consequently, to get a more accurate abrasive waterjet model, this subsection deals with the extension of the momentum transfer model. Also, the reaction force of the abrasive waterjet is carried out.

6.4.2.1 Reaction force of abrasive waterjet

• Experimental design

The setup of the experiment for measurement of the reaction forces of the abrasive waterjet is shown in Figure 4.4b. The jet parameters used in the experiment are presented in Table 6.4.

As it was done for the reaction force of the pure waterjet, to avoid a full factorial experimental design at all levels (5x3x8x12x4=5768 measurements), a selected subset of the full factorial including 685 individual measurements with two replications has been done. This is because the full factorial design requires a lot of time-consuming. Also, the jet-parameters should be combined properly with each other such as large sizes of abrasives should be tested with a large orifice and nozzle diameter, a small diameter of orifice should be used for tests with high water pressures,

71 small diameters of orifices and nozzles should be used for small abrasive mass flow rates etc.

Table 6.4: Experimental parameters for the abrasive reaction force

Parameter Unit Level number Values

Orifice diameter mm 5 0.125; 0.15; 0.175; 0.2 and 0.25 Nozzle diameter mm 3 0.5; 0.82 and 0.92 Water pressure MPa 8 150; 210; 260; 320; 360; 430; 510 and 610 Abrasive mass flow g/min 12 0; 50; 100; 150; 200; 250; 300; 350; 400; rate 450; 500 and 540 Abrasive 4 #50; #80; #120 and #150 HPX

• Results and discussions:

Figure 6.16 shows the relation between the abrasive mass flow rate and the abrasive waterjet reaction force with different jet parameters. As noted in [Claud89] and [Susu06], the reaction force decreases with the increase of the abrasive mass flow rate. This is because higher abrasive mass flow rate causes a larger loss during the acceleration process.

It is in agreement with [Susu06] that the increase in the water pressure, the nozzle diameter and the orifice diameter will lead to an increase of the reaction force (see Figure 6.16a, Figure 6.16b and Figure 6.16c, respectively). As the water pressure increases the water stream velocity and therefore the particle velocity increase as well. Therefore, the reaction force will rise. Also, with the enlargement of the nozzle diameter the reaction force goes up since the particle velocity increases. The reason is that increasing nozzle diameter will reduce the friction of the jet with the nozzle wall. Finally, with the increase of the orifice diameter the water flow rate and therefore the reaction force increase.

The effect of the abrasive size on the reaction force is shown in Figures 6.16d and 6.16e. It is found that with an increase of the abrasive size the abrasive reaction force decreases (Figure 6.16d). The reason of that is larger particle diameters cause lower velocities [Momb98]. However, in the case of bigger orifice diameters, the reaction force is nearly not affected by the abrasive particle size (Figure 6.16e). It can be explained that in this case the increased water flow rate is dominating the loss, thereby reducing the relative influence of the abrasive particle size.

The data of the measurements of the abrasive waterjet reaction force, together with those of the pure waterjet reaction force, will be used to determine the momentum transfer efficiency in the following subsection.

72 #120 HPX; p =360 MPa #80 HPX; d /d =0.25/0.92 w ori f 20 22 p =360 MPa (N)

(N) w 20 18 p =310 MPa awj awj w p =210 MPa w 18 16

16 14

d /d =0.175/0.92 14 ori f 12 d /d =0.25/0.92 ori f 12

10 10

8 8

6 0 50 100 150 200 250 300 350 6 0 100 200 300 400 500 600

Abrasive mass flow rate (g/min) Abrasive waterjet reaction force F Abrasive mass flow rate (g/min) Abrasive waterjet reaction force F

a) b) #120 HPX; p =360 MPa w 8 (N) awj 7

6 c) d /d =0.125/0.82 ori f 5 d /d =0.125/0.5 ori f

2 0 50 100 150 200 250 300 Abrasive waterjet reaction force F Abrasive mass flow rate (g/min) p =310 MPa; d /d =0.176/0.92 p =210 MPa; d /d =0.25/0.92 w ori f w ori f

9 (N) 12 (N) aWj awj 11.5 #150 HPX 8.5 #120 HPX 11 #80 HPX

8 10.5

7.5 9.5

#150 HPX 9 7 #120 HPX #80 HPX 8.5

6.5 8

0 50 100 150 200 250 300 Abrasive waterjet reaction force F 0 100 200 300 400 Abrasive waterjet reaction force F Abrasive mass flow rate (g/min) Abrasive mass flow rate (g/min) d) e) Figure 6.16: Reaction force of the abrasive waterjet

73 6.4.2.2 Extension of momentum transfer efficiency model

In an abrasive entrainment waterjet system, the acceleration of abrasive particles is accomplished by a momentum transfer from the high-speed waterjet to the abrasive particles. Using the same methodology as in [Hoog02b], the following equation can be written for the impulse balance:

Fmvmv=⋅−⋅ (6.38) ∑∑()in ∑ ()out

Orifice High pressure water Abrasive particle

Vpa,01 Vair,0 Vwj,0 Vpa,01x Abrasive particles Vpa,02x Vpa,02 Vpa,0ix Vpa,0i

Mixing chamber

Focusing tube Vpa,1 Vpa,1x Vpa,2x Vpa,2

Vpa,jx Vpa,j vwj vwj vwj

Figure 6.17: AWJ cutting head schema and mechanism of mixing and acceleration process

The resulting force on the AWJ cutting head is equal to the change in momentum of the incoming and outgoing mass fluxes:

N N ⎛⎞⎛⎞pa ,0 pa Fmvmvmvmvmvmv=⎜⎟⎜⎟ ⋅ + ⋅ +⋅ − ⋅ + ⋅+⋅ (6.39) fr⎜⎟⎜⎟∑∑ pa,0 i pa ,0 i air ,0 air ,0 w wj ,0 pa , j pa , j air air w wj ⎝⎠⎝⎠i =1 j =1 Where, mv ⋅ and mv ⋅ are equal to the forces at the positions at which the particles ∑ ()in ∑ ()out enter the water stream and leave the focusing tube, respectively.

For the impulse balance in the direction of the water jet, the Equation 6.39 can be rewritten as:

N N ⎛⎞⎛⎞pa ,0 pa Fmvmvmvmvmvmv=⎜⎟⎜⎟ ⋅ + ⋅ +⋅ − ⋅ + ⋅+⋅ (6.40) fr⎜⎟⎜⎟∑∑ paipaixairairwwj,0 ,0 ,0 ,0 ,0 pajpajxairairwwj , , ⎝⎠⎝⎠i =1 j =1

Equation 6.40 is simplified by neglecting the air mass flow rate because the amount of the air on the entire mass of the abrasive waterjet is negligibly small (about 3%) [Momb98]. Also, the initial velocity of abrasive particles entering the water stream is neglected as it is much smaller than that

74 of particles leaving the nozzle [Momb98]. In addition, it is stated:

vvvpajx, ≤= wj awj (6.41)

This results in

NNpp ∑∑mvpaj,,⋅< pjx mv paj , ⋅ wj (6.42) jj==11

N p ∑ mvpaj,,⋅=⋅⋅ pajx kmv a wj (6.43) j =1

From the above analysis, Equation 6.40 can be rewritten as Fmvfr=⋅ w wj,0 −⋅+( kmmv a w) ⋅ awj =⋅ mv w wj ,0 ⋅−(1 η ) (6.44)

Where, η is momentum transfer efficiency coefficient:

(6.45) η =⋅(kmawawjwwj + m) ⋅ v/ ( m ⋅ v ,0 )

So far, there have been various studies on momentum transfer efficiency. Hashish [Hash89b] investigated the effect of the water pressure on the mixing efficiency by measuring the reaction force of a pure and of an abrasive waterjet. Miller [Mill91] calculated the momentum transfer efficiency based on the measured particle velocity. Claude et al. [Clau98] investigated the effect of the water pressure and the mixing chamber diameter on the momentum transfer. Hoogstrate [Hoog00] introduced a model for the momentum transfer efficiency:

η =−⋅ccR12 (6.46)

Where, R=ṁa/ṁw is defined as the abrasive load ratio; c1=1 and c2=1.6 are empirical constants. In a further study, Susuzlu and Hoogstrate [Susu06] investigated the momentum transfer efficiency for AWJ cutting beyond 400 MPa.

Although there have been many studies on this subject, there is still not a reliable model for the determination of the momentum transfer efficiency. Also, the influence of the abrasive size on the momentum transfer has not been studied.

In the next sub-section, the extension of the momentum transfer efficiency model is carried out. The effects of many jet-parameters such as the water pressure, the abrasive mass flow rate, the focusing tube diameter and the abrasive size have been investigated.

• Proposed model for momentum transfer efficiency

From Equation 6.45 it can be deducted:

η =+⋅⋅=vvawj/// wj,0 kRvv awj wj ,0 FF awj wj ,0 (6.47)

75 Where, Fwj,0 and Fawj are the reaction forces of the pure and of the abrasive waterjet:

Fkmmvawj =⋅( awaw +) ⋅j (6.48)

Fmvwj ,0=⋅wwj ,0 (6.49)

Equation 6.47 can be expressed as follows

η =+⋅=ccRF12awj/ F wj ,0 (6.50)

Where, c1 and c2 are coefficients:

cv1,0,0= awj/ v wj (6.51)

ckvv2,0=⋅awj / wj (6.52)

In the above equations, vawj,0 and vawj are the velocities of the abrasive waterjet, with the abrasive ratio R=0 and R>0, respectively. From Equation 6.50, the momentum transfer efficiency can be determined directly from the measurement of the reaction forces of the plain and abrasive waterjet.

Consequently, the coefficients c1 and c2 can be found through a regression analysis of the measured data.

Modeling of coefficient c1:

From Equation 6.50, c1 equals the momentum transfer efficiency when the abrasive load ratio R=0, i.e. there are no abrasive particles involved in the process. Consequently, c1 does not depend on the abrasive mass flow rate, nor on the initial particle diameter, but only on the velocities of the pure water jet entering and leaving the focusing tube. Therefore c1 depends on the water pressure, the orifice diameter, and on the diameter, the length and the alignment of the focusing tube [Susu06]. The nozzle length has been kept constant in this study and the effect of alignment has not been taken into account explicitly. Therefore, c1 is described as a function of water pressure, orifice diameter and focusing tube diameter:

cfpdd1 = ()worif,, (6.53)

Based on the Buckingham Pi theorem, c1 can be expressed in a dimensionless form as follows:

kk23 ckdd11=⋅()()ori// f ⋅ pp w a (6.54)

Where, pa is the ambient pressure (pa ≈1.01325 bar).

Modeling of coefficient c2:

From Equation 6.52, c2 depends on the velocity of the pure water and the abrasive waterjet (vwj,0 and vawj). As mentioned above, these velocities depend on the water pressure, the orifice diameter and the nozzle diameter. In addition, the velocity of the abrasive waterjet also depends on the

76 abrasive mass flow rate and the particle diameter. Therefore, c2 is represented by:

cfpdddm2 = ()worif,,,, p a (6.55)

Using Buckingham Pi theorem c2 is expressed in a dimensionless form as

k 5 k 6 ckdd24=⋅()pf/ ⋅ R (6.56)

6.4.2.3 Results and discussions

• Momentum transfer efficiency:

Figure 6.18 shows the relationship between the momentum transfer efficiency η and the abrasive load ratio R. As noted in [Susu06], the momentum transfer decreases with the increase of the abrasive load ratio. The reason is that with a certain set up, the abrasive waterjet reaction force decreases when the abrasive load ratio (or the abrasive mass flow rate) increases (see Subsection 6.4.2.1). Since the reaction force of the pure waterjet is constant, the momentum transfer efficiency will decrease (see Equation 6.50).

The effect of the orifice diameter on the momentum transfer is shown in Figure 6.18a. It is detected that smaller orifice diameters lead to higher momentum transfer efficiency. It can be explained that with the increase of the orifice diameter, both the reaction force of the pure and the abrasive waterjet will rise (see Subsections 6.4.1.1 and 6.4.2.1). However, the increase in the pure reaction force is stronger than that in the abrasive reaction force. For example, at the water pressure of 360 MPa and the nozzle diameter of 0.92 mm, the pure waterjet reaction force increases from 11.62 to 23.92 N (2.06 times) while the abrasive waterjet reaction force (at the abrasive mass flow rate of 0 g/min) rises from 10.02 to 19.69 N (1.97 times) (see Figures 6.7 and 6.16a). This is because in the second case, with the mixing chamber and the focusing tube, the friction loss increases with the increase of the water volume flow rate (since the orifice diameter is enlarged). It will lead to the reduction of the increasing rate of the waterjet velocity and therefore the abrasive reaction force.

It is observed that with the increase of the water pressure the momentum transfer efficiency increases (Figure 6.18b). Like the influence of the orifice diameter, both the pure and the abrasive waterjet reaction force increase when the water pressure increases. However, in this case, the increase in the pure waterjet reaction force is weaker than that in the abrasive waterjet reaction force. For example, with the orifice diameter of 0.25 mm and the nozzle diameter of 0.92 mm, the pure waterjet reaction force increases 1.59 times (from 15.01 to 23.92 N) while the abrasive waterjet reaction force increases 1.8 times (from 11.39 to 20.49 N) when the water pressure rises from 210 to 360 MPa (see Figures 6.7 and 6.16b). As a result, the momentum transfer efficiency goes up when the water pressure increases.

77 #120 HPX; p =360 MPa #80 HPX; d /d =0.25/0.92 w ori f 0.9 0.9

0.85 p =360 MPa d /d =0.175/0.92 w 0.85 ori f p =310 MPa d /d =0.25/0.92 w ori f 0.8 p =210 MPa w

0.75 0.8

0.7

0.75 0.65

0.6 0.7

0.55 Momentum transfer efficiency (−) Momentum transfer efficiency (−) 0.65 0.5 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Abrasive load ratio R (−) Abrasive load ratio R (−)

a) b) #120 HPX; p =360 MPa w 0.95

d /d =0.125/0.82 0.9 ori f d /d =0.125/0.5 ori f 0.85

0.8

c) 0.75

0.7

0.65

0.6

0.55 Mometum transfer efficiency (−)

0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Abrasive load ratio R (−) p =310 MPa; d /d =0.176/0.92 p =210 MPa; d /d =0.25/0.92 w ori f w ori f 0.95 0.8

#150 HPX #150 HPX 0.9 #120 HPX 0.75 #120 HPX #80 HPX #80 HPX

0.85 0.7

0.8 0.65

0.75 0.6 Momentum transfer efficiency (−) Momentum transfer efficiency (−) 0.7 0.55 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Abrasive load ratio R (−) Abrasive load ratio R (−) d) e)

Figure 6.18: Momentum transfer efficiency versus abrasive load ratio

As indicated in [Susu06], an increase of the focusing tube diameter leads to an enlargement of the momentum transfer efficiency (Figure 6.18c). This is caused by the increase of the abrasive

78 waterjet reaction force when the nozzle diameter increases (see Subsection 6.4.2.1).

The effect of the abrasive size on η is shown in Figure 6.18d. With the increase of the abrasive size the momentum transfer efficiency decreases. The reason of that is the reaction force of the abrasive waterjet decreases with the increase of the abrasive size (see Subsection 6.4.2.1). Also, with large orifice diameters, the abrasive waterjet reaction force (Subsection 6.4.2.1) and therefore the momentum transfer efficiency is nearly not affected by the abrasive size (Figure 6.18e).

• Regression analysis:

1 0.9

0.85 0.9 0.8 0.8 0.75

0.7 0.7

0.65 0.6 0.6 Calculated momentum Calculated momentum transfer efficiency (−) 0.5 GMA garnet #80 transfer efficiency (−) 0.55 Olivine #60

0.4 0.5 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 0.6 0.7 0.8 0.9 Experimental momentum Experimental momentum transfer efficiency (−) transfer efficiency (−) Figure 6.19: Experimental and calculated Figure 6.20: Verification of the momentum momentum transfer efficiency transfer efficiency model

By conducting a regression analysis on the experimental data (consisting of 685 individual measurements), the coefficients c1 and c2 (Equations 6.54 and 6.56), and the momentum transfer efficiency (Equation 6.50) are determined as follows:

−0.21 0.09 cddpp1 =⋅0.3151()()ori / f ⋅ w / a (6.57)

0.2 −0.24 cddR2 =−0.6817 ⋅()pf / ⋅ (6.58)

0.2 −0.21 0.09 0.76 η =⋅0.3151()()ddori / f ⋅ pp w / a −⋅ 0.6817( dd p / f ) ⋅ R (6.59)

In which, dp is the average diameter of the new abrasive particles, determined by [Momb98]:

−1.0315 dmeshp =⋅17.479 (6.60)

Figure 6.19 shows the correlation between the experimental and the calculated momentum transfer efficiency (with R2=0.88). For verification of the model, experiments with the same setup were carried out with Olivine #60 and GMA garnet #80. The experimental data fit well with the calculated momentum transfer efficiency (R2=0.86) (Figure 6.20). From these results, although the

79 model is built based on the experimental data for Barton garnet, it can be used to determine the momentum transfer efficiency when cutting with other abrasives such as Olivine and GMA garnet.

p =360 MPa; d =0.25 mm; #80 HPX w ori 0.1

0.08

0.06

0.04 d =0.76 mm f d =1 mm 0.02 f d =1.27 mm

Power transfer efficiency (−) f

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Abrasive load ratio (−)

Figure 6.21: Power transfer efficiency with different nozzle diameters

6.4.2.4 Conclusions

In this section, the extension of the abrasive waterjet model has been done by investigation of the reaction force of the abrasive waterjet and by modeling the momentum transfer efficiency.

In the extension model, the effects of jet-parameters, e.g. the water pressure, the orifice diameter, the nozzle diameter etc., have been taken into account. Also, the influence of the abrasive particle diameter has been taken into account. Moreover, the model has been verified with different types of abrasives. The power transfer efficiency (see Equation 6.11 and Figure 6.21) and therefore the abrasive waterjet model become more accurate. The model can be used for different size and types of abrasives.

The model allows the development of an accurate AWJ cutting model, which can be found by the incorporation of the discharge coefficient model, the momentum transfer model, and the cutting efficiency model.

6.4.3 Abrasive - work material interaction modeling

The abrasive – work material interaction model is used to identify the relation between the work material characteristics, the abrasive characteristics and the cutting efficiency in the process of using the kinetic energy of the abrasive particles to remove the chips of the work material. The model is an extension of the cutting efficiency model in [Hoog00]. In the model, the effect of the process parameters, the effect of work materials as well as the influence of the abrasive materials on the cutting efficiency is taken into account.

80 6.4.3.1 Introduction

As mentioned in Section 6.3, the volume removal rate of a work material is determined as follows [Hoog00]:

ξ ⋅ Pabr Qmat = (6.13 repeated) ec

Where, Pabr is the power of abrasive particles (W) (see Equation 6.10), ec is the specific cutting energy (J/m3), and ξ is the cutting efficiency coefficient considering the energy loss in the material removal process.

Hoogstrate [Hoog00] found that in AWJ machining, the specific cutting energy ec of a work material equals the specific melting energy. From this, the author calculated the specific cutting energy of work materials (see Table 6.2). A relation between the specific cutting energy and the machinability number Nm defied by Zeng [Zeng92] was also established (see Equation 6.16).

It is noted that the cutting efficiency coefficient ξ is affected by the feed speed and the abrasive types. Also, the cutting efficiency depends on the ratio of the hardness and the toughness of the abrasive and the work material [Hoog00]. However, as the cutting efficiency was not the main objective of his study, only the feed speed v f was taken into account for the calculation of the coefficient ξ (see Equation 6.15).

In practice, the energy dissipation as well as the cutting efficiency depend not only on the above parameters but also on many other process parameters such as the nozzle diameter, the abrasive mass flow rate, the water pressure etc. [Momb99]. Therefore, a more reliable model for calculation of the cutting energy efficiency which takes these effects into account is needed. In the present study such a model has been developed as shown in the following sub-sections. The model is extended from the existing model of Hoogstrate [Hoog00].

6.4.3.2 Proposed model for abrasive – work material interaction

From Equation 6.13, the cutting efficiency coefficient can be calculated as follows: eQ⋅ ξ = cmat (6.61) Pabr

Using the same assumptions as those in [Hoog00], viz. the cutting width is constant over the depth of cut and equals the abrasive jet diameter dawj which is assumed to be the same as the nozzle diameter df, the volume removal rate of work material can be determined by:

Qmat= dh f⋅⋅max v f (6.62)

Substituting (6.62) into (6.61) gives:

81 edh⋅⋅ ⋅ v ξ = cfmax f (6.63) Pabr

It follows from Equations 6.61 and 6.10 that the cutting efficiency coefficient ξ is a function of momentum transfer efficiency η, which depends on various parameters such as the water pressure, the orifice diameter, the nozzle diameter, the abrasive mass flow rate and the particle diameter [Pi07b]. Consequently, the cutting efficiency coefficient ξ can also be expressed as a function of these parameters.

As already mentioned in Section 6.3, the types and the shape of the abrasives as well as the ratio of hardness and toughness of the abrasive to the work material also affect the cutting efficiency [Hoog00]. This effect can be expressed through the effect of the abrasive type, abrasive particle diameter, and the work material.

From the above arguments, the general function for the cutting efficient coefficient can be expressed as follows:

ξ = fv()fworifp,, p d ,,,,, d d m ama k k (6.64)

In which, km is the work material coefficient considering the effect of the work material on the cutting efficiency; ka is the abrasive material coefficient considering the effect of the abrasive type on the cutting efficiency.

In a dimensionless form, the function of ξ reads:

kk12k 3 k 4 ⎛⎞⎛⎞vpd ⎛⎞d ⎛ ⎞ ξ =⋅⋅kkfw ⋅p ⋅ori ⋅ ⋅ Rk 5 (6.65) am⎜⎟⎜⎟⎜⎟ ⎜ ⎟ ⎝⎠⎝⎠vduf⎝⎠ d f ⎝ p a ⎠

Where, vu is the unit velocity (vu =1 m/s); pa is the ambient pressure (pa = 1 at or 101325.01 Pa); Rmm= aw/ is the abrasive load ratio.

Choosing ka=1 for Barton garnet, Equation 6.65 can be rewritten as follows:

ξ =⋅kaBaξ (6.66)

In which, ξBa is the cutting efficiency coefficient when cutting with Barton garnet:

kk12k 3 k 4 ⎛⎞⎛⎞vpd ⎛⎞d ⎛ ⎞ ξ =⋅kRfw ⋅p ⋅ori ⋅ ⋅k 5 (6.67) Ba m ⎜⎟⎜⎟⎜⎟ ⎜ ⎟ ⎝⎠⎝⎠vduf⎝⎠ d f ⎝ p a ⎠

As the cutting efficiency ξBa is calculated by Equation 6.63, the coefficients km through k5 in

Equation 6.67 can be determined by the regression analysis based on the experimental data. The abrasive material coefficient ka will be predicted in the Subsection 6.4.3.4.

82 6.4.3.3 Cutting efficiency experiments

For modeling the cutting efficiency coefficient 209 data from [Zeng92] were used. The data were obtained from experiments which were done for Al6061T6 with the process parameters in Table 6.5.

Table 6.5: Experimental parameters for Zeng’s data

Parameter Unit Level number Values

Orifice diameter mm 6 0.229; 0.305; 0.356; 0.381; 0.406 and 0.457 Nozzle diameter mm 5 0.762; 1.016; 1.27; 1.524 and 1.778 Water pressure MPa 5 138; 173; 207; 242 and 276 Abrasive mass flow g/s 9 3.02; 3.78; 4.54; 5.76; 6.05; 7.56; 9.45; rate 11.34 and 13.23 Feed speed mm/s 5 2; 3; 4; 5 and 6 Abrasive 2 #80 and #120 HPX

As can be seen from above setting, experiments from [Zeng92] were conducted with only two abrasive sizes (#120 and #80 HPX) and with low values of the water pressure (less than 280 MPa). In order to investigate the effect of abrasive sizes as well as of higher water pressure further experiments were carried out.

Table 6.6: Experimental parameters for the cutting efficiency

Parameter Unit Level number Values

Orifice diameter mm 2 0.175 and 0.255 Nozzle diameter mm 1 0.92 Water pressure MPa 2 300 and 360 Abrasive mass flow g/s 9 0.73; 1; 1.5; 2.5; 3.25; 4.17; 4.53; 5 and 7.5 rate Feed speed mm/s 10 0.7; 1; 1.2; 1.5; 1.8; 2; 2.2; 2.5; 3 and 3.5 Abrasive 4 #50; #80; #120 and #150 HPX

Figure 4.5a and 4.5b respectively show the setup and the cutting sample of the experiment for the maximum depth of cut in order to determine the cutting efficiency coefficient according to Equation 6.63. The process parameters used in the experiments (for the work material Al6061T6) were shown in Table 6.6.

Instead of doing a full factorial experimental design at all levels (2x1x2x9x10x4=1440 measurements), a selected subset of 120 individual cutting tests from the full factorial was chosen to perform. This is because testing the full design requires a lot of time-consuming. Also, the process parameters should be combined properly with each other such as the large size of

83 abrasives should be tested with large orifice, small diameters of orifice should be used for small abrasive mass flow rates, a large feed speed should be used with the low water pressure, etc.

6.4.3.4 Results and discussions

• Relations between process parameters and cutting efficiency

There are several process parameters that affect the cutting efficiency such as the feed speed, the abrasive size, the water pressure, the abrasive load ratio, and so on. The effects of these parameters are discussed as follows (see also Figure 6.22 and 6.23).

Full jet can be used to cut Fraction of jet can be used to cut

Abrasive waterjet Work material Figure 6.22: Influence of the velocity on the cutting efficiency [Hoog00]

It is known that high feed speeds cause strong bending of the front of the kerf [Hoog00]. The total cross section of the jet is therefore in contact with the work material over a short depth of cut (Figure 6.22). In contrast, low feed speeds result in a nearly straight cutting front, which causes more energy loss in the jet itself because of the collisions among the abrasive particles and the internal friction in the jet. Hence, in general, the cutting efficiency increases with the increase of the feed speed (or the ratio of the feed speed to unit speed) (Figure 6.23a).

Figure 6.23b shows the effect of the ratio of abrasive particle to nozzle diameter on the cutting efficiency. With an increase of the abrasive particle diameter (or an increase of the ratio of abrasive particle to nozzle diameter) the momentum transfer efficiency decreases [Pi07b]. This leads to a reduction of the power of abrasive particles (see Equation 6.10) and therefore a decrease of the cutting efficiency (see Equation 6.63).

84 0.7 0.7

0.6 Experimental 0.6 Experimental Predicted Predicted

0.5 0.5

0.4 0.4

0.3 0.3

Cutting efficiency coefficient (−) 2 2.5 3 3.5 0.1 0.15 0.2 0.25 0.3 0.35

Ratio of feed speed −3 Cutting efficiency coefficient (−) Ratio of abrasive particle to unit speed v /v (−) x 10 to nozzle diameter d /d (−) f u p f a) b) 0.7

0.6 Experimental Predicted

0.5

0.4

0.3 1500 2000 2500 3000 3500

Cutting efficiency coefficient (−) Ratio of water pressure to ambient pressure p /p (−) w a c) 0.7 0.7

Experimental 0.6 0.6 Predicted

0.5 0.5

0.4 Experimental 0.4 Predicted

0.3 0.15 0.2 0.25 0.3 0.35 0.4 0.3 Ratio of orifice 0.1 0.15 0.2 0.25 0.3 0.35 Cutting efficiency coefficient (−) to nozzle diameter d /d (−) Cutting efficiency coefficient (−) Abrasive load ratio R (−) ori f d) e) (data from [Zeng92]) Figure 6.23: Relation between effected factors and cutting efficiency

85 It is clear that both the power of the abrasive particles and the maximum depth of cut increase as the water pressure (or the ratio of water pressure to ambient pressure) increases. However, the rate of the increase of the maximum depth of cut always prevails. Consequently, the cutting efficiency decreases with the increase of the water pressure (Figure 6.23c).

The effect of the abrasive load ratio on the power transfer efficiency is shown in Figure 6.4. It is known that there is an optimum value of the abrasive load ratio (or the abrasive mass flow rate) for the maximum power transfer efficiency (Figure 6.21) and therefore for the power of abrasive particles. Besides, there is an optimum abrasive mass flow rate (or abrasive load ratio) for the maximum depth of cut (see Subsection 2.1.3). Theoretically, with a certain setup, the optimum value of the abrasive ratio for the maximum power of abrasive particles is the optimum value for the maximum depth of cut. Also, the effects of the abrasive load ratio on the abrasive particle power and on the maximum depth of cut have the same trend. However, as the loss of the power during the mixing process, the effect of the abrasive load ratio on the maximum depth of cut is less than that on the power of abrasive particles. From this and from Equation 6.63, the cutting efficiency decreases with the increase in the abrasive load ratio (see Figure 6.23d).

Figure 6.23e shows the relationship between the ratio of orifice to nozzle diameter and the cutting efficiency. It is observed that there is an optimum value of the ratio for the cutting efficiency. This is because there exists an optimum value of the ratio of the orifice to nozzle diameter (in this case it was 0.305) to achieve the maximum depth of cut [Blic90]. Although this effected trend is not taken into the model (Equation 6.65) it is not a problem as the effect of the ratio on the cutting efficiency is not dominant (see Figure 6.23e).

• Regression analysis

0.8

0.7 This study [Zeng92] 0.6

0.5

0.4 Calculated cutting 0.3 efficiency coefficient (−)

0.2 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Experimental cutting efficiency coefficient (−)

Figure 6.24: Experimental and calculated cutting efficiency

86 The coefficients in Equation 6.67 are determined using a regression analysis of the data from 329 cutting tests including 209 tests for Barton garnet from [Zeng92] and 120 tests for Barton garnet from this study (see Subsection 6.4.3.3). Equation 6.67 can be rewritten as follows:

0.254−− 0.15550.3104 0.2318 ⎛⎞vp ⎛d ⎞⎛⎞d ⎛ ⎞ ξ =⋅kRfw ⋅p ⋅ori ⋅ −0.1236 (6.68) Ba m ⎜⎟ ⎜ ⎟⎜⎟ ⎜ ⎟ ⎝⎠vduf ⎝ ⎠⎝⎠ d f ⎝ p a ⎠

Where, km=12.056 (cutting with Al6061T6).

Figure 6.24 shows the correlation between the experimental and the calculated cutting efficiency (with R2=0.91).

• Determination of the work coefficient

From Equation 6.68, the work material coefficient km can be expressed by:

0.254−− 0.15550.3104 0.2318 ⎡⎤⎛⎞vp ⎛d ⎞⎛⎞d ⎛ ⎞ kR=⋅⋅⋅ξ / ⎢⎥⎜⎟fw ⎜p ⎟⎜⎟ori ⎜ ⎟ −0.1236 (6.69) mBa⎢⎥⎜⎟vd ⎜ ⎟⎜⎟ d ⎜ p ⎟ ⎣⎦⎝⎠uf ⎝ ⎠⎝⎠ f ⎝ a ⎠

From Equations 6.63 and 6.69, the following procedure for finding the work material coefficient for a work material is given:

-Conducting cutting tests: cutting tests should be done with different process parameters. The more numbers of cuts the higher the accuracy of the work material coefficient will be.

-Calculating the cutting efficiency coefficient by Equation 6.63;

-Calculating the work material coefficient km by Equation 6.69. The average value of km of all cutting tests is the work material coefficient.

To find the work material coefficient of SS304, 90 cutting tests with the same setup for Al6061T6 (see Section 6.4.3.3) were conducted.

For determining the work material of AD99.5, SS316L, mild steel, and titanium data from previous studies ([Zeng92], [Ma93]) were used. These data were obtained from the cutting tests with the following process parameters: the water pressure from 102 to 355 MPa, the orifice diameter from 0.152 to 0.457 mm, the nozzle diameter from 0.762 to 1.86 mm, the abrasive mass flow rate from 1.36 to 13.23 g/s, and the feed speed from 0.42 to 6 mm/s.

Based on the data from the above mentioned studies, the work material coefficients of various work materials are predicted (see Table 6.7).

87 Table 6.7: Work material coefficient of different engineering materials

Work material Source Number of cuts km

AD99.5 [Zeng92] 12 14.561 SS304 This study 90 14.287 SS316L [Zeng92] 13 13.77 Mild steel [Ma93] 72 13.42 Titanium [Ma93] 20 13.477 Al6061T6 This study and [Zeng92] 329 12.056

It is found that there is a relation between the machinability number Nm (see Subsection 6.2.2) and the work material coefficient (Figure 6.25). This relation can be described by the following equation (with R2=0.91):

kNmm=−14.86 0.0128 (6.70) (−) m 20 AD99.5 SS304 15 Al6016T6 SS316L Titanium 10 Mild steel

5 k =14.86−0.0128N m m

R2=0.91 0 0 50 100 150 200 250 Machinability number N (−) m

Work material coefficient k

Figure 6.25: Machinability number versus work material coefficient

For verification of the model (Equations 6.68 and 6.70), 42 cutting test data from a previous study [Chal91] were used. These data were obtained from the experiments for cutting mild steel with various process parameters: 1 size of Barton garnet (#80 HPX), 1 level of the water pressure (310 MPa), 1 level of the feed speed (2.5 mm/s), 9 levels of the orifice diameter (0.2, 0.25, 0.28, 0.3, 0.35, 0.38, 0.46, 0.53, and 0.61 mm), 3 levels of the focusing tube diameter (0.762, 1.14, and 1.65 mm), and 9 levels of the abrasive mass flow rate (1.83, 3.75, 5.65, 7.56, 9.5, 11.4, 13.25, 15, 18.93 g/s).

Figure 6.26 shows a rather good correlation (R2=0.85) between the experimental and the predicted cutting efficiency using the model.

88 0.7

0.65

0.6

0.55

0.5

Calculated cutting 0.45 efficiency coefficient (−) 0.4 0.4 0.45 0.5 0.55 0.6 0.65 Experimental cutting efficiency coefficient (−)

Figure 6.26: Verifying of the cutting efficiency coefficient model

• Determination of abrasive material coefficient

From Equation 6.63, the maximum possible depth of cut can be calculated as follows:

Pabr hmax =⋅ξ (6.71) edvcff⋅⋅ Substituting (6.66) into (6.71) gives

Pabr hkmax =⋅⋅aBaξ (6.72) edvcff⋅⋅ And results in:

hmax ka = (6.73) hmax,Ba

With

Pabr hmax,Ba=⋅ξ Ba (6.74) edvcff⋅⋅

In which, hmax,Ba is the maximum depth of cut when cutting with Barton garnet; ξBa is the cutting efficiency coefficient when cutting with Barton garnet determined by Equation 6.68.

From Equation 6.73, the following procedure for determination of the abrasive material coefficient is proposed:

-Carrying out experiments for the maximum depth of cut for the used abrasive. The experiments should be done for different process parameters.

89 -Determining the abrasive material coefficient ka by Equation 6.73. The average value of ka of all cutting tests is the coefficient.

As mentioned in Section 4.2, in practice, most of the AWJ shops (90%) use garnet, followed by olivine (15% of the shops). Among garnet types, Barton and GMA garnet are the most common abrasives. Therefore, GMA garnet and olivine have been chosen for determining their abrasive material coefficients in the present study.

To determine the abrasive material coefficient of GMA garnet and olivine, experiments with the same setup as that for Barton garnet (see Subsection 6.4.3.3) were conducted.

For GMA abrasives, 166 cutting tests were performed with the following process parameters: three types of GMA garnet (#50, #80, and #120), 2 levels of water pressure (300 and 360 MPa) 2 levels of the orifice diameter (0.175, and 0.255 mm), one level of the nozzle diameter (0.92 mm), 8 levels of the abrasive mass flow rate (0.83, 1.67, 2.5, 3.33, 4.17, 5, 6.67, and 7.5 g/s), 6 levels of the feed speed (0.7, 1, 1.5, 2, 2.5 and 3 mm/s), and two types of work materials (Al6061T6 and SS304).

0.6 0.5

Al6061T6 Al6061T6 SS304 SS304 0.5 0.45

0.4 0.4

0.3 0.35 Calculated cutting Calculated cutting efficiency coefficient (−) efficiency coefficient (−)

0.2 0.3 0.2 0.3 0.4 0.5 0.6 0.3 0.35 0.4 0.45 0.5 Experimental cutting Experimental cutting efficiency coefficient (−) efficiency coefficient (−) a) Cutting with GMA garnet b) Cutting with Olivine Figure 6.27: Experimental and calculated cutting efficiency coefficient

For olivine, 56 cutting tests were carried out with two types of olivine (#60 and #90), 5 levels of abrasive mass flow rate (0.83, 1.67, 3.33, 4.17, and 5 g/s), and 4 levels of the feed speed (0.7, 1, 1.5, and 2 mm/s). Other process parameters were the same as those used for GMA garnet tests.

The calculated results give the coefficient ka=0.92 for GMA garnet and ka=0.96 Olivine. With this value, the predicted cutting efficiency fit well with the experimental cutting efficiency for both GMA garnet (with R2=0.83) and Olivine (with R2=0.80) (Figure 6.27).

From Equations 6.66 and 6.68, with the abrasive material coefficient, the cutting efficiency

90 coefficient can be expressed by the following equation:

0.254−− 0.15550.3104 0.2318 ⎛⎞vp ⎛d ⎞⎛⎞d ⎛ ⎞ ξ =⋅⋅kkfw ⋅p ⋅ori ⋅ R−0.1236 (6.75) am⎜⎟ ⎜ ⎟⎜⎟ ⎜ ⎟ ⎝⎠vduf ⎝ ⎠⎝⎠ d f ⎝ p a ⎠

In which, ka=1 for Barton garnet; ka=0.92 for GMA garnet and ka=0.96 for Olivine.

6.4.3.5 Conclusions

In this section, an abrasive – work material interaction model has been proposed. The model is based on the extension of the cutting efficiency model which has been found by combining the physical-mathematical model and experimental analyses. In the model, the effects of the process parameters such as the water pressure, the orifice diameter, the nozzle diameter, the abrasive mass flow rate etc. were taken into account. Also, the influence of the abrasive sizes is determined.

By introducing the work material coefficient, the effect of work materials on the cutting efficiency has been investigated. In addition, since the relation between the machinability number and the work material coefficient has been found, the cutting efficiency model can be used for various work materials.

By introducing the abrasive material coefficient, the model can be applied for different types of abrasives.

The model enables to develop a more reliable AWJ cutting model, which can be derived by the integration of the pure waterjet model, the abrasive waterjet model and the abrasive – work material interaction model.

6.5 Modeling the AWJ cutting process

As addressed in Section 6.4, the AWJ cutting process model, extended from Hoogstrate’s model, includes of three sub-models: the pure waterjet model, the abrasive waterjet model, and the abrasive – work material interaction model. The maximum possible depth of cut can be calculated by the following equation [Hoog00]:

Pabr hmax =⋅ξ (6.76) edvcff⋅⋅ For the sake of consistency, the determinations of the parameters are summarized as follows:

The power of abrasive particles Pabr is determined according to [Hoog00]:

R π 2 Pcpd=⋅η 23/22 ⋅⋅⋅ ⋅ ⋅ (6.10 repeated) abr2 d4 ρ w ori ()1 + R w

91 The coefficient of discharge cd is calculated by the following equation of the first sub-model:

−−0.2343 1 1 / n ⎛⎞ρ ⋅⋅vd E ⎡⎤ ⎛⎞np⋅ c =⋅10.9638⎜⎟0,wth ori ⋅w 0 ⋅+−⎢⎥ ⎜⎟ 1w 1 (6.37 repeated) d ⎜⎟μ pn−1 ⎢⎥ ⎜⎟E ⎝⎠00w ()⎣⎦ ⎝⎠w

The momentum transfer efficiency η is determined using the second sub-model:

0.2 −0.21 0.09 0.76 η =⋅0.3151()()ddori / f ⋅ pp w / a −⋅ 0.6817( dd p / f ) ⋅ R (6.59 repeated)

The cutting efficiency coefficient ξ is calculated by the third sub-model:

0.254−− 0.15550.3104 0.2318 ⎛⎞vp ⎛d ⎞⎛⎞d ⎛ ⎞ ξ =⋅⋅kkfw ⋅p ⋅ori ⋅ R−0.1236 (6.75 repeated) am⎜⎟ ⎜ ⎟⎜⎟ ⎜ ⎟ ⎝⎠vduf ⎝ ⎠⎝⎠ d f ⎝ p a ⎠

In which, ka is the abrasive material coefficient; ka=1 for Barton garnet; ka=0.92 for GMA garnet and ka=0.96 for Olivine. km is the work material coefficient determined by the following equation:

kNmm=−14.86 0.0128 (6.70 repeated)

70 50 AD99.5 [Zeng92] SS316L [Zeng92] 60 Al6061T6 Mild steel [Ma93] SS304 40 Titanium [Ma93] 50 Al6061T6 [Zeng92] (mm) (mm) 30 40 max,cal 30 max,cal 20

20 of cut h

of cut h 10 10 Calculated maximal depth Calculated maximal depth

0 0 0 10 20 30 40 50 60 70 0 10 20 30 40 50 Experimental maximal depth Experimental maximal depth of cut h (mm) of cut h (mm) max,ex max,ex a) b) Figure 6.28: Verification of the maximum depth of cut model

To verify the above cutting process model for Barton garnet abrasives, two experimental data sets were used: one from 210 tests of the present study including 120 tests for Al6061T6 (see Sub- section 6.4.3.3) and 90 tests for SS304 with the same set up, and one from 326 tests of previous studies including 12 tests for AD99.5 [Zeng92], 13 tests for SS316L [Zeng92], 209 tests for Al6061T6 [Zeng92], 20 test for titanium [Ma93], 72 tests for mild steel [Ma93] (see Subsection 6.4.3.4). Figure 6.28 shows the calculated versus the experimental maximum depths of cut for both data sets. The results indicate that the model predictions are in very good agreement with the

92 experimental data (R2=0.97 for the present data and R2=0.98 for the data from [Zeng92], [Ma93]).

To verify the cutting process model for other abrasives, 166 data for GMA garnet and 56 data for olivine were used (see Subsection 6.4.3.4). Figure 6.29 shows that the predicted maximum depth of cut fit very well with the experimental depth of cut for both GMA garnet (with R2=0.97) and Olivine (with R2=0.99).

60 60

Al6061T6 Al6061T SS304 50 SS304

40 40 (mm) (mm)

30 max,cal max,cal

20 20 of cut h of cut h 10 Calculated maximal depth Calculated maximal depth 0 0 0 20 40 60 0 20 40 60 Experimental maximal depth Experimental maximal depth of cut h (mm) of cut h (mm) max,ex max,ex a) Cutting with GMA garnet b) Cutting with Olivine

Figure 6.29: Experimental and calculated maximum depth of cut for cutting with different abrasives

70 70 #150 HPX; Al6061T6; d =0.92 mm f #80 HPX; Al6061T6 60 60

50 50 (mm) 40 (mm) 40 max,cal

30 max,cal 30

20 20 of cut h Extended model of cut h 10 10 Extended model Calculated maximal depth Hoogstrate’s model Zeng’s model Calculated maximal depth 0 0 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 Experimental maximal depth Experimental maximal depth of cut h (mm) of cut h (mm) max,ex max,ex a) b) Figure 6.30: Experimental and calculated maximum depth of cut when calculating with different cutting process models

93 For comparing the extended model with other AWJ cutting process models, Figure 6.30 shows the experimental and the predicted maximum depth of cut when calculating with different models. The comparison between the extended model and Hoogstrate’s model (see Section 6.3) is done with 32 data of cutting with #120HPX abrasive and the work material of Al6061T6 (see Figure 6.30a). It is observed that the extended model is more accurate than the Hoogstrate’s model (R2=0.94 and R2=0.88 for the extended model and Hoogstrate’s model, respectively). The reason could be that in this case, the tests were done with the nozzle diameter of 0.9 mm and Hoogstrate’s model is based on data when cutting with only one value of the nozzle diameter (0.8 mm). Also, as mentioned in Section 6.3, in Hoogstrate’s model, the coefficient of discharge is constant and only the effect of the feed speed has been taken into the model for the momentum transfer efficiency.

The comparison between the extended model and Zeng’s model (see Section 6.2) was shown in Figure 6.30b. In this case, 58 data when cutting with #80HPX and the work material of Al6061T6 with different jet parameters were considered. The results calculated by the extended model (with R2=0.95) is much better than that when calculated with Zeng’s data (with R2=0.57). It could be explained that as Zeng’s model has been built based on the experiments with water pressure less than 280 MPa (see Section 6.2), it may not be suitable for the high values of the water pressure (in this case the water pressure values range from 300 MPa to 360 MPa).

6.6 Conclusions

In this chapter, Hoogstrate’s model (or the cutting process model) has been extended by improvements of three sub-models, i.e. the pure waterjet model, the abrasive waterjet model, and the abrasive – work material interaction model. The model is more reliable and can be used for a wide range of applications.

The effects of various process parameters, such as the water pressure, the abrasive mass flow rate, the orifice diameter, the nozzle diameter, etc. are investigated. Also, the influences of the abrasive sizes on the maximum depth of cut are taken into account.

The interaction of the abrasive and the work material in the cutting process is investigated by introducing the work material coefficient and the abrasive material coefficient. Hence, the model can be applied for various work materials as well as for different types of abrasives.

As the model predicts the optimum trend of the effect of the abrasive mass flow rate on the maximum depth of cut (see Section 6.3), it can be used for the AWJ optimization problem.

94 7 Optimization in AWJ cutting process

This chapter deals with the optimization in AWJ cutting. First, a cost and profit analysis is carried out. Many cost elements such as machine tool cost, orifice cost, nozzle cost, abrasive cost etc. are taken into account. Then optimizations are performed in order to get optimum nozzle lifetime and optimum abrasive mass flow rate. Finally, solutions to select optimally other jet parameters are suggested.

7.1 Cost and profit analysis

As mentioned in Chapter 5, AWJ economical optimization problems consist of two objective functions including the minimum total cutting cost and the maximum profit rate. Consequently, to perform the economical optimizations it is necessary to have the analysis of the cost and profit in AWJ machining. Hence in this section a cost analysis and a profit analysis are carried out.

7.1.1 Cost analysis

In practice, the cutting cost per unit length is normally used for evaluating the cost effectiveness of an AWJ cutting process (see Section 2.2). The cutting cost per length can be calculated from the cutting cost per hour and the feed speed [Hoog06]. Therefore, in order to solve the economical optimization problems, we need to determine the AWJ cutting cost per hour and thereafter the AWJ cutting cost per unit length.

7.1.1.1 AWJ cutting cost per hour

In AWJ machining, the total cutting cost per hour Ch (€/h) when cutting with multiple jet formers can be determined as follows [Karp04]:

CChfixhh=+,var, C (7.1)

Where Cfix,h is the fixed cost per hour (€/h) including the operation independent cost components;

Cvar,h are the variable costs per hour (€/h) including the operation dependent cost components.

The fixed cost per hour Cfix,h consists of the hourly machine tool cost Cmt, h (€/h) and the wages including overhead cost per hour Cwa,h (€/h). Cfix,h is calculated by the following equation:

95 CCCfix,,, h=+ mt h wa h (7.2)

The variable cost includes all direct cost of consumables such as orifices, focusing tubes, water and abrasives. The cost of maintenance based on consumables will be included in the machine tool cost [Hoog06].

CCCCCvar,,,h=+++ orihfhwhah ,, (7.3)

In which, Cori,h is the orifice cost per hour (€/h), Cf,h is the nozzle cost per hour (€/h), Cw,h is the water cost per hour (€/h), and Ca,h is the abrasive cost per hour (€/h).

Substituting (7.2) and (7.3) into (7.1) gives:

CCh=+++++ mthwahorihfhwhah,,,,,, C C C C C (7.4)

In the next sub-sections each of the cost elements will be addressed in more detail.

7.1.1.2 Hourly machine tool cost

The hourly machine tool cost Cmt,h is determined by [Karp04]:

CCCCde,, y+++ in y ro , y ma , y + C en , y C mt, h = (7.5) Tuse

Where, Cde,y is the annual cost of depreciation (€/y), Cin,y is the annual cost of interest (€/y), Cro,y is the annual cost of occupied room (€/y), Cma,y is the annual cost of maintenance (€/y), Cen,y is the annual cost of energy (€/y), and Tuse is the annual time of use (h/y).

The above cost components can be calculated as follows [Karp04]:

CCTde, y= rpl/ tot (7.6)

CCxin,int y=⋅ rpl /2 (7.7)

CCAro, y=⋅ sqm mt (7.8)

CxCma, y=⋅ ma rpl (7.9)

CTCPden, y=⋅⋅⋅ use e tot op (7.10)

Tuse=⋅⋅⋅xtd sh sh wor x ut (7.11)

In which, Crpl is AWJ system’s replacement cost (€);Ttot is the number of years of depreciation (y);

2 xint is the rate of interest (1/y); Csqm is the annual room cost per squared meter (€/(m .y)); Amt is 2 the occupied area of the machine tool (m ); xma is the maintenance rate (usually xma= 38… %)

[Karp04]; Ce is the energy cost (€/kWh); Ptot is the installed machine tool power (kW); dop is the operation duration rate (%); xsh is the number of shifts per day; tsh is the duration of one shift

(h/d); dwor is the number of working days per year (d/y).

96 In Equation 7.11, xut is the utilization rate. In practice, the utilization rate is usually between 0.7 and 0.8 [Karp04]. When cutting with multiple jet formers, this rate decreases with the increase of the time for changing nozzles tcn (hours). In this case, the utilization rate can be calculated as follows [Hoo06]:

xtnLut=−⋅()0.7… 0.8 cn f / f (7.12)

Where, nf is the number of jet formers, Lf is the nozzle lifetime (h). The calculation of the nozzle lifetime will be discussed in Sub-section 7.2.1.

7.1.1.3 Wages including overhead cost per hour

One of the benefits of AWJ cutting is that it can operate safely without manual supervision for a long period of time. Hence, it is possible to reduce the cost of wages significantly by running unmanned shifts. This can be taken into account by introducing of the kmsh coefficient which is defined as [Hoog06]:

x msh kmsh = (7.13) x sh

Where, xmsh is the number of manned shifts per day.

The wages including overhead cost per hour, Cwa,h (€/h), can be determined as follows [Hoog06]:

kCC⋅+ ⋅⋅⋅ xtd Cwa, y msh() la,, h ov h sh sh wor Cwa, h == Txtdxuse sh⋅⋅ sh wor ⋅ ut

kCCmsh⋅+() la,, h ov h C wa, h = (7.14) x ut

In which, Cla,h is the labor costs per hour (€/h), and Cov,h is the overhead cost per hour (€/h).

7.1.1.4 Orifice cost per hour

The orifice wear cost per hour, Cori,h (€/h), is calculated by:

CnCLori,, h=⋅ f ori p/ ori (7.15)

Where, Cori,p is the orifice cost per piece (€/piece); Lori is the orifice lifetime (h).

7.1.1.5 Nozzle cost per hour

The nozzle wear cost per hour, Cf,h can be predicted as follows:

CnCLfh,,=⋅ f fp/ f (7.16)

Where, Cf,p is the nozzle cost per piece (€/piece).

97 7.1.1.6 Water cost per hour

The water cost per hour, Cw,h (€/h), can be determined by the following equation:

CnmCwh,,=⋅⋅⋅3600 f w wm (7.17) Where, Cw,m is water cost per kilogram (€/kg); mw is water mass flow rate (kg/s/jet former). Using

Bernoulli’s equation, the water mass flow rate is determined by:

π ⋅d 2 2 ⋅ p or w mcwdw=⋅⋅ρ ⋅ (7.18) 4 ρw

In which, ρw is water density (see Equation 6.25), and cd is the coefficient of discharge (see

Equation 6.37).

7.1.1.7 Abrasive cost per hour

The abrasive cost per hour, Ca,h (€/h), is calculated by:

CnmCah,,=⋅⋅⋅3600 f a am (7.19) Where, Ca,m is abrasive cost per kilogram (including disposal cost) (€/kg) and ma is the abrasive mass flow rate (kg/s). Substituting (7.15), (7.16), (7.17) and (7.19) into (7.4) gives:

⎛⎞CC CC=++⋅++⋅⋅+⋅⋅ C nor,, p f p 3600 mC 3600 mC (7.20) hmthwahf,,⎜⎟ wwm , aam , ⎝⎠LLor f

7.1.1.8 AWJ cutting cost per length

The cutting cost per meter length can be calculated from the cutting cost per hour by the following equation:

C h C l = (7.21) 3600 ⋅⋅nvffa,

Where, vf,a is the average feed speed (m/s).

Substituting (7.20) into (7.21) gives

1 ⎛⎞CC+⋅+⋅CC mCmC C =+++mth,, wahor,, p f p w wm , a am , (7.22) l ⎜⎟ 3600 ⋅vnLLfa, ⎝⎠ f or f v fa, It follows from Equation 7.22 that for the same set up (i.e. the same water pressure, the same abrasive mass flow rate, the same nozzle diameter, etc.) the variable cost per unit length is the same for both cutting with n f jet formers and cutting with one jet former. However, the fixed cost

98 per hour when cutting with multiple jet formers is n f times smaller than that when cutting with one jet former. Hence, the fixed cost and therefore the total cutting cost can be reduced significantly by increasing the number of the jet formers as noted in [Ande05].

The nozzle lifetime and the average cutting speed will be discussed in more detail in the Sub- section 7.2.1.

7.1.2 Profit analysis

Generally, the maximum profit rate or the profit per unit time is the main economical objective of a company. Therefore, as mentioned in Chapter 5, this is also the objective of the AWJ optimization problem in the present study. To perform the AWJ optimization for getting the maximum profit rate it is necessary to have a profit rate model. For that reason, this sub-section deals with the profit analysis in AWJ machining.

Theoretically, the profit or net income per product Prp can be calculated as follow:

Prp =−CCsal, p net (7.23)

Where, Csal,p is the sale price per product before tax (€/piece), Cnet is the net cost per product

(€/piece); Cnet can be calculated by the following equation [Karp04]:

CCCCnet=++ mtt pc Co&& De + C Ad Di (7.24)

In which, Cmtt is the material total cost (€/piece), CCo&De is the cost for concept and design (€/piece),

CAd&Di is the cost for administration and distribution (€/piece), and Cpc is the manufacturing cost per piece (€/piece). The manufacturing cost per piece Cpc can be calculated by the following equation [Ka04]:

CC CCC=+++pre rep (7.25) pcsusNB⋅ B sin

Where, Cpre is the preparation cost (€), Crep is the cost for repetition of order (€), Csin is the manufacturing single cost per piece (€/piece), Csus is the succeeding cost per piece (€/piece), N is the number of orders, B is the batch size (pieces per order).

Substituting (7.24) and (7.25) into (7.23) we have:

CC Pr =−−−CCCC −−−−pre rep CC (7.26) p sal,&& p mtt Co De Ad DiNB⋅ B sus sin

In AWJ machining, the cutting cost per unit length Cl (see Equation 7.22) is commonly used to calculate the economical problem. Therefore, we can rewrite Equation 7.26 for the profit per unit length of cutting Prl as follows:

99 CC Pr =−−−−−−−CCC Cpre rep CC (7.27) l sal,&& l mtt Co De Ad DiNB⋅ B sus l

In Equation 7.27, the cutting cost per unit length Cl depends on the cutting regime and it can be various. The rest cost components including the sale price per unit length before tax Csal,l, Cmtt, ,

CCo&De, CAd&Di, Cpre, Crep, and Csus are not affected by the cutting regime. However, they can also be CC different and the sum of CCC−−−−−− Cpre rep Ccan be expressed as sal,&& l mtt Co De Ad DiNB⋅ B sus follows:

CC CCC−−−−−−=⋅ Cpre rep CkC (7.28) sal,&& l mtt Co De Ad DiNB⋅ B sus p l ,0 And Equation 7.27 becomes

Prlpll=⋅kC,0 − C (7.29)

In which, Cl,0 is basic cutting cost per unit length (€/m) which is determined with a specific cutting regime; kp is a factor named the profit coefficient. The profit coefficient kp depends on various parameters such as the material total cost Cmtt, the cost for concept and design CCo&De, the cost for administration and distribution CAd&Di, etc. It also depends on the company business policy as it is affected by the sale price per unit length before tax Csal,l (see Equation 7.27).

From Equation 7.29, the profit rate or the profit per hour Prh (€/h) can be determined by the following equation:

Prhfapll=⋅⋅⋅− 3600 vkCC,,0() (7.30)

Where, vf,a is the average feed speed (m/s).

7.2 Optimization for determining optimum nozzle lifetime

In this section two optimization problems are investigated. The first problem (see Subsection 7.2.3) is to determine the optimum nozzle exchange diameter for minimum AWJ cutting cost per unit length, and the other problem (see Subsection 7.2.4) is to find it for maximum profit per hour. Based on the results of the optimization problem, the term “optimum nozzle lifetime” or “optimum nozzle diameter” is introduced in AWJ machining. By regression analyses, two models for calculations of the “optimum nozzle diameter” for getting minimum cutting cost and maximum profit rate are proposed. Also, the effectiveness of cutting with “optimum nozzle diameter” is presented.

To determine the optimum nozzle lifetime, considerations of the nozzle lifetime should be carried out. In addition, it is necessary to have a model for calculation of the nozzle wear. Therefore, the following subsection deals with the AWJ nozzle lifetime and the nozzle wear modeling.

100 7.2.1 Nozzle lifetime and nozzle wear in AWJ machining

In this subsection, considerations of the nozzle lifetime are presented. Thereafter, the modeling of the nozzle wear is carried out. Based on many test data from previous studies, a model for the nozzle wear rate is proposed. The model can be used for the optimization problem to find the optimum nozzle lifetime as well as for the prediction of the AWJ cutting regime.

7.2.1.1 AWJ nozzle lifetime considerations

(mm) 1.8 f

ROCTEC 100 1.6 ROCTEC 500

1.4

p =379 MPa; #80HPX w 1.2 m =453 g/min; l =76 mm a f d =0.33; d =1.07 mm ori f 1 0 50 100 150 200 Nozzle lifetime L (h) f

Nozzle exit bore diameter d

Figure 7.1: Nozzle exit bore diameter versus nozzle lifetime [Kenn06] In practice, the nozzle wear can be quantified by the increase of the nozzle exit bore diameter. It is noted that the relation between the growth of the exit bore diameter of a composite carbide nozzle and its lifetime is linear or almost linear [Pi05]. This linear trend is also observed in the data of many long term wear tests for composite carbide nozzles by various companies [Mort91]. Figure 7.1 illustrates an example of this trend according to [Kenn06]. Consequently, the nozzle lifetime can be determined as follows:

Lddfffdf=−()/,0 δ (7.31) Where, df,0 is the initial nozzle diameter (m); df is the exchange nozzle diameter (m); δdf is the nozzle exit bore wear rate (m/h). The nozzle wear modeling will be presented in the following subsection.

It follows from Equation 7.31 that an increase in the nozzle lifetime will result in a growth of the nozzle diameter. This leads to a decrease of the cutting quality. To maintain the quality, the feed speed should be reduced [Hoog06]. In practice, the nozzle lifetime depends on the policy of each company. Technically, the lifetime of a nozzle can be 100 hours or more. It is reported that for composite carbide nozzles, the average lifetime is approximately 96 hours [Mort95].

101 140

( /h) 120 f,h 100

0 0 10 20 30 40 50 60 70 Nozzle lifetime L (h) f Nozzle wear cost per hour C

Figure 7.2: Nozzle wear cost per hour versus nozzle lifetime (with Cf,p = 70 €/piece)

From the above analysis, a short nozzle lifetime allows cutting with high feed speed and therefore high productivity. However, with a nozzle cost per piece, the nozzle cost per hour is very high if the nozzle lifetime is very small (e.g. less than 10 h). In contrast, the nozzle wear cost per hour is very low if the nozzle lifetime is long ( L f ≥ 20 h) (Figure 7.2). Therefore, there will be an optimum value for the nozzle lifetime by which the total cutting cost will be minimum [Hoog06]. In addition, there will be a value of the nozzle lifetime at which function (kCp ⋅−ll,0 C) and therefore the profit rate is maximum (see Equation 7.30).

7.2.1.2 AWJ nozzle wear modeling

• Reviews on AWJ nozzle wear modeling

In an entrainment AWJ system, abrasive particles are mixed with a high-velocity stream of water for making the cutting beam. The velocity of abrasive particles in the nozzle depends on many parameters (e.g. water pressure, abrasive mass flow rate, orifice diameter and nozzle diameter) and it can reach 500 m/s or more [Himm91]. The high velocity particles generate rapid wear of the nozzles. The nozzle exit bore growth leads to a decrease of the feed speed and thus to an increase of AWJ cutting cost. The exit bore growth also affects the cutting precision as the kerf width increases. Consequently, the effects of jet-parameters on the nozzle wear and the nozzle wear modeling have been the objectives of many research activities.

Up to now, there have been several studies on the AWJ nozzle wear. K. A. Schwetz et al. [Schw95] carried out a study on the wear rate of boron carbide ceramics by abrasive waterjets at different impact angles. The effects of abrasive types such as olivine, garnet and alumina on the maximum

102 lifetime for AWJ nozzles (for B4C-5%C, 60 B4C - 40 TiB2 and hard metal WC-6%Co nozzles) were investigated [Schw95].

M. Hashish [Hash94] investigated the effects of the abrasive size, the orifice diameter, and the focus length on the nozzle exit wear rate. The influence of the nozzle materials on the nozzle wear was also included. It was concluded that the wear mechanisms along the nozzle change from erosion by the impact of particles at the upstream sections to abrasion at the downstream sections [Hash94].

In a later study, Hashish [Hash97b] summarized the effects of various jet parameters on the nozzle wear rate as shown in Table 7.1. Various wear patterns (both in cross section and axial wear patterns) are described.

Table 7. 1: Effects of jet parameters on the nozzle wear rate [Hash97b]

Parameter Effects on the nozzle wear rate

Water pressure pw -Wear rate varies linearly with the water pressure;

Orifice diameter dori -Wear rate increases rapidly with the increase of orifice diameter; 2 proportion to d ori ; -There is a specific abrasive mass flow rate for maximum wear Abrasive mass flow rate ma rate;

Initial nozzle diameter df,0 -Wear rate decreases with the increase of the initial nozzle diameter;

Nozzle length lf -Wear rate decreases with the increase of the nozzle length.

G. Mort [Mort91] studied the nozzle wear of ROCTEC 100 nozzles using various test data. He noted that the lifetime of the composite carbide nozzles is 20 times more than that of tungsten carbide / cobalt. The average of the lifetime of ROCTEC 100 nozzles is approximately 70 hours for cutting with abrasive garnet.

In another study, G. Mort [Mort95] noted that about 62% of AWJ shops use Boride ROCTEC composite carbide nozzles and 90% of the shops use abrasive garnet for cutting.

M. Nanduri et al. [Nand97 and Nand00] carried out experimental studies on the effect of nozzle geometry on the nozzle wear. The influences of inlet angle [Nand97], inlet depth, nozzle length, and nozzle diameter [Nand00] on the nozzle wear were derived graphically. It is noted that the nozzle wear strongly depends on the diameter ratio of orifice to nozzle and becomes maximum if the ratio is between 0.33 and 0.42. Hence, both the cutting performance and the nozzle wear should be taken into account in the AWJ cost optimization problem.

The effects of other parameters such as the water pressure, the abrasive mass flow rate, and the

103 orifice diameter on the wear of WC/Co nozzles were also investigated [Nand02]. It is observed that an increase of the abrasive mass flow rate leads to a linear rise of the nozzle weight loss rate and of the exit bore diameter. Also, the nozzle weight loss rate and the exit diameter wear rate can reach maximum values when the water pressure increases. Unfortunately, this trend was still not taken into account in their model for WC/Co nozzles [Nand02] as follows:

0.9 0.38 0.7 −4 pdworia m WN =×8.07 10 0.5 0.8 (7.32) dLff,0

Where, WN is the nozzle weight loss rate (g/min/mm).

In conclusion, the effects of various nozzle parameters and jet-parameters on the nozzle wear have been investigated for both boron carbide and tungsten carbide nozzle materials. It is apparent that composite carbide has become the most important nozzle material because of its long-lifetime. Also, garnet is the most common abrasive in AWJ cutting. However, the effects of jet-parameters, e.g. the water pressure, the abrasive mass flow rate, the orifice diameter etc. on the wear of composite carbide nozzles for cutting with abrasive garnet are still not well understood. Moreover, there is still a lack of a model for prediction of the nozzle wear of composite carbide nozzles.

In order to have a nozzle wear model, the following part deals with the modeling of the nozzle wear of composite carbide nozzles when cutting with abrasive garnet. The model can be used for determination of the nozzle lifetime and therefore for the AWJ optimization problem.

• Nozzle wear modeling

From Equation 7.31 the wear rate of composite carbide nozzles can be determined as follows:

dd ff− ,0 δdf = (7.33) Lf

To build a model for composite carbide nozzles, many long-term wear test data were analyzed. These tests were done with composite carbide nozzles (ROCTEC 100) when cutting with abrasive garnet # 80 with various jet parameters such as the water pressure, the abrasive mass flow rate, the initial nozzle diameter, and the orifice diameter. The exit bore growth rate with various jet- parameters was shown in Table 7.2.

It follows from Table 7.1 and Table 7.2 that the nozzle exit bore wear rate, δdf , is a function of jet parameters in general:

δdf= fp() w,,,, mdld a f f ori (7.34)

104 Table 7. 2: Jet parameters and nozzle exit bore wear rate

pw ma dori d f ,0 l f L f δdf Reference (MPa) (g/s) (mm) (mm) (mm) (hr) (mm/h)

A1 241.36 4.54 0.305 1.168 70 67 0.0038 A2 241.36 4.54 0.305 1.194 70 67 0.0038 A3 241.36 4.54 0.305 1.194 70 83 0.0040 B 324.05 3.78 0.254 0.762 76 40 0.0038 C1 303.37 7.56 0.356 1.473 70 70 0.0044 C2 303.37 7.56 0.356 1.473 70 70 0.0047 D 379.21 7.56 0.330 1.016 76 80 0.0053 E1 241.36 11.34 0.457 1.575 70 23 0.0077 E2 241.36 11.34 0.457 1.118 70 23 0.0088

Note: A1, A2, A3 - Trade-A-blade Indianapolis, IN (Job Shop #2) [Mort91]; B - Ingersoll-Rand, Baxter Spring, KS [Mort91]; C1, C2 - Sugino Corp. Schaumburg, IL [Mort91]; D - David G. Taggart et al. [Tagg97]; E1, E2 - Flow International Corp., Kent WA [Mort91].

−3 x 10 10

Calculated nozzle wear rate (mm/h) 2 4 6 8 10 −3 Actual nozzle wear rate (mm/h) x 10

Figure 7.3: Correlation between actual and predicted nozzle wear rate

By conducting a regression analysis of the nozzle wear test data (Table 7.2), the following equation is obtained [Pi07a]:

7 0.24 0.13 1.8 4.167×⋅ 10 pwaori ⋅md ⋅ δdf = 0.67 0.05 (7.35) dlff,0 ⋅

Figure 7.3 shows the correlation between the experimental and predicted nozzle wear rates. The data fit quite well with the calculated results (with R2=0.97).

105 • Effects of jet parameters on AWJ nozzle wear

There are many jet parameters that affect the nozzle wear rate such as the water pressure, the abrasive mass flow rate, the orifice diameter, the initial nozzle diameter, the nozzle length, and the abrasive load ratio on the exit bore wear rate. Based on the model calculation, the influences of these parameters are discussed as follows (see also Figure 7.4).

The relation between water pressure and the wear rate is almost linear as shown in Figure 7.4a which is in agreement with [Hash97b]. Figure 7.4b shows the relation between the abrasive mass flow rate and the nozzle wear rate. Like Hashish noted [Hash97b], with an increase of the initial nozzle diameter the wear rate decreases (Figure 7.4c). Also, the influence of the nozzle length on the wear rate is very small (Figure 7.4d). This is because the effect of the nozzle length (for nozzle length from 70 to 100 mm) on the water velocity as well as on the particle velocity is very small [Tazi96]. In contrast, the wear rate increases significantly with the increase of the orifice diameter

1.8 (proportion to d ori ) (Figure 7.4e).

• Conclusions

In this subsection, a model for prediction of the AWJ nozzle exit bore wear rate has been proposed. The model is based on various long term wear test data for composite carbide nozzles which are the most common nozzles in AWJ machining.

The effects of jet-parameters, e.g. the initial nozzle diameter, the orifice diameter, the abrasive mass flow rate, and the water pressure, on the wear rate have been taken into account.

The wear model can be used to determine the nozzle wear rate and the nozzle lifetime for the prediction of the AWJ cutting regime as well as for an AWJ optimization program.

7.2.2 Relation between the nozzle lifetime and the feed speed

As mentioned in Subsection 7.2.1, in AWJ cutting, the nozzle wear leads to an increase in the nozzle diameter and therefore the decrease of the cutting quality. As a result, the feed speed should be reduced in order to maintain the cutting quality [Hoog06]. It is impractical to have an ideal feed speed which will be changed gradually with the increase of the nozzle diameter. In practice, the feed speed is kept constant in a period of time or it is not even changed during the whole nozzle lifetime, depending on the company policy. Figure 7.5 shows the ideal feed speed and two policies for the feed speed: policy I with the feed speed vi is kept constant in every 8 hours, and policy II

* with the feed speed v i is not changed for every 40 hours.

If the feed speed is kept constant in every tck hours, the average feed speed is determined as follows:

106 −3 −3 x 10 x 10 10 10

8 8

6 6

4 4

2 Nozzle exit bore wear rate (mm/h) 200 250 300 350 400 2 Nozzle exit bore wear rate (mm/h) 200 400 600 800 1000 Water pressure p (MPa) w Abrasive mass flow rate (g/min) a) b)

−3 x 10 10

C) 6

Nozzle exit bore wear rate (mm/h) 2 0.6 0.8 1 1.2 1.4 Initial nozzle diameter d (mm) f

−3 −3 x 10 x 10 10 10

8 8

6 6

4 4

2 Nozzle exit bore wear rate (mm/h) 2 Nozzle exit bore wear rate (mm/h) 0.2 0.25 0.3 0.35 0.4 0.45 60 70 80 90 100 Nozzle length l (mm) Orifice diameter d (mm) f ori d) e) Figure 7.4: Jet parameters versus calculated nozzle wear rate

107 nnn vvttvnfa, =⋅∑∑∑ i ck// ck = i (7.36) iii===111

Where, tck is the time of keeping the same feed speed (tck =4; 6; 8…hours); n is the number of periods of keeping the feed speed constant; vi is the minimum feed speed in the period of time i (Figure 7.5).

3.5 v 1 3.4 v 2 3.3

3.2 v i (mm/s)

f 3.1

3 v n

2.9 * * v v 2 1 2.8 Feed speed v 2.7

2.6 0 20 40 60 80 Nozzle lifetime L (h) f

Figure 7.5: Feed speed versus nozzle lifetime

With Barton garnet #80 HPX; dori=0.38 mm; df,0=1.14 mm; SS 304; m a =13 g/s; hmax=25 mm; δdf =0.0072 mm/h; pw=350 MPa.

To determine vi, Equation 6.75 for the calculation of the cutting efficiency is rewritten as follows:

0.254 ⎛⎞v * f (7.37) ξξ=⋅⎜⎟ ⎝⎠v u In which, ξ does not depend on the feed speed and is defined as:

−−0.15550.3104 0.2318 ⎛⎞d ⎛⎞d ⎛⎞p ξ * =⋅⋅kkp ⋅ori ⋅w R−0.1236 (7.38) am⎜⎟⎜⎟ ⎜⎟ ⎝⎠ddpffa⎝⎠ ⎝⎠

From Equations 7.37 and 7.38, vi can be calculated by:

1.3405 ⎛⎞ξ * ⋅P v =⎜abr ⎟ (7.39) i ⎜⎟ed⋅⋅ h ⎝⎠cfi,max

In which, df,i (m) is the nozzle diameter after the time period i and hmax is the maximum depth of cut (m) (see Equation 6.76).

From Equation 7.39, it follows:

108 1.3405 ⎛⎞d vv=⋅⎜⎟f ,0 (7.40) io⎜⎟d ⎝⎠fi,

Where, v0 is the feed speed when cutting with a new nozzle, i.e. at i=0 with the initial nozzle diameter df,0, v0 is calculated as follows:

1.3405 ⎛⎞ξ * ⋅P v =⎜abr ⎟ (7.41) 0 ⎜⎟ed⋅⋅ h ⎝⎠cf,0 max

Substituting (7.40) into (7.36) gives

vd⋅ 1.3405 n 1 v =⋅0,0f (7.42) fa, ∑ 1.3405 ndi =1 fi,

7.2.3 Optimization for determining optimum nozzle lifetime for minimum cutting cost

7.2.3.1 Optimization problem

The objective of the optimization problem is to determine the optimum value of the nozzle- exchange diameter for getting the minimum cutting cost per unit length. Mathematically, the cutting cost per length Cl is minimum if the derivative of its function (see Equation 7.22) with respect to the variable df equals zero:

dC l = 0 (7.43) dd f The optimum nozzle exchange diameter can be obtained by solving the above equation. However, it is rather difficult to solve Equation 7.43 analytically. Hence, a so-called “Golden ratio search” is chosen for resolving the cost optimization problem.

The optimization problem can be expressed by the following objective function:

minCfdlf= ( ) (7.44)

With a constraint: ddd≤≤ (7.45) f ,0 ff,max

In practice, for a specific company, the cost elements such as the hourly machine tool cost, the wages including overhead cost per hour, the abrasive cost per kilogram, the nozzle cost per piece etc. are usually constant. Therefore, with a certain setup (i.e. the water pressure, the initial nozzle diameter, the orifice diameter, etc.), one value of the optimum nozzle-exchange diameter can be determined by solving the optimization problem. However, the cost elements are different from

109 various companies and they can also differ depending on time and location. For example, the abrasive price in the USA can be 0.6 (€/kg) while it is only 0.2 ÷ 0.3 (€/kg) in Europe. The labor cost per hour can be 20 (€/h) or more in Europe while it is less than 1 (€/h) in many developing countries in Asia. Also, the process parameters can be varied by AWJ users. As a result, the optimum nozzle-exchange diameter should be found as a function of many variables of the cost elements as well as the process parameters.

Based on Equations 7.22, 7.44, and 7.45, a computer program was built to determine the optimum nozzle-exchange diameter for getting the minimum cutting cost. At first, seventeen variables were investigated to determine the optimum nozzle-exchange diameter with different set ups of the AWJ process. These are the abrasive particle diameter dp, the abrasive mass flow rate m a , the water pressure pw, the orifice diameter dori, the initial nozzle diameter df,0, the nozzle length lf, the maximum depth of cut hmax, the time for keeping cutting speed constant tck, the machine tool hourly rate Cmt,h, the wages including overhead per hour Cwa,h, the abrasive cost per kilogram Ca,m, the water cost per kilogram Cw,m, the orifice cost per piece Cori,p, the orifice lifetime Lori, the nozzle cost per piece Cf,p, the nozzle exit bore wear rate δdf , and the number of cutting heads nf. However, it is detected that the optimum exchange nozzle diameter is directly affected by only twelve of these variables, viz. pw, dori, df,0, m a , dp, tck, Cmt,h, Cwa,h, Ca,m, Cf,p, δdf , and nf. Hence, the effects of the process parameters on the optimum values of the nozzle-exchange diameter df,op are investigated in terms of these twelve variables by the following function:

dfpdddmCCCCtnf ,,0,,,,op= (, w ori , f ,,, p a mt h , wa h , a m , f p ,δ df ,,) ck f (7.46)

To create different setups, the following data were chosen as input to the optimization program: pw=150…400 (MPa), dori=0.15…0.45 (mm), df,0=0.6…1.6 (mm), ma =2…20 (g/s), 4 sizes of Baton garnet (#50, #80, #120, and #150HPX), Cmth=10…100 (€/h), Cwa,h=5…50 (€/h), Cf,p=50…210 (€/piece), Ca=0.2…1.2 (€/kg), tck=2…29 (h), δdf =0.002…0.014 (mm/h), nf=1…4, and

ddff,max=+ ,0 0.5 mm.

7.2.3.2 Results and discussions

Figure 7.6 shows the relation between the nozzle-exchange diameter and the total cutting cost per unit length. The calculation was made with the following set up parameters: nf = 1, Crpl= 200000 €,

2 Ttot= 5 y, dwor= 250 d/y, xint= 10 %, Csqm= 50 €/m2, Amt= 35 m , xma= 6 %, Ptot= 38 kW, e = 0.06

€/kWh, dop=30 %, xsh=2, xmsh=2, tsh = 8 h/d, xtLut=− 0.8 cn / f , tcn= 0.15 h; workpiece material:

SS 304, hmax= 25 mm, pw=360 MPa, m a =13 g/s, df,0=1.14 mm, dor=0.38 mm, tck= 8 h, Cori,p= 12

€/piece, Lori=40 h, Cf,p=70 €/piece, Lf=90 h, #80HPX, Ca,m= 0.7 €/kg, Cla,h= 20 €/h, Cov,h=15 €/h

110 and Cw,m=0.004 €/kg.

It is observed that the cutting cost per unit length depends strongly on the exchange nozzle diameter. The cost is minimum when the exchange nozzle diameter equals a certain value of d f ,op

(Figure 7.6), namely the “optimum diameter” [Pi05]. It is interesting that the “optimum lifetime” determined by the “optimum diameter” is much smaller than the conventional lifetime. In this example (Figure 7.6), the “optimum lifetime” was only 21.07 hours whereas the average nozzle lifetime was approximately 96 hours according to [Mort95] (or 50, 80, and 100 hours according to [Zeng93], [Hash04], and [Sing93], respectively).

20 60 (EUR/m) l 15 45 (EUR/h) h

10 30

Cost per metre 5 Profit per hour 15

d fop,Cl d fmax Profit per hour Pr 0 0 1 d 1.2 d 1.4 1.6 1.8 2 fmin fop,pr Nozzle−exchange diameter d (mm) Cutting cost per unit length C f Figure 7.6: Nozzle-exchange diameter versus total cutting cost and profit per hour

As aforementioned, there are various parameters affecting the optimum nozzle-exchange diameter. These parameters are the initial nozzle diameter df,0, the nozzle wear rate δdf , the number of jet formers nf, the nozzle cost per piece Cf,p, the hourly machine tool cost Cmt,h, the wages including overhead cost per hour Cwa,h, the abrasive cost per kilogram Ca,m, the water cost per kilogram Cwa,m, and the nozzle changing time tck. The effects of these parameters (calculated with #80HPX; nf=1; dori=0.38, df,0 = 1.14 mm; δdf = 0.0072 mm/h; Cmt,h = 40 €/h; Cwa,h = 40 €/h; Cf,p = 70 €/piece; Ca,m

= 0.7 €/kg; tck = 8 h) are discussed as follows (see also Figure 7.7a to Figure 7.7l).

The effects of the water pressure and the orifice diameter on the optimum nozzle diameter are small (Figure 7.7a and 7.7b). Also, the abrasive mass flow rate does not affect the optimum diameter directly. However, the optimum diameter depends on the water pressure, the abrasive mass flow rate as well as on the orifice diameter and the initial nozzle diameter indirectly because of their influence on the nozzle wear rate δdf (see Subsection 7.2.1.2).

111 The initial nozzle diameter df,0 and the nozzle wear rate δdf (see Figure 7.7c and Figure 7.7d, respectively) are most influential on the optimum diameter compared to the abrasive particle diameter dp and the time period of constant feed speed tck (Figure 7.7e and Figure 7.7f, respectively). This is because the effects of dp and tck on the average cutting speed are much less than those of the initial nozzle diameter and the nozzle wear rate.

It is observed that the optimum diameter decreases with the increase of the hourly machine tool cost (Figure 7.7g), the wages including overhead cost per hour (Figure 7.7h), and the abrasive cost per hour (Figure 7.7i). As these costs increase the cutting cost per unit length increases (see Equation 7.22). Therefore, to reduce the cutting cost per unit length, the optimum diameter has to be decreased correspondingly in order to increase the average feed speed. In contrast, with the increase of the nozzle cost per each Cf,p the optimum diameter increases (Figure 7.7j). The reason behind this is that a higher nozzle cost per piece leads to a higher the cutting cost per hour. Therefore, the optimum diameter increases to augment the nozzle lifetime in order to reduce the cutting cost.

It was found out that the optimum diameter increases when the number of cutting heads increases (Figure 7.7k). This is because with more cutting heads, the total time for replacing the nozzles increases and the annual time of use Tuse decreases. Therefore, the machine tool hourly rate Cmt,h goes up (see Equation 7.5). As a result, the optimum diameter has to be raised in order to increase the nozzle lifetime for reducing the machine tool hourly rate.

The water cost per kilogram Cw,m does not affect the “optimum diameter” since it is negligibly small compared with the total cutting cost.

The nozzle replacement time tc,n affects the “optimum diameter” indirectly through the annual time of use Tuse when calculating Cmt,h (see Subsection 7.1.1.1).

Modeling for the optimum diameter has been carried out in order to have a quick method for determining it. To do that, from the above analysis, Equation 7.46 can be expressed by the following dimensionless form using Buckingham Pi theorem:

kk12 k 3k 4 d ⎛⎞⎛⎞⎛Ct⋅⋅⋅⋅⋅ Ct3600 mCt ⎞⎛⎞d fop, =⋅k ⎜⎟⎜⎟⎜mt,, h ck ⋅ wa h ck ⋅ a a , m ck ⎟ ⋅⎜⎟ori ⋅ dnCnCC0 ⎜⎟⎜⎟⎜⋅⋅ ⎟⎜⎟ d fffpffpfpf,0⎝⎠⎝⎠⎝ , , , ⎠⎝⎠ ,0 (7.47) k 5 k 6 kk78 ⎛⎞d ⎛⎞p ⎛⎞⎛⎞δ ⋅tm ⋅⋅⋅⎜⎟f ,0 ⎜⎟w ⎜⎟⎜⎟df ck ⋅ a ⎜⎟dp⎜⎟⎜⎟⎜⎟ d m ⎝⎠pa⎝⎠⎝⎠⎝⎠ f,0 a ,0

Where, ma,0 is the unit abrasive mass flow rate ( ma ,0 = 1 kg/s).

112 1.4 1.4 (mm) (mm) 1.35 1.35 f,op f,op

1.3 1.3

1.25 1.25

1.2 1.2

1.15 For minimal cutting cost 1.15 For minimal cutting cost For maximal profit For maximal profit Optimal nozzle diameter d

1.1 Optimal nozzle diameter d 1.1 150 200 250 300 350 400 0.25 0.3 0.35 0.4 0.45 Water pressure p (MPa) Orifice diameter d (mm) w ori a) b) 1.8 1.4 For minimal cutting cost (mm) For maximal profit (mm) For minimal cutting cost 1.35 1.6 For maximal profit f,op f,op

1.3 1.4

1.25 1.2 1.2

1 1.15 Optimal nozzle diameter d

Optimal nozzle diameter d 0.8 1.1 0.8 1 1.2 1.4 1.6 2 4 6 8 10 Initial nozzle diameter d (mm) Nozzle exit bore wear rate (mm/h) −3 f,0 x 10 c) d) 1.4 1.4 (mm) (mm) 1.35 1.35 f,op f,op

1.3 1.3

1.25 1.25

1.2 1.2

For minimal cutting cost 1.15 For minimal cutting cost 1.15 For maximal profit For maximal profit

1.1 Optimal nozzle diameter d 1.1 Optimal nozzle diameter d 0 0.1 0.2 0.3 0.4 2 4 6 8 10 12 Abrasive particle diameter d (mm) Time for keeping feed speed constant t (h) p ck e) f) Figure 7.7a-f: Effects of factors on the optimum diameter

113 1.4 1.4 (mm) 1.35 1.35 (mm) f,op f,op 1.3 1.3

1.25 1.25

1.2 1.2

For minimal cutting cost For minimal cutting cost 1.15 1.15 For maximal profit For maximal profit Optimal nozzle diameter d

Optimal nozzle diameter d 1.1 1.1 0 20 40 60 80 100 0 10 20 30 40 50 Hourly machine tool cost C (EUR/h) Wagee including overhead cost C (EUR/h) mt,h wa,h g) h) 1.4 1.4 (mm) 1.35 1.35 (mm) f,op f,op 1.3 1.3

1.25 1.25

1.2 1.2

For minimal cutting cost For minimal cutting cost 1.15 1.15 For maximal profit For maximal profit Optimal nozzle diameter d

1.1 1.1

Optimal nozzle diameter d 0.2 0.4 0.6 0.8 50 60 70 80 90 Abrasive cost per kilogram C (EUR/kg) Nozzle cost per piece C (EUR/piece) a,m f,p i) j) 1.4 1.4

1.35 1.35 (mm) (mm) f,op f,op 1.3 1.3

1.25 1.25

1.2 1.2

For minimal cutting cost 1.15 1.15 For maximal profit Optimal nozzle diameter d Optimal nozzle diameter d

1.1 1.1 1 2 3 4 0 1 2 3 4 Number of jet former n (−) Profit coefficient k (−) f p k) l) Figure 7.7g-l: Effects of factors on the optimum diameter

114 Through a regression analysis of the data of the optimization program (11952 computer generated data points), the coefficients k0 through k8 in Equation 7.47 are determined and the following regression model was found (with R2=0.99) for the prediction of the optimum diameter:

−−0.0239 0.0144 − 0.0288 −0.0126 ⎛⎞⎛⎞⎛Ct⋅⋅⋅⋅⋅ Ct3600 mCt ⎞⎛⎞d d =⋅1.9464 ⎜⎟⎜⎟⎜mt,, h ck ⋅ wa h ck ⋅ a a , m ck ⎟ ⋅⎜⎟ori ⋅ fop, ⎜⎟⎜⎟⎜nC⋅⋅ nC C ⎟⎜⎟ d ⎝⎠⎝⎠⎝ffp,, ffp fp , ⎠⎝⎠ f ,0 (7.48) −0.0092 −0.0184 0.0585 0.0314 ⎛⎞d ⎛⎞p ⎛⎞δ ⋅t ⎛⎞m ⋅⋅⋅⎜⎟⎜⎟f ,0 ⎜⎟w df ck ⋅⋅⎜⎟a d ⎜⎟dp⎜⎟ ⎜ d⎟⎜⎟m f ,0 ⎝⎠pa⎝⎠ ⎝⎠ f,0 ⎝⎠a ,0 It is noted that for the prediction of the optimum diameter using Equation 7.48 the nozzle lifetime can be taken as 25 and 49 hrs for ROCTEC100 and ROCTEC500 nozzles (ROCTEC100 and

ROCTEC500 are trademarks of Kennametal Inc.) respectively for rough calculation of Tuse by Equation 7.11. This is because the optimum nozzle lifetime usually is from 12 to 38 hours for ROCTEC100 nozzles (calculated with Cf,p=50…110 (€/piece) and δdf = 0.003… 0.009 (mm/h)) or from 29 to 69 for ROCTEC500 nozzles (calculated with Cf,p=120…210 (€/piece) and δdf = 0.002… 0.004 (mm/h)). Moreover, almost the same value of the optimum diameter is found when the calculation is made with Lf =25 hrs for ROCTEC100 or Lf =49 hrs for ROCTEC500, or with the optimum lifetime.

7.2.4 Optimization for finding optimum nozzle lifetime for maximum profit rate

7.2.4.1 Optimization problem

To determine the optimum nozzle lifetime in order to get the maximum profit rate, the optimization problem can be expressed as follows:

The objective function:

max Prhf= fd ( ) (7.49)

With a constraint: ddd≤≤ (7.50) f ,0 ff,max

Based on Equations 7.30, 7.49, and 7.50, a computer program was developed to determine the optimum nozzle-exchange diameter for getting the maximum profit rate. As the profit rate depends on the cutting cost per unit length (see Equation 7.30), the process parameters and the cost elements that affect the optimum nozzle exchange diameter for maximum profit rate are the same as those for the minimum cutting cost (see Subsection 7.2.3.1). Besides, the profit rate depends on

115 the profit coefficient k p . Therefore, the optimum nozzle-exchange diameter for maximum profit rate can be expressed as:

dfpdddmCCCCtnkfop,,0,,,,= (, w ori , f ,, p a , mth , wah , am , fp ,δ df ,,,) ck f p (7.51)

The data used in the program for creating various set-ups are the same as those in the program for finding the optimum nozzle exchange diameter (see Subsection 7.2.3.1). The cutting cost per unit length when cutting with optimum exchange nozzle diameter for minimum cutting cost was chosen as the basic cutting cost per unit length Cl,0 . In addition, the profit coefficient varied in the range between 0.5 and 4.5.

7.2.4.2 Results and discussion

The relation between the nozzle exchange diameter and the profit per hour is shown in Figure 7.6.

(calculated with the data in Subsection 7.2.3.2 and with the profit coefficient kp=0.5). It is observed that the effect of the nozzle exchange diameter on the profit rate is noticeable, even much larger than that on the cutting cost per unit length (depending on the profit coefficient). The reason of that is the nozzle exchange diameter affects both the feed speed and the cutting cost per unit length which are two influencing factors on the profit rate (see Equation 7.30).

It is found that the optimum nozzle lifetime for maximum profit rate (in this example 18.82 hours) is smaller than that for the minimum cutting cost (in this case 21.07 hours) and both of these optimum values are much smaller than the conventional nozzle lifetime (see Subsection 7.2.3.2). Also, the profit rate can increase significantly (in this example 21.5 %) when cutting with the optimum nozzle lifetime in comparison with cutting with the conventional nozzle lifetime (with the average nozzle lifetime of 90 hours [Mort95]).

As mentioned in Subsection 7.2.4.1, the process parameters and the cost elements affecting the optimum nozzle exchange diameter for maximum profit rate are the same as those for the minimum cutting cost. The effects of these parameters (12 parameters – see Subsection 7.2.3.1) on the optimum nozzle exchange diameter are analyzed in Subsection 7.2.3.2 (see also Figure 7.7a to Figure 7.7k).

The relation between the profit coefficient and the optimum diameter is shown in Figure 7.7l. With the increase of the profit coefficient the optimum diameter reduces. The reason is that with a large value of the profit coefficient the effect of the cutting cost per unit length Cl on the profit rate

Prh is smaller than that of the average feed speed vf,a (Equation 7.30). Consequently, the optimum diameter will reduce when the profit coefficient kp increases in order to raise the average feed speed and therefore to increase the profit rate.

116 From the above analysis and by using the Buckingham Pi theorem, Equation 7.51 can be expressed as follows:

kk12 k 3 k 4 d ⎛⎞⎛⎞Ct⋅⋅⋅⋅⋅ Ct⎛⎞3600 mCt ⎛⎞d fop, =⋅k ⎜⎟⎜⎟mt,, h ck ⋅ wa h ck ⋅⎜⎟ a a , m ck ⋅⎜⎟ori ⋅ dnCnCC0 ⎜⎟⎜⎟⋅⋅⎜⎟⎜⎟ d fffpffpf,0⎝⎠⎝⎠ , ,⎝⎠⎝⎠ f ,0 (7.52) k 5 k 6 kk78 ⎛⎞d ⎛⎞p ⎛⎞⎛⎞δ ⋅tm ⋅⋅⋅⎜⎟f ,0 ⎜⎟w ⎜⎟⎜⎟df ck ⋅ a ⋅k k 9 ⎜⎟dp⎜⎟⎜⎟⎜⎟ d m p ⎝⎠pa⎝⎠⎝⎠⎝⎠ f,0 a ,0

In which, ma ,0 is the unit abrasive mass flow rate ( ma ,0 = 1 kg/s).

A regression analysis was done with the data of the optimization program (consisting of 11952 computer generated data) to get the coefficients k0 through k9 in Equation 7.52. The optimum diameter for the maximum profit can then be determined by the following regression model (with R2=0.99):

−−0.0144 0.0098 −0.0198 −0.0076 ⎛⎞⎛⎞Ct⋅⋅⋅⋅⋅ Ct⎛⎞3600 mCt ⎛⎞d d =⋅1.5603 ⎜⎟⎜⎟mt,, h ck ⋅ wa h ck ⋅⎜⎟ a a , m ck ⋅⎜⎟ori ⋅ fop, ⎜⎟⎜⎟nC⋅⋅ nC⎜⎟ C⎜⎟ d ⎝⎠⎝⎠ffp,, ffp⎝⎠ f⎝⎠ f ,0 (7.53) −0.0061 −0.0136 0.0342 0.0218 ⎛⎞d ⎛⎞p ⎛⎞δ ⋅t ⎛⎞m ⋅⋅⋅⎜⎟⎜⎟f ,0 ⎜⎟w df ck ⋅⋅⋅⎜⎟a kd0.0192 ⎜⎟dp⎜⎟ ⎜⎟ d⎜⎟m pf,0 ⎝⎠pa⎝⎠ ⎝⎠ f,0 ⎝⎠a ,0

It was found that the average optimum nozzle lifetime for maximum profit was 23 hours for ROCTEC100 nozzles (calculated with Cf,p=50…110 (€/piece) and δdf = 0.003… 0.009 (mm/h)) and was 44 hours for ROCTEC500 (calculated with Cf,p=120…210 (€/piece) and δdf = 0.002… 0.004 (mm/h)). Therefore, as noted in Subsection 7.2.3.2, for the prediction of the optimum diameter (Equation 7.53) the nozzle lifetime should be chosen as Lf=23 (h) for

ROCTEC100 or Lf=44 for ROCTEC500 nozzles for the rough calculation of Tuse by Equation 7.11.

7.2.5 Benefits of cutting with optimum nozzle lifetime

To clarify the benefits of cutting with optimum nozzle lifetime two examples are illustrated as follows. The first example was calculated with the same data as those in the example in Figure 7.6 (see Subsection 7.2.3.2) and with the total length of cut of 950 m. In the second example, the following data were used: nf =4, Crpl=300000, Ptot=90 (kW), and the other data were the same as those in the first example.

Table 7.3 shows the calculated results of the first example with three cutting scenarios: cutting with the old regime (with the nozzle lifetime is Lf=90 hours), cutting with the optimum nozzle lifetime for minimum cutting cost, and cutting with the optimum lifetime for maximum profit rate.

117 Table 7.3: Calculated results of the first example

Optimum regime

Parameter Unit Old regime For minimum For maximum cutting cost profit rate Number of jet formers - 1 1 1 Nozzle-exchange diameter mm 1.788 1.292 1.276 Nozzle lifetime hours 90 21.07 18.82 Average feed speed mm/s 2.93 3.22 3.23 Number of used nozzles per - 1 3.89 4.34 jet former Total cutting time hours 90 81.96 81.67 Total working time hours 90.15 82.54 82.32 Hourly cutting cost €/h 108.52 111.57 112.04 Cutting cost per unit length €/m 10.28 9.62 9.63 Profit per hour €/h 43.88 55.78 55.89 Total profit € 3948.81 5020.47 5030.16 Nozzle saving € 0 208.6 240.1 Total profit including nozzle € 3948.81 5229.07 5270.26 saving Saving of cutting cost % 0 6.38 6.31 Saving of total profit % 0 24.48 25.08 Total saving time h 0 7.61 7.83 Saving of working time % 0 8.44 8.68

It is observed that a number of 3.89 nozzles were used in only 21.07 hours for a jet former when cutting with the optimum regime for minimum cutting cost and a number of 4.34 nozzles were used in only 18.82 hours when cutting for the maximum profit rate. These nozzles can be used for cutting with abrasives of bigger size or cutting with the traditional regime. If they are used with the traditional regime, they can be used for additional 68.93 hours (90-21.07) when cutting for the minimum cutting cost and for additional 81.18 hours (90-18.82) when cutting for the maximum profit rate. As a result, they will result in a profit of (3.89x68.93)/90=2.98 new nozzles when cutting for the minimum cutting cost and (4.34x71.19)/90=3.43 new nozzles when cutting for the maximum profit rate. With Cf,p= 70 (€/piece), the nozzle saving is 2.98x70=208.6 (€) when cutting for the minimum cutting cost and 3.43x70=240.1 (€) when cutting for the maximum profit rate. Therefore, the total profit including nozzle saving is 5020.45+208.6=5229.07 (€) (increased by 24.48 % in comparison with cutting with the traditional regime) when cutting for the minimum cutting cost and is 5030.16+240.1=5270.26 (€) (increased by 25.07 % in comparison with the traditional regime). The saving of working time is 7.61 hours (8.44 %) when cutting for the minimum cost and 7.83 hours (8.69 %) when cutting for the maximum profit rate compared with the old regime. Figure 7.8 illustrates the benefit of cutting with optimum nozzle diameter in the

118 example 1.

Table 7.4 presents the calculated results for the second example. As is done in the first example, the calculated results were found for three different cutting scenarios: cutting with the old regime

(the nozzle lifetime Lf is 90 hours), cutting with the optimum nozzle lifetime for minimum cutting cost, and cutting with the optimum lifetime for maximum profit rate.

130

Old regime 120 Minimum cutting cost Maximum profit rate

110

100

80 Normalized presentation (%)

60 Cutting time Cutting cost Profit rate per meter

Figure 7.8: Comparison of the benefit of cutting with optimum nozzle diameter (first example)

130

120 Old regime Minimum cutting cost Maximum profit rate 110

100

70 Normalized presentation (%)

60 Cutting time Cutting cost Profit rate per meter

Figure 7.9: Comparison of the benefit of cutting with optimum nozzle diameter (second example)

It follows from the calculated results that although the saving of total profit when cutting with the optimum regime with 4 jet formers (19.29 % for the minimum cutting cost and 21.21 % for the

119 maximum profit) is less than that when cutting with one jet former, but the total of profit is much higher (10843.34 (€) for the minimum cutting cost and 11107.67 (€) for the maximum profit) than that when cutting with one jet former (5228.98 (€) for the minimum cutting cost and 5270.48 (€) for the maximum profit). This is because cutting with multiple jet formers helps reduce the cutting cost per length significantly. In this case, the cutting cost per length is only 5.07 (€/m) for cutting for the minimum cutting cost and 5.09 (€/m) for cutting for the maximum profit rate whereas it is 9.62 (€/m) for the minimum cutting cost and 9.63 (€/m) for the maximum profit rate when cutting with one jet former. Figure 7.9 illustrates the benefit of cutting with optimum nozzle diameter in the example 2.

Table 7.4: Calculated results of the second example

Optimum regime

Parameter Unit Old regime For minimum For maximum cutting cost profit rate Numer of jet formers - 4 4 4 Nozzle-exchange diameter mm 1.788 1.385 1.334 Nozzle lifetime hours 90 34.05 26.98 Average feed speed mm/s 2.93 3.16 3.19 Number of used nozzles per - 1 2.45 3.06 jet former Total cutting time hours 90 83.57 82.70 Total working time hours 90.60 83.94 83.16 Hourly cutting cost €/h 56.29 57.87 58.53 Cutting cost per unit length €/m 5.33 5.07 5.09 Profit per hour €/h 97.24 115.74 116.74 Total profit € 8571.35 10416.11 10506.76 Nozzle saving € 0 427.23 600.91 Total profit including nozzle € 8571.35 10843.34 11107.67 saving Saving of cutting cost % 0 4.54 4.45 Saving of total profit % 0 19.29 21.21 Total saving time h 0 6.66 7.44 Saving of working time % 0 7.35 8.22

7.2.6 Conclusions

A new approach to use the AWJ nozzles has been proposed. Instead of using the nozzles for a long period of time as in “traditional method” (with the nozzle lifetime average is 96 hours [Mort95]) they can be used in much shorter time (determined by the “optimum diameter” by using Equation 6.40 for cutting for the minimum cutting cost or Equation 6.62 for cutting for the maximum profit

120 rate). After this time period, the nozzles can be reused for other cutting applications such as cutting with bigger abrasive sizes, or used for cutting with the traditional method.

Two objectives of the optimization problems to predict the optimum nozzle lifetime (or nozzle- exchange diameter) have been used. These are the minimum cutting cost per unit length and the maximum profit rate.

By using regression analyses and using Buckingham Pi theory, two dimensionless models for calculation of the optimum nozzle-exchange diameter for the above objectives have been proposed. The effects of various process parameters (e.g. the water pressure, the initial nozzle diameter, the abrasive particle diameter, etc.) as well as the influences of the cost elements (such as the hourly machine tool cost, the wages including overhead cost, the nozzle cost per piece etc.) on the optimum nozzle diameter are effectively taken into account.

Using the cost variables considering the nozzle exit bore wear rate also as a variable in the optimization problem, the results of this calculation become quite general and flexible.

Cutting with the optimum nozzle lifetime can save a lot of the profit rate (up to 25… 30 %), the cutting cost per unit length (up to 79… %), and the cutting time (up to 810… %). Moreover, nozzle companies can also benefit from increasing the sale amount of nozzles (about 2 to 4 times).

7.3 Optimization for determining the optimum abrasive mass flow rate

In this section three optimization problems are addresses respectively. The first problem (see Subsection 7.3.1) deals with seeking the optimum abrasive mass flow rate for maximum cutting performance (or maximum depth of cut). In the second problem (see Subsection 7.3.2), the optimum abrasive mass flow rate for minimum cutting cost per unit length is determined. The last problem (see Subsection 7.3.3) is concerned with the optimum abrasive mass flow rate for the maximum profit rate. From the results of these optimization problems, three respective models are derived using regression analyses. Also, the benefits of cutting with the optimum abrasive mass flow rate are discussed.

7.3.1 Optimization for determining the optimum abrasive mass flow rate for maximum cutting performance

7.3.1.1 Optimization problem

The objective of this optimization problem is to determine the optimum abrasive mass flow rate for getting the maximum cutting performance (or the maximum depth of cut). As is done for the

121 optimization for the optimum nozzle lifetime, the optimization problem can be expressed by the following objective function:

max hfmmax = ()a (7.54)

Where, hmax is the maximum depth of cut determined by Equation 6.76.

With the constraint:

mmmaaamin≤≤ max (7.55)

Practically, with a certain setup of the AWJ process parameters (i.e. the water pressure, the orifice diameter, the nozzle diameter etc.) there is an optimum value of the abrasive mass flow rate for the maximum depth of cut. Also, the process parameters can be varied by AWJ users. Therefore, the optimum abrasive mass flow rate should be determined as a function of various variables of the process parameters.

Based on Equations 6.76, 7.54, and 7.55, a computer program was built to find the optimum abrasive mass flow rate for getting the maximum depth of cut. At first, an investigation was carried out with seven variables to evaluate their dependency on the optimum abrasive mass flow rate.

These were the water pressure pw, the orifice diameter dori, the nozzle diameter df, the abrasive particle diameter dp, the number of cutting heads nf, the feed speed vf, the work material coefficient km, and the maximum depth of cut hmax. However, it is found that only four out of seven variables have clear influence on the optimum abrasive mass flow rate. These four variables are jet parameters including the water pressure, the orifice diameter, the nozzle diameter, and the abrasive particle diameter.

From the above finding, the influence of the process parameters on the optimum values of the abrasive mass flow rate can be expressed as follow:

mfpdddaophm, = (, w ori ,,) f p (7.56)

To investigate the effects of jet parameters on the optimum abrasive mass flow rate with various setups, the following data were used in the optimization program: pw =150… 400 (MPa), dori=0.15…0.45 (mm), df=0.15…1.6 (mm), 4 sizes of Baton garnet (#50, #80, #120, and #150HPX), and 1≤≤ma 40 .

7.3.1.2 Results and discussions

Figure 7.10 shows the relation between optimum abrasive mass flow rate and the maximum depth of cut. The calculation was made with the following set up: nf= 4; work material is SS304; hmax =

122 25 mm; vf=2.5 mm/s; pw=360 MPa; df0=1.14 mm; dori=0.38 mm; and abrasive material is #80HPX. It was observed that the maximum depth of cut depends strongly on the abrasive mass flow rate. As mentioned previously (see Subsection 2.1.3), there is an optimum value of the abrasive mass flow rate maophm, for the maximum depth of cut (Figure 7.10).

20 40 Cutting cost per meter Maximal depth of cut (mm) (EUR/m) max

l 15 30

10 20

5 10

m Maximal depth of cut h a,ophm 0 0 Cutting cost per meter C m 0 5 a,opCl 10 15 20 Abrasive mass flow rate (g/s)

Figure 7.10: Abrasive mass flow rate versus maximum depth of cut and cutting cost per meter

The effect of the jet parameters on the optimum abrasive mass flow rate is shown in Figure 7.11 which agrees with [Guo94a] in that the optimum abrasive mass flow rate increases with the increase of the water pressure (Figure 7.11a), of the orifice diameter (Figure 7.11b), and of the nozzle diameter (Figure 7.11c). Besides, the optimum abrasive mass flow rate decreases with the increase of the abrasive particle diameter (Figure 7.11d). This can be explained physically as follows. Larger abrasive particles cause larger friction between the solid-particle surface and the flowing water [Momb98] and therefore result in slower particle velocity. The reduction of the particle velocity leads to the decrease of the impact frequency of abrasive particles. Because at the optimum abrasive mass flow rate the benefit of the particle impact frequency is balanced with the loss in the particle velocity [Zeng94], the decrease of the particle impact frequency leads to the reduction in the optimum abrasive mass flow rate.

From the above arguments and using the Buckingham Pi theorem, Equation 7.56 can be expressed in the following dimensionless form:

k 1 k 2 k 3 m ⎛⎞d ⎛⎞dp⎛⎞ aophm, =⋅k ⎜⎟ori ⋅⎜⎟fw ⋅⎜⎟ (7.57) mddp 0 ⎜⎟⎜⎟⎜⎟ wfpa⎝⎠⎝⎠⎝⎠

The coefficients k0 through k3 in Equation 7.57 are determined by using a regression analysis of the data of the optimization program (consisting of 10224 data). Consequently, the following regression

123 model was found (with R2=0.999) for the prediction of the optimum abrasive mass flow rate for the maximum depth of cut:

−0.1484 0.1461 0.0656 ⎛⎞d ⎛⎞dp⎛⎞ mm=⋅0.1088 ⎜⎟ori ⋅⎜⎟fw ⋅⎜⎟ ⋅ (7.58) aophm, ⎜⎟ddp⎜⎟ ⎜⎟ w ⎝⎠fpa⎝⎠ ⎝⎠

20 20 For maximal depth of cut For minimal cutting cost For maximal profit rate 15 15

10 10

5 5 For maximal depth of cut For minimal cutting cost For maximal profit rate 0 0 200 250 300 350 400 0.2 0.25 0.3 0.35 0.4 Orifice diameter d (mm) Water pressure p (MPa) ori Optimal abrasive mass flow rate (g/s) w Optimal abrasive mass flow rate (g/s) a) b) 20 20

15 15

10 10

5 5 For maximal depth of cut For maximal depth of cut For minimal cutting cost For minimal cutting cost For maximal profit rate For maximal profit rate 0 0 0.8 1 1.2 1.4 1.6 0 0.1 0.2 0.3 0.4 Nozzle diameter d (mm) Arasive particle diameter d (mm) f p Optimal abrasive mass flow rate (g/s) Optimal abrasive mass flow rate (g/s) c) d) Figure 7.11: Effects of jet parameters on optimum abrasive mass flow rate

7.3.2 Optimization for determining the optimum abrasive mass flow rate for minimum cutting cost

7.3.2.1 Optimization problem

To determine the optimum abrasive mass flow rate for the minimum cutting cost per unit length, this optimization problem can be expressed as follows:

124 The objective function:

minCfmla= () (7.59)

With a constraint:

mmmaaamin≤≤ max (7.60)

As usual, the optimization problem is also resolved using the algorithm of the “Golden ratio search”. A computer program based on Equations 7.22, 7.59, and 7.60 was built to determine the optimum abrasive mass flow rate. It was found that the optimum abrasive mass flow rate for the minimum cutting cost depends on many process parameters and also on the cost elements. The dependency of the optimum abrasive mass flow rate on these parameters can be expressed as:

mfpdddnCCCCaopCl,,,,= (, w ori ,,,, f p f mth , wah ,, a f h ) (7.61)

The following data were used to create various setups in the optimization problem: pw=150…400

(MPa), dori=0.15…0.45 (mm), df=0.6…1.6 (mm), 4 sizes of Baton garnet (#50, #80, #120, and

#150HPX), Cmt,h=10…100 (€/h), Cwa,h=5…50 (€/h), Cf,p=50…210 (€/piece), Ca=0.2…1.1 (€/kg), tck=2…29 (h), δdf = 0.002… 0.014 (mm/h), nf=1…4, and 140≤≤ma .

7.3.2.2 Results and discussions

The relation between the abrasive mass flow rate and the cutting cost per meter was shown in Figure 7.10. It is observed that there is an optimum value of the abrasive mass flow rate for the minimum cutting cost per unit length. In addition, the optimum abrasive mass flow rate for the minimum cutting cost is much less than that for the maximum depth of cut.

As mentioned in Subsection 7.3.2.1, the optimum abrasive mass flow rate depends on various process parameters as well as the cost elements. Figure 7.11 shows the relation between the jet parameters and the optimum abrasive mass flow rate. It follows that the trend of the relation between the jet parameters and the optimum abrasive mass flow rate for the minimum cutting cost is similar to that for the maximum depth of cut but with smaller optimum values (see Subsection 7.3.1.2).

Figure 7.12 shows the relation between the cost elements and the optimum abrasive mass flow rate. It appears that with an increase of the hourly machine tool cost, the wages including overhead cost per hour, and the nozzle cost per hour, the optimum abrasive mass flow rate for the minimum cutting cost per hour increases (Figure 7.12a to 7.12c). This is because with an increase of the hourly machine tool cost, the hourly wages including overhead cost, and the hourly nozzle wear cost, the cutting cost per unit length increases (see Equation 7.22). Therefore, to reduce the cutting

125 cost per unit length, the optimum abrasive mass flow rate needs to be reduced to get the average feed speed increased. In contrast, the optimum abrasive mass flow rate decreases with the increase of the abrasive cost per kilogram (Figure 7.12d). This is because higher abrasive cost per kilogram leads to higher cutting cost per hour. Consequently, the optimum abrasive mass flow rate decreases to reduce the cutting cost per hour and therefore to reduce the cutting cost per meter.

As mentioned in Chapter 2, the abrasive cost is usually the largest cost element and can be 20% to 70% of the total cost. A higher number of jet formers result in higher abrasive cost and therefore in higher total cost. Therefore, with an increase of the jet former number the optimum abrasive mass flow rate needs to be reduced to lower the cutting cost (Figure 7.12e).

Based on the above analysis, Equation 7.61 can be expressed in the following dimensionless form:

kkk133 m ⎛⎞⎛⎞⎛⎞CCC aopCl, =⋅k ⎜⎟⎜⎟⎜⎟mt,,, h ⋅ wa h f h ⋅ mmCmCmC0 ⎜⎟⎜⎟⎜⎟3600⋅⋅ 3600 ⋅⋅ 3600 ⋅⋅ waamaamaam⎝⎠⎝⎠⎝⎠,0 , ,0 , ,0 , (7.62) k 4 k 5 k 6 ⎛⎞d ⎛⎞dp⎛⎞ ⋅⋅⋅⋅⎜⎟ori ⎜⎟fw⎜⎟n k 7 ⎜⎟ddp⎜⎟⎜⎟ f ⎝⎠fpa⎝⎠⎝⎠

The coefficients k0 through k7 in Equation 7.62 have been determined by using a regression analysis on the data of the optimization program (consisting of 10224 data). Consequently, the following regression model has been found (with R2=0.95) to predict the optimum abrasive mass flow rate for the minimum cutting cost per unit length:

0.0937 0.0684 0.0214 ⎛⎞⎛⎞⎛⎞CCC m =⋅0.4738 ⎜⎟⎜⎟⎜⎟mt,,, h ⋅ wa h f h ⋅ aopCl, ⎜⎟⎜⎟⎜⎟3600⋅⋅mC 3600 ⋅⋅ mC 3600 ⋅⋅ mC ⎝⎠⎝⎠⎝⎠aam,0 , aam ,0 , aam ,0 , (7.63) −0.3341 0.0332 −0.0416 ⎛⎞d ⎛⎞dp⎛⎞ ⋅⋅⋅⋅⋅⎜⎟ori ⎜⎟fw⎜⎟ nm1996 ⎜⎟ddp⎜⎟ ⎜⎟ fw ⎝⎠fpa⎝⎠ ⎝⎠

7.3.3 Optimization for determining optimum abrasive mass flow rate for maximum profit rate

7.3.3.1 Optimization problem

This optimization problem is to determine the optimum abrasive mass flow rate for the maximum profit rate. The objective of the optimization problem can be expressed as follows:

max Prha= fm() (7.64)

With a constraint:

mmmaaamin≤≤ max (7.65)

126 20 20

For minimal cutting cot 16 16 For minimal cutting cost For maximal profit rate For maximal profit rate

12 12

8 8

4 4 0 20 40 60 80 100 0 10 20 30 40 50 Hourly wages including Hourly machine tool cost C (EUR/h) overhead cost C (EUR/h) Optimal abrasive mass flow rate (g/s) Optimal abrasive mass flow rate (g/s) mt,h wa,h a) b) 20

For minimal cutting cost 16 For maximal profit rate

c) 12

4 0 3 6 9 12 Hourly nozzle wear cost C (EUR/h) f,h Optimal abrasive mass flow rate (g/s) 20 20

For minimal cutting cost For minimal cutting cost 16 16 For maximal profit rate For maximal profit rate

12 12

8 8

4 4 0.2 0.4 0.6 0.8 1 1 2 3 4 Abrasive cost per kilogram C (EUR/kg) a,m Number of jet former n (−) Optimal abrasive mass flow rate (g/s) Optimal abrasive mass flow rate (g/s) f d) e) Figure 7.12: Cost elements and number of jet formers versus optimum abrasive mass flow rate

127

Similarly, to determine the optimum abrasive mass flow rate, a computer program was created based on Equations 7.30, 7.64, and 7.65. The Golden search method was used again as in the previous section. It was found that the optimum abrasive mass flow rate for the maximum profit rate depends on many process parameters and also on the cost elements. These are the water pressure, the orifice diameter, the nozzle diameter, the abrasive particle diameter, the number of jet former, the machine tool hourly rate, the wages including overhead per hour, the abrasive cost per kilogram, the nozzle cost per hour, and the profit coefficient. The relation among these parameters and the optimum abrasive mass flow rate can be expressed as:

mfpdddnCCCCkaop,Pr= (, w ori ,,,, f p f mth , , wah , ,, a f , h ,) p (7.66)

The same data set in the program for the optimum abrasive mass flow rate for the minimum cutting cost (see Subsection 7.3.2.1) were used to investigate the above relation with different setups. The cutting cost per unit length when cutting with optimum abrasive mass flow rate for getting maximum cutting performance was chosen as the basic cutting cost per unit length Cl,0. The profit coefficient kp was varied between 0.5 and 4.5.

7.3.3.2 Results and discussions

Figure 7.13 shows the relation between the abrasive mass flow rate and the profit rate, and also the maximum depth of cut. It is apparent that there exists an optimum value of the abrasive mass flow rate for the maximum profit rate.

200 40

150 30 (mm) max (EUR/h) h 100 20

Profit rate 50 Maximal depth of cut 10

Profit rate Pr m a,ophm 0 0 m Maximal depth of cut h 0 5 10 a,opPr 15 20 Abrasive mass flow rate (g/s)

Figure 7.13: Abrasive mass flow rate versus maximum depth of cut and profit rate

It was found that the effecting trends of the jet parameters on the optimum abrasive mass flow

128 rate for the maximum profit rate were the same as those for the maximum depth of cut and for the minimum cutting cost (see Subsection 7.3.1.2 and Figure 7.9). Also, it is observed that the optimum values of the abrasive mass flow rate for the maximum profit rate are smaller than those for the maximum depth of cut but larger than those for the minimum cutting cost (Figure 7.9).

The effects of the cost elements and the number of jet formers on the optimum abrasive mass flow rate for the maximum profit rate were shown in Figure 7.10. It follows that the effecting trends of these factors on the optimum abrasive mass flow rate for the maximum profit rate were the same as those for the maximum depth of cut and for the minimum cutting cost (see Subsection 7.3.2.2).

40 20

15 30

20 5 Maxiaml depth of cut (mm) Increase of profit rate (%)

10 0 0.2 0.25 0.3 0.35 0.4 0 1 2 3 4 Profit coefficient k (−) Orifice diameter (mm) p

Figure 7.14: Profit coefficient versus optimum Figure 7.15: Profit coefficient versus increase of abrasive mass flow rate profit rate

Figure 7.14 shows the relation between the profit coefficient and the optimum abrasive mass flow rate. The optimum abrasive mass flow rate is proportional to the profit coefficient. This is because with a large value of the profit coefficient the effect of the cutting cost per meter Cl on the profit rate Prh is less than that of the average feed speed vf,a (see Equation 7.30). As a result, if the profit coefficient increases, the optimum abrasive mass flow rate will go down in order to increase the average feed speed and therefore to increase the profit rate.

Figure 7.15 shows the relation between the profit coefficient and the increase of the profit rate when comparing cutting for maximum profit rate and for maximum cutting performance. Like above, as the profit coefficient increases, the optimum abrasive mass flow rate for the maximum profit rate increases and the increase of the profit rate diminishes. If the profit coefficient is larger than 4 the benefit by cutting for the maximum profit rate is very small (less than 1%). In this case, the optimum abrasive mass flow rate for cutting for the maximum profit rate is nearly the same as that for cutting for the maximum cutting performance.

129 Based on the above analysis and using the Buckingham Pi theory, Equation 7.75 can be expressed as follows:

kkk133 m ⎛⎞⎛⎞⎛⎞CCC aop,Pr=⋅k ⎜⎟⎜⎟⎜⎟mt,,, h ⋅ wa h f h ⋅ mmCmCmC0 ⎜⎟⎜⎟⎜⎟3600⋅⋅ 3600 ⋅⋅ 3600 ⋅⋅ w⎝⎠⎝⎠⎝⎠ a,0 am , a ,0 am , a ,0 am , (7.67) k 4 kk57k 6 ⎛⎞d ⎛⎞dpn⎛⎞ ⎛⎞ ⋅⋅⋅⋅⎜⎟ori ⎜⎟fwf⎜⎟ ⎜⎟ ⎜⎟ddpk⎜⎟⎜⎟ ⎜⎟ ⎝⎠fpap⎝⎠⎝⎠ ⎝⎠

A regression analysis (based on 10224 computer generated data points) is used to determine the

2 coefficients k0 through k7 resulting in the following regression model (with R =0.99) for the prediction of the optimum abrasive mass flow rate for the maximum profit rate:

0.0282 0.0194 0.0163 ⎛⎞⎛⎞⎛⎞CCC m =⋅0.1585 ⎜⎟⎜⎟⎜⎟mt,,, h ⋅ wa h f h ⋅ aop,Pr ⎜⎟⎜⎟⎜⎟3600⋅⋅mC 3600 ⋅⋅ mC 3600 ⋅⋅ mC ⎝⎠⎝⎠⎝⎠aam,0 , aam ,0 , aam ,0 , (7.68) −0.201 0.11210.0346 − 0.0708 ⎛⎞d ⎛⎞dpn⎛⎞ ⎛⎞ ⋅⋅⋅⋅⋅⎜⎟ori ⎜⎟fwf⎜⎟ ⎜⎟ m ⎜⎟ddpk⎜⎟⎜⎟ ⎜⎟ w ⎝⎠fpap⎝⎠⎝⎠ ⎝⎠

7.3.4 Benefits of cutting with the optimum abrasive mass flow rate

To delineate the benefits of cutting with the optimum abrasive mass flow rate two illustrative examples are presented as follows. The first example was calculated with the same data as those in the example in Figure 7.6 (see Subsection 7.2.3.2) and with Lf=90 hours, the profit coefficient kp=0.5. The second example was also calculated with the same data as those in the first example but with nf=4, Crpl=300000, Ptot=90 (kW).

Table 7.5: Calculated results of the first example

Optimum regime Parameter Unit Chalmers’ For For For optimum maximum minimum maximum regime depth of cut cutting cost profit rate

Number of jet formers - 1 1 1 1 Optimum abrasive g/s 7.75 15.79 11.59 12.89 mass flow rate Average feed speed mm/s 2.77 3.43 3.28 3.36 Hourly cutting cost €/h 112.74 115.55 104.95 108.22 Cutting cost per €/m 9.55 9.36 8.89 8.94 unit length Profit rate €/h 44.76 57.77 60.74 61.66 Increase of profit rate % 0 22.53 26.31 27.41

130 Table 7.5 shows the calculated results for the first example with four cutting cases: the first is cutting with Chalmers’ optimum regime (with the optimum abrasive mass flow rate for getting 0.85 maximum depth of cut [Chalm91] (see Subsection 2.1.3)), the other three are cutting with the optimum abrasive mass flow rate for the maximum depth of cut, the minimum cutting cost, and for the maximum profit rate. It follows that, compared with the cutting result in the first case [Chalm91], the profit rate increases by 22.61% when cutting for the maximum depth of cut, by 26.37% for the minimum cutting cost, and by 27.48% for the maximum profit rate. Figure 7.16 shows the benefit of cutting with optimum abrasive mass flow rate.

140

Chalmers’ regime 130 For maximum depth of cut For minimum cutting cost For maximum profit rate 120

110

100

80 Normalized presentation (%)

70 Cutting time Cutting cost Profit rate per meter

Figure 7.16: Benefit of cutting with optimum abrasive mass flow rate (first example)

In practice, sometimes the objective of cutting for the maximum cutting performance (or for the minimum cutting time) is chosen. In such a situation, with the data of this example, the cutting time can be saved 1.98%, 4.40%, and 19.19% when comparing with cutting for the maximum profit rate, cutting for the minimum cutting cost, and cutting for 0.85 maximum depth of cut (suggestion in [Chalm91]), respectively.

Table 7.6 shows the calculated results for the second example. As is done in the first example, the calculated results were found when cutting with four different cases: cutting with Chalmers’ optimum regime (for getting 0.85 maximum depth of cut), cutting with the optimum abrasive mass flow rate for the maximum depth of cut, for the minimum cutting cost, and for the maximum profit rate. As shown in Table 7.6, in comparison with cutting for the maximum depth of cut, the profit rate increases by 11.59% when cutting with the optimum abrasive mass flow rate for the minimum cutting cost, by 16.91% for maximum profit rate, and only by 5.75% when cutting with the optimum abrasive mass flow rate as suggested in [Chalm91]. The benefit of cutting with optimum abrasive mass flow rate is illustrated in Figure 7.17.

131 Table 7.6: Calculated results of the second example

Optimum regime Parameter Unit Chalmers’ For For For optimum maximum minimum maximum depth of cut cutting cost profit rate regime

Number of jet formers - 4 4 4 4 Optimum abrasive g/s 7.75 15.79 8.64 11.09 mass flow rate Average feed speed mm/s 2.77 3.43 2.93 3.24 Hourly cutting cost €/h 183.6 254.37 182.64 207.13 Cutting cost per €/m 4.32 5.13 4.23 4.42 unit length Profit rate €/h 134.72 126.64 143.31 152.62 Increase of profit rate % 6.00 0 11.63 17.02

140 Chalmers’ regime For maximum depth of cut 130 For minimum cutting cost For maximum profit rate 120

110

100

Mormalized presentation (%) 70

60 Cutting time Cutting cost Profit rate per meter

Figure 7.17: Benefit of cutting with optimum abrasive mass flow rate (second example)

When cutting for the maximum cutting performance, the time of cutting can be saved by 5.62%, 14.58%, and 19.19% when comparing with cutting for the maximum profit rate, cutting for the minimum cutting cost, and cutting for 0.85 maximum depth of cut (after [Chalm91]), respectively.

7.3.5 Conclusions

In this section, three objectives for the optimum abrasive mass flow rate are suggested. They are the maximum cutting performance, the minimum cutting cost per unit length and the maximum

132 profit rate.

Based on the Buckingham Pi theory and regression analyses, three dimensionless models for the predictions of the optimum abrasive mass flow rate for the above objectives have been proposed. Through modeling the effects of various process parameters (e.g. the water pressure, the orifice diameter, the nozzle diameter etc.) and of the cost elements (e.g. the hourly machine tool cost, the wages including overhead cost per hour, the hourly nozzle cost etc.) on the optimum abrasive mass flow rate, the optimum values of the abrasive mass flow rate can be easily predicted.

By considering the cost elements such as the hourly machine tool cost, the wages including overhead cost per hour etc. as well as the profit coefficient as variables, the results of these calculations become very general and flexible.

Cutting with the optimum abrasive mass flow rate can save a lot of both the profit rate (up to 30%) and the time of cutting (up to 25 %).

7.4 Selection of process parameters for the optimum cutting regime

In practice, an ordinary cutting for a certain type of work material with a specified depth of cut and cutting quality is determined by the following process parameters: the water pressure, the abrasive mass flow rate, the abrasive type and size, the standoff distance, the orifice diameter, the nozzle diameter, and the nozzle lifetime. The feed speed then can be calculated according to an AWJ cutting process model such as Equation 6.76. However, all these parameters need to be selected appropriately in order to come up with an optimum cutting regime.

Generally speaking, a higher water pressure will lead to a higher the maximum depth of cut (see Figure 7.14). However, there are very few AWJ systems that operate above 380 MPa regularly, because of high maintenance cost due to early failures at high water pressures [Olse03]. Therefore, the highest possible water pressure among its commonly-used low magnitudes, i.e. less than 380 to 400 MPa, should be chosen.

As already discussed in Subsection 2.1.5, the optimum standoff distance is about 2 mm [Guo94b], which is also chosen here for AWJ cutting.

The predictions of the optimum values of the nozzle exchange diameter (or the optimum nozzle lifetime) and the abrasive mass flow rate have already been discussed in the preceding sections.

The selection of remaining parameters, i.e. the orifice diameter, the nozzle diameter, and the number of jet formers is treated in the following subsection. A procedure to select a proper abrasive material is also discussed.

133 7.4.1 Optimum selection of the number of jet formers, the orifice diameter, and the nozzle diameter

As already mentioned in Subsection 2.1.1, the optimum diameter ratio of the nozzle to the orifice is between 3 to 4 [Blic90]. In practice, this range is further narrowed to between 3 and 3.3 and can be considered fixed (see Subsection 2.1.1). Therefore, we need to select the optimum value of the orifice diameter only.

The relations between the orifice diameter and the maximum depth of cut, the minimum cutting cost per meter, and the maximum profit rate are shown in Figure 7.18a, Figure 7.18b, and Figure

7.18c, respectively (calculated using the data of the example shown in Figure 7.6, with df=3.dori, and with the optimum abrasive mass flow rate; see Section 7.3). It follows that the higher the orifice diameter is, the higher the benefit of the cutting can be (i.e. the higher the maximum depth of cut, the lower the cutting cost per meter, and the higher the maximum profit rate). Also, with an increase of the number of jet formers the benefit of the cutting increases (the minimum cutting cost decreases and the maximum profit rate increases - see Figure 7.18b and Figure 7.18c). However, with an increase of the orifice diameter as well as an increase of the number of jet formers the electric power of the AWJ system increases (see Figure 7.18d). Therefore, the jet former number and the orifice diameter must be selected with considering the electric power of the system.

Figure 7.18e and Figure 7.18f show the relation between the number of the jet formers and the minimum cutting cot per meter, and the maximum profit rate, respectively with different orifice diameter values (the data were also the same as those in the example in Figure 7.6, and with the optimum abrasive mass flow rate – see Section 7.3 and df=3.dori). It is observed that with the same electric power, the cutting benefit increases significantly when the number of jet formers increases

(e.g. with Pelec=45 kW the minimum cutting cost decreases from 8.52 to 6.54 (€/m) (decreases 23.24%) and the maximum profit rate increases from 64.38 to 73.61 (increases 12.5%) when the number of jet formers increases from 1 to 4). Therefore, with a certain electric power of the system, the optimum value of the number of jet formers is its possible maximum value.

The necessary electric power Pelec (w) of an AWJ system can be predicted as follows:

nPfwj⋅ Pelec = (7.69) η0

Where

ηo is the overall efficiency considering the power losses due to the disturbances of the flow, the internal friction losses, as well as the compressibility of the water. The overall efficiency increases as the water pressure increases and ranges from 0.6 to about 0.85 [Momb03].

134 40 18

C n =1 16 lmin f

(mm) C n =2 lmin f

max 30 C n =3 14 lmin f C n =4 lmin f 12 20 10

p =300 MPa 8 w 10 p =350 MPa w p =400 MPa 6 w Maximum depth of cut h 0 4 0.2 0.25 0.3 0.35 0.4 0.2 0.25 0.3 0.35 0.4 Orifice diameter d (mm) Orifice diameter d (mm) ori inimum cutting cost per meter (EUR/m) ori M a) b) 200 200 P n =1 r,hmax f P n =1 P n =2 elec f r,hmax f P n =2 P n =3 elec f r,hmax f P n =3

150 (kW) 150 P n =4 elec f r,hmax f P n =4 elec f elec

100 100

50 50 Electric power P Maximum profit rate (EUR/h) 0 0 0.2 0.25 0.3 0.35 0.4 0.2 0.25 0.3 0.35 0.4 Orifice diameter d (mm) Orifice diameter d (mm) ori ori c) d) 10 100 d =0.257 mm ori

P =45 kW d =0.364 mm elec ori 9 P =75 kW 90 elec d =0.514 mm d =0.4 mm ori d =0.297 mm ori ori

8 80 d =0.23 mm d =0.282 mm ori ori d =0.23 mm d =0.514 mm ori ori d =0.4 mm 7 70 ori d =0.2 mm ori

d =0.364 mm d =0.2 mm per meter (EUR/m) ori ori P =45 kW 6 60 elec Minimum cutting cost d =0.282 mm P =75 kW ori elec d =0.297 mm ori

d =0.257 mm Maximum profit rate (EUR/h) 5 ori 50 1 2 3 4 1 2 3 4 Number of jet former n (−) Number of jet former n (−) f f e) f) Figure 7.18: Orifice diameter versus various parameters

135 Pwj is the theoretical power of the water which can be determined by the following equation [Hoog00]:

Ppqwj=⋅ w w (7.70)

In which, qwth, is the theoretical water volume flow rate calculated as follows:

π ⋅d 2 2 ⋅ p ori w qw = (6.23 repeated) 4 ρ0

From Equations (7.69), (7.70) and (6.23), for cutting with nf jet formers with a certain abrasive type and abrasive size, and with the electric power of the system Pelec, the maximum orifice diameter (also the optimum orifice diameter) can be calculated as follows:

η ⋅P d = oelec (7.71) ori ,max 1.5 npfw⋅⋅πρ/4 2/ ⋅

7.4.2 Optimum selection of abrasive type and size

Optimum selection of the abrasive type and size is based on prescribed objectives by AWJ users such as the cutting performance, the abrasive cost, the total cutting cost [Pi07a], and the profit rate. Therefore, the following procedure is suggested for optimum selecting the abrasive type and size:

-Optimum selecting other parameters such as the water pressure, the number of jet formers, the orifice diameter, and the nozzle diameter.

-Comparing for selecting the abrasive size and type: The objectives used for comparing can be the maximum depth of cut, the minimum cutting cost per length, or the maximum profit rate, depending on AWJ users.

The following example illustrates the optimum selection of the abrasive type and size. The same data of the example shown in Figure 7.6 were used together with nf=4, Lf=25 hours,ηo = 0.8 , and

Pelec=75 kW. Three different types of abrasives, viz. #80 HPX, #80 GMA, and #120 GMA, were used for comparison to select the optimum abrasive. Table 7.7 shows the calculated results for three distinct objectives: the maximum cutting performance (or minimum cutting time), the minimum cutting cost per unit length, and the maximum profit rate.

It follows that among given abrasives, the abrasive #120 GMA (with the abrasive cost per kilogram including disposal cost Ca=0.35 €/kg) has the highest value of the average feed speed (vf,a=1.98 mm/s), the lowest value of the cutting cost per meter (Cl=4.78 €/m), and the highest value of the profit rate (Pr,h=117.98 €/h). Therefore, this abrasive is simply the best considering all the

136 objectives.

As the cutting performance of #80 HPX is higher than that of #80 GMA (see Subsection 6.4.3.4), it is therefore more effective to use #80 HPX instead of #80 GMA for cutting for the maximum cutting performance (the average feed speeds were 1.9 mm/s and 1.7 mm/s when cutting with #80 HPX and #80 GMA, respectively). Nevertheless, #80 HPX (Ca=0.7 €/kg) is much more expensive than

#80 GMA (Ca=0.3 €/kg), hence the abrasive #80 GMA is better than #80 HPX in terms of the minimum cutting cost and the maximum profit rate (see Table 7.7).

Table 7.7: Calculated results for optimum abrasive selecting

For maximum cutting For minimum For maximum Abrasive Abrasive performance cutting cost profit rate cost (€/kg) P maophm, v fa, maopCl, C l maopPr, rh,

#80HPX 0.7 7.11 1.90 4.92 5.93 5.54 80.25 #80GMA 0.3 7.11 1.70 5.79 5.32 6.28 88.59 #120GMA 0.35 7.57 1.98 5.93 4.78 6.61 117.98

7.4.3 Procedure for determination of the optimum AWJ cutting regime

The procedure for determining the optimum cutting regime is as follows:

-Selection of the water pressure;

-Selection of the standoff distance (usually about 2 mm);

-Selection of the number of jet formers (the possible maximum);

-Calculation of the orifice diameter (Equation 7.71);

-Calculation of the initial nozzle diameter: df,0=(3…3.3).dori;

-Calculation of the optimum abrasive mass flow rate (Equation 7.58, Equation 7.63, and Equation 7.68);

-Calculation of the nozzle exchange diameter (Equation 7.48, and Equation 7.53);

-Selection of the abrasive type and size by comparing the average feed speed (Equation 7.42 for the maximum cutting performance), comparing the cutting cost per unit length (Equation 7.22 for the minimum cutting cost per meter), and comparing the profit rate (Equation 7.30 for the maximum profit rate).

It is noted that the optimum abrasive mass flow rate (for both the minimum cutting cost and for the maximum profit rate – see Subsection 7.3.2 and 7.3.3) depends on the nozzle lifetime. At first, for

137 rough estimation of the optimum abrasive mass flow rate, the nozzle lifetime can be chosen Lf=25

(h) for ROCTEC100 or Lf=49 (h) for ROCTEC500 nozzles (see Subsection 6.2.3.2). Next, this first estimation of the abrasive mass flow rate is used for rough calculation of the optimum nozzle exchange diameter. Finally, the optimum abrasive mass flow rate is re-estimated using the calculated optimum nozzle lifetime.

7.5 Conclusions

In this chapter, the AWJ optimization problems have been resolved to determine the optimum nozzle exchange diameter (or the optimum nozzle lifetime) and the optimum abrasive mass flow rate. The use of the “Golden ratio search” algorithm for the optimization problems has proven to be very efficient and relatively easy to incorporate in a computer program.

Two objectives have been proposed in the optimization problems to predict the optimum nozzle lifetime (or nozzle-exchange diameter. These are the minimum cutting cost per unit length and the maximum profit rate. For the optimum abrasive mass flow rate three objectives have been suggested which include the maximum cutting performance, the minimum cutting cost per unit length and the maximum profit rate.

The results of the problems have been effectively formulated through regression analyses and the use of the Buckingham Pi theory. Two models for the determination of the optimum nozzle- exchange diameter and three models for the calculation of the optimum abrasive mass flow rate (for the above objectives) have been derived.

The effects of various process parameters (e.g. the water pressure, the initial nozzle diameter, the abrasive particle diameter, etc.) and the cost elements (such as the hourly machine tool cost, the wages including overhead cost, the nozzle cost per piece etc.) on the optimum nozzle diameter as well as on the optimum abrasive mass flow rate have been taken into account.

By considering the cost elements as well as many other parameters e.g. the nozzle exit bore wear rate, the profit coefficient, etc. as variables in the optimization problems, the results of these calculations are generalized and can be used for a variety of practical applications.

Cutting with the optimum nozzle lifetime as well as with the optimum abrasive mass flow rate can be very beneficial, saving a lot of the profit rate, the cutting cost per unit length, and of the cutting time.

The optimum selection of other parameters including the water pressure, the orifice diameter, the nozzle diameter and the number of jet formers has been discussed. Moreover, a procedure for the prediction of the optimum AWJ cutting regime has been proposed.

138 8 Recycling and recharging of abrasives

In this chapter, the recycling and recharging of GMA abrasive, the world’s most popular abrasive for cleaning and AWJ cutting, is investigated. The investigation includes the reusability of GMA abrasive, the cutting performance and the cutting quality of the recycled and recharged abrasive. Also, the optimum particle size for getting the maximum cutting performance for both recycling and recharging of GMA abrasive is determined.

8.1 Reusability of abrasives

In AWJ machining, the abrasives after cutting can be reused for cutting several times (one, two, or three times). The reusability of an abrasive is the percentage of the abrasive amount before cutting that can be reused. In this section, the reusability of GMA garnet after first cut (or first recycling) and after second cut (or the second recycling) is investigated.

8.1.1 Experimental setup

To determine the reusability of the first recycling of GMA garnet, the abrasives after cutting are collected, washed, dried, chips-separated, sieved and sorted. Figure 4.6 shows the experimental setup for collecting the abrasives. The experimental parameters were: GMA garnet #80, the water pressure of 360 MPa, the orifice diameter of 0.255 mm, the focusing tube diameter of 0.92 mm, the abrasive mass flow rate of 400 g/min, the feed speed of 60 mm/min (for getting the rough cut) and the workmaterial was mild steel.

In the experiment, eight samples of abrasives were collected from 63 kg of new GMA abrasives. To collect the abrasives, a big tank was used as a special catcher. To slow down the abrasive particles without breaking them any further, the catcher was filled with water. After collecting, the abrasives were washed and then dried. The mild-steel chips were separated by using magnetic separation. For abrasive sieving, a sieve shaker and thirteen sieves (International standard -ISO3310-1) were used (see Figure 4.7). The nominal aperture sizes of the sieves were 45, 63, 75, 90, 106, 125, 150, 180, 215, 250, 300, 355 and 425 micrometer.

To investigate the reusability of the abrasive after the second cut (or the second recycling), a sample of the collected abrasive after the first cut was used for cutting and collecting the abrasives. The setup and the procedure of the experiment was the same as the one for the first recycling (see

139 above).

8.1.2 Results and discussions

Table 8.1: Reusability after first cut of GMA#80

Sieve Reusability (%) size Sample Sample Sample Sample Sample Sample Sample Sample Average (μm) 1 2 3 4 5 6 7 8

>300 1.71 1.38 1.89 1.91 1.13 1.99 3.21 3.94 2.15 >250 6.73 5.78 7.73 6.82 5.55 7.41 9.43 11.61 7.63 >212 16.38 14.78 19.22 17.02 15.11 18.26 21.49 25.65 18.49 >180 24.02 21.29 26.56 23.91 22.72 24.53 27.60 34.29 25.61 >150 33.79 31.39 37.26 34.39 34.38 31.98 34.39 41.53 34.89 >125 41.86 40.24 48.02 44.44 44.09 41.63 44.83 50.55 44.46 >106 46.56 45.05 53.86 49.73 49.38 47.66 50.64 55.49 49.80 >90 50.06 48.69 58.14 54.41 53.17 51.40 54.19 59.05 53.64 >75 53.49 52.48 62.23 57.52 57.11 54.59 56.99 62.50 57.11

70 57.11 60 53.64 49.8 50 44.46

40 34.89 25.61 30 18.49

Reusability (%) 20 7.63 10 2.15 0 >300 >250 >212 >180 >150 >125 >106 >90 >75 (µm ) Sieve nominal aperture size (µm) Figure 8.1: Reusability of GMA garnet after the first cut

To determine the reusability of GMA garnet using various particle sizes, after sieving, the abrasives were sorted in size range from >75 to >300 microns. Table 8.1 shows the results of the reusability after the first cut from 8 collected samples. The average reusability of these samples is also calculated. Figure 8.1 illustrates the average reusability of GMA garnet from various abrasive sizes. It follows that the reusability decreases with the increase of the recycled particle size. For example, the average reusability after the first cut of particles more than 75 µm was 57.11% while it was only 7.63% for particles size of over 250 µm.

From the results, the reusability of particles over 212 µm or larger is too small (less than 18.5%) to be profitable. Therefore, the particle sizes of over 75, 90, 106, 125, 150 and 180 µm are chosen to

140 determine the optimum size for the recycling and recharging. In this chapter, the optimum particle size for getting the maximum cutting performance will be investigated. For the minimum total cutting cost and the maximum profit rate, the optimum particle size will be discussed in the next chapter.

Table 8.2 presents the reusability after the first and the second cut (or the first and the second recycling). The cumulative retained mass of the first recycling, the second recycling and #80 GMA abrasives are shown in the Figure 8.2. It follows that the reusability of the first and the second recycling is almost the same for the abrasive size of more than 75, 90 and 106 µm (Table 8.2). This is because the main particle breakdown process may be a shift from big particles into medium size particles and then to very small size [Labu91]. In this case, about 10% of the mass of particles from 212 to 300 µm are shifted to the sizes from 75 to 180 µm (Table 8.2).

Table 8.2: Reusability after the first and the second cut

First recycling Second recycling Sieve size (μm) Absolute retained Reusability Absolute retained mass Reusability mass (%) (%) (%) (%)

>300 3.86 3.94 0.33 0.36 >250 7.67 11.61 2.77 3.13 >212 14.04 25.65 9.90 13.03 >180 8.64 34.29 9.83 22.85 >150 7.25 41.53 9.85 32.70 >125 9.01 50.55 11.53 44.23 >106 4.94 55.49 7.34 51.57 >90 3.56 59.05 4.61 56.18 >75 3.45 62.50 4.03 60.21

8.2 Cutting performance and cutting quality of recycled abrasives

8.2.1 Experimental setup

The setup of the experiment for the comparison of the cutting performance (or the maximum depth of cut) of the recycled abrasives was shown in Figure 4.5a. The workpiece made of Al6061T6 is shown in Figure 4.5b. Six samples of recycled abrasives (with particles larger than 63, 75, 90, 106, 125, 150 and 180 µm) and a sample of new GMA #80 were used for the test. Each sample of abrasives was tested with six cuts and with two replications, and with the following regime: the water pressure of 360 MPa, the feed speed of 120 mm/min, the orifice diameter of 0.255 mm, the nozzle diameter of 0.92 mm, three levels of the abrasive mass flow rate (50, 150 and 300 g/min).

141 100

40 #80 GMA 20 I−recycled ab. II−recycled ab. Cumulative retained mass (%) 0 425 355 250 180 125 90 63 Sieve nominal aperture size (µm) Figure 8.2: Cumulative retained mass of recycled abrasives

To compare the cutting performance of the first and the second recycled abrasives with the new GMA#80, the same setup was used. The first and second recycled samples were chosen with particles larger than 90 µm. Nine cuts were done for each sample with the same process parameters mentioned above and with two levels of the water pressure (300 and 360 MPa). Cuts with the water pressure of 360 MPa were done with two replications and with 300 MPa were done with one.

To investigate the influence of the recycled abrasives on the surface roughness, 5 samples of recycled abrasives (with the particles larger 63, 75, 90, 106 and 125 µm) and new GMA#80 were cut with the following regime: the water pressure of 300 MPa, the abrasive mass flow rate of 300 g/min, the orifice diameter of 0.255 mm, the focusing tube diameter of 0.92 mm, the feed speed of 120 mm/min, the work material Al6061T6, and with two replications.

8.2.2 Results and discussions

Figure 8.3 shows the cutting performance of the recycled abrasives. It follows that the cutting performance of all samples is higher than that of new GMA#80. This is because the average particle diameter (calculated by Equation 2.11 [Momb98]) of recycled abrasives is smaller than that of new abrasives (Table 8.3). Consequently, with a unit time, the number of particles in the recycled samples and then the number of particles taking part in the cutting process increases. Therefore, the volume of removed workmaterial and consequently the cutting performance increases. This result agrees with observations from Ohlsen [Ohl97] in which the recycled abrasives with the size in the range 125 to 150 microns can lead to the maximum depth of cut (or the maximum cutting performance). In addition, it is found that there is an optimum value of the recycled abrasive size with which the cutting performance is maximum [Pi08]. In this case, the optimum value is found for the sample with particles larger than 90 µm (Figure 8.3).

142 1.2 1.173 1.147 1.136 1.15 1.117 1.126

1.1

1.03 1.05 1 1 Normalized

0.95 cutting performance (−) 0.9 #80 GMA >63 >75 >90 >106 >125 >150 (µm) Abrasive sample

Figure 8.3: Cutting performance of recycled abrasives

Table 8.3: Average particle diameter of first recycled abrasives

Abrasive #80 >180 >150 >125 >106 >90 >75 >63 sample GMA µm µm µm µm µm µm µm Average particle diameter 246.32 215.3 195.6 177.66 165.59 155.32 145.59 137.69

The cutting performance of the first and second recycled abrasives is also investigated (Figure 8.4). It is observed that the cutting performance of both first and second recycled abrasive samples is higher than that of the new GMA#80. The reason is that, as indicated in the above analysis, both the average particle diameters of the first recycled particles (181.1 µm in this case) and the second recycled particles (156.3 µm) are smaller than that of the new (246.32 µm).

The above results are not in agreement with the observations from M. Kantha Babu et al. [Babu03] who noted that the cutting performance of the recycled abrasives is much less than that of the new abrasives (only 0.82 and 0.79 for the first and the second recycled abrasives, respectively). The reason could be the effect of the shape of the new abrasive particles on the shape and thus on the cutting performance of the recycled abrasives. Figure 8.5 shows the breaking mechanism of abrasive particles [Pi08]. In practice, particles tend to break into a few large and many small pieces. If a particle is broken into two big and into many smaller pieces, a round particle will usually be decomposed into two long particles (Figure 8.5a); a long particle will result in two round particles (Figure 8.5b). That means that new round particles can lead to long sharp recycled abrasives while new long particles can lead to round recycled abrasives. In this study the roundness of the new particles (#80 GMA) is 0.4 to 0.8 and in Babu’s study (Indian garnet #80) it is 0.2 to 0.6 [Fold01]. Therefore, after cutting process, the roundness of the recycled Indian particles is larger than that the new particles (Figure 8.6) and therefore their cutting performance reduces. In contrast, as the roundness is larger than that of new Indian particles, GMA particles after cutting can be broken into shaper particles (see Figure 8.7) and their cutting performance is better than that of the new.

143 1.3

1.173 1.151 1.2

1.1 1

1 Normalized 0.9

cutting performance (−) 0.8 #80 HPX I−Recycling II−Recycling Abrasive sample

Figure 8.4: Cutting performance of the first and second recycled abrasives

a) b)

Figure 8.5: Mechanism of particle breaking

a) New Indian abrasives #80 b) First recycled abrasives (>90 μm) Figure 8.6: Effect of the initial shape on the recycled shape - for Indian garnet [Babu03]

144 200μm/div 200μm/div

a) New GMA#80 b) First recycled abrasives (>90 µm)

b) Second recycled abrasives (>90 µm)

200μm/div

Figure 8.7: Effect of the initial shape on the recycled particle shape - for GMA garnet

• Kerf width and kerf taper:

1.2

1.1

0.9

0.8 Top kerf width (mm)

Kerf parameter Bottom kerf width (mm) 0.7 kerf taper (−)

0.6 >63 >75 >90 >106 >125 >150 #80 µm µm µm µm µm µm GMA Recycled abrasive sample Figure 8.8: Effect of recycled abrasives on kerf parameters with different samples

Figure 8.8 describes the top width, the bottom kerf widths (at the position of the rough cut -

145 0.8xhmax) and the taper (equals the ratio of the top and the bottom width) when cutting with 6 samples of recycled abrasives (with the particles larger than 63, 75, 90, 106, 125, and 150 µm) and #80 GMA abrasives (The process parameters used in the tests were: the water pressure of 360 MPa, the orifice diameter of 0.255 mm, the focusing tube diameter of 0.92 mm, the abrasive mass flow rate of 300 g/min, the feed speed of 2 mm/s). It is observed that the kerf widths (at the top and the bottom) when cutting with recycled abrasives are smaller than those with new GMA abrasives (Figure 8.8). This result is in agreement with those when cutting with recycled Indian abrasives [Babu03]. This is because of the reduction of the particle size of the recycled abrasives compared to the new. However, the particle size reduction is not significant and the kerf tapers are almost unchanged (Figure 8.8).

1.25

1.2

1.15

1.1 Top kerf width (mm) Bottom kerf width (mm) 1.05 kerf taper (−)

1 Kerf parameter 0.95

0.9 I−recycled ab. II−recycled ab. #80 GMA >90 µm >90 µm Abrasive sample Figure 8.9: Effect of multiple recycled abrasives on kerf parameters

The effect of the first and the second recycled abrasives on the kerf parameters is shown in Figure 8.9. As noted by M. Kantha Babu [Babu03], continuous recycling can lead to a decrease of the kerf widths (at the top and the bottom) as well as of the kerf taper.

• Surface roughness:

The surface roughness (Ra) was measured by a tactile stylus tip device (Talysurf 2, Taylor Hobson – see Section 4.8) and at the depth of 10 mm with the following parameters: the measurement speed was 0.5 mm/s, the cut-off was 2.5 mm, the bandwidth was 300:1, and the evaluation length was 12.5 mm. The average of the surface roughness of two replications of cutting is shown in Figure 8.10. It follows that the surface roughness decreases with the decrease of the size of the recycled abrasives (see Figure 8.10). This can be explained smaller particle diameter leads to smaller value of Ra.

146 9.5

9 (µm)

a 8.5

7.5

6.5

Surface roughness R 6

5.5 Recycled Recycled Recycled Recycled Recycled #80 GMA >63 µm >75 µm >90 µm >106 µm >125 µm Abrasive sample Figure 8.10: Effect of recycled abrasives on the surface roughness

8.3 Cutting performance and cutting quality of recharged abrasives

As mentioned in Chapter 2, abrasive recharging is a process in which new abrasives are added to recycled abrasives. The aim of the recharging is to maintain the amount of input abrasives as well as to ensure the maximum cutting performance is maintained at all time. This section is concerned with the cutting performance and the cutting quality of the recharged abrasives.

8.3.1 Experimental setup

To determine the optimum particle size of the recharged abrasives, seven samples of the first recycled abrasives (with the particles larger than 45, 75, 90, 106, 125, 150 and 180 µm) were recharged with new abrasives (#80 GMA) in order to get the same amount of abrasives (100%) (Table 8.4). The cutting performances of these recharged abrasive samples were then investigated by doing an experiment with the same setup as the experiment for the cutting performance of recycled abrasives (Figure 4.5). Each recharged abrasive sample was tested with 6 cuts with two replications and with the following setting: one level of the water pressure (300 MPa), three levels of the abrasive mass flow rate (50, 150 and 300 g/min), one level of feed speed (120 mm/min), one level of the orifice diameter (0.255 mm), one level of the nozzle diameter (0.92 mm), and the workmaterial of Al6061T6. The results of this experiment were also used to study the effect of the recharged abrasives on the kerf widths of the cuts.

To investigate the influence of the recharged abrasives on the surface roughness, 5 samples of recharged abrasives (with the abrasive size larger than 90, 106, 125, 150 and 180 µm) and GMA#80 were cut with two replications and with the following regime: the water pressure of 300 MPa, the abrasive mass flow rate of 300 g/min, the orifice diameter of 0.255 mm, the nozzle diameter of 0.92 mm, the feed speed of 120 mm/min, and the workmaterial of Al6061T6.

147 Table 8.4: Recharging abrasives

Sample Sieve nominal Reusability Recharged abrasive aperture size (µm) (%) (%)

1 180 25.61 74.39 2 150 34.89 65.11 3 125 44.46 55.54 4 106 49.80 50.20 5 90 53.64 46.36 6 75 57.11 42.89 7 45 65.77 34.23

8.3.2 Results and discussions

Figure 8.11 describes the cutting performances of the recharged abrasives. It follows that the cutting performances of all recharged samples are higher than that of the new GMA#80. In addition, it is observed that there is an optimum particle size for the recharging with which the cutting performance is maximum [Pi08]. In this case, recharging with recycled particles larger than 90 µm is optimum for the cutting performance (Figure 8.11). This result is also in agreement with the optimum size of the recycled abrasives (particles larger than 90 µm) for the maximum cutting performance.

1.12 1.09 1.1 1.075 1.08

1.06 1.029 1.032 1.04 1.028 1.02 1.004 Normalized 0.996 1

cutting performance (−) 0.98

0.96 >45 >75 >90 >106 >125 >150 >180(µm) Abrasive sample

Figure 8.11: Cutting performance of recharged abrasives

• Kerf width and kerf taper:

Figure 8.12 demonstrates the top width, the bottom kerf widths (at the position of the rough cut -

* 0.8 hmax ) and the taper (equals the ratio of the top and the bottom width) when cutting with 5 samples of recharged abrasives and new GMA#80. It is shown that cutting with recharged

148 abrasives does not reduce the kerf widths much when compared to cutting with the new abrasive as cutting with the recycled abrasives does (see Subsection 8.2.2). The top and the bottom kerf widths in this case are only slightly smaller than those when cutting with the new abrasive (Figure 8.12). The reason is that when adding more new abrasives (about 46.36% to 74.39%) the number of big particles in the recharged abrasives is much higher than that in the recycled abrasives. Therefore, the kerf widths when cutting with recharged abrasives can not be reduced significantly.

1.2

1.1

0.9

Kerf parameter 0.8 Top kerf width (mm) Bottom kerf width (mm) 0.7 Kerf tape (−)

0.6 Recharged Recharged Recharged Recharged Recharged #80 GMA >90 µm >106 µm >125 µm >150 µm >180 µm Abrasive sample (−)

Figure 8.12: Effect of recharging on kerf parameters

• Surface roughness:

7.5

6.5

5.5 Surface roughness (µm)

5 Recharged Recharged Recharged Recharged Recharged Recharged #80 GMA >63 µm >90 µm >106 µm >125 µm >150 µm >180 µm Abrasive sample (−) Figure 8.13: Effect of recharging on surface roughness

As was done for the recycled abrasives, the surface roughness (Ra) when cutting with the recharged abrasives was measured at the measurement depth of 10 mm, the measurement speed of 0.5 mm/s, the cut-off of 2.5 mm, the bandwidth of 300:1, and the evaluation length of 12.5 mm. The average surface roughness of two replications of cutting was shown in Figure 8.13. It follows that

149 the surface roughness when cutting with the recharged abrasives (for all samples) is generally smaller than that when cutting with the new abrasive. However, there is no clear trend and the average surface roughness values when cutting with recharged abrasives are around 6 µm.

8.3.3 Multi-recharging of abrasive

8.3.3.1 Experimental setup

As addressed in Chapter 2, abrasive recharging adds new abrasives to recycled abrasives in order to maintain the amount of input abrasives as well as the cutting performance. It is expected that the recharging process can be done continuously without significant effects on the cutting performance. In order to verify this as well as to determine the reusability of the recharged abrasives after the first, the second and the third cut (or multi-recharging), experiments are carried out. Since the recharged abrasive with particles larger than 90 µm is the optimum, the experiments for the reusability and the cutting performance of multiple recharged abrasives were performed with this size. A sample of recycled abrasives after the first cut (with the particles larger than 90 µm) was used for making the first recharged abrasives. The first recharged abrasives are used for cutting in order to collect the abrasives for making the second recharged abrasives. The third recharged abrasives are also created using the same method. With each recharged abrasive of the multi- recharging, the reusability is measured by sieve analysis. In addition, to compare the cutting performance of these recharged abrasives (I, II and III-recharged abrasives) with the new abrasive, an experiment (with the setup in Figure 4.3) was performed with the following process parameters: two levels of the water pressure (300 and 360 MPa), three levels of the abrasive mass flow rate (50, 150 and 300 g/min), one level of the feed speed (120 mm/min), one level of the orifice diameter (0.255 mm), one level of the nozzle diameter (0.92 mm), one replication with the water pressure of 300 MPa, and two replications with 360 MPa.

8.3.3.2 Results and discussions

Table 8.5 shows the reusability of the first, the second and the third recharging. It is observed that the reusability is nearly equal for three levels of the recharging. For the recharging with particles larger than 90 µm, the reusability is 52.8, 53.06 and 52.85 % with the first, the second and the third recharging, respectively.

Figure 8.14 plots the cutting performance of the first, the second and the third recharged abrasives in comparison with new GMA#80. It follows that the cutting performance coefficients of recharged abrasives are almost the same with different levels of the recharging and they are all higher than those of the new abrasive (The average of the cutting performance coefficients of the first, second and the third recharged abrasives was 1.09). The reason could be the reusability values of all levels are nearly unchanged. Therefore, the amounts of new and recycled abrasives in the recharged

150 abrasives are not much different. In addition, the cutting performance coefficients of the recycled abrasives are also almost the same with different levels of the recycling (see Subsection 8.2.2). Consequently, the cutting performance coefficients of the recharged abrasives are constant after multiple recharging.

From the above results, the multi-recharging can be done continuously without affecting the reusability and the cutting performance.

Table 8.5: Reusability of multi-recharged abrasives

Reusability (%) Sieve size (μm) I-recharged II-recharged III-recharged >300 1.13 1.67 1.52 >250 5.51 6.03 6.25 >212 15.01 15.91 16.51 >180 22.56 21.64 22.46 >150 34.14 28.52 29.05 >125 43.78 41.33 41.86 >106 49.04 48.92 49.32 >90 52.80 53.06 52.85 >75 56.71 56.54 56.01

1.2

1.105 1.071 1.085 1.1

1 1

0.9 Cutting performance (−)

0.8 I−recharged II−recharged III−recharged #80 GMA Abrasive sample Figure 8.14: Cutting performance of multiple recharged abrasives

8.4 Conclusions

In this chapter, the reusability of GMA garnet, including the reusability of the first and the second recycling have been investigated. In addition, the reusability of the first, the second, and the third recharged abrasives have been analyzed.

151 The cutting performance as well as the cutting quality of the recycled and the recharged abrasives (with different abrasive size) was investigated. Also, it has been found that the optimum particle size for the cutting performance is particles larger than 90 µm for both the recycling and the recharging.

The cutting performance when cutting with recycled and recharged abrasives is higher than that when cutting with new abrasives (about 17% and 10% when cutting with recycled and recharged abrasives with particles larger than 90 µm, respectively).

The effects of the recycled and recharged abrasives on the cutting quality have been evaluated. It is found that the kerf width and the surface roughness when cutting with recycled abrasives and recharged are slightly smaller than those when cutting with new abrasives.

It has been concluded that the multi-recharging can be done continuously without affecting the reusability and the AWJ cutting performance.

152 9 Economics of abrasive recycling

This chapter is concerned with the economics of the recycling of GMA abrasives. For doing this, firstly, a cost analysis for the recycled and recharged abrasives is carried out. The cost model takes into account the effects of various cost elements such as the hourly machine tool cost, the wages including overhead cost per hour, etc. Next, two ways for economic comparisons in order to select suitable abrasives is introduced. They are compared of the minimum cutting cost per unit length and the maximum profit rate. Based on these results, the economics of cutting with recycled and recharged abrasives is investigated. Finally, suggestions for an effective abrasive recycling process are proposed.

9.1 Cost calculation for recycled and recharged abrasives

9.1.1 Cost analysis

In this section, a cost analysis for recycled abrasives as well as for recharged abrasives is conducted. In the cost analysis, the influences of many cost elements on the recycle abrasive cost are investigated. These effects are illustrated by the calculated results with the data of WARD 1 and WARD 2 (WARD 1 and WARD 2 are recycling systems of WARDJet, Inc. – see Appendix).

9.1.1.1 Cost analysis for recycled abrasives

The recycled abrasive cost per kilogram Ca,m,recy (€/kg) can be calculated by:

C hrecy, C amrecy,, = (9.1) kGut, recy⋅ recy

Where

Grecy-recycling capacity per hour of the recycling system (kg/h);

kut,recy-utilization coefficient of the recycling system which considers the abrasive input for recycling is less than the recycling capacity of the recycling system ( kut, recy ≤ 1 );

Ch,recy-the total recycled cost per hour (€/h); Ch,recy is calculated by the following equation:

CCCChrecy,,,,,=++ mth wah whrecy (9.2)

In the above equation, Cmt,h is the hourly machine tool cost (€/h) (see Equation 7.5); Cwa,h is the

153 wages including overhead cost (€/h) (Equation 7.14); Cw,h,recy is the hourly water cost (€/h).

It is noted that when calculation of the hourly machine tool cost (Equation 7.5) and the wages including the overhead cost per hour (Equation 7.14), the value of the utilization rate in recycling process (see Subsection 7.1.1) is different from that in AWJ machining process. The reason is that in practice, most of the recycling systems are directly connected with the AWJ cutting system and they can run automatically. Therefore, the recycling system can be operated with the same worker who also runs the AWJ system. Consequently, coefficient kmsh (see Equation 7.13) can be chosen from 0 to 0.2, depending on the policy of each individual company.

The water cost per hour can be determined by the following equation:

CCmwhrecy,,=⋅⋅3600 wm , wrecy , (9.3)

Where, Cw,m is the cost per kilogram of water (€/kg); mwrecy, is water mass flow rate of the recycling system (kg/s).

9.1.1.2 Cost analysis for recharged abrasives

As mentioned in Chapter 2, recharging of abrasives adds new abrasives to recycled abrasives in order to maintain the amount of input abrasives as well as the cutting performance. If the reusability of the abrasives is denoted as rabr (%) then the new abrasives added to make the recharged abrasives is (100-rabr) (%). Therefore, based on the cost of the new and the recycled abrasives, the recharged abrasive cost can be calculated as follows:

CrCr⋅−+(100 ) ⋅ C = am,,, abr amrecy abr (9.4) amrech,, 100

Where, Ca,m, Ca,m,rech, and Ca,m,recy are the cost per kilogram (€/kg) of the new abrasives (including disposal cost), the recharged abrasives (including disposal cost) and the recycled abrasives, respectively.

9.1.2 Results and discussions Based on the cost model (Equation 9.1), the effects of cost components on the cost of the recycled abrasives were investigated with the following assumed inputs: recycling systems WARD1 and WARD2; For WARD1, Crpl=50000 (€), Grecy =81.65 kg/h, mwrecy, =0.051 kg/s [Ward05a]; For

WARD2, Crpl=37000 (€), Grecy =36.29 kg/h, mwrecy, =0.032 kg/s [Ward05b]; kut,recy=1; xsh=2; tsh=8

2 2 hours/day, dwor=250 days/year, xut=0.8, xint=10 %, Ttot=5 years, Csqm=50 €/m , Amt=10 m , xmail=0.04, Ce=0.06 €/kWh, dop=100%, Cw,m=0.004 €/kg.

154 0.25 0.6

k =0.1 msh k =0.2 0.5 WARD1 msh WARD2 0.2 0.4 (EUR/kg) (EUR/kg) 0.3 0.15 a,m,recy a,m,recy C C 0.2 Recycled abrasive cost Recycled abrasive cost

0.1 10 20 30 40 50 0.1 Wages including overhead C (EUR/h) 1 2 3 4 5 wag,h Depreciation year T (−) tot a) b)

0.5 0.3 WARD1 WARD2 0.25 0.4

0.2

0.3 (EUR/kg) (EUR/kg) 0.15 a,m,recy a,m,recy 0.2 C C 0.1 WARD1 WARD2 Recycled abrasive cost Recycled abrasive cost

0.1 0.05 0.6 0.7 0.8 0.9 1 0.04 0.06 0.08 0.1 Utilization coefficient k (−) Electric power cost C (EUR/kWh) ut,recy e c) d)

0.3 0.4

0.35 C =50000 Eu rpl 0.25 C =100000 Eu 0.3 rpl C =150000 Eu rpl 0.25 0.2 0.2 (EUR/kg) (EUR/kg)

0.15 0.15 a,m,recy a,m,recy C WARD1 C 0.1 0.1 WARD2 Recycled abrasive cost Recycled abrasive cost 0.05

0.05 0 2 4 6 8 10 50 100 150 200 250 Water cost C (EUR/kg) −3 Drying capacity G (kg/h) w,m x 10 dc

e) f) Figure 9. 1: Cost components versus recycled abrasive cost

155 Figure 9.1 shows the effect of cost elements on the recycled abrasive cost. It follows that the recycled abrasive cost depends strongly on the wages including overhead cost and the coefficient kmsh (Figure 9.1a). For example, for WARD1, an increase of the wages from 10 to 50 (€/h) causes the recycled abrasive cost to increase by 36.5% for kmsh=0.1 and by 50% for kmsh=0.2. Hence, the abrasive recycling can give much more profit in regions where the labour cost is cheap such as developing countries in Asia or Africa [Pi06].

The effects of the depreciation years and the utilization coefficient of the recycling system kut,recy were shown in Figure 9.1b and 9.1c. As a result, increasing the depreciation years as well as the utilization coefficient (by cutting with multiple cutting heads or using one recycling system for several AWJ cutting machines) is a good way to reduce the cost of recycling. It is found that the electric power cost and the water cost affect lightly the cost of recycled abrasives. This is because the demand for the electric power and the water used for recycling one kilogram of abrasives is very small. Figure 9.1f describes the effect of the drying capacity on the recycled-abrasive cost. It follows that the recycled abrasive cost depends strongly on the drying capacity of the recycling system. Therefore, to reduce the recycled abrasive cost, the drying capacity of a recycling system should be as large as possible [Pi07a].

From Figure 9.1, it can be seen that with every value of the cost components the cost of recycled abrasives done with WARD 1 system is usually much less than done with WARD 2 system. This is because the cost of the recycled abrasives depends strongly on the recycling capacity (Figure 9.1f) and the recycling capacity of WARD1 (81.65 kg/h) is much larger than that of WARD 2 (32.69 kg/h) (see Subsection 9.1.2). As a result, use of WARD1 is the best among the considered systems.

0.4

0.35

0.3 (EUR /kg) 0.25

a,m,rech WARD1 C 0.2 WARD2 Recharged abrasive cost 0.15 20 30 40 50 60 70 80 Reusability (%)

Figure 9. 2: Recharged abrasive cost versus reusability

From Equation 9.4, it becomes clear that the trend of cost components on the recharged abrasive cost is the same as that on the recycled abrasive cost. In addition, the relationship between the

156 reusability and the cost of recharged abrasives, as described by Equation 9.4, is shown in Figure

9.2 (calculated with Ca,m=0.4 €/kg; other parameters were the same when calculating the recycled abrasive cost- Subsection 9.1.1). It is observed that even when the reusability is very low (for example 20%), the recharged abrasive cost is still less than the cost of new abrasives (for both recycling systems WARD1 and WARD2).

9.2 Economic comparisons for selecting abrasives

As mentioned in Chapter 7, to select an abrasive properly, it is needed to compare not only the cutting performance and the cost of abrasive itself, but also the total cutting cost [Pi07a]. In addition, the profit rate, a very important objective, has to be compared as well.

For cost comparison when cutting with new and recycled (or recharged) abrasives, the cutting cost per length is used. Using Equation 7.21 for cutting with new abrasive, the cutting cost per unit length Cl,recy (€/m) when cutting with recycled abrasives can be calculated as follows:

C h C lrecy, = (9.5) 3600 ⋅⋅nvffarecy,,

In which

vkvfarecy,,=⋅ c fa , (9.6)

Where

Ch -AWJ cutting cost per hour (€/h) (see Equation 7.20);

vf,a, vf,a,recy -the average feed speed when cutting with the new and the recycled abrasives, respectively (m/s);

nf -number of the jet formers;

kc -cutting performance coefficient, considering the difference between the cutting performance of the new abrasives (kc=1) and the cutting performance of the recycled (or recharged) abrasives.

For comparison of the profit rate, using Equation 7.36 for cutting with new abrasive, the following equation will be used for calculation of the maximum profit rate when cutting with recycled abrasives:

Prrecy,,,,0, h=⋅ 3600 vkCC f a recy ⋅⋅−( p l l recy ) (9.7)

Where, the cutting cost per unit length Cl,recy and the average feed speed vf,a,recy are determined as mentioned above.

157 20 100 k =1.2 c k =1.1 c k =1 c 15 75 k =0.9 13.83 c

10 50

(EUR/m) 48.14 k =1.2 c k =1.1 5 c 25 k =1 c Minimal cutting cost k =0.9 c Maximal profit rate (EUR/h) 0 0 0 0.3 0.38 0.6 0.9 1.2 0 0.161 0.5 0.9 1 1.5 Abrasive cost per kilogram C (EUR/kg) Abrasive cost per kilogram C (EUR/kg) a,m a,m

a) b) Figure 9. 3: Economic comparisons for selecting abrasives

Figure 9.3 shows the effects of abrasive cost per kilogram Ca,m and the cutting performance coefficient kc on the minimum cutting cost (Figure 9.3a) and on the maximum profit rate (Figure

9.3b). The assumed inputs for this calculation were: nf=1, Crpl= 200000 €, df,0= 0.76 mm, dori= 0.25 mm, Lf =90 h, and kp=0.5. Other data were the same as those in the example in Figure 7.6.

It is observed that the effect of the cutting performance coefficient on the cutting cost is much higher than that of the abrasive cost per kilogram [Pi06]. For example, to have the same cutting cost of 13.83 €/m, the total cost when cutting with a new abrasives having kc=1and Ca,m=0.38

€/kg, a recycled abrasive having kc=1.1 can have the abrasive cost of 0.9 €/kg (2.37 times of the cost of new abrasives) (see Figure 9.3a).

It is also found that the influence of the cutting performance on the maximum profit rate is much larger than that on the minimum cutting cost. As shown in Figure 9.3b, for having the same profit rate of 48.14 €/h, the profit rate when cutting with a new abrasives having kc =1 and Cam, =0.161

€/kg, a recycled abrasive having kc =1.1 can have the abrasive cost of 0.9 €/kg (5.59 times of the cost of new abrasives).

From these observations, it can be concluded that to compare the recycled and the new abrasives as well as two types of new or recycled abrasives in order to select a proper abrasive, it is needed to compare the total cutting cost per unit length and the maximum profit rate. Also, the abrasive cost per kilogram and especially the cutting performance are two main elements of the comparison [Pi07a].

In the following sections the economics of cutting with recycled and recharged abrasives will be

158 investigated based on this economic analysis.

9.3 Economics of cutting with recycled and recharged abrasives

In order to find the economics of cutting with recycled and recharged abrasives, several following terms of using abrasives are suggested:

-Cutting with new abrasives is a process in which only new abrasives are used for cutting;

-Cutting with first recycled abrasives (or I-recycled abrasives) is a process in which both new abrasives and I-recycled abrasives are used. The first recycled abrasives are made from the new abrasives. In addition, the total mass of the new and the recycled abrasives must be equal to that of new abrasives in the process of cutting with new abrasives when comparing.

-Cutting with multi-recycled abrasives (or with first and second recycled abrasives) is a process in which new abrasives, I-recycled abrasives and II-recycled abrasives are used. The first recycled abrasives are made from the new abrasives and the second recycled abrasives are made from the I-recycled abrasives. The total mass of the new, I-recycled and II-recycled abrasives must also equal the mass of new abrasives in the process of cutting with new abrasives when comparing.

The above terms will be used for comparisons of the minimum cutting cost and the maximum profit rate for finding the optimum particle size of recycled and recharged abrasives as well as for evaluating the economics of cutting with recycled abrasives, multi-recycled abrasives, and recharged abrasives.

9.3.1 Economics of cutting with recycled abrasives

9.3.1.1 Finding optimum recycled abrasive size

Table 9.1: Parameters of recycled abrasive samples

Abrasive Reusability Mass of new Mass of recycled Cost of recycled

sample (%) abrasives (kg) abrasives (kg) kc abrasives(€/kg)

>75 57.11 1122.78 641.22 1.147 0.135 >90 53.64 1148.14 615.86 1.173 0.144 >106 49.8 1177.57 586.43 1.136 0.155 >125 44.46 1221.10 542.90 1.126 0.174

To find the optimum abrasive size for recycling a cost comparison (see Section 9.2) was carried out. In this investigation, four samples of recycled abrasives (with particle sizes more than 75, 90, 106 and 125 µm) were used for calculating the AWJ cutting cost. The assumed data used in the

159 calculation were the same as those in Section 9.2. In addition, the parameters of the recycled samples were given in Table 9.1, and assuming that the cost per kilogram of the new abrasives

(including disposal cost) Cam, was 0.4 (€/kg).

Figure 9.4 illustrates the effect of the recycled abrasive size on the cutting cost per meter with different number of jet formers. It is observed that the minimum cutting cost reduces strongly when the number of the jet formers increases. In contrast, the influence of the recycled size on the total cutting cost is negligible (Figure 9.4). The reason is that the amount of the new abrasives used for cutting is much higher than that of the recycled abrasives (from 1.75 to 2.24 times when the recycled abrasive size is changed from >75 to >125 µm - see Table 9.1). Also, the mass of the new abrasives is not altered too much when the recycled size is changed (only about 8% when the recycled size varied from >75 to >125 µm). Therefore, the abrasive cost per meter is only affected slightly by the recycled abrasive size.

13 85

12 80 11 n =1 f n =2 75 10 f n 1 n =3 = f n =2 9 n =4 70 f f n =3 f n =4 8 65 f 7 60 6 55 5 Maximal profit rate (EUR/h) Minimal cutting cost (EUR/m) 4 50 >75 µm >90 µm >106 µm >125 µm >75 µm >90 µm >106 µm >125 µm Recyled abrasive sample Recyled abrasive sample Figure 9. 4: Recycled particle size versus Figure 9. 5: Recycled particle size versus cutting cost per meter profit rate

It is found that the particle size larger than 90 µm is the optimum value for the recycling. With this value, the cutting cost per unit length is minimum with all values of the number of the jet formers (Figure 9.4). The reason is that with this size the recycled abrasives have the maximum cutting performance (see Subsection 8.3.2).

The influence of the recycled abrasive size on the profit rate was shown in Figure 9.5. The maximum profit rate, like the minimum cutting cost per meter, depends strongly on the number of the jet formers. It increases considerably with the increase of the jet former number. The effect of the recycled abrasive size on the profit rate is not significant but it is stronger than that on the cutting cost per meter. This is because in this case the cutting performance affects both the cutting cost per meter and the average feed speed which are two main effecting factors on the profit rate

160 (see Equation 9.7).

It is found that the optimum recycled abrasive size for the maximum profit rate is larger than 90 µm (see Figure 9.5), like for the minimum cutting cost per unit length. Also, the reason of that is with this size the cutting performance is maximum.

9.3.1.2 Economics of cutting with first recycled abrasives

Being the optimum recycled abrasives (see the above subsection), the recycled abrasives with particles larger than 90 µm are used for investigating the economics of cutting with the I-recycled abrasives. In order to do this, comparisons of the minimum cutting cost per unit length and the maximum profit rate (see Section 9.2) when cutting with the new and the I-recycled abrasives were carried out.

Table 9.2: Minimum cutting cost per meter when cutting with new and I-recycled abrasives

Minimum cutting cost per meter (€/m) n f Cam, =0.2 (€/kg) Cam, =0.3 (€/kg) Cam, =0.4 (€/kg) Cam, =0.5 (€/kg) New I-recycled New I-recycled New I-recycled New I-recycled

1 13.14 12.07 13.53 12.32 13.91 12.57 14.29 12.82 2 7.09 6.48 7.46 6.73 7.82 6.96 8.17 7.2 3 5.74 5.24 6.1 5.47 6.45 5.71 6.79 5.93 4 4.55 4.15 4.9 4.38 5.24 4.6 5.57 4.82

Table 9.3: Total cost saving when cutting with I-recycled abrasives

Total cutting cost saving (%) n f Cam, =0.2 (€/kg) Cam, =0.3 (€/kg) Cam, =0.4 (€/kg) Cam, =0.5 (€/kg) 1 8.16 8.94 9.63 10.26 2 8.54 9.84 10.94 11.87 3 8.72 10.26 11.52 12.57 4 8.96 10.8 12.25 13.42

Table 9.2 shows the results of the cutting cost comparison when cutting with new and I-recycled abrasives. It is observed that the cutting cost when cutting with the I-recycled abrasives is much less than that when cutting with new abrasives. The total cost saving, when cutting with the I- recycled abrasives, can be 8.16 to 13.42 % in comparison with cutting with the new abrasives depending on the number of the jet formers and the cost of the new abrasives (Table 9.3). Figure 9.6 shows the comparisons of the normalized cutting cost per meter between cutting with new and first recycled abrasives.

The maximum profit rates when cutting with the new and the I-recycled abrasives were shown in Table 9.4. The profit rate when cutting with the I-recycled abrasives is much higher than that when

161 cutting with the new abrasives. Cutting with the I-recycled abrasive can save the profit rate 20.58 to 27.61 % in comparison with cutting with the new abrasives depending on the jet former number and the cost per kilogram of the new abrasives (Table 9.5). The comparisons of the normalized profit rate between cutting with new and first recycled abrasives are shown in Figure 9.7.

120

115 GMA#80 I−recycled abrasive with n =1 f I−recycled abrasive with n =2 110 f I−recycled abrasive with n =3 f I−recycled abrasive with n =4 105 f

100

per meter (%) 85

80 Normalized cutting cost 75

70 0.2 0.3 0.4 0.5 Abrasive cost C (EUR/kg) a,m

Figure 9. 6: Normalized cutting cost per meter when cutting with first recycled abrasives

160 GMA#80 I−recyled abrasive with n =1 150 f I−recyled abrasive with n =2 f I−recyled abrasive with n =3 140 f I−recyled abrasive with n =4 f

130

120

110

100

90 Normalized profit rate (%)

80 0.2 0.3 0.4 0.5 Abrasive cost C (EUR/kg) a,m

Figure 9. 7: Normalized profit rate when cutting with first recycled abrasives

162 Table 9.4: Maximum profit rate when cutting with new and I-recycled abrasives

Maximum profit rate (€/h) n f Cam, =0.2 (€/kg) Cam, =0.3 (€/kg) Cam, =0.4 (€/kg) Cam, =0.5 (€/kg) New I-recycled New I-recycled New I-recycled New I-recycled

1 40.11 50.5 41.25 52.65 42.34 54.71 43.4 56.7 2 43.16 54.68 45.25 58.65 47.22 62.4 49.09 65.98 3 52.29 66.47 55.3 72.22 58.12 77.61 60.78 82.7 4 55.16 70.4 58.97 77.73 62.51 84.52 65.83 90.93

Table 9.5: Profit rate saving when cutting with I-recycled abrasives

Profit rate saving (%) n f Cam, =0.2 (€/kg) Cam, =0.3 (€/kg) Cam, =0.4 (€/kg) Cam, =0.5 (€/kg) 1 20.58 21.65 22.6 23.45 2 21.08 22.85 24.32 25.59 3 21.33 23.43 25.11 26.5 4 21.64 24.13 26.05 27.61

9.3.1.3 Economics of cutting with multi-recycled abrasives

As done for the economics of cutting with the I-recycled abrasive, to investigate the economics of cutting with the multi-recycled abrasives, comparisons of the minimum cutting cost per unit length and the maximum profit rate when cutting with new and multi-recycled abrasives were conducted. The recycled size more than 90 µm was used since it is the optimum recycled particle size (see Subsection 9.3.1.1).

The minimum cutting cost per meter when cutting with the new and the multi-recycled abrasives was given in Table 9.6. It is found that the cutting cost per meter, when cutting with the multiple- recycled abrasives, is less than that when cutting with I-recycled abrasives. In this case, the cutting cost saving is from 11.7 to 18.39 % in comparison with cutting with the new abrasive, depending on the jet former number and the new abrasive cost (Table 9.7). Figure 9.8 shows the comparisons of the normalized cutting cost per meter between cutting with new and I and II-recycled abrasives.

Table 9.8 shows the calculated results of the maximum profit rates when cutting with the new and the multi-recycled abrasives. It is observed that the profit rate when cutting with the multi-recycled abrasives is much higher than that when cutting with the new. The profit rate, when cutting with the multi-recycled abrasives, can be saved by 27.81 to 36.74 % in comparison with cutting with the new abrasives depending on the jet former number and the cost per kilogram of the new abrasive when cutting with multi-recycled abrasive (Table 9.9). The comparisons of the normalized profit rate between cutting with new and I and II-recycled abrasives are shown in Figure 9.9.

163

Table 9.6: Minimum cutting cost when cutting with new and multi-recycled abrasives

Minimum cutting cost per meter (€/m) n f Cam, =0.2 (€/kg) Cam, =0.3 (€/kg) Cam, =0.4 (€/kg) Cam, =0.5 (€/kg) New I-II-recycled New I-II-recycled New I-II-recycled New I-II-recycled

1 13.14 11.6 13.53 11.79 13.91 11.98 14.29 12.16 2 7.09 6.22 7.46 6.4 7.82 6.58 8.17 6.75 3 5.74 5.02 6.1 5.19 6.45 5.37 6.79 5.54 4 4.55 3.97 4.9 4.14 5.24 4.31 5.57 4.47

Table 9.7: Total cutting cost saving when cutting with multi-recycled abrasives

Total cost saving (%) n f Cam, =0.2 (€/kg) Cam, =0.3 (€/kg) Cam, =0.4 (€/kg) Cam, =0.5 (€/kg) 1 11.7 12.86 13.91 14.87 2 12.25 14.22 15.89 17.33 3 12.52 14.85 16.77 18.39 4 12.88 15.65 17.87 19.69

Table 9.8: Maximum profit rate when cutting with new and multi-recycled abrasives

Maximum profit rate (€/h) n f Cam, =0.2 (€/kg) Cam, =0.3 (€/kg) Cam, =0.4 (€/kg) Cam, =0.5 (€/kg) New I-II-recycled New I-II-recycled New I-II-recycled New I-II-recycled

1 40.11 55.56 41.25 58.25 42.34 60.83 43.4 63.34 2 43.16 60.33 45.25 65.32 47.22 70.06 49.09 74.61 3 52.29 73.43 55.3 80.7 58.12 87.54 60.78 94.02 4 55.16 77.91 58.97 87.21 62.51 95.86 65.83 104.07

Table 9.9: Profit rate saving when cutting with multi-recycled abrasives

Profit rate saving (%) n f Cam, =0.2 (€/kg) Cam, =0.3 (€/kg) Cam, =0.4 (€/kg) Cam, =0.5 (€/kg) 1 27.81 29.18 30.39 31.48 2 28.46 30.73 32.61 34.2 3 28.79 31.47 33.6 35.35 4 29.2 32.38 34.8 36.74

164 120 GMA#80 I−II recycled abrasives with n =1 115 f I−II recycled abrasives with n =2 f 110 I−II recycled abrasives with n =3 f I−II recycled abrasives with n =4 105 f

100

per meter (%) 85

80 Normalized cutting cost 75

70 0.2 0.3 0.4 0.5 Abrasive cost C (EUR/kg) a,m

Figure 9. 8: Normalized cutting cost per meter when cutting with I-II recycled abrasives

180 GMA#80 I−II recycled abrasive with n =1 170 f I−II recycled abrasive with n =2 f I−II recycled abrasive with n =3 160 f I−II recycled abrasive with n =4 f 150

140

130

120

110

100

Normalized profit rate (%) 90

80 0.2 0.3 0.4 0.4 Abrasive cost C (EUR/kg) a,m

Figure 9. 9: Normalized profit rate when cutting with I-II recycled abrasives

165 9.3.2 Economics of cutting with recharged abrasives

9.3.2.1 Finding optimum recharged abrasive size

Using the same way as it was done for the recycled abrasives, comparisons of the minimum cutting cost per unit length and the maximum profit rate were performed in order to find the optimum recharged abrasive size. In these comparisons, four samples of the recharged abrasives (with the particle size larger than 75, 90, 106 and 125 µm) were used to calculate the minimum cutting cost and the maximum profit rate. The data used in the calculations were the same as those in Section 8.2. In addition, the parameters of the recharged samples were given in Table 9.10 and assuming that the cost of the new abrasives Cam, was 0.4 (€/kg).

Table 9.10: Parameters of recharged abrasive samples

Abrasive Reusability Mass of new Mass of recycled Cost of recharged sample (%) abrasives (kg) abrasives (kg) kc abrasives(€/kg)

>75 57.11 756.58 1007.42 1.029 0.249 >90 53.64 817.79 946.21 1.090 0.263 >106 49.80 885.53 878.47 1.075 0.278 >125 44.46 979.73 784.27 1.032 0.300

Figure 9.10 presents the relation between the recharged abrasive size and the minimum cutting cost per meter with different numbers of the jet formers. It is observed that the cutting cost depends strongly on the number of the jet formers. In addition, unlike cutting with recycled abrasives (Figure 9.4), the total cutting cost is affected significantly by the recharged particle size (Figure 9.10). The reason is that the amount of the new abrasives used for the recharging and the amount of the recycled abrasives are not much different (Table 9.10). Moreover, the mass of the new abrasives is changed significantly when changing the recharged particle size (about 22.8% when the recharged abrasive size is changed from >75 to >125 µm). Consequently, the abrasive cost per meter and therefore the total cutting cost per meter depend considerably on the recharged size.

It is found that the particle size more than 90 µm is the optimum value for the recharging. With this value, the cutting cost per meter is minimum with all values of the number of the jet formers (Figure 9.10). This is because with this particle size the recycled abrasives and therefore the recharged abrasives have the maximum cutting performance (see Subsection 8.3.2).

The effect of the recharged abrasive size on the maximum profit rate is shown in Figure 9.11. Like the minimum cutting cost, the maximum profit rate depends significantly on the jet former number. Also, the influence of recharged particle size on the profit rate is quite large.

166 The optimum recycled abrasive size for the maximum profit rate, as for the minimum cutting cost per unit length, is larger than 90 µm (see Figure 9.11). It could be explained that with this size the cutting performance is maximum.

13 100

11 80 n =1 10 f n =2 f 9 n =3 f 60 n =4 8 f n =1 f 7 n =2 f 40 n =3 6 f n =4 f 5 Maximal profit rate (EUR/h)

Minimal cutting cost (EUR/m) 4 20 >75 µm >90 µm >106 µm >125 µm >75 µm >90 µm >106 µm >125 µm Recharged abrasive sample Recharged abrasive sample Figure 9. 10: Recharged particle size versus Figure 9. 11: Recharged particle size versus cutting cost per meter profit rate

9.3.2.2 Economics of cutting with recharged abrasives

As was found above, the particle size 90 µm is the optimum value for the abrasive recharging. Therefore, recharged abrasives, with particles are larger than 90 µm, are used for investigating the economics of cutting with recharged abrasives. In order to do this, comparisons of the minimum cutting cost per unit length and the maximum profit rate were carried out.

The calculated results of the minimum cutting cost when cutting with the new and the recharged abrasives were presented in Table 9.11. From the results, cutting with the recharged abrasives can reduce significantly the cutting cost per meter. In this case, depending on the number of the jet formers and the cost of the new abrasive, the cutting cost can be saved from 11.72 to 21.01 % in comparison with cutting with the new abrasives (Table 9.12). The comparisons of the normalized cutting cost per meter between cutting with new and recharged abrasives are shown in Figure 9.12.

Table 9.13 shows the calculated results of the maximum profit rates when cutting with the new and the recharged abrasives. The profit rate, when cutting with the recharged abrasives, is much higher than that when cutting with the new. It is clear that the profit rate when cutting with the recharged abrasives is super in comparison with cutting with the new abrasives (the saving was from 27.85 to 38.47 % depending on the number of the jet formers and the cost per kilogram of the new abrasives – see Table 9.14). Figure 9.13 presents the comparisons of the normalized profit rate between cutting with new and recharged abrasive.

167 Table 9.11: Minimum cutting cost when cutting with new and recharged abrasives

Minimum cutting cost (€/m) n f Cam, =0.2 (€/kg) Cam, =0.3 (€/kg) Cam, =0.4 (€/kg) Cam, =0.5 (€/kg) New Recharged New Recharged New Recharged New Recharged

1 13.14 11.6 13.53 11.77 13.91 11.93 14.29 12.08 2 7.09 6.21 7.46 6.37 7.82 6.52 8.17 6.68 3 5.74 5.01 6.1 5.17 6.45 5.32 6.79 5.46 4 4.55 3.96 4.9 4.11 5.24 4.26 5.57 4.4

Table 9.12: Total cutting cost saving when cutting with recharged abrasives

Total cost saving (%) n f Cam, =0.2 (€/kg) Cam, =0.3 (€/kg) Cam, =0.4 (€/kg) Cam, =0.5 (€/kg) 1 11.72 13.04 14.27 15.41 2 12.35 14.58 16.54 18.26 3 12.65 15.29 17.56 19.5 4 13.04 16.19 18.81 21.01

Table 9. 6: Maximum profit rate when cutting with new and recharged abrasives

Maximum profit rate (€/h) n f Cam, =0.2 (€/kg) Cam, =0.3 (€/kg) Cam, =0.4 (€/kg) Cam, =0.5 (€/kg) New Recharged New Recharged New Recharged New Recharged

1 40.11 55.59 41.25 58.45 42.34 61.25 43.4 63.97 2 43.16 60.48 45.25 65.85 47.22 71.02 49.09 75.99 3 52.29 73.69 55.3 81.5 58.12 88.99 60.78 96.16 4 55.16 78.3 58.97 88.33 62.51 97.88 65.83 106.98

Table 9.13: Profit rate saving when cutting with recharged abrasives

Profit rate saving (%) n f Cam, =0.2 (€/kg) Cam, =0.3 (€/kg) Cam, =0.4 (€/kg) Cam, =0.5 (€/kg) 1 27.85 29.43 30.86 32.15 2 28.65 31.28 33.51 35.4 3 29.04 32.14 34.69 36.79 4 29.55 33.23 36.14 38.47

168 120 GMA#80 115 Recharged abrasive with n =1 f Recharged abrasive with n =2 110 f Recharged abrasive with n =3 f Recharged abrasive with n =4 105 f

100

per meter (%) 85

80 Normalized cutting cost 75

70 0.2 0.3 0.4 0.5 Abrasive cost C (EUR/kg) a,,m

Figure 9. 12: Normalized cutting cost per meter when cutting with recharged abrasives

190 GMA#80 180 Recharged abrasive with n =1 f Recharged abrasive with n =2 f 170 Recharged abrasive with n =3 f Recharged abrasive with n =4 160 f

150

140

130

120

110

100 Normalized profit rate (%) 90

80 0.2 0.3 0.4 0.5 Abrasive cost C (EUR/kg) a,m

Figure 9. 13: Normalized profit rate when cutting with recharged abrasives

169 9.3.3 Comparisons among cutting with new, recycled and recharged abrasives

Based on the calculated results of the comparisons of the minimum cutting cost and the maximum profit rate when cutting with the new, the I-recycled, the multi-recycled, and the recharged abrasives (see Subsection 9.3.2 and Subsection 9.3.3), economic comparisons among cutting with them are shown in Figure 9.8 (for the minimum cutting cost) and Figure 9.9 (for the maximum profit rate). The comparisons are calculated with the optimum particle size (>90 μm) for both the recycled and the recharged abrasives, and with the cost of the new abrasive is 0.4 (€/kg).

It is certain that cutting with the recycled and the recharged abrasives can save a lot of both the cutting cost per unit length and the profit rate. In addition, although cutting with the recharged abrasives is the best for both the minimum cutting cost per unit length and the maximum mal profit rate, cutting with the multi-recycled and the recharged abrasives can have nearly the same benefits (see Figure 9.8 and Figure 9.9).

14 100 New ab. I−recycled ab. 12 I and II−recycled ab. Recharged ab. 80

10 60

40 New ab. 6 I−recycled ab. I and II−recycled ab. Recharged ab. Maximal profit rate (EUR/h) Minimal cutting cost (EUR/m) 4 20 1 2 3 4 1 2 3 4 Number of jet former n (−) Number of jet former n (−) f f

Figure 9.14: Cutting cost comparison Figure 9.15: Profit rate comparison when cutting when cutting with different abrasives with different abrasives

9.4 Suggestions for abrasive recycling process

From the results of the research on the recycling and recharging, some suggestions for a GMA abrasive recycling process are proposed:

-The particle size larger than 90 µm is the optimum size (for both the cutting performance and the economics) for recycling as well as for recharging.

-Abrasive recharging is preferred over abrasive recycling because it is the best for both the

170 minimum cutting cost and the maximum profit rate.

-The proposed scheme of the recharged-recycling process is shown in Figure 9.10. The system includes two sub-systems: recycling system and recharging system. The recycling system can be used not only for the recycling and the recharging process but also for recycling process. Although dry-sieving has been used in the experiments, wet-sieving is suggested for the recycling system because it can sieve faster. The wet-sieving has been used successfully in commercial recycling systems such as WARD Jet recycling system and Jet Edge (Jet Edge is the name of a Waterjet Company) recycling system. The recharging system has to mix the new and the recycled abrasives in a certain rate depending on the reusability.

-If recycling is chosen, multiple recycling is better than only first recycling.

-As the cost of recycled abrasives depends strongly on the utilization coefficient of the recycling system kut,recy (see Figure 9.1c), the recycling system should be run with kut,recy as large as possible.

That means if the amount of abrasives after cutting is not large enough (kut,recy <<1), the system should run with non-continuous shifts. For example, the AWJ machines and the recycling system can be with the rate 2 shifts of AWJ machines and 1 sift of the operated recycling system.

-To reduce the recycled-abrasive cost the drying capacity of the recycling system should be as large as possible (if the abrasive input is enough). This is because the recycled abrasive cost reduces significantly if the drying capacity increases (Figure 9.1f).

Recharging Recharged system abrasives Mixing Jet New former abrasives Recycled abrasives

Drying

Recycling system Washing and wet-sieving

Water Abrasives

Figure 9. 16: Scheme of abrasive recharged-recycling system

9.5 Conclusions

In this chapter, the economics of the recycling and recharging of GMA garnet has been

171 investigated.

A cost analysis to predict the cost of recycled and recharged abrasives has been introduced. The effects of various cost elements, i.e. hourly machine tool cost, wages including overhead cost per hour, and the water cost per hour, have been taken into account. Also, the drying capacity of the recycling system has been mentioned.

It has been found that to evaluate the economics of cutting with new and recycled abrasives (or with two types of abrasives), it is needed to compare the minimum cutting cost per unit length as well as the maximum profit rate. Based on that, for understanding the economics of the recycling and recharging of GMA garnet, the comparisons when cutting with the first recycled, the multi- recycled, the recharged and the new abrasives have been carried out. From the results, the optimum particle sizes for the recycled abrasives as well as for the recharged abrasives have been found. In addition, the economics of cutting with the recycled and recharged abrasives is shown.

It can be concluded that the particle size larger than 90 μm is the optimum size for the recycled abrasives and the recharged abrasives for both the cutting performance and the economics. Also, it has been found that the recharged abrasive is the best for both the minimum cutting cost as well as the maximum profit rate.

Several suggestions for a GMA abrasive recycling process have been proposed. The proposed process promises that the recycling process will be more effective.

172 10 Conclusions and recommendations for further research

This final chapter deals with the conclusions of the study. In addition, for more accurate models and better results in AWJ optimization, several recommendations for further research are also suggested.

10.1 Conclusions

The goal of this research is to carry out an optimization in AWJ machining for getting the minimum cutting time, the minimum total cutting cost and the maximum profit rate. To do this, a lot of works have been carried on the modeling of the cutting process for AWJ optimization, on the optimization in AWJ machining for the above objectives, and on the abrasive recycling. The results and conclusions of this research can be summarized as follows:

1. An extension of the AWJ cutting process model (Hoogstrate’s model) has been suggested. This work consists of the modeling of three sub-models including the pure waterjet model, the abrasive waterjet model, and the abrasive - workmaterial interaction model. The model has been verified for various work materials as well as for different abrasive types and sizes with different process parameters. There are several advantages of the extension model:

-Since the model has the optimum trend of the effect of the abrasive mass flow rate on the maximum depth of cut, it can be used for the AWJ optimization problems.

-The model becomes more accurate as the effects of various process parameters, e.g. the water pressure, the orifice diameter, the nozzle diameter, the abrasive mass flow rate, etc., have been taken into consideration.

-By using the workmaterial coefficient, the model can be applied for various work materials such as aluminum, stainless steel, titanium, mild steel, etc.

-The model can be used for different abrasive sizes (with any size) and types (Barton garnet, GMA garnet and Olivine).

2. The optimization in AWJ machining has been carried out. By solving the optimization problems, the optimum nozzle lifetime and the optimum abrasive mass flow rate for different objectives have been determined. The following conclusions can be drawn as the results of the optimization work:

173 -There are two types of AWJ optimization problems including the technical problem and the economical problem. Depending on the users, the objective function of an AWJ optimization problem can be the maximum depth of cut (for maximum cutting performance or minimum cutting time) (for the technical problem), the minimum total cutting cost, and the maximum profit rate (for the economical problem).

-A model for AWJ nozzle exit bore wear rate has been proposed. The effect of many jet parameters on the nozzle wear have been taken into account. The model can be used to predict the nozzle wear rate and the nozzle lifetime in AWJ optimization problems as well as for the calculation of the AWJ cutting regime.

-The influences of various process parameters (as above-mentioned) and cost elements (e.g. the hourly machine tool cost, the wages including overhead cost, the nozzle cost per piece, etc.) on the optimum nozzle lifetime (or the nozzle-exchange diameter) as well as on the optimum abrasive mass flow rate have been taken into account.

-By using process parameters, cost elements and using the nozzle exit bore wear rate, the profit coefficient as variables, the results of the optimization problems are very general and flexible.

-A new way for using the AWJ nozzles has been proposed. Instead of using the nozzles in a long time they can be used in much shorter time (determined by the optimum nozzle lifetime). After using for optimum time, they can be reused for other cutting applications such as cutting with bigger abrasive size, or using for cutting with traditional strategies.

-Cutting with the optimum nozzle lifetime can save a lot of the profit rate (up to 25…30 %), the cutting cost per unit length (up to 7…9 %), and the cutting time (up to 8…10 %). Moreover, nozzle companies can also benefit from increasing the sale amount of nozzles (about 2 to 4 times).

-The application of the optimum nozzle lifetime is not only in AWJ machining but also in other area such as in abrasive blasting. It is also found that there is an optimum value for nozzle exchange diameter in abrasive blasting. Cleaning with optimum nozzle diameter can save a lot of both cleaning cost and cleaning time [Pi07c].

-It is found that there are optimum values of the abrasive mass flow rate not only for the maximum cutting performance and the minimum cutting cost (as it is stated in many previous studies) but also for the maximum profit rate. In addition, the optimum values of the abrasive mass flow rate for the maximum profit rate are smaller than those for the maximum depth of cut but larger than those for the minimum cutting cost.

-Cutting with the optimum abrasive mass flow rate can save a lot of the cutting time (up to 25%), the total cutting cost (up to 7…10 %), and the profit rate (up to 30%).

174 -In order to determine simply the optimum nozzle exchange diameter and the optimum abrasive mass flow rate, two models for the prediction of the optimum nozzle-exchange diameter (or optimum nozzle lifetime) and three models for determination of the optimum abrasive mass flow rate have been proposed. The models for the optimum nozzle-exchange diameter help to determine the minimum total cutting cost and the maximum profit rate. The models for the optimum abrasive mass flow rate are developed to identify the maximum depth of cut, the minimum cutting cost, or the maximum profit rate.

-The optimum selection of other parameters consisting of the water pressure, the orifice diameter, the nozzle diameter and the number of the jet formers has been drawn. In addition, a procedure for optimum prediction of the AWJ cutting regime has been proposed.

3. The recycling and recharging of GMA abrasive have been investigated. The investigation includes the reusability of the abrasive, the cutting performance, and the cutting quality of the recycled and recharged abrasives. In addition, the economics of cutting with the recycled and recharged abrasives is evaluated. Several conclusions can be obtained from the investigation as follows:

-The particle size of larger than 90 μm is the optimum size for the recycling and the recharging of GMA abrasives for getting the maximum cutting performance, the minimum total cutting cost, and the maximum profit rate.

-The reusability of recycled and recharged abrasives with different particle sizes has been investigated. With the above optimum particle size, the reusability of both the first and the second recycled abrasives is larger than 50%. In addition, the reusability (with the optimum size) is nearly the same for three levels of the recharging including the first, the second and the third recharging (it is also larger than 50%).

-The cutting performance when cutting with recycled and recharged abrasives is higher than that when cutting with the new abrasive (about 17% when cutting with recycled abrasives and 10% when cutting with recharged abrasive).

-Cutting with recycled and recharged GMA abrasives can reduce the total cutting cost and increase the profit rate significantly.

-Cutting with the recycled, multi-recycled and recharged abrasives can reduce the total cutting cost per unit length 8…13 %, 11…19 % and 11…21 %, respectively, in comparison with cutting with the new abrasives. These values depend on the number of the jet formers and the cost per kilogram of the new abrasives.

-Cutting with the recycled, multi-recycled and recharged abrasives can increase the profit rate 20…27 %, 27…36 % and 27…38 %, respectively, depending on the number of the jet formers and the cost per kilogram of the new abrasives in comparison with cutting with the new abrasives.

175 -The multi-recharging can be done continuously without adverse effects on the reusability and the cutting performance.

-Suggestions for an effective process of abrasive recycling have been proposed.

10.2 Recommendations

Although this research has given several useful results, the following recommendations for further studies are suggested in order to further improve AWJ processes:

1. Optimization in AWJ machining with very high water pressure should be investigated. In practice, there are very few AWJ cutting systems which operate above 380 MPa. The reason is that very high pressure can lead to early failures and therefore very high maintenance cost [Olse03]. However, cutting with the pressure higher than 400 MPa may have many advantages such as it can increase the feed speed and the depth of cut, reduce the total cutting cost, etc. [Hoog05]. Thus optimization problems when cutting with very high water pressure (above 400 MPa) should be done.

To do the optimization in very high water pressure AWJ cutting, the following works must be done:

-As mentioned in Chapter 6, the AWJ cutting process model (Hoogstrate’s model) has been extended for cutting with the water pressure of lower than 380 MPa. Consequently, the process model should be extended for cutting with very high water pressure.

-As mentioned above, the high pressure water can cause a high maintenance cost. Therefore, the effect of the high water pressure on the maintenance cost must be found.

-Since the nozzle wear model has been found for ROCTEC100 nozzles and for cutting with the water pressure less than 380 MPa, it needs to be extended for the cases of the high pressure. Also, a model for the prediction of the wear of ROCTEC500 nozzles should be found.

2. The abrasive material coefficient should be extended for more abrasive types. In this research, the abrasive material coefficient has been found for three most famous abrasive types including Barton garnet, GMA garnet, and olivine. As mentioned in Chapter 6, there are some more abrasive types such as slag, aluminum oxide, silica sand, etc. Therefore, the coefficient should be extended for more abrasive types in order to use the cutting process model for more applications.

3. The economics of abrasive recycling should be carried out for other abrasive types. Until now, as mentioned in Chapter 2, the recycling of other abrasive types such as Barton garnet, Indian garnet, etc. have been investigated in previous studies [Ohls97], [Babu02], [Babu03], etc. However, the economics of cutting with recycled and recharged abrasives has been done only in the present research and for only GMA abrasive. For that reason, the economics of the recycling of other abrasive types should be evaluated.

176 References

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182 Appendix: Recycling system

The recycling system which used in the calculation of the economics of abrasive recycling was the Waterjet Abrasive Recycling Dispenser (WARD) from WARDJet, Inc. Two types of WARD system including WARD 1 and WARD 2 were used in the calculation for economic comparing.

A.1 WARD 1

Figure A.1 shows a picture of WARD 1 and Table A.1 presents the characteristics of the recycling system.

Figure A.1: WARD 1 system [Ward08]

183 Table A.1: Parameters of WADR 1 [Ward05a]

Parameter Unit Value

Drying capacity per hour Kg/h 81.65 2 Machine dimensions mm 1524x1524 Fresh water requirements Kg/s 0.051 Power requirements 480 volt, 3 phase, 60 amp breaker Air requirements 12 cfm for removal system Dryer type Fluid bed dryer Controller Full PLC and programmable controller Washing and screening system 760 mm diameter vibrating screen Secondary screen 30 mesh to remove all oversize Removal nozzles and hosing Four standard nozzles and hosing System price € 50000

A.2 WARD 2

Figure A.2: WARD 2 system [Ward08]

184 A picture of the WARD 2 recycling system is shown in Figure A.2. The characteristics of WARD 2 are listed in Table A.2.

Table A.2: Parameters of WADR 2 [Ward05b]

Parameter Unit Value

Drying capacity per hour Kg/h 36.29 2 Machine dimensions mm 1070x1070 Fresh water requirements Kg/s 0.032 Power requirements 220 volt, single phase, 60 amp breaker Air requirements 6 cfm for removal system Dryer type Direct contact with heating block Controller Simple dials, timers and switches Washing and screening system 610 mm diameter vibrating screen Secondary screen 30 mesh to remove all oversize Removal nozzles and hosing Two standard nozzles and hosing System price € 37000

185

186 About the author

Vu Ngoc Pi was born in Thai Binh province, Vietnam on September 12th, 1964. After finishing his secondary school in 1980, he started studying Mechanical Engineering at Thai Nguyen University of Technology. In 1985, he obtained his BSc degree and become a lecturer and researcher of the Mechanical Engineering Faculty of the Thai Nguyen University.

From 1990 to 2002, he was a vice-dean of the Mechanical Engineering Faculty of the Thai Nguyen University of Technology.

In 1997, he received his Master of Engineering from Hanoi University of Technology, Vietnam on a thesis entitled “About Optimum Design of Geared Motors”.

In December of 2003, he started his PhD. research at the Laboratory for Precision Manufacturing and Assembly of Delft University of Technology on Abrasive Waterjet Cutting under the supervision of Prof. Dr.-Ing. habil Bernhard Karpuschewski and Dr. ir. André Hoogstrate, which resulted in the present thesis. The research was supported by Training Scientific and Technical Cadres in Institutions Overseas with the State Budget (Project 322) (from Vietnamese government) and the Management Centre for International Cooperation (CICAT) (from Delft University of Technology).

187