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Gas and Oil Reliability Engineering. Modeling and AnalysisEduardo Calixto
Gas and Oil Reliability Engineering: Modeling and Analysis, Second Edition, provides the latest tactics and processes that can be used in oil and gas markets to improve reliability knowledge and reduce costs to stay competitive, especially while oil prices are low.
Updated with relevant analysis and case studies covering equipment for both onshore and offshore operations, this reference provides the engineer and manager with more information on lifetime data analysis (LDA), safety integrity levels (SILs), and asset management.
New chapters on safety, more coverage on the latest software, and techniques such as ReBi (Reliability-Based Inspection), ReGBI (Reliability Growth-Based Inspection), RCM (Reliability Centered Maintenance), and LDA (Lifetime Data Analysis), and asset integrity management, make the book a critical resource that will arm engineers and managers with the basic reliability principles and standard concepts that are necessary to explain their use for reliability assurance for the oil and gas industry.
- Provides the latest tactics and processes that can be used in oil and gas markets to improve reliability knowledge and reduce costs
- Presents practical knowledge with over 20 new internationally-based case studies covering BOPs, offshore platforms, pipelines, valves, and subsea equipment from various locations, such as Australia, the Middle East, and Asia
- Contains expanded explanations of reliability skills with a new chapter on asset integrity management, relevant software, and techniques training, such as THERP, ASEP, RBI, FMEA, and RAMS
Gas and Oil Reliability Engineering Modeling and Analysis Second Edition Dr. Eduardo Calixto Federal Fluminense University - LATEC AMSTERDAM • BOSTON • HEIDELBERG • LONDON NEW YORK • OXFORD • PARIS • SAN DIEGO SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO Gulf Professional Publishing is an imprint of Elsevier Gulf Professional Publishing is an imprint of Elsevier 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, USA The Boulevard, Langford Lane, Kidlington, Oxford, OX5 1GB, UK First Edition 2013 Second Edition 2016 Copyright © 2016, 2013 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher's permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress ISBN: 978-0-12-805427-7 For information on all Gulf Professional Publishing visit our website at https://www.elsevier.com/ Publisher: Joe Hayton Acquisition Editor: Katie Hammon Editorial Project Manager: Kattie Washington Production Project Manager: Sruthi Satheesh Designer: Victoria Pearson Typeset by TNQ Books and Journals To my parents, Jose de Arimatea and Maria Auxiliadora Calixto and to my son, Leonardo Calixto Preface The oil and gas industry is a competitive market that requires high-performance assets that can be translated into high operational availability, production efficiency, reliability, and maintainability of all assets. Nowadays, to achieve such high performance, it is necessary to extend this expectation to vendors. Therefore reliability engineering tools are very important to this industry and have contributed greatly to its success during the last several decades. Reliability engineering should be applied systematically in the oil and gas industry to support the assets to achieve and maintain high performance. To meet this goal, it is necessary to establish the reliability management program, which must start at the very beginning of the asset life cycle phase and must be part of daily operations. The reliability management program includes the application of different qualitative and quantitative methods throughout the asset life cycle such as ALT (accelerated life test), HALT (high accelerated life test), RGA (reliability growth analysis), DFMEA (design failure mode and effects analysis), PFMEA (process failure mode and effects analysis), SFMEA (system failure mode and effects analysis), WA (warranty analysis), FRACAS (failure report analysis and correction actions system), PDA (probabilistic degradation analysis), RCM (reliability-centered maintenance), RBI (risk-based inspection), ReBI (reliability-based inspection), ReGBI (reliability growth-based inspection), ORT (optimum replacement time), RAM (reliability, availability, and maintainability) analysis, FTA (fault tree analysis), ETA (event tree analysis), LOPA (layers of protection analysis), SIL (safety integrity level) analysis, bow tie analysis, HRA (human reliability analysis), and APO (asset performance optimization). Indeed, for companies with assets such as operational plants, platforms, and other facilities, quantitative and qualitative techniques are required during different phases of the life cycle. In project (concept, predesign, and design) and operational phases, DFMEA, SFMEA, RCM, RBI, WA, FRACAS, ReBI, ReGBI, ORT, RAM, APO, and HRA can be applied to support decisions for achieving and maintaining high performance in plant facilities and equipment. In addition, safety is one of the most important performance aspects of the oil and gas industry, and quantitative risk analysis methods, such as FTA, ETA, LOPA, SIL, and bow tie analysis, as well as qualitative risk analysis such as HAZOP, HAZID, PHA, and FMEA, can be implemented during the project (concept, predesign, and design) and operational phases. Moreover, when equipment is being developed by vendors, ALT, HALT, RGA, and DFMEA are highly important for supporting product development and achieving the performance targets defined in WA by oil and gas companies. This book discusses all of these techniques and includes examples applied to the oil and gas industry. In addition, reliability engineering program implementation as well as asset management is introduced with examples and case studies. Asset management aims to drive action to achieve high-performance assets during the life cycle phases. Asset management is integrated with the company's strategy, business plan, and performance at all organization levels. Therefore reliability engineering plays an important role in an asset management program that encompasses asset integrity and integrated logistic support programs, as will be described in this book. To present all reliability engineering methods as well as reliability engineering program and asset management concepts and applications, this book is organized into different chapters as follows. xv xvi Preface Chapter 1: The main objective of lifetime data analysis (LDA) is to predict the reliability and failure rate for a specific period of time based on PDF (probability density function) parameters. Therefore the description of historical failure and repair data, the type of data (complete, censored, and interval), and how to obtain information from the specialists are relevant and will be discussed. Such PDF plots the reliability and failure rate function. However, the main issue when performing LDA is to define which PDF fits better with the historical data. Thus different goodness of fit methods, such as the plot, rank regression, maximum likelihood, chi square, SmirnoveKolmogorov, and Cramérevon Mises are discussed in this chapter with examples applied to the oil and gas industry. In addition, the different types of PDFs, such as exponential, Weibull, lognormal, loglogistic, normal, logistic, Gumbel, gamma, and R, generalized Gama, Rayleigh, and their parameter characteristics, are also discussed including the importance of confidence limits based on the Fisher matrix concept. At the end of this chapter, LDA case studies applied to oil and gas equipment such as pumps, valves, compressors, heat exchangers, pipelines, and furnaces are presented to clarify the concept of LDA in real applications based on software tools. Chapter 2: This chapter describes the importance of qualitative (HALT and HASS) and quantitative (ALT, RGA, and PDA) reliability engineering methods to understand product weakness under high operational stress and also to predict reliability concerning stress factors that influence equipment performance. During the design phase, ALT, HALT, HASS, and RGA are applied to predict asset failures and reliability in early design stages as well as to support equipment development and performance target achievement. The PDA is applied during the operation phase to predict equipment reliability based on equipment degradation caused by failure cumulative effects. The ALT methodology will be described based on different methods such as Arrhenius, Eyring, inverse power law, temperaturee humidity, temperatureenonthermal, general loglinear, proportional hazard model, and the cumulative risk model. In addition, reliability growth analysis methodology will also be described based on different methods such as Duane, CroweAMSAA (NHPP), LloydeLipow, Gompertz, logistic, Crow extended, and power law. The PDA methodology will present different methods such as linear, exponential, power, logarithmic, and the phase exponential model. All these methods are described mathematically with examples and graphs applied to the oil and gas industry. Chapter 3: This chapter begins with the concepts of failure and different FMEA examples. Furthermore, the qualitative approach to define critical equipment based on different criteria such as safety, environment, production, and cost will be discussed. To clarify the different maintenance strategies applied to RCM the concept of preventive maintenance as well as different types of predictive maintenance will be introduced with examples. The aim of this chapter is to introduce reliability engineering, qualitative methods related to maintenance such as FMEA, RCM, and RBI as well as quantitative methods such as ReBI, RGBI, and ORT. In addition, FRACAS and WA will be discussed as a baseline for a reliability database to enable the LDA discussed in Chapter 1. At the end of the chapter, several FMEA, RCM, and RBI case studies will be demonstrated related to the main critical equipment for the oil and gas industry such as pumps, valves, compressors, vessels, pipelines, tanks, and flexible risers. Chapter 4: The aim of this chapter is to demonstrate the concept of sensors, performance indexes such as reliability, availability, and production efficiency as well as the approach to predict such performance indexes based on RAM methodology. Therefore RAM analysis is the basis of complex system performance analysis. To demonstrate such methodology, RAM analysis steps such as scope definition, LDA, modeling, simulation, critical analysis, sensitivity analysis, and conclusions will be discussed. In fact, RAM analysis simulates the system behavior in terms of subsystem and Preface xvii equipment failures over the life cycle. To perform such simulation the reliability block diagram model is necessary, as will be demonstrated. In addition, rather than predicting the performance index the main object of RAM analysis is to analyze the influence of external factors on system performance, such as logistics, spare parts, redundancy configuration, as well as the effect of preventive maintenance in performance indexes. Each of such external factor influences will be demonstrated by example in this chapter. Moreover, to predict the necessary performance improvement, improvement allocation and optimization concepts will be discussed. The chapter will present 10 different case studies applied to oil and gas onshore and offshore assets during projects and the operation phase. Chapter 5: This chapter aims to present different HRAs to model human error during the different asset life cycles, such as design, manufacturing, commissioning, operation, as well as transportation and maintenance. In addition to human error probability assessment, the human performance factors will be analyzed to minimize the factors that influence human error such as internal (psychological, social, physical, and mental) and external (technology, procedures, ergonomic, and layout). HRA can be performed by different methods such as THERP (technique for human error rate prediction), OAT (operator action tree), ASEP (accident sequence evaluation program), HEART (human error assessment reduction technique), STAHR (social technical analysis of human reliability), SPAR-H (standardized plant analysis risk human reliability), SLIM (success likelihood index method), SHERPA (systematic human error reduction and prediction approach), and Bayesian networks. The advantages and disadvantages of implementing each HRA method will also be discussed. In addition, case studies applied to safety, operational, and maintenance related to the oil and gas industry will be presented. Chapter 6: This chapter aims to present the concept of risk management over the asset life cycle based on hazard identification, risk analysis, risk evaluation, and risk mitigation. To proceed with the risk analysis, different qualitative and quantitative methods will be presented with examples applied to the oil and gas industry. Qualitative risk methods such as HAZOP, HAZID, PHA, and FMEA will demonstrate examples applied to the oil and gas industry to identify different types of hazard and analyze the risk qualitatively. In addition, quantitative risk analysis methods applied to predict the risk, such as FTA, ETA, LOPA, SIL, and bow tie analysis, will be presented with oil and gas examples. At the end, case studies will demonstrate the different combinations of qualitative and quantitative risk methods as well as the combination of risk analysis with other methods such as RAM analysis and HRA. Chapter 7: The aim of this chapter is to discuss the process of building up a successful reliability engineering program in the oil and gas industry. In this way, the first step will be to understand the competitiveness of the oil and gas industry based on different factors such as customers, suppliers, competitors, regulators, and substitute products. A further step will be to understand the application of different reliability methods over enterprises' life cycles. Once the importance of each method application is understood, the successful factors that influence reliability engineering programs, such as culture, organizational framework, resource, and work routine, will be presented. In addition, 10 reliability engineering pitfalls will be presented to avoid common mistakes that have been made for different organizations around the word. Despite these important factors, there are many barriers to implementing reliability engineering programs in many organizations, such as leader profile, fast food culture, and a standard approach. Finally, successful cases of reliability engineering organizations as well as organizations that promote reliability engineering around the world, including Bayer, USNRC, ESRA, ESReDA, SINTEF, Karlsruhe Institute Technology, Indian Institute of Technology xviii Preface Kharagpur, University of Strathclyde Business School, and University of Stavanger, will be presented. Chapter 8: The aim of this chapter is to introduce the concept of asset management by including reliability engineering, asset integrity management, and integrated logistics support programs. Asset management has the main objective of supporting the assets to achieve high performance. Therefore different methods based on reliability engineering, risk management, human reliability, as well as life cycle cost must be performed in a different asset life cycle as defined by the asset management plan. The chapter also presents standards such as PAS 55 and ISO 5500 and additional references such as KP3 asset integrity program and the concepts of JP 886 standards related to ILS. Moreover, the chapter proposes the quantitative asset management evaluation methodology based on ISO 55000 and quality award methodology. To clarify the asset management concepts, four case studies related to asset management are presented. The first one describes asset integrity management during the design phase applied to a subsea asset. The second case study describes the recovery sulfur plant asset integrity implementation during the predesign phase. The third case study describes the integrated logistics support program applied during the design phase for subsea assets. The fourth case study describes an asset management program applied to an integrated offshore system. The benefits of this book include: • • • • • • • • • Understanding how to use failure and repair historical data to predict reliability based on examples and case studies; Understanding how to predict reliability during the design phase, considering stressor factors as well as how to improve such reliability; Predicting reliability based on the cumulative effect of failures; Understanding the concept of different methods applied to repairable equipment and systems with several examples of FMEA, RCM, and RBI applied to the oil and gas industry; Predicting and optimizing complex asset performance, such as refinery plants, utility facilities, subsea, and platform, considering different factors such as reliability, maintainability, preventive maintenance, spare parts, logistics, and life cycle cost; Understanding the application of HRA by considering the influence of human error on asset safety, operation, and maintenance task performance; Understanding the effect of safety on asset performance by considering the relation between safety, reliability, and maintenance; Understanding the importance of a reliability program over the asset life cycle as well as the application of different reliability engineering methods in different asset phases: the 10 reliability pitfalls as well as the reliability program barriers will also be clearly understood; Understanding the asset management program concepts and the relation of asset management to other important programs such as reliability engineering, asset integrity, and integrated logistic support. The new edition of this book is based on the author's knowledge and experience over the past 13 years as a reliability engineer in the oil and gas industry by applying consultant services and proposing solutions for different oil and gas companies from the United States, South America, Europe, Asia, Africa, Australia, and the Middle East. Acknowledgment Thank you to my masters from Fluminense Federal University, Gilson Brito Alves Lima and Oswaldo Luiz Gonçalves Quelhas, for supporting and sponsoring my engineering career. I also thank the following teammates: Joao Marcus Sampaio Gueiros, Jr., Carlos Daniel, Wilson Alves dos Santos, Geraldo Alves, Cid Atusi, Darlene Paulo Barbosa, Leonardo Muniz Carneiro, Aneil Souza, Michael Sabat, Carlos Hanriot, Willyane Castro, Paulo Ricardo, Ronaldo Priante, Oswaldo Martins, Istone R., Alexandre Nunes, Milton Igino, Jose Luiz Nunes, Fernando Sigiliao, Joao Eustaquio, Carlos Eustaquio Soukef, Fabio Franca, Nelmo Furtado, Carlos Andre, Delton Correa, Miguel Ricardo, Rafael Ribeiro, Marcio Bonfa, Jorge Fernandes, Claudio Garcia, Joseane Garcia, Paulo Rijo, Manoel Coelho, Antônio Ribeiro Louzana, Wilson Antunes, Jr., Marco Evangelista, Mariano Pacholok, Mauricio Requiao, Ricardo Alexandre, Amauri dos Santos Cardoso, Helio Goes, Manoel Jose Gomez, Romeu Wachburger, Gustavo de Carvalho, Jorge Luiz Ventura, Douglas Tirapani, Borges Ezequiel, Adelci Menezes, Aldo Silvestre, Tony Lisias, Frederico Vieira, Mario Barros, Paulo Rosemberg, Francisco Bezerra, Eduardo Guerra, William Frederic Schmitt, Antonio Carlos Freitas Araujo, Luiz Eduardo Lopes, Gustavo Furtado, Emerson Vilela, Carlos Frederico Eira, Marcelo Ramos, Atila R., and Professor Carlos Amadeu Palerosi for supporting my reliability analysis over the years. For contributing to this book and to my reliability engineering career, I thank Joao Marcus Sampaio Gueiros, Jr. from Petrobras, Claudio Spano from Reliasoft Brazil, Cid Augusto from Reliasoft Brazil, Pauli Adriano de Amada Garcia from Fluminense Federal University, and Paulo Renato Alves Firmino from UFRPE. I have also to thank Stephen Laurie from Frontica Business Solutions (UK), David Thompson from RAMsoft Ltd. (UK), Doug Ogden from Reliasoft Corporation (USA), Yzak Bot from BQR Reliability Ltd. (Israel), and Andrzej Kozak from UDT, Safety Inspection Office (Poland) for all their support on my international career as a reliability engineer. To my wife Isabel Katrin Calixto: thank you for all your support during this time. xix CHAPTER LIFETIME DATA ANALYSIS 1 CHAPTER OUTLINE 1.1 Quantitative Failure Data Analysis .....................................................................................................2 1.2 Probability Density Functions ..........................................................................................................11 1.2.1 Exponential Probability Density Function....................................................................17 1.2.2 Normal Probability Density Function ..........................................................................20 1.2.3 Logistic Probability Density Function .........................................................................23 1.2.4 Lognormal Probability Density Function......................................................................25 1.2.5 Loglogistic Probability Density Function .....................................................................27 1.2.6 Gumbel Probability Density Function .........................................................................29 1.2.7 Weibull Probability Density Function..........................................................................31 Mixed Weibull Probability Density Function..................................................................... 33 1.2.8 Gamma Probability Density Function..........................................................................35 1.2.9 Generalized Gamma Probability Density Function........................................................37 1.2.10 Rayleigh Probability Density Function ........................................................................39 1.3 Goodness of Fit Methods: How to Define PDF Parameters and Choose PDF that Fits Better in Failures Data ....................................................................................................................41 1.3.1 Plot Method .............................................................................................................42 1.3.2 Rank Regression ......................................................................................................47 1.3.3 Maximum Likelihood Methods ...................................................................................50 1.3.4 Chi Square Methods .................................................................................................52 1.3.5 KolmogoroveSmirnov Method....................................................................................54 1.3.6 Cramerevon Mises Tests...........................................................................................57 1.4 How Reliable is Reliability: Confidence Bound Will Tell You About !!! ................................................59 1.5 Lifetime Data Analysis Cases ..........................................................................................................64 1.5.1 Pump Lifetime Data Analysis.....................................................................................64 1.5.2 Screw Compressor Lifetime Data Analysis Case...........................................................67 1.5.3 Valve Lifetime Data Analysis Case..............................................................................75 1.5.4 Sensor Lifetime Data Analysis Case ...........................................................................76 1.5.5 Heat Exchanger Lifetime Data Analysis Cases .............................................................80 1.5.6 Pipeline Lifetime Data Analysis Cases ........................................................................83 1.5.7 Furnace Lifetime Data Analysis Cases ........................................................................87 References ............................................................................................................................................92 Gas and Oil Reliability Engineering. http://dx.doi.org/10.1016/B978-0-12-805427-7.00001-4 Copyright © 2016 Elsevier Inc. All rights reserved. 1 2 CHAPTER 1 LIFETIME DATA ANALYSIS 1.1 QUANTITATIVE FAILURE DATA ANALYSIS The reliability concept is the probability a piece of equipment, product, or service being successful until a specific time. To define equipment, product, and service reliability it is necessary to collect historical failure data. Therefore the first step in the life cycle analysis study is to know how failures occur over time, which is critical to defining an index like failure rate, reliability, availability, and mean time to failure (MTTF) to support decisions in defining the best time for inspection, maintenance, to check if the equipment has achieved a reliability requirement, and to supply information to new projects. To conduct lifetime data analysis it is necessary to have historical data about failure modes. The failure mode is the way that a piece of equipment or product loses part or all of its capacity to conduct its function. Many companies in the oil and gas and other industries do not have historical data for their equipment, and a number of equipment suppliers have no historical failure data for their products. Therefore the first step in lifetime data analysis is to collect data, but in many cases the engineer who needs data for lifetime data analysis is not the same person who fixes or performs maintenance on the equipment and collects the data. In summary, some companies have historical failure data and others do not. An environment for assessing root cause analysis and solving problem as well as making decisions based on reliable information makes the data collection process and the creation of historical data reports very important. For companies that do not have data to make decisions, the first step is to create a historical data report before carrying out lifetime data analysis. When doing so, managers must be aware of the importance of collecting equipment failure data and also introducing and supporting employees to do so. Moreover, employees must be trained in collecting data and take decisions based on reliable data. This is a big challenge for most companies, because even when procedures and programs are established, it is necessary to collect, assess, and store failure data in files and reports for access later. Depending on the system, collecting failure data depends on maintenance and inspection routines, and this data collection process often competes with other activities. In the oil and gas industry, equipment generally does not have a high frequency of failure, which enables employees to more easily collect and work with equipment failure data. For many reliability professionals, historical failure data means a reliability index, which includes failure rate, reliability, availability, efficiency, MTTF, or probability density function (PDF) parameters. For inspection and maintenance professionals, historical failure data means files with services described by type of failure of occurrence, time to repair data, and recommendations. In fact, if there are no reliability index and PDF parameters for conducting lifetime data analysis, this data must be created by reliability specialists based on available data. In reality, creating the data is the first step of lifetime data analysis, and then defining the reliability index based on this data. The best scenario, of course, is that there is a reliability index and PDF parameters available for reliability professionals, but this is not usually the case. Thus two points of view among reliability professionals are discussed all over the world: the reliability index and PDF parameters must be defined in a report to make analysis easier, or index and PDF parameters must be calculated and updated for the specialist. When reliability professionals assess PDF parameters from reports, the chance of error is greater than when comparing 1.1 QUANTITATIVE FAILURE DATA ANALYSIS 3 them with defined parameters based on historical data. Despite the time required to assess files, creating historical data reports before and then creating the PDF parameters, reliability index, lifetime data analysis based on historical data, and failure root cause analysis are more reliable because they are better understood and updated more frequently. Another important point is that equipment PDF characteristics change over time and PDFs must be assessed whenever a failure occurs, even though there is a reliability index. Thus the failure data reports must be updated from time to time. Additionally, new equipment has different lifetime data cycles over time, and this information needs updating, which makes the reliability index cumbersome. To conduct life cycle analysis the following data, classified by configuration, is required: • • • • • Individual or grouped data; Complete data; Right suspension data; Left suspension data; Data in interval. Individual data is data from one piece of equipment only and grouped historical data comes from more than one piece of similar equipment. In the first case the main objective is to assess equipment for lifetime data analysis, and historical failure data from one piece of equipment is enough, but such equipment should have a certain quantity of data for reliable lifetime data analysis. In some cases there is not enough historical data and it is necessary to look at a similar piece of equipment with a similar function and operational condition to create the historical failure data. In real life, it is not always easy to find similar equipment because in many cases, maintenance operations, and process conditions interfere with the equipment life cycle. In cases where reliability analysis is conducted during the project phase, similarity is easier to obtain because operational conditions, processes, and maintenance procedures are similar to project requirements. However, to increase the reliability of lifetime data analysis historical grouped data must be used and in this case requires considering more than one piece of similar equipment to create PDFs for the equipment being assessed. It is also necessary to validate equipment similarities and in projects this is also easier. When historical data is defined when the failure occurs, the data is called complete and in this case it is necessary to establish a time measure (hours, days, months, years). It is essential to know the initial operation time, that is, when the equipment life cycle began. Caution must be used when defining the initial operation time because in some equipment there is a different start time because it has changed over time. Maintenance and operational data in many cases help to validate the initial operation time. In some cases, equipment has no failure data reports and it appears that the first failure occurred after 5 or 10 years, but in reality no failures have been reported. Fig. 1.1 shows different failure mode data for pumps. Such information is assessed from failure data reports, which include root cause of failure, repair time, and recommendations, as shown in Fig. 1.2. There are many types of reports, but when failure modes are defined it is easier for everyone to understand what happened, why it happened, and to assess if the recommendations conducted solved the failures. When defining failure modes, all employees should understand what each of the failure modes means otherwise some failure modes will be described incorrectly. Sometimes it is difficult to define the failure mode, and in this case it is easier to put the general failure mode as "other" for the classification. However, this must be avoided whenever possible because it does not help identify and solve problems or improve equipment. 4 CHAPTER 1 LIFETIME DATA ANALYSIS FIGURE 1.1 Pump complete data failure modes. The other possibility is to use electronic failure reports, which have the following advantages: • • • • Can be consulted for different sites; Can be updated automatically; Support life cycle analysis automatically; Save maintenance and reliability specialists time in life cycle analysis. Despite those advantages it is necessary to train people to input data in electronic reports. Additionally, electronic reports often do not have the same details as paper reports and in some cases this can influence important decisions. Information security is another concern because electronic reports are easier to access and copy than paper reports. The disadvantages of using electronic reports include: • • • • They have fewer details; May have errors because in some cases the person who inputs the data is not the one who assesses the equipment failure; If there is an electronic system failure the electronic report cannot be accessed; Depending on the particular case, if there is an error in the index, such as failure rate or reliability, it is necessary to check the mathematics used to compile the report. For data configuration, in some cases, when some of the equipment used to create the PDFs has not failed in the observed time, it is considered right censured data and must be considered in the analysis. In real life, in many cases, this data is often not taken into account, but it can influence the reliability index. The other type of data configuration is when there is some data that failure occurred before a specific time, and there is no information about when such failure occurred. This happens most often when failure reports are configured after equipment operation start time. While it may seem that the 1.1 QUANTITATIVE FAILURE DATA ANALYSIS Data: Ref: Equip Tag: B-114001A 10/12/2004 R033 Professional : Alexandre Nunes Type of intervention: Inspection 5 Managment: Dinamic Maintanance Corretive Maintanance Programmed Maintanance Predictive Maintanance x 10/12/2004 10/12/2004 12/12/2004 Data of itervention : Data of start service : Data of Finish service : x Item 1 - Seal leakage 2 - Bearing 3 - Shaft 4 - Rotor 5 - Eletric Motor 6 - Vibration 7 - Impeller 8 - Rings suction 9 - Gaskets 10 - Specify other Time of itervention : Time of start service : Time of Finish service : 8h 9h 10h Failure Mode Types Root Cause Pump operation over than specified in procedure Pump Draw Discharge nozzle Volute Casing Bearings Impeller Seal Signature : Suction nozzle Shaft Oil Rings FIGURE 1.2 Equipment failure reports. equipment had high reliability, in reality there were unreported failures at the beginning of the equipment life cycle. A good example is what happens in one critical equipment life cycle analysis, for example, coke formation in a furnace is expected to happen every 6 months. After looking at the failure report, the PDF that indicates frequency of failures over time was concentrated in 4 years; totally different from what project engineers expected. After consulting the operator it was confirmed 6 CHAPTER 1 LIFETIME DATA ANALYSIS FIGURE 1.3 Furnace PDF (coke formation). that the equipment failed every 6 months, but the failures were not reported at the beginning of the life cycle. Fig. 1.3 shows different PDFs from reported failures data and real data. The PDF characteristic will be discussed in the following sections, but looking at Fig. 1.3 it is possible to see how different the PDF is and how it cannot be used to make decisions. Another historical data configuration is when there is no exact information about when equipment failure occurred but the interval of failure time and this type of data is called interval data. In many cases this is considered enough information to do life cycle analysis, but in some other cases it is not. Fig. 1.4 shows equipment failure occurrences in different intervals over time. In many cases this kind of failure data configuration can be obtained from maintenance and operation specialist opinions even when data is not reported. This is most often the case when equipment failures are not reported, but when they occur the impact on the system is great. The big challenge in life cycle analysis is working with data when there is not much available, or the data available is not reliable enough to be considered. In this case, specialist opinion can be used to define the PDF parameters, and there are some techniques to estimate the variable values from specialist opinion: • • Aggregated individual method: In this method, experts do not meet but make estimates individually. These estimates are then aggregated statistically by taking the geometric mean of all the individual estimates for each task. Delphi method: In this method, experts make their assessments individually and then all the assessments are shown to all the experts and then the parameter values are defined for the group. 1.1 QUANTITATIVE FAILURE DATA ANALYSIS 7 FIGURE 1.4 Turbine failures in interval data. • • • Nominal group technique: This method is similar to the Delphi method, but after the group discussion, each expert makes his or her own assessment. These assessments are then statistically aggregated. Consensus group method: In this method, each member contributes to the discussion, but the group as a whole must then arrive at an estimate upon which all members of the group agree. Bayesian inference methodology: This method is a mathematical approach applied to estimating variable values (a posteriori variable values) based on prior knowledge (ie, taking into account all specialist opinions and prior knowledge to estimate variable values as explained next). The aggregated individual method requires mathematic treatment using the geometric mean to define the final variable value. In this way the weight of each individual opinion will highly influence the results. Such an approach is indicated when there is heterogeneous knowledge among specialists about the estimated variable value. This approach is helpful when it is difficult to get a value consensus of the specialist group, but caution is required when defining specialist opinion weight. The Delphi method requires that specialists know other specialists' opinions and assess until the discussed point is agreed upon (in this case a variable value). With this approach, files are sent to specialists in different places to get opinions, and this process is repeated until consensus is achieved. This approach can be difficult because there is no discussion about conflicting opinions, and it is not always clear why a specialist defines a different value. Despite this, this method is a good option when it is not possible for all specialists to meet. In some cases, for example, specialists send back the questionnaires after the third sequence, and it is necessary to take into account their opinion and decide the value of the variable. The nominal group technique is similar to the Delphi method but after a group discussion specialists give their own opinions about variable values and then those values are statistically assessed. Depending on the variance between variable values there might be a higher or lower error expectation. When specialists have similar opinions, there is not significant variance in the variable value result. The consensus group method requires that specialists discuss the values (after their own individual analyses) with other specialists and then come to a conclusion about ideal parameter values. This approach is helpful because all specialists are given the opportunity to discuss their opinion and details 8 CHAPTER 1 LIFETIME DATA ANALYSIS can be discussed. This approach is most common when there is known equipment but no failure data analysis. In such an approach it is necessary to pay attention to the operational and maintenance conditions the specialist is basing his or her opinion on. A good example is when one specialist states his opinion about heat exchanger incrustation and says it happens in 3 years with half-year deviation. In an effort to better understand his opinion, other specialists ask about which period of time he was taking into account, and he describes what he saw in the last 5e10 years. But this equipment had been in use for 20 years. Fig. 1.5 shows the difference of specialist opinions in terms of frequency of incrustation in the heat exchanger. The difference in these results is very influential. Bayesian inference methodology is a mathematical approach applied to defining variable values based on a priori knowledge to estimate a posteriori variable values (ie, all specialist opinions are considered and prior knowledge is used to estimate variable values). This state is represented by the Bayes equation, as follows: PðAjBÞ ¼ PðAXBÞ PðBjAÞ PðAÞ ¼ PðBÞ PðBÞ This equation can be represented also like: pðqjEÞ ¼ PðqXEÞ PðEjfÞ p0 ðfÞ ¼R PðEÞ PðEjfÞ p0 ðfÞ FIGURE 1.5 Specialist A (normaldm ¼ 5; s ¼ 0.5) and other specialists' opinion (Gumbeldm ¼ 20; s ¼ 0.5). 1.1 QUANTITATIVE FAILURE DATA ANALYSIS 9 where p(qjE) ¼ a posteriori knowledge, which represents uncertainty about q after knowing the value of E; p0(q) ¼ is prior knowledge, before knowing the value of E; and P(Ejq) ¼ is maximum likelihood of specialist opinion. Applying specialist opinion it is possible to estimate the value of q. Such an approach is often used in drilling projects in Brazil to define the probability of an event when risk analysis is being conducted. In this case, that approach is adequate because historical failure data from other drills is not reliable because of the existing different conditions for each drill. After looking at different types of data configuration and specialist opinion techniques the next step is to create the PDFs and assess the data characteristics. Thus data characteristics can be individual, grouped, complete, right suspension, left suspension, at interval, or a combination of these configurations. In addition, data can also be multicensored. This happens when because of any reason a component or piece of equipment under life cycle analysis is censored (maintenance, change in policy, energy breakdown, etc.) without the necessary analysis time. This type of data is common for standby equipment where the main equipment is operated for a period of time and then the standby equipment is substituted for operation. Consequently, there will be a failure and suspense data for different periods of time. Another important difference in data characteristics is between repairable and nonrepairable equipment. When we are considering nonrepairable equipment or components, when such a failure occurs, a new piece of equipment is introduced and a new initial time has to be established to calculate failure time. This happens only when a component or piece of equipment is considered as good as new. Such an assumption is hard to do in real life, even though a component is new, because the processes, maintenance, and operational actions still affect the equipment life cycle. When a human error in component assembly occurs, for example, it is common to have a failure in a couple of hours after replacement even when failure, based on historical data, is expected after some years. Such failure times cannot be considered in life cycle analysis, but when a "good as new" assumption is taken into account, such data will influence PDF shape. In some cases, equipment that would be represented for a PDF shape with failure at the end of the life cycle will be represented by a PDF shape with failure at the beginning of the life cycle. For repairable equipment or components common in the oil and gas industry, each failure must consider the initial time (T0) when the equipment began operation to calculate time to failure, as shown in Fig. 1.1. When repairable equipment or components replace old ones, a new initial time will be defined. The Laplace test, also known as the centroid test, is a measure that compares the centroid of the observed arrival times with the midpoint of the period of observation. Such a method determines whether discrete events in a process have a trend. In this way the Laplace score can be mathematically calculated by: PN T i¼0 i Tn N qffiffiffiffiffiffiffi 2 Ls ¼ 1 Tn 12N where Ls ¼ Laplace score; Tn ¼ period of failure of initial time; T ¼ observation period; and N ¼ number of failure time data. 10 CHAPTER 1 LIFETIME DATA ANALYSIS When the last event occurs at the end of the observation period (Tn ¼ T), use N 1 despite N, so the formula will be: PN1 Ti T i¼1 N ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 Ls ¼ q T 12ðN1 1Þ A Laplace score greater than zero means that there is an increasing trend, and a score less than zero means there is a decreasing trend. When the Laplace score is zero, there is no trend, and in this case it is stationary. When determining the reliability of a repairable system under life cycle analysis, the Laplace test can be used to check failure trends. Table 1.1 shows an example of a seal pump failure over a long period. The first column shows the time when the seal leakage occurred. The second column shows the time between seal leakage, and if this time is used, the seal is considered to be as good as new after repair. This is similar to having an initial time after any seal repair. The last column gives the time between failures, from smallest to highest value. To prove that data cannot be treated as good as new because there is an increase trend in failure, the Laplace test is applied as follows: PN1 Ti i¼1 T2 N 1 Ls ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T 12ðN1 1Þ P51 Ls ¼ T T i¼0 i 5q ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 T 12ð51 1Þ ¼ 42:58 12:92 4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 12:92 12ð51 1Þ ¼ 2:24 If there is an increase in failure rate, this means wear on the seal. Thus failure data has to be fitted in the first column of the table, and it is not correct to consider that after seal repair the seal is as good as new. Fig. 1.6 shows the big difference between the data from the first column and third column of Table 1.1. The PDF on the left shows that most failures occur at the beginning of the life cycle (data from the third column of Table 1.1) and the gray PDF on the right shows that most failures occurs at the end of Table 1.1 Time to Failure (Seal Leakage) T TBF TBF Sequence 6.42 10.67 12.67 12.83 12.92 6.42 4.25 2 0.17 0.08 0.08 0.17 2 4.25 6.42 TBF, Time between failures. 1.2 PROBABILITY DENSITY FUNCTIONS 11 FIGURE 1.6 Wrong versus correct approach PDF (seal leakage failures). the life cycle (data from the first column of Table 1.1). The next section describes the types of PDFs and other reliability parameters. 1.2 PROBABILITY DENSITY FUNCTIONS The PDFs describe graphically the possibility of events occurring over time; in equipment life cycle analysis, this means failure or repair time occurrence over time. This allows maintenance and reliability professionals to make decisions about maintenance policies, inspection policies, and failure behavior. Actuality, another index is necessary, such as failure rate or reliability function to make these decisions, but PDFs are the first step to better understanding how failures occur over time. Fig. 1.7 shows different shapes of PDFs that represent different types of equipment in the oil and gas industry. In fact, failures have a greater chance of occurring at the beginning, during a specific period of time, at the end, or randomly during the equipment life cycle. In some cases, equipment has an expected behavior in terms of failure. Electrical devices have expected constant failure rate and mechanical components have expected increasing failure rate. Sometimes, process conditions or even human actions chance failure equipment's behavior. This is what happens, for example, with an electronic actuator valve that, despite random failure over time, is suspected of failure, because of the effects of water whenever it rains, of having a normal PDF despite exponential PDF behavior on it, as shown in Fig. 1.8. In doing so, equipment PDF behavior is only an expectation of occurrence, because the only way to find out the equipment PDF is to conduct life cycle analysis. It must be noted that no matter what the PDF shape is, it is important to try to understand clearly why the equipment PDF has such a shape. It is also important to validate this information with FIGURE 1.7 PDFs and equipment. FIGURE 1.8 Pressure swing adsorption system valve actuator PDF. 1.2 PROBABILITY DENSITY FUNCTIONS 13 maintenance professionals and operators who know the equipment. In some cases, some data may be missed or not reported in the historical data. PDFs for reliability engineering are represented mathematically in most cases for the following functions: • • • • • • Exponential Normal Logistic Lognormal Loglogistic Weibull The exponential PDF describes random behavior over time and fits well to electrical and electronic equipment best. The normal PDF describes some dynamic equipment failures or failures that occur in specific periods of time with some deviation. The logistic PDF is similar in shape to the normal PDF. The lognormal PDF best describes failure that occurs at the beginning of the life cycle that mostly represents failure in a project, startup, installation, or operation. The loglogistic PDF is similar in shape to lognormal. The Weibull PDF is a generic function and depends on parameters that represent exponential, lognormal, or normal PDFs. The gamma and generalized gamma are also generic PDFs but can represent exponential, lognormal, normal, and Gumbel PDFs, depending on parameter characteristics. Gumbel PDFs represent equipment failures that occur at the end of the life cycle such as in a pipeline, vessel, and towers, and in some cases before the end of the life cycle because a process or facility has influenced the failure mechanism. Despite being used intensively to describe failure over time, PDFs may also describe repair time, costs, or other variables. For repair time, the lognormal and normal PDFs are most often used by reliability professionals. For lognormal PDFs most of the repairs are made for short periods of time when performed by experienced employees and take considerably more time when repair is carried out by an inexperienced employee or logistic issues cause repair delays. A normal PDF is used to represent repair failure for a repair that is made mostly in one period of time with a deviation. The following section explains each PDF mathematically to best illustrate reliability concepts. The PDF shows the behavior of the variable in a time interval, in other words the chance of such an event occurring in a time interval. So a PDF is mathematically represented as follows: Z b Pða x bÞ ¼ f ðxÞdx a This is represented graphically by Fig. 1.9, which is the area between interval a and b. The probability cumulative is PDF integration that represents the chance to failure occurring until time t and is represented by the equation: Z t Pðx tÞ ¼ f ðxÞdx ¼ FðtÞ 0 The cumulative probability of failure is represented by Fig. 1.10. As discussed, reliability is the probability of a piece of equipment, product, or service operating successfully until a specific period of time, and is mathematically complementary of cumulative FIGURE 1.9 Probability density function. FIGURE 1.10 Cumulative density function. Probability of failure (from 0 to t ¼ 2.9). 1.2 PROBABILITY DENSITY FUNCTIONS 15 failure probability. Thus the following equation represents the relation between cumulative failure and reliability (if the two values are added, the result is 100% or 1): RðtÞ þ FðtÞ ¼ 1 RðtÞ ¼ 1 FðtÞ Z t RðtÞ ¼ 1 f ðxÞdx 0 This equation shows that failure rate varies over time. To have a constant value the relation between the PDF and reliability must be constant. Failure rate function analysis is a very important tool for maintenance and reliability professionals because it provides good information about how failure rates change over time. The classic failure rate representation is the bathtub curve, as shown in Fig. 1.11. In fact, equipment failure rate is represented for one or two bathtub curve periods. When three periods of equipment-life shapes exist, such as the bathtub curve, Weibull 3P is being represented. In Weibull 3P (three parameters), three pieces of equipment form a common system or three components form one piece of equipment. Thus the bathtub curve is represented for mixed Weibull, which comprises more than one population; in this case the data of three components. In Fig. 1.12, early life (b ¼ 0.45; h ¼ 2.45; g ¼ 0.45) occurs from 0 to 3.8 years, useful life (b ¼ 1.06; h ¼ 0.063; g ¼ 0.39) FIGURE 1.11 Cumulative density function. Reliability (from 0 to t ¼ 2.9). 16 CHAPTER 1 LIFETIME DATA ANALYSIS occurs from 3.8 to 7.9 years, and wear-out (b ¼ 49.95; h ¼ 8.92; g ¼ 0.14) occurs from 7.9 years on. Generally, lognormal or loglogistic PDF represents well early failures. The exponential PDF represents well random failures. The normal or logistic PDF represents well wear-out failures. The Weibull 3P may be performing different bathtub curve characteristics. Therefore, if equipment, component, or product failure rate shapes the early life characteristic, in most cases, this means failure caused by a design or manufacturing error, installation error, or even human error during operation. Whether the failure rate shapes as a constant line, it's means that the failures occur randomly. Finally, if the failure rate shapes such as an increasing failure rate, that means wear-out. The other important concept in reliability engineering is MTTF, which means the expected time to failure, represented by: Z N MTTF ¼ t$f ðtÞdt 0 In many cases, MTTF is calculated as an arithmetic average, which is correct only for normal, logistic, or PDFs with such normal characteristics, because in this case mean, mode, and expected time are all the same. Another important concept is the mean time between failure (MTBF) value, which is similar to the MTTF value, but repair time is included in the MTBF case. In many cases in the oil and gas industry, expected time to failure is represented in years and expected time to repair is represented in hours. Sometimes repair time is less than 1 day and in most cases less than 1 month. In some cases it takes more than a month to repair a piece of equipment, but in these cases there is mostly a logistical delay issue, such as purchase or delivery delay. In these cases logistical delays are included in the MTBF calculation. The MTBF function can be represented as follows: MTBF ¼ MTTF þ MTTR Z T Z t MTBF ¼ T$f ðxÞdx þ t$f ðyÞdy 0 0 where T is time to failure and t is time to repair. When time to repair is too small compared to time to failure, the MTBF is approximately the MTTF as follows: MTTF >>>> MTTR MTBF z MTTF Z T MTBF z T$f ðxÞdx 0 Another important index is failure rate, which is defined by relations between PDF and reliability functions, as shown in the equation: lðtÞ ¼ f ðtÞ RðtÞ This represents that the failure rate varies over time. To have a constant value the relation between PDF and reliability must be constant. Failure rate function analysis is a very important tool for maintenance and reliability professionals because it gives reliable information as to how the failure rate behaves over time. The classic failure rate representation is a bathtub curve, as shown in Fig. 1.12. 1.2 PROBABILITY DENSITY FUNCTIONS 17 FIGURE 1.12 Bathtub curve. 1.2.1 EXPONENTIAL PROBABILITY DENSITY FUNCTION To further explain reliability engineering concepts we will begin with the exponential PDF because of its simple mathematics compared to other PDFs. The exponential PDF represents a random occurrence over time and best represents electronic, electrical, or random events. However, in some cases, electrical and electronic equipment does not have random failure occurrences over time. The exponential PDF equation is: f ðtÞ ¼ lelt Fig. 1.13 shows the exponential PDF (l ¼ 1.68; g ¼ 0.46), which represents a failure in the temperature alarm. Notice that in the figure the curve begins with a range at 0.46. This means the position parameter (g) represents how long one piece of equipment operates without failure; in other words, how long one piece of equipment has 100% reliability. This means that before parameter position value (g), equipment has 100% reliability. In this case, g ¼ 0.46 (year). When there is a position parameter, it is represented in the PDF equation by: f ðtÞ ¼ lelðtgÞ 18 CHAPTER 1 LIFETIME DATA ANALYSIS FIGURE 1.13 Exponential PDF. This means that failure occurs randomly after a period of time and that it is observed in some electrical equipment. In some cases, parameter position (g) may represent a guaranteed time during which no equipment failures are expected; in other words, 100% reliability until time t ¼ g. After understanding the exponential PDF it is necessary to define the reliability function, the cumulative density function (CDF), and then the failure rate and MTTF as follows: RðtÞ þ FðtÞ ¼ 1 Z t Z t l FðtÞ ¼ f ðxÞdx ¼ lelt ¼ 1 elt ¼ 1 elt l 0 0 RðtÞ ¼ 1 FðtÞ ¼ 1 1 elt ¼ elt The exponential reliability function depends only on the failure rate parameter, therefore the equation is simple. Whenever the exponential reliability function is applied to calculate equipment, product, service, or event reliability, the main assumption is that events occur randomly over time; otherwise it makes no sense to use it. Another important index is failure rate, which is obtained by dividing the PDF and reliability functions to define the failure rate, as follows: lðtÞ ¼ f ðtÞ lelt ¼ lt ¼ l RðtÞ e 1.2 PROBABILITY DENSITY FUNCTIONS 19 The failure rate is constant over time, as shown in Fig. 1.11. The failure rate was calculated based on the PDF and reliability function of Fig. 1.14. In doing so it is possible to see the range of time without value, which represents the position parameter (g ¼ 0.46). The failure rate is constant if events occur randomly over time. To calculate the MTTF applying the following equation, it is possible to see that the MTTF is the inverse of the failure rate in the exponential PDF case: Z t Z t MTTF ¼ t$f ðxÞdx ¼ t lelt dt 0 0 l 1 ¼ l2 t l This happens only for the exponential PDF. Many reliability and maintenance professionals incorrectly consider the MTTF the inverse of the failure rate when the PDF is not exponential. This fact influences decisions because the MTTF cannot be constant over time if failure is not represented by the exponential PDF, which means failures are not random. In wear-out failure phases, the MTTF is lower than the previous phase, and if it has been considered constant, failure will likely occur before the time expected. MTTF ¼ t$ FIGURE 1.14 Failure rate (g ¼ 0.46). 20 CHAPTER 1 LIFETIME DATA ANALYSIS FIGURE 1.15 Gas compressor and component failure rates. Many specialists consider the system PDF as exponential because they believe that by regarding different PDFs for each component and equipment, the system PDF shape will be exponential. In fact, this does not always happen, because depending on the life cycle time assessed, it will have different PDF configurations for the system's equipment. For example, a gas compressor with many components (eg, electric motor, bearing, valve, and seal) with a compressor failure rate is comprised of different component failure rates and will result in an increased compressor failure rate and not a constant failure rate shape, as shown in Fig. 1.15. In a gas compressor there are components with increased failure rates, such as the seal and bearing, constant failure rates, such as the electric motor, and decreased failure rates, such as the gas valve. Including all the data to define the gas compressor failure rate the result is an increased failure rate, as shown in red in Fig. 1.15. The following section describes the normal PDF, which is used in many cases by maintenance and reliability specialists. 1.2.2 NORMAL PROBABILITY DENSITY FUNCTION The normal PDF is a frequently used function because it describes the process under control, which means the variable values occur around the mean with deviation. In fact, many variables from many analyses are treated like normal distributions, but are not always well represented. Once there exists a higher deviation it is harder to predict the variable value; in the reliability case it is either failure time or repair time. In other 1.2 PROBABILITY DENSITY FUNCTIONS 21 words, the less reliable the variable prediction, the less accurate the failure time or repair time value will be. Different from the usual exponential PDF, the normal PDF has two parameters: average (m) and deviation (s). These are called position and scale parameters, respectively. It is important to notice that whenever s decreases, the PDF is pushed toward the mean, which means it becomes narrower and taller. In contrast, whenever s increases, the PDF spreads out away from the mean, which means it becomes broader and shallower. The normal PDF is represented mathematically by: 1 Tm 1 f ðTÞ ¼ pffiffiffiffiffiffi e2 ð s Þ s 2p Fig. 1.16 shows the PDF configuration in a pump seal leakage failure. As discussed, failure time averages around (m ¼ 3.94) with deviation (s ¼ 0.59). The whole figure area represents a 100% chance of failure, and there will always be more chance of seal leakage occurring around the average. The normal reliability function is represented by: Z N 1 Tm 1 pffiffiffiffiffiffi e2 ð s Þ dt RðTÞ ¼ T s 2p There are two remarkable characteristics in the normal PDF. First, the failure rate increases from one specific period of time, which represents the wear-out life characteristic in the bathtub curve. Fig. 1.17 shows an example seal leakage increased failure rate over time. In fact, there will be a constant failure rate during part of the life cycle, and such a constant failure rate before an increase in FIGURE 1.16 Pump seal leakage (normal PDF). 22 CHAPTER 1 LIFETIME DATA ANALYSIS FIGURE 1.17 Seal pump failure rate. failure rate is the main objective of preventive maintenance. This means that by applying preventive maintenance it is possible to avoid increased failure rate or wear-out for a period of the equipment life cycle. To prevent the wear-out life cycle, inspections and preventive maintenance must be conducted before the increased failure rate time begins, and that is a good contribution the reliability engineer can give to maintenance equipment policies. The second remarkable point is that in the normal PDF the MTTF is similar to the mean. Only in this case, the mean average is similar to the expected number of failures. The pump seal leakage failure rate is increased, which represents the wear-out life cycle. Nevertheless, the wear-out life cycle does not mean the equipment has to be replaced. In fact, after repair, depending on equipment degradation and maintenance efficiency, most equipment can recover to almost 100% of initial reliability. To define inspection and maintenance periods of time, the failure rate must be assessed, and in the seal leakage example, 2.7 years is the time during which the failure rate starts to increase, so a specific time must be defined before 2.7 years to perform an inspection of the seal, and if necessary conduct preventive maintenance. In fact, inspection and maintenance will be conducted for different component failure rates and such data will provide input information for maintenance professionals to plan their inspection and maintenance routines over time. In addition to recovering reliability in equipment, there will be one period of time during which operational costs will increase, and in this case such equipment must be replaced. This is the replacement optimum time approach, explained in Chapter 6. 1.2 PROBABILITY DENSITY FUNCTIONS 23 1.2.3 LOGISTIC PROBABILITY DENSITY FUNCTION The logistic PDF is very similar to the normal PDF in shape and also describes the process under control, with a simple mathematical concept. The logistic parameters are mean (m) and deviation (s), and the variable values, failure, or repair time, for example, vary around deviation. Despite having a similar shape the logistic PDF shape looks like a normal PDF. As in the normal PDF case, the less reliable the variable prediction is, the less reliable the failure time or repair time value will be. Similar to the normal PDF the logistic PDF has two parameters: mean (m) and deviation (s), which are also called position and scale parameters, respectively. It is important to notice that whenever s decreases, the PDF is pushed toward the mean, which means it becomes narrower and taller. In contrast, whenever s increases, the PDF spreads out away from the mean, which means it becomes broader and shallower. The logistic PDF is represented mathematically by the equation: f ðtÞ ¼ ez sð1 þ ez Þ2 where: tm s Fig. 1.18 shows the PDF configuration, for example, in seal pump leakage failure data, this time using the logistic PDF (m ¼ 3.94; s ¼ 0.347). As discussed, failure varies around the average. The z¼ FIGURE 1.18 Pump seal leakage (logistic PDF). 24 CHAPTER 1 LIFETIME DATA ANALYSIS whole figure area represents 100% chance of failure, and there will always be more chance of seal leakage occurring around the average. The reliability logistic PDF is represented by: 1 1 þ ez Similar to the normal PDF the failure rate increases from one specific period of time t, which represents the wear-out life characteristic bathtub curve. In fact, there will be part constant failure rate, and this is better than an increased failure rate, and preventive maintenance tries to keep equipment in useful life and avoid wear-out. To prevent the wear-out life cycle, inspection and preventive maintenance must be conducted beforehand, and this is a good way reliability engineers can enhance maintenance policies. As before, the MTTF is similar to the mean. Only in this case the mean average is similar to the expected number of failures. Fig. 1.19 shows the pump seal failure rate as an example increasing over time from 2.1 to 5.6 years and then staying constant. Despite the similarity in the PDF shapes presented in Figs. 1.16 and 1.18, the failure rate presented in Fig. 1.17 is different from the failure rate presented in Fig. 1.19. In the logistic case the point to expect inspection and maintenance is 2.1 years, earlier than the 2.7 years presented in the normal failure rate figure. Despite the same failure data there are some differences between the results that can influence decisions when different PDFs are taken into account, even when they are very similar, as with the normal and logistic PDFs. RðtÞ ¼ FIGURE 1.19 Pump seal failure rate (logistic PDF). 1.2 PROBABILITY DENSITY FUNCTIONS 25 1.2.4 LOGNORMAL PROBABILITY DENSITY FUNCTION The lognormal PDF shapes tell us that most failures occur at the beginning of the life cycle and happen most often because the project was not good, the startup equipment was incorrect, operation of the equipment capacity was poor, or the equipment was built incorrectly. All this has great influence on equipment failure occurring at the beginning of a piece of equipment's life cycle. The lognormal PDF has two parameters: average (m) and deviation (s), which are called position and scale parameters, respectively. It is important to notice that whenever s decreases, the PDF is pushed toward the mean, which means it becomes narrower and taller. In contrast, whenever s increases, the PDF spreads out away from the mean, which means it becomes broader and shallower. Different from normal and logistic distribution the lognormal PDF is skewed to the right, and because of the effect of scale parameters, equipment has more of a chance of failing at the beginning of the life cycle. Mathematically, the lognormal PDF is represented by the function: 0 0 1 T m 1 2 sT 0 0 f ðT Þ ¼ e sT 0 where: sT 0 ¼ LnsT ; T 0 ¼ LnT; m0 ¼ m. The lognormal reliability function is represented by the equation: Z N 0 0 1 T m 1 0 pffiffiffiffiffiffi e2 s0 dt RðT Þ ¼ 0 T s 2p A real example of lognormal can be applied to repair time. In fact, using lognormal to represent repair time suggests that repair is most often performed for a shorter period of time by experienced employees and takes longer for inexperienced employees. Fig. 1.20 shows valve repair time represented by the lognormal PDF. In fact, in many cases a valve is repaired in a warehouse and replaced with a new one so as to not shut down any process. The lognormal PDF describes well the repair time because represent that most of repairs are done in a short period of time and a small number of repair takes a longer time. Even though, it is also possible to describe repair time based on another PDF functions. The repair time can be represented by the normal PDF for example. That means, the repair time is always done during an average time with a standard deviation. In this case, the employees take a similar amount of time to repair equipment. As discussed the lognormal PDF best represents failures at the beginning of the life cycle and those that occur, for example, in tower distillation in refinery plants when oil specification has changed. In this case the lognormal PDF represents the corrosion in the tower, as shown in Fig. 1.21. This is why it is so important to understand exactly why failures occur. In this example the failure occurred before the expected time because of a bad decision to accept a different oil specification in the distillation plant. The expected PDF for corrosion is Gumbel, which means that such a failure mode would happen only in a wear-out life cycle, or in other words at the end of equipment life. Another important point to understand is failure rate behavior over time. The lognormal failure rate increases over time, and after a specific period of time decreases, as shown in Fig. 1.22. That time represents control time and it reduces failures at the beginning of the equipment life cycle. The best way to stop failures is to use proper startup equipment and keep a quality procedure in place to detect quality failure. 26 CHAPTER 1 LIFETIME DATA ANALYSIS FIGURE 1.20 Valve repair time PDF. FIGURE 1.21 Furnace corrosion lognormal PDF. 1.2 PROBABILITY DENSITY FUNCTIONS 27 FIGURE 1.22 Furnace corrosion failure rate time (lognormal). The lognormal distribution is unwanted in all systems because it means equipment failure at the beginning of the life cycle. To avoid this it is necessary to be careful in assembly and startup, and operate and perform maintenance at the beginning of the equipment life cycle. Some equipment has lower reliability than expected and has lognormal distribution. In many cases, when this happens there is not enough time to perform maintenance. In other situations, to save money and reduce costs, maintenance and operation services from suppliers are not taken into account in purchase orders, even when the maintenance team is not familiar with new equipment; consequently repair quality is poor and longer shutdown as well as equipment failing come sooner than expected. 1.2.5 LOGLOGISTIC PROBABILITY DENSITY FUNCTION The loglogistic PDF, like the lognormal PDF shape, shows that most failures occur at the beginning of the life cycle and happen for the same reasons discussed before. The loglogistic PDF has two parameters: average (m) and deviation (s), which are called position and scale parameters, respectively. Again, whenever s decreases, the PDF is pushed toward the mean, which means it becomes narrower and taller. And again, in contrast, whenever s increases, the PDF spreads out away from the mean, 28 CHAPTER 1 LIFETIME DATA ANALYSIS which means it becomes broader and shallower. The loglogistic PDF is also skewed to the right, and because of this, equipment will often fail at the beginning of the life cycle, as in the lognormal PDF case. Mathematically, loglogistic PDFs are represented by: f ðtÞ ¼ ez stð1 þ ez Þ2 where: z¼ ln t m s and t ¼ life cycle time. The loglogistic reliability function is represented by the equation: 1 1 þ ez For example, Fig. 1.23 shows the loglogistic PDF that also represents corrosion in a furnace. Note that there is little difference between the lognormal PDF (gray) and the loglogistic PDF (black). Also note the loglogistic failure rate and its behavior over time. The loglogistic failure rate as well as the lognormal failure rate increase over time, and after specific periods of time decrease, as shown in Fig. 1.24. Comparing the loglogistic failure rate (black line) with the lognormal failure rate (gray line) it is possible to see in Fig. 1.24 that the loglogistic failure rate decreases faster than the logistic failure RðTÞ ¼ FIGURE 1.23 Furnace corrosion loglogistic PDF. 1.2 PROBABILITY DENSITY FUNCTIONS 29 FIGURE 1.24 Furnace corrosion failure rate time (loglogisticdblack line). rate, even using the same historical failure data. Thus it is important to pay attention and choose the PDF that fits the historical failure data better to make the best decisions. We will now discuss the Gumbel PDF, which is skewed to the left, having the opposite mean of the lognormal and loglogistic distributions. 1.2.6 GUMBEL PROBABILITY DENSITY FUNCTION The Gumbel, or smallest extreme value, PDF is the opposite of the lognormal PDF in terms of shape. The curve shape is skewed to the left because most of the failures occur at the end of the life cycle, which represents the robustness of equipment such as vessels and tanks. The Gumbel PDF has two parameters: average (m) and deviation (s), which are called position and scale parameters, respectively. Whenever s decreases, the PDF is pushed toward the mean and becomes narrower and taller. Whenever s increases, the PDF spreads out away from the mean and becomes broader and shallower. Different from the lognormal and loglogistic distributions, the Gumbel PDF is skewed to the left and scale parameters are on the right, so the equipment has a higher chance of failing at the end of the life cycle. Mathematically, the Gumbel PDF is represented by: z 1 f ðtÞ ¼ ee s 30 CHAPTER 1 LIFETIME DATA ANALYSIS where: T m s The Gumbel reliability function is represented by: z¼ RðTÞ ¼ ee z An example of a Gumbel failure is when a tower in a hydrogen generation unit has external corrosion. Such failures occur around 18 years of operation, despite maintenance during the life cycle. Fig. 1.25 shows corrosion in the tower. The failure rate behavior over time has some similarity to the normal and logistic PDFs because after constant value the failure rate starts to increase in a specific period of time. Despite the similarity, in the Gumbel failure rate function, when the failure rate starts to increase it is mostly during the wearout period. In other words, in normal and logistic PDFs, if maintenance was conducted before the increased failure rate period, the equipment will recover some reliability. That does not usually occur in the Gumbel PDF, mainly for vase and tank failure modes. Some pumps and compressors have component failures skewed to the left, and they are well represented by the Gumbel PDF. It is possible to perform maintenance in such equipment and recover part of its reliability. The Gumbel failure rate is constant most of time, and after a specific period of time increases, as shown in the example of external corrosion in the tower of the hydrogen generation plant in Fig. 1.26. FIGURE 1.25 Furnace external corrosion Gumbel PDF. 1.2 PROBABILITY DENSITY FUNCTIONS 31 FIGURE 1.26 Furnace corrosion failure rate time (Gumbel). The following PDFs are generic, which means such functions can represent different PDFs depending on the characteristic and combination parameters. The generic PDFs are Weibull, gamma, and generalized gamma. The first one is more predicable than the last two, as will be shown. 1.2.7 WEIBULL PROBABILITY DENSITY FUNCTION The first generic PDF to be discussed is the Weibull function, which can represent exponential, lognormal, or normal shape characteristics. The Weibull PDF can have any of those characteristics, which means a random failure occurrence over the life cycle, or failure occurrence at the beginning of the life cycle with failure time skewed to the right on average with deviation or failure occurrence around a specific period of time centralized in the average with deviation. The Weibull PDF shape behavior depends on the shape parameter (b), which can be: 0 < b < 1 (asymptotic shape) b ¼ 1 (exponential asymptotic shape) 1 < b < 2 (lognormal shape) b > 2 (normal shape) Regarding shape parameter, as the beta value becomes higher, the PDF shape starts to change from normal shape to Gumbel shape. Fig. 1.27 shows the lognormal PDF characteristic when failures 32 CHAPTER 1 LIFETIME DATA ANALYSIS FIGURE 1.27 Furnace burner damage (Weibull PDF). occur at the beginning of the life cycle. In this case, furnace burner damage may occur, as it did in the distillation furnace after 2 years of operation because of high temperatures in the furnace operation. This is another example of why it is so important to know exactly why a failure is occurring. In this case the failure occurred before the expected time because a different oil specification was used in the distillation plant. The expected PDF for burner damage is Gumbel, which means that such a failure would happen only in the wear-out life cycle, or in other words at the end of a piece of equipment's life. The Weibull PDF has three parameters: a shape parameter (b), a characteristic life parameter (h), and a position parameter (g). If the position parameter is zero, the Weibull PDF has two parameters. The characteristic life or scale parameter means that 63.2% of failures will occur until the h value, that is, a period of time. The position parameter represents how long equipment has 100% reliability; in other words there will be no failure until the g value, which is a certain period of time. In doing so the Weibull PDF is represented by: b1 b T g b1 Tg f ðtÞ ¼ e h h h where b > 0, h > 0, and g > 0. 1.2 PROBABILITY DENSITY FUNCTIONS 33 The Weibull reliability function is represented by the equation: Tgb RðTÞ ¼ e h The Weibull failure rate function is represented by the equation: b T g b1 lðTÞ ¼ h h The Weibull two-parameter PDF in Fig. 1.27 has parameter values b ¼ 1.06 and h ¼ 3.87. Look at the shape parameter. When 1 < b < 2 the PDF shape looks like the lognormal PDF and the characteristic life is h ¼ 3.87, which means that until 3.87 years 63% failure will occur. The failure rate shape in the Weibull function depends on the shape parameter (b), which can be constant over time, constant part of the time, and increasing from a specific time (t), or decreasing part of the time and after a specific time (t) to be constant; that is, respectively, exponential, normal, and lognormal behavior. The advantage of Weibull over other generic PDFs such as gamma and generalized gamma is that by looking at the parameters it is easy to have a clear idea about the shapes. Despite this it represents different PDFs well, but that is not to say that Weibull distributions are best in all cases. In some cases, despite the similarity with other PDFs, it would be better to use other PDFs that best fit the data. This will be discussed in the next section. Mixed Weibull Probability Density Function The mixed Weibull density function has the special characteristic to provide information of different subpopulations inside the sample data assessed. In other words, if equipment historical data is assessed for lifetime data analysis, different subpopulations, such as components, can be characterized based on Weibull parameters. Therefore depending on the K number of subpopulations, there will be a K number of parameters as follows: The K subpopulation is represented by: N1 ; N2 ; N3 ; .NK The Weibull parameters for K subpopulations are represented by: ðb1 ; h1 ; g1 Þ; ðb2 ; h2 ; g2 Þ; ðb3 ; h3 ; g3 Þ; .; ðbK ; hK ; gK Þ In doing so the Weibull density function is represented by the equation: bi 1 K i X Ni bi T gi bi 1 Tg hi f ðtÞ ¼ e Nhi hi i¼1 where b > 0, h > 0, and g > 0. The Weibull reliability function is represented by the equation: bi K i X Ni Tg hi e RðTÞ ¼ N i¼1 34 CHAPTER 1 LIFETIME DATA ANALYSIS The Weibull failure rate function is represented by the equation: lðTÞ ¼ f1;2;3.K ðTÞ R1;2;3.K ðTÞ N1 N2 N3 NK f1 ðTÞ þ f2 ðTÞ þ f3 ðTÞ þ . þ fK ðTÞ N N N N lðTÞ ¼ N1 N2 N3 NK R1 ðTÞ þ R2 ðTÞ þ R3 ðTÞ þ . þ RK ðTÞ N N N N An example of mixed Weibull analysis is demonstrated in Fig. 1.28, when a lifetime data analysis was performed for a refinery pump that shows three subpopulations represented by black, red, and green colors plotted on an unreliability time function. The parameters for each population are, respectively: • • • Population 1 (b ¼ 1, h ¼ 1, g ¼ 1) Population 2 (b ¼ 2, h ¼ 2, g ¼ 2) Population 3 (b ¼ 3, h ¼ 3, g ¼ 1) The population 1 parameters define the early life failure based on a b value lower than 1 (b ¼ 0.85). In addition, it is possible to define 24.5% of the chance that failure will happen in years based on the h value (h ¼ 45.43). The population 2 parameters define the wear out failure pattern based on a b value lower than 1 (b ¼ 5.1). In addition, it is possible to define 36.73% of the chance that failure will happen in years based on the h value (h ¼ 315.14). The population 3 parameters define the early life failure based on a b value lower than 1 (b ¼ 0.72). In addition, it is possible to define 38.77% of the chance that failure will happen in years based on the h value (h ¼ 4100.8). FIGURE 1.28 Pump mixed Weibull PDF. 1.2 PROBABILITY DENSITY FUNCTIONS 35 1.2.8 GAMMA PROBABILITY DENSITY FUNCTION The second generic PDF is the gamma, which, like the Weibull distribution, can represent exponential, lognormal, or normal shape characteristics. The gamma PDF can have any of those characteristics, which means random failure occurrence over the life cycle, or failure occurrence at the beginning of the life cycle with failure time skewed to the right on average with deviation or failure occurrence around a specific period of time centralized in the average with deviation. The gamma PDF shape behavior depends on the shape parameter (k), which can be: k < 1 (asymptotic shape) k ¼ 1 (exponential asymptotic shape) k > 1(lognormal shape) Fig. 1.29 shows different PDF shapes for the gamma PDF depending on shape parameters. From top to bottom, the first line shape looks like an asymptotic shape compared to the Weibull PDF (0 < b < 1), and in this case the shape parameter of the gamma PDF is k ¼ 0.2. The second line shape looks like the exponential PDF shape compared to the exponential or Weibull PDF (b ¼ 1), and in this case the shape parameter of the gamma PDF is k ¼ 0.98 (y1). The third line shape looks like lognormal, and in this case the shape parameter of the gamma PDF is k ¼ 1.2. In all three cases the location parameter is m ¼ 3.8. As with other functions, the gamma PDF has the scale parameter and the highest of such parameters in the PDF is stretched out to the right and its height decreases. In contrast a lower value of FIGURE 1.29 Gama PDF with different shape. 36 CHAPTER 1 LIFETIME DATA ANALYSIS the scale parameter PDF is stretched out to the left and its height increases. The gamma PDF is represented by: z ekze f ðtÞ ¼ tGðkÞ where z ¼ Lnt e m; em ¼ scale parameter; and k ¼ shape parameter. The reliability function is represented by the equation: RðtÞ ¼ 1 G1 ðk; ez Þ The failure rate shape in the gamma PDF also depends on the shape parameter (k), and it can be constant over time (k ¼ 0.9852) (Fig. 1.29), decreasing part of the time, and constant from a specific time (k ¼ 0.21) (Fig. 1.29), or increasing part of the time and then being constant (k ¼ 1.2) (Fig. 1.29). This means, respectively, exponential, asymptotic, and lognormal behavior. Fig. 1.30 shows different failure rate shapes, which depend on the shape parameter (k) value. This means, respectively, exponential, asymptotic, and lognormal behavior. A good example of the gamma PDF (k ¼ 2 and m ¼ 0.57) is a compressor in a propylene plant that fails during the first 2 years because of an incorrect startup procedure after energy shutdown. Despite energy shutdown downtime, the compressor downtime was critical because of the increased total downtime, and the situation only improved after the maintenance manager hired supplier compressor operation services. Fig. 1.31 shows the compressor failure PDF, which means energy shutdown and FIGURE 1.30 Failure rate time (gamma). 1.2 PROBABILITY DENSITY FUNCTIONS 37 FIGURE 1.31 Energy shutdown in the compressor (gamma PDF). human error in the startup compressor. After 3 years the startup compressor procedure conducted properly reduced downtime in energy shutdown cases. After 8 years, cogeneration energy started to supply energy to the propylene plant and energy shutdown slowed down over the years. The next PDF is generalized gamma, which represents different PDFs; however, it is complex, and the shape depends on parameter combinations. 1.2.9 GENERALIZED GAMMA PROBABILITY DENSITY FUNCTION The third generic PDF to be discussed is the generalized gamma function, which can represent different PDF distributions such as exponential, lognormal, normal, or Gumbel shape characteristics; random failures occurring over the life cycle; failures occurring at the beginning of the life cycle with the shape skewed to the right with average and deviation; failures occurring around the specific period of time centralized in the average with deviation; as well as failures occurring at the end of the life cycle with the shape skewed to the left with average and deviation, respectively. The gamma PDF shape behavior depends not only on the shape parameter (k) value, but on a combination of shape parameters and scale parameters (q), including: l ¼ 1 and s ¼ 1 (exponential asymptotic shape) l ¼ 0 (lognormal shape) 38 CHAPTER 1 LIFETIME DATA ANALYSIS When l ¼ s is approximately the gamma distribution shape, it can be the exponential, lognormal, or normal shape (Pallerosi, 2007). In fact, the combination stated previously is very rare when working with data to create a gamma PDF and this makes the generalized gamma hard to predict looking at only parameter values, so it is better to look at the PDF shape by itself. Fig. 1.32 shows the generalized gamma PDF (m ¼ 1.52; s ¼ 0.58; l ¼ 0.116), which represents turbine blade damage failure caused by a component from a cracking catalyst plant. The gamma PDF is represented by: kb1 t b b t $ f ðtÞ ¼ $eðqÞ GðkÞ$q q where q ¼ scale parameter; k ¼ shape parameter; b ¼ shape parameter; and q, k, b > 0. If: 1 1 m ¼ Lnq þ $Ln 2 b l 1 s ¼ pffiffiffi b k 1 l ¼ pffiffiffi k FIGURE 1.32 Turbine blade damage PDF (generalized gamma). 1.2 PROBABILITY DENSITY FUNCTIONS 39 To l s 0, f(t) is: f ðtÞ ¼ 1 jlj $ $e s$t G 12 ln tm l$ s þln 1 l2 l2 el$ ln tm s l To l ¼ 0, f(t) is: 2 1 ln tm 1 pffiffiffiffiffiffi$e2 ð s Þ s$t 2p The generalized gamma reliability function is represented by the following equations: If l < 0: ! ln tm el ð s Þ 1 RðtÞ ¼ GI ; 2 l2 l f ðtÞ ¼ If l ¼ 0: RðtÞ ¼ 1 Q ln t m s If l > 0: RðtÞ ¼ 1 GI ln t m elð s Þ 1 ; 2 l2 l ! The failure rate shape in the gamma function depends on shape parameter (l) and other parameters. Fig. 1.33 shows the compressor blade failure rate function shape, which increases at the beginning and decreases after a specific period of time (t ¼ 7). After discussing the main PDF, which describes equipment failure over time and repair time, it is necessary to know how to define PDF parameters and which is the better PDF for failure or repair data, that is, the PDF that best fits the data. 1.2.10 RAYLEIGH PROBABILITY DENSITY FUNCTION The fourth generic PDF function to be presented is the Rayleigh function, which may also represent different PDF distributions like exponential, lognormal, normal, or Gumbel shape characteristics. The Rayleigh probability density function is represented by the equation: t t22 e 2s s2 t0 f ðtÞ ¼ where s ¼ scale parameter and X ¼ location parameter. The Rayleigh CDF function is described by the equation: t2 FðtÞ ¼ 1 e 2s2 FIGURE 1.33 Turbine blade damage failure rate function (generalized gamma). Rayleigh PDF- f(t) x T 100.00% 90.00% 80.00% 70.00% PDF f(t) 60.00% σ=4.5 σ=2.5 50.00% σ=1.5 σ=1.0 40.00% σ=3.5 30.00% 20.00% 10.00% 0.00% 0 2 4 6 Time (years) FIGURE 1.34 Probability density function f (t). 8 10 12 1.3 GOODNESS OF FIT METHODS 41 FIGURE 1.35 Pump PDF (Rayleigh). Therefore: FðtÞ þ RðtÞ ¼ 1 RðtÞ ¼ 1 FðtÞ t2 RðtÞ ¼ 1 1 e2s2 t2 RðtÞ ¼ e2s2 The Rayleigh CDF shape behavior depends on the scale parameter (s), as shown in Fig. 1.34. Fig. 1.34 shows Rayleigh PDF (s ¼ 65,927 hours (7.5 years)) plotted on CAFDE software from BQR Reliability Engineering Consultant Ltd, which represents pump failures that happen because of bearing and seal failures. Section 1.5 will demonstrate several case studies to show how to perform lifetime data analysis. Case 1.5.1 will describe the components PDF of the pump related to Fig. 1.35. 1.3 GOODNESS OF FIT METHODS: HOW TO DEFINE PDF PARAMETERS AND CHOOSE PDF THAT FITS BETTER IN FAILURES DATA After understanding the different PDF types that represent repair time or failure time, some main questions arise, such as: • • • After collecting data, how do you create the PDF? How do you define the PDF parameters? If you have PDF parameters, how do you determine the best PDF for the failure data? How do you compare two or more PDFs to determine the best one for the failure or repair data? 42 CHAPTER 1 LIFETIME DATA ANALYSIS The first question is answered with different methodologies and the most known are the plot method, the minimum quadratic approach, and the maximum likelihood method. Thus before choosing one method to define the best PDF that fits better on failure or repair data, it is necessary to define PDF parameters. There are two strategies for choosing the best PDF: the first one is to choose the PDF that best fits your data, and the second strategy is to choose the generic PDF (eg, Weibull 2P) and then look into the parameter characteristics and compare them with similar PDFs. When using the first strategy, choosing the correct PDF depends on failure or repair data frequency over the life cycle. So the best PDF will be: • • • • At the beginning of the life cycle: lognormal or loglogistic; During a specific period of time with some equal variance on both sides: normal or logistic; At the end of the life cycle: Gumbel; Randomly occurring over the life cycle: exponential. Generic PDFs, such as Weibull, gamma, and the generalized gamma, may also be chosen. Indeed, the strategy for choosing the PDF that best fits the frequency occurrence may be limited because the specialist may have to choose the PDF based on the equipment or component characteristics. A reliability engineer, for example, assessing the electrical equipment may be likely to choose the exponential PDF based on experience that such equipment has random failures over time or from knowledge gained from literature. The second strategy is to define a generic PDF and look into PDF parameter characteristics; then it is possible to compare it to a similar PDF to find the one that fits better, but this is the second step after defining the PDF parameters. If a generic PDF is chosen, it is important to remember the limitations it represents for other PDFs. For example, Weibull and gamma represent the exponential PDF well, as well as the normal and lognormal PDFs, but not the Gumbel PDF very well. The generalized gamma PDF represents most PDF distributions well, but it is mathematically difficult to work with. To give an example of how to approach PDF parameters, we discuss an electric compressor motor that operates in a drill facility. The electric motor historical failure data does not have exponential shape characteristics, as shown in Fig. 1.36, but the Weibull PDF is used to define parameters. So the next step is to apply the plot method, rank regression method, or the maximum likelihood method. The electric motor histogram gives an idea of the PDF shape, but when a specialist does not have software available or has to make a fast decision, he or she will go with the generic PDF. 1.3.1 PLOT METHOD The first method to define PDF parameters is the plot method, and the first step is to define the rank of the failure. Then it is necessary to define the cumulative probability of failure values for each failure time, and with plotting functions, it is possible to define the PDF parameter values. To define the cumulative probability of failure values for each failure time it is necessary to apply a median rank method with 50% confidence. The median rank equation is: n X n ðMRÞk ð1 MRÞnk ¼ 0:5 k k¼i 1.3 GOODNESS OF FIT METHODS 43 FIGURE 1.36 Electric motor failure histogram. The Bernard equation gives approximately the same values as the median rank method, and the equation is: i 0:3 n þ 0:4 Using data from Fig. 1.36 and applying the Bernard equation, the probability of failure in each time is given in Fig. 1.37. Using data from Fig. 1.33 and applying the Bernard equation, the probability of failure in each time is given in Fig. 1.37. The next step is to plot data on Weibull probability paper, for example, to obtain Weibull parameters. In doing so we are assuming that Weibull distribution will be used like PDFs and the further step is to find out which PDF fits better to the electric motor failure data. Consequently, when plotting cumulative probability failure values on Weibull paper it is possible to define the PDF parameters, as shown in Fig. 1.37. The Weibull paper is obtained by applying Log on X values and LogLog on Y values Y and X axes. The shape parameter (b) is a slope of linear function. The scale parameter, or characteristic life (h), is defined when it goes to 63% of failure and graphically when the Y axis meets the function with a direct line from 63% in the Y axis and then meets the value in the X axis, which is the life characteristic Fðti Þ ¼ 44 CHAPTER 1 LIFETIME DATA ANALYSIS Time to Failure (h) Failure Order F(t) 58 1 6.7% 180 2 16.3% 216 3 26% 252 4 35.6% 421 5 45.2% 515 6 54.8% 571 7 64.4% 777 8 74% 817 9 83.7% 923 10 93.3% FIGURE 1.37 Cumulative probability of failure (Bernard equation). parameter value (h). The position parameter is defined by the difference of the first X value from the first curve (X1) and the first X value from the adjusted curve (X2), as shown in Fig. 1.38. If the adjusted line is on the right, the position value is negative, and if it is on the left, the position parameter value is positive. The plotted methodology can be applied for all PDFs; it depends on the strategy used to define the PDF that best fits the failure or repair data. As discussed, when a generic PDF such as Weibull is FIGURE 1.38 Plotted Weibull 3P CFD and parameters. 1.3 GOODNESS OF FIT METHODS 45 chosen, look at the parameter characteristics to more easily identify the PDF shape. In the electric motor case the three Weibull parameters are b ¼ 1.64, h ¼ 622, and g ¼ 63.73. This means the PDF shape looks like a lognormal PDF (1 b 2). The characteristic parameter (h) means that 63% of failure will occur until 622 h, and the position parameter (g) means that at 63.73 in the time period the equipment starts to degrade. Because it is a negative value, degradation will begin before equipment starts to work. In real life, this means degradation will occur while equipment is in stock or transported to the warehouse. If we do not consider Weibull 3P, it means we are considering Weibull 2P, the shape parameter. (b) will be 1.27 and the characteristic life (h) parameter will be 549. In this case the Weibull 2P parameters are b ¼ 1.27 and h ¼ 549. This means the PDF shape also looks like the lognormal PDF (1.2 b 2.5). The characteristic parameter (h) means that 63% of failures will occur until 549 hours. The Weibull 2P parameter value is very similar to the Weibull 3P parameter value. Fig. 1.39 shows the Weibull 2P plotted. In the end, because it is electrical equipment, it would be helpful to think in regard to exponential distribution, so in this case there is only one parameter to estimate when the CDF is plotted, and that is MTTF. To define the MTTF value it is necessary to define the value of R(t). So regarding t ¼ MTTF it is possible to define R(t) when substituting t in the reliability equation. Further, looking at the graph shown in Fig. 1.39 and regarding such an R(t) value, then dropping down to the X scale in the FIGURE 1.39 Plotted Weibull 2P CFD and parameters. 46 CHAPTER 1 LIFETIME DATA ANALYSIS reliability curve on the graph, it is possible to define the MTTF, which is a time value, as shown in Fig. 1.40. So when t ¼ MTTF: FIGURE 1.40 Plotted exponential CFD and parameter. In fact, because it is electrical equipment, it feasible to regard to exponential distribution to represents this equipment. Therefore, in this case, there is only one parameter to estimate when the CDF is plotted, and that is MTTF. To define the MTTF value plotted in the exponential CDF it is necessary to define the value of F(t). Drawing a line from such value to meet the CFD equation and then dropping down to the X scale results in MTTF, as shown in Fig. 1.40. So when t ¼ MTTF: RðtÞ ¼ elt l¼ 1 MTTF If t ¼ MTTF: RðtÞ ¼ e MTTF t ¼ e MTTF MTTF ¼ e1 ¼ 0:368 1 1 The MTTF value is 476 hours. To define the parameter in the normal and logistic PDFs after plotting the CDF, it is necessary to go to the Y axis line in 50% of failure probabilities and when the line drops down to the X axis in the X scale. That is, the average (m) in the normal distribution meets the CDF line and drops down until meeting the X-valued average (m) in the normal distribution. In the lognormal distribution it is necessary to apply ln in such a value. 1.3 GOODNESS OF FIT METHODS 47 The plot method is a good first step, because it is not possible to compare two or more PDFs to know which one best fits the failure or repair data. That is only possible using the following methods, because the plot method is only a visual representation of how well data is adjusted to linear functions and gives PDF parameters. 1.3.2 RANK REGRESSION Rank regression is often used instead of least squares or linear regression because values of Y come from median rank regression in the Y scale. The rank regression method defines the best straight line that has the best distance between the setup point and the line (function) having the Y or X scale as a reference. This methodology is not applied to Weibull 3P, gamma, and generalized gamma, because in those cases it is not possible to use linear regression. The first step of the rank regression method is to create a linear function applying ln on both the CDF equation sides and then to define the linear parameters for the Y (or X) value. The rank regression equations are: N X i¼1 ðA þ Bxi yi Þ2 ¼ minða; bÞ N X ða þ bxi þ yi Þ i¼1 where A and B are the estimation of the a and b values based on the following equations: PN PN xi i¼1 yi þ B i¼1 ¼ Y BX A¼ N N PN PN PN xi yi xi yi i¼1 N i¼1 B ¼ i¼1 P 2 N x i¼1 i PN 2 x i¼1 i N Applying such an equation to Weibull 2P, as discussed previously, the first step is to turn the Weibull 2P into a linear equation, so: b Th FðTÞ ¼ 1 e b Th 1 FðTÞ ¼ e b T lnð1 FðTÞÞ ¼ ln e h b T lnð1 FðTÞÞ ¼ ln e h b T lnð1 FðTÞÞ ¼ h b T lnðlnð1 FðTÞÞÞ ¼ ln h T nðlnð1 FðTÞÞÞ ¼ b ln h lnðlnð1 FðTÞÞÞ ¼ b ln T b ln h 48 CHAPTER 1 LIFETIME DATA ANALYSIS By applying CDF values of the electric motor in rank regression methodology it is possible to estimate the Weibull 2P parameters. Table 1.2 makes obtaining such parameters easy. So turns out in linear equation the linear function parameters are: Y ¼ lnðlnð1 FðTÞÞÞ A ¼ b ln h B¼b In doing so, applying CDF values of the electric motor in rank regression methodology it is possible to estimate Weibull 2P parameters. Table 1.2 allows obtaining such parameters easily. Observing the values in Table 1.2 and substituting in the following equations, the parameters are: PN PN PN PN PN x y i¼1 i i¼1 i x y yi xi i i B ¼ i¼1 A ¼ i¼1 B i¼1 ¼ Y BX N 2 N N PN x i¼1 i PN 2 x i¼1 i N B¼ 5:23Þ 22:31 58:98ð 10 354:61 ð58:98Þ2 10 ¼ 22:31 þ 30:84 8:53 ¼ ¼ 1:26 354:61 347:86 6:75 and: A¼ 5:23 58:98 1:26$ ¼ 0:523 7:42 ¼ 7:94 10 10 A ¼ b