Dynamics and Control of Nuclear Reactors 0128152613, 9780128152614

Table of contents :
Cover
Dynamics and Control
of Nuclear Reactors
Copyright
Dedication
Preface
Communication with authors
Acknowledgments
1
ntroduction
Introduction
System dynamics and control design
References
2
Nuclear reactor designs
Introduction
Generation I reactors
Generation II reactors
Generation III reactors
Generation III+ reactors
Generation IV reactors
Advanced reactors
Early twenty-first century construction
References
Further reading
3
The point reactor kinetics equations
Neutronics
Delayed neutrons
Delayed neutrons from fission products
Photoneutrons from nuclei excited by gamma rays
Development of the point reactor kinetics equations
Alternate choices for the neutronic variable
Perturbation form of the point kinetics equations
Transfer functions
Frequency response function
Stability
Fluid-fuel reactors
References
Further reading
4
Solutions of the point reactor kinetics equations and interpretation
Evolution of simulation methods
Numerical analysis
Maneuvers in a zero power reactor
Analytical solutions
Solutions for small perturbations
Sinusoidal reactivity and frequency response
Fluid fuel reactor response
The inhour equation
References
Further reading
5
Subcritical operation
The neutron source
Relation between neutron flux and reactivity in a subcritical reactor
The inverse multiplication factor
Responses during startup
Power ascension
Further reading
6
Fission product poisoning
The problem
Dynamics of xenon-135
Xe-135 production
Xe-135 losses
Equations for Xe-135 behavior
Steady state Xe-135
Xe-135 poisoning
Behavior of Xe-135 after Startup
Xe-135 after Shutdown
Xe-135 poisoning after a power increase
Xe-135 poisoning after power maneuvers
Coupled neutronic-xenon transients
Xenon-induced spatial oscillations
Xenon in molten salt reactors
Samarium-149 poisoning
Summary
References
7
Reactivity feedbacks
Basics
Fuel temperature feedback in thermal reactors
Moderator temperature feedback in thermal reactors
Pressure and void coefficients in thermal reactors
Fission product feedback
Combined reactivity feedback
Power coefficient of reactivity and the power defect
Reactivity feedback effect on the frequency response
Destabilizing negative feedback: A physical explanation
Explanation of stability using state-space representation
References
8
Reactor control
Introduction
Open-loop and closed-loop control systems
Basic control theory
Manual control
On-off controller
Proportional controller
Integral controller
Differential controller
Combined controllers
An example of proportional and integral controller for a first order system
Proportional controller
Integral controller
Advanced controllers
Control of a zero-power reactor
Control options in power reactors
Effect of inherent feedbacks on control options
Load following operation
The role of stored energy
Steady-state power distribution control
Important reactivity feedbacks and control strategies for various reactor types
References
9
Space-time kinetics
Introduction
Diffusion theory
Multi-group diffusion theory
Calculation requirements
Computer software
Models and computational methods
Finite difference methods
Finite element method (FEM)
Modal methods
Quasi-static methods
Nodal methods
References
10
Reactor thermal-hydraulics
Introduction
Heat conduction in fuel elements
Heat transfer to liquid coolant
Boiling coolant
Plenum and piping models
Pressurizer
Heat exchanger model
Steam generator model
U-Tube steam generator (UTSG)
Once-through steam generator (OTSG)
Balance-of-Plant (BOP) system models
Reactor system models
References
Further reading
11
Nuclear reactor safety
Introduction
Reactor safety principles
Early accidents with fuel damage
Accidents
Assessment
Analysis of potential reactor accidents
Accidents in Generation-II power reactors
Three mile Island [14]
Chernobyl [15]
Fukushima Dai-ichi [16]
Consequences and lessons learned
References
12
Pressurized water reactors
Introduction
PWR characteristics [1-3]
The reactor core
The pressurizer
Steam generators
U-tube steam generator (UTSG)
Once-through steam generator (OTSG)
Horizontal steam generator
Reactivity feedbacks
Power maneuvering
Steady-state programs for PWRs
Heat transfer in a steam generator
Fuel-to-coolant heat transfer
Equivalence between reactor power and power delivered to the steam generator at steady state
Energy change in the coolant
Development of a steady-state program
Steady-state program for PWRs with once-through steam generators (OTSG)
Control rod operating band and control rod operation
Feedwater control for PWR with U-tube steam generators [2, 4, 5]
Control of a PWR with once-through steam generators [3]
Turbine control
Summary of main PWR controllers
PWR safety systems
Example of a PWR simulation
References
Further reading
13
Boiling water reactors
Introduction
History of BWR design evolution
BWR-1
BWR-2
BWR-3
BWR-4
BWR-5
BWR-6
ABWR
Characteristics of BWRs
General features of a BWR
Recirculation flow and jet pumps
Other features of BWRs
Reactivity feedbacks in BWRs
Reactivity and recirculation flow
Total reactivity balance
BWR dynamic models
BWR stability problem and impact on control
The power flow map and startup
On-line stability monitoring
Power maneuvering
BWR control strategy
BWR safety
Advantages and disadvantages
References
Further reading
14
Pressurized heavy water reactors
Introduction
PHWR characteristics
Neutronic features [3]
Temperature feedback in heavy water reactors
The void coefficient
Reactivity control mechanisms
Control systems
Unit power regulator
Reactor regulating system
Pressure and inventory control
Steam generator level control
Steam generator pressure control
Maneuvering
Reactor dynamics
Modeling strategy
Reactor power response to reactivity insertion
References
15
Nuclear plant simulators
Introduction
Types of simulators and their purpose
Simulator games
Desk-top simulators
Control room simulators
Desk-top simulators
Introduction
PC simulation
Using an IAEA simulator
Simulation of PWR and BWR plant transients
PWR simulation
BWR simulation
How to obtain an IAEA simulator?
Internet-based desk-top simulators
Control room simulators
References
16
Nuclear plant instrumentation
Introduction
Sensor characteristics
Neutron and gamma ray detectors
Ionization chambers
Fission detectors
Self-powered neutron detectors
Scintillation detectors
Gamma thermometers
Nitrogen-16 measurement
Temperature sensors
Resistance thermometers
Thermocouples
Thermowells and bypass installation
Advanced temperature sensors
Pressure sensors
Flow sensors
Flow vs. pressure drop
Advanced flowmeters
Level sensors
Differential pressure
Bubbler
Actuator status sensors
PWR instrumentation
BWR instrumentation
CANDU (PHWR) reactor instrumentation
High temperature reactor instrumentation
Liquid metal fast breeder reactor (LMFBR) instrumentation
High temperature gas-cooled reactor (HTGR) instrumentation
Molten salt reactor instrumentation
References
Further reading
APPENDIX A
Generation II reactor parameters
Pressurized water reactor (PWR)
Boiling water reactor (BWR)
Pressurized heavy water reactor (PHWR): CANDU reactor
References
APPENDIX B
Advanced reactors
Introduction
Design possibilities
A note about reactors that use thorium
Advanced reactor marketplace
Large evolutionary reactors
Pressurized water reactors
Boiling water reactors
Pressurized heavy water reactors
Large developmental reactors
Gas-cooled reactors
Liquid metal fast breeder reactors
Molten salt reactors
Heavy water reactors
Small reactors
Introduction
Incentives
Small reactor list
Dynamics of advanced reactors
References
APPENDIX C
Basic reactor physics
Introduction
Neutron interactions
Reaction rates and nuclear power generation
Nuclear fission
Fast and thermal neutrons
Relation between specific power and neutron flux
Neutron lifetime and generation time
Multiplication factor and reactivity
Computing effective multiplication factor
Neutron transport and diffusion
References
APPENDIX D
Laplace transforms and transfer functions
Introduction
Defining the Laplace transform
Calculating Laplace transforms
The inverse Laplace transform
Method of residues
Inverse transform using partial fractions
Transfer functions
Feedback transfer functions
The convolution integral
Laplace transforms and partial differential equations
References
APPENDIX E
Frequency response analysis of linear systems
Frequency response theory
Computing frequency response function
Systems with oscillatory behavior
Systems with time delay dynamics
Frequency response of distributed systems
Frequency response measurements
References
APPENDIX F
State variable models and transient analysis
Introduction
State variable models
General solution of the multiple-input multiple-output (MISO) linear
Definition of multiple-input multiple output (MIMO) systems
Transfer function representation of MIMO systems
Transient response of MIMO systems
The state transition matrix
The matrix exponential solution
Sensitivity analysis
Numerical solutions of ordinary differential equations
Euler's method
Runge-Kutta order-two method
Solutions for partial differential equations
Examples of partial differential equations
Solution of partial difference equations using the finite difference method
Introduction
Formulation of grids and nodes [8]
FDM solution of the two-dimensional heat conduction problem [8]
Solution of partial difference equations using the finite element method
References
APPENDIX G
Matlab and Simulink: A brief tutorial
Introduction
Getting started with simulink
Simulation of a single-input single-output (SISO) system
Simulation of a closed-loop system with P-I controller
Solving linear differential equations using state-space models
Computing step response using a transfer function
Computing eigenvalues and eigenvectors
References
APPENDIX H
Analytical solution of the point reactor kinetics equations and the prompt jump approximation
Introduction
Analytical solution of the point kinetics equations
The prompt jump
An example
APPENDIX I
A moving boundary model
Introduction
Development of a moving boundary model
APPENDIX J
Modeling and simulation of a pressurized water reactor
Introduction
Linearized isolated core neutronic model
Numerical values of coefficients in the isolated core neutronic model
Fuel-to-coolant heat transfer
Numerical values of coefficients in the isolated core thermal-hydraulic model
State space representation of dynamic equations
Simulation of PWR isolated core dynamics response
Frequency response characteristics of reactor core dynamics
PWR NSSS dynamics
Neutronics
Core thermal-hydraulics
T-average controller
Piping and plenums
Pressurizer and its controller
U-tube steam generator modeling and control
NSSS model
Plant system parameters used in the models
NSSS simulated response to a steam valve perturbation
References
Further reading
APPENDIX K
Modeling and simulation of a molten salt reactor
Introduction
Molten salt reactor experiment (MSRE)
Lumped parameter model of the MSRE
Sub-system models and characteristics
Nodal model of the MSRE system
Equations describing neutronics and reactor heat transfer
Parameters used in simulation models
Results of simulation of MSR dynamics
References
Further reading
Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Z
Back Cover

Citation preview

Dynamics and Control of Nuclear Reactors

Dynamics and Control of Nuclear Reactors

Thomas W. Kerlin Belle R. Upadhyaya

Academic Press is an imprint of Elsevier 125 London Wall, London EC2Y 5AS, United Kingdom 525 B Street, Suite 1650, San Diego, CA 92101, United States 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom © 2019 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Designations used by companies to distinguish their products are often claimed as trademarks or registered trademarks. In all instances in which Academic Press Publishing is aware of a claim, the product names appear in initial capital or all capital letters. Readers, however, should contact the appropriate companies for more complete information regarding trademarks and registration. Matlab® and Simulink® are registered trademarks of Mathworks, Inc. in the United States and/or other countries. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN 978-0-12-815261-4 For information on all Academic Press publications visit our website at https://www.elsevier.com/books-and-journals

Cover image courtesy of the Tennessee Valley Authority (TVA), Knoxville, TN, USA., all rights reserved Publisher: Katie Hammon Acquisition Editor: Maria Convey Editorial Project Manager: Lindsay Lawrence Production Project Manager: Surya Narayanan Jayachandran Cover Designer: Matthew Limbert Typeset by SPi Global, India

Dedication A good teacher is like a candle – it consumes itself to light the way for others. Dedicated to Nancy Kerlin and Nimmi Upadhyaya

Preface The transient behavior of nuclear power reactors depends on the inherent characteristics of the system and the performance of control systems. These issues must be analyzed to predict and understand the performance of a power reactor. This book deals with the methods needed for this analysis. The book is designed to be useful as an introductory text and as a text for advanced study. Customizing the presentation by an instructor to match the level of the student is accomplished by selection of appropriate chapters and appendices. Analysis of reactor dynamics is presented both in the time domain and in the frequency domain. The basics of frequency response analysis of linear systems are described in the appendix. It is assumed that the student has introductory knowledge of nuclear reactor physics. Appendices that deal with basic material may be used as needed for introduction or review. Advanced students with appropriate prior knowledge can omit these Appendices. The authors strongly recommend that the student review the Appendices; these are an important supplement to the book chapters. Appendices that deal with advanced topics are suitable for advanced readers. These appendices are probably too detailed for use by introductory readers. Placing advanced material in appendices avoids inclusion of topics in the body of the book that would be confusing to introductory readers. The book contains numerous problems and exercises. Some are designed to familiarize readers with application of techniques presented in the book. They are appropriate for readers of all levels. Some exercises and problems are more involved and are intended for advanced readers. These include assignments that require reporting on topics covered in the literature. It is hoped that the approach used in this book will provide instructors with the flexibility needed to adapt to the prior knowledge of the students in his/her class. One might ask why a new book on nuclear power is needed. Many people think that nuclear power is unsafe, too expensive or unnecessary because of other environment-friendly options for producing power, mainly solar or wind power. The reader deserves to know the opinion of the authors on the need for nuclear power. It is as follows: Ø Wind and solar power are intermittent due to dependence on nature. The authors support the use of wind and solar to its maximum capability. Ø Even though locally-produced power will increase, there will always be a need for large power plants. Ø Power generation from plants that burn fossil fuels will decrease because of pollution and climate change concerns. Ø Nuclear power plants are safe because of lessons learned, emphasis on safety in reactor design and operation, and a strong safety culture. But, operators must fully understand the reactor and its operational features in order to avoid mistakes.

xv

xvi

Preface

Communication with authors The authors wish to encourage the users of Dynamics and Control of Nuclear Reactors, to provide feedback on the contents of the book and suggestions for improvement in the on-line editions. This would assist us in updating the technical content as the nuclear reactor designs evolve, especially with the development of advanced reactors. Thomas W. Kerlin [email protected] Belle R. Upadhyaya [email protected]

Acknowledgments We wish to acknowledge the role of scores of students who helped refine the development of an understandable and comprehensive treatment of course material on reactor dynamics and control. Graduate students provided innovative research results that contributed to the book’s content. The University of Tennessee Nuclear Engineering Department provided a supportive environment that facilitated work on the book. We are thankful to Professor Wes Hines, Head, Department of Nuclear Engineering, University of Tennessee, for his assistance and thoughtful suggestions. Several colleagues helped through discussing various technical issues. Most noteworthy were the following individuals: Syd Ball, retired scientist, Oak Ridge National Laboratory; Larry Miller, Professor Emeritus, University of Tennessee; Jamie Coble, Associate Professor, University of Tennessee; Ondrej Chvala, Research Professor, University of Tennessee; Lubomir Sklenka, Professor, Department of Nuclear Reactors, Czech Technical University, Prague. A special thanks to Vik Singh, MS graduate in Nuclear Engineering, University of Tennessee, for his diligence in checking the technical contents of the book and in assisting in the development of graphics and simulation plots used in the book. Finally, the preparation of this book would not have been possible without the patience and support of our wives, Nancy Kerlin and Nimmi Upadhyaya. We deeply appreciate their encouragement and sacrifice while we poured over a large amount of technical material. We are grateful to numerous organizations, including reactor suppliers, scientific societies, government agencies, and technical publishers for providing permission to use copyrighted material in the book. The following is a list of organizations to whom we are deeply indebted: •

• • • • • • • •

American Association of Arts and Sciences, Boston, MA: Nuclear Reactors: Generation to Generation, by Stephen M. Goldberg and Robert Rosner, published by the American Association of Arts and Sciences, 2011. Analysis and Measurement Services Corporation, Knoxville, TN. AREVA NP, France. The Babcock and Wilcox Company, Barberton, OH: Steam, its generation and use, Edition 42, 2015. Elsevier Woodhead Publishing, Cambridge, U.K.: Handbook of Small Modular Reactors, edited by M.D. Carelli and D.T. Ingersoll, 2015. GE Hitachi Nuclear Energy Americas LLC, Wilmington, NC. Micro-Simulation Technology, Montville, NJ: Dr. Li-Chi Cliff Po, President. Rosatom Overseas, The VVER today: Evolution, Design, Safety, www.rosatom.ru. Vik Singh: Study of the Dynamic Behavior of Molten Salt Reactors, MS Thesis, University of Tennessee, Knoxville, May 2019.

xvii

xviii

Acknowledgments

• • •

• • • •

Taylor and Francis, Boca Raton, FL: Nuclear Technology, Vol. 165, No. 2, February 2009. Tennessee Valley Authority (TVA). University Network of Excellence in Nuclear Engineering (UNENE), McMaster University, Hamilton, Ontario, Canada: The Essential CANDU, A Textbook on the CANDU Nuclear Power Plant Technology, Editor-in-Chief Wm. J. Garland. U.S. Department of Energy. U.S. Nuclear Regulatory Commission. Western Simulation Corporation, Frederick, MD. Westinghouse Electric Company LLC, 1000 Westinghouse Drive, Cranberry Township, PA: Copyright 1984 Westinghouse Electric Corporation; reprinted with permission of Westinghouse Electric Company LLC.

We are grateful to Ms. Lindsay Lawrence, Elsevier Editorial Project Manager, for her continued and valuable assistance during the preparation of the manuscript. We appreciate the supervision provided by the Elsevier Publishing personnel during the production of this book.

CHAPTER

Introduction

1

1.1 Introduction The power level of a nuclear reactor at any time is expected to be as desired by the plant operator. These plants undergo transients that are induced by operator actions, by actions initiated by an automatic control system, or by a component failure. Designers and operators must understand the transient behavior in order to achieve desired operation and safety. The first step is to understand the transient operation of a reactor that operates at a low power level, so low that there are negligible increases in temperature because of fission heating. Such a reactor is usually called a zero-power reactor. The power is not actually zero, but is so low that significant heating does not occur, and temperature related feedback effects are negligible. Many research reactors are zero-power reactors. A power reactor, on the other hand, operates at power levels high enough to cause major temperature increases. The temperatures of reactor components change along with reactor power during transients and these temperature changes, in turn, affect reactor power (a feedback loop). Also, power reactors that contain a compressible fluid undergo pressure changes during a transient. These pressure changes also affect reactor power (another feedback loop). Transients are usually accompanied by control actions. Control systems monitor selected plant parameters (such as power, temperature, pressure, flow rate) and change appropriate controllable actions (such as control rods and valve positions). Creating a set of mathematical equations and parameters (coefficients) in those equations to be used to analyze reactor transients is called modeling. Creating a solution to those equations is called system simulation. Nuclear reactor simulations generally have one of three purposes: obtaining a basic understanding of reactor behavior during transients, analysis of transients during normal maneuvering and response during accident conditions, and operator training. Each of these functions has different requirements for the level of detail in the model. Reactor simulation efforts started in the early days of reactor operation. Early simulation involved hand calculations and rudimentary calculators. Shortly thereafter, reactor simulation turned to computers for implementation, and simulation technology matured as the capabilities of computer technology evolved. Analog computers were used extensively in early simulations. These computers used Dynamics and Control of Nuclear Reactors. https://doi.org/10.1016/B978-0-12-815261-4.00001-9 # 2019 Elsevier Inc. All rights reserved.

1

2

CHAPTER 1 Introduction

electrical circuits to mimic reactor operation. Next came hybrid computers. They used digital computations along with analog components. The digital components handled computations that were not possible or practical with analog components. As digital computers became more powerful and faster, they came to dominate reactor simulation activities. Computer simulations may be performed on personal computers for some applications. Some solve model equations and provide numerical and/or graphical results. Other, more sophisticated personal computer simulations provide screen displays that mimic actual reactor control room displays. Simulators for operator training include full-scope simulators that duplicate the control room for the reactor being simulated. The displays provide computed results for all of the variables monitored in the actual plant and include capability for simulating all operator actions. Reactor accident analyses involve very detailed models that are implemented on large, high-performance computers. Simulations deal with large disturbances with potentially large consequences. Analyses include major accident scenarios such as loss of coolant and control rod ejection. This book addresses modeling and simulation of nuclear reactors, both zeropower reactors and power reactors. Modeling options include a wide range of possibilities, each with very different levels of complexity. Modeling and simulation is not a “do it and be finished” activity. Reactor constituents change continuously during operation and immediately at restart after refueling. These changes cause changes in the quantities that determine the reactor’s dynamic behavior. So, there is no such thing as model or simulation that defines the reactor at all times. Furthermore, even trying to evaluate the parameters needed in a model is complicated by the need to know neutronic and heat transfer properties that depend on position in the core and the burnup history of the fuel and are difficult to evaluate. The importance of simulation is to provide a way to understand what goes on in a reactor and why it happens rather than a precise determination of reactor dynamic response behavior for a specific disturbance on a specific day. There is very little in the book that requires detailed knowledge of reactor physics, but familiarity with reactor physics at least at the introductory level is helpful.

1.2 System dynamics and control design Power generating units (such as a nuclear power plants, fossil-fueled power plants, etc.) and large industrial facilities are complex systems. The design of these systems requires extensive analysis that uses dynamic models and simulation of their operation under various conditions. Because of mathematical methods developed over the past two centuries and computer capability developed since the 1950s, powerful techniques exist for analysis of dynamic systems and for design of control systems. It is now possible to predict the way a system will respond to external disturbances and to develop a control strategy that will cause the system to perform as desired.

1.2 System dynamics and control design

The ability to describe the system dynamics using a variety of models is crucial to achieve a good engineering design. The control or regulation of a power plant requires critical measurements of process parameters (and neutron power measurements in a nuclear power plant). As a result, a typical large nuclear plant employs a few thousand measurements. These are used by control systems, plant safety (protection) systems, and by monitoring systems. Thus, instrumentation and control play a critical role in safe and reliable operation of commercial nuclear power plants. Dynamic performance is an important issue in many industrial systems. The key issues in dynamic system performance are the following: •

•

Can the system be moved from one desired value (set point) to another in an acceptable manner? That is, without deviating from limits of variation and within an acceptable time interval. Can the system respond in a stable manner without exceeding safety limits when subjected to unplanned disturbances (possibly due to an accident, an external disturbance, failure of a component, or human error)?

The latest development to enhance the power and usefulness of digital simulation is modular modeling software. Modular modeling software provides a menu of models of commonly encountered systems (reactor kinetics, fuel-to-coolant heat transfer, hot and cold leg volumes, steam generators, feed water heaters, pressurizers, steam turbine, condenser, moisture separators, steam reheaters, pumps, valves, etc., including their control modules) and an automated means for linking them together and running simulations. Because the model for each component is used in many different analyses, great effort by highly qualified experts to develop and check the software is warranted. Several vendors market new simulation and control design software systems. The International Atomic Energy Agency (IAEA) also provides simulators for most types of power reactors to qualified organizations in member countries. These are used quite extensively for training in industry and universities. Ref. [1] provides the procedure for requesting IAEA simulation software. Recommendation by the IAEA representative for the country of the requester is required. In this book, we emphasize the use of the software system MATLAB and its Toolboxes. A companion system called Simulink is used for the simulation of large processes, such as a nuclear power plant. These software systems are designed for implementation in personal computers (PCs) and (larger) mainframe computers. The MATLAB, Simulink, and the toolboxes are comprehensive collections of functions (software modules) and are developed and marketed by The MathWorks, Inc. We recommend strongly that the students familiarize themselves with MATLAB, Simulink and the associated Toolboxes [2, 3]. ‘MATLAB is a high-performance language for technical computing. It integrates computation, visualization, and programming in an easy-to-use environment. The name, MATLAB, stands for matrix laboratory’ [4]. An open source simulation platform based on the Modelica modeling language is a popular resource for system modeling and simulation [5].

3

4

CHAPTER 1 Introduction

This book addresses the approaches for the analysis of dynamic systems and control system design. In addition to the discussion of current reactor systems, an overview of next generation nuclear plants (NGNP), small modular reactors (SMR), and instrumentation systems are presented. Treatments of individual topics progress from introductory to advanced levels. For use in undergraduate engineering courses, the coverage may be limited to the simpler and less rigorous portions that appear in pertinent chapters. Several chapters are totally devoted to introductory topics and some to advanced topics. Appendices are included to provide details of subjects whose inclusion in the text would interrupt the flow of information needed for a student’s learning. The appendices are an integral part of the book and the reader is encouraged to review the material. Sample problems are solved in the text and exercises are provided for students to solve. Some problems require computer solutions, including student-prepared computer codes.

Exercises 1.1.

Go to the IAEA web site (see Ref. [1]), determine all of the reactors for which simulation software is available, and document your review

References [1] International Atomic Energy Agency: Website: https://www.iaea.org/topics/nuclearpower-reactors/nuclear-reactor-simulators-for-education-and-training/, Email: Simulators. [email protected]. [2] H. Klee, Simulation of Dynamic Systems with MATLAB and Simulink, CRC Press, Boca Raton, FL, 2007. [3] MATLAB and Simulink User Guides, The MathWorks, Inc., Natick, MA. [4] C. Moler, The Origins of MATLAB, MathWorks Technical Articles, 2004. [5] OpenModelica, Open Source Modelica Consortium, www.openmodelica.org.

CHAPTER

Nuclear reactor designs

2

2.1 Introduction Familiarity with the features of reactors that affect dynamic characteristics and control strategies is necessary for the purpose of this book. It is assumed that the reader knows about general reactor characteristics, but this chapter reviews reactor features that are pertinent for the study of reactor dynamics and control. The evolution of commercial nuclear power is often described in terms of different generations characterized by the dominant reactor designs for each generation. Designs being built in some parts of the world in the early twenty-first century are Generation III designs. Later improved designs constitute Generation III + and Generation IV. Generation V systems are radically new, highly speculative designs that are being investigated for possible later use. Goldberg and Rosner [1] identify six major factors that influence the development and deployment of nuclear power reactors. These are: cost effectiveness, safety, security and non-proliferation, grid appropriateness, commercialization roadmap, and the fuel cycle.

2.2 Generation I reactors The first nuclear reactor, designed by Enrico Fermi and built at The University of Chicago in 1942, served to demonstrate the feasibility of operating a system based on Uranium fission. The success of the “Chicago Pile” led to construction of reactors in Oak Ridge, Tennessee and Hanford, Washington whose purpose was to produce Pu-239 for nuclear weapons by transforming U-238 into Pu-239 by neutron capture. Additional reactors for Plutonium and Tritium production were subsequently built at the Savannah River Site in South Carolina. Following the initial focus on producing a nuclear bomb, interest developed in using nuclear fission reactors to produce useful energy. The U.S. government, through the Atomic Energy Commission, embarked on a reactor development program. The first nuclear electricity was produced with a small generator and used to power a light bulb at the Experimental Breeder Reactor, EBR-1, in Arco, Idaho in 1951. In 1954, the U.S. Congress removed the ban on nuclear activities by nongovernment organizations. Dynamics and Control of Nuclear Reactors. https://doi.org/10.1016/B978-0-12-815261-4.00002-0 # 2019 Elsevier Inc. All rights reserved.

5

6

CHAPTER 2 Nuclear reactor designs

The first successful application of a nuclear reactor for energy production was military, the propulsion plant for the USS Nautilus, a submarine commissioned in 1955. The Nautilus used a pressurized water reactor supplied by Westinghouse for the U.S. Navy. The success of the Nautilus led to the construction and operation of the first U.S. land-based power reactor, the Shippingport 60 MWe reactor in Pennsylvania, commissioned in 1957. The Shippingport reactor was the prototype of the large pressurized water reactors that were built subsequently in the U.S. and other countries. Another power reactor that was to be the prototype of large-scale commercial reactor systems was the 200 MWe Dresden boiling water reactor, commissioned in Illinois in 1960. While the U.S. focused on development of large pressurized water reactors (PWRs) and boiling water reactors (BWRs), Canada focused on power reactors using heavy water. These plants are called CANDU (acronym for CANada Deuterium Uranium) reactors. The development of gas-cooled reactors in the United Kingdom started with the Calder Hall Magnox reactor, commissioned as a Generation I reactor in 1956. The Magnox reactor used natural Uranium as the fuel with graphite as the moderator and carbon dioxide gas as the coolant. The term magnox comes from the magnesiumaluminum cladding used to fabricate the fuel elements. Reactors built in the early years (1950s through early 1960s) are usually referred to as Generation I reactors.

2.3 Generation II reactors Generation II reactors are the large commercial reactors (typified by PWRs and BWRs) built in the U.S. and elsewhere in the 1960s up to the end of the 1990s. Since analysts often need to acquire plant parameters, Appendix A provides typical design data for Generation II PWRs, BWRs and CANDU reactors. In addition to PWRs, BWRs and CANDU reactors, Generation II reactors include, Advanced Gas-cooled Reactors (AGRs) in the UK, Russian Vodo-Vodyanoi Energetichesky Reactors (VVERs), and the Russian RBMK reactors. In a PWR, water flows over fuel rods in the reactor core and extracts heat. The heated water passes through piping and into tubes inside the steam generators. A second stream of water flows outside of the tubes of the steam generators (on the shell side) where boiling occurs. There are two main types of steam generators in U.S. designs: the U-tube recirculation type and the once-through type. The Russian VVERs use horizontal steam generators. Figs. 2.1, 2.2, and 2.3 show the main components of the two types of U.S. PWRs. Details pertinent to PWR dynamics and control are presented in Chapter 12. In a BWR water flows over fuel rods, extracts heat and boils. After separating the steam from liquid water, the saturated 100% quality steam passes to the turbine. Fig. 2.4 shows a schematic of a BWR. Details pertinent to BWR dynamics and control appear in Chapter 13.

2.3 Generation II reactors

Steamline Walls made of concrete and steel 3–5 feet thick (1–1.5 meters)

Containment Cooling System

4

3

Steam Generator

Reactor Control Vessel Rods Turbine Generator

Condenser

Heater

Condensate Pumps

Coolant Loop

2 Core

1

Feed Pumps

Demineralizer

Pressurizer

Reactor Coolant Pumps

Containment Structure

Emergency Water Supply Systems

FIG. 2.1 Layout of a typical pressurized water reactor plant. Source: U.S. NRC, www.nrc.gov/reactors/pwrs.html.

PWRs and BWRs are designated as light water reactors (LWRs) to distinguish them from reactors that use other coolants. Design parameters of a typical commercial PWR and a BWR are given in Appendix A. Canada developed the CANDU reactor that uses natural Uranium fuel and heavy water as a coolant and neutron moderator. In a CANDU reactor, heavy water flows over fuel rods and extracts heat. The heated heavy water passes into a U-tube steam generator similar to the type used in PWRs. Fig. 2.5 shows a schematic of a CANDU reactor. CANDU reactors are pressurized heavy water-moderated reactors, often

7

8

CHAPTER 2 Nuclear reactor designs

STEAM GENERATOR

PRESSURIZER

RCP

REACTOR VESSEL

FIG. 2.2 A four-loop pressurized water reactor system with recirculation-type (U-tube) steam generators. Courtesy of AREVA.

called PHWRs. Design parameters of a typical commercial CANDU reactor are given Appendix A. Future sections address the processes that dictate reactor dynamics for all reactor types, but the details of the reactor dynamics and control strategies are presented only for those Generation II types developed and employed in North America (PWR, BWR, and CANDU reactors). Potential advanced reactors are addressed, but in less detail than currently operating reactors. Other power reactor types were developed and deployed, but so far have never enjoyed the widespread application of the Generation II reactors described above. Gas-cooled reactors (GCRs) using carbon dioxide or helium coolant and graphite moderator have been built and operated, mostly in Great Britain. These high temperature gas-cooled reactors (HTGRs) deployed in Great Britain are referred to as the advanced gas-cooled reactors (AGRs).

2.3 Generation II reactors

FIG. 2.3 A pressurized water reactor with once-through steam generators. Courtesy of The Babcock & Wilcox Company (Steam, Its Generation and Use, Edition 42, The Babcock & Wilcox Company, 2015).

The sodium fast reactor (SFR) is a fast breeder reactor (FBR) that uses liquid sodium as the coolant. Both experimental and commercial sodium-cooled fast breeder reactors have been designed, built, and operated. EBR-1, EBR-2 and Fermi operated in the U.S., and Phenix and SuperPhenix reactors operated in France. Russia operates two commercial SFRs—BN-600 and BN-800. A 500 MWe Indian prototype fast breeder reactor (PFBR) was developed by the Indira Gandhi Center for Atomic Research. A 600 MWe Chinese SFR (CFR-600) was developed by the China Institute of Atomic Energy.

9

10 CHAPTER 2 Nuclear reactor designs

Containment/Drywell

Reactor Vessel Steam Line

Turbine Building

Throttle Valve Steam Dryer & Moisture Separator

Electrical Generator

Turbine

Reactor Core

Condenser

Jet Pump

To/From River Recirculation Pump Pump Containment Suppression Chamber

FIG. 2.4 A boiling water reactor plant with Mark-III containment (U.S. NRC).

2.3 Generation II reactors Steam pipes

Steam generators (Boilers) Primary pumps (Heat transport pumps) Pressurizer

Feedwater return

Feedwater return

Headers

Headers

Calandria

Reactor

Fuel Light water (H2O) steam

Fuel channel (Pressure tube) Moderator pump

Light water condensate Heavy water (D2O) coolant Heavy water moderator

Moderator heat exchanger

FIG. 2.5 A CANDU reactor nuclear steam supply system. Courtesy of UNENE (W.J. Garland (Editor-in-Chief), The Essential CANDU: A Textbook on the CANDU Nuclear Power Plant Technology, University Network of Excellence in Nuclear Engineering (UNENE), Hamilton, Ontario, Canada, 2014).

11

12

CHAPTER 2 Nuclear reactor designs

The VVER is the Russian version of the PWR. It has similarities and some significant differences from western PWRs. VVERs have horizontal steam generators, hexagonal fuel assemblies, and have no vessel bottom penetrations. VVERs range in power ratings from 440 MWe to 1200 MWe. Ref. [2] provides a summary of evolution of VVER designs. Fig. 2.6 shows the schematic of a four-loop 1200-MWe VVER plant [2]. The RBMK is a graphite-moderated, water cooled reactor of Russian design. It is known to have severe stability problems which unfortunately manifested themselves in the Chernobyl accident. RBMK reactors are no longer being built.

Emergency core cooling systems

Steam generator feed water inlet Spray line

Steam generator

Surge line

Primary pump

Pressurizer Reactor vessel

FIG. 2.6 Layout of a 1200-MWe VVER plant. Courtesy of Rosatom Overseas, The VVER Today: Evolution, Design, Safety, 2018, www.rosatom.ru.

2.6 Generation IV reactors

2.4 Generation III reactors New designs, called Generation III reactors, have been developed and constructed. They are evolutionary improvements over Generation II reactors. Features of Generation III reactors include: • • •

• •

Standardized designs that expedite licensing, reduce capital cost, and reduce construction time Passive safety features that reduce the need for actuation and proper operation of engineered safety systems Longer intervals between refueling achieved by use of higher fuel enrichment and burnable poisons (materials that absorb neutrons, but whose strength declines along with fuel consumption) Ability to operate with mixed oxide fuels (Uranium and Plutonium) Simpler designs that expedite construction and operation.

The first Generation III reactor to begin operation was the Kashiwazaki-6 ABWR (Advanced Boiling Water Reactor) in 1996 in Japan. AP600 is a Generation III PWR design by Westinghouse Electric Company.

2.5 Generation III+ reactors Generation III+ reactors offer modular design and a significant improvement in safety over Generation III reactors. These designs incorporate passive safety features such as natural circulation and gravity coolant feed, and rely less on actuators (valves, pumps, etc.) and operator actions. Examples of Gen-III+ designs are AP1000 (Westinghouse), Advanced CANDU reactor (ACR1000), APR-1400 (Korean Advanced Pressurized Water Reactor), VVER-1200, and the European Pressurized Reactor (EPR). The 1650 MWe EPR is also known as Evolutionary Power Reactor in the international market. Ref. [3] provides important design features of the Generation III + VVER-1200 system. Included in Generation III+ reactors are small modular reactors (SMRs). These range in capacity from 25 MWe to 200 MWe. Light water SMRs dominate current design and development. Further details of integral and small modular reactors are given in Appendix B. The International Atomic Energy Agency (IAEA) refers to small and medium reactors as SMRs, with a power range of 25–350 MWe.

2.6 Generation IV reactors Work is under way to develop new reactor designs for possible construction in the future. Currently (2019), six concepts are under consideration. The objective is to achieve improved safety, improved proliferation resistance, reduced waste, and reduced cost. Generation IV reactor designs provide high temperature operation.

13

14

CHAPTER 2 Nuclear reactor designs

Higher temperatures can enable higher thermodynamic efficiencies in electricity production and can provide process heat for applications such as high-temperature electrolysis of water to produce hydrogen. The designs under consideration include both new concepts and resurrection of some older concepts that previously failed to reach commercialization. Ref. [4], prepared by OECD Nuclear Energy Agency, provides a technology roadmap update on generation IV (gen-IV) reactors. The gen-IV reactor concepts under consideration are the following [4] • • •

• •

•

Gas-Cooled Fast Reactor (GCFR): The GCFR is a breeder reactor that uses helium gas to carry the heat from fuel elements comprising the reactor core. Lead-cooled Fast Reactor (LFR): The LFR is a breeder reactor that uses liquid lead to carry heat from fuel elements comprising the reactor core. Molten Salt Reactor (MSR): The MSR is a fluid-fuel reactor. Fuel is dissolved in a molten salt that flows through the core. The MSR concept permits continuous processing to remove fission products and add fresh fuel, but this is accomplished at the cost of complex plumbing and materials handling. MSR designs include fast reactors (no moderator) and thermal reactors (prismatic graphite block with channels for fluid fuel-bearing salts) Sodium-Cooled Fast Reactor (SFR): The SFR is a breeder reactor that uses liquid sodium to carry heat from fuel elements comprising the reactor core. Supercritical Water-Cooled Reactor (SCWR): The SCWR is an outgrowth of current designs. It operates at much higher temperatures and pressures than current light water reactors. Very-High Temperature Reactor (VHTR): The VHTR uses helium coolant that flows through a core consisting of graphite impregnated with fuel particles. Designs include plants that use the Brayton Cycle instead of a steam cycle. Hot gas passes directly to a gas turbine in a Brayton cycle.

2.7 Advanced reactors At the time when this book was written, many new reactor designs were under consideration and a few were experiencing prototype construction. Since the focus of this book is reactor dynamics and control, most readers would consider inclusion of detailed descriptions of these designs to be superfluous and unnecessary for their study. But, since some of these designs will likely be implemented, some readers will find that they are involved in their analysis and are consequently interested in information about them. Information about sodium fast reactors, gas-cooled reactors, molten salt reactors, and integral small modular reactors is presented in Appendix B. This appendix is not to be considered as essential material for every reader. Rather it should be considered as a resource that can be accessed when some of these reactor designs reach maturity and implementation. It is expected that instructors in university courses will choose to alert students that this appendix exists, but probably to skip it in course presentation.

References

2.8 Early twenty-first century construction As this book was being written (2019) a number of plants had closed or are scheduled to close (mainly in the U.S., Eastern Europe, Germany, and Japan), but many new nuclear power plants are reaching operation and being built around the world in the first half of the twenty-first century. Over fifty units are scheduled for completion by 2020. This includes two plants in the U.S., but most of the activity is outside the U.S., including Russia, India, South Korea, United Arab Emirates, but mainly in China. Most Chinese plants are Generation III or III + PWRs. Generation IV plants are also under construction, a high-temperature gas-cooled reactor (a Pebble Bed Modular Reactor (PBMR)) and a 600 MWe sodium-cooled fast reactor. China embarked on a massive program with cooperation with foreign suppliers initially (mainly the U.S. and France) and subsequently with a growing indigenous industry. China has even entered the market to supply plants in other countries. India completed the construction of a 500 MWe prototype SFR, and is developing an Advanced Heavy Water Reactor (AHWR) using Thorium-based oxide fuel [5]. Generation V reactors are speculative concepts that may be suitable as successors to Generation IV systems. It is too early to identify leading candidates, their technical feasibility or their economic competitiveness. Conceptually, these would incorporate features such as remote and near-autonomous operation, ‘walk-away’ safety, and safeguard from external threats. It will be several years before Generation V receives serious consideration.

Exercises 2.1.

Make a review of literature, find information about Generation III + and Generation IV reactors, and document their unique features. See References.

References [1] S.M. Goldberg, R. Rosner, Nuclear Reactors: Generation to Generation, American Academy of Arts and Sciences, 2011. [2] Rosatom Overseas, The VVER Today: Evolution, Design, Safety, www.rosatom.ru, 2018. [3] Status report—108, VVER-1200 (V-491), www.iaea.org, 2018. [4] OECD Nuclear Energy Agency, Technology Roadmap for Generation IV Nuclear Energy Systems, January 2014. [5] M. Todosow, A. Aronson, L.-Y. Cheng, R. Wigeland, C. Bathke, C. Murphy, B. Boyer, J. Doyle, B. Fane, B. Ebbinghaus, The Indian Advanced Heavy Water Reactor (AHWR) and Non-Proliferation Attributes, Brookhaven National Laboratory, August 2012. BNL98372-2012.

15

16

CHAPTER 2 Nuclear reactor designs

Further reading [6] W.J. Garland, Editor-in-Chief, The Essential CANDU: A Textbook on the CANDU Nuclear Power Plant Technology, University Network of Excellence in Nuclear Engineering (UNENE), Hamilton, Ontario, Canada, 2014. [7] The Babcock & Wilcox Company (Ed.), Steam, Its Generation and Use, Edition 42, 2015.

CHAPTER

The point reactor kinetics equations

3

3.1 Neutronics The neutron population in a nuclear reactor is a function of time, position, direction of motion, and energy. Neutrons appear at some position in the reactor as a result of a fission reaction between uranium or plutonium and a neutron from a previous generation. The neutron emerges from the fission reaction with a large kinetic energy (an energy of around 3 MeV or a speed of around 3
109 cm/s). These neutrons undergo elastic and inelastic scattering events with materials in the reactor core (fuel, structure, cladding, coolant, moderator, etc.) and, as a result, lose energy. Most current-generation reactors include a moderator, a material with a high probability of slowing neurons by scattering collisions while absorbing few neutrons. Typical moderators are water, heavy water, and graphite. Reactors with moderators are called thermal neutron reactors. In these reactors, most of the neutrons slow to energies less than 0.1 eV (a speed of around 4
105 cm/s). Fast reactors have no moderator and rely on fissions with fast neutrons. In a thermal reactor, the time between a neutron’s birth and eventual absorption by a target nucleus is typically 10 to 30 microseconds, and even faster in a reactor with a fast neutron spectrum. The most complete neutronic description of a reactor is the Boltzmann transport equation. This equation gives the neutron population as a function of seven independent variables (time, three position coordinates, energy and two direction vectors). The neutron diffusion model is one-step simpler. In diffusion theory, the direction dependence is removed, leaving a model with five independent variables. Further simplification occurs with eliminating the spatial dependence (treating the reactor as a “point”) and reducing the energy treatment to a single energy group. These simplifications may seem extreme, but the simplified neutronics model has proved suitable for a wide range of reactor simulations. Appendix C provides a brief discussion of basic reactor physics for those who need familiarization or review.

Dynamics and Control of Nuclear Reactors. https://doi.org/10.1016/B978-0-12-815261-4.00003-2 # 2019 Elsevier Inc. All rights reserved.

17

18

CHAPTER 3 The point reactor kinetics equations

3.2 Delayed neutrons 3.2.1 Delayed neutrons from fission products Not all of the neutrons resulting from a fission reaction emerge immediately. In a reactor, a small fraction of the neutrons appears upon radioactive decay of a fission product. Some fission products have too many neutrons for nuclear stability. They can relieve this excess by either of two decay possibilities: emission of a beta particle or emission of a neutron. Delayed neutrons come from neutron emissions from elements that result from previous beta decay, but still have too many neutrons. See Fig. 3.1 for the general scheme for producing delayed neutrons. As long as the contribution of delayed neutrons is essential to maintain the chain reaction, transients must “wait” for release of the delayed neutrons. If reactor operation relied entirely on fission neutrons, transients would be too fast to tolerate. Nuclear bombs are designed to rely on fission neutrons and clearly things happen very quickly. Delayed neutrons appear with lower kinetic energy than fission neutrons (around 0.5 MeV). This lower energy at birth influences the importance of delayed neutrons in causing subsequent absorption by target nuclei and fission reactions. Also, delayed neutrons have a smaller probability of leaking out of the reactor core compared to prompt neutrons. We shall see that delayed neutrons are very important in making reactors controllable. Delayed neutrons may be treated with a model that includes six neutron precursor groups. A delayed neutron precursor group is the collection of certain fission fragment that decays to a stable isotope by giving rise to a delayed neutron. The yield of each precursor group depends on which fuel material is involved. The total fraction of delayed neutrons ranges from 0.0022 to 0.007, depending on the fissioning isotope involved. We shall see that these small fractions have a major impact on reactor operation. Delayed neutron data for thermal fissions in three fissile materials and for fast fissions in U-238 (fissionable only with fast neutrons) appear in Tables 3.1–3.4.

Fission product

Prompt neutrons

Neutron Fissile material Precursor

Delayed neutron Delayed β-decay Emitter Stable nucleus

FIG. 3.1 Production of delayed neutrons.

3.2 Delayed neutrons

Table 3.1 Delayed neutron data for U-235 (thermal fission). Group

Decay constant, λi (s21)

Delayed neutron fraction (βi)

1 2 3 4 5 6

0.0126 0.0337 0.139 0.325 1.13 3.50

0.000224 0.000777 0.000655 0.000723 0.000133 0.000088

Total delayed neutron fraction: 0.0067.

Table 3.2 Delayed neutron data for U-233 (thermal fission). Group

Decay constant, λi (s21)

Delayed neutron fraction (βi)

1 2 3 4 5 6

0.0126 0.0337 0.139 0.325 1.13 3.50

0.000224 0.000777 0.000655 0.000723 0.000133 0.000088

Total delayed neutron fraction: 0.0026.

Table 3.3 Delayed neutron data for Pu-239 (thermal fission). Group

Decay constant, λi (s21)

Delayed neutron fraction (βi)

1 2 3 4 5 6

0.0128 0.0301 0.124 0.325 1.12 2.69

0.000077 0.000656 0.000464 0.000717 0.000189 0.000097

Total delayed neutron fraction: 0.0022.

Table 3.4 Delayed neutron data for U-238 (fast fission). Group

Decay constant, λi (s21)

Delayed neutron fraction (βi)

1 2 3 4 5 6

0.0132 0.0321 0.139 0.358 1.41 4.02

0.000213 0.002247 0.002657 0.006363 0.003690 0.001230

Total delayed neutron fraction: 0.0164.

19

20

CHAPTER 3 The point reactor kinetics equations

There are differences between the delayed neutron data for thermal fissions and fast fissions in U-235, U-233 and Pu-239, but they are small. The reader should be aware that different references report slightly different values for delayed neutron data. These differences have little influence on practical reactor simulations. Also, note that Pu-239 and U-233 have much smaller delayed neutron fractions than U-235. This has an impact on dynamic behavior. A reactor fueled with low-enrichment U-235 produces Pu-239 by neutron capture in U-238. Consequently, the ratio of Pu-239 to U-235 increases as the reactor operates and the “effective” delayed neutron fraction decreases. A single delayed neutron group is often used for approximate calculations. In this case, approximate average values for the effective delayed neutron fraction and effective delayed neutron decay constant are required. Typical values for a U-235 fueled reactor are 0.0067 for the effective delayed neutron fraction and 0.08 s1 for the effective decay constant (see Section 4.6 for calculating the effective decay constant). The main utility of the model with a single delay group is that it provides a simple, easy-to-implement tool for illustrating the features of reactor transients. We will use the model with a single delayed neutron precursor group for illustrating some features of a reactor transient, but the full six-group model will be used in simulations to illustrate reactor maneuvers.

3.2.2 Photoneutrons from nuclei excited by gamma rays Several isotopes can produce photoneutrons by interaction with high energy gamma rays produced in fission reactions. The materials that can produce neutrons by interacting with fission gamma rays are deuterium, lithium, beryllium, and carbon. These materials have a small enough binding energy (all have binding energies of 7.25 MeV or less) to enable photoneutron-producing reactions with gamma rays having energies that occur during fission reactions. Fission gamma rays are not energetic enough to cause (γ, n) reactions in other materials. In photoneutron production, the high energy gamma ray causes the target nucleus to transition to an excited state. This excited state persists until the nucleus emits a neutron. The process of neutron emission follows the law of radioactive decay as characterized by a half-life. The different photoneutron producing nuclei have half-lives ranging from around 2.5 s to 12.5 days. The photoneutron yield is much smaller than the delayed neutron yield from fission products, but the half-lives of some of the photoneutron precursors are much longer than the precursor half-lives of delayed neutrons from fission products. Consequently, photoneutron production can become larger than delayed neutron production from fission products after time passage following shutdown of the reactor. This occurs after several minutes after shutdown in CANDU reactors. Photoneutrons are produced in reactors that use Deuterium by the (γ, n) reaction between Deuterium and 2.225 MeV gamma rays. The coolant and moderator in CANDU reactors are almost pure D2O. Even light water reactors have a small

3.3 Development of the point reactor kinetics equations

fraction of naturally-occurring Deuterium. Deuterium is also produced in LWRs by neutron absorption in Hydrogen. Consequently, they have a similar, but smaller photoneutron effect than CANDU reactors [1]. Molten salt reactors have three components (Li, Be, and C) that are susceptible to photoneutron production with fission gamma rays.

3.3 Development of the point reactor kinetics equations The purpose here is to arrive at reactor kinetics equations using the simplest and most intuitive approach. We want to develop a set of ordinary differential equations describing the time evolution of neutron density or reactor power. Many developments have been published that derive the equations from first principles of reactor physics. But all approaches give the same final results and all have the details swept into simple quantities such as keff and reactivity. Simulations involve the use of specified values of keff or reactivity as input disturbances or as feedback terms in power reactor simulations where keff or reactivity feedback is proportional to various process variables such as temperature or pressure. The basic idea in formulating dynamic equations is as follows: ðRate of change of some quantityÞ ¼ ðRate of productionÞ  ðRate of lossesÞ

(3.1)

For the point kinetics model, the equations are presented below. dn ¼ Fp PΤ + Pd  La + L dt

(3.2)

dCi ¼ Fd PΤ  Ldecay i ¼ 1,2,…, 6 dt

(3.3)

where n ¼ neutron density (neutrons per unit volume) Fp ¼ fraction of neutrons that are released promptly PT ¼ total rate of neutron production by fission Pd ¼ rate of release of delayed neutrons La+L ¼ rate of neutron losses by absorption and leakage Fd ¼ fraction of neutrons that are released after a delay Ci ¼ concentration of ith delayed neutron precursor Ldecay ¼ rate of decay of delayed neutron precursors (equal to Pd) Let β be defined as the total fraction of neutron production that is delayed and βi be the fraction of fissions that result in production of delayed neutron precursor, Ci. The loss of precursor, Ci, is given by λiCi, where λi is the radioactive decay constant, and is the contribution of the ith precursor to the delayed production term, Pd. Since there are multiple precursor species, typically represented by six groups, the equations become

21

22

CHAPTER 3 The point reactor kinetics equations

6 X dn λi C i ¼ ð1  βÞPΤ  La + L + dt i¼1

(3.4)

dCi ¼ βi PΤ  λi Ci i ¼ 1, 2,…, 6 dt

(3.5)

Eq. (3.4) may be written as.   6 X dn La + L + λi C i ¼ P T ð1  β Þ  PT dt i¼1

(3.6)

Note that LPa +T L is simply the ratio of total neutron productions (by fission) to total neutron losses (by absorption and leakage) and is equal to the effective multiplication factor, keff. So Eq. (3.6) becomes.   6 X dn 1 + λi C i ¼ PT ð1  βÞ  dt keff i¼1

(3.7)

  6 X keff  1 dn β + λi C i ¼ PT keff dt i¼1

(3.8)

or

or 6 X dn λi Ci ¼ PT ðρ  βÞ + dt i¼1

(3.9)

where ρ¼

keff  1 keff

(3.10)

The total neutron production, PT, is given by. PT ¼ ν Σf Φ

or PT ¼ ν Σf n v

(3.11)

where ν ¼ average number of neutrons produced per fission Σf ¼ macroscopic fission cross section Φ ¼ neutron flux v ¼ neutron velocity As shown in elementary reactor physics books [2, 3], the reciprocal of (ν Σf v) is the time between production of a neutron and the production of new neutrons by fission, and is called the generation time, Λ. Therefore, the final form of the point kinetics equations is

3.3 Development of the point reactor kinetics equations

6 X dn ðρ  βÞ λi Ci ¼ n+ dt Λ i¼1

(3.12)

dCi βi ¼ n  λi C i dt Λ

(3.13)

An alternate form of the kinetics equations may also be developed. The development begins by re-writing Eqs. (3.4) and (3.5) as follows: 6 X dn PT 1+ λi Ci ¼ L a + L ð1  β Þ La + L dt i¼1

(3.14)

dCi PT β  λi Ci ¼ La + L dt La + L i

(3.15)

Using the same procedure and definitions as above gives   6 X keff  1  β dn n+ λi Ci ¼ l dt i¼1

(3.16)

dCi βi ¼ keff n  λi Ci dt l

(3.17)

This is called the lifetime formulation, with l ¼ 1/{(Σa + leakage operator)v}. Note that the generation time and lifetime are equal for a critical reactor. Upon a change in the reactor’s absorption cross section (as in motion of a control rod) the generation time is expected to stay unchanged while the lifetime would undergo a small (essentially negligible) change. The generation time formulation will be used in subsequent sections of this book. The simple derivations provided above are rigorous and independent of reliance on specific quantities from reactor physics other than the generation time or lifetime. Of course, reactivity or keff could be computed from first principles for simulating a specific situation. But, that is not how transient simulations are normally done. Transients are calculated for specified changes in reactivity or keff, not for some change in a fundamental nuclear property. It should be noted that there are several different ways to express the neutron multiplication factor, keff or reactivity, ρ. Different authors chose among the following measures: keff ρ (reactivity) mk (¼ 0.001 keff) per cent mill or pcm (¼ 0.00001 keff) Δk/k %Δk/k ¼ (0.01*Δk/k)

23

24

CHAPTER 3 The point reactor kinetics equations

Dollar, $ (¼ ρ/β) Cent, ¢ (¼ 0.01$) Note that one dollar ($) of reactivity is numerically equal to β, the magnitude of the total delayed neutron fraction. All of these measures are suitable, but a potential cause for confusion. Reactivity, ρ, and dollar ($) or cent (¢), is used throughout this book.

3.4 Alternate choices for the neutronic variable In the above derivations of the point kinetics equations, the neutron density, n, was chosen as the neutronic variable. Here, we show that the equations may be written with neutron flux, reactor power or relative reactor power. First use the relation, Φ ¼ nv, to replace the neutron density, n, with neutron flux, Φ. The result is 6 X dΦ ðρ  βÞ λi C0i ¼ Φ+ dt Λ i¼1

(3.18)

dC0i βi ¼ Φ  λi C0i dt Λ

(3.19)

where Ci 0 ¼ v Ci

Note that the precursor terms in this formulation are no longer actual precursor concentrations, but are the non-physical quantity, (v Ci). But since the solution variable of interest is the neutron flux, the physical interpretation of the precursor variable is inconsequential. Now reformulate again with power as the variable of interest. Multiply Eqs. (3.18) and (3.19) by (F Σf V) where F is the conversion from fission rate to power ( 3.2
1011 watt seconds per fission) and V is the reactor volume to obtain power, P ¼ F Σf Φ V. The result is 6 X dP ðρ  βÞ λi C00i ¼ P+ dt Λ i¼1

(3.20)

dC00i βi ¼ P  λi C00i dt Λ

(3.21)

where Ci 00 ¼ F Σf V Ci

Finally reformulate with relative power, P/P(0), as the variable of interest. Here P(0) is a nominal power; for example, the 100% power level. The result is

3.5 Perturbation form of the point kinetics equations

6 X dP=Pð0Þ ðρ  βÞ 00 0 λi Ci ¼ P=Pð0Þ + dt Λ i¼1

  dC0 00i βi P 00 ¼  λi C0 i dt Λ Pð0Þ

(3.22)

(3.23)

where Ci 000 ¼ ðF Σf V Ci =Pð0ÞÞ Ci

So, the form in each case is the same, different only in the quantity used to represent neutron density, neutron flux, power or relative power and the inconsequential definition of the precursor term. In subsequent appearances, the precursor term will simply be labeled, Ci, even though, as we have just demonstrated, the physical definition is different in each formulation. If only one delayed neutron group is used for more approximate simulations, the equations are as shown below (using the relative power or fractional power formulation for this illustration).     d P ðρ  β Þ P ¼ + λC dt Pð0Þ Λ Pð0Þ

(3.24a)

  dC β P ¼  λC dt Λ Pð0Þ

(3.24b)

β¼

6 X

βi

(3.25a)

i¼1

1 λ¼0 6 1 ,li is the mean lifetime of delayed neutron precursor group i: X β i li C B C B i¼1 C B B β C A @

(3.25b)

λ is the one delayed neutron precursor decay constant.

3.5 Perturbation form of the point kinetics equations It is sometimes useful to formulate the point kinetics equations in terms of deviations from steady state. Such a formulation facilitates development of linear models for reactors with reactivity feedback and the development of reactor transfer functions. Development of the perturbation form of the point kinetics equations (version with power, P, as the neutronic variable) begins with the following definitions:

25

26

CHAPTER 3 The point reactor kinetics equations

P ¼ Pð0Þ + δP C ¼ Cð0Þ + δC ρ ¼ ρð0Þ + δρ

where P(0) ¼ initial steady state value of P δP ¼ deviation of P from its initial steady state value C(0) ¼ initial steady state value of C δC ¼ deviation of C from its initial steady state value ρ(0) ¼ initial steady state value of ρ (¼ zero because the initial state is a critical reactor) δρ ¼ deviation of ρ. Substituting these definitions into the point kinetics equations gives   6 X dPð0Þ dδP δρ  β λi Ci ð0Þ + δCi Þ + ¼ ðPð0Þ + δPÞ + dt dt Λ i¼1

(3.26)

  dCi ð0Þ dδCi βi ¼ ðPð0Þ + δPÞ  λi Ci ð0Þ + δCiÞ + dt Λ dt

(3.27)

Note that (at steady state) dPð0Þ ¼0 dt dCð0Þ ¼0 dt ρð0Þ ¼ 0 βi Pð0Þ  λi Ci ¼ 0 Λ

This leads to the following equations: 6 X dδP δρ β δρ λi δCi ¼ Pð0Þ  δP + δP + dt Λ Λ Λ i¼1

(3.28)

dCi βi ¼ δP  λi δCi dt Λ

(3.29)

In general, one may also arrive at Eqs. (3.28) and (3.29) by subtracting the point reactor kinetics equations at a specified nominal condition from the two Eqs. (3.26) and (3.27).

3.6 Transfer functions

This equation is the perturbation form of the kinetics equations. It is totally equivalent to the standard form. In the perturbation form, all initial conditions are zero. Note that the perturbation form of Eqs. (3.28) and (3.29) are also true for any non-steady state nominal power P(0). All of the terms in the perturbation equations are linear, constant coefficient terms except (δρδP). If the use of the perturbation form is restricted to small perturbations, then (δρ.δP) is small compared to {δρ.P(0)} and can be ignored. The final result for the “small perturbation” form of the point kinetics equations is: 6 X dδP δρ β λi δCi ¼ Pð0Þ  δP + dt Λ Λ i¼1

(3.30)

dCi βi ¼ δP  λi δCi i ¼ 1,2, …,6 dt Λ

(3.31)

Eqs. (3.30) and (3.31) may be rewritten by dividing them by P(0), thus using fractional changes in the power. This representation will come in handy in the discussion of fuel-to-coolant heat transfer dynamics.

3.6 Transfer functions A transfer function is defined as the Laplace transform of a system’s output deviation divided by the Laplace transform of the system’s input deviation. Appendix D addresses Laplace transform theory. Laplace transform is a convenient tool for transforming from a differential equation to an algebraic equation for expressing the relationship between two variables. Generally, one variable is considered as a system output and the second variable as a system input. The analysis can be easily extended to multiple-input multiple-output systems. For a reactor, the transfer function that relates power to reactivity is δP(s)/δρ(s). We derive this transfer function by Laplace transforming the small perturbation equations. Initial conditions are zero because there is no deviation from steady state initially. The result is sδP ¼

6 X δρ β λi δCi Pð0Þ  δP + Λ Λ i¼1

(3.32)

βi δP  λi δCi Λ

(3.33)

sδCi ¼

where all variables are now functions of the Laplace transform parameter, s. Solving for δP(s)/δρ(s) gives the desired transfer function: δP ¼ δρ

0 B Λs@1 +

1

1 βi Λ C A i¼1 ðs + λ Þ i

X6

(3.34)

27

28

CHAPTER 3 The point reactor kinetics equations

The transfer function can be recast to give per cent power deviation per cent of reactivity as follows: β δ%P Λ ¼ 0 δc B X6 s@1 + i¼1

1 βi Λ C A ðs + λ i Þ

(3.35)

This form will be useful in subsequent discussions about reactor frequency response. The transfer function for the model with a single group of delayed neutrons is. β ðs + λ Þ δP ðper cent power Þ Λ   ¼ β δρ ðcentÞ s s+λ+ Λ

(3.36)

Transfer functions can be useful in cases where the output of one subsystem serves as the input to another subsystem. For example, consider the arrangement shown in Fig. 3.2. The output of subsystem 1 with transfer function, G1, serves as the input to subsystem 2 with transfer function, G2. The overall transfer function is simply the product of the subsystem transfer functions. Transfer functions have served in classical control system design. A control system involves measurement of a system output and processing that output to add some quantity to the input to achieve desired dynamic response. The configuration of a closed-loop system is as shown in Fig. 3.3, where H is the feedback transfer function. Since the control action generally serves to reduce the effect of the input disturbance, the usual convention is to show the feedback as a negative contribution to the input. In this case, the overall transfer function is given by Eq. (3.37). δOðsÞ GðsÞ ¼ δIðsÞ 1 + GðsÞH ðsÞ

(3.37)

The configuration shown in Fig. 3.3 also applies for systems with inherent feedback. In this case, it is customary to let the feedback, H, determine the sign of the signal entering the summing junction. In this case, the overall transfer function is as follows: δOðsÞ GðsÞ ¼ δIðsÞ 1  GðsÞH ðsÞ

Input X

G1(s)

G2(s)

FIG. 3.2 A Series (cascade) configuration of transfer functions.

(3.38)

Output Y

3.8 Stability

Input

x(t) + X(s)

–

Plant

y(t) Output

G(s)

Y(s)

Feedback H(s)

FIG. 3.3 Transfer functions in a feedback configuration. Note that, in some cases, the feedback effect may be positive.

3.7 Frequency response function The frequency response of a linear system is defined as the sinusoidal deviations in a system’s output resulting from a sinusoidal deviation in the system’s input. There is an initial transient following start of a sinusoidal input followed by a sinusoidal evolution of the output. The frequency response is characterized by the ratio of the output’s amplitude to the input’s amplitude and the phase shift between the two sinusoids. Appendix E addresses frequency response theory. As shown in Appendix E, the frequency response pffiffiffiffiffiffiffi may be calculated by substituting s ¼ jω in the transfer function (where j ¼ 1 and ω is the frequency in rad/s). Performing the complex manipulation provides the real and imaginary parts of the solution.

3.8 Stability Stability is an issue for any dynamic system. Stability analysis methods are welldeveloped for linear systems and are computationally simple. Before the advent of modern digital computers and simulation software, formal stability analysis methods were easier to perform than transient analysis (which would also reveal stability problems). Stability analysis methods (those other than time-domain simulations) currently find little use in reactor analysis and are not addressed here. The exception is in analyzing coupled thermal-hydraulic/neutronic instabilities in BWRs (see Chapter 13). Linear stability is a universal concept. That is, a stable linear system demonstrates bounded outputs for bounded inputs. Linear stability analysis also provides a tool for assessing the suitability of a candidate control system. One purpose of a controller is to cancel out the effect of an input disturbance. To emphasize this fact, the block diagram usually shows the feedback subtracted from the input. In a system with feedback due to processes within the system, it is more

29

30

CHAPTER 3 The point reactor kinetics equations

logical to add the feedback to the input. The feedback may be positive or negative, depending on the process responsible for the feedback. For example, in a reactor with a positive temperature coefficient of reactivity, the feedback from the process that determines that a temperature increase would be positive and destabilizing. However, feedbacks from other processes in the system would also contribute positive or negative feedback. The total feedback would be the net feedback from all of the system feedbacks. Therefore, a positive feedback from one of the processes is acceptable for a system if other stabilizing negative feedbacks dominate. Later chapters show that some reactors have a positive feedback component, but are stable because of other, stronger negative feedbacks. Negative feedbacks are usually stabilizing and, therefore, desirable. However, negative feedbacks can also be destabilizing. This happens if the negative feedback is delayed before adding to the input and if the feedback gain exceeds a certain limit. Section 7.6 addresses the issue of destabilizing negative feedback and Section 13.6 addresses the issue of destabilizing negative feedback in BWRs. Nonlinear systems are a different story. They can have multiple equilibrium points. A transient can involve jumping from one equilibrium point to another and each equilibrium point can be stable or unstable. Nonlinear stability can depend on the magnitude of the input disturbance. Nonlinear systems can demonstrate limit cycles, a continuous oscillatory response that may be non-symmetric in shape. Efforts to develop useful and practical stability analysis methods for nonlinear systems have occupied well-qualified mathematicians for many years, but the efforts failed. Stability analysis, especially analysis of nonlinear systems, usually relies on numerical solutions of the governing differential equations. Because the nature of the response can depend on the magnitude and form of the input disturbance, it is necessary to simulate a number of different scenarios. The characteristics of linearization of a nonlinear system depend on the nominal state about which the model is linearized. Thus, the linearized form of a nonlinear equation generally varies from one equilibrium point to another equilibrium point. In a reactor, it is possible for the flux shape to be unstable. That is, a disturbance in one part of the reactor can stimulate cyclical flux increases in one part of the reactor while other parts experience flux decreases. Generally, spatial stability is a concern mainly in large, loosely-coupled reactors (reactors in which individual regions are almost independently critical). Spatial stability is addressed in Chapter 9. Flow instabilities are an issue in some reactors, especially boiling water reactors. BWRs are operated so as to avoid conditions in which flow instability are likely Chapter 13 addresses this issue.

3.9 Fluid-fuel reactors In the 1960s, two types of fluid-fuel reactors (aqueous homogeneous and molten salt) were considered as candidates for future power reactors, and small test reactors were built and operated. Subsequently, these concepts were abandoned in favor

3.9 Fluid-fuel reactors

of other concepts and interest in simulating them disappeared. Now, the molten salt reactor, a fluid-fuel reactor, is under consideration for possible implementation as a Generation IV reactor. Since fluid-fuel reactors are a future possibility and because they require a different form of reactor kinetics equations, they are discussed here. For dynamic characteristics, the important distinction between solid-fuel reactors and fluid-fuel reactors is the treatment and consequence of delayed neutron behavior [3, 4]. In a solid fuel reactor, the delayed neutron precursors stay in place until they release a neutron. In fluid-fuel reactors, the fuel is dissolved in a solvent that carries the fuel through the reactor core, then through an external loop where heat is extracted before returning to the core. The delayed neutron precursors are created in the core, but they emit neutrons while in the core and while in the external loop. This reduces the delayed neutron contribution to the neutron population in the core. Thus, transients depend more on fission neutrons than delayed neutrons compared to a solid-fuel reactor. Because of the circulation effect on delayed neutrons, the point kinetics equations must be modified as follows: 6 X dPðtÞ ðρ  βÞ λi CCi ¼ PðtÞ + dt Λ 1

(3.39)

dCCi βi CCi CCi ðt  τL Þeλi τL ¼ PðtÞ  λi CCi  + dt Λ τC τC

(3.40)

where the new terms are CCi ¼ the i-th precursor concentration in the core τc ¼ fluid residence time in the core τL ¼ fluid residence time in the external loop Solving this set of equations requires special attention to the initial conditions. For steady state, offset reactivity must be added to compensate for the effect of losing delayed neutrons during out-of-core decays. The offset reactivity for steady state is given by. ρ0 ¼ βT 

6 X i¼1

βi  

1 1+ 1  eλi τL λi τ c

(3.41)

For example, a U-235 fueled fluid fuel reactor with a core residence time of 8.46 s and an external loop residence time of 16.73 s has an offset reactivity of 0.00247. This is the reactivity that must be added (as by withdrawal of control rods) to maintain steady state when flow begins. Of course, this rather large reactivity is introduced if flow stops. Further discussion of molten-salt reactor dynamics is presented in Appendix K.

31

32

CHAPTER 3 The point reactor kinetics equations

Exercises 3.1.

Explain why transfer function models facilitate combining subsystem models to obtain a model for the whole system.

3.2.

Explain the physical basis for the new terms in the kinetics equation for fluidfuel reactor.

3.3.

Verify that Eq. (3.41) follows from Eqs. (3.39) and (3.40).

3.4.

Evaluate Eq. (3.41) for a stationary-fuel reactor. Discuss your result.

References [1] W.J. Garland (Ed.), The Essential CANDU: A Textbook on the CANDU Nuclear Power Plant Technology, University Network of Excellence in Nuclear Engineering (UNENE), Hamilton, Ontario, Canada, 2014 (Editor-in-Chief). [2] J.J. Duderstadt, L.J. Hamilton, Nuclear Reactor Analysis, John Wiley, New York, 1976. [3] J.K. Shultis, R.E. Faw, Fundamentals of Nuclear Science and Engineering, second ed., CRC Press, Boca Raton, FL, 2007. [4] T.W. Kerlin, S.J. Ball, R.C. Steffy, Theoretical dynamics analysis of the molten-salt reactor experiment, Nucl. Technol. 10 (1971) 118–132.

Further reading [5] T.W. Kerlin, S.J. Ball, R.C. Steffy, M.R. Buckner, Experience with dynamic testing methods at the molten-salt reactor experiment, Nucl. Technol. 10 (1971) 103–117.

CHAPTER

Solutions of the point reactor kinetics equations and interpretation

4

4.1 Evolution of simulation methods In earlier days, analysts had to perform modeling, computer programming, and numerical analysis to accomplish a reactor simulation. Often, one person had to handle all three tasks. Now software packages are available that eliminate the need for expertise in computer programming and numerical analysis. Even software that provides simulations for specific cases is available. These packages expand the ability to perform reliable simulations, but the analyst still needs basic understanding in order to interpret the results correctly.

4.2 Numerical analysis Mathematically, the point reactor kinetics equations are a “stiff system”. Solutions of the reactor kinetics equations have a very fast component (due to the small value of generation time or lifetime) and a slow component (due to the much larger values of the precursor half-lives). Consequently, any numerical method used to solve the equations must have the ability to handle stiff systems. A number of differential equation solvers are available in easy-to-use software packages, and most have good methods for handling stiff systems. Generally, stiff systems require small time steps in the solution. The more sophisticated solution methods can achieve suitable accuracy with varying time steps, small values at the beginning of a transient when the response is changing rapidly, and larger time steps later when the speed of response slows down. However, it should be noted that the speed of modern computers sometimes makes it feasible to use small computation increments needed in simpler methods. The MATLAB/Simulink software platform is often used for numerical solutions. The simplest simulation involves a model consisting of linear, constantcoefficient differential equations. The point reactor kinetics equations with constant reactivity are a linear constant-coefficient set of equations. For simulations of the response to time-varying reactivity perturbations, the point kinetics equations are a set of variable-coefficient equations. As we shall see in later sections, reactivity is a function of other dependent variables (such as component temperatures and system pressure) in a power reactor. In this case, the model is nonlinear. Dynamics and Control of Nuclear Reactors. https://doi.org/10.1016/B978-0-12-815261-4.00004-4 # 2019 Elsevier Inc. All rights reserved.

33

34

CHAPTER 4 Solutions of the point reactor kinetics equations

Available numerical methods can handle all of the model categories. The software packages commonly require the model to be expressed as a set of coupled, first-order differential equations (like we have seen for the point reactor kinetics equations). These software packages require a model with n first-order differential equations to be expressed as follows: dx1 ¼ a11 x1 + a12 x2 + ⋯a1n xn + f1 dt

(4.1)

dx2 ¼ a21 x1 + a22 x2 + ⋯a2n xn + f2 dt

(4.2)

dxn ¼ an1 x1 + an2 x2 + ⋯ann xn + fn dt

(4.3)

where, xi ¼ a dependent variable (i ¼ 1, 2, …, n), aij ¼ a constant coefficient (i ¼ 1, 2, …, n; j ¼ 1, 2, …, n), fi ¼ a forcing function. In matrix notation, dX ¼ AX + f dt

(4.4)

X is (n  1) state vector, f is a (n  1) vector of forcing terms, and A is the (n  n) “system matrix” containing the system parameters. A model with n equations is called an n-th order state variable model. The formulation in Eq. (4.4) is often called the state-space representation of system dynamics. This representation, in general, can also involve nonlinear functions of the state variables by adding a vector of nonlinear terms, g(X). See Appendix F for a description of state variable models. Implementation requires supplying values for the elements in the A matrix and f vector to appropriate software. For example, the matrix representation for constant reactivity (as in a step change) is as follows: 3 ρβ λ λ λ λ λ λ 1 2 3 4 5 6 7 6 Λ 7 6 7 6 β 6 1 λ 0 0 0 0 0 7 7 6 1 7 6 Λ 7 6 7 6 β2 7 6 0 λ 0 0 0 0 2 7 6 Λ 7 6 7 6 7 6 β3 A¼6 0 0 7 0 0 λ3 0 7 6 Λ 7 6 7 6 β 7 6 4 0 0 0 λ 0 0 4 7 6 Λ 7 6 7 6 7 6 β5 7 6 0 0 0 0 λ 0 5 7 6 Λ 7 6 5 4 β 6 0 0 0 0 0 λ6 Λ 2

(4.5a)

4.2 Numerical analysis

Note that ρ is a constant and most of the elements of the matrix are zero for a simulation of a step change in reactivity. Matrices with most elements equal to zero are called sparse matrices. Note also that all of the diagonal elements are negative if ρ < β, and all but one is positive if ρ > β. Matrices with all negative diagonal elements are more likely to indicate stability of the response than matrices with one or more positive diagonal elements. But the negativity of diagonal elements does not ensure stability. The lesson here is that a modeler should recheck the formulation if positive diagonal elements appear in a matrix representing a system that is thought to represent a stable system. The solution vector (state variables) and the initial condition vector are defined as follows: 2

3 P 6 P0 7 6 7 6 7 6 C1 7 6 7 6 7 6 C2 7 6 7 7 X¼6 6 C3 7 6 7 6 7 6 C4 7 6 7 6 7 6 C5 7 4 5 2

C6

1

3

7 6 7 6 7 6 7 6 6 β1 7 7 6 6 λ1 Λ 7 7 6 7 6 6 β2 7 7 6 6 λ2 Λ 7 7 6 6 β 7 7 6 3 Xð0Þ ¼ 6 7 6 λ3 Λ 7 7 6 6 β 7 6 4 7 7 6 6 λ4 Λ 7 7 6 6 β 7 6 5 7 6λ Λ7 6 5 7 7 6 4 β6 5

(4.5b)

λ6 Λ

Note that the vector, f, is null (or zero) in this case. The model is an initial value problem. It is noted that a matrix formulation of a perturbation model has zero initial conditions, but a non-zero forcing vector. Formulating a state variable perturbation model is left as an exercise for the reader. The solution for a linear system is a sum of exponentials. The exponential coefficients may be positive or negative, real or complex. (Complex coefficients indicate an oscillatory response). The coefficients are called eigenvalues and are the same as poles of the system transfer function. It is important to note that the above linear system is absolutely stable if all the eigenvalues (or poles of system transfer function) are negative or have negative real parts (for the case of complex eigenvalues). The point kinetics model for a constant step change in reactivity is a 7-th order linear dynamic system. Therefore, its solution is a sum of seven exponential terms. As a transient unfolds, the most positive (or least negative term) eventually dominates the response of interest, n(t). That is. nðtÞ  constant  est , for large t:

(4.6)

35

36

CHAPTER 4 Solutions of the point reactor kinetics equations

For a positive reactivity step, s is positive, and the neutron density increases indefinitely. The reciprocal of s is the time required for the response to increase by a factor of e. This quantity is called the reactor period, T, where T ¼ 1/s. For a negative reactivity step, all of the exponentials are negative and the neutron density decreases indefinitely. In this case the terms with large negative exponentials decrease faster with time than the slower exponentials, leaving a decrease determined largely by the slowest exponential term. The smallest value for s (in a light water reactor) is around 1/80 s1. So, the response to a negative reactivity step eventually settles out to an exponential with a negative period of around 80 s. But this result is not quite exact because delayed neutron precursors continue to be produced as the neutron density decreases. But the resulting delayed neutron production is small compared to production from precursors that existed before the initiation of the reactivity decrease. Available software packages usually solve state variable formulations of a dynamic model. These include MATLAB/Simulink, MAPLE, MATHEMATICA, Modelica, and others. Students would benefit by familiarity with one of these or a similar software system. Appendix G provides a brief description of MATLAB/ Simulink.

4.3 Maneuvers in a zero power reactor Simulations presented here are for a U-235 fueled reactor with a generation time of 105 s. This is called the reference model in subsequent sections of this chapter. Numerical solutions for the reference model and various perturbations are obtained and are shown in subsequent sections. A table of parameters related to delayed neutron precursors is given in Chapter 3. The most common perturbation used to illustrate the behavior of a zero-power reactor is the step change in reactivity. Fig. 4.1 shows the responses of the reference model to a positive step change of ρ ¼ 0.00067 (10¢), as the solution of Eqs. (3.22) and (3.23). This simulation illustrates the initial prompt jump, the sudden increase in reactor power that follows a step increase. This is the initial response to an immediate and constant change in reactivity. In order to demonstrate the effect of the magnitude of reactivity on response, responses at a selected time into the transient following a reactivity step are calculated. Fig. 4.2 shows the response at one second into the transient following various reactivity steps. The dramatic increase in the response as the reactivity approaches a value of ρ ¼ β, the delayed neutron fraction, is apparent. The rapid rise when reactivity exceeds the delayed neutron fraction occurs because the reactor no longer depends on delayed neutron contributions to grow. The prompt neutrons alone are sufficient to cause a positive rate of change. This condition is called prompt critical. However, it should be noted that delayed neutrons still influence the response. They are no longer essential, but they reduce the rate of change relative to the response if there were no delayed neutrons.

4.3 Maneuvers in a zero power reactor

4

3.5

P/P(0)

3

2.5

2

1.5

1

0

10

20

30

40

50 Time (s)

60

70

80

90

100

FIG. 4.1 Response of a zero-power reactor to a + 10 cent step reactivity perturbation.

P/P(0), 1 second into transient

150

125

100

75

50

25

0 0

1

2

4 3 Step reactivity magnitude

5

6

7 × 10−3

FIG. 4.2 Effect of step reactivity magnitude on reactor response at t ¼ 1 s, for different reactivity values.

The reactor period is an important parameter in the operation of a reactor. A large value of the reactor period indicates that reactor power is remaining at or near a constant value. Short period indicates that a reactor excursion is under way. Reactor period is a safety-related parameter and must be monitored so that it does not fall below a specified value.

37

CHAPTER 4 Solutions of the point reactor kinetics equations

As demonstrated in Fig. 4.2, there is a significant difference in the response when ρ < β and when ρ > β. As a result, it has become common to express reactivity as a fraction of the value when ρ ¼ β. This unit is called a dollar of reactivity. That is, one dollar of reactivity is numerically equal to the magnitude of total delayed neutron fraction, β, or 1dollar ($) ¼ β. In general, reactivity (dollars) ¼ ρ/β. Likewise, reactivity in cents is defined as reactivity (cents) ¼ 100 [ρ/β]. Clearly, it is necessary to limit reactivity inputs to a few cents. In future illustrations in this book, reactivity is usually expressed in cents. If there were no delayed neutrons, the kinetics model becomes.      d P ρ P ¼ dt Pð0Þ Λ Pð0Þ

(4.7)

The response in this case is given by.

ρt P ¼ exp Pð0Þ Λ

(4.8)

This equation indicates a much greater rate of increase than that seen with a model that appropriately includes delayed neutrons. Fig. 4.3 shows responses to negative reactivity steps. As for positive reactivity steps, there is a prompt jump. The subsequent decrease in power is strongly influenced by delayed neutrons. The precursors decay according to their individual decay constants. The decaying precursors come from two sources: those present before the reactivity decrease and those produced by the decreasing (but continuing) fissions. Actual reactivity changes in a reactor are usually gradual rather than sudden. Solutions for ramp changes illustrate typical behavior. Fig. 4.4 shows a response for a 0.1 cent/s reactivity ramp followed by a step reactivity decrease to zero. 1

-5 cents -10 cents -25 cents

0.9 0.8 0.7 P/P(0)

38

0.6 0.5 0.4 0.3 0.2 0

10

20

30 Time (s)

40

FIG. 4.3 Fractional power response to negative reactivity step perturbations.

50

60

4.4 Analytical solutions

1.35 1.3

P/P(0)

1.25 1.2 1.15 1.1 1.05 1

0

20

40

60 Time (s)

80

100

120

FIG. 4.4 Fractional power response to a ramp reactivity input, followed by a step change to zero reactivity.

4.4 Analytical solutions Analytical solutions for the six-delay neutron precursor group zero power model and for more detailed power reactor models are too tedious for practical implementation. However, an analytical solution to the point kinetics equation with one delay neutron group is easily performed for a step change in reactivity. This solution is useful in understanding the nature of reactor transients. Laplace transforms are a convenient tool for solving linear, constant coefficient differential equations. Appendix D provides a brief summary of Laplace transform theory. Appendix H shows the derivation of the analytical solution of the point reactor kinetics equations for a step change in reactivity. The parameters used in the simulation are as follows ρ ¼ +10 cents ¼ 0.1*0.0067 ¼ 0.00067 β ¼ 0.0067 Λ ¼ 0.00001 s. λ ¼ 0.08 s-1 The Laplace transform of the fractional power is given by   β λ+ +s PðsÞ Λ ¼ Pð0Þ Λs2 + ðβ  ρ0 + λΛÞ s  λρ0 Λ

(4.9)

The roots of the denominator polynomial are used to solve for P(t)/P(0). The two roots of this polynomial are given by.

39

40

CHAPTER 4 Solutions of the point reactor kinetics equations

s1 , s2 ¼

 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðβ  ρ + λΛÞ  ðβ  ρ + λΛÞ2 + 4λρΛ 2Λ

The general solution has the form PPðð0tÞÞ ¼ Aes1 t + Bes2 t . The solution for the example case is given by PðtÞ ¼ 1:111e0:0088t  0:111e603:09t Pð0Þ

(4.10)

Note that the solution for reactor power contains two exponential terms, one of the exponents is positive and the other is negative. The term with the negative exponent goes to zero rapidly, leaving the term with the positive exponent to define the continuing transient. The reactor period for this example is given by T ¼ 1/ 0.0088 ¼ 113.6 s. Now consider the response to a negative step change in reactivity. The following equation is the solution for the same reactor as above for a  10¢ reactivity step. PðtÞ ¼ 0:909e0:0073t + 0:0909e737:07t Pð0Þ

(4.11)

Note that for the positive reactivity step change, one of the exponents is positive and the other is negative; and for the negative reactivity step change, both the exponents are negative. As shown in Section 4.8, for a six-delayed neutron group model, there is one positive exponential and six negative exponentials for a positive step and seven negative exponentials for a negative step. The behavior described above can be deduced by examining the equation for neutron density. 6 X dn ðρ  βÞ λi Ci ¼ n+ dt Λ i¼1

(4.12)

Note that the first term on the right is negative for ρ < β and the second term is always positive. Thus, delayed neutrons are essential to maintain an increasing neutron density if ρ < β. If ρ > β, the rate of increase is positive even if there were no delayed neutron contribution.

4.5 Solutions for small perturbations The small perturbation form of the zero-power reactor equations is sometimes used. But the reader must remember that it is applicable only for small perturbations with respect to a nominal. Fig. 4.5 shows simulations performed with the full, sixdelay group model and the small perturbation model (also with six delay groups). The discrepancies in the results obtained with the small perturbation model are apparent.

4.6 Sinusoidal reactivity and frequency response

2.5

2

P/P(0)

1.5

1

0.5

0 0

Exact solution Perturbation solution

10

20

30 Time (s)

40

50

60

FIG. 4.5 Comparison of fractional power response using the exact model and the perturbation form of the model.

4.6 Sinusoidal reactivity and frequency response Recall that the frequency response of a linear system is based on the response to a sinusoidal reactivity input (see App. E for frequency domain analysis of linear systems). The perturbation form of the point reactor equations is used to calculate these responses. Fig. 4.6 shows the response to a 100 rad/s sinusoidal reactivity input. Note that there is an initial transient. (The first cycle is different than the subsequent cycles). Fig. 4.7 shows the response to a much lower frequency sinusoid (0.1 rad/s). In this case, there is an offset (different positive and negative swings), but an initial transient is too small to be apparent. These results show that the offset varies as the inverse of the frequency and that the initial transient decays very quickly (apparent only at high frequency). The response to a sinusoidal reactivity depends on the amplitude of the sinusoid. Figs. 4.8 and 4.9 show the responses to sinusoids with amplitudes of 1 cent and 50 cents. Clearly, the positive swings are much larger than the negative swings for large amplitude sinusoidal reactivity inputs. These figures show why the frequency response applies only for small perturbations. Experimental frequency response tests must use small amplitude reactivity inputs. Of course, feedbacks affect the response in reactors operating at significant power levels. Fig. 4.10A shows the frequency response magnitude for a zero-power reactor with several different values for the generation time. The higher break frequency increases as the generation time decreases. The plot of phase angle vs. frequency is shown in Fig. 4.10B.

41

CHAPTER 4 Solutions of the point reactor kinetics equations

1.008

1.006

1.004

P/P0

1.002

1

0.998

0.996

0.994 0

0.02

0.04

0.06

0.08

0.1 Time (s)

0.12

0.14

0.16

0.18

0.2

FIG. 4.6 Fractional power response to a sinusoidal reactivity perturbation of frequency 100 rad/s. 0.02

0.015

0.01 Δ P/P(0)

42

0.005

0 −0.005 −0.01 0

20

40

60

80

100 Time (s)

120

140

160

180

200

FIG. 4.7 Fractional power response to a sinusoidal reactivity perturbation of frequency 0.1 rad/s.

This figure shows several important features: 1. The amplitude of the power response is high at low frequencies (and approaches infinity as the frequency decreases). This simply means that inserting reactivity and leaving it inserted for a long time creates a large power response. This is consistent with behavior observed for a step change in reactivity. Note that this low frequency response is independent of generation time. 2. There is a mid-frequency plateau where power changes by 1% / cent of reactivity. The width of the plateau depends on the generation time, but the magnitude

4.6 Sinusoidal reactivity and frequency response

0.015

0.01

ΔP/P(0)

0.005

0 −0.005 −0.01 −0.015 0

5

10

15 Time (s)

20

25

30

FIG. 4.8 Fractional power response to a 1 rad/s sinusoidal reactivity perturbation of amplitude 1 cent.

3.5 3 2.5

ΔP/P(0)

2 1.5 1 0.5 0 −0.5

0

5

10

15 Time (s)

20

25

30

FIG. 4.9 Fractional power response to a 1 rad/s sinusoidal reactivity perturbation of amplitude 50 cent.

is independent of generation time. A smaller generation time results in a larger plateau width. 3. The high frequency response decreases at a rate of one decade per decade of frequency increase. The frequency of onset of this effect decreases with increasing generation time. This means that reactors with shorter generation times have greater high frequency response. The intersection of the plateau line and the high frequency asymptote (break frequency) is approximately equal to β/Λ rad/s.

43

CHAPTER 4 Solutions of the point reactor kinetics equations

100

1E-5 1E-6 1E-7

%power/cent

10

1

0.1

0.01

10−3

10−2

10−1

100

10−2

10−1

100

(A)

101 102 Frequency (rad/s)

103

104

105

103

104

105

0

−20 Phase (deg)

44

1E-5 1E-6 1E-7

−40 −60 −80 10−3

(B)

101

102

Frequency (rad/s)

FIG. 4.10 (A) Frequency response magnitudes for different generation times. (B) Frequency response phase angles for different generation times.

4. The phase shift is 90 degrees for low frequencies and for high frequencies. This means that the power response lags behind a sinusoidal reactivity perturbation by one fourth of a cycle. In the mid-range plateau, the phase shift approaches zero degrees, indicating that the power variations “keep up” with reactivity perturbations in this frequency range. See App. E for details of frequency response analysis of linear constant coefficient systems. It should be noted that eliminating the large amplitude at low frequencies is a design objective for power reactors. This is accomplished by design to achieve appropriate reactivity feedback from quantities such as moderator temperature, fuel temperature or fluid pressure, and design of control systems. These issues are addressed in later chapters.

4.6 Sinusoidal reactivity and frequency response

Comparisons of frequency response results for a model with six precursor groups and one with a single precursor group provide a means to identify an effective 1-group precursor decay constant that causes the best agreement between the sixgroup model and the one-group model. Fig. 4.11 shows such a comparison for an effective one-group decay constant of 0.08 s1. The magnitudes for the two models are in good agreement except for small differences for frequencies between 0.01 and 1 rad/s. The phase differences are more apparent, showing differences between 0.001 and 10 rad/s. The agreement for a one-delay group decay constant of 0.08 s1 is about as good as possible. 103

1 precursor group 6 precursor groups

%power/cent

102 101 100 10−1 10−2 10−3 10−4

10−2

(A)

100 Frequency (rad/s)

102

104

0 1 precursor group 6 precursor groups

Phase (deg)

−20

−40

−60

−80 10−4

(B)

10−2

100

102

104

Frequency (rad/s)

FIG. 4.11 (A) Comparison of frequency response magnitudes for one delayed neutron precursor group and six delayed neutron precursor groups. (B) Comparison of frequency response phase angles for one delayed neutron precursor group and six delayed neutron precursor groups.

45

CHAPTER 4 Solutions of the point reactor kinetics equations

Consequently, this value is recommended for one-group calculations for a U-235 fueled reactor. However, since the frequency response is based on a small perturbation model, good agreement is expected only for small perturbations.

4.7 Fluid fuel reactor response The equations for a fluid fuel reactor, described in Section 3.9, are used for simulations in this section. No reactivity feedback effects are considered in this simulation. Fluid fuel reactor responses have a unique dependence on the flow rate of the liquid fuel. The contribution of delayed neutrons depends on the residence time of the fuel in the reactor. The circulating fuel causes a change in the effective delayed neutron fraction by an amount equal to ρ0, which is the steady state reactivity in the system. The effective delayed neutron fraction is equal to (β – ρ0). Therefore, a dollar of reactivity is equal to (β – ρ0), rather than the value β used in solid fuel reactors. See Section 3.9 for neutronic equations of a fluid fuel reactor. The most extreme case of flow reduction in a fluid-fuel reactor is a flow stoppage in which case the in-core residence time goes to infinity. Fig. 4.12 shows the response of a fluid-fuel reactor following a flow stoppage. Fig. 4.13 shows the responses to a step increase in reactivity for two different loop transit times. It should be noted that in an operating fluid-fuel power reactor, reduced flow results in increased fluid temperature, and an increased feedback effect causing the power to level off. Clearly, the reduction of the delayed neutron contribution due to out-of-core precursor decays makes a fluid-fuel-reactor more responsive than an equivalently fueled stationary-fuel reactor. 9 8 7 6 P/P(0)

46

5 4 3 2 1 0 0

1

2

3

4

5 Time (s)

6

7

8

9

10

FIG. 4.12 Fractional power response of a U-235 fueled fluid fuel reactor to flow stoppage at t ¼ 2 s.

4.8 The inhour equation

5 5 s loop time 7.5 s loop time

P/P(0)

4

3

2

1

0

5

10

15

20

25 Time (s)

30

35

40

45

50

FIG. 4.13 Fractional power response of a fluid fueled reactor for two different loop transit times. Magnitude of step reactivity insertion ¼ 0.1(β – ρ0). Core resident time ¼ 2 s. Loop residence times are 5 and 7.5 s.

The transfer function for a zero-power fluid-fuel reactor is as follows: δn ¼ nð0Þδρ

1 6 X

λðiÞβðiÞ Λs + β  ρð0Þ  1 1 ðs + λðiÞÞτðLÞ i¼1 s + λðiÞ +  e τðcÞ τðcÞ

(4.13)

pffiffiffiffiffiffiffi The frequency response is obtained by substituting s ¼ jω (j ¼ 1) in the transfer function and performing the complex algebra. See App. E for a description of frequency response magnitude and phase plots. Recall the Euler formula, e-jωτ ¼ cos(ωτ) – j sin(ωτ). The frequency response plots (magnitude and phase angle) are shown in Fig. 4.14. Note the “bump” in the amplitude and the “dip” in the phase around frequencies of 0.1–1 rad/s. This feature in the zero-power frequency response occurs in the same frequency range where feedback effects would dominate in a power reactor.

4.8 The inhour equation The Inhour equation is a relationship between the magnitude of a positive step reactivity change and the exponential coefficients in the response. Of course, evaluation of the eigenvalues provides the same information. The development begins with Laplace transforming Eqs. (3.12) and (3.13) as follows: sn  n0 ¼

  X 6 ρβ λi Ci n+ Λ i¼1

(4.14)

47

Magnitude (% change in power to per cent change in reactivity)

CHAPTER 4 Solutions of the point reactor kinetics equations

5 s loop time 7.5 s loop time

101

100

10−1 −2 10

10−1

100 101 Frequency (rad/s)

(A)

102

103

0 5 s loop time 7.5 s loop time

−20 Phase (deg)

48

−40 −60 −80 −100 10−3

10−2

(B)

10−1

100 101 Frequency (rad/s)

102

103

104

FIG. 4.14 (A) Fluid fuel reactor frequency response magnitude for two different loop residence times of 5 s and 7.5 s. (B) Fluid fuel reactor frequency response phase angle for two different loop residence times of 5 s and 7.5 s.

sCi  Ci0 ¼

Solving for n(s) gives

βi n  λi Ci Λ

! 6 6 X ρβ X λi βi =Λ λi Ci0 n ¼ n0 + s  s + λi Λ s + λ i i¼1 i¼1

(4.15)

(4.16)

The seven roots of the bracketed term on the left are the “characteristic values”, si, of the equation. The solution has the form nðtÞ ¼

7 X

Ai esi t

i¼1

where, Ai ¼ coefficients derived from the initial conditions.

(4.17)

4.8 The inhour equation

The multiplier on the left-hand side of Eq. (4.16) is the characteristic polynomial in the variable s. Equating this to zero and rearranging gives the following equation: 6 X βi ρ¼s Λ+ s + λi i¼1

! (4.18)

The characteristic roots of Eq. (4.18) may be obtained graphically. That is, plot the right-hand side and the constant value of ρ on the same graph. The values of s at the intersections are the roots. This graph serves to illustrate the form of the solution as is the purpose of this exercise. Graphs of the right-hand side of Eq. (4.18) appear in many references [1]. These graphs are usually qualitative sketches since the roots span over three decades and include both positive and negative values. Here we illustrate the form of the solution of Eq. (4.18) quantitatively using two graphs. The first is a semi-logarithmic plot for the roots that are always negative. This solves the problem of covering three decades. The ordinate chosen for this plot is the period, the time T, required for the terms to decrease by a factor of e (Ti ¼ 1/si). The second is a plot that spans the positive and negative values around the origin. Fig. 4.15 illustrates the graphs for a U-235 fueled reactor with a generation time of 105 s. The roots are the values of the ordinate at which a plot of the right-hand side (a line representing a constant value of ρ) intersects the plots for the left-hand side. These graphs reveal the following: • • •

All roots are negative when ρ < 0 Six roots are negative and one is positive when ρ > 0 The root with the least negative value has a period of around 80 s.

The least negative root limits the rate of power decrease following a negative reactivity insertion. However, this does not mean that the power decrease will necessarily change with a simple exponential drop with a period of 80 s. The more negative roots will still contribute and their contribution will depend on continuing precursor production during the power decrease as well as contributions from precursors present at the start of the transient. There is a practical aspect to the behavior described above. The inhour equation is often cited as a relation for evaluating reactivity changes (for example in control rod calibrations). This assumption leads to a form of the inhour equation with s ¼ 1/T, where T is the reactor period observed after initial transients have died away. 6 X 1 βi ρ¼ Λ+ 1=T + λi T i¼1

! (4.19)

49

CHAPTER 4 Solutions of the point reactor kinetics equations

3.0E-2

Right Hand Side, r

2.0E-2

1.0E-2

0 −1.0E-2 −2.0E-2 −3.0E-2 10−1

100

(A)

101

102

Period, T 0.01

0.005 Right Hand Side

50

0

−0.005

(B)

−0.01 −0.05

−0.04

−0.03

−0.02

−0.01 0 0.01 Laplace parameter, s

0.02

0.03

0.04

0.05

FIG. 4.15 (A) Right hand side of the Inhour equation. The x-axis (period) uses logarithmic scale. (B) Right hand side of the Inhour equation in the neighborhood of zero.

An alternate, frequently used, measure of reactivity is the “INHOUR”. It is defined as the ratio of reactivity to the reactivity that would cause a stable period of one hour (T ¼ 3600 s). That is 1 6 X 1B βðiÞ C ÞA @Λ + 1 T i¼ + λðiÞ T 1 0 ρ¼ 6 X 1 B βðiÞ C A @Λ + 1 3600 i¼1 + λðiÞ 3600 0

(4.20)

Further reading

Exercises 4.1.

Show the state variable model for the perturbation form of the point kinetics equations (six delayed neutron groups) for a step change in reactivity and for a variable reactivity input.

4.2.

What would be the response at one second after a step change in reactivity of 10 cents if there were no delayed neutrons? Use data for a U-235 reactor with a generation time of 105 s.

4.3.

Delayed neutron precursors are a reservoir of future neutrons. How does this relate to the shutdown of a reactor?

4.4.

Estimate the magnitude of the response of a reactor to a sinusoidal reactivity with an amplitude of 10 cents and a frequency of 10 rad per second. Why is your answer not exact?

4.5.

Show that the transfer function of a circulating fuel reduces to the transfer derived in Chapter 3 when the in-core residence time approaches infinity.

4.6.

Why is a period meter useful for a control room display? Why does a reactor operating at power not fail to have a stable period?

4.7.

A U-235 fueled zero power reactor with a generation time of 105 experiences a stable period of ten minutes. What reactivity caused this transient?

References [1] J.J. Duderstadt, L.J. Hamilton, Nuclear Reactor Analysis, John Wiley & Sons, New York, 1976.

Further reading [2] H. Klee, Simulation of Dynamic Systems with MATLAB and Simulink, CRC Press, Boca Raton, FL, 2007. [3] MATLAB and Simulink User Guides, The MathWorks, Inc., Natick, MA.

51

CHAPTER

Subcritical operation

5

5.1 The neutron source Reactors require a source of neutrons during startup in order to cause flux levels that are readily measurable. Reactors have naturally-occurring neutron sources: spontaneous fissions in Uranium, neutrons from cosmic ray interactions with reactor constituents and photoneutrons from high energy gamma ray interactions with certain light isotopes present as reactor constituents. The gamma rays for photoneutron production are from decay of precursors nuclei that were left in an excited state due to gamma ray interaction during prior reactor operation. But artificial sources are used to ensure that the resulting neutron flux is large enough to cause the necessary measurable signals. Generally, these sources are a neutron emitter (for example, Californium-252 or a mixture of an alpha particle emitter and a material that undergoes (α-n) reactions, such as plutonium-beryllium or americium-beryllium).

5.2 Relation between neutron flux and reactivity in a subcritical reactor The modification to the point kinetics equations needed for modeling a subcritical reactor is simply to add a source term, S, to Eq. (3.12) or one of its other formulations. The neutron density (or a quantity proportional to the neutron density) after reactivity is increased, but remains subcritical, reaches a new steady state. The steady state is described by setting the time derivatives equal to zero in the modified equations (for example, Eqs. (3.12) and (3.13)). The steady-state value of neutron density is given by SΛ ρ

(5.1)

1 ρ ¼ n SΛ

(5.2)

n¼

or

Note that reactivity is negative because a subcritical reactor is under consideration here. Eq. (5.2) shows that the reciprocal of the steady-state neutron density (or a Dynamics and Control of Nuclear Reactors. https://doi.org/10.1016/B978-0-12-815261-4.00005-6 # 2019 Elsevier Inc. All rights reserved.

53

CHAPTER 5 Subcritical operation

proportionally-measured quantity) increases linearly with decreasing negative reactivity in a subcritical reactor. An example of a quantity that is measured is the neutron detector count rate, pulses/s.

5.3 The inverse multiplication factor During startup, nuclear instrumentation measures a signal that is proportional to neutron density. This measured quantity, M, increases as reactivity increases (becomes less negative) and would start to increase exponentially when that reactivity became positive. The reciprocal of M would decrease with increasing reactivity, approaching zero as criticality is approached. Operators observe the trend of 1/M to indicate the approach to criticality (typically by control rod withdrawal). The data are extrapolated to estimate the point at which 1/M equals zero (criticality). The extrapolation is repeated as new data become available. The operator would decrease the size of reactivity increments as 1/M approaches zero in order to avoid an overshoot. Note that the units for M do not matter, so long as they are consistent. Fig. 5.1 shows a 1/M plot for a hypothetical reactor. In operating reactors, the 1/M plot may be somewhat nonlinear because of inaccuracies in the point kinetics model, and the actual neutronic behavior in an operating reactor deviates from theoretical predictions. Therefore, it is necessary to extrapolate this quantity successively during approach to critical in order to predict the point at which criticality occurs during startup.

5.4 Responses during startup Startup of a reactor begins with the reactor at subcritical and a source present in the reactor. Operators begin startup by increasing reactivity slightly, usually by partial withdrawal of a control rod. Fig. 5.2 shows responses of a subcritical reactor to 80 70 60 1/M

54

50 40 30 20 10

0

2

4

8 10 6 Reactivity increase increment number

FIG. 5.1 Plot showing the approach to critical as a function of reactivity insertion.

12

14

5.5 Power ascension

15

Power

10

5

0 0.5

1

1.5

2

2.5

3 3.5 Time (min)

4

4.5

5

5.5

6

FIG. 5.2 Responses of a subcritical reactor to reactivity step increases of 10 cents per step, starting at a subcritical reactivity of 50 cents.

reactivity step increases. Note the larger increase in the power change and the longer time required to stabilize following a reactivity change as the reactor gets closer to critical. In this example, the initial subcritical reactivity has a value of 50 cents, with subsequent step reactivity insertions of 10 cents.

5.5 Power ascension After criticality is achieved, the rise to desired power begins. An increment of reactivity is added and the power ascent begins. See Chapter 4 for illustrations of power increases in a critical reactor. Operators monitor the reactor power and the reactor period (the time required for the power to increase by a factor of e) to ensure that the power is increasing at an acceptable rate. As the desired power level is approached the operators reduce the reactivity in order to reach criticality (reactivity ¼ 0) at the desired power level.

Exercises 5.1.

Confirm that Eq. (5.1) follows from the point kinetics equations (as modified by adding a source term, S).

5.2.

Consider the kinetics Eqs. (3.25) and (3.26) for one delayed neutron group. Insert a source term S(t) in Eq. (3.25) equal to 1.0/s. Calculate the fractional power for successive step reactivity insertions of 5, 10, 15, 20, and 25 cents.

55

56

CHAPTER 5 Subcritical operation

5.3.

a.

Plot the time history of the fractional power (as a function of time), making sure that the power comes to a near steady-state value before the next reactivity insertion.

b.

Plot the steady-state inverse power ratio P(0)/P, as a function of reactivity. Explain the behavior of this parameter as a function of reactivity.

Table 5.1 represents approximate numerical data from an approach to critical experiment at a training reactor. The table shows the ratio of neutron detector count rate and control rod position. Table 5.1 Experimental data showing neutron detector count rate and control rod position Normalized detector pulse rate (n0/ni)

Control rod position (mm)

1.0 0.98 0.87 0.47 0.20 0.12 0 (extrapolated)

0.0 100 200 350 425 460 ?

Use a linear fit to successive sets of three points, and extrapolate to determine the value of the control rod position to reach n0/ni ¼ 0. What is the extrapolated control rod position as the ratio n0/ni approaches zero? Make a plot of n0/ni vs. control rod position and comment on the behavior of the normalized neutron density with reactivity.

5.4.

Reactivity determines the nuclear power for subcritical reactors and for critical reactors, but in fundamentally different ways. Explain.

5.5.

We have seen that steady state conditions are achieved for subcritical reactors with any level of negative reactivity. In contrast, show (analytically) that the only steady state achievable in a critical reactor is for reactivity equal to zero.

Further reading [1] G. Gedeon, U.S. Department of Energy Fundamentals Handbook, Nuclear Physics and Reactor Theory, Module 4, Reactor Kinetics and Operation, 1993. [2] J. Rataj, et al., Reactor Physics Course at VR-1 Reactor, Czech Technical University, Prague, 2017.

CHAPTER

Fission product poisoning

6

6.1 The problem Fission reactions produce primary fission products directly (immediately when a fission reaction occurs) and through radioactive decay of primary fission products. Some of these isotopes have very large absorption cross sections. Their presence in a reactor causes a substantial reactivity decrease that must be canceled by adding reactivity (as by withdrawing a control rod or diluting a dissolved neutron poison in the reactor coolant). There are many fission products, but two are especially important because of their impact on reactor operation. These fission products are Xenon135 and Samarium-149. The effect of Xe-135 has three components: steady-state global poisoning, transient global poisoning, and spatial oscillations in reactor power.

6.2 Dynamics of xenon-135 Xenon-135 has a very large absorption cross section for thermal neutrons (σa
3.5  106 b).

6.2.1 Xe-135 production Xenon-135 is produced directly as a fission product and by decay of another fission product, Iodine-135. Xenon-135 has a fission yield (atoms produced per fission) of 0.003. That is, a Xe-135 atom is produced in 0.3% of fissions. I-135 is the result of decay of another fission product, Tellurium-135. But Te-135 decays very rapidly, so I-135 appears at essentially the same time as the Te-135 atom. I-135 has a fission yield of 0.063. I-135 decays into Xe-135 with a half-life of 6.7 h (decay constant of 2.87  105 s1). The I-135 absorption cross section is small, resulting in negligible absorption losses compared to decay losses.

6.2.2 Xe-135 losses Xe-135 disappears as a result of radioactive decay and by neutron absorption. The half- life of Xe-135 is 9.2 h (decay constant of 2.09  105 s1). Neutron absorptions in Xe-135 cause disappearance of Xe-135 atoms at a rate of X σaX Φ, where X is the Dynamics and Control of Nuclear Reactors. https://doi.org/10.1016/B978-0-12-815261-4.00006-8 # 2019 Elsevier Inc. All rights reserved.

57

58

CHAPTER 6 Fission product poisoning

Xe-135 concentration, σaX is the Xe-135 absorption cross section and Φ is the neutron flux. The absorptions in Xe-135 consume neutrons that could otherwise be available for fissions in the fuel, thereby reducing reactivity.

6.2.3 Equations for Xe-135 behavior The differential equations for I-135 and Xe-135 are as follows: X dI Φ  λI I ¼ γI f dt

(6.1)

dX ¼ γ X Σ f Φ + λI I  Xσ aX Φ  λX X dt

(6.2)

where I ¼ I-135 concentration (number of atoms/cm3) γ I ¼ I-135 fission yield (0.063 I-135 atoms per fission) Φ ¼ neutron flux (number of neutrons/(cm2-sec)) λI ¼ I-135 decay constant (2.87  105 s1) X ¼ Xe-135 concentration (number of atoms/cm3) γ X ¼ Xe-135 fission yield (0.003 Xe-135 atoms per fission) λX ¼ Xe-135 decay constant (2.09  105 s1) σ aX ¼ Xe-135 absorption cross section (3.5  106 b at 0.0253 eV) To solve these equations, it is necessary to know the cross sections and the neutron flux or the reactor specific power (kW/kg of fuel), which uniquely defines the neutron flux. It should be noted that reported cross sections are for mono-energetic neutrons at an energy of 0.0253 eV or a speed of 2200 m/s. This energy corresponds to a temperature of 20 °C or 293 K. In a reactor, the absorption and fission cross sections must be corrected for the actual temperature of the moderator. In a thermal reactor it is usually assumed that the Maxwell distribution applies for neutrons in equilibrium with moderator atoms and that the cross section varies as the inverse of the neutron velocity. As shown in elementary reactor physics books [1], the “effective” cross section for a material at temperature, T (Kelvin), is 1 σ ðTÞ ¼ 1:128

rffiffiffiffiffiffiffiffi! 293 σ ð0:0253eVÞ T

For example, the fission cross section for U-235 is 649 b for neutrons at 2200 m/s, but the effective neutron fission cross section is 411 b at 300 °C (a typical moderator temperature in a pressurized water reactor). It is possible to reformulate the equations for Xe-135 transients with various definitions for the quantities in the equations. One such reformulation is as follows. dI=Nf ¼ γ I σ f Φ  λI I=Nf dt

(6.3)

6.2 Dynamics of xenon-135

dX=Nf ¼ γ X σ f Φ + λI I=Nf  X=Nf σ aX Φ  λX X=Nf dt

(6.4)

dI 0 ¼ γ I σ f Φ  λI I0 dt

(6.5)

dX0 ¼ γ X σ f Φ + λI I 0  X0 σ aX Φ  λX X0 dt

(6.6)

or

where I/Nf ¼ I0 (the number of I-135 atoms per fissile atom in the reactor) X/Nf ¼ X0 (the number of Xe-135 atoms per fissile atom in the reactor)

6.2.4 Steady state Xe-135 The steady state quantities of I-135 and Xe-135 are obtained by setting the derivative terms in the equations equal to zero. The result is γI σf Φ λΙ

(6.7)

ðγ X + γ I Þσ f Φ ðλX + σ aX ΦÞ

(6.8)

0 Iss ¼

0 Xss ¼

Note that I0 ss increases in proportion to the neutron flux, while X0 ss increases in proportion to the neutron flux at low flux levels (when λX » σaX Φ is small) and reaches saturation at high flux when λX « σaX Φ is large. Since I-135 becomes Xe-135 upon radioactive decay, its steady state value is a reservoir of future Xe-135. Therefore, the ratio of steady state I-135 to steady state Xe-135 serves as an indicator of future Xe-135. The ratio is 0 Iss γ ðλX + σ aX ΦÞ ¼ I 0 Xss λ I ðγ X + γ I Þ

(6.9)

Inserting values for fission yields and decay constants gives (using a moderator temperature of 300 °C in evaluating the Xe-135 absorption cross section resulting in 2.22  106 b and using fission yields and decay constants for fission in U-235) 0 Iss ¼ 0:695 + 0:738  1013 Φ 0 Xss

(6.10)

Fig. 6.1 shows the ratio of steady-state I-135 concentration to the steady-state concentration of Xe-135. Note that X0 ss is slightly greater than I0 ss for small flux levels, but I0 ss exceeds Xe135 at higher flux levels.

59

CHAPTER 6 Fission product poisoning

10 Ratio of lodine to Xenon

60

8 6 4 2 0 1012

1013 Neutron flux (#/ cm

1014 –2

s)

FIG. 6.1 Ratio of steady-state I-135 to Xe-135.

6.2.5 Xe-135 poisoning Fission products increase non-fission absorptions. The six-factor formula of reactor physics permits evaluation of the reactivity loss due to increased non-fission absorptions. The six-factor formula is: keff ¼ η ε p f Lf Ls

(6.11)

where η ¼ neutrons produced per absorption in fuel ε ¼ fast fission factor p ¼ resonance escape probability f ¼ thermal utilization factor Lf ¼ fast neutron non-leakage probability Ls ¼ thermal neutron non-leakage probability The thermal utilization (f), is the term affected by an increase in fission product quantities as follows: f¼

Σaf Σaf + ΣaO

(6.12)

where Σaf ¼ macroscopic cross section for absorptions in fissile material ΣaO ¼ macroscopic cross section for absorptions in all non-fissile material The presence of fission products alters the thermal utilization as follows: f0 ¼

Σaf Σaf + ΣaO + Σap

where Σap ¼ macroscopic cross section for absorptions in fission products.

(6.13)

6.2 Dynamics of xenon-135

Therefore keff  1 f 0  f ¼ 0 keff f

Define the following:

(6.14)

X z ¼ XaO af

X P¼X

ap af

Therefore ρ¼

keff  1 P ¼ keff 1+z

(6.15)

where ρ ¼ reactivity decrease due to fission product absorptions. Since z « 1 in a reactor, a rough approximation (good enough to illustrate the approximate magnitude of fission product poisons on reactivity) is. ρ ¼ P ¼ 

Σap Σaf

(6.16)

Note that the reactivity loss is given by ρ ¼ X0

σ aX σ af

(6.17)

and X0 (recall X0 ¼ X/Nf) is the solution variable in Eq. (6.6). The ratio of X-135 absorption cross section to U-235 absorption cross section is (3.5  106)/650 or 5380. Therefore ρ ¼ 5380 ðX0 Þ

(6.18)

At steady state, the Xenon value is given by 0 Xss ¼ 5380 

ðγ X + γ I Þσ f Φ ðλX + σ aX ΦÞ

(6.19)

Using the cross sections for a moderator temperature of 300 °C (σf ¼ 348 barns σaX ¼ 2.22  106 barns) gives 0 Xss ¼ 5380  

ð0:066Þ  348  1024 Φ  2:09  105 + 2:22  1018 Φ

(6.20)

Fig. 6.2 shows the Xe-135 steady state poisoning as a function of neutron flux. Note that the maximum steady state reactivity loss is around eight dollars at high flux levels for this example.

61

CHAPTER 6 Fission product poisoning

Reactivity loss due to Xe-135 ($)

62

10 8 6 4 2 0 1012

1013

1014 –2

Neutron flux (#/ cm

1015

s)

FIG. 6.2 Xenon steady-state poisoning reactivity loss as a function of neutron flux.

6.2.6 Behavior of Xe-135 after Startup Eqs. (6.5) and (6.6) apply for reactor startup (with initial conditions, I0 ¼ 0 and X0 ¼ 0). For a hypothetical step change to full power (in actual operation, power would be increased gradually), the equations can be solved readily by analytical or numerical methods. A reference reactor is defined here. It is also the basis for illustrations appearing later in this chapter. The pertinent characteristics of the reference reactor are as follows: Reactor power ¼ 3000 MW (3  109 W) Fuel loading ¼ 100 metric tons (108 g of enriched fuel) Fuel enrichment ¼ 3% U-235 Loading of U-235 ¼ 3  106 g Moderator temperature ¼ 300 °C Effective microscopic fission cross section for U-235 at 300 °C ¼ 348  1024 cm2. Neutron flux at full power ¼ 3.47  1013 #neutron/cm2-sec (See Appendix I for calculation of the flux.) The number of fissile atoms in the reactor is given by (6.023  1023) x 3  106 ¼ 1.8069  1029 The effective absorption cross section for Xe-135 at 300 °C is given by   rffiffiffiffiffiffiffiffi 3:5x1018 293 ¼ 2:22  1018 cm2 σ aX ¼ 1:128 573

Therefore, the terms, σf Φ and σaX Φ, in the equations become

    σf Φ ¼ 348  1024  3:47  1013 ¼ 1:208  108

and

    σaX Φ ¼ 2:22  1018  3:47  1013 ¼ 7:70  105

6.2 Dynamics of xenon-135

Reactivity loss ($)

8 6 4 25% step change in reactor power 50% step change in reactor power 100% step change in reactor power

2 0

0

10

20

30

40

50

60

70

80

90

100

Time (h)

FIG. 6.3 Xe-135 poisoning transients for step changes in reactor power from zero power.

Fig. 6.3 shows Xe-135 reference reactor poisoning transients for several step changes in reactor power from zero power.

6.2.7 Xe-135 after Shutdown Consider a reactor in which I-135 and Xe-135 have reached their steady state levels as given by Eqs. (6.7) and (6.8), respectively. Now assume that the neutron flux suddenly goes to zero. The transient responses of I-135 and Xe-135 are given by 0 I 0 ðtÞ ¼ Iss exp ðλI tÞ

0 0 exp ðλX tÞ + Iss  X0 ðtÞ ¼ Xss

 λI  ð exp ðλX tÞ  exp ðλI tÞÞ λI  λX

(6.21)



(6.22)

Fig. 6.4 shows Xe-135 poisoning after shutdown of the reference reactor for four different initial power levels. The data for these simulations are the same as for the reactor described in Section 6.2.6. The results show that the level of poisoning increases with initial reactor power and is large. The ability to override this poisoning depends on the amount of reactivity that can be added by other means (usually control rods or diluting dissolved neutron poisons). For large initial flux levels, the poisoning reaches levels that exceed the ability to override with available operator actions. Therefore, restart is impossible until the Xe-135 falls low enough to permit restart. Fig. 6.5 shows a hypothetical case. The reference reactor shuts down from full power and operator actions can add no more than 9 dollars of reactivity. The figure shows the time interval during which restart is impossible, and is from around 1.5 to 21 h. Fig. 6.6 shows I-135 as well as Xe-135 for a step change in reference reactor power from zero to 100%, then a step decrease back to zero power. Note that at steady state I-135 level at 100% power is larger than the Xe-135 level. As the I135 decays after shutdown, it causes the peak in Xe-135 level.

63

CHAPTER 6 Fission product poisoning

14 100% power initally

Reactivity loss ($)

12 75%

10

Specific power = 30 kW/kg at 100% or Neutron flux = 3.47e13

8

50%

6 25%

4 2 0

0

10

20

30

40 50 60 Time after shutdown (h)

70

80

90

100

90

100

FIG. 6.4 Xe-135 poisoning of the reference reactor for four different initial power levels. 14 12 Reactivity loss ($)

64

10

Maximum available reactivity override

8 6 4 2 0 0

10

20

30

50 60 40 Time after shutdown (h)

70

80

FIG. 6.5 Reactivity loss due to Xe-135 poisoning after shutdown from full power, with a maximum override reactivity of 9 dollars.

6.2.8 Xe-135 poisoning after a power increase After an increase to a higher power level, the increased flux initially causes a reduction in poisoning through removal of Xe-135 by neutron absorption. This effect occurs because the higher flux decreases the Xe-135 concentration faster than the production from decay of I-135. Subsequently, the poisoning grows to its new equilibrium level as I-135 concentration increases to its new, higher level. Fig. 6.7 shows

6.2 Dynamics of xenon-135

25 Reactor shutdown at 100 hr Iodine-135

Reactivity loss ($)

20

I-135 reactivity to be lost upon decay to Xe-135

15

10

Xe-135 reactivity loss Xenon-135

5

0 0

20

40

60

80

100 Time (h)

120

140

160

180

200

FIG. 6.6 Changes in I-135 and Xe-135 for a step change in reactor power from zero to 100% and then back to zero. 7 Higher power level, more xenon

6

Reactivity loss ($)

Lower power level, less xenon

5

4

3

2

1

0

10

20

30

40

50 Time (h)

60

70

80

90

100

FIG. 6.7 Xe-135 poisoning as power increases by a step from 10% level.

reference reactor transients for increases from 10% power to two different, higher power levels. Note that the initial response to a power increase is a decrease in Xenon poisoning and a corresponding increase in reactivity. The increase in the burnout of Xe-135 is faster than production of new Xe-135 following buildup of I-135. Fig. 6.8 shows the initial response to power increases. The initial response to a power change is a

65

CHAPTER 6 Fission product poisoning

2.5 2.4 2.3 Reactivity loss ($)

66

Lower power level, less initial xenon burnup

2.2 2.1 2 1.9 Higher power level, more initial xenon burnup

1.8 0

0.5

1

1.5

2

2.5 Time (h)

3

3.5

4

4.5

5

FIG. 6.8 Initial Xenon reactivity change following step change in power of 10%.

component of reactivity feedback in a power reactor. See Chapter 7 for a discussion of feedback effects.

6.2.9 Xe-135 poisoning after power maneuvers Of course, maneuvers occur that involve various changes in reactor power, various time intervals between changes and various rates of changes. Two maneuvers were simulated to illustrate Xe-135 poisoning during maneuvers in the reference reactor. First consider the case of a ramp increase in power from zero to full power, then a step decrease back to zero power. Fig. 6.9 shows such a transient for the reference reactor. Fig. 6.10 shows the variations in Xe-135 and I-135 for a hypothetical daily pattern of power changes. The reference reactor operates at full power for twelve hours then 50% for twelve hours. This pattern repeats day after day. This type of behavior might be expected in which a reactor provides more power during the day when demand is high and less power at night when demand is low. The Xe-135 poisoning might appear counterintuitive. Explaining the behavior is left to the reader (see the exercises.).

6.2.10 Coupled neutronic-xenon transients Previous sections describe the response of Xe-135 to prescribed changes in reactor power. Of course, a change in Xe-135 concentration causes a change in reactivity and the reactivity change causes a change in reactor power. The neutronics and the Xe135 poisoning affect each other. They are coupled. Fig. 6.11 shows reactivity (1 cent step + Xe-135 poisoning).

6.2 Dynamics of xenon-135

20

Reactivity loss ($)

I-135

15

10 Xe-135

5

0 0

20

40

60 Time (h)

80

100

120

FIG. 6.9 Xe-135 and I-135 transients during power ramp from zero to full power for 50 h and then back to zero power. 25

Reactivity loss ($)

20 I-135

15

10 Xe-135

5

0 0

10

20

30 Time (h)

40

50

60

FIG. 6.10 Xe-135 and I-135 reactivity changes for daily power level variations between 100% and 50%.

Note the large reactivity loss due to the increasing Xe-135 concentration due to decay of I-135. The poisoning reaches its greatest negative effect at around 8.5 h after the start of the transient when the poisoning reaches around 6 dollars. The poisoning then decreases as Xe-135 burns out until criticality is re-established at around 26.5 h. After 26.5 h the reactivity continues to rise due to continuing burnout of Xe-135, causing a rapid increase in reactor power.

67

CHAPTER 6 Fission product poisoning

1 0 –1 Reactivity ($)

68

–2 –3 –4 –5 –6 0

5

10

15 Time (h)

20

25

30

FIG. 6.11 Xe-135 poisoning after 1 cent reactivity step.

6.2.11 Xenon-induced spatial oscillations Xe-135 can cause spatial oscillations in reactors. Consider a case in which reactivity is simultaneously increased in one region and decreased in the other. The following scenario would ensue. Increased Power Region

Decreased Power Region

P

"

P

#

I Xe ρ P Xe ρ P

" (increase future Xe) # (initial burnout) " (reduced Xe poisoning) " (caused by initial Xe burnout) " (from increased decay of I) # #

I Xe ρ P Xe ρ P

# " # # # " "

Note that power oscillates and the regional powers are 180 degrees out of phase. The oscillations continue because each region now finds itself where the other region began. In a power reactor, negative reactivity feedback from temperature changes and control action would dampen or eliminate the oscillations. Operating procedures serve to suppress any Xenon oscillations in power reactors.

6.3 Samarium-149 poisoning

6.2.12 Xenon in molten salt reactors Fission product gases can be removed continuously in molten salt reactors by simple sparging. Consequently, these reactors do not have a major issue with Xe-135.

6.3 Samarium-149 poisoning Samarium-149, also has a very large absorption cross section for thermal neutrons (σa
4.2  104 b). Sm-149 results from decay of Promethium-149. Pm-149 is a daughter product of Neodymium-149. Nd-149 has a fission yield of 0.014 and it decays into Pm-149 with a half-life of 1.7 h. Since the half-life of Pm-149 is much longer (47 h), it is acceptable to treat Pm-149 as the direct fission product. The equations for Pm-149 and Sm-149 variations are (modified from [2]) dP0 ¼ γ P σ f Φ  λP P0 dt

(6.23)

dS0 ¼ λP P0  S0 σ aS Φ dt

(6.24)

where P0 ¼ P/Nf S0 ¼ S/Nf P ¼ Promethium-149 concentration S ¼ Samarium-149 concentration Nf ¼ concentration of fissile material γ P ¼ fission yield of Promethium-149 (0.014) λP ¼ decay constant of Promethium-149 (4.1  106 s1) σ aS ¼ Samarium-149 absorption cross section (5.3  104 b at 2200 m/s and 3.36  104 b effective for moderator at 300 °C) As in the treatment of Xe-135, the steady state poisoning due to Sm-149 is given by Poisoning ¼ S0 (σ aS/σ af) The steady state value of S0 is given by S0 ss ¼ 0.014 (σ f/σ aS) Therefore Samarium poisoning, S0 ss ¼ 0.014 (σ f/σ af) ¼ 0.012 Poisoning of 0.012 reactivity units is equal to 1.8 dollars. Note that the steady state Sm-149 poisoning is independent of the flux level. Sm-149 buildup after startup is very slow because of the long half-life of Pm-149. However, once Sm-149 reaches equilibrium, it remains unchanged thereafter. Fig. 6.12 shows poisoning due to Sm-149 after startup of the reference reactor.

69

CHAPTER 6 Fission product poisoning

2

Reactivity loss ($)

1.5

1

0.5

0 0

200

400

600 Time (h)

800

1000

1200

FIG. 6.12 Sm-149 poisoning after reactor startup. 2

1.95 Reactivity loss ($)

70

1.9

1.85

1.8 0

10

20

30

40

50 60 Time (days)

70

80

90

100

FIG. 6.13 Sm-149 poisoning after a power step decrease from 100% to 50%.

Fig. 6.13 shows Sm-149 poisoning in the reference reactor after a power decrease. The decreased flux initially causes the absorptions in Sm-149 to increase, then a gradual reduction in Pm-149 concentration and decay causes the Sm-149 poisoning to return to its initial value. The notable feature of Sm-149 transients is the long-time scales involved. Fig. 6.14 shows Samarium-149 after shutdown.

References

2.5

Reactivity loss ($)

2

1.5

1

0.5

0 0

500

1000

1500 Time (h)

2000

2500

3000

FIG. 6.14 Rise in Sm-149 poisoning after power step increase to 100%, followed by reactor shutdown.

6.4 Summary This chapter has shown the behavior and consequences of Xe-135 and Sm-149 production. Results for an actual reactor would depend on the unique characteristics of that reactor, but the general behavior would be similar.

Exercises 6.1.

Why were operators of early reactors with high neutron flux surprised to find it necessary to remove control rods continually in order to maintain criticality?

6.2.

Why does the Xenon poisoning dip briefly following a step increase in reactor power?

6.3.

Some people will find the Xe-135 behavior shown in Fig. 6.9 to be counterintuitive. Why might they think that? Give a physical explanation of the behavior depicted.

References [1] A.R. Foster, R.L. Wright Jr., Basic Nuclear Engineering, fourth ed., Allyn and Bacon, Boston, 1983. [2] J.J. Duderstadt, L.J. Hamilton, Nuclear Reactor Analysis, John Wiley & Sons, New York, 1983.

71

CHAPTER

Reactivity feedbacks

7

7.1 Basics Changes in power reactor temperature or pressure cause reactivity changes. The causes for these reactivity changes are the temperature dependence of nuclear properties, changes in the quantity of material present in the reactor core because of density changes or changes in the dimensions or shape of core components. Evaluating the feedback reactivity involves the use of feedback coefficients (change in reactivity per unit change in a process variable). Feedback reactivity effects are very important in achieving suitable reactor performance. This chapter addresses a qualitative description of relevant effects. Quantitative evaluation of feedback coefficients requires use of detailed neutronic models. These methods are described in books on reactor physics [1] and are beyond the scope and purpose of this book. Here, it will be assumed that analysts will be provided with reactivity coefficients needed to perform a dynamic simulation. The following sections address reactivity feedbacks in general for thermal reactors, specifically for thermal reactors that are so-called Generation II and Generation III reactors.

7.2 Fuel temperature feedback in thermal reactors Fuel temperature affects reactivity through changes in Doppler broadening of absorption resonances. Some fission neutrons are lost during slowing down as a result of absorption in heavy isotopes (principally U-238 and Pu-240 in thermal reactors with low-enrichment uranium fuel). Resonances in their absorption cross sections cause a decrease in the number of neutrons reaching thermal energy (the resonance escape probability decreases). Changes in temperature change the relative motion between the resonance absorber and a neutron. This causes a reduction in the peak of the absorption cross section and a broadening over a wider range of energies. See Fig. 7.1. Since the cross section is still very large in the broadened region, the heavy isotope absorbs more neutrons than would be absorbed in an unbroadened resonance. This increased resonance absorption causes reactivity to decrease. The effect occurs in U-238 and is even stronger in Pu-240 as it builds up by absorptions in Dynamics and Control of Nuclear Reactors. https://doi.org/10.1016/B978-0-12-815261-4.00007-X # 2019 Elsevier Inc. All rights reserved.

73

CHAPTER 7 Reactivity feedbacks

UNBROADENED ABSORPTION CROSS SECTION sa

74

DOPPLER BROADENED

E0 ENERGY

FIG. 7.1 Doppler broadening at a resonance cross section due to increasing temperature, such as in U-238 and Pu-240 isotopes.

U-238 and Pu-239. The fuel temperature coefficient of reactivity is always negative in a thermal reactor. Because fuel temperature changes before changes in any other process variable following a power change, fuel temperature reactivity feedback occurs quickly after a power change. It is sometimes called the prompt reactivity feedback. This negative reactivity helps to limit power changes. The fuel temperature feedback following a power change depends on the magnitude of the fuel temperature change as well as the magnitude of the feedback coefficient. Fuel temperature changes per unit change in power depend on the heat capacity of the fuel and the heat transfer resistance between the fuel and the coolant. See Chapter 10.

7.3 Moderator temperature feedback in thermal reactors The neutron moderator in a reactor may be a liquid as in a moderator/coolant in light water reactors, as liquid separate from the coolant as in CANDU reactors, as a solid as in gas-cooled thermal reactors, fluid-fuel thermal reactors and Russian RBMK reactors, or absent as in fast reactors. Moderator temperature reactivity feedback is always slower to respond to a power change than fuel temperature feedback. But moderator/coolant temperature feedback changes first for a reactor inlet temperature change in reactors with liquid that serves both as the coolant and the

7.3 Moderator temperature feedback in thermal reactors

Fractional neutron density

4

× 10–4 293 K 493 K 593 K

3

2

1

0 0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

v, Neutron velocity (m/s)

FIG. 7.2 Maxwell-Boltzmann distribution of thermal neutrons for three temperatures.

moderator. Such a change follows a change in heat removal in the secondary system. The moderator temperature coefficient may be positive or negative and its value can change significantly with reactor operation. An increase in the moderator temperature of a thermal reactor causes the thermal energy spectrum of thermalized neutrons to shift to a higher energy. Thermal neutrons are in equilibrium with the thermal energy of the moderator. Higher moderator temperature means greater thermal motion of moderator atoms and a consequent higher energy of neutrons that interact with the moderator atoms. The energy spectrum of moderator atoms, and consequently the energy spectrum of thermalized neutrons is given by the Maxwell-Boltzmann distribution. Fig. 7.2 shows distributions for three different temperatures. Many reactor constituents have absorption and fission cross sections that vary as the reciprocal of neutron velocity. This is called a “1/v” dependence of the cross section, σ. The reaction rate is given by R¼Nσφ

(7.1)

R¼Nσnv

(7.2)

or where R ¼ reaction rate (number of interactions/(cm3 s) N ¼ absorber number density (number of nuclei/cm3) σ ¼ microscopic cross section (cm2) n ¼ neutron density (number of neutrons/cm3) v ¼ neutron velocity (cm/s)

75

76

CHAPTER 7 Reactivity feedbacks

FIG. 7.3 Fission cross sections of fissile isotopes, U-235 and Pu-239 (ENDF, open access, evaluation number ENDF/B-VIII.0).

For 1/v absorbers, the v in the denominator (σ  constant/v) cancels the v in the numerator, causing the reaction rate to be independent of moderator temperature. However, some important reactor constituents are not 1/v absorbers. Fig. 7.3 shows cross sections for the fissile isotopes, U-235 and Pu-239. The U-235 cross section shows a slight negative departure from 1/v behavior. This means that the spectrum effect causes U-235 absorptions and fissions to decrease when moderator temperature increases (a negative component of moderator temperature coefficient of reactivity). This effect is small compared to other effects caused by moderator temperature changes. The Pu-239 effect is a different story. Pu-239 has a low energy (around 0.3 eV) capture and fission resonance. This causes a positive departure from 1/v behavior. This means that the spectrum effect causes Pu-239 captures and fissions to increase when moderator temperature increases (a positive component of moderator temperature coefficient of reactivity). This effect is significant in reactors with significant Pu-239 inventories (for example in thermal reactors that are fueled with lowenrichment uranium and consequent Pu-239 production by neutron captures in U-238). Coolant (or moderator/coolant) temperature changes cause density changes that cause changes in neutron absorptions in the coolant. Since increased coolant temperature always causes fluid expansion and consequent removal of a neutron absorber from the core, the neutron absorption component of the coolant temperature coefficient is always positive. The change in moderator density also changes neutron scattering which, in turn, changes neutron mean-free paths. Increase in mean free paths enables more neutrons

7.3 Moderator temperature feedback in thermal reactors

to travel to core boundaries and leak from the core and the opposite occurs with decreases in mean free path. The effect of a temperature increase is to decrease moderator density, increase neutron mean-free path, increase neutron leakage and reduce reactivity. The scattering component of the moderator temperature coefficient of reactivity is always negative. Temperature increases in coolant/moderator cause a density increase and, consequently, fewer coolant/moderator atoms in the reactor core. Since coolant/ moderators have non-zero absorption cross sections, temperature increase decrease neutron absorptions. The absorption component of the coolant/moderator coefficient of reactivity is always positive. Reactor design includes evaluation of the effect of moderator population on available reactivity. Fig. 7.4 illustrates the effect. Too little moderator is sub-optimal in producing available reactivity by slowing neutrons to thermal energy. A reactor with too little moderator is said to be undermoderated. Too much moderator is sub-optimal in producing available reactivity because neutron absorptions override the effect of additional slowing down in the moderator. A reactor with too much moderator is said to be over-moderated. Light water reactors are designed to be under-moderated. That is, moderator removal causes a reactivity decrease. Thus, a temperature increase in a liquid moderator-coolant causes a decrease in density, a decrease in neutron slowing down

under-moderated

over-moderated

REACTIVITY

keff

Resonance escape probability (p)

Thermal utilization factor (f)

MODERATOR-TO-FUEL RATIO (Nm/NU) AS TEMPERATURE INCREASES

FIG. 7.4 Effect of moderator-to-fuel ratio on reactivity in under-moderated and over-moderated reactors. Adapted from www.nuclear-power.net.

77

78

CHAPTER 7 Reactivity feedbacks

and a reactivity decrease. The moderator temperature coefficient of reactivity is always negative in under-moderated reactors with liquid moderator-coolant. However, poison (typically boric acid) is sometimes dissolved in a coolant/moderator. Thermal expansion causes removal of the dissolved poison from the core along with the coolant. This poison removal is a positive reactivity effect. Therefore, the coolant/ moderator temperature coefficient may be positive or negative, depending on the concentration of dissolved poison. This can happen, especially at the beginning of cycle when the dissolved boron concentration is the highest, and an increase in the moderator temperature decreases the boron nuclei per unit volume, thus have a positive effect on reactivity. CANDU reactors have pressurized coolant channels embedded in a large tank (calandria) filled with heavy water. Heavy water also serves as a coolant for the fuel in the channels. Both heavy water regions contribute to neutron thermalization. CANDU reactors are over-moderated. Therefore, the moderator temperature coefficient of reactivity is positive.

7.4 Pressure and void coefficients in thermal reactors Thermal reactors with liquid coolant experience reactivity changes when voids in the coolant undergo concentration changes (even liquid-cooled reactors can have small bubbles in the coolant). The quantity of coolant in the core decreases as liquid boils, thereby reducing absorptions in the coolant. This is a positive reactivity feedback. Also, the slowing down of neutrons to thermal energy decreases because of a reduced moderator concentration. This is a negative reactivity feedback. In under-moderated reactors the net effect is a negative void coefficient. There are two types of existing power reactors that are over-moderated and have positive void coefficients, The Soviet RBMK (see Chapter 11) and the Canadian CANDU reactor (see Chapter 14). Both types have moderator (graphite in the RBMK and heavy water in the CANDU) that is physically separated from the coolant channels. The coolant is boiling light water in the RBMK and liquid heavy water in the CANDU. In reactors with in-core boiling (BWRs) voids decrease when pressure increases. Therefore, they have positive pressure coefficients of reactivity. Active, full-blown boiling in liquid coolants is an abnormal occurrence. Voiding in this situation is a factor in determining the response to the abnormal occurrence.

7.5 Fission product feedback A change in reactor power causes a change in the fission product inventory. The new equilibrium inventory after a power increase causes increased fission product production, increased neutron absorptions and reduced reactivity, but effects on reactivity vary during the transition to equilibrium.

7.6 Combined reactivity feedback

Some fission products appear immediately upon a fission reaction (primary fission products) and some appear after decay of primary fission products (secondary fission products). The short-term effect of a power change is a prompt change in primary fission product production, a delayed change in secondary fission product production, and a prompt change in burnup of fission products present before the power change. Therefore, differences occur in the path to equilibrium for different fission products. Xe-135, the most important fission product, undergoes an important trajectory to equilibrium. Most of the Xe-135 comes from decay of I-135 and a smaller production as a primary fission product. Consider the response to a power increase. The shortterm effect is an increase in Xe-135 burnout by neutron absorption. This is a positive reactivity feedback. The increased flux also causes an increase in I-135 production. As I-135 concentration increases, more Xe-135 appears due to decay of I-135. The increase in Xe-135 is a negative reactivity feedback. The contribution to Xe-135 production by I-135 decay eventually results in a higher Xe-135 concentration than existed before the power increase. Thus, the Xe-135 feedback reactivity coefficient is positive immediately after a power change and becomes negative as time passes. The positive portion lasts for a few hours and the negative portion reaches equilibrium in many hours. See Chapter 6 for details of Xe-135 poisoning and its influence on neutron dynamics. Note that Xe-135 effects are much slower (hours) than temperature and pressure effects (seconds).

7.6 Combined reactivity feedback A reactor’s response characteristics depend on the net effect of all of the feedback reactivities. Fig. 7.5 shows the situation in block diagram form. Table 7.1 summarizes the ways that reactor power influences reactivity.

FIG. 7.5 Block diagram showing reactivity feedback paths.

79

80

CHAPTER 7 Reactivity feedbacks

Table 7.1 Influence of reactor power on reactivity

7.8 Reactivity feedback effect on the frequency response

7.7 Power coefficient of reactivity and the power defect Changes in reactivity induced by external means (such as a control rod) in a power reactor trigger power changes and reactivity feedback effects. If the net effect of all feedback effects is negative, reactor power settles out at a power level in which feedback reactivity cancels the externally induced reactivity change, giving a net reactivity of zero as required for steady state. A measure of the net reactivity change following a power change (Δρ/ΔP) is called the power coefficient of reactivity. It defines the total feedback reactivity at equilibrium after a power change. The individual feedbacks follow different trajectories to equilibrium and this time dependence is not addressed in the power coefficient. The power coefficient defines the new steady state power level after a change in external reactivity. Clearly the power coefficient of reactivity must be negative for a stable reactor. Note that the power coefficient of reactivity involves processes that operate on different time scales and depend on thermal-hydraulic processes that determine the magnitude of various reactivity changes. For example, the Doppler reactivity feedback is the product of fuel temperature change, ΔTf and the Doppler coefficient of reactivity, Δρ/ΔTf. While the power coefficient defines the incremental change in reactivity per unit change in power, also of interest is the total reactivity change as the reactor moves from one steady power level to another steady power level. This measure is called the power defect (it should be noted that some authors define the term, power coefficient, as the power defect as defined above). The power defect from zero power to full power in a LWR is ΔρPD  0.01 to 0.03 (ρ) [1].

7.8 Reactivity feedback effect on the frequency response As shown in Chapter 3, the overall transfer function for a system with inherent (negative) feedback may be written as follows: δO G ¼ δI 1 + GH

(7.3)

where G is the process (feedforward) transfer function and H is the feedback transfer function. Now consider the effect that negative reactivity feedback has on the overall power to reactivity frequency response. As shown earlier, the frequency response magnitude for a reactor without feedback (given by G) is large at low frequencies. Rewrite Eq. (7.3) as δO 1 ¼ 1 δI +H G

(7.4)

81

CHAPTER 7 Reactivity feedbacks

At low frequencies, the magnitude of G is very large, and Eq. (7.4) may be approximated as δO 1  δI H

(7.5)

Note that G and H are functions of frequency, ω (rad/s). So, the low frequency response is determined by the feedback reactivity. The units of H are reactivity over power (typically, cents/% power). So, the units of 1/H are % power/cent of reactivity, or simply the inverse of the power coefficient. At low frequencies, the feedback, H, “keeps up” with a reactivity perturbation. That is, the magnitude is constant at low frequencies. As frequency increases, the feedback becomes unable to “keep up” with reactivity perturbations. That is, the magnitude of H starts to decrease at some frequency. The frequency at which this change occurs is characterized by the “break frequency”, typically the reciprocal of the fuel-to-coolant heat transfer time constant in reactors with dominant fuel reactivity feedback. The feedback has a decreasing effect on the overall response at higher frequencies. Now consider frequencies much higher than the break frequency of H where the magnitude of H is small and G dominates the transfer function. At high frequencies the transfer function may be approximated by the forward transfer function, G. In this case δO G δI

(7.6)

That is, the reactor responds as if it has no feedback. The arguments presented above should be intuitive to the reader. Fig. 7.6 illustrates the net effect of feedback on the reactor frequency response for a typical PWR. A rough approximation to the feedback transfer function is H ¼ 1/(s + 0.2). H(s) has a break frequency of 0.2 rad/s, indicating a fuel to coolant heat transfer time constant of 5 s. The low and high break frequencies are at  0.2 rad/s and  700 rad/s, respectively. This behavior matches with the approximations indicated by Eqs. (7.5) and (7.6).

% Power/Cent

1 0.1 0.01 0.001 90 Phase (deg)

82

45 0 –45 –90 10–2

10–1

100

101 102 Frequency (rad/s)

103

104

105

FIG. 7.6 Frequency domain plots of transfer function magnitude and phase for a typical PWR with feedback. The feedback transfer function H ¼ 0.01/(s + 0.2).

7.9 Destabilizing negative feedback: A physical explanation

Note that the low frequency response (below around 1 rad/s) is determined by the reactivity feedback (1/H). At higher frequencies, the feedback is small and the response is identical to the response of a zero-power reactor (see Chapter 4 for the zero-power frequency response). Ref. [2] describes the dynamic modeling and experimental analysis of a commercial PWR. The experimental data, generated by a pseudo-random binary sequence (PRBS) perturbation of the control rod reactivity, were used to calculate the power-to-reactivity transfer function. The frequency domain plots shown in Fig. 7.6 match very well with the reactor operational data analysis, albeit using a point kinetics model with a simple feedback transfer function.

7.9 Destabilizing negative feedback: A physical explanation Since destabilizing negative feedback can occur in power reactors, it is important to understand the physical basis for this phenomenon. Destabilizing negative feedback is an important issue for BWRs (see Chapter 13). Feedback in a system can either augment or diminish the effect of input disturbances. One might think that negative feedback is always stabilizing, but this is not true. Negative feedback can be stabilizing or destabilizing. In this section, we will show the physical basis for destabilizing negative feedback. The timing of negative feedback is the crucial issue. For example, if the process that causes negative feedback is shifted in time, it can be experiencing a negative part of an oscillation when the process being affected by the feedback is experiencing a positive part of an oscillation. The result of negative feedback when the feedback variable is negative is positive. That is, for instability to be caused by negative feedback, the feedback phase shift must be such that it changes the sign of the quantity being fed back. This occurs when the feedback causes the phase shift to lie between 90o and 270o. If the feedback is a single first order lag (H ¼ K/(s + a)), the phase shift varies between 0° and 90°, the feedback cannot alter the sign of the feedback. So negative feedback always is stabilizing in a system with a first order lag as feedback. So, the feedback must be second order or higher for negative feedback to destabilize a system. An example of a system with second order feedback will help in understanding the phenomenon. Consider a simple third order system defined by the following equations: dx ¼ x  K z + f dt

(7.7)

dy ¼ 2x  2y dt

(7.8)

dz ¼ 3y  3z dt

(7.9)

The magnitude of the feedback from variable, z, in Eq. (7.7) is given by the coefficient, K. Eqs. (7.8) and (7.9) define the feedback. So, the feedback is second order and the phase shift lies between 0° and  180°.

83

CHAPTER 7 Reactivity feedbacks

FIG. 7.7 Block diagram representation of destabilizing negative feedback example. Note that when gain K is less than zero, the feedback becomes negative.

Fig. 7.7 shows the block diagram representation of the system defined by the above equations with K ¼ 5. The feedback variable is z, so the feedback transfer function is given by δZ(s)/ δX(s). The frequency response function for δZ(s)/δX(s) is plotted in Fig. 7.8. The phase shift is more negative than 90° for all frequencies above around 2.3 rad/s. The magnitude is 0.5 or less for these frequencies. So, the feedback is positive with a magnitude of (5)  (0.5) ¼ + 2.5 or less for frequencies above 2.3 rad/s. Fig. 7.9 shows the response of the variable, x, for a step forcing function, f ¼ 1 and for several values of feedback gain, K. For a value, K ¼ 5, the system has damped oscillatory behavior and is stable. For K ¼ 10, the system has sustained oscillation, and for K ¼ 15, the system is unstable. Clearly, destabilization depends on the feedback phase shift and the magnitude of the feedback. Now consider Fig. 7.10. It shows the response of the variable, x, and the variable, z that is the feedback variable to the first equation. Note that x begins responding immediately and z

Magnitude (abs)

100

10–1

10–2

10–3 0 Phase (deg)

84

–45 –90 –135 –180 10–2

10–1

100 Frequency (rad/s)

101

102

FIG. 7.8 Magnitude and phase angle of the transfer function, G(s) ¼ δZ(s)/δX(s), G ðs Þ ¼ ðs + 2Þ6ðs + 3Þ.

7.9 Destabilizing negative feedback: A physical explanation

Step response (variable x in the model)

0.8 K=5

0.6

K=10 K=15

0.4 0.2 0 –0.2 –0.4 –0.6 –0.8 0

0.5

1

1.5

2

2.5 Time (s)

3

3.5

4

4.5

5

FIG. 7.9 Step response of variable x for feedback gains K ¼ 5, K ¼ 10, and K ¼ 15. 0.8 x

Step response of variable x, and feedback variable, z

0.6

z

0.4 0.2 0 –0.2 –0.4 –0.6 –0.8 0

0.5

1

1.5

2

2.5 Time (s)

3

3.5

4

4.5

5

FIG. 7.10 Step response of variable, x and the feedback variable, z for gain K ¼ 15.

initially shows a negligible response. After a short time, z starts an oscillatory response, but the waveform is shifted relative to the x waveform. Note that the feedback variable (z) lags behind the variable affected by the feedback (x). The lag is around 108° in this example. It should be obvious that the largest destabilizing effect would occur if the lag is 180o. Destabilization depends on both the phase shift and the magnitude of the negative feedback effect. Even small phase shifts are destabilizing if the feedback magnitude is large enough. So, the commonly-stated concern about the destabilizing effect of 180° phase shifts is a bit misleading. Any lag can be destabilizing if the feedback is second order or higher and the magnitude of negative feedback is large enough. Alternatively, the destabilizing effect in a negative feedback system can be seen with the main system of order two or higher and first order feedback transfer function with increasing feedback gain.

85

86

CHAPTER 7 Reactivity feedbacks

7.10 Explanation of stability using state-space representation Express the three differential Eqs. (7.7)–(7.9) in the state-space form as. dX dt ¼ AX + Bf

(7.10)

X is the vector of state variables and f is a forcing term. Matrices A (3  3) and B (3  1) are given by 2 3 3 1 1 0 K A ¼ 4 2 2 0 5, B ¼ 4 0 5 0 0 3 3 2

(7.11)

Note that the coefficient K is redefined as (K). Now calculate the eigenvalues of matrix A for different values of ‘gain’ K. The characteristic polynomial is given by gðλÞ5λ3 + 6λ2 + 11λ + 6 + 6K

(7.12)

For K 5 5, the eigenvalues are: 20.3928 + 2.5980j, 20.3928–2.5980j, 25.2145. All the eigenvalues have negative real parts, and the system is stable. For K 5 10, the eigenvalues are: 3.3166j, 2 3.3166j, 2 6.0. There are two imaginary eigenvalues, and the real eigenvalue is negative. The system is marginally stable. For K 5 15, the eigenvalues are: 0.2779 + 3.8166j, 0.2779–3.8166j, 26.5558. The complex eigenvalues have positive real parts, and the system is unstable. •

•

If we consider K as a feedback gain, changing the value of K changes the stability characteristics of the system. This is true even though the system is inherently a negative feedback system. This is true of all negative feedback systems with the overall system order greater than or equal to three; this is a well-known fact. As the feedback gain is increased, the system tends towards instability. The phase lag between any two state variables does not have to be 180° for the system to become unstable.

Consider an open-loop transfer function of order 3 with stable poles and of the form GðsÞ ¼

1 as3 + bs2 + cs + d

(7.13)

If the closed-loop system has negative feedback with a simple gain K (as the feedback transfer function), the closed-loop transfer function becomes Gc ðsÞ ¼

1 as3 + bs2 + cs + d + K

(7.14)

As the magnitude of gain K increases, the closed loop poles tend to be less negative and eventually become positive for a limiting value of gain K.

References

Exercises 7.1.

Explain the shape of the curve in Fig. 7.4 in terms of the thermal utilization and the resonance escape probability in the six-factor formula For exercises 7.2–7.4: The transfer function of a zero-power reactor is given by G0 ¼

δN ðsÞ=n0 ¼ δρðsÞ

ðs + λ Þ   β s Λ s+λ+ Λ

7.2.

Make a Bode plot of G0(s) and determine the low-frequency and highfrequency break points of the dB-magnitude plot. Use the following values for the parameters of a typical PWR: λ ¼ 0:08 s1 , β ¼ 0:0067, Λ ¼ 105 s.

7.3.

Now consider a negative feedback effect, similar to that in a power reactor. Assume a feedback transfer function of the form H ðsÞ ¼

(a) (b)

(c)

0:03 s + 0:25

Determine the closed-loop transfer function for this case. Make a Bode plot of the closed-loop transfer function and determine the low-frequency and high-frequency break points of the dB-magnitude plot. Compare these with the results of part (1).

7.4.

For the reactor with feedback transfer function specified in the previous exercise, compute the response δn(t)/n(0) for a step change in the reactivity of 0.001. Plot the responses for tmax in the range 10–100 s. You may use the MATLAB command step(sys), with appropriate value for the step magnitude. Is the time response stable or unstable? Explain.

7.5.

Calculate the eigenvalues of matrix A, for K ¼ 8 in Eq. (7.11)

References [1] J.J. Duderstadt, L.J. Hamilton, Nuclear Reactor Analysis, John Wiley & Sons, New York, 1983. [2] T.W. Kerlin, E.M. Katz, J.G. Thakkar, J.E. Strange, Theoretical and experimental dynamic analysis of the H.B. Robinson nuclear plant. Nucl. Technol. 30 (3) (1976) 299–316, https://doi.org/10.13182/NT76-A31645.

87

CHAPTER

Reactor control

8

8.1 Introduction All control systems have two purposes: to hold a process at a desired condition (or set point) when experiencing external disturbances and to move a process from one condition to another desired condition. Since the primary purpose of a power reactor is to produce power, usually electricity but sometimes process heat, the first job of the control system is to cause the delivered power to match the desired power. The control system also causes various plant variables (such as temperature, pressure, liquid level, etc.) to attain desired values and to reach these values in a timely fashion. These plant controllers have the additional function of establishing conditions that enhance economics, avoid undesirable plant conditions, maximize durability of components, and establish initial conditions that ensure a manageable and safe response to abnormal occurrences.

8.2 Open-loop and closed-loop control systems Fig. 8.1 shows an open-loop system, in which the system input, X(t), is not directly affected by the system output. A perturbation is applied to the system with the intention that the output would achieve the desired value. The input is “calibrated” to generate an output which is close to the desired output. Some examples of systems with open-loop control are. • • •

Opening or closing a kitchen sink tap to provide desired water flow. Operation of a toaster. Hitting a golf ball towards a target.

Note that in each of the above examples, the response of the system (water flow rate, cooking time, flight and final position of the golf ball) is dependent on a pre-set command and cannot be modified once the command is executed. A closed-loop system is one in which the control action, X(t), depends on the system output, Y(t). Closed-loop control systems are commonly called feedback control systems. See Fig. 8.2. This controller/actuator module is designated by the block, Gc(s). Dynamics and Control of Nuclear Reactors. https://doi.org/10.1016/B978-0-12-815261-4.00008-1 # 2019 Elsevier Inc. All rights reserved.

89

90

CHAPTER 8 Reactor control

Output

Input Process

Controller

X(t)

Y(t)

FIG. 8.1 An open-loop control system. Reference Input Based on desired output

Comparison X(t)

e(t)

Gc Controller

Gp Process

Y(t)

Measurements H

FIG. 8.2 A closed-loop (feedback) control system.

Following are some of the features of a closed-loop control system: 1. When feedback is introduced, certain controlled systems can go unstable. 2. The introduction of feedback control reduces the sensitivity of a system to disturbances and changes due to component aging. 3. Feedback control reduces the effect of natural fluctuations in process variables and measurements. These fluctuations are usually called noise. 4. Since a closed-loop controller performs comparison of a desired (set point) value and an actual value of the output, and provides feedback through measurements, closed-loop control systems are always more expensive than open-loop control systems. The components of a system that affect the controller performance are: (1) The dynamics of the process being controlled; (2) actuators such as valves, thrusters, motors, heaters, sprays, etc.; (3) various sensors perform process measurements such as temperature, pressure, flow, level, position, velocity; yaw, pitch and roll of an airplane; metal sheet thickness, sheet tension, force on a stand in a rolling mill, etc. Chapter 16 addresses sensors that are important in nuclear reactors. The complexity of control systems varies widely and depends on the complexity of the process and the required control actions. Automatic control generally means that all control actions are taken by the system, with minimum or no human interference. This is especially true when the process under study is large and complex, the desired accuracy of the output or the quality of the product is high, or when

8.3 Basic control theory

actions must be taken quickly, making it then necessary to fully automate the system. Such is the case with reactor control. Some examples of common closed-loop control actions are •

•

•

Temperature control in a room using a heater or an air-conditioner. The thermostat measures the room temperature, compares it with the desired (set point) temperature, and the error provides a signal to switch on or off the heater or the air-conditioner. Control of automobile motion by the driver. The driver compares (by visual observation) the heading of the vehicle against the markings on the road and determines an error. Based on this information, the steering wheel is manipulated to maintain proper position of the vehicle on the road. A wide receiver catching a football thrown by the quarterback. Explain why this is an example of closed-loop control. How does this differ from hitting a golf ball towards a target?

8.3 Basic control theory Controller design and control actions vary from being simple to highly complex. There are five types of feedback control systems: • • • • • •

Manual On-off Proportional Integral Differential Advanced.

Feedback control action uses an error signal, which is the difference between the desired output (set point) and the actual system output (measured value). The error signal, e(t), is defined as Error Signal ¼ Set point, Xset ðdesired conditionÞ  Measurement, Xm ðactual conditionÞ eðtÞ ¼ Xset ðtÞ  Xm ðtÞ

(8.1)

The control system makes use of the error signal to initiate an action by the actuator.

8.3.1 Manual control In this control action the necessary actuation is performed by a person in the loop. He/she observes the system’s output and takes control action. This may be based on the experience and/or the intuition of the person.

91

92

CHAPTER 8 Reactor control

8.3.2 On-off controller A common and simple controller is an on-off controller. The control action takes on two values. Examples of the on-off controller include applications in space heating, air-conditioning systems, and a household oven. Fig. 8.3 shows the actions of an onoff controller. The control action is on when the error is positive and is off when the error is negative. To avoid frequent actuation of the switch near zero error, a dead-band may be provided. In the dead-band region, the controller action does not change. This modification is shown in Fig. 8.4. With a dead-band controller, the control action does not change for a small, specified band of error on either side of the zero error. An example is a room thermostat controller with a dead band of 2 °C. This modification could avoid serious problems in the actuators, such as, frequent starting and stopping of electric motors or frequent actuation and turning off of a heater.

f(t)

A

0

Error, e(t)

FIG. 8.3 A typical on-off controller without a dead-band.

f(t)

0 Dead Band

FIG. 8.4 A typical on-off controller with a dead-band.

Error, e(t)

8.3 Basic control theory

8.3.3 Proportional controller A controller whose action is proportional to the error signal is called a proportional controller. This controller has the form f ðtÞ ¼ Kp fXset  Xm g ¼ Kp eðtÞ

(8.2)

where Kp is called the proportional constant. Fig. 8.5 shows an illustration of proportional control action. In general, both positive and negative control actions may occur. A dead-band may be used to avoid frequent switching of the control action. Fig. 8.6 illustrates a proportional controller with a dead-band and a cut-off at the controller limit.

f(t)

f(t) = K.e(t)

0

e(t)

FIG. 8.5 A typical proportional controller.

f(t) Controller limit

0 Dead Band

FIG. 8.6 A proportional controller with a dead-band and a limit.

Error, e(t)

93

94

CHAPTER 8 Reactor control

In this case, the control action is turned off when the error falls in the dead-band region. Note that, in general, the constant (or gain) Kp can be made to change in some controllers, being large for large errors and small for small errors (often referred to as a nonlinear gain). A feature of proportional control is that an error signal must be non-zero for control action to occur. Therefore, proportional control cannot drive a variable to its set point following an external disturbance. For example, consider what would happen if a home thermostat used proportional control rather than on-off control. If the house cooled to Tm because of a drop in outside temperature, the control action would be Kp (Ts - Tm). For successful return to the set point, (Ts - Tm) ¼ 0, the control action would have to be zero rather than a necessary change in heating or cooling input.

8.3.4 Integral controller An integral controller (also called reset controller) can eliminate the steady-state error that occurs with a proportional controller. Integral control action is expressed as follows: Zt fI ðtÞ ¼ Ki

eðvÞdv

(8.3)

o

fI(t) is the integral control action and Ki is the integral constant. Because of the continuous change in the control action (caused by integration) the constant Ki is often referred to as the reset constant. Integral controllers can reduce the error to zero, thereby eliminating the problem with proportional controllers. Caution must be taken in implementing an integral controller to avoid continuous increase in the control action, caused by the integration of the error. This can be achieved by limiting the actuator action beyond a certain level. As the error approaches zero, the magnitude of the integral constant may be reduced, thus ensuring smaller fluctuations in the system output.

8.3.5 Differential controller As the name suggests, differential control action is proportional to the time rate of change of the error signal. Because of its sensitivity to fluctuations in the measured process variable, derivative control is seldom used, but is sometimes applied successfully by using a low-pass filter to reduce high-frequency noise in the error signal. Differential controllers are useful in systems where there is a considerable lag time between the control action and its effect on the system output. This time lag can result in an incorrect error term being supplied to the controller, and the system may go into instability. By combining the proportional and differential components of the error term, the controller can anticipate the future changes taking place in the output, in addition to the error itself. Thus, a differential controller could help stabilize a closed-loop system. This controller is expressed as.

8.3 Basic control theory

  de fD ðtÞ ¼ Kd dt

(8.4)

8.3.6 Combined controllers Controllers employing more than one mode of control are often used. The most common is proportional plus integral (PI control) [1, 2]. The designer of a PI control system must identify and implement proper selection of the coefficients, Kp and Ki. The PI control action is expressed as. Zt f ðtÞ ¼ Kp eðtÞ + Ki

eðvÞdv

(8.5)

o

8.3.7 An example of proportional and integral controller for a first order system To illustrate the specific properties of proportional and integral control actions, the closed-loop system shown in Fig. 8.2 is considered here. The process Gp(s) is selected for this example is a first order system given by. G p ð sÞ ¼

1 s+a

(8.6)

8.3.7.1 Proportional controller In this type of controller, the control action is proportional to the error. Thus, Gc ðsÞ ¼ Kp , Kp > 0 and FðsÞ ¼ Kp EðsÞ

As shown in Appendix D, Section D.5, the closed-loop transfer function is. Y ðs Þ Gc ðsÞGp ðsÞ  ¼ XðsÞ 1 + Gc ðsÞGp ðsÞ

(8.7)

For our example, Eq. (8.7) becomes Y ðsÞ Kp  , Kp > 0 ¼ X ð sÞ s + a + Kp

(8.8)

The Laplace transform of the error, e(t), is given by (See Section 3.7). EðsÞ ¼ 

X ð sÞ  1 + Gc ðsÞGp ðsÞ

(8.9)

For our example, E(s) becomes. ðs + aÞXðsÞ  EðsÞ ¼  s + a + Kp

For a unit step input, X(s) ¼ 1/s, and ðs + a Þ  EðsÞ ¼  s s + a + Kp

(8.10)

95

96

CHAPTER 8 Reactor control

The final, steady-state value of the error, e(t), when x(t) is a unit step, is given by. eðtÞ ¼

lim a  sEðsÞ ¼  s!0 a + Kp

(8.11)

Note that the steady-state error is not equal to zero, and may be decreased by increasing Kp, the proportional gain. For a ¼ 0.02 and Kp ¼ 1, the steady-state error is e(∞) ¼ 1/51 ¼ 0.0196. If Kp is increased to 10, then the steady-state error is e(∞) ¼ 0.002.

Remarks 1. As the proportional gain constant Kp increases, the steady-state error decreases. However, the steady-state error is never zero. 2. As the gain Kp increases, the time constant of the system decreases, and the rate of system response increases. For the open-loop system in the above example (a ¼ 0.02), the time constant, τ ¼ 50 s. For the closed-loop system with a proportional gain Kp, the time constant is given by τ¼

1  a + Kp

(8.12)

Thus, for Kp ¼ 0.1, the time constant decreases to τ ¼ 8.33 s. 3. An increased Kp results in a faster response of the control system. Since there are limitations on the achievable responses of control devices, Kp should not be increased arbitrarily. Furthermore, an increase in Kp above a certain value can make the system unstable. The stability margin will decrease with increasing values of Kp.

All these points must be considered in choosing the proportional gain. The proportional controller increases the speed of response of a system. The steady-state error is nonzero.

8.3.7.2 Integral controller An integral control is expressed as: Zt f ðt Þ ¼ K i

eðvÞdv o

In the Laplace domain, the control action has the form. FðsÞ ¼

Ki EðsÞ s

(8.13)

A very important caution to be exercised in using integral controllers is the consideration of actuator limits. When an actuator reaches its lower or upper limit (because of actuator saturation), the error may remain large and the integral action will continuously increase the error term. This characteristic is called integral windup and must be avoided by changing the sign of the error using appropriate logic. A large transient may be observed if the actuator reaches its limit.

8.4 Control of a zero-power reactor

The closed-loop transfer function of a system with transfer function, 1/(s + a) and with integral controller is given by Y ðsÞ Ki ¼ XðsÞ ðs2 + as + Ki Þ

(8.14)

Note that the closed-loop system becomes a second-order dynamic system, with the potential for the system response to have an oscillatory characteristic, as Ki becomes large. The system error is given by EðsÞ ¼

sðs + aÞXðsÞ ðs2 + as + Ki Þ

(8.15)

The steady-state error for a unit step input is eðtÞ ¼

lim sEðsÞ ¼ 0 s!0

(8.16)

The integral control thus results in a zero steady-state error for the above system. For a ¼ 0.02, the closed-loop transfer function is given by. Y ðs Þ Ki ¼ XðsÞ ðs2 + 0:02s + Ki Þ

(8.17)

For Ki > 0.0001, the roots of the denominator polynomial are complex, resulting in an oscillatory response of to a step input. Thus, extreme caution must be exercised in choosing Ki for a general system. A large value of Ki results in a highly oscillatory behavior, and quite likely makes the system unstable. Thus, the choice of an appropriate value of Ki is important, rather than the use of an arbitrary absolute value.

8.3.8 Advanced controllers Advanced control strategies use computed estimates of future consequences of current control actions rather than current consequences as in classical control. Advanced model-based controllers are necessary in some applications (such as controlling missile trajectories). Sophisticated advanced control software is available, but is not currently needed for reactor control. The next generation reactors may find use for advanced control implementation.

8.4 Control of a zero-power reactor This section illustrates system control by simulation of a zero-power reactor with proportional control, integral control, and proportional plus integral (PI) control. Included are responses to a reactivity step input and a power set point change. The reader should note the effectiveness of the controller in driving the response to the desired final value.

97

98

CHAPTER 8 Reactor control

The model used here is a U-235 fueled zero-power reactor with a generation time of 105 s. The proportional controller for reactivity control selected for this example has a gain of Kp ¼ 1 cent/% power change. That is Δρp ðtÞ ¼ Kp ½PSet  PðtÞ

(8.18)

where Δρp(t)¼proportional control (reactivity) Kp ¼ proportional coefficient (in this example Kp ¼ 1  0.0067/100 ¼ 6.7 105) P(t)¼reactor power at time, t PSet ¼ reactor power set point. The integral controller selected for this example has a gain of Ki ¼ 0.1 cents/% power second. That is Zt Δρi ðtÞ ¼ Ki

ðPset  PðvÞÞdv

(8.19)

o

Δρi(t)¼integral control (reactivity) Ki ¼ integral coefficient (in this example Ki ¼ 0.1  0.0067/100 ¼ 6.7  106) The coefficients used here were selected to illustrate control characteristics. Actual performance could be improved by optimizing the coefficient values. First consider a transient initiated by a step increase in external reactivity. Fig. 8.7 shows the responses for a reactor with proportional only, integral only, and proportional plus integral control actions. Note that proportional only control limits the power increase, but the final power is different than the set point. This is as expected since there must be an error signal for non-zero control action as needed to cancel the external reactivity that initiated the transient. Now consider a transient initiated by a step increase in the reactor power set point. Fig. 8.8 shows the responses for a reactor with proportional only, integral only, and proportional plus integral control actions. Note that all three control options successfully bring the power to the new set point. This is as expected since there is no external reactivity for the proportional controller to cancel. Note also that there is a prompt jump for the cases with proportional control and proportional plus integral control. The explanation of this behavior is left as an exercise for the reader. Note that proportional control only fails to return reactor power to its (unchanged) set point for a reactivity disturbance, but succeeds in bringing reactor power to its new set point following a change in set point. This happens because achieving steady state requires that control reactivity changes to cancel the reactivity perturbation. For

8.4 Control of a zero-power reactor

P I Pl

1.14 1.12 1.1

P/P(0)

1.08 1.06 1.04 1.02 1 0.98 0.96

0

10

20

30 Time (s)

40

50

60

FIG. 8.7 Response of a zero-power reactor to a reactivity disturbance with proportional (P) only, integral (I) only, and proportional + integral (PI) control actions.

2.4 P I Pl

2.2

P/P(0)

2 1.8 1.6 1.4 1.2 1 0

10

20

30 Time (s)

40

50

60

FIG. 8.8 Response of a zero power reactor to a change in the power set point with proportional (P) only, integral (I) only, and proportional + integral (PI) control actions.

99

100

CHAPTER 8 Reactor control

this to happen with proportional control, there must be a non-zero value of (PSet - P). For changes in reactor power set point, there is no such occurrence.

8.5 Control options in power reactors In a system such as a power reactor, the control engineer must begin by asking, “What do I need to control?” The main possibilities include: • • • • • • •

Reactor power (total and local) Power mismatch between reactor and turbine Temperatures at various points in the system Pressures (primary pressure in systems with steam generators and steam pressure in BWRs or in steam generators) Water level in BWRs and in U-tube steam generators Feedwater flow rate Frequency of generator electrical output.

These are the main possibilities, but, of course ancillary equipment such as feedwater heaters, steam reheaters, and condensers also must be monitored and controlled. Then the control engineer asks, “What control actions are available?” Here are the main ones: • • • • • • • • •

Control rod position (full length, full strength; full length, part strength; part length (gray)) Dissolved neutron absorber (slow) Burnable poisons (very slow and the effect is pre-determined) Water in in-core liquid chambers (used in CANDU reactors) Main steam valve Heater and spray in the pressurizer Feedwater flow rate to steam generators Recirculation flow rate in BWRs. Primary and secondary liquid sodium flow rates in sodium fast reactors.

This list encompasses the main choices available to the control engineer.

8.6 Effect of inherent feedbacks on control options The usual form of a control system appears in Fig. 8.2 Because of the strong inherent feedback effects in power reactors, the control system might be viewed as operating in parallel with inherent feedbacks. But in power reactors, the strategy is often to control variables (such as flow rates or pressures) that affect feedbacks. For example, reactivity might be adjusted to achieve a coolant temperature set point. In this case, reactivity change causes a power change and this power change is driven to the level that causes coolant temperature to reach a new set point.

8.7 Load following operation

The inherent feedback in power reactors creates opportunities and challenges for the control engineer. For example, molten salt reactors have strong temperature reactivity feedback effects, and to a great extent they are self-regulating systems. Control action can alter a reactor input directly (for example, using control rods to change reactivity) or indirectly (for example using control action to change a process variable such as temperature or pressure to change reactivity feedback). Control engineers select control strategies that provide the most effective responses.

8.7 Load following operation Load following is the adjustment of a power plant production to match grid electrical power demand. In a network of power plants, some may be assigned load following duties and other plants do not respond to demand changes (so-called base load plants). Decisions on assignment of load following duties depend on economic and technical considerations. The economic consideration is based on the operating cost contribution to power cost versus the capital cost contribution. Nuclear plants have lower operating cost and higher capital cost than plants fueled by coal, natural gas, or oil. Therefore, it is desirable to run nuclear plants at maximum capacity at all times and to assign load following to plants with smaller cost penalties for operating at reduced or zero power. The technical consideration relates to the ability of a plant to change production quickly and safely. For example, a hydroelectric plant can increase production almost instantaneously by opening valves that admit water to the turbines. Even though hydroelectric power costs are almost entirely capital costs, the excellent load following capability often results in assigning them load following functions. The earlier nuclear power plants (Generation II and somewhat in Generation III) were usually expected to serve as base-load plants and there was little consideration for designing load following capability into these plants. But in systems where nuclear plants provide a large fraction of total production, load following capability became a necessity. The situation in France, where nuclear power production is over 70% of the total electrical generation, is a good example of a case where load-following nuclear plants are needed. Load-following is also needed in hybrid power networks where a mix of nuclear energy and renewable energy (wind and solar whose output depends on nature) is used. A base-load plant’s power output can be changed by operator action (changing the power set point manually). When this occurs, other load-following plants automatically change their output to maintain the desired electrical frequency. One consideration is whether the load following strategy is based on “reactor following turbine” or “turbine following reactor”. In reactor following schemes, the plan is to adjust the main steam valve first after a load imbalance to satisfy electrical power demand. Adjustment of reactor power follows until reactor power equals demanded power. In turbine following schemes, reactor power adjustment occurs first following a load imbalance. Steam is admitted to the turbine after the reactor power adjustment.

101

102

CHAPTER 8 Reactor control

The total power production in all of the power plants on a grid must exactly match the power demand by consumers on the grid. A mismatch between power production and power demand causes a change in the frequency of the alternating current. This change in frequency causes load following plants on the grid to change power production until the frequency returns to its set point (60 Hz in the U.S.). Note that an operator induced change in the power production of a base-load plant (like most nuclear plants) causes a grid frequency change and, consequently a change in power production in load following plants on the grid.

8.8 The role of stored energy Power reactors can supply added steam upon demand before the reactor power changes. This is through use of energy stored in reactor fluid and metal components. Fluids boil and metal components cool down to provide energy. For example, stored energy in a typical PWR can provide around 10 full power seconds per psi of pressure drop. Even in a BWR, stored energy can be used temporarily. Even though opening the BWR main steam valve causes pressure reduction, increased boiling, reduced reactivity and reduced power, this can be tolerated if control action inserts reactivity to cancel the temporary reactivity decrease. The steam delivered to the turbine during this temporary episode is provided mostly by energy stored in saturated water.

8.9 Steady-state power distribution control It is desirable for the power distribution to be uniform throughout the reactor core. A uniform power distribution would cause equal fuel consumption throughout the reactor, and consequently, better economic performance. But neutron leakage at the periphery, moderator density variations in some reactors and position of control rods used to suppress available reactivity all cause non-uniform power distributions. There are naturally occurring effects, refueling procedures, and control actions that can make the power distribution more uniform. The naturally-occurring effect is higher fuel burnup and fission production in regions with higher power densities. This causes a power density reduction in those regions as the reactor operates. The refueling procedure in light water reactors involves shuffling old fuel towards the center of the core and adding fresh fuel at the periphery. This causes a flattening of the power distribution. Control actions involve positioning control rods where they can aid in flattening the power distribution. Control rods include full-length full-strength rods, full-length part-strength rods, part-length rods, and fixed-position burnable poison rods. Fulllength full-strength control rods have a strong neutron poison throughout the length of the rod. Therefore, they cause a localized reduction in power density along their whole insertion length.

8.10 Important reactivity feedbacks and control strategies

Full-length part-strength control rods have lower poison concentrations throughout the rods than full strength rods. Like full-length full-strength control rods, they affect local power density, but not as strongly. Part-length control rods have neutron absorber only near the tip. They can reduce local power density at in-core regions without strongly affecting the power density in regions behind the absorber region. They are useful in controlling the power distribution. Heavy water reactors (CANDU) use in-core chambers where light water can be introduced. Increasing light water in the chambers decreases reactivity and reduces the local power production. Fixed-position burnable poison rods are installed in high power density regions before the reactor goes into operation. They affect the local power density and their strength decreases as the reactor operates and the poison is transmuted by neutron absorption. These are common in BWRs. BWR fuel assemblies also contain “water rods” that help increase moderation and thus reactivity.

8.10 Important reactivity feedbacks and control strategies for various reactor types Chapters 12–14 address various reactor types and their important reactivity feedbacks and control strategies.

Exercises 8.1.

Consider the closed-loop system shown in Fig. 8.9 (a) For Gc(s) ¼ 1, determine the steady-state error in the output when the input is a unit step function. (b) Repeat part (a) for the case when Gc(s)is a PdI controller given by Gc ðsÞ ¼ 1 +

Controller X(s) + _

FIG. 8.9 Problem 8.1.

K p + Ki /s

0:1 s

Plant 200 (s + 1) (s + 2)

Y(s)

103

104

CHAPTER 8 Reactor control

WATER IN DESIRED LEVEL

VALVE ACTUATOR

WATER OUT FIG. 8.10 Problem 8.4, a water level control system.

8.2.

Verify Eq. (8.10).

8.3.

Verify that the system defined by Eq. (8.17) becomes oscillatory for Ki > 0.0001.

8.4.

An example of a feedback control action is shown in Fig. 8.10. The objective is to maintain the level of water in a tank. The water level is measured by a level gauge and the controller adjusts the valve based on the error between the set point (desired) value of the level and the actual (measured) level. The flow out of the tank is allowed to change, so that the water in the tank is maintained at a desired level. Develop a block diagram of the tank level control system showing the system, controller, measurement, and actuator function.

8.5.

Explain why two of the cases shown in Fig. 8.8 experience a prompt jump and one case does not.

8.6.

Formulate equations for a zero-power reactor with proportional and integral control. Use the one-delay group model for neutronics. Define all terms.

8.7.

Explain why the initial response to a reactivity disturbance (as shown in Fig. 8.7) is so different for an integral controller then the response for proportional or proportional plus integral control.

References [1] C.L. Phillips, J.M. Parr, Feedback Control Systems, fifth ed., Prentice Hall, Upper Saddle River, NJ, 2011. [2] R.C. Dorf, R.H. Bishop, Modern Control Systems, twelfth ed., Prentice Hall-Pearson, Upper Saddle River, NJ, 2011.

CHAPTER

Space-time kinetics

9

9.1 Introduction Point kinetics models have proven their value in reactor dynamic simulation, but are extreme simplifications of what goes on in a nuclear reactor. Changes in the local neutron flux during a transient are often important. Consequently, space-time neutronic models have been developed and implemented. The most complete description of the spatial distribution of neutrons in a reactor is given by neutron transport theory. Transport theory defines a reactor in terms of seven independent variables: three position coordinates, two direction vectors, energy and time. The transport theory equation is called the Boltzmann equation. Computer codes have been developed for neutron transport, but they suffer from complexity and long computing time. Diffusion theory provides a simpler, yet often satisfactory, approach.

9.2 Diffusion theory Most reactor studies treat neutron motion as a diffusion process—that is, neutrons tend to diffuse from regions of high neutron density to regions of low neutron density. Diffusion theory ignores the direction dependence of the neutrons. Other processes besides neutron diffusion have diffusion theory models. See Chapter 10 for heat conduction theory based on heat diffusion theory. Diffusion theory models use partial differential equations. Such models are called distributed parameter models. Models involving ordinary differential equations are called lumped parameter models. Exact solutions are available for some distributed parameter models. For example, exact solutions are available for heat conduction in a homogeneous solid in slab, cylindrical, or spherical geometry. Solutions are even available for a layered solid, but they are complex. Exact solutions for highly inhomogeneous media (like a reactor core) are intractable. A typical approach for inhomogeneous media simulations is to treat the space as an array of segments with internally averaged properties and coupling terms to treat segment-to-segment transfers. This approach results in a set of ordinary differential equations that can be solved by standard ordinary differential equation solvers. Dynamics and Control of Nuclear Reactors. https://doi.org/10.1016/B978-0-12-815261-4.00009-3 # 2019 Elsevier Inc. All rights reserved.

105

106

CHAPTER 9 Space-time kinetics

9.3 Multi-group diffusion theory Neutron diffusion theory is more complicated than many other distributed parameter processes because of the energy dependence of the neutron population and the nuclear reaction rates. The usual approach for neutron diffusion is to invoke multiple energy groups with average, energy-dependent properties in each group. Use of few (two to four) energy groups is common for thermal reactor space-time analysis. Since the nuclear reactions in fast reactors span a significant range of neutron energies, space time analysis usually employs more energy groups than thermal reactors. We begin the development of the neutron diffusion approximation by writing the balance for an elemental volume in terms of the various reaction rates. The development will be for group of neutrons with an average energy. In practice, multiple coupled energy groups are used to handle the energy dependence. The equations for the group of neutrons with average energy, E, contains eleven terms in the neutron diffusion equation and the delayed neutron precursor equations. Note that the symbol, E, stands for an energy group rather than a specific energy. Note also that the geometry and material properties are constant. For clarity, each term is numbered and defined below. ð1Þ ð2Þ ð3Þ ðX 4Þ ðX 5Þ X X ∂n χ ð E, E Þ υ ð E Þ ð 1  β Þ ð r, E Þ Φ ð r, E , t Þ + S ð r, E, t Þ + ðr, Ei ! EÞ Φðr, Ei , tÞ ðr, E, tÞ ¼ i i i i i f i R ∂t ð6Þ ð7Þ ð8Þ X X X  a ðr, EÞ Φðr, Ei , tÞ  ð r, E Þ Φ ð r, E, t Þ  L ð r, E, t Þ + g ðEÞ λj Cj ðr, tÞ R j j

(9.1)

The delayed neutron precursor equation contains three terms, numbered as follows: ð9Þ ð10Þ ð11Þ X X ∂Cj ðr, Ei Þ Φðr, Ei , tÞ  λj Cj ðr, tÞ ðr, tÞ ¼ βj i υðEi Þ f ∂t

(9.2)

Following are the physical interpretations of each term: (1) Rate of change of the neutron density at position, r, energy group, E, and time, t. (2) Rate of production of fission neutrons at position, r, in all energy groups that produce neutrons in energy group, E, at time, t. (3) Rate of neutron production from an external source at position, r, energy group, E, and time, t. (4) Rate of scattering of neutrons from other energy groups at position, r, at time, t, that appear in group, E. (5) Rate of neutron absorptions at position, r, energy group, E, at time, t. (6) Rate of scattering of neutrons at position, r, energy group, E, and time, t, that fall into another energy group. (7) Rate of neutron leakage at position, r, energy group, E, at time, t. (8) Rate of release of delayed neutrons in all precursor groups at position, r, energy group, E, and time, t. (9) Rate of change of the j-th delayed neutron precursor. (10) Rate of production of the the j-th delayed neutron precursor. (11) Rate of decay of the j-th delayed neutron precursor.

9.3 Multi-group diffusion theory

The factors in the equation are as follows: n(r, E, t) ¼ neutron density at position, r, energy group, E, and time, t. It is also equal to Φ/v, where v ¼ average neutron velocity in the group. χ(E, Ei) ¼ fraction of the neutrons that are born in energy group, E, because of fissions in energy group, Ei. r ¼ the position vector. E ¼ neutron energy group. υ(Ei) ¼ number of neutrons produced per fission caused by neutrons in energy group, Ei. β ¼ total delayed neutron fraction. βj ¼ delayed neutron fraction for the j-th delayed neutron group. Ʃf(r, Ei) ¼ macroscopic fission cross section at position, r, in energy group, Ei. Φ(r, Ei, t) ¼ neutron flux at position, r, for energy group, Ei at time, t. S(r, E, t) ¼ rate of neutrons released from an artificial source in energy group, E, at position, r, at time, t. ƩR(r, Ei ! E) ¼ macroscopic removal cross section for scattering from energy group, Ei, to energy group, E, at position, r. Ʃa(r, Ei) ¼ macroscopic absorption cross section for energy group, Ei at position, r. ƩR(r, E) ¼ macroscopic removal cross section for neutron scattering out of energy group, E, at position, r. L(r, E, t) ¼ leakage of neutrons from energy group, E, at position, r, at time, t. gj(E) ¼ fraction of delayed neutrons from precursor group, j, that appear in energy group, E. λj ¼ decay constant for the j-th delayed neutron precursor group. Cj(r, t) ¼ concentration of the j-th delayed neutron precursor group at position, r, and at time, t. Two quantities that have not been encountered in previous discussions need explanation. These are the removal cross section, ƩR, and the neutron leakage term, L(r, E, t). The removal cross section is the probability that a scattering event in energy group, i, results in delivering the scattered neutron into energy group, j. Reactor physics books [1] provide formulas for the removal cross section. The neutron leakage term, L, is given by the following: Lðr, E, tÞ ¼ rDr Φðr, Ei , tÞ

(9.3)

D ¼ the diffusion coefficient ¼ 1/(3Ʃs). If the diffusion coefficient is constant, Eq. (9.3) becomes Lðr, E, tÞ ¼ Dr2 Φðr, Ei , tÞ

(9.4)

r is the Laplacian operator, shown below for radial and axial dependence. 2

r2 ¼

  1 ∂ ∂ ∂2 r + 2 ∂z r ∂r ∂r

(9.5)

107

108

CHAPTER 9 Space-time kinetics

9.4 Calculation requirements Planning and operating a nuclear power reactor requires a detailed knowledge of the spatial neutron flux and reaction rates. Both steady-state and transient analyses are required. Enumerating the conditions to be analyzed and the information needed for performing the analyses illustrates the enormity of the problem. A typical light water reactor contains as many as 40,000 or more individual fuel rods. As the reactor operates, the composition of the rods changes significantly. Operation depletes fissile material, produces new fissile material, produces fission products, and produces actinides. The concentrations of all of these materials depend on neutron flux at all positions during prior reactor operation. The flux depends on the localized isotopic concentrations, history of power generation, and history of control rod positions. The fuel rods typically stay in the reactor for four years in light water reactors and are moved during refueling. It is necessary to calculate the local isotope concentrations before a new neutron diffusion calculation of the new flux distribution can begin. The nuclear cross sections of all of the isotopes must be known because they influence the neutron population and must be addressed in a detailed analysis. It is clear that the analyst must know the concentrations of hundreds of isotopes at hundreds of positions. Reactivity feedback must be addressed along with neutron diffusion in a power reactor space-time analysis. Space-dependent feedbacks occur simultaneously with space-dependent power changes and these feedbacks alter the power distribution. Therefore, a comprehensive simulation would require modeling the feedback processes simultaneously with the neutronics.

9.5 Computer software Powerful software provides computational capability for reactor analysis using Boltzmann transport theory or neutron diffusion theory. The elementary theory presented above serves to help understand the phenomena that occur in a reactor rather than to provide a basis for practical computations. But users of the available software must understand the underlying phenomena for proper interpretation of computed results.

9.6 Models and computational methods Space-time models usually rely on the multi-group neutron diffusion equations. Independent variables include time, group energies, and one to three position coordinates. Various methods have been developed to solve the equations. These methods include the following:

References

9.6.1 Finite difference methods Finite difference methods use discrete approximations to the space derivatives. This results in a set of ordinary differential equations that can be solved numerically. See Appendix F for a description of the finite difference method.

9.6.2 Finite element method (FEM) See Appendix F for a brief discussion of the finite element method. The finite element method was developed initially for structural analysis of objects with complex geometry. Its use later expanded to include other disciplines, including heat transfer and fluid mechanics analysis. FEM has also been used for reactor analysis.

9.6.3 Modal methods The neutron population is represented with specified shape functions multiplied by time-dependent amplitudes. The algorithm solves for the amplitude functions.

9.6.4 Quasi-static methods The flux shape is assumed to be slowly varying. Point kinetics solutions provide the response between re-evaluations of the flux shape using steady-state diffusion theory.

9.6.5 Nodal methods The reactor is divided into sub-regions, each of which is assumed to have uniform nuclear properties. Node-to-node coefficients define the neutron flow [2]. There have been many reported developments of methods for space-time kinetics. The literature contains many reports of efforts to reduce computation times, increase accuracy, and provide accuracy assessments.

Exercises 9.1 Explain why there is no such thing as an exact space-time simulation. 9.2 How does the flux shape affect economics? 9.3 How would a space-time model be modified to deal with Xe-135? 9.4 What would happen to the local Xe-135 concentration if an off-center control rod were moved out by a small distance? How would it affect the local power? How would it affect total power? If the reactor returned to a constant power without any control action, explain how this happened.

References [1] J.H. Ferziger, P.F. Zweifel, The Theory of Neutron Slowing Down in Nuclear Reactors, The MIT Press, Cambridge, MA, 1966. [2] J.J. Duderstadt, L.J. Hamilton, Nuclear Reactor Analysis, John Wiley & Sons, New York, 1976.

109

CHAPTER

Reactor thermalhydraulics

10

10.1 Introduction Temperature and pressure of reactor fluids and solids are important variables in steady state and transient operation. Along with associated coefficients of reactivity, they determine the magnitude of reactivity feedbacks. Conservation of mass, energy, and momentum are the basis for thermal-hydraulics models. However, since pressure transients reach a new steady state so much faster in a transient than mass or energy, the differential equation for momentum is usually unnecessary. Much of the information in this chapter comes from Ref. [1]. This chapter includes short descriptions of major heat transport systems in nuclear power reactors. Chapters 12, 13, and 14 provide more detailed information about power reactor characteristics and control systems.

10.2 Heat conduction in fuel elements Most of the reactors addressed in this book use cylindrical UO2 fuel rods clad in zircaloy and have a gas-filled gap between the UO2 and the cladding. Fuel heat transfer in reactors with non-cylindrical fuel elements (such as high temperature gas cooled reactors and molten salt reactors).is not addressed here. A complete heat transfer model would require a partial differential equation in two spatial dimensions (axial and radial … azimuthal is not required) and time. Solutions of such a model equation are known, but they are not suitable for coupling with models for cooling fluid. Instead, lumped parameter models (also, sometimes referred to as nodal models) must be used. Lumped parameter models involve breaking the system into regions with uniform internal properties and coupling to adjacent regions. It is possible to model fuel rods with concentric radial lumps, but the simplest and most common lumped parameter model for cylindrical fuel elements uses a single radial node. Consider a single radial node model for a fuel element with heat transfer to fluid coolant. The fuel has mass, Mf, and specific heat capacity, Cf. The model equation is as follows: Mf Cf


dTf ¼ UA Tf  θavg + Pf dt

Dynamics and Control of Nuclear Reactors. https://doi.org/10.1016/B978-0-12-815261-4.00010-X # 2019 Elsevier Inc. All rights reserved.

(10.1)

111

112

CHAPTER 10 Reactor thermal-hydraulics

where Mf ¼ mass of fuel Cf ¼ specific heat capacity of fuel Tf ¼ fuel temperature Pf ¼ power released in the fuel node U ¼ overall fuel-to-coolant heat transfer coefficient A ¼ fuel cylinder surface area (fuel-to-coolant heat transfer area) θavg ¼ average coolant temperature in the adjacent coolant node Eq. (10.1) may be rewritten as
dTf Pf UA ¼ Tf  θavg + dt Mf Cf Mf Cf

(10.2)

The quantity, (Mf Cf /UA) has the units of time. It is the time constant for fuel-tocoolant heat transfer. Typical values for LWRs and CANDU reactors are 4 to 5 s.

10.3 Heat transfer to liquid coolant The core heat transfer model also requires heat balance equations for the coolant. A general model requires mass and energy balances. If the coolant density and node volume are constant, a mass balance is not needed (see Section 10.4 for a discussion of heat transfer in a model with a moving boundary). As with the fuel model, a nodal model for the coolant is needed. Consider the system shown in. Fig. 10.1 The figure shows that there are five variables as defined below: Pc ¼ power generated within the node (as by interaction of radiation with coolant atoms) Tf ¼ temperature of adjacent fuel node θin ¼ inlet coolant temperature θout ¼ outlet coolant temperature θavg ¼ average coolant temperature in the node θout

Fuel Node

Tf

θavg

θin

FIG. 10.1 Heat transfer to a liquid coolant lump (node).

Pc

Coolant Node

10.3 Heat transfer to liquid coolant

The nodal internal power generation, Pc, the fuel temperature, Tf, and the inlet coolant temperature, θin, are defined by other subsystem equations. That leaves two variables, but the coolant equation provides only one. An assumption is required to eliminate θout. θavg must be retained because it appears in the equation for heat transfer from fuel to coolant. The average temperature is given by the following: θavg ¼ ðθin + θout Þ=2

(10.3)

θout ¼ 2 θavg  θin

(10.4)

or There is a problem with this formulation. Note in Eq. (10.4) that a sudden increase in inlet temperature would cause a sudden decrease in outlet temperature. This is an unphysical feature, causing consideration of an alternate formulation. Another possibility is the “well-stirred-tank” formulation. That is, the outlet temperature from the node is set equal to the average node temperature. This solves the problem in the previous formulation, but equating average and outlet temperatures does not represent actual behavior very well. Ray Mann of Oak Ridge National Laboratory addressed the problem [2]. Mann’s formulation uses two coolant nodes adjacent to a single fuel node; Fig. 10.2 shows this arrangement. Well-stirred-tank models represent each pair of coolant nodes. The outlet temperature of the first coolant node (assumed equal to the average temperature of that node) serves as the coolant temperature that provides the driving force for heat transfer from the fuel. Each coolant node receives half of the heat transfer from the fuel node. Therefore, the model equations for Mann’s formulation are as follows: Mc C c


dθ1 UA ¼ W Cc ðθin  θ1 Þ + Tf  θ1 + Pc1 dt 2

(10.5)

Mc Cc


dθ2 UA ¼ W C c ðθ 1  θ 2 Þ + Tf  θ1 + Pc2 dt 2

(10.6)

∼ Θ2

∼ Θ2 ∼ Θ1

Tf ∼ Θ1

Qin

FIG. 10.2 Schematic of Mann’s model with one fuel node and two coolant nodes.

113

114

CHAPTER 10 Reactor thermal-hydraulics

where Mc ¼ mass of coolant in a node Cc ¼ specific heat capacity of coolant θ1 ¼ temperature in the first coolant node θ2 ¼ temperature in the second coolant node W ¼ coolant mass flow rate U ¼ overall heat transfer coefficient from fuel to coolant A ¼ total fuel surface area for fuel to coolant heat transfer Pc1 ¼ heat generation rate in the first coolant node Pc2 ¼ heat generation rate in the second coolant node. Note that the fuel-to-coolant heat transfer is given by the difference between the fuel temperature and the temperature of the first coolant node (assumed equal to the outlet temperature of that coolant node). The fuel-to-coolant heat transfer is divided equally to both coolant nodes in this formulation. The simplest formulation involves representing all of the fuel in a single node and all of the coolant as a pair of Mann’s nodes. A more detailed formulation uses a series of axial fuel nodes, each coupled with a pair of Mann’s nodes. See Fig. 10.3 for the case of two fuel nodes. ∼ ∼ Θ4 = Θout

∼ Θ4

Tf2

∼ Θ3

∼ Θ3 ∼ Θ2

∼ Θ2 Tf1

∼ Θ1

∼ Θ1 ∼ Θin

FIG. 10.3 Series of Mann’s models for more than one fuel node (two fuel nodes) representation.

10.5 Plenum and piping models

10.4 Boiling coolant Modeling channels with boiling fluid is more complex than modeling channels with single-phase fluid. Boiling heat transfer models are needed for BWRs and steam generators. Consider a heated channel into which liquid fluid is introduced into a channel with boiling fluid. The fluid temperature in the sub-cooled region at the entrance increases as it travels through the channel. At some point, the hotter fluid adjacent to the heated surface reaches saturation temperature even though the fluid farther from the heated surface remains sub-cooled. Boiling begins near the surface. The heat transfer coefficient for this region is greater than in the sub-cooled region. As the fluid continues through the channel, the bulk fluid temperature reaches saturation and vigorous boiling throughout the fluid occurs. High heat transfer coefficients occur in this region. A mixture of liquid and vapor exists in this region. This liquid-vapor mixture exits the core in BWRs and in U-tube steam generators. If all of the liquid boils before exiting the heated region (as in once-through steam generators), continuing heat transfer results in superheating of the steam. In a transient, the boundaries between the different regions move. Models have been developed in which these boundaries are state variables. Moving boundary models for BWRs and steam generators are quite complex, too complex for inclusion in this introductory book. But the concept is important. Appendix F illustrates the approach with a moving boundary model for the subcooled node in a heated channel.

10.5 Plenum and piping models Plenums are regions in which fluid enters and mixes with the existing fluid inventory. Plenums exist at core inlets and outlets and at steam generator inlets and outlets. Well-stirred-tank models are usually used for plenums. The model is as follows: Mp Cp


dθp ¼ W Cp θpin  θp dt

(10.7)

where Mp ¼ mass of fluid in the plenum Cp ¼ specific heat capacity of the fluid W ¼ fluid flow rate θp ¼ fluid temperature in the plenum (equal to plenum outlet temperature because of well-stirred-tank approximation) θpin ¼ temperature of fluid entering the plenum Piping carries fluid from one reactor subsystem to another. Piping models are used for hot leg and cold leg piping in PWRs and CANDU reactors, and feedwater for BWRs and steam generators.

115

116

CHAPTER 10 Reactor thermal-hydraulics

Pure time delay models are often used. That is the formulation is simply as follows: θout ðtÞ ¼ θin ðt  τÞ

(10.8)

where τ ¼ residence time in the piping. Well-stirred-tank models are also used. The rationale for a well-stirred-tank model is that some axial mixing does occur in piping. The more general dynamic formulation of the temperature of the liquid in the piping has the form dθout 1 ¼ ðθin  θout Þ dt τ

(10.9)

The resident time is approximated as the ratio between the mass of fluid (Mp) in the piping and the fluid flow rate (Wp). τ ¼ Mp =Wp

(10.10)

10.6 Pressurizer A PWR pressurizer is a vessel with liquid water in the bottom section and saturated steam in the top section. A pressurizer is used to regulate the primary coolant pressure ( 150 bars) in PWRs and CANDU reactors. The pressurizer is connected to one of the hot leg pipings with a long surge line. Fig. 10.4 shows a typical pressurizer. Because of the contact between steam and liquid water, the water is also at the saturation temperature at steady state. Spray of cooler water enters from the top and electrical heaters at the bottom heat the liquid water. The steady state can be disturbed by water inflow or outflow, changes in inlet water temperature, changes in spray flow or changes in heater power. A PWR pressurizer control system can alter the pressure by modulating heater power and/or spray flow. A schematic representation of a pressurizer model structure appears in Fig. 10.5. As shown in Chapter 14, PWR and CANDU reactor pressurizers are slightly different.

10.7 Heat exchanger model Mann’s formulation also may be used in liquid-liquid heat exchangers. Fig. 10.6 shows the model. The equations are as follows: Mp1 Cp


UA
dθp1 ¼ Wp Cp θpin  θp1 + Tt  θp1 dt 2

(10.11)

Mp2 Cp


UA
dθp2 ¼ Wp Cp θp1  θp2 + Tt  θp1 dt 2

(10.12)

10.7 Heat exchanger model

FIG. 10.4 Schematic of a pressurizer used in a PWR. Courtesy of Westinghouse Electric Company (The Westinghouse Pressurized Water Reactor Nuclear Power Plant, Westinghouse Electric Corporation, Pittsburgh, 1984).

117

118

CHAPTER 10 Reactor thermal-hydraulics

FIG. 10.5 Schematic of a pressurizer model.

Wp Θp,in

Mp1 Θp1

Θp1

Ms2 Θs2

Mt Tt

Mp2 Θp2

Θs1

Ms1 Θs1

Ws Θs,in

FIG. 10.6 Schematic of a liquid-liquid heat exchanger model.

10.8 Steam generator model


dTt ¼ U A Tt  θp1 + U A ðTt  θs1 Þ dt

(10.13)

Ms1 Cs

dθs1 UA ðTt  θs1 Þ ¼ Ws Cs ðθsin  θs1 Þ + dt 2

(10.14)

Ms2 Cs

dθs2 UA ðTt  θs1 Þ ¼ Ws Cs ðθs1  θs2 Þ + 2 dt

(10.15)

Mt Ct

where Mp1 ¼ mass of primary fluid node-1 Mp2 ¼ mass of primary fluid node-2 Mt ¼ mass of metal node (heat exchanger tubing) Ms1 ¼ mass of secondary fluid node-1 Ms2 ¼ mass of primary fluid node-2 Wp ¼ flow rate of primary fluid Ws ¼ flow rate of secondary fluid Cp ¼ specific heat capacity of the primary fluid Cs ¼ specific heat capacity of the secondary fluid Ct ¼ specific heat capacity of the tube metal U ¼ overall heat transfer coefficient from primary fluid to metal, or from metal to secondary fluid A ¼ heat transfer area from primary fluid to metal node, or from metal node to secondary fluid θpin ¼ temperature of primary fluid inflow θsin ¼ temperature of secondary fluid inflow θp1 ¼ temperature of primary fluid node-1 θp2 ¼ temperature of primary fluid node-2 θs1 ¼ temperature of secondary fluid node-1 θs2 ¼ temperature of secondary fluid node-2 Tt ¼ temperature of metal (tube) node

10.8 Steam generator model Steam generator modeling can be simple or complex. Fig. 10.7 shows a typical U-tube steam generator. Different requirements for secondary system modeling determine the detail required for steam generators. For example, if the simulation focuses on reactor behavior, then a simple steam generator model that adequately simulates heat removal is adequate. A simple model, called the “teakettle model” represents steam generator dynamics with only three Eqs. A schematic representation of a teakettle model appears in Fig. 10.8. This approach has been found to represent heat transfer to the secondary fluid quite well in overall system simulations [1].

119

120

CHAPTER 10 Reactor thermal-hydraulics

STEAM OUTLET TO TURBINE GENERATOR DEMISTERS SECONDARY MOISTURE SEPARATOR

SECONDARY MANWAY

UPPER SHELL

FEEDWATER RING

ORFICE RINGS SWIRL VANE PRIMARY MOISTURE SEPARATOR

FEEDWATER INLET

ANTIVIBRATION BARS

TUBE BUNDLE

LOWER SHELL

WRAPPER

TUBE SUPPORT PLATES

SECONDARY HANDHOLE

BLOWDOWN LINE TUBE SHEET

TUBE LANE BLOCK PRIMARY MANWAY

PRIMARY COOLANT OUTLET

PRIMARY COOLANT INLET

FIG. 10.7 Schematic of a typical U-tube steam generator (UTSG). Courtesy of Westinghouse Electric Company (The Westinghouse Pressurized Water Reactor Nuclear Power Plant, Westinghouse Electric Corporation, Pittsburgh, 1984).

10.8 Steam generator model

FIG. 10.8 Schematic representation of a teakettle model of a steam generator.

10.8.1 U-Tube steam generator (UTSG) More detailed models are needed if steam generator performance itself is under consideration. For example, feedwater control system simulation is often an objective. Finite difference models with parameter updates in each node as a transient proceeds are used, but are quite complex. The moving boundary approach described in Section 10.4 also be used for steam generator modeling and is somewhat simpler to implement [3–6]. The nodal structure for a moving boundary model for a U-tube steam generator appears in Fig. 10.9. The model uses fourteen coupled differential equations and additional algebraic equations. The complete model is described in Ref. [6].

10.8.2 Once-through steam generator (OTSG) Once-through steam generators for PWRs are shell and tube heat exchangers with liquid water inside the tubes and water and steam on the shell side. Fig. 10.10 is a schematic of a once-through steam generator (OTSG) [7]. A schematic for a moving boundary model appears in Fig. 10.11 [8]. Note that two boundaries move (the boundary between the subcooled water section and the boiling section, and the boundary between the boiling section and the superheating section). A detailed model of an OTSG was developed in Ref. [8].

121

122

CHAPTER 10 Reactor thermal-hydraulics

FIG. 10.9 Lumped parameter (nodal) structure of a moving boundary U-tube steam generator model [6].

10.9 Balance-of-Plant (BOP) system models The components of a balance-of-plant (BOP) system are similar in nuclear power plants that generate steam using reactor thermal power. These include PWRs, BWRs, CANDU reactors, sodium fast reactors, gas-cooled reactors, molten salt reactors, and small modular reactors. Fig. 10.12 shows the major components of a BOP system and associated state variables [8]. The figure shows the following BOP components: • • • • • • •

Steam chest High pressure and low pressure turbines Moisture separators and steam reheaters Condenser Feedwater heaters Condenser pressure controller Feedwater temperature controller.

10.9 Balance-of-Plant (BOP) system models

FIG. 10.10 Schematic of a once-through steam generator (OTSG). Courtesy of The Babcock & Wilcox Company (Steam, its generation and use, The Babcock & Wilcox Company, forty second ed., 2015).

123

124

CHAPTER 10 Reactor thermal-hydraulics

FIG. 10.11 A moving boundary model of OTSG [8].

The dynamics of these components are represented by using mass and energy conservation Eqs. A complete description of the governing equations is given in Refs. [6, 8] with associated plant parameters. Valve and pump models are not included in this representation. For a complete representation of the BOP dynamics it is necessary to include dynamic models of these components (and actuators).

10.10 Reactor system models Chapter 12 addresses PWR dynamics. An appendix to Chapter 12 illustrates the formulation of a complete PWR reactor system model using models described in Chapter 3 for neutronics and this chapter for heat transfer. Another appendix (Appendix K) addresses molten salt reactor dynamics. Appendix K provides an illustration of the use of modeling techniques for a reactor with very different properties than current commercial reactors.

Wv Hs

From OTSG

Win Hs

Ws Hs

Nozzle chest

Wnc Hnc Pnc

Whpt Hhpt Phpt

Reheater Moisture separator

Wms Wrh Hrh Prh

Steam valve

Wlpt Hlpt Plpt

HP turbine IP & LP turbine

Wbhp

Generator TCWin

TCWout

WlIq WbIp

Condenser

To OTSG

Wfw HP feedwater heater Whp→lp

FIG. 10.12 Schematic of the BOP system showing important components and associated state variables [8].

Wco Hco LP feedwater heater

10.10 Reactor system models

cooling water

Wvrh

125

126

CHAPTER 10 Reactor thermal-hydraulics

Exercises 10.1. Reformulate Mann’s model for an assumption that the average coolant temperature is the average of the temperature in each node rather than the temperature in the first node. How would this affect simulation results? 10.2. Compare the computational differences for modeling a boiling channel with the moving boundary approach and a model with fixed boundaries and updating of coefficients during a transient. 10.3. Consider a moving boundary model for a once through steam generator with superheat. How many boundaries are needed for a dynamic model? How would the boundaries move (up or down) following an increase in primary side fluid temperature? 10.4. Consider a PWR with dissolved boron in the coolant. Describe differences in the reactor power response to an increase in inlet coolant temperature for a condition of low boron concentration and a condition of high boron concentration. 10.5. A reactor with a negative fuel temperature coefficient of reactivity and a positive coolant temperature coefficient of reactivity can be stable even if the magnitude of the positive coefficient is larger than the magnitude of the negative coefficient. Explain how this is possible and why it might be counterintuitive. 10.6. A change in boiling rate caused by a disturbance in a Boiling Water Reactor causes changes in coolant density as it moves along the channel. BWRs have negative coolant density coefficients. This effect can be destabilizing. a. Explain how this happens. b. Would increasing the coolant flow rate make the system more stable or less stable? Explain.

References [1] T.W. Kerlin, Dynamic analysis and control of pressurized water reactors, in: C.T. Leondes (Ed.), Control and Dynamic Systems, vol. 14, Academic Press, 1978. [2] S.J. Ball, Approximate models for distributed parameter heat transfer systems, ISA Trans. 3 (1) (1964) 38–47. [3] M.R.A. Ali, Lumped-Parameter, State Variable Dynamic Model for U-Tube Recirculation Type Steam Generators, PhD dissertation, The University of Tennessee, 1976. [4] A. Ray, H.F. Bowman, A nonlinear dynamic model for a once-through subcritical steam generator, J. Dyn. Syst. Meas. Control 98 (3) (1976) 332–339. Series G. [5] T.W. Kerlin, E.M. Katz, J. Freels, J.G. Thakkar, Dynamic modeling of nuclear steam generators, in: Proceedings of the Second International Conference sponsored by the British Nuclear Energy Society, Bournemouth, England, 1979. October.

Further reading

[6] M. Naghedolfeizi, B.R. Upadhyaya, Dynamic Modeling of a Pressurized Water Reactor Plant for Diagnostics and Control, University of Tennessee Research Report, 1991. DOE/NE/88ER12824-02, June. [7] Steam, its generation and use, The Babcock & Wilcox Company, forty second ed., 2015. [8] V. Singh, Study of Dynamic Behavior of Molten Salt Reactors, MS Thesis, The University of Tennessee, Knoxville, 2019.

Further reading [9] A.T. Chen, A Digital Simulation for Nuclear Once-Through Steam Generators, PhD dissertation, The University of Tennessee, 1976. [10] The Westinghouse Pressurized Water Reactor Nuclear Power Plant, Westinghouse Electric Corporation, Pittsburgh, 1984.

127

CHAPTER

Nuclear reactor safety

11

11.1 Introduction Reactor safety is an important issue, deserving of a complete treatment. However, a full treatment is beyond the scope and purpose of this book. But all nuclear engineers need to know what can happen when things go bad. There have been accidents in several research reactors and in three nuclear power reactors. Reactor dynamic characteristics played a major part in determining the way one of the nuclear power plant accidents unfolded.

11.2 Reactor safety principles Two safety functions must occur following an abnormal safety-related event in a power reactor. The fission reaction must be quenched (usually by immediate insertion of safety rods, a reactor “scram”). This stops most of the fission power production, but some fissions continue and the inventory of radioactive elements continues to decay and produce heat. The decay heat power level depends on the operating history of the reactor, but initially is typically around 6% of previous reactor power. The second essential function is continued cooling of the reactor core. Cooling is required to prevent a temperature rise due to the continued power production and to prevent fuel melting. Backup power systems (emergency Diesel generators) are available in Generation II reactors to provide electricity to keep the coolant pumps running in the event of the loss of normal electrical power. Also, plants use auxiliary pumps, whose function is to come online if normal feedwater pumps are inoperable or have been valved out, which is a serious violation of procedure. It should be noted that newer reactor designs include emergency cooling systems that do not require electrical power. Cooling water enters the reactor by gravity or from pressurized vessels.

11.3 Early accidents with fuel damage Accidents occurred in the early days of reactor deployment. The worst were those in which fuel melting occurred. Accidents with fuel melting include those described briefly in the following section. Dynamics and Control of Nuclear Reactors. https://doi.org/10.1016/B978-0-12-815261-4.00011-1 # 2019 Elsevier Inc. All rights reserved.

129

130

CHAPTER 11 Nuclear reactor safety

11.3.1 Accidents NRX: An accident occurred on December 12, 1952 at the NRX reactor in Canada [1]. NRX is a heavy-water moderated, light-water cooled reactor. During shutdown, while working on the pneumatically operated control rod drives, the operator wrongly opened valves, causing control rods to be withdrawn. While acting to correct the problems, there was a miscommunication between operators, and control rods were further withdrawn. This caused a rapid increase in reactor power, with resulting fuel melting. SRE: An accident occurred in July 1959 at the Sodium Reactor Experiment (SRE) in California. SRE was a sodium cooled, graphite moderated reactor [2]. Tetralin (an organic liquid used as a sealant in the SRE pumps) leaked into the sodium coolant. Solid carbon resulted from degradation of the tetralin. The carbon restricted coolant flow through the core. Some of the fuel overheated and melted. SL-1: An accident occurred on December 21, 1960 at the SL-1 reactor in Idaho [3]. During preparations to restart the shutdown reactor, a control rod was withdrawn manually to attach to its drive mechanism. It was withdrawn too far and prompt criticality occurred. Power increased rapidly, causing an explosion in the reactor. Windscale: An accident occurred on October 10, 1957 at the Windscale reactor in Britain [4]. Windscale was an air cooled, graphite moderated reactor. A graphite fire occurred because of release of Wigner energy in the graphite. Wigner energy is energy stored in a material because of dislocation in the atomic structure of atoms caused by neutron absorption. This stored energy is released as heat when the atoms spontaneously relax from the dislocated condition. Wigner energy caused a fire in the graphite that burned for several days, destroying the reactor. Fermi-I: An accident occurred on October 5, 1966 at the Fermi-I reactor in Michigan [5]. Fermi-I was a sodium cooled fast reactor. A metal plate dislodged from its position beneath the reactor core. It flowed to the bottom of the core, blocking flow to several fuel elements. Some fuel overheated and melted. Lucens: An accident occurred on January 29, 1969 at the Lucens reactor in Switzerland [6]. Lucens was a carbon dioxide cooled, heavy water moderated reactor. During shutdown moisture condensed on fuel elements, causing corrosion. Corrosion products accumulated in the flow path, restricting coolant flow. The fuel overheated and melted. Russian Submarines: Russian nuclear submarines experienced an astonishing number of accidents (too many to describe here) [7].

11.3.2 Assessment One of the lessons learned from this limited experience is the importance of avoiding a loss-of-coolant event. Generation-II and -III reactors feature back-up cooling systems. But even they have failed in subsequent reactor accidents. Some of the accidents described above resulted in the death of plant workers, and direct exposure to radiation or radioactive contamination outside of the plant

11.4 Analysis of potential reactor accidents

boundary (with the possibility of causing cancer in exposed residents). These occurred because of an inadequate safety culture, design deficiencies, and unsafe operation (often because of failure to understand how to deal with abnormal reactor behavior). These events were disasters, but they were an unfortunate part of the path to better understanding of how to build and operate reactors safely. An analogy is the accidents that occurred during the age of early space exploration. A slower pace of early reactor implementation might have avoided the accidents, but there was a strong motivation to hurry in the interest of producing weapons materials or advancing nuclear power for generating electricity.

11.4 Analysis of potential reactor accidents Analyses are performed to assess reactor safety before they are licensed for construction and operation. These analyses include deterministic assessments and probabilistic assessments. The deterministic assessments involve simulation of postulated reactor accidents. The probabilistic assessments involve evaluation of the likelihood of component failures resulting in an accident. The deterministic simulations use all of the modeling methods described in previous chapters, but also model phenomena that do not occur in normal operation (such as loss of coolant or rapid control rod ejection). The reactor safety simulations must address extreme conditions such as boiling in normally liquid coolant, fuel melting, pressurization in the primary system or the containment building, and structural failures. Great effort at a number of government and private organizations to prepare various computer codes to simulate reactor accidents has occurred. For example, the reactor safety computer codes at the U.S. Nuclear Regulatory Commission (NRC) include the following [8]: • • • • • • • •

Probabilistic risk assessment codes Fuel behavior codes Reactor kinetics codes Thermal-hydraulics codes Severe accident codes Protection codes Radionuclide transport codes Materials performance codes

Safety analysis involves analysis of a Design Basis Accident (DBA) [9]. The DBA analysis is a simulation of the worst conceivable accident (typically a loss of coolant). The assumption is that if the reactor can tolerate a DBA, then it can tolerate other accidents considered less serious than the DBA. But experience has shown us the accidents that are worse than the DBA can occur because of unanticipated natural events (like a larger than anticipated tsunami), equipment failure (like a stuck valve), or operator errors. Such events are called Beyond Design Basis Accidents (BDBA) [10]. There are now active analyses of BDBAs underway and the U.S.

131

132

CHAPTER 11 Nuclear reactor safety

Nuclear Regulatory Commission (NRC) rule requires that these analyses be conducted for all nuclear plants – new reactor builds, reactors requiring life extension, and operating reactors. The NRC considers information provided by an applicant in a Safety Analysis Report (SAR). The SAR must address NRC requirements published in 10CFR50, a code of federal regulations (CFR) document [11]. The NRC reviews the SAR using procedures defined in NUREG800 (NUREG stands for Nuclear Regulation). Originally, applicants had to submit two SARs, a Preliminary SAR (PSAR) for permission to construct and a Final SAR (FSAR) to operate. Subsequently, the NRC simplified the application process, requiring only a single SAR. The applicant’s proposal is reviewed by the NRC staff and an independent group of reactor safety experts (The Advisory Committee on Reactor Safeguards or ACRS). The proposal also undergoes a public hearing where opponents are given an opportunity to express their concerns. The NRC can impose new requirements after the reactor goes into operation through Nuclear Regulatory Guides. These are imposed as a result of further NRC study, new developments or experience. For example, NRC decided that it is necessary to confirm that the response time of safety system sensors is as short as assumed in the SAR. The Electric Power Research Institute (EPRI) sponsored a research project that resulted in an in-situ response time testing of resistance thermometers [12] or resistance temperature detectors (RTD). The NRC approved the test and it is used routinely in PWRs. Probabilistic risk assessment (PRA) provides the calculated likelihood of an accident [13]. Likelihood of failures in safety-related components are combined to provide the probability of an overall failure and an accident. PRA results are presented as the likelihood of an accident in Y years. PRAs are done very carefully, but the possibility of inaccurate component failure probabilities or failure to realize and include an important component failure means that PRA results cannot be judged as perfect. Also, even a low probability does not mean that an event could not occur. A likelihood of one failure in a million years means that a failure in the first year is very unlikely, but not impossible. So, the issue is “How good is good enough?”

11.5 Accidents in Generation-II power reactors 11.5.1 Three mile Island [14] On March 28, 1979, an accident occurred at Three Mile Island unit 2 near Harrisburg, PA. TMI-2 was an 800 MWe PWR supplied by Babcock and Wilcox. The initial factor in causing the accident was a problem introduced during routine maintenance of a component in the secondary system called the condensate polisher. It is a filter used to purify the secondary water. Forcing water through the filter to clean it found its way into other secondary systems. The crucial effect was disabling the feedwater pumps that fed water into the steam generators.

11.5 Accidents in Generation-II power reactors

Without feedwater, the steam generators lost the ability to remove heat from the primary water. The resulting increase in primary temperature and pressure caused an automatic reactor scram. Auxiliary feedwater pumps were activated, but a closed valve blocked flow. This valve closure was a serious violation of procedure. The increased primary loop pressure caused opening of the pilot-operated relief valve at the top of the pressurizer. This allowed fluid to flow to the pressurizer relief tank. Opening the relief valve caused reduction of primary pressure. The pressure reduction should have caused the relief valve to close, but it was stuck open. Primary water continued to escape through the open valve. The operators had no indication of a stuck-open relief valve. The only indication was that a “close valve” signal had been sent, but there was no instrumentation to reveal the actual valve condition. As the outflow from the pressurizer continued, the operators acted on the indication that the pressurizer level was too high. The prescribed response is to reduce primary loop flow. At this point, coolant was escaping and no replacement coolant was entering. The result was fuel overheating and melting. Molten Zirconium reacted with water to produce hydrogen. Since hydrogen is an explosion hazard, it was crucial to remove it. Careful venting eliminated the hydrogen. Eventually, operators closed a block valve in the pressurizer relief piping and restored coolant flow. These actions resulted in ending the incident. In summary, four main missteps and design deficiencies caused the accident. Maintenance personnel performed a procedure that caused disabling of feedwater flow. Maintenance personnel closed a valve (a violation of plant procedures) that prevented flow from auxiliary feedwater pumps. A mechanical flaw caused the pressurizer relief valve to stick open. Inadequate instrumentation on relief valve position that left operators unaware of actual valve condition.

11.5.2 Chernobyl [15] An accident occurred at Unit 4 of the Chernobyl boiling water nuclear power plant near Pripyat, Ukraine on April 26, 1986. The1000 MWe reactor was a Soviet design called an RBMK plant. It uses water coolant flowing in vertical pressure tubes imbedded in a graphite moderator. The fuel is 2% enriched Uranium in UO2 pellets enclosed in Zircaloy cladding. The coolant boils and, after passing through steam separators, the resulting dry, saturated steam flows into the turbine. Neutrons are slowed down by the graphite moderator and the in-core water moderator/coolant. The reactor is over-moderated. That is, removal of moderator increases reactivity. In an over-moderated reactor, the increase in reactivity occurs because the reduction of moderator absorptions is greater than the decrease in reactivity because of reduction in neutron thermalization. Boiling of the water in an RBMK decreases the quantity of water in the core, thereby reducing the neutron absorptions in the water. Of course, absorptions and slowing down in the graphite are unaffected by a change in water density, so the water density effect dominates, giving the RBMK a positive void coefficient.

133

134

CHAPTER 11 Nuclear reactor safety

The Doppler coefficient in the fuel is always negative, so fuel temperature is a stabilizing effect. The power coefficient is the net effect of the negative Doppler effect and the positive void reactivity effect. Since at low power levels, fuel temperature feedback is smaller than at high power levels, the positive void effect dominates. The RBMK has a very undesirable large positive power coefficient at low power. Another important safety-related feature of the RBMK reactor is the design of its control rods. Insertion drives them into water-filled channels. The positive reactivity associated with removal of water would be overwhelmed if the control rods contained a strong neutron absorber along its full length. But the bottom section of the control rods contains graphite, a much weaker neutron absorber than water. Consequently, the bottom of the control rod introduces positive reactivity upon insertion. The part of the control above the graphite section contains a strong neutron absorber and it introduces negative reactivity when that portion enters the core. The accident occurred because of problems initiated by an experiment designed to evaluate the potential of a way to improve emergency cooling of the reactor. In the event of a reactor scram and simultaneously losing electrical power, diesel generators start to provide electricity to power cooling pumps, but the rise to full generator power is slow. So, the possibility of getting temporary electrical power from the turbine as it coasted down was to be evaluated. The experiment was to be performed with the reactor at a power level of 700 to 1000 MWth. This would have avoided the high positive power coefficient at lower power levels. However, stabilizing the power level at the desired value did not occur as planned. Furthermore, delays caused an operator shift change to operators who were not as well informed about the test procedure. Power reduction from full power began and had reached about 50% power when the dispatcher prohibited further power reduction because of grid power requirements. After a delay, permission was granted to continue with power reduction. In the attempt to reduce the power level to that needed for the experiment, power was inadvertently reduced to a very low level (around 30 MWth). At this point the operator started removing control rods to increase power. Because of Xe-135 buildup at low power due to I-135 decay, a number of control rods were withdrawn to compensate for the reactivity loss due to Xe-135. Power eventually stabilized at around 200 MWth. It was decided to proceed with the test even though the power level was far below that prescribed for the test. It was a condition in which the reactor had a very strong void coefficient. It was a fatal decision. The steam flow to the turbine was stopped to start the test. Pumping power decreased as their electrical power from the slowing turbine—generator decreased. Coolant flow to the reactor decreased, boiling of coolant increased, reactivity increased, and fission power increased rapidly. The increased power caused further boiling and further increased reactivity. This void reactivity (along with reactivity increases caused by burnout of Xe-135) caused strong reactivity feedback and continuing uncontrolled power increase. Because key personnel were killed and records were lost, there is confusion about events that occurred at this point. It is possible that an attempt to stop the power rise by inserting control rods made the problem worse

11.5 Accidents in Generation-II power reactors

because insertion introduced graphite at the tip of the rods, introducing reactivity. In any case, a runaway power transient occurred and destroyed the reactor. Explosions occurred, exposing the damaged reactor to the atmosphere and releasing radionuclides. Radioactive debris littered the reactor site and fires erupted. Heroic workers risked their lives to bring things under control. Thirty-two people died immediately or shortly after the accident. The radiation contamination in surrounding regions created the likelihood of eventual health problems for residents.

11.5.3 Fukushima Dai-ichi [16] On March 11, 2011 a major accident occurred at the Fukushima Dai-ichi power plant on the eastern shore of Japan near the city of Sendai. The plant has six boiling water reactors built by General Electric. Unlike the prior accidents at Three Mile Island and Chernobyl, neither operator errors nor reactor design flaws caused the accident. The cause was a huge earthquake-caused Tsunami and unsafe location of emergency power systems and heat exchangers. At the time of the earthquake, units 1, 2 and 3 were operating and units 4, 5 and 6 were shut down. The operating reactors were scrammed immediately after the earthquake struck. Fifty minutes after the earthquake a 13 m (42.6 ft) high tsunami struck. The reactor buildings survived the earthquake and tsunami, but the offsite power was disrupted and the backup power sources (diesel generators and batteries were disabled) and heat exchangers used to transfer heat to seawater were destroyed. The unsafe positioning of crucial safety-related backup systems close to the ocean was clearly a mistake. The potential problem with the location of these systems was recognized and some work had started on relocating these systems to higher ground, but little had been done prior to the accident. Without cooling, decay heat caused fuel melting in units 1, 2 and 3. Additionally, a chemical reaction between Zirconium and water produced hydrogen gas, which eventually exploded in units 1, 2, 3 and 4 (which received hydrogen leaked from unit 3). Units 1–4 were destroyed and radionuclides were released into the air and water, creating long-term health concerns for residents in nearby areas. This increased the already large impact on the human population (almost 19,000 died immediately due to the tsunami). After the initial chaos subsided, a long process of stabilization and cleanup began. As mentioned above, the Fukushima event is a Beyond-Design-Basis Accident. A near-term task force established by the U.S. Nuclear Regulatory Commission published a report on the lessons learned from the Fukushima Dai-ichi accident [17]. The NRC recommendations included twelve major points encompassing regulatory framework, ensuring protection, enhancing mitigation, strengthening emergency preparedness, and improving the efficiency of NRC programs. These must be applied to both operating reactors and new reactors.

135

136

CHAPTER 11 Nuclear reactor safety

11.6 Consequences and lessons learned Many people who received radiation doses from Chernobyl and Fukushima Dai-ichi live in fear of disease or death. Passage of many years will reveal their fates. The three accidents described above had a large impact on the nuclear industry around the world. Reactors had been or would be shut down, construction was stopped, and plans for new plants were abandoned. Rebirth of the nuclear seemed imminent in the early twenty-first century until the Fukushima Dai-ichi accident again created new concerns and cancellation or postponement of plans for new reactors. But some countries were not dissuaded in their new reactor construction programs. New reactor programs continue in many countries, especially China, Russia and India. The causes of the accidents ranged from operator error to equipment malfunction to natural catastrophe to reactor design flaws to ignorance about the dangerous dynamics of reactors under certain operating conditions. These problems led to an increasing operator training, system modifications and stronger regulatory oversight. Reactors are certainly safer as a result of the painful lessons learned from the accidents, but there is no room for complacency. Ref. [18] is a good book on light water reactor safety. The book describes the thermal-hydraulics of severe accidents and contains case studies of reactor accidents

Exercises 11.1. Prepare of a flow chart showing the sequence of events and decisions in the Three-Mile Island accident. 11.2. Repeat Exercise 11.1 for the Chernobyl accident. 11.3. Repeat Exercise 11.1 for the Fukushima Dai-ichi accident. 11.4. There have been many articles written about the reactor accidents described above. Find one of these articles on the internet and write a review, citing the accuracy or inaccuracy of the article and providing a critique of any conclusions in the article. 11.5. Write a review of the NRC report on Fukushima Dai-ichi lessons-learned (Ref. [17]), and expand on the important recommendations in the report. 11.6. Find a website that says that nuclear power is unsafe. Write a short (1000 to 1200 words) review. Be objective, citing claims that you consider are true as well as those that you consider are false. Cite reasons for your assessments. Include a copy of the article that you reviewed as well as your review.

References

References [1] NRX Reactor Accident Description at https://nuclear-energy.net/nuclear-accidents/ chalk-river.html. [2] SRE Reactor Accident description at www.etec.energy.gov/Library/Main/Pickard% 20SRE%20presentation.pdf. [3] SL-1 Reactor Accident Description at https://timeline.com/arco-first-nuclear-accidentf16ec1105b9c. [4] Windscale Reactor Accident Description at www.nucleartourist.com/events/windscal. htm. [5] Fermi-1 Reactor Accident Description at mragheb.com/NPRE%20457%20CSE% 20462%20Safety%20Analysis%20of%20Nuclear%20Reactor%20Systems/ Fermi%20I%20Fuel%20Meltdown%20Incident.pdf. [6] Lucens Reactor Accident Description at https://www.revolvy.com/page/Lucens-reactor. [7] Russian Submarine Reactor Accident Description at NRX Reactor at https://www. theguardian.com/world/2000/aug/15/kursk.russia1. [8] 19CFR50 Description at www.nrc.gov/reading-rm/doc-collections/cfr/part050. [9] DBA Description at www.nrc.gov/.../basic-ref/glossary/design-basis-accident.html. [10] BDBA Description at www.nrc.gov/.../glossary/beyond-design-basis-accidents.html. [11] NRC Codes Description at www.nrc.gov/about-nrc/regulatory/research/safetycodes. html. [12] T.W. Kerlin, L.F. Miller, H.M. Hashemian, In-situ response time testing of platinum resistance thermometers, ISA Trans. 17 (4) (1978) 71–88. [13] EPRI, Basics of Nuclear Power Plant Probabilistic Risk Assessment, Electric Power Research Institute (EPRI) report, Available at www.mydocs.epri.com/docs/ publicmeetingmaterials/1108/J7NBS83L7MY/E, 2011. [14] TMI reactor accident description at www.history.com/this-day-in-history/nuclearaccident-at-three-mile-island. [15] Chernobyl Reactor Accident Description at www.history.com/this-day-in-history/ nuclear-disaster-at-chernobyl. [16] Fukushima Reactor Accident Description at www.world-nuclear.org/information-library/ safety-and-security/safety-of-plants/fukushima-accident.aspx. [17] U.S. Nuclear Regulatory Commission, Recommendations for Enhancing Reactor Safety in the 21st Century, the NRC Near-Term Task Force Review of Insights from the Fukushima Dai-Ichi Accident, June 2011, 2011. [18] B.R. Segal (Ed.), Nuclear Safety in Light Water Reactors, Academic Press, 2012.

137

CHAPTER

Pressurized water reactors

12

12.1 Introduction This chapter addresses generation II and III pressurized water reactors (PWRs). Pressurized water reactors are the most common type in the U.S. They are also used extensively in other countries. Commercial PWR development followed successful development of PWRs for naval applications. There has been little change in the basic design of PWR power plants and their control systems since their initial implementation in the mid-twentieth century. The early designers knew what they were doing.

12.2 PWR characteristics [1–3] Three different PWR manufacturers in the U.S. supplied Generation II and III nuclear power plants. Westinghouse Electric Company and Combustion Engineering provided plants with U-tube steam generators (UTSG), and the Babcock and Wilcox (B&W) Company provided plants with once through steam generators (OTSG). The Russian VVER pressurized water reactor uses horizontal shell-and-tube steam generators. Fig. 12.1 shows the layout of a typical PWR plant. A PWR with U-tube steam generators is chosen to show the components in a PWR plant. The primary system consists of a reactor vessel, hot leg and cold leg piping, a pressurizer, and reactor coolant pumps (RCP). The U-tube steam generator (UTSG) connects the primary system with the secondary system or the balance-of-plant (BOP) system. The BOP system includes the steam turbine, an electrical generator, steam re-heaters, moisture separators, feedwater heaters, and a condenser. The reactor vessel and the steam generators are located in a large containment building. The turbine-generator system and the BOP components are located in a separate building. Design parameters for a typical 1150 MWe four-loop PWR (Westinghouse PWR) appear in Appendix A. The primary water (
550 °F) from cold leg pipes enters the reactor vessel near the top, flows down through an annular region between the inner wall of the reactor vessel and a shroud around the core (called the core barrel) to the bottom of the vessel, and then flows up through the core, thereby removing heat from the fuel. Dynamics and Control of Nuclear Reactors. https://doi.org/10.1016/B978-0-12-815261-4.00012-3 # 2019 Elsevier Inc. All rights reserved.

139

140

CHAPTER 12 Pressurized water reactors

FIG. 12.1 Layout of a typical PWR plant. Source: U.S. Nuclear Regulatory Commission, www.nrc.gov/reactors/pwrs.html.

The nominal UO2 fuel centerline temperature at full power is 4140 °F. The water exits the reactor vessel (
615 °F) at the top and flows through hot leg piping into steam generators. The primary water enters the steam generator at the inlet plenum and flows through the steam generator tubing. The total water flow rate through the core in a typical 1150 MWe plant is approximately 138  106 lbm/h. The average core coolant velocity is 15.5 ft./s. The 1150 MWe Westinghouse PWR has four U-tube steam generators (UTSGs). Sections 10.6 and 12.5.1 provide details of UTSG characteristics.

12.4 The pressurizer

The pressure in the primary system is maintained at about 2250 lb./in2 (
153 bars, 1 bar ¼ 105 Pa) so that there is no boiling in the primary system. This is accomplished by a pressurizer, which is connected to one of the hot legs. Sections 10.7 and 12.4 provide details of the pressurizer. The combination of reactor core (thermal power) and the steam generator system is called the Nuclear Steam Supply System (NSSS).

12.3 The reactor core All three U.S. PWR manufacturers used similar reactor core designs. Differences included fuel bundle designs and control rod designs. A cylindrical vessel contains the fuel assemblies in PWRs. Fig. 12.2 depicts a typical reactor vessel, showing reactor internals and control rod drive mechanisms. A fuel assembly is shown in Fig. 12.3. The reactor vessel diameter is about 14 ft., and the height is about 44 ft. The reactor vessel material is carbon steel (8-in. thick) with an inside stainless-steel cladding (nominal thickness 7/32 in.). The weight of the uranium oxide fuel is approximately 82,000 kg. A core support barrel is attached to the reactor vessel flange, and it supports the core. There are approximately 200 fuel assemblies in a typical 1150 MWe PWR. A fuel assembly is a bundle of fuel pins (or fuel rods), typically in a 17  17 or 19  19 array. The fuel pins form an open lattice structure. The fuel rod contains pellets of Uranium oxide (UO2) (
0.4 in. diameter x 0.6 in. long) contained in Zircaloy cladding tubes with a He-filled gap of
0.002 in. The active core length is 12 ft. and has a diameter of about 11 ft. Water flows upwards through the core, to the vessel upper plenum, and to hot leg piping that leads to the steam generator. Mechanically driven control rods enter from above (see Fig. 12.4). There are about 60 control rod assemblies (clusters) that occupy spaces in the fuel assemblies.

12.4 The pressurizer A pressurizer is connected to the hot leg through a pipe (or a surge line). Also see Section 10.6. Fig. 12.5 shows a typical pressurizer. The purpose of the pressurizer is to control the pressure in the primary loop at a nominal coolant pressure of 2250 lb./in2 (
153 bars). The primary pressure is regulated by modulating heater power and spray flow from a cold leg. Fig. 12.6 shows a pressure controller. The overall height and diameter of the pressurizer in a four-loop plant are approximately 52 ft., 9 in. (16.1 m) and 7 ft., 8 in. (2.3 m), respectively and a total volume of 1800 ft3 (31 m3). The pressurizer has electric immersion heaters with a total power of 1800 kw. The maximum spray flow rate is 900 gpm (57 l/s) with a continuous flow

141

142

CHAPTER 12 Pressurized water reactors

FIG. 12.2 A typical PWR reactor vessel. Courtesy of Westinghouse Electric Company (Westinghouse Electric Company, The Westinghouse Pressurized Water Reactor Nuclear Power Plant, Westinghouse Electric Company, Water Reactor Division, Pittsburgh, 1984).

rate of 1 gpm (63 ml/s) [2]. The pressurizer has two power-operated relief valves and three self-actuating safety valves. The water in the pressurizer is the only free surface in the primary coolant system. At full power the pressurizer contains 60% of its volume full of water. Changes in

12.4 The pressurizer

FIG. 12.3 A PWR fuel bundle. Courtesy of Westinghouse Electric Company (Westinghouse Electric Company, The Westinghouse Pressurized Water Reactor Nuclear Power Plant, Westinghouse Electric Company, Water Reactor Division, Pittsburgh, 1984).

143

144

CHAPTER 12 Pressurized water reactors

FIG. 12.4 A PWR control rod assembly. Courtesy of Westinghouse Electric Company (Westinghouse Electric Company, The Westinghouse Pressurized Water Reactor Nuclear Power Plant, Westinghouse Electric Company, Water Reactor Division, Pittsburgh, 1984).

12.4 The pressurizer

FIG. 12.5 A typical pressurizer. Courtesy of Westinghouse Electric Company (Westinghouse Electric Company, The Westinghouse Pressurized Water Reactor Nuclear Power Plant, Westinghouse Electric Company, Water Reactor Division, Pittsburgh, 1984).

145

146

CHAPTER 12 Pressurized water reactors

On/Off Heater

Pressure Setpoint Pressure Measurement

–

Pressure Controller

Proportional Heater

Spray

+

FIG. 12.6 A pressurizer pressure controller, showing both heater and spray actuation.

pressurizer water level are usually the result of water density changes caused by changes in average coolant temperature. A system called the Chemical and Volume Control System (CVCS) [1] or the Makeup and Purification System controls the water level in the pressurizer. For simplicity, the term CVCS will be used here to designate both types. Water is injected into the primary coolant system to increase the pressurizer water level to the set point. A let down flow system decreases the water level. As the name suggests, the CVCS controls more than pressurizer level. Two other functions of the CVCS are water purification using filters and demineralizers and controlling soluble poison concentration by adding or removing boric acid.

12.5 Steam generators 12.5.1 U-tube steam generator (UTSG) Fig. 12.7 shows a U-tube steam generator. Also see Section 10.8. Hot leg water enters a plenum at the bottom of the U-tube steam generator, then flows upward in tubes to the bend at the top then down into an outlet plenum (hence a U-tube). There are approximately 6000 tubes in a typical UTSG, with a tube OD of 0.875-in. (22.1 mm). A typical UTSG is approximately 68-ft tall, with the upper and lower shell diameters 15-ft and 11-ft, respectively. See Table A.1 for additional information about UTSGs. The hot water inside the tubes transfers heat to water flowing outside the tubes. The water outside of the tubes, in the shell region, is called secondary water. A mixture of secondary steam and water exits the heated section and flows into steam separators and driers. The steam separators contain vanes that induce swirling flow. Centrifugal force drives the liquid water outward. The partially dried steam then passes into stream driers where channels cause a 180° change in flow. This further separates steam and water. The water from the steam separator and drier flows down into an annular region between the steam generator vessel inner wall and a shroud that encircles the tube region. This annular region is called the downcomer.

12.5 Steam generators

FIG. 12.7 A U-tube steam generator. Courtesy of Westinghouse Electric Company (Westinghouse Electric Company, The Westinghouse Pressurized Water Reactor Nuclear Power Plant, Westinghouse Electric Company, Water Reactor Division, Pittsburgh, 1984).

147

148

CHAPTER 12 Pressurized water reactors

Feedwater enters the steam generator through a nozzle above the tubes and flows into the downcomer where it mixes with the recirculation flow. The saturated steam exiting the steam separators and driers passes through piping and a nozzle chest to the turbine. A slightly different version of the U-tube steam generator also exists. This system, called an integral economizer steam generator, features a chamber around the tubes where they exit the steam generator. Part of the feedwater enters this chamber, called the economizer, rather than flowing into the downcomer region. The water entering the economizer is cooler than the recirculated water entering the downcomer from the steam separators and driers. The cooler water enhances heat transfer from the primary water. Heated water flows from the economizer and mixes with water from the downcomer. The improved heat transfer increases the efficiency of the steam generator. Section 10.8 addresses U-tube steam generator modeling.

12.5.2 Once-through steam generator (OTSG) A once-through steam generator is basically a vertical shell and tube heat exchanger. Also see Section 10.8. Fig. 12.8 shows a once-through steam generator. A typical OTSG is about 73-ft tall (22.25 m) and has a 13-ft (3.96 m) shell diameter. It has 15,000 to 16,000 vertical tubes with an OD of 0.625-in. (15.9 mm). The tube material is Inconel-600 with a tube thickness of 0.034-in. (0.864 mm) [3]. More details of OTSGs may be found in Ref. [3]. Hot leg water enters at the top, flows downward through the tubes then exits at the bottom. Secondary water flows upward outside of the tubes, in the shell side of the steam generator. Heat transfer in the lower part of the tube region brings the secondary water to the saturation condition and boiling ensues. As secondary steam continues upward, heat transfer raises the steam temperature above the saturation temperature. The resulting superheated steam passes through piping to the turbine. Section 10.8 addresses once through steam generator modeling.

12.5.3 Horizontal steam generator There is also a third type of PWR steam generator. It is used in the Russian VVER reactor. VVERs use horizontal shell-and-tube steam generators.

12.6 Reactivity feedbacks Reactivity feedbacks occur in a PWR as a result of fuel temperature changes and moderator/coolant temperature changes. The fuel temperature coefficient, due to the Doppler effect, is always negative (see Section 7.2). The fuel temperature reactivity dominates feedback in PWRs. Reactivity change is due to the product of fuel temperature change and the fuel temperature

FIG. 12.8 A once-through steam generator. Courtesy of the Babcock & Wilcox Company (The Babcock & Wilcox Company, Steam: Its Generation and Use, The Babcock & Wilcox Company, 42nd ed., 2015).

150

CHAPTER 12 Pressurized water reactors

coefficient of reactivity. Since fuel temperature changes much more rapidly than changes in coolant temperature following a power change, the fuel temperature reactivity dominates prompt reactivity feedback following a reactor power change. The feedback reactivity due to coolant temperature changes occurs because temperature changes cause density changes and because temperature changes cause changes in the thermal neutron spectrum (see Section 7.3). Coolant density changes affect the quantity of coolant in the core, and, consequently, parasitic absorptions in the coolant and thermalization of fission neutrons. Increased temperature decreases the rate of parasitic absorptions, increases reactivity and causes a positive component of the coolant temperature coefficient of reactivity. Increased temperature reduces the quantity of moderator per unit volume in the core, reduces neutron thermalization, decreases reactivity and causes a negative component of the coolant temperature coefficient of reactivity. The thermalization effect dominates when dissolved poison concentration is low or zero. In other words, the reactor is under-moderated. Dissolved neutron absorber (boric acid) is used to reduce available reactivity in PWRs. Early in core life, available reactivity is large. Using dissolved neutron absorber has two advantages: it reduces the need for control rod reactivity strength and it controls reactivity without causing local power spikes or dips. However, the presence of a strong absorber in the coolant/moderator affects the coolant temperature coefficient of reactivity. A coolant temperature increase reduces coolant density, thereby forcing coolant and contained boric acid out of the core. Consequently, the coolant temperature coefficient of reactivity is positive when the boric acid concentration is high. This is the case early in the life of a fuel loading when a large boric acid concentration is used to offset a large available reactivity. This affects transient behavior, but the fuel temperature coefficient of reactivity (always negative) ensures satisfactory transient behavior. Increases in moderator temperature cause hardening of the thermal neutron spectrum. This causes a negative component of the reactivity change due to changes in U-235 absorptions and a positive component due to changes in Pu-239 absorptions. See Section 7.3. As in all thermal spectrum power reactors, burnup and production of Xe-135 causes a reactivity feedback effect. See Section 6.2 for the Xe-135 effect on power variations. The power coefficient of a PWR is always negative. The Doppler coefficient is dominant, including when the moderator/coolant reactivity coefficient is positive. Consequently, the response to an external reactivity change is a transition to a new steady state power level.

12.7 Power maneuvering Power may be changed by operator action or, in load following plants, by changes specified by changes in the demand from the electrical grid. A logical place to start in designing power maneuvering capability is to consider the way a plant

12.8 Steady-state programs for PWRs

would respond to a change in power demand with no control action. For example, run a simulation with opening or closing the main steam valve and observe what happens. Then the designer could undertake design of control strategies that exploit desirable features of the inherent response or overcome undesirable features. First, consider the inherent (no control action) response for a PWR. The scenario following opening of the steam valve is as follows: – – – – – – – – –

steam valve opening " steam flow to turbine " turbine power " steam pressure # steam production in steam generator " temperature on secondary side of steam generator # heat transfer rate from primary to secondary " primary side fluid temperature # reactor inlet moderator/coolant temperature #.

Reactor with negative moderator/ coolant temperature coefficient of reactivity

Reactor with positive moderator/ coolant temperature coefficient of reactivity

reactivity " reactor power "

reactivity # reactor power # fuel temperature # reactivity " reactor power "

This scenario shows that reactors with negative coolant and fuel temperature coefficients inherently respond by increasing reactor power. Also, reactors with positive coolant temperature coefficients but negative fuel temperature coefficients in which the feedback from fuel temperature is stronger, inherently respond by increasing reactor power. The net result in both cases is an inherent reactor power increase in response to a steam flow increase. It can be shown that the ultimate reactor power change exactly matches the turbine power change. Therefore, the control engineer’s job is to enhance this performance to achieve desired values for other process variables besides power.

12.8 Steady-state programs for PWRs The next step after understanding a power reactor response with no control action is to develop a control strategy that ensures that all process variables achieve desired values. This involves establishing set points and identification of control actions that will drive

151

152

CHAPTER 12 Pressurized water reactors

selected process variables to their set points in a timely and safe manner. The variation of process variable set points with power level is called a steady-state program.

12.8.1 Heat transfer in a steam generator Relationships that must exist at steady state can be used to define a steady-state program. Simplified and approximate versions of these relations suffice here since the purpose is explaining the development of a steady-state program. The relationship for heat transfer in the steam generator is as follows: PSG ¼ USG ASG θavg  TS




(12.1)

where PSG ¼ power extracted from the steam generator θavg ¼ average primary coolant temperature in the core (and the steam generator) TS ¼ average steam temperature USG ¼ steam generator overall heat transfer coefficient from primary coolant to secondary side ASG ¼ steam generator heat transfer surface area. Eq. (12.1) shows that average coolant temperature and steam temperature cannot both be constant if the value of USG is constant. Reactors with U-tube steam generators produce saturated steam and almost all of the heat transfer surface produces boiling. The steam generator overall heat transfer coefficient varies only slightly with changes in power level. Therefore, the choice is to hold either average coolant temperature or steam temperature constant or to cause both to change in a prescribed way. Here, we make the assumption that the overall heat transfer coefficient is constant at all power levels. This is not strictly true, but it is close enough for facilitating our understanding of steady state programs. Other relationships are necessary at steady state condition. They are stated in the following.

12.8.2 Fuel-to-coolant heat transfer PR ¼ UFC AFC TF  θavg




where UFC ¼ average fuel-to-coolant heat transfer coefficient AFC ¼ fuel-to-coolant heat transfer surface area TF ¼ average fuel temperature PR ¼ reactor power (thermal power transferred from fuel to coolant).

(12.2)

12.8 Steady-state programs for PWRs

12.8.3 Equivalence between reactor power and power delivered to the steam generator at steady state At steady state operation the reactor power is equal to the heat transferred in the steam generator. PR ¼ PSG

(12.3)

The average temperature in Eqs. (12.1) and (12.2) is defined as θavg ¼

ðθHL + θCL Þ 2

(12.4)

θHL ¼ 2 θavg  θCL

where θHL ¼ average hot leg temperature. θCL ¼ average cold leg temperature.

12.8.4 Energy change in the coolant The energy carried by the flowing primary coolant is given by P ¼ W C ðθHL  θCL Þ

or P ¼ W C 2θavg  2θCL

(12.5)


(12.6)

Solving for the cold leg temperature gives θCL ¼ θavg 

P ð2WCÞ

(12.7)

where W ¼ mass flow rate of coolant C ¼ specific heat capacity of coolant. Now consider the case in which the average coolant temperature for power level, P, is specified. Then θCL is given by Eq. (12.7). The hot leg temperature, θHL, is given by θHL ¼ 2 θavg  θCL

(12.8)

Control rod action permits operation with constant average coolant temperature, constant average steam temperature or prescribed changes in average coolant temperature and steam temperature as a function of steady state reactor power.

12.8.5 Development of a steady-state program The development of a steady state program in which it is desired to specify average coolant temperature proceeds as follows:

153

CHAPTER 12 Pressurized water reactors

– – – –

Specify the desired values of average coolant temperature at power, P. Calculate the average steam temperature using Eq. (12.1). Calculate the cold leg temperature using Eq. (12.7). Calculate the hot leg temperature using Eq. (12.8).

The above development shows that specification of reactor power and one other variable (average coolant temperature in this example) permits calculation of the other variables in a steady-state program (TS, θHL and θCL). The required control rod reactivity is normally not shown in a steady-state program, but it is readily calculated using a reactivity balance:

αf Tf  Tf 0 + αc θavg  θavg0 + δρcont ¼ 0

(12.9)

αf ¼ fuel temperature coefficient of reactivity αc ¼ coolant temperature coefficient of reactivity Tf0 ¼ fuel temperature at zero power θavg0 ¼ average coolant temperature at zero power. A steady-state program with constant average coolant temperature is preferred for the primary coolant loop because it limits the duty on the pressurizer. A steady state program with constant average steam temperature is preferred for the steam temperature because it permits optimization of turbine performance. The control policy being used in practice is a compromise. The coolant average temperature set point increases with power level; and steam temperature automatically decreases, but not by a large amount. A steady-state program for a PWR with a U-tube steam generator is shown in Fig. 12.9. The steady-state temperature changes are shown from zero power (hot functional condition) to 100% power level for a typical 1100 MWe Westinghouse PWR. 610

900

600 880 Psteam

590

860 580 570

840

Tout

560

Tavg

820

Steam pressure (psia)

Reactor coolant temperature (°F)

154

550 Tin

800

540 530 0

10

20

30

40

50

60

70

80

90

780 100

FIG. 12.9 Steady-state program (sliding average temperature program) for a typical 1100MWe PWR with U-tube steam generators.

12.8 Steady-state programs for PWRs

12.8.6 Steady-state program for PWRs with once-through steam generators (OTSG) The situation for PWRs with once through steam generators is quite different. The significant difference between U-tube and once-through steam generators is the difference in heat transfer coefficients in the steam generator. We have seen that the heat transfer coefficient in U-tube steam generators is nearly constant at different power levels. In once-through steam generators, there are regions with very different heat transfer coefficients. The overall heat transfer coefficient has three components: the primary side coolant to metal, the internal component in the metal tube, and the surface or film component. The primary side and internal component heat transfer coefficients do not change when secondary fluid conditions change, but the film component depends strongly on secondary fluid conditions. Feedwater enters the heated region at close to the boiling point. So, the heat transfer near the bottom of the steam generator is boiling heat transfer with a large film heat transfer coefficient (typically 6000 to 7000 BTU/(hr-ft2-°F). All of the feedwater boils at some point and the remainder of the heated region causes superheat. The film heat transfer coefficient on the secondary side in the superheating region is much lower than in the boiling region (typically 200 to 300 BTU/(hr-ft2-°F)). Since more heat transfer is required to boil the feedwater at higher power levels, more of the heat transfer surface is used to boil secondary water at higher power levels. We can see the effect on steam temperature of increasing heat transfer using a simple and approximate assumption. That is, assume that the heat transfer coefficient increases linearly with power as indicated below. USG ASG ¼ USG ASG0 ð1 + bPÞ

(12.10)

where b ¼ slope of USGASG vs. power. USGASG0 ¼ value when P ¼ 0. Substituting Eq. (12.10) into Eq. (12.1) gives TS ¼ θavg 

P ðUSG ASG0 ð1 + bPÞÞ

(12.11)

Since all quantities in the second term on the right-hand side are positive, the steam temperature change is smaller when b is non-zero than when it is zero (as with a U-tube steam generator). Therefore, steam temperature in a steady state program (with constant average temperature) decreases relative to the steam temperature variation in a system with a constant steam generator heat transfer coefficient. In reactors with once through steam generators, it is possible to implement a steady state program with constant average coolant temperature over most of the range of power levels. Eq. (12.1) is satisfied by increases in the effective heat transfer coefficient and by decreases in steam temperature. Since there is no fixed relation between temperature and pressure for superheated steam, the pressure can be

155

156

CHAPTER 12 Pressurized water reactors

FIG. 12.10 Steady-state program for a PWR with once-through steam generators.

controlled separately by adjusting the steam flow rate (or feedwater flow rate). Fig. 12.10 shows a steady-state program for a PWR with a once-through steam generator, similar to that used in a B&W reactor. Eq. (12.11) provides the resulting steam temperature changes. Reactivity control uses the coolant average temperature specified in the steady state program as the set point used for control rod motion. The control action causes a sequence of events that eventually affects the quantities of interest. The sequence of events is as follows: – change in coolant average temperature or change in reactor power # – deviation from set point from the steady-state program causes control rod motion # – control rods change reactivity # – reactivity change causes power change # – power change causes fuel temperature change # – fuel temperature change causes coolant temperature change #

12.9 Control rod operating band and control rod operation Two considerations dictate the allowable positioning of control rods: – There must be enough out-of-core and partially-inserted rods to bring the reactor subcritical. – There must be enough control rods in the reactor to be able to introduce reactivity needed in power change maneuvers. These constraints dictate that there is an allowable band of control rod positions.

12.9 Control rod operating band and control rod operation

FIG. 12.11 Reactivity controller and control rod speed program used in a typical Westinghouse PWR. Adapted from the U.S. Nuclear Regulatory Commission (Westinghouse Electric Company, The Westinghouse Pressurized Water Reactor Nuclear Power Plant, Westinghouse Electric Company, Water Reactor Division, Pittsburgh, 1984.). SPM: steps per minute.

The rod controller receives the temperature error signal and responds as shown in Fig. 12.11. This is similar to reactivity control in a Westinghouse PWR. During loadfollowing maneuver, the total error signal may also include a rate of change of power mismatch, converted to an equivalent temperature value. During a load-following maneuver, the total error signal may also include a rate of change of power mismatch, converted to an equivalent temperature value. Thus, the total error is a sum of T-average error and power mismatch error [2]. The full-length rod control cluster (RCC) assemblies are operated by a magnetic jack mechanism. The rod motion has a dead-band of 1.5 °F. The rod motion goes to

157

158

CHAPTER 12 Pressurized water reactors

zero for errors in the range  1.0 °F. The dead-band includes a 0.5 °F lock-up [2]. This avoids a continuous chatter in the control rod motion due to small errors in a small range adjacent to a 1 °F temperature error. Fig. 12.4 shows the magnetic jack control rod drive mechanism (CRDM) used in Westinghouse PWRs. The minimum rod speed is 8 SPM (steps per minute) and the maximum speed is 72 SPM. Each step corresponds to a motion of approximately 5/8 in.

12.10 Feedwater control for PWR with U-tube steam generators [2, 4, 5] U-tube steam generators produce a mixture of saturated water and steam. The water and steam are separated in the steam separators and driers in the top of the steam generator. The separated steam goes to the turbine and the separated water flows into the downcomer, an annular region inside the outer wall of the steam generator. The water level in the downcomer is measured and controlled. This level remains constant at all power levels. A phenomenon called shrink-and-swell occurs in U-tube steam generators and it dictates the type of control used for feedwater. (This phenomenon also occurs in socalled drum-type boilers in fossil-fuel plants and in BWRs and also necessitates a similar control strategy). Consider what happens when the main steam valve opens. Pressure in the steam generator drops, causing saturated water to flash and the two-phase volume to swell. This drives more steam-water mixture into the steam separators and driers and, consequently, more liquid water flowing into the downcomer. So, the downcomer level temporarily increases, signaling a need to decrease feedwater flow. But since the steam flow has increased to start the transient, this is the wrong direction for feedwater flow changes. So, feedwater level alone is an inadequate measurement to determine feedwater flow control. The opposite happens when the there is a reduction in the steam flow rate resulting in a shrinking of the steam generator level. Using a controller that uses three measurements to determine control action solves the problem of shrink and swell. The downcomer level measurement is augmented by measurement of steam flow and feedwater flow rates. The difference between steam flow rate and feedwater flow rate, along with downcomer water level error, provides the inputs to a controller as shown in Fig. 12.12. Such a controller is called a three-element controller. The error signals used in the controller are defined as Flow error,EW ¼ Steam flow rate  Feedwater flow rate Level error,EL ¼ Level set point  Measured level

(12.12)

Since the two error terms have different units, the flow error is divided by the full power steam flow rate and the level error is divided by the allowable level range. Thus the error terms are non-dimensional. Note that there is no direct steam pressure

12.10 Feedwater control for PWR with U-tube steam generators

Set point

Water level

Steam flow

Feedwater flow

L

WST

WFW

– +

–

+

DL Allowable level range

DW Full power steam flow

K1

K2

(1 + Ki/S)

Control signal to feedwater valve

Control Signal =

K1

DL Lrange

+ K2

DW Wsteam,100%

• 1+

Ki S

FIG. 12.12 Three-element U-tube steam generator controller.

control in a PWR with a U-tube steam generator. The turbine controller adjusts the steam valve to admit steam as needed to provide required turbine power. The reactivity controller responds to changes caused by a change in steam flow rate. Resulting processes in the system then determine the resulting steam pressure as specified in the steady state program.

159

160

CHAPTER 12 Pressurized water reactors

12.11 Control of a PWR with once-through steam generators [3] PWRs with once-through steam generators employ steam pressure control. The pressure controller uses two input signals: the measured steam flow demand indicated by the turbine load and measured steam pressure deviation from its set point. Consequently, the pressure controller is called a two-element controller. Note that a PWR with once-through steam generators operates in the turbine-followingreactor mode.

12.12 Turbine control The turbine control system modulates the steam valve to achieve desired steam flow. For PWRs with U-tube steam generators, the controller modulates the valve to admit steam until the turbine power reaches the power set point. For PWRs with once through steam generators the controller modulates the valve to maintain constant steam pressure.

12.13 Summary of main PWR controllers There are many control systems for auxiliary systems in PWRs. But there are five main control systems. These systems and their function are summarized as follows: Reactivity: Controller actuates control rods using set points from the steady state program. Primary Loop Pressure: Controller actuates heaters and spray flow in the pressurizer based on departure from the pressure set point. Pressurizer Level: CVCS actuates flows into and out of pressurizer based on departure from the level set point. This maintains the primary system coolant inventory. Feedwater Flow: • PWR with a UTSG uses a 3-element controller (downcomer level, steam flow rate and feedwater flow rate). • PWR with an OTSG uses a 2-element controller (feedwater demand and steam pressure) Turbine: Manually specified demand actuates steam valve or electrical frequency measured by a governor actuates steam valve in load following PWRs. Other controllers include – main condenser pressure, feedwater temperature, feedwater heater water level, etc.

12.15 Example of a PWR simulation

12.14 PWR safety systems PWRs are operated so as to avoid accidents. But the probability of an accident is not zero. Consequently, PWRs employ a “defense-in-depth” design approach and build engineered safety systems into the plant. The defense-in-depth approach is to incorporate multiple barriers to prevent release of radionuclides into the environment. The first barrier is the UO2 fuel pellet. UO2 contains most of the radionuclides. The migration of radionuclides into the gas space inside the fuel pin increases as fuel temperature increases. But the high melting point of the UO2 (around 5000 °F) reduces total release relative to the total release as would occur upon melting. The second barrier is the Zircaloy cladding (sheath). It contains the radionuclides that have leaked into the gap between the sheath and the fuel pellets, unless high temperature or high internal pressure causes failure of the cladding. It must be noted that melting of Zircaloy (melting point of Zircaloy-4 is 3360 °F) causes a new problem: hydrogen production from the reaction between liquid zirconium and water. Hydrogen is a serious explosion hazard. The third barrier is the primary system piping and vessels. The fourth barrier is the containment building. The engineered safety systems serve to limit temperature and pressure increases. PWRs employ water injection systems to provide continued cooling of the fuel in the event of a loss of primary coolant. Separate systems provide cooling water for pipe breaks ranging from small leaks (with continuing full or partial primary pressure) to complete loss of coolant (with complete primary system depressurization). Containment cooling engineered systems limit containment pressure. They use a water spray system or stored ice to reduce temperature and pressure in the containment. Electrical power is required to operate safety systems in the event of an accident in Generation II and III reactors. Backup systems (batteries and Diesel generators) provide power in the event of loss of off-site power. Failure of emergency power systems during an accident is catastrophic in these systems. New designs avoid the need for electrically-driven emergency cooling systems by employing emergency cooling that is gravity-driven or provided by flow from a pressurized tank.

12.15 Example of a PWR simulation Appendix J describes two lumped parameter models of a PWR plant [5]. It includes a simple, linearized reactor core model and a complete nonlinear model for the primary and secondary components. This chapter provides several simulation results for the linearized reactor core model from Appendix J. The model includes neutronics and core heat transfer (no representation of steam generators or BOP). Available input disturbances in a

161

CHAPTER 12 Pressurized water reactors

simulation include reactivity and core inlet temperature. Selected simulation results are provided here to illustrate PWR dynamic characteristics. Fig. 12.13 shows the response to a reactivity perturbation of δρ ¼ 0.001. For the PWR under consideration this reactivity is equivalent to 14.5 cents. The purpose of this example is to illustrate the reactor response to a control rod perturbation and the effect of both fuel temperature and moderator temperature feedback effects. Fig. 12.12 shows an initial prompt jump in the fractional power. As the fuel temperatures increases quickly, the Doppler effect provides a negative reactivity and decreases the power change. As the coolant temperature increases due to fuel heat-up, the moderator temperature change starts making additional changes in reactivity feedback. The reactivity feedbacks bring the power to a steady-state value. Thus, the PWR is stable under a reactivity insertion without any control action. The steady state power change for the simulation results shown in Fig. 12.12 is
5.2% for a reactivity insertion of 14.5 cents. Thus, a reactivity insertion of 2.8 cents is required to change the power by 1%. The power coefficient for this hypothetical reactor is 14.5 cents/5.2% or Δρ/ΔP ¼ 2.8 cents/% power. Fig. 12.14 shows the response to a + 5 °F core inlet temperature perturbation.

Fractional reactor power: delta-rho = 0.001 0.16

0.14

0.12

0.1 deltaP/Po

162

0.08

0.06

0.04

0.02

0

0

10

20

30

40

50 Time (s)

60

70

80

FIG. 12.13 Example of a PWR response to a reactivity step perturbation of δρ ¼ 0.001.

90

100

12.15 Example of a PWR simulation

Linearized Model Response: delta-Tcl = 5 deg F 0 –0.01

Fractional reactor power, delta-P/Po

–0.02 –0.03 –0.04 –0.05 –0.06 –0.07 –0.08 –0.09 –0.1 0

10

20

30

40

50 Time (s)

60

70

80

90

100

FIG. 12.14 Example of a PWR response to a + 5 °F core inlet coolant temperature step.

The purpose of this example is to illustrate the reactor response to a perturbation in the coolant/moderator inlet temperature and the resulting effect of fuel temperature and moderator temperature feedback effects. Fig. 12.13 shows an initial decrease in the fractional reactor power due to a quick increase in the coolant temperature caused by the coolant temperature reactivity feedback effect. The decrease in power causes the fuel temperature to decrease and as a result the power starts to increase due to Doppler feedback. The combination of increase in moderator temperature and decrease in fuel temperature brings the reactor power to a steady state value. Note that this final steady-state power level is less than the initial level by
5.5%. The PWR is again stable under a coolant inlet temperature perturbation without any control action. This example illustrates the dynamic behavior of reactor power, fuel temperature, and coolant node temperatures and the effect of temperature feedbacks in stabilizing the reactor response. Frequency responses were also calculated for the linear core model. Details appear in Appendix J. Figs. 12.15 and 12.16 (from Appendix J) show the power/ reactivity frequency response gain and phase.

163

CHAPTER 12 Pressurized water reactors

1.2

%Power/Cent

1

0.8

0.6

0.4

0.2

0 10–3

10–2

10–1

100 101 Frequency (rad/s)

102

103

104

102

103

104

FIG. 12.15 Power-to-reactivity frequency response gain for a PWR. 40 20 0 Phase (deg)

164

–20 –40 –60 –80

–100 10–3

10–2

10–1

100 101 Frequency (rad/s)

FIG. 12.16 Power-to-reactivity frequency response phase for a PWR.

Time response and frequency response provide alternate ways to characterize system dynamics. Basically, they tell the same story in different ways. Exercises are provided for the reader to identify and explain the equivalence of the time domain and frequency representations of specific important phenomena in the system’s dynamic behavior.

Exercises

Exercises 12.1. Verify Eqs. (12.5), (12.7) and (12.8). 12.2. Consider a PWR with a U-tube steam generator with the following properties: αf ¼ 1:3∗105 ðρ=o FÞ αc ¼ 2:0∗104 ðρ=o FÞ
Af ¼ 42; 460 ft2

ASG ¼ 133,290 (ft2) in three steam generators Uf ¼ 200 BTU=hr ft2 Þ
USG ¼ 1000 BTU=hr  ft2  o F W ¼ 28; 200 ðlbm=sÞ C ¼ 1:0 ðBTU=ðlbm  o FÞ

a. Prepare a steady state program for the reactor prescribed above for no change in reactivity. b. Estimate δθavg for the reactor prescribed above for δTS ¼ 0 and a reactivity change of one cent. c. Estimate δTs for the reactor prescribed above for δθavg ¼ 0 and a reactivity change of one cent. 12.3. The steady state program for a PWR with U-tube steam generators is shown in Fig. 12.9 a. Sketch a steady state program for a PWR with a U-tube steam generator and constant steam temperature. b. Explain why the variables other than steam temperature are as you show in your sketch. c. What are the relative advantages and disadvantages of the steady state program of Fig. 12.9 and your constant steam temperature program? 12.4. Section 12.6 includes a diagram that uses arrows to show a sequence of events following an initial disturbance. Create a similar diagram for a PWR with a U-tube steam generator. Include control actions and show all important system variables. Note that some actions influence more than one downstream action and some actions depend on more than one upstream action. Show these actions in your diagram. 12.5. Repeat Exercise 12.3 for a PWR with a once-through steam generator. 12.6. The frequency response for a PWR appear in Figs. 12.15 and 12.16. The plots demonstrate three distinct regions (0 to 0.1 rad/s, 0.1 to 2 rad/s and above

165

166

CHAPTER 12 Pressurized water reactors

2 rad/s). Explain why these are considered distinct regions and describe the physical basis for the behavior shown in each region. 12.7. A PWR is a closed loop feedback system. Recall that a feedback system can be represented as follows: Gc ¼

Go 1 + Go H

where Gc ¼ closed loop transfer function Go ¼ open loop transfer function H ¼ feedback transfer function a. Show that the low frequency response is given by 1/H. What is the physical basis for this behavior? b. Show that the high frequency response is given by Go. What is the physical basis for this behavior?

References [1] U.S. Nuclear Regulatory Commission, www.nrc.gov/reading-rm/basic-ref/students/foreducators/04.pdf. [2] Westinghouse Electric Company, The Westinghouse Pressurized Water Reactor Nuclear Power Plant, Westinghouse Electric Company, Water Reactor Division, Pittsburgh, 1984. [3] The Babcock & Wilcox Company, Steam: Its Generation and Use, 42nd ed., The Babcock & Wilcox Company, 2015. [4] T.W. Kerlin, Dynamics and control of pressurized water reactors, in: C.T. Leondes (Ed.), Control and Dynamic Systems, Academic Press, New York, 1978, pp. 103–212. [5] M. Naghedolfeizi, B.R. Upadhyaya, Dynamic modeling of a pressurized water reactor plant for diagnostics and control, in: Research Report, University of Tennessee, 1991. DOE/NE/88ER12824–02, June.

Further reading [6] F. Bevilaqua, J.F. Gibbons, System-80 combustion engineering’s standard 3800-MWt PWR, Combustion 45 (14) (1974). [7] M.R. Ali, Lumped Parameter State Variable Models for U-Tube Recirculation Type Nuclear Steam Generators, PhD dissertation, Nuclear Engineering Department, The University of Tennessee, 1976. August. [8] D.C. Arwood, T.W. Kerlin, A mathematical model for an integral economizer U-tube steam generator, Nucl. Technol. 35 (1977) 12–32.

CHAPTER

Boiling water reactors

13

13.1 Introduction Numerous Boiling Water Reactors (BWRs) are operated in the U.S. and other countries. About 30% of the commercial nuclear reactors in the U.S. are BWRs. Several different generations of BWRs have been built or planned. This chapter addresses the important common features and their influence on the dynamic characteristics of BWRs.

13.2 History of BWR design evolution Unlike PWR development, BWRs have undergone an evolution of designs with significant changes along the way. Two experimental boiling water reactors were built at Argonne National Laboratory to test the viability of this reactor type (Borax-1 in 1953 and EBWR, the Experimental Boiling Water Reactor in 1956). General Electric (GE) entered BWR development with construction of the Vallecitos prototype BWR in 1957. GE then embarked on design and construction of commercial BWR power plants. As of this writing, seven designs of commercial power plants by General Electric, some with significant changes from its predecessors, have been built. These are designated as BWR-1 through BWR-6 and ABWR (Advanced Boiling Water Reactor). A new BWR (the ESBWR or Economic Simplified BWR) has been designed. The main differences in the various designs are containment features, forced circulation vs. natural circulation, and in-vessel jet pumps with flow driven by external pumps vs. integral mechanical pumps. The evolution of GE power plants is described below:

13.2.1 BWR-1 These were early low-power BWRs (all less than 200 MWe). Three versions were built and had different design features, but all are designated as BWR-1. Differences in BWR-1 designs includes direct cycle (reactor steam goes directly to the turbine) or indirect cycle (reactor steam feeds a separate steam generator) and natural circulation of water flow into the core region or forced circulation. Dynamics and Control of Nuclear Reactors. https://doi.org/10.1016/B978-0-12-815261-4.00013-5 # 2019 Elsevier Inc. All rights reserved.

167

168

CHAPTER 13 Boiling water reactors

13.2.2 BWR-2 BWR-2 produces greater power levels (greater than 500 MWe) than BWR-1. It uses mechanical recirculation pumps and Mark I containment (See below). BWR-2 and four subsequent designs (BWR-3 through BWR-6)) are all considered to be Generation II reactors.

13.2.3 BWR-3 BWR-3 produces greater power levels (800 MWe) than BWR-2 and was the first BWR to use jet pumps for recirculation flow. It uses Mark I containment.

13.2.4 BWR-4 BWR-4 is similar to BWR-3, but it operates at higher power (1100 MWe). BWR-4 reactors use either Mark I or Mark II containment.

13.2.5 BWR-5 BWR-5 is similar to BWR-4. It operates at the same power level as BWR-4 power (1100 MWe). BWR-5 reactors use either Mark I or Mark II containment.

13.2.6 BWR-6 BWR- 6 is available in different configurations having power levels of 600– 1400 MWe. BWR-6 uses Mark III containment.

13.2.7 ABWR ABWR is a Generation III reactor. The ABWR employs internal mechanical recirculation pumps. It uses Mark III containment and the power level is 1500 MWe.

13.3 Characteristics of BWRs 13.3.1 General features of a BWR Since there are five different Generation II BWRs, it is necessary to pick one for providing an overview of BWR characteristics. BWR-6 was chosen, but its dynamic behavior and control strategy is typical of all Generation II BWRs. The main difference is that it operates at a higher power level than earlier designs. Fig. 13.1 shows a typical BWR-6 system. Fig. 13.2 shows a BWR-6 reactor vessel and internals [1]. Subcooled water enters the bottom of the core. The flow rate and pressure are such that boiling begins near the entrance. Boiling continues along the rest of the passage through the core. A steam-water mixture exits the core region. This mixture

13.3 Characteristics of BWRs

Containment Cooling System

Steam Line Reactor Vessel

Turbine Generator Separators & Dryers

Heater Condenser

Feedwater 3

Condensate Pumps

Core 1&2

Feed Pumps

Control Rods

Demineralizer

Recirculation Pumps Emergency Water Supply Systems

FIG. 13.1 Schematic of a typical boiling water reactor system. U.S. Nuclear Regulatory Commission: www.nrc.gov/reactors/bwrs.html.

then passes through steam separators and steam driers located above the core. These systems remove water by centrifugal force and by sudden reversals in flow direction. The removed water flows downward into an annular region between the vessel and a core shroud. This annular region is called the downcomer. Fig. 13.3 shows a BWR-6 fuel bundle. The fuel is Uranium oxide contained in Zircaloy tubes. A typical 1220 MWe BWR-6 core consists of about 750 fuel assemblies. Each assembly is enclosed in a fuel box with an 8
8 or a 10
10 array of fuel rods (pins). The fuel pins are similar to those in a PWR, and have an active length of 12 ft. The fuel ‘box’ constrains the flow of coolant in the assembly. Upper and lower tie plates provide structural support for the assembly, along with a few tie rods. Some spaces in the fuel assembly are taken by water rods to provide additional neutron moderation. A typical BWR contains 60–70 thousand fuel rods and

169

170

CHAPTER 13 Boiling water reactors

STEAM DRYER LIFTING LUG VENT AND HEAD SPRAY

STEAM DRYER ASSEMBLY STEAM OUTLET

STEAM SEPARATOR ASSEMBLY

CORE SPRAY INLET

FEEDWATER INLET FEEDWATER SPARGER

LOW PRESSURE COOLANT INJECTION INLET

CORE SPRAY LINE

CORE SPRAY SPARGER TOP GUIDE JET PUMP ASSEMBLY CORE SHROUD

FUEL ASSEMBLIES

CONTROL BLADE

CORE PLATE JET PUMP/RECIRCULATION WATER INLET

VESSEL SUPPORT SKIRT

RECIRCULATION WATER OUTLET

SHIELD WALL

CONTROL ROD DRIVES

CONTROL ROD DRIVE HYDRAULIC LINES IN-CORE FLUX MONITOR

FIG. 13.2 BWR vessel and reactor internals. Courtesy of GE Hitachi Nuclear Energy Americas LLC (GE Nuclear Energy, BWR-6: General Description of a Boiling Water Reactor).

160 metric tons (160,000 Kg) of UO2. Design parameters of a typical BWR-6 are given in Appendix A. Fig. 13.3 also shows a BWR-6 control assembly. It is a cruciform-shaped structure that enters the core from below and is inserted in the space between four fuel assemblies. The below-core location is necessary since the region above the core

13.3 Characteristics of BWRs

FOUR-BUNDLE FUEL MODULE FUEL ROD WATER RODS Care Lattice

TIE RODS

FIG. 13.3 Four fuel assemblies (boxed channels) showing the control rod cruciform shaped assembly in the center of fuel four fuel assemblies. Courtesy of GE Hitachi Nuclear Energy Americas LLC (GE Nuclear Energy, BWR-6: General Description of a Boiling Water Reactor).

contains steam separators and driers. The location of control rod assemblies means that gravity-based insertion is impossible. Insertion from below places the control rod assembles enter through the subcooled region where control rod worth is greatest and facilitates refueling from above. The control assemblies contain boron carbide and are used for reactivity control and power flattening. Burnable poison (oxide of gadolinium – gadolinia) is mixed with the fuel for shim control.

171

172

CHAPTER 13 Boiling water reactors

13.3.2 Recirculation flow and jet pumps The core flow in a BWR-6 is controlled by two recirculation pumps that distribute the water to a set of jet pumps surrounding the core. Each recirculation pump distributes water to one of two manifolds. Each manifold supplies water via pipes to jet pumps. A pair of jet pumps receives water from a single pipe. There is a total of 20 jet pumps, with a typical jet pump overall length of 19 ft. Mechanical pumps (recirculation pumps) withdraw water from the downcomer and pump it at elevated pressure into the jet pumps. The jet pumps have no moving parts, making them maintenance-free reactor components. Fig. 13.4 illustrates the principle of operation of a jet pump [2]. The recirculation flow enters the jet pump nozzle at a high pressure and increases to a high velocity as it flows through the narrow throat, and results in a pressure drop. The suction flow in the downcomer region enters the inlet nozzle at a low pressure. The pressure decreases as the suction flow passes through the converging section of this nozzle. The driving flow and the suction flow mix in the throat region (mixing

FIG. 13.4 Operation of a jet pump. Courtesy of GE Hitachi Nuclear Energy Americas LLC (GE Nuclear Energy, BWR-6: General Description of a Boiling Water Reactor).

13.4 Reactivity feedbacks in BWRs

section with a constant diameter) resulting in an increase in the fluid pressure due to a change in the fluid velocity in this section. A long diffuser is connected at the end of the mixing section, causing an increase in the fluid pressure that drives the coolant into the lower plenum and then up through the reactor core.

13.3.3 Other features of BWRs The following are typical core parameters for a BWR-6: • • • • • • • • • • •

Total coolant flow rate: 105
106 lbm/h Plant efficiency: 34% Core diameter: 193 in. Number of control rods: 177 Coolant pressure: 1040 psia Core-exit (steam) temperature: 551°F Feed water temperature: 420°F Average coolant exit quality: 15% Vessel diameter: 19 ft. Wall thickness: 5.7 in./6.46 in. Vessel height: 71 ft. Core power density: 54 kW/l.

The steam produced in the core and separated from liquid water passes through a control valve into the turbine. The steam pressure is maintained at a constant value by throttling the steam valve. Exhaust steam passes into a condenser and the condensate passes through a series of feedwater heaters before returning to the reactor vessel. Note that a BWR system shares general features with U-tube steam generators used in most PWRs (a heated riser, steam separators and driers, and a downcomer). A BWR containment consists of a concrete “drywell” that encloses the reactor. If steam escapes from the reactor vessel or related piping it flows into the drywell. The drywell has piping that connects it to a large pool of water called the suppression pool. The suppression pool water condenses the steam and reduces pressure in the drywell. Three different types of suppression pool are used, Mark I, II and III [3]. See Fig. 13.5.

13.4 Reactivity feedbacks in BWRs The BWR fuel temperature coefficient of reactivity is due to the Doppler effect (fuel temperature feedback reactivity) and is always negative. Typically, the Doppler coefficient in BWRs is around 2
105 Δρ/°C. The magnitude of the negative Doppler coefficient increases as fuel temperature increases. BWRs are under-moderated. Increases in moderator/coolant temperature cause increased boiling and reduced in-core liquid water density. Thus, an increase in

173

174

CHAPTER 13 Boiling water reactors

moderator/coolant temperature decreases neutron moderation (a negative reactivity mechanism) and decreases neutron absorptions (a positive reactivity mechanism). The moderation effect is larger, causing a net negative moderator/coolant temperature coefficient of reactivity. Typically, the moderator/coolant temperature coefficient is around 3
104 Δρ/°C. The void coefficient depends strongly on reactor conditions, but it also is always negative. Typically, the void coefficient is around 1.4
103 Δρ/% voids. Pressure also affects reactivity because pressure affects boiling. For example, a pressure increase causes a reduction in core voids. Because the void coefficient is negative, a pressure increase causes a reactivity increase. That is, the pressure coefficient of reactivity is positive. The control strategy in BWRs is to maintain constant pressure by modulating the steam valve.

DRYWELL HEAD

DRYWELL FLANGE

DRYWELL SHEAR LUG SUPPORT

REACTOR PRESSURE VESSEL

DRYWELL SHIELD WALL

CORE

RADIAL BEAM

MANWAY RADIAL BEAM VACUUM BREAKER

JET DEFLECTOR VENT

VENT HEADER

DOWNCOMERS WATER LEVEL

(A)

SUPPRESSION CHAMBER (TORUS)

FIG. 13.5 (A) BWR Mark I containment with toroidal pressure suppression chamber. (continued)

13.4 Reactivity feedbacks in BWRs

The moderator temperature also affects the thermal neutron spectrum. Increases in moderator temperature cause hardening of the thermal neutron spectrum. This causes a negative component of the reactivity change due to changes in U-235 absorptions and a positive component due to changes in Pu-239 absorptions. See Section 7.3. As in all thermal spectrum power reactors, burnup and production of Xe-135 causes a reactivity feedback effect. See Section 6.2. The power coefficient for BWRs is negative, thereby ensuring that reactor power achieves a new steady state level following a change in external reactivity.

DRYWELL HEAD

DRYWELL

REACTOR VESSEL

SACRIFICIAL SHIELD WALL STEEL LINER REACTOR PEDESTAL

DRYWELL DECK

S/R VALVE TAILPIPE (18) EQUIPMENT HANDLING PLATFORM DOWNCOMER (VENT)

VACUUM BREAKERS (5) SUPPORT COLUMN (12)

PRESSURE SUPPRESSION CHAMBER

WATER LEVEL

QUENCHER (18) REINFORCED CONCRETE

(B) FIG. 13.5, cont’d (B) BWR Mark II containment chamber. (continued)

175

176

CHAPTER 13 Boiling water reactors

CONTAINMENT SPRAY SHIELD BUILDING 125 TON CRANE W/15 TON AUX HOOK

CONTAINMENT UPPER POOL

DRYWELL HEAD FUEL TRANSFER POOL

REACTOR VESSEL REACTOR SHIELD DRYWELL BOUNDRY

WEIR WALL S/R VALVE LINE

DRYWELL FUEL TRANSFER TUBE

SUPRESSION POOL HORIZONTAL VENT

(C) FIG. 13.5, cont’d (C) BWR Mark III containment. Courtesy of GE Hitachi Nuclear Energy Americas LLC (General electric advanced technology manual, Chapter 6.2, BWR Primary Containments, U.S. Nuclear Regulatory Commission, https://www.nrc.gov/docs/ ML1414/ML14140A181.pdf).

13.5 Reactivity and recirculation flow BWRs can use control rods to change reactivity, but an alternate way is to change recirculation flow rate. Increasing the recirculation flow rate increases the amount of liquid water in the core relative to steam. Since the BWR is under-moderated, this increases reactivity, thereby increasing reactor power and steam production. Thus, BWRs have two ways to change reactivity by external means, whereas PWRs have one (control rod motion). In forced circulation BWRs recirculation pumps are used to draw water from the lower downcomer region and distribute the water to a set of jet pumps at an elevation above the pump suction location. Thus, a BWR is a variable flow system, with the flow modulation facilitating start-up and load-following operations. Two recirculating pumps distribute water to jet pumps, through a sparger ring. Changing the pumping power (hence, the coolant flow rate) causes a change in reactivity through a change in core voids. See Section 13.3.2 for a description of jet pump operation.

13.7 BWR dynamic models

13.6 Total reactivity balance For steady state, zero reactivity is required. The total reactivity balance is as follows: ρ ¼ Control rod reactivity + Recirculation flow reactivity + Feedback reactivity ¼ 0

The externally-controlled reactivity may be achieved by a combination of control rod reactivity and recirculation flow reactivity. Thus, a desired reactivity setting for either externally-controlled reactivity can be achieved by adjusting the other externally-controlled reactivity.

13.7 BWR dynamic models Detailed BWR dynamic models include treatment of all of the complex neutronic and thermal-hydraulic effects that contribute to the dynamics of the system. Both linear and nonlinear models exist. Detailed models are too complex for inclusion here. Interested readers can find information in the literature [4, 5]. Linear models provide estimates of the small-perturbation time response and frequency response. An approximate, low-order model provides simple simulation capability. It accounts for the neutronic and thermal-hydraulic processes that determine feedback reactivity. A low-order model [4] was developed by fitting a loworder transfer function to match the closed-loop frequency response calculated with a detailed model [5]. The results obtained with the low-order model are essentially identical with results from the detailed model. Note that the author of Ref. [4] chose to express the frequency in Hz rather than in rad/s as used elsewhere in this book. Figures in this chapter use rad/s for frequency units. The low-order closed loop transfer function used in the fit [4] is Gc ðsÞ ¼

K ðs2 + as + bÞ ðs + cÞ + ds + eÞ ðs + f Þ ðs + gÞ

ðs 2

(13.1)

The following values of the low-order model parameters are typical for BWR simulation [4]: K ¼ a gain that varies with power level a ¼ 0.36 b ¼ 0.1055 c ¼ 0.03 d ¼ 0.09 e ¼ 0.1044 f ¼ 0.25 g ¼ 21.0 The low-order model from Ref. [4] provides the capability to further investigate BWR dynamics and the cause for potential stability issues. The low-order model

177

178

CHAPTER 13 Boiling water reactors

of Ref. [4] was modified to provide the amplitude in units of % power/cent of reactivity as used throughout this book. Fig. 13.6 shows the power-to-reactivity frequency response obtained with the low-order model. Note the resonance at around 0.3 rad/s. The resonance grows if changes in conditions cause the system to move toward instability. Recall that the closed loop transfer function for a system with feedback is given by Gc ¼

Go 1 + Go H

(A)

(B) FIG. 13.6 (A) BWR frequency response amplitude, (B) BWR frequency response phase.

(13.2)

13.8 BWR stability problem and impact on control

where Gc ¼ closed-loop transfer function (feedback effects included). Go ¼ open-loop transfer function (the zero-power transfer function). H ¼ feedback (cents/% power). At low frequencies, the magnitude of Go is large. Consequently, Gc ¼ 1/H at low frequencies. The feedback frequency response can be calculated using H¼

1 1  Gc Go

(13.3)

The resulting feedback frequency response appears in Fig. 13.7. Note that the phase lag is more than 90 deg. at frequencies above around 0.1 rad/s. As shown in Section 3.8 such phase shifts in system feedback can cause instability if the feedback gain is large enough. The frequency response for various feedback conditions can be deduced by applying a multiplicative factor, K to the feedback term (GoH in Eq. (13.2)). Gc ¼

Go 1 + K Go H

(13.4)

Note that K ¼ 1 for the original low-order model. Fig. 13.8 shows the closed-loop gain for various values of K. Clearly the resonance at around 0.3 rad/s grows as K increases and shifts to higher frequencies. Ref. [4] shows that the system becomes unstable at a value of K  2.25. The above discussion reveals that useful insights can be deduced if basic principles of dynamic analysis of feedback systems are understood and employed.

13.8 BWR stability problem and impact on control BWRs exhibit instability at conditions of high power and low recirculation flow. This instability is caused by a complex coupling of neutronics and thermalhydraulics. The basic cause of BWR instability is time lagged flow and reactivity feedbacks. Recall that positive feedbacks usually cause stability problems, but negative feedbacks also can cause instability if their effect is delayed (and the feedback gain is large enough). See Section 3.8. Subcooled water enters a BWR channel at the bottom. As it flows upward, boiling occurs and the void fraction increases. A disturbance (typically inlet flow change, inlet subcooling change, or power change) causes a localized change in steam bubble concentration in the lower portion of the channel. The propagation of this bubble packet as it travels up the channel is called a density wave [2]. This density wave causes changes in the local pressure drops as it propagates. Consider a disturbance that causes an increase in steam bubble formation in the lower portion of the channel. The resulting density wave travels upward, and it travels faster than the flow prior to the disturbance. As a result, the total channel pressure drop

179

180

CHAPTER 13 Boiling water reactors

(A)

(B) FIG. 13.7 (A) Feedback frequency response amplitude. (B) Feedback frequency response phase shift.

increases, but it is delayed relative to the input disturbance. If the pressure drop is large and it lags the input disturbance by 180 degrees, it destabilizes the channel. The density wave also affects reactor power by the void feedback effect on reactivity. The local reactivity and reactor power change follow the change in void density, but the heat transfer from the fuel to the fluid lags because of the fuel’s heat transfer lag. The fuel heat transfer time constant in a BWR is typically 6–10 s. So, the neutronic response is also lagged, thereby introducing its additional component of the lagged response. The coupled thermal-hydraulic and neutronic response can cause instability, especially at low flow and high power conditions. At low flow and high power, resulting lagged thermal-hydraulic and neutronic feedbacks have values that induce instability.

13.9 The power flow map and startup

FIG. 13.8 Closed-loop frequency response amplitude for various values of K.

Fig. 13.9 shows the feedback paths that determine stability. Instabilities of different nature can occur. These include single channel instability, in-phase core-wide instability (the whole core responds in unison) and out-ofphase core instability (different regions respond out-of-phase with one another). The processes involved in coupled thermal-hydraulic and neutronics are very complex and not amenable to a simple analysis. Very detailed computer codes are necessary and several analysis codes have been developed. Many publications have addressed the BWR stability problem, its analysis and its mitigation. The magnitude of the effort to deal with BWR instability illustrates the importance of the problem. Details may be found in the literature (see Refs. [2, 4, 6–10]). The typical conditions for instability are power levels of 35–60% and core flow rates of 30–45%. The strategy for avoiding instability is to avoid operation in the range of reactor power levels and core flow rates where instability occurs. Specifically, the reactor power is kept below the instability threshold at low core flows by using control rods to adjust power.

13.9 The power flow map and startup A reactor with a negative power coefficient (such as a BWR) experiences a specific new steady state power following a reactivity change (see Chapter 7). Since BWR externally controlled reactivity depends on control rod positions and recirculation flow there are many different paths to change reactor power. Control rods are used for reactivity adjustment to keep power low at low flows to avoid the instability

181

182

CHAPTER 13 Boiling water reactors

Reactivity

NEUTRONICS

Doppler

Power

Fuel temp

FUEL

Void reactivity CORE T-H

Direct heat

Inlet enthalpy Outlet pressure

RECIRCULATION LOOP

Inlet flow

FIG. 13.9 BWR dynamics block diagram showing the various feedback paths.

region. When flow increases beyond around 30% of full flow, control rods are used to increase power. Subsequently, recirculation flow changes are used to induce reactivity changes and, consequently, power changes. Explanation of the power flow map for flows above around 30% conveniently begins with consideration of the required control rod reactivity needed to achieve some desired power at 100% flow. For example, achieving 100% power at 100% flow requires a specific amount of control rod reactivity. If the control rod reactivity remains constant while flow decreases, the power follows a fixed trajectory down to the flow control minimum (around 30%). This trajectory is called the 100% line. The same logic applies for other lines. For example, there is a specific required control rod reactivity needed to achieve 50% power at 100% flow. This trajectory is called the 50% line, and the power again follows a fixed trajectory down to the flow control minimum. Fig. 13.10 shows a typical BWR power-flow map. This map is for the Advanced Boiling Water Reactor. Similar maps apply for other BWR designs. The map shows the strategy for using control rods and core flow to achieve specific reactor power levels. The shaded area is the region to be avoided because of instability problems for the indicated range of core flows and power levels. The startup trajectory provides a useful means to explain the power-flow map. First consider a hypothetical reactor startup for a BWR with a power flow map as shown in Fig. 13.10. Startup involves achieving criticality by withdrawal of control rods while natural circulation provides core flow (slightly above 30% of full flow). Then control rods are withdrawn and core flow is increased to around 40% using recirculation pumps until the reactor power reaches around 65% of full power. Subsequently, core flow is used to induce positive reactivity and power increases.

13.10 On-line stability monitoring

130 NINE OF TEN INTERNAL PUMPS OPERATING

120

PERCENT PUMP SPEED 0 0 NATURAL CIRCULATION 1 30 2 35 3 40 100% POWER = 3926 MWt 4 50 5 60 100% FLOW = 52.2 X 106 kg/h 6 70

110 100

PERCENT POWER

90

7 80 100% SPEED = 157 rad/s 8 90 9 95 10 100 3 2 PERCENT ROD LINE 1 A 102 B 100 C 80 0 D 60 E 40 F 20 REGION III

80 70 60 50 40

8

9

10

7 6 5 4 A B C

REGION IV

D

E

30

REGION II

REGION I

20

F

TYPICAL STARTUP PATH

10

STEAM SEPARATOR LIMIT

0 0

10

20

30

40

50 60 70 PERCENT CORE FLOW

80

90

100

110

120

FIG. 13.10 A BWR power-flow map. Courtesy of GE Hitachi Nuclear Energy Americas LLC (ABWR Design Control Document, prepared by GE Nuclear Energy for the U.S. Nuclear Regulatory Commission, 1997).

This scenario involves the use of control rods to reach the 100% line before switching to recirculation flow changes to increase power. Now consider an alternate scenario. In this case use control rods to reach the 50% line. Then increase core flow to some power level at or below 50%. Then withdraw control rods until the reactivity increase corresponds to the 100% line. The power flow map for BWRs is roughly analogous to the steady state program for PWRs. The BWR situation is more complicated because of the need to avoid instability and because two reactivity control measures are available in BWRs. The BWR power flow map indicates a range of acceptable conditions while a PWR steady state program indicates desired conditions.

13.10 On-line stability monitoring Stability in an operating BWR may be monitored by analyzing the natural fluctuations in measured signals. This process is usually called reactor noise analysis. Two analysis methods are available: spectrum analysis and time-series analysis. Spectrum analysis uses Fourier transforms of the measured fluctuations to provide the power spectrum (signal energy vs. frequency). Time series analysis involves the estimation using the following model, called an auto-regression (AR) model, from the measured data.

183

184

CHAPTER 13 Boiling water reactors

xðtÞ ¼

n X

ai xðt  iΔtÞ + vðtÞ

(13.5)

i¼1

where x(t) is a stationary random signal (measured by neutron detectors). v(t) is a model prediction error. {ai, i ¼ 1, 2, …, n} is a set of model parameters. Δt ¼ data sampling interval (sec). n ¼ model order. The model parameters {ai, i ¼ 1, 2, …, n}and the AR model order n are estimated using the measurements such that the model prediction error is minimized. The least-squares approach uses the given sampled measurements {x(1), x(2), …, x(N)}, where N is the total data points. More general forms of the AR model, called an auto-regression moving average (ARMA) model, are used in some applications. See Ref. [11] for details. An example illustrates the evaluation of a time series model. Consider the form of Eq. (13.5) for a second-order fit. The analysis begins with the third measured value of x(t). xð3Þ ¼ a1 xð2Þ + a2 xð1Þ + vð3Þ xð4Þ ¼ a1 xð3Þ + a2 xð2Þ + vð4Þ xð5Þ ¼ a1 xð4Þ + a2 xð3Þ + vð5Þ ⋮ xðN Þ ¼ a1 xðN  1Þ + a2 xðN  2Þ + vðN Þ

(13.6)

Note that a1 and a2 are unknown parameters whose values are sought by using all the available measurements {x(1), x(2), …, x(N)}. An efficient approach for estimating the parameters is to minimize the error between the left hand side and the right hand side (also called the model prediction error by minimizing a squared error function shown below, with respect to (a1, a2): Min J ¼

N X

ðxðkÞ  a1 xðk  1Þ  a2 xðk  2ÞÞ2 ða1, a2Þ

(13.7)

k¼3

The two parameters are estimated by solving the two equations obtained from ∂J ∂J ¼ 0 and ¼0 ∂a1 ∂a2

(13.8)

The two equations are then simplified by collecting the terms multiplying a1 and a2 and solving for the 2-dimensional vector (a1, a2) to give the following solution: 31 2 N 3 N X X xðk  1Þxðk  2Þ 7 6 xðkÞxðk  1Þ 7   6 7 6 k¼3 7 6 a1 k¼3 k¼3 7 6 7 ¼6 7 7 6 6 N N N X a2 5 4X 5 4X 2 xðk  1Þxðk  2Þ xðk  2Þ xðkÞxðk  2Þ 2

N X xðk  1Þ2

k¼3

k¼3

k¼3

(13.9)

13.11 Power maneuvering

As the number of data points, N, increases, the estimates of (a1, a2) converge to the actual values with least error. Once the AR coefficients are determined, the resulting model can be used to compute the impulse response of the system. The above discussion shows the basic idea of time series modeling from observed data. Recursive parameter estimation techniques are available to compute the (n + 1)-th order model from the n-th order model; these do not require the inversion of large matrices. These are computationally fast and more accurate than methods using direct matrix inversion [12]. An AR analysis of neutron power fluctuations using average power range monitor (APRM) detector signals from two operating BWRs provides the powerto-reactivity impulse response. The developed modeled may be used directly to compute this impulse response in a time recursive fashion. The impulse response may then be used to estimate a decay ratio of the neutron power response to a change in the reactivity. The decay ratio is defined as the ratio between successive positive peaks or successive negative peaks calculated from the impulse response function. For stable reactor operation the decay ratio must be less than 1, and must be less than a value specified by the regulatory agency. An increased power-to-flow ratio indicates a system with a smaller stability margin. A case study [10] provides a stochastic time series model of a measured neutron signal. The developed model was then used to generate the response to an impulse change in the reactivity as its input. Fig. 13.11 shows impulse response results for the case study. Data from two BWRs operating at different power-to-flow ratios were processed using the AR model. The upper plot in Fig. 13.11 shows the power-to-reactivity impulse response of a BWR-4 plant operating at 100% power and 100% recirculation flow rate. The calculated decay ratio is, DR ¼ 0.024. The lower plot in Fig. 13.11 shows the impulse response of a BWR-4 plant operating at 100% power and 65% recirculation flow rate (this is a test case). The calculated decay ratio is, DR ¼ 0.37. The decay ratio of the impulse response function is less than one, and thus both the systems are stable. As the power-to-flow ratio increases this stability margin decreases, indicating a change in the reactor operating characteristic. The flow-topower ratio for the two operating cases are 100% and 65%, respectively. Both BWRs are rated around 1100 MWe. This method of stability monitoring is recommended as a criterion applied to operating reactors by the U.S. Nuclear Regulatory Commission [13].

13.11 Power maneuvering The scenario following opening of the main steam valve in an uncontrolled BWR is as follows: • •

Main steam valve opening " Steam flow to turbine "

185

CHAPTER 13 Boiling water reactors

2 DECAY RATIO,DR = 0.024 DAMPING COEFFICIENT,

IMPULSE RESPONSE

x = 0.51

1

0

–1 0

2

4

6

4

6

TIME (S)

4 DECAY RATIO,DR = 0.37 DAMPING COEFFICIENT,

IMPULSE RESPONSE

186

x = 0.16

2

0

–2

0

2 TIME (S)

FIG. 13.11 Impulse response to reactivity perturbation of two BWR-4 systems at different power-to-flow operations. The estimated decay ratios are: 0.024 (top), 0.37 (bottom).

• • • • • •

Turbine power " Steam pressure # Moderator/coolant boiling " In-core voids " Reactivity # Reactor power #

This scenario shows that the inherent initial response in a BWR is a reduction in reactor power in response to an increase in steam flow. The control engineer’s job is to overcome this behavior through appropriate control action.

13.13 BWR safety

13.12 BWR control strategy We have seen that a BWR with no control action responds initially in the wrong direction following an increase in steam flow. So, the basic idea in BWR control is to increase reactor power before releasing more steam to the turbine following an increase in demand. A reactor operated in this way is called a “turbine following boiler”. That is, following a power demand maneuver, the reactor power is adjusted first. The turbine waits until the reactor changes power level before experiencing a change in steam flow. The main control systems in a BWR are the reactor power controller, the feedwater controller, and the pressure controller. The reactor controller uses control rod motion and core flow adjustment to control reactivity. As shown above, the choice of control action (control rods or core flow) depends on reactor condition. The power-flow map provides information on allowable flows at all power levels. The feedwater controller is a so-called three element controller. Measurements provide the downcomer level, the feedwater flow rate and the steam flow rate. Feedwater flow rate is adjusted to eliminate a deviation in level from its set point and to eliminate a mismatch between feedwater flow rate and steam flow rate. This type of control is necessary because shrink and swell occur in BWRs (just like in U-tube steam generators). The pressure controller adjusts the steam valve to maintain constant steam pressure. Called the electro-hydraulic controller, it modulates the steam valve to achieve constant pressure. Consider the response to an increase in power demand. The first action is an increase in core flow. This increases reactivity, power level and steam production. The resulting pressure increase causes the pressure controller to open the steam valve and consequently provide the steam flow needed to satisfy the increase in power demand. The feedwater flow is regulated to maintain the downcomer level at a set point. This scenario illustrates the “turbine following boiler” approach used in BWRs.

13.13 BWR safety Like PWRs, generation II plants require emergency power supplies to provide coolant water pumping in the event of an accident. As shown in Section 11.5.3 emergency power was lost due to a tsunami at the Fukushima Dai-ichi power plant in Japan. This led to catastrophic failures, but it was a result of placing emergency power facilities in a vulnerable location rather than a failure of BWR safety philosophy. Changes at other generation II BWRs provides increased security of emergency power supplies. New BWR designs eliminate the need for electrically driven emergency coolant pumps by using gravity feed and flow from pressurized tanks.

187

188

CHAPTER 13 Boiling water reactors

13.14 Advantages and disadvantages BWRs are sometimes touted as being simpler than PWRs (fewer components, including no steam generators or pressurizers), operation at lower pressure, and being well-suited for power maneuvers. The main disadvantage is the need to operate in a way that avoids instability. Procedures are in place to deal with this problem, but they certainly represent a departure from simplicity. Also, because steam entering the turbine contains radioisotopes produced in the core (mainly nitrogen 16), the turbine becomes radioactive and maintenance and repairs are affected, but the radiation drops rapidly after reactor shutdown. Since N-16 has a half-life of 7.1 s, it decays quickly, permitting safe access to the turbine for maintenance. BWRs and PWRs are competitors around the world. Since both are in operation and being built, the relative advantages and disadvantages of these two types must be viewed as comparable.

Exercises 13.1. Explain why Fig. 13.7 for the feedback phase shift indicates that large feedback gains can cause instability. 13.2. Compare the frequency response plots for PWRs and BWRs and discuss the implications of any major differences.

References [1] GE Nuclear Energy, BWR-6: General Description of a Boiling Water Reactor. [2] R.T. Lahey Jr., F.J. Moody, The Thermal-Hydraulics of a Boiling Water Nuclear Reactor, American Nuclear Society, LaGrange Park, 1977. [3] NRC, General electric advanced technology manual, Chapter 4.3, Power Oscillations, U.S. NRC, n.d. https://www.nrc.gov/docs/ML1414/ML14140A074.pdf. [4] J.A. March-Leuba, Dynamic Behavior of Boiling Water Reactors, Doctoral Dissertation, The University of Tennessee, Knoxville, 1984. available at: http://trace.tennessee.edu/ utk_graddiss/1655. [5] P.J. Otaduy, Modeling of the Dynamic Behavior of Large Boiling Water Reactor Cores, PhD Dissertation, University of Florida, 1979. [6] J.A. March-Leuba, Density-wave instabilities in boiling water reactors, Published as Oak Ridge National Laboratory Report ORNL/TM-12130 and as U.S. Nuclear Regulatory Commission report NUREG/CR-6003, October, 1992. [7] C. Kao, A Boiling Water Reactor Simulator for Stability Analysis, PhD dissertation, The Massachusetts Institute of Technology, 1996. February. [8] R. Hu, Stability Analysis of the Boiling Water Reactor: Methods and Advanced Designs, Doctoral dissertation. MIT, 2010. June. [9] J. March-Leuba, A reduced-order model of boiling water reactor linear dynamics, Nucl. Technol. 75 (1986) 15–22.

Further reading

[10] B.R. Upadhyaya, M. Kitamura, Stability monitoring of boiling water reactors by time series analysis of neutron noise, Nucl. Sci. Eng. 77 (1981) 480–492. [11] B.R. Upadhyaya, T.W. Kerlin, Estimation of response time characteristics of platinum resistance thermometers by the noise analysis technique, ISA Trans. 17 (1978) 21–38. [12] G.E.P. Box, G.M. Jenkins, Time Series Analysis: Forecasting and Control, Holden-Day, San Francisco, 1970. [13] U.S. Nuclear Regulatory Commission Standard Review Plan, Boiling Water Reactor Stability, NUREG-0800, March, 2007.

Further reading [14] International Atomic Energy Agency, Boiling Water Reactor Simulator Training Course Series No. 23, available at, www.pub.iaea.org/MTCD/publications/PDF/TCS-23_web. pdf. [15] General electric advanced technology manual, Chapter 6.2, BWR Primary Containments, U.S. Nuclear Regulatory Commission, https://www.nrc.gov/docs/ML1414/ ML14140A181.pdf. [16] ABWR Design Control Document, prepared by GE Nuclear Energy for the U.S. Nuclear Regulatory Commission, 1997.

189

CHAPTER

Pressurized heavy water reactors

14

14.1 Introduction This chapter contains a brief description of the characteristics of pressurized heavy water reactors (PHWRs), and their dynamics and control [1, 2]. But the PHWR itself is very different than the PWR described previously. The discussion of PHWRs deserves inclusion here because a number of units operate in Canada and in other countries. The dynamic characteristics and control system designs of PHWRs differ significantly from those of the more common light water reactors. The PHWR was developed in Canada where the design was called the CANDU reactor (for CANada Deuterium Uranium). Canada’s nuclear power program and its development provide an interesting and informative story. Upon deciding to “go nuclear”, Canada could have easily decided to depend on the U.S. to sell them reactors as was done in many other countries. But Canada decided to develop its own reactors. Canada innovatively developed a design that could be realized with their own resources and manufacturing infrastructure. Canada might not fully succeed in the world-wide competition for reactor sales, but its accomplishments deserve admiration. CANDU reactors were built in Canada and were exported to several other countries. References [3, 4] provide additional details related to reactor physics and CANDU-6 design features.

14.2 PHWR characteristics Fig. 14.1 shows a typical PHWR nuclear steam supply system [1]. The reactor is contained in a large cylindrical vessel, called the calandria that is positioned horizontally. The calandria vessel has a diameter of 7.6 m (25 ft) and the wall is made of stainless steel (3 cm thick). The D2O moderator in this vessel is maintained at low pressure and temperature. The calandria contains an array of about 380 horizontal tubes (Zircaloy) that contain bundles of UO2 fuel rods surrounded by heavy water coolant. The pressure tubes are the boundary between high temperature, high pressure heavy water inside the tubes and low temperature, low pressure heavy water outside the tubes. Thus, a PHWR reactor has separate moderator and coolant and both produce reactivity feedbacks. Dynamics and Control of Nuclear Reactors. https://doi.org/10.1016/B978-0-12-815261-4.00014-7 # 2019 Elsevier Inc. All rights reserved.

191

192

CHAPTER 14 Pressurized heavy water reactors

Steam pipes

Steam generators (Boilers) Primary pumps (Heat transport pumps) Pressurizer

Feedwater return

Feedwater return

Headers

Headers

Calandria

Reactor

Fuel Fuel channel (Pressure tube)

Light water (H2O) steam Moderator pump Light water condensate Heavy water (D2O) coolant Heavy water moderator

Moderator heat exchanger

FIG. 14.1 A PHWR (CANDU reactor) nuclear steam supply system. Courtesy of UNENE, Ontario, Canada (W.J. Garland (Editor-in-Chief), The Essential CANDU: A textbook on the CANDU Nuclear Power Plant Technology, University Network of Excellence in Nuclear Engineering (UNENE), McMaster University, Hamilton, Canada, 2019, Retrieved from https://www.unene.ca/education/candutextbook, Chapter 8).

14.2 PHWR characteristics

The heavy water coolant is pressurized (around 10 MPa or 1450 lb./in2 at the bundle exit) to permit high temperature without boiling. Coolant inlet temperature is typically around 266 °C and the outlet temperature is around 312 °C. Coolant flow in half of the channels flows from left-to-right and the other half flows from right-to-left. The PHWR fuel is natural Uranium. A typical 600-MWe CANDU reactor consists of about 4500 fuel bundles, weighing about 90 Te (90,000 Kg). Refueling of CANDU reactors occurs daily while the reactor operates at power. The refueling is an online loading system, controlled remotely. The fuel is Uranium oxide pellets enclosed in Zircaloy cladding and arranged in bundles as shown in Fig. 14.2. A typical bundle has 37 fuel pins. Fuel bundles are short (around 49.5 cm or 19.5 in.). A channel contains a string of these bundles arranged end-to-end. Refueling involves use of a special refueling machine that opens a channel, removes an old fuel bundle and inserts a new fuel bundle. This is an automated on-line refueling machine. Approximately 15 fuel bundles are replaced every day. Refueling occurs while the plant operates at power. Fuel residence time in the reactor is typically one year.

Pressure Tube Inter Element Spacers

Zircaloy End Cap Zircaloy Fuel Sheath Canlub Graphite Interlayer Uranium Dioxide Pellets Zircaloy End Support Plate

Zircaloy Bearing Pads

FIG. 14.2 A PHWR fuel bundle showing the fuel elements. The fuel bundle is in a pressure tube and is surrounded by the calandria tube. The calandria vessel contains the D2O moderator. Courtesy of UNENE, Ontario, Canada (W.J. Garland (Editor-in-Chief), The Essential CANDU: A textbook on the CANDU Nuclear Power Plant Technology, University Network of Excellence in Nuclear Engineering (UNENE), McMaster University, Hamilton, Canada, 2019, Retrieved from https://www.unene.ca/education/candutextbook, Chapter 8).

193

194

CHAPTER 14 Pressurized heavy water reactors

Insulation on the fuel tubes reduces heat transfer between coolant in the tubes and surrounding heavy water moderator in the calandria. In addition, a cooling system keeps the calandria heavy water at a low temperature (60–65 °C) and pressure slightly above atmospheric. Several considerations influenced the design decisions for PHWR reactors: First was the choice of tubes and pipes to contain the high-pressure coolant. Highpressure tubes and pipes require less advanced fabrication facilities than used in manufacturing large vessels as those used in light water reactors. Countries without large vessel manufacturing capability can produce pressure tubes and pipes. Second was the decision to use natural Uranium fuel. This decision eliminated the need for Uranium enrichment and the need for large, complex and expensive facilities. This decision also led to the need for heavy water as the moderator and coolant. Heavy water enables operation with natural Uranium. Of course, producing heavy water involves isotopic separation of D2O from ordinary water, but this is easier than separation of Uranium isotopes. The design decisions facilitated implementation in developing countries as well as in Canada.

14.3 Neutronic features [3] A PHWR is large in a neutronic sense. Consequently, it is susceptible to spatial instabilities. The control system addresses power shape control as well as global variables. PHWR reactors are over-moderated. That is, reduction of moderator or coolant causes reactivity increases. See Section 7.3. Since new fuel replaces some of the depleted fuel every day, there is a small reactivity “bump” and a small change in power distribution every day.

14.4 Temperature feedback in heavy water reactors The temperature feedback effect on reactivity in PHWRs has three components: fuel, heavy water moderator, and heavy water coolant. PHWRs have strong negative fuel temperature coefficients (Doppler effect), positive moderator and coolant temperature coefficients of reactivity, and positive void coefficient of reactivity. PHWRs are over-moderated. That is, a decrease in moderator increases reactivity. Since PHWRs are over-moderated, the moderator and coolant temperature coefficients of reactivity are positive. The large inventory of heavy water moderator in the calandria is insulated and independently cooled, so its temperature does not change much during a power transient. The Doppler coefficient in PHWR reactors is always negative and is the dominant temperature effect on reactivity. Typical feedback coefficients for PHWRs are as follows [3]: Fuel:
810
6 Δρ/ ° C

14.6 Reactivity control mechanisms

Coolant +4010
6 Δρ/ ° C Moderator + 8010
6 Δρ/ ° C In the U.S., the power coefficient is defined as the reactivity change per per-cent change in power and the power defect is defined as the total reactivity change when going from zero power to full power. In Canada, the power coefficient definition is the same as the U.S. definition of the power defect. This confuses the issue a bit. Here the U.S. definition of the power defect is used. In PHWR reactors, the power defect is around Δρ ¼
0.003. The negative fuel effect dominates over positive coolant and moderator effects because fuel temperature changes much more than coolant or moderator temperature when power changes.

14.5 The void coefficient Like the RBMK, PHWR reactors have a positive void coefficient of reactivity. Decreasing in-core moderator, as by boiling a normally liquid moderator, in over-moderated reactors causes a reactivity increase. The coolant in PHWRs contributes to neutron thermalization in addition to the heavy water moderator in the calandria. Because of insulation from fuel channels, independent cooling, and the large volume, moderator boiling is not plausible. So, the issue is boiling in the coolant. Complete voiding of the coolant channels would cause a large reactivity increase (Δρ ¼ 0.007 to 0.013). The positive reactivity for complete voiding exceeds prompt criticality and is unacceptable. Fortunately, complete voiding is unlikely, and should it occur in an abnormal event, it progresses slowly enough to permit countermeasures (reactor scram).

14.6 Reactivity control mechanisms Heavy water reactors have four reactivity control mechanisms. The calandria contains chambers into which light water can be added or removed. Since light water is a much stronger neutron absorber than heavy water, introduction of light water reduces reactivity. The light water chambers constitute the main reactivity control system. The light water chambers also serve for controlling the flux shape in the reactor core. The water level in individual chambers can be adjusted to influence the neutron flux in the region around the chamber. “Adjuster rods” are absorber rods that serve to flatten the flux distribution and are usually fully inserted. They can also provide positive reactivity when adding light water to the in-core light water chambers is inadequate. They can also be removed to help with Xenon override. “Control absorber rods” are normally positioned outside of the core and are inserted vertically into the core. They can be used to insert negative reactivity when the light water chambers are inadequate. Dissolved poison, typically gadolinium or boron, may be added or removed from the moderator in the calandria to reduce or increase reactivity.

195

196

CHAPTER 14 Pressurized heavy water reactors

14.7 Control systems The control systems for PHWR are more complex than LWR control systems, but also very flexible in providing multiple options for reactivity control. PHWR reactors employ computer control. All nuclear station control and monitoring functions are performed by redundant digital computer systems [4]. The digital control system performs the following functions [4]: • • • •

Maneuvering during normal and abnormal plant conditions Automatic control of normal reactor operation during startup and at any power level Automatic reactor shut down if a reactor safety is issue is encountered Fault tolerant to instrumentation failures and continued safe operation.

There are five main control systems, described below.

14.7.1 Unit power regulator The Unit Power Regulator modulates steam flow to the turbine in order to cause the electrical power to equal set point power.

14.7.2 Reactor regulating system The computer uses an array of neutronic and thermal measurements to determine the reactor power. The difference between the measured power and the power set point causes the computer to introduce reactivity changes by the reactivity control devices. Note that there is no direct control of coolant temperature as in a PWR.

14.7.3 Pressure and inventory control A PHWR pressurizer performs control of primary pressure and the inventory of primary D2O coolant. PHWR pressurizers use electrical heaters that can be modulated to increase or decrease pressure. In a PHWR pressurizer, pressure reduction can be initiated by bleeding steam to a condenser. The D2O level in the pressurizer is monitored. Differences between the measured level and the level set point causes D2O addition or withdrawal by a feed and bleed system.

14.7.4 Steam generator level control PHWRs use U-tube steam generators to transfer heat from the heavy water primary coolant inside tubes to secondary light water outside the tubes. Integral preheaters inside the steam generator brings the feedwater to saturation, then boiling ensues as the feedwater continues its upward path. The heated section is called the riser.

14.9 Reactor dynamics

A mixture of around 10% steam and 90% liquid water emerges from the heated section. Steam separators separate the steam from the water. The steam, containing less than 0.1% of water flows to the turbine. The separated liquid water passes to the downcomer region where it mixes with feedwater before it passes into the riser region. PHWRs use three-element feedwater control during power operation. The threeelement controller involves the measurement of steam generator level, steam flow rate, and feedwater flow rate. Level set point mismatch error and flow mismatch error are used in the controller to adjust the feed flow rate to minimize the combined errors. As in U.S. reactors, this approach overcomes problems with shrink and swell (see Chapter 10).

14.7.5 Steam generator pressure control Steam pressure is held constant during power operation. There are two modes of steam generator pressure control. In the “NORMAL” mode, pressure is controlled by changing reactor power. In the “ALTERNATE” mode, pressure is controlled by changing steam flow to the turbine.

14.8 Maneuvering PHWR computers enable flexibility in maneuvering strategies. Operators can choose to operate in a reactor-follow-turbine mode or a turbine-follow-reactor mode. In the reactor-following or “NORMAL” mode, the steam flow to the turbine responds first after a change in power demand. This causes a change in steam pressure that leads to calculation of a change in the reactor power set point. The reactor regulating system then changes reactivity with the reactivity control mechanisms until the delivered reactor power equals to power set point. In the turbine-following or “ALTERNATE” mode, the reactor power changes first. The new power demand is used as the reactor power set point in the reactor regulating system. Reactivity control mechanisms are activated to change reactivity until reactor power equals the set point. The change in reactor power causes changes in heat transfer in the steam generators, and, consequently, changes in steam pressure. The turbine controller will then adjust steam flow until steam pressure returns to its set point. Fig. 14.3 is a block diagram of the CANDU reactor regulation system (RRS) [5].

14.9 Reactor dynamics An example of a CANDU reactor modeling and dynamic simulation is presented in this section. For details of modeling see Refs. [5, 6].

197

198

CHAPTER 14 Pressurized heavy water reactors

Reactivity Control Mechanisms

To steam generators

Mechanical Control Absorbers (MCA) Adjuster Rods (AR)

Reactor

Liquid Zone Controllers (LZC)

Ion Chambers

Neutron flux and rate

Digital Control Algorithms

Reactivity Control Mechanisms Algorithms

Power Error

Plantinum Detectors

Demand Power Routine

Reactor Power

Power Measurement and Calibration Thermal Power Measurement

Reactor Power Setpoint (From the operator in alternate mode)

FIG. 14.3 Block diagram of the CANDU reactor regulation system. Used with permission from Nuclear Technology, Francis & Taylor (H. Javidnia, J. Jiang, M. Borairi, Modeling and simulation of a CANDU reactor for control system design and analysis, Nucl. Technol. 165 (2) (2009) 174–189).

14.9.1 Modeling strategy Reference [5] presents the development of a lumped parameter CANDU reactor core dynamics model. Because of the arrangement of the fuel in the calandria vessel, incorporating coolant and moderator flows, and the large reactor core size, the reactor core is divided into 14 zones. A set of coupled nonlinear neutron kinetic equations were developed by extending a single point reactor kinetics equation, such as the one used for light water reactor modeling. The coupling between the zones was accomplished by describing the effect of neutrons in zone on the fission reactions in an adjacent zone. The authors also include fission product poisoning using equations for Xenon and Iodine production and decay. Nonlinear ordinary differential equations were used to describe the multi-zone neutronic behavior and the use of various reactivity control features of the CANDU reactor. See Section 14.6. Core reactivity control was achieved by liquid zone controllers, mechanical control absorbers, and adjustable rods. Mechanical control rods are located external to the core and are also used for reactor scram by gravity insertion. See Ref. [5] for definition of the 14 zones, including the number of fuel channels in each zone and volume. Fig. 14.4 shows the division of the CANDU reactor core into 14 zones, as defined in Ref. [5]. Note that the zones in a group have the same physical dimensions and physics properties.

14.9 Reactor dynamics

FIG. 14.4 Division of the CANDU reactor core into 14 zones. Used with permission from Nuclear Technology, Francis & Taylor (H. Javidnia, J. Jiang, M. Borairi, Modeling and simulation of a CANDU reactor for control system design and analysis, Nucl. Technol. 165 (2) (2009) 174–189).

14.9.2 Reactor power response to reactivity insertion Reactor power response to a ramp insertion of reactivity in a zone and its effects on other zones was described in Ref. [5]. A ramp reactivity was introduced from zero to 0.00015 over a time period of 1000 s. The ramp insertion is slow at a rate of 1.5  10
7/s. Fig. 14.5 shows the variation of fractional power change in zone 1. The power is expressed in fractional power unit (FPU). Fig. 14.6 shows reactor power transients in zones 2 through 7. Power transients in zones 8 through 14 are shown in Fig. 14.7. Power levels in various zones are adjusted by control actions to minimize power tilts in the core and to maintain the total fractional power equal to 1 (100%). See Ref. [5] for more information. These simulations show that the CANDU reactor has unique digital control actions to satisfy set point regulation and to maintain proper power distribution in the core.

199

CHAPTER 14 Pressurized heavy water reactors

1.0045 1.004

zone 1 power (FPU)

1.0035 1.003 1.0025 1.002 1.0015 1.001 1.0005 1 0

5

15

10

20

25

time (h)

FIG. 14.5 Power variation in zone 1 due to ramp reactivity insertion in zone 1: FPU: Full Power Unit. Used with permission from Nuclear Technology, Francis & Taylor (H. Javidnia, J. Jiang, M. Borairi, Modeling and simulation of a CANDU reactor for control system design and analysis, Nucl. Technol. 165 (2) (2009) 174–189).

1.0002 zone 2 zone 3 zone 4 zone 5 zone 6 zone 7

1.0001

1 zonal powers (FPU)

200

0.9999

0.9998

0.9997

0.9996

0.9995 0

5

10

15

20

25

time (h)

FIG. 14.6 Power variation in zones 2-7 due to ramp reactivity insertion in zone 1. FPU: Full Power Unit. Used with permission from Nuclear Technology, Francis & Taylor (H. Javidnia, J. Jiang, M. Borairi, Modeling and simulation of a CANDU reactor for control system design and analysis, Nucl. Technol. 165 (2) (2009) 174–189).

References

1

zonal powers (FPU)

0.9999

0.9998

0.9997 zone 8 zone 9 zone 10 zone 11 zone 12 zone 13 zone 14

0.9996

0.9995

0

5

10

15

20

25

time (h)

FIG. 14.7 Power variation in zones 8-14 due to ramp reactivity insertion in zone 1. FPU: Full Power Unit. Used with permission from Nuclear Technology, Francis & Taylor (H. Javidnia, J. Jiang, M. Borairi, Modeling and simulation of a CANDU reactor for control system design and analysis, Nucl. Technol. 165 (2) (2009) 174–189).

Exercises 14.1.

Write a sales pitch that might be used to convince a potential reactor buyer to choose a PHWR rather than a PWR.

14.2.

Write a sales pitch that might be used to convince a potential reactor buyer to choose a PWR rather than a PHWR.

14.3.

How would the sales pitches for China and Mexico differ?

14.4.

PHWRs are touted for use in developing countries. Why is load-following a major issue in those countries?

14.5.

The authors of Reference 5 used fourteen zones for the PHWR core dynamic model while PWR core models often use one zone. Explain.

References [1] W.J. Garland, Editor-in-ChiefThe Essential CANDU: A textbook on the CANDU Nuclear Power Plant Technology, University Network of Excellence in Nuclear Engineering (UNENE), McMaster University, Hamilton, Canada, 2019. Retrieved fromhttps://www. unene.ca/education/candu-textbook.

201

202

CHAPTER 14 Pressurized heavy water reactors

[2] G.T. Bereznai, G. Harvel, Introduction to PHWR Systems and Operation, Available at https://www.iaea.org/.../2011-Oct-Simulators-WS/Harvel-PHWR.pdf. [3] Reactor Physics, Available at https://canteach.PHWR.org/Content Library/20030101.pdf. [4] CANDU 6 Technical Summary, AECL, Canada, 2005. [5] H. Javidnia, J. Jiang, M. Borairi, Modeling and simulation of a CANDU reactor for control system design and analysis, Nucl. Technol. 165 (2) (2009) 174–189. [6] A.P. Tiwari, Modeling and Control of a Large Pressurized Heavy Water Reactor, PhD Dissertation Indian Institute of Technology, Bombay, 1999.

CHAPTER

Nuclear plant simulators

15

15.1 Introduction Many computer-based simulators are available for training and/or education. They provide students and/or trainees with experience that helps them to understand reactor behavior during normal operation and during postulated accident conditions. The student or trainee initiates control actions to achieve desired reactor conditions, including actions that stop accident conditions or minimize undesirable consequences of an accident.

15.2 Types of simulators and their purpose 15.2.1 Simulator games The simplest simulator is a game simulator on a personal computer. Numerous game simulators are available as free downloads on the internet. They provide the user with an ability to drive the simulated response to a desired objective condition. Game simulators provide results that illustrate the way a reactor behaves. They do not provide high-fidelity simulation of a specific reactor design. The emphasis is a qualitative illustration of how a reactor works. They are intended for use by students and interested citizens. Refs. [1–7] show some simulator games that are available on the Internet (in 2019).

15.2.2 Desk-top simulators Desk-top simulators include those that run on software loaded onto a personal computer and those that run on personal computers connected to software via the Internet. Desk-top simulators for training and education provide high-fidelity simulations of specific reactor types. Refs. [8–14] show some simulators for education and training that are available on the internet (in 2019). They include part-task simulators and whole-plant simulators. These include simulators that are commercially available and tailored for a specific reactor. The International Atomic Energy Agency (IAEA) provides whole-plant and parttask desk-top simulators (A part-task simulator simulates only a reactor sub-system) that run on software loaded onto a personal computer. IAEA simulators are widely Dynamics and Control of Nuclear Reactors. https://doi.org/10.1016/B978-0-12-815261-4.00015-9 # 2019 Elsevier Inc. All rights reserved.

203

204

CHAPTER 15 Nuclear plant simulators

used and are described in Section 15.3.1. IAEA provides free simulation software to authorized users in IAEA member countries [8]. Desk-top simulators that access software via the internet are also available and are addressed in Section 15.3.2.

15.2.3 Control room simulators Control room simulators are whole-plant simulators with control rooms that mimic a reactor control room. Trainees see plant condition indications and take operator actions exactly like in an actual reactor control room. Section 15.4 addresses control room simulators.

15.3 Desk-top simulators 15.3.1 Introduction Desk-top simulators are those that run on software loaded onto personal computers (PC). Internet-based desk-top simulators are those in which the user’s personal computer is connected to simulator software via the internet.

15.3.1.1 PC simulation Whole-plant desk-top PC simulators are available from several sources. PC simulators from the IAEA are often used for training and education and are described below. IAEA desk-top simulators were available when this section was written (June 2019) and are as follows: IAEA PWR Simulators • • • • •

Two-loop PWR Advanced two-loop large PWR Russian-type PWR (VVER) Advanced passive PWR Integral PWR

IAEA BWR Simulators • •

Conventional BWR with active safety systems Advanced BWR with passive safety systems

Pressurized Heavy Water Reactor (PHWR or CANDU) • •

Conventional PHWR (for example, CANDU reactor) Advanced PHWR

A part-task IAEA simulator, called the Micro-Physics Nuclear Reactor Simulator is also available. A High Temperature Gas Cooled Reactor (HTGR) simulator is also available.

15.3 Desk-top simulators

15.3.1.2 Using an IAEA simulator Figs. 15.1 and 15.2 show typical screen displays for an IAEA PWR simulator and a BWR simulator. These PC-based simulators were developed by Micro-Simulation Technology [9]. The schematics display important plant parameters during a transient. Both numerical values and plots of selected process variables and reactor power may be displayed as needed and the information may be stored in data files. Both normal operation and accident scenarios can be simulated. The simulation platform, PCTRAN, was developed by Micro-Simulation Technology [15, 16] and is available from IAEA for member nations.

15.3.2 Simulation of PWR and BWR plant transients 15.3.2.1 PWR simulation The computer platform used by the IAEA for training purposes has the ability to simulate various dynamic features of a PWR. The variables that can be changed fall under the following categories: basic, thermal, and rad (radiation) data. These variables affect the initial conditions and are monitored using the transient plots during the run. The plots are displayed in real time, showing different variables as the simulation proceeds. The graphing options include axes scaling, labeling, selection of variables, and saving the choices. An example of load reduction transient in a typical PWR is shown in Fig. 15.3. The simulation shows that reactor power follows the turbine demand. The effect of temperature feedback is seen in the transient when the demand power changes. Various accident scenarios can be simulated with this PC-based software. The effect of the selected malfunction can be displayed using process variables that are relevant to the selected scenario. While the malfunction is in place, new graphics will appear to show the conditions of process variables. It is possible to take actions to prevent the continuation of the malfunction, and thus avoid a severe accident condition. The simulation may be performed either in the manual or automatic mode by selecting appropriate action buttons. It is necessary provide necessary information to the program if run in the manual mode. It is possible to revert to automatic mode at any time during the transient. The plant schematic visual aid displays variations in certain parameters, such as water levels, valve actuation, control rod motion, etc., during plant transient. All the graphical plots and raw data can be saved for future use.

15.3.2.2 BWR simulation BWR simulation is more involved than the PWR simulation. One of the key parameters is the initial reactor power level, since this value determines the approach for further power changes, either an increase or a decrease. The “Power/Flow Map & Controls” option contains a graph that displays recirculation flow rate (%) versus reactor power (%). The program allows the choice of

205

206 CHAPTER 15 Nuclear plant simulators

FIG. 15.1 Operational schematic of a PWR plant using a PC-based simulator (Courtesy of Micro-Simulation Technology).

15.3 Desk-top simulators

FIG. 15.2

207

Operational schematic of a BWR plant using a PC-based simulator. (Courtesy of Micro-Simulation Technology).

208

CHAPTER 15 Nuclear plant simulators

Load Reduction 120 110 100 90 80 70 60

Power Core thermal (%) Power Neutron Flux (%) Power Turbine load (%)

50 40 30 20 10 0 0

500

1000

1500 Time (s)

2000

2500

3000

PCTran 6/27/2011 7:19:50 PM

FIG. 15.3 Simulation of load reduction transient in a 4-loop PWR. (Courtesy of Micro-Simulation Technology [16]).

set points, desired power level, reactivity insertion, and flow rate variation. Multiple screens are available to obtain information during simulation. The display options that can be selected are similar to those for the PWR simulation. The BWR simulation has many screens and different settings to adjust in order to create different models for real-life situations.

15.3.3 How to obtain an IAEA simulator? The International Atomic Energy Agency (IAEA) sponsors the development and distribution of nuclear plant simulators for training and education. The simulation platforms are developed for IAEA by commercial organizations. These simulators are available to university professors and engineers of IAEA member states for use in teaching the safety aspects and operation of nuclear power plants. IAEA distributes the software and relevant documentation at no cost to requesting parties from member states and is made available through a SharePoint located within IAEA’s Nucleus website. The requesting person/organization must register at the following website to obtain a Nucleus login ID: https://nucleus.iaea.org/Pages/default.aspx Mail the completed Request Form to: [email protected]

Example of an industrial control room simulator (Courtesy of Western Services Corporation [17]).

15.3 Desk-top simulators

FIG. 15.4

209

210

CHAPTER 15 Nuclear plant simulators

15.3.4 Internet-based desk-top simulators Several Internet-based nuclear reactor simulators are available. An example may be found at the site www.nuclearpowersimulator.com. This on-line simulator performs both normal operation and reactor anomaly transients. A more advanced simulator for a variety of PWRs and BWRs was developed by Micro-System Technology, called the PCTRAN [16]. PCTRAN is an interactive simulation platform. Graphic icons and pull-down menu facilitate plant simulation, including normal operation and accident transients. Manual pump trips, valve operation, set point changes, power maneuver, and others can be easily performed. Manual override of emergency features is possible and is useful for simulating plant accidents such as the event at Three Mile Island-Unit 2 (TMI-2).

15.4 Control room simulators Control room simulators mimic an actual reactor control room. They include the same displays that exist in the reactor control room and the layout is identical. The actuators and displays are interfaced with a computer that operates reactor software. An instructor has the capability to introduce both normal and accident events in the simulation. Operator trainees respond with actions intended to terminate the event safely. Their responses are judged by the instructors. A trainee must demonstrate proficiency in order to be certified as a reactor operator. Fig. 15.4 shows a typical control room simulator by Western Services Corporation [17].

References Game simulators [1] [2] [3] [4] [5] [6] [7]

www.ae4rv.com/store/nuke_pc.htm (game). download.cnet.com (game). nuclearconnect.org › In the Classroom › For Students (ANS…game) https://steamcommunity.com/sharedfiles/filedetails/?id¼725812726 (game). https://play.google.com/store/apps/details?id¼ru.DmitryLomakin (game). nuclearpowersimulator.com (game). https://www.techspot.com/.../3969-nuclear-power-plant-simulator.htm (game).

Education and Training Simulators [8] [9] [10] [11] [12] [13] [14]

https://www.iaea.org/topics/nuclear-power-reactors/nuclear-reactor. Micro-System Technology: http://microsimtech.com furryelephant.com/.../radioactivity/nuclear-reactor-power-simulation. nuclear.playgen.com (University of Manchester simulator). https://www.nuclearinst.com/Nuclear-Reactor-Simulator (The Nuclear Institute). https://www.gses.com/simulation-technology (commercial systems for sale). getintopc.com/softwares/simulators/boiling-water-reactor-nuclear.

References

General [15] L.C. Po, PC-based simulator for education in advanced nuclear power plant construction, in: International Symposium on the Peaceful Applications of Nuclear Technology in the GCC Countries, Jeddah, 2008http://www.microsimtech.com. [16] PCTRAN, Manual for a 4-loop Pressurized Water Reactor, Micro-Simulation Technology, http://www.microsimtech.com/pctran/, 2016. [17] Western Services Corporation: n.d. https://www.ws-corp.com

211

CHAPTER

Nuclear plant instrumentation

16

16.1 Introduction Nuclear power plants require several thousand measurements during operation. These instrumentation systems supply the information needed by safety systems (protection systems), control systems, and plant monitoring systems. Reactor instrumentation includes the following devices: •

• •

•

Ex-vessel neutron detectors, gamma ray detectors, resistance temperature detectors (RTDs) and thermocouples to measure primary and secondary fluid temperatures; pressure transducers for primary and secondary side pressures; flow meters for primary and secondary flow measurements; and liquid level detectors. Some PWRs have in-core neutron detectors. All BWRs have in-core neutron detector strings. Important safety and control system instrumentation have redundant sensors in order to maintain the reliability of measurements, and to minimize forced plant shutdowns caused by a few malfunctioning instrument channels. Measurements based on multiple redundant sensors use a sort of voting system. Consider a measurement that uses three redundant sensors. If all three sensors provide the same output (within tolerances) the sensors are judged to be functioning properly and the measurement is accepted. If one sensor output disagrees with the other two, it is judged to have malfunctioned. The disagreeing sensor is ignored and the two agreeing sensors are used. This is called two out of three logic. If less than two sensors agree, the measurement is judged to be unreliable and appropriate action is undertaken, including shutdown. A large nuclear power plant uses as many as 10,000 sensors and detectors in the control, safety, and monitoring instrumentation systems.

Refs. [1, 2] provide a comprehensive description of nuclear power plant instrumentation. These handbooks were written many years ago, but most of the information is still pertinent. Refs. [3, 4] provide information on state-of-the-art nuclear plant instrumentation and their performance.

Dynamics and Control of Nuclear Reactors. https://doi.org/10.1016/B978-0-12-815261-4.00016-0 # 2019 Elsevier Inc. All rights reserved.

213

214

CHAPTER 16 Nuclear plant instrumentation

16.2 Sensor characteristics 16.2.1 Neutron and gamma ray detectors Since neutrons have no electrical charge, they cannot be detected directly. Rather, detectors contain material that undergoes reactions with neutrons and releases charged reaction products or light that can be detected. In some sensors, charged particles create a measurable electrical current in detectors that maintain a voltage difference between conductors. The sensor’s calibration converts the measured current into neutron flux. In some sensors, the charged particle migrates from its source to a metallic sheath, causing a measurable voltage. The sensor’s calibration converts the measured voltage into neutron flux. In some sensors, the neutron interaction produces light. The light intensity can be measured, and calibration is used to convert the measured light intensity into neutron flux. Another measurement of reactor power uses ex-core monitoring of radionuclides produced in the core by neutron absorption. Reactor power can also be measured by sensing temperature changes due to gamma ray heating inside a sheathed probe. No single detector can satisfy all of the requirements for measuring reactor power. Three different detectors are used to cover the full range of reactor power: startup, mid-range and full power. Ex-core sensors provide total power measurements and in-core detectors monitor local power. Brief descriptions of various detector types are as follows:

16.2.1.1 Ionization chambers Ionization chambers are cylindrical tubes with a centrally located wire (see Fig. 16.1).

FIG. 16.1 An ionization chamber.

16.2 Sensor characteristics

A voltage is applied between the tube metal and the wire. The tube is filled with a neutron-absorbing gas, typically BF3 or has boron coated on the inside wall of the cylinder. Neutron absorptions in boron produce charged particles that migrate under the voltage difference and produce a measurable current. Ionization chambers can monitor individual events (pulse mode) when the neutron flux is small. It can provide a continuous current when the neutron flux is large.

16.2.1.2 Fission detectors A fission detector is an ionization chamber containing fissile material coated on the inner wall of a metallic sheath (see Fig. 16.2). Incident neutrons cause fissions and result in release of charged particles. The charged particles migrate due to the applied voltage and produce a measurable current.

16.2.1.3 Self-powered neutron detectors Self-powered detectors contain a neutron absorber (typically Rhodium, Vanadium or Platinum) inside an insulated metallic sheath (see Fig. 16.3). Neutron absorptions produce an isotope that decays by beta emission. The beta particle (electron) migrates to the sheath, creating a potential difference between the sheath and a small electrical current. This current (microamperes) is measured to provide the neutron flux at the sensor. Self-powered detectors are small and require no external power supply. They are used for in-core measurement of local neutron flux.

FIG. 16.2 A fission chamber.

215

216

CHAPTER 16 Nuclear plant instrumentation

FIG. 16.3 A self-powered neutron detector.

Since the appearance of the beta particle in a self-powered detector must wait for radioactive decay of the emitter, the measurement is not instantaneous. So, the measurement experiences a time lag that depends on the half-life of the emitter. Also, the sensor calibration changes as the absorber is consumed by neutron absorptions.

16.2.1.4 Scintillation detectors Scintillation detectors contain material that emits light upon absorbing a neutron (see Fig. 16.4). The sensor monitors the light intensity and converts that signal into neutron flux.

16.2.1.5 Gamma thermometers Fig. 16.5 shows the general layout of a gamma thermometer. The sensor contains a differential thermocouple that measures the temperature difference between two locations inside the sensor. One location is insulated and the other is un-insulated. Heat deposited (mainly by gamma rays) inside the sensor causes the temperatures at the two locations to be different because of different heat transfer resistances. Following equations describe the principle of a gamma thermometer.

FIG. 16.4 A scintillation detector.

16.2 Sensor characteristics

FIG. 16.5 A gamma thermometer.

1 ðTh  θÞ, for the insulated region Rh

(16.1)

1 ðTc  θÞ, for the un-insulated region Rc

(16.2)

P¼

P¼

where P ¼ power deposited (equal in both regions) Th ¼ temperature in the hotter insulated region Tc ¼ temperature in the cooler un-insulated region Rh ¼ heat transfer resistance in the insulated region Rc ¼ heat transfer resistance in the un-insulated region θ ¼ temperature of the fluid around the sensor. Eqs. (16.1) and (16.2) yield the following: ðTh  Tc Þ ¼ ðRh  Rc Þ P

(16.3)

That is, the temperature difference is proportional to the power.

16.2.1.6 Nitrogen-16 measurement N-16 is produced in the coolant in light water reactors by neutron absorption in oxygen. The N-16 flows with the coolant into the hot leg where gamma rays from N-16 decay are measured. Since N-16 production is proportional to reactor power, monitoring the ex-core gamma rays from N-16 provides the N-16 concentration and, therefore, the reactor power.

217

218

CHAPTER 16 Nuclear plant instrumentation

16.2.2 Temperature sensors Temperature sensors in a presently-operating reactor are either resistance temperature detectors (RTDs) or thermocouples inside a stainless-steel sheath.

16.2.2.1 Resistance thermometers RTDs have sensing elements made of metal, typically platinum. The platinum metal in some reactor RTDs is in the form of a wire wrapped around a mandrel (typically magnesium oxide) inside a stainless-steel tube with magnesium oxide insulator between the mandrel and the inner wall of the sheath. See Fig. 16.6. Another RTD design uses a platinum wire coil cemented to the inside wall of a hollow section of a metallic tube. This approach provides a very fast-responding temperature measurement because the heat transfer resistance between the coil and the sheath is small. See Fig. 16.7. Platinum has a well-defined temperature resistance relationship. Instrumentation measures the resistance and converts it to a temperature measurement using temperature vs. resistance calibration data. The resistance increases with temperature and the temperature-resistance relation is almost linear. But the readout instrumentation accounts for the small non-linearity.

FIG. 16.6 A resistance temperature detector.

16.2 Sensor characteristics

FIG. 16.7 A fast response RTD.

16.2.2.2 Thermocouples Thermocouples have two dissimilar wires that are joined to form the thermocouple junction. Reactor thermocouples are contained inside a stainless-steel sheath. Insulator (magnesium oxide) fills the space between the thermocouple and the inner wall of the sheath. See Fig. 16.8. A thermocouple creates a voltage that depends on the temperature difference between the junction and open end of the wires. So, a thermocouple voltage is a function of the temperature difference between the open end and the junction. The junction temperature is obtained as follows: – Instrumentation measures the thermocouple voltage.

FIG. 16.8 A sheathed thermocouple.

219

220

CHAPTER 16 Nuclear plant instrumentation

– Instrumentation measures the open-end temperature using a different type of sensor (usually a thermistor or integrated circuit sensor operating at the instrumentation which operates at ambient temperature). – Instrumentation calculates the voltage that would occur for a thermocouple with a junction at the ambient temperature measured at the instrumentation and the open end at 0 °C. – Instrumentation adds the measured thermocouple junction-to-open end voltage and the calculated ambient-to 0 °C voltage. – Instrumentation uses the summed voltages to calculate junction temperature using standard calibration data for a thermocouple with the open end at 0 °C. Thermocouples may be used for in-core coolant temperature measurements. Type-K thermocouples (chromel - alumel) or type-N (nicrosil - nisil) are suitable. Type-N is usually preferred for high temperature measurements because of a decalibration tendency in Type-K. Thermocouples may have the junction insulated from the sheath or have the thermocouple wires attached to the sheath (a grounded-junction thermocouple). Grounded-junction thermocouples have faster time response than insulated junction thermocouples.

16.2.2.3 Thermowells and bypass installation It is necessary to be able to replace temperature sensors if they malfunction, decalibrate, or develop unsatisfactory time response. Therefore, coolant temperature sensors are placed in thermowells (see Fig. 16.9) or in bypass lines that can be valved out of the primary system. Thermowells and sheathed sensors are sometimes modified to reduce the heat transfer resistance and increase the speed of response. One approach is to taper the end of the sensor and taper the bore of the thermowell with a matching taper near the end. This provides a snug fit in the region containing the sensing element. But caution is required. If foreign material collects at the tip of the thermowell or if the sensor is not fully inserted, then an air gap between the sensor and the thermowell occurs and the speed of response decreases. Another approach is to use a silver bushing around the sensor sheath in the section where the sensing element is located. The silver is threaded to provide contact points between the sensor and the thermowell. The silver is soft and will conform to make contact with the thermowell. But caution is required here too. If the sensor is withdrawn and reinserted, some of the silver will be rubbed off and the enhanced contact with the thermowell will diminish.

FIG. 16.9 A thermowell.

16.2 Sensor characteristics

Use of a thermowell or a bypass line installation causes a slower time response following a coolant temperature change than would occur with a bare sensor inserted in the fluid. Regulations require verification of suitable sensor time response. A method for in-situ response time measurement of RTDs is available [5].

16.2.2.4 Advanced temperature sensors Pyrometry is potentially useful for temperature measurement in high-temperature conditions. Pyrometers measure the heat radiation from an object. Pyrometers could be used to measure pipe wall temperature or fluid temperature by focusing into wells immersed in the fluid. A new temperature measuring system is based on monitoring of Johnson noise in the sensing element. Johnson noise is naturally occurring temperature-dependent fluctuations in material. Johnson noise thermometry systems do not decalibrate and they are suitable for high temperature measurements.

16.2.3 Pressure sensors Reactor pressure sensors have an elastic element (either a diaphragm, bellows or a Bourdon tube) that expands or contracts when experiencing a pressure difference across the element. See Fig. 16.10. Pressure is determined by measuring the displacement of the elastic element or by measuring the force required to push the element back to its rest position (called a force-balance sensor).

FIG. 16.10 Elastic components for pressure sensors.

221

222

CHAPTER 16 Nuclear plant instrumentation

16.2.4 Flow sensors Flow measurements are very important in nuclear reactors and other process industries. Consequently, there have been major efforts to develop flow sensors for various process conditions, resulting in numerous ways to measure fluid flow in processes. Some of the methods are old and widely used and some are recently developed. Described below are several methods with applicability in current or future reactors.

16.2.4.1 Flow vs. pressure drop A common reactor fluid flow measurement involves measurement of the pressure drop across a constriction in the flow path. This method has seen widespread use in process industries for many years. The constriction can be an orifice or a venturi section (see Fig. 16.11) or even just a bend in a pipe.

16.2.4.2 Advanced flowmeters Advanced flow measurement technology is also available, and some will likely be used in advanced reactors. These include ultrasonic sensors, magnetic flowmeters for liquid metal flow measurement, and eddy current sensors for liquid metal flow measurement. These sensors exist in various configurations. Examples that illustrate the technology are described briefly below: One type of ultrasonic flowmeter employs two sound transducers mounted at an angle to the pipe and mounted so as to face one another. One transducer is aimed in the direction of flow and one aimed in opposition to the flow in a pipe. When the upstream transducer emits an ultrasonic pulse, the signal reaches the downstream transducer after passing through fluid traveling in the same direction as the pulse. When the downstream transducer emits an ultrasonic pulse, the signal reaches the upstream transducer after passing through fluid flowing in the opposite direction of the pulse. The transit times of the pulses depends on the velocity of the fluid

FIG. 16.11 A venturi device for flow measurement.

16.2 Sensor characteristics

through which it passes. These transit times are measured, and their difference is proportional to the fluid flow rate. In a magnetic flowmeter, a pipe containing flowing liquid metal is placed between the poles of a magnet. A voltage is generated by the motion of a conductor through a magnetic field. Note that electric generators operate on the same principle (motion of a conductor through a magnetic field). Eddy current flowmeters are devices containing three coils positioned in the direction of flow. The middle coil is excited with a constant alternating current. This current creates induced voltages in the other two coils. When immersed in flowing liquid metal, the upstream coil’s voltage is less than the downstream coil’s voltage and this voltage difference is proportional to the fluid flow rate. Coriolis flowmeters measure mass flow. They are based on the twisting of oscillating flexible tubes caused by the inertia of fluid flowing inside the tubes. The twisting is proportional to the mass flow rate of the fluid.

16.2.5 Level sensors 16.2.5.1 Differential pressure Level measurements (as water level in the downcomer in a U-tube steam generator or in a BWR, or the pressurizer water level) use the differential pressure above and below the water columns to provide the level. See Fig. 16.12. The pressure at the bottom of a column of liquid is given by the following: P ¼ ρgh

where P ¼ pressure ρ ¼ density of the fluid g ¼ acceleration due to gravity h ¼ height of the fluid column.

FIG. 16.12 A differential pressure level measurement.

(16.4)

223

224

CHAPTER 16 Nuclear plant instrumentation

Therefore, the differential pressure is proportional to the fluid column height and provides quantitative height measurement if the fluid density is known. In level measurement by differential pressure measurement, the fluid is in contact with the pressure sensor. Therefore, if the fluid is radioactive or otherwise hazardous, maintenance on or replacement of the pressure sensor is problematical.

16.2.5.2 Bubbler A bubbler system also provides level measurement capability. See Fig. 16.13. The figure shows a bubbler system for a tank that is open to the atmosphere. A regulator maintains constant gas flow. The gas flow induces bubble release at the bottom of the tank. As the fluid level increases, the flow regulator increases pressure in order to maintain flow. To maintain bubble flow, the pressure at the bubble outlet must equal the fluid pressure at the outlet location. Therefore, the gas line pressure equals the fluid pressure at the location of the bubbler outlet. So, the pressure on the gas line equals the liquid pressure at the location of the gas outlet. Bubbler fluid level measurements for a closed tank require a pressure regulator to maintain constant pressure in the space above the fluid. Bubbler fluid level measurements are useful in limited-space situations (only the gas tube must occupy space in and near the tank) and it is applicable in hot or otherwise hostile environments. Only gas is in the sensing line adjacent to the pressure sensor, thereby facilitating maintenance.

FIG. 16.13 A bubbler level measurement system.

16.4 BWR instrumentation

16.2.6 Actuator status sensors As demonstrated in the Three Mile Island - II accident, it is not enough to rely on signals sent for actuators to operate. Actuators, such as valves, can fail to operate in response to signals from the control system or an operator. An independent measurement of actuator status is needed to confirm operation.

16.3 PWR instrumentation Fig. 16.14 shows the major sensors and their placement in a typical pressurized water reactor power plant. Instrumentation in both primary and secondary sides of the plant are shown. Specific instrumentation channels include the following: •

• • • • • • •

In-core and ex-vessel neutron detectors. In-core sensors are on a drive system that moves the sensors to desired core locations to provide neutron flux distribution (mapping) information. Ex-vessel neutron detectors are placed in instrument wells in the biological shield surrounding the reactor vessel. Typically, there are eight detector assemblies (long ionization chambers). The nuclear instrumentation monitors reactor power from source range to intermediate range, to 120% of full power [6]. N-16 detectors are located on hot leg piping and are used to measure both reactor power and primary coolant flow rate (the latter has an accuracy of 1.5%). Core-exit thermocouples located above the core, typically at about 45 locations. Sensors for digital rod position indication. Hot leg and cold leg RTDs (resistance temperature detectors) needed for both control and safety system functions. Pressurizer pressure and water level. Primary coolant flow rate. Steam generator feedwater flow rate, steam flow rate, water level.

Redundancies are included in all these measurements. Fig. 16.15 shows in-core instrumentation in a typical PWR [6]. Fig. 16.15 shows flux detector thimble guide tubes (for flux mapping) introduced from the vessel bottom and core-exit thermocouples that are inserted from the top of the reactor vessel. Major balance of plant instrumentation includes steam flow rate and temperature at different turbine stages, condenser pressure, feedwater temperature, and feed pump conditions.

16.4 BWR instrumentation Instrumentation for a typical BWR appear in Fig. 16.16.

225

226

R

P

LP turbine

CP spray Control rod drive mechanism

F

Steam generator heater

Generator

HP turbine

Pressurizer P L

Reactor core

T coolant pump

CP Control rod position

P Pressure

feedwater control valve

F Flow rate

R

L

T

Rotation speed

feedwater heater

feedwater pump

T F

Water level

Condenser

condenser dump valve

neutron flux

Fluid flow Measurement

Temperature

FIG. 16.14 Schematic of important PWR plant instrumentation.

CHAPTER 16 Nuclear plant instrumentation

turbine control valve

P P

L

16.4 BWR instrumentation

FIG. 16.15 In-core instrumentation in a typical PWR. Courtesy of Westinghouse Electric Company ([6]).

The following are the important measurements in a typical BWR system: • • • • • • • •

Local power range monitor (LPRM) detector signals (each LPRM string or tube has 4 fission chambers). Average power range monitor (APRM) signals. An APRM signal is measured by averaging about 20 LPRM detector signals from across the core. Reactor coolant flow rate. This is measured by venturi meters at the recirculation pump discharge locations. Downcomer water level. Steam flow rate. Reactor pressure (steam pressure). Feed water flow rate. Feed water temperature.

A typical BWR has about 45 strings of in-core detectors (fission chambers). Fig. 16.17 shows a BWR core map indicating detector locations. The in-core

227

228

P

Main steam flow rate F Water level L

P

R

Reactor core

LP turbine

neutron flux

bypass valve

Recirculation pump

Generator

HP turbine

F

Condenser Feedwater pump

F Control rod drive mechanism

Feedwater heater

T F Flow rate

P Pressure

L Water level

R Rotation speed

FIG. 16.16 Schematic of important BWR plant instrumentation.

T Temperature

Fluid flow Measurement

P

CHAPTER 16 Nuclear plant instrumentation

T

16.5 CANDU (PHWR) reactor instrumentation

CORE POSITION MAP 57

49

41 INSTRUMENT TUBE 33

25

17

09

08

16

24

32

40

48

56

CORE TOP 144 in. 126 in.

D

90 in.

C

54 in.

B

18 in.

A

235

U-COATED IONIZATION CHAMBER (1 in. LONG BY ¼ in. DIAM)

CORE BOTTOM 0 in.

FIG. 16.17 A BWR core map showing in-core detector string locations [7].

instrument tube is placed among four fuel channels, with the central space available for a traversing in-core probe (TIP) detector used for calibrating LPRM detectors.

16.5 CANDU (PHWR) reactor instrumentation Much of the instrumentation in typical CANDU reactors is similar to instrumentation in pressurized light water reactors. The balance-of-plant is essentially the same in both types and the instrumentation is essentially the same. Likewise, many of the

229

230

CHAPTER 16 Nuclear plant instrumentation

measurements in the nuclear measurements in the nuclear steam supply system are similar. The steam generator is the U-tube type like the type used in most pressurized light water reactors and required measurements are pressure, temperature, level, and flow. Primary coolant temperature is measured with resistance temperature detectors and neutron flux is measured with ex-core detectors and in-core detectors. In-core neutron detectors are located in different regions of the core. Approximately 102 Vanadium self-powered detector are inserted vertically and are used for flux mapping. These detectors have slow response time and provide excellent flux measurements at steady-state conditions and are not used for reactor control. Twenty-eight (2 for each of the 14 zones) platinum-clad Inconel detectors are inserted vertically from the reactivity mechanism deck and are used in the reactor regulation system (RRS) [8]. Additionally, 34 platinum-clad Inconel detectors are inserted vertically and 24 are inserted horizontally at various channels as part of the reactor shutdown system (SDS). One set of in-core BF3 ionization chambers is used for flux measurement during start-up. A set of ex-core BF3 detectors is used for power measurement. The in-core BF3 detectors are withdrawn after the initial start-up phase. The reactor core and its operation are more complex in PHWRs than in PWRs, requiring more measurements than in a pressurized light water reactor. The CANDU reactor core is large (both physically and neutronically) and on-line fuel insertion and removal occurs frequently. Local changes in the neutron flux occur at refueling sites. Therefore, more in-core detectors are needed to monitor the flux shape. CANDU reactors also include measurement of flow, inlet temperature and outlet temperature in selected channels, requiring more pressure sensors and resistance temperature detectors. CANDU reactors also need tritium tracking because neutron captures in deuterium produces tritium.

16.6 High temperature reactor instrumentation There are three high temperature reactors under consideration for deployment. They are the liquid metal fast breeder reactor (LMFBR), the high temperature gas cooled reactor (HTGR) and the molten salt reactor (MSR). Coolant temperatures are high compared to water-cooled reactors and reach as high as 1000 °C in gas-cooled reactors. Many sensors used in reactors with water or heavy water coolant will not work at the higher temperature experienced in the advanced reactors. Either new sensors must be employed or developed, or the systems must be designed so as not to need certain measurements. Sensors used to measure coolant temperature must operate at temperatures above the limit for RTDs (661 °C). Type-N thermocouples with an upper limit of 1200 °C can tolerate conditions in high temperature reactors. But thermocouples are known to drift at high temperature. (Type-N was developed as a more stable replacement for

16.6 High temperature reactor instrumentation

Type-K). Other advanced temperature systems, such as Johnson noise thermometers or pyrometers, might augment thermocouple measurements. Not only are the temperatures high, but the reactor characteristics also determine sensor requirements. In-core and primary loop sensors must be made of materials that are chemically compatible with the coolant (especially liquid metal in LMFBRs and molten salt in MSRs). Chemicals present in the reactors dictate the need to monitor chemical processes and sensors must be capable of operating in the chemical environments. The speed with which reactor conditions can change in a reactor determine the required speed of response of sensors or lack thereof.

16.6.1 Liquid metal fast breeder reactor (LMFBR) instrumentation Instrumentation needed for LMFBRs is influenced by the plant characteristics (very high temperature, chemically active coolant, fast neutron spectrum and fast response in a transient). Following is a list of instrumentation used in a typical sodium fast reactor (SFR) [9–11]. • • • • • • • • • • • • • • • • • • • •

Neutron power (in-core and ex-core neutron detectors) Primary pump (sodium) flow rate and pump differential pressure (ΔP) Cold pool sodium temperature Hot pool sodium temperature IHX pump (sodium) flow rate and pump differential pressure (ΔP) Pump motor electrical signatures Core inlet plenum sodium temperature Core outlet plenum sodium temperature Control rod position/reactivity IHX primary sodium outlet temperature IHX secondary sodium inlet temperature Steam generator sodium inlet temperature Steam generator steam temperature Superheated steam pressure Steam flow rate to the turbine Feedwater flow rate to the steam generator Feedwater temperature Condenser pressure, condenser coolant inlet and outlet temperatures Feedwater heater water level and inlet/outlet temperatures. High-pressure and low-pressure extraction steam flow rates.

These measurements span the following systems: reactor core, primary heat transport system, intermediate heat transport system (IHTS), steam generator system (SGS), and balance-of-plant (BOP). Depending on the system design, other measurements and corresponding instrumentation would be necessary.

231

232

CHAPTER 16 Nuclear plant instrumentation

See Refs. [9–11] for more details of the liquid flow system and instrumentation in a SFR. Some specific sensors for the 500 MWe prototype fast breeder reactor (PFBR) include the following [9]: • • • • •

Core inlet temperature is measured by six thermocouples. Each sub-assembly coolant outlet temperature is measured by two thermocouples. High-temperature in-core fission chambers are used for neutron flux measurements. Sodium flow rate is measured by eddy current flow meters at the outlet of the primary pumps. Level of sodium in the reactor pool is measured by continuous level probes. In addition, there is leak detection instrumentation in the steam generators.

Other instrumentation includes sensors for cover gas hydrogen metering, electrochemical in-sodium hydrogen measurement, electrochemical carbon meter, sodium ionization detector, etc.

16.6.2 High temperature gas-cooled reactor (HTGR) instrumentation Instrumentation needed for HTGRs is influenced by the plant characteristics (very high temperature, inert coolant, and large heat capacity that causes slow response in a transient). A summary of the various sensors used for measurements follows without reference to a specific commercial HTGR plant. Following are typical measurements for HTGRs [12]. • • •

• •

• •

•

Primary coolant core inlet and outlet temperatures, along with core internals are measured using Type-N thermocouples. Graphite block temperatures are measured using tungsten-rhenium thermocouples. The mass flow rate of the primary coolant is calculated from data of the pressure head of the helium blower, coolant temperature, coolant pressure, and the compressor rotational speed. This is an inferred quantity. High-temperature Coriolis gas flow meters may be used in the future for coolant flow measurement. Two types of neutron detectors are used: fission counters for wide-range monitoring system (WRMS) and uncompensated ionization chambers for power range monitoring system (PRMS), which can detect a low neutron flux level outside the reactor pressure vessel. WRMS are installed on the internal surface of the pressure vessel. Primary coolant pressure is monitored by safety-grade pressure transmitters. The primary coolant flow through the reactor core is measured by a differential pressure transmitter that monitors the core inlet and out let pressure difference. This measurement is also used by the safety (protection) system. Other measurements include gas properties, moisture ingress (from circulation through the steam generator), gas leak detection, and strain gauges on the internal vessel structure and on the vessel external surface.

Certain sensors are unique to HTGRs as compared to water-cooled reactors.

Entrainment separator F

Secondary salt pump

Primary salt pump

F

F

F

F

Tertiary salt pump

F F

P L

Gas separator

F

F

F

P

P

P

T

T

T

N

C C F P

F

F

P

P

P

T

T

T

C

Chemical composition

F

Flow rate

L

Level

P

Pressure

T

Temperature

F T

Heat rejection tank

P L

C

T F

Balance-of-plant

F F

Gas cleanup system

Primary drain tank

P L F

FIG. 16.18

T

C

F

T

Freeze plug

P

P

T

16.6 High temperature reactor instrumentation

Primary HX

Reactor

F

Tertiary HX

Bubble generator

Secondary HX

T

Instrumentation layout for a molten salt demonstration reactor plant.

233

234

CHAPTER 16 Nuclear plant instrumentation

16.6.3 Molten salt reactor instrumentation Molten salt power reactors are candidates for future implementation, but at the time of preparing this book, they were still in the design and development stage. Fig. 16.18 shows the conceptual layout of the main instrumentation in a molten salt demonstration reactor (MSDR) [13]. The schematic shows placement of process sensors in the primary, secondary, and tertiary loops, as well as ex-vessel neutron detectors. Most of the required measurement are the same as in earlier plants, but MSR conditions (high temperature, salt chemistry, and use of thorium) dictate the use of some different sensor technologies. Various options for chemical processes to separate uranium, thorium and protactinium into different streams and handling are under consideration. The chemical processes selected in an MSR design will require appropriate instrumentation. Temperature measurements will rely on thermocouples (type-N) because temperatures are above the limit for RTD use. Pressure and flow measurements are made using standard instrumentation with sensing lines from process points to transmitter locations. Modern magnetic flow meters are sensitive enough for use in molten salt systems. Ex-vessel neutron detectors as used in other nuclear power plants would be applicable in a similar configuration in MSRs. Most ex-vessel neutron detectors span the active fuel length and are placed around the top half and bottom half of the reactor vessel. Generally, there is a set of four detectors in the top half and another set of four detectors in the bottom half, placed symmetrically around the reactor vessel; thus, the reactor power can be monitored radially as well as axially. Monitoring of chemical processes occurs in all types of power reactors but is much more important in MSRs. MSRs may be viewed as chemical plants in which nuclear fission occurs and produces heat. Fig. 16.18 shows locations in various salt loops where chemical composition measurements are made. A review of instrumentation for advanced reactors is presented in Ref. [14].

References [1] J.M. Harrer, J.G. Beckerley, Nuclear Power Reactor Instrumentation Systems Handbook, vol. 1, TID-25952-P1, National Technical Information Service, U.S. Department of Commerce, Springfield, VA, 1973. [2] J.M. Harrer, J.G. Beckerley, Nuclear Power Reactor Instrumentation Systems Handbook, vol. 2, TID-25952-P2, National Technical Information Service, U.S. Department of Commerce, Springfield, VA, 1974. [3] H.M. Hashemian, Maintenance of Process Instrumentation in Nuclear Power Plants, Springer-Verlag, Berlin, Germany, 2006. [4] H.M. Hashemian, Monitoring and Measuring I&C Performance in Nuclear Power Plants, International Society of Automation, Research Triangle Park, NC, 2014. [5] T.W. Kerlin, L.F. Miller, H.M. Hashemian, In-situ response time testing of platinum resistance thermometers, ISA Trans. 17 (4) (1978) 71–88.

Further reading

[6] Westinghouse Electric Company, The Westinghouse Pressurized Water Reactor Nuclear Power Plant, Westinghouse Electric Company, Pittsburgh, 1984. [7] D.N. Fry, J. March-Leuba, F.J. Sweeney, Use of Neutron Noise for Diagnosis of InVessel Anomalies in Light Water Reactors, Oak Ridge National Laboratory, 1984, ORNL/TM-8774 (NUREG/CR-3303, January). [8] W.J. Garland, The Essential CANDU: A Textbook on the CANDU Nuclear Power Plant Technology, UNENE, McMaster University, Canada, 2019. [9] P. Swaminathan, Modeling of Instrumentation and Control System of Prototype Fast Breeder Reactor, Doctoral Dissertation, Sathyabama University, Chennai, India, 2008. [10] G. Vaidyanathan, et al., Sensors in sodium cooled fast breeder reactors, Natl. J. Electron. Sci. Syst. (India) 3 (2) (2012) 78–87. [11] K. Velusamy, et al., Overview of pool hydraulic design of Indian prototype fast breeder reactor, Sadhana 35 (2) (2010) 97–128. [12] HTGR Technology Course for the Nuclear Regulatory Commission, Module 12, Instrumentation and Controls (I&C) and Control Room Design, INL and General Atomics, May 24–27, 2010. [13] E.S. Bettis, L.G. Alexander, H.L. Watts, Design Studies of a Molten-Salt Reactor Demonstration Plant, Oak Ridge National Laboratory, ORNL-TM-3832, 1972. [14] K. Korsah, et al., Assessment of Sensor Technologies for Advanced Reactors, ORNL/ TM-2016/337, August, 2016.

Further reading [15] S.J. Ball, D.E. Holcomb, S.M. Cetiner, HTGR Measurements and Instrumentation, Oak Ridge National Laboratory, ORNL/TM-2012/107, 2012. [16] T.W. Kerlin, R.L. Shepard, Industrial Temperature Measurement, Instrument Society of America, Research Triangle Park, NC, 1982. [17] T.W. Kerlin, M. Johnson, Practical Thermocouple Thermometry, second ed., International Society of Automation, Research Triangle Park, NC, 2012.

235

APPENDIX

A

Generation II reactor parameters

This Appendix contains tables of important parameters of typical pressurized water reactors, boiling water reactors, and CANDU reactors. The tables list the parameters for representative reactor systems.

A.1 Pressurized water reactor (PWR) Tables A.1 and A.2 summarize important parameters of a typical four-loop pressurized water reactor [1, 2]. Table A.1 Typical four-loop PWR nuclear steam supply system (NSSS). Reactor thermal power Approximate electrical power generation Number of U-tube steam generators Steam generator tube material Steam pressure Steam flow rate per steam generator Steam generator overall height Steam generator upper shell outer diameter Steam generator lower shell outer diameter Steam generator shell material Reactor coolant pump motor horsepower Hot leg inner diameter Cold leg inner diameter Total coolant flow rate Primary coolant pressure

3411 MW 1150 MWe 4 Thermally coated Inconel-600 1000 psia (69 bar) 3,813,000 lb./h (480 kg/s) 67 ft 8 in. (20.6 m) 14 ft 7–3/4 in. (4.5 m) 11 ft 3 in. (3.4 m) Mn-Mo steel 7000 29 in. (73.7 cm) 27.5 in. (69.9 cm) 138.4  106 lb./h (17,438 kg/s) 2250 psia (155 bar)

Adapted from The Westinghouse Pressurized Water Reactor Nuclear Power Plant, Westinghouse Electric Company, Nuclear Operations Division, Pittsburgh, 1984; M. Naghedolfeizi, B.R. Upadhyaya, Dynamic Modeling of a Pressurized Water Reactor Plant for Diagnostics and Control, Research Report, University of Tennessee, DOE/NE/88ER12824-02, June 1991.

.

237

238

APPENDIX A Generation II reactor parameters

Table A.2 Typical PWR pressure vessel and core parameters. Overall length of assembled vessel Inside diameter of vessel Average vessel thickness Nominal vessel cladding thickness Vessel material Cladding material Coolant volume in vessel Number of vessel material surveillance capsules Number of fuel assemblies Active fuel length Fuel assembly array Fuel weight in the core (Uranium) Average fuel burnup Equivalent core diameter Nominal core coolant inlet temperature Core coolant outlet temperature Average coolant temperature rise in vessel Heat generated in fuel Total number of fuel rods in core Fuel rod cladding material Number of rods in the control banks Number of rods in the shutdown banks

44 ft, 7 in. (13.6 m) 173 in. (4.4 m) 8 in. (20.3 cm) 7/32 in. (0.56 cm) Low-alloy steel Stainless steel 4885 ft3 (138 m3) 6 193 12 ft (365.8 cm) 17  17 81,639 kg 32,000 MWd/Te of heavy metal 11.06 ft (338 cm) 557.5 °F (291.9 °C) 618.5 °F (325.8 °C) 61 °F (33.9 °C) 97.4% 50,952 Zircaloy-4 25 28

Adapted from The Westinghouse Pressurized Water Reactor Nuclear Power Plant, Westinghouse Electric Company, Nuclear Operations Division, Pittsburgh, 1984; M. Naghedolfeizi, B.R. Upadhyaya, Dynamic Modeling of a Pressurized Water Reactor Plant for Diagnostics and Control, Research Report, University of Tennessee, DOE/NE/88ER12824-02, June 1991.

A.2 Boiling water reactor (BWR) Important parameters of a typical boiling water reactor are listed in Tables A.3 and A.4 [3–6].

A.3 Pressurized heavy water reactor (PHWR): CANDU reactor Tables A.5 and A.6 summarize important parameters of a typical CANDU (CANada Deuterium Uranium) reactor [3, 7].

APPENDIX A Generation II reactor parameters

Table A.3 Typical BWR nuclear steam supply system (NSSS). Reactor thermal power x Coolant pressure (two-phase) Feedwater flow rate Total core coolant flow rate Steam (coolant) temperature Feedwater temperature Steam flow rate

3579 M W 1220 MWe 1040 psia (70 bar) 14,414,400 lb./h (1,820 kg/s) 104  106 lb./h (13,131 kg/s) 551 °F (288 °C) 420 °F (216 °C) 14,414,400 lb./h (1,820 kg/s)

General Electric Company, BWR/6: General Description of a Boiling Water Reactor, San Jose, 1980; U.S. NRC Technical Training Center, Boiling Water Reactor (BWR) Systems, Chapter 3, Revisions 0200, 0400. Adapted from A.V. Nero, Jr., A Guidebook to Nuclear Reactors, University of California Press, Berkeley, 1979; J. Buongiorno, BWR Description, MIT OpenCourseWear, 22.06: Engineering of Nuclear Systems, Massachusetts Institute of Technology, Cambridge, MA, 2010.

Table A.4 Typical BWR pressure vessel and core parameters. Overall vessel height Inside diameter of vessel Average vessel thickness Nominal vessel cladding thickness Vessel material Vessel cladding material Coolant volume in vessel Number of fuel assemblies Core (fuel rod) active length Fuel assembly array Fuel weight in the core (Uranium) Average fuel burnup Equivalent core diameter Heat generated in fuel Total number of fuel rods in core Fuel rod cladding material Number of rods (drives)

71 ft (21.7 m) 238 in. (6.05 m) 6 in. (15.4 cm) 1/8 in. (0.32 cm) Mn-Mo-Ni steel Austenitic stainless steel 4520 ft3 (128 m3) 748 150 in. (381 cm) 8  8 (enclosed fuel array) 155,000 kg 28,400 MWd/Te of heavy metal 15 ft 13 in. (490 cm) 97.4% 46,376 Zircaloy-4 185 (193 in BWR-6)

General Electric Company, BWR/6: General Description of a Boiling Water Reactor, San Jose, 1980; U.S. NRC Technical Training Center, Boiling Water Reactor (BWR) Systems, Chapter 3, Revisions 0200, 0400. Adapted from A.V. Nero, Jr., A Guidebook to Nuclear Reactors, University of California Press, Berkeley, 1979; J. Buongiorno, BWR Description, MIT OpenCourseWear, 22.06: Engineering of Nuclear Systems, Massachusetts Institute of Technology, Cambridge, MA, 2010.

239

240

APPENDIX A Generation II reactor parameters

Table A.5 Typical CANDU-600 reactor nuclear steam supply system (NSSS). Reactor thermal power Approximate electrical power generation Number of U-tube steam generators Steam pressure Steam outlet temperature Steam flow rate (total) Feedwater inlet temperature Steam generator shell material Steam generator tube material Reactor coolant pump motor horsepower Total coolant flow rate Average coolant flow rate per channel Coolant pressure (entrance to channel) Coolant pressure (exit of channel)

2060 MW 600 MWe 4 680 psia (46.9 bar) 500 °F (260 °C) 8.3  106 lb./h (1047 kg/s) 369 °F (187 °C) Mn-Mo steel Inconel-800 7040 60  106 lb./h (7,600 kg/s) 158,400 ib./h (20 kg/s) 1602 psia (111 bar) 1493 psia (103 bar)

W.J. Garland (Editor-in-Chief), The Essential CANDU: A textbook on the CANDU Nuclear Power Plant Technology, University Network of Excellence in Nuclear Engineering (UNENE), McMaster University, Retrieved from https://www.unene.ca/education/candu-textbook, 2019. Adapted from A.V. Nero, Jr., A Guidebook to Nuclear Reactors, University of California Press, Berkeley, 1979.

Table A.6 CANDU pressure vessel and core parameters. Overall length of calandria vessel Calandria outer diameter Calandria wall thickness Nominal vessel cladding thickness Calandria material Cladding material Coolant volume in vessel Number of fuel channels in core (calandria tubes) Number of fuel bundles per channel Fuel bundle diameter Fuel bundle length Number of fuel elements (rods) per bundle Number of fuel pellets per rods Active fuel length Fuel assembly array (circular)

25 ft (7.6 m) 25 ft (7.6 m) 1.125 in. (3 cm), 2 in. (5 cm) ends 7/32 in. (0.56 cm), Stainless steel Stainless steel 4885 ft3 (138 m3) 380 (Zircaloy-Niobium) 12 4 in. (10 cm) 19.5 in. (49.5 cm) 37 30 234 in. (6.3 m) Rods with Zircaloy bearing pads

APPENDIX A Generation II reactor parameters

Table A.6 CANDU pressure vessel and core parameters. Continued Fuel weight in the core (Uranium) Average fuel burnup Equivalent core diameter Nominal core coolant inlet temperature Core coolant outlet temperature Average coolant temperature rise in core Moderator Total heavy water inventory Moderator inlet temperature Moderator outlet temperature Moderator pressure Total number of fuel rods in core Fuel rod cladding material Number of control rods or compartments

95,000 kg 7000 MWd/Te of Uranium 248 in. (6.3 m) 512 °F (267 °C) 594 °F (312 °C) 82 °F (45 °C) Heavy water (99.75% D2O) 1.02  106 lb. (463,000 kg) 110 °F (43 °C) 160 °F (71 °C) Approximately atmospheric (1 bar) 168,720 Zircaloy-4 Between 4 and 21 of each of light water compartments, adjustable absorbers, shutdown absorber rods or poison injection ports.

W.J. Garland (Editor-in-Chief), The Essential CANDU: A textbook on the CANDU Nuclear Power Plant Technology, University Network of Excellence in Nuclear Engineering (UNENE), McMaster University, Retrieved from https://www.unene.ca/education/candu-textbook, 2019. Adapted from A.V. Nero, Jr., A Guidebook to Nuclear Reactors, University of California Press, Berkeley, 1979.

References [1] The Westinghouse Pressurized Water Reactor Nuclear Power Plant, Westinghouse Electric Company, Nuclear Operations Division, Pittsburgh, 1984. [2] M. Naghedolfeizi and B.R. Upadhyaya, Dynamic Modeling of a Pressurized Water Reactor Plant for Diagnostics and Control, Research Report, University of Tennessee, DOE/ NE/88ER12824-02, June 1991. [3] A.V. Nero Jr., A Guidebook to Nuclear Reactors, University of California Press, Berkeley, 1979. [4] General Electric Company, BWR/6: General Description of a Boiling Water Reactor, San Jose, 1980. [5] U.S. NRC Technical Training Center, Boiling Water Reactor (BWR) Systems, Chapter 3, Revisions 0200, 0400, 2018. [6] J. Buongiorno, BWR Description, MIT OpenCourseWear, 22.06: Engineering of Nuclear Systems, Massachusetts Institute of Technology, Cambridge, MA, 2010. [7] W.J. Garland, Editor-in-Chief, The Essential CANDU: A Textbook on the CANDU Nuclear Power Plant Technology, University Network of Excellence in Nuclear Engineering (UNENE), McMaster University, 2019. Retrieved from, https://www.unene.ca/educa tion/candu-textbook.

241

APPENDIX

Advanced reactors

B

B.1 Introduction Advanced reactor designs (labeled Generation III, III+ and IV reactors) have been prepared and a few advanced reactors are operating at the time when this book was prepared (2019). Features of advanced reactors were summarized in Ref. [1] and are quoted below: “A more standardized design for each type to expedite licensing, reduce capital cost and reduce construction time. A simpler and more rugged design, making them easier to operate and less vulnerable to operational upsets. Higher availability and longer operating life—typically 60 years. Further reduced possibility of core melt accidents. Substantial grace period, so that following shutdown the plant requires no active intervention for (typically) 72 h. Stronger reinforcement against aircraft impact than earlier designs, to resist radiological release. Higher burn-up to use fuel more fully and efficiently, and reduce the amount of waste. Greater use of burnable absorbers (‘poisons’) to extend fuel life.”

Advanced reactor designs include the use of passive safety systems. These systems operate if an accident occurs without actuation by an operator, without actuation in response to measured signal by an engineered system and without a need for electrical power. Passive systems depend on natural processes such as gravity, natural circulation, relief valve operation and melting of freeze valves. Advanced reactors achieve lower cost and faster construction by use of shop fabricated components rather than on-site fabrication of those components. This is called modular design. Many of the reactors employ “integral” designs. That is, they position other components, such as steam generators and pressurizers, as well as the reactor core inside the same vessel. This enhances safety but increases the required size of the vessel.

243

244

APPENDIX B Advanced reactors

This Appendix emphasizes the features of advanced reactors that affect dynamics, control, and safety.

B.2 Design possibilities There are many possibilities for the constituents of a power reactor and for different operational requirements and capabilities: The main choices include the following: • • • • • • •

Fuel: U-235, U-233, Pu-239 Form of fuel: metal, metal oxide, metal carbide, fluid Fertile material: U-238, Th-232 Moderator: water, heavy water, graphite, none (in fast reactors) Reactor coolant: liquid water, boiling water, liquid heavy water, helium, carbon dioxide, molten salt, molten sodium, molten lead, molten lead-bismuth Secondary coolant: none (boiling water reactors), saturated steam, superheated steam, helium, carbon dioxide Electricity production: Rankine cycle (steam turbine) Brayton cycle (gas turbine)

Designers have chosen various combinations of these constituents and operational features for potential advanced reactors. Designs for domestic use and export have been prepared in several countries including the United States, Canada, France, Japan, China, South Korea, India and Russia. Some of the various designs differ only slightly from designs from other countries. But the number of different possibilities is large and too extensive for inclusion of descriptions of every potential advanced reactor in this book.

B.3 A note about reactors that use thorium Some of the new reactor designs use Th-232 as a fertile material. They produce U-233 by the following reactions: Th-232 + n ! Th-233 ! Pa-233 + β ! U-233 + β

(B.1)

Th-233 has a half-life of 22 min and a very large capture cross section (about 2.7 times the U-233 fission cross section). Neutron losses to Th-233 would be major if it had a longer half-life. The relatively short half-life causes the residence time of Th-233 to be short and of rather small consequence. The Pa-233 is a different story. It has a half-life of 27 days and a significant appetite for neutrons (absorption cross section for thermal neutrons is around 7.5% as large as the U-233 cross section). Therefore Th-232/U-233 reactors experience a significant loss of bred U-233 if the Pa-233 stays in the reactor. Reducing this problem requires removal and sequestration of the Pa-233 in a low neutron flux region or

APPENDIX B Advanced reactors

creating a neutron spectrum with few neutrons at energies where the Pa-233 cross section is large. This is essential in order to achieve breeding (producing as much or more fissile material that is being consumed). Reactors that produce Pu-239 from U-238 also have intermediate isotopes that can absorb neutrons. The reactions are as follows: U-238 + n ! U-239 ! Np-239 + β ! Pu-239 + β

(B.2)

The Np-239 is the isotope of concern. But it has a fairly short half-life (at least as compared to Pa-233 in the Th-232/U-233 case) and a small appetite for neutrons compared to fissile isotopes. Therefore, there is no incentive to separate and sequester Np-239 in U-238/Pu-239 reactors. It is also noted that separation and sequestration of Pa-233 yields pure U-233 that represents a proliferation risk.

B.4 Advanced reactor marketplace The marketplace for advanced reactors is one of intense competition and change. Companies have merged, international cooperative partnerships have been forged and some countries that previously imported reactors have developed domestic capability and even export capability. The story of these developments is long, complex and beyond the scope and purpose of this book. The following list shows the 2018 status of advanced rector activity, including reactors in operation, reactors under construction and reactors in development (see Refs. [1, 2] unless otherwise noted for detailed information.): • • • • • • •

Pressurized Water Reactors: at least 12 (note predominance). Boiling Water Reactors: at least 3 Pressurized Heavy Water Reactors: at least 2 Liquid Metal Fast Reactors: at least 4 High Temperature Gas-Cooled Reactors: at least 1 (see Ref. [3, 4]. Also much work on small HTGRs) Molten Salt Reactors: at least 2 (See Ref [5]. Also much work on integral MSRs) Advanced Heavy Water Reactor: at least 1.

The information presented above is only a snapshot of the situation in 2018. The list will surely change in the future. Interested readers can find extensive up-to-date information about nuclear plant suppliers and their designs in the literature (including information on the internet). The World Nuclear Association is an excellent source of information. The way that the development evolves will determine which reactors are built, where reactors are built, who will build them and who operates them. All of these developments will create a growing need for engineers with capabilities in dynamic analysis and control system design.

245

246

APPENDIX B Advanced reactors

B.5 Large evolutionary reactors Large reactors feature large power production capability, including units with power production of as great as 1500 MWe. Each employs all or some of the features described in Section B.1. Numerous large advanced power reactors are offered by reactor vendors and several have been built and operated at the time of preparation of this book (early 2019). But there have been problems with cost overruns and construction delays, causing problems for some vendors and for buyers of these advanced systems. For example, Westinghouse Electric Company and General Electric have undergone major restructuring. Nevertheless, enthusiasm for advanced large reactor implementation continues in the United States and especially in other countries. Russia, China and India have robust programs that are much larger than U.S. programs. These countries offer Generation III, III+ and even Generation IV for domestic and export markets.

B.5.1 Pressurized water reactors Generation III and III+ pressurized water reactors are offered by companies in several countries, including The United States, France, Japan, China and Russia. These PWRs have many common features and all strive to provide the features described in Section B.1. Describing the unique features of each of these reactors is beyond the scope and purpose of this book. Rather, one system, the Westinghouse AP 1000 has been chosen for a brief description here. The AP1000 (for Advanced Passive) is a Generation III+ reactor with a thermal power of 3400 MW and an electrical power of 1117 MW [6, 7]. Two AP1000 units are operating in China (as of June 2019). Two AP1000 reactors are under construction in the U.S. The reactor core of the AP1000 is similar to preceding Westinghouse PWRs. The new features include fewer components in the heat removal systems, especially the safety systems. The safety systems depend entirely on natural phenomena: gravity, natural circulation, and compressed gas to induce coolant flow from reservoirs of water. There are no safety-related pumps, fans, chillers or other rotating machinery. The safety systems operate automatically. These require no operator action or electrical power. The safety systems are designed to protect the fuel from overheating indefinitely with no human intervention and without a need for electrical power. The spent fuel pool provides indefinite cooling for spent fuel. The safety injection systems are as follows: •

Core Makeup Tanks (CMT). These tanks provide flow to the reactor coolant system while the pressure is still high after an accident. They contain borated water at normal reactor coolant system pressure. Water from the CMT flows into the reactor coolant system by gravity when isolation valves open automatically.

APPENDIX B Advanced reactors

•

•

Accumulators. Accumulators provide flow to the reactor coolant systems when the pressure drops to medium levels. They are tanks that are pressurized with nitrogen to 700 psig. In-Core Refueling Water Storage Tank (IRWST). The IRWST is a large tank containing borated water (590,000 gal) and is open to the interior of the containment building. Water flows by gravity from the IRWST into the reactor coolant system after system depressurization. The Automatic Depressurization System reduces system pressure by releasing steam into the IRWST or directly into the containment building. Water spray on the outside of the steel containment vessel cools and condenses in-containment steam. The condensate returns to the IRWST. The IRWST also serves as the heat sink for long-term cooling. A heat exchanger located in the IRWST receives hot fluid from the reactor coolant system and returns cooler water by natural circulation. If the IRWST water boils, the steam is condensed by the containment cooling system and the condensate returns to the IRWST.

B.5.2 Boiling water reactors Here, we consider the Economic Safe Boiling Water Reactor (ESBWR), a Generation III+ design [8, 9]. The ESBWR employs passive heat transfer in the reactor cooling system and in the safety system. The reactor coolant flows through the core by natural circulation, thereby eliminating coolant pumps as used in the Advanced Boiling Water Reactor (ABWR) and most of the earlier BWRs. Two design modifications enhance natural circulation. The fuel region is shorter than in earlier BWRs, thereby reducing the core pressure drop. A chimney above the core region further enhances natural circulation. The ESBWR design eliminates coolant pumps and associated piping, thereby simplifying construction, reducing cost and enhancing safety. The safety system has four main components: The Isolation Condenser System (ICS), the Gravity Driven Cooling System (GDCS), the Passive Containment Cooling System (PCCS) and The Standby Liquid Control System (SLCS). Each employs a reservoir containing water and no electrical power or operator action is required for operation. The ICS operates when the reactor coolant system remains intact and at elevated pressure. The ICS is connected to a heat exchanger located in a water reservoir located above the reactor. Heat transfer in the heat exchanger condenses steam released from the reactor cooling system and the resulting liquid water returns to the reactor cooling system by natural circulation. The GDCS transfers water by gravity from a large pool above the reactor after depression by the Automatic Depressurization System that automatically releases steam to the Suppression Pool below the reactor. The PCCS uses a heat exchanger to transfer heat from the containment volume to water in a pool above the reactor. The SLCS injects borated water into the reactor.

247

248

APPENDIX B Advanced reactors

B.5.3 Pressurized heavy water reactors Canada offers two advanced pressurized heavy water reactor designs. The reactors differ in design and purpose. Brief descriptions follow: •

•

Advanced CANDU Reactor (ACR): The ACR is a Generation III+ reactor, designed to compete with other Generation III+ reactors for service as large central station power plants (Ref. [10]). The main difference from earlier CANDU reactors in the use of light water coolant instead of heavy water. Heavy water moderator in a calandria remains. The ACR employs the principles described in Section B.1. However, the ACR has proved to be non-competitive compared to other Generation III+ designs and development has been stopped. Advanced Fuel CANDU Reactor (AFCR): The Generation III AFCR appears to offer the CANDU reactor a new lease on life. It offers the options of using recycled spent fuel from light water reactors and using low enriched uranium/ thorium (LEU//Th) fuel. The AFCR design is basically the same as the CANDU6, but with new fuel designs [11–13].

Spent fuel from light water reactors contains too little fissile material for continued use in the LWR. But the LWR spent fuel contains enough fissile material for use in a CANDU reactor because of CANDU’s better neutron economy (due to heavy water’s small neutron absorption probability). The LEU/Th fuel version enables use of the thorium resource for power production. The AFCR fuel bundle is cylindrical with 43 concentric fuel rods. The recycled spent fuel version uses 42 rods containing recycled spent fuel and one (centrally located) rods containing a mixture of recycled spent fuel and dysprosium oxide. The LEU/Th version uses eight inner rods containing thorium oxide and thirty-five outer rods containing low enrichment uranium.

B.6 Large developmental reactors Several large reactor designs in the developmental stage are potential options for wide-spread implementation. Each design draws on experience with earlier prototypes. Brief descriptions follow:

B.6.1 Gas-cooled reactors Magnox, CO2 cooled power reactors have been operating in Britain for many years, but they were overshadowed by light water reactors developed in other countries. No advanced gas-cooled reactors have been built, but technology development and prototype reactor implementation has been pursued. Advanced gas-cooled reactors include prismatic and pebble bed designs (Refs. [14–17]) Prismatic designs have fuel particles embedded in graphite blocks and pebble bed designs have fuel particles in graphite balls.

APPENDIX B Advanced reactors

Both reactor designs use TRISO (Tristructural Isotropic) small fuel particles (0.5 mm) that can withstand high temperatures and contain fission products. A TRISO coated fuel particle consists of an outer pyrolytic carbon layer (0.92 mm outer diameter), followed by a silicon carbide and an inner pyrolytic carbon layers, and an inner porous carbon buffer. The fuel kernel is in the center of the particle. The silicon carbide’s high melting temperature (1600 °C) protects against particle failure. Regardless of the specific reactor design, all advanced gas-cooled reactors have the following features: • • • • • • •

Chemically inert Helium coolant Single phase Helium coolant (no issues due to boiling) Negligible neutron absorptions in Helium coolant (zero coolant temperature coefficient of reactivity High thermal conductivity in graphite (avoids hot spots) Slow response to perturbations due to large heat capacity Dominant negative fuel temperature coefficient of reactivity due to the Doppler effect Power stabilizes after a perturbation with no control action.

All of these features combine to enhance gas-cooled reactor safety. Both large and small versions of gas-cooled reactors are candidates for implementation.

B.6.2 Liquid metal fast breeder reactors The potential for breeding in fast reactors has been realized since the early days of nuclear power and a number of prototype reactors have been operated (Refs. 18–20). Two large fast reactors (BN 600 and BN 800) are operating in in Russia as this book is being written (2018) and fast reactors are under construction in China [21] and India [22]. Sodium fast reactors (SFRs) are a class of advanced reactor design that uses sodium to remove heat from the reactor core, and transfer to heat exchangers and steam generators. SFRs typically fall into one of two design categories: pool-type and loop-type. Pool-type SFRs feature primary coolant system components such as pumps and intermediate heat exchangers positioned inside the reactor vessel. In loop-type reactors, these components are exterior to the reactor vessel. Pool-type configurations reduce the possibility of many accident scenarios present in light water reactors. Since sodium (Na-23) atoms are heavier than hydrogen and oxygen atoms, neutrons lose less energy in collisions with sodium atoms than hydrogen and oxygen in LWRs, thus enabling fast fission reactions. Sodium has a large range of temperatures (371–1156 K) in the liquid phase enabling the absorption of significant heat in the liquid phase. A large sodium reservoir provides excellent thermal inertia against overheating. Because of the high boiling point of sodium (compared to operating temperatures), it is not necessary to pressurize the primary loop, thus enabling the

249

250

APPENDIX B Advanced reactors

use of reactor vessels with smaller thickness compared to LWRs. Any chemical interaction of sodium with water must be avoided and is a safety issue to be considered in the design of heat exchangers.

B.6.3 Molten salt reactors In a molten salt reactor (MSR) the system uses molten salt to transfer heat from the reactor primary side, to an intermediate heat exchanger, and then to the balance-ofplant side of the system for electricity production and/or industrial heat applications [23, 24]. In a salt-cooled reactor, the core is stationary with solid fuel and liquid salt is used as the coolant. In a salt-fueled reactor, the fuel is dissolved in a liquid salt and the liquid fuel circulates through the primary and the intermediate loops. The Department of Energy adopted the term “fluoride salt-cooled hightemperature reactor” (FHR) in 2010 to distinguish fluoride salt-cooled MSRs from salt-fueled MSRs. Liquid salt-fueled reactors are called molten salt reactors or MSRs. Potential future molten salt reactors include thermal reactors and fast reactors. The thermal version uses graphite as the moderator. In both types, the fuel is dissolved in a molten salt. MSRs operate at low pressure with the ability for on-line refueling, thus avoiding the need for reactor shut down for regular refueling. In the molten salt breeder reactor (MSBR) design, the conversion of Th-232 first results in Protactinium-233 and, consequently, has the problem of loss of U-233 production through neutron captures in Pa-233 as described in Section B.1.1. The MSBR addresses this problem by removing and sequestering the molten salt stream that contains Pa-233. The MSBR’s unique use of fertile material dissolved in a fluid enables a solution to the problem of U-233 losses by captures in Pa-233.

B.6.4 Heavy water reactors The Advanced Heavy Water Reactor (AHWR) is a new heavy water design from India [25–27]. It uses light water coolant that boils in vertical channels located in a tank of heavy water moderator. Coolant flows by natural circulation. The AHWR is designed to use thorium for power production. The cylindrical fuel bundle contains 54 rods, an inner circle with 30 rods containing ThO2 and an outer circle with 24 rods containing ThO2 and PuO2. Since other reactors with similar features had stability problems, detailed stability studies have been performed for the AHWR. The AHWR has a physically separate and stationary moderator from the boiling coolant like the Russian RBMK (Chernobyl) reactor. The AHWR has boiling coolant as in BWRs. Both of these reactors have stability issues. Stability studies for the AHWR indicate adequate stability margins.

APPENDIX B Advanced reactors

B.7 Small reactors B.7.1 Introduction Small reactors are those with power levels of 300 MWe or less. All of the reactors discussed in Section B.3 also have small reactor versions. Since the designs are generally smaller, but with similar components as large reactor designs, design features are not provided here. The design of small modular reactors is integral in nature, with reactor core, heat exchangers/steam generators, coolant pumps, and control rod drive mechanisms contained in a large vessel. A summary description of instrumentation and control systems in small reactors is given in Ref. [28].

B.7.2 Incentives The incentive for small reactors is faster construction and lower incremental cost to the purchaser, thereby adding capacity as needed and adding debt incrementally. The smaller components allow even greater shop fabrication than large advanced reactors. The smaller components also permit placing more components inside the primary containment, thereby enhancing safety. Small reactors can be placed in a single-reactor facility in regions where power demand is low or in multi-reactor sites where reactors are added as demand increases.

B.7.3 Small reactor list Many small reactor designs are under development (too many for descriptions here). Table B.1 shows a partial list of small reactors adapted from Refs. [29, 30].

Table B.1 Summary of small reactors: water cooled, gas cooled, liquid metal cooled fast reactors, and molten salt reactors.

MR design

Country

Technology developer

Type

Module output (MWe)

Water cooled reactors CAREM-25

Argentina

CNEA

ACP-100 Flexblue PHWR-220

China France India

CNNC DCNS NPCIL

iPWR, natural circulation iPWR Loop (marine based) HW-cooled, HW-moderated CANDU design

27 100 160 235

Continued

251

252

APPENDIX B Advanced reactors

Table B.1 Summary of small reactors: water cooled, gas cooled, liquid metal cooled fast reactors, and molten salt reactors. Continued

MR design

Country

Technology developer

AHWR300LEU

India

BARC

SMART

KAERI

KLT-40S

Republic of Korea Russia

RITM-200 NuScale

Russia USA

SMR-160 mPower

USA USA

W-SMR

USA

OKBM NuScale Power Holtec Generation mPower Westinghouse Electric Co.

OKBM

Type

Module output (MWe)

LW-cooled, HWmoderated (pressure tubes) iPWR

304

Compact loop (marine based) Integral iPWR, natural circulation (HCSG) Compact loop iPWR (OTSG)

35

100

50 45 160 180

iPWR (steam separation outside vessel)

225

HTGR - Pebble Bed (2 modules)

105

HTGR—Pebble Bed HTGR-Prismatic

100 150

HTGR-Prismatic

240

Gas-cooled reactors HTR-PM

China

PBMR GT-MHR

RSA USA

EM2

USA

INET, Tsinghua University Eskom PBMR General Atomics General Atomics

Liquid metal cooled fast reactors and molten salt reactors CEFR 4S (Super Safe Small & Simple) SVBR-100 PRISM

China Japan

CIAE Toshiba

Pool type, LMFR SFR

20 10 or 50

Russia USA

AKME GE Nuclear

Lead-Bismuth SFR

101 311

Adapted from M.D. Carelli, D.T. Ingersoll (Eds.), Handbook of Small Modular Nuclear Reactors, Elsevier Woodhead Publishing, 2015; S.M. Goldberg, R. Rosner, Nuclear Reactors: Generation to Generation, American Academy of Arts and Sciences, 2011.

B.8 Dynamics of advanced reactors The first step in the dynamic analysis of a reactor is simulation of the inherent dynamics (the natural dynamic behavior in the absence of any control action). This behavior is determined by natural feedbacks of the reactor. Most of the feedbacks

APPENDIX B Advanced reactors

depend on fuel temperature, coolant temperature, moderator temperature (in thermal reactors), pressure (primarily in reactors with boiling coolant and temperature of structural components (feedback due to dimensional changes and primarily in fast reactors). All of the advanced reactors have a negative fuel temperature coefficient of reactivity. That is, a fuel temperature increase causes a decrease in fission rate. Most, but not all, of the other feedbacks in the various advanced reactors are also characterized by negative coefficients of reactivity. For example, the pressure coefficient is positive in reactors with boiling coolant and the temperature coefficient of the coolant is positive in reactors with significant concentrations of a neutron poison in the coolant. The rates of change of the conditions (temperature, boiling, etc.) of reactor components determine the speed of associated reactivity feedbacks. These rates of change depend on heat transfer coefficients and the heat capacity of the component and is characterized by the time constant. Long time constants mean that the associated response lags behind the initiator of a transient. Time constants of reactivity feedbacks vary significantly among the advanced reactors. The time response of reactor components has two impacts on system dynamics: reactivity feedbacks are delayed and reactor component temperatures change more slowly if time constants are large. For example, graphite-moderated, gas-cooled reactors have large heat capacities and they respond slowly to a disturbance. It should be noted that negative feedbacks can be destabilizing if lagged behind the transient initiator.

References [1] Large Advanced Nuclear Power Reactors, World Nuclear Association, October. Available at www.world-nuclear.org/.../advanced-nuclear-power-reactors.aspx, 2018. [2] Nuclear Power Reactors, World Nuclear Association, October. available at www.worldnuclear.org/small-nuclear-power-reactors.aspx, 2018. [3] Medium Size HTGR: www.us.areva.com/home/liblocal/docs/Nuclear/HTGR/HTGRInfoKit-2014-03.pdf. [4] CNNC’s New Reactor Set to Go into Global Market, www.ecns.cn/business/2017-07-19/ detail-ifytetzm3037842.shtml. [5] D.E. Holcomb, History, Background, and Current MSR Developments, www.gain.inl. gov/SiteAssets/MoltenSaltReactor/Module1-History. [6] W.E. Cummins, M.M. Corletti, T.L. Schultz, Westinghouse AP-1000 advanced passive plant, in: Proceedings of ICAPP, Cordoba, Spain, May 4–7, 2003. Available at www. nuclearinfo.net/.../WebHomeCostOfNuclearPower/AP1000Reactor.pdf. [7] AP1000 Passive Safety Systems, Available at: www.nrc.gov/docs/ML1523/ ML15230A043.pdf W.E. [8] GE Hitachi’s ABWR and ESBWR, Safer, Simpler, Smarter, Available at www.oecd-nea. org/ndd/workshops/innovtech/presentations/. [9] The Functions of the ESBWR Emergency Core Cooling System, Available at www.auto mationenergy.com/2017/07/16/the-functions-of-the-esbwr (n.d.). [10] Advanced CANDU reactor, Available at www.revolvy.com/topic/AdvancedCANDU reactor. [11] Advanced Fuel CANDU Reactor, SNC-Lavalin, Available at www.snclavalin.com/en/ files/documents/publications/afcr-technical.

253

254

APPENDIX B Advanced reactors

[12] AFCR, Transcending natural uranium fuel cycles - Nuclear, Available at www. neimagazine.com/features/...natural-uranium-fuel-cycles-4432140. [13] The AFCR and China’s fuel cycle, World Nuclear News, Available at www.worldnuclear-news.org/E-The-AFCR-and-Chinas-fuel-cycle. [14] Gas-Cooled Reactor, Advanced Heavy Water Reactor, Available at www.nptel.ac.in/ courses/103106101/Module - 3/Lecture - 4.pdf. [15] Gas-Cooled Reactors—IAEA Scientific and Technical, Available at www.pub.iaea.org/ MTCD/publications/PDF/CSPS-14-P/CSP-14_part2.pdf. [16] Very-High Temperature Reactor (VHTR), Available at https://www.gen-4.org/gif/jcms/ c_42153/very-high-temperature-reactor-vhtr. [17] The High Temperature Gas-Cooled Reactor (HTGR)—Safe, Clean and Sustainable Energy for the Future, Available at www.ngnpalliance.org/index.php/htgr. [18] Sodium-Cooled Fast Reactor (SFR), Available at https://www.gen-4.org/gif/jcms/c_ 9361/sfr. [19] Liquid Metal Fast Breeder Reactors (LMFBR), Available at www.ati.ac.at/fileadmin/ files/research_areas/ssnm/nmkt/11_LMFBR.pdf. [20] Fast Neutron Reactors—World Nuclear Association, Available at www.world-nuclear. org/.../fast-neutron-reactors.aspx. [21] China Building a 600 MWe Fast Neutron Reactor which Will Become Workhorse in the 2040s, Available at www.nextbigfuture.com. [22] India Nearing Completion of 500 MW Commercial Fast Breeder Reactor, Available at www.nextbigfuture.com (n.d.). [23] Molten Salt Reactors—World Nuclear Association, Available at www.world-nuclear. org/.../molten-salt-reactors.aspx (n.d.). [24] Overview of MSR Technology and Concepts, Available at https://gain.inl.gov/ SiteAssets/MoltenSaltReactor/Module2-Overview (n.d.). [25] Advanced Heavy Water Reactor, Bhabha Atomic Research Centre (BARC), India. Available at: www.barc.gov.in/publications/eb/golden/reactor/toc/chapter1/1.pdf. [26] Analytical Studies and Experimental Validation of AHWR, BARC, India. Available at: www.barc.gov.in/publications/eb/golden/reactor/toc/chapter1/1_9.pdf. [27] M. Todosow, A. Aronson, L.-Y. Cheng, R. Wigeland, C. Bathke, C. Murphy, B. Boyer, B. Fane, B. Ebbinghaus, The Indian Advanced Heavy Water Reactor and NonProliferation Attributes, Brookhaven National Laboratory, BNL-98372-2012, August, 2012. [28] IAEA, Instrumentation and Control Systems for Advanced Small Reactors, IAEA Nuclear Energy Series No. NP-T-3.19, IAEA, Vienna, 2017. [29] M.D. Carelli, D.T. Ingersoll, Handbook of Small Modular Nuclear Reactors, Elsevier Woodhead Publishing, 2015. [30] S.M. Goldberg, R. Rosner, Nuclear Reactors: Generation to Generation, American Academy of Arts and Sciences, 2011.

APPENDIX

Basic reactor physics

C

C.1 Introduction It is assumed that most readers are familiar with basic reactor theory. But for those lacking that familiarity or those needing a refresher, this appendix provides a brief overview of basic concepts relevant to a study of reactor dynamics. Greater detail may be found in pertinent references [1–4].

C.2 Neutron interactions Interaction of neutrons with matter results in various outcomes depending on the neutron energy and the nature of target material (target nuclei). The interactions that are important in reactor operation are as follows: Elastic collision: This occurs when the neutron shares its energy with the target nucleus without exciting the nucleus. In a collision between a target nucleus and a neutron, the target nucleus recoils and the neutron continues with lower energy. This is analogous to a “billiard ball collision”. The total kinetic energy (KE) is conserved. This is the primary mode of slowing down of neutrons to thermal energies by interactions with light nuclei of the moderator. Inelastic collision: In an inelastic collision the target nucleus becomes excited, emits a gamma ray, and emits a neutron with lower energy than the incident neutron. Radiative capture: In radiative capture the neutron is absorbed by the target nucleus, produces an excited nucleus that becomes stable by emitting gamma rays. Transmutation: In a transmutation reaction, neutron absorption yields new isotopes. For example 10

B + 1 n ! 7 Li + 4 He ðαÞ

(C.1)

Fission: The essential reaction that takes place in a nuclear reactor is the fission reaction, which occurs in certain heavy nuclei. 92U-235 is the only naturally occurring isotope of uranium that has this property for reactions with slow neutrons. The other main isotopes that undergo fission by slow (or thermal) neutrons are 92U-233, 94Pu-239, and 94Pu-241. The natural abundance of the isotopes of uranium (as it is found in nature) is as follows: U-234 ¼ 0:006%

255

256

APPENDIX C Basic reactor physics

U-235 ¼ 0:714% U-238 ¼ 99:28%

C.3 Reaction rates and nuclear power generation In this section, some of the interactions between neutrons and atomic nuclei, are reviewed. Since neutrons have no electrical charge, they can enter into nuclear reactions even when their velocities are low. A brief review of neutron cross sections, neutron flux, reaction rates, and power generation follows in this section. In a nuclear reactor, the issue is the fate of fission neutrons. Fission neutrons result in new fissions, non-fission neutron captures, and neutron leakage. The average energy of fission neutrons is around 2 MeV. These fast neutrons interact with the core materials (structure, fuel, moderator, etc.) by absorption and scattering reactions. Collisions resulting in scattering will slow down the neutrons. Neutron cross sections are the basic data used for determining nuclear reaction rates. The microscopic cross section is represented by the symbol, σ. Microscopic cross sections are basically target areas for incident neutrons. The units for cross sections are cm2. Typical values for cross sections are 1022 to 1026 cm2. To simplify specification of cross section values, a new unit, called the barn is used. A barn is defined as 1024 cm2. Early workers, apparently a jocular bunch, said that, to a neutron, a target with area, 1024 square centimeters, is as big as a ‘barn door’. Reactions of importance in nuclear reactors are fission, capture, absorption (fission + capture), elastic scattering and inelastic scattering. Cross sections are energy dependent. Fission and capture cross sections decrease with increasing neutron energy. For many isotopes these cross sections vary as the reciprocal of the neuron velocity at low energies (1/v or as the reciprocal of the square root of the neutron energy). Isotopes that follow the 1/v law are called 1/v absorbers. The total microscopic cross section, σT, available for interaction between a neutron and a target nucleus is σT ¼ σa + σs

(C.2)

σa ¼ microscopic absorption cross section σs ¼ microscopic scattering cross section. These may be further classified as σa ¼ σf + σc ðfission + captureÞ σs ¼ σse + σsi ðelastic + inelasticÞ

For a given concentration of target nuclei, the number of collisions in a given time interval is proportional to the distance traveled by the neutrons in the volume. Some important relationships follow: •

Neutron flux: ϕ ¼ nv (number of neutrons/cm2-s)

APPENDIX C Basic reactor physics

N ¼ Density of target nuclei (number of nuclei/cm3) Macroscopic cross section: Σ ¼ σN (cm2/cm3 or cm1) Reaction rate: R ¼ Σϕ (number of interactions/cm3-s) Fission reaction rate: Rf ¼ Σf ϕ Rf ¼ [Nσf] [n(t)v] Rf is the number of fission reactions/cm3-s σf is the microscopic fission cross section

• • • • • • •

The reaction rate in a 1/v absorber is given by R¼Nσϕ

or R ¼ Nðc=vÞ nv

where c is a constant. Note that the velocity terms cancel. Therefore, the reaction rate for 1/v absorbers is independent of the neutron energy. Example C.1 The energy of neutrons in an experimental reactor is approximately equal to 0.0253 eV. This corresponds to a speed of about 2200 m/s for neutrons. As an exercise, let us let flux ϕ ¼ 2  1012/(cm2 s). Calculate the neutron density. n ¼ ϕ=v ¼

2  1012 cm2 s1 ¼ 9  106 =cm3 2200  100m s1

In the above example, use the microscopic absorption cross section, σa ¼ 694 b for U-235. The density of target nuclei is N ¼ 0.05  1024/cm3. Then Σa ¼ Nσa ¼ 34 cm1

The reaction rate is given by the following: R ¼ ϕ Σa ¼ 6:8  1013 =cm3 s

R is also the rate at which U-235 nuclei are consumed. Note that the microscopic fission cross section of U-235 at E ¼ 0.025 eV is, σf ¼ 582 b. The capture cross section is σc ¼ 112 b and the absorption cross section is σa ¼ σf + σc ¼ 694. The ratio σf/σa ¼ 582/694 ¼ 0.84. That is, 84% of thermal neutron absorptions in U-235 result in a fission reaction. If an absorber cross section decreases slower with increasing neutron energy (and neutron velocity) than 1/v, then the absorption rate relative to 1/v absorbers increases as neutron energy increases. The reverse is true for absorbers whose cross section decreases faster than 1/v. The energy dependence of low energy absorptions plays an important role in determining the dynamics of power reactors. The power produced in a reactor is given by the following: P ¼ ðNVÞ σf nv F Watt

(C.3)

257

258

APPENDIX C Basic reactor physics

where V ¼ Total volume of core. F ¼ Energy produced per fission (3.225  1011 Watt s/fission). If we assume that all the terms on the right-hand side of Eq. (C.3) are constant, except neutron density, n, the reactor power is directly proportional to the neutron density or the neutron flux. P/n/ϕ

(C.4)

This relationship is important because in the neutronics equations we can replace the neutron density with actual reactor power within a multiplication factor.

C.4 Nuclear fission The nuclei of U-235, U-233, Pu-239 and Pu-241 can undergo fission by low-energy neutrons. Such materials are called fissile materials. Isotopes such as U-238 and Th-232 can undergo fission with fast neutrons and are said to be fissionable (as opposed to fissile) and are referred to as fertile materials. Note that U-238 and Th-232 can be converted to Pu-239 and U-233, respectively, by absorption of neutrons and final decay into the respective fissile isotopes. The reactions are as follows 92 U

238

+0 n1 !

92 U

239

!

239 93 Np

#

93 Pu

+1 β0

239

+1 β0

(C.5)

The half-life for U-239 is 23 min and for Np-238 is 2.3 days 232 90 Th

+0 n1 !

233 90 Th

!

91 Pa

233

#

92 U

233

+1 β0 +1 β0

(C.6)

The half-life for Th-233 is 22 min and it decays to Protactinium (Pa-233). Pa-233 has a half-life of 27 days and has a quite large neutron absorption cross section (around 7.5% as large as the U-233 fission cross section). Therefore, Pa-233 is a significant neutron absorber and its presence diminishes potential U-233 production. The residence of Pa-233 in an operating reactor core results in a significant loss in U-233 production. Numerous fission fragments are released during the fission reaction. These are classified according to the percent of fission yield. Fission yield is the percentage of a given isotope atoms in the total of all fission fragments. The reactions are as follows: 235 236 ∗ 1 92 U + 0 n ! 92 U

A2 1 ! A1 z1 F + z2 F + υ0 n + Energy

where υ ¼ number of neutrons produced in the fission reaction (2 or 3).

APPENDIX C Basic reactor physics

The most probable fission fragments are Cs-140 (Cesium) and Rb-93 (Rubidium). For example, the reactions involving these fission fragments are given as follows: 235 236 ∗ 1 92 U + 0 n ! 92 U

93 1 !140 55 Cs + 37 Rb + 30 n + Energy ð 200 MeVÞ

(C.7)

The high-speed fission fragments lose energy by interaction with the molecules of the surrounding medium (fuel, structure, moderator, etc.), thus converting kinetic energy (KE) to thermal energy. There is also heating due to gamma radiation and slowing down of neutrons to lower energy levels. An average of υ ¼ 2.43 neutrons are produced per fission induced by thermal neutrons in U-235. Avogadro’s Number (AN): is the number of molecules per gram mole or atoms per gram atom of a substance. One gram-atom is the quantity of substance in grams, numerically equal to its atomic mass. Avogadro’s Number is numerically equal to AN ¼ 6.023  1023 atoms per gram-atom. Example C.2 The number density (N) of U-235 atoms in natural uranium is the number of atoms of U-235 per cm3. The number density is given by N ¼ ðANÞ ρ e=m where ρ ¼ material density, gm/cm3 e ¼ enrichment [U-235/(U-235 + U-238)] m ¼ gram U per gram atom U.

The number density of U-235 in 1% enriched uranium (with a material density of 19.0 g/cm3 is [3]. N235 ¼


   19:0 gramU=cm3  6:023  1023 ðatomsU=gram  atomUÞ atomsU  235  0:01 238 ðgramU=gram  atomUÞ atomU

¼ 4:80  1020 atoms U-235=cm3 :

Some energy equivalents are as follows: 1 eV is the amount of kinetic energy imparted to an electron when accelerated through a potential difference of 1 V. 1 eV ¼ 1.602  1019 J 1 cal ¼ 4.184 J. The absorption of a neutron by the U-235 nucleus has the form 235 236 ∗ 1 92 U + 0 n ! 92 U

(C.8)

This results in extra internal energy of the product, because the sum of masses of the two interacting particles is greater than that of a normal U-236 nucleus (at ground state). This excess energy is sufficient to cause nuclear fission (electrostatic repulsion dominates nuclear attraction).

259

260

APPENDIX C Basic reactor physics

Table C.1 Energy distribution for fission induced by thermal neutrons in U-235. Source

Energy (MeV)

Fission product KE Neutron KE Energy in gamma radiation (instantaneous and delayed) Energy in β-decay of fission products • Total (available as thermal energy): Energy not available as heat • Total energy created in one fission reaction:

168 5 11 7 191 11 202

The neutron binding energies of U-235, U-233, and U-239 nuclei (odd number of neutrons) are about I MeV higher than the nuclei of Th-232 and U-238 (even number of neutrons). This additional binding energy is sufficient to exceed the critical energy for fission, with low-energy neutrons. For example: for U-235, critical energy for fission to occur is 5.5 MeV and the binding energy of an extra neutron is 6.6 MeV. The critical and binding energies for U-238 are 5.9 MeV and 4.9 MeV. Out of two to three neutrons released per fission (depending on the fissioning isotope involved), one of these is used to produce the next fission reaction in a steadystate chain reaction. The remaining neutrons are consumed by • • • •

Leakage from the core Capture by non-fuel reactor constituents (such as coolant, moderator and structural materials) Non-fission capture in the fuel (radiative capture) Capture by the fertile nuclei (such as U-238, resonance capture).

The distribution of fission energy in various forms is shown in Table C.1.

C.5 Fast and thermal neutrons Immediately following fission, the neutrons possess high kinetic energy, in the million-eV range (0.1–15 MeV). Most current-generation reactors include a material (called a moderator) whose purpose is to slow down neutrons while capturing few neutrons. Fast neutrons lose their energy due to scattering collisions with various nuclei in the medium (especially in the moderator, see Section C.2), and become slow neutrons (energy 1, the number of neutrons from one generation to the next increases without bound, and such a reactor is said to be super-critical. In summary: • • •

k ¼ 1, critical reactor k < 1, sub-critical reactor k > 1, super-critical reactor

C.9 Computing effective multiplication factor The following factors determine the magnitude of the multiplication factor k: 1. Thermal fission factor, η: The factor, η, is defined as the number of fast neutrons produced per thermal neutron absorption in the fuel. That is: η¼ν

σfuel f σfuel a

2. ν ¼ number of neutrons produced per fission. A typical value for η is around 1.65 for a U-235-fueled thermal reactor. 3. Thermal utilization factor, f: The factor, f, is defined as the number of neutron absorptions in the fuel per total number of neutron absorptions. That is: f¼

Σfuel a Σtotal a

A typical value for f is around 0.71 for a U-235-fueled thermal reactor. 4. Resonance escape probability, p: The factor, p, is equal to the number of neutrons that reach thermal energy per fast neutron born. It accounts for neutron losses in resonances during slowing down. A typical value for p is around 0.87 for a U-235-fueled thermal reactor. 5. Fast fission factor, ε: This factor is defined as the total number of neutrons from both thermal and fast fissions per number of neutrons from thermal fissions. A typical value for ε is around 1.02 for a U-235-fueled thermal reactor Using the above four factors we define the effective multiplication of an infinite size core as k∞ ¼ η f p ε

(C.14)

265

266

APPENDIX C Basic reactor physics

The above formula is referred to as the four-factor formula. This does not take into account the probability of neutron leakage from finite size cores. In order to include this effect, two more factors are added to the above calculation [1–4]. 6. Fast non-leakage probability, PFnl: The fast non-leakage factor is the number of fast neutrons do not leak out of the core during slowing down to thermal neutron per fast neutron produced. A typical value for PFnl0.97 is around 0.97 for a U-235-fueled thermal reactor. 7. Thermal non-leakage probability, PTnl: The thermal non-leakage factor is the number thermal neutrons do not leak out of the core per thermal neutron produced. A typical value for PTnl is around 0.99 for a U-235-fueled thermal reactor. 8. Effective multiplication factor, keff: With definition of non-leakage probabilities, we can now calculate the effective multiplication factor as keff ¼ k∞ PFnl PTnl ¼ η f p ε PFnl PTnl

(C.15)

The formula in Eq. (C.15) is generally referred to as the six-factor formula. The reader can find more details in Ref. [1, 2].

C.10 Neutron transport and diffusion The most complete description of the spatial distribution of neutrons in a reactor is given by neutron transport theory. Transport theory defines a reactor in terms of seven independent variables: three position coordinates, two direction vectors, energy and time. The transport theory equation is called the Boltzmann equation. Computer codes have been developed for transport theory analysis, but they suffer from complexity, long computing time, and difficulty in determining detailed finemesh parameters needed for implementation. Most reactor studies treat neutron motion as a diffusion process – that is, neutrons tend to diffuse from regions of high neutron density to regions of low neutron density. Diffusion theory ignores the direction dependence of the neutrons.

Exercises C.1. The neutron flux in a certain reactor is 2  1013 neutrons/(cm2-sec). If the neutrons have a mean velocity of 3100 m/s, calculate the neutron density. Indicate the units. C.2. The neutron flux in a commercial pressurized water reactor (PWR) is 2  1013/ (cm2-s). The macroscopic cross section for fission is 30 cm2/cm3. Calculate the rate of fission reactions per cm3. Indicate the units. Simplify your answer.

APPENDIX C Basic reactor physics

C.3. The neutron flux in a high flux reactor is approximately 1015 neutrons/cm2-sec. The microscopic neutron cross section for fission in U-235 for this reactor is 500  1024 cm2. The density of U-235 target nuclei is 1024 nuclei/cm3. (a) Calculate the rate at which fission reactions take place in this reactor. Simplify your answer and indicate the units. (b) If the energy generated per fission is approximately 200 MeV, calculate the total power produced in the reactor per cm3. Indicate the units. 1 MeV ¼ 1.60  1013 J. C.4. Match the following neutron energies (E) with thermal neutrons, fast neutrons, and neutrons in the resonance energy range. E: 0.1 MeV - 15 MeV _________________. E < 1 eV _________________. E: 1 eV – 0.1 MeV _________________. C.5. The energy of neutrons in a light water reactor is approximately equal to 0.0253 eV. This corresponds to a speed of about 2200 m/s for neutrons at 20 deg. C. (a) If the neutron flux is 2  1012 neutrons/(cm2-sec) calculate the neutron density. Indicate the units. 1 m ¼ 100 cm. (b) If the microscopic absorption cross section is σa ¼ 694  1024 cm2 for U-235 and the density of target nuclei of U-235 is N ¼ 0.048  1024/cm3, calculate the total reaction rate. Indicate units. (c) What is the rate at which U-235 nuclei are consumed in this reactor?

References [1] J.J. Duderstadt, L.J. Hamilton, Nuclear Reactor Analysis, John Wiley, New York, 1976. [2] J.K. Shultis, R.E. Faw, Fundamentals of Nuclear Science and Engineering, second ed, CRC Press, Boca Raton, FL, 2007. [3] A.R. Foster, R.L. Wright, Basic Nuclear Engineering, Allyn and Bacon, Boston, 1983. [4] J.R. Lamarsh, Introduction to Nuclear Reactor Theory, Addison-Wesley, Reading, MA, 1983. [5] https://www.nuclear-power.net/glossary/, 2018.

267

APPENDIX

Laplace transforms and transfer functions

D

D.1 Introduction Laplace transforms have an important role in the analysis of dynamic systems. They permit conversion of a linear differential equation into an algebraic equation. This algebraic equation may then be manipulated so that the solution to the original differential equation is obtained by inspection using Laplace transform tables and/or some simple rules. The important concept of a system transfer function is developed in this chapter. Transformation to the Laplace domain facilitates the simplification of large systems with several dynamic subsystems and thus solve for the overall system response. The transfer function representation of a dynamic system may be used to develop the frequency response of the system.

D.2 Defining the Laplace transform The Laplace transform of a time function, f(t), is defined as follows [1, 2]: ∞ ð

FðsÞ ¼ Lff ðtÞg ¼

f ðtÞest dt

(D.1)

0

The Laplace transform exists when the above integral is finite ( 0 F(s + a) 1 1 s 1 s2 n! sn + 1 1 s+a n! ðs + aÞn + 1 ω s2 + ω2 s 2 s + ω2

D.4 The inverse Laplace transform Inversion of Laplace transforms is the determination of functions (functions of time in system dynamics applications) corresponding to Laplace transformed quantities. A method, called the method of residues provides a simple procedure by which the Laplace transform may be inverted by inspection. If the denominator polynomial roots (poles) of the Laplace transform appear as first order terms, then the method of partial fractions can be applied easily. Both techniques are illustrated.

D.4.1 Method of residues L1 fFðsÞg ¼

X

½residues of FðsÞest 

(D.9)

all poles

where the residue of an n-th order pole at s ¼ s1, is given by Rs1 ¼

 n1  1 d n st ð ð s  s Þ F ð s Þe Þ 1 ðn  1Þ! dsn1 s¼s1

(D.10)

APPENDIX D Laplace transforms and transfer functions

This formidable-looking formula is quite easy to apply, especially for a first order pole (n ¼ 1). For the first order case, the residue becomes Rs1 ¼ ½ðs  s1 ÞFðsÞest s¼s1

(D.11)

This can be applied to the general transform with all first order poles (a1, a2, …) FðsÞ ¼

ðs  b1 Þðs  b2 Þ…ðs  bm Þ ðs  a1 Þðs  a2 Þ…ðs  an Þ

(D.12)

Inversion by the residue theorem gives (by inspection) f ðtÞ ¼

ða1  b1 Þða1  b2 Þ…ða1  bm Þ a1 t e +⋯ ða1  a2 Þða1  a3 Þ…ða1  an Þ ðan  b1 Þðan  b2 Þ…ðan  bm Þ an t + e ðan  a1 Þðan  a2 Þ…ðan  an1 Þ

(D.13)

Note that inversion for a specific pole involves removal of the term containing that pole and substituting that pole’s value for s in the other terms. A little experience will permit one to perform such an inversion very quickly. Example D.4 Let us use the method of residues to invert the following Laplace transform FðsÞ ¼

ðs + 1Þðs + 4Þ ðs + 2Þðs + 6Þðs + 7Þ

(D.14)

The inversion gives: ð2 + 1Þð2 + 4Þe2t ð6 + 1Þð6 + 4Þe6t ð7 + 1Þð7 + 4Þe7t + + ð2 + 6Þð2 + 7Þ ð6 + 2Þð6 + 7Þ ð7 + 2Þð7 + 6Þ 1 2t 5 6t 18 7t f ðtÞ ¼ e  e + e 10 2 5 f ðtÞ ¼

(D.15)

Example D.5 Consider a second order pole FðsÞ ¼

ðs + 1Þðs + 4Þ

(D.16)

ðs + 2Þðs + 6Þ2

The residue at the pole, s ¼ 2, is given by Rs¼2 ¼

ð2 + 1Þð2 + 4Þ ð2 + 6Þ2

Rs¼2 ¼

1 2t e 8

e2t

273

274

APPENDIX D Laplace transforms and transfer functions

The residue at the pole, s ¼ 6, is given by " ( )# 1 d ð21Þ 2 ðs + 1Þðs + 4Þ st ð s + 6 Þ e ðs  1Þ! dsð21Þ ðs + 2Þðs + 6Þ2 s¼6    d ðs + 1Þðs + 4Þ st Rs¼6 ¼ e ds ðs + 2Þ s¼6 " # ðs + 4Þ st ðs + 1Þ st est tðs + 1Þðs + 4Þ st Rs¼6 ¼ + e + e  ðs + 1Þðs + 4Þ e s+2 ðs + 2Þ ðs + 2Þ ðs + 2Þ2

Rs¼6 ¼

s¼6

ð6 + 4Þ 6t ð6 + 1Þ 6t ð6 + 1Þð6 + 4Þ 6t Rs¼6 ¼ e e + e  ð6 + 2Þ ð6 + 2Þ ð6 + 2Þ2 tð6 + 1Þð6 + 4Þ 6t e ð6 + 2Þ   1 5 5 6t 5 6t Rs¼6 ¼ +  e  te 2 4 8 2 +

9 5 Rs¼6 ¼ e6t  te6t 8 2 Therefore, the complete inversion gives 1 9 5 f ðtÞ ¼  e2t + e6t  te6t 8 8 2

(D.17)

These examples show that the inversion of transforms with multiple order poles is considerably more tedious than the inversion of transforms with all first order (simple) poles. However, most dynamic systems of interest do not have multiple poles. The poles of the transfer function determine the stability of a linear system. If any pole has a positive real part, the response will increase indefinitely. So, the requirement for stability is that the real part of evert pole be negative.

D.4.2 Inverse transform using partial fractions The method of partial fractions for inverting the Laplace transform involves rewriting the transform as sum of individual terms corresponding to the denominator factors. Once this is accomplished, each of the terms are inverted to the time domain by simple table lookup. The method is illustrated in Example D.6. Example D.6

Solve the following second-order differential equation when f(t) is a unit step function. Assume zero initial conditions. Use the method of Laplace transform d2 x dx + 3 + 2x ¼ f ðtÞ dt dt2

(D.18)

APPENDIX D Laplace transforms and transfer functions

Solution Take the Laplace transform of both sides of Eq. (D.18). Use Table D.1 for appropriate transforms s2 XðsÞ  sxð0Þ  x_ ð0Þ + 3 ½sXðsÞ  xð0Þ + 2XðsÞ ¼ FðsÞ With F(s) ¼ 1/s and zero initial conditions, the above simplifies as XðsÞ ¼

FðsÞ 1 ¼ s2 + 3s + 2 sðs + 1Þðs + 2Þ

Express X(s) in the partial fraction form as 1 k1 k2 k3 + ¼ + sðs + 1Þðs + 2Þ s s + 1 s + 2

XðsÞ ¼

(D.19)

Apply the following steps successively to solve for the constants k1, k2, and k3. Multiply both sides of Eq. (D.19) by s, set s ¼ 0, and solve for k1. Multiply both sides of Eq. (D.19) by (s + 1), set s ¼ 1, and solve for k2. Multiply both sides of Eq. (D.19) by (s + 2), set s ¼ 2, and solve for k3. The numerical values of k1, k2, and k3 are shown in Eq. (D.20) 1 1 k1 ¼ , k2 ¼ 1, k3 ¼ : 2 2

(D.20)

1 1 1  + 2s s + 1 2ðs + 2Þ

(D.21)

Thus X(s) simplifies as XðsÞ ¼

To find x(t), take the inverse Laplace transform of the three terms in Eq. (D.21). See Table D.1 for Laplace transform pairs. Thus, 1 1 xðtÞ ¼  et + e2t 2 2 Note that Eq. (D.22) satisfies the assumed zero initial conditions

(D.22)

1 1 xð0Þ ¼  1 + ¼ 0 2 2 dx ¼ et  e2t and x_ ð0Þ ¼ 0 dt One last point to note—the steady-state value of x(t), as t ! ∞, is xss ðtÞ ¼ 12

D.5 Transfer functions Laplace transforms permit formulation of transfer functions. For transfer functions, the input and output are considered as perturbation variables or system with zero initial conditions. That is, they are deviations from steady state. Consider the representation of a linear system shown in Fig. D.3. x(t) Input X(s)

FIG. D.3 Linear system representation.

G(s)

y(t) Output Y(s)

275

276

APPENDIX D Laplace transforms and transfer functions

The input and output are denoted by δx(t) and δy(t), respectively. The corresponding Laplace transforms are δX(s) and δY(s). The transfer function, G(s), is defined as the ratio between the Laplace transform of the output and the Laplace transform of the input. G ðs Þ 

δY ðsÞ δXðsÞ

(D.23)

In general, the numerator (order m) and denominator (order n) of the transfer function are polynomials in ‘s’ G ð sÞ ¼

ðs  z1 Þðs  z2 Þ…ðs  zm Þ ðs  p1 Þðs  p2 Þ…ðs  pn Þ

(D.24)

The parameters {p1, p2, …, pn} are called the poles of G(s). The parameters {z1, z2, …, zm} are called the zeros of G(s). If r of the poles have the same value, then that pole is called an rth-order pole. In general, the poles and zeros of G(s) are complex numbers. It is more common for the poles to be complex values than the zeros, because the poles of G(s) represent the system dynamics. Also, the order of the denominator polynomial is always greater than the order of the numerator polynomial. The transfer function reflects the basic characteristics of a system (such as a differential equation) and is not dependent on initial conditions.

D.6 Feedback transfer functions A common and important case is a system in which the output is fed back to the input as shown in Fig. D.4. Note that, in some cases, the feedback effect may be positive. In this case δY ðsÞ ¼ GðsÞδXðsÞ + GðsÞH ðsÞδY ðsÞ

Input

Plant

x(t) + X(s)

-

G(s)

y(t) Output Y(s)

Feedback H(s)

FIG. D.4 Transfer functions in a feedback configuration. Note that, in some cases, the feedback effect may be positive.

APPENDIX D Laplace transforms and transfer functions

Simplifying δY ðsÞ G ðs Þ ¼ δXðsÞ 1  GðsÞH ðsÞ

(D.25)

The feedback is often subtracted from instead of added to the input (negative feedback system). In this case, the closed-loop transfer function is given by δY ðsÞ G ðs Þ ¼ δXðsÞ 1 + GðsÞH ðsÞ

(D.26)

Since industrial controllers usually provide a signal that is subtracted from the input, the form of Eq. (D.26) is applicable in that case.

D.7 The convolution integral If the input (x) and output (y) variables are related by a linear time-invariant system, and if G(s) is the transfer function relating x(t) and y(t), then Y ðsÞ ¼ GðsÞXðsÞ

(D.27)

If the input is a unit impulse function, then X(s) ¼ 1, and Y(s) ¼ G(s). Thus the impulse response function, g(t), of this system is simply the inverse Laplace transform of G(s). That is, yI ðtÞ  gðtÞ ¼ L1 ½GðsÞ

(D.28)

Thus, once the impulse response of a linear system is known, the response to any other input is determined from inverting Eq. (D.27). This inversion of the product of two Laplace transforms is called the convolution integral and is given by yðtÞ ¼ L1 ½GðsÞXðsÞ ¼

∞ ð

gðt  τÞxðτÞ dτ

(D.29)

0

This is a fundamental property of a linear system. By using the property that for a physically realizable system, the impulse response is zero for time t < 0, integral (D.29) may be rewritten by changing the upper limit to the current time, t. ðt y ðt Þ ¼

gðt  τÞxðτÞ dτ

(D.30)

0

Eq. (D.30) is the form of the convolution integral used in numerical calculations. The convolution integral is directly applicable to multivariate state variable equations.

277

278

APPENDIX D Laplace transforms and transfer functions

D.8 Laplace transforms and partial differential equations Partial differential equations are those that have more than one independent variable. For reactor engineering applications, the independent variables are time and position. Applications include one, two, or three position variables. The most common and most easily solved are one-dimensional models. Recall that Laplace transformation reduces an ordinary differential equation to an algebraic equation. Laplace transformation (with respect to time) of a onedimensional partial differential equation results in an ordinary differential equation with position as the independent variable. Laplace transformation of multidimensional models eliminates the time derivative term, but the resulting model is still a partial differential equation in the position variables. The solution of the Laplace-transformed equation provides a transfer function. The space- and frequency-dependent frequency response may be obtained by substituting jω for s in the transfer function. An example illustrates the procedure. Consider the following simple, onedimensional partial differential equation: ∂u ∂u + + buðx, tÞ ¼ 0 ∂t ∂x

(D.31)

Variable u(x, t) is a function of position (x) and time (t). The initial condition is assumed to be zero. That is, u (x, 0) ¼ 0. Laplace transformation of Eq. (D.31) yields dU + ðs + bÞU ¼ 0 dx

(D.32)

where U(x, s) is the Laplace transform of u(x, t). The solution of Eq. (D.32) is Uðx, sÞ ¼ U0 eðs + bÞx

(D.33)

The frequency response for a specified value of a x is given by (setting s ¼ jω) U ðx, ωÞ ¼ ebx ejωx U0

(D.34)

The magnitude and phase angle of the frequency response function at (x, ω) are given by Magnitude of U ðx, ωÞ ¼ ebx

(D.35)

Phase angle of U ðx, ωÞ ¼ ðω xÞ

(D.36)

Note that the magnitude of U(x, ω) is not a function of frequency, ω; but changes with variable, x. The phase angle, for a given value of ω, changes as a linear function of x.

APPENDIX D Laplace transforms and transfer functions

Exercises D.1. Determine the inverse Laplace transforms of the following: s+1 ðs + 2Þðs + 3Þ s+1 ðs + 2Þ2 s+1 s2 + 4

D.2. The Laplace transform of a time function x(t) is given by X ðs Þ ¼

6 s ðs + 1Þðs + 3Þ

(a)

Determine the time function x(t).

(b)

Calculate the value of x(t) for t ¼ 0.

(c)

Calculate the value of x(t) as t goes to infinity (same as the steady-state value).

D.3. A time domain function f(t) is given by f ðt Þ ¼ sin ðt Þ + cos ðt Þ + e t

Determine the Laplace transform F(s) of f(t), and simplify your answer in the form of a ratio of two polynomials in s. D.4. Consider the following transfer function of a second order system: Gðs Þ ¼

Y ðs Þ 2 ¼ X ðs Þ s 2 + s + 2

(a)

Calculate the roots (poles) of the denominator polynomial.

(b)

If the input x(t) is a unit step function, determine the response y(t) to this input. You may use the method of residues or partial fraction.

(c)

Make a plot of this step response. You may use the MATLAB command step (sys) where ‘sys’ is defined by the transfer function G(s). Comment on the characteristics of this second order system response.

(d)

Is this system stable or unstable? Explain.

D.5. A system has a transfer function given by G ðs Þ ¼

1 s+1

279

280

APPENDIX D Laplace transforms and transfer functions

The input to the system is given by x ðt Þ ¼ sin t 0

0t T t >T

Derive the system output function, y(t). Use the convolution integral to determine y(t). Note that you must derive two expressions for y(t), one for 0  t  T, and another for t > T.

D.6. If F(s) is the Laplace transform of a time function f(t), and If G(s) is the Laplace transform of another time function g(t), state the Laplace transform of {a.f(t) + b.g(t)}, where a and b are constant parameters. D.7. Determine the Laplace transform of the pulse function f(t) shown in the figure below f ðt Þ ¼ 2, ¼ 0,

1t 5 elsewhere

(a)

Solve this problem by using the definition of the Laplace transform integral.

(b)

Verify your answer using the method of superposition of a delayed positive unit step function and a delayed negative unit step function. f(t)

2

0

1

5

t

References [1] R. Saucedo, E.E. Schiring, Introduction to Continuous and Digital Control Systems, MacMillan Company, New York, 1968. [2] C.L. Phillips, J.M. Parr, Feedback Control Systems, fifth ed, Prentice Hall, Upper Saddle River, NJ, 2011.

APPENDIX

E

Frequency response analysis of linear systems E.1 Frequency response theory

The frequency response of a linear time-invariant system is defined as the response of a selected system output resulting from a sinusoidal perturbation in a selected system input. The output of a linear system to a sinusoidal input is also a sinusoidal function with the same frequency as the input sinusoid, but shifted by phase angle, Φ. The ratio of the amplitude of the output sinusoidal function to the amplitude of the input sinusoidal function, and the phase angle completely define the frequency response of the system. After perturbing the system by a sine function of a certain frequency, an initial, non-sinusoidal output occurs. After initial transient components decay, a continuing sinusoidal output occurs. This approach is valid only for stable systems; that is for systems with all the poles of the transfer function with negative real parts. The frequency response function is widely used in the study of linear systems, in system design to achieve desired characteristics, and in the stability analysis of linear systems. Certain frequency domain parameters can be directly related to system characteristics in the time domain. In addition, the frequency domain analysis can provide quick insight into the dynamic nature of a system by interpreting the significance of the magnitude and/or phase in certain frequency bands. The basic contributions to this area of systems analysis were made by Bode, Nyquist, Nichols, and others [1]. Bode plots are the most common graphical depictions of a system’s frequency response. They show the response magnitude versus frequency (plotted on a log-log scale); and phase angle versus frequency (plotted on a semi-log scale) with the phase angle in degrees. Now, let us show how frequency responses are calculated. Consider the transfer function, G(s), of a stable system [2]. See Fig. E.1 G ðs Þ ¼

δYðsÞ δXðsÞ

(E.1)

δY ðsÞ ¼ GðsÞ δXðsÞ

Input

dx(t) dX(s)

G(s)

(E.2)

dy(t) dY(s)

Output

FIG. E.1 An open-loop system with transfer function G(s).

281

282

APPENDIX E Frequency response analysis of linear systems

Consider an input δx(t) of the form δxðtÞ ¼ Α sin ðωtÞ

(E.3)

As shown in Table D.1 in Appendix D, the Laplace transform of δx(t) is δXðsÞ ¼

Αω s2 + ω2

(E.4)

A ¼ amplitude of the sinusoidal function. ω ¼ frequency ðrad=secondÞ of sin ðωtÞ:

Substituting for δX(s), Eq. (E.2) becomes. δY ðsÞ ¼ GðsÞ

Αω s2 + ω2

(E.5)

If G(s) is stable and has n distinct poles, pi, one may write the inverse Laplace transform, δy(t), as follows:


δyðtÞ ¼ k1 ep1 t + k2 ep2 t + … + kn epn t + inverse Laplace transform of

Aω s2 + ω 2



(E.6)

The time-domain terms corresponding to the poles of G(s) go to zero as t ! ∞, because all the poles have negative real parts in a stable system. Thus, for the steady-state response, it is sufficient to consider the poles of s2 +1 ω2 . Since these are given by s ¼  jω, δY(s) is expressed as (we are not interested in the terms containing the poles of G(s)). δY ðsÞ ¼

GðsÞΑω ðs + jωÞðs  jωÞ

(E.7)

The inverse Laplace transform of Eq. (E.7) is as follows: δyðtÞ ¼

GðjωÞΑωejωt Gð jωÞΑωejωt + ð2jωÞ 2jω

(E.8)

G(jω) is a complex number and may be expressed using its magnitude and phase angle: Gð jωÞ ¼ jGðjωj exp ðjϕðωÞÞ

(E.9)

where n o1=2 jGð jωÞj ¼ ½ ReGðjω Þ2 + ½ ImGðjω Þ2

(E.10a)

tan φðωÞ ¼ ImGð jωÞ= ReGð jωÞ

(E.10b)

This complex plane representation is illustrated in Fig. E.2. The time dependent response then becomes δyðtÞ ¼

Or

ΑjGð jωÞjejφ ejωt ΑjGð jωÞjejφ ejωt  2j 2j

(E.11)

APPENDIX E Frequency response analysis of linear systems

Im G(jw)

|G(jw)|

φ(ω) Re G(jw)

0

FIG. E.2 Representation of a complex number G(jω) in terms of its magnitude and phase.

δyðtÞ ¼ ΑjGð jωÞj jðω t + φÞ

e Note that e 2j sinusoidal input is

jðω t + φÞ

ejðω t + φÞ  ejðω t + φÞ 2j

(E.12)

¼ sin ðωt + φÞ. Therefore, the steady-state response to a δyðtÞ ¼ Α jG ð jωÞjsin ð ωt + φ Þ

(E.13)

This development shows that when a linear system is perturbed by a sinusoidal input of amplitude A and frequency ω, its steady-state response is also a sinusoidal function of same frequency (ω) and shifted by an angle, Φ, and the amplitude is the product of jG(jω)j and the input amplitude. The theoretical frequency response is obtained simply by substituting jω for s in the transfer function and carrying out the complex arithmetic.

E.2 Computing frequency response function Now let us illustrate the calculation of a system frequency response. Example E.1 Consider the following transfer function: GðsÞ ¼ Gð jωÞ ¼

1 s+1

(E.14)

1 1  jω ¼ jω + 1 1 + ω2

RefGð jωÞg ¼

1 1 + ω2

(E.15a)

ImfGð jωÞg ¼

ω 1 + ω2

(E.15b)

 1=2 1 jGð jωÞj ¼ ð ReGÞ2 + ð ImGÞ2 ¼ pffiffiffiffiffiffiffiffiffiffiffiffi 1 + ω2

(E.16)

Continued

283

APPENDIX E Frequency response analysis of linear systems

Example E.1—Cont’d φðωÞ ¼ tan 1



 ImGð jωÞ ¼ tan 1 ðωÞ ReGð jωÞ

(E.17)

These magnitude and phase angle are calculated for selected values of ω. For example, for ω ¼ 0.1 rad/s, 1 RefGð jωÞg ¼ ¼ 0:99 (E.18a) 1:01 ImfGð jωÞg ¼

0:1 ¼ 0:099 1:01

(E.18b)

h i1=2 ¼ 0:995 jGð jωÞj ¼ ð0:99Þ2 + ð0:099Þ2

(E.19)

φ ¼ tan 1 ð0:1Þ ¼ 5:7 degrees

(E.20)

Note that the frequency response magnitude can be easily calculated by dividing the magnitude of the numerator of G(jω) by the magnitude of the denominator. This calculation may be repeated at a number of frequencies. The results of such a calculation are shown as a Bode plot in Fig. E.3.

Bode plots of G(jw) = 1/(1+jwT), T = 1

Magnitude (dB)

0 –5 –10 –15

Break Frequency

–20 –25 0

Phase (deg)

284

–45

–90 10–1

FIG. E.3 Bode plot for Example E.1.

100 Frequency (rad/s)

101

APPENDIX E Frequency response analysis of linear systems

Note that the magnitude is constant at low frequencies and it decreases at a rate of one decade per decade of frequency at high frequencies. The frequency, at which the low frequency and high frequency asymptotes intersect, is called the break frequency, ωb, of the frequency response magnitude plot. The magnitude at ω ¼ 1 rad/s is equal to p1ffiffi2 ¼ 0:707. This frequency is called the half-power frequency since jGð jωÞj2 ¼ 12. The magnitude of 0.707 is often referred to as the RMS (root-meansquared) value. The units for frequency in the above developments are rad/s, but it is also common practice to use frequency in cycles/s or Hertz (Hz). Frequency ω (rad/s) is converted to frequency f (Hz or cycles/s) using the relationship. ω ¼ 2π f

(E.21)

There is a direct relationship between system poles and zeroes and the asymptotic magnitude and phase for systems described by ordinary differential equations. For example, consider the contribution of a term, 1/(s + a) in a transfer function. Substitute s ¼ jω to obtain 1/(jω + a). The term goes to a constant value, 1/a, at low frequencies and to 1/jω or equivalently, jω/(ω2), at high frequencies. The term, jω/(ω2), is a negative imaginary number for all values of ω, corresponding to a phase shift of 90 degrees. Other poles and zeroes would likewise have asymptotic contributions to the total system frequency response. Table E.1 shows asymptotic magnitudes and phases for various terms in system transfer functions. These systems, characterized by the asymptotic relations as shown in Table E.1 [2], are called minimum phase systems. All systems described by ordinary differential equations are minimum phase systems except for those containing pure time delays. Some systems require partial differential equations for their description and they are not minimum phase systems.

Remark 1. In the literature on control systems analysis, it is common practice to define the magnitude in terms of the deciBel (dB) given by jGð jωÞj ðin deciBelsÞ ¼ 10 log 10 Gðjωj2

(E.22)

dB magnitude ¼ 20 log 10 jGð jωÞj

(E.23)

One deciBel or 1 dB ¼ one-tenth of a Bel. The unit Bel was named in honor of Graham Bell and is used to express a power level with respect to a standard power level. Thus the magnitude in Bels is log10 j G(jωj2. 2. If the transfer function has the general form G(s) ¼ 1/(s + a), the parameter a is the break frequency of the magnitude plot. The inverse of this parameter is called the time constant of the first order system. That is, time constant, τ ¼ 1/a sec.

285

286

APPENDIX E Frequency response analysis of linear systems

Table E.1 Asymptotic magnitudes and phases of common frequency response functions. Transfer function G(s) (G(jω)) s ð jωÞ s+a ðjω + aÞ n

ðs + aÞ n  ðjω + aÞ 1 s 1 jω 1

s+a   1 jω + a 1 ðs + aÞn
 1 n ðjω + aÞ

Asymptotic amplitude

Asymptotic Phase (deg)

Low frequency

High frequency

Low frequency

High frequency

j G(jω)j ∝ ω

jG(jω)j ∝ ω

+90

+90

j G(jω)j ¼ constant

jG(jω)j ∝ ω

0

+90

j G(jω)j ¼ constant

jG(jω)j ∝ ωn

0

+90n

jGð jωÞj∝ ω1

jGð jωÞj∝ ω1

90

90

j G(jω)j ¼ constant

jGð jωÞj∝ ω1

0

90

j G(jω)j ¼ constant

jGð jωÞj∝ ω1n

0

90n

Example E.2 In order to illustrate the use of asymptotic magnitudes and phases, consider the transfer function.

GðsÞ ¼

ðs + 1Þ ðs + 0:1Þðs + 10Þ

(E.24)

Each term contributes to the magnitude and phase. The break frequencies of the terms (s + 1), (s + 0.1) and (s + 10) are respectively at 1, 0.1 and 10 rad/s. Fig. E.3 shows the asymptotic magnitude plots of each term and the composite magnitude plot. The exact Bode plots are shown in Fig. E.4. (See Fig. E.5.) The asymptotic values of the phase angle are: φ(ω) ! 0 as ω ! 0, φ(ω) !  90 deg for large ω

APPENDIX E Frequency response analysis of linear systems

FIG. E.4 Asymptotic Bode plots of G(s) ¼ (s + 1)/(s + 0.1)(s + 10). Bode plots of G(s) = (s + 1)/[(s + 0.1)(s + 10)] 0

Magnitude (dB)

–10 –20 –30 –40 –50

Phase (deg)

0

–45

–90 10–2

10–1

100 Frequency (rad/s)

FIG. E.5 Exact Bode plots of G(s) ¼ (s + 1)/[(s + 0.1)(s + 10)].

101

102

287

288

APPENDIX E Frequency response analysis of linear systems

Certain other useful information can be obtained from a Bode plot. For instance, we observe in Fig. E.2 that the amplitude of the frequency response of s +1 a is essentially constant out to about a radians per second. This simply means that the inertia of the system is sufficiently small for the output to keep up with the input for these frequencies. At higher frequencies, the system output is unable to keep up with the system input and the amplitude decreases. This is also observed in the phase plot. The frequency range over which the amplitude is at least p1ffiffi ¼ 0:707as large as the steady state ðω ¼ 0Þ amplitude is called the bandwidth 2 of the system.

E.3 Systems with oscillatory behavior A system frequency response may also be used to provide information about the oscillatory characteristics of the system. If the amplitude peaks at some frequency, this means that the system will greatly amplify any component of the input at that frequency. This phenomenon is called resonance. In general, a Bode plot with a tall, narrow peak in the magnitude indicates that the system tends to be highly oscillatory. Damped oscillatory behavior is also noticed in neutron power response in a BWR, primarily caused by the void reactivity feedback.

Example E.3 As an example, consider the system with the following transfer function GðsÞ ¼

ω2n s2 + 2ζωn s + ω2n

(E.25)

Rewrite G(s) in the form GðsÞ ¼

1 s2 2ζ s + +1 ω2n ωn

(E.26)

The frequency response function is given by 1  Gð jωÞ ¼  ω2 2ζ 1 2 +j ω ωn ωn Fig. E.6 shows the frequency response for ωn ¼ 1.0 and ζ ¼ 0.1.

(E.27)

APPENDIX E Frequency response analysis of linear systems

Bode plots of G(s) = 1/(s**2 + 0.2s + 1), zeta = 0.1 20

Magnitude (dB)

10 0 –10 –20 –30 –40 0

Phase (deg)

–45

–90

–135

–180 10–1

100

101

Frequency (rad/s)

FIG. E.6 Bode plot for Example E.3.

Because of the resonance at 1.0 rad per second, an input disturbance would cause an oscillatory response with that frequency. Example E.4 An example of application to a BWR is the frequency response characteristic of the neutron power as a function of reactivity. The neutron power to reactivity transfer function in a BWR demonstrates frequency response behavior, similar to a second order system with oscillatory behavior. A sharper peak in the magnitude plot indicates a more sustained oscillation of the neutron power for a change in the reactivity. This measurement is used to monitor the stability of BWRs during plant operation. Fig. E.7 shows frequency response functions of neutron detector signals from two BWRs, indicating the characteristic peak frequencies. Continued

289

APPENDIX E Frequency response analysis of linear systems

Example E.4—Cont’d

POWER SPECTRUM (1/Hz)

101

100

10–1 PEAK FREQUENCY –0.38 Hz.

10–2

10–3 0.01

0.1

1.0

10

FREQUENCY (Hz)

101

POWER SPECTRUM (1/Hz)

290

100

10–1 PEAK FREQUENCY – 0.35 Hz.

10–2

10–3 0.01

0.1

1.0

10

FREQUENCY (Hz)

FIG. E.7 Frequency response functions calculated using measurements from Average Power Range Monitor (APRM) detectors in two different BWRs. The maximum or peak frequencies are approximately equal to 0.40 and 0.35 Hz.

APPENDIX E Frequency response analysis of linear systems

E.4 Systems with time delay dynamics Many dynamic systems have an inherent time delay. For example, in pipe flow, a downstream point experiences a disturbance at some time later than an upstream disturbance. In a nuclear power plant, the hot water from the vessel in a pressurized water reactor is carried through by hot leg piping and delivered to steam generators. When a temperature change takes place in the water leaving the vessel, this change is carried through the flowing water in the hot leg. Depending on the length of the hot leg, a certain transport time is needed to sense this change at some other point. This time delay or transport delay (dead time) between two variables δx(t) and δy(t) is expressed as δyðtÞ ¼ cδxðt  DÞ,c is a constant parameter:

(E.28)

y(t) detects the changes in x(t) after D seconds. The transfer function of a pure time delay is G ðs Þ ¼

δY ðsÞ ¼ cesD δXðsÞ

(E.29)

The frequency response function is given by Gð jωÞ ¼ c exp ðjωDÞ

(E.30)

Thus j G(jω)j ¼ constant for all ω and the phase angle φ (ω) ¼ - ω D; D is the slope of the linear phase angle plot.

Example E.5 An example of the use of pure delay dynamics between two detector signals is the measurement of flow velocity in a BWR. The phase angle between two in-core detectors, placed parallel to the core axis, has a linear form, and the slope of the phase angle corresponds to the time delay in the flow passage from the upstream detector to the downstream detector. Fig. E.8 is the core map of a typical BWR, showing the location of in-core neutron detector strings. The detectors in each string are labeled A, B, C, and D from bottom (upstream) to top (downstream) of core. Fig. E.9 is a plot of the phase angle between the signals from detectors B and C. As seen in this plot, the phase angle between the detector signals B and C is linear, indicating that the relationship between these two detector signals may be approximated by a pure time delay in the frequency band of interest. Properties such as these, including the characteristic frequency response of neutron detectors in BWRs, are used to monitor reactor performance (stability, flow pattern, etc.) in an on-line manner. Continued

291

Example E.5—Cont’d PLAN TOP VIEW 0° 57

49

41

33 90°

25

17

09

16

08

24

32

40

48

56

LPRM Detector String Locations. BWR–4 Core map.

FIG. E.8 Core map of a BWR-4 showing in-core neutron detector strings. There are 40–50 LPRM (local power range monitor) detector strings in a typical 1200 MWe reactor core [3]. 0

TRANSIT TIME = 0.167 S. B–C VELOCITY = 5.48 M/S.

PHASE (DEG)

–200 LINEAR FIT D

C –400 B

A –600 0

2

4 FREQUENCY (Hz)

6

8

FIG. E.9 Phase angle between detectors B and C, and the linear fit to the phase. Flow transit time or delay time from B to C, τ ¼ 0.17 s. Estimated two-phase flow velocity ¼ 18 ft/s (5.48 m/s).

APPENDIX E Frequency response analysis of linear systems

A linear fit to the phase plot is shown in Fig. E.9. The slope of this straight line corresponds to the transport time between the two detector locations. The knowledge of the distance between the two detectors (3 ft ( 0.91 m)) can then be used to calculate an average two-phase flow velocity in this BWR at the location of the detector string. The estimated transport time (delay time) is 0.17 s and the average flow velocity is 18 ft/s (5.48 m/s).

E.5 Frequency response of distributed systems As shown in Appendix D, Section D.8, Eq. (D.34) gives the transfer function for a fluid channel for temperature at position, x, per change in the channel inlet temperature. The transfer function is U ðx, sÞ ¼ U0 eðs + bÞx

(E.31)

With b ¼ 1, Eq. (E.31) is rewritten as Uðx, sÞ ¼ eðs + 1Þx U0

(E.32)

The frequency response for this transfer function is determined by substituting s ¼ jω. U ðx, ωÞ ¼ ex ejωx U0

(E.33)

The magnitude and phase angle of the frequency response function at (x, ω) are given by Magnitude of U ðx, ωÞ ¼ ex

(E.34)

Phase angle of U ðx, ωÞ ¼ ðω xÞ

(E.35)

Note that the magnitude is independent of frequency and decreases as the position, x, increases. The phase angle decreases as the frequency and position increase. This illustrates that distributed parameter frequency responses are not minimum phase systems as are lumped parameter frequency responses.

E.6 Frequency response measurements Frequency response measurements are sometimes used to aid in the understanding of a dynamic process or to check the validity of a theoretical dynamic model. The most obvious approach is to introduce an input sinusoid. But many systems (including nuclear reactors) lack hardware that can produce a sinusoidal input. They usually are better able to introduce step changes. Therefore, methods have been devised and implemented that can employ binary inputs (an input that takes two values and steps up and down in a prescribed sequence). The simplest binary signal is the square wave. Fourier analysis shows that the square wave can be represented as a sum of sinusoids. The square wave has most

293

APPENDIX E Frequency response analysis of linear systems

of its strength in a single frequency (its first harmonic). Other binary sequences with their strengths distributed over a range of frequencies are available. One such sequence is the pseudo random binary sequence (PRBS). Fig. E.10 shows a PRBS. This is a seven-bit sequence. Note that the sequence repeats after seven shifts. Each bit has a certain specified time interval. The frequency spectra for PRBS sequences are as follows [4] Pk ¼

A2 Z2

Pk ¼


 2 ðZ + 1ÞA2 sin ðkπ=ZÞ 2 for k 6¼ 0 Z2 kπ=Z

for k ¼ 0 (E.36)

The frequency spectra for several PRBS signals are shown in Fig. E.11 [4]. Note that the spectrum is quite flat, especially for the longer sequences. This feature is an approximation to the spectrum of a wide-band random signal (constant level at all frequencies). Note also that the PRBS is periodic (signal strength is concentrated at discrete frequencies). Fourier analysis at harmonic frequencies can provide the sinusoidal response at that frequency. The frequency response is therefore the ratio of Fourier transform of output to Fourier transform of input at selected harmonic frequencies. An algorithm for generating PRBS signals uses a digital shift register with modulo-two adder feedback. Modulo-two addition is as follows: 0 + 0 ¼ 0. 0 + 1 ¼ 1. 1 + 0 ¼ 1. 1 + 1 ¼ 0. 2

1 PRBS Amplitude

294

0

–1

–2 0

2

4

6

8

PRBS Bit Number

FIG. E.10 A seven bit PRBS input.

10

12

14

APPENDIX E Frequency response analysis of linear systems

1.0 DENOTES HARMONIC FREQUENCY BASIS – SAME TOTAL DURATION FOR ALL SIGNALS

0.5

ENERGY/[T X (PULSE AMPLITUDE)2]

0.2

0.1

0.05 63 – BIT 0.02

0.01

31 – BIT

7 – BIT

0.005

0.002

0.001

1

10

100

Harmonic FIG. E.11 Frequency spectra for several PRBS signals [4].

A shift register operates by moving the contents of each register (0 or 1) to the next register to the right at each shift. This operation empties the left-most register. Feedback involves moving the sum (modulo two) of contents to the left-most register. For example, consider a three-register arrangement with adder feedback from registers 1 and 3. The initial contents of the registers are all set equal to 1 in this example. Operation of the system results in the following:

295

296

APPENDIX E Frequency response analysis of linear systems

Register 1

2

3

1 0 1 0 0 1 1 1

1 1 0 1 0 0 1 1

1 1 1 0 1 0 0 1

Note that the content of the registers repeats after seven shifts. The sequence in each register is a 7-bit pseudo-random binary sequence. The feedback configurations for several PRBS signals is as follows: Number of Registers

Registers in Feedback

Length of PRBS

2 3 4 5 6 7

1, 1, 1, 2, 1, 1,

3 7 15 31 63 127

2 3 4 5 6 7

PRBS sequences may be used as inputs to an operating reactor using available hardware such as control rod position and steam valve opening. Signal strength is increased by using multiple periods of the sequence. Fourier analysis of input and output signals provides the frequency response at the harmonic frequencies of the PRBS used. Note that PRBS sequences have an odd number of bits. There is always one more stage of one sign than the other. Consequently, there will be a drift in the output in the direction caused by the input stage that has one more use than the other. That is, the first harmonic is non-zero. Other binary signals are also available. The “n sequence” is obtained by simply changing the sign of every other bit in a PRBS. The n-sequence has an even number of stages in the sequence and there is the same number of stages of each sign. The fundamental harmonic is zero and there is no drift. Another binary test sequence is the multi-frequency binary sequence. It is obtained by a computer optimization of a bit pattern that optimizes signal strength in selected frequencies. All of the binary sequences described above have been used in tests on research reactors (Molten Salt Reactor Experiment with U-235 fuel, Molten Salt Reactor Experiment with U-233 fuel, High Flux Isotope Reactor, EBR II) and power reactors (H.B. Robinson, Oconee, and Millstone PWRs). All of these tests served to check the validity of theoretical models [5].

APPENDIX E Frequency response analysis of linear systems

Exercises E.1

Construct asymptotic Bode plots for the open-loop transfer functions given below. Verify your plots by comparison with the Bode plots generated by using MATLAB function “bode”. (a) GðsÞ ¼ s2 ðs40+ ð8sÞð+s2+Þ 10Þ (b) GðsÞ ¼ ðs + 1Þ9ðs + 3Þ2 s

(c) GðsÞ ¼ 10 s e+ 10 E.2

Consider the following frequency response function G ð jωÞ ¼

1 1 + jω=4  ðω=4Þ2

Construct a log-log plot of j G(f)j vs. f , for a frequency range of 0.01–10 Hz. (Substitute ω ¼ 2πf, and obtain an expression for j G(f)j; compute and plot j G(f)j as a function of f). Estimate the frequency at which a peak is seen in the j G(f)j graph. E.3

Consider the following transfer function of a linear system.

GðsÞ ¼ ðs +10:2Þ, where s is the Laplace variable; the unit of time is ‘second’. pffiffiffiffiffiffiffiffiffiffiffiffiffi Hint: If z ¼ x + jy, is a complex number, the magnitude of z is jzj ¼ x2 + y2 , and

the phase angle is φ ¼ tan 1 yx . (a) State the frequency response function of G(s). (b) Compute the magnitude of the frequency response function G(jω) for ω ¼ 0 rad/s. (c) Compute the phase angle (degrees) of G(jω) for ω ¼ 0 rad/s. (d) Compute the magnitude of G(jω) for ω ¼ 0.2 rad/s. Simplify your answer. (e) Compute the phase angle (degrees) of G(jω) for ω ¼ 0.2 rad/s. Simplify your answer. (f) Make a Bode plot of G(jω) using the MATLAB function “bode”. (g) Determine the break frequency (rad/s) of the device defined by the above transfer function G(s). (h) If this device is a resistance temperature detector (RTD), determine its time constant (sec). (i) What is the meaning of the magnitude of the frequency response function, G(jω), of a linear system at any frequency, ω? E.4

The neutron power to reactivity transfer function model of a BWR has the folpffiffiffiffiffiffi ffi lowing zeros and poles. [j ¼ 1].

Zeros: 0.03, 0.18 + j0.27, 0.18 – j0.27. Poles: 0.25, 21.7, 0.045 + j0.32, 0.045 – j0.32.

297

298

APPENDIX E Frequency response analysis of linear systems

(a) Compute the transfer function as a ratio of two polynomials. (b) Make a Bode plot of the frequency response function, and indicate salient features of the magnitude plot.

References [1] C.L. Phillips, J.M. Parr, Feedback Control Systems, fifth ed., Prentice-Hall, Upper Saddle River, NJ, 2011. [2] R. Saucedo, E.E. Schiring, Introduction to Continuous and Digital Control Systems, MacMillan Company, New York, 1968. [3] B.R. Upadhyaya, M. Kitamura, Stability monitoring of boiling water reactors by time series analysis of neutron noise, Nucl. Sci. Eng. 77 (4) (1981) 480–492. [4] T.W. Kerlin, Frequency Response Testing in Nuclear Reactors, Academic Press, New York, 1974. [5] T.W. Kerlin, E.M. Katz, J.G. Thakkar, J.E. Strange, Theoretical and experimental dynamic analysis of the H.B. Robinson nuclear plant, Nucl. Technol. 30 (September 1976) 299–316.

APPENDIX

State variable models and transient analysis F.1 Introduction

F

Models of real-world systems often require large sets of coupled differential equations to represent all of the interconnected processes. Models involving hundreds of equations are not uncommon. This naturally leads to the use of vector-matrix notation and to the formulation of computer solution software packages that accept matrix formulations of system models. These packages can accept models with any number of system equations (up to some maximum dictated by computer capability) and generate solutions. These simulation software packages include MATLAB (see Appendix G), MAPLE, MATHEMATICA, MODELICA and others. Vector-matrix formulations and matrix-oriented solution techniques are important in dynamic modeling and simulation. The state variable representation of systems (both linear and nonlinear) is described in this appendix with emphasis on linear system models. The state variable representation was first used by Kalman in describing the general theory of control systems [1, 2]. Both time-domain analysis of multi-input multi-output (MIMO) linear timeinvariant systems and their analysis in the Laplace domain are described in this Appendix.

F.2 State variable models Consider n linear algebraic equations in n variables (x1, x2, …xn). a11 x1 + a12 x2 + … + a1n xn ¼ b1 a21 x1 + a22 x2 + … + a2n xn ¼ b2 :…………………………………… :…………………………………… an1 x1 + an2 x2 + … + ann xn ¼ bn

(F.1)

The set of equations may be rewritten as

32 3 2 3 x1 b1 a11 a12 … a1n 6 a21 a22 … a2n 76 x2 7 6 b2 7 76 7 6 7 6 6 ::……………… 76 : 7 ¼ 6 : 7 76 7 6 7 6 4 ::……………… 54 : 5 4 : 5 an1 an2 … ann xn bn 2

(F.2)

Using matrix notation, Eq. (F.2) may be written as Ax¼b

(F.3)

299

300

APPENDIX F State variable models and transient analysis

A is an (n x n) square matrix; and x and b are (n x 1) column vectors. A matrix is shown by an upper-case letter and a vector by a lower-case letter with an underline. Laplace transform of a variable, z(t), is indicated by the upper-case letter, Z(s). The element of a matrix, A, in row i and column j is denoted by aij. If the number of rows (m) is equal to the number of columns (n), m ¼ n, then A is called a square matrix. If m 6¼ n, then A is a rectangular matrix. If most of the matrix elements are zero, the matrix is said to be sparse. State variable matrices in nuclear reactor simulations are generally sparse, with non-zero elements clustered around the matrix diagonal. The A matrix for most nuclear reactors have negative diagonal elements. A positive diagonal element may exist in a model for a stable system, but instability is often encountered for such a model. A positive diagonal element causes a suspicion that the solution will indicate instability. If the analyst has reason to believe that the reactor is stable, it is wise to check the correctness of a positive diagonal element. A column vector x of variables is an ordered array of scalars. A column vector x with n elements has the dimension (n x 1) as shown in Eq. (F.4a). 3 x1 6 x2 7 6 7 6: 7 7 Column Vector x ¼ 6 6 : 7 ðnx1Þ 6 7 4: 5 xn 2

(F.4a)

A row vector is defined as the transpose of a column vector. x T ¼ ½x1 , x2 , …xn 

(F.4b)

Dynamic analysis involves the use of matrix differential equations as shown in Eq. (F.5). dx1 ¼ a11 x1 + a12 x2 + … + a1n xn + f1 + g1 dt dx2 ¼ a21 x1 + a22 x2 + … + a2n xn + f2 + g2 dt … …

(F.5)

… dxn ¼ an1 x1 + an2 x2 + … + ann xn + fn + gn dt

There are n equations in n solution variables, {xi, i ¼ 1, 2, …, n}. There are n2 constant coefficients. The {fi, i ¼ 1, 2, …, n} represent the external forcing functions (including zero values). The {gi, i ¼ 1, 2, …, n} represent nonlinear terms in the state variables (including zero values). These equations may be expressed in matrix notation as follows. dx ¼ Ax + f + g dt

(F.6)

APPENDIX F State variable models and transient analysis

where x ¼ solution vector (state variables). A ¼ system matrix of coefficients. f ¼ vector of forcing functions. g ¼ vector of nonlinear terms in the state variables.

F.3 General solution of the multiple-input multiple-output (MISO) linear State variable model The application of the Laplace transform method for representing the system transfer function and for obtaining the time-domain response to an external perturbation can be easily extended to multivariate system [3]. The following topics are discussed in this section: (i) General representation of time-invariant systems in the state-space form. (ii) Transfer function representation of multiple-input multiple output (MIMO) systems. (iii) Transient response of MIMO systems. (iv) The state transition matrix.

F.3.1 Definition of multiple-input multiple output (MIMO) systems We define a multivariate linear system in the following form. dx ¼ Ax + Bf dt

(F.7a)

y ¼ Cx

(F.7b)

x¼ fx1 , x2 , …, xn g is an n-dimensional state vector. f ¼ f1, f2 , …, fp is an p-dimensional input vector. y ¼ fy1 , y2 , …, ym g is an m-dimensional output or measurement vector. A is an (n x n) time-independent (or time-invariant) system matrix. B is an (n x p) time-independent input matrix. C is an (m x n) time-independent output matrix. If C ¼ I, the identity matrix, then y ¼ x. If p ¼ 1, then matrix B becomes an (n x 1) vector and f is a scalar input. In general, the number of measurements (m) is less than the number of state variables (n). In actual systems, it is often not possible to measure all the state variables in the model. If y is a scalar measurement, then matrix C is a row vector (1 x n).

301

302

APPENDIX F State variable models and transient analysis

F.3.2 Transfer function representation of MIMO systems Assume the initial conditions are known and are given by X (0). Taking the Laplace transform of Eq. (F.7a) gives. sXðsÞ  xð0Þ ¼ AXðsÞ + BFðsÞ

Simplification yields. ðsI  AÞXðsÞ ¼ Xð0Þ + BFðsÞ

(F.8)

XðsÞ ¼ ðsI  AÞ1 xð0Þ + ðsI  AÞ1 BFðsÞ

(F.9)

Solve for X(s) to give.

Eq. (F.9) may be used to solve for X(t). Assuming x(0) ¼ 0, we can express X(s) using the form. XðsÞ ¼ GfX ðsÞFðsÞ

(F.10)

Gf X ¼ ðsI  AÞ1 B

(F.11)

where

GfX is an (nxp) matrix and is the transfer function matrix between f (input variables) and X (state variables). To derive the transfer function between Y(s) and f(s), take the Laplace transform of Eq. (F.7b). Y ðsÞ ¼ CXðsÞ

(F.12)

Substituting Eq. (F.10) in Eq. (F.12) gives. Y ðsÞ ¼ CðsI  AÞ1 BFðsÞ

(F.13)

From Eq. (F.13) define the transfer function matrix between input f and output Y as   GfY ðsÞ ¼ C sI  A1 B

(F.14)

Using Eq. (F.14) in Eq. (F.13) gives Y ðsÞ ¼ Gf Y ðsÞFðsÞ

(F.15)

Remark Note that the roots of the denominator of the transfer functions in Eq. (F.14) are the roots of the polynomial in the Laplace variable, s, and is given by the determinant of (sI – A). These roots are also the poles of the transfer functions. For an n-th order matrix, there are n-poles. These poles are also the eigenvalues of system matrix, A. Thus, the system stability can be easily checked by calculating the eigenvalues of matrix, A.

APPENDIX F State variable models and transient analysis

Example F.1 Consider a system with two inputs (f) and two outputs (y). Then Eq. (F.15) may be written as      y1 ðsÞ G G f1 ðsÞ ¼ 11 12 (F.16) y2 ðsÞ G21 G22 f2 ðsÞ Thus, the transfer function between output yi and input fj is given by the matrix element. yi ðsÞ ¼ Gij ðsÞ; i ¼ 1,2; j ¼ 1,2 fj ðsÞ

(F.17)

F.3.3 Transient response of MIMO systems The response x(t) for a given initial condition x(0) and forcing term f(t) is determined by taking the inverse Laplace transform of Eq. (F.9). Thus, h i h i xðtÞ ¼ L1 ðsI  AÞ1 xð0Þ + L1 ðsI  AÞ1 BFðsÞ

(F.18)

h i Define ϕðtÞ  L1 ðsI  AÞ1

(F.19)

The inverse transform in Eq. (F.18) can be accomplished by using the convolution integral (see Appendix D). If Y(s) ¼G(s)X(s), the inverse transform is achieved by the following convolution integral. yðtÞ ¼ L1 ½GðsÞXðsÞ ¼

∞ ð

gðt  τÞxðτÞ dτ 0

Using the convolution integral and the definition (F.19), x(t) becomes. ðt xðtÞ ¼ φðtÞxð0Þ + φðt  τÞBf ðτÞdτ

(F.20)

0

(The convolution between two time functions p(t) and q(t) is defined by the following integral: Ðt pðtÞ∗qðtÞ ¼ pðt  τÞ qðτÞ dτ, p(t) and q(t) are causal functions). 0

Eq. (F.20) is similar the solution of a linear first order system with initial condition. Furthermore, the solution of a first order linear time invariant system. dx ¼ ax + f ðtÞ dt

(F.21)

is given by ðt xðtÞ ¼ eat xð0Þ + eaðtτÞ f ðτÞdτ 0

(F.22)

303

304

APPENDIX F State variable models and transient analysis

Therefore, for a MIMO system described by Eq. (F.7a), the solution X(t) is written as. ðt xðtÞ ¼ exp ðAtÞxð0Þ +

exp ½Aðt  τÞBf ðτÞdτ

(F.23)

0

For the linear time-invariant system, the matrix φ(t) in Eq. (F.20) is the same as the matrix function exp. (At). Thus, for this time-invariant linear system. φðtÞ ¼ exp ðAtÞ

Example F.2 A linear system is described by

     dx 0 2 3 x1 + f ¼ 1 1 4 x2 dt y ¼ x1 + x 2 :

(a) Determine the transfer function Y(s)/F(s). (b) Determine the response y(t) for a unit step f(t). Assume zero initial conditions.

Solution XðsÞ ¼ ðsI  AÞ1 bFðsÞ   s + 2 3   ¼ ðs + 2Þðs + 4Þ  3 ¼ ðs + 1Þðs + 5Þ jsI  Aj ¼  1 s + 4 ðsI  AÞ1 ¼

  1 s + 4 3 ðs + 1Þðs + 5Þ 1 s + 2

Y ðsÞ ¼ ½1 1XðsÞ ¼ ½ 1 1 ðsI  AÞ1

  0 FðsÞ 1

Y ðsÞ ðs  1Þ ¼ FðsÞ ðs + 1Þðs + 5Þ (a) For a unit step input, F(s)¼1/s

Y ðsÞ ¼

s1 sðs + 1Þðs + 5Þ

(F.24)

APPENDIX F State variable models and transient analysis

Using partial fractions (or the method of residues) gives. Y ðsÞ ¼ 

1 1 3 1 +  5s 2ðs + 1Þ 10 ð2 + 5Þ

3 5t e Time response yðtÞ ¼  15 + 12 et  10

F.3.4 The state transition matrix The matrix φ(t) in Eq. (F.20) is often called the State Transition Matrix (STM) because it provides the solution x(t) at time t from an initial time t ¼ 0 (or t ¼ t0). In general Eq. (F.20) may be written in the form. ðt

xðtÞ ¼ φðt, t0 Þxðt0 Þ +

φðt, τÞBf ðτÞdτ

(F.25)

t0

We showed that for the special case of time-invariant system. h i φðtÞ ¼ L1 ðsI  AÞ1

(F.26a)

φðtÞ ¼ exp ðAtÞ

(F.26b)

and For time-invariant systems the state transition matrix φ(t) satisfies the following properties [3]. dφ ¼ AφðtÞ dt

(F.27)

φðt0 , t0 Þ ¼ I, φð0Þ ¼ I

(F.28)

φðt, t0 Þ ¼ φðt  t0 Þ

(F.29)

These properties satisfy the solution.

ðt xðtÞ ¼ φðtÞxð0Þ + φðt  τÞBf ðτÞdτ 0

Differentiate Eq. (F.30) with respect to t. ðt dx dφðtÞ xð0Þ + φð0ÞBf ðtÞ + Aφðt  τÞBf ðτÞdτ ¼ dt dt 0

Using Eqs. (F.27) and (F.28) in the above gives. ðt dx ¼ AφðtÞxð0Þ + Bf ðtÞ + A φðt  τÞBf ðτÞdτ dt 0

2

ðt

3

¼ A 4 φðtÞxð0Þ + φðt  τÞBf ðτÞdτ5 + Bf ðtÞ 0

(F.30)

305

306

APPENDIX F State variable models and transient analysis

or dx ¼ AxðtÞ + Bf ðtÞ dt

This is the original set of differential equations. The state transition matrix φ(t) ¼ exp(At) also has the following properties. φðt1 Þφðt2 Þ ¼ φðt1 + t2 Þ

(F.31)

φðtÞ1 ¼ φðtÞ

(F.32)

ðt xðtÞ ¼ φðt  t0 Þxðt0 Þ +

φðt  τÞBf ðτÞdτ t0

Example F.3 For the system defined in Example F.2, determine the state transition matrix.

Solution h i φðtÞ ¼ L1 ðsI  AÞ1  A¼

   2 3 s+2 3 , ðsI  AÞ ¼ 1 4 1 s+4

ðsI  AÞ1 ¼

  1 s + 4 3 ðs + 1Þðs + 5Þ 1 s + 2

3 s+4 3 6 ðs + 1Þðs + 5Þ ðs + 1Þðs + 5Þ 7 7 ¼6 5 4 1 s+2 ðs + 1Þðs + 5Þ ðs + 1Þðs + 5Þ 2

3 3 1 1 1 3 1 3 1 6 4 ðs + 1Þ + 4 ðs + 5Þ  4 ðs + 1Þ + 4 ðs + 5Þ 7 7 ¼6 4 1 1 1 1 1 1 3 1 5 + +  4 ðs + 1Þ 4 ðs + 5Þ 4 ðs + 1Þ 4 ðs + 5Þ 2

Taking Laplace inverse transform gives the state transition matrix. 2 3 3 t 1 5t 3 t 3 5t +  + e e e e 6 7 4 4 4 φðtÞ ¼ 4 41 1 1 t 3 5t 5  et + e5t e + 3 4 4 4 4

(F.33)

APPENDIX F State variable models and transient analysis

Example F.4 For the system defined in Example F.2, solve for x1(t) and x2(t) for a unit step input f(t). Assume zero initial conditions.

Solution XðsÞ ¼ ðsI  AÞ1 bFðsÞ ¼

   1 s + 4 3 0 1 ðs + 1Þðs + 5Þ 1 s + 2 1 s XðsÞ ¼

  1 3 sðs + 1Þðs + 5Þ s + 2

X1 ðsÞ ¼

3 sðs + 1Þðs + 5Þ

3 3 3 and x1 ðtÞ ¼  + et  e5t 5 4 20 X2 ðsÞ ¼

s+2 sðs + 1Þðs + 5Þ

2 1 3 and x2 ðtÞ ¼  et  e5t 5 4 20

Remark

Note that y(t) ¼ x1(t) + x2(t). Substituting for x1 and x2. 1 1 3 yðtÞ ¼  + et  e5t 5 2 10 Compare this with the answer in Example F.2.

F.4 The matrix exponential solution This section addresses the matrix exponential solution method. This method uses a clever application of matrix properties to obtain an efficient and simple solution technique [4]. For a model that is an inhomogeneous (f 6¼ 0) linear system (g ¼ 0) with an inhomogeneous term, (f) that is constant or can be represented as piecewise constant at each time step, the solution is as follows.

307

308

APPENDIX F State variable models and transient analysis

Using the linear system response given in Eq. (F.23), and iterative solution of X(t) at time step {i} is given in terms of its value at time step {i – 1} by   x ðiÞ ¼ eAΔt x ði  1Þ + A1 eAΔt  I f ðiÞ

(F.34)

where Δt ¼ time step between solution evaluations. I ¼ the identity matrix (value of 1 on the diagonal sand zeroes elsewhere). The matrix exponential is defined as eAΔt ¼ I + AΔt + ðAΔtÞ2 =2! + ðAΔtÞ3 =3! +

(F.35)

Note the similarity to the scalar exponential definition. Also note that it is necessary to evaluate eAΔt only once for use in every calculation of X(i). For simplicity, define C1 ¼ eAΔt

(F.36)

Now consider the other term in Eq. (F.34).

h i   A1 eAΔt  I ¼ A1 I + AΔt + ðAΔtÞ2 =2! + ðAΔtÞ3 =3! + …  I

Simplifying gives

h i   A1 eAΔt  I ¼ Δt I + ðAΔtÞ=2! + ðAΔtÞ2 =3! + …

(F.37)

It is also necessary to evaluate this quantity only once and apply it at every time step. Define this constant matrix as   C2 ¼ A1 eAΔt  I

(F.38)

We now may write the recursive solution as x ðiÞ ¼ C1 x ði  1Þ + C2 f ðiÞ

(F.39)

Thus, the solution involves repeated matrix multiplications using the same values of C1 and C2 at each time step. The matrix exponential method is exact for linear homogeneous models (initial value problems) or inhomogeneous models with a constant forcing vector. The matrix exponential method can provide approximate solutions by treating variable forcing vectors or nonlinear terms as piecewise constant terms. The accuracy of these solutions increases as time steps get smaller. Other numerical solution methods are considered in Sections F.6 and F.7.

F.5 Sensitivity analysis Sensitivity analysis is sometimes used to determine the importance of specified system parameters in determining the system response. Since sensitivity analysis is most easily performed with the matrix exponential approach, it is considered here before addressing other numerical methods later in this appendix [5].

APPENDIX F State variable models and transient analysis

The sensitivity for parameter, P, is given by Δ(x (t))/ΔP; that is, change in the response x(t) for a change in the parameter from a nominal value. The quantity, P, may appear in more than one matrix element. The sensitivity is a time-dependent vector quantity that gives the change in calculated response per unit change in the parameter of interest. Sensitivity information enables the analyst to evaluate the consequences of uncertainties in design parameters and to determine changes needed in specified design parameters in order to obtain a desired change in system response. Sensitivities may be obtained by “brute force”. That is, change a parameter in the model, repeat the simulation and see what happens. This means that the whole analysis must start over with a new coefficient matrix. Efficient sensitivity analysis is possible for linear model simulations using the matrix exponential approach. It avoids the need to re-analyze by using the same C1 and C2 matrices that were used to obtain the solution for x. The sensitivity to a matrix element, ajk, is dx/dajk. An equation for sensitivities is obtained by differentiating Eq. (F.7a) with respect to ajk as follows:       d dx dx dA ¼A + x dt dajk dajk dajk

Define the sensitivity vector as.

 Sjk ¼

dx dajk

(F.40)

 (F.41)

The differential equation of the sensitivity vector becomes.   dSjk dA x ¼ ASjk + dt dajk

(F.42)

Note that (dA/dajk) is a matrix with 1 in the j-th row and the k-th column and zeroes in all other matrix locations. Also, note that Eq. (F.42) has exactly the same form as Eq. (F.7a). Therefore, the solution of Eq. (F.42) for the sensitivity vector at time step, (i + 1), is   S jk ði + 1Þ ¼ C1 S jk ðiÞ + C2 dA=dajk x ði + 1Þ

(F.43)

Note that (dA/dajk) is a simple matrix with one non-zero element, and C1 and C2 are the same as used in the solution for X; and X(i + 1) is known from the solution for X. Therefore, the solution for the sensitivity equation only involves matrix multiplications of known quantities. It is common for a design parameter, P, to appear in several matrix elements. The sensitivity to P is obtained from.    dajk dx X dx ¼ dP dP j, k dajk

(F.44)

Sensitivity analysis can be tedious, requiring inspection of large sets of timedependent sensitivity results, but providing important and useful results for the reactor designer. If the system description is nonlinear, involving several parameters

309

310

APPENDIX F State variable models and transient analysis

(sometimes multiplying each other), the sensitivity analysis can be performed systematic perturbation of one or more parameters and making several simulation runs. Because of the availability of fast personal computing, the latter approach is popular for sensitivity analysis. Examples of typical parameters include, heat transfer coefficients, thermal properties, feedback coefficients, areas and masses used in thermalhydraulic equations, and time constants.

F.6 Numerical solutions of ordinary differential equations Numerous numerical techniques have been developed for solving ordinary differential Equations [6, 7]. There are three main concerns in numerical solutions: 1. Truncation: This is the error due to approximating an infinite series with a finite number of terms. 2. Round-off: This is the error due to the necessary carrying of a finite number of digits. 3. Stability: Stability is concerned with the tendency of errors to increase or decrease as the solution proceeds. Numerical techniques are classified as one-step methods or multi-step methods. In one-step methods, the approximation depends only on information available at time, t. In multi-step methods, the solution at time, (t + Δt), depends on information at more than one previous time steps. Some of the more important techniques are the following. One-step methods: 1. 2. 3. 4.

Euler Runge-Kutta order two Runge-Kutta order four Runge-Kutta Fehlberg

Multi-step methods: 1. 2. 3. 4. 5.

Adams-Bashforth Adams-Moulton Adams fourth-order predictor-corrector Milne Simpson

Available solution software packages use one or more of these techniques. Users do not need to program a solution method when using prepared computer software, but they should know what goes on in a computer solution. Two of the methods are described below to illustrate the general procedure. The basic idea is to estimate the average slope of the solution between time step i and time step (i + 1).

APPENDIX F State variable models and transient analysis

F.6.1 Euler’s method Euler’s method, the simplest formulation, uses the slope at time step i to estimate the change in going from i to (i + 1) using Eq. (F.7a). That is xði + 1Þ ¼ xðiÞ +

dxðiÞ ðΔtÞ dt

h i xði + 1Þ ¼ xðiÞ + AxðiÞ + f ðiÞ + gðiÞ Δt

(F.45)

(F.46)

F.6.2 Runge-Kutta order-two method In the Runge-Kutta2 method, the average slope between time step i and time step (i +1) is estimated and used to evaluate x(i + 1). The value of x(i + 1) is first approximated (as in Euler’s method) and used to obtain an estimate of the slope at time step i + 1 as follows: i dxði + 1Þ h  AxðiÞ + f ðiÞ + gðiÞ Δt dt

(F.47)

Then an estimate of the average slope for the interval between time step i and time step (i + 1) is S ¼ Average slope 



1 dxðiÞ dxði + 1Þ + 2 dt dt

(F.48)

d xðiÞ dxði + 1Þ Note that dt is known from the solution at time step i and dt is estimated using the condition at time step i. Then the estimated value of x(i + 1) is

x ði + 1Þ ¼ x ðiÞ + SΔt

(F.49)

F.7 Solutions for partial differential equations Some partial differential equations are amenable to analytic solutions. For example, as will be shown in Chapter 10, analytic solutions are possible for some heat transfer problems. But most real-world simulations require numerical solutions. For nuclear reactor simulations, the neutron diffusion equations have position, neutron energy and time as the independent variables. The solutions typically use discrete energy groups for the energy dependence and use numerical approximations for the position variables. The result is a set of ordinary equations with time as the independent variable. Techniques discussed above then provide the solution.

311

312

APPENDIX F State variable models and transient analysis

F.7.1 Examples of partial differential equations A few examples of partial differential equations encountered in engineering applications are stated. Neutron diffusion equation: The one-speed neutron diffusion equation, as a function of space (r) and time (t) is given by   X 1 ∂φ ðr Þφðr, tÞ ¼ Sðr, tÞ  r:Dðr Þrφ + a v ∂t

(F.50)

The parameters in this equation are defined as in Appendix C. The important variable in Eq. (F.50) is the space and time-dependent neutron flux φ (r, t). One-dimensional heat conduction: One-dimensional heat conduction through a thin plate as a function of distance from a surface (x) and time (t) has the form. ∂2 T 1 ∂T ¼ ∂x2 α ∂t

(F.51)

T (x, t) is the temperature variation across the plate and α is the thermal conductivity of the plate material. Two-dimensional heat conduction: The partial differential equation for heat conduction of a flat plate, for the temperature distribution is T(x, y, t), has the form. ∂2 T ∂2 T 1 ∂T + ¼ ∂x2 ∂y2 α ∂t

(F.52)

Three-dimensional heat conduction: The temperature distribution {T(x, y, z, t)}in an isotropic body (thermal conductivity at any point in the body is independent of the direction of heat flow) is given by [6, 8]. ∂2 T ∂2 T ∂2 T 1 ∂T + + ¼ ∂x2 ∂y2 ∂z2 α ∂t

(F.53)

One dimensional wave equation: If a string is fixed between two points and vibrates in a vertical plane, its vertical displacement x at time t is given by the differential equation. A

∂2 uðx, tÞ ∂2 uðx, tÞ ¼ ∂x2 ∂t2

(F.54)

F.7.2 Solution of partial difference equations using the finite difference method Advances in numerical computing and techniques facilitate accurate and fast solution of complex boundary value problems. These methods are used to solve problems encountered in diffusion theory, heat transfer, fluid mechanics, structural analysis, electrostatics, magnetism, and other engineering fields. This section provides a brief overview of the finite difference method (FDM) which is a popular and

APPENDIX F State variable models and transient analysis

well-developed technique for numerical solution of partial differential equations (PDEs). Some of the material in this section is adapted from Ref. [6, 8].

F.7.2.1 Introduction A simple and clear description of the FDM is given using the following twodimensional equation which is typical of the two-dimensional heat conduction problem [6, 8]. ∂2 T ðx, yÞ ∂2 T ðx, yÞ + ¼ qðx, yÞ ∂x2 ∂y2

(F.55)

The boundary condition has the form. T ðx, yÞ ¼ gðx, yÞ, a  x  b and c  y  d

(F.56)

In the FDM, derivatives are approximated by differences. For example, for a function of one variable, the approximations have the form (as given in Ref. [6, 8]) df ðxÞ f ðx + ΔxÞ  f ðxÞ d 2 f ðxÞ f ðx + ΔxÞ  2f ðxÞ + f ðx  ΔxÞ ¼ ¼ , and dx Δx dx2 ðΔxÞ2

(F.57)

F.7.2.2 Formulation of grids and nodes [8] With the choice of integers n and m, define the step sizes. Δx ¼

ðb  aÞ ðd  cÞ and Δy ¼ n m

(F.58)

The partitioning of the intervals [a, b] and [c, d] enables the formation of grids with vertical and horizontal lines and the definition of nodes as the intersection of the grid lines. Thus, xi ¼ a + iΔx and yj ¼ c + jΔy; i ¼ 0,1, 2,…, n; j ¼ 0,1, 2,…,m

(F.59)

An example of grids and nodes for n ¼ m ¼ 4 is illustrated in Fig. F.1. The intersections of the grid lines x ¼ xi and y ¼ yj are the nodes or the mesh points (i, j) of the finite difference grids.

F.7.2.3 FDM solution of the two-dimensional heat conduction problem [8] The two-dimensional heat conduction equation is given in Eq. (F.55) with boundary conditions in Eq. (F.56). The second partial derivatives of temperature T(x, y) are approximated as follows [8].         ∂2 T xi , yj T xi + 1 , yj  2T xi , yj + T xi1 , yj  ∂x2 ðΔxÞ2

(F.60)

        ∂2 T xi , yj T xi , yj + 1  2T xi , yj + T xi , yj1  ∂y2 ðΔyÞ2

(F.61)

The heat conduction Eq. (F.56) at node (xi, yj) is then written in discrete form as.

313

314

APPENDIX F State variable models and transient analysis

y

y4 y3 y2 y1 y0

x0

x1

x2

x3

x4

x

FIG. F.1 Grids and nodes for a two-dimensional finite difference mesh with n ¼ m ¼ 4. Adapted from R.L. Burden, J.D. Faires, Numerical Analysis, third ed, PWS-Kent Publishing Co., Boston, 1985.

      T xi + 1 , yj  2T xi , yj + T xi1 , yj ðΔxÞ

2

+

      T xi , yj + 1  2T xi , yj + T xi , yj1 ðΔyÞ

2

  ¼ q xi , yj (F.62)

For i ¼ 1, 2, …, (n-1) and j ¼ 1, 2, …, (m-1) with appropriate boundary conditions. Eq. (F.62) is rewritten as a difference equation (with indices i and j) in the form.

"

# 2      Δx 2 Δx  2 + 1 Ti, j  Ti + 1, j + Ti1, j  Ti, j + 1 + Ti, j1 ¼ ðΔxÞ2 q xi , yj (F.63) Δy Δy

For i ¼ 1, 2, …, (n-1) and j ¼ 1, 2, …, (m-1). For the case of n ¼ m ¼ 4, the boundary conditions are given by the following.   T0, j ¼ gx0 , yj , Tn, j ¼ g xn , yj , Ti, 0 ¼ gðxi , y0 Þ, Ti, m ¼ gðxi , ym Þ,

for j ¼ 0, 1,2, 3,4 for j ¼ 0, 1,2, 3,4 for i ¼ 1,2,3 for i ¼ 1,2, 3

(F.64)

Note that the Ti,j values are approximations to the actual values T(xi, yj) in both Eqs. (F.63) and (F.64). Also note that each Eq. (F.63) involves four adjacent nodes with respect to the central node (xi, yj). This approximation is often called the central difference method. Using the boundary conditions defined by Eq. (F.64), Eq. (F.63) represents a set of (n-1) x (m-1) linear equations in (n-1) x (m-1) unknowns. The matrix representation has the form. A T ¼q

(F.65)

Matrix A is sparse with elements in a diagonal band; T is the vector of temperature values at all nodes except the boundary values, and q is a vector of known energy

APPENDIX F State variable models and transient analysis

input to the system. The solution vector T is then solved iteratively (Eq. (F.65)) for a large sparse system of equations. The technique described here can be generalized to a three-dimensional body with appropriate nodalization and the knowledge of boundary conditions. See Ref. [6, 8] for more details of solutions to partial differential equations.

F.7.3 Solution of partial difference equations using the finite element method Solution of partial differential equations of irregular geometries and boundary conditions with derivatives of the spatial variables lead to increased complexity in problem formulation. Irregular shapes of boundaries and the need to approximate derivative conditions by finite differences require establishing proper grid points [6]. The finite element method (FEM) can overcome some of the numerical issues faced by the FDM, and has been used for multi-dimensional neutron diffusion solutions [9]. The FEM is used for solutions of problems in various engineering disciplines, including analysis of civil and aerospace structures, fluid mechanics, heat transfer, electrostatics, electromagnetics, wave propagation, and others [10].

Exercises F.1

Calculate the Laplace transform vector, X(s), for the state variable matrix equation. dx ¼ Ax + f dt  A¼

F.2

     1 2 1 3 , f¼ , X ð0Þ ¼ 3 4 2 4

A state variable matrix differential equation is defined as follows. dx ¼ Ax + b f dt  x¼

     1 2 x1 1 , A¼ , b¼ , 3 4 x2 2

Determine the transfer function vector. Simplify your answer. 3 2 X1 ðs Þ 6 U ðs Þ 7 7 6 7 6 4 X2 ðs Þ 5 U ðs Þ

315

316

APPENDIX F State variable models and transient analysis

F.3

Determine the transfer function vector for a system with an additional delay terms. dx ¼ Ax + d + b f dt         1 2 x x ðt  1Þ + 2x2 ðt  3Þ 1 x ¼ 1 A¼ d¼ 1 b¼ 3 4 3 4x1 ðt  2Þ x2

F.4

Consider the second order system with two inputs. dx 1 ¼ x1 + 2x2 + f1 + 2f2 dt dx 2 ¼ 3x1  4x2 + f2 dt

(a)

Compute the transfer function matrix 

G ðs Þ ¼

G11 ðs Þ G12 ðs Þ G21 ðs Þ G22 ðs Þ



where Gij ðs Þ ¼ Xi ðs Þ=fj ðs Þ

(b)

Compute the response of x1 and x2 due to an impulse in f1 for the above system. Assume that f2 ¼ 0.

F.5

Verify Eq. (F.37).

F.6

Calculate eAt for the following matrix. Truncate the solution after a few terms. 

A¼

1 2 3 4



F.7

What is the maximum number of sensitivities to matrix elements for a dynamic system with a (n x n) coefficient matrix? Explain.

F.8

Consider the following first-order differential equation. dx ¼ 3x + 5 dt

b. c.

Calculate the solution at t ¼ 1 with a Δt ¼ 0.5 using the Euler method and the Runge-Kutta2 method. Repeat the calculations for a Δt ¼ 0.2. Discuss your results.

F.9

Calculate the eigenvalues of the matrix used in Exercise F.6.

a.

APPENDIX F State variable models and transient analysis

References [1] R.E. Kalman, On the general theory of control systems, IRE Transactions on Control Systems (1959) 481–492. [2] T. Kailath, A.H. Sayed, B. Hassibi, Linear Estimation, Prentice-Hall, Upper Saddle River, NJ, 2000. [3] D.G. Shultz, J.L. Melsa, State Functions and Linear Control Systems, McGraw-Hill, New York, 1967. [4] S.J. Ball, R.E. Adams, MATEXP, a General Purpose Digital Computer Program for Solving Ordinary Differential Equations by the Matrix Exponential Method, Oak Ridge National Laboratory (ORNL), ORNL-TM-1933, 1967. [5] T.W. Kerlin, Sensitivities by the state variable method, Simulation 8 (6) (1967) 337–345. [6] R.L. Burden, J.D. Faires, Numerical Analysis, third ed, PWS-Kent Publishing Co, Boston, 1985. [7] G.W. Gear, Numerical Initial-Value Problems in Ordinary Differential Equations, Prentice-Hall, Englewood Cliffs, NJ, 1971. [8] Two-Dimensional Conduction: Finite-Difference Equations and Solutions. www. visualslope.com/Library/FDM-for-heat-transfering.pdf, 2018. [9] J.J. Duderstadt, L.J. Hamilton, Nuclear Reactor Analysis, John Wiley & Sons, New York, 1976. [10] Introduction to Finite Element Analysis (FEA) or Finite Element Method (FEM). https:// www.engr.uvic.ca/mech410/lectures/FEA_Theory.pdf, 2018.

317

APPENDIX

Matlab and Simulink: A brief tutorial

G

G.1 Introduction Simulink is a companion program to MATLAB, and is a software system for simulating dynamic systems, both linear and nonlinear. “It is a graphical menu-driven program that allows the user to model a system by drawing a block diagram on the computer screen and manipulating it to simulate various system responses”. It can be used for linear, nonlinear, continuous-time, discrete-time, single-input single-output (SISO) and multiple-input multiple-output (MIMO) systems. Linear time-invariant system models can be easily constructed using transfer functions [1]. State-space and transfer function models and their solutions are discussed in this tutorial. System response plots can be made during simulation and can be integrated with MATLAB commands. This tutorial discusses the following: • • • •

Features of Simulink. Example of using Simulink for a first-order system. Designing a proportional-integral (P-I) controller using Simulink. Solving state-space equations

G.2 Getting started with simulink Start the MATLAB program by double-clicking on the MATLAB icon. At the MATLAB prompt ▶ type simulink. • •

In the Simulink Library Browser window click on create a new model icon. This creates a new window that is untitled. Open the Simulink directory by clicking on the +Simulink icon. Several items are displayed, including the following: ▪ Continuous ▪ Discontinuous ▪ Discrete ▪ Look-up Tables ▪ Math Operations

319

320

APPENDIX G Matlab and Simulink: A brief tutorial

▪ ▪ ▪ ▪ ▪ ▪ ▪

Model Verification Ports & Subsystems Signal Attributes Signal Routing Sinks Sources User-defined Functions.

We will use some of these features to build our models and display the results. Explore the details of these features by clicking on each. Use the help feature in each to learn more about Simulink functions.

G.3 Simulation of a single-input single-output (SISO) system Consider the following example to illustrate some of the features of Simulink. •

Click on +Continuous. Drag the Transfer Fcn icon into the work space. A default transfer function is shown. Double click on this block. Change the second coefficient in the denominator to 0.02. Now the transfer function block has the form Gp ðsÞ ¼

•

•

•

•

•

12 s + 0:02

(G.1)

Click on the +Sources button and drag the function Step to the work area. Connect the step block to the Gp(s) block. In the step block, use step time equal to zero, so that there is no delay in the step input. Click on the +Sinks button and drag Scope and To Workspace buttons to the work area. Connect the output of the main block to both Scope and To Workspace blocks. If necessary, adjust the parameters of the Scope and To Workspace blocks. For example, change the name of the workspace block to level. Changes to any of the blocks may be made by double-clicking on the block and adjusting the parameters. Use ‘array’ for structure of data. Fig. G.1 shows the model diagram and the above blocks (this is not an executable block). After the model is completed you may save this in a file. Hint: Connecting one block to another: Select the source block; then hold down the Ctrl key while left clicking on the destination block. Run the simulation by clicking on simulation and then click start. Set the stop time equal to 300. By clicking on the Scope you may view the simulation. The output may also be plotted using the MATLAB plot command plot (tout, level) as shown in Fig. G.2. Note that the time points are stored in the array ‘tout’.

APPENDIX G Matlab and Simulink: A brief tutorial

12 s+0.02 Step

Transfer Fcn

Scope

level

To Workspace

FIG. G.1 Example Simulink model block diagram.

Example simulation for TF = 12/(s+0.02) 600

500

Level

400

300

200

100

0

0

50

100

150

200

250 300 Time (s)

350

400

450

FIG. G.2 Plot of the variable, level, as obtained from the Simulink simulation block.

500

321

322

APPENDIX G Matlab and Simulink: A brief tutorial

G.4 Simulation of a closed-loop system with P-I controller The Simulink model for controlling the variable, level, using a proportional-integral (P-I) controller is shown in Fig. G.3. The transfer function of the P-I controller is given by Gc ðsÞ ¼ Kp +

Ki Kp s + Ki ¼ s s

(G.2)

Kp ¼ proportional constant, Ki ¼ integral constant. Gc ðsÞ ¼

The summing block

0:01s + 0:0025 s

(G.3)

is available in the +Math library as Sum. Click on

this, and drag it to the work space and make connections as shown in Fig. G.3. If the summing block has a positive feedback, convert the feedback signal to a negative value by inserting the gain block with Gain 5 2 1. Fig. G.4 shows the response of the system to a unit step change in the input (or set point).

+ + Step

1

num(s) s

s+0.02

Transfer fcn

Transfer fcn1

–1 Gain

Scope

level To Workspace

FIG. G.3 Simulink model with level control using a P-I controller. Enter the values of Kp and Ki in the first transfer function block [num(s)]. Use Kp ¼ 0.01, Ki ¼ 0.0025.

APPENDIX G Matlab and Simulink: A brief tutorial

Level response with a P-I controller 1.4

1.2

level (ft)

1

0.8

0.6

0.4

0.2

0 0

50

100

150

200

250

300

350

Time (s)

FIG. G.4 Response of the level dynamics for a set point change of unity. Note that the the settling time (for the level to be within 5% of steady-state value) is about 200 s.

G.5 Solving linear differential equations using state-space models The state-space model of a linear dynamic system is written as. dx ¼ Ax + Bf dt y ¼ Cx + Df

(G.4)

x ¼ (n x 1) vector of state variables, y ¼ (r x 1) vector of outputs, f ¼ (m x 1) vector of inputs. A(n x n), B(n x m), C(r x n) and D(r x m) are matrices. In most applications matrix D is zero. Also, matrix B may be just a column vector of dimension (n x 1), with a single input variable. In the Simulink library browser, click on continuous and select the state-space model. Click and drag the state-space icon to the work space. If you double-click on this box, a state-space block parameter window appears. You may define the

323

324

APPENDIX G Matlab and Simulink: A brief tutorial

. x = Ax + Bu y = Cx + Du Step

State-Space

Scope

simout To Workspace

FIG. G.5 Block diagram of the state-space model.

matrices in this block, or you may simply define the names of the matrices (such as A, B, C, D) and specify them in the MATLAB workspace. If these matrices are saved in a file, load this file on to MATLAB. Define the initial condition vector as [0 0 0 0 0]. The length of the vector is equal to the number of state variables in the vector x. See remarks at the end for additional information on saving and plotting arrays. Connect step (from the sources), scope and To Workspace (from sinks) blocks to the state-space block as shown in Fig. G.5. Connecting one block to another: Select the source block, then hold down the Ctrl key while left clicking on the destination block. Verify each of the blocks for appropriate values of its parameters. Double-click on the To Workspace block. The output variables are defined in the vector simout. You may change the variable name. The decimation value allows you to write data at every n-th sample. Use default values for both decimation (1) and sample time (1). Use array for structure of data. After setting and applying these parameters, go back to the main work space. Click on Simulation on the control bar. Set start time (0.0) and stop time (10.0). If you are satisfied with all the settings click on Simulation and then click on Start. This completes the simulation. You may plot the vector simout using MATLAB.

Example G.1

A ¼ [0 1; -1 –1] B ¼ [0 1]’; the magnitude of the step may be changed by changing the entry in (B) C ¼ [1 0; 0 1] D ¼ [0 0]’ Initial Condition ¼ [0 0]’ Save your model (use a filename) for future applications.

APPENDIX G Matlab and Simulink: A brief tutorial

Simulation of a state-space model 1.2

1

X1 and X2

0.8

0.6

0.4

0.2

0.0

–0.2 0

2

6

4

8

10

Time (s)

FIG. G.6 Response of the state variables x1 () and x2 () for a step input.

Fig. G.6 shows the plots of the two state variables. The graph is created by using MATLAB commands for plotting. You may also save this plot in a file.

G.6 Computing step response using a transfer function The transfer function of a linear system is defined as the ratio between the Laplace transform of its output {Y(s)}and the Laplace transform of its input {X(s)}. Consider a transfer function of the form. GðsÞ ¼

Y ðs Þ b1 s + b0 ¼ XðsÞ a2 s2 + a1 s + a0

(G.5)

The numerator is a first order polynomial and the denominator is a second order polynomial. The following are the MATLAB commands for computing the unit step response of the above system; x(t) is a unit step input. num ¼[b1 b0]; den ¼[a2 a1 a0]; sys ¼ tf(num,den); step(sys, tfinal)

325

APPENDIX G Matlab and Simulink: A brief tutorial

Unit step response of a second order system: G(s) = 2/(s2 + s + 2) 1.4 1.2 1 Step Response

326

0.8 0.6 0.4 0.2 0 0

2

4

6 Time (s) (seconds)

8

10

12

FIG. G.7 Step response of a second order linear system with transfer functionG ðs Þ ¼ YX ððssÞÞ ¼ s 2 +2s + 2.

Remarks ‘num’ and ‘den’ are numerator and denominator polynomials defined by the coefficients. ‘tf’ is the transfer function that defines the system ‘sys’. After invoking the command ‘step(sys)’, the plot may be better presented by labeling the x- and y-axes, a title for the plot, and grids. ‘tfinal’ is the total time for computation.

Example G.2 Compute and plot the unit step response of the system defined by the transfer function GðsÞ ¼

Y ðsÞ 2 : ¼ XðsÞ s2 + s + 2

MATLAB Commands: num¼[2]; den¼[1 1 2]; sys ¼tf(num,den); step(sys, 100) The step response function is shown in Fig. G.7.

G.7 Computing eigenvalues and eigenvectors To determine the eigenvalues of a square matrix A, use the command eig(A). To find the eigenvalues and unit eigenvectors, use the command: [U, D] ¼ eig(A). The columns of U are unit eigenvectors and the diagonal elements of D are the eigenvalues of A. Try the matrix A ¼ [2 1;1 2] to compute its eigenvalues and eigenvectors.

APPENDIX G Matlab and Simulink: A brief tutorial

Remarks •

Plotting the simout variable #n, where n is the integer defining the variable number.

plot(tout,simout(:,n)) Two plots on the same graph: plot(tout, simout(:,1),’-‘, tout, simout(:,2),’–‘). (solid ‘-‘, and dashed ‘–‘lines). Same approach holds for multiple plots on the same graph. • A plot may be edited by using the Tools option or by clicking on the edit icon. You may want to change the line style (dotted, dashed, etc.), add symbols, or add color to the graph. • Saving the system matrices and other parameters. Define all the matrices, for example, A, B, C, and D. Then save as follows in MATLAB: Save filename A B C D Note that there are no commas separating the variable names. If you do not specify variable names, then save filename would save all the variables that are in the MATLAB workspace. Example: save F:\Example\SecondOrderSystem A B C D • Before starting the simulation load the stored file. You may want to check the parameter values in this file before running the simulation. Example: load F:\Example\SecondOrderSystem A B C D.

Useful Hint: You need to ensure that the length of the variable ‘tout’ (number of time points) matches the length of the system state variables. Do the following before you run the simulation. • • • • •

In your Simulink graphical model, click on ‘Simulation’. Then click on ‘Configuration Parameters’. On the left column of this window, select ‘Data Import/Export’ and click on it. Then go down to the ‘save options’. Disable the block marked ‘list data points to last’. This may not be seen in the newer version of MATLAB-Simulink.

This would ensure that you are not limiting the number of simulation points that are generated. For further reading refer to Ref. [2].

References [1] MATLAB and Simulink User’s Guide, MathWorks, Inc, Natick, MA, 2018. [2] H. Klee, Simulation of Dynamic Systems with MATLAB and Simulink, CRC Press, Boca Raton, FL, 2007.

327

APPENDIX

Analytical solution of the point reactor kinetics equations and the prompt jump approximation

H

H.1 Introduction This appendix addresses an approximate solution of the point reactor kinetics equations and compares it with an exact solution. The derivation of the approximate solution is tedious, but the final result is very simple. The prompt jump is the initial rapid response in reactor power following a step change in reactivity. A simple formula for the prompt jump is presented. The development uses one delayed neutron group model to illustrate the nature of the response without the complication of using multiple delayed neutron groups.

H.2 Analytical solution of the point kinetics equations Begin with the point kinetics equations with one group of delayed neutrons. dn ðρ  βÞ ¼ nðtÞ + λCðtÞ dt Λ

(H.1)

dC β ¼ nðtÞ  λCðtÞ dt Λ

(H.2)

Laplace transformation of Eqs. (H.1) and (H.2) yields the following: sN ðsÞ  nð0Þ ¼

ðρ  β Þ N ðsÞ + λCðsÞ Λ

(H.3)

β N ðsÞ  λCðsÞ Λ

(H.4)

sCðsÞ  Cð0Þ ¼

The initial condition,

dC dt ¼ 0,

yields the following: Cð0Þ ¼

β nð0Þ λΛ

(H.5)

329

330

APPENDIX H Analytical solution of the point reactor kinetics equations

Substituting in Eq. (H.4) gives. C ð sÞ ¼

β β N ð sÞ + nð0Þ Λðs + λÞ λΛðs + λÞ

(H.6)

Substituting Eq. (H.6) into Eq. (H.3) and solving for the Laplace transform of neutron density yields:


s

or

 ðρ  βÞ λβ β N ðsÞ ¼ nð0Þ + N ð sÞ + nð0Þ Λ Λ ðs + λ Þ Λ ðs + λ Þ

(H.7)


 ðρ  β Þ λβ β  N ð sÞ ¼ 1 + nð0Þ Λ Λðs + λÞ Λ ðs + λ Þ

(H.8)


s

This equation reduces to



 
 βρ λρ β s2 + λ + s N ð sÞ ¼ s + λ + nð0Þ Λ Λ Λ

(H.9)

The Laplace transform of the solution N(s)/n(0) is given by


 β N ð sÞ Λ
  nð0Þ ¼
β  ρ λρ nð0Þ s s2 + λ + Λ Λ s+λ+

(H.10)

The roots of the denominator polynomial are

0 1
 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
 1@ βρ βρ 2 λρA s1 , s2 ¼ +4  λ+  λ+ 2 Λ Λ Λ

(H.11)

The roots may be rewritten as s1 , s2 ¼

 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðβ  ρ + λΛÞ  ðβ  ρ + λΛÞ2 + 4λρΛ 2Λ

(H.12)

Rewrite Eq. (H.11) as follows:

1 v ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

 1B βρ βρ u 4λρ=Λ C C B u1 + s1 ,s2 ¼ B λ +  λ+  C u
2@ Λ Λ t β  ρ 2A λ+ Λ 0

(H.13)

The square root term in Eq. (H.13) is of the form (1 + x)1/2. This can be expanded using the binomial theorem to give ð1 + xÞ1=2 ¼ 1 + ½ x  1=8 x2 + … Let x ¼


4λρ=Λ λ+

 βρ 2 Λ

(H.14)

APPENDIX H Analytical solution of the point reactor kinetics equations

Then

11 0

 C 1B βρ 1 βρ B 4λρ=Λ C B CC B s1 ,s2 ¼ B λ +  λ+ B1 +
2 CC 2@ Λ 2 Λ @ β  ρ AA λ+ Λ 0

(H.15)

Rewriting Eq. (H.15) gives


  1 βρ 1 βρ 2λρ s1 , s2 ¼  λ +  λ+ + 2 Λ 2 Λ ðλΛ + β  ρÞ

(H.16)

βρ βρ λρ Because βρ Λ ≫λ, Λ + λ  Λ :Because, ðλΛ + βρÞ ≪1, higher order terms in the series expansion are small compared to the first term (n ¼ 1). The two roots, s1 and s2, may be approximated as.

s1 ¼ 

s2 ¼

ðβ  ρÞ Λ

λρ ðβ  ρÞ

(H.17)

(H.18)

The approximate Laplace transform solution is.


 β s+λ+ N ð sÞ Λ 
 ¼
ðβ  ρÞ λρ nð0Þ s s+ Λ ðβ  ρÞ

(H.19)

The time response n(t)/n(0) is obtained by taking the inverse transform of Eq. (H.19):

 nðtÞ ρ ðβ  ρÞ β λρ ¼ exp  t + exp t nð0Þ ðβ  ρÞ Λ ðβ  ρÞ ðβ  ρÞ

(H.20)

H.3 The prompt jump The second term represents the prompt jump in the response to a step change in reactivity. Since the response is very fast, the approximate instantaneous change is given β by ðβρ Þ. This fast initial response is called the prompt jump.

H.4 An example We now obtain two solutions for a + 10 cent reactivity step. The approximation derived above (Eq. (H.15)) provides the first solution. A solution without the simplifying assumptions (using actual values from Eq. (H.11)) provides the second solution. Both solutions use the following values for the model coefficients:

331

APPENDIX H Analytical solution of the point reactor kinetics equations

ρ ¼ + 10 cents ¼ 0:1 ð0:0067Þ ¼ 0:00067 β ¼ 0:0067 Λ ¼ 0:00001 s: λ ¼ 0:08 s1

Substituting these values in the approximate solution gives nðtÞ ¼ 1:111e0:0088t  0:111e603t nð0Þ

(H.21)

Now consider a solution without simplifying assumptions. The kinetic equations are solved for a 10 cent reactivity step and the following initial conditions Cð0Þ ¼

β nð0Þ ¼ 8375 nð0Þ λΛ

Laplace transforms are used to solve the equations. A bit of algebra gives the Laplace transform of the solution as. N ð sÞ 670:08 ¼ nð0Þ ðs  0:00888Þ ðs + 603:0889Þ

(H.22)

Inverting Eq. (H.22) gives nðtÞ ¼ 1:111e0:0088 t  0:111e603:09t nð0Þ

(H.23)

Fig. H.1 shows the response for both models. The figure shows only one response plot because the two solutions are almost identical (maximum difference  0.45%). 1.4

1.35

1.3

1.25 n(t)/n(0)

332

1.2

1.15

1.1

1.05

1 0

2

4

6

8

10

12

14

16

18

20

Time (s)

FIG. H.1 Prompt jump and steady state response of a zero-power reactor to a + 10 cents change in reactivity.

APPENDIX H Analytical solution of the point reactor kinetics equations

Exercises H.1. Verify Eqs. (H.11) through (H.15). H.2. The following data are used in analyzing the thermal fission of U-233 nuclei. Delayed neutron data for U-233 (thermal fission) Group decay 1 2 3 4 5 6

Constant, λi (s1) 0.0126 0.0337 0.139 0.325 1.13 3.50

Delayed Neutron Fraction (βi) 0.000224 0.000777 0.000655 0.000723 0.000133 0.000088

Mean neutron generation time, ^ equals; 105 s.

(a)

Calculate the total delayed neutron fraction (β), and the decay constant (λ) for the case of one delayed neutron group approximation.

(b)

Simplify the Laplace transform N(s)/n(0), and express it as a ratio of two polynomials. Simplify the transfer function using the above data and for a constant reactivity of ρ0 50.001.

(c)

Calculate the two roots of the denominator polynomial. Determine the time response n(t)/n(0) and calculate the stable reactor period (sec) for this case.

(d)

Make a semi-log plot of n(t)/n(0) versus time for the case when ρ0 50.001. Use a time scale to cover 10 s of response.

333

APPENDIX

A moving boundary model I.1 Introduction

I

Dynamic models often must represent phenomena in which properties along a channel change during a transient. Two approaches are available for modeling this situation. One nodal model approach uses fixed node sizes with update of node coefficients as conditions change. This approach is often used, but standard equation solvers cannot be used. Another approach is to represent nodal boundaries as model variables that change during a transient. In this case standard equation solvers can be used. The development of a moving boundary model for a subcooled channel node adjacent to a fuel node serves to illustrate the moving boundary modeling approach.

I.2 Development of a moving boundary model Consider the schematic shown in Fig. I.1. The downstream node is defined as a node in which boiling begins. Note that the boundary will move when there is a change in heat added to or removed from the coolant and when the downstream pressure changes and thereby changes the saturation temperature (and exit temperature) of the fluid. Proceed by identifying quantities that are constant, quantities that are dependent variables in other sub-systems and quantities that are to be dependent variables for the sub-system under consideration here. Quantities that are constant are as follows: ρf ¼ density of fuel S ¼ circumference of the fuel rod (area per unit length) U ¼ overall fuel-to coolant heat transfer coefficient ρc ¼ density of coolant A ¼ cross sectional area of the coolant channel Quantities that are either constant or variables in other subsystem models are as follows: Pf ¼ power supplied to the fuel (as provided by neutronic model) Win ¼ inlet flow rate to the channel θin ¼ coolant inlet temperature θout ¼ coolant outlet temperature (equal to the saturation temperature which is a function of pressure, a variable determined by a downstream model)

335

336

APPENDIX I A moving boundary model

θout

Boiling

Fuel L

Tf

SubCooled

θin FIG. I.1 A moving boundary model.

Cf ¼ fuel specific heat capacity Cc ¼ coolant specific heat capacity Quantities remaining unspecified are the variables to be determined in the subsystem under consideration. They are as follows: Tf ¼ fuel temperature L ¼ length of the channel (determines the boundary location of the sub-section under consideration) Wout ¼ coolant flow rate out of the sub-section θav ¼ average coolant temperature in the sub-section First note that Θav is given by θav ¼ ðθin + θout Þ=2

(I.1)

Both θin and θout are specified elsewhere. So θav is eliminated as a variable to be determined by the sub-system model under consideration. Next consider a coolant mass balance. ρA

dL ¼ Win  Wout dt

(I.2)

dL dt

(I.3)

or Wout ¼ Win  ρA

APPENDIX I A moving boundary model

Note that this equation indicates that the outflow decreases as the node length increases, an intuitive fact. Since L is a variable in the sub-system model, Wout is known if L is known. Now consider an energy balance for the fuel. The equation is as follows: Cf ρf S

or



d LTf ¼ Pf  U SL Tf  θav dt

 
d Tf dL + Tf Cf ρf S L ¼ Pf  U SL Tf  θav dt dt

(I.4)

(I.5)

This equation contains all known quantities, or quantities from other subsystem models except for Tf and L. S is the heat transfer area. Thus, Eq. (I.5) is an equation in two unknowns. An energy balance on the coolant provides the required additional equation. The energy balance for the coolant is as follows: ρc A Cc


d ðLθav Þ ¼ U SL Tf  θav + Win Cc θin  Wout Cc θout dt

Rewriting, Eq. (I.6) becomes. 

ρc A Cc

d θav dL + θav L dt dt




¼ U SL Tf  θav + Win Cc θin  Wout Cc θout

(I.6)

(I.7)

Every quantity in this equation is either constant or defined by other sub-system equations except for L. Eqs. (I.5) and (I.7) provide two nonlinear equations in two unknowns (L and Tf). So the sub-system model is complete. A complete system simulation would involve coupling the sub-system model developed here with other sub-system models and solving the overall model. Various solution methods, ranging from numerical solution of the nonlinear equations to linearization and using a linear solution method, are available. The development presented here illustrates that moving boundary models are nonlinear and rather complex.

337

APPENDIX

Modeling and simulation of a pressurized water reactor

J

J.1 Introduction This appendix describes the modeling and simulation of a pressurized water reactor (PWR). Two versions are presented: one a linearized model for an isolated core and one a nonlinear model for a nuclear steam supply system (NSSS) with U-tube steam generators, but no balance-of-plant components (turbine, condenser, reheaters, moisture separators, or feedwater heaters). Modeling and simulation of a nuclear power plant involves the following steps: 1. Define the purpose of the simulation (such as inherent behavior of an un-controlled plant, optimization of controllers, demonstration of load-following behavior, training, etc.) 2. Select a nodal structure for the model. 3. Specify the equations for each node. 4. Evaluate numerical values for the coefficients in the equations and formulate in matrix form. 5. Run the simulation (possibly using an equation solver such as MATLABSimulink). The purpose of the linearized isolated core model is to illustrate the development of model equations and their combination to create a model for a simple system representation and to show typical responses to external perturbations. Of course, this model is an abstraction because PWR cores do not operate in isolation. The purpose of the NSSS simulation is to illustrate the results achievable with a more realistic plant representation (nonlinear, six delayed neutron groups, U-tube steam generator, piping and plenums, but no BOP representation). A large system consists of multiple sub-systems. In order to facilitate effective dynamic modeling, it is prudent to develop models of individual components, and verify the models independently. These interchangeable component models are then integrated to form the whole plant model. The modeling of sub-systems or modules and their integration to form the entire system model is called modular modeling. This approach is common in practice, especially for a plant with multiple sub-systems, and a need for interchangeability of modules.

339

340

APPENDIX J Modeling and simulation of a pressurized water reactor

A whole plant simulation (NSSS and BOP) would include the following modular models: • • • • • • • • • • •

Reactor core neutronics and heat transfer Hot leg and cold leg volumes and connecting plena. Steam generator and its controller Pressurizer and its controller Steam chest High pressure and low-pressure turbines Steam reheaters Moisture separators Condenser and its controller Feedwater heaters and controllers Reactor power controller.

Other components include valves, pumps, and sensors. Chapters 3 (point kinetics equations), 10 (thermal-hydraulics) 12 (PWR characteristics) provide information needed for modeling.

and

J.2 Linearized isolated core neutronic model The point kinetics equations require a specification of the reactivity. In the isolated core model, the total reactivity is a sum of reactivity due to changes in fuel and moderator temperatures, and external reactivity. Assuming two coolant nodes with temperatures θ1 and θ2 (See Section J.4 below), the total reactivity and the feedback reactivity are expressed as δρtotal ¼ δρfeedback + δρexternal

(J.1)

δ θ1 + δ θ2 + αf δ Tf 2

(J.2)

δ ρf ¼ αc

where δρtotal ¼ total reactivity change δρfeedback ¼ feedback reactivity change δρexternal ¼ external reactivity change δρf ¼ fuel temperature reactivity feedback αc ¼ coolant temperature coefficient of reactivity αf ¼ fuel temperature coefficient of reactivity δθ1 ¼ temperature change in first coolant node δθ2 ¼ temperature change in second coolant node.

APPENDIX J Modeling and simulation of a pressurized water reactor

The reactor power and precursor concentration equations become (see Chapter 3).

 d δP β δ Ρ αc αf δρ ¼ + ðδ θ1 + δ θ2 Þ + δ Τf + λδC + ext 2Λ Λ Λ dt Po Λ Ρo

(J.3)

A single delayed neutron group is used in this simplified model.
 d β δP  λ δC δC ¼ dt Λ Ρo

(J.4)

The variables and parameters in the equations are defined in Table J.1. Table J.1 Isolated core variables and values of constants. Variables P0 δP δC θ1 θ2 Tf THL TCL ρext

¼ Nominal reactor thermal power ¼ Variation of reactor power about the nominal value ¼ Fractional variation in the delayed neutron precursor concentration ¼ Coolant node-1 temperature ¼ Coolant node-2 temperature ¼ Fuel node temperature ¼ Hot leg coolant temperature ¼ Cold leg coolant temperature ¼ External reactivity

Constant parameters β λ Λ αc αf mf mc1 mc2 m_ c Afc Ufc Cpf Cpc f (1-f)

¼ Total delayed neutron fraction (0.006898) ¼ Decay constant for one group delayed neutron precursors (0.0822 s1) ¼ Neutron mean generation time (1.79  105 s) ¼ Moderator (coolant) temperature feedback coefficient of reactivity (2.0  104 δρ/oF) ¼ Fuel temperature feedback coefficient of reactivity (1.1  105 δρ/oF) ¼ Mass of fuel (222,739 lbm) ¼ Mass of coolant in node-1 (12,341 lbm) ¼ Mass of coolant in node-2 (12,341 lbm) ¼ Total core coolant mass flow rate (1.5  108 lbm/h) ¼ Total fuel-to-coolant heat transfer area (59,900 ft2) ¼ Overall fuel-to-coolant heat transfer coefficient (200 BTU/lbm-oF) ¼ Specific heat capacity of fuel (0.059 BTU/lbm-oF) ¼ Specific heat capacity of coolant (1.39 BTU/lbm-oF) ¼ Fraction of nuclear power deposited in the fuel (0.974) ¼ Fraction of nuclear power deposited directly in the coolant (0.026)

Note: The coolant temperature coefficient of reactivity changes during operation of the reactor. Boric acid is typically used early after fueling to suppress the large available reactivity. This causes the temperature coefficient to be positive or less negative than with un-borated coolant. Therefore, the value cited above is representative of conditions after boron removal.

341

342

APPENDIX J Modeling and simulation of a pressurized water reactor

Note that the last term in Eq. ( J.3) is the applied (external) reactivity change. It is a forcing function to be specified by the analyst in a simulation.

J.3 Numerical values of coefficients in the isolated core neutronic model Inserting values from Table J.1 in Eqs. ( J.3) and (J.4) gives.


 d δP ¼ 385:36 δP=P0 + 0:0822 δC  0:6145 δTf  5:5866 δθ1 dt P0  5:5866 δθ2 + 5:59X104 δρext d ðδCÞ ¼ 385:36 δP=P0  0:0822 δC dt

J.4 Fuel-to-coolant heat transfer The core heat transfer model is a nodal approximation of the continuous temperature variation of the fuel and coolant in the axial direction. See Fig. J.1. for the lumped parameter approximation (see Chapter 10). The lumped parameter model of Fig. J.1 may be used for a number of axial and radial regions in the core. The simplest approximation uses a single fuel node coupled to two adjacent coolant nodes. The fluid transport in each section is represented by a “two well-stirred (well-mixed) tanks in series” approximation (Mann’s model, see Section 10.3). The resulting fuel temperature equation is as follows: d f Ρ0 δ Ρ Ufc Αfc ðδ Τ f Þ ¼  ðδ Τf  δ θ1 Þ: mf CΡf Ρ0 mf CΡf dt

(J.5)

In Eq. ( J.5), f is the fraction of the total reactor power deposited directly in the fuel. Note that the heat transfer time constant is denoted by τ, and it indicates the rate at which heat is transferred from fuel to coolant. 1 Ufc Αfc ¼ τ mf CΡf



sec 1



(J.6)

An estimate of τ is calculated using the data in Table J.2, and it has the following value. τ¼

mf CΡf 222739x0:059x3600 ¼  4 sec Ufc Αfc 200x59900

(J.7)

APPENDIX J Modeling and simulation of a pressurized water reactor

Hot Leg

θ2 Coolant Node-2 θ2

Direct Heating

Nuclear Power

Fuel Tf

THL

θ1

Driving Force For Heat Transfer

Coolant Node-1 θ1

TCL

Cold Leg

TP

FIG. J.1 One fuel-node, two coolant-node model of fuel-to-coolant heat transfer.

The model for heat transfer to and from the first coolant node is as follows:
  Ufc Αfc1 d ð1  f ÞΡ0 δ Ρ m + ðδθ1 Þ ¼ ðδ Τf  δ θ1 Þ + c ðδ ΤCL  δθ1 Þ: 2mc1 Cpc Ρ0 mc1 Cpc mc1 dt

(J.8)

In Eq. ( J.8), {(1-f)δP} is the fraction of the reactor power deposited in the coolant and is referred to as direct heating. Direct heating of the coolant is due to the deposition of energy from gamma radiation and particle (neutron) absorption in the coolant. The model for heat transfer to and from the second coolant node is as follows:
  Uf c Αf c2 ð1  f ÞΡ0 δ Ρ d mc + ðδ θ 2 Þ ¼ ðδ Τf  δ θ1 Þ + ðδ θ1  δ θ2 Þ: 2mc2 Cp c Ρ0 mc2 Cp c mc2 dt

(J.9)

Note that δTCL, is the cold leg temperature perturbation. It is a forcing function to be specified by the analyst in a simulation.

Remarks In the above equations, mc1 and mc2 are each chosen as half the total coolant mass. Similarly, Afc1 and Afc2 are each chosen as half the total fuel-to-coolant heat transfer area.

343

344

APPENDIX J Modeling and simulation of a pressurized water reactor

Table J.2 Neutronics and heat transfer parameters for a typical four-loop PWR [3]. Parameter

Value

1. Core diameter (inches) 2. Core height (inches) 3. First delayed neutron group fraction 4. Second delayed neutron group fraction 5. Third delayed neutron group fraction 6. Fourth delayed neutron group fraction 7. Fifth delayed neutron group fraction 8. Sixth delayed neutron group fraction 9. Total delayed neutron group fraction 10. First group decay constant (1/s) 11. Second group decay constant (1/s) 12. Third group decay constant (1/s) 13. Fourth group decay constant (1/s) 14. Fifth group decay constant (1/s) 15. Sixth group decay constant (1/s) 16. Moderator temperature coefficient of reactivity (1/∘ F) 17. Fuel temperature coefficient of reactivity (1/∘ F) 18. Mean prompt neutron generation time (sec) 19. Nominal power output (MWth) 20. Fraction of total power deposited in fuel 21. Coolant volume in upper plenum (ft3) 22. Coolant Volume in Lower Plenum (ft3) 23. Coolant volume in hot leg pipings (ft3) 24. Coolant volume in cold leg pipings (ft3) 25. Coolant volume in the core (ft3) 26. Mass of fuel (lbm) 27. Total coolant mass flow rate (lbm/h) 28. Effective fuel-to-coolant heat transfer area (ft2) 29. Specific heat capacity of fuel (BTU/lbm-∘ F) 30. Specific heat capacity of moderator (BTU/lbm -∘ F) 31. Overall fuel-to-coolant heat transfer coefficient (BTU/Hr -ft2  ∘ F)

119.7 144 0.000209 0.001414 0.001309 0.002727 0.000925 0.000314 0.006898 0.0125 0.0308 0.1140 0.307 1.19 3.19 -2.0  104 -1.1  105 1.79  105 3436 0.974 1376 1791 1000 2000 540 222,739 1.5  108 59,900 0.059 1.39 200

ρwater ¼ 45.71 lbm/ft3, Total mass of coolant, mc ¼ 24, 683 lbm. All units are in the English system: 1bm, sec, ft., F, BTU. 1 W ¼ 3.4121 BTU/h.

J.5 Numerical values of coefficients in the isolated core thermal-hydraulic model Data in Table J.1 were used to obtain numerical values for the three thermalhydraulics (fuel-to-coolant heat transfer) equations. The results are as follows:

APPENDIX J Modeling and simulation of a pressurized water reactor

d   δTf ¼ 241:87 δP=Po  0:2532 δTf + 0:2532 δθ1 dt d ðδθ1 Þ ¼ 2:437:8 δP=Po + 0:097 δTf  3:4731 δθ1 + 3:4731 δTCL dt d ðδθ2 Þ ¼ 2:437:8 δP=Po + 0:097δTf + 3:2791 δθ1  3:3761 δθ2 dt

J.6 State space representation of dynamic equations The above set of five first order linear differential equations may be written in a matrix form. Generally, the state space representation is given by the system equations as shown in Eq. (J.10). System Equations :

dX ¼ AX + BU dt

(J.10)

X (n x 1) is the vector of state variables U (p x 1) is the (external) input vector A (n x n) is the system matrix B (n x p) is the input matrix. The elements of the X vector are as follows: x1 x2 x3 x4 x5

¼ δP/P0 ¼ δC ¼ δTf ¼ δθ1 ¼ δθ2

The elements of the U vector are as follows: u1 u2 u3 u4 u5

¼ δρext ¼0 ¼0 ¼ δTCL ¼0

Eq. (J.10) was used in MATLAB-Simulink for computing the time response of the system for a given input function U(t), and for a specified initial value X(0). The system matrix A reflects the characteristics of the system. For the stability of the above linear system, all the eigenvalues of A must have negative real parts.

345

346

APPENDIX J Modeling and simulation of a pressurized water reactor

The system matrix, A, for the linearized, isolated core model is shown below: 3 385:36 0:0822 0:6145 5:5866 5:5866 7 6 385:36 0:0822 0 0 0 7 6 7 6 0 0:2532 0:2532 0 A ¼ 6 241:87 7 5 4 2:473 0 0:097 3:4731 0 2:473 0 0:097 3:2791 3:3761 2

(J.11)

Computation of the eigenvalues of matrix A provides an indication of the stability of the linear system. For a stable system, all the eigenvalues of matrix A must have negative real parts. This is left as an exercise. For the case of two inputs, the B matrix is as follows: 2

5:59 x 104 60 6 B¼6 60 40 0

0 0 0 0 0

0 0 0 0 0

0 0 0 3:4731 0

3 0 07 7 07 7 05 0

(J.12)

If we use one input at a time, either δρext or δTCL, then matrix B may be represented as a (5  1) column vector with a non-zero entry in the appropriate row location corresponding to that input. The simulations were performed for the case of one external perturbation at a time.

J.7 Simulation of PWR isolated core dynamics response The equations developed above represent an “isolated” PWR core model. That is, the steam generator and balance-of-plant are not modeled. The external reactivity and inlet coolant temperature are input quantities to be specified by the analyst. In a full-plant model, the coolant inlet temperature would be the result of steam generator and BOP dynamics. Two simulations of the linearized isolated core system were performed. Fig. J.2 through J.6 show the response to an external reactivity perturbation of 0.001 ( 14.5 cents). The plots show fractional reactor power, normalized delayed neutron precursor concentration, fuel node temperature, coolant node-1 temperature and coolant node-2 temperature. As expected, the reactor power shows a prompt jump. It then settles at  5.2% above its initial value. This indicates a steady-state power change of 0.36% / cent of reactivity insertion. The fuel and coolant temperature nodes show increases from their initial, steady-state values. The steady state change in the fuel temperature is  51 °F, giving a steady-state fuel temperature change of 3.5 °F / cent of reactivity. Note that the power rises rapidly then decreases to a new steady state as shown in Fig. J.2. Quenching the power increase is due to reactivity feedbacks, mainly due to the Doppler effect in the fuel. Note the fuel temperature transient in Fig. J.4. Figs. J.7 through J.11 show the responses of the core dynamics for a 5 °F increase in the cold leg temperature.

Fractional reactor power: delta-rho=0.001 0.16

0.14

0.12

deltaP/Po

0.1

0.08

0.06

0.04

0.02

0

0

10

20

30

40

50 Time (s)

60

70

80

90

100

80

90

100

FIG. J.2 Fractional power change for △ρ ¼ 0.001. Linearized Model Response: delta-rho=0.001

Total delayed neutron precursor conc. (fraction)

250

200

150

100

50

0

0

10

20

30

40

50 Time (s)

FIG. J.3 Fractional precursor concentration for △ ρ ¼ 0.001.

60

70

Linearized Model Response: delta-rho=0.001 60

Fuel node temperature (F)

50

40

30

20

10

0

0

10

20

30

40

50 Time (s)

60

70

80

90

100

FIG. J.4 Fuel node temperature for △ρ ¼ 0.001. Linearized Model Response: delta-rho=0.001

Node-1 coolant temperature (F)

1.5

1

0.5

0

0

10

20

30

40

50 Time (s)

FIG. J.5 Coolant node-1 temperature for △ρ ¼ 0.001.

60

70

80

90

100

Linearized Model Response: delta-rho=0.001 3

Node-2 coolant temperature (F)

2.5

2

1.5

1

0.5

0

0

10

20

30

40

50 Time (s)

60

70

80

90

100

FIG. J.6 Coolant node-2 temperature for △ρ ¼ 0.001. Linearized Model Response: delta-Tcl=5 deg F 0 –0.01

Fractional reactor power, delta-P/Po

–0.02 –0.03 –0.04 –0.05 –0.06 –0.07 –0.08 –0.09 –0.1

0

10

20

30

40

50 Time (s)

FIG. J.7 Fractional power change for △TCL ¼ 5 °F.

60

70

80

90

100

Linearized Model Response: delta-Tcl=5 deg F 0

Total delayed precusrsor conc. (fraction)

–50

–100

–150

–200

–250

–300

0

10

20

30

40

50

60

70

80

90

100

80

90

100

Time (s)

FIG. J.8 Fractional precursor concentration for △ TCL ¼ 5 °F. Linearized Model Response: delta-Tcl=5 deg F 10

Fuel node temperature (F)

0

–10

–20

–30

–40

–50

0

10

20

30

FIG. J.9 Fuel node temperature for △TCL ¼ 5 °F.

40

50 Time (s)

60

70

Linearized Model Response: delta-Tcl=5 deg F 4.5

4

Node-1 coolant temperature (F)

3.5

3

2.5

2

1.5

1

0.5

0

0

10

20

30

40

50 Time (s)

60

70

80

90

100

FIG. J.10 Coolant node-1 temperature for △TCL ¼ 5 °F. Linearized model Response: delta-Tcl=5 deg F 4

3.5

Node-2 coolant temperature (F)

3

2.5

2

1.5

1

0.5

0

0

10

20

30

40

50 Time (s)

FIG. J.11 Coolant node-2 temperature for △TCL ¼ 5 °F.

60

70

80

90

100

352

APPENDIX J Modeling and simulation of a pressurized water reactor

The increased moderator temperature results in a negative feedback reactivity, thus causing a decrease in the steady-state reactor power level as shown in Fig. J.7. The fuel node temperature follows the power change as shown in Fig. J.9. A decrease in fuel temperature causes a reactivity increase as needed to offset the negative reactivity from the increased coolant temperature. The two coolant node temperatures initially increase because of the increased cold leg temperature, and then decrease and settle to lower steady-state values as shown in Figs. J.10 and J.11. The final coolant temperature changes are positive, causing negative reactivity feedback. These two examples demonstrate the effect of external reactivity and temperature perturbations on core dynamic response, and the influence of the feedback effects to bring the process to a steady-state value without an active control intervention in the system.

J.8 Frequency response characteristics of reactor core dynamics For completeness of the discussion of the isolated core model, the frequency response was calculated. The fractional power-to-reactivity frequency response magnitude is shown in Fig. J.12A. As expected, both low frequency and high frequency responses have distinct break frequencies. The low frequency break is at  0.2 rad/s (due to temperature reactivity feedback) and the high frequency break is at  400 rad/s ( β/Λ). Fig. J.12A shows that the steady-state (low-frequency) fractional power is equal to 0.36% / cent of reactivity insertion. This matches with the fractional power change observed in the time domain response. This shows a one-to-one equivalence between time domain and frequency domain responses of linear dynamics. The power-to-reactivity frequency response phase angle is shown in Fig. J.12B. Note that as frequency approaches zero (steady-state condition), reactor power and reactivity are in phase with each other (phase angle ¼ 0). Fig. J.13A and B are the frequency domain plots of magnitude and phase angle of fuel temperature-to-reactivity dynamics, respectively. The magnitude plot indicates a low frequency (steady-state) change in fuel temperature of 3.5 °F / cent of reactivity. Again, these results show the equivalence between the time-domain and frequencydomain responses. Also, as frequency approaches zero (steady-state condition), fuel temperature and reactivity are in phase with each other (phase angle ¼ 0).

J.9 PWR NSSS dynamics The material in this section was drawn from Ref. [3]. The author of Ref. [3] was Dr. Upadhyaya’s graduate student. His work culminated decades of work on PWR subsystem and whole-plant modeling and simulation by a number of graduate students working with Dr. Kerlin or Dr. Upadhyaya at The University of Tennessee. This section presents simulation of a typical four-loop PWR with U-tube steam generators (UTSG). The simulation focused on coupling the reactor core dynamics

APPENDIX J Modeling and simulation of a pressurized water reactor

1.2

1

%Power/Cent

0.8

0.6

0.4

0.2

0 10–3

10–2

10–1

(A)

100

101

102

103

104

102

103

104

Frequency (rad/s)

40

20

Phase (deg)

0

–20

–40

–60

–80

–100 –3 10

10–2

(B)

10–1

100

101

Frequency (rad/s)

FIG. J.12A, B Fractional power-to-reactivity frequency response magnitude (A) and phase angle (B) plot for the isolated core PWR.

with the steam generator dynamics. In general, This NSSS model is nonlinear and requires the use of advanced simulation software tools, such as the MATLABSimulink platform. Following are the sub-system models considered in this simulation.

J.9.1 Neutronics The model used point kinetics with six delayed neutron groups. Reactivity inputs were from fuel temperature, coolant temperature, and Tavg control. The nonlinear terms consisting of products of reactivity and reactor power in the point kinetics equation were included.

353

APPENDIX J Modeling and simulation of a pressurized water reactor

4 3.5 3 dTf/dr (∞F/Cent)

354

2.5 2 1.5 1 0.5 0 10–3

10–2

10–1

(A)

100 100 Frequency (rad/s)

102

103

104

FIG. J.13A, B Fuel temperature-to-reactivity frequency response magnitude (A) and phase angle (B) plot for the isolated core PWR.

J.9.2 Core thermal-hydraulics The core thermal-hydraulic model used three axial sections, each formulated using the Mann’s model approach (one fuel node adjacent to two coolant nodes for each axial section). This resulted in nine differential equations for the core thermal-hydraulics.

J.9.3 T-average controller The model includes representation of the Tavg controller described in Section 12.8. It uses the average coolant temperature as the input to the controller that actuates reactivity introduced by control rods.

APPENDIX J Modeling and simulation of a pressurized water reactor

J.9.4 Piping and plenums Coolant in the upper core region, the hot leg piping and the steam generator inlet plenum was combined and treated as a well-stirred volume. Likewise, the coolant in the steam generator outlet plenum, the cold-leg piping and the water between the core inlet nozzle and plenum beneath the core was combined and treated as a well-stirred volume.

J.9.5 Pressurizer and its controller The formulation uses a pressurizer model as described in Sections 10.6 and a controller (see Section 10.6).

J.9.6 U-tube steam generator modeling and control The formulation uses a moving-boundary model as described in Section 10.7. A three-element feedwater controller model was used for the steam generator.

J.9.7 NSSS model The sub-system models were coupled to provide a 49-th order nonlinear model for the NSSS. Note that the balance-of-plant (turbine, condenser, reheaters, feedwater heaters and their controllers) are not included. Therefore, simulations are initiated by modulating the steam flow to the turbine. Feedwater flow is controlled by the three-element controller. But feedwater temperature is not a variable. It would be determined in the balance of plant. The author of the whole plant model presented above also developed another model that includes all balance-of-plant components, but it was not presented here in the interest of brevity.

J.10 Plant system parameters used in the models Neutronics and heat transfer parameters for a typical 1140 MWe four-loop PWR are given in Table J.2. The integration of core dynamic model, and models of UTSG and pressurizer requires the knowledge of additional parameters. Tables J.3 and J.4 list design parameters for a UTSG and a pressurizer [2].

J.11 NSSS simulated response to a steam valve perturbation The steam flow perturbation is caused by closing the steam valve by 15% from its nominal 100% power level condition. Figs J.14–J.17 show the response of selected state variables [2]. These are as follows: • •

Primary coolant average temperature Hot leg temperature

355

356

APPENDIX J Modeling and simulation of a pressurized water reactor

• • • • • • • • •

Cold leg temperature Average temperature of second fuel node (note that three fuel nodes are used in the neutronic equations) Reactor core normalized power Control rod reactivity perturbation Steam flow rate Feedwater flow rate Steam generator water level Pressurizer pressure Steam generator steam pressure.

Table J.3 Design parameters of a U-tube steam generator (UTSG) [3]. Parameter

Value

1. Number of U-tubes 2. Tube outside diameter (inch) 3. Tube metal thickness (inch) 4. Height of U-tubes (ft) 5. Total height of steam generator (ft) 6. Effective flow area in tube region (ft2) 7. Effective flow area in down-comer region (ft2) 8. Effective flow area in riser region (ft2) 9. Effective flow area in drum section (ft2) 10. Riser height (ft) 11. Primary water mass flow rate (lbm/h) 12. Volume of primary water in steam generator (ft3) 13. Specific heat capacity of primary water (BTU/lbm-°F) 14. Inlet temperature of primary water (°F) 15. Outlet temperature of primary water (°F) 16. Average pressure on the primary side (psia) 17. Average density of primary water (lbm/ft3) 18. Outlet steam flow rate (lbm/h) 19. Steam pressure (psia) 20. Steam temperature at saturation pressure (°F) 21. Inlet temperature of feedwater (°F) 22. Average density of secondary sub-cooled water (lbm/ft3) 23. Effective heat transfer area (ft2) 24. Film heat transfer coefficient of primary water in tubes (BTU/h-ft2-°F) 25. Film heat transfer coefficient of secondary sub-cooled water (BTU/h-ft2-°F) 26. Film heat transfer coefficient of secondary boiling water (BTU/h-ft2-°F) 27. Metal tube thermal conductivity (btu/h-ft-°F)

3388 0.875 0.05 35.54 67.67 60.87 32.0 48.7 110.74 9.63 3.939  107 1077 1.39 592.5 542.5 2250 45.71 3.731  106 849.7 521.9 434.3 52.32 51,500 4500 1972 6000 15

APPENDIX J Modeling and simulation of a pressurized water reactor

Table J.4 Design parameters of a pressurizer [3]. Parameter

Value

1. Operating pressure of primary side system (psia) 2. Saturation temperature (°F) 3. Steam volume at full power (ft3) 4. Water volume at full power (ft3) 5. Initial water level (ft) 6. Effective cross section area (ft2) 7. Average density of water (lbm/ft3) 8. Density of steam (lbm/ft3) 9. Electric heater output (kW) 10. Continuous spray flow rate (gpm) 11. Specific heat capacity of saturated water (BTU/lbm-°F) 12. Gain factor of the heater PI controller (kW/psi) 13. Time constant of the heater PI controller (s)

2250 653 720 1080 28.06 38.48 37.06 6.45 1800 1.0 2.12 250 900

FIG. J.14 Primary coolant temperatures for a 15% decrease in steam valve opening. Adapted from M. Naghedolfeizi, B.R. Upadhyaya, Dynamic Modeling of a Pressurized Water Reactor Plant for Diagnostics and Control, Research Report, University of Tennessee, DOE/NE/88ER12824-02, 1991.

357

FIG. J.15 Fuel temperature, reactor power, and reactivity from Tavg controller for a 15% decrease in steam valve opening. Adapted from M. Naghedolfeizi, B.R. Upadhyaya, Dynamic Modeling of a Pressurized Water Reactor Plant for Diagnostics and Control, Research Report, University of Tennessee, DOE/NE/88ER12824–02, 1991.

FIG. J.16 Steam generator steam flow rate, feedwater flow rate, and downcomer level for a 15% decrease in steam valve opening. Adapted from M. Naghedolfeizi, B.R. Upadhyaya, Dynamic Modeling of a Pressurized Water Reactor Plant for Diagnostics and Control, Research Report, University of Tennessee, DOE/NE/88ER12824–02, 1991.

APPENDIX J Modeling and simulation of a pressurized water reactor

FIG. J.17 Pressurizer pressure and steam pressure for a 15% decrease in steam valve opening. Adapted from M. Naghedolfeizi, B.R. Upadhyaya, Dynamic Modeling of a Pressurized Water Reactor Plant for Diagnostics and Control, Research Report, University of Tennessee, DOE/NE/88ER12824-02, 1991.

As the steam valve is closed by 15% the steam flow rate initially experiences a sudden decrease and the steam pressure initially increases. This results in the steam generator downcomer level to fall below its set point value, due to the shrink phenomenon. The three-element controller, by a combination of level mismatch and steam flow – feedwater flow mismatch, initially causes a sudden decrease in feedwater flow, then brings the level back to the set point. At steady-state both the steam flow rate and the feedwater flow rate reach the same value. As the steam flow rate and the feedwater flow rate decrease, the average primary coolant temperature increases initially, causing the reactor power to decrease. The T-average controller provides negative control rod reactivity to overcome T-average and set point mismatch. The reactor power settles at 88% with a negative external reactivity of 24 cents. The actual average temperature is off by about 1 °F at steady-state due to the dead-band in the control rod motion, causing the rod bank to remain stationary while the error is in the dead-band. During the early part of the transient, the average primary temperature increases causing a surge flow into the pressurizer. This results in a pressure increase and activates the spray controller to reduce the pressurizer pressure. Later in the transient, the pressure falls below the set point due to a decrease in the average coolant temperature. This activates the heater banks and the pressurizer pressure goes back to the set point.

359

360

APPENDIX J Modeling and simulation of a pressurized water reactor

The simulations show that a lumped parameter model effectively describes the PWR NSSS system and the transient behavior can be explained in a phenomenological fashion. It is suggested that the reader review Ref. [2–4] for detailed description of modeling and transient simulation.

Exercises J.1

Consider the system matrix, A, given by Eq. (J.23) describing core neutronics and heat transfer in a PWR. Calculate the eigenvalues of the system matrix. Is this dynamic system stable or unstable? Explain.

J.2

What is the physical significance of the first term on the right-hand side of Eqs. J.8 and J.9? Why does the denominator contain the factor, 2?

J.3

Show the BU term in Eq. J.10 for the case of a reactivity increase of 10 cents.

J.4

The dynamic modeling of a typical commercial PWR is presented in Ref. [4]. Make a critical review of this reference and state the important dynamic characteristics of a PWR. Compare the experimental and simulation results presented in the reference. Provide your comments.

J.5

How would the A matrix for the isolated core model change if the coolant temperature coefficient were positive (as soon after fueling)?

J.6

Examine each of the plots for the NSSS simulation. Explain the physical basis for the initial response for each plot.

References [1] T.W. Kerlin, Dynamic analysis and control of pressurized water reactors, in: C.T. Leondes (Ed.), Control and Dynamic Systems, vol. 14, 1978, pp. 103–212. [2] M. Naghedolfeizi, B.R. Upadhyaya, Dynamic Modeling of a Pressurized Water Reactor Plant for Diagnostics and Control, Research Report, University of Tennessee, DOE/NE/ 88ER12824–02, (1991). [3] T.W. Kerlin, E.M. Katz, J.G. Thakkar, J.E. Strange, Theoretical and experimental dynamic analysis of the H.B. Robinson nuclear plant, Nucl. Technol. 30 (September 1976) 299–316.

Further reading [4] S.J. Ball, Approximate models for distributed-parameter heat transfer systems, ISA Trans. 3 (1) (1964) 38–47.

APPENDIX

Modeling and simulation of a molten salt reactor

K

K.1 Introduction This appendix describes the modeling and simulation of a molten salt reactor (MSR). Since valid design parameters are available for the Molten Salt Reactor Experiment (MSRE), this system is used to demonstrate the modeling approach, transient response, and frequency response characteristics [1–5]. The model includes the reactor dynamics and the secondary salt loop. The goal is to demonstrate the modeling approach used for a liquid fuel reactor, specifically, the molten salt reactor (MSR) and the response of the reactor to external perturbations. The following topics are described briefly. The reader is encouraged to review relevant publications listed in References to this appendix. • • • • •

A brief description of the MSRE Systems that are modeled, model assumptions, and a few important dynamic equations Tables listing sub-system parameters—reactor core, primary loop, and the heat exchanger. Results of simulation. These include response to reactivity perturbation and frequency response characteristics of a molten salt reactor Summary, remarks, and suggested open problems.

K.2 Molten salt reactor experiment (MSRE) A schematic of the MSRE system is shown in Fig. K.1. [1]. The MSRE operated from June 1965 through December 1969. The design power rating of the MSRE was 10 MWth. The carrier salt circulated in the final pre-critical tests contained depleted uranium that was enriched with 61 wt% UF4-LiF eutectic salt [1]. The MSRE operated at a maximum power level of
8 MWth [1]. Material for this appendix are adapted from Refs. [5–7]. U-235 was later replaced with U-233, making it the first reactor ever operated primarily using U-233. The MSRE was operationally stable over the entire range of power (1, 5, and 8 MWth), with the measured dynamic characteristics showing close agreements with design estimates for both U-233 and U-235 fuel loadings. The reactor core was composed of a matrix of rectangular graphite blocks for neutron moderation. The molten, fuel-bearing carrier salt of 7LiF-BeF2-ZrF4-UF4 (65-29.1-5-0.9 mol%) at 632 °C was pumped through the core, generating 8 MW

361

362

APPENDIX K Modeling and simulation of a molten salt reactor

FIG. K.1 Schematic of the Molten Salt Reactor Experiment (MSRE) system [1].

of heat through fission, and raising the temperature of the salt by
22 °C. The fuel salt then circulated through a heat exchanger where it transferred the primary heat to a non-fueled secondary coolant salt of 7LiF-BeF2 (66–34 mol%) before returning to the core. The secondary coolant salt circulated through an air-cooled radiator and rejected heat to the atmosphere. Three flexible control rods consisting of gadolinium in the form of Gd2O3-Al2O3 ceramic clad with Inconel were driven on a chain and clutch system to raise and lower the rods. The rods were used to maintain power level and outlet fuel temperature. Criticality in MSRE was first achieved with U-235 fuel (33% enriched) and later with U-233 (91% enriched) fuel [1]. The power generated in the reactor was estimated using the product of mass flow rate of the salt, specific heat capacity (at constant pressure), and the difference in the inlet and outlet temperatures of the salt. Temperature measurements were made using thermocouples placed on the outside of piping in the secondary system. Mass flow rates of the salt streams did not change in MSRE. Only the temperature difference between the inlet and outlet salt changed during operation at different power levels.

K.3 Lumped parameter model of the MSRE K.3.1 Sub-system models and characteristics The MSRE dynamic model includes the following [5]: (i) (ii) (iii) (iv)

Reactor core neutronics and heat transfer Fuel salt circulation affecting delayed neutron production Heat exchanger from primary salt to secondary salt Heat exchanger from secondary salt to an air-cooled radiator.

APPENDIX K Modeling and simulation of a molten salt reactor

The last component is not included in the current simulation. See Ref. [5] for details. The two important characteristics of MSRE that contribute to the dynamic behavior are its heterogeneous core and the fluid fuel that circulates continuously. The fuel circulation reduces the delayed-neutron precursors decaying in the core, reduces the rate of temperature change in the fuel during a power change, and introduces delayed fuel temperature and neutron production effects. The heterogeneity of the core induces delayed feedback effects due to slow temperature variations in the graphite [2]. The main difference in the two fissile fuel types used in MSRE and considered here are the relative delayed-neutron fractions. U-233 has a total delayed neutron fraction β ¼ 0.00264, and for U-235, β ¼ 0.0065. Also, temperature changes in the salt have a much larger effect on the reactivity feedback than temperature variations in the graphite moderator. The heat transfer and heat capacities of the core components are such that fast changes in temperature of the fuel salt do not translate to fast changes in graphite temperature. Thus, short-term feedback is dominated by fuel temperature changes. When operating at very low power, MSRE tends to be sluggish. At these power levels, the fission chain reaction is controlled using control rods alone, and the dynamic behavior depends on the prompt neutron lifetime and the effective delayed-neutron fraction. When operating at high power, the kinetic behavior of MSRE is dependent on the fuel and graphite temperature coefficients of reactivity, the power density of the fuel, heat capacities of the various components, heat transfer coefficients, and transport lags in the salt circuits [2,3]. The MSRE system has an overall negative reactivity coefficient. The dominant effect here is the change in the fuel salt density due to temperature increase that reduces the amount of fissile material present in the core leading to more neutron leakage. The graphite also has a negative feedback coefficient, due to a combination of leakage and spectral effects.

K.3.2 Nodal model of the MSRE system Nonlinear models of the MSRE dynamics are outlined in this section. One-region and multi-region nodal models are described in Ref. [5]. The modeling approach is adapted from Kerlin et al. [3] and Singh [5]. The importance weighting for each of the regions is adapted from the final ORNL model [3], and elaborated originally for a 10 MWth U-235 fueled MSRE system in Ball and Kerlin [8]. This modeling approach uses nonlinear neutron kinetics equations [7]. The model is validated using experimental data from the MSRE [4,9]. A modified point kinetics model with six delayed-neutron groups is the basis for the one-region and multi-region models, and for U-233 and U-235 fueled cores. This neutron dynamics model accounts explicitly for delayed-neutron losses in the external loop through the heat exchanger. The neutron density and precursor concentrations are expressed as a fraction of nominal power. The nine-region model accounts for spatial variation in power production and feedback effects by using importance weighting. The masses of the nine regions along with residence times and power

363

APPENDIX K Modeling and simulation of a molten salt reactor

COOLANT

COOLANT

Tubes FUEL

FUEL

tau

364

tau FUEL

FUEL

FUEL

FUEL

Graphite #1 RADIATOR

CORE tau

Tubes

COOLANT

FUEL HEAT EXCHANGER

Indicates a pure time delay

COOLANT

tau

FIG. K.2 One-region lumped-parameter model of the MSRE [4,5].

generation fractions are adapted from Ref. [8]. No spatial variations of neutron flux are considered in the one-region model. Fig. K.2 shows the one-region model for the MSRE. Except for the core, the flow described in Fig. K.2 applies to both the models. The core heat transfer description for the one-region model consists of one graphite node and two fuel flow nodes. For the nine-region model, the core is divided radially from the center into nine regions for a total of nine graphite nodes and eighteen fuel flow nodes [8].

K.3.3 Equations describing neutronics and reactor heat transfer The neutron dynamics is described by the modified point reactor kinetics equations, similar to those employed in earlier studies of MSRE [3]. The main difference here is that the model is inherently nonlinear. Eqs. (K.1) and (K.2) show a system of seven, coupled, nonlinear, delayed-differential equations. X6 dnðtÞ ðρðtÞ  βÞ λ C ðtÞ + SðtÞ ¼ nðtÞ + i¼1 i i dt Λ

(K.1)

dCi ðtÞ βi Ci ðtÞ Ci ðt  τL Þeλi τL + ¼ nðtÞ  λi Ci ðtÞ  Λ τC dt τC

(K.2)

where n(t) is neutron density, Ci(t) represents the concentration of the i-th delayedneutron precursor (where, i ¼ 1…6), ρ(t) is the total reactivity as a function of time (input), βi is the delayed-neutron fraction of the i-th delayed group, β is the total

APPENDIX K Modeling and simulation of a molten salt reactor

delayed-neutron fraction, S(t) is the source perturbation term, τC is the fuel transit time in the core (8.46 s), and τL is the fuel transit time in the external loop (16.73 s). With the above modifications, the reactivity necessary for steady state operation ρo is non-zero, unlike in the case of solid-fuel reactors. It is obtained by setting the derivatives on the left side of Eqs. (K.1) and (K.2) equal to zero and solving for ρ(t 5 0). ρo ¼ β 

X6 i¼1

βi    1  1+ 1  eλi τL λi τ C

(K.3)

The ρo term is the reactivity change due to circulating fuel and accounts for delayedneutrons lost in circulation out of the core. It is dependent on the fissile material and has a value of ρo
0.00247 for U-235 fuel, and ρo
0.00112 for U-233 fuel. When the reactor model operates at steady-state, the natural reactivity feedbacks from the fuel and graphite sum up to this value. Thus, ρo can be viewed as the reactivity needed to achieve steady state in going from a stationary solid fuel to a circulating fluid fuel. The total reactivity for the system is expressed as: ρðtÞ ¼ ρo + ρfb ðtÞ + ρext ðtÞ

(K.4)

The feedback reactivity, ρfb(t), has contributions due to changes in fuel salt and graphite temperatures. The total heat from fission is deposited in the core components according to importance factors, and all the heat is carried away by the fuel salt which is the primary coolant. The temperature change equation for the fuel salt contains a fractional power generation term, a fuel-to-fuel node heat transfer term, and a fuel-to-graphite heat transfer term, as shown in Eqs. (K.6) and (K.7). The steady-state temperature of the graphite node is higher than the fuel nodes because all the heat generated in the graphite eventually needs to be carried away by the fuel salt.   n    Kg1 hAfg  n0 + Tg  Tf 1 mf 1 Cpf Kg1 + Kg2 mf 1 Cpf

 dTf 1 Wf  ¼ Tfin  Tf 1 + dt mf 1

K1 P0

 dTf 2 Wf  ¼ Tf 1  Tf 2 + dt mf 2

K2 P0

  n    Kg2 hAfg  n0 + Tg  Tf 1 Kg1 + Kg2 mf 2 Cpf mf 2 Cpf

(K.6)

(K.7)

In Eqs. (K.6) and (K.7), Wf is the mass flow rate of fuel salt, mf1 and mf2 represent the mass of fuel nodes ‘1’ and ‘2’ respectively, Cpf is the fuel salt specific heat capacity, K1 and K2 are the fraction of total power generated in fuel nodes ‘1’ and ‘2’, Kg1 and Kg2 are the fraction of power generated in the graphite transferred to each fuel node, hAfg is the product of area and heat transfer coefficient for the fuel-graphite interface, Po is the nominal power which multiplied with fractional neutron density n/no gives the instantaneous power, and the Ts represent the temperatures of the

365

366

APPENDIX K Modeling and simulation of a molten salt reactor

various nodes. Note that the direction of heat transfer depends on the instantaneous temperature of the various nodes. The graphite node contains a power generation term, since 7% of the power is deposited in the graphite, and a fuel to graphite heat transfer term, as in Eq. (K.8).   n P  0 n0  dTg  hAfg  + Tf  Tg ¼ Kg1 + Kg2 mg Cpg dt mg Cpg

(K.8)

Here, mg is the mass of the graphite node, Kg is the fraction of heat deposited in the graphite node, and Cpg is the graphite specific heat capacity. All other terms have the same meaning as before. Similar equations are constructed for each region in the nine-region model. Heat exchanger and radiator nodes use analogous equations as described above, but without the power generation terms. Steady-state temperatures throughout the model were adopted from MSRE design documents. In the one-region model, the mass of the various components is distributed equally among the nodes, and intermediate nodal temperatures are calculated by dividing the difference in the inlet and outlet temperatures equally.

K.3.4 Parameters used in simulation models A large number of documents and reports related to various aspects of the design and operation of the MSRE was published by ORNL over a period of ten years. These documents included MSRE parameters that evolved over this period. A consistent set of MSRE parameters is established as part of a research project at the University of Tennessee [5], and are representative of the reactor operation. These parameters correspond to the MSRE system when the experimental data were collected. Neutronics parameters used for the two fuel types (U-235, U-233) are presented in Table K.1. Delayed-neutron data for the two fuel types are given in Table K.2. The columns titled U-233 and U-235 correspond to values for the enrichments stated in Section K.2. A set of physical modeling parameters is listed in Table K.3. A digital version of the entire set of parameters is available at the University of Tennessee. Table K.1 Neutronics parameters for U-233 and U-235 [3,8]. Parameter

U-235

U-233

Prompt neutron lifetime Λ Delayed-neutron fraction β Core transit time τC External loop transit time τL Fuel salt reactivity coefficient αf Graphite reactivity coefficient αg

4  105 s 0.0065 8.46 s 16.73 s 8.71  105 δρ/°C 6.66  105 δρ/°C

2.4  105 s 0.00264 8.46 s 16.73 s 11.034  105 δρ/°C 5.814  105 δρ/°C

APPENDIX K Modeling and simulation of a molten salt reactor

Table K.2 Delayed-neutron group data for U-233 and U-235. Group

λi (s21)

U-233

U-235

1 2 3 4 5 6 Total

0.0126 0.0337 0.139 0.325 1.13 2.50

0.00023 0.00079 0.00067 0.00073 0.00013 0.00009 0.00264

0.000215 0.00142 0.00127 0.00257 0.00075 0.00027 0.00650

Table K.3 Model parameters used in MSRE simulation. Nominal power

8 MW(th)

Fraction of power generated in the salt Fraction of power generated in the graphite Mass of salt in core Mass of graphite in core Mass flow rate in the primary circuit Mass flow rate in the secondary circuit Specific heat capacity of fuel salt Specific heat capacity of graphite Specific heat capacity of coolant salt Mass of fuel salt in the heat exchanger Mass of coolant salt in the heat exchanger Heat transfer coefficient between fuel and graphite Heat transfer coefficient between fuel salt and metal Heat transfer coefficient between coolant salt and metal Core inlet temperature Core outlet temperature (nominal power) Heat exchanger inlet temperature for coolant salt Heat exchanger outlet temperature for coolant salt Time delay—core to heat exchanger Time delay—heat exchanger to core Time delay—heat exchanger to radiator Time delay—radiator to heat exchanger

0.93 0.07 1374 kg 3634 kg 162 kg/s 100 kg/s 1966 J kg1 °C1 1773 J kg1 °C1 2390 J kg1 °C1 348 kg 100 kg 0.036 MW °C1 0.648 MW °C1 0.306 MW °C1 632 °C 657 °C 546 °C 579 °C 3.77 s 8.67 s 4.71 s 8.24 s

K.4 Results of simulation of MSR dynamics This section presents the results of simulation of transient response of the MSRE and its frequency domain characteristics. Fig. K.3 shows the response of the U-233 fueled MSRE model to a step reactivity perturbation of +10 cents, using the one-region model.

367

368

APPENDIX K Modeling and simulation of a molten salt reactor

FIG. K.3 Transient response of reactor power for the U-233 fueled MSRE system to a + 10 cent step reactivity perturbation at 8 MWth power level.

The core power has a prompt jump immediately following the step insertion accompanied by a rapid increase in the temperature of the fuel salt. After reaching a peak in the first couple of seconds, the negative feedback reactivity of the fuel stops any further increase in power. During this period “cold” salt from the heat exchanger at a constant temperature continues to flow into the core. The negative feedback of the salt at this time is sufficient to counteract the step input, and the power begins to level off briefly before “warmer” fuel salt from the initial reactivity insertion reenters the core after making a loop through the heat exchanger. This increase in average core temperature introduces further negative feedback, thus decreasing the power. The transient responses of fuel node-2 and graphite node temperatures are shown in Fig. K.4. The power to reactivity frequency response (transfer function) of the MSRE system at 8 MWth is plotted in Fig. K.5. The frequency range is extended up to 1000 rad/s to illustrate the high frequency behavior of the MSR. The 1% power/cent plateau is clearly seen in the plot, as the frequency increases and the feedback effect decreases. The high frequency break point is around 200 rad/s which is similar to the the openloop response of the MSR, as seen in Chapter 4, Fig. 4.14A. A dip in the magnitude around 0.3 rad/s is attributed to fuel recirculation time (combined fuel core resident time and the loop circulation time). Furthermore, the low frequency response of the frequency domain magnitude plot has a value of
0.6% power/cent. This value matches with the steady-state value of reactor power (Fig. K.3) for a + 10 cent reactivity perturbation. Fig. K.6 shows the corresponding phase vs. frequency plot. At low frequencies, the phase angle goes to zero indicating that the reactor power and input reactivity change together. At high frequencies, reactor power lags external reactivity by 90 deg.

APPENDIX K Modeling and simulation of a molten salt reactor

FIG. K.4 Transient responses of fuel and graphite temperatures to a + 10 cent reactivity insertion.

FIG. K.5 Power-to-reactivity frequency response magnitude plot of the molten salt reactor at a nominal power of 8 MWth.

The frequency response analysis demonstrates the general similarity in dynamic behavior of reactor systems with feedback, from light water reactors to molten salt reactors. This analysis also serves as a verification of lumped parameter dynamic models for application to control design.

369

370

APPENDIX K Modeling and simulation of a molten salt reactor

FIG. K.6 Power-to-reactivity frequency response phase plot of the molten salt reactor at a nominal power of 8 MWth.

Remarks An important feature of reactor operation is the open-loop load-following capability of the MSR system to changes in demand power. The natural temperature feedback of the fuel salt and graphite are sufficient to control the reactor. This self-regulation can be leveraged in well-designed MSR systems to be used as on-demand and grid-stabilizing power generation. The reactor is inherently “self-regulating” in its response to external power demand. This feature is important in the design of truly “walk away” safe next generation reactors. Furthermore, temperatures in the salt carrying sections of the plant are observed to be varying around an average value. This indicates that the temperature of the hottest node decreases while the temperature of the coldest node increases during a load-following maneuver. Thus, salts in the system do not freeze when load-following leading to safe and stable operation [5]. The self-regulation and load-following characteristics are seen in both molten salt reactors and molten salt breeder reactors [8,9]. These are unique features of MSRs. For the interested reader, multi-physics modeling approach for MSRs is presented in Ref. [10].

Exercises K.1. Review references [7,10]. Compare the results using lumped-parameter modeling and more detailed multi-physics modeling approaches. Comment on the two approaches.

APPENDIX K Modeling and simulation of a molten salt reactor

References [1] P.N. Haubenreich, J.R. Engel, Experience with the molten-salt reactor experiment, Nucl. Appl. Technol. 8 (1970) 118–136. [2] P.N. Haubenreich, J.R. Engel, B.E. Prince, H.C. Claiborne, MSRE Design and Operations Report, Part III: Nuclear Analysis, ORNL-TM-0730, 1964. [3] T.W. Kerlin, S.J. Ball, R.C. Steffy, Theoretical dynamic analysis of the molten-salt reactor experiment, Nucl. Technol. 10 (1971) 118–132. [4] T.W. Kerlin, S.J. Ball, R.C. Steffy, M.R. Buckner, Experiences with dynamic testing methods at the molten-salt reactor experiment, Nucl. Technol. 10 (1971) 103–117. [5] V. Singh, Study of the Dynamic Behavior of Molten Salt Reactors, MS Thesis, University of Tennessee, Knoxville, 2019. [6] V. Singh, M.R. Lish, O. Chva´la, B.R. Upadhyaya, Dynamic modeling and performance analysis of a two-fluid molten-salt breeder reactor system, Nucl. Technol. 202 (1) (2018) 15–38. [7] V. Singh, A.M. Wheeler, M.R. Lish, O. Chvala, B.R. Upadhyaya, Nonlinear dynamic model of molten-salt reactor experiment—validation and operational analysis, Ann. Nucl. Energy 113 (March 2018) 177–193. [8] S.J. Ball, T.W. Kerlin, Stability Analysis of the Molten Salt Reactor Experiment, ORNL-TM-1070, Oak Ridge National Laboratory, 1965. [9] R.C. Steffy Jr., Experimental Dynamic Analysis of MSRE with 233U Fuel, ORNLTM-2997, Oak Ridge National Laboratory, 1970. [10] A. Cammi, C. Fiorina, C. Guerrieri, L. Luzzi, Dimensional effects in the modeling of MSR dynamics: Moving on from simplified schemes of analysis to a multi-physics modeling approach, Nucl. Eng. Des. 246 (2012) 12–26.

Further reading [11] H.G. MacPherson, Molten Salt Reactor Program: Quarterly Progress Report for the Period Ending July 31, 1960, ORNL-3014, 1960. [12] O.W. Burke, MSRE Analog Computer Simulation of a Loss-of-Flow Accident in the Secondary System, ORNL-CF-11-20, 1960.

371

Index Note: Page numbers followed by f indicate figures, t indicate tables, and b indicate boxes.

A Accidents in generation-II power reactors Chernobyl, 133–135 Fukushima Dai-ichi, 135 Three Mile Island, 132–133 nuclear reactor safety, 130 Accumulators, 247 Actuator, 91, 94, 225 Adjuster rods, pressurized heavy water reactors, 195 Advanced boiling water reactor (ABWR), 167–168, 247 Advanced CANDU Reactor (ACR), 248 Advanced flow measurement technology, 222 Advanced fuel CANDU reactor (AFCR), 248 Advanced gas-cooled reactors (AGRs), 6, 8 Advanced heavy water reactor (AHWR), 15, 250 Advanced reactors design, 243–244 dynamics of, 252–253 large developmental reactors gas-cooled reactors, 248–249 heavy water reactors, 250 liquid metal fast breeder reactors, 249–250 molten salt reactor, 250 large evolutionary reactors boiling water reactors, 247 pressurized heavy water reactors, 248 pressurized water reactors, 246–247 marketplace, 245 small reactors incentives, 251 water cooled, gas cooled, liquid metal cooled fast reactors and molten salt reactors, 251–252t using thorium, 244–245 AGRs. See Advanced gas-cooled reactors (AGRs) AHWR. See Advanced heavy water reactor (AHWR) Analytical solutions, point reactor kinetics equations, 39–40 AP600, 13 AP1000, 246 Asymptotic magnitudes, 286t

Atomic Energy Commission, 5 Automatic Depressurization System Reduces system, 247 Automobile motion, control of, 91 Auto regression (AR) model, 183–185 Auxiliary feedwater pumps, 133 Average power range monitor (APRM) detector signals, 185, 227, 290f Avogadro’s Number (AN), 259

B Balance-of-plant (BOP) system, 122–124, 125f, 139, 229–230 Basic reactor computing effective multiplication factor, 265–266 fast and thermal neutrons, 260–262, 261–263f multiplication factor and reactivity, 264–265 neutron interactions, 255–256 neutron lifetime and generation time, 264 neutron transport and diffusion, 266–267 nuclear fission, 258–260, 260t reaction rates and nuclear power generation, 256–258 specific power and neutron flux, 263–264 Beta particle, 215–216 Beyond Design Basis Accidents (BDBA), 131–132 Binomial theorem, 330 Bode plots, 281, 288–289 Boiling coolant, 115 Boiling water reactors (BWRs), 6, 10f, 115 advanced boiling water reactor, 168 advanced reactors, 247 advantages and disadvantages, 188 BWR-1, 167 BWR-2, 168 BWR-3, 168 BWR-4, 168 BWR-5, 168 BWR-6, 168 parameters, 173 control strategy, 187 dynamic models, 177–179, 178f, 180–181f features of, 168–171, 169–171f General Electric power plants, evolution of, 167

373

374

Index

Boiling water reactors (BWRs) (Continued) generation II reactor parameters, 238, 239t instrumentation, 225–229, 228–229f nuclear plant simulators, 205–208 on-line stability monitoring, 183–185, 186f power flow map and startup, 181–183, 183f power maneuvering, 185–186 reactivity and recirculation flow, 176 reactivity feedbacks in, 173–175 recirculation flow and jet pumps, 172–173, 172f safety, 187 stability problem and impact on control, 179–181 total reactivity balance, 177 Boltzmann transport equation, 17, 105 Break frequency, 285 Brute force, 309 Bubbler, 224, 224f BWRs. See Boiling water reactors (BWRs) Bypass installation, 220–221

C Calandria, 191, 194–195 Calder Hall Magnox reactor, 6 Canada Deuterium Uranium (CANDU) reactors, 6–8, 11f, 20–21, 78, 191 instrumentation, level sensors, 229–230 pressurized heavy water reactors, 197, 198f Chemical and Volume Control System (CVCS), 146 Chernobyl accident, 12, 133–135 Chicago Pile, 5 Closed-loop control systems, 89–91 configuration of, 28, 29f features of, 90 with P-I controller simulation, 322, 322–323f Closed-loop transfer function, 86, 95 Code of federal regulations (CFR) document, 132 Combined controllers, 95 Combined reactivity feedback, 79–80 Commercial nuclear power, evolution of, 5 Complete system simulation, 337 Computer simulations, 2 Computing effective multiplication factor, 265–266 Computing frequency response function, 283–288 Containment cooling engineered systems, 161 Control absorber rods, pressurized heavy water reactors, 195 Control rod drive mechanism (CRDM), 157–158 Control room simulators, 204 Control theory, 1, 53–54, 91–97 advanced controllers, 97 combined controllers, 95

complexity of, 90–91 control options, 100–101 differential controller, 94–95 integral controller, 94, 96–97 manual control, 91 on-off controller, 92 proportional controller, 93–96 steady-state power distribution control, 102–103 of zero-power reactor, 97–100 Convolution integral, Laplace transforms, 277 Coolant density, 150 Coolant temperature, 76 Core-exit thermocouples, 225 Core heat transfer model, 112, 342, 364 Core inlet temperature, 232 Core Makeup Tanks (CMT), 246 Core thermal-hydraulics, 354 Coriolis flowmeters, 223 Coupled neutronic-xenon transients, 66–67

D Damped oscillatory behavior, 288–289 Dead-band controller, 92, 286–287 Decay ratio, 185 Defense-in-depth approach, 161 Delayed neutrons circulation effect on, 31 decay constant, 20 from fission products, 18–20, 18f Pu-239, 19t t0025, 19t U-233, 19t for U-235, 19t nuclei excited photoneutrons, by gamma rays, 20–21 precursors, 31 transfer function, 28 Design Basis Accident (DBA), 131–132 Desk-top simulators, nuclear plant simulators, 203–204 International Atomic Energy Agency simulator, 205, 206–207f, 208–209 personal computers simulation, 204 Destabilizing effect, 85 Deterministic simulations, 131 Differential controller, 94–95 Differential pressure, 223–224, 223f Diffusion theory, 105, 266–267 Dissolved neutron absorber, 150 Dissolved poison, pressurized heavy water reactors, 195

Index

Distributed systems, frequency response analysis, 293 Doppler broadening, 73, 74f, 262 Doppler coefficient, 134, 173, 194 Doppler effect, 148–150 Downstream node, 335 Downstream transducer, 222–223 Dresden boiling water reactor, 6 Dynamic analysis, 300 Dynamic equations, of pressurized water reactor, 345–346 Dynamic models, 177–179, 178f, 180–181f, 335 Dynamic performance, 3

E Economic Safe Boiling Water Reactor (ESBWR), 247 Eddy current flowmeters, 223 Effective decay constant, 20 Effective multiplication factor, 266 Eigenvalues, 326–327 Eigenvectors, 326–327 Elastic collision, 255 Electrical power, 161 Electric Power Research Institute (EPRI), 132 Error signal, 91 Euler’s method, 311 Evolutionary power reactor (EPR), 13 Experimental Breeder Reactor (EBR-1), 5 Exponential coefficients, 35 Ex-vessel neutron detectors, 225, 234

F Fast fission factor, 265 Fast neutrons, basic reactor, 260–262, 261–263f Fast non-leakage probability, 266 Feedback coefficients, pressurized heavy water reactors, 194 Feedback control systems, p0145, 91 Feedback reactivity, 365 Feedback transfer functions, Laplace transforms, 276–277, 276f Feedwater controller, 187 Feedwater control system simulation, 121 Feedwater flow, pressurized water reactors, 160 Fermi-I, 130 Final SAR (FSAR), 132 Finite difference method (FDM), 109, 312–315 Finite element method (FEM), 109, 315–316 Fissile isotopes, cross section, 76f Fissile materials, 258

Fission, 255 cross sections, 76f detector, 215, 215f fragments, 258 gamma rays, 20 neutrons, 18–20 Fission product feedback, 78–79 Fixed-position burnable poison rods, 103 Flow sensors, 222–223 advanced flowmeters, 222–223 flow vs. pressure drop, 222, 222f Fluid fuel reactor response, point reactor kinetics equations, 46–47, 46–48f Fluid-fuel reactors, 30–32 Fluoride salt-cooled high-temperature reactor (FHR), 250 Four-factor formula, 266 Fourier analysis, 293–294 Frequency, 285 Frequency domain response, 81–83 Frequency range, 368 Frequency response analysis, of linear systems computing frequency response function, 283–288 distributed systems, 293 frequency response theory, 281–283, 281f, 283f measurements, 293–298 systems with oscillatory behavior, 288–290 systems with time delay dynamics, 291–293 Frequency response function, 29 Frequency response, point reactor kinetics equations, 41–46, 42–45f Frequency response theory, 281–283, 281f, 283f Fuel bundles, 193 Fuel circulation, 363 Fuel elements, heat conduction in, 111–112 Fuel temperature coefficient, 148–150 Fuel temperature feedback, 73–74 Fuel-to-coolant heat transfer, 114, 152 pressurized water reactor, 342–343, 343f, 344t Fukushima Dai-ichi power plant, 135 Full-length full-strength control rods, 102 Full-length part-strength control rods, 103

G Game simulators, 203 Gamma thermometers, 216–217, 217f Gas-cooled fast reactor (GCFR), 14 Gas-cooled reactors (GCRs), 6, 8, 248–249 General Electric (GE), 167 Generation I reactors, 5–6

375

376

Index

Generation II reactors accidents in chernobyl, 133–135 Fukushima Dai-ichi, 135 Three Mile Island, 132–133 advanced gas-cooled reactors, 6, 8 boiling water reactors, 6, 10f CANDU reactor, 6–8, 11f gas-cooled reactors, 8 high temperature gas-cooled reactors, 8 parameters boiling water reactor, 238, 239t pressurized heavy water reactor, 238–240, 240–241t pressurized water reactor, 237, 237–238t pressurized water reactors, 6, 7–9f RBMK reactors, 6, 12 sodium fast reactor, 9 steam generators, 6 vodo-vodyanoi energetichesky reactors, 6, 12 Generation III reactors AP600, 13 Kashiwazaki-6 ABWR, 13 Generation III + reactors, 13 Generation IV reactors gas-cooled fast reactor, 14 lead-cooled fast reactor, 14 molten salt reactor, 14 sodium-cooled fast reactor, 14 supercritical water-cooled reactor, 14 very-high temperature reactor, 14 Generation time, basic reactor, 264 Graphite node, 366 Gravity Driven Cooling System (GDCS), 247

H Half-power frequency, 285 Heat conduction, in fuel elements, 111–112 Heat exchanger model, 116–119, 366 Heat transfer, 344t Heavy water coolant, 193 Heavy water reactors, 103, 194–195, 250 High frequency response, 43 High temperature gas-cooled reactors (HTGRs), 8, 232–233 High-temperature in-core fission chambers, 232 High temperature reactor instrumentation, 230–234 high temperature gas-cooled reactor instrumentation, 232–233 level sensors, 230–234

liquid metal fast breeder reactor instrumentation, 231–232 molten salt reactor instrumentation, 233f, 234 Horizontal steam generator, 148 Hydroelectric plant, 101

I IAEA. See International Atomic Energy Agency (IAEA) In-core neutron detectors, 230, 292f In-Core Refueling Water Storage Tank (IRWST), 247 In-core sensors, 225, 227f Inelastic collision, 255 Inhour equation, 47–51, 50f Integral controller, 94, 96–97 Integral windup, 96 International Atomic Energy Agency (IAEA), 3, 203–204 simulator, 205, 206–207f, 208–209 Internet-based desk-top simulators, 210 Inverse multiplication factor, 54 Inverse transform multiple-input multiple-output, 303 using partial fractions, 274–275 Iodine-135 (I-135), 57 behavior equations, 58–59 fission product feedback, 79 steady state quantities of, 59 Ionization chambers, 214–215, 214f Isolated core dynamics response simulation, 346–352, 347–348f Isolated core neutronic model, 342 Isolation Condenser System (ICS), 247 Isotopes, 258, 262

J Jet pumps, 172–173, 172f Johnson noise, 221

K Kashiwazaki-6 ABWR, 13

L Laplace transformation, 27, 47–48, 269, 329 calculation, 270–272, 270–272b, 272t convolution integral, 277 definition, 269–270 feedback transfer functions, 276–277, 276f integral, 269–270

Index

inversion of inverse transform using partial fractions, 274–275 method of residues, 272–274 and partial differential equations, 278–280 transfer functions, 275–276, 275f Lead-cooled fast reactor (LFR), 14 Level sensors, 223–224 boiling water reactor instrumentation, 225–229, 228–229f bubbler, 224, 224f CANDU reactor instrumentation, 229–230 differential pressure, 223–224, 223f high temperature reactor instrumentation, 230–234 high temperature gas-cooled reactor instrumentation, 232–233 liquid metal fast breeder reactor instrumentation, 231–232 molten salt reactor instrumentation, 233f, 234 pressurized water reactor instrumentation, 225, 226–227f Light water chambers, pressurized heavy water reactors, 195 Light water reactors (LWRs), 7, 77–78 Linear differential equations, using state-space models, 323–325 Linearized isolated core neutronic model, 340–342, 341t Linear systems, frequency response analysis of computing frequency response function, 283–288 distributed systems, 293 frequency response theory, 281–283, 281f, 283f measurements, 293–298 systems with oscillatory behavior, 288–290 systems with time delay dynamics, 291–293 Linear time-invariant system, 304–305 Liquid coolant, heat transfer to, 112–114, 112–114f Liquid metal fast breeder reactor (LMFBR) instrumentation, 231–232 Liquid metal fast breeder reactors, 249–250 Liquid salt-fueled reactors, 250 Load following operation, 101–102 Local power range monitor (LPRM) detector signals, 227 Low-order model, boiling water reactors, 177 Lucens, 130 Lumped parameter models, 342 fuel element, heat conduction in, 111 molten salt reactor neutronics and reactor heat transfer, equations, 364–366

nodal model, 363–364 simulation models, parameters, 366, 366–367t sub-system models and characteristics, 362–363

M Magnetic flowmeter, 223 Magnox reactor, 6, 248 Mann’s model, 113, 113f MATLAB, 33, 319, 325 closed-loop system with P-I controller simulation, 322, 322–323f computing eigenvalues and eigenvectors, 326–327 single-input single-output, 320–321 state-space models, solving linear differential equations using, 323–325, 324–325f transfer function, computing step response using, 325–326 Matrix exponential solution, 307–308 Matrix-oriented solution techniques, 299 Maxwell-Boltzmann distribution, 75, 75f, 261, 261f Mechanical pumps, boiling water reactors, 172–173, 172f Micro-Simulation Technology, 205 Mid-frequency plateau, 42 Minimum phase systems, 285–288 Modal methods, 109 Moderator density, change in, 76–77 Moderator temperature feedback, 74–78 Modified point kinetics model, 363–364 Molten salt breeder reactor (MSBR), 250 Molten salt reactor (MSR), 14, 21, 101, 250, 361 experiment, 361–362 instrumentation, 233f, 234 lumped parameter model neutronics and reactor heat transfer, equations, 364–366 nodal model, 363–364 simulation models, parameters, 366, 366–367t sub-system models and characteristics, 362–363 simulation of, 367–370, 369–370f xenon in, 69 Molten salt reactor experiment (MSRE), 361–366, 362f Molten Zirconium, 133 Moving boundary model, 335–337, 336f MSR. See Molten Salt Reactor (MSR) Multi-group diffusion theory, 106–107

377

378

Index

Multiple-input multiple-output (MIMO), 27, 301–307 definition of, 301 linear time-invariant systems, 299 state transition matrix, 305–307 transfer function representation, 302–303 transient response, 303–305 Multiplication factor, basic reactor, 264–265

N Nautilus, 6 N-16 detectors, 225 Near-term task force, 135 Neutron, 133 absorptions, 57–58 and gamma ray detectors, 214–217 fission detector, 215, 215f gamma thermometers, 216–217, 217f ionization chambers, 214–215, 214f scintillation detectors, 216, 216f self-powered neutron detector, 215–216, 216f interactions, 255–256 moderator, 74–75 population, 17, 22 source, 53 Neutron diffusion equation, 106, 312 Neutron dynamics, 364–365 Neutron flux, 58–59 vs. reactivity, 53–54 Xenon steady-state poisoning, 62f Neutron generation time, 22–23 Neutronics, 17, 344t, 353, 364–366 Neutron lifetime, basic reactor, 264 Neutron transport, basic reactor, 266–267 Next generation nuclear plants (NGNP), 4 NGNP. See Next generation nuclear plants (NGNP) Nodal methods, 109, 363–364 NRX reactor, 130 n sequence, 296 Nuclear bombs, 18 Nuclear fission, basic reactor, 258–260, 260t Nuclear power plant, 5, 256–258 instrumentation, 54 actuator status sensors, 225 flow sensors, 222–223 level sensors, 223–224 neutron and gamma ray detectors, 214–217 pressure sensors, 221, 221f temperature sensors, 218–221 modeling and simulation of, 339

simulators BWR simulation, 205–208 control room simulators, 204 desk-top simulators, 203–204 Internet-based desk-top simulators, 210 PWR simulation, 205, 208f simulator games, 203 Nuclear reactor safety, 129 accidents, 130 assessment, 130–131 consequences, 136 generation- II power reactors, accidents in chernobyl, 133–135 Fukushima Dai-ichi, 135 Three Mile Island, 132–133 potential reactor accidents, analysis of, 131–132 principles, 129 Nuclear Regulatory Commission (NRC), 132 Nuclear steam supply system (NSSS) boiling water reactor, 239t CANDU-600 reactor, 240t pressurized water reactor, 237t core thermal-hydraulics, 354 neutronics, 353 piping and plenums, 355 simulated response, to steam valve perturbation, 355–360, 357–359f Tavg controller, 354 U-tube steam generator modeling and control, 355 Nuclei excited photoneutrons, 20–21 Numerical analysis, point reactor kinetics equations, 33–36

O OECD Nuclear Energy Agency, 14 Once-through steam generator (OTSG), 121, 123–124f pressurized water reactors, 155–156, 156f, 160 steam generators, 148, 149f One delayed neutron group model, 329 One-dimensional heat conduction, 312 One dimensional wave equation, 312 One nodal model approach, 335 One-region lumped-parameter model, 364f On-line stability monitoring, boiling water reactors, 183–185, 186f On-off controller, 92 Open-loop control systems, 89–91 Open-loop transfer function, 86 Ordinary differential equations, numerical solutions of, 310–311

Index

Euler’s method, 311 Runge-Kutta order-two method, 311 Oscillatory behavior, 288–290

P Partial differential equations laplace transforms and, 278–280 neutron diffusion equation, 312 one-dimensional heat conduction, 312 one dimensional wave equation, 312 three-dimensional heat conduction, 312 two-dimensional heat conduction, 312 using finite difference method, 312–315 grids and node formulation, 313 two-dimensional heat conduction problem, finite difference method, 313–315 using the finite element method, 315–316 Part-length control rods, 103 Passive Containment Cooling System (PCCS), 247 Personal computers simulation, desk-top simulators, 204 Perturbation equations, 27, 36, 37f Phase shift, 44 Photoneutron production, 20 Piping models, 115–116, 355 Platinum, 218 Plenum, 115–116, 355 Point reactor kinetics equations, 105, 329–333 analytical solutions, 39–40 calculation requirements, 108 computer software, 108 delayed neutrons from fission products, 18–20, 18f, 19t nuclei excited photoneutrons, by gamma rays, 20–21 diffusion theory, 105 dynamic equations, formulating, 21 finite difference methods, 109 finite element method, 109 fluid-fuel reactors, 30–32, 46–47, 46–48f frequency response function, 29 inhour equation, 47–51, 50f modal methods, 109 multi-group diffusion theory, 106–107 neutronics, 17 neutronic variable, 24–25 nodal methods, 109 numerical analysis, 33–36 perturbation form, 25–27 prompt jump, 331 quasi-static methods, 109 simulation methods, 33

sinusoidal reactivity and frequency response, 41–46, 42–45f small perturbations, solutions for, 40, 41f stability analysis, 29 feedback, 29–30 nonlinear systems, 30 spatial stability, 30 transfer functions, 27–28 zero power reactor, maneuver in, 36–38, 37–39f Power ascension, 55–56 Power coefficient, pressurized heavy water reactors, 195 Power flow map, boiling water reactors, 181–183, 183f Power maneuvering, 150–151, 185–186 Power range monitoring system (PRMS), 232 Power-to-reactivity frequency response, 163, 164f, 178, 368, 369–370f PRBS. See Pseudo-random binary sequence (PRBS) Pressure drop, 222 Pressure sensors, 221, 221f Pressure tubes, 191 Pressurized heavy water reactors (PHWRs), 191 advanced reactors, 248 characteristics, 191–194, 192–193f control systems, 196–197 generation II reactor parameters, 238–240, 240–241t heavy water reactors, temperature feedback in, 194–195 maneuvering, 197 neutronic features, 194 reactivity control mechanisms, 195 reactor dynamics modeling strategy, 198, 199f reactor power response, 199–201 void coefficient, 195 Pressurized water reactor (PWR), 6, 7–9f, 139, 339 advanced reactors, 246–247 characteristics, 139–141, 140f Chemical and Volume Control System, 146 control of, 160 control rod operating band and control rod operation, 156–158, 157f dynamic equations, state space representation of, 345–346 feedwater control for, 158–159, 159f frequency response characteristics, reactor core dynamics, 352, 353–354f fuel-to-coolant heat transfer, 342–343, 343f, 344t generation II reactor parameters, 237, 237–238t

379

380

Index

Pressurized water reactor (PWR) (Continued) instrumentation, 225, 226–227f isolated core dynamics response, simulation, 346–352, 347–348f isolated core neutronic model coefficients, numerical values, 342 linearized isolated core neutronic model, 340–342, 341t lumped parameter models of, 161 nuclear plant simulators, 205, 208f nuclear steam supply system core thermal-hydraulics, 354 neutronics, 353 piping and plenums, 355 simulated response, to steam valve perturbation, 355–360, 357–359f Tavg controller, 354 U-tube steam generator modeling and control, 355 plant system parameters, 355, 356t power maneuvering, 150–151 pressure controller, 146f reactivity feedbacks, 148–150 reactor core, 141, 142–144f safety systems, 161 steady-state programs coolant, energy change in, 153 development of, 153–154 equivalence, 153 fuel-to-coolant heat transfer, 152 with once-through steam generator, 155–156, 156f steam generator, heat transfer in, 152 steam generators horizontal, 148 once-through steam generator, 148, 149f U-tube steam generator, 146–148, 147f turbine control, 160 typical pressurizer, 141, 145f Pressurizer, 116, 117–118f Primary loop pressure, pressurized water reactors, 160 Probabilistic risk assessment (PRA), 132 Promethium-149 (Pm-149), 69–70 Prompt jump, 331, 368 Proportional controller, 93–96, 98–100 Proportional-integral (P-I) controller simulation, 322, 322–323f Protactinium-233, 250 Prototype fast breeder reactor (PFBR), 232 Pseudo random binary sequence (PRBS), 83, 293–294, 294f, 296

Pu-239 cross section, 76, 76f Pyrometry, 221

Q Quasi-static methods, 109

R Radiative capture, 255 Radiator nodes, 366 Radioactive debris, 135 Radioactive decay, 20, 57–58 Ramp reactivity, 199 RBMK reactors, 6, 12, 78, 133–134 Reaction rates, reactor, 256–258 Reactivity, 24 basic reactor, 264–265 boiling water reactors, 176 control mechanisms, pressurized heavy water reactors, 195 and power defect, power coefficient of, 81 Reactivity feedbacks, 103–104, 148–150 combined reactivity feedback, 79–80 destabilizing negative feedback, 83–85 fission product feedback, 78–79 on frequency response, 81–83 fuel temperature feedback, 73–74 moderator temperature feedback, 74–78 pressure and void coefficients, 78 Reactivity loss, 61 Reactivity, pressurized water reactors, 160 Reactor accident analyses, 2 Reactor coolant flow rate, 227 Reactor coolant pumps (RCP), 139 Reactor core, 141, 142–144f, 361–362 Reactor dynamics, pressurized heavy water reactors modeling strategy, 198, 199f reactor power response, 199–201 Reactor heat transfer, 364–366 Reactor noise analysis, 183–184 Reactor physics, six-factor formula of, 60 Reactor power influence, on reactivity, 80t Reactor power response, pressurized heavy water reactors, 199–201 Reactor regulation system (RRS), 196, 230 Reactor simulation efforts, 1–2 Reactor system models, 124–126 Reactor thermal-hydraulics balance-of-plant system, 122–124, 125f boiling coolant, 115 fuel elements, heat conduction in, 111–112 heat exchanger model, 116–119

Index

liquid coolant, heat transfer to, 112–114, 112–114f once-through steam generator, 121, 123–124f plenum and piping models, 115–116 pressurizer, 116, 117–118f reactor system models, 124–126 steam generator modeling, 119–121, 120–121f U-tube steam generator, 121, 122f Reactor thermocouples, 219 Real-world systems, models, 299 Recirculation flow, boiling water reactors, 172–173, 172f, 176 Rectangular matrix, 300 Reference model, 36 Refueling, 193 Residue theorem, inversion, 272–274 Resistance thermometers (RTDs), 218, 218–219f Resonance escape probability, 265 Resonance frequency, 288–289 Riser region, pressurized heavy water reactors, 196–197 Rod control cluster (RCC), 157–158 Rod controller, 156–158 Runge-Kutta order-two method, 311 Russian Submarines, 130

S Safety Analysis Report (SAR), 132 Safety injection systems, 246–247 Salt-cooled reactor, 250 Salt-fueled reactor, 250 Samarium-149 poisoning, 69–70 Scintillation detectors, 216, 216f SCWR. See Supercritical Water-Cooled Reactor (SCWR) Self-powered neutron detector, 215–216, 216f Sensitivity analysis, 308–310 SFR. See Sodium fast reactor (SFR) Shippingport reactor, 6 Shrink-and-swell, 158 Simout, 324–325 Simulation methods, point reactor kinetics equations, 33 Simulator games, 203 Simulin, 3 Simulink, 33, 319–320, 325 closed-loop system with P-I controller simulation, 322, 322–323f computing eigenvalues and eigenvectors, 326–327 single-input single-output, 320–321

state-space models, solving linear differential equations using, 323–325, 324–325f transfer function, computing step response using, 325–326 Single-input single-output (SISO) system, 320–321 Single radial node model, 111–112 Sinusoidal reactivity, point reactor kinetics equations, 41–46, 42–45f Six-factor formula, 266 Slowing down time, 264 SL-1 reactor, 130 Small modular reactors (SMR), 4, 13 Sodium (Na-23) atoms, 249–250 Sodium fast reactor (SFR), 9, 14–15, 249 Sodium flow rate, 232 Sodium Reactor Experiment (SRE), 130 Solid fuel reactor, 31 Sparse matrices, 35 Spatial oscillations, 68 Spatial stability, 30 Spectrum analysis, 183–184 Square matrix, 300 Stability analysis methods, 29–30 Stable reactor period, 50–51 Standby Liquid Control System (SLCS), 247 State-space models, solving linear differential equations using, 323–325, 324–325f State transition matrix, multiple-input multiple-output, 305–307 State variable models, 299–301 Steady-state neutron density, 53 Steady-state power distribution control, 102–103 Steady-state programs, pressurized water reactors coolant, energy change in, 153 development of, 153–154 equivalence, 153 fuel-to-coolant heat transfer, 152 with once-through steam generator, 155–156, 156f steam generator, heat transfer in, 152 Steam flow perturbation, 355–359, 358f Steam flow rate, 359 Steam generators, 6, 8–9f level control, pressurized heavy water reactors, 196–197 modeling, 119–121, 120–121f pressure control, 197 pressurized water reactors horizontal, 148 once-through steam generator, 148, 149f U-tube steam generator, 146–148, 147f

381

382

Index

Steam valve, 359 Steam-water mixture, 168–169 Stiff system, 33 Subcritical reactor inverse multiplication factor, 54 neutron flux vs. reactivity, 53–54 neutron source, 53 power ascension, 55–56 responses of, 54–55, 55f Sub-system model, 337 Supercritical Water-Cooled Reactor (SCWR), 14 System bandwidth, 288 System dynamics, 2–4 System frequency response, 288–290 System response plots, 319

T Tavg controller, 354 Teakettle model, 119, 121f Temperature control, 91 Temperature feedback effect, heavy water reactors, 194–195 Temperature sensors Johnson noise, 221 pyrometry, 221 resistance thermometers, 218, 218–219f thermocouples, 219–220, 219f thermowells and bypass installation, 220–221, 220f Thermal fission factor, 265 Thermal lifetime, 264 Thermal neutrons basic reactor, 260–262, 261–263f maxwell-Boltzmann distribution of, 261, 261f reactors, 17 U-235, 260t Thermal non-leakage probability, 266 Thermal reactors fuel temperature feedback, 73–74 moderator temperature feedback, 74–78 pressure and void coefficients, 78 Thermal utilization factor, 60, 265 Thermocouples, 219–220, 219f Thermowells, 220–221, 220f Thorium, 244–245 Three-dimensional heat conduction, 312 Three element controller, 187 Three-element feedwater control, pressurized heavy water reactors, 197 Three-element U-tube steam generator controller, 158, 159f Three Mile Island, 132–133

Time delay dynamics, frequency response analysis, 291–293 Time-domain response, 301 Total reactivity balance, boiling water reactors, 177 Transfer Fcn icon, 320 Transfer function, 27–28 computing step response using, 325–326 Laplace transforms, 275–276, 275f multiple-input multiple-output, 302–303 Transient response, multiple-input multiple-output, 303–305 Transmutation, 255 Tristructural Isotropic (TRISO) small fuel particles, 249 Turbine control system, 160 Turbine following boiler, 187 Turbine, pressurized water reactors, 160 Two-dimensional heat conduction, 312 Type-K thermocouples, 220 Type-N thermocouples, 220

U U-233, 367t U-235, 361 cross section, 76, 76f delayed-neutron group data for, 367t thermal neutrons, 260t U-238 neutron, 262f Ultrasonic flowmeter, 222–223 Unit power regulator, pressurized heavy water reactors, 196 Unit step function, 274 UO2 fuel pellet, 161 Upstream transducer, 222–223 Uranium, 193–194 Uranium oxide, 169 U.S. land-based power reactor, 6 U.S. Nuclear Regulatory Commission (NRC), 131–132, 135 U-tube steam generator (UTSG), 119, 120f, 121, 122f, 140, 355 pressurized heavy water reactors, 196–197 pressurized water reactor, 158–159, 159f, 356t steam generators, 146–148, 147f

V Vector-matrix formulations, 299 Very-High Temperature Reactor (VHTR), 14 VHTR. See Very-High Temperature Reactor (VHTR) Vodo-Vodyanoi Energetichesky Reactors (VVERs), 6, 12

Index

Void coefficient, pressurized heavy water reactors, 195 VVERs. See Vodo-Vodyanoi Energetichesky Reactors (VVERs)

W Well-stirred-tank formulation, 113, 116 Western Services Corporation, 209f, 210 Wide-range monitoring system (WRMS), 232 Windscale, 130

coupled neutronic-xenon transients, 66–67 fission product feedback, 79 losses, 57–58 in molten salt reactors, 69 poisoning, 60–61, 64–66 production, 57 steady state quantities, 59 Xenon-induced spatial oscillations, 68 Xenon steady-state poisoning, 60–61

X

Z

Xenon-135, 150 after shutdown, 63 behavior, after startup, 62–63 behavior equations, 58–59

Zero-power reactor, 1 control theory of, 97–100 maneuver, 36–38, 37–39f Zircaloy cladding, 161

383