Pagina principale Nuclear Reactor Engineering: Reactor Systems Engineering

Nuclear Reactor Engineering: Reactor Systems Engineering


Dr. Samuel Glasstone, the senior author of the previous editions of this book, was anxious to live until his ninetieth birthday, but passed away in 1986, a few months short of this milestone. I am grateful for the many years of stimulation received during our association, and in preparing this edition have attempted to maintain his approach. Previous editions of this book were intended to serve as a text for students and a reference for practicing engineers. Emphasis was given to the broad perspective, particularly for topics important to reactor design and oper­ ation, with basic coverage provided in such supporting areas as neutronics, thermal-hydraulics, and materials. This, the Fourth Edition, was prepared with these same general objectives in mind. However, during the past three decades, the nuclear industry and university educational programs have matured considerably, presenting some challenges in meeting the objec­ tives of this book. Nuclear power reactors have become much more complex, with an ac­ companying growth in supporting technology. University programs now offer separate courses covering such basic topics as reactor physics, thermal­ hydraulics, and materials. Finally, the general availability of inexpensive xv xvi Preface powerful micro-and minicomputers has transformed design and analysis procedures so that sophisticated methods are now commonly used instead of earlier, more approximate approaches.

Categories: Technique\\Energy
Anno: 1994
Edizione: 4
Editore: Springer US
Lingua: english
Pagine: 381 / 394
ISBN 13: 978-1-4615-2083-2
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Springer-Sc ience+Business Media, B.V.

© 1994 Springer Science+Business Media Dordrecht
Originally published by Chapman & Hall,lnc. in 1994
Softcover reprint of the hardcover 4th edition 1994

AII rights reserved . No part of this book may be reprinted or reproduced or utilized
hereafter invented, including photocopying and recording, ar by an information
starage or retrieval system, without permission in writing from the publishers.
Library of Congress Cataloging-in-Publication Data
Glasstone, Samuel , 1897Nuclear reactor engineering / Samuel Glasstone and Alexander
Sesonske. - 4th ed.
p. cm.
Includes bibliographical references and index.
Contents: v. 1. Reactor design basics - v. 2. Reactor systems
ISBN 978-1-4613-5866-4
ISBN 978-1-4615-2083-2 (eBook)
DOI 10.1007/978-1-4615-2083-2
1. Nuclear reactors. 1. Sesonske, Alexander , 1921- .
II. Title.
TK9202.G55 1994











Introduction, 488; PWR Nuclear Steam Supply
System, 489.
System Modeling, 490; System Interactions, 490;
Design Interactions and Intersystem Dependencies,
491; Sensitivity Analysis and Design Parameter
Interactions, 491; Feedback, 491; Optimization, 492;
Expert Systems and Artificial Intelligence, 493;
Computer Code Sources, 494.






Introduction, 495; Abstracts, 496; Nonarchival
Literature Sources, 496; Data Centers, 497; Data
Access Networks, 498; Information Age Challenges,








The Role of Energy Transport in Reactor Design, 501;
Thermodynamic Viewpoint, 502; Design Methods,


Fission Energy, 503; Spatial Distribution of Energy
Sources in Reactor Core, 505; Average and Maximum
Power in Single Fuel Channel, 507; Power and Flux
Flattening, 508; Other Heat Sources, 509.


Introduction, 509; Conduction of Heat, 509;
Convection of Heat, 511; Conduction with Convection
Boundary Conditions, 512; Radiation Heat Transfer,
515; Systems with Internal Heat Sources, 517;
Conduction in Irregular Geometries, 529; Transient
Heat Conduction, 530; Transient Heat Transfer, 534.


Introduction, 535; Laminar and Turbulent Flow, 535;
Heat- Transfer Coefficients of Ordinary Fluids, 539;
Heat-Transfer Coefficients of Gases, 541.


Introduction, 541; Heat Transfer in Reactor Rod
Bundles, 543.


Pool Boiling, 543; Flow Boiling, 546; Boiling Crisis,
548; Prediction of Burnout Conditions, 549; Boiling
Heat- Transfer Coefficients, 550.

Introduction, 552; Flow Pressure Drop, 552; Pressure
Drop in Turbulent Flow, 554; Velocity Head Losses,




556; Two-Phase Flow, 558; Two-Phase Pressure
Drop, 558; Limiting Flow with Compressible Fluids,
561; Natural Circulation Cooling, 562.




General Considerations, 565; Peaking and
Hot-Channel Factors, 566; Idealized Axial
Temperature Distributions, 567; Heat-Flux-Related
Limitations in Pressurized- Water Reactors, 573;
Factors and Subfactors, 576; Enthalpy Rise
Hot-Channel Factor, 578; Statistical Core Design
Techniques, 580; Boiling-Water Reactors, 581;
Gas-Cooled Reactors, 583; Fast Liquid-Metal-Cooled
Reactors, 584.









Production, 590; Isotopic Enrichment, 590; Isotopic
Feed Material and Separative Work Requirements,
591; Fabrication of Fuel Assemblies, 594.


Introduction, 594; Fuel Burnup, 595; Staggered
Refueling, 596; Fuel Management Terminology, 598.


The Initial PWR Core and Subsequent Reload
Patterns, 598; Burnable Absorber Rods, 602; PWR
Fuel Assembly Design Trends, 603; The Fuel Reload
Design Process, 604; Levels of Core Modeling, 604;
Neutronic Analysis Methods, 605; Reload Core
Pattern Design Considerations, 605; Automation and
Optimization, 607; The Haling Principle, 607.

Introduction, 608; Control Cell Core, 610; BWR Fuel
Assembly Design Trends, 610; BWR Spectral Shift
Operation, 611; BWR Core Modeling Methods, 611.





Introduction, 612; The Conversion Ratio, 613; The
Breeding Ratio, 615; Thorium Utilization, 618;
Plutonium Utilization, 619; Proliferation Risk, 620.


Introduction, 620; Time Value of Money, 621; Capital
Costs, 623; Operation and Maintenance Costs, 625;
Fuel Costs, 625; Electric Power Generation Costs,
626; Role of Rate Regulation, 627.




Introduction, 629; Design Approaches and Analysis,







Environmental Concerns, 632; Emissions from
Fossil-Fueled Power Plants, 633; The Greenhouse
Effect, 634; Overview of Nuclear Power Effects, 634.


Introduction, 635; Regulatory Bases for Exposure
Pathways, 636; Radiation Exposure Pathways, 636;
Radiation Levels "As Low As Is Reasonably
Achievable," 638.




Introduction, 640; Spent-Fuel Logistics, 640; Pool
Storage Capacity Enhancement, 641.




Introduction, 644; Retrievable Storage, 645;
Permanent Disposal, 646.

Lessons from the Oklo Reactor Waste, 648.





Introduction, 648; Head-End Treatment, 649;
Solvent-Extraction Separation Processes, 650; Other
Separation Processes, 654; Fuel Reprocessing Waste
Management, 656; Characteristics of Solidified
High-Level Waste, 657.

Sources of Radioactivity, 659; Reactor Radwaste
Systems, 660; Pressurized- Water Reactors, 661;
Boiling-Water Reactors, 663.

Condenser Cooling Requirements, 663; Regulation of
Thermal Discharges, 666; Treatment of Thermal
Discharge, 666.








Technological Risk and Public Perception, 673; Public
Acceptance of Nuclear Power Plants, 674; Defense in
Depth, 675.


Introduction, 675; Quality Assurance: Codes and
Standards, 676; Redundancy and Diversity, 677;
Inherent Reactor Stability, 678; Reactor Protection
System, 678; Reactor Trip Signals, 680; Shutdown
Cooling, 680.


Introduction, 681; The Emergency Core-Cooling
System, 682; Containment Systems, 685.


Categories of Abnormal Events, 691; Events of
Moderate Frequency, 693; Events of Low Probability,

Control Element Ejection, 698; Spent-Fuel Handling
Accident, 699; Loss-of-Coolant Accident, 699;
Emergency Core-Cooling Criteria, 703.





PWR Sequences, 706; BWR Sequences, 707.


Introduction, 707; Barriers to the Escape of
Radioactivity, 708; Radionuclide Importance Factors,
708; Fission Product Transport Overview, 708; Fission
Product Chemistry, 709; Fine Particle Dynamics, 712;
Aerosols, 713; Explosions, 714.


Introduction, 715; Licensing "Evaluation" Models,
716; System Modeling Methods, 717; Representative
Best Estimate System Modeling Codes, 720; Modeling
of Fluid and Structure Interactions, 721; Severe
Accident Modeling, 721.


Introduction, 723; Radiological Criteria of Site
Acceptability, 723; Radiation Dose Calculations, 725;
Emergency Response Planning, 730; Seismic Design
Criteria, 730; Other Natural Events, 731.


Introduction, 732; The Three Mile Island Accident,
732; Impact of the Three Mile Island Accident, 733;
Chernobyl, 734; Impact of the Chernobyl Accident,


Introduction, 738; Information and Analysis, 738;
Supporting Instrumentation, 739; Accident
Management Strategy Development, 739; Equipment
Modification and Personnel Training, 739.


Introduction, 739; Deterministic and Probabilistic
Analysis, 740; Elementary Binary State Concepts, 741;
Fault Tree Analysis, 743; Quantitative Fault Tree
Analysis, 744; Event Trees, 745; Computer Modeling,
747; Risk Assessment Studies, 748.

Introduction, 750; Design Certification, 751; Early Site
Permit, 751; Combined Construction and Operating
License, 752; State-Level Regulation, 752.





Introduction, 753; Protection against Sabotage, 753.









Introduction, 760; Reactor Vessel and Core, 760;
Control and Safety Systems, 765; Coolant Circulation
and Steam Generating Systems, 766.


Introduction, 769; Design Features, 770.


Introduction, 774; Core and Vessel, 776; Coolant
Recirculation System, 778; Control System, 779;
Feedwater Temperature and Fuel Cycle Length, 781.


Introduction, 781; System Features, 783;
Standardization and Certification, 783.


Introduction, 783; Design Specifications and Core
Features, 785; Heat Removal, 785; Control System,
788; Safety Features, 789; The Evolutionary CANDU









Generation Dispatching, 792; Operating Cycle Length
and Outage Management, 792.

Normal Operational Maneuvers, 793; The Control
Room, 797.





Introduction, 801; Expert Systems for Operator
Support, 801; Expert Systems Development, 802;
Neural Network Development, 802.


Introduction, 802; Plant Aging, 803; Life-Cycle
Management, 804.




Introduction, 805; Technical Options, 805;
Decommissioning Experience, 806.




What Is Needed, 808; Plant Size, 810; Advanced
Systems Common Design Features, 810; Types of
Passive Systems, 811.


Introduction, 812; Passive Features, 814; Other
Innovations, 815.


Introduction, 816; Natural Circulation, 816; Other
Passive Features, 819; Other Innovative Features, 820.


Introduction, 820; Modular Concept, 821; Fuel
Microspheres, 822; Prismatic Core, 822; Nuclear
Steam Supply System, 824; Gas- Turbine Option, 824;
Passive Features and System Safety, 824; Economic
Potential, 826.


Introduction, 827; Plant Description, 827; Fuel
System, 830; Concept Potential, 830.

Introduction, 832; The PIUS Reactor, 832; The Safe
Integral Reactor (SIR), 833.





Introduction, 834; The Size Issue, 834; Other Issues,





years of stimulation received during our association, and in preparing this
edition have attempted to maintain his approach.
perspective, particularly for topics important to reactor design and operation, with basic coverage provided in such supporting areas as neutronics,
decades, the nuclear industry and university educational programs have
matured considerably, presenting some challenges in meeting the objectives of this book.
Nuclear power reactors have become much more complex, with an accompanying growth in supporting technology. University programs now
offer separate courses covering such basic topics as reactor physics, thermalhydraulics, and materials. Finally, the general availability of inexpensive



powerful micro- and minicomputers has transformed design and analysis
procedures so that sophisticated methods are now commonly used instead
of earlier, more approximate approaches.
In light of this picture, giving priority to needed perspective, even at the
expense of some depth which is now available elsewhere, was considered
appropriate. Also, since it was important to keep the length of the book
about the same, necessary new material could only be accommodated by
deleting some old material. Significant new material has been added, particularly in the areas of reactor safety, fuel management, plant operations,
and advanced systems. However, material that generally continues to meet
the objectives of the book has been retained, both to preserve its flavor
and to keep the revision effort within reasonable bounds. Also, after the
passing of Dr. Glasstone, I felt it inappropriate to change the basic approach of the book.
Readers of the book will want to use computer-based methods to supplement the text material, as appropriate. Space did not permit a meaningful presentation of the methodology required. All problems listed at
the end of the chapters may be solved with hand calculations. Although I
have continued the use of SI units, as begun in the Third Edition, complete
adoption by industry has been slower than anticipated. Therefore, some
problems utilize English units.
A two-volume format has been adopted for this edition to provide readers with some flexibility. The chapters have been rearranged somewhat to
provide volume coherence, with basic material concentrated in the first
volume. An Instructor's Manual is also available for qualified instructors,
to be ordered directly from the publisher.
The suggestions made by A. L. B. Ho, L. E. Hochreiter, B. K. Malaviya,
V. H. Ransom, 1. R. Redding, G. R. Odette, and T. G. Theofanous are
gratefully acknowledged.
Thanks are due to the Chapman & Hall team that published the book,
particularly Marielle Reiter for production administration and Barbara
Zeiders for editorial assistance.
Finally, I wish to thank my wife for her help and encouragement during
the preparation of the book.
Alexander Sesonske
San Diego, California
March 1994



The Systems Concept,
Design Decisions, and
Information Tools

8.1. The availability of new, powerful digital computers in recent years
has resulted not only in increased sophistication in nuclear reactor engineering but also in the development of other disciplines involving systems
and optimal control theory. These subjects have potential applications in
reactor engineering. In nuclear power plant design and operation, decisions
must be made in applying engineering principles to the problems to be
solved. Since plants are complex, aids to the decision-making process are
essential. One aid is to recognize certain portions of the plant having a
common function as a system, a term that we will explain further shortly.
Decisions regarding dependencies between systems can be expedited by
this type of representation.
8.2. The systems concept as a decision tool has its roots in a subject
known as operations research [1], which evolved from the strategic planning
needs of World War II. As powerful computers became available, sophisticated methods for systems analysis, modeling, and simulation were developed for engineering and management applications. In this computer487


The Systems Concept, Design Decisions, and Information Tools

based information age, we are witnessing a trend toward greater and greater
general use of such systems-related tools for decision making. Computer
modeling of nuclear power plant behavior is essential for normal operation
as well as for the analysis of postulated accidents to meet licensing requirements (§12.231 et seq.).
8.3. Space here does not permit even an introduction to operations
research and modeling procedures. Reliability analysis applications will be
mentioned in Chapter 12. However, we do wish to alert the reader to the
possibility of using available operations research-based methods for the
analysis and design of nuclear reactor systems. A first step here is to identify
some typical systems.
8.4. In modern engineering practice which utilizes the power of the
computer to accomplish most tasks, information of various kinds is a necessary tool for decision making. Nuclear engineers have available to them
a variety of data bases, computer codes, and other information from many
sources. An introduction to these resources is useful as a complement to
the presentation of relevant reactor engineering topics in this book. Also,
we have drawn attention to appropriate information sources in various
other chapters.

8.5. A nuclear power plant consists of a number of systems which interact with one another in various ways. Knowledge of these intersystem
interactions is essential for the design and operation of the plant. The
systems themselves represent applications of the principles described in
the previous and subsequent chapters of this book. We believe that it is
useful at this point to introduce the system concept as applied to nuclear
power plants and to indicate some of the intersystem dependencies. As
energy transport, fuel management, safety, and other topics are presented
in subsequent chapters, the reader can then be alert to the integration of
the subject matter in plant applications.
8.6. The term system may be defined as a group of interrelated elements,
or subsystems, forming the "whole" in a regular, organized manner [2].
In engineering design, the concept lends itself to mathematical modeling
in which the interrelationships can be expressed formally for purposes of
design optimization. Nuclear engineers also use such a systems modeling
approach for probabilistic risk analysis (§12.206 et seq.).
8.7. Just what constitutes a system or subsystem depends on the need
for analysis of the interdependencies that may exist. A system may have



many subsystems or parts which, in turn, could have within them "subsubsystems," usually referred to merely as subsystems. Where does this
subdivision stop? The key question is whether the needs of the problem
at hand require further subdivision. If not, the constituent part is referred
to as a component.
8.8. Unfortunately, the term system is often used loosely to refer to any
collection of components even if the grouping is, in fact, a subsystem. For
example, manufacturers' literature for some pressurized-water reactors describes the reactor coolant "system" as within the Nuclear Steam Supply
System. Also described are auxiliary fluid systems and electrical, instrumentation, and control systems. Systems for waste processing, fuel storage,
and ventilation are also covered. To be consistent with such practice, we
will refer to both specific systems and subsystems as "systems" but indicate
their hierarchial relationship as appropriate. The nature of some of these
systems will be reasonably clear based on the material already presented.
Others will become clearer as subsequent chapters are read.

PWR Nuclear Steam Supply System
Reactor core system

8.9. Although not normally labeled as such, the reactor core is indeed
a system, with such components as the fuel assemblies and control elements. Here, we see an example of the need sometimes to define the
boundaries of a system arbitrarily since the control elements could be
included in the control system.
Reactor coolant system

8.10. Strictly speaking, the reactor coolant system includes those components associated with the flow of the primary coolant, such as the coolant
pumps, pressurizer, and associated piping and valves. However, the reactor
vessel and steam generators could also be included arbitrarily.
Instrumentation and control system

8.11. This system includes the following major subsystems:


Nuclear instrumentation that indicates power level
In-core instrumentation to provide nuclear flux distribution
Process instrumentation for nonnuclear measurements in stearn system supply components
Reactor protection system for initiating safety in response to abnormal
conditions (§12.22)
Control room (§14.25)


The Systems Concept, Design Decisions, and Information Tools

Engineered safeguards system

8.12. Countermeasures in the event of an accident are provided for by
this system. Included in this system are several subsystems, such as the
safety injection systems, containment sprays, the emergency feedwater
system, and other safety features (§12.33 et seq.).
Other systems

8.13. The Nuclear Steam Supply System also includes a number of other
subsystems that we need not consider here, such as those for auxiliary
fluids, containment, fuel handling, waste management, and chemical control.
System Modeling

8.14. The usefulness of the systems concept as a decision tool is enhanced if the system and its behavior can be described in a clear manner.
Graphical representation is a classical approach. However, with a wide
variety of computers now available, some type of representation that permits mathematical manipulation is preferred. This could take various forms,
which could involve graphics, analytical descriptions, or combinations of
each. For example, in risk management, graphical representation is useful
but manipulation follows the rules of Boolian algebra (§12.218).
8.15. System behavior is normally described by quantitative relationships. For example, coolant circulating pumps may be described in terms
of performance specifications such as flow rate, pressure differential, etc.
On the other hand, some systems may involve qualitative or subjective
inputs in addition to quantitative relationships. For example, in developing
a model for a radioactive waste storage facility, technical requirements
such as the desired amount and activity of radioactive isotopes are essential.
However, there is also the likely need to input items such as local acceptance considerations and regulatory requirements. Economic factors may
also require input into the model.
8.16. In analytically modeling a system in such a way that associations
between systems can be manipulated by computer, quantitative relationships can be handled in a straightforward manner. However, to permit
similar computer manipulation of subjective modeling inputs, they must
be quantified in some way, often a significant challenge.
System Interactions

8.17. The systems concept is a useful design tool for the analysis of how
component and subsystem failures can propagate and affect performance.

The Computer As a Decision Tool


For example, in a PWR, a reactor coolant pump could fail as a result of
an electrical fault. The resulting reduction in flow would, in turn, raise the
core water temperature, possibly cause boiling, then raise the fuel temperature, all of which would affect the reactivity as determined by the
appropriate coefficients. This, and similar chains of events, can be treated
analytically. Thus, a systematic study of intersystem dependencies provides
the designer with a thorough examination of various modes of system
response which would not otherwise be apparent.
8.18. Fault tree and event tree analyses (§12.213) are related analytical
procedures for assessing the reliability of safety-related systems. As will
be discussed in Chapter 12, these approaches are valuable decision tools
for identifying "weak links" in intersystem functional dependencies.
Design Interactions and Intersystem Dependencies
8.19. In the design of various parts of the Nuclear Steam Supply System
(NSSS), the effect of one parameter on another must be considered. For
example, in a PWR, the fuel rod diameter and spacing affect both the
moderator/fuel ratio and the coolant pressure loss through the core (§9.110).
Throughout the NSSS, there are numerous design parameters which form
an interrelated matrix which must be addressed systematically during the
design process in order to develop specifications. A simplified indication
of the need for integrating various parametric contributions in the design
process is shown in Fig. 8.1. To simplify the figure, we have omitted the
equally important areas of safety design and economics-related design,
which also interrelate with the three areas indicated.
Sensitivity Analysis and Design Parameter Interactions

8.20. After the system has been modeled and intersystem dependencies
identified, it is desirable to carry out a sensitivity analysis. Assume that the
system in question is described by input parameters that may be varied
and by output functions which characterize the behavior of the system and
might, in turn, serve as input to another system. We would like to know
how sensitive the output functions are to adjustments to the input parameters. Should some parameters have little effect on the output, the model
could be simplified. Also, should one or more input parameters be found
to have an overriding effect, such input parameters would be identified as
being critical to the design. At any rate, the sensitivity analysis step provides
insight to the workings of the system and is a decision tool.

8.21. As part of our study of reactor control, we learned how feedback
could affect the behavior of a system, particularly its stability (§5.132).


The Systems Concept, Design Decisions, and Information Tools










--- - - l






_ _ _ _ _ _ _ _ _ _ _ _ _ JI

Fig. 8.1.

Integration of parametric contributions.

Should the output of our system serve as the input to a second system
or subsystem and then have one of the second system outputs return as
input to the original system, we have a feedback effect. In the case of
reactor control, the second system could either reinforce or inhibit an
original disturbance signal, leading to an unstable or a stable condition,
8.22. In the reactor design logical flow diagram shown in Fig. 8.1, the
required fuel loading, flux pattern, and power distribution evolve from the
nuclear design, and some feedback is provided to both the nuclear data
and thermal-hydraulic design activities. Thus, we have a typical procedure
initiated by a trial design which is then refined by feedback. Although not
shown in the figure, the process involves an evaluative stage in which results
are compared with goals. This procedure lends itself to automation using
standard computer programming practice.

8.23. In engineering design, there is often the desire not only to use
the systems concept to develop a feasible solution to a given problem, but

The Computer As a Decision Tool


to use various formal methods to find an optimum or best solution. To
accomplish this, the various parameters controlling the system are adjusted
to yield "behavior" that is best in relation to a prescribed so-called "figure
of merit" or design goal.
8.24. Application of various formal optimization approaches to nuclear
engineering problems, particularly to those in fuel management, has attracted the attention of researchers for many years [3]. However, since
most methods require extensive iteration by computer, approximations in
modeling the core in fuel reloading problems (§1O.40) have been necessary
to maintain a practical level of calculation effort. Therefore, the resulting
uncertainty in the results has limited commercial interest in applying existing methods. This picture could change as improved core modeling methods are developed utilizing newer computers, particularly those based on
parallel processing. Separate from the fuel reloading problem, the optimum
design of the total plant to provide energy at minimum cost has been
standard practice for many years. In the case of the plant, the number of
significant parameters is modest compared with the reload core. Thus,
computer modeling of the plant design problem is easily managed.
Expert Systems and Artificial Intelligence [4]

8.25. As the power of computers increased over the years, their use to
accomplish so-called "thinking" processes which simulate certain human
activities became the subject of a subdiscipline known as artificial intelligence (AI). Applications have varied from simple game playing (chess) to
speech recognition. Normally required are a bank of stored knowledge
and a search procedure in accordance with specified rules that lead to a
decision. The boundaries of these efforts are not well defined. For example,
many optimization methods follow a similar approach, although optimization is not normally considered an AI procedure. Our initial attention
will be focused on "expert systems," a so-called single-purpose AI procedure that is now widely used commercially [5].
8.26. An expert system is a computer program that incorporates the
knowledge and reasoning of human experts as a decision tool. A typical
expert system includes a knowledge base, and inference engine, and a systemhuman interface. The knowledge base includes available relevant information from data bases as well as various rules-of-thumb facts developed
from experience by so-called "heuristic" processes. The inference engine
manipulates the knowledge base information by programmed rules of logic
to reach the desired conclusions. Translation of input into computer language and then from the computed results to output is accomplished by
the interface feature.
8.27. The major advantage of this type of program is the ability to apply


The Systems Concept, Design Decisions, and Information Tools

the wisdom of experts systematically in a repeatable manner to the solution
of a problem without the experts being present. Prescribed logic paths for
decision making are incorporated in the inference engine. "Reasoning"
techniques may be forward "chaining" starting from a set of known facts
or backward chaining in which the system works backward from goals and
tries to establish needed supporting evidence. Common features are "ifthen," "and," and "or" decision gates. Inprecise or "fuzzy" information
may be incorporated into the program through the use of "certainty factors," which represent judgments of the validity of a fact on a numerical
scale of 0 to 1, where 1 means complete certainty.
8.28. In nuclear reactor control rooms, expert systems can provide the
operator with guidance regarding measures to be taken in the event of
unanticipated incidents, thus reducing operator error. However, the applicability of an expert system is limited by the scope of information in the
knowledge base. Therefore, development of expert systems to cover a very
wide range of emergency situations is taking place. Other uses include the
monitoring of the performance of various plant components and recommending operating changes; various training situations, including the analysis of trainee performance; and many management-related applications.
8.29. Neural network procedures utilize a type of computer architecture
in which the processing elements are interconnected so that a great many
calculations can be carried out in parallel, imitating the way that neuron
cells in the brain process information. The theory and descriptions of some
neural network applications are available in a number of references [6].
In general, neural networks consist of processing units, arranged in layers,
with connections between units. These connections have "weights" which
have a memory capability and may be "trained" to adjust to changing
conditions. Information is processed in a distributed manner rather than
by a conventional computer serial approach. Hence a general characteristic
is the ability of a network quickly to recognize the various conditions or
states of a complex system once it has been suitably "trained." Thus,
applications in pattern recognition, language translation, and speech understanding are typical.
8.30. Applications to the operation of nuclear power plants are being
developed. For example, efficient pattern recognition can be a useful diagnostic tool in interpreting control room data. Another area is the modeling of nonlinear systems with application to process dynamics [7].
Computer Code Sources
8.31. The computer is an essential tool for modern nuclear engineering
practice. Although hand calculations are valuable for instruction, conceptual visualization, scoping, and preliminary design, the complexity of nu-

Information As a Decision Tool


clear systems demands computer calculations for final design. Hence, there
is a need for computer programs to meet a wide variety of requirements.
Although writing a custom program may be necessary for some applications, it is first desirable to explore the availability of developed codes that
will meet the need. Therefore, familiarity with the sources of available
computer codes is a useful engineering tool.
8.32. An essential first step in code use is to become thoroughly familiar
with the problem to be solved and to determine what codes are best for
the given task. In addition to developing background from the literature,
consultation with experts experienced in solving similar problems may be
necessary. In the process, it is likely that code availability will be determined. Our purpose here is merely to point out some common code sources
should it be necessary to use them.
8.33. The National Energy Software Center at Argonne, Illinois serves
as a central computer program library in the areas of reactor physics,
reactor engineering, and design. Periodically, it publishes a compilation of
code abstracts. The center has available for distribution to qualified users
well over 1000 programs.
8.34. A second major resource is the Radiation Shielding Information
Center at Oak Ridge, Tennessee (§6.159). Although the emphasis in the
computer code collection is on radiation transport and shielding, much
information is also provided on data analysis. User code packages are
provided by the Center, which has a collection of several hundred programs.
8.35. The Electric Power Research Institute (EPRI), the research arm
of the electric utility industry, is responsible for significant code development. Software packages for EPRI-sponsored work are available to member utilities and others under a licensing arrangement from the Electric
Power Software Center, operated by the Power Computing Co., a commercial information service in Dallas, Texas.
8.36. For software not available from central collections, it is often
possible to obtain packages directly from the code developer by making
appropriate arrangements. However, proprietary considerations may limit

8.37. Familiarity with information resources is an important decision
tool for the nuclear reactor engineer. Traditionally, the published literature
has been the first step in developing background to solve a given problem.
Although an understanding of relevant principles is still essential, conven-


The Systems Concept, Design Decisions, and In/ormation Tools

ient access to a variety of data bases and other information can expedite
problem solving. Our purpose here is to draw attention to some of these
resources. A useful guide to nuclear engineering resources is given by
ledruch [8].
8.38. Most practicing engineers recognize the need to update their expertise continually through routine scanning of professional society news
magazines and journals devoted to their speciality. We assume that they
have access to a technical library and know how to use it to find articles
in the archival literature and books of interest. Computerized catalogs are
available in most major libraries. Therefore, extensive searches can be
carried out quickly. Our objective here is to remind the nuclear reactor
engineer of several other important information sources.

8.39. Atomindex, published by the International Atomic Energy Agency,
provides comprehensive abstract coverage of the entire field of nuclear
engineering and science. Energy Research Abstracts published by the U.S.
Department of Energy provides coverage of nuclear and other energyrelated activities with particular attention to those is the United States.
Research reports, conference proceedings, and various safety-related regulations are covered. It is issued semimonthly and is indexed annually.
Therefore, it represents a convenient first step for a literature search. Since
it is arranged by subject matter, with detailed subcategories, it is easy to
browse through contributions in areas of interest. Sources for the literature
cited are given. Another resource is the Engineering Index, which covers
all areas of engineering.
Nonarchival Literature Sources

8.40. In addition to archival journals, some other literature sources are
of particular interest to nuclear engineers.

8.41. It has been customary over the years to publish the results of work
sponsored by the U.S. government in the form of both progress reports
and final reports. Although some of this work eventually makes its way
into the archival journals, the report literature provides a picture of ongoing
activities. Various types of reports are also required as part of the reactor
licensing process (§12.237). For example, Safety Analysis Reports are an
excellent source of information regarding power reactor systems. The Electric Power Research Institute in Palo Alto, California also maintains a
Research Report Center for its sponsored work. The EPRI Journal, a
periodical that describes EPRI activities, lists newly issued reports.

Information As a Decision Tool


Standards and codes

8.42. Standards and codes have played an important role in engineering
practice for many years. The American Society of Mechanical Engineers
Boiler Code had its beginnings in the nineteenth century, for example.
Presently, a large body of standards developed under the auspices of the
respective engineering professional societies, including the American Nuclear Society, are coordinated by the American National Standards Institute (ANSI). However, the American Nuclear Society maintains the Information Center on Nuclear Standards (ICONS) for all nuclear standards,
including those developed by other societies.
Federal regulations, guides, and periodicals

8.43. A great deal of technical information is provided in the regulatory
literature. Title 10 of the Code of Federal Regulations (10 CFR) is devoted
to nuclear energy and consists of a number of parts. For example, Part 20
(lOCFR20) deals with the standards for protection against radiation. The
various parts of the Code of Federal Regulations, which have the force of
law, are supplemented by Nuclear Regulatory Guides. These are advisory
and are coordinated in some cases with engineering society standards and
codes. There are several hundred published guides divided into ten categories. They are revised periodically.
8.44. The U.S. Department of Energy publishes several periodicals and
newsletters. For example, Nuclear Safety is a technical progress review
issued four times per year.

Data Centers
8.45. Once he or she has developed the methods needed to solve a given
problem, the nuclear engineer frequently needs such specialized information as neutron cross sections, physical constants, material properties,
etc. To meet this need, a number of centers have been established to
maintain special collections termed data bases. In many cases, the information is stored in a computerized retrieval system that can be searched
on-line by the end user. Another approach has been to collect formulations
for properties and other information so that they can be made available
on a suitable computer tape. Only several of the major data centers will
be mentioned here. Other information sources are indicated elsewhere in
the text, as appropriate.
8.46. The National Technical Information Service (NTIS) in Springfield, Virginia maintains a very large collection of reports and other materials resulting from activities of the federal government. Data bases, such
as the Energy Data Base, are available for on-line searching.
8.47. The Technical Information Center at Oak Ridge, Tennessee, op-


The Systems Concept, Design Decisions, and Information Tools

erated for the U.S. Department of Energy, is responsible for a wide variety
of services, including abstracting of reports and publication of technical
progress reviews. It prepares the Energy Data Base available from NTIS.
8.48. The earliest center, originally devoted primarily to nuclear cross
sections, is the Brookhaven National Nuclear Data Center at Upton, New
York. Data evaluation remains a major responsibility. Results are available
through publications and computer tapes.
8.49. The Center for Information and Numerical Data Analysis and
Synthesis (CINDAS) at West Lafayette, Indiana is an example of a data
center providing engineering information. Published data describing thermop hysical and other properties of many materials are collected and evaluated. A data base with on-line interactive capability is maintained.

Data Access Networks
8.50. The ability to access various data bases in information centers by
computerized communications networks is a major convenience. In most
cases, the computerized data base with interactive capabilities is maintained
by the responsible center, while access is obtained by means of a commercial communications network such as SPRINTNET or BT TYMNET
[9]. Generally, a commercial service provider serves as an intermediate
link between the communication network, which is "dialed up" by the
user, and the various data bases. Such providers, which offer numerous
information and search services, generally provide catalogs of data bases
that are available through them [10]. A given data base may be offered
by more than one provider. In some cases, a provider may also offer
communication network service.
8.51. On-line interactive access to some data bases may be available
directly by telephone. For example, the U.S. Department of Energy provides an on-line search service known as FEDIX for universities and other
research organizations. No registration or access fees are required. The
Federal Telecommunications Service (FTS) also provides network access
to many data bases to U.S. government contractors. RECON, developed
by the Technical Information Service for Department of Energy contractors
and other agencies, is an interactive search system for a large number of
data bases of interest to the nuclear engineering community. Included are
such areas as safety information, codes available at the National Energy
Software Center (§8.33), and electric power data.
8.52. The NUCLEAR NETWORK, operated by the Institute of Nuclear Power Operations (INPO), provides direct communication between
reactor licensees, reactor vendors, and other nuclear power-related groups
such as the Nuclear Safety Analysis Center of the Electric Power Research
Institute. Most use is devoted to communications rather than literature



8.53. The INTERNET, when it began in 1969 as ARPANET, linked a
few universities, government agencies, and defense-related laboratories.
During recent years, it has grown at a rapid rate to accommodate commercial as well as institutional users and has become the world's largest
computer network. Since most networks are connected to the INTERNET,
it has become in a sense a worldwide network of computer networks. A
large variety of services are available through the INTERNET, including
electronic mail, access to data bases, and on-line discussion groups [11].
Public access is available through commercial providers (§8.50).
Information Age Challenges

8.54. Making effective use of all of the information resources available
to help the engineer solve a given problem is somewhat of a challenge.
The discussion above is intended only as an introduction to the types of
services available, which are being continually expanded and updated.
Therefore, guidance from an information service professional or technical
librarian is desirable.

BEAM, W. R., "Systems Engineering: Architecture and Design," McGraw-Hill
Book Co., 1990.
BLANCHARD, B. S., and W. J. FABRYCKY, "Systems Engineering and Analysis,"
2nd Ed., Prentice Hall, 1990.
BOARDMAN, J., "Systems Engineering: An Introduction," Prentice Hall, 1990.
LEWIS, W. P., "Fundamentals of Engineering Design," Prentice Hall, 1989.
PAPALAMBROS, P. Y., and D. J. WILDE, "Principles of Optimal Design: Modeling
and Computation," Cambridge University Press, 1988.
RALSTON, A., and E. D. REILLY, eds., "Encyclopedia of Computer Science," 3rd
Ed., Van Nostrand Reinhold Co., 1993.
RA Y, M. S., "Elements of Engineering Design: An Integrated Approach," Prentice



J. G. Ecker and M. Kupferschmid, "Introduction to Operations Research,"
John Wiley & Sons, 1988.
J. Boardman, "Systems Engineering: An Introduction," Prentice Hall, 1990.
T. J. Downar and A. Sesonske, "Light Water Reactor Fuel Cycle Optimization: Theory versus Practice," Adv. Nucl. Sci. Technol., 20, 71 (1988).
D. T. Pham, Ed., "Artificial Intelligence in Design," Springer-Verlag, 1991;
M. Sharples et al., "Computers and Thought," MIT Press, 1989.
M. D. Rychener, "Expert Systems for Engineering Design," Academic Press,




The Systems Concept, Design Decisions, and Information Tools

1988; J. A. Bernard and T. Washio, "Expert System Applications within the
Nuclear Industry," American Nuclear Society, 1989.
R. E. Uhrig, Nuc!. Safety, 32, 68 (1991); P. D. Wasserman, "Neural Computing: Theory and Practice," Van Nostrand Reinhold Co., 1990.
A. G. Parlos et ai., Trans. Am. Nucl. Soc., 63,109 (1991).
J. Jedruch, "Nuclear Engineering Data Bases, Standards, and Numerical
Analysis," Van Nostrand Reinhold Co., 1985; D. N. Chorafas and S. J. Legg.,
"The Engineering Data Base," Butterworth, 1988; M. Edelhart and O. Davies,
"Omni Online Database Directory," Collier Macmillan Publishers, 1984.
U.S. Sprint, 12490 Sunrise Valley Drive, Reston VA 22096 (800) 736-1130;
BT Tymnet, Inc., P.O. Box 49019, San Jose, CA 95161, (800) 872-7654.
DIALOG Information Services, Inc., 3460 Hillview Avenue, Palo Alto, CA
94304, (800) 334-2564; ORBIT Search Service, 8000 Westpark Drive, McLean, VA 22102, (800) 456-7248.
T. L. LaQuey, "Internet Companion: A Beginner's Guide to Global Networking," Addison-Wesley Publishing Co., 1993.



Energy Transport

The Role of Energy Transport in Reactor Design
9.1. Most of the fission reaction energy deposited in fuel is immediately
converted to heat. If the fuel is to remain at steady state (constant temperature), the heat must be transported away at the same rate as it is
generated. Although we have seen that negative reactivity feedback, if
present, tends to limit power increases, it is essential for the heat generationheat removal rate balance to be maintained, to prevent temperatures that
might result in the failure of fuel and structural materials. Thus, the design
of the core depends just as much on adequate heat removal as on nuclear
9.2. Effective heat removal is a function of many design parameters,
including the fuel geometry, coolant flow characteristics, and properties
of materials, as well as related neutronic behavior. In many aspects of the
design, conventional engineering principles of heat transfer and fluid mechanics are applicable. The term thermal-hydraulics is commonly used to
describe the effort involving the integration of heat-transfer and fluid me501


Energy Transport

chanics principles to accomplish the desired rate of heat removal from the
core under both operating and accident conditions. The purpose of this
chapter is to introduce this area of nuclear reactor engineering.
9.3. Various accidental causes of a mismatch between the fuel heat
generation and heat removal capabilities of the coolant leading to reactor
damage are carefully analyzed as part of the required safety analysis to be
discussed in Chapter 12. Of particular concern are events that lead to a
reduction in the coolant mass flow rate. Since each unit mass of coolant
would then receive more energy, boiling is a probable consequence. Such
a transition to a two-phase system would result in an increase in flow
resistance, further aggravating the unstable situation. Therefore, the thermalhydraulics of two-phase systems plays an important role in reactor design.
Analysis is required for the design of safety features which are provided
in a reactor plant to reduce the consequences of a serious accident in which
the core might be damaged. Although the relevant thermal-hydraulics basics
will be discussed in this chapter, treatment of the design applications will
be deferred until Chapter 12.
Thermodynamic Viewpoint

9.4. As background for the thermal-hydraulics aspects of reactor design,
it is helpful to consider the path of energy transport from the thermodynamic viewpoint. In a nuclear power plant, heat flows from the fuel elements to a coolant, then perhaps to a secondary coolant, as in a PWR, to
form steam. Work is done by expanding the steam in a turbine. The turbine
exhaust steam is then condensed and recycled. Heat extracted during the
condensation process is rejected to the environment. Thus, we have a
movement of energy from the heat generation at high temperature in the
fuel elements progressively downward on the temperature scale until the
environment is reached. Along the way, a portion of the original energy
is converted to useful work by a steam expansion in the turbine that is
almost thermodynamically reversible. However, at each heat flow step, an
irreversible temperature "drop" is needed to accomplish the heat transfer
at a practical rate. As we shall see, this temperature difference acts as the
driving force for the thermal transport rate process.
9.5. The thermodynamic efficiency of the conversion to work is improved if the heat input from the steam to the turbine is at as high a
temperature as possible and the heat rejection in the condenser is at as
Iowa temperature as possible. Since the "high" temperature in a watercooled reactor is usually limited by materials and pressure considerations,
and the "sink" or rejection temperature by environmental and other factors, the irreversible temperature "drops" required for heat transfer between these two limits must be expended carefully by the thermal system

Heat Sources in Reactor Systems


Design Methods

9.6. Engineers have used design methods describing the transfer of heat
and the movement of fluids for many years. In a typical reactor core, the
transport of heat from the fuel to the moving coolant involves the traditional
processes of conduction and convection. Further steps in the thermodynamic energy path described above also involve the behavior of fluids in
motion, particularly the effects of a second phase, as is produced by boiling.
Over the years, a substantial body of literature has been developed describing the principles relevant to thermal system design. Early, largely
empirical design methods have been supplemented by models having some
theoretical basis. Finally, the general availability of powerful computers
has made a reasonably sophisticated description of transport processes
practical for design purposes.
9.7. Since it is not our purpose here to treat the principles of conductive
and convective heat transfer from the beginning, readers who have not
been introduced to this area are advised first to consult an elementary text.
Of importance is the concept of a rate process and the role of a mathematical representation of the conservation of mass, energy, and momentum
in the description of such processes. With this background, we are then
able to introduce the highlights of nuclear reactor thermal-hydraulics. Discussions of sophisticated analysis methods are beyond the scope of this
book but are available in standard sources [1].
9.S. Our objective here is merely to provide a picture of energy transport considerations in a reactor system using simple analysis methods. Such
methods may be useful for preliminary design or scoping, but appropriate
computer codes are needed for subsequent more detailed design. However,
it is important to recognize that every method has a level of confidence or
range of error that should be identified by the user.

Fission Energy

9.9. The energy released in the core by fission appears in various forms,
but mainly as the kinetic energy of the fission fragments, the fission neutrons, and the beta particles resulting from radioactive decay of the fission
products (see Table 1.2). The fission fragments are usually stopped within
the fuel elements themselves; the small fraction that escapes into the cladding penetrates only about 0.01 mm. The beta particles of high energy may
travel up to 2 mm in a cladding material such as zircaloy (§7.117), and so
a large fraction of these particles may escape from the fuel element into
the moderator or coolant, but they will not get out of the reactor core.


Energy Transport

The fission neutrons lose most of their energy in the first few collisions
with moderator atoms, and they travel distances ranging from some centimeters to a few feet. It is seem, therefore, that most of the heat from
the three sources under consideration, comprising about 90 percent or more
of the total energy generated, will be released within the reactor core.
9.10. The remaining 10 percent, or less, of the energy produced in
fission appears as gamma rays which are distributed throughout the reactor
core and surrounding components in a manner dependent on the specific
materials and geometries involved. Thermal stresses as a result of gamma
heating of the reactor vessel and surrounding radiation shielding must be
considered by the designer (§7.30).
9.11. As we have seen (§2.213), heat generation by radioactive decay
of the fission fragments continues after the fission reaction ceases. Therefore, provision must be made for cooling the fuel elements after shutdown.
Particularly important are emergency core cooling features to be effective
in the case of an accidental loss of normal cooling capability (§ 12.33). Also,
the role of gamma heating affects the after-shutdown spatial distribution
somewhat. For example, 1 hour after shutdown, the heat generation rate
in the fuel elements will be about 1.5 percent of the operating value,
whereas in the reflector and shield it will be approximately 10 percent of
the rate during operation.
9.12. It was seen in Chapter 1 that the total energy released in fission,
which ultimately appears as heat, is made up of contributions from a number of sources. In general, the total energy, exclusive of the neutrino energy
which is lost to the reactor system, may be expressed by

= 191 +

Ec (in MeV),

where Ec is the energy liberated as a result of various parasitic neutron
capture processes, e.g., nonfission capture in uranium-235 and uranium238, and capture in moderator, coolant, structure, etc.; this includes the
energy of the capture gamma radiations and the decay energies, i.e., the
energies of the alpha and beta particles and gamma rays, of any radioactive
species that are formed by parasitic neutron capture. * Since the value of
Ec will obviously depend upon the nature of the materials present in the
reactor core, it is evident that the total amount of heat produced by fission
will vary, to some extent, from one type of reactor to another.
* Although Ec should include the decay energies of all radioactive species formed as a
result of nonfission neutron reactions, only those of relatively short half-life contribute to
the energy release in the state of pseudo-equilibrium attained by the reactor soon after startup.

Heat Sources in Reactor Systems


Example 9.1. Determine the total energy release in the core of a
pressurized-water reactor (PWR) having volume fractions of uranium oxide
(U0 2 ), water, and iron of 0.32, 0.58, and 0.10, respectively. The oxide
fuel (density = 10.2 x 103 kg/m 3 ) has an average enrichment of2.8 percent
235U, and the average cooling water density is 0.69 X 103 kg/m 3 . The energy
released per neutron captured in uranium, water, and iron may be taken
as 6.8, 2.2, and 6.0 MeV, respectively.
The solution requires the calculation of the macroscopic neutron capture
cross section for each component (except for the oxygen in U02 which is
very small) as well as the 235U macroscopic fission cross section. The final
result depends on ratios of cross sections rather than absolute values; hence,
the tabulated values for 0.0253-eV neutron microscopic cross sections may
be used in the calculations. The capture-to-fission ratios are then as given
in the following summary:
N (nuclei/m3 x 1028 )

2:c (m-l)
2:t (m- 1)
2: c P52:f




H2 O












The energy released as a result of nonfission captures is


(6.8 x 0.16) + (6.8 x 0.17) + (2.2 x 0.076) + (6.0 x 0.19)

= 3.6 MeV
The total energy released per fission in this core is thus


191 + 3.6

= 195 MeV


3.1 x 10- 11 J.

Spatial Distribution of Energy Sources in Reactor Core
9.13. As already noted, a large proportion of the energy from fission
is available in the form of heat within a very short distance of the fission
event. The total rate of heat generation is proportional to the fission rate,
i.e., to ~r<\> or NfIr<\>' where fIr is the microscopic fission cross section, <\>
is the neutron flux, and N is the number of fissile nuclei per unit volume
of fuel. If N remains uniform throughout a particular reactor or reactor
region, as an initial approximation, the thermal source function, expressed


Energy Transport

as the power density, may be taken to be the same as the spatial distribution
of the neutron flux.
9.14. The overall flux distribution in the reactor core can be calculated
by the methods described in Chapter 3 and 4. For a reflected cylindrical
reactor in which the fuel is distributed uniformly, the flux distribution in
the core and reflector can be represented approximately by

where <l>max is the flux at the center of the core where it is assumed to have
its maximum value; 10 is the zero-order Bessel function of the first kind,
rand z are the radial and vertical coordinates, as in Fig. 3.11, and R' and
H' are the effective radius and height, respectively, of the reactor, including
an allowance for the reflector.
9.15. The average neutron flux in the core is given by

= 'ITR2H





<l>2'lTr dr dz,


where Rand H are the actual radius and height of the fuel-containing
region. Upon substituting the expression for <I> from equation (9.1) and
performing the integration, the result is



2RR'J1 (2.405 :,)

2H' Sin(¥ . :;,)

= _ _ _ _ __




where 11 is the first-order Bessel function.
9.16. If the concentration of fissile material is uniform throughout the
reactor core, the power density may be taken to be proportional to the
neutron flux. * It then follows that the reciprocal of equation (9.3) is equal
to Pma,jPav , i.e., the ratio of the maximum to the average power density.
For a reflected cylindrical reactor, RIR' and HIH' may be taken as roughly
5/6, and then



= 2.4.

'Commercial power reactors have radial fuel regions with different enrichments in uranium235; as a result, the power distribution differs from that for uniform enrichment assumed
here. Also, the axial power density is likely to be skewed for reload core loadings as a result
of previous operational history (see Chapter 10).

Heat Sources in Reactor Systems


For a bare reactor, i.e., one without a reflector, RIR' and HIH' are both
unity; then equation (9.3), after inversion, yields
Pmax ( b )




2J 1(2.405) .


"2 = . 4.

If in a bare cylindrical reactor the arrangement of fuel elements or control

rods (or both) is such that the flux is uniform in the radial direction, the
factor containing the Bessel function is unity, and PmaxlPav is -rr/2, i.e.,
1.57. Similar calculations have been made for reactors of other shapes,
and the results of some core geometries are given in Table 9.1. It may be
noted that, because of the arguments presented in §3.54, based on the
similarity of the curves in Fig. 3.12, a cosine distribution of the flux may
be assumed in all cases, e.g., even in the radial direction of a cylindrical
Average and Maximum Power in Single Fuel Channel
9.17. In reactor cooling problems, it is often of interest to estimate the
maximum heat-generation rate (or power density) at a point in a given
fuel channel, rather than for the whole reactor core. In equation (9.1), the
first (Bessel function) term gives the radial flux distribution, whereas the
second (cosine) term represents the axial distribution in a cylindrical reactor. In any specified axial fuel channel, at a fixed radial distance r from
the reactor center, the neutron flux distribution is

where (<Pmax)axial> the maximum flux at the center of the given channel, is
equal to ;<PmaxJo(2.405rIR').
TABLE 9.1. Ratio of Maximum to Average Power
Densities in Reactors With Uniform Fuel Distribution
Core Geometry
Sphere (bare)
Rectangular parallelepiped (bare)
Cylinder (bare)
Cylinder (bare, flat radial flux)
Cylinder (reflected)
Pool type (water reflected)




Energy Transport

9.18. The average flux in an axial fuel channel in a cylindrical core is
given by

2H' Sin(~2 .H'H)

= (<I>max)axial


If the fuel enrichment is uniform in the axial direction, as is sometimes

the case, the axial power density will be proportional to the neutron flux;
( P max )





. _.2H ' sm
2 H'

9.19. The result expressed by equation (9.5), or the simplification for
the core without end reflectors, is also applicable to fuel channels in a
rectangular parallelepiped reactor. For this geometry, the flux distribution
in any direction parallel to one of three principal axes is represented by
an expression analogous to equation (9.4). The only change necessary is
to replace H by the actual length of the reactor core in the given direction,
and H' by the effective length including allowance for the reflector.

Power and Flux Flattening
9.20. In most reactor designs, the power output is limited by the maximum permissible temperature of the fuel elements, i.e., by the value of
Pmax' In some cases, it is possible to increase the power output by decreasing
Pmax/Pay' In other words, the power output of the reactor can be increased
by a more uniform distribution of the power density. This is sometimes
referred to as a "flattening" of the power distribution or of the neutron
9.21. We have seen that the presence of the reflector provides some
flux flattening (§3 .136). In addition, reduction in the concentration offissile
atoms in fuel that would otherwise have a high neutron flux could provide
a flattening effect. Such fuel management schemes, described in Chapter
10, may also provide improved utilization of the available neutrons and
decrease the cost of the electricity produced by the reactor plant.

Heat-Transmission Principles


Other Heat Sources

9.22. About 3 percent of the heat generated in the reactor system is
produced in the moderator as the result of the slowing down of fission
neutrons, the stopping of beta particles from the fission products, and the
absorption of gamma rays from various sources. In light-water reactors,
this presents no special cooling problem. Cooling requirements in other
systems may be a function of the design. The need for cooling should also
be recognized in the design of control elements. Heat is generated in such
components outside the core as the thermal shield and reactor vessel in
water reactors and the radiation shield in all reactors. This heat is a result
of the absorption of escaping neutrons and gamma radiation. The subject
of heat generation in shields has been treated in Chapter 6.

9.23. Three general mechanisms are distinguished whereby heat is transferred from one point to another, namely: (1) conduction, (2) convection,
and (3) radiation. These three methods for heat removal will be reviewed
and their general characteristics outlined. The subsequent discussion, however, will be devoted to the first two, since they are of major importance
in existing reactor designs. The preliminary treatment will refer to heat
transmission in systems without internal sources, and subsequently the
problems associated with internal sources, such as exist in reactor fuel
elements, will be taken up.
Conduction of Heat

9.24. The term conduction refers to the transfer of heat by molecular
(and sometimes electronic) interaction without any accompanying macroscopic displacement of matter. The flow of heat by conduction is governed by the relationship known as the Fourier equation, i.e.,

q = -kA dx'


where q is the rate (per unit time) at which heat is conducted in the x
direction through a plane of area A normal to this direction, at a point
where the temperature gradient is dtldx. * The quantity k, defined by equa'The quantity qlA, having the dimensions of heat/(time)(area), i.e., W/m2, is called the
heat flux.


Energy Transport

tion (9.6), is the thermal conductivity. In SI units, q is expressed in J/s
(or W), A in m2 , and dtldx in °C/m (or Kim); hence, the units of k are
(W/m2)(m/K), usually written in the form W/m . K. In English units, k
will be in Btu/(hr)(ftzWF/ft).
9.25. The thermal conductivity k is a physical property of the medium
through which the heat conduction occurs. For anisotropic substances, the
value of k is a function of direction; although methods are available for
making allowance for such variations, they are ignored in most analytical
solutions of conduction problems. The thermal conductivity is also temperature dependent and can generally be expressed in a power series; thus,

which, in many cases, may be approximated to the simple linear form


k o(1

+ at).

Where considerable accuracy is desirable (and possible), allowance must
be made for the variation of thermal conductivity with temperature. But
very frequently, especially when the temperature range is not great, k is
taken to be constant. However, when uranium oxide is used as fuel, as it
is in most power reactors, the temperature gradients are quite large, and
allowance must be made for the variation of the thermal conductivity with
temperature (§9.46). Some values of k of interest in reactor design are
given in the Appendix.
9.26. Upon integration of equation (9.6) over the x direction, it is found,
for unidirectional heat flow by conduction in a slab of constant cross section,
with k independent of temperature, that
q= -kA~,
Xl -



where t1 and t2 are the temperatures at two points whose coordinates are
Xl and X2. The result means that the temperature gradient at a point, i.e.,
dtldx, in the Fourier equation (9.6) may be replaced by the average gradient
over any distance, i.e., by (t1 - t2)/(Xl - x 2).
9.27. If t1 - t2 is replaced by at, the temperature difference, and X2 Xl by L, the length of the heat-flow path, equation (9.7) upon rearrangement takes the form




Heat-Transmission Principles


This expression is analogous to Ohm's law, I = EIR; hence the quantity
LlkA is often called the thermal resistance for a slab conductor. The analogy
between conduction of heat and electricity is the basis of the thermal circuit
concept which is very useful in solving heat-transfer problems. In general,
the rate of heat flow q is equivalent to the current I; the temperature
difference fl.t is the analogue of the potential difference (or EMF) E; and
the thermal resistance replaces the electrical resistance.
Convection of Heat

9.28. Heat transmission by convection, which is usually concerned with
the transfer of heat across a solid-fluid interface, involves macroscopic
motion of the fluid. In free convection, the motion is a consequence of the
buoyant forces generated in the fluid due to temperature differences within
it. In forced convection, on the other hand, the fluid is moved by mechanical
means, e.g., by a pump. When applied to heat removal in reactors, two
aspects of convection heat transmission must be considered; there is, first,
transfer of heat from the material which is being cooled, e.g., the fuel
element, to the coolant; and, second, the transport of this heat, usually in
sensible form, by flow of the coolant from one point in the system to
another. Fluid flow is thus an important problem, as will be seen later, in
the study of reactor cooling systems.
9.29. The fundamental equation of convection heat transfer, for both
free and forced motion of the fluid, is the so-called Newton's law of cooling,
which may be written as

where q is the rate of convection heat transfer to or from a surface of area
Ah when the temperature difference is fl.t. The quantity h, defined by
equation (9.6), is commonly called the heat-transfer coefficient and is expressed in units of W/m 2 • K. It should be noted that equation (9.9) applies
to convection heat flow in either direction, i.e., from solid to fluid or from
fluid to solid; the actual direction of the flow depends, of course, upon
the sign of fl.t.
9.30. Although the value of h is dependent upon the physical properties
of the fluid medium, it is also a function of the shape and dimensions of
the interface, and of the nature, direction, and velocity of the fluid flow.
Thus, the heat-transfer coefficient is a property of the particular system
under consideration. Another factor which determines h is the exact definition of fl.t, i.e., the temperature difference between the surface and the
fluid. Whereas the surface temperature of the solid is uniquely defined,


Energy Transport

that of the fluid is subject to several arbitrary definitions. The latter is
usually taken as the "mixed-mean (or bulk) fluid temperature" tm given

where p, cp ' and v are the density, specific heat, and flow velocity, respectively, of the fluid, and At is the cross-sectional (or transverse) flow
area of the fluid. The value of the heat-transfer coefficient in any given
circumstances can be determined experimentally, but for design purposes,
the general practice is to use results predicted by means of various theoretical and semiempirical expressions (see §9.78 et seq.).
9.31. By slight rearrangement, equation (9.9) can be written in the
Ohm's law form


lIhA h '

so that, as before, q and At are analogous to current and potential difference, respectively, but lIhAh now represents the thermal resistance to
convection-heat transfer. Thermal circuits can thus be constructed with one
or more stages of thermal conduction combined with convection for the
solution of problems involving both types of heat transfer, as will be shown
in the succeeding paragraphs.
Conduction With Convection Boundary Conditions
Infinite Slab
9.32. Although thermal problems in reactors generally involve internal
sources, the first (and simplest) case to be considered is that of an infinite
slab of finite thickness with no internal heat source. It is postulated that
heat is transferred at both faces by convection and that the heat-transfer
coefficients are ha and hb' respectively. The idealization of an infinite slab
is a good approximation to real (finite) systems in which the transverse
dimension is large compared with the thickness, e.g., the cladding of a
parallel-plate type fuel element or the flat plate of a heat exchanger. In
the following treatment, steady-state conditions with the thermal conductivity independent of temperature will be postulated.
9.33. The characteristics of the physical system are shown at the left in
Fig. 9.1, and the equivalent thermal-circuit diagram is given at the right.

Heat- Transmission Principles



R1 =_1_


= kA



= hb









• x





Heat conduction in an infinite slab with convection boundaries.

The total thermal resistance is the sum of three terms, namely, two (Rl
and R 3 ) for the convection resistances and the third (R 2 ) for the conduction
resistance of the slab; hence, if A is the heat-flow area, the total thermal
resistance, R, is





= h a A + kA + hb A = VA'


The relationship between the heat-flow rate and the temperature difference
t:.t = ta - tb, where ta and tb are the mixed-mean fluid temperatures at
the two faces, is thus given by
q = -R- = VA ta - tb
fa -





The quantity U, defined by equation (9.10), is called the overall coefficient
of heat transfer; it has the same dimensions as the heat-transfer coefficient.
9.34. These equations may be used to determine the steady-state difference in temperature ta - tb between two fluids in a heat exchanger if
the heat-flow rate q is given. Alternatively, if ta - tb is fixed, the corresponding value of q can be calculated for specified conditions. The indi-


Energy Transport

vidual temperature differences, i.e., the two fluid-solid differences and that
between the two faces of the slab, can be determined by the use of equations
(9.9) and (9.7), respectively.
Hollow Cylinder

9.35. Tubular heat-transfer surfaces are used in many heat exchangers,
and the performance of these can be analyzed on the basis of heat conduction through hollow circular cylinders with convection at both interior
and exterior surfaces. The physical system in cross section and the thermalcircuit diagram are shown in Fig. 9.2; it is assumed that all the heat flow
is in a radial direction. The inner and outer radii of the cylinder are a and
b, respectively; ha and hb are the heat-transfer coefficients for the inner
and outer fluids, and ta and tb are the mixed-mean temperatures of the
inner and outer fluids, respectively.
9.36. The convection thermal resistances are equal to lIhaAa and
lIhbAb' where Aa and Ab are the areas of the inner and outer surfaces.
The thermal conduction resistance of the cylinder wall remains to be determined. If lis the length of the cylinder, then the heat-flow rate, expressed
at an arbitrary radius r, due to conduction within the wall is given by
equation (9.6) as



k(27Trl) dr'

where k is the thermal conductivity and 27frl is equivalent to A, the area
through which heat flow occurs. This expression may be rearranged and
integrated over the wall thickness, i.e., from r = a to r = b; the result

Fig. 9.2.

Heat conduction in a hollow cylinder with convection boundaries.

R. =_1_



_ In(blal
w - 2"kl


Rb = hbAb









Heat-Transmission Principles




= In(b/a)/21Tkl'


where at is the difference in temperature across the cylinder wall.
9.37. It is evident from equation (9.11) that the thermal resistance Rw
of the wall is given by


21Tkl .


The total thermal resistance of the system under consideration is consequently

where Ub is the overall heat-transfer coefficient based on the outer surface
of the cylinder. According to the thermal-circuit concept



Radiation Heat Transfer
9.38. As a result of changes in the thermal motions of its constituent
particles, which are a function of the temperature, every body emits energy
'For values of b/a < 2, the quantity In(b/a), which appears here and in heat-transfer
equations for clad fuel rods (§9.47 et seq.), is well approximated by 2(b - a)/(b + a).


Energy Transport

in the form of electromagnetic radiations over a range of wave lengths;
this is called thermal radiation. The amount of energy carried by the radiation is unchanged as it passes through a vacuum and is largely unaffected
by dry air and many other gases, with the notable exceptions of carbon
dioxide and water vapor. But when the radiation falls on a solid body or
in its passage through the last-mentioned gases, part or all of the energy
is absorbed. The fraction of the incident radiation that is absorbed is called
the absorptivity. An ideal material which absorbs all the radiation falling
upon it, and thus has an absorptivity of unity, is designated a black body.
The emissive power or thermal radiation flux, i.e., the energy radiated in
unit time per unit area of a black body, is given by the Stefan-Boltzmann
law, namely,
Thermal radiation flux



where the constant IT has the value of 5.68 x 10- 8 W/mz . K4. The ratio
of the emissive power of an actual surface to that from a black body is
referred to as the emissivity, and for a body in thermal equilibrium, the
emissivity and absorptivity are equal (Kirchhoff's law).
9.39. The emissivity (and absorptivity) of surfaces vary with the nature
of the material and its temperature as well as with its physical condition,
e.g., roughness, cleanliness, etc. For metals, the emissivity is relatively
low; the values generally range from about 0.05 or less for a highly polished
surface to 0.2 or 0.3 for a roughened surface. If the metal is covered with
an oxide film, the emissivity is greatly increased. Nonmetals have fairly
high emissivities, although, in contrast to metals, the values decrease with
increasing temperature. The emissivity of graphite, for example, approaches unity at temperatures up to about 800 K (530°C), so that it
approximates a black body. It is therefore a good emitter and absorber of
thermal radiation. When radiation falls on a body, the proportion which
is not absorbed is partly transmitted, e.g., through a "transparent" material
such as air, and partly reflected, e.g., by a polished metal surface.
9.40. If two surfaces at different temperatures are separated by a nonabsorbing medium, there is an interchange of radiation between them since
they both act as absorbers and emitters. However, the net result is the
transfer of energy from the hotter to the colder surface, the rate of energy
transfer, e.g., in watts, being given by

where TI and T z are the absolute temperatures of the hotter and colder
bodies, respectively, Al is the surface area of the former, and £1,2 is an
interchange factor which is related to the emissivities (or absorptivities) of

Heat-Transmission Principles


the two surfaces; if both bodies were black, i.e., perfect absorbers and
emitters, £1,2 would be unity. Certain geometrical factors should be included in this expression, but they may be disregarded here.
9.41. Even though radiation does not transfer any appreciable amount
of energy directly to the coolant in a reactor, it may do so indirectly. In
gas-cooled reactors operating at high temperatures, heat is transferred by
convection from the fuel to the coolant, which is generally helium gas. The
latter is transparent to thermal radiation and so does not absorb any radiant
energy directly. However, radiation is transferred through the gas to the
moderator (graphite), and this then loses energy by convection to the
gaseous coolant. Thus, radiative transfer provides a means for conveying
heat from the fuel element to the gas in an indirect manner. Radiation
may also be significant in the transfer of heat from one reactor component,
such as a fuel element, to another component, thus leading to thermal
gradients which must be taken into account in stress analysis.
Systems With Internal Heat Sources

One-Dimensional Slab

9.42. The treatment of heat transmission in the steady state will now
be extended to systems with internal sources; for this purpose, the simple
one-dimensional case of an infinite slab will be examined (Fig. 9.3). Consider a thin section of the slab of thickness dx at x; then in the steady state
the heat-rate balance requires that
Heat conducted out of A dx

Fig, 9.3.

Heat generated in A dx + Heat conducted into A dx,

Heat transmission in an infinite slab with internal source.





t, "'"'"



x x + dx






Energy Transport

where A is the heat-conduction area and A dx is the volume of the thin
section. If the thermal conductivity is taken to be independent of temperature (and hence of x), it follows from equation (9.6) that

qin = - kA(!)x

qout = - kA (:t)



assuming the heat flow is from left to right. Upon taking the difference of
these terms and expanding the expression for qout> it is found that

qout - qin -kA{[ (!)x + (~~)x dx + ... J - (:~)J




higher-order terms being neglected.
9.43. If Q(x) represents the volumetric thermal source strength, expressed as heat generated per unit time per unit volume at x, e.g., in
W/m 3 , then Q(a)A dx is the heat generated per unit time in the section of
the slab under consideration. It follows, therefore, from the heat-balance
equation and from equation (9.14) that in the steady state

-kA ( dxd t) dx Q(x)A dx



so that

where Q(x) mayor may not vary with x, but k is assumed to be constant.
General Geometry

9.44. It can be shown that equation (9.15) is a particular form of the
general equation for the steady state in a system with an internal source,
_ \J2t = Q(r)
k '


Heat-Transmission Principles


where the thermal conductivity k is the same in all directions, i.e., for a
truly isotropic medium, and \7 2 is the Laplacian operator (§3.1l). It is
apparent that in the one-dimensional case in Cartesian coordinates, equation (9.16) reduces to equation (9.15). For conduction in a cylinder in the
radial direction, equation (9.16) becomes
_ (d 2t + !

. dt)






and for a sphere
_ (d 2t

+ ~ . dt)




One-Dimensional Conduction in Cylindrical Fuel Rod

9.45. In most power reactors, the fuel consists of cylindrical pellets
arranged end to end. This can be idealized as an infinitely long cylindrical
rod with a uniformly distributed heat source. Because of the idealized
infinite length, heat may be assumed to be conducted in the radial direction
only; hence, equation (9.17) is applicable with Q(r}, represented by Q,
independent of the radial coordinate r since the source is uniform. Provided
that k is independent of temperature, this equation may be written in the






which, if integrated twice, leads to
for 0:::; r :::; a,
where a is the outer radius of the cylinder (Fig. 9.4). The constants C1 and
C2 can be evaluated from the boundary conditions which are

-dr = 0' for r = 0 and t =



for r


The appropriate solution is therefore



Energy Transport


Fig. 9.4.

Radial heat transmission in an infinitely long fuel rod.

where t is the temperature at the radial distance rand tl is that at the
surface. If to is the temperature along the central axis where r = 0, it
follows that

for the temperature drop across the fuel rod.
9.46. When the thermal conductivity is not independent of temperature,
equation (9.19) may be written as



k(t) dt

Qa 2

= -,



where the quantity on the left side is characteristic of the fuel material for
the given temperature range. A related useful parameter is the heatgeneration rate per unit length of the cylindrical fuel element (or linear
heat rate) qL. Since qL = QCrra2), it is seen that




k(t) dt





= --1=..


Equation (9.20) shows that the temperature difference between the axis
and surface of a long cylindrical fuel rod with uniform heat generation
depends only on the linear heat rate and is independent of the rod diameter,
even if the thermal conductivity varies with temperature. The linear heat
rate is therefore a useful design parameter. For example, a linear heat rate
that would lead to central melting of the fuel material is usually listed as
a limiting core design specification.

Heat- Transmission Principles


Example 9.2. Determine the steady-state temperature difference between the center and the surface of a cylindrical uranium dioxide fuel
pellet, enriched to 2.9 percent in 235U, which forms part of a long PWR
fuel rod (§1.57). What is the heat flux at the surface of the pellet? (At the
specified low enrichment, weight and atomic percentages are essentially
the same.)
The outer diameter of the pellet is 8.19 mm and its density is 10.2 x
103 kg/m 3 • An average thermal-neutron flux of 4.1 x 10 17 neutrons/m 2 • s
is applied to the pellet and the conditions are such that the effective microscopic fission cross section of 235U is 360 b.

The first step is to determine the thermal source strength Q; this is equal
to Lf<PEf, where Lf is the effective macroscopic fission cross section in the
fuel, <P is the thermal-neutron flux, and Ef is the energy released in the
fuel per fission, which may be taken to be 200 MeV. Since <p and Ef
are known, Lf = N(If remains to be evaluated; N is the number of 235U
nuclei/m 3 of the fuel, and (If is the effective microscopic fission cross section
given above. This leaves N to be determined.
Since the fuel contains 97.1 percent of uranium-238, it is a good approximation to take the average molecular weight of the uranium dioxide
as 270 (0.270 kg/mole). The number of molecules in a mole is 6.02 x 1023
(the Avogadro number); hence,
Number of V0 2 molecules per m 3 of fuel


(10.2 x 103)(6.02 x 1023 )




1028 ,

where 10.2 x 103 kg/m3 is the density of the uranium oxide. Each molecule
contains one atom of uranium; hence, this is also the number of atoms of
uranium per m3 of fuel. Of this number, 2.9 percent are uranium-235 atoms;
it follows, therefore, that



= (0.029)(2.28 x 1028 )(360 x 10- 28 ) = 24 m- I •

The fission rate is Lf<P, and since fissions at the rate of 3.1 x 1010 s - 1
produce 1 W of power (§1.48),

Q = (24)(4.1

X 10 17)
3.1 x 1010

= 320


108 W/m3.

The temperature difference between the center and surface of the pellet
is determined from equation (9.19) where a, the pellet radius, is 8.19/2


Energy Transport

mm (0.0041 m), and k may be taken to be 2.80 W/m . K for D0 2 in this
case. Hence,
to -



Qa 2


(3.20 x 108 )(0.0041)2




Consider a unit of length of a fuel pellet; this has a volume of 7l'a2 , and
the heat generated within it is 7l'a 2 Q. The heat is transmitted in a radial
direction out of the surface area of unit length, i.e., 27l'a. Hence, the heat
flux qlA is given by

!1. = 7ra 2 Q = aQ = (0.0041)(3.20 x 108 ) = 656000 W/m 2




, .

These values are typical for averages in a PWR core. However, as will
be seen later, the maximum heat-generation rates, with corresponding
temperature differences and thermal fluxes, may be several times the average values.
One-Dimensional Conduction in Clad Fuel Rod

9.47. The cylindrical fuel rods in power reactors are usually clad with
a zirconium alloy or in some cases with stainless steel. The heat path is
then from the inner fuel cylinder (pellets), through the interface between
the fuel and cladding, by conduction through the cladding, and finally by
convection from the outer surface of the cladding to the coolant (Fig. 9.5).
9.48. Consider first the fuel cylinder, radius a; if to is the temperature
along the central axis and tl is the temperature at the fuel-cladding inter-

Fig. 9.5.

Radial heat transmission in a clad cylindrical fuel rod.


Heat- Transmission Principles


face, it follows from equation (9.19) that

If tm is the mixed-mean (or bulk) temperature of the coolant surrounding

the rod, the transmission of heat through the cladding to the coolant can
be represented by the thermal-circuit method; thus,

where Rclad , the thermal resistance of the cladding, is identical with Rw in
§9.37 and is given by equation (9.12); Rconv' the convection thermal resistance at the cladding-coolant interface (heat-transfer coefficient = h), is
lIhA, where A is the outside area of the cylinder of length I, i.e., 27Thl. It
follows, therefore, that
t1 -



q(1k,: In ;:;b+ hb1) '


where kc is the thermal conductivity of the cladding. At the interface
between fuel and cladding, where r = a, dtldr = - Qal2k; since the surface
area is here 27Tal, it follows from a heat-rate balance that

and, hence,

This gives the temperature difference between the fuel-cladding interface
and the coolant. The first term on the right (including Qa 2/2) represents
the temperature drop t1 - t2 across the cladding, and the second term
(including Qa 212) is the drop t2 - tm at the cladding-coolant interface.
9.49. Upon adding equations (9.21) and (9.22), the temperature drop
from the center of the fuel element to the coolant is found to be

assuming negligible thermal resistance at the fuel-cladding interface.
Equations for the temperature distribution in the cladding; or for the dif-


Energy Transport

ference between the temperature at any point in the fuel or the cladding
and the coolant can be readily derived, if required.
Gap Conductance

9.50. As shown in Fig. 9.6, fuel rod designs normally provide for a small
annular gap between the uranium dioxide fuel pellet and the cladding. The
gap is initially filled with helium, but during irradiation there is some
dilution of the helium with fission-product gases. More important, however, is the swelling and cracking of the pellet (§7 .172) which tends to close
the gap in a nonuniform manner. The interface heat-transfer problem
therefore progresses from one involving conduction through the gas to a
combination of conduction through the gas and across the interface between two surfaces in partial contact. Also, the gap conductance changes
with the power level and material effects related to fuel exposure (Chapter
7). Although analytical models have been developed based on the thermal
conductance of a gas film of the thickness indicated by the roughness of
the two surfaces, the results tend to be unreliable. Semiempirical computer
models based on test data are therefore used for design purposes [2]. The
gap conductance is also important in the calculation models required for
safety evaluation (§12.128). For orientation purposes, the order of magnitude of the heat-transfer coefficient at the interface of irradiated oxide
pellets and metallic cladding may be taken to be 6000 W/m 2 • K.
Example 9.3. A PWR fuel rod (see Example 9.2), with pellets 8.19
mm in diameter, is clad with zircaloy 0.57 mm thick; the outer diameter
of the clad rod is 9.5 mm. If the bulk (mixed-mean) coolant temperature
is 315°C and the volumetric heat generation rate (power density) in the
fuel is 3.20 x 108 W/m 3 , determine the temperature at the center of the
fuel and at the outer surface of the cladding. The heat-transfer coefficient

Fig. 9.6.

Section through fuel pellet with gas gap and cladding.

Heat- Transmission Principles


at the cladding-coolant interface may be taken to be 34 kW/m2 - K. The
thermal conductivity of the zircaloy is 13 W1m . K.
The temperature drop between the coolant and the cladding is obtained
from the second term on the right of equation (9.22); thus,





108 W/m 3

a = 1/2 (8.19 mm) = 4.095 mm


112 (9.50 mm)


4.75 mm

t2 -

tm =

(3.20 X 108 )(0.004095)2
(2)(3.4 x 104 )(0.00475)


17 C.

Since tm is 315°C, the cladding surface temperature is 315 + 17 = 332°C.
The temperature drop across the cladding is obtained from the first term
on the right of equation (9.22); that is,

where, as shown in Fig. 9.6, a' is the inner radius of the cladding, i.e.,
112(9.50) - 0.57 = 4.18 mm. Hence,


= (3.20


108 )(0.004095)2 I 0.00475
n 0.00418

= 260C


(Use of the approximation for In (bla') in §9.36 footnote leads to essentially
the same result.)
Next, consider the temperature drop at across the gas gap between the
fuel and the cladding. This is given by an expression equivalent to the
second term on the right of equation (9.22), except that b is replaced by
a and h by the gap conductance. As stated in §9.50, the latter may be taken
to be 6000 W/m 2 • K; hence,

= Qa 2 = (3.20 x 108 )(0.004095) = 1090C




Energy Transport

The temperature drop across the fuel itself has been determined in
Example 6.2 as 480°C; hence, the central fuel temperature is given by



+ 109 + 26 + 332

= 950°C.

This is an average temperature; it is expected that in some fuel rods the
central temperature will be significantly higher. The results of the foregoing
calculations are plotted in Fig. 9.7 with the central temperature value
rounded off.
Exponential Heat Source

9.51. In all the preceding examples it has been postulated for simplicity,
since it is a good approximation, that the heat source is uniformly distributed within a conductor. A situation will now be considered in which the
heat source has an exponential distribution. This is sometimes the case in
certain external reactor components, such as thermal shields (§6.90) and
pressure vessels, where the heat generated is due to the absorption of
gamma rays and the slowing down of neutrons. Although the heat-generation
processes are actually quite complex, it may happen that, as the result of
a combination of circumstances, the volumetric heat source can be represented approximately by

where IJ. is an effective linear attenuation coefficient (or macroscopic cross
section) of the radiations (d. §6.199).
Fig. 9.7.

Approximate radial temperature distribution in a fuel rod of a water-cooled




;:) 700





w 600




Heat- Transmission Principles


9.52. If the shield or pressure vessel is a sphere or cylinder of large
dimensions or is rectangular in shape, it may be treated as a slab; equation
(9.15) then takes the form

Provided k is constant, the general solution is

The boundary conditions are (see Fig. 9.8),


Fig. 9.8.


for x






t2 for x



Heat transmission in a slab with exponential source.




Energy Transport

where L is the thickness of the slab. These lead to the solution

for the steady-state temperature distribution in the slab. It may be noted
that the first term on the right represents the linear temperature distribution
due only to the difference in temperature between the two faces of the
slab, whereas the second gives the effect of the exponential heat source.
9.53. Under certain conditions, the combination of the two terms just
mentioned leads to a temperature maximum within the slab. The point at
which this occurs may be obtained by setting the derivative of equation
(9.24) with respect to x equal to zero. The result is

The maximum temperature can be obtained by inserting this value for x
into equation (9.24).
Example 9.4. A water-cooled and water-moderated power reactor is
contained within a thick-walled pressure vessel. This vessel is protected
from excessive irradiation (and, thus, excessive thermal stress) by steel,
cylindrical thermal shields between the reactor core and the pressure vessel.
One of these shields, 50 mm thick, whose surfaces are both maintained at
300°C, receives a gamma-ray energy flux of 1018 MeV/m 2 • s. Calculate the
location and magnitude of the maximum temperature in this shield, which
may be treated as a slab. The coefficient f.L of the gamma radiation (§9.51)
in the steel may be taken to be 27 m -1, and the thermal conductivity as
40 W/m . K.

In this case, t1 and
equation (9.25),


are both 300°C, so that

t1 -


is zero. Hence, by

1 - e - (27)(0.05)
(X)'m.x = - 27 In (27)(0.05)

= 0.022 m (22 mm).
The maximum temperature is thus attained slightly inside from the midradius of the shield.
The value of the maximum temperature is calculated from equation
(9.24) with t1 = 300°C, k = 40 W/m . K, and x = 0.022 m. The value of

Heat- Transmission Principles


Qo is derived from §6.22 as <P'YE'Yf.La, where f.La> the energy absorption
coefficient, is 16.4 m- 1 for steel. Since <p'YE'Y is 10 18 MeV/m 2 • s, it follows

= (1Q18)(16.4) = 1.64 x 1019 MeV/m3



2.63 MJ/m 3 • s.

Hence, from equation (9.24),
t max


t max

_ 300






106 )




-(27)(0.05) _

1}0.022 _ -(27)(0.022)



is consequently 311°C.
Conduction in Irregular Geometries

9.54. The combination of nonuniform heat-source distribution and irregular geometries, which occur in many reactor components, complicates
the calculation of thermal-conduction rates and temperature distributions.
The problem is an important one because high thermal stresses frequently
result from large temperature gradients. A study of the temperature stresses
in reactor components, which frequently have irregular shapes, depends
upon a knowledge of the temperature pattern within the materials. A
number of relatively simple codes suitable for microcomputers are available
to solve multidimensional conduction problems under both steady-state
and transient conditions. A stepwise, finite-difference approach, as described in the following sections for transient heat conduction, is effective.
An example of such a situation is found in graphite-moderated, gas-cooled
reactors. These generally consist of massive graphite blocks pierced by long
holes with cylindrical fuel elements and passages for coolant. Pressure
vessels containing the reactor and thermal and biological shields frequently
present similar difficulties in the determination of temperature distribution
and thermal-stress analysis. For such cases, a rigorous solution of the basic
heat-conduction equation (9.16) is usually impossible. However, approximate methods have been developed in some instances; thus, a circular
harmonics series solution has been proposed for the case of a massive
graphite block treated as equivalent to a number of adjacent cylinders [3].
9.55. Temperature patterns for nonregular boundary conditions may be
determined using numerical "relaxation" methods. The procedure is first
to divide the irregular volume, so far as is possible, into a number of
regular subvolumes. The heat-conduction equation may now be applied to


Energy Transport

these subvolumes in a systematic manner based on an assumed temperature
distribution compatible with the boundary conditions. By carrying out a
heat-balance calculation for each of the subvolumes, the inaccuracy of the
assumed temperature pattern is determined and expressed as a so-called
residual for each subvolume. On the basis of these residuals, the assumed
temperatures are changed in a systematic manner to reduce (or "relax")
the residuals from point to point. The heat-balance procedure and temperature adjustment are repeated until the residuals have been reduced to
zero; the corresponding temperature distribution is then taken to be the
correct one.

Transient Heat Conduction
9.56. For a reactor operating in the steady state, the spatial distribution
of temperature in any component, e.g., a fuel rod, is given by equation
(9.16), assuming the medium to be isotropic and the thermal conductivity
independent of temperature. In a transient situation, however, such as
when the reactor is being started up or shut down, the steady-state condition
is not applicable. In deriving the steady-state equation (9.15) for onedimensional heat flow, of which equation (9.16) is the general form, a heat
(or energy) balance was obtained by equating the difference between the
rates of heat flow into and out of a volume element to the rate of internal
heat generation.
9.57. For the transient situation, the energy balance (or conservation)
must include the time rate of change of internal energy in the volume
element. This is given by cpp(A dx)(dtldO), where cp is the specific heat of
the material and p is its density; A dx is the volume of the element under
consideration (§9.42) and dtldO is the rate of increase of temperature t with
time o. Hence, equation (9.15) becomes
tPt) dx = Q(x)A dx - cppA dx de

-kA ( dx2


where «, called the thermal diffusivity of the material, is defined by

Heat-Transmission Principles


The general form of equation (9.26) for the spatial distribution of temperature in a conducting medium with internal heat generation under transient conditions is
_ 'Pt


Q(r) _ 1. dt


which may be compared with the steady-state equation (9.16).
9.58. In the analysis of transient behavio