the linear reactivity model for nuclear fuel management 0894480359

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The Linear Reactivity Model for Nuclear Fuel Management

The Linear Reactivity Model for Nuclear Fuel Management M. J. Driscoll Massachusetts Institute of Technology

T. J. Downar Purdue University

E. E. Pilat Yankee Atomic Electric Company

American Nuclear Society La Grange Park, Illinois USA

Library of Congress Cataloging-in-Publication Data Driscoll, M. J. The linear reactivity model for nuclear fuel management / M.J. Driscoll, T.J. Downar, E.E. Pilat. p. cm. Includes bibliographical references. ISBN 0-89448-035-9 1. Nuclear reactors-Reactivity-Mathematical models. 2. Nuclear fuels-Management. I. Downar, Thomas Joseph. 11. Pilat, E. E., 1937. 111. Title. TK9202.D74 1990 62 1.48'3 1-dc20 ISBN: 0-89448-035-9 Library of Congress Catalog Card Number: 89-1 8324 ANS Order No. 350014 Copyright O 1990 American Nuclear Society 555 North Kensington Avenue La Grange Park, Illinois 60525 All rights reserved. No part of this book may be reproduced in any form without the written permission of the publisher. Printed in the United States of America

Contents Preface

-

ix

-Nomenclature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. xiii Chapter 1: Introduction

1.1 Foreword 1.2 Background 1.3 The Steady State with Equal Power Sharing 1.3.1 Cycle and Discharge Burnup 1.3.2 Mean Control Poison Concentration 1.3.3 Core Fission Product Inventory 1.3.4 Fixed Discharge Burnup 1.3.5 Fixed Cycle Burnup 1.3.6 Equivalence in Space and Time 1.4 Parametric Behavior of Po and A 1.5 Core Leakage 1.6 Unequal Batch Size 1.7 Alternative Formulations 1.7.1 Burnup Potential 1.7.2 Effective Enrichment 1.8 Summary References Problems Chapter 2: Nonsteady-State Applications 2.1 Introduction 2.2 Perturbations from the Steady State 2.2.1 Cycle Burnup 2.2.2 Reload Enrichment

2.2.3 Reload Batch Size 2.2.4 Cycle Burnup Versus Batch Burnup

~

. . . . .. 1 1 2 7

7 10

11 13 15 17 17 21 27 34 34 35 36 36 38

42 42 42 43 44

48 53

2.3 The Cumulative Effect of Cycle Burnup Perturbations 2.3.1 Single-Cycle Events: One-Time-Only Coastdown 2.3.2 Repeated Coastdown 2.4 The Linear Reactivity Model Applied to Startup Batches 2.5 Summary References Problems

53 53 58 59 65 65 66

Chapter 3: Unequal Power Sharing. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 69 3.1 Introduction 69 3.2 Power Weighting of the Reactivity Balance 69 3.3 A Reactivity-Based Power-Sharing Relation 73 3.4 An Analytical Method for Unequal Power Sharing 77 3.5 The Effects of Burnable Poison Use on Discharge Burnup 81 3.6 The Effects of Neutron Leakage on Discharge Burnup 84 3.7 Combining Several Options 87 3.8 An Application to Various Strategies of Core Reactivity Control 88 3.9 Burnable Poison Physics: The Ash Layer Model 90 3.10 An Application to In-Core Fuel Management: Burndown of Power Peaking 96 3.11 Computer Applications: Unequal Power Sharing and Nonsteady-State Conditions 98 3.12 Summary 99 References 100 Problems 101 Chapter 4: Assembly Power Mapping, . . . . . . . . . . . . . . . . . . . . . . . .. 4.1 Introduction 4.2 Assembly Power Estimation 4.2.1 Power/Reactivity Algorithm 4.2.2 Simple Illustrative Derivation of Power-Sharing Relations 4.2.3 Extension to Two Dimensions 4.2.4 Relation to Conventional Nodal Methods 4.3 -R eload Pattern Calculations 4.4 Suppression of Assembly Power Using Burnable Poisons

103 103 103 103 105 107 109 111 119

4.5 Intra-Assembly Power Profiles 4.6 Thermal Flux and Power Corrections 4.6.1 Thermal Leakage Correction 4.6.2 Power-Related Feedback 4.6.3 Power Versus Source Normalization 4.7 Reload Pattern Optimization 4.8 BWR Fuel Management 4.9 Summary 4.10 Bibliography on Reload Optimization References Problems

Chapter 5: Uranium Utilization 5.1 Introduction 5.2 Uranium Utilization Defined and Illustrated 5.3 Consistent Comparisons 5.4 Test Case Requiring Less Mass per Assembly

120

121 121 122 124 125 128 129 , 130 132

132 . 137

137 137 143 148

5.5 Another Nurnerical Example: Pin Pulling and Bundle Reconstitution 5.6 Other Options for Improving Core Performance 5.7 Adjusted Test Case Approach 5.8 Summary References Problems

Chapter 6.1 6.2 6.3 6.4 6.5

6: Nonlinear Analysis and Other Topics Introduction Derivation of a Nonlinear Reactivity Model CANDU Fuel Management Error Analysis Applications Using k in Place of p 6.6 Other Applications: Spectral Shift Control and the Tandem Fuel Cycle

148

154 155 157 157 159 . 162

162 162 164

167 171

6.6.1 Spectral Shift Control

173 173

6.6.2 The Tandem Fuel Cycle

175

6.7 Other Sources of Error 6.7.1 Spectral Ambiguity: Coarse-Group Aspects 6.7.2 Spectral Ambiguity: Fine-Group Aspects 6.7.3 Parametric Invariance 6.8 Sumrnary References Problems

176 176 180 183 184 184 186

Appendix A: Mathematical Supplement. . . . . . . . . . . . . . . . . . . . . . .. 190 Appendix B: Burnup Isotopics: The BRICC Program . . . . . . . . . . . .. 201 Appendix C: Cycle-by-Cycle Burnup: The BRACC Program. . . . . . .. 212 Appendix D: Core Power Distributions: The RPM Program. . . . . . .. 221 Index

'. . . . . .. 229

Preface The generation of electricity in central station nuclear power units has been a global reality for more than 25 years, and the provision of fuel to these devices has grown into an industry having multibillion dollar annual receipts. During this time, a state of relative maturity has been achieved in fueling practices and methods of analysis. As in so many other fields of endeavor, complex computer programs are now used to carry out state-ofthe-art design calculations. While this improves precision, it does not necessarily foster understanding. Thus, there is a role for simpler models describing nuclear reactor core behavior, such as the one that is the subject of this book-a collection of algorithms and methods collectively designated "the linear reactivity model" or LRM. We have found it useful in the following ways: 1. as a teaching tool, to explain the basic principles of nucl ear fuel management 2. in research, as a computational module in intermediate- and longrange fuel management studies, where detailed methods are too cumbersome and slow; it also facilitates making comparisons on an "all-else-being-equal" basis, which is sometimes difficult to do with more elaborate methods (where, for example, it may be difficult to ascertain that one has " equally optimized" core loading patterns) 3. as a scoping technique to evaluate alternatives and reduce the number of cases subjected to detailed scrutiny using more expensive methods 4. to infer parametric behavior, for correlating the results of more complex computations, and for interpolation among them-which again saves time and money; for example, by decreasing the number of cases needed to map out a region of interest, and by decreasing the number of iterations in a trial-and-error analysis 5. as a quality control device, to help spot errors in the use of complicated design codes, which can be particularly helpful if one prevents mistakes early in the sequence of computations. Just the first of these applications provides an adequate rationale for looking into the LRM; and, indeed, simple tutorial versions of this general approach can be traced back to the 1950s. However, with the notable exception of Egan's lengthy paper in the 1984 issue of Progress in Nuclear Energy, previous publications have been either limited in scope or focused on quite specialized aspects such as the convergence of refueling sequences. ix

This lack has helped motivate the preparation of this text, which organizes and expands on developments hitherto recorded in somewhat obscure research reports and in an unpublished set of notes used in a graduate-level subject at the Massachusetts Institute of Technology. Several points should be made with regard to th e intended audience and scope of coverage of this text. It is written for both students of nuclear engineering (at the senior or graduate level) and for practitioners in the fuel management field. It presupposes completion of an elementary course in reactor physics and a general knowledge of reactor design and the ex-core fuel cycle. Such background material had to be excluded to permit the indepth treatment of in-core behavior. We have similarly omitted the important subject of fuel cycle economics, even though a primary application of the LRM is unquestionably to develop input to such calculations. The format is in the general style of a textbook, suitable for use in a fuel management course (when appropriately supplemented) or for selfstudy. Thus, many solved examples are incorporated, and we have also included problem sets at the end of each chapter. (A solutions manual is available to instructors from the publisher upon request on official departmental letterhead.) Furthermore, because of this general pedagogical predisposition, parametric variations and design or operational options beyond the narrower (and narrowing) range of current practice are examined. While the LRM is more widely applicable, attention is focused on the pressurized water reactor since it accounts for some two-thirds of the nuclear capacity installed or on order in the United States and half of the capacity worldwide. Most of the material is also directly applicable to the boiling water reactor (BWR), the next most prevalent reactor type, although more remains to be done on specific aspects of BWR/LRM neutronics. Part of Chapter Six deals with the pressurized heavy water (i.e., CANDU) reactor-the third leading candidate for global reactor sales. Too little has been done to apply the LRM to other reactors of current interest, such as the high-temperature gas-cooled reactor and the liquid-metal fast breeder reactor, to merit attention here. Likewise, attention is focused on the once-through fuel cycle because it dominates current practice. Finally, three small, interactive microcomputer programs (IBM-PC compatible diskette included with the text) have been coded to provide the reader with a convenient means for further evaluation and application of the basic principles outlined in the main text. The authors gratefully acknowledge the essential contributions of many colleagues to both the research base for this work and all of the subsequent stages in its evolution into a finished book. Particular thanks are due Galal Abu-Zaied, Manson Benedict, Khalid Elmediouri, Altamash Kamal, PinWu Kao, Peter M. Lang, Samuel H. Levine, Wee Tee Loh, Mushtaq A. Malik, Regis A. Matzie, Eduardo Montaldo-Volachec, Dave Phegley, Ildo L. Sauer, Matthew J: Siegel, John Stevens, Wendell D. Wagner, and Alan L. Wight. Finally, special thanks are due to those who reviewed and conx

structively critiqued the prepublication draft on behalf of ANS: J. D. Lewins, Sam Levine, and Dan Meneley. Our apologies to others, too numerous to mention, but whose sum total 'of contributions was certainly of comparable benefit. We, the authors, of course bear the sole responsibility for whatever shortcomings remain. M. J. Driscoll T. J. Downar E. E. Pilat

February 1990

xi

Nomenclature

Major variables used in the body of this monograph are defined below, and references given to the chapter/section/table/equation that discusses the item in question in the most definitive manner. Occasionally the same symbol is used for more than one variable, in which event only the principal uses are noted. Refer to Appendixes B, C, and D for variables employed in the BASIC programs described there. Entries are in alphabetical order, first Roman, then Greek. Dimensions (if applicable) are indicated in parentheses following the definition.

Definition

Symbol

A

B

B2 D

f F

g

h H 1

(1) slope constant in linear p(B) approximation, p = Po - AB (kg/Mwd) (2) neutron loss rate by absorption (1) burnup, thermal energy generated per unit mass of heavy metal charged (MWd/kg); with subscripts: a.b,c, . .. = values appropriate to successive cycles, a,b,c, ... c = cycle; average burnup increment of entire core over one operating cycle d = discharge, burnup of fuel batch discharged 1 = burnup of a batch-loaded core (2) buckling (cnr-') neutron diffusion coefficient (ern) power fraction; fraction of core power delivered by batch in question; or power of assembly relative to that of core-average assembly (1) neutron production rate by fission in fuel batch in question (2) natural uranium feed to enrichment plant (kg) grams width of node (usually an assembly) (em) active height of fuel assembly (ern) index (subscript) used to denote j'th batch or

cycle xiii

Refer to:

Sec. 1.2 Sec. 3.2 Sec. 1.3

Sec. 2.2.1

Sec. 4.2.3 Sec. 3.3 Sees. 3.2, 3.3 Sec. 3.2 Sec. 5.2 Sec. 3.9 Sees. 3.3, 4.2.1 Sec. 1.5 Throughout

I J

J

k L

M

n N p p

q

uranium utilization index, ratio of uranium utilization in test case to that in a reference case index (subscript) used to denote j'th batch or cycle net neutron current (n cm? S-I) neutron multiplication ratio, usually with subscript " 00" to denote infinite medium thermal neutron diffusion length (em) (1) fast neutron migration length (em) (2) mass (kg) number of batches used in staggered reload refueling scheme (1) same as n, or (2) number of assemblies in a given batch subscript denoting batch (1) mass of heavy metal in fuel batch (metric . tons, MT) (2) probability thermal power generated by fuel batch of assembly in question (MW) or power density

Sec. 5.3 Throughout Sees. 1.5, 3.9 Sec. 1.2 Sec. 4.6.2 Sec. 3.3, Eq. (3.14) Sec. 1.3.2 Sec. 1.3.1

Sec. 1.6 Table 2.6 Sec. 1.3.3 Eq. (1.32) Sec. 3.2

(MWjcm 3 )

Q r r2 R

s

S t V V

JiV

total thermal power generated by core (MW) radial position within fuel or burnable poison pIn coefficient of determination: measure of goodness of least-squares curve fit (1) number of faces exposed to reflector nodes (2) core radius (em) volumetric neutron source rate (n S-I per unit volume) total neutron source strength in region in question (n S-I) subscript denoting test case uranium utilization (MWdjkg U N AT ) volume fraction of core constituent, designated by subscript: f = fuel, C = cladding, m = moderator FLARE kernel xiv

Sec. 3.2 Sec. 3.9 Sec. A.3 Table 4.1 Eq. (1.30) Sec. 1.6 Sec. 1.6 Sec. 5.3 Sec. 5.2 Sec. 1.4

Sec. 4.2.4

x

y

enrichment of fissile species (usually 235U) in heavy metal charged to reactor (wt%) with subscripts: f = natural uranium feed p = reactor reload fuel w = enrichment plant tails o = intercept of linear (Po/A) versus x p relation yield, number of nuclei per fission of fission product species (xenon, samarium, nonsaturating)

Sec. 1.4

(1) volumetric expansion coefficient (2) proportionality constant between enrichment and reactivity (3) capture-to-fission ratio (4) albedo coefficient in burnup versus enrichment correlation (1) proportionality constant: mass of fission products per unit mass of heavy metal fissioned (2) conversion factor (short tons U 30g/kg uranium) (1) fraction of batch removed (2) composite loss fraction in processing (3) net increment operator (4) fertile/fissile fission ratio difference operator fractional increment in burnup due to perturbation in core design characteristics or fuel management tactics position coordinate in leakage kernel adjustable parameter characterizing interassembly power sharing energy yield per fission (MeV/fission) radioactive decay constant, In 2/half life (yr ') proportionality constant: mass of control ab-

Eq. (1.26) Eq. (2.41)

Sec. B.3

Greek: a

~

E

S (0

K

A j..t

Eq. (6.27) Sec. 4.2.4 Eq. (1.25) Sec. 1.3.3 Eq. (5.1) Sec. 2.2.3 Sec. 5.2 Eq. (2.25) Eq. (6.27) Throughout Sec. 3.4

Eq. (1.27) Sec. 3.3 Sec. 4.6.3 Appendix B Sec. 1.3.2

sorber per unit of reactivity control v

neutron yield per fission (neutrons/fission) xv

Sec. 4.6.3

p

a L

cp

(0

reactivity, fractional excess of neutron production; usually with subscripts: o = extrapolated value at zero burnup 00 = value appropriate to infinite medium c = cycle L = component associated with leakage from active core region s = mean reactivity of system microscopic cross section for neutron interaction (em 2) or (barns = 10-24 cm 2) (1) macroscopic cross section for neutron interaction (cm') (2) summation over indices indicated neutron flux (n cm? S-I) with subscripts: I or f = fast group 2 or th = thermal group burnable absorber loading (wt% or g/pin)

Special Symbols: Z' denotes perturbed value of Z Z denotes average value of parameter Z Z denotes maximum value of parameter Z Z denotes average value of Z over entire in-core exposure until present Laplacian operator

xvi

Sec. 1.2

Appendix B Appendix B Throughout Sec. 3.3

Sec. 3.9 Sec. 2.2.2

Sec. 3.4 Sec. 3.3

Chapter One Introduction 1.1 Foreword As far back as the late 1950s the observation was made that reactivity (or multiplication factor) was a very nearly linear function of burnup for light water reactor (LWR) lattices, and this fact was put to use to develop a simple prescription relating steady-state discharge burnup and the number of staggered-reload fuel batches. 1 Analysis seldom went beyond this elementary application, with the notable recent exception of Egan's lengthy article. 2 Even more can be done, however, and this book's purpose is to document how the linear reactivity model (LRM) can be extended to demonstrate most of the important principles involved in fuel management strategy. In this initial chapter the groundwork for subsequent embellishment is laid. This book focuses on LWRs and, in particular, on pressurized water reactors (PWRs), because of the dominant position of this type of reactor in the current market, as shown in Table 1.1. In Chapter 6 the methodology is extended to include pressurized heavy water (PHWR) units-the next most prevalent concept-and there is no obvious limitation preventing applications to other designs. As we progress, attention is called to the solution of three basic problems: 1. estimation of burnup given batch enrichment and core fraction (Section 1.3.1) 2. determination of the reactivity (enrichment) of a fixed number of reload fuel assemblies to achieve a prescribed cycle or discharge burnup (Section 1.3.4) 3. specification of the number of assemblies (not necessarily an integer subdivision of the in-core total) of fixed enrichment to achieve a prescribed cycle or discharge burnup (Section 1.6). In the course of addressing these tasks, several subtopics will be examined with the objective of providing insight into the nature of reactor core behavior. ~ . 1

2

The Linear Reactivity Model

TABLE 1.1 Nuclear Power Plants of the World* T ype (Coolant/Moderator)

PWR LWR

(H 2O)

BWR

United States

84 }

99%

40

World 323 }

77%

106

PHWR (CANDU) PHWR (PV) (D 2O) LWCHWR HWBLWR

42 2 1 2

GCR AGR

32 14 (Graphite)

LGR HTGR LMFBR

1

8

(Sodium)

Total units TotaIGW(e) Total operable G r¥(e) Operable

28 2

125 114

560 434

108 95

417 308

* Op erable, under construction, or on order [> 30 MW(e)] as of December 31, 1988. Data from Nuclear Ne ws, Vol. 32, No.2 (Feb. 1989). Key. PWR = pressurized water reactor BWR = boiling water reactor PHWR = pressurized hea vy-water-moderated and cooled reactor; PV = pressure vessel type; CANDU = pressure tube type LWCHWR = light-water-cooled heavy-water-moderated reactor HWBLWR = heavy-water-moderated boiling light-water-cool ed reactor GCR = gas-cooled reactor AGR = advanced gas-cooled reactor LGR = light-water-cooled, graphite-moderated reactor HTGR = high-temperature gas-cooled reactor LMFBR = liquid-metal-cooled fast breeder reactor

1.2 Background Reactivity is a fraction defined as the excess of neutron production (production minus destruction) per total neutron production: (1.1 )

Introduction

where k is the effective neutron multiplication factor and the cross sections refer to fission and absorption (including leakage). Reactivity is an integral quantity, and the cross sections in Eq. (1.1) must be interpreted as appropriate averages over the neutron energy spectrum and, strictly speaking, over all space. In what follows, however, a "reactivity" will be associated with a finite mass of fuel. This assignment is a source of ambiguity and error, since only in an infinite medium are we sure that a unique self-determined neutron energy spectrum is present; conversely, a very small mass of material may have its spectrum wholly determined by the properties ofits surroundings. Fortunately, LWR assemblies (typically the smallest fuel mass unit used in fuel management) are large enough that local and infinite medium reactivities are sufficiently close in magnitude. Radkowsky has carried out studies of seed-and-blanket core configurations that support this approximation.' Hence, reactivity can be computed for an infinite medium and the global effect of leakage accounted for separately (see Section 1.5). Alternatively, part of the leakage-for example, axial leakage-can be incorporated into the reactivity computation and only the radial portion treated explicitly. A second major assumption is that reactivity is a state function of burnup-in other words, a unique single-valued function of the specified burnup, and not of the detailed history of the manner/in which the burnup (B) was accumulated. This is a fairly good approximation for PWR fuel, but less so for boiling water reactor (BWR) assemblies, whose neutron spectrum is a function of void history. Fortunately, many competing neutronics effects cancel over the entire in-core history of a fuel assembly, such that accurate estimates of the discharge burnup (although not necessarily cycle-by-cycle values) can still be made. Even for PWR assemblies, the state-function approximation is subject to imprecision. A reactor is operated such that its reactivity is always identically zero; the concentration of a neutron control poison is adjusted to ensure that this is so. The LRM, however, is based on the use of the potential reactivity of an assembly-that is, the value computed in the absence of control poison. Strictly speaking, then, one should not compute pCB) with poison removed, but with poison present in the correct, variable amount up to the time at which B is achieved. This is a tedious prospect and would be application-specific (hence, violating the concept of a "state function"); thus it is common to compute pCB) in the presence of cycle-average poison, or in its complete absence. Sefcik" shows that the use of an estimated invariant steady-state cycle-average poison value is acceptable. Aside from the niceties of definition, the most prominent assumption of the LRM is, as its name suggests, the assertion that reactivity is a linear function of burnup," a

See Chapter 6 for an extension to nonlinear pCB) behavior.

3

4

The Linear Reactivity Model

P

=

(1.2)

Po - AB,

where A is some constant. The intercept Po in Eq. (1.2) is determined by extrapolation; it corresponds to the value after saturating fission products (xenon and samarium) have come to equilibrium. The slope A of the linear function is a strong function of the fuel conversion ratio and hence unique for each combination of initial fuel enrichment and fuel-to-moderator ratio (see Section 1.4). .All smooth curves can be approximated by a linear segment over a sufficiently short interval; the remarkable fact is that Eq. (1.2) holds quite well for LWR fuel over the entire burnup span of practical interest. This observation is best regarded as empirical; indeed, Loh? cites reasons for nonlinearity in certain components such as the reactivity penalty imputed to fission product accumulation-all of which implies a fortunate compensation of competing effects. This is substantiated by the pCB) plots for total fission products and end-of-chain heavy nuclides in Ref. 6, reproduced here as Fig. 1; 1. Figures 1.2 and 1.3 from Ref. 5 show pCB) plots for various

0.140 Total Fission Products

0.120

J::

1: 0.100

o

~

->-

.~

0.016

o m

Q)

c:

~ 0.012

m

C)

Q)

z

0.008

End-of-Chain Heavy Nuclides

0.004

1

234 5 Burnup (10 MWd/kg)

6

Fig. 1.1 Reactivity worth of fission products and end-of-chain heavy nuclides versus burnup (from Ref. 6).

Introduction

fissile/fertile combinations. As can be seen, all are linear with the exception of plutonium/thorium. Because the once-through fuel cycle appears likely to dominate commercial practice for the next several decades, 235U fuel is

0.3

Fuel Type

.A. 233UjU02 0.2

e235UjU02

a.

-

.Pu/U0 2

>-

~ 0.1 (.)

ctS

(1)

a: 0.0

-0.1

t-------+------+---="""":::----+----""ooo,....--------1

0

10

20 Burnup, B (MWd/kg)

30

40

Fig. 1.2 Reactivity as a function of burnup for various fissile materials in V0 2 for a representative PWR lattice.

Fuel Type

0.2

0.1 a.

>::::

.A.

233UjThO

•

235U/Th0

•

PuITh0 2

2 2

>

U 0.0 ctS

r---------+-----.,;:::..-~:::.....~.____--+-----____l

(1)

c: -0.1

-0.2

L-

o

....L-

--'-

10

20 Burnup, B (MWd/kg)

---L...

30

---l

40

Fig. 1.3 Reactivity as a function of burnup for various fissile materials in Th0 2 for a representative PWR lattice.

5

6

The Linear Reactivity Model

of the most interest here. Figure 1.4 shows pCB) histories for various enrichments of such fuel in a representative PWR lattice. It should be noted that other authors prefer to correlate k(B) instead of pCB) as a linear function of B. This is very nearly equivalent, since k = 1/(1 - p) ~ 1 + p, for small p. However, visual examination and goodnessof-fit computations have consistently shown Eq. (1.2) to be superior and, lacking other reasons to prefer k(B) , all of our work is based on reacti vity. Finally, space does not permit an extended discussion of the means by which pCB) traces are to be computed. Almost all of the results discussed in this text were calculated using an Electric Power Research Institute version of the LEOPARD program"; the cross-section set supplied with this program was developed from ENDF/B-IV. Despite its antiquity, a recent intercomparison has shown that LEOPARD remains competitive with newer state-of-the-art methods." Table A.4 of Appendix A demonstrates the excellent linearity ofa representative pCB) trace generated using LEOPARD. A simplified fuel isotopics program, BRICC, is described in Appendix B with which pCB) traces can be generated using a microcomputer. Table B.l of Appendix B demonstrates the excellent linearity of a wide range of pCB) traces generated by BRICC. However, it is assumed that more sophisticated methods will be available when using the LRM for other than pedagogical purposes. 0.3

0.2

0.1 Q.

~

>

:;:: 0

co Q)

0.0

c::

5.0wt%

-0.1

-0.2

4.34wt% 3.5wt% 3.0wt% 2.0 wt% 2.5 wt% ~----::.l::------:~-----:::l=----=-=---*,---;:

Fig. 1.4 Reactivity as a function of burnup and lattice.

mu enrichment for a representative PWR

Introduction

1.3 The Steady State with Equal Power Sharing We will return to examine various details later, but at this point it seems appropriate to derive the "classical" result for which the LRM was first conceived. 1.3.1 Cycle and Discharge Burnup In the following, we assume that a given batch of fuel is inserted into a host core maintained at k 1.0, and that the new fuel is itself kept at P == a by inclusion of sufficient control poison. As burnup proceeds, poison is removed until none remains; the core has then reached the reactivity-limited end-of-cycle condition, and part of the burned fuel must be replaced. The excess reactivity available from the new fuel batch (and the equal-in-magnitude, but opposite-in-sign control poison reactivity) is assumed to vary linearly with burnup. It is ofinterest to analyze the steady-state behavior of a core made up of n equal batches of fuel, l/n'th of which (the most highly burned batch) is replaced by fresh fuel at the end of a burnup cycle. Total fuel residence time in the core is thus n cycles. First, it is instructive to note the simple case where the core consists of a single batch of fuel (n = 1). The reactivity potential of the fuel assembly decreases linearly with burnup, that is:

=

p

o B

or

P

=

Po - AB .

(1.3)

In a simple one-batch irradiation, the end-of-reactivity-life burnup, B), is attained when the reactivity of the batch, P, falls to zero in Eq. (1.3): p

=

0

=

Po --:- AB

(104)

7

8

The Linear Reactivity Model

or

where Po = reactivity to be held down by control poison at beginning of life. For the 'general case, it is assumed that all fuel operates at the same, core-average power density. Then for an n-batch core at steady state, at the end of a burnup cycle the freshest batch will be burned, B o the next older 2Be, etc.; and the oldest batch, which is ready for discharge, is nli, = B d , whcre .zi, is the burnup of the batch at discharge. The mean reactivity of the mixture, p; can be computed merely by averaging batch reactivities": (1.5)

The core reactivity, Ps, from applying Eqs. (1.3) and (1.5) to an n-batchcore is then

PI = Po - ABc pz = Po - 2ABe

Pn = Po - nABe Ps -_ n1 [ npo - n( n 2+ 1) AB] e -- 0

(1.6)

n

. whiICh t h e summation . ~ . n(n + 1) bid In ~} = - - - - has een emp aye . j=1 2 Solving for the burnup, (1.7)

Note that this result applies to the use of a fixed assembly design (same Po and A) in cores subjected to different multibatch management schemes. As is shown in Section 1.4, both Po and A are functions of the assembly initial enrichment and fuel-to-moderator ratio, An alternative tactic would be to In Chapter 3 we show that source (= power) weighting is more appropriate, but generally does not have a large effect on discharge burnup.

b

Introduction

fix cycle burnup and adjust enrichment and hence Po (an option discussed in more detail in a later section of this chapter) or to adjust the number of assemblies changed. In a real life situation, a number of practical considerations govern the choice of strategy: 1. To allow refueling in low peak load seasons (spring or fall), 12-, 18-, or 24-month refueling intervals are preferred. 2. There is an economic optimum burnup, which, in the case of fuel cycle cost, depends on the prevailing costs for the various fuel cycle transactions (e.g., cost of uranium, enrichment, fabrication, and back-end costs) and, in the case of system costs, depends on the makeup energy costs during refueling downtime. 3. License limits in the near term and ultimately water side corrosion of cladding must be respected (usually specified by assembly discharge burnup limits). When the most-burned batch is replaced with a fresh batch, the beginning-of-cycle (BOC) reactivity (hence, the reactivity swing over that cycle and the corresponding control poison requirement) can be obtained by redoing the previous sum with one less cycle of exposure for each fuel batch, 1 {[ npo - n(n 2+

~Pc = ~

1)] ABc + nABc}=

ABc

or 2

Ap, = n

+

(1.8)

1 Po .

For fixed reload enrichment the discharge burnup is increased by the factor (2nln + 1); hence, the length of time between refue1ings (and cycle energy) varies as (21n + 1) = .1Pc!Po- Thus, we have the relative behavior:

1

Number of Batches, n ') 3 4 5

Discharge burnup, BnlB I

1

1.33

1.5

1.6

1.67

2

Cycle length and poison requirement, ~pclPo

1

0.67

0.5

0.4

0.33

0

~

Figures 1.5 and 1.6 present this information in graphical form.

00

9

10

The Linear Reactivity Model

Dis charge Burnup : Bdn = . Cycle Burnup:

(;:1) Bdl

Bdn 2 BC=-n- = n+1 B,

Q...

- p* 0

Cycle Reac tiv ity Swing : f::,.p = 2+ P C n 1 0

>. ..-

>

..u o .

>

*

P03

..u 0

Q)

n::

Burnup, B Fig. 1.7

Adj ustment of initial reactivity to keep discharge burnup co nstant.

B 1/3. For the limiting case of continuous refueling, n = 00, the same discharge burnup, BJ, is achieved with an initial reactivity of Po", or a decrease compared to the single-batch core of

Ap;

=

lim ( n_oo

1

n -

2

n

P OI

)

=

POl/2

(1.22)

The enrichment savings achieved by increasing the number of fuel batches can be computed directly from the reduction in initial reactivity. In the range of interest, initial reactivity varies approximately linearly with reload enrichment. For a representative PWR lattice,

Po

~

0.1 (x, - 1.0) ,

(1.23)

where x p = reload enrichment, 235U by weight percent. Equations (1.20) and (1.23) are readily applied to estimate the enrichment savings achievable by increasing th e number of batches. For example,

Introduction

consider a three-batch core having x p 0.20. From Eq. (1.20), *

P04 =

*

P03

=

3.0 wt%. From Eq. (1.23), P03

+ 1) (42·4 (32 +. 31)

Hence P~4 = (30/32) (0.20) = 0.1875, and from Eq. (1.23), x p4 = 2.875 wt% 235U, a savings of 0.125 wt% 235U. Note that, for simplicity, the assumption of a constant reactivity versus burnup slope, A , has been retained, when in fact it is also a function of the fuel enrichment. This assumption will be . relaxed in Section 1.4. 1.3.5 Fixed Cycle Burnup Since utility and operating staff generally prefer to keep refueling intervals to one year or longer (and in six-month increments), it is likely that cycle length (hence, cycle burnup) will be kept fixed when other changes in fuel management tactics are contemplated. Figure 1.8 illustrates the manner in which the same cycle burnup can be achieved for one- and three-batch cores. For the single-batch core, Be = B d l = B and therefore Be = PallA. From Eq. (1.6), Be = (21n + 1) P~nIA; therefore, * Pan

=

(n +2 1)

POI

where again we have assumed that A is not a function of enrichment. For

> .u o Q)

0::

POI

Cycle Burnup

Burnup, B Fig. 1.8 Adjustment of initial reactivity to keep cycle burnup constant.

15

16

The Linear Reactivity Model

the three-batch core (n = 3), Eq. (1.24) gives P~3 = 2pOb which is shown in Fig. 1.8. Thus, for the three-to-four-batch transition of Section 1.3.4, we would now have

* P04

-=

* P03

Hence P~4 = (5/4)(0.20) = 0.25 and x p 4 = 3.5 wt% 235U; the enrichment would have to be increased 0.5 wt% to keep cycle length constant-and, of course, discharge burnup would be increased by a factor of 4/3. The linear reactivity model results for this and the preceding sections are summarized in Table 1.2.. TABLE 1.2 Summary of LRM Results for Steady-State Cores Constraint

Parameter

Fixed Po

Discharge burnup, B d

BOC EOC

-
-

..

Fig. 4.2

continued Top: N o. 9, in (A)-C(A)-OUL Bottom: NO.1 0, in (A)-out-C(A).

117

118

The Linear Reactivity Model

TABLE 4.3 Summary of RPM Results* Reload Configuration Results: IIV C Characteristics

Number of Assemblies in Batch #

Arrangem ent Number

Description

1 2 3 4 5 6 7 8 9 10

Ou t-In (S) Low Le ak age Pure Scatter Very Low Le aka ge Out-In(A )-C(A ) C(A)-In(A)- Out C(A )-Out-In (A ) Out(S)-CCA ) In(A)-CCA)-Out In(A )-Out-C (A )

1 65 65 65 64 65 64 64 65 64 65

2 64 64 64 64 64 64 64 64 65 64

3 64 64 64 65 64 65 65 64 64 64

Average Assembly Pow er in Batch #

1 0.899 1.115 0.956 0.2 65 0 .429 0 .104 0.327 0.4 73 0.4 60 0.552

2 0.908 0.286 1.002 1.379 1.237 0.507 0.089 1.183 1.05 6 0.77 9

3 1.194 1.597 1.04 3 1.350 1.342 2.367 2.5 16 1.352 1.483 1.676

Peak Assembly Po ison

1.482 2.735 1.854 2.478 1.827 3.8 14 4. 339 1.809 1.965 2.21 4

p

0 .091 3 0 . 1303 0 . 1068 0 .1574 0. 1126 0.209 1 0 .204 8 0 .120 5 0 .14 85 0 .14 18

Lea ka ge

p

0.0399 0.00 82 0.0 16 1 0.0066 0.0435 0.0024 0.0021 0.0342 0.0 123 0.0226

Reload Confi guration Results: EVC Characteristics

Arran gement Number

Description

1 2 3 4 5 6 7 8 9 10

Out-In (S) Lo w Leakage Pure Scatt er Very Low Leak age Out-In (A)-C (A) C(A )-In (A)-Out CCA )-Out-In(A ) Out(S )-C(A) In (A )-C(A )-Out In(A )-Out-C (A )

Batch Bumup MWd/kg

1 9.22 12.11 10.29 5.4 0 5.74 3.6 1 6.73 6.44 6.60 7.75

Avera ge Assem bly Power in Batch #

2 1 2 3 3 10.94 1.037 1.007 9.67 0 .956 5 .69 16.04 1.031 0.723 1.246 10.89 11.56 0.93 1 0.994 1.076 17.11 17.46 0.544 1.185 1.267 12.45 12.7 9 0.685 1.179 1.141 10.39 25. 13 0.447 1.078 1.468 3.49 23.30 0.827 0.533 1.631 12.43 13.80 0.71 2 1.105 1.188 15.24 18.34 0 .524 1.219 1.253 9.71 18.53 0.743 0.8 4 3 1.418

Core Average Cycle Bumup Le ak age p MWd/kg

9.94 11.29 10.91 13.34 10.30 13.11 11.24 10.86 13.40 11.97

0. 034 2 0 .0229 0.0261 0 .015 3 0.038 2 0 .0126 0.0 164 0 .033 1 0 .0 147 0.0255

• T he RP M prog ram acco m pa nying thi s le xI m a y nOI exa c tly re prod uce the res ults give n here.

Although the RPM results exhibit all of th e qualitative features expected of the various loading patterns, the relatively crude nature of its algorithms must be kept in mind. Beyond that, no attempt has been made to optimize each pattern; other arrangements exist that would also satisfy one's conception of an "annular ring" or "checkerboard" zone consistent with the definition of the ten categories examined here. Finally, the foregoing should also convey the immense complexity of the process involved in arriving at the " best" loading pattern, particularly if the additional degree of freedom of variable burnable poison loading in fresh assemblies is permitted.

Assembly Power Mapping

4.4 Suppression of Assembly Power Using Burnable Poisons The function of burnable poison (BP) has already been discussed in Chapter 3 in some detail. Here only the aspects associated with core power profiles are examined. Consider the loading pattern of Fig. 4.2(No. 3), in which the fresh fuel assemblies do not contain BP; as can be seen, peak assembly powers are higher than usually tolerable, particularly in the fresh assembly of row I , column 3. This suggests that BP be added to suppress the beginning of cycle (BOC) powers. The power-sharing relation of Table 4.1 can be manipulated to yield an approximate value for the BP reactivity needed to suppress the power of assembly i such that it equals the average of its neighbors: LlpBPi ~

"8I (J; \l

- I)

(4.29)

This equation is derived neglecting the effect of changes in assembly i on its neighbors. Nonetheless, for isolated changes it can be quite accurate. Figure 4.3 shows a rerun of the core of Fig. 4.2(No. 3), this time with BP in the fresh assembly of position (1,3) computed from Eq. (4.29). The BOC power peaking is considerably reduced, but as indicated by the end of cycle (EOC) power, the BP defers peak conditions until later in the cycle, in contrast to the behavior of the unpoisoned fresh assemblies. Because of the approximate nature of Eq. (4.29), the results exhibited in Fig. 4.3 are not the best that can be attained. If desired, Eq. (4.29) could be applied to all the fresh assemblies to fine-tune the PBP loadings in an iterative fashion and further improve the power distribution in Fig. 4.3. In practice, all BP loadings must be adjusted to fit into a few standard Assembly Type -

BOC Powe r - - - EOC Power----

1 0 .93 8 1.080

2 1.065 1.193

3' 1.229 1.428

1 1.265 1.156

2 1.368 1.1 60

3 1.552 1.3 04

1 0.973 0 .9 64

2 0.552 0.663

1 1.014 1.076

2 1.240 1.2 11

3 1.54 5 1.308

1 1.264 1.078

2 1.291 1.14 4

3 1.142 1.134

1 0.538 0.645

1 1.061 0.979

1 1.18 2 1.03 0

3 1.489 1.258

1 1.077 1.02 1

2 0.877 0.958

3 0.585 0.759

2 1.237 1.085

2 1.244 1.138

3 1.195 1. 195

1 0 .6 65 0.791

2 0.382 0.525

1 0.914 0.941

2 0.830 0.95 8

0 .640 0.8 55

1 0.5 34 0.706

3 0.437 0.653

' bp 8 P = 0 .12

Fig. 4.3

3

The effect of BP on the scatter power distribution (rerun of pattern 3 from Fig. 4.2).

119

120

The Linear Reactivity Model

categories (e.g., four , eight, or twelve BP pins/assembly) and then we must make the best of the resulting loading patterns. Combined with the use of two or more enrichments per reload batch, th e resulting flexibility is generally adequate, and very little advantage is surrendered compared to the use of continuously variable enrichments and poison loadings.

4.5 Intra-Assembly Power Profiles While the present analysis has been terminated at the assembly-average lev el of detail, the reader should be aware that licensing-level analyses require computations down to the pin-by-pin level. Furthermore, such analyses must also be carried out for various transients, which more often than not prove more limiting than the steady-state conditions that have virtually monopolized our attention in this text. While beyond the scope of this book, several points are worthy of note with regard to the extensibility of the present work: 1. Most fuel integrity limits, steady state or transient, can be related

to pin linear power in kilowatts per meter. To a good first approximation, peak pin power at interior core positions can be estimated as the product of the precomputed, isolated assembly peak-to-average pin power ratio and the assembly relative power from the . RPM calculation. At the core periphery, steep power gradients invalidate such a technique. However, for core reload design purposes, the peak pin power usually occurs at interior core positions (especially in low-leakage core designs). Intra-assembly pin power peaking ratios are shown in Fig. 4.4 for a large PWR at BOC. Note, however, that RPM is not a licensing qualified program, and that man y details (such as the number of BP pins) affect the pin-toaverage power ratio appropriate to a given assembly. Fuel Type -_ Ratio of Peak Pin_ to Assembly Average Power

Fig. 4.4

1

1

1

2

1

2

1

3

1.05

1.06

1.06

1.18

1.06

1.18

1.08

1.4 5

1

2

1

2

1

2

3

1.05

1.17

1.06

1.16

1.07

1.16

1 .40

1

2

1

2

1

3

1.07

1.17

1.06

1.1 6

1.09

1.51

1

2

1

2

3

1.06

1.16

1.07

1.20

1.64

1

2

3

1.07

1.20

1.35

3

3

1.20

1.85

Intra-assembly PWR pin power peaking ratios at BOC (from Ref. 7).

Assembly Power Mapping

2. Finally, note that the algorithm of Table .4.1 can be applied to clusters of unit cells by going to the limit e = 1.0 (i.e., h2 < < AP). Hence, a modified version of RPM could be developed (i.e., using "mirror" boundary conditions all around) that would be able to estimate pin-by-pin powers within a single assembly. The results are only qualitatively significant, however, since on a unit cell level thermal neutron leakage is significant; thereby invalidating the basic assumption of one-and-a-half-group theory to an extent for which our crude post hoc correction cannot compensate. Reference 9 provides a useful review of licensing-related considerations for PWR cores.

4.6 Thermal Flux and Power Corrections It may appear remarkable that something as simple as the FLARE approach

described in Section 4.2.4 can be as accurate as it has pro ven to be. Since their inception, considerable work has been done to improve the capabilities of nodal methods; some in the nature of post hoc, add-on corrections, and, more recently, approaches that are more rigorous in an analytical sense. Nevertheless, it should be appreciated that much of the success of nodal methods in the past has come through fine-tuning the nodal models' adjustable parameters against more sophisticated computations or experimental data. The same holds true for RPM. The following analyses suggest how and why this can be so.

4.6.1 Thermal Leakage Correction The nodal model of Table 4.1 was derived under the approximation that thermal neutrons are absorbed immediately upon thermalization. Their thermal diffusion length, L , however, is on the order of 1.6 em , so that a non-negligible net current can exist between assemblies. Corrections for this effect have been introduced into many state-of-the-art nodal codes, which encourages a similar effort here. Recall from Section 1.5 of Chapter 1 that leakage from a uniform slab source imbedded in an infinite medium of similar neutronic properties is just a fraction, L/h, of the source neutrons, where L is the diffusion length (migration length in Sec. 1.5, where fast group leakage was considered) and h the node width. In two dimensions the leakage is doubled. Thus, adjacent node powers with thermal leakage (denoted by an asterisk) and without are approximately related by - 2L , J; * =J; ( I - I2L) ; + fsl;

[,' ~ ], (1 _2~) · + ,r, 2~

(4.30)

121

122

The Linear Reactivity Model

These relations can be manipulated to obtain (4.31 ) But since j,

~

-

IsI(l - 8Pi), Eq. (4.31) becomes (4.32)

This correction is in actuality a degenerate form of Becker's 10 more elaborate treatment. It has been incorporated in RPM as a post-convergence correction. Kao I 1 has found that this considerably improves agreement between RPM and a more accurate two-group nodal code, and that this crude correction gives results that are not inferior overall to Becker's correction, at least when applied to RPM. Note that Eq. (4.32) can be rewritten as

1

(4.33)

so that an alternative would be to adjust 8 to account for thermal leakage effects, i.e., (4.34) In fact, even with a separately applied correction for thermal leakage, it has been found useful to adjust 8 and PL empirically in RPM to optimize agreement with a benchmark computation if that is available for the core in question, a common practice in other nodal codes, where kernel parameters and reflector albedoes are often adjusted. Kao ' I found that, once adjusted, the values remain valid for shuffled patterns. 4.6.2 Power-Related Feedback An assembly's reactivity status is affected by its power level. An increase in power level leads to an increase in moderator and fuel temperature, both of which reduce the reactivity slightly through the negative moderator temperature coefficient of reactivity and the Doppler

Assembly Power Mapping

reactivity coefficient of the fuel (primarily that of its fertile species) . In addition, a higher power density corresponds to a higher concentration of saturating fission products (xenon and samarium), which also reduces reactivity. To first order, small feedback effects may not affect power level, as the following reasoning suggests. The perturbed and original power are related by

j; *

=

j;

+

aj;) ( api _I

~Pi

+ . .. . ."

(4.35)

The reactivity perturbation may, in turn, be proportional to the change in " power level:

Ap, ~ - y(f/ - j; ) .

(4.36)

Hence,

L" ~ j; + aaj;

[-y(l;* - j; )] ,

(4.37)

Pi

which rearranges to

J/

(1 + Y:~) = J, (1 + Y:~,)

(4.38)

Hence j;* -- j; to first order. Nevertheless, strong feedback effects such as xenon poisoning and boiling do have a profound effect on core power profiles-one that is often beneficial (e.g., flattening the steady-state distribution) but sometimes, especially in transients, can lead to increased local peaking. If the reactivity perturbation is measured relative to a core-average" assembly at zero reactivity, it follows that

j; *

=

j; [ 1

+

aj; (Pi - 0) api

1I' Ji

+ ...

]

or

Ii *

j;

~ ----=-~-

1 -

(1; )Pi 1 aj;

"api

,

(4.39)

123

124

The Linear Reactivity Model

which, when multiplied by the expression for J: in Table 4.1, gives (again to first order) the original equation with modified e: (4.40) Hence, a first-order built-in correction for feedback can be assumed if e is adjusted to match the results of more sophisticated calculations that explicitly consider the coupling between thermal-hydraulic and neutronic effects. If the derivative term in Eq. (4.40) is negative, then e decreases and the difference between the power in node i and its neighbors becomes less pronounced, as expected.

4.6.3 Power Versus Source Normalization Although more a limitation of the crude nodal algorithm used in the present work than a problem in general, the assumption that source and power normalization are equivalent bears examination. Recall that in the one-and-one-half-group derivation the assumption was made that (K/v)/(D 1/M2 ) was constant, i.e., q

=

-K -D

en en

128

The Linear Reactivity Model

Octo.nt SYMMetry~ Quo.e/ro.nt __ SyMMetry

~Centro.l As s enbly

,/

0 0 0 0 0 0 0 ® 0 0 0 0 0 p

X

X

X

x

P

X

= Control Roe/ Loco. t ion

P

X p

P = Per-iphe ro.l Loco. tion

p

Toto.l As ser-oue s in Core

=

193

Fig. 4.5 Constrained assembly positions in the Zion PWR core .

from Ref. 1 with some recent additions, categorizes this work according to the general methodology employed.

4.8 BWR Fuel Management In some respects the management of boiling water reactor (BWR) fuel is simpler than that ofPWR fuel despite the fact that BWR cores have roughly four times as many assemblies. This is, in part, because of the extra degrees of freedom available in the form of control rods and flow control (hence, void distribution). This permits the BWR fuel manager to leave fuel assemblies in a given position for their entire in-core residence. As shown in Fig. 4.6, the use of a four-batch fuel management scheme, with staggered reloading, gives rise to a checkerboard arrangement of assemblies of different degrees of exposure; and in successive cycles, as the oldest fuel is replaced, the pattern rotates, while the other fuel assemblies remain in place until their turn for discharge comes along. One of the traditional practices in BWR reload design is to use a constant power, Haling' ?distribution throughout the burnup cycle as a target for choosing the fuel arrangement and control rod insertion sequence. However, recent trends l 3 are toward more creative fuel management schemes such as a form of spectral shift control where the flow toward EOe is increased to take advantage of the higher conversion ratio in the top of the core.

Assembly Power Mapping

CONVENTIONAL LOADING

-$

* ** 3

3

2

2

3 2

3

2

Note: No shuffling needed . Pattern rotates but no t assembl ies j for successive cycles :

*

~~_2E.._~_~_~ 013110

-21 1

-3T2

J

L One assembly of batch ' 3 ' CONTROL CELL CORE

-* -$-o o -$-$1 3

2

2

1 3

Note: Diagonal assembly loading; control rad pattern changes next cyc le when: 0-11-2 2 - 3 3 - 0

IMPROVED

* -$* 3 1

o

Fig. 4.6

2

-$2 0

+

, I --~_.

I

I

cce

Note: Fresh assembl ies , not all an same diagonal

1 3

BWR assembly loading patterns.

Also shown in Fig. 4.6 are control cell core (CCC) loadings," which avoid having control rods inserted next to fresh assemblies. This type of loading has become the reference scheme for essentially all BWRs. It is also now standard practice to employ a row of natural uranium or high-burnup fuel assemblies around the periphery of the core-the BWR equivalent of the PWR low-leakage configuration. BWR fuel designers have also pioneered the use of natural uranium blankets at the top and bottom of their fuel assemblies.

4.9 Summary In this chapter the connection between batch-wise and assembly-wise fuel management decisions has been sketched out at a somewhat elementary level. A crude nodal approach suitable for pedagogical applications has been used to demonstrate cause and effect in the selection of core layout. Finally, the focus is narrowed to behavior on the pin-by-pin level, following which some appreciation is provided as to why simple nodal methods can successfully accommodate some rather complex aspects of reactor core neutronies.

129

130

The Linear Reactivity Model

4.10 Bibliography on Reload Optimization D. H. Ahn and S. H. Levine, "Direct Placement of Fuel Assemblies Using the Gradient Projection Method," Tran s. Am. Nucl. Soc. , 46, 123 (1984). D. Chang and S. H. Levine, "Optimal Placem ent and Poison Control Depletion of Individual Assemblies Using the Gradient Projection Method," Trans. Am. Nucl. Soc., 43, 164 (1982). K . Chitkara and J. Weisman, "An Equilibrium Approach to Optimal In-Core Fuel Management for Pressurized Water Reactors," Nucl. Technol., 24, 33 (1974). W. Ciechanowiecz, "A Multilevel Approach to Nuclear Fuel Burnup Optimization ," Nucl. Sci. Eng., 57, 39 (1975).

J. P. Colletti, S. H . Levine, and J. B. Lewis, " Iterative Solution to the Optimal Poison Management Problem in Pressurized Water Reactors," Nucl. Technol. , 63, 415 (1983). J. R. Fagan and A. Sesonske, "Optimal Fuel Replacement in Reactivity Limited Systems," J. Nucl. En ergy, 23, 683 (1969). G. Goertzel, "Minimum Critical Mass and Flat Flux," J. Nucl. Energy, 2, 193 (1956). P. Goldschmidt, "Minimum Critical Mass in Intermediate Reactors Subject to Constraints on Power Density and Fuel Enrichment," Nucl. Sci. Eng., 49, 263 (1972). L. W. Ho and A. F. Rohach, "Perturbation Theory in Nuclear Fuel Management Optimization," Nucl. Sci. Eng., 82, 15 I (1982).

A. Ho and A. Sesonske, "Extended Burnup Fuel Cycle Optimization for Pressurized Water Reactors," Nucl. Techno/., 58, 422 (198 2). T. Hoshino, "In-Core Fuel Management Optimization by Heuristic Learning Technique," Nucl. Sci. Eng. , 49, 59 (1972).

H. Y. Huang and S. H. Levine, "A New Method for Optimizing Core Reloads," Trans. Am. Nucl. Soc., 30, 339 (1978). M. G. Izenson, "Automated PWR Reload Design Optimization," SM Thesis, Department of Nuclear Engineering, Massachusetts Institute of Technology (June 1983). Y. J. Kim, T. Downar, and A. Sesonske, "Optimization of Core Reload Design for Low-Leakage Fuel Management in Pressurized Water Reactors," Nuclear Science and Engineering, 96, 85 (1987). B. 1. Lin, B. Zolotar, and J. Weisman, " An Automated Procedure for Selection of Optimal Refueling Policies for Light Water Reactors," Nucl. Technol., 44 , 258 (1979).

M. Melice, "Pressurized Water Reactor Optimal Core Management and Reactivity Profiles," Nucl. Sci. Eng. , 37, 451 (1969). J. O. Mingle, "In-Core Fuel Management Via Perturbation Theory," Nucl. Technol., 27, 248 (I 975). H. Motoda, J. Herczeg, and A. Sesonske, "Optimization of Refueling Schedule for Light Water Reactors," Nucl. Technol., 25, 477 (I975).

Assembly Power Mapping

H . Motoda and O. Yokomizo, "Method to Minimize Power Peaking in Refueling Schedule of Boiling Water Reactor," J. N ucl. Sci. Techn o!., 14, 108 (1977). B. N. Naft and A. Sesonske, "Pressurized Water Reactor Optimal Fuel Management," N ucl. Techno!., 14, 123 (1972). T. A. Rieck et al., "The Effect of Refueling Decisions and Engineering Constraints on the Fuel Management for a Pressurized Water Reactor," MITNE-158, Department of Nuclear Engineering, Massachusetts Institute of Technology (Jan. 1974). B. M. Rothleder et al., "The Potential for Expert System Support in Solving the Pressurized Water Reactor Fuel Shuffling Problem," Nucl. Sci. Eng., 100, 440450 (1988) .

T. O. Sauar, " Application of Linear Programming to In-Core Fuel Management Optimization in Light Water Reactors," Nucl. S ci. Eng. , 46, 274 (1971). I. L. Sauer and M. J . Driscoll, "A Core Reload Pattern and Composition Optimi-

zation Methodology for Pressurized Water Reactors," MITNE-266 , Department of Nuclear Engin eering, Massachusetts Institute of Technology (Mar. 1985). J. A. Stillman, Y.A. Chao, T. J. Downar, "The Optimum Fuel and Power Distribution for a PWR Burnup Cycle," Nucl. Sci. Eng., 103, 321-333 (1989) . G. Sto ry and R. Grow, "A Mi crocomputer Workstation for Des ign Analysis ofPWR Fuel Loading Patterns," Trans. A m . N ucl. S oc., 50, 97-98 (Nov. 1985). R. B. Stout and A. H. Robinson, "Determination of Optimum Fuel Loading in Pressurized Water Reactors Using Dynamic Programming," N ucl. Techn o!., 20, 86 (1973). A. Suzuki and R. Kiyose, "Maximizing the Average Fuel Burnup Over Entire Cycle: A Poison Management Optimization Problem for Multizone Light-Water Reactor," N ucl. S ci. Eng. , 44, 121 (1971). A. Suzuki and R. Kiyose, " Applica tion of Linear Programming to Refueling Optimization for Light Water Moderated Power Reactors," Nucl. Sci. Eng., 46, 112 (1971 ). W. B. Terney and E. A. Williamson, Jr. , " T he Design of Reload Cores Using Optimal Control Theory," Nu cl. Sci . Eng., 82, 260 (198 2). P. Turinsky and G. Hobson, "Automatic Determination of Pressurized Water Reactor Core Loading Patterns That Maximize Beginning-of-Cycle Reactivity Within Power-Peaking and Burnup Constraints," N ucl. Technol., 74, 5 (July 1986). D. C. Wade and W. B. Terney, "Optimal Control of Nuclear Reactor Depletion," N ucl. Sci . E ng., 45, 199 (1971). I. Wall and H. Fenech , " T he Application of Dynamic Programming to Fuel Management Optimization," Nucl. S ci. Eng., 22, 285 (1965).

J. White et al., "Fuel Management Optimization Based on Generalized Perturbation Theory," Proc. Topl. Mtg. A dvances in Fue! Man agem ent, Pinehurst, North Carolina, March 2- 5, 1986, pp. 525-532, Am erican Nuclear Societ y, La Grange Park,

Illinois (1986).

131

132

The Linear Reactivity Model

References 1. 1. L. Sauer and M. J. Driscoll, "A Core Reload Pattern and Composition O ptimi zation Methodology for Pressurized Wat er Reactors," MITNE-266, Department of Nuclear Engineering, Mas sachusetts Institute of Technology (M ar. 1985). 2. 1. L. Sauer, M. J. Driscoll, and P.-W. Kao, " A Nodal Formulation Relating Assem bly Power and Rea cti vit y," Trans. A m . N ucl. Soc., 50, 86 (1985). 3. A. Radkowsky, " Seed Blanket Reactors," in CR e Handbook ofN uclear R eactors Calculations, Y. Ronen, Ed., Vol. 3, CRC Press Inc. , Boca Raton, Florida (1986). 4. D. L. Delp et al., "FLARE-A Three-Dimensional Boiling Water Reactor Simulator," GEAP-4598 , General Electric Co. (Jul y 1964). 5. R. J. J. Stamm'ler and M. J. Abbate, M ethods of St eady-State R eactor Physics in Nuclear Design, Academic Press, New York (1983). 6. K. Elmediouri, "Development and Evaluation of Algorithms for PWR Fu el Assembly Shuffling," Nucl ear Engineering Thesis, Department of Nuclear Engineering, Massachusetts Institute of Technology (Jul y 1987). 7. A. Anacona et al., "Coarse Mesh Techniques for Multi-dimensional Core Analysis," Proc. ANS Topl. Mtg. Advances in Reactor Physics, Gatlinburg, Tennessee, April 10-12, 1978, CONF-78041 , pp. 145-155, American Nuclear Society ( 1978). 8. S. R. Specter et al., " Contro l Cell Nuclear Reactor Core," U.S. Patent No. 4,285,769 (Aug. 1981). 9. R. A. Matzie, "Licensing Assessment of PWR Extended Burnup Fuel Cycles," CEND-381 , Combustion Engin eering Co. (Mar. 1981). 10. M. Becker, "Incorporation of Spectral Effects into One-Group Nodal Simulators ," Nucl. Sci. Eng., 59, 276 (1976). 11. P.-W. Kao and M. J. Driscoll, " An Evaluation of the RPM Nodal Program ," Internal Report, Department of Nuclear Engineering, Massachusetts Institute of Technology (Aug. 1985). 12. R. K. Haling, "Operational Strategy for Maintaining an Optimum Power Distribution Throughout Life," Proc. Top!. M tg. N uclear Performance Power R eactor Cores, TID-7672, U.S. Atomic Energy Commission (1964). 13. R. L. Crowther and A. P. Rees e, " L\VR Innovative Core Loading Strategies," Proc. ANS Topl. Mtg. Advances in Fuel Managem ent, Pinehurst, North Carolina, March 2-5, 1986, pp. 45-54, Am erican Nuclear Society, La Grange Park, Illinois (1986).

Problems 4.1 For a one-dimensional slab reactor, the nodal power coupling relation between assembly i and the average of its two surrounding neighbors IS

Assembly Power Mapping

where 8 = 1 + 1/3 (hlM)2. Test this relation for a homogeneous medium of total width H, a node width h = HilS, and a migration length }J = h12; recall that the exact analytical expression for the power profile IS

. f( x)

~

cos

(~)

.

(a) Show, by equating materials and geometric' buckling, that system reactivity is given by

P= I

+

(IT:)'

(b) Integrate cos (nxlH) over successive 10-degree increments to find the average nodal powers and compare their ratios to the ratios predicted using the nodal coupling relation above. (c) Repeat for a coarse mesh, h = 6M. 4.2 Consider an infinite pin cell lattice of two types of otherwise identical pins, one containing gadolinium burnable poison (type i in the figure below): j

k-

1

-j-

I

j

-k

I

I

I 1

-

repeating element consists of one i, four half-j's, and four quarter-k's

I

I

j

}h

j

1

j

I

k-

1

j

I

-j-

1

-k j

j

1

The reactivity of poison-free cells is P = 0.02 and that of poisoned cells, Pi' Assuming that M2> > h2 such that 8 = 1, the system average cell power is 1.0, and the system is just critical, use the nodal power coupling relation to find the powers of the three different cell types in the subject pattern. In the process, find a numerical value for Pi- Comment on your

133

134

The Linear Reactivity Model

findings with respect to the computed powers of type j and k cells , as they relate to the shortcomings of one-and-a-half-group theory. 4.3 Repeat problem 4.2, but the cells are now assemblies, for which 8 = 3. 4.4 Consider a cylindrical assembly of diameter h and reactivity Pi' surrounded by an annulus of assemblies of diameter 3h and reactivity p. Assume a fast flux shape cp

ao

=

+

a 2r2

+

a 3r3

Relate region-average power and fluxes by .

> u

~

\

-0.02

\ \

\

0 1) to run hotter than older fuel ; when n > 1, assembly power density is scaled by the factor (1 + 9ilp), and the fuel temperature is adjusted 201

202

The Linear Reactivity Model

TABLE

n.i

Features of the BRICC Program Neutronics Models • Assembly-average unit cells (UO b H 20 , Zy) treated • Two-group zero-dimensional diffusion theory • (J values, fission yield library for A = 234 through 243 • Uranium/plutonium fuel cycle • Spectrum and (J values recomputed each burn up step • Single effective resonance model• Thermal disadvantage, fast advantage factors" • Maxwellian temperature correction of thermal (Ju • Non-L' v correction of thermal (Ja for heavy metals • Doppler broadening of resonance (Je • Dancoff effect correction • Steady-state xenon and samarium • Empirical prescriptions for: H 20 downscatter (J(q>1!q>2) Fission product (Ju (B) Water density as function of temperature Input • Volume fractions fuel and moderator • Fuel pin diameter and fuel density • Initial heavy-metal isotopic composition • Fuel moderator temperature • Power density • Burnup increment and number of steps • Geometric buckling • Cycle-average boron (optional) Output • Summary of input • Built-in cross section and other ph ysics data • Burnup accumulated • Mean energy per fission • Total and component-wise reactivities • Heavy-metal isotopic composition • Conversion ratio • Other integral parameters: 6-28, 6-25, p-28 • Fast-to-thermal flux ratio • Inverse boron worth a b

From Ref. 3. From Ref. 4.

accordingly. Finally, by specifying a zero or finite buckling, global core leakage can be included or excluded.

Burnup Isotopics: The BRiCe Program

B.3 Programming Details Figure B.I is a program flowchart for BRICC, displaying its major features. Numbers in parentheses indicate the corresponding line numbers in the listing. Coding is very straightforward; the only items requiring more than cursory attention are the steps needed to have the code calculate and print out the zero'th burnup step (initial core status) and the iteration needed to force the computed burnup step to match the specified step length. The basic equation utilized in BRICC is a finite difference version of the differential equation for isotopic transmutation: dN·

-d I t

=

N i- I

(J c i-I

'

- N,

(Ja i

,

(B.l)

As indicated in Eq. (B. I), isotope i is produced only by neutron capture in a nucl eus having a mass number one less than i, and is destroyed only by neutron absorption. Thus, radioactive decay is ignored-an item of consequence onl y for 24 1p U, and even then a tolerable simplification. This same equation can be applied to nuclei such as 235U and 238U, which are not produced (to any significant extent) by neutron capture merely by setting Nand/or o , of their precursors equal to zero. This strategem is used in BRICC to simplify the coding; thus mass number 234 in BRICC is a fictitious zero-property entry. Finite data are included for nuclei having A = 237 (Np) and 243 (Am) , but these trace elements have virtually no effect on the neutron balance. Using E and S to denote the end and start of a step , respectively, and switching to fluence or flux-time, 8, as the independent variable, where (B.2)

Eq. (B. I) can be written as NE I. - NS-I

=

i 1 (NE i - 1 + 2 NS - )

. (J c,l -l

i 68 _ (NE i + 2 NS )

.

A8

(Ja ,1 Ll.

(B.3)

On the right side of Eq. (B.3), the arithmetic average concentrations of nuclides i and i -lover the time step have been used. Note that both NE and NS will be known for isotope i - I if we calculate compositions in the order of increasing atomic mass. Equation (B.3) can be solved for NE i and converted from kilogram moles to kilograms by multiplying the former by the atomic mass, MA:

203

204

The Linear Reactivity Model

Input Data: Cross Sections and Core Parameters

Cross Section Preparation Thermal, Epithermal, and Collapse

Calculate End of Step Masses

Adjust Flux-Time Step

Set Up Zero·· Step Conditions

No

Compute Reactivity Balance Print Step Results

No

··Zero step is repeated several times before proceeding Fig. B.1 Flowchart for the BRICC burnup isotopics program.

Burnup Isotopics: The BRICC Program

ME . = M E i- 1 + MSi- 1 I 2

(Jc,i-l

1

1

+ MSi