Principles and Practices of Engineering PE Nuclear Reference Handbook [1.2 ed.]

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PE Nuclear Reference Handbook Version 1.2

This document may be printed from the NCEES Web site for your personal use, but it may not be copied, reproduced, distributed electronically or in print, or posted online without the express written permission of NCEES. Contact [email protected] for more information.

Copyright ©2019 by NCEES®. All rights reserved. All NCEES material is copyrighted under the laws of the United States. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means without the prior written permission of NCEES. Requests for permissions should be addressed in writing to [email protected]. Third posting August 2020

CONTRIBUTORS The PE Nuclear Reference Handbook was developed by members of the American Nuclear Society.

Editors Chapters 1 and 5 Nathan A. Carstens, Ph.D., P.E. Pacific Northwest National Laboratory Chapter 2 Paul G. Edelmann, Ph.D., P.E. Los Alamos National Laboratory Chapter 3 Joshua L. Vajda, P.E., CHP AECOM/URS Federal Services Rebecca L. Steinman, Ph.D., P.E. Exelon Generation Chapter 4 (Nuclear Criticality/Neutronics) Joseph W. Nielsen, Ph.D., P.E. Idaho National Laboratory Chapter 4 (Kinetics) Necdet Kurul, Ph.D., P.E. GE Hitachi Nuclear Energy

Santiago Parra, Ph.D., P.E. Los Alamos National Laboratory Jon D. McWhirter, Ph.D., P.E. TerraPower Harold E. Williamson, P.E. HEW Enterprises Mathew Merten, P.E. ATC Nuclear David S. Orr, P.E. Duke Energy Glenn E. Sjoden, Ph.D., P.E. U.S. Air Force Alexandra L. Siwy, P.E. U.S. Nuclear Regulatory Commission Andrew D. Siwy, P.E. U.S. Nuclear Regulatory Commission

Reviewers

Mark I. Drucker, P.E. Structural Integrity Associates

John S. Bennion, Ph.D., P.E., CHP GE Hitachi Nuclear Energy

Mathew Panicker, Ph.D., P.E. U.S. Nuclear Regulatory Commission

Stanley H. Levinson, Ph.D., P.E. Independent Contractor

Tracy E. Stover, Ph.D., P.E. Savannah River Nuclear Solutions

Jay Z. James, Ph.D., P.E. University of California, Berkeley

Steven A. Arndt, Ph.D., P.E. U.S. Nuclear Regulatory Commission

Brian A. Collins, P.E. Pacific Northwest National Laboratory

Martin W. Schleehauf, P.E. Naval Nuclear Laboratory

Nathan S. Huffman, P.E. Duke Energy

Randy C. Bunt, P.E. Southern Nuclear

Mark W. Peres, P.E. Fluor Nuclear Power

Robert Hayes, Ph.D., P.E., CHP North Carolina State University

Zhegang Ma, Ph.D., P.E. Idaho National Laboratory

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PREFACE About the PE Nuclear Reference Handbook The Principles and Practices of Engineering (PE) Nuclear exam is computer-based, and NCEES will supply all the resource material that you may use during the exam. Reviewing the PE Nuclear Reference Handbook before exam day will help you become familiar with the charts, formulas, tables, and other reference information provided. The PE Nuclear Reference Handbook does not contain all the information required to answer every question on the exam. Basic theories, conversions, formulas, and definitions that examinees are expected to know have not been included. You will not be allowed to bring your personal copy of the PE Nuclear Reference Handbook into the exam room. Instead, the computer-based exam will include a PDF version of the Handbook for your use. The PDF version of the PE Nuclear Reference Handbook that you use on exam day will be very similar to this one. Pages not needed to solve exam questions—such as the cover, introductory material, and references—may not be included in the exam version. In addition, NCEES will periodically revise and update the Handbook, and each exam will be administered using the updated version. No printed copies of the Handbook will be allowed in the exam room.

Other Supplied Exam Material In addition to the PE Nuclear Reference Handbook, the exam will include CFRs and regulatory guides for your use. A list of the material that will be included in your exam is available at ncees.org along with the exam specifications. Any additional material required for the solution of a particular exam question will be included in the question itself. You will not be allowed to bring personal copies of any material into the exam room.

Updates on Exam Content and Procedures NCEES provides PE exam information at ncees.org. Included there are updates on everything exam-related, including specifications, exam-day policies, scoring, and practice exams.

Errata To report errata in this book, log in to your MyNCEES account and send a message. Examinees are not penalized for any errors in the Handbook that affect an exam question.

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CONTENTS 1 NUCLEAR POWER SYSTEMS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Design and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Energy Generation and Conversion . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.2 Probabilistic Risk Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Common Cause Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Quantitative Risk Assessment. . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Event Tree. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Fault Tree. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.3 Heat Transfer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Basic Heat-Transfer Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Convection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Radiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conduction Through a Plane Wall. . . . . . . . . . . . . . . . . . . . . . . 6 Conduction Through a Cylindrical Wall. . . . . . . . . . . . . . . . . . . . 6 Critical Insulation Radius . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Thermal Resistance (R) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Composite Plane Wall. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Transient Conduction Using the Lumped Capacitance Model. . . . . . . . . 8 Fins. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 External Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Internal Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Boiling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Quality and Void Fraction. . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Nucleate Pool Boiling Critical Heat Flux. . . . . . . . . . . . . . . . . . . 22 Minimum Heat Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Film Boiling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Film Condensation of a Pure Vapor. . . . . . . . . . . . . . . . . . . . . . 24 vii

Natural (Free) Convection . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Heat Exchangers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Types of Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Shape Factor (View Factor, Configuration Factor) Relations. . . . . . . . . 29 Net Energy Exchange by Radiation Between Two Bodies. . . . . . . . . . 30 Peaking Factors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Limiting Factors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Peaking Factor Correction Terms . . . . . . . . . . . . . . . . . . . . . . . 34 1.1.4 Fluid Mechanics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Density, Specific Volume, Specific Weight, and Specific Gravity. . . . . . 35 Stress and Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Surface Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 Characteristics of a Static Liquid . . . . . . . . . . . . . . . . . . . . . . . . . 36 The Pressure Field in a Static Liquid. . . . . . . . . . . . . . . . . . . . . 36 Manometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Forces on Submerged Surfaces and the Center of Pressure . . . . . . . . . 37 Archimedes Principle and Buoyancy . . . . . . . . . . . . . . . . . . . . . 38 Principles of One-Dimensional Fluid Flow . . . . . . . . . . . . . . . . . . . 38 The Continuity Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . 38 The Energy Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 The Field Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 Hydraulic Gradient (Grade Line) . . . . . . . . . . . . . . . . . . . . . . . 39 Energy Line (Bernoulli Equation) . . . . . . . . . . . . . . . . . . . . . . 39 Critical Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 Fluid Flow Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Reynolds Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Laminar Flow in Circular Pipe. . . . . . . . . . . . . . . . . . . . . . . . 41 Consequences of Fluid Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Head Loss Due to Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Minor Losses in Pipe Fittings, Contractions, and Expansions. . . . . . . . 44 Pressure Drop for Laminar Flow. . . . . . . . . . . . . . . . . . . . . . . 45 Flow in Noncircular Conduits . . . . . . . . . . . . . . . . . . . . . . . . 45 Drag Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 viii

Characteristics of Selected Flow Configurations. . . . . . . . . . . . . . . . . 46 Open-Channel Flow and/or Pipe Flow of Water. . . . . . . . . . . . . . . 46 Submerged Orifice Operating Under Steady-Flow Conditions . . . . . . . . 46 Orifice Discharging Freely into Atmosphere. . . . . . . . . . . . . . . . . 47 Multipath Pipeline Problems. . . . . . . . . . . . . . . . . . . . . . . . . 47 The Impulse-Momentum Principle. . . . . . . . . . . . . . . . . . . . . . . . 48 Pipe Bends, Enlargements, and Contractions . . . . . . . . . . . . . . . . . 48 Compressible Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 Mach Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 Isentropic Flow Relationships . . . . . . . . . . . . . . . . . . . . . . . . 49 Fluid Flow Machinery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 Centrifugal Pump Characteristics. . . . . . . . . . . . . . . . . . . . . . . 50 Pump Power Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Fan Characteristics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Compressors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Turbines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Performance of Components. . . . . . . . . . . . . . . . . . . . . . . . . 54 Fluid Flow Measurement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 The Pitot Tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 Venturi Meters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 Orifices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Dimensional Homogeneity. . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 Dimensional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 Similitude. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 1.1.5 Thermodynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 Properties of Single-Component Systems . . . . . . . . . . . . . . . . . . . . 58 Nomenclature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 State Functions: Properties. . . . . . . . . . . . . . . . . . . . . . . . . . 58 Properties for Two-Phase (Vapor-Liquid) Systems . . . . . . . . . . . . . . 62 PVT Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Ideal Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Ideal Gas Mixtures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Equations of State (EOS). . . . . . . . . . . . . . . . . . . . . . . . . . . 65 First Law of Thermodynamics. . . . . . . . . . . . . . . . . . . . . . . . . . 66 Closed Thermodynamic System . . . . . . . . . . . . . . . . . . . . . . . 66 ix

Open Thermodynamic System. . . . . . . . . . . . . . . . . . . . . . . . 67 Steady-Flow Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Basic Cycles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Psychrometrics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 Second Law of Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . 77 Kelvin-Planck Statement of Second Law. . . . . . . . . . . . . . . . . . . 77 Clausius' Statement of Second Law. . . . . . . . . . . . . . . . . . . . . . 77 Entropy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 Exergy (Availability). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 Closed-System Exergy (Availability) . . . . . . . . . . . . . . . . . . . . . 78 Open-System Exergy (Availability). . . . . . . . . . . . . . . . . . . . . . 78 Gibbs Free Energy, DG. . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 Helmholtz Free Energy, DA. . . . . . . . . . . . . . . . . . . . . . . . . . 79 Irreversibility, I. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 1.1.6 Reactor Safety Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 Design-Basis Accidents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 Loss of Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 Fuel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 Chemical Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 1.1.7 Reliability Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 Parts-Count Approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 Failure Modes and Effects Analysis. . . . . . . . . . . . . . . . . . . . . . . . 88 Stress Margin Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 1.1.8 Severe Accident Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 LWR Loss-of-Coolant Accidents. . . . . . . . . . . . . . . . . . . . . . . . . 89 Leakage from Containment [39]. . . . . . . . . . . . . . . . . . . . . . . . . . 91 1.1.9 System Interactions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 BWR Power Instability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 MCPR Safety Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 1.1.10 Thermal Hydraulics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 1.1.11 Engineering Economics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 Risk. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Nonannual Compounding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Break-Even Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Inflation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 x

Depreciation [86]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Book Value. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Taxation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 Capitalized Costs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 Bonds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 Rate-of-Return . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 Benefit-Cost Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 ALARA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 Factors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 1.1.12 Uncertainty Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Setpoint Determination [22] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 1.2 Components and Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 1.2.1 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 1.2.2 Instrumentation and Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 1.2.3 BWR Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 Reactor Trip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Steam Line Valves and Safety/Relief Valve (SRV) Operation. . . . . . . . . . 129 Emergency Core Cooling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 Heat and Radioactivity Removal. . . . . . . . . . . . . . . . . . . . . . . . . 135 BWR Chemistry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 Standby Liquid Control System. . . . . . . . . . . . . . . . . . . . . . . . . 141 Containment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 BWR Controls. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 1.3 PWR Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 Reactor Makeup System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 Steam Generator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 PWR ECCS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 PWR Containment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 PWR Reactor Trip Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . 167 PWR Controls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 1.3 Regulations, Codes, and Standards. . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 1.3.1 Standards. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 1.3.2 Regulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 xi

1.3.3 Licensing Documentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 Safety Analysis Reports for Light Water Reactors. . . . . . . . . . . . . . . . 175 Technical Specifications for Light Water Reactors. . . . . . . . . . . . . . . . 176 Deviations from the FSAR and Technical Specifications. . . . . . . . . . . . 177 Safety Analysis Reports for New Plant Designs. . . . . . . . . . . . . . . . . 177 Safety Analysis Reports for Light Water Reactors . . . . . . . . . . . . . . . 178 Technical Specifications for Light Water Reactors . . . . . . . . . . . . . . . 179 Deviations from the FSAR and Technical Specifications . . . . . . . . . . . . 179 Safety Analysis Reports for New Plant Designs . . . . . . . . . . . . . . . . . 180 2 NUCLEAR FUEL CYCLE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 2.1 Fuel Design and Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 2.1.1 Fuel Cladding. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 2.1.2 Depletion, Burnup, and Transmutation. . . . . . . . . . . . . . . . . . . . . . 182 Used Fuel Assay. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 Requirements for Assay Data of Used Nuclear Fuel. . . . . . . . . . . . . . . 188 Nuclear Criticality Safety Considerations for Used Fuel Storage . . . . . . . . 189 2.1.3 Fuel Cycle Processes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 2.1.4 Fuel Bundle Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 Fuel Assembly for a PWR . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 Fuel Assembly for a BWR . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 2.1.5 Conversion and Enrichment Processes. . . . . . . . . . . . . . . . . . . . . . 199 Purification of U3O8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 Conversion of U3O8 to UF6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 Enrichment Balance of Materials. . . . . . . . . . . . . . . . . . . . . . . . 202 Separation Factor [46] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 Separation Work Units. . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 Enrichment by Gaseous Diffusion . . . . . . . . . . . . . . . . . . . . . . . . 205 Enrichment by Gas Centrifuge. . . . . . . . . . . . . . . . . . . . . . . . . . 206 Enrichment by Laser Excitation. . . . . . . . . . . . . . . . . . . . . . . . . 208 2.2 Handling, Shipping, and Storage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 2.2.1 Nuclear Material Accountability and Control. . . . . . . . . . . . . . . . . . 208 2.2.2 Radioactive Materials Storage . . . . . . . . . . . . . . . . . . . . . . . . . . 209 2.2.3 Transport and Storage Design. . . . . . . . . . . . . . . . . . . . . . . . . . 210 3 INTERACTION OF RADIATION WITH MATTER . . . . . . . . . . . . . . . . . . . . 214 3.1 Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 xii

3.1.1 Terms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 3.1.2 Buildup Factors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 Intensity Without Buildup Factors . . . . . . . . . . . . . . . . . . . . . . . . 217 Intensity With Buildup Factors. . . . . . . . . . . . . . . . . . . . . . . . . . 217 Point Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 Line Source. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 Line Source Without Shield. . . . . . . . . . . . . . . . . . . . . . . . . 220 Line Source Without Shield and Offset Receptor. . . . . . . . . . . . . . 221 Infinite Line Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 Line Source With a Shield . . . . . . . . . . . . . . . . . . . . . . . . . . 222 Sievert Integral. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 Disk Source. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 Disk Source With Offset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 Disk Source With Shield. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 E1 Integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 3.1.3 Using the Chart of the Nuclides. . . . . . . . . . . . . . . . . . . . . . . . . 227 Decay Constant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 Mean Life [43] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 Binding Energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 Energy Release. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 3.1.4 Energy Deposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 Stopping Power. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 Range and Energy of Alpha Particles in Air . . . . . . . . . . . . . . . . . . . 235 Range of Alpha Particles in a Medium [43] . . . . . . . . . . . . . . . . . . . . 235 Range of Beta Particles in a Medium [43]. . . . . . . . . . . . . . . . . . . . . 235 Energy Absorbed by Photons [43] . . . . . . . . . . . . . . . . . . . . . . . . . 235 3.1.5 Interaction Coefficients [83] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 3.1.6 Interaction of Photons With Matter. . . . . . . . . . . . . . . . . . . . . . . 243 Photoelectric Effect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 Pair Production. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 Compton Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 Total Microscopic Cross Section. . . . . . . . . . . . . . . . . . . . . . . . . 246 Corrections for Radiative Energy Loss [83] . . . . . . . . . . . . . . . . . . . . 246 3.1.7 Radiation Effects on Materials [30] . . . . . . . . . . . . . . . . . . . . . . . . 247 Creep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 xiii

Embrittlement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 Radiolytic Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 3.1.8 Neutron Activation Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 Number Density Calculation. . . . . . . . . . . . . . . . . . . . . . . . . . . 250 Production of Activation Products . . . . . . . . . . . . . . . . . . . . . . . . 250 Production of Activation Products With Product Removal . . . . . . . . . . . 251 3.1.9 Neutron Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 Neutron Flux From Point Source in Nonattenuating Medium. . . . . . . . . . 251 Neutron Flux from Point Source in Absorbing Medium. . . . . . . . . . . . . 252 3.1.10 Nuclear Reaction Rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 Total Microscopic Cross Section. . . . . . . . . . . . . . . . . . . . . . . . . 252 Absorption Microscopic Cross Section . . . . . . . . . . . . . . . . . . . . . 252 Macroscopic Cross Section (at a Predefined Energy). . . . . . . . . . . . . . 252 Neutron Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 Neutron Reaction Rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 3.1.11 Shield Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 Neutron Absorber. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 Skyshine [83]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 High-Z Materials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 Low-Z Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 3.1.12 Radioactive Waste Classification. . . . . . . . . . . . . . . . . . . . . . . . . 254 Low-Level Radioactive Waste (LLRW) [65] . . . . . . . . . . . . . . . . . 254 Classes of Waste . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 Classification Determined by Long-Lived Radionuclides. . . . . . . . . . 255 The Sum-of-Fractions Rule for Mixtures of Radionuclides. . . . . . . . . 256 Classification Determined by Short-Lived Radionuclides. . . . . . . . . . 256 Classification Determined by Both Long- and Short-Lived Radionuclides. 256 Default LLRW Classification . . . . . . . . . . . . . . . . . . . . . . . . 257 High-Level Radioactive Waste [62] . . . . . . . . . . . . . . . . . . . . . . 257 3.2 Protection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 3.2.1 Dose Assessment and Personnel Safety. . . . . . . . . . . . . . . . . . . . . 258 Biological Effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 Acute Radiation Exposure [18] . . . . . . . . . . . . . . . . . . . . . . . . . . 258 Chronic Radiation Exposure [18] . . . . . . . . . . . . . . . . . . . . . . . . . 259 External Doses of Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 xiv

Internal Doses of Radiation [88] . . . . . . . . . . . . . . . . . . . . . . . . . . 260 Effective-Dose Equivalent: Quality Factor. . . . . . . . . . . . . . . . . . . . 261 Effective Dose: Weighting Factor . . . . . . . . . . . . . . . . . . . . . . . . 263 3.2.2 Dosimetry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 Thermoluminescent Dosimeter [54] . . . . . . . . . . . . . . . . . . . . . . . . 264 3.2.3 Emergency Plans. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 Criticality Alarm System [11], [54]. . . . . . . . . . . . . . . . . . . . . . . . . . 266 Effluent Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 3.2.4 Radiation Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 Mode of Operation [40] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 Gas Ionization Detectors [87] . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 Current Mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 Pulse Mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 Ionization Chambers [87] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 Proportional Counters [87]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 Geiger-Müller (GM) Counters [87] . . . . . . . . . . . . . . . . . . . . . . . . 275 Gas Flow Counters [87] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 Semiconductor Detectors [87] . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 Solid-State Detectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 Scintillation Detectors [87]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 Inorganic Scintillation Detectors . . . . . . . . . . . . . . . . . . . . . . 277 Organic Scintillation Detectors. . . . . . . . . . . . . . . . . . . . . . . . 278 Gaseous Scintillation Detectors. . . . . . . . . . . . . . . . . . . . . . . 278 Neutron Detectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 3.2.5 Counting Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 Error Propagation – Sum or Difference . . . . . . . . . . . . . . . . . . . . . 280 Error Propagation – Multiplication/Division by Constant. . . . . . . . . . . . 280 Error Propagation – Multiplication/Division of Independent Counts. . . . . . 280 Error Propagation – Mean Value of Independent Counts . . . . . . . . . . . . 280 Detection Efficiency Solid Angle. . . . . . . . . . . . . . . . . . . . . . . . . 280 Nonparalyzable Dead-Time Model. . . . . . . . . . . . . . . . . . . . . . . . 281 Paralyzable Dead-Time Model. . . . . . . . . . . . . . . . . . . . . . . . . . 281 4 NUCLEAR CRITICALITY/KINETICS/NEUTRONICS. . . . . . . . . . . . . . . . . . 282 4.1 Criticality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 4.1.1 Diffusion Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 xv

4.1.2 Six-Factor Formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 4.1.3 Extrapolation Distance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 4.1.4 Adjoint Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 4.1.5 Perturbation Theory (Spatial Effects) . . . . . . . . . . . . . . . . . . . . . . 294 4.1.6 Criticality Safety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 Double Contingency Principle [3]. . . . . . . . . . . . . . . . . . . . . . . . . 295 Control Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 Hierarchy of Controls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 4.2 Kinetics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 4.2.1 Reactivity, Neutron Lifetime, and Delayed Neutrons. . . . . . . . . . . . . . 297 4.2.2 Prompt Jump/Drop Approximation . . . . . . . . . . . . . . . . . . . . . . . 300 4.2.3 One-Delayed-Neutron-Group Approximation . . . . . . . . . . . . . . . . . . 300 4.2.4 Fission Product Poisoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 4.2.5 Xenon-135 Poisoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 4.2.6 Samarium-149 Fission Product Poisoning. . . . . . . . . . . . . . . . . . . . 301 4.2.7 Temperature Effects on Reactivity. . . . . . . . . . . . . . . . . . . . . . . . 302 Fuel Temperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 Moderator Temperature and Void Reactivity Coefficients. . . . . . . . . . . . 303 4.2.8 Xenon-135 Oscillation [17] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 4.3 Neutronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 4.3.1 Neutron Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 4.3.2 Chart of the Nuclides. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 4.3.3 Neutron Cross Sections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 Scattering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 Elastic Scattering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 Inelastic Scattering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 Total Scattering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 Fission. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 Total . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 Neutrons Produced per Fission. . . . . . . . . . . . . . . . . . . . . . . . . . 328 Neutrons Produced per Absorption. . . . . . . . . . . . . . . . . . . . . . . . 331 4.3.4 Absorbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 Burnable Poison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 Control Rod Worth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 xvi

4.3.5 Energy Release. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 4.3.6

Core Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 Monte Carlo Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339

4.3.7 Reactivity Transients. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 Anticipated Transient Without Scram (ATWS) . . . . . . . . . . . . . . . . . 341 4.3.8 Thermalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 Maxwellian Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344 Resonance Capture. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 Doppler Broadening. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 5 GENERAL. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 5.1 Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 5.1.1 Algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 Quadratic Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 Exponents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 Logarithm Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 Polar Coordinate System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 Euler's Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352 5.1.2 Trigonometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352 Law of Sines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352 Law of Cosines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352 Identities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 5.1.3 Analytical Geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 Straight Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 Quadratic Surface (Sphere). . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 Nomenclature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 Circular Segment [27]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 Circular Sector [27] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 Sphere [27]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 Right Circular Cylinder [27]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 5.1.4 Differential Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 The Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 Test for a Maximum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 Test for a Minimum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 xvii

Test for a Point of Inflection . . . . . . . . . . . . . . . . . . . . . . . . . 357 The Partial Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 The Curvature of Any Curve [90]. . . . . . . . . . . . . . . . . . . . . . . . . 358 Curvature in Rectangular Coordinates. . . . . . . . . . . . . . . . . . . . 358 The Radius of Curvature. . . . . . . . . . . . . . . . . . . . . . . . . . . 358 L'Hôpital's Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 5.1.5 Integral Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360 Indefinite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 Centroids and Moments of Inertia . . . . . . . . . . . . . . . . . . . . . . . . 362 5.1.6 Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 First-Order Linear Homogeneous Differential Equations with Constant Coefficients. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 First-Order Linear Nonhomogeneous Differential Equations. . . . . . . . . . 364 Second-Order Linear Homogeneous Differential Equations with Constant Coefficients. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 5.1.7 Vector Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 Gradient, Divergence, and Curl . . . . . . . . . . . . . . . . . . . . . . . . . 366 5.1.8 Special Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 Difference Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 First-Order Linear Difference Equation. . . . . . . . . . . . . . . . . . . . . 367 Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 5.2 Probability and Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 5.2.1 Definitions [60], [56], [61], [5], [6]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 5.2.2 Dispersion, Mean, Median, and Mode Values . . . . . . . . . . . . . . . . . . 372 5.2.3 Permutations and Combinations . . . . . . . . . . . . . . . . . . . . . . . . . 374 5.2.4 Sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374 De Morgan's Law. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374 Associative Law. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374 Distributive Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374 5.2.5 Laws of Probability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374 Property 1. General Character of Probability . . . . . . . . . . . . . . . . . . 374 Property 2. Law of Total Probability. . . . . . . . . . . . . . . . . . . . . . . 375 Property 3. Law of Compound or Joint Probability. . . . . . . . . . . . . . . 375 Bayes' Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 xviii

5.2.6 Probability Functions, Distributions, and Expected Values. . . . . . . . . . . 375 Probability Density Function. . . . . . . . . . . . . . . . . . . . . . . . . . . 375 Cumulative Distribution Functions. . . . . . . . . . . . . . . . . . . . . . . . 376 Expected Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 Sums of Random Variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 Binomial Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 Normal Distribution (Gaussian Distribution) . . . . . . . . . . . . . . . . . . 381 The Central Limit Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 Poisson Distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 t-Distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384 F-Distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386 Beta Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386 Gamma Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386 χ2 – Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386 5.2.7 Linear Regression and Goodness of Fit . . . . . . . . . . . . . . . . . . . . . 392 Least Squares. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 Standard Error of Estimate. . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 Confidence Interval for Intercept at . . . . . . . . . . . . . . . . . . . . . . . 393 Confidence Interval for Slope bt . . . . . . . . . . . . . . . . . . . . . . . . . 393 Sample Correlation Coefficient (R) and Coefficient of Determination (R2) . . . 393 5.2.8 Hypothesis Testing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 One-Way Analysis of Variance (ANOVA). . . . . . . . . . . . . . . . . . . . 393 5.2.9 Confidence Intervals, Sample Distributions, and Sample Size. . . . . . . . . . 394 Confidence Interval for the Mean of a Normal Distribution. . . . . . . . . . . 394 Confidence Interval for the Difference Between Two Means. . . . . . . . . . 394 Confidence Intervals for the Variance of a Normal Distribution. . . . . . . . . 395 Sample Size. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 5.2.10 Test Statistics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 5.2.11 Statistical Quality Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396 Average and Range Charts. . . . . . . . . . . . . . . . . . . . . . . . . . . . 396 5.2.12 Standard Deviation Charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 5.2.13 Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398 Tests for Out of Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398 5.3 Material Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 xix

5.4 Mechanics of Materials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408 5.4.1 Uniaxial Stress-Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408 Engineering Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408 Percent Elongation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408 Percent Reduction in Area (RA) . . . . . . . . . . . . . . . . . . . . . . . . . 408 Shear Stress-Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 Uniaxial Loading and Deformation . . . . . . . . . . . . . . . . . . . . . . . 409 Thermal Deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 Cylindrical Pressure Vessel. . . . . . . . . . . . . . . . . . . . . . . . . . . . 410 Hooke's Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410 5.5 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 5.5.1 Ideal Gas Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416 5.5.2 Fundamental Constants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416 5.5.3 Conversion Factors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 5.6 Water Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 5.6.1 English Units. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 Subcooled Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 Saturated Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435 Superheated Properties (Steam Tables). . . . . . . . . . . . . . . . . . . . . 447 5.6.2 Metric Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458 Subcooled Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458 Saturated Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472 Superheated Properties (Steam Tables). . . . . . . . . . . . . . . . . . . . . 481 5.7 Piping Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492 5.8 Periodic Table. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498 NOMENCLATURE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506

xx

1

NUCLEAR POWER SYSTEMS

The use of U.S. Customary System (USCS) units in many equations in the areas of fluid mechanics and heat transfer requires the inclusion of a gravitational conversion constant gc (gc = 32.174 lbm-ft/ lbf-sec2) in order to properly cancel and/or obtain consistent units. For example, ft d 2n g sec lbm lbf ^fth 5=? d 2 n, and ∆p = ρ g ∆z 5=? d 3 n ft ft c lbm-ft d 2n lbf -sec lbm ft 2 nc m ρV ft3 sec lbf 5=? d 2 n ∆p = K 2g 5=? ft lbm-ft c d n lbf -sec 2 2

d

For SI units, gc = 1 kg•m/N•s2. Be careful when solving problems using USCS units to ensure correct magnitude and dimensional homogeneity.

1.1 Design and Analysis 1.1.1 Energy Generation and Conversion A few common formulas used in energy conversion are: 1 • Kinetic energy KE = 2 m0 v 2 • Planck's Law • Einstein's Law • Relativistic Mass

• Relativistic KE where

hc = E h= o m E = mc2 m0  m= v2 − 1 c2 KE = m0 c 2 f

1 − 1p 1 − b2

v b=c

1

CHAPTER 1. NUCLEAR POWER SYSTEMS

1.1.2

Probabilistic Risk Assessment

Common Cause Modeling Common cause failures occur when more than one component, subsystem, or system fails due to shared causes. Three parametric common-cause-factor (CCF) models are: 1. Beta-factor (original CCF) model where: [68] b=

Number of common cause failures Total number of failures

2. Multiple Greek letter (MGL) model (expanded on beta-factor model) 3. Alpha-factor model (addressed uncertainty concerns in MGL)

Quantitative Risk Assessment It is important to develop perspective on the relative risk of reactor operation, e.g., as compared to alternative reactor designs or to other energy production, industrial operations, general human activities, and natural events. Quantitative risk assessment, including the collection of methods known specifically as probabilistic risk assessment (PRA), has been increasingly important to reactor safety since first applied in detail through the WASH-1400 [81] "Reactor Safety Study." With subsequent expansion and refinement, similar methods are now applied to most reactors and a variety of other technological activities. Regulatory Guide 1.174 [55] provides an approach for using probabilistic risk assessment in riskinformed decisions on plant-specific changes to the licensing basis. Event Tree

Definition of the reactor accident sequences that can lead to core meltdown is the important first step in risk assessment. Various outcomes of given initiating events are identified using a logic system based on event trees—an inductive technique in which a set of successive failures is assumed and the final outcome is determined. A simple example of event-tree methodology is shown in Figure 1.1 for failure of engineered safety systems in a loss-of-coolant accident. The initiating event for the sequence is a large break LOCA. Offsite power, emergency diesel generators, and ECCS systems with successes or failures are then considered sequentially. (This formulation does not allow partial successes or failures in individual systems.) If all events were completely independent, there would be a total of 8 distinct possibilities (generally 2i outcomes where i is the number of independent events that follow the initiating event). This situation is shown by Figure 1.1, where λA is the estimated frequency of the LOCA. The systems are designed for high reliability, therefore the conditional failure probabilities Pi for system i are generally much less than unity and the success probabilities, 1 − Pi may be assumed to be roughly equal to 1 (as has been incorporated in Figure 1.1). The overall probability associated with complete paths, then, is simply the product of the individual failure probabilities included in it. Event-tree methods were employed in WASH-1400 for system-failure and containment-release analyses. Low-risk paths were eliminated to leave the higher-risk paths for more detailed analysis.

2

CHAPTER 1. NUCLEAR POWER SYSTEMS

Figure 1.1: Event-Tree Logic for LOCA in a LWR Fault Tree

Quantification of the likelihood of a system failure uses a logic system based on fault trees—a deductive technique in which a final outcome is assumed and the failure(s) leading to it are determined. Starting from the system as a whole, subsystems and then individual components are analyzed to identify the underlying failure mechanisms and to develop a basis for determining failure probabilities. Fault-tree logic illustrating key elements for loss of electric power to the engineered safety systems is shown in Figure 1.2. The primary event is initiated by loss of either ac or dc power, e.g., for pumps and instrumentation, respectively. Because failure data are not readily available for these functions, the next "lower" level needs to be considered. Failure of ac power implies loss of both off-site and on-site power. Further subdivision of the ac power function accounts for multiple tie-ins to the off-site power grid, redundant diesel-generator systems for on-site power, dc battery sources, and then individual components of each system. A similar procedure is applied for analysis of the dc network. The models can become quite complex, especially in terms of interactions such as use of ac power to charge the dc batteries and requirement for dc instrumentation to control the diesel generators. Among other issues applicable to the system represented by Figure 1.2 are the limited lifetime and load capacity of the batteries. Data used with the fault trees include those for component failures, human error, and testing and maintenance time. Human error was found to have a probability up to 100 times greater than component failure. The testing and maintenance were included in recognition that the related down-time is equivalent to system failure. One hour off-line per week, for example, is equivalent to a 6 × 10−3 per year nonavailability or "failure" rate. Although the most reliable data come from comparison of similar systems, the lack of reactor experience required use of data from fossil power plants and chemical operations. The unique LWR problems of radiation damage, activation, and high-temperature wet steam required that the uncertainties be adjusted. Sensitivity of the results to variations in failure rates (especially related to common modes among components, human error, and testing) was analyzed to a limited extent using Monte Carlo techniques and log-normal uncertainty distributions. 3

Power to emergency safety systems OR

DC power

AC power

AND

Emergency on-site AC power

Offsite AC power

AND

AND

Battery chargers

DC batteries

AND

AND

4 Diesel A

Inverter A from DC batteries

Diesel B

Inverter B from DC batteries

Grid connection 1

Grid connection 2

Battery charger set A

Battery charger set B

Battery bank A

Battery bank B

Rectifier A from AC power

Rectifier B from AC power

Figure 1.2: Fault Tree Logic for Loss of Electric Power to Engineered Safety Systems in a PWR [39]

CHAPTER 1. NUCLEAR POWER SYSTEMS

AND

CHAPTER 1. NUCLEAR POWER SYSTEMS

1.1.3 Heat Transfer Heat transfer occurs in three modes: conduction, convection, and radiation. Basic Heat-Transfer Equations Conduction

Fourier's Law of Conduction: dT Qo =− kA dx where Qo = rate of heat transfer, in W W = thermal conductivity, in m : K A = cross-sectional area perpendicular to direction of heat transfer, in m2 K dT dx = change in temperature per unit length, in m k

Convection

Newton's Law of Cooling: Qo = hA _Tw − T3 i where

h = convection heat-transfer coefficient of the fluid, in A = convection surface area, in m2 Tw = wall surface temperature, in K T∞ = bulk fluid temperature, in K Radiation

The radiation emitted by a body is given by Qo = εσAT 4 where ε = emissivity of the body

σ = Stefan-Boltzmann constant W σ = 5.67 × 10−8, in 2 4 m :K A = body surface area, in m2 T = absolute temperature, in K

5

W m2 : K

CHAPTER 1. NUCLEAR POWER SYSTEMS

Conduction Conduction Through a Plane Wall

Electrical circuit analogue of conduction through a plane wall is shown in Figure 1.3.

Figure 1.3: Electrical Circuit Analogue of Conduction Through a Plane Wall Qo = where

− kA _T1 − T2 i L

W k = thermal conductivity, in m : K A = wall surface area normal to heat flow, in m2 L = wall thickness, in m T1 = temperature of one surface of the wall, in K T2 = temperature of the other surface of the wall, in K Conduction Through a Cylindrical Wall

Electrical circuit analogue of conduction through a cylindrical wall is shown in Figure 1.4.

Figure 1.4: Electrical Circuit Analogue of Conduction Through a Cylinder Wall Qo =

− 2rkL _T1 − T2 i r ln r2 1 6

CHAPTER 1. NUCLEAR POWER SYSTEMS Critical Insulation Radius

Electrical circuit analogue of critical insulation radius through a cylindrical wall is shown in Figure 1.5.

Insulation

rcr

Figure 1.5: Electrical Circuit Analogue of Critical Cylinder Insulation Radius rcr = where

kinsulation h3

h∞ = convection heat-transfer coefficient of the fluid, in

W m2 : K

Thermal Resistance (R)

DT Qo = R total Resistances in series are added, so: R total = RR where

K L R = kA , plane wall conduction resistance, in W where L = wall thickness, in m r ln d r2 n 1 K R = 2rkL , cylindrical wall conduction resistance, in W where L = cylinder length, in m K 1 R = hA , convection resistance, in W

7

CHAPTER 1. NUCLEAR POWER SYSTEMS Composite Plane Wall

A composite plane wall is shown in Figure 1.6.

Figure 1.6: Composite Plane Wall To evaluate surface or intermediate temperatures: T −T T −T Qo = 1R 2 = 2R 3 A B Transient Conduction Using the Lumped Capacitance Model

The lumped capacitance model is valid if h6 Biot number, Bi = kA 11 1 s where h = convection heat-transfer coefficient of the fluid, in 6 = volume of the body, in m3

W k = thermal conductivity of the body, in m : K As = surface area of the body, in m2

8

W m2 : K

CHAPTER 1. NUCLEAR POWER SYSTEMS Fourier Number

The thermal Fourier number is the ratio of thermal diffusion to thermal storage and may be defined as at Fo = 2 L where k m2 α = tC , thermal diffusivity, in s p where

W k = thermal conductivity, in m : K kg t = density, in 3 m J Cp = heat capacity of the body, in kg : K

t = characteristic time, in s

L = conduction length, in m Constant Fluid Temperature

If the temperature may be considered uniform within the body at any time, the heat-transfer rate at the body surface is given by dT Qo = hAs _T − T3 i = − t6C p dt where T = body temperature, in K T∞ = fluid temperature, in K t

= time, in s

The temperature variation of the body with time is T (t) − T∞ = (T − T∞) e−βt where

hA b = t6Cs _s −1 i p

1 b = x _s −1 i τ = time constant, in s The total heat transferred (Qtotal) up to time t is Qtotal = ρ∀Cp [Ti − T (t)] where Ti = initial body temperature, in K Fins

For a straight fin with uniform cross section (assuming negligible heat transfer from tip) Qo = hPkAc _Tb − T3 i tanh _mLc i where

h

= convection heat-transfer coefficient of the fluid, in 9

W m2 : K

CHAPTER 1. NUCLEAR POWER SYSTEMS

P = perimeter of exposed fin cross section, in m W k = fin thermal conductivity, in m : K Ac = fin cross-sectional area, in m2 Tb = temperature at base of fin, in K T∞ = fluid temperature, in K m = hP kAc A Lc = L + Pc , corrected length of fin, in m Rectangular Fin

A rectangular fin is shown in Figure 1.7. T∞ , h

P = 2w + 2t Ac = wt t

Tb

w

L

Figure 1.7: Rectangular Fin [47] Pin Fin

A pin fin is shown in Figure 1.8. T∞ , h

P= π D D

Tb

L

Figure 1.8: Pin Fin [47]

10

Ac =

πD 2 4

CHAPTER 1. NUCLEAR POWER SYSTEMS

Convection External Flow Flat Plate of Length L in Parallel Flow 1 1 h= L 2 3 = Nu . Re Pr , where Re L 1 10 5 0 6640 L L k 1 h= L = Nu 0.0366Re 0.L8 Pr 3 , where Re L 2 10 5 L k

where, in all cases, fluid properties should be evaluated at the average temperature between that of the body and that of the flowing fluid, and where hL Nu = average Nusselt number, Nu = k Cpn Pr = Prandtl number = k h = average convection heat-transfer coefficient of the fluid, in L = length, in m

W m2 : K

The Reynolds number is tu L Re L = n3 where m u3 = free stream velocity of fluid, in s kg µ = dynamic viscosity of fluid, in s : m kg ρ = density of fluid, in 3 m Cylinder of Diameter D in Cross Flow

Re D =

tu 3 D n

where D = diameter, in m 1 hD = = CRe Dn Pr 3 , in which the coefficients are defined in Table 1.1. Nu D k

Table 1.1: Coefficients for Cylinder in Cross-Flow [47] ReD

C

n

1−4 4 − 40 40 − 4000 4000 − 40,000 40,000 − 250,000

0.989 0.911 0.683 0.193 0.0266

0.330 0.385 0.466 0.618 0.805

11

CHAPTER 1. NUCLEAR POWER SYSTEMS Flow Over a Sphere of Diameter D

where

1 1 hD Nu D = k = 2.0 + 0.60 Re D2 Pr 3

1 < ReD < 70,000 0.6 < Pr < 400 Internal Flow Laminar Flow in Circular Tubes

For laminar flow (ReD < 2,300), fully developed conditions: 48 NuD = 11 (uniform heat flux) NuD = 3.66 (constant surface temperature) For laminar flow (ReD < 2,300), combined entry length with constant surface temperature: 1

where

Re D Pr 3 n 0.14 = Nu D 1.86 f L p d n b n s D L = length of tube, in m D = tube diameter, in m

kg µb = dynamic viscosity of fluid, in s : m , at bulk temperature of fluid, Tb kg µs = dynamic viscosity of fluid, in s : m , at inside surface temperature of the tube, Ts Turbulent Flow in Circular Tubes

A commonly used correlation for turbulent, fully developed, forced flow is the Dittus-Boelter correlation [15]: Nu D = 0.023Re 0D.8 Pr n where n = 0.3 for cooling n = 0.4 for heating For turbulent flow (ReD >104, Pr > 0.7) for either uniform surface temperature or uniform heat flux condition, the Sieder-Tate equation offers a good approximation: 1 n Nu D = 0.023Re 0D.8 Pr 3 d n b n

0.14

s

Noncircular Ducts

In place of the diameter, D, use the equivalent (hydraulic) diameter (DH ) defined as 4 # ^cross-sectional areah DH = wetted perimeter

12

CHAPTER 1. NUCLEAR POWER SYSTEMS Circular Annulus (Do > Di)

In place of the diameter D use the equivalent (hydraulic) diameter (DH ) defined as D H = Do − Di Liquid Metals (0.003 < Pr < 0.05) Nu D = 6.3 + 0.167Re 0D.85 Pr 0.93 (uniform heat flux)

Nu D = 7.0 + 0.025Re 0D.8 Pr 0.8 (constant wall temperature) Boiling

Evaporation occurring at a solid-liquid interface when Tsolid > Tsat,liquid q′′ = h (Ts − Tsat) = h∆Te where

∆Te = excess temperature

Pool Boiling: Liquid is quiescent; motion near solid surface is due to free convection and mixing induced by bubble growth and detachment. Forced Convection Boiling: Fluid motion is induced by external means in addition to free convection and bubble-induced mixing. Sub-Cooled Boiling: Temperature of liquid is below saturation temperature; bubbles forming at surface may condense in the liquid. Saturated Boiling: Liquid temperature slightly exceeds the saturation temperature; bubbles forming at the surface are propelled through liquid by buoyancy forces. The boiling curve in SI units is illustrated in Figure 1.9. A similar curve in USCS is in Figure 1.10. BOILING REGIMES

FREE CONVECTION

NUCLEATE

ISOLATED BUBBLES

107

C CRITICAL HEAT FLUX, q"max

q"s (W/m2)

P B

105 q"min

104

D A

1

5

LEIDENFROST POINT, q"min

ONB

ΔTe, A ΔTe, B 103

FILM

J ETS AND COLUMNS

q"max

106

TRANSITION

10

ΔTe, C

ΔTe, D

30 120 ΔTe =Ts − Tsat (°C)

1,000

Figure 1.9: Typical Boiling Curve for Water at One Atmosphere: Surface Heat Flux qms as a Function of Excess Temperature, ∆Te = Ts − Tsat [37]

13

CHAPTER 1. NUCLEAR POWER SYSTEMS

Figure 1.10: Typical Boiling Curve in USCS [94] Figure 1.11 shows the different flow patterns that can exist in a BWR fuel bundle during normal operation. As coolant (single-phase liquid) enters the fuel bundle, it is slightly subcooled and begins to gain heat from the forced-flow convection mechanism. Because of subcooling, there is little or no bubble formation. As energy is gained, the coolant temperature increases until nucleate boiling with its attendant bubble formation begins. Early states of nucleate boiling occur while the bulk coolant in the bundle is below liquid saturation enthalpy, and the bubbles readily collapse as the turbulent flow and their buoyancy sweeps them away from the clad surface. A point will be reached where the bulk coolant enthalpy is at liquid saturation (bulk boiling) and the bubbles will no longer collapse in the coolant as they are swept away. The bubbles now begin to exist separately throughout the bulk coolant, causing a significant steam fraction to be present in the coolant. From this point to the bundle outlet, the bubbles continue to form at the fuel rod surface (nucleate boiling) and be swept into the coolant and begin to coalesce into larger and larger slugs of steam (slug flow). At the outlet of the very highest powered fuel bundles, steam may fill most of the bundle flow area between fuel rods, but a thin annulus of water adheres to the fuel rod surfaces (annular flow). In this annular flow region, the wetted rod surface is still transferring heat through the nucleate boiling mechanism. The critical heat flux (CHF) must be avoided at all core locations to prevent cladding oxidation and/or melting. In pressurized systems, the formation of a layer of individual bubbles may lead to coalescence and the isolation of a section of the clad from the coolant, as shown by part (a) of Figure 1.12. The boiling coolant, by contrast, can have a central vapor core with an annular region of liquid for cooling. When the annular region is thinned excessively by a high heat flux, dryout may occur, as shown in by part (b) of Figure 1.12. (Gaseous or supercritical coolants such as the helium in the HTGR design, of course, are not subject to these phenomena.)

14

CHAPTER 1. NUCLEAR POWER SYSTEMS

Figure 1.11: Fuel Channel Boiling Conditions [24]

15

CHAPTER 1. NUCLEAR POWER SYSTEMS

Figure 1.12: Critical Heat Flux Effects for (a) PWR and (b) BWR Reactors [39] Partial film boiling is not permitted in a reactor because the insulating effect of the steam bubbles causes rapid increases in cladding temperature that can lead to cladding failures. Since the cladding prevents the escape of fission products, failure causes a release of fission products to the coolant. It would be quite simple to prevent cladding failures by not allowing any boiling to occur. However, the advantages of the high heat transfer from nucleate boiling would be lost. The problem now becomes one of allowing nucleate boiling and its associated benefits while preventing the detrimental effects of partial film boiling. In other words, the departure from nucleate boiling (DNB) must be prevented. Many correlations focus on predicting CHF. Due to the complexity of predicting CHF, it is often necessary to perform experiments at controlled conditions that measure the CHF. One manner of expressing the accuracy of the correlations is to plot the predicted versus measured CHF, as shown in Figure 1.13. 1.3 1.2

PREDICTED=MEASURED 1.1 1.0 0.9 0.8

NON-CONSERVATIVE

0.7 SLOPE = .85

0.6 0.5 0.4

CONSERVATIVE 0.3 0.2 0.1 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

2 (Q CHF/1O6) PRED, PREDICTED CRITICAL HEAT FLUX (BTU/HR-FT )

Figure 1.13: Example Predicted versus Measured CHF in USCS [94]

16

1.3

CHAPTER 1. NUCLEAR POWER SYSTEMS

The CHF in the fuel may depend upon several things including the temperature profile and the velocity profile in the fuel bundle. An example set of PWR conditions is shown in Figure 1.14. A

FUEL ROD

A

A

CLADDING

CLADDING

PELLET

A PELLET

COOLANT CHANNEL

COOLANT VELOCITY PROFILE

20 16 12 8 4

1700 1600 Tc

TEMPERATURE PROFILE

1500 1400 1300 1200 1100 1000 900 800

Ts

Tm

700 600 500

Tc - Pellet Centerline Temperature

Ts - Rod Surface Temperature

Tm - Moderator Temperature

Figure 1.14: Example PWR Local Radial Temperature and Velocity [94] The temperature profiles for the fuel, clad, and coolant in Figure 1.15 represent a system that has a cosine-shaped flux. Thus, while the power density and the linear heat rate follow the flux shape, the temperature distributions are skewed by the changing capacity of the coolant to remove the heat energy. Axially symmetric locations, for example, have the same value of linear heat rate, q′, and the same temperature difference, ∆T , between the coolant and the cladding. Because the coolant increases in temperature as it flows up the channel, the clad and, thus, the fuel temperature are relatively higher in the upper axial region of the core. The characteristic relationships among the core-average, average-channel, hot channel, and critical linear heat fluxes for a PWR are shown by Figure 1.16. The heat fluxes are peaked toward the bottom of the core. However, the critical linear heat flux decreases as the coolant is heated while flowing up the channel. The result is a minimum value of the DNBR at the position shown. The essential difference between hot-spot (i.e., qlmax ) and DNB limits is emphasized by the axial separation between the two, shown in Figure 1.16. The linear heat generation rates in Figure 1.16 are shown as being peaked toward the bottom of the core. Negative fuel (Doppler) and moderator feedbacks cause the local reactivity to decrease with increasing coolant temperature. Thus, the linear heat profile in fresh PWR fuel is peaked toward the region of lower coolant temperatures in the bottom of the core. 17

CHAPTER 1. NUCLEAR POWER SYSTEMS Top ΔT

P(z)~q'(z)~φ(z) Tc Tclad

Axial Position, z

Tcool

Coolant Flow

Midcore

ΔT

Bottom Temperature, T

Figure 1.15: Axial Temperature for the Coolant (Tcool), the Clad (Tclad), and the Fuel Pellet Centerline (TC), Based on a Cosine Flux Distribution [39] Top

Axial Position, z

Minimum DNBR

Coolant Flow

Midcore q'max E0. Consequently, ss,0 decreases to the left of the resonance peak and increases to the right, which together lead to the asymmetry in the curve.

346

CHAPTER 4. NUCLEAR CRITICALITY/KINETICS/NEUTRONICS

Figure 4.67: Reaction and Scattering Cross Sections vs. Neutron Energy in Vicinity of a Resonance [7] Suppose that a solution is being sought for the flux f(E) in the vicinity of a narrow resonance at Ei. If the resonances are well separated, it is reasonable to assume that other resonances do not affect the integral very much. Neglect of the effect of these other resonances is referred to as the flux recovery approximation, because it implies that the flux "recovers" to an asymptotic value between resonances. σm + σpοt 1 φ _Ei = σ _Ei E

It is evident, therefore, that since the numerator is constant, the flux will have a pronounced dip in the energy corresponding to a resonance, as shown in Figure 4.68.

Figure 4.68: Cross Sections and Neutron Flux in the Vicinity of a Narrow Resonance [7]

347

CHAPTER 4. NUCLEAR CRITICALITY/KINETICS/NEUTRONICS

Doppler Broadening Cross sections are derived for nuclei at rest. When nuclei are in thermal motion, the resonances will broaden due to the change in relative velocity or the Doppler effect. The velocity of the nuclei may be characterized as a function of direction Vx, Vy, Vz, where Z is defined in the direction of neutron travel. The neutron speed is 2E v= m If the temperature of the medium is above the Debye temperature (roughly 200K for metallic uranium and thorium or 500K in U3O8), then it is a good approximation to assume a Maxwellian distribution of the nuclear velocities.

Using the Maxwellian distribution and reasonable assumptions, one may derive the nuclear cross section of reaction of type x as a function of energy:

where

Γ σx ^Eh = σ0 Γx

E0 E Ψ _ζ, Y i

Ψ _ζ, Y i =

3

2 ^ − h exp =− ζ X Y G 4 dX 1 + X2 2

ζ 2 π

#

−3

2 X = Γ _Er − E0i 2 Y = Γ _E − E0i 4kTE0 4kTE ∆ = A . A Γ ζ =∆ 2 1 1 Er = 2 m :_v − Vzi + V x2 + V y2D . 2 m `v 2 − 2vVzj An example of the effect of Doppler broadening is shown in Figure 4.69.

348

CHAPTER 4. NUCLEAR CRITICALITY/KINETICS/NEUTRONICS

Figure 4.69: Doppler Broadening of a Resonance With Increasing Temperature [42]

349

5

GENERAL

This chapter provides background information that may be of use throughout the examination.

5.1 Mathematics 5.1.1

Algebra

Quadratic Equation ax2 + bx + c = 0 − b± b 2 − 4ac where a ! 0 2a

x = Roots =

Exponents anam = an+m an/am = an−m (an)m = anm (ab)m = ambm m

m c a m = am b b

a

d pn q

= aa

k

4b, real and equal for a2 = 4b, and complex for a2 < 4b. If a2 > 4b, the solution is of the form (overdamped) y = C1 e r1 x + C2 e r2 x If a2 = 4b, the solution is of the form (critically damped) y = _C1 + C2 x i e r1 x

If a2 < 4b, the solution is of the form (underdamped) where

y = eαx `C1 cos βx + C2 sin βxj

a a =− 2 b=

4b − a 2 2 364

CHAPTER 5. GENERAL

5.1.7 Vector Analysis A graphical representation of vectors is shown in Figure 5.9. A = ax i + a y j + az k Addition and subtraction: A + B = _a x + b x i i + _a y + b y i j + _a z + b z i k A − B = _a x − b x i i + _a y − b y i j + _a z − b z i k

The dot product is a scalar product and represents the projection of B onto A times |A|. It is given by A : B = a x b x + a y b y + a z b z = | A || B | cos i = B : A The cross product is a vector product of magnitude |B||A| sin θ which is perpendicular to the plane containing A and B. The product is i

j

k

A # B = a x a y a z =− B # A bx b y bz

Figure 5.9: Vectors [47]

365

CHAPTER 5. GENERAL

A graphical representation of a cross product is shown in Figure 5.10.

Figure 5.10: Cross Product [47] The sense of A × B is determined by the right-hand rule. A # B = A B n sin i n = unit vector perpendicular to the plane of A and B.

Gradient, Divergence, and Curl 2 2 2 dz = d 2x i + 2y j + 2z k n z 2 2 2 d : V = d 2x i + 2y j + 2z k n : `V1 i + V2 j + V3 k j 2 2 2 d # V = d 2x i + 2y j + 2z k n # `V1 i + V2 j + V3 k j The Laplacian of a scalar function z is d2 z = f

22 z 22 z 22 z + + p 2x 2 2y 2 2z 2

Identities A • B = B • A; A • (B + C) = A • B + A • C A • A = |A|2 i•i=j•j=k•k=1 i•j=j•k=k•i=0 If A • B = 0, then either A = 0, B = 0, or A is perpendicular to B. A × B = −B × A A × (B + C) = (A × B) + (A × C) (B + C) × A = (B × A) + (C × A) i×i=j×j=k×k=0 i × j = k = −j × i; j × k = i = −k × j k × i = j = −i × k If A × B = 0, then either A = 0, B = 0, or A is parallel to B. d2z = d • (dz) = (d • d)z d × dz = 0 366

CHAPTER 5. GENERAL

d • (d × A) = 0 d × (d × A) = d(d • A) − d2A

5.1.8

Special Equations

Difference Equations Any system whose input v (t) and output y(t) are defined only at the equally spaced intervals y + −y f (t) = yl = t i 1 − t i i+1 i can be described by a difference equation. First-Order Linear Difference Equation Dt = ti + 1 − ti yi + 1 = yi + yl^Dt h Bessel Functions Bessel functions are solutions to the differential equation: 2 dy 2d y + x + _x 2 − α 2i y = 0 x dx dx 2 Bessel functions for integer values of α appear in the solution to the Laplace equation in cylindrical coordinates. The Bessel function of the first kind, Jα, begin at the origin for positive, integer values of α. They may be represented by: + 3 ( − 1) m x m2m α = / c Jα (x) 2 m = 0 m!Γ _m + α + 1i The Bessel function of the second kind, Yα, are singular (negative infinity) at the origin. For non–integer values, they may be represented as: Jα ^xh cos ^απh − J− α ^xh sin ^απh For integer values n, they are the limit as non–integer α approaches n: Yα ^xh =

Yn ^xh = lim Yα ^xh α"n

Modified Bessel functions are valid for complex arguments of x. The modified first kind Bessel function is: 3 2m + α 1 − c xm Iα ^xh = i α Jα (ix) = / 2 m = 0 m!Γ _m + α + 1i The modified second kind Bessel function is: π I− ^xh − Iα ^xh Kα ^xh = 2 α sin ^απh

Approximate inputs to the various common Bessel functions which yield a value of 0 are shown in Table 5.2.

367

CHAPTER 5. GENERAL

Table 5.2: Bessel Function Roots f(x) x

J0 0.0 2.4048

J1 Y0 Y1 0.0 0.0 0.0 3.8317 0.8936 2.1971

I0 – –

I1 0.0 0.0

K0 0.0

K1 0.0

3

3

A table of Bessel Functions is shown in Table 5.3. The four Bessel functions of 0 kind are shown in Figure 5.11.

X 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60

J0(x) 1.0000 0.9994 0.9975 0.9944 0.9900 0.9844 0.9776 0.9696 0.9604 0.9500 0.9385 0.9258 0.9120 0.8971 0.8812 0.8642 0.8463 0.8274 0.8075 0.7868 0.7652 0.7428 0.7196 0.6957 0.6711 0.6459 0.6201 0.5937 0.5669 0.5395 0.5118 0.4838 0.4554

J1(x) 0.0000 0.0250 0.0499 0.0748 0.0995 0.1240 0.1483 0.1723 0.1960 0.2194 0.2423 0.2647 0.2867 0.3081 0.3290 0.3492 0.3688 0.3878 0.4059 0.4234 0.4401 0.4559 0.4709 0.4850 0.4983 0.5106 0.5220 0.5325 0.5419 0.5504 0.5579 0.5644 0.5699

Table 5.3: Bessel Functions Y0(x) Y1(x) I0(x) –∞ –∞ 1.000 –1.9793 –12.790 1.001 –1.5342 –6.4590 1.003 –1.2708 –4.3637 1.006 –1.0811 –3.3238 1.010 –0.9316 –2.7041 1.016 –0.8073 –2.2931 1.023 –0.7003 –2.0004 1.031 –0.6060 –1.7809 1.040 –0.5214 –1.6095 1.051 − 1.4715 1.063 –0.4445 –0.3739 –1.3572 1.077 –0.3085 –1.2604 1.092 –0.2476 –1.1768 1.108 –0.1907 –1.1032 1.126 –0.1372 –1.0376 1.146 –0.0868 –0.9781 1.167 –0.0393 –0.9236 1.189 0.0056 –0.8731 1.213 0.0481 –0.8258 1.239 0.0883 –0.7812 1.266 0.1262 –0.7388 1.295 0.1622 –0.6981 1.326 0.1961 –0.6590 1.359 0.2281 –0.6211 1.394 0.2582 –0.5844 1.430 0.2865 –0.5485 1.469 0.3131 –0.5135 1.510 0.3379 –0.4791 1.553 0.3610 –0.4454 1.599 0.3824 –0.4123 1.647 0.4022 –0.3797 1.697 0.4204 –0.3476 1.750 368

I1(x) 0.0000 0.0250 0.0501 0.0752 0.1005 0.1260 0.1517 0.1777 0.2040 0.2307 0.2579 0.2855 0.3137 0.3425 0.3719 0.4020 0.4329 0.4646 0.4971 0.5306 0.5652 0.6008 0.6375 0.6754 0.7147 0.7553 0.7973 0.8409 0.8861 0.9330 0.9817 1.0322 1.0848

K0(x) ∞ 3.1142 2.4271 2.0300 1.7527 1.5415 1.3725 1.2327 1.1145 1.0129 0.9244 0.8466 0.7775 0.7159 0.6605 0.6106 0.5653 0.5242 0.4867 0.4524 0.4210 0.3922 0.3656 0.3411 0.3185 0.2976 0.2782 0.2603 0.2437 0.2282 0.2138 0.2004 0.1880

K1(x) ∞ 19.9097 9.8538 6.4775 4.7760 3.7470 3.0560 2.5591 2.1844 1.8915 1.6564 1.4637 1.3028 1.1668 1.0503 0.9496 0.8618 0.7847 0.7165 0.6560 0.6019 0.5534 0.5098 0.4703 0.4346 0.4021 0.3725 0.3455 0.3208 0.2982 0.2774 0.2583 0.2406

CHAPTER 5. GENERAL

X 1.65 1.70 1.75 1.80 1.85 1.90 1.95 2.00 2.05 2.10 2.15 2.20 2.25 2.30 2.35 2.40 2.45 2.50 2.55 2.60 2.65 2.70 2.75 2.80 2.85 2.90 2.95 3.00 3.05 3.10 3.15 3.20 3.25 3.30 3.35 3.40 3.45 3.50 3.55 3.60

J0(x) 0.4268 0.3980 0.3690 0.3400 0.3109 0.2818 0.2528 0.2239 0.1951 0.1666 0.1383 0.1104 0.0827 0.0555 0.0288 0.0025 –0.0232 –0.0484 –0.0729 –0.0968 –0.1200 –0.1424 –0.1641 –0.1850 –0.2051 –0.2243 –0.2426 –0.2601 –0.2765 –0.2921 –0.3066 –0.3202 –0.3328 –0.3443 –0.3548 –0.3643 –0.3727 –0.3801 –0.3865 –0.3918

Table 5.3: Bessel Functions (cont'd) J1(x) Y0(x) Y1(x) I0(x) I1(x) 0.5743 0.4370 –0.3159 1.806 1.1395 0.5778 0.4520 –0.2847 1.864 1.1963 0.5802 0.4655 –0.2540 1.925 1.2555 0.5815 0.4774 –0.2237 1.990 1.3172 0.5818 0.4879 –0.1938 2.057 1.3814 0.5812 0.4968 –0.1644 2.128 1.4482 0.5794 0.5043 –0.1355 2.202 1.5180 0.5767 0.5104 –0.1070 2.280 1.591 0.5730 0.5150 –0.0791 2.361 1.666 0.5683 0.5183 –0.0517 2.446 1.745 0.5626 0.5202 –0.0248 2.536 1.828 0.5560 0.5208 0.0015 2.629 1.914 0.5484 0.5201 0.0272 2.727 2.004 0.5399 0.5181 0.0523 2.830 2.098 0.5305 0.5148 0.0767 2.937 2.196 0.5202 0.5104 0.1005 3.049 2.298 0.5091 0.5048 0.1236 3.167 2.405 0.4971 0.4981 0.1459 3.290 2.517 0.4843 0.4902 0.1675 3.419 2.633 0.4708 0.4813 0.1884 3.553 2.755 0.4566 0.4714 0.2084 3.694 2.883 0.4416 0.4605 0.2276 3.842 3.016 0.4260 0.4487 0.2460 3.996 3.155 0.4097 0.4359 0.2635 4.157 3.301 0.3928 0.4223 0.2802 4.326 3.453 0.3754 0.4079 0.2959 4.503 3.613 0.3575 0.3927 0.3108 4.688 3.779 0.3391 0.3769 0.3247 4.881 3.953 0.3202 0.3603 0.3376 5.083 4.136 0.3009 0.3431 0.3496 5.294 4.326 0.2813 0.3253 0.3607 5.516 4.526 0.2613 0.3071 0.3707 5.747 4.734 0.2411 0.2883 0.3798 5.989 4.953 0.2207 0.2691 0.3879 6.243 5.181 0.2000 0.2495 0.3949 6.508 5.420 0.1792 0.2296 0.4010 6.785 5.670 0.1583 0.2094 0.4061 7.075 5.932 0.1374 0.1890 0.4102 7.378 6.206 0.1164 0.1684 0.4133 7.696 6.493 0.0955 0.1477 0.4154 8.028 6.793 369

K0(x) 0.1763 0.1655 0.1554 0.1459 0.1371 0.1288 0.1211 0.1139 0.1071 0.1008 0.0948 0.0893 0.0840 0.0791 0.0745 0.0702 0.0662 0.0623 0.0588 0.0554 0.0522 0.0493 0.0465 0.0438 0.0413 0.0390 0.0368 0.0347 0.0328 0.0310 0.0292 0.0276 0.0261 0.0246 0.0232 0.0220 0.0207 0.0196 0.0185 0.0175

K1(x) 0.2244 0.2094 0.1955 0.1826 0.1707 0.1597 0.1494 0.1399 0.1310 0.1227 0.1151 0.1079 0.1012 0.0950 0.0892 0.0837 0.0786 0.0739 0.0694 0.0653 0.0614 0.0577 0.0543 0.0511 0.0481 0.0453 0.0426 0.0402 0.0378 0.0356 0.0336 0.0316 0.0298 0.0281 0.0265 0.0250 0.0236 0.0222 0.0210 0.0198

CHAPTER 5. GENERAL

X 3.65 3.70 3.75 3.80 3.85 3.90 3.95 4.00

J0(x) –0.3960 –0.3992 –0.4014 –0.4026 –0.4027 –0.4018 –0.4000 –0.3971

Table 5.3: Bessel Functions (cont'd) J1(x) Y0(x) Y1(x) I0(x) I1(x) 0.0746 0.1269 0.4165 8.375 7.107 0.0538 0.1061 0.4167 8.739 7.436 0.0332 0.0853 0.4159 9.119 7.780 0.0128 0.0645 0.4141 9.517 8.140 –0.0074 0.0439 0.4114 9.933 8.518 –0.0272 0.0234 0.4078 10.369 8.913 –0.0468 0.0031 0.4033 10.825 9.326 –0.0660 –0.0169 0.3979 11.302 9.759

370

K0(x) 0.0165 0.0156 0.0148 0.0140 0.0132 0.0125 0.0118 0.0112

K1(x) 0.0187 0.0176 0.0166 0.0157 0.0148 0.0140 0.0132 0.0125

Bessel Functions: 0 Order 1.2 1

J0 Y0

0.8

I0 K0

0.4 0.2 0 0.0

0.5

1.0

1.5

2.0

-0.2 -0.4 -0.6 -0.8

X Value ()

Figure 5.11: Bessel Functions of 0 Order

2.5

3.0

3.5

4.0

CHAPTER 5. GENERAL

371

Bessel Function at Value ()

0.6

CHAPTER 5. GENERAL

5.2 Probability and Statistics 5.2.1 Definitions [60], [56], [61], [5], [6] Core Damage Frequency (CDF): expected number of core damage events per unit of time. Level 1 Analysis: identification and quantification of the sequence of events leading to the onset of core damage; the typical metric is CDF. Level 2 Analysis: evaluation of reactor and containment response to severe accident challenges (e.g., accident sequences following physical damage to reactor fuel) and quantification of the mechanisms, amounts, and probabilities of subsequent radioactive material releases to the environment; the typical metric is release category frequency. Level 3 Analysis: estimation of the consequences of the release to the environment from radioactive materials, as identified in Level 1 analysis and Level 2 analysis. Importance Measure: a metric that evaluates a feature's importance in further reducing the risk and its importance in maintaining the present risk level; features include, but are limited to: safety functions, safety systems, components, surveillance tests, human activities, mitigation functions. Mean Time to Failure (MTTF): the mean time expected until the first failure of a system or component. Minimum cut sets: the smallest combination of component failures that, if they all occur, will cause the top event (of a fault tree) to occur. Path sets: the complement of a minimum cut set, and therefore defined as the “success modes” by which the top event (of a fault tree) will not occur; path sets are a common way of showing success states in a reliability block diagram. Reliability: the complement of unreliability. Risk Achievement Worth (RAW): an importance measure showing the increase in risk if a plant feature (e.g., system, component, or human action) is assumed to be failed all the time. Risk Reduction Worth (RRW): an importance measure showing the decrease in risk if the plant feature (e.g., system, component, or human action) is assumed to be perfectly reliable. SSC: a structure, system or component. Unavailability: the probability that a structure, system, or component is not capable of supporting its function including, but not limited to, the time it is disabled for test or maintenance. Unreliability: the probability that a structure, system, or component will not perform its specified function. under given conditions upon demand or for a prescribed time.

5.2.2 Dispersion, Mean, Median, and Mode Values If X1, X2, ..., Xn represent the values of a random sample of n items or observations, the arithmetic mean of these items or observations, denoted X , is defined as n 1 1 X = an k_X1 + X2 + ... + Xni = an k / Xi i=1 X " µ for sufficiently large values of n

372

CHAPTER 5. GENERAL

The weighted arithmetic mean is /w X X w = /wi i i where Xi = the value of the ith observation wi = the weight applied to Xi The variance of the population is the arithmetic mean of the squared deviations from the population mean. If μ is the arithmetic mean of a discrete population of size N, the population variance is defined by 2 2 2 1 σ 2 = N :`X1 − µj + `X2 − µj + ... + `XN − µj D N

2 1 σ 2 = N / `Xi − µj i=1

Standard deviation formulas are c 1 m/ `Xi − µj N

2

σpopulation =

σsum = σ12 + σ 22 + ... + σ n2 σseries = σ n σ σmean = n σproduct =

A 2 σb2 + B 2 σ a2

The sample variance is n 2 1 2 s = c n − 1 m / _ Xi − X i = i 1

The sample standard deviation is s=

c 1− m n 1

n

/ _ Xi − X i2

i=1

The sample coefficient of variation = CV =

s X

The sample geometric mean = n X1 X 2 X3 ...X n 1 2 n / Xi When the discrete data are rearranged in increasing order and n is odd, the median is the value of th the b n + 1 l item. 2 th n th n When n is even, the median is the average of the a 2 k and a 2 + 1 k items. The mode of a set of data is the value that occurs with greatest frequency. The sample root–mean–square value =

The sample range R is the largest sample value minus the smallest sample value.

373

CHAPTER 5. GENERAL

5.2.3 Permutations and Combinations A permutation is a particular sequence of a given set of objects. A combination is the set itself without reference to order. 1. The number of different permutations of n distinct objects taken r at a time is P _ n, r i =

n! _n − r i !

nPr is an alternative notation for P(n, r) 2. The number of different combinations of n distinct objects taken r at a time is P _ n, r i n! C _n, r i = r! = r! _ n − r i ! nCr and ` nr j are alternative notations for C(n, r)

3. The number of different permutations of n objects taken n at a time, given that ni are of type i, where i = 1, 2, . . . , k and /ni = n, is n! P _n; n1, n 2, ..., n k i = n !n !...n ! 1 2 k

5.2.4 Sets A,B is defined as the union of A and B. A+B is defined as the intersection of A and B.

De Morgan's Law A,B = A+B A+B = A,B Associative Law A , _ B , C i = _ A , Bi , C A + _ B + C i = _ A + Bi + C Distributive Law A , _ B + C i = _ A , Bi + _ A , C i A + _ B , C i = _ A + Bi , _ A + C i 5.2.5

Laws of Probability

Property 1. General Character of Probability The probability P(E) of an event E is a real number in the range of 0 to 1. The probability of an impossible event is 0 and that of an event certain to occur is 1.

374

CHAPTER 5. GENERAL

Property 2. Law of Total Probability P(A + B) = P(A) + P(B) − P(A, B) where P(A + B) = the probability that either A or B occur alone or that both occur together P(A)

= the probability that A occurs

P(B)

= the probability that B occurs

P(A, B) = the probability that both A and B occur together

Property 3. Law of Compound or Joint Probability If neither P(A) nor P(B) is zero, P(A, B) = P(A)P(B|A) = P(B)P(A|B) where P(B|A) = the probability that B occurs given the fact that A has occurred P(A|B) = the probability that A occurs given the fact that B has occurred If either P(A) or P(B) is zero, then P(A, B) = 0.

Bayes' Theorem P _ B j A) =

P _ B j i P _ A B j)

n

/ P _ A B i) P _ B i i

i=1

where P(Aj) is the probability of event Aj within the population of A P(Bj) is the probability of Bj within the population of B

5.2.6 Probability Functions, Distributions, and Expected Values A random variable X has a probability associated with each of its possible values. The probability is termed a discrete probability if X can assume only discrete values, or X = x1 , x2 , x3 , ..., xn The discrete probability of any single event, X = xi, occurring is defined as P(xi) while the probability mass function of the random variable X is defined by f(xk) = P(X = xk), k = 1, 2, ..., n

Probability Density Function If X is continuous, the probability density function, f, is defined such that P _a # X # b i =

b

#

a

f _ x i dx

375

CHAPTER 5. GENERAL

Cumulative Distribution Functions The cumulative distribution function, F, of a discrete random variable X that has a probability distribution described by P(xi) is defined as = F_ xm i

m

= / P_ xk i

P _ X # x m i, m = 1, 2, ..., n

k=1

If X is continuous, the cumulative distribution function, F, is defined by x

F (x) =

#

f (t) dt

−3

which implies that F(a) is the probability that X ≤ a.

Expected Values Let X be a discrete random variable having a probability mass function f (xk), k = 1, 2, . . . , n The expected value of X is defined as n

= 6X @ n E=

/ xk f _ xk i

k=1

The variance of X is defined as σ 2 = V 6X@ =

/ (xk − µ) 2 f _xki n

k=1

Let X be a continuous random variable having a density function f(X) and let Y = g(X) be some general function. The expected value of Y is: E 6Y @ = E 7g (X)A =

3

#

−3

g ^ x h f ^ x h dx

The mean or expected value of the random variable X is now defined as µ = E 6X@ =

3

# xf ^xhdx

−3

while the variance is given by σ = V 6X@ = E :`X − µj D = 2

2

3

# `x − µj2 f ^xhdx

−3

The standard deviation is given by σ = V 6X@

σ The coefficient of variation is defined as µ .

376

CHAPTER 5. GENERAL

Sums of Random Variables Y = a1X1 + a2X2 + . . . + anXn The expected value of Y is: μy = E(Y ) = a1E(X1) + a2E(X2) + . . . + anE(Xn) If the random variables are statistically independent, then the variance of Y is: σ 2y = V ^Y h = a12 V _X1i + a 22 V _X2i + ... + a n2 V _Xni σ 2y = a12 σ12 + a 22 σ 22 + ... + a n2 σ 2n

Also, the standard deviation of Y is: σ y = σ 2y

Binomial Distribution P(x) is the probability that x successes will occur in n trials. If p = probability of success and q = probability of failure = 1 − p, then n! − − Pn _ x i = C _n, x i p x q n x = p xqn x x! _n − x i ! where x

= 0, 1, 2, . . . , n

C(n, x) = the number of combinations n, p

= parameters

The variance is given by the form: p `1 − qj & np `1 − pj = npq n The cumulative values of the binomial distribution are shown in Table 5.4. σ2 =

377

CHAPTER 5. GENERAL

n 1 2 2 3 3 3 4 4 4 4 5 5 5 5 5 6 6 6 6 6 6 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8

x 0 0 1 0 1 2 0 1 2 3 0 1 2 3 4 0 1 2 3 4 5 0 1 2 3 4 5 6 0 1 2 3 4 5 6 7

0.1 0.9000 0.8100 0.9900 0.7290 0.9720 0.9990 0.6561 0.9477 0.9963 0.9999 0.5905 0.9185 0.9914 0.9995 1.0000 0.5314 0.8857 0.9842 0.9987 0.9999 1.0000 0.4783 0.8503 0.9743 0.9973 0.9998 1.0000 1.0000 0.4305 0.8131 0.9619 0.9950 0.9996 1.0000 1.0000 1.0000

Table 5.4: Cumulative Binomial Distribution Probabilities, P(X ≤ x) 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.8000 0.7000 0.6000 0.5000 0.4000 0.3000 0.2000 0.1000 0.6400 0.4900 0.3600 0.2500 0.1600 0.0900 0.0400 0.0100 0.9600 0.9100 0.8400 0.7500 0.6400 0.5100 0.3600 0.1900 0.5120 0.3430 0.2160 0.1250 0.0640 0.0270 0.0080 0.0010 0.8960 0.7840 0.6480 0.5000 0.3520 0.2160 0.1040 0.0280 0.9920 0.9730 0.9360 0.8750 0.7840 0.6570 0.4880 0.2710 0.4096 0.2401 0.1296 0.0625 0.0256 0.0081 0.0016 0.0001 0.8192 0.6517 0.4752 0.3125 0.1792 0.0837 0.0272 0.0037 0.9728 0.9163 0.8208 0.6875 0.5248 0.3483 0.1808 0.0523 0.9984 0.9919 0.9744 0.9375 0.8704 0.7599 0.5904 0.3439 0.3277 0.1681 0.0778 0.0313 0.0102 0.0024 0.0003 0.0000 0.7373 0.5282 0.3370 0.1875 0.0870 0.0308 0.0067 0.0005 0.9421 0.8369 0.6826 0.5000 0.3174 0.1631 0.0579 0.0086 0.9933 0.9692 0.9130 0.8125 0.6630 0.4718 0.2627 0.0815 0.9997 0.9976 0.9898 0.9688 0.9222 0.8319 0.6723 0.4095 0.2621 0.1176 0.0467 0.0156 0.0041 0.0007 0.0001 0.0000 0.6554 0.4202 0.2333 0.1094 0.0410 0.0109 0.0016 0.0001 0.9011 0.7443 0.5443 0.3438 0.1792 0.0705 0.0170 0.0013 0.9830 0.9295 0.8208 0.6563 0.4557 0.2557 0.0989 0.0159 0.9984 0.9891 0.9590 0.8906 0.7667 0.5798 0.3446 0.1143 0.9999 0.9993 0.9959 0.9844 0.9533 0.8824 0.7379 0.4686 0.2097 0.0824 0.0280 0.0078 0.0016 0.0002 0.0000 0.0000 0.5767 0.3294 0.1586 0.0625 0.0188 0.0038 0.0004 0.0000 0.8520 0.6471 0.4199 0.2266 0.0963 0.0288 0.0047 0.0002 0.9667 0.8740 0.7102 0.5000 0.2898 0.1260 0.0333 0.0027 0.9953 0.9712 0.9037 0.7734 0.5801 0.3529 0.1480 0.0257 0.9996 0.9962 0.9812 0.9375 0.8414 0.6706 0.4233 0.1497 1.0000 0.9998 0.9984 0.9922 0.9720 0.9176 0.7903 0.5217 0.1678 0.0576 0.0168 0.0039 0.0007 0.0001 0.0000 0.0000 0.5033 0.2553 0.1064 0.0352 0.0085 0.0013 0.0001 0.0000 0.7969 0.5518 0.3154 0.1445 0.0498 0.0113 0.0012 0.0000 0.9437 0.8059 0.5941 0.3633 0.1737 0.0580 0.0104 0.0004 0.9896 0.9420 0.8263 0.6367 0.4059 0.1941 0.0563 0.0050 0.9988 0.9887 0.9502 0.8555 0.6846 0.4482 0.2031 0.0381 0.9999 0.9987 0.9915 0.9648 0.8936 0.7447 0.4967 0.1869 1.0000 0.9999 0.9993 0.9961 0.9832 0.9424 0.8322 0.5695

378

0.95 0.0500 0.0025 0.0975 0.0001 0.0073 0.1426 0.0000 0.0005 0.0140 0.1855 0.0000 0.0000 0.0012 0.0226 0.2262 0.0000 0.0000 0.0001 0.0022 0.0328 0.2649 0.0000 0.0000 0.0000 0.0002 0.0038 0.0444 0.3017 0.0000 0.0000 0.0000 0.0000 0.0004 0.0058 0.0572 0.3366

0.99 0.0100 0.0001 0.0199 0.0000 0.0003 0.0297 0.0000 0.0000 0.0006 0.0394 0.0000 0.0000 0.0000 0.0010 0.0490 0.0000 0.0000 0.0000 0.0000 0.0015 0.0585 0.0000 0.0000 0.0000 0.0000 0.0000 0.0020 0.0679 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0027 0.0773

CHAPTER 5. GENERAL

n 9 9 9 9 9 9 9 9 9 10 10 10 10 10 10 10 10 10 10 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15

x 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Table 5.4: Cumulative Binomial Distribution Probabilities, P(X ≤ x) (cont'd) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.95 0.3874 0.1342 0.0404 0.0101 0.0020 0.0003 0.0000 0.0000 0.0000 0.0000 0.7748 0.4362 0.1960 0.0705 0.0195 0.0038 0.0004 0.0000 0.0000 0.0000 0.9470 0.7382 0.4628 0.2318 0.0898 0.0250 0.0043 0.0003 0.0000 0.0000 0.9917 0.9144 0.7297 0.4826 0.2539 0.0994 0.0253 0.0031 0.0001 0.0000 0.9991 0.9804 0.9012 0.7334 0.5000 0.2666 0.0988 0.0196 0.0009 0.0000 0.9999 0.9969 0.9747 0.9006 0.7461 0.5174 0.2703 0.0856 0.0083 0.0006 1.0000 0.9997 0.9957 0.9750 0.9102 0.7682 0.5372 0.2618 0.0530 0.0084 1.0000 1.0000 0.9996 0.9962 0.9805 0.9295 0.8040 0.5638 0.2252 0.0712 1.0000 1.0000 1.0000 0.9997 0.9980 0.9899 0.9596 0.8658 0.6126 0.3698 0.3487 0.1074 0.0282 0.0060 0.0010 0.0001 0.0000 0.0000 0.0000 0.0000 0.7361 0.3758 0.1493 0.0464 0.0107 0.0017 0.0001 0.0000 0.0000 0.0000 0.9298 0.6778 0.3828 0.1673 0.0547 0.0123 0.0016 0.0001 0.0000 0.0000 0.9872 0.8791 0.6496 0.3823 0.1719 0.0548 0.0106 0.0009 0.0000 0.0000 0.9984 0.9672 0.8497 0.6331 0.3770 0.1662 0.0473 0.0064 0.0001 0.0000 0.9999 0.9936 0.9527 0.8338 0.6230 0.3669 0.1503 0.0328 0.0016 0.0001 1.0000 0.9991 0.9894 0.9452 0.8281 0.6177 0.3504 0.1209 0.0128 0.0010 1.0000 0.9999 0.9984 0.9877 0.9453 0.8327 0.6172 0.3222 0.0702 0.0115 1.0000 1.0000 0.9999 0.9983 0.9893 0.9536 0.8507 0.6242 0.2639 0.0861 1.0000 1.0000 1.0000 0.9999 0.9990 0.9940 0.9718 0.8926 0.6513 0.4013 0.2059 0.0352 0.0047 0.0005 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.5490 0.1671 0.0353 0.0052 0.0005 0.0000 0.0000 0.0000 0.0000 0.0000 0.8159 0.3980 0.1268 0.0271 0.0037 0.0003 0.0000 0.0000 0.0000 0.0000 0.9444 0.6482 0.2969 0.0905 0.0176 0.0019 0.0001 0.0000 0.0000 0.0000 0.9873 0.8358 0.5155 0.2173 0.0592 0.0093 0.0007 0.0000 0.0000 0.0000 0.9978 0.9389 0.7216 0.4032 0.1509 0.0338 0.0037 0.0001 0.0000 0.0000 0.9997 0.9819 0.8689 0.6098 0.3036 0.0950 0.0152 0.0008 0.0000 0.0000 1.0000 0.9958 0.9500 0.7869 0.5000 0.2131 0.0500 0.0042 0.0000 0.0000 1.0000 0.9992 0.9848 0.9050 0.6964 0.3902 0.1311 0.0181 0.0003 0.0000 1.0000 0.9999 0.9963 0.9662 0.8491 0.5968 0.2784 0.0611 0.0022 0.0001 1.0000 1.0000 0.9993 0.9907 0.9408 0.7827 0.4845 0.1642 0.0127 0.0006 1.0000 1.0000 0.9999 0.9981 0.9824 0.9095 0.7031 0.3518 0.0556 0.0055 1.0000 1.0000 1.0000 0.9997 0.9963 0.9729 0.8732 0.6020 0.1841 0.0362 1.0000 1.0000 1.0000 1.0000 0.9995 0.9948 0.9647 0.8329 0.4510 0.1710 1.0000 1.0000 1.0000 1.0000 1.0000 0.9995 0.9953 0.9648 0.7941 0.5367

379

0.99 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0034 0.0865 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0043 0.0956 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0004 0.0096 0.1399

CHAPTER 5. GENERAL

n 20 20 20 20 20 20 20 20 20 20 20

Table 5.4: Cumulative Binomial Distribution Probabilities, P(X ≤ x) (cont'd) x 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.95 0 0.1216 0.0115 0.0008 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1 0.3917 0.0692 0.0076 0.0005 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 2 0.6769 0.2061 0.0355 0.0036 0.0002 0.0000 0.0000 0.0000 0.0000 0.0000 3 0.8670 0.4114 0.1071 0.0160 0.0013 0.0000 0.0000 0.0000 0.0000 0.0000 4 0.9568 0.6296 0.2375 0.0510 0.0059 0.0003 0.0000 0.0000 0.0000 0.0000 5 0.9887 0.8042 0.4164 0.1256 0.0207 0.0016 0.0000 0.0000 0.0000 0.0000 6 0.9976 0.9133 0.6080 0.2500 0.0577 0.0065 0.0003 0.0000 0.0000 0.0000 7 0.9996 0.9679 0.7723 0.4159 0.1316 0.0210 0.0013 0.0000 0.0000 0.0000 8 0.9999 0.9900 0.8867 0.5956 0.2517 0.0565 0.0051 0.0001 0.0000 0.0000 9 1.0000 0.9974 0.9520 0.7553 0.4119 0.1275 0.0171 0.0006 0.0000 0.0000 10 1.0000 0.9994 0.9829 0.8725 0.5881 0.2447 0.0480 0.0026 0.0000 0.0000

0.99 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

20 20 20 20 20 20 20 20 20

11 12 13 14 15 16 17 18 19

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0010 0.0169 0.1821

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.9949 0.9987 0.9997 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.9435 0.9790 0.9935 0.9984 0.9997 1.0000 1.0000 1.0000 1.0000

0.7483 0.8684 0.9423 0.9793 0.9941 0.9987 0.9998 1.0000 1.0000

0.4044 0.5841 0.7500 0.8744 0.9490 0.9840 0.9964 0.9995 1.0000

380

0.1133 0.2277 0.3920 0.5836 0.7625 0.8929 0.9645 0.9924 0.9992

0.0100 0.0321 0.0867 0.1958 0.3704 0.5886 0.7939 0.9308 0.9885

0.0001 0.0004 0.0024 0.0113 0.0432 0.1330 0.3231 0.6083 0.8784

0.0000 0.0000 0.0000 0.0003 0.0026 0.0159 0.0755 0.2642 0.6415

CHAPTER 5. GENERAL

Normal Distribution (Gaussian Distribution) This is a unimodal distribution, the mode being x = μ, with two points of inflection (each located at a distance σ to either side of the mode). The averages of n observations tend to become normally distributed as n increases. The variate x is said to be normally distributed if its density function f(x) is given by an expression of the form f (x) =

− 2 1 exp >− 1 d x µ n H 2 σ σ 2π

where μ = the population mean σ = the standard deviation of the population −∞ ≤ x ≤ ∞ When μ = 0 and σ2 = σ = 1, the distribution is called a standardized or unit normal distribution. Then 2 1 f (x) = exp d − x n , where − 3 # x # 3. 2 2r x−µ It is noted that Z = σ follows a standardized normal distribution function. A unit normal distribution table is included as Table 5.5. In the table, the following notations are used: F(x) = the area under the curve from −∞ to x R(x) = the area under the curve from x to ∞ W(x) = the area under the curve between −x and x F(−x) = 1 − F(x)

381

CHAPTER 5. GENERAL

Table 5.5: Unit Normal Distribution

f x x 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0

f(x) 0.3989 0.3970 0.3910 0.3814 0.3683 0.3521 0.3332 0.3123 0.2897 0.2661 0.2420 0.2179 0.1942 0.1714 0.1497 0.1295 0.1109 0.0940 0.0790 0.0656 0.0540 0.0440 0.0355 0.0283 0.0224 0.0175 0.0136 0.0104 0.0079 0.0060 0.0044

x

x

F(x) 0.5000 0.5398 0.5793 0.6179 0.6554 0.6915 0.7257 0.7580 0.7881 0.8159 0.8413 0.8643 0.8849 0.9032 0.9192 0.9332 0.9452 0.9554 0.9641 0.9713 0.9772 0.9821 0.9861 0.9893 0.9918 0.9938 0.9953 0.9965 0.9974 0.9981 0.9987

R(x) 0.5000 0.4602 0.4207 0.3821 0.3446 0.3085 0.2743 0.2420 0.2119 0.1841 0.1587 0.1357 0.1151 0.0968 0.0808 0.0668 0.0548 0.0446 0.0359 0.0287 0.0228 0.0179 0.0139 0.0107 0.0082 0.0062 0.0047 0.0035 0.0026 0.0019 0.0013

382

-x

+x

2R(x) 1.0000 0.9203 0.8415 0.7642 0.6892 0.6171 0.5485 0.4839 0.4237 0.3681 0.3173 0.2713 0.2301 0.1936 0.1615 0.1336 0.1096 0.0891 0.0719 0.0574 0.0455 0.0357 0.0278 0.0214 0.0164 0.0124 0.0093 0.0069 0.0051 0.0037 0.0027

-x +x W(x) 0.0000 0.0797 0.1585 0.2358 0.3108 0.3829 0.4515 0.5161 0.5763 0.6319 0.6827 0.7287 0.7699 0.8064 0.8385 0.8664 0.8904 0.9109 0.9281 0.9426 0.9545 0.9643 0.9722 0.9786 0.9836 0.9876 0.9907 0.9931 0.9949 0.9963 0.9973

CHAPTER 5. GENERAL

Table 5.5: Unit Normal Distribution (cont'd)

f x x

f(x)

x

x

F(x)

-x

+x

-x +x

R(x)

2R(x)

W(x)

R(x) 0.1000 0.0500 0.0250 0.0200 0.0100 0.0050

2R(x) 0.2000 0.1000 0.0500 0.0400 0.0200 0.0100

W(x) 0.8000 0.9000 0.9500 0.9600 0.9800 0.9900

Fractiles x 1.2816 1.6449 1.9600 2.0537 2.3263 2.5758

f(x) 0.1755 0.1031 0.0584 0.0484 0.0267 0.0145

F(x) 0.9000 0.9500 0.9750 0.9800 0.9900 0.9950

The Central Limit Theorem Let X1, X2, ..., Xn be a sequence of independent and identically distributed random variables each having mean μ and variance σ2. Then for large n, the Central Limit Theorem asserts that the sum Y = X1 + X2 + . . . + Xn is approximately normal. ny = n and the standard deviation σ σy = n

Poisson Distribution Given a Poisson process, the probability of n success in N trials is given by [1]

where

^vt hx P _ Xt = x i = x! exp ^- vt h for x = 0, 1, 2... ν is the mean (average per unit time) occurrence rate Xt is the number of occurrences in time (or space) t is the time (or space) interval

if one sets λ = νt then the Poisson distribution becomes λx − P ^x events in th = x! e λ The mean and variance are μ = λ

σ2 = λ

383

CHAPTER 5. GENERAL

t-Distribution Student's t-distribution has the probability density function given by: + v+1 Cc v 2 1 m 2 − 2 t c + m f^ t h = v 1 v vr C c 2 m where ν = number of degrees of freedom n = sample size ν=n−1 Γ = gamma function t=

x−n s n

−3 # t # 3 Table 5.6 gives the values of tα,ν for values of α and ν. Note that, in view of the symmetry of the t-distribution, t1−α,ν = − tα,ν The function3for α follows: α=

# f ^t hdt

tα,v

384

CHAPTER 5. GENERAL

Table 5.6: Student's t-Distribution F(t)

0

α= v 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 ∞

33.33%

25%

20%

0.5774 0.5000 0.4759 0.4642 0.4573 0.4527 0.4495 0.4471 0.4452 0.4438 0.4426 0.4415 0.4407 0.4400 0.4394 0.4388 0.4383 0.4379 0.4375 0.4372 0.4369 0.4366 0.4363 0.4361 0.4359 0.4357 0.4355 0.4353 0.4352 0.4350 0.4307

1.000 0.8165 0.7649 0.7407 0.7267 0.7176 0.7111 0.7064 0.7027 0.6998 0.6974 0.6955 0.6938 0.6924 0.6912 0.6901 0.6892 0.6884 0.6876 0.6870 0.6864 0.6858 0.6853 0.6848 0.6844 0.6840 0.6837 0.6834 0.6830 0.6828 0.6745

1.376 1.061 0.9785 0.9410 0.9195 0.9057 0.8960 0.8889 0.8834 0.8791 0.8755 0.8726 0.8702 0.8681 0.8662 0.8647 0.8633 0.8620 0.8610 0.8600 0.8591 0.8583 0.8575 0.8569 0.8562 0.8557 0.8551 0.8546 0.8542 0.8538 0.8416

α

t(α,ν)

Values of a 15% 10% 5% Values of t(α, ν) 1.963 3.078 6.314 1.386 1.886 2.920 1.250 1.638 2.353 1.190 1.533 2.132 1.156 1.476 2.015 1.134 1.440 1.943 1.119 1.415 1.895 1.108 1.397 1.860 1.100 1.383 1.833 1.093 1.372 1.812 1.088 1.363 1.796 1.083 1.356 1.782 1.079 1.350 1.771 1.076 1.345 1.761 1.074 1.341 1.753 1.071 1.337 1.746 1.069 1.333 1.740 1.067 1.330 1.734 1.066 1.328 1.729 1.064 1.325 1.725 1.063 1.323 1.721 1.061 1.321 1.717 1.060 1.319 1.714 1.059 1.318 1.711 1.058 1.316 1.708 1.058 1.315 1.706 1.057 1.314 1.703 1.056 1.313 1.701 1.055 1.311 1.699 1.055 1.310 1.697 1.036 1.282 1.645 385

2.5%

1%

0.5%

12.71 4.303 3.182 2.776 2.571 2.447 2.365 2.306 2.262 2.228 2.201 2.179 2.160 2.145 2.131 2.120 2.110 2.101 2.093 2.086 2.080 2.074 2.069 2.064 2.060 2.056 2.052 2.048 2.045 2.042 1.960

31.82 6.965 4.541 3.747 3.365 3.143 2.998 2.896 2.821 2.764 2.718 2.681 2.650 2.624 2.602 2.583 2.567 2.552 2.539 2.528 2.518 2.508 2.500 2.492 2.485 2.479 2.473 2.467 2.462 2.457 2.326

63.66 9.925 5.841 4.604 4.032 3.707 3.499 3.355 3.250 3.169 3.106 3.055 3.012 2.977 2.947 2.921 2.898 2.878 2.861 2.845 2.831 2.819 2.807 2.797 2.787 2.779 2.771 2.763 2.756 2.750 2.576

CHAPTER 5. GENERAL

F-Distribution The F-distribution is a continuous probability distribution given by: _d1 αi 1 d 2d2 d

_d1 α + d2i 1

d + d2

f _α, d1, d2i =

d d αB d 1 , 2 n 2 2

where α = is the input variable which is zero or a positive, real number d1 = is distribution function 1 which is a positive, real number d2 = is distribution function 2 which is a positive, real number B = is the beta function Table 5.7 gives the values of f (a, d1, d2) where a = 0.05.

Beta Function The beta function is B _ x, y i =

1

# t x − 1 _1 − t iy − 1 dt

0

The beta function is related to the gamma function by B _ x, y i =

C^ xhC_ y i C` x + yj

where x and y are positive integers this simplifies to B _ x, y i =

_ x − 1i!` y − 1j!

` x + y − 1j!

Table 5.8 gives selected values of the Beta Function

Gamma Function C ^n h =

3

#

tn

− 1 −t

e dt, n 2 0

0

χ2 – Distribution If Z1, Z2, ..., Zn are independent unit normal random variables, then x 2 = Z12 + Z 22 + ... + Z n2 is said to have a chi–square distribution with n degrees of freedom. Table 5.9 gives values of x a2,n for selected values of a and n.

386

Table 5.7: F-Distribution F(f)

α=0.05 0

1

2

3

4

5

6

7

d2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

161.4 18.51 10.13 7.709 6.608 5.987 5.591 5.318 5.117 4.965 4.844 4.747 4.667 4.600 4.543 4.494 4.451

199.5 19.00 9.552 6.944 5.786 5.143 4.737 4.459 4.256 4.103 3.982 3.885 3.806 3.739 3.682 3.634 3.592

215.7 19.16 9.277 6.591 5.409 4.757 4.347 4.066 3.863 3.708 3.587 3.490 3.411 3.344 3.287 3.239 3.197

224.6 19.25 9.117 6.388 5.192 4.534 4.120 3.838 3.633 3.478 3.357 3.259 3.179 3.112 3.056 3.007 2.965

230.2 19.30 9.013 6.256 5.050 4.387 3.972 3.687 3.482 3.326 3.204 3.106 3.025 2.958 2.901 2.852 2.810

234.0 19.33 8.941 6.163 4.950 4.284 3.866 3.581 3.374 3.217 3.095 2.996 2.915 2.848 2.790 2.741 2.699

236.8 19.35 8.887 6.094 4.876 4.207 3.787 3.500 3.293 3.135 3.012 2.913 2.832 2.764 2.707 2.657 2.614

f

Numerator d1 8 9

10

15

20

30

60

120

∞

241.9 19.40 8.786 5.964 4.735 4.060 3.637 3.347 3.137 2.978 2.854 2.753 2.671 2.602 2.544 2.494 2.450

245.9 19.43 8.703 5.858 4.619 3.938 3.511 3.218 3.006 2.845 2.719 2.617 2.533 2.463 2.403 2.352 2.308

248.0 19.45 8.660 5.803 4.558 3.874 3.445 3.150 2.936 2.774 2.646 2.544 2.459 2.388 2.328 2.276 2.230

250.1 19.46 8.617 5.746 4.496 3.808 3.376 3.079 2.864 2.700 2.570 2.466 2.380 2.308 2.247 2.194 2.148

252.2 19.48 8.572 5.688 4.431 3.740 3.304 3.005 2.787 2.621 2.490 2.384 2.297 2.223 2.160 2.106 2.058

253.3 19.49 8.549 5.658 4.398 3.705 3.267 2.967 2.748 2.580 2.448 2.341 2.252 2.178 2.114 2.059 2.011

254.3 19.50 8.526 5.628 4.365 3.669 3.230 2.928 2.707 2.538 2.404 2.296 2.206 2.131 2.066 2.010 1.960

238.9 19.37 8.845 6.041 4.818 4.147 3.726 3.438 3.230 3.072 2.948 2.849 2.767 2.699 2.641 2.591 2.548

240.5 19.38 8.812 5.999 4.772 4.099 3.677 3.388 3.179 3.020 2.896 2.796 2.714 2.646 2.588 2.538 2.494

CHAPTER 5. GENERAL

387

Denom intor

f(α,d1,d2)

Table 5.7: F-Distribution (cont'd) F(f)

α=0.05 0

1

2

3

4

5

6

7

d2 18 19 20 21 22 23 24 25 26 27 28 29 30 40 60 120 ∞

4.414 4.381 4.351 4.325 4.301 4.279 4.260 4.242 4.225 4.210 4.196 4.183 4.171 4.085 4.001 3.920 3.841

3.555 3.522 3.493 3.467 3.443 3.422 3.403 3.385 3.369 3.354 3.340 3.328 3.316 3.232 3.150 3.072 2.996

3.160 3.127 3.098 3.072 3.049 3.028 3.009 2.991 2.975 2.960 2.947 2.934 2.922 2.839 2.758 2.680 2.605

2.928 2.895 2.866 2.840 2.817 2.796 2.776 2.759 2.743 2.728 2.714 2.701 2.690 2.606 2.525 2.447 2.372

2.773 2.740 2.711 2.685 2.661 2.640 2.621 2.603 2.587 2.572 2.558 2.545 2.534 2.449 2.368 2.290 2.214

2.661 2.628 2.599 2.573 2.549 2.528 2.508 2.490 2.474 2.459 2.445 2.432 2.421 2.336 2.254 2.175 2.099

2.577 2.544 2.514 2.488 2.464 2.442 2.423 2.405 2.388 2.373 2.359 2.346 2.334 2.249 2.167 2.087 2.010

f

Numerator d1 8 9

10

15

20

30

60

120

∞

2.412 2.378 2.348 2.321 2.297 2.275 2.255 2.236 2.220 2.204 2.190 2.177 2.165 2.077 1.993 1.910 1.831

2.269 2.234 2.203 2.176 2.151 2.128 2.108 2.089 2.072 2.056 2.041 2.027 2.015 1.924 1.836 1.750 1.666

2.191 2.155 2.124 2.096 2.071 2.048 2.027 2.007 1.990 1.974 1.959 1.945 1.932 1.839 1.748 1.659 1.571

2.107 2.071 2.039 2.010 1.984 1.961 1.939 1.919 1.901 1.884 1.869 1.854 1.841 1.744 1.649 1.554 1.459

2.017 1.980 1.946 1.916 1.889 1.865 1.842 1.822 1.803 1.785 1.769 1.754 1.740 1.637 1.534 1.429 1.318

1.968 1.930 1.896 1.866 1.838 1.813 1.790 1.768 1.749 1.731 1.714 1.698 1.683 1.577 1.467 1.352 1.221

1.917 1.878 1.843 1.812 1.783 1.757 1.733 1.711 1.691 1.672 1.654 1.638 1.622 1.509 1.389 1.254 1.000

2.510 2.477 2.447 2.420 2.397 2.375 2.355 2.337 2.321 2.305 2.291 2.278 2.266 2.180 2.097 2.016 1.938

2.456 2.423 2.393 2.366 2.342 2.320 2.300 2.282 2.265 2.250 2.236 2.223 2.211 2.124 2.040 1.959 1.880

CHAPTER 5. GENERAL

388

Denom intor

f(α,d1,d2)

x

2 5.00E−01 1.67E−01 8.33E−02 5.00E−02 3.33E−02 2.38E−02 1.79E−02 1.39E−02 1.11E−02 9.09E−03

3 3.33E−01 8.33E−02 3.33E−02 1.67E−02 9.52E−03 5.95E−03 3.97E−03 2.78E−03 2.02E−03 1.52E−03

7 1.43E−01 1.79E−02 3.97E−03 1.19E−03 4.33E−04 1.80E−04 8.33E−05 4.16E−05 2.22E−05 1.25E−05

8 1.25E−01 1.39E−02 2.78E−03 7.58E−04 2.53E−04 9.71E−05 4.16E−05 1.94E−05 9.71E−06 5.14E−06

9 1.11E−01 1.11E−02 2.02E−03 5.05E−04 1.55E−04 5.55E−05 2.22E−05 9.71E−06 4.57E−06 2.29E−06

389

Table 5.9: χ2 – Distribution f(χ2)

0

α= n

0.995

1 2 3 4 5 6

0.00003927 0.01003 0.07172 0.2070 0.4117 0.6757

0.990

0.975

0.950

α

χ2(α,n)

χ2

0.900

0.100

0.050

0.025

0.010

0.005

2.706 4.605 6.251 7.779 9.236 10.64

3.841 5.991 7.815 9.488 11.07 12.59

5.024 7.378 9.348 11.14 12.83 14.45

6.635 9.210 11.34 13.28 15.09 16.81

7.879 10.60 12.84 14.86 16.75 18.55

Values of x (a, n) 0.003932 0.01579 0.1026 0.2107 0.3518 0.5844 0.7107 1.064 1.145 1.610 1.635 2.204 2

0.0001571 0.02010 0.1148 0.2971 0.5543 0.8721

0.0009821 0.05064 0.2158 0.4844 0.8312 1.237

10 1.00E−01 9.09E−03 1.52E−03 3.50E−04 9.99E−05 3.33E−05 1.25E−05 5.14E−06 2.29E−06 1.08E−06

CHAPTER 5. GENERAL

1 2 3 4 5 6 7 8 9 10

1 1.00E+00 5.00E−01 3.33E−01 2.50E−01 2.00E−01 1.67E−01 1.43E−01 1.25E−01 1.11E−01 1.00E−01

Table 5.8: Beta Function y 4 5 6 2.50E−01 2.00E−01 1.67E−01 5.00E−02 3.33E−02 2.38E−02 1.67E−02 9.52E−03 5.95E−03 7.14E−03 3.57E−03 1.98E−03 3.57E−03 1.59E−03 7.94E−04 1.98E−03 7.94E−04 3.61E−04 1.19E−03 4.33E−04 1.80E−04 7.58E−04 2.53E−04 9.71E−05 5.05E−04 1.55E−04 5.55E−05 3.50E−04 9.99E−05 3.33E−05

Table 5.9: χ2 – Distribution (cont'd) f(χ2) Cindy,

0

0.995

0.990

0.975

0.9893 1.344 1.735 2.156 2.603 3.074 3.565 4.075 4.601 5.142 5.697 6.265 6.844 7.434 8.034 8.643 9.260 9.886 10.52 11.16

1.239 1.646 2.088 2.558 3.053 3.571 4.107 4.660 5.229 5.812 6.408 7.015 7.633 8.260 8.897 9.542 10.20 10.86 11.52 12.20

1.690 2.180 2.700 3.247 3.816 4.404 5.009 5.629 6.262 6.908 7.564 8.231 8.907 9.591 10.28 10.98 11.69 12.40 13.12 13.84

0.950 0.900 Values of x2(a, n) 2.167 2.833 2.733 3.490 3.325 4.168 3.940 4.865 4.575 5.578 5.226 6.304 5.892 7.042 6.571 7.790 7.261 8.547 7.962 9.312 8.672 10.09 9.390 10.86 10.12 11.65 10.85 12.44 11.59 13.24 12.34 14.04 13.09 14.85 13.85 15.66 14.61 16.47 15.38 17.29

χ2

0.100

0.050

0.025

0.010

0.005

12.02 13.36 14.68 15.99 17.28 18.55 19.81 21.06 22.31 23.54 24.77 25.99 27.20 28.41 29.62 30.81 32.01 33.20 34.38 35.56

14.07 15.51 16.92 18.31 19.68 21.03 22.36 23.68 25.00 26.30 27.59 28.87 30.14 31.41 32.67 33.92 35.17 36.42 37.65 38.89

16.01 17.53 19.02 20.48 21.92 23.34 24.74 26.12 27.49 28.85 30.19 31.53 32.85 34.17 35.48 36.78 38.08 39.36 40.65 41.92

18.48 20.09 21.67 23.21 24.72 26.22 27.69 29.14 30.58 32.00 33.41 34.81 36.19 37.57 38.93 40.29 41.64 42.98 44.31 45.64

20.28 21.95 23.59 25.19 26.76 28.30 29.82 31.32 32.80 34.27 35.72 37.16 38.58 40.00 41.40 42.80 44.18 45.56 46.93 48.29

CHAPTER 5. GENERAL

390

α= n 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

α

χ2(α,n)

Table 5.9: χ2 – Distribution (cont'd) f(χ2) Cindy,

0

0.995

0.990

0.975

11.81 12.46 13.12 13.79 20.71 27.99 35.53 43.28 51.17 59.20 67.33

12.88 13.56 14.26 14.95 22.16 29.71 37.48 45.44 53.54 61.75 70.06

14.57 15.31 16.05 16.79 24.43 32.36 40.48 48.76 57.15 65.65 74.22

0.950 0.900 Values of x2(a, n) 16.15 18.11 16.93 18.94 17.71 19.77 18.49 20.60 26.51 29.05 34.76 37.69 43.19 46.46 51.74 55.33 60.39 64.28 69.13 73.29 77.93 82.36

χ2

0.100

0.050

0.025

0.010

0.005

36.74 37.92 39.09 40.26 51.81 63.17 74.40 85.53 96.58 107.6 118.5

40.11 41.34 42.56 43.77 55.76 67.50 79.08 90.53 101.9 113.1 124.3

43.19 44.46 45.72 46.98 59.34 71.42 83.30 95.02 106.6 118.1 129.6

46.96 48.28 49.59 50.89 63.69 76.15 88.38 100.4 112.3 124.1 135.8

49.64 50.99 52.34 53.67 66.77 79.49 91.95 104.2 116.3 128.3 140.2

CHAPTER 5. GENERAL

391

α= n 27 28 29 30 40 50 60 70 80 90 100

α

χ2(α,n)

CHAPTER 5. GENERAL

5.2.7

Linear Regression and Goodness of Fit

Least Squares t y = at + bx where tr y-intercept = at = yr − bx S slope = bt = xy S xx S xy = S xx =

n

n

n

i=1

i=1

i=1

/ xi yi − b 1n lf / xi pf / yi p n

/

i=1

n

2

f / xi p

x i2 − b 1n l

i=1

n

yr = b 1n lf / yi p = i 1 n

xr = b 1n lf / xi p i=1

where n = sample size Sxx = sum of squares of x Sxy = sum of x-y products

Standard Error of Estimate S e2

=

S yy =

2 S xx S yy − S xy = MSE S xx _n − 2 i n

/

i=1

n

2

f / yi p

y i2 − b 1n l

i=1

where Syy = sum of squares of y The mean square root of error is SS MSE = N error −k where N = / ni i

k = number of populations

392

CHAPTER 5. GENERAL

Confidence Interval for Intercept at 2 d1n + Sxr n MSE

at ! tα , n − 2

xx

2

Confidence Interval for Slope bt MSE bt ! t a , n − 2 S xx 2 Sample Correlation Coefficient (R) and Coefficient of Determination (R2) S xy R= S xx S yy S xy2 R2 = S S xx yy

5.2.8

Hypothesis Testing

One-Way Analysis of Variance (ANOVA) Given independent random samples of size ni from k populations, then: k

/

nj

k

2

/ ` xij − xr j =

i=1 j=1

/

nj

2

k

/ ` xij − xi j + / ni _ xi − xr i

i=1 j=1

2

or

i=1

SS total = SSerrqr + SS treatments Let T be the grand total of all N = Σini observations and Ti be the total of the ni observations of the ith sample. T2 C= N SS total =

k

nj

/ / x ij2 − C

i=1 j=1

SS treatments =

k

/ Tnii

i=1

2

−C

SSerror = SS total − SS treatments Consider an unknown parameter θ of a statistical distribution. Let the null hypothesis be H 0: n = n 0 and let the alternative hypothesis be H1: n = n1 Rejecting H0 when it is true is known as a type I error, while accepting H0 when it is wrong is known as a type II error. Furthermore, the probabilities of type I and type II errors are usually represented by the symbols α and β, respectively: α = probability (type I error) β = probability (type II error) The probability of a type I error is known as the level of significance of the test. 393

CHAPTER 5. GENERAL

Assume that the values of α and β are given. The sample size can be obtained from the following relationships. In (A) and (B), μ1 is the value assumed to be the true mean. (A) H0 : μ = μ0; H1 : μ ≠ μ0 JK n − n NO JK n − n NO 0 KK 0 O K a a + − Z OO − U KK v Z OOO b = U KK v 2 2 OO O K K n O K n L P L P An approximate result is a Z a2 + Zb k v 2 2

n-

_n1 − n 0 i

2

(B) H0: n = n 0; H1: n 2 n 0 JKµ − µ NO KK 0 + ZαOOO β = Φ KK σ OO K n L P

`Zα + Zβj σ 2 2

n=

5.2.9

`µ1 − µ2j

2

Confidence Intervals, Sample Distributions, and Sample Size

Confidence Interval for the Mean of a Normal Distribution (A) Standard deviation s is known σ σ X − Zα2 # µ # X + Zα2 n n (B) Standard deviation s is not known s s X − ta # n # X + ta 2 n 2 n where t a corresponds to n − 1 degrees of freedom. 2

Confidence Interval for the Difference Between Two Means (A) Standard deviations σ1 and σ2 are known σ2 σ2 σ2 σ2 X 1 − X 2 − Zα2 n1 + n 2 # µ1 − µ2 # X 1 − X 2 + Zσ2 n1 + n 2 1 2 1 2 (B) Standard deviations σ1 and σ2 are not known X1 − X 2 − t a 2

c n1 + n1 m9_n1 − 1 i S12 + _n2 − 1 i S 22C 1 2 # n1 − n 2 n1 + n2 − 2

n1 − n 2 # X 1 − X 2 + t a 2

c n1 + n1 m9_n1 − 1 i S12 + _n2 − 1 i S 22C 1 2 n1 + n 2 − 2

where t a corresponds to n1 + n2 − 2 degrees of freedom. 2

394

CHAPTER 5. GENERAL

Confidence Intervals for the Variance of a Normal Distribution _n − 1i s 2

x 2α2 , n − 1

#σ # 2

_n − 1i s 2

x12− α2 , n − 1

Sample Size z=

X−µ σ n

n =f

2

Zα2 σ p xr − µ

5.2.10 Test Statistics The following definitions apply. Zvar =

X − µ0 σ n

t var =

X − n0 s n

where Zvar = standard normal Z score tvar = sample distribution test statistic s = known standard deviation μo = population mean X = hypothesized mean or sample mean n = sample size s = computed sample standard deviation The Z score is applicable when the standard deviation (s) is known. The test statistic is applicable when the standard deviation (s) is computed at time of sampling. Zα corresponds to the appropriate probability under the normal probability curve for a given Zvar. tα,n−1 corresponds to the appropriate probability under the t–Distribution with n − 1 degrees of freedom for a given tvar.

395

CHAPTER 5. GENERAL

Table 5.10: Values of Z a 2

[47]

Confidence Interval

Za

80% 90% 95% 96% 98% 99%

1.2816 1.6449 1.9600 2.0537 2.3263 2.5758

2

5.2.11 Statistical Quality Control Average and Range Charts Table 5.11: Average and Range Charts [47] n A2 D3 D4 2 1.880 0 3.268 3 1.023 0 2.574 4 0.729 0 2.282 5 0.577 0 2.114 6 0.483 0 2.004 7 0.419 0.076 1.924 8 0.373 0.136 1.864 9 0.337 0.184 1.816 10 0.308 0.223 1.777 Xi = individual observation n = the sample size of a group k = the number of groups R = (range) the difference between the largest and smallest observations in a sample of size n. X=

X1 + X2 + ... + Xn n

X=

X1 + X 2 + ... + X k k

R1 + R 2 + ... + R k k The R Chart formulas are: R=

CL R = R UCL R = D 4 R LCL R = D3 R 396

CHAPTER 5. GENERAL

The X Chart formulas are: CL X = X UCL X = X + A2 R LCL X = X − A2 R

5.2.12 Standard Deviation Charts Table 5.12: Standard Deviation Charts [47] n A3 B3 B4 2 2.659 0 3.267 3 1.954 0 2.568 4 1.628 0 2.266 5 1.427 0 2.089 6 1.287 0.030 1.970 7 1.182 0.119 1.882 8 1.099 0.185 1.815 9 1.032 0.239 1.761 10 0.975 0.284 1.716 UCL X = X + A3 S CL X = X LCL X = X − A3 S UCLS = B 4 S CLS = S LCLS = B3 S

397

CHAPTER 5. GENERAL

5.2.13 Approximations The following table and equations may be used to generate initial approximations of the items indicated. Table 5.13: Approximations [47] n c4 d2 d3 2 0.7979 1.128 0.853 3 0.8862 1.693 0.888 4 0.9213 2.059 0.880 5 0.9400 2.326 0.864 6 0.9515 2.534 0.848 7 0.9594 2.704 0.833 8 0.9650 2.847 0.820 9 0.9693 2.970 0.808 10 0.9727 3.078 0.797 R σt = d 2 S σt = c 4 σR = d3 σt σS = σt 1 − c 42 where σt = estimate of s sR = estimate of the standard deviation of the ranges of the samples sS = estimate of the standard deviation of the standard deviations of the samples

Tests for Out of Control 1. A single point falls outside the (three sigma) control limits. 2. Two out of three successive points fall on the same side of and more than two sigma units from the center line. 3. Four out of five successive points fall on the same side of and more than one sigma unit from the center line. 4. Eight successive points fall on the same side of the center line.

398

CHAPTER 5. GENERAL

Source of Variation

Table 5.14: One–Way ANOVA Table [47] Degrees of Freedom Sum of Squares Mean Square

Between Treatments

k−1

SStreatments

Error

N−k

SSerror

Total

N−1

SStotal

Source of Variation

SS treatments k−1 SS MSE = N error −k MST =

Table 5.15: Two–Way ANOVA Table [47] Degrees of Freedom Sum of Squares Mean Square

Between Treatments

k−1

SStreatments

Between Blocks

n−1

SSblocks

Error

(k − 1)(n − 1)

SSerror

Total

N−1

SStotal

SS treatments k−1 SS MSB = n blocks −1 MST =

MSE =

SSerror _ k − 1 i_ n − 1 i

Table 5.16: Tests on Means of Normal Distribution – Variance Known [47] Hypothesis Test Statistic Criteria for Rejection H 0: n = n 0 Z0 2 Z a 2 H1: n ! n 0 H 0: n = n 0 H1: n 1 n 0

Z0 /

X − n0 v n

Z0 1 − Za

H 0: n = n 0 H1: n 2 n 0

Z0 2 Za

H 0: n 1 − n 2 = c H1: n1 − n 2 ! c

Z0 2 Z a

H 0: n 1 − n 2 = c H1: n1 − n 2 1 c

2

Z0 =

X1 − X 2 − c v12 v 22 n1 + n 2

H 0: n 1 − n 2 = c H1: n1 − n 2 2 c

Z0 1 − Za

Z0 2 Za

399

F

MST MSE

F

MST MSE MSB MSE

CHAPTER 5. GENERAL

Table 5.17: Tests on Means of Normal Distribution – Variance Unknown [47] Hypothesis Test Statistic Criteria for Rejection H 0: n = n 0 t0 2 t a ,n − 1 2 H1: n ! n 0 t0 /

H 0: n = n 0 H1: n 1 n 0 H 0: n = n 0 H1: n 2 n 0 H 0: n 1 − n 2 = c

H1: n1 − n 2 ! c __ __ __ __ __ __

H 0: n 1 − n 2 = c H1: n1 − n 2 1 c __ __ __ __ __ __

H 0: n 1 − n 2 = c H1: n1 − n 2 2 c

X − n0 S n

t0 1 − ta ,n − 1

t0 2 ta,n − 1 _b Z] ]] Variances equal: bb ]] bb − − c X X 2 ]] t = 1 bb ]] 0 bb 1 1 ]] bb Sp n + n 1 2 ]] bb ]] v = n + n − 2 bb 1 2 ]] b ]] −−−−−−−−−−−−−−− bbb ]] b ]] Variances unequal: bbb ]] bb ] b − − X X c `b [] t = 1 2 ]] 0 bb 2 2 S1 S 2 ]] bb + ]] bb n n 1 2 ]] bb ]] bb 2 2 2 S1 S 2 o ]] bb e ]] bb n1 + n 2 ]]v = bb ]] 2 2 2 2 b b S S ]] e 1 o e 2 o bb ]] bb n1 n2 ]] b + n1 − 1 n 2 − 1 bb ] a \ S p2

=

t0 2 t a ,v 2

__ __ __ __ __ __ __

t0 1 − ta,v __ __ __ __ __ __ __

_n1 − 1 iS12 + _n2 − 1 iS 22 v

400

t0 2 ta,v

CHAPTER 5. GENERAL

Table 5.18: Tests on Variances of Normal Distribution With Unknown Mean [47] Hypothesis Test Statistic Criteria for Rejection 2

H0: v 2 = v 02

| 02 2 | a , n − 1 or

H1: v 2 ! v 02

| 02

H0: v 2 = v 02 H1: v 2 1 v 02

| 02 =

_n − 1 iS 2

v 02

| 02 2 | a2 , n − 1

H1: v 2 2 v 02

H1: v12 ! v 22

2

| 02 1 |1 − a , n − 1 2

H0: v 2 = v 02

H0: v12 = v 22

1

2 2 |1 − a , n − 1 2

F0 =

S12 S 22

F0 =

S 22 S12

F0 > Fa, n 2 − 1, n1 − 1

F0 =

S12 S 22

F0 > Fa, n1 − 1, n 2 − 1

F0 2 Fa , n1 − 1, n 2 − 1 2

F0 1 F1 − a , n1 − 1, n 2 − 1 2

H0: v12 = v 22 H1: v12 1 v 22 H0: v12 = v 22 H1: v12 2 v 22

A list of common probability and density functions with formulas for their means and variances are shown in Table 5.19.

401

Table 5.19: Probability and Density Functions: Means and Variances [47] Variable

Binomial Coefficient Binomial Exponential Gamma

402

Geometric Lognormal Multinomial Negative Binomial Normal Poisson

` nx j =

Variance

np

np(1-p)

b

b2

ab

ab2

nr N

r_ N − r in_ N − n i N 2 _ N − 1i

1 p

`1 − p j

n!

x! _ n − x i !

b _ x; n, p i = ` nx j p x `1 − p j f^ xh =

n−x

1 − bx e b

−x

−

Mean

xa 1 e b f^ xh = a ; a 2 0, b 2 0 b C ^a h a nN−−xr k r `xj N `n j

h _ x; n, r, N i =

g _ x; p i = p `1 − p j

x−1

f _ x; n, v i =

1 xv 2r

e

−

8ln ^ x h − nB

2v

2

;x20

2

n! f _ x1, ..., x k i = x !, ..., x ! p1x1 ...p kx k 1 k f _ y; r, p i = b r − 1

y+r−1

f^ xh =

l p r `1 − p j

y

1 −1 d x−n n e 2 v v 2r

f _ x; m i =

2

−m

mxe x!

en

+ 1 v2 2

p2

`e v − 1 j e 2 n + v 2

npi

npi `1 − pi j

r p

r `1 − p j p2

n

v2

m

m

2

CHAPTER 5. GENERAL

Hyper Geometric

Equation

Table 5.19: Probability and Density Functions: Means and Variances (cont'd) [47] Variable

Equation

Mean

Variance

Triangular

RS 2_ x − a i VW SS if a # x # m W WW _ i i _ − − b a m a f ^ x h = SS 2_b − x i W SS if m 1 x # bW WW S_b − a i_b − m i T X 1 f^ xh = ; a#x#b _b − a i

a+b+m 3

a2 + b2 + m2 - ab - am - bm 18

_a + b i

_b − a i

1 _ + i b a C< a 1 F a

+ 1 m − C 2 c a + 1 mF b a