Dec 10, 2004 - requirements of ASTM B 306, ASTM B 280, ASTM B 819 and ASTM B 837 ...... which will yield from Eq. (3-25), a positive free energy change, G 0 ...... of j (not in the input layer) is obtained by the use of a transfer function f. the.
Decision Support Tool for Optimal Replacement of Plumbing Systems
Juneseok Lee
Thesis submitted to the Faculty of the Virginia Polytechnic Institute and State University in partial fulfillment of the requirements for the degree of
Master of Science in Civil Engineering
Dr. G.V. Loganathan, Chair Dr. Tamim Younos Dr. Vinod Lohani
Dec 10, 2004 Blacksburg, Virginia
Keywords: Copper corrosion, decision support tool, geographic information system (GIS), home plumbing, life cycle assessment (LCA), non-homogeneous poisson process, optimal replacement, simulation, neural network
Copyright 2004, Juneseok Lee
Decision Support Tool for Optimal Replacement of Plumbing Systems By Juneseok Lee (Abstract)
Pinhole corrosion leak in home plumbing has emerged as a significant issue. In the major water distribution system managed by municipalities and water utilities the costs are distributed among all subscribers. The home plumbing repair/replacement cost and possible water damage cost must be addressed by the home owner. There are also issues of the value of home, insurance rates, health consequences, and taste and odor problems. These issues have become major concerns to home owners. Cradle to grave life cycle assessment is becoming an integral part of industrial manufacturing. In this thesis comprehensive details pertaining to life cycle assessment are presented. Copper tubing for plumbing installations is mainly obtained from recycled copper. Various stages of copper plumbing pipe manufacturing are explained. A comprehensive synthesis of various corrosion mechanisms is presented. Particular reference is given to copper plumbing pipe corrosion. A decision support tool for replacing copper plumbing pipes is presented. The deterioration process is grouped into early, normal and late stages. Because available data reflects late stage process, an optimization, neural network and curve fitting models are developed to infer early and normal stage behavior of the plumbing system. Utilizing the inferred leak rates a non-homogeneous poisson process model is developed to generate leak arrival times. An economically sustainable replacement criterion is adopted to determine optimal replacement time.
Acknowledgement I would like to sincerely appreciate the people and organization that made possible to finish the work on this thesis. z Dr. Loganathan has always been helpful from the moment I was in Virginia Tech. Without his support, this work has not been possible. I admire his perfection and being meticulous in work, resourceful ideas, and his sincere character. He’s my mentor whom I can respect. z Dr. Younos has been willing to serve as a member of my committee. I always respect his critical work and soul witness. z Dr. Lohani has also been willing to serve as one of my committee. I once again appreciate his advice and help. z I’m deeply indebted to the webmaster of the Copper Development Association for permitting me to use material available on their web-sites (www.copper.org) and available technical references. z I also thank for CEE 2814 team members (which made my campus life very busy but exciting), Dr. Loganathan (1st part professor), Jeff Connor (2nd part professor), Tim Bayse (Master Teaching Assistant), Qasem Abdelal, and Jonathan Ladd. I also thank for all the Measurement students whom we had lab time together. z I also thank for Dr. Borinara Park who is assistant professor at Illinois state university. He’s Korea university alumni, and his endless enthusiasm and achievements always look great to me. Also I thank for his criticisms and lessons in life. He should be one of my role models. z My parents, Sang-Ok Lee, Jung-In Kim, have supported me endlessly. Without their grand love and help to me, I couldn’t do anything by myself. I deeply express my gratitude to them. Also my little blood bro, June-Kie Lee, I firmly believe we both iii
will lead a wonderful life. We both are marine, you remember? z Lastly, whenever I was in difficult situations, I could overcome with the Spirit of Republic of Korea Marine Corps. I always miss the period I was in Yeon-pyung Island (the island of Republic of Korea which is nearest to North Korea in West Coast) and the blue oceans and the sky. Once Marine, Forever Marine!
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Table of contents ABSTRACT........................................................................................................................................... ii ACKNOWLEDGEMENTS ................................................................................................................ iii TABLE OF CONTENTS ......................................................................................................................v LIST OF TABLES................................................................................................................................vii LIST OF FIGURES............................................................................................................................ viii
CHAPTER 1 - INTRODUCTION ..........................................................................................1 1.1 Motivation .......................................................................................................................1 1.2 Major System and Minor System considerations.................................................. ..... 3 1.3 Objectives............................................................. ............................................................7 1.4 Organization of the thesis……………………............................................................ 7 CHAPTER 2 - LIFE CYCLE ASSESSMENT OF COPPER PLUMBING PIPES.............9 2.1 Introduction.................................................................................................... ..................9 2.2 Life Cycle Assessment (LCA)...................................................... ...............................10 2.2.1 LCA application .......................................................................................10 2.2.2 Main Framework of LCA .........................................................................12 2.3 Characteristics and Origination of Copper….............................................................17 2.3.1 Properties of copper element………..........................................................17 2.3.2 Sources of copper .......................................................................................18 2.4 Recycling….....................................................................................................................19 2.5 Process technology….....................................................................................................21 2.6 Uses of copper products…………...............................................................................24 2.7 Characteristics of copper pipe and manufacture…………………….....................25 2.8 Environmental aspect of copper pipe ........................................................................32 2.9 LCA application to copper plumbing pipe…………………………......................33 CHAPTER 3 - REVIEW OF CORROSION………………....................................................40 3.1 Introduction ....................................................................................................................40 3.2 General types of corrosion .......................................................................................41 3.2.1 Uniform Corrosion ......................................................... ………………..41 3.2.2 Erosion Corrosion .......................................................................................41 3.2.3 Pitting Corrosion .......................................................................................42 3.2.4 Galvanic Corrosion……….........................................................................43 3.2.5 Crevice Corrosion .......................................................................................45 3.2.6 Inter-granular corrosion………………….................................................45 3.2.7 Stress Corrosion Cracking (SCC)…………………................................46 3.3 Corrosion rate measurement .......................................................................................47 3.4 Corrosion and scale formation index ........................................................................48 v
3.5 Copper corrosion................... ........................................................................................50 3.5.1 Electrochemistry of copper-water pipes................... ................................50 3.5.2 Copper Chemistry ............................................ ................... ................... ..56 3.5.3 Uniform corrosion ..................................................... ................... .............57 3.5.4 Pitting corrosion .......................................................... ................... ........58 3.5.5 Empirical models .................................................. ................... ................59 3.5.6 Erosion corrosion ............................................................. ................... .....61 3.5.7 Microbiologically Induced Corrosion................... ....................................61 3.6 Summary................................... ................... ................... ............................…….........61 CHAPTER 4 - WSSC DATA ANALYSIS................... ................... ............................................62 4.1 Introduction....................................................... .............................................................62 4.2 Data Analysis................... ................... ...........................................................................63 4.3 GIS Analysis of spatial distribution of pinhole leaks................... ............................68 CHAPTER 5 - ANALYSIS OF RANDOM LEAK ARRIVALS ...........................................72 5.1 Introduction .................. ................... ................... ................... .....................................72 5.2 Repair/Replacement analysis ..................................................... ................... .............75 5.3 Leak rate models.............................................................. ................... ................... ......76 5.3.1 Shamir and Howard’s model.......................................................................76 5.3.2 Neural network model..................................................................................77 5.4 Simulation of leak occurrences.....................................................................................79 5.4.1 Non-homogeneous poisson process (NHPP) .......... .................................79 5.4.2 Simulation of Non-Homogeneous Poisson Processes (Thinning process)....................................................................................................................80 5.5 Application to WSSC data................... ................... .....................................................82 CHAPTER 6 - THESIS SUMMARY............................................................................................93 CHAPTER 7 – BIBLIOGRAPHY................... ................... ........ ................................................94 GLOSSARY APPENDIX A – Virginia Tech drinking water survey APPENDIX B – Neural networks PREDICT software tutorial APPENDIX C – MATLAB source code VITA
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List of tables
Table 1.1 Characteristics of Major and Minor distribution system…………………………….4 Table 2.1 World Copper Production…………………………………………………………..19 Table 2.2 Wall thickness and diameter of K, L, and M pipe………………………………….26 Table 2.3 Standard environmental data sheet………………………..……….. ...……………35 Table 2.4 Inventory table…………………….…………………………………………….....36 Table 2.5 Principal environmental problems…………………………….. ...………………..37 Table 2.6 Classification and characterization…………….………….. ...……………………38 Table 3.1 Types of Failures in Copper Pipe…………………………………………………..40 Table 3.2 Galvanic series……………………………………………………………………..44 Table 3.3 Pitting corrosion………………..…………………………………………………..60 Table 4.1 Leak Rate by Temperature………………….……………………………………...65 Table 4.2 Leak Rate by Orientation…………………………………………………………..66 Table 4.3 Leak Rate by Pipe Size…………………………………………………………….67 Table 4.4 pinhole leak analysis for the WSSC distribution area……………………………...71 Table 5.1 Results of Shamir- Howard Model (observed data only) ………………………… 89 Table 5.2 Results of Shamir-Howard Model (Modified data) ……………………………….90 Table 5.3 Thinning procedure (1 iteration) ……………………………………….………….91 Table 5.4 Predicted leak rate from Neural Networks…………………………………………91 Table 5.5 Possible leak scenario………………………………………………………………92
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List of figures Figure 1.1 National distribution of pinhole leaks……………………..……………………….2 Figure 1.2 Decision making process…………………………………………………………...2 Figure 1.3 Major and Minor distribution system …………………………………………….. 3 Figure 1.4 Relations on Consumer decisions…………………………………………………..5 Figure 2.1 Main framework of LCA (SETAC LCA framework) …………………………….11 Figure 2.2 Impact assessment……………………………………………… ………………...16 Figure 2.3 Vertical continuous casting (Source: www.copper.org) …………………………..29 Figure 2.4 Extrusion (Source: www.copper.org) …………………………………………….30 Figure 2.5 Tube drawing over fixed mandrel (Source: www.copper.org) …………………...31 Figure 2.6 Tube drawing over a floating plug mandrel (Source: www.copper.org) …………31 Figure 3.1 Uniform Corrosion………………………………………………………………..41 Figure 3.2 Erosion Corrosion…………………………………………………………………41 Figure 3.3 Pitting Corrosion…………………………………………………………………..42 Figure 3.4 Galvanic Corrosion………………………………………………………………..43 Figure 3.5 Crevice Corrosion…………………………………………………………………45 Figure 3.6 Inter-granular Corrosion…………………………………………………………..45 Figure 3.7 Stress Corrosion Cracking………………………………………………………...46 Figure 3.8 Pitting corrosion scheme………………………………………………………….56 Figure 3.9 Corrosion Rate vs. Temperature…………………………………………………..58 Figure 4.1 Number of Reports vs. Pipe Installed Year………………………………………..63 Figure 4.2 Leak Rate vs. Pipe Installed Year………………………..………………………..64 Figure 4.3 LRFF vs. Pipe Installed Decade…………………………………………………..64 Figure 4.4 Number of reports vs. Temperature……………………………………………… 65 Figure 4.5 Number of Reporting vs. Pipe Orientation………………………………………..66 Figure 4.6 Number of Report by Pipe size……………………………………………………67 Figure 4.7 Number of Reports by Pipe Type…………………………………………………68 Figure 4.8 Area that’s covered by WSSC…………………………………………………….70 Figure 4.9 Leak Number on the basis of GIS Analysis………………………………………70 Figure 5.1 Plot of Present worth cost…………………………………………………………84 Figure 5.2 Hybrid methods…………………………………………………………………...84 Figure 5.3 Observed leak rate………………………………………………………………...85 Figure 5.4 Feed forward back propagation neural network with two hidden layers…………85 Figure 5.5 Prediction comparison…………………………………………………………….86 Figure 5.6 Distribution of leak occurrences…………………………………………………..88
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CHAPTER 1 - INTRODUCTION 1. 1 Motivation Recent outbreaks of pinhole leaks nationally are a significant issue. Figure 1.1 shows the national distribution of pinhole leaks based on the data gathered by the Copper Development Association and Plumber’s Survey conducted by Dr. Edwards (2004). In the major water distribution system managed by municipalities and water utilities the costs are distributed among all subscribers. However, home plumbing repair/replacement cost and possible water damage cost must be addressed by the homeowner. When a pipe has a corrosion hole, the homeowner is typically faced with the following issues: water damage cost, repair cost, service disruption, lowering of home value, home insurance premium increase, and health consequences such as resulting from brown mold growth and mental stress from both concentrated nature of individual repair damage costs and health effects. Home owners near the hot spot areas are considering additional treatment to water in terms of corrosion inhibition, different plumbing materials such as plastics and stainless steel, and coating the interior of the pipe. Homeowners are also seeking advice on whether to continue to repair or replace the plumbing system. A repairable asset is considered to be in continued use but can be restored to a new state, better, about the same, or worse for a cost at the time of repair. Assume W0 be the initial value of the home plumbing system and W1 is its value after a corrosion incident with a repair/ replacement cost of R. The loss is therefore due to repair and the value lost [R+ (W0 -W1)]. Now, the same construct is repeated for N repairs accounting for proper discounting. At the Nth repair a decision has to be made whether to replace the system with an alternative or continue to repair it. This decision is obtained by minimizing the total discounted loss over N times. At each repair, the available alternatives are examined in place of retaining the old pipe. (Loganathan, MUSES proposal, 2002) 1
Also a decision on using other pipe material such as plastic, stainless steel, and copper with increased wall thickness has to be made. These different materials have to be considered for cost, consumer preference from the point of view of selling the house, corrosion susceptibility, strength, behavior in the case of a fire, and health effects. Different materials show different leak behaviors. These considerations are shown in Figure 1.2.
Figure 1.1 National distribution of pinhole leaks
Figure 1.2 Decision making process
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1. 2 Major System and Minor System considerations Water distribution system is composed of a major system and a minor system. The major system brings treated water to the street level of homes and the minor system carries the water from street level mains into homes (Figure 1.3). Table 1.1 lists the major differences between the major and the minor systems.
Figure 1.3 Major and Minor distribution system
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Table 1.1 Characteristics of Major and Minor distribution system (Loganathan, 2002 MUSES proposal)
Characteristic
Major System
Minor System
Pipe diameter
4in – 14 in
½ in – 1 in
Pipe material
Ductile iron, Plastic, Cast iron
Copper, Plastic, Galvanized iron
Pipe length
In miles (10s to several 100s)
In feet (several 100s)
Pipe wall thickness
Ductile iron 6.6 mm and above
Copper: K 1.25 – 1.7 mm; L 1.02 – 1.3 mm; M 0.71 – 0.89 mm
Corrosion
Both internal and external
Internal
Velocity
3 to 6 ft/sec
~ 4 ft/sec
Life expectancy
Ductile iron ~ 80 years
Ownership
Utility
Regulation
Government
Copper ~ 80 years; Galvanized iron 40 – 50 years End user (Homeowner, companies, industries) Individual/insurance; Replace piping
Cost
Property damage Service response Customer Dissatisfaction Availability of the data
Distributed by water rate Distributed – few 100s to several 1000s in dollars
$4,000 - $6,000
Few 100s to few 1000s in dollars
Immediate
Delayed
Marginal to serious
Serious
Records kept – computerized
May not have records
In decision making process, consumers are influenced by various factors. First, the regulations and standards of the federal, state, and local governments have major impacts. These regulations influence plumbers, material producers, insurance companies, and water utility companies; consequently, consumers are influenced by all of the above service providers. Figure 1.4 shows these overall relationships.
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-Federal Regulations -State Regulations -Local Regulations
Plumbers and Material Producers
Insurance companies (CLUE Report, difficult to be covered…)
Water utility companies
Households/ Consumers
Figure 1.4 Relations on Consumer decisions
Insurance companies have access to CLUE reports (Comprehensive Loss Underwriting Report) which form a comprehensive database of personal property information relating to insurance claims on private property. It is managed by ChoicePoint Asset Co. located in Alepharetta, GA. It contains personal information on the insured, the name of the insurance company involved, the policy number, the claim number, the accident, as well as the amount paid out to the customer. Finally, it lists the customer’s claim history report. If a customer calls the insurance company just for an inquiry of home plumbing issues, their call history is recorded on the CLUE report. Therefore, customers may have difficulty in purchasing home insurance (www.franscona.com/resource/jag403clue.htm). The cost of corrosion to public infrastructure is estimated to be in billions of dollars without including loss of about 15% of treated water through leaks. Estimates are that in the USA alone 10~32% of the drinking water is lost due to corrosion at a cost of $22 billion per year. The cost related to corrosion in private drinking water infrastructure, or plumbing in residential, commercial, and school buildings, is nearly twice that of the public infrastructure
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(Edwards, 2003). A survey in 1999 revealed that copper was selected in 90% of new homes, followed by PEX at 7%, and PVC at 4%. The length of private plumbing in homes, residential facilities, schools, and commercial buildings is 10 times longer than that of the public infrastructure. Copper pipes can result in copper concentration change in drinking water significantly. World Health Organization recommends copper concentration not to exceed 2.0mg/L and USEPA established an action level of 1.3 mg/L for drinking water (Dietrich, 2003). Regarding alternative drinking water sources, developing a well introduces its own water quality related issues and costs. Bottled water is typically expensive and does not address the issue of home use of water for bathing, cleaning, and other domestic activities. The quality of the bottled water has been brought into question regarding the source of that water and the shelf life from processing to consumption. Dual systems supplying potable and non-potable water require the maintenance of two networks and are clearly expensive. Point of use treatment alternatives require separate installation and maintenance costs. Pipe layout can influence the flow patterns that may have impact on corrosion. These alternatives have different life times, impacts and risks (Loganathan, 2002 MUSES proposal). The public’s perception of risk and reaction to hazards while hard to measure, play a fundamental role in this problem. Risk is an estimator of uncertainty defined as the probability of failure, which potentially affects consumer’s well-being. Objective risks are based on relative frequencies of occurrence obtained from historical or experimental studies. Perceived or subjective risk involves personal or subjective judgment and is a function of confidence. Information should be provided on the implications of risks for consumers as individuals rather than as a group. Incorporation of consumers’ preferences for water quality improvements in modeling can be performed by contingent valuation, conjoint analysis and decision analysis. Consumer risk averseness and proneness play a dominant role in developing a preference or utility function (Loganathan, Dietrich, 2002 MUSES proposal). 6
1.3 Objectives The objectives of this thesis are to:
1) Develop a detailed description of life cycle assessment procedure and its application to copper plumbing pipe 2) Provide a comprehensive synthesis of aqueous corrosion in plumbing pipes. 3) Develop a decision support tool for repair/replacement of plumbing systems
The plumbing system deterioration has emerged as a critical issue for a significant number of home owners throughout the country. The home owners are looking for sound advice on plumbing system repair and replacement. The repair issue involves installing a new pipe along with an older pipe without aggravating the problem. For example, it has been pointed out, copper pipes should never be installed upstream of galvanized iron pipes as it can cause corrosion in galvanized iron pipes. Therefore in hot water recirculation systems, copper galvanized iron combination should be avoided. Protective interior coating of copper pipes and use of plastic pipes are being considered to prolong the life of a plumbing system. In this thesis, the focus is towards the replacement issue rather than the alternative methods of repairing.
1.4 Organization of the thesis In chapter 2, a detailed description of life cycle assessment procedure is given along with details pertaining to copper pipe manufacturing. In chapter 3, a comprehensive synthesis of aqueous corrosion theory is given. Chapter 4 contains a preliminary analysis of observed plumbing pipe leak data. It is found that locations close to water treatment plant suffer the greatest number of leak incidents. It is conjectured that high pressure and relatively high concentration of chlorine might be the causal factors. The leak rates are also calculated 7
corresponding to the various periods in which the houses were built. These leak rates display a late stage behavior and serve as the basis for model development in chapter 5. A decision support tool for finding optimal replacement time for plumbing systems is developed in chapter 5. In this chapter an optimization model that minimizes absolute deviations between theoretical and observed leak rates is given. Also a neural network model for leak rates is developed. These leak rates display age dependence. A non-homogeneous poisson process that accounts for time dependent leak rate is developed to generate leak arrival times. These leak arrival times in turn are used within an economically sustainable replacement scheme for determining the optimal replacement time. Chapter 6 summarizes key contributions of this thesis.
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CHAPTER 2 - LIFE CYCLE ASSESSMENT OF COPPER PLUMBING PIPES 2.1 Introduction In this chapter, copper plumbing pipe is analyzed from a life cycle assessment (LCA) perspective. Life cycle assessment is a system-wide or “cradle to grave” analysis encompassing all aspects including the final disposal. The characteristics and procedures of standard LCA are fully described in this chapter. Though LCA methodology has now been standardized by the industrial Standards Organization (ISO 14000), the physical data are not generally available for mines and mineral processing industries. The processes of mining, extracting, smelting/refining are very diverse across the world and the manufacturing processes of the pipes also vary in different plants and countries. Therefore, in this chapter, instead of a formal LCA, a complete overview of the copper pipe from the life cycle perspective is presented. It’s known that a major part of drinking water copper pipes is recycled as well as made from recycled copper. This chapter includes discussions related to copper mining, manufacturing process, and recycling. The LCA methodology related sections in this chapter closely follow Heijungs, Huppes, Udo de Haes, Van den Berg, and Dutilh (1996; Life Cycle Assessment: what it is and how to do it, United Nations Environmental Programme, Paris, France).
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2.2 Life Cycle Assessment (LCA) Life cycle assessment (LCA) is the process of evaluating the effects that a product has on the environment over the entire period of its life cycle according to the definition of ISO 14000. LCA considers the whole life cycle of the product or its functions, covering all the processes: extraction and processing; manufacturing, transport and distribution; use, reuse, and maintenance; recycling and final disposal are included in LCA. These characteristics enable LCA to be an environmental decision making tool that provides additional information. This complicated procedure attempts to provide better objective answers that can yield sustainable types of production and consumption.
2.2.1 LCA application LCA’s results regarding environmental impacts of product’s life cycle will provide decision rules combined with economic effects, convenience to customers, and safety. It facilitates communication regarding the environmental aspects of products, production design, production improvements, eco-labeling criteria, and policy strategies. When there are disputes related to environmental impacts between governments and Non-governmental organizations (NGO), LCA serves as a facilitator in terms of its unbiased scientific and quantitative nature.
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A. Goal Definition and Scope 1. Goal Definition and Scope
B. Inventory Analysis 2.
3.
4.
Constructing
Collecting
Defining
Processing
Process Flow
Data
System
Data
Chart
5.
Boundaries
C. Impact Assessment 6. Classifications and
7. Valuation
Characterization
D. Improvement Assessment 8. Reporting and Improvement Assessment
Figure 2.1 Main framework of LCA (SETAC LCA framework)
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2.2.2 Main Framework of LCA SETAC (Society of Environmental Toxicology and Chemistry) is the lead agency responsible for coordination and harmonization of international LCA activities. The standard LCA framework is shown in Figure 2.1. As shown in Figure 2.1, LCA is composed of four processes: defining the goal and scope of the study, inventory analysis, impact assessment, and improvement assessment. LCA is usually an iterative process. At first, a superficial analysis is performed with approximate data. Then with this rough assessment, performance enhancement is achieved by focusing on specific points where one desires improvement. The improved results are used as the basis for even more detailed study. As the types of applications and degrees of detail can be different, it’s important to know what level of sophistication is required.
Goal definition and scope Goal definition and scope of the LCA framework involves drawing up a specification of the study, how it is investigated, and how it’ll be performed. Usually the LCA can have objectives as decisions on pros and cons of a product, production improvement, and the comparison of the products. As mentioned above, the scope of an LCA is related to the level of sophistication required for the goal of the study. If LCA results are intended for external use, a meticulous study including independent quality assurance is needed. The unit of product or function to be compared is referred to as the ‘functional unit’.
Inventory analysis The inventory analysis elaborates detailed procedures required in the manufacturing, use and eventual disposal of a product. These processes constitute the life cycle of that product. All the processes require inputs and produce outputs.
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Constructing process flow chart The first task in inventory analysis is to construct a process flow chart of all the processes involved in the product life cycle. All processes start with the extraction of raw materials and energy from the environment. They go through several stages of production and consumption. Eventually, they end at disposal sites. But growing rate of material recycling causes an increasing number of product systems that involve their own life cycle processes within a larger LCA study.
Collecting the data Next step is to collect data on each process. Quantified data of inputs and outputs from each process need to be obtained from the literature and published data. Collecting the data is known to be time consuming and difficult task in an LCA activity.
Defining the system boundaries After flow charting and data collection are done, the next step is to define the system more precisely by defining its boundaries. LCA can be reduced to a manageable size by omitting processes that fall outside of the boundaries. For example, copper pipes can be made from the scrap metals or from copper mining. Depending on the purpose of the LCA, the system boundaries can be defined differently. Much care is needed when defining the system as slight differences can have substantial influence on the results.
Processing data The last stage of the inventory analysis is to adjust the values of the inputs and outputs from each process which is related to each functional unit using an appropriate transformation into a convenient form. For example, all emissions of CO2 are identified for the whole system, added up 13
and entered in the inventory table as ‘y kg of CO2’. This table contains the environmental inputs and outputs. This inventory analysis is the main component of the LCA. It requires the most data and takes significant amount of time (Heijungs, 1996). The standard forms for inventory analysis can be found in Table 2.3.
Impact assessment Impact assessment is for the interpretation of the impacts from the inventory analysis on the environment. To thoroughly assess the products investigated, the environmental effects must be compared. Impact assessment is composed of three steps; classification, characterization, and valuation.
Classification In classification step, all the inputs and outputs are classified according to the different environmental problems they can cause. The significant problems usually are: resource depletion, energy depletion, global warming, acidification of soil and lake, photochemical oxidation, human toxicity, ozone depletion, and nitrification. Each input or output can contribute to several types of problems. For example, NO2 emissions can influence global warming, acidification, and human health simultaneously. Various kinds of environmental problems and their characteristics can be found in Table 2.5.
Characterization Next step is characterization. In this process, quantification of the contribution to each major environmental problem is done. For this, equivalency factors which indicate how much substances contribute to a problem compared to a reference substance are used. For example, the
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reference substance of the global warming is CO2. Methane’s effect on global warming can be expressed in terms of equivalent amount of CO2 which would have the same effects. If the value is 9, then 1kg of methane has the same global warming effects of 9kg of CO2. Regarding the location (certain substances impacts on specific sites) and the fate of the substance is the area that’s still being researched. After this step, next is the normalization which is optional. This is done by dividing the score for environmental problem such as energy depletion or global warming by the annual rate of it (Heijungs, 1996). The basic scheme for classification and characterization can be found in Table 2.6.
Valuation The last step of the impact assessment is the valuation. It involves the total comparison of the environmental problems to which each product contributes. By weighting each environmental problem in terms of its importance, the results can be a single environmental index. In assigning the weighting factor, there’s much subjectivity involved. For example, which has heavier impacts between global warming and acid rain? Characterization is designed to incorporate scientific or empirical knowledge on environment, but valuation is more based on social preferences at that time.
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Figure 2.2 Impact assessment (Modifed from Heijungs, 1996)
Improvement assessment The final goal of the LCA is to improve environment through better production, consumption, and sustainable development. This analysis starts with the most important areas where improvements can be made. At this stage, it can involve an assessment of economic, ergonomic and other aspects of the products. By repeating these aforementioned procedures, the product system can be improved and optimized.
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2.3 Characteristics and Origination of Copper Historically and technologically, copper has been very important metallic element to humans. Copper’s ability to conduct electricity, corrosion resistance, malleability, ductility, and strength has made it as the indispensable element. From ancient times, copper has been used for weapons, tools, and copper alloys. Even today, copper has its vitality in various areas such as electroplating, plumbing, electric motors, and so on (CRC Handbook of Chemistry and Physics, 1994).
2.3.1 Properties of copper element Copper’s atomic number is 29 and it belongs to the 1B group in periodic table. Silver and gold are also in 1B group, and they have some properties in common with copper. In thermal and electrical conductivity, silver and copper are the first and second. Third and fourth are gold and aluminum respectively, but copper excels these elements significantly. Though copper oxidizes in air, it is known to be highly corrosion resistant. It has a high melting point of 1083 D C with a high boiling point of 2595 D C. Pure copper is very ductile and can be easily drawn into wire. Copper alloys are formed in order to increase its strength, but it is known that most alloying elements reduce the thermal or electrical conductivity of copper significantly. Copper has several valence states, ranging from 0 to +3. Pure copper Cu (0) is very stable. It oxidizes to Cu (I)2O (cuprite), which is black and unstable. With heating, the cuprite oxidizes to normal Cu (II)O (cupric oxide), which is quite stable and insoluble. Some other salts notably the sulfate, CuSO4, are extremely soluble. The attractive blue-green patina on exposed copper surfaces consists of several compounds, depending on the surroundings. For example, in marine environment where there is salt in the air, atacamite Cu2(OH)3Cl is likely to be formed. Corrosion of copper will be explained in more detail in Chapter 3. Copper is known to be quite biologically active (Ayres 2002, AWWA 1996).
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2.3.2 Sources of copper Several kinds of ores provide most of the copper in this world. The dominant copper ores are sulfides (sulfur (S) compounds), bornite (Cu5FeS4), chalcocite (Cu2S), and chalcopyrite (CuFeS2) . Azurite (2CuCO3. Cu(OH)3), cuprite (Cu2O), and malachite (CuCO3.Cu(OH)2) are oxidizes ores that also yield significant amounts of copper. These ores also contain lead, zinc, gold, cobalt, and platinum in addition to copper. Usually copper ores contain less than 4 percent of the metal. On world-wide scale, approximately 12 million tons of copper are mined each year. Copper deposits are located in almost every continent and especially mountain ranges extending from Alaska to the tip of South America where copper is most abundant. In most places nowadays, large open-pit mining is used (Ayres, 2002).
Leading producers Chile is the world’s largest copper-producing nation, mining about a third of the world’s supply. The Unites States mines about a sixth of the world’s copper and ranks second in the world production. US uses more copper than it mines, and it imports copper from other countries such as Canada, Chile, and Peru. Most of the copper mined in the Unites States comes from Arizona. Other countries with large deposits of copper are Indonesia, Australia, Canada, China, Kazakhstan, Mexico, Poland, and Zambia. In Table 2.1, the leading copper producing countries are listed (Ayres, 2002).
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Table 2.1 World Copper Production (in Million Metric tons Cu content) (Original Source: USGS Mineral Commodity Summaries, “Copper”, 2001)
Country
Production
Chile
4.382
United States
1.660
Indonesia
0.740
Australia
0.735
Canada
0.614
Peru
0.536
Russia
0.530
China
0.500
Poland
0.460
Total
12.600
2.4 Recycling Copper and copper alloys have been recycled with the knowledge that copper objects could be melted and cast into new objects. Weapons were recycled from decorative and household goods during the war, and after that time, they were turned back into household or other non-war related products. Since early days, recycled copper has remained a major source of copper in the United States. Today, the process of transforming unalloyed copper scrap into new copper products begins with purchasing copper scrap from scrap processors. There are mainly 2 kinds of scraps which can be utilized. First is No. 1 scrap and second is No. 2 scrap. No. 1 scrap contains more than 99% copper and is simply re-melted. It usually consists of clean, unalloyed, and uncoated
19
copper solids. No.2 copper scrap must be re-refined and consists of unalloyed copper having a nominal 96% copper content, may include oxidized or coated/plated pieces and copper wire free of excessive oxidation. When copper scrap is about to be recycled, it is visually assayed and graded, and analyzed chemically when necessary. No. 1 scrap material is directly melted and in some cases brought to higher purity with fire refinement. Chemical analysis checks the purity level of the copper when the charge (material other than coke in a blast furnace) is fully melted, and the molten copper is deoxidized and cast into intermediate shapes such as billets, cakes, ingots for further processing. No. 2 scrap is usually electrolytically refined to attain the desired purity level. Before that, the scrap material is fire refined, melted and cast into anodes. These anodes are the raw materials used in cathode production. The anodes are then electrolytically refined, essentially an electroplating process in which the anode is electrolytically dissolved into a bath of sulfuric acid and then electroplated out of the solution onto a stainless steel sheet. Thin sheets of pure copper are pulled off the stainless sheet and placed between anode plates in other electrolytic cells where further electroplating transforms these anodes into 99.98% pure copper which builds up into cathodes as it plates out on the thin pure copper sheets. This pyrometallurgical process has been widely used. There’s new hydrometallurgical method which is increasingly used today that includes SX/EW (Solvent Extraction/ Electro Winning) (source: CDA Technical Report: The U.S. Copper-base scrap industry and its by-products-2001 and website www.copper.org). Copper alloys are also recycled. Alloy scrap has to be segregated and scrap of unknown composition may be melted and analyzed to determine its composition. Then, alloy recycling is done by melting together scrap of known composition. From CDA (Copper Development Association) statistics, it’s known that scrap consumption over the past 20 years has ranged between 44 and 54.7% of the total copper consumed in the US each year. The largest category of
20
scrap is customer returned new scrap which is directly remelted during the process. The recycling rate of old scrap tends to fluctuate depending on copper prices and other economic situations. Recovery rates of old scrap decline when copper prices are low. Old scrap consists of discarded electric cable, junked automobile radiators and air conditioners and countless other products. (CDA Technical Report: Copper-base Scrap Industry and its By-Products, 2001)
2.5 Process technology In the mining spot, copper ore in the form of large boulders are loaded with huge power shovels into trucks. And they’re carried to the mills. Generally, all the processes are designed to separate valuable minerals from the ore and waste rock to extract copper and other metals that may be in the resulting mixture, and to purify these metals even though all the processes are not same. In a typical process, the ore is sent to the mill, where it is crushed and the waste rock removed. The resulting material is then sent to the smelter, where the metallic copper is separated from impurities. This copper may contain other metals, such as gold, silver, and nickel that must be removed by refining. There are two ways to extract copper, one is pyrometallurgical and the other is hydrometallurgical. Pyrometallurgical method follows milling, smelting, and electrolytic refining whereas hydrometallurgical method involves milling, leaching, and SX/EW (Solvent Extraction, Electrowinning). The former is used about 80 percent and the latter is used around 20 percent in US for copper processing. But due to environmental issues, the combination of pyrometallurgical and hydrometallurgical methods is in increasing use (Ayres, 2002). The detailed procedures are given in the following and are adopted from Ayres (2002) and CDA website www.copper.org.
21
Milling Milling starts in a crusher, where the ore is broken into small pieces. Then water is added to the crushed ore to form a slurry. The slurry passes into ball mills which are drum-shaped cylinders partially filled with iron balls. As the cylinders rotate, the balls grind the ore into particles small enough to pass through a screen with 10,000 openings per square inch. The slurry next goes through a flotation process that concentrates the mineral-bearing particles. There, chemicals and oil are added, and the entire mixture is agitated with air to make it bubble. The bubbles rise to the top of the cell with the particles and form froth. This froth is skimmed off and dried. The product, called copper concentrate, may contain from 15 to 33 percent copper. The waste material, called tailings are not attached to the bubbles. It is emptied from the lower part of the flotation cell and sent to storage ponds (source: www.copper.org).
Smelting Smelting removes most of the remaining impurities from the copper. In smelting, copper concentrate is dried, and then blown with air and pure oxygen into a flash smelting furnace. In the furnace, the concentrate burns and melts, releasing some impurities in the form of sulfur dioxide gas. The molten material falls to the bottom of the furnace, where it separates into slag and copper matte. Slag, which contains iron oxide, silica, and other impurities, rises to the surface. Then the slag is discarded. Copper matte is heavier and collects under the slag. Copper matte contains 50 to 75 percent copper. It also contains some impurities in the form of iron sulfide and other metals. In the next stage of the process, the molten matte goes through a converter. In the converter, blowers force air through a converter and silica is added. The silica combines with the impurities, forming slag. The slag is again skimmed from the top. The new mixture is called blister copper which is 97 to 99.5 percent pure. The blister copper is refined in a fire-refining
22
furnace. This furnace removes most of the remaining impurities, mainly oxygen. When natural gas is blown to melt copper and the natural gas (mostly methane) burns, oxygen and other gases are removed from the copper. The resulting copper is 99.9 percent pure (source: www.copper.org).
Electrolytic refining Copper to be used in electrical conductors must be electrolytically refined to a purity of more than 99.9 percent. To do this, fire-refined copper is cast into cakes about 3 feet square and 2 inches thick. The cakes serve as anodes in the electrolytic process. The copper anodes are put into tanks containing a solution of copper sulfate and sulfuric acid. They are suspended alternately with cathodes, which are thin sheets of pure copper. When an electric current passes through the tank, the anode bars gradually dissolve, depositing copper more than 99.99 percent pure on the cathodes. Most of the remaining impurities in the anodes settle to the bottom of the tank and form sludge. After electrolysis, the copper cathodes are usually melted in a furnace and cast into various shapes and sizes (source: www.copper.org).
Leaching It is a method of dissolving metal out of ore with a chemical solvent. Leaching recovers copper from ores that do not react to the chemicals used in the flotation process. In leaching, water containing sulfuric acid or other chemicals circulates through the ore and dissolves the copper. The solution is then mixed with a kerosene solvent containing chemicals that extract the copper. The mixture separates and the copper-bearing chemicals flow into a sulfuric acid solution. This solution is put into a tank to undergo solvent extraction-electro winning, a process similar to electrolytic refining. The resulting copper is about 99.9 percent pure (Ayres,2002).
23
Solvent Extraction and Electro Winning (SX/EW)
The copper-laden solution is treated and transferred to an electrolytic process tank. When electrically charged, pure copper ions migrate directly from the solution to cathodes made from pure copper foil. SX-EW is different from pyrometallurgical methods in that the anodes are inert. Hydrometallurgy (known by SX/EW) is the separation of a desired metal from an ore or concentrate by dissolution and later precipitation or electro-winning. This method is in rapid growth as of environmental and economic aspects. (Ayres, 2002)
2.6 Uses of copper products Fabricating plants for brass and wire mills make semi-finished forms including sheets, tubes, and wires. They make these forms from copper rods, cakes, ingots, and billets. Manufacturers of copper products buy the semi-finished forms from these plants. About 35% of copper consumption in the United States is for electrical equipment. 32% is for fabricated metal products which include plumbing and pipe fittings. Machinery other than electrical is around 13 %, and transportation equipment constitutes 12 %. Miscellaneous uses account for less than 8% of the sales. In electrical equipment, high conductivity copper is indispensable. Copper is widely used in plumbing materials because of corrosion resistance, malleability, fire safety, and economy (www.copper.org). More details about copper plumbing are given in a later section.
Alloys and compounds Copper can be mixed well with other elements, and there are more than 1,000 different alloys. The presence of the other element or elements can modify the tensile strength, corrosion fatigue, and wear resistance of the copper. One of the most important groups of copper alloys is brass
24
which is the combination of the copper with zinc. And with the addition of tin, the base metal bronze can be made. As with the zinc in brass, the percentage of tin in bronze is variable, and according to different compositions, the resultant property varies.
2.7 Characteristics of copper pipe and manufacture In the US, more than 90% of the domestic plumbing system is made up of copper. It came into widespread use since 1930. Through many centuries, light, strong, corrosion-resistant copper tube has had a proven history of reliable service in installations. Here, main characteristics and methods of copper pipe production are explained. In this section, contents from CDA Copper tube hand book (2002) and website www.copper.org are adopted.
Types of pipes Copper plumbing pipe is manufactured from Copper No. C12200 (99.9% Copper), in accordance with the requirements of ASTM (American Society for Testing and Materials) Standard B 88. Most provincial regulatory authorities require that copper tube for use in plumbing systems be Third-Party Certified for compliance with ASTM B 88. Types DWV(Drain, waste, ventilate), ACR(Air conditioning and refrigeration), Medical Gas, and Type G/GAS tube meet the requirements of ASTM B 306, ASTM B 280, ASTM B 819 and ASTM B 837 respectively.
Types K, L, and M tubes have actual outside diameters which are 1/8-inch larger than the nominal (standard) sizes which the tube is commonly called. For example, a 1/2-inch Type M tube has an actual outside diameter of 5/8-inch. Type K tube has thicker walls than Type L tube, and Type L walls are thicker than Type M for any given diameter. Table 2.2 provides the dimensions for Types K, L, and M tubes. Temper denotes the hardness and strength of a tube.
25
Straight lengths are primarily drawn temper, or as more commonly known, hard tube. Annealed temper tube is referred to as soft tube. It is usually in coiled form, but certain sizes are also available in straight lengths (CDA Copper tube handbook, 2002).
Table 2.2 Wall thickness and diameter of K, L, and M pipe (Source: Copper Tube Handbook, CDA (2002))
Nominal Or
Outside
Standard Size
Diameter
(Inch)
(Inch)
1/4
Inside Diameter (Inch)
Wall Thickness (Inch)
K
L
M
K
L
M
3/8
0.305
0.315
-
0.035
0.030
-
3/8
1/2
0.402
0.430
0.450
0.049
0.035
0.025
1/2
5/8
0.527
0.545
0.569
0.049
0.040
0.028
5/8
3/4
0.652
0.666
-
0.049
0.042
-
3/4
7/8
0.745
0.785
0.811
0.065
0.045
0.032
1
1 1/8
0.995
1.025
1.055
0.065
0.050
0.035
1 1/4
1 3/8
1.245
1.265
1.291
0.065
0.055
0.042
1 1/2
1 5/8
1.481
1.505
1.527
0.072
0.060
0.049
2
2 1/8
1.959
1.985
2.009
0.083
0.070
0.058
2 1/2
2 5/8
2.435
2.465
2.495
0.095
0.080
0.065
3
3 1/8
2.907
2.945
2.981
0.109
0.090
0.072
Copper plumbing pipe production
Manufacturing of the copper plumbing pipe has been improved for energy-efficiency and environmentally acceptable quality under stringent standards.
26
Raw materials
The raw material for the production of plumbing pipe is mainly copper scrap, newly refined copper or copper ingots. The choice of the raw material is dependent on the plant’s technical capabilities, economic factors. The most common form of copper scrap is the recycled copper wire that has been stripped of its insulation or baled copper pipe that has been removed from destructed buildings. Another common form is the runaround scrap or home scraps which are generated usually in the mill itself. Only the No. 1 copper scrap can be used to make copper pipe. Industry-wide, about 64% of the copper in plumbing tube is derived from recycled scrap, although the percentage varies drastically among different mills. This type of high-quality scrap costs around 90% of the value of newly refined cathode as very little refining is needed for processing the metal to the required purity of the plumbing pipe (source: www.copper.org).
Melting
The charge of raw materials is melted in a furnace whose function is to melt the copper charge. If the raw materials are only in the form of cathode, refined ingot or home scrap, a shaft furnace is sufficient. But if the raw materials are scrap, reverberatory or other hearth-type furnaces are used because they have the ability to refine the copper prior to casting. In a typical operation using scrap as the raw material, the charge is melted and brought to temperatures between 2300 °F and 2400 °F, which are several hundred degrees above copper's melting point of 1981 °F. The copper is then fire-refined by contacting the melt with oxygen, which preferentially reacts with impurities to form oxides. These oxides, being lighter than the liquid metal, float to the surface, where they become trapped in slag. After the refining process and when the slag is skimmed off, only the pure fire-refined copper remains. This copper is now 99.9%+ Cu. It is of essentially the
27
same purity as fire-refined copper produced from the ore. Samples are taken to check the progress and when purity reaches the level required by the specification ASTM B88, the metal can be cast. At the end of refining process, controlled amount of phosphorous is added to limit copper’s oxygen content to remain within a reasonable range. This process is called the phosphorous deoxidization. It bears the designation C12200 under the Unified Numbering System (UNS) used to identify metals and alloys (source: www.copper.org). Casting In holding furnace or tundish, the molten metal is transferred from the melting or refining furnace. Holding furnace or tundish works as a casting process, and it is heated enough to maintain the molten metal at an appropriate temperature. Also, to protect the oxidation, liquid metal surface is covered with a blanket of graphite powder. In this process, copper is cast into large logs by continuous or semi continuous methods. For continuous casting, metal is poured into horizontally oriented cylindrical molds. And these force the copper to freeze quickly with cool water. Then the solidified chilled molds are withdrawn by gripping devices. At that time, a moving saw cuts the log into two-foot long sections as it emerges from the casting machine. These sections are called billets which weigh usually 400 pounds. Semi-continuous casting is the process when it is done vertically. When the length of the log reaches the depth of the pit beneath the molds, this process is interrupted. Molten metal is then added to the mold at the same rate that the floor is withdrawn downward. When the resulting logs reach the desired length, the mold is withdrawn upward, allowing the logs to be removed from the pit (source: www.copper.org). These processes are shown in Figure 2.3.
28
Figure 2.3 Vertical continuous casting (Source: www.copper.org)
Piercing To make the copper pliable, the billets are reheated about 1535 °F. A piercing mandrel is driven lengthwise through the center of the billets what will become the inside wall of the plumbing tube.
29
Figure 2.4 Extrusion (Source: www.copper.org)
Extrusion The billet is placed in the chamber of an extrusion process, heated to the proper hot temperature. The chamber contains a die at one end and a hydraulic ram at the other end. The front face of the ram is fitted with a dummy block that is slightly smaller than the billet in diameter. As the ram moves forward, the copper is forced over the hole in the die and it causes a long hollow tube. These procedures are shown in Figure 2.4. The diameter and length can vary according the capacities of the mill. Metal near the surface of the billet extrudes backwards over the undersized dummy block, and this forms a shell which contains copper oxide, which is recycled to the refining furnace. Rollers carry the extruded tube emerging from the die so it remains straight until it is cool. And the tube is cleaned to remove surface oxide scale for the next stage (source: www.copper.org).
30
Drawing Drawing process is for pulling the hollow tube through a series of hardened steel dies to reduce its diameter. The tube is pointed at one end to fit the next die and it is held by automatic jaws attached to a rotating, drawing machine. A tapered plug mandrel is placed inside the tube. And depending on the process used, plug mandrel can be fixed or floating. As the tube is drawn into the drawing machine, the mandrel and die act together to reduce outside diameter and wall thickness. This process is iterated until desired wall thickness is obtained (source: www.copper.org). In figure 2.5 and 2.6, the different types of drawing are shown.
Figure 2.5 Tube drawing over fixed mandrel (Source: www.copper.org)
Figure 2.6 Tube drawing over a floating plug mandrel (Source: www.copper.org)
31
Final Steps Now the tube is ready to be shipped. At regular intervals, the samples of the finished tubes are taken and ensured that it meets all the requirements of the ASTM B88.
2.8 Environmental aspect of copper pipe In the US, the usage of copper in plumbing systems has risen nearly 5 percent per year since 1992, and in 1997, it reached approximately two thirds of a billion pounds. More than 90 percent of domestic plumbing systems are composed of copper. And of all copper consumed in US, copper water tube accounts for about 8 percent. 65 percent of the copper tube is derived from the recycled scrap. The plumbing pipe is never deserted to landfill. It’s known that almost all the copper pipe is recycled. For economic and technical reasons, new copper is used even though the scrap as a raw material is helpful to the environment. The scrap usually costs less. But copper market fluctuation causes the price of refined copper and that of the scrap almost the same. If the distance from a tube plant that produces scrap is far away, then the refined copper becomes the better option to avoid the additional transportation fee. Some manufacturers cannot remove impurities from scrap in their plant. Then they have no choice but to buy refined scrap. Sometimes tube makers are forced to use refined copper as the furnaces are shut down due to maintenance problems. These problems are quite frequent nowadays in the US. That’s why still refined copper is used for plumbing (source: www.copper.org).
32
2.9. LCA application to copper plumbing pipes
The purpose of the Life Cycle Assessment for copper plumbing pipe is to understand the impacts of it on the environment. LCA should be performed for other plumbing materials such as plastics and stainless steel as well for comparing economic effects, material performance, and customer satisfaction. A 100 ft copper L pipe can be considered as the functional unit. Next, inventory analysis should be carried out. The flow chart for each process is constructed. As mentioned in the previous section, copper mining, milling, smelting, and copper pipe manufacturing including melting, casting, piercing, extrusion, drawing, and usage, maintenance, and disposal or recycling are listed. For each step, inputs resources, and energy consumption and product outputs involving emission to air, and other impacts water are required. However, the aforementioned processes are too diverse for different plants and countries. The standardized data are in general hard to obtain. The next step is to define the system boundaries. From CDA (Copper Development Association) statistics, it’s known that 64% of the copper plumbing pipe is made from recycled copper which means 64 % do not need the processes of mining, milling, and fire-refining. The materials are directly obtained from available copper scrap by recycling. For each process, the sum of emissions and inputs are identified for the whole system. For example, the amount of energy consumed in mining, milling, smelting, and pipe manufacturing (casting, piercing, extrusion, drawing, and transporting) are added to yield the energy consumption value (Table 2.4). With the inventory analysis, the effects on the environment are interpreted in impact assessment. As shown in Table 2.6, all the inputs and outputs are classified according to the environmental problems they result in. For instance, CO2, CH4, or O2 can cause global warming and NOx, NH4, or P can be the culprit for nitrification. It’s known that each input or output contributes to several
33
problems. With an equivalency factor, total points for each principal environmental problem are summed. As shown in Table 2.6, energy depletion potential (EDP), global warming potential (GWP), photochemical oxidant formation (POCP) have total scores. This yields the load on the environment by each process. Next step is valuation process. By assigning weighting factors to each of the environmental problems, the results are reduced to a different single index which is easier to interpret when comparing the environmental problems of different pipe materials. The weighting factor typically involves subjectivity and social preferences at that time. Final step is improvement assessment. With the characterization of the process, production, consumption, and development can be recommended for better environment. After these processes, results can be combined with economic and other aspects of the manufacturing as a whole.
The LCA forms a part of the decision making process. Regulative processes within the local, state, and federal, governments require consideration of impacts on the environment. The key benefit of the LCA process is the framework for an analytical impact assessment of manufacturing process as whole.
34
Table2.3 Standard environmental data sheet
Process
Environmental Data sheet
Data source:
Date:
Inputs (data per tonne of main product)
Outputs (data per tonne of main product)
Raw Materials
Main products
Extracted from the environment By-products (kg/tonne)
Bought in
Solid waste to be processed (kg/tonne)
Fluid waste to be processed (kg/tonne) Energy Environmental outputs (kg/tonne)
Emission to air Transport Services (means, load, distance) Emission to water
Emission to land
Other inputs
35
Table 2.4 Inventory table
Inventory table for 100ft of Copper L pipe Copper Mining
Grinding
Energy resource (GJ)
Emission to air (kg) CO2 CO hydrocarbons NOX SO2 Particles Liquid particles in air
Emissions to water (kg) nitrogen phosphates potassium oxide calcium oxide magnesium oxide insecticides herbicides oil hexan
Solid waste (kg) industrial high risk
36
Concentration
…..
Table 2.5 Principal environmental problems
abiodic depletion potential(ADP) energy depletion potential (EDP) global warming potential (GWP) photochemical oxidant formation(POCP)
measured relative to global supplies measured as MJ/kg or MJ/m2 measured relative to the effect of 1kg CO2 measured relative to the effect of 1Kg ethylene
acidification potential(AP)
measured relative to the effect of 1Kg SO2
human toxicity potential
measured as the human body weight that would be exposed to the
(HT)
toxicologically acceptable limit by 1kg of the substance
ecotoxicity, aquatic (ECA) nutrification potential (NP) ozone depletion potential (ODP)
volume of water that would be polluted to a critical level by 1kg of substance measured relative to the effect of 1kg phosphate measured relative to the effect of 1Kg of CFC-11
37
Table 2.6 Classification and characterization
Classification and Characterization for 100ft copper L pipe Resources
Inventory Amounts
Emission to Air
Emission to water
GJ
CO2
CO
CXHY
NOX
3.95
250.0
0.1
1.1
0.75
SO2
C6H14
herbicide
insecticide nitrogen
oil
PO43-
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
AP
…
…
…
…
…
…
…
HT
…
…
…
…
…
…
…
ECA
…
…
…
…
…
…
…
ECT
…
…
…
…
…
…
…
NP
…
…
…
…
…
…
…
ODP
…
…
…
…
…
…
…
Equivalency Factors ADP EDP GWP POCP
1.0 1.0 0.377
38
…
Table 2.6 Classification and characterization (cont’d)
Multiplied characterization results
Total
ADP EDP
3.95
…
…
…
…
…
3.95
…
…
…
…
…
250.0
…
…
…
…
…
0.95
AP
…
…
…
…
…
…
HT
…
…
…
…
…
…
ECA
…
…
…
…
…
…
ECT
…
…
…
…
…
…
NP
…
…
…
…
…
…
ODP
…
…
…
…
…
…
GWP POCP
250.0 0.407
0.541
39
CHAPTER 3 - REVIEW OF CORROSION 3.1 Introduction A distinction is drawn between pinhole formation and corrosion. Corrosion is generally taken to be a large scale loss of metal while pinhole is highly localized, small diameter hole that goes through the pipe wall. Nevertheless, because the underlying causal mechanism is the same, in this chapter, a review of corrosion is presented. Recent handbook on corrosion by Roberge (2000) contains comprehensive details. First, general types of corrosion are introduced. Details regarding corrosion specific to copper pipe namely, uniform, pitting, and erosion corrosion are explained in a later section. Corrosion indices for determining the corrosiveness of water are also introduced. Second, the crucial concepts, thermodynamics and electrode kinetics are introduced. In addition, various kinds of corrosion rate measurement methods and techniques currently used are given. Most of the contents in this chapter are the summary of the currently available corrosion references focusing on the corrosion of home plumbing copper pipe. Table 3.1 contains the distribution of the copper pipe failure types that occurred during 1980’s (Copper Development Association, 1983).
Table 3.1 Types of Failures in Copper Pipe ((Original source: Statistics from US Copper Development Association and (CDA))
Frequency (%)
Cause of failure
U.S.A. (1983)
Pitting Corrosion
58
Erosion Corrosion
24
Faulty Workmanship*
5
Outside Corrosion
7
Fatigue
2
Other
4
40
3.2 General types of corrosion Various forms of general corrosion that not only can take place in water distribution system but also in the atmosphere are described. Usually, corrosion is classified as uniform, galvanic, crevice, pitting, inter-granular, erosion, and stress corrosion. These are identified visually or with the aid of inspection tools and classified according to their characteristics and mechanisms.
3.2.1 Uniform Corrosion
Figure 3.1 Uniform Corrosion
Uniform corrosion is the most frequent form in corrosion. Electrochemical reaction proceeds uniformly over the surface. From Figure 3.1, it is seen that general thinning of metal leads to failure. Uniform corrosion accounts for the greatest loss of metal. But from engineering considerations, this is not critically important as they rarely result in leaks or failures of infrastructure even though they lead to the shortage of design life and/or loss of efficiency of the infrastructure. 3.2.2 Erosion Corrosion
Figure 3.2 Erosion Corrosion
41
By definition, “erosion corrosion is the cumulative damage induced by electrochemical corrosion reactions and mechanical effects from relative motion between the electrolyte and the corroding surface or accelerated degradation in the presence of this relative motion caused by high velocity, with mechanical wear and abrasion effects”(Roberge, 2000). Pipe bends, elbows, joints, valves, pumps, nozzles, heat exchangers, and turbine blades are the most frequent places where the erosion corrosion takes place. Figure 3.2 shows the mechanism schematically.
3.2.3 Pitting Corrosion
Figure 3.3 Pitting Corrosion
Pitting is considered to be one of the most dangerous and destructive forms of corrosion. Even though the weight loss of the material is very small compared to the other forms corrosion, it causes the system fail easily. From Figure 3.3, it is shown that pitting corrosion does little harm in weight loss, but it deteriorates the system locally. From an engineering viewpoint, pitting corrosion is significant as their occurrence is dominant as shown in Table 3.1. The size of the pitting corrosion on metal is so small that it’s very difficult to detect and to measure the variable depth and number of pits.
42
3.2.4 Galvanic Corrosion
Figure 3.4 Galvanic Corrosion
When two adjacent and dissimilar metals are immersed in corrosive or conductive solution, as shown in Figure 3.4, cathodic and anodic areas are determined. As there exists potential difference between the different metals, electrochemical reaction occurs. This is called galvanic corrosion. In this process, less noble metal will corrode becoming anodic (negative electrode) and more noble metal becomes cathodic (positive electrode). In Table 3.2, the galvanic series is shown. The more noble metals (non-corrosive) are listed towards the bottom of the table; the upper part of the table has the less noble metals which are more easily corroded. For example, if gold and steel are immersed in a conductive solution, gold will remain un-corroded and steel corrodes according to the galvanic series table. As shown in Table 3.2, copper is nobler than any of the materials that are commonly used in water distribution systems. The galvanic series shown in Table 3.2 provides the basis for assessing corrosion tendencies (Fontana, 1986). The farther apart in the series, the greater is the potential difference and the possibility for galvanic corrosion.
43
Table 3.2 Galvanic series
Corroded End (Anodic, or Least Noble) Magnesium Magnesium Alloy Zinc Aluminum Cadmium Steel or Iron Cast Iron Iron alloys Lead-tin solders Lead Tin Nickel Brasses Copper Bronzes Titanium Monel Silver solder Silver Carbon (graphite) Gold
Non-corroded end (cathodic, or most noble)
44
3.2.5 Crevice Corrosion
Figure 3.5 Crevice Corrosion
As shown in Figure 3.5, in some locations such as crevices, oxygen may be depleted during corrosion reaction. So, the oxygen reduction reaction cannot be continued and this results in the anodic characteristic. This specific area can be highly corrosive. The characteristics of the metal and the concentration of the aqueous solution which is involved in the reaction are the most crucial factors that determine the potential of the corrosion. Corrosion reaction is due to the concentration differences and this reaction can be said to be the equalizing potential process. For estimating the potential, Nernst equation can be used. (explained in a later section). Crevice corrosion forms under gaskets, washers, lap joints, and crevices.
3.2.6 Inter-granular corrosion
Figure 3.6 Inter-granular Corrosion
Microstructure of the metals or alloys is called as grain. And grain boundary separates the grain. When there’s localized attack along the grain boundaries or adjacent to the grain boundaries, it is called inter-granular corrosion as shown in Figure 3.6. So, the metal disintegrates and loses its
45
strength. Impurities at the grain boundaries or diminution, enrichment of one of alloys in the boundary of the grain may result in inter-granular corrosion.
3.2.7 Stress Corrosion Cracking (SCC)
Crack
Figure 3.7 Stress Corrosion Cracking
Tensile stress in corrosive medium can create cracking in metals and this is called the Stress corrosion cracking (SCC). Unlike other forms of corrosion, it is only identifiable with microscopic examination. Usually the fine cracks occur into the material but not on the surface as shown in Figure 3.7. It’s hard to find location of the fine cracks, and prediction of the damage is not easy.
46
3.3 Corrosion rate measurement There are several ways to assess the rate of corrosion. Before assessing the rate measurement methods, corrosion rate expression is presented. A commonly used expression for corrosion (Fontana, 1986) is Mils per year (mpy) = Where;
534 ⋅ W D⋅A⋅T
(3-1)
W = weight loss, (mg) D = density of specimen, (g/cm3) A = area of specimen, (cm2) T = exposure time, (hr) Mil = milli inch =
1 inch 1000
Also, corrosion rate can be expressed in terms of current density.
Faraday’s law for charge passed through a conductor, Q is given by (Jones, 1996) t
Q = ∫ Idt = 0
mnF a
(3-2)
in which; I = current in amperes, F = Faraday’s constant = 96,500 coulombs/ mole of electrons, n = number of moles of electrons, a= atomic mass of the anode material, m = mass of anode lost, t = time duration (seconds) Dividing both the sides of Eq. (3.2) by (tA) where A = surface area, we obtain Q I m nF nF = =i= ( ) = r( ) tA A At a a
where; r =
(3-3)
m = rate of corrosion in depth units and At r=
(3-4)
ia nF
47
3.4 Corrosion and scale formation index A method of corrosion prevention is to have the correct amount of CaCO3 scale to form within a pipe providing a protective coating. The Langelier and Ryznar indices are widely used (Tchobanoglous and Schroeder, 1985). The Langelier index (LI) is given by LI = pHmeasured – pHsat
(3-5)
in which; pHmeasured = actual pH value measured in the water pHsat = pH of the water in equilibrium with solid CaCO3 (Saturated pH) LI > 0 water is scale forming (supersaturated with respect to calcite CaCO3) LI = 0 neutral LI < 0 water is corrosive (unsaturated with respect to CaCO3) The Ryznar index (RI) is given by RI = 2pHsat – pHmeasured with
(3-6)
RI < 5.5 heavy scale will form 5.5 < RI < 6.2 scale will form 6.2 < RI < 6.8 no difficulties 6.8 < RI < 8.5 water is corrosive RI > 8.5 water is very corrosive
The value of pHsat is obtained from pH sat = − log10 [
K 2γ Ca +2 [Ca 2 + ]γ HCO− [HCO3− ] 3
K sp
(3-7) ]
in which; γ (⋅) = activity coefficient (3-8)
1
log10 γ i = −
0.5(z i ) 2 µ 2 1+ µ
1 2
for µ < 0.1
where; γ i = activity coefficient for ionic species i 48
zi = charge of ionic species i
µ = ionic strength of solution =
1 ci zi2 ∑ 2 i
ci = concentration of ionic species i (moles/ liter) K2 = second dissociation constant of carbonic acid from bicarbonate to carbonate =
[H + ][CO3−2 ] [HCO3− ]
and Ksp = solubility product constant = [Ca+2][ CO3−2 ] [⋅] = concentration in moles/ liter (molar concentration)
an estimate for µ = 2.5 (10-5) TDS g/m3 and TDS = Total dissolved solids
49
3.5 Copper corrosion
3.5.1 Electrochemistry of copper-water pipes The term corrosion is exclusively used for metals (Basalo, 1992). Electrochemistry is the interconversion of electrical energy and chemical energy (Chang, Raymond,: Chemistry, Mcgraw Hill, New York, 1998). Electrochemical processes are redox [oxidation (giving up electrons) – reduction (accepting electrons)] reactions. In redox reactions pertaining to metals carrying water, the metal gives up electrons and therefore corrodes forming the anode- negative electrodepositive node for the external flow of electrons. The cations from the metal flow internally through the solution to the cathode-positive electrode – negative node for the current; the anions from the solution move toward the anode. The following discussion is based on Basalo (1992). A metal atom can be oxidized as M → M n + + ne −
(3-9)
Specifically for copper we have Cu → Cu 2 + + 2e −
(3-10)
The scheme is shown in Figure 3.8. The loss of electrons from the metal indicates that the metal serves as a micro-anode. The related electrons are captured by several mechanisms resulting in the respective micro-cathodes within the water carrying pipe. If the solution is lacking dissolved oxygen, the available H+ ions in the water react with the released electrons to form the hydrogen gas by 2H + + 2e − ⇔ H 2
(3-11)
The location at which the mechanism takes place is called the hydrogen electrode. If the solution has sufficient dissolved oxygen, we have O 2 + 2H 2O + 4e − ⇔ 4OH −
(3-12)
forming an oxygen electrode along with hydroxide ions. If the solution contains the disinfectant chlorine in the form of chlorine gas Cl2, two reactions occur - hydrolysis and ionization (Tchobanoglous and Schroeder, 1987). In the hydrolysis reaction, hypochlorite, HOCl is formed; the chlorite ion OCl- is formed in the ionization reaction. The two reactions and the corresponding equilibrium constants are given by
50
Cl 2 + H 2O ⇔ HOCl + H + + Cl− and
K1 = With [H2O] =
[HOCl][H + ][Cl− ] = 4.5(10−4 ) at 25oC [Cl2 ]
(3-13) (3-14)
1000g = 55.5 moles and the concentration of water, [H2O] is mol/l or 55.5 18.02g / mo l
M (in molarity). This is a large value and compared to the other concentrations in the expression, it is generally taken that [H2O] does not change. This constant [H2O] is absorbed into the reported equilibrium constant values. The ionization reaction is given by HOCl ⇔ H + + OCl −
K2 =
[H + ][OCl− ] = 3.7(10−8 ) at 25oC [HOCl]
(3-15) (3-16)
For pH less than 7.5, the preponderant residual component is HOCl and the corresponding reduction equation is
HOCl + H + + 2e− → Cl− + H 2 O
(3-17)
For pH greater than 7.5, the preponderant residual component is OCl- ion and
OCl− + 2H + + 2e− → Cl− + H 2 O
(3-18)
The standard electromotive force (emf) of a cell, Ecell is given by E ocell = E ocathode − E oanode = E ored + E ooxid
(3-19)
in which E oanode must be used as a reduction potential from the reduction potential table. For the anode reduction of Cu → Cu + + + 2e − from the standard reduction potentials we have (Chang, 1998) o E oCu / Cu ++ = −E oCu ++ / Cu = −0.34V = E oxid
(3-20)
For the cathode reactions from Eq. (3-11) E oH + / H = 0V
(3-21)
From Eq. (3-12) E ored =0.40
(3-22)
and from Eq.(3-13) E ored = 1.482
(3-23)
2
51
Therefore, the reactions of HOCl or OCl- are more significant in copper corrosion. Nernst equation (3-24) given by E = Eo −
(3-24)
RT ln Q nF
enables the calculation of E when the reaction quotient, Q, is not equal to the equilibrium constant K; further in Eq.(3-24) R = universal gas constant ( J / o K ⋅ Kg ⋅ mol ), T= absolute temperature (oK), n= number of moles of electrons that pass through the circuit (mol), F= Faraday’s constant (J/V kg mol).
It is pointed out that Eq. (3-24) provides only the potential. It does not provide the current or rate of reaction. It is also pointed out that the Faraday’s law simply states that if the amount of charge transmitted over certain time is available, only then with the aid of Eq. (3-24) one can obtain estimates of the rate of corrosion. Therefore, the above so called thermodynamic analysis does not provide information on the rate of corrosion or the corrosion current. A positive E from Eq.(3-24) when plugged into
∆G = free energy change = -nFE
(3-25)
yields a negative value for ∆G which indicates that the reaction is spontaneous or will occur; but does not say at what rate. Following Jones (1996), consider 3Pb + 2Al3 + = 3Pb 2 + + 2Al
(3-26)
Pb 2 + + 2e − = Pb
(3-27)
and Al3+ + 3e-=Al
(3-28)
with the two half-cells
From the standard reduction potential table Epb/pb2+= +0.126 V= Eoxid
(3-29)
and E Al3+ / Al = −1.662V = E red
(3-30)
Assuming that the reaction (3-26) proceeds from the left to the right E cell = −1.662 − (−0.126) = −1.532V
(3-31)
which will yield from Eq. (3-25), a positive free energy change, ∆G > 0 . It indicates that the left to right equation is not possible and the reaction must proceed in the opposite direction with
52
3Pb 2 + + 2Al → 3Pb + 2Al3 +
(3-32)
Corrosion rate The following discussion is based on Fontana (1986). He points out that the standard reduction potentials are measured when the metal and the metal ions in the solution are in equilibrium. However, during corrosion this ideal condition does not exist. There will be initially current flow between the anode and the cathode. To satisfy the charge conservation hypothesis, the process has to seek the condition of the total rate of oxidation must equal the total rate of reduction. The current density corresponding to this point is called corrosion current density icorr since it represents the rate of metal dissolution. To obtain the exact amount of metal loss icorr should be used in place of i in Faraday’s law. The corrosion current should be calculated as follows. (Callister, W.D, Material Science and Engineering, John Wiley, New York, NY, 2000). The reduction potential is obtained as
E red,corr = E corr = E 0red + β red log(
(3-33)
icorr ) i0,red
Similarly the oxidation potential is given by
E oxid,corr = E corr = E 0oxid + β oxid log(
icorr i0,oxid
)
(3-34)
in which: E0 = standard potential, red = reduction, oxid = oxidation, β = Tafel coefficient, and E = potential. By setting Ered, corr = Eoxid, corr, we obtain icorr. For zinc corroding in an acid solution we can use (Callister, 2000) Zn + 2H+ → Zn2+ + H2
(3-35)
with E 0red = - 0.763V, E 0oxid = 0 V, i0,red = 10-7 A/cm2, i0,oxid = 10-10 A/cm2,
β red = + 0.09 and β oxid = - 0.08. We are interested in calculating the corrosion rate of zinc. By setting i i ] −0.763 + 0.09log[ corr−7 ] = 0 − 0.08log[ corr 10 10−10
(3-36)
we have icorr = 1.19(10-4) A/cm2 From Fraday’s law (Jones, 1996) corrosion rate, r, in mpy (milli inches per year) r =
53
0.129ai nD
in which: r (mpy), i = current density ( µ A / cm 2 ), D = density (g/cm3), n = number of moles, a = gram atomic mass r=
(0.129)(65.37)119 = 70.3mpy 2(7.14)
(3-37)
For copper corroding in an acid solution we can use (Callister, 2000) Cu + 2H+ → Cu2+ + H2
(3-38)
with E 0red = 0.337 V, E 0oxid = 0 V, i0,red = 2* 10-7 A/cm2, i0,oxid = 10-10 A/cm2, For β red =+0.09, +0.10, +0.11, +0.12, +0.13, +0.14, and +0.15, β oxid = - 0.08. We are interested in calculating the corrosion rate of copper. By setting 0.337 + 0.09 ⋅ log(
icorr i ) = 0 − 0.08 log( corr ) −7 2 * 10 10 −10
(3-39)
we have icorr = 5.8244* (10-11) A/cm2 when β red is +0.09, in the same way, icorr =9.1555* (10-11) A/cm2 when β red is +0.10, icorr = 1.3722 * (10-10) A/cm2 when β red is +0.11, icorr= 1.9752 * (10-10) A/cm2 when β red is +0.12, icorr= 2.7462 * (10-10) A/cm2 when β red is +0.13, icorr= 3.7054 * (10-10) A/cm2 when β red is +0.14, icorr= 4.8711 * (10-10)A/cm2 when β red is +0.15.
From Faraday’s law (Jones, 1996) corrosion rate, r, in mpy (milli inches per year) r =
0.129ai nD
in which: r (mpy), i = current density ( µ A / cm 2 ), D = density (g/cm3), n = number of moles, a = gram atomic mass r =
(0.129) ⋅ (63.55) ⋅ (icorr ) 2 ⋅ (8.92)
For different tafel coefficient, r can be calculated as, 54
(3-40)
r1 = 2.67647* (10-5) mpy, (when β red : 0.09) r2 = 4.20721* (10-5) mpy, (when β red : 0.10) r3 = 6.30593* (10-5) mpy, (when β red : 0.11) r4 = 9.07671* (10-5) mpy, (when β red : 0.12) r5 = 0.0001262 mpy, (when β red : 0.13) r6 = 0.0001703 mpy, (when β red : 0.14) r7 = 0.0002238 mpy, (when β red : 0.15) For the maximum values of corrosion rate for copper, i0,red = 2* 10-7 A/cm2 can be plugged in Eq. (3) instead of icorr. Then, r is 0.091905 mpy. If
we
convert
0.091905 *10−3 *
it,
0.091905
mpy
=
0.091905
*
10-3
inch/
year
=
inch 25.4(mm) * = 0.002334 mm/ year. From this, we can infer that year 1(inch)
penetrating a hole through 2 mm thickness of plumbing pipe takes (2/0.0233 = 856.76) about 857 years.
Compared to the corrosion rate of zinc (70.3 mpy), the corrosion rate of copper is much slower than that of the zinc.
55
Figure 3.8 Pitting corrosion scheme (From Edwards, 2002 MUSES seminar)
3.5.2 Copper Chemistry When copper corrodes, it is oxidized to Cu (I) (cuprous) and Cu (II) (cupric) states. They form insoluble precipitates and soluble complexes. They can form scale but often found as suspensions. The soluble complexes influence the solubility of passivating solids and contribute to copper concentration in drinking water. Oxidation of Cu (m) forms Cu+ as the initial product followed by precipitation of cuprite (Cu2O). Cu (m ) → Cu + + e −
(3-41)
2Cu + + H 2O → Cu 2O( S ) + 2 H +
(3-42)
Cuprite may be further oxidized to Cu (II) that can precipitate as tenorite (CuO). 1 1 ⋅ Cu 2 O( s ) + ⋅ H 2 O → Cu 2+ + OH − + e − 2 2
(3-43)
Cu 2+ + H 2O → CuO( s ) + 2 H +
(3-44)
56
It is widely known that dissolved oxygen (DO) or residual chlorine can act as electron acceptors, and significantly affect copper oxidation and corrosion rates. The various copper solids that may form in water distribution systems are given below. Cu + + Cl − → CuCl (Cuprous chloride: Nantokite)
(3-45)
Cu + + H 2 0 − H + → CuOH (Cuprous hydroxide)
(3-46)
Cu + + CO3−2 → Cu 2CO3 (Cuprous carbonate)
(3-47)
2Cu + + SO42− → Cu 2 SO4 (Cuprous Sulfate) 2Cu 2+ + 2 H 2O → Cu (OH ) 2 + 2 H + (Cupric hydroxide)
(3-48) (3-49)
3.5.3 Uniform corrosion Copper is much more resistant to uniform attack than other metals. Evans (1982) showed that the corrosion rate of copper is the slowest as compared to that of iron and zinc. As pH value drops below 6, copper corrosion rate increases in uniform corrosion. This kind of uniform corrosion will shorten the life cycle of the material. The uniform corrosion is typically the result of copper oxidation. According to AWWA (1996), however, at high pH levels, the usual form of corrosion is non-uniform corrosion. The amount of copper corrosion is inversely proportional to the pH. As the pH of the water increases, the corrosion release decreases.
Calcium carbonate scale formation protects copper pipe. It is known that chloride, sulfate, and bicarbonate concentrations have much influence on copper release and corrosion. In addition, water velocity and temperature have significant role in copper corrosion. Velocity influences erosion corrosion. High velocity water can impact the scale and the rate of attack is proportional to temperature and affects the hot water system. In Figure 3.9, it is seen that as the temperature increases, corrosion rate increases and the velocity affects the corrosion rate.
57
Figure 3.9 Corrosion Rate vs. Temperature (Source: Internal Corrosion of Water Distribution Systems by AWWA, 1996)
3.5.4 Pitting corrosion When specific areas are targeted, pitting corrosion occurs. Pitting corrosion of copper is occurring at a significant rate across the US. The combination of high residual aluminum, relatively high total chlorine residuals, and pH above 8.2 are observed when pitting is associated with aluminum deposits. It’s known that copper pitting is composed of 1) initiation of pitting, 2) propagation of pitting, and 3) revitalization of the pitting. Serious pitting requires these processes. Rushing (2002) asserts that in order to reduce the pitting corrosion pH should be reduced, add phosphate (PO4), NOM should be present, and use ferric coagulants (FeCl3). Main causes of pitting are water quality, hydraulics, and microbiology. Rather than just one factor, a combination of these factors influences pitting.
58
According to AWWA (1996), pH, pE (measure of redox potential), chloride, sulfate, and other organic/inorganic ions; hardness and other organic/inorganic cations; oxidant type and concentration; alkalinity should be considered.
3.5.5 Empirical models J.R.Myers (1991, from Lane 1993) developed an equation for pit depth in copper tubing exposed to cold aggressive water using experiments. P = 0.04 t1/3
(3-50)
Where; P = the pit depth in inches t = time in years The requirements are that pH of 7.0 ~ 7.7, DO ≥ 3mg/L, Carbon Dioxide ≥ 15mg / L, Chloride (Cl-) ≥ 15 mg/ L, and SO42- ≥ 15 mg/L. And in hot water P = 0.0148 t0.5
(3-51)
Where; P = the pit depth in inches t = time in years With a temperature greater than 130° F and containing > 0.1 mg/L aluminum, bicarbonate < 75mg/L, and pH < 7.6, and bicarbonate – sulfate ratio < 1.5.
Illinois State Water Survey (Neff, Sollo, and Lane 1975) determined the corrosion rate for copper pipe in cold water based on data from 21 different locations.
Corrosion rate (mdd) = 2.993 – {0.03084 * [mg/L carbon dioxide (as CO2)]} + [0.001857 * (mg/L TDS)] – [0.3268 * (pH)]
(3-52)
(mdd: milligrams per square decimeter per day, so if multiplied by 0.16, that would be exprssion in mils per year(mpy)).
59
The requirements are that TDS (Total Dissolved Solids) is 115 ~ 1,312 mg/L, CO2 (as CO2) is 0.0 ~ 27 mg/L, and pH is 7.1 ~ 9.7. Edwards, Ferguson, and Reiber (1994) classified pitting corrosion into type I (cold water), type II (hot water), and type III (soft water). Their findings for copper corrosion are shown in Table 3.3.
Table 3.3 Pitting corrosion (Original from Edwards, 1994, summarized from AWWA)
Effects
Type I Pitting (Cold Water)
Type II Pitting (Hot water)
Type III Pitting (Soft Water)
Water quality
Failure of pipe
Cold water, pH above 7.
Failure of pipe
Hot water, pH below 7.2
Blue water, release of voluminous by-product
Soft water, pH above 8.0
Main causes
Reducing methods
Deposits on pipe, Stagnation at early NOM/ increase bicarbonate and period, chlorine pH, concentration, orthophosphate Aluminum coagulants High temp, chlorine Reduce residuals, temperature, Aluminum pH. coagulation Stagnation, high pH values, Aluminum coagulation
NOM, avoid stagnation
It takes approximately three to four years, but sometimes within a few months for type I pitting to penetrate the pipe. Pitting is more frequent in Type I than Type II or III. Multiple pits are likely to lie along longitudinal lines. In addition, most pits and leaks are found in horizontally installed pipes. Some of the known conditions for initiation of pitting are associated with the condition of the metal before it is exposed to water and water quality has significant impact related to the propagating tendency. Type II is much slower in corroding than type I and rarely causes failures in a system. If failure happens, it occurs in water temperature greater than 60oC and in soft water (water that has scale-forming impurities removed) areas.
60
3.5.6 Erosion corrosion Copper is a soft metal and readily damaged than other metals. Copper pipe is susceptible to erosion corrosion. High velocity (greater than 4 ft/sec) and temperature are the main reasons for erosion corrosion. High velocity usually results from faulty workmanships. There are national standards that restrict the velocity. National Association of Corrosion Engineer’s recommendation is 4fps for K copper pipe. Temperature is also an important factor for erosion corrosion. Majority of the erosion corrosion is reported from hot water tubing system. In this situation, the velocity should be less than 1.5fps by national recommendation (AWWA, 1996).
3.5.7 Microbiologically Induced Corrosion It is known that in interior pipe surfaces, bacteria can be the cause of copper corrosion. Bacteria often grow in biofilms with scales and corrosion products. Experiments show that after disinfections, the rate of problems has decreased.
3.6 Summary In this chapter, a general review of corrosion is presented. First, the various types of corrosion that can occur in water contacting metals are described. Corrosion rate is crucial in the decision making process. As the focus of the thesis is on drinking water copper pipe, corrosion of copper pipe contacting potable water is addressed in detail. Corrosion scale, copper chemistry, and frequent forms of corrosion in copper pipe are explained.
61
CHAPTER 4 - WSSC DATA ANALYSIS 4.1 Introduction Washington Suburban Sanitary Commission (WSSC) is among the 10 largest water and wastewater utilities in the United States. WSSC serves 420,000 customer accounts and provides water sewer services to 1.6 million resident customers in D.C. and Maryland. The pinhole leak (which is the same as copper pitting corrosion) problem has emerged as a significant problem in the source area of WSSC. Some of the complaints shown below from WSSC customers reflect the nature of the problem in terms of where the leaks occur and the extent of the damage (From WSSC website survey): “We had so many pinhole leaks that we finally had a plumber come in and replace all of the cold-water pipes”
“There were multiple pinhole leaks which required the entire pipe between water main and house intake to be replaced.”
“We had so many pinhole leaks that caused so much damage to drywall, possessions, and carpet last year we replaced all the copper at both the plumber's and WSSC's suggestion.” The average cost of repairing domestic copper pipe is around $500 ~ 1000, and the replacement of the system is approximately $ 4,000~ 6,000 (From national plumber survey). In this chapter, a statistical analysis is performed to obtain the failure rates. In addition, with the aid of GIS (Geographic Information System), spatial distribution of pinhole leaks are displayed and analyzed for selecting top 3 zip codes of detailed customer survey.
62
4.2 Data Analysis WSSC provided basic data on pinhole leaks without revealing any consumer information. Figure 4.1 shows the frequency distribution of number of leaks reported vs. pipe installed year.
Figure 4.1 Number of Reports vs. Pipe Installed Year
Figure 4.2 shows the calculated leak rates. The leak rate is given by
Leak rate =
Total number of leaks in the reported period Number of years in the reported period
A box-whisker plot of the data is shown in Figure 4.3.
63
Figure 4.2 Leak Rate After First Leak vs. Pipe Installed Year
Figure 4.3- LRFF vs. Pipe Installed Decade
64
Figure 4.4 shows that most of the pinhole leaks are reported from Cold water pipes. The leak rate is higher in cold water pipes as shown in Table 4.1.
Figure 4.4 Number of reports vs. Temperature
Table 4.1 Leak Rate by Temperature
Avg. Leak Rate Hot
0.31
Cold
0.40
65
Figure 4.5 shows that most of the leaks are in horizontal pipes. However, the leak rate does not show any trend.
Figure 4.5 Number of Reporting vs. Pipe Orientation
Table 4.2 Leak Rate by Orientation Average of
Orientation
Leak Rate
Vertical
0.40
Horizontal
0.57
Both
1.21
66
Figure 4.6 shows that most leaks occur in thinner pipes. The leak rates however, show no appreciable trends as shown in Table 4.3.
Figure 4.6 Number of Report by Pipe size
Table 4.3 Leak Rate by Pipe Size Avg. Leak Rate After First Failure 0.5inch
0.65
0.75inch
0.73
1inch
0.55
67
From figure 4.7, it is seen that M pipe has the maximum leak rate. As explained in Chapter 2, among K, L, M pipe, M pipe has the thinnest thickness.
Figure 4.7 Number of Reports by Pipe Type
4.3 GIS Analysis of spatial distribution of pinhole leaks There are 4 reservoirs, namely, Triadelphia, Rocky Gorge, Little Seneca and Jennings Randolph with total holding capacity of 14 billion gallons in Maryland. Both Little Seneca and Jennings Randolph are regionally shared. And there are 2 water treatment plants. The Patuxent (max 56 MGD) and the Potomac (max 285 MGD) plants produce an average of 167 million gallons per day (MGD) of safe drinking water. Figure 4.8 represents the area that’s covered by the WSSC. The pinhole leak locations are shown by zip codes in Figure 4.9.
68
Table 4.4 contains zip codes corresponding to the ranked number of leaks per person and absolute number of leaks. Column (1) contains zip codes for the ranked number of leaks per person. Column (2) has the population, column (3) has the number of leaks, column (4) has the ranked number of leaks per person, column (5) has the ranks for the number of leaks per person, and column (6) has the ranks for the number of leaks.
From Table 4.4 data it is seen that the areas that have significant pinhole leak problems are located in close proximity with the water treatment plants. Based on this observation, zip codes 20817, 20707, and 20815 are selected as top three choices for detailed customer survey. The area corresponding to zip code 20817 has the maximum number of pin hole leaks and has the second rank in the number leaks per person. Column (4), number of pinhole leaks per person, is important as the population in each zip code area is different. The region is in close proximity to the Potomac water treatment plant. Zip code area of 20707 is selected as the third choice in the top 3 locations because of its close proximity to the Patuxent water treatment plant with rank 1 in the number of leaks and rank 2 for the number of leaks for that treatment plant. It has overall rank of 8 in both the categories when both the treatment plants are considered. It is also recommended zip code area 20815 as the third choice because it experiences significant number of pinhole leaks. When detailed data become available from this survey, more elaborate calculations are possible. The survey questionnaire (made by Dr. Bosch et al.) is shown in the Appendix A.
69
Figure 4.8 Area that’s covered by WSSC
Figure 4.9 Leak Number on the basis of GIS Analysis
70
Table 4.4 pinhole leak analysis for the WSSC distribution area
Zip
(1)
Rank by number
Rank by
Population
# of
Pinhole/
of leaks per
number
2001
Leaks
person
person
of leaks
(2)
(3)
(4)
(5)
(6)
20816
15143
173
0.011424421 1
6
20817
33954
380
0.011191612 2
1
20815
28270
284
0.010045985 3
2
20814
25916
218
0.008411792 4
4
20903
18039
127
0.007040302 5
12
20895
19713
134
0.006797545 6
10
20705
20551
129
0.006277067 7
11
20707
25815
149
0.005771838 8
8
20901
36935
205
0.005550291 9
5
20854
48226
261
0.005412018 10
3
20905
17295
93
0.005377277 11
15
20853
26432
119
0.004502119 12
13
20910
37144
155
0.004172949 13
7
20740
21503
79
0.003673906 14
16
20708
25633
94
0.003667148 15
14
20852
40252
140
0.003478088 16
9
71
CHAPTER 5 - ANALYSIS OF RANDOM LEAK ARRIVALS 5.1 Introduction As discussed in the previous chapters, the pinhole corrosion leak in home plumbing has emerged as a significant issue. In the major water distribution system managed by municipalities and water utilities the costs are distributed among all subscribers. However, the home plumbing repair/replacement cost and possible water damage cost must be addressed by the home owner. For most homeowners, the home is their most valuable asset. The possibility of falling home value, losing home insurance, water damage, frequent repairs, taste and odor, and health concerns are some key issues involved in home plumbing maintenance. The homeowner has to decide at the time of pinhole leak whether to repair or replace the system. If the owner decides to replace the system, another decision on which material to use should be made. Local regulations may not permit use of certain materials. In new homes, the plumbing contractor may decide the material. The replacement decision depends on three factors: (1) financial affordability, (2) hydraulics of flow within the plumbing pipes, and (3) quality of water. Figure 5.1 shows the effect of financial viability. The accelerated replacement refers to replacing the plumbing system well in advance of the optimal replacement time. Delayed replacement ideally includes all the consequences of neglecting repairs or just performing repairs amounting to paying penalties to compensate for high replacement cost. Hydraulics related to plumbing pipes addresses velocity, pressure and flow through numerous appurtenances along with temperature for hot water. The wall thickness of plumbing pipes rarely exceeds 2mm and diameter varies between 0.5 inches to 1 inch. The street level pressure of the major system is the key determinant in designing a plumbing system. If the pressure is high, it must be dissipated by frictional and minor losses with a maximum velocity range of 4-8 ft/sec (Nielsen, 1990). The velocity limitations are chosen to protect against 72
corrosion and cavitation and pressure rise due to stoppage of flow (water hammer). The time between leaks depends on the flow behavior but not well understood. From the Washington Suburban Sanitary Commission (WSSC) pinhole leak data (see chapter 4), it’s found that leaks tend to cluster near treatment plants. High pressure and chlorine regions typically surround treatment plants and it’s anticipated that high pressures and chlorine might play a role in pinhole leaks. It’s also observed that if a corrosion pit is initiated due to high stress or water quality, velocity can aggravate and dominate pit growth. Water quality poses two kinds of threats: initiation of corrosion and possible further degradation of quality from interaction with corroded elements. Copper reacts with oxygenated water to form a thin corroded layer of copper oxide which inhibits further corrosion and protects the metal underneath. If the water velocity is greater than 4 ft/sec, the oxide layer may be destroyed and the metal can corrode. (Taber, G. “corrosion in open and closed systems” PM engineer 2000; available at http://www.pmengineer.com/CDA/ArticleInformation/features/BNP_Features_Item/0,2732,87 83,00.html). Taber (2000) cites amount of oxygen, chemicals present in water, presence of dissimilar metals, temperature, pressure, and flow rate to be the key factors in corrosion.
Rushing et al. (2004) report that the combination of pH greater than 8.2, aluminum deposits greater than 50 ppb and chlorine concentrate of 2 mg/l will result in copper pipe corrosion. A 90 degree bend in the pipe will accelerate corrosion due to this recipe. As shown in Figure 5.2, both hydraulic and water quality effects determine the nature of leak arrival times. In this thesis the net effect of these two causal phenomena in the form of the failure time data is considered. Unfortunately, the only available data is as shown in Figure 5.3, the leak rate data. The homeowner data is obtained from WSSC. It does not contain leak arrival times which are crucial in a decision model. In this chapter, we present a model that can yield leak arrival times from specified leak rates. 73
The repair/ replacement process is visualized as follows. The pinhole leaks arrive randomly. The leaks occur at random locations and at random time intervals. From a cost analysis point of view the random locations may not be that critical than the expenditure incurred. Therefore, at a broader level in minimizing the cost, the arrival times of pinhole leaks will dictate the repair/ replacement decision. It is also possible that the arrival of the first leak serves as a harbinger of subsequent, immediate leaks. However, present data suggest that such a uniform corrosion behavior is not evident. Another point of view is based on time; whether the leak occurs at an early stage of installation perhaps indicating installation flaws or occurring at a late stage indicating a deteriorating system; or within the anticipated normal design life. The early, normal, late stage of a plumbing system, possibility of clustering of leak occurrences in time, point to leak arrival patterns in time. The leak rate itself is a time dependent function. The rather high leak rates shown in Figure 5.3 suggest late stage behavior. In this chapter, several models that fit this late stage break rate are formulated. These models are used to infer the early and normal stage leak rate behavior. A non-homogeneous poisson process (NHPP) model is considered for modeling the leak arrivals. This construct enables the development of a minimum cost model in terms of when the expenditures are incurred. The minimum cost model is based on the following notion. A risk-averse homeowner may decide to replace the plumbing system at a time well ahead of the optimal replacement called an accelerated replacement with the convenience of avoiding any major damages. It is not prudent to wait beyond the optimal replacement time called delayed replacement. Clearly, the advantage lies in exercising the replacement at the optimal time. Such an optimal replacement time is dictated by the arrival pattern of the leaks. In the following section 5.2, a minimum cost model is adopted from Loganathan, Park, and Sherali (2002). In section 5.3, several models for the observed leak rate are fitted. From these models, the complete leak rate behavior is synthesized. It’s pointed out that only a subset of these fitted models is suitable for a plumbing system. In section 5.4, the NHPP model is presented 74
in detail. Section 5.5 contains an example application that displays the variability in the leak arrival patterns.
5.2 Repair/Replacement analysis At the time of the nth leak, a decision has to be made whether to replace the plumbing system at a cost of Fn or to repair it at a cost of Cn. The scenario also implies that for the previous (n 1) leaks only repairs have been performed. If we assume that the plumbing will be replaced (Cn included in the sum should be adjusted for Fn when necessary) at the time of nth leak, tn, we can write the present worth of the total cost of the pipe as: n
Tn = ∑ ( i =1
Ci Fn ) + ti (1 + R) (1 + R) t n
(5-1)
in which: R = discount rate, ti = time of ith leak measured from the installation year (year), Ci = repair cost of ith break, Fn = replacement cost at time, tn, Tn = total cost at time ‘0’ (present worth). When the system is new, it tends to experience very few leaks. An old system experiences more leaks under the same conditions. Therefore, the combination of varying time interval between leaks (accelerated leak arrivals towards the end), relatively smaller repair cost, and a generally large replacement cost leads to the existence of a “U” shaped present worth of the total cost curve over time (Figure 5.1). The derivation of the threshold break rate seeks the time of the minimum present worth total cost. Loganathan et al. (2002) have presented the following threshold break rate (Brkth) equation Brk th >
ln(1 + R ) ⎞ ⎛C ln ⎜⎜ n +1 + 1⎟⎟ ⎝ Fn ⎠
(5-2)
in which: Cn+1 = repair cost at (n+1)th leak and Fn = replacement cost.
From the observed data for any given system we can derive a current leak rate. Whenever the 75
current leak rate, Brkcur equals or exceeds Brkth for the first time, the plumbing system should be replaced. Figure 5.3 shows the observed behavior for leak rate as a function of plumbing system installation period (WSSC data).
5.3 Leak rate models 5.3.1 Shamir and Howard’s model Note that for the optimal replacement time, the earliest time at which Eq. (5-2) is satisfied must be chosen. From the optimal time onwards for any time, t, Eq. (5-2) will be satisfied. In order to infer the earliest time at which Eq. (5-2) holds it is necessary to generate leak arrival times. To mimic the actual occurrence of leaks, we adopt the following model due to Shamir and Howard (1979) given by
N (t ) = N (t0 )e A( t −t ) 0
(5-3)
Where N (t) = number of leaks in year t; t = time in years; t0= base year for the analysis (pipe installation year, or the first year for which leak data are available); and A= growth rate coefficient (1/year).
Problem 1 Setting the leak rate N (t) to be equal to the Brkth given in the previous section, we obtain the optimal time of replacement
t * = t0 +
1 ⎡ ln(1 + R ) Fn ⎤ ln ⎢ ⎥ A ⎣ N (t0 )Cn +1 ⎦
(5-4)
For suitably chosen values of t* of replacement time, initial values for N (t0), A, and t0 can be obtained. From Eq. (5-4), t* is assumed to be 25 and 35 years which could be optimal replacement times. The parameters R, Fn, Cn+1, and t0 are constants. For each t* value, the N (t0) and A can have different pairs of values.
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Another procedure is to optimally select the parameters N (t0), A, and to for a suitably chosen objective function. We choose to minimize the sum of the absolute deviations between the calculated and observed leak rates shown in Figure 5.3. That is,
Problem 2 Minimize
∑e
+ t
+ e −t
t
Subject to; where
At + b + e −t − e +t = ln [observed break rate] for all t.
b = ln [N(t0)] and t0 =0 e−t and e +t or slack (convert less than or equal to type constraints into an equation)
and surplus (convert greater than or equal to type constraints into an equation) variables For non-negativity restrictions we write A= A+- A-, b= b+- b-, and A+, A-, b+, b-, e +t ,e −t ≥ 0 Problem P2 is a linear program. From the optimal values for N (t0) and A, we can determine N (t) for any t. In the WSSC data it’s not clear when the plumbing systems were installed. Therefore, it was arbitrarily decided that the WSSC leak rates apply for houses that are 30 years and older. When the on going survey from MUSES project in Virginia Tech is incorporated, more satisfactory results can be expected.
5.3.2 Neural network model Our discussion on neural network modeling closely follows the discussion in Winston and Venkataramanan (2003). Consider Figure 5.4. Each square in Figure 5.4 is a cell of the network also called a neuron.
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The first column of cells is the input layer and the cells are set at the input values. The network in Figure 5.4 is a feed forward network in the sense that the results go to only the higher layers; there are no feed back loops called “recurrent networks”. Cell 0 is called the bias that constitutes a constant input to the next layer. Cell 10 in the last column is the output cell and the last layer is the output layer. All other columns between the input and output columns are called the hidden layers. For any cell j not in the input layer, its input INP(j) is given by INP( j) = ∑ w ij (output from cell i)
(5-5)
i
The output of j (not in the input layer) is obtained by the use of a transfer function f. the output of j, OUT(j) is given as OUT( j) = f[INP( j)]
(5-6)
The optimal weights wij are determined by minimizing the sum of squared differences between the calculated outputs at cell 10 (see Figure 5.4) OUTk(10) and the corresponding observed values Ok(10) given by
∑[OUT (10) − O k
k
(10)]2 for the output layer. While the
k
above description clearly parallels regression analysis, it is the ability to introduce hidden layers that enables neural networks to solve complicated problems. Also, standard functions such as the logistic sigmoidal function given by f (a) =
1 1 + exp(−a)
(5-7)
ea − e− a ea + e − a
(5-8)
and the tan h function given by f (a) =
are used as the transfer functions. In this chapter, the software PREDICT (produced by NEURALWARE) is used to fit the neural network model.
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NeuralWorks PREDICT Software As mentioned above, neural networks are useful when an unknown relationship exists between a set of input data and output data. They can detect trends in data, generalize the data, and predict the outcome. To build a model with PREDICT software involves 3 steps: (1) collection of data and pre-processing, (2) building and training (to learn the structure of the data) the network, (3) testing and validating the network. Based on the input and the corresponding output provided, PREDICT automatically selects, trains, tests, and validates the model. A tutorial on using PREDICT is given in the Appendix B.
5.4 Simulation of leak occurrences In section 5.2, repair/replacement analysis Eq. (5-2) yields optimal replacement as the earliest time at which the condition is satisfied (i.e. satisfied for the first time). Unfortunately, the leak rates given in Figure 5.3 for the WSSC display a late stage behavior (bathtub curve is composed of early, stable, and late stage behavior). In section 5.3, we develop two types of leak rate models namely, an optimization model, Problem P2, and a neural network model. The purpose of these models is to provide an ability to backtrack through leak rate behavior to identify the potential early and normal stage leak rates. These two models provide leak rates throughout the life of a home plumbing system. Potentially, one can assess the leak rates at various times from these two models to use in Eq. (5-2). However, we find generating possible sequences of leak occurrence times has value in terms of ranges (descriptive statistics) for the leak times generated. Also, certain cost details such as total cost can be obtained. Moreover, it presents a comprehensive analysis of leak occurrence behavior.
5.4.1 Non-homogeneous poisson process (NHPP) In the non-homogeneous poisson process, we permit the poisson parameter λ to be a function of time, λ (t). Because leak rates vary with the age of the plumbing system, such a 79
model is necessary. The NHPP also known as non-stationary poisson process is given by (Law and Kelton, 2000). P[ N( t+s ) – N(t) = k ] =
e − b(t,s) [b(t,s)]k k!
(for k=0,1,2.. and t,s ≥ 0, and b(t,s) = θ (t + s) − θ (t) =
(5-9) t +s
∫ λ (y)dy ) t
in which θ (t) is called the expectation function and λ (t) is the rate or intensity function. Using the leak rate models developed in section 5.3 for λ (t) , we present the following procedure for generating the leak occurrence times.
5.4.2 Simulation of Non-Homogeneous Poisson Processes (Thinning process) 1. Let λ = max{λ (t )} be finite. 2. Generate a stationary Poisson process with constant rate λ and arrival times {ti}. For the Poisson process with parameter λ , the inter-arrival times are exponentially distributed with parameter λ . To generate exponentially distributed inter-arrival times randomly, we observe the following F ( x) = 1 − e − λ x . From uniform random number generator obtain U ~ U (0, 1) for F(x). For U = 1 − e − λ x , we obtain X = − we also have x = −
1
λ
1
λ
ln(1 − U ) . Because (1-U) ~ U (0, 1),
ln U . Because x represents the inter-arrival times only, we set
the arrival time ti = ti-1 +xi in which ti is the arrival time of the ith arrival and xi is the inter-arrival time between the (i-1)st and ith arrivals. 3. Following Ross (1996), we make the following observation. Consider a homogeneous
Poisson process with parameter λ , we count an event that
occurs at time t from the homogeneous process with probability
λ (t ) . The λ
counted events then follow a non-homogeneous process with parameter λ (t ) . Using
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the above results, for each arrival time ti, we generate a
Ui ~ U (0, 1). If U i ≤
λ (ti ) , λ
we consider the event at ti as counted. Note we have ti from step 2. More formally let
Ii =
λ (ti ) λ
1
if U i ≤
0
if otherwise
Let the counted indices be denoted by the set J = {i: Ii =1} The events at times ti for i ∈ J , constitute the non-homogeneous Poisson
arrivals.
Following Law and Kelton (2000), the above steps can be implemented as: Assume that ti-1 has been validly generated and want to generate the next arrival time ti: 1. Let t0 = 0. 2. Set t = ti-1 3. Generate U1 and U2 as IID (Independent Identically Distributed) U(0,1) independent of any previous random variates. 4. Replace t by t − (1/ λ ) ln U1 . 5. If U 2 ≤ λ (t ) / λ , return ti = t. Otherwise, go back to step 2.
If the evaluation of λ (t ) is slow, ( λ (t ) is a complicated function involving exponential and trigonometric calculations), computation time might be saved in step 4 by adding an acceptance pretest; i.e., the current value for t is automatically accepted as
the
next
arrival
time
if
U 2 ≤ λ* / λ
,
where λ* = min{λ (t )} and λ = max{λ (t )} . This would be useful especially when λ (t ) is fairly flat.
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5.5 Application to WSSC data The methodology described in this chapter is applied to the observed data from Washington Suburban Sanitary Commission (WSSC). The leak rates are shown in Figure 5.3. Unfortunately, it is not clear at what ages of a plumbing system the leak rates shown apply. Therefore, we have chosen to apply various lag periods starting at 30 years for the leak rate behavior. Assuming a design life of 50 years for copper plumbing systems, we assume lags of 30, 35… years would capture late stage deterioration behavior. Problem P2, optimization formulation in terms of WSSC data is given in Tables 5.1 and 5.2. The results along with the observed data are shown in Table 5.1. Clearly, the match is very good. Next we apply the neural network methodology. Unfortunately with very few data points we cannot fit a neural network model. Therefore, we append data points arbitrarily that are very close to the observed data. These points are obtained by interpolation between the observed data. Linear optimization results along with these data are shown in Table 5.2. Using the expanded data set, we fit both the optimization model and the neural network model. The model results are shown in Figure 5.5. As the LP results shows, it is seen that 30 year lagged observed values and Shamir and Howard’s graph match well. For the expanded data set, as the Shamir and Howard’s equation is exponential, it fits earlier parts, but does not fit well the normal and late stages. On the contrary, neural network fit the data well. It’s showing an Sshape curve and mimics all stages very well. Various curves are fitted in Figure 5.5 with the neural network model performing the best. Possible leak scenarios are generated as shown in Table 5.3 using Table 5.4 neural network leak rates. In Table 5.3 the first, second, and third leak arrival times are 2, 8, and 16 years respectively. The thinning process is simulated 1,000 times and the mean, median, maximum, minimum, and standard deviation statistics for each leak arrival time are computed in Table 5.5. This is done with MATLAB and the source code is given in the Appendix C. For the rejection probabilities, λ (t) values are obtained from the neural network model as shown in 82
Table 5.4. The results of 1,000 simulations are summarized in Table 5.5. The distributions of leak arrival times are shown in Figure 5.6. From the shape of the distribution, each leak occurrence appears to fit a gamma distribution. From the simulated possible leak scenarios, economic analysis is performed to obtain the
ln( optimal replacement time. Following Loganathan et al. (2002), t n +1 − t n
