Mathematical Methods in Nuclear Reactor Dynamics

465 Pages·1971·5.507 MB·English

by  Ziya Akcasuh (Auth.)

NUCLEAR SCIENCE AND TECHNOLOGY
A Series of Monographs and Textbooks
CONSULTING EDITOR
V. L. PARSEGIAN
Chair of Rensselaer Professor
Rensselaer Polytechnic Institute
Troy, New York
1. John F. Flagg (Ed.)
CHEMICAL PROCESSING OF REACTOR FUELS, 1961
2. M. L Yeater (Ed.)
NEUTRON PHYSICS, 1962
3. Melville Clark, Jr., and Kent F. Hansen
NUMERICAL METHODS OF REACTOR ANALYSIS, 1964
4. James W. Haffner
RADIATION AND SHIELDING IN SPACE, 1967
5. Weston M. Stacey, Jr.
SPACE-TIME NUCLEAR REACTOR KINETICS, 1969
6. Ronald R. Mohler and C. N. Shen
OPTIMAL CONTROL OF NUCLEAR REACTORS, 1970
7. Ziya Akcasu, Gerald S. Lellouche, and Louis M. Shotkin
MATHEMATICAL METHODS IN NUCLEAR REACTOR DYNAMICS, 1971
In preparation:
John Graham
FAST REACTOR SAFETY
MATHEMATICAL METHODS IN
NUCLEAR REACTOR DYNAMICS
Ziya Akcasu
UNIVERSITY OF MICHIGAN
ANN ARBOR, MICHIGAN
Gerald S. Lellouche
Louis M. Shotkin
BROOKHAVEN NATIONAL LABORATORY
UPTON, LONG ISLAND, NEW YORK
A C A D E M IC PRESS New York and London
COPYRIGHT © 1971, BY ACADEMIC PRESS, INC.
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To the memory of Jack Chernick
Preface
This book is derived from lectures given at the University of Michigan
for the past several years by the first author. As such it has the qualities
of a graduate text as well as a reference work. The book will prepare the
reader to follow the current literature and approach practical and
theoretical problems in reactor dynamics.
The reader is assumed to have adequate background in reactor physics
and operational mathematics. Applied mathematicians and control
engineers who are interested in the stability analysis and control of
nuclear reactors will find sufficient background material on the physical
aspects of reactor dynamics. Nuclear engineers interested in the more
theoretical aspects of reactor dynamics are introduced to the basic
concepts and tools of the mathematical theory of stability.
Because of the limitations on the size of the book, space-dependent
kinetics, numerical methods for solving kinetic problems, and statistical
methods in reactor analysis are not included. The topics which are
treated in this book constitute a fairly complete study of point-reactor
kinetics and linear and nonlinear reactor dynamics. The authors have
tried not to duplicate the subject matter and approach of existing
texts and have attempted to develop a consistent point of view in
xi
xii Preface
the treatment of each topic and in the derivation of the pertinent
results.
The problems at the end of each chapter are an integral part of the
text; in many cases the details of derivations are left as exercises. Further­
more, the problems have been chosen to make specific points where,
in many cases, a discussion of a given point would have required a
digression from the main argument. Practical problems in terms of the
kinetic study of a particular reactor type should supplement these
problems when the book is used as a text.
The authors thank Professor R. K. Osborn of the University of
Michigan for allowing them to use his lecture notes in the discussion
of feedback in Chapter 1. They are also grateful to the Reactor Theory
Group at Brookhaven National Laboratory headed by Jack Chernick
for providing the opportunity to complete this work. They gratefully
acknowledge the typing of Miss Ellie Mitchell and Mrs. Elaine Taylor,
as well as the other technical assistance received from Brookhaven
National Laboratory.
C H A P T ER 1
Kinetic Equations
Reactor dynamics is concerned with the time behavior of the neutron
population in an arbitrary medium whose nuclear and geometric proper­
ties may vary in time. The first step in reactor dynamics is to introduce
and define the macroscopic physical quantities and the dynamical
variables that describe the medium and the neutron population in
sufficient detail. The second step is to find time-dependent equations
that interrelate the various dynamical variables in terms of the nuclear,
thermal, and mechanical properties of the medium, and to determine the
time evolution of the neutron population. These equations are the
kinetic equations which we wish to discuss in this chapter. The last
and most difficult step is to introduce analytical and numerical techniques
in order to solve the kinetic equations either rigorously or approximately,
and to extract all the information relevant to the performance and safety
of the reactor as well as the power plant as a whole. The remaining
chapters of this book are concerned with these aspects of reactor
dynamics.
In obtaining the kinetic equations, we shall present the description
of various physical phenomena influencing the temporal behavior of
neutrons in detail to provide an adequate understanding of the physics
of reactor dynamics, and to point out the interrelationships between
various diverse phenomena in a sufficiently precise manner.
2 1. Kinetic Equations
1.1. Transport Equation
The expected temporal behavior of the neutron population in an
arbitrary medium is completely described by introducing a function of
seven variables «(r, v, t) which we shall refer to as the angular neutron
density. It is defined as
3 3
«(r, v, t) dr dv
s
i.e., the expected nu3mber of neutrons in the volume element d r about r
with velocities in d v about v at time t.
By defining the angular density as the expected, rather than the actual,
number of neutrons in an element of volume in the phase space, we
have excluded the possibility of describing the fluctuations in the
neutron population. The actual number of neutrons is always an integer,
and hence is discontinuous in time, whereas the expected number does
not have to be an integer and is a continuous function of time. The
description of fluctuations of the neutron population, commonly called
reactor noise, requires probabilistic concepts. We shall not introduce
these concepts in this book.
The next step after having introduced the angular density is to find
an equation to determine its time evolution in terms of the nuclear and
geometric properties of the medium. In vacuum, this equation is simple
and readily obtained [1]:
[dn(r, v, t)jdt] + v • Vw(r, v, t) = S(r, v, t) (1)
where S(r, v, i) is the neutron source, defined as
3 3
S(r, v, t) dr dv dt
z
i.e., the number of neutrons inserted in dh at r and in d v at v in the
time interval dt about t.
Equation (1) describes the streaming of neutrons on straight lines,
and is the simplest fo3rm sof the transport equation. It is a balance relation
in whisch — v • Vn d r dv is the rate of removal of neutrons in sdh at r
and dv at v due to streaming (leakage), and S(r, v, t) dh d v is the
production rate of neutrons in the same volume element. The solution
of (1) is straightforward (see Problem 1 at the end of this chapter) [1]:
«(r, v, t) = n(r - vt, v, 0) + Jf dt' S[r - v(t - t'), v, t'] (2)
o
where n(r, v, 0) is the initial distribution of neutrons.
1.1. Transport Equation 3
It is assumed in (1) that there are no forces acting on the neutrons (no
gravity), and no collisions between neutrons (Problem 2). We shall make
these two assumptions consistently in this book unless stated otherwise.
As a result of the collisions of neutrons with atomic nuclei, the
equation satisfied by the angular density in a material medium is
extremely complicated. Before attempting to derive this equation, we
must first introduce the appropriate functions to describe the effect of
collisions on the evolution of neutrons quantitatively. These functions
are related to the macroscopic nuclear properties of the medium.
We assume that the collisions of neutrons are instantaneous. Then,
their effect on the motions of the neutrons can be described by specifying
the mean free path between collisions as a function of the neutron
velocity. The inverse of the mean free path is the macroscopic cross
section E(r, v, t), which is defined as
3 3
E(r, v, t) vn(r, v, t) dr dv
i.e., the mean number of collisionss per second (collision rate) in dh at r
for neutrons with velocities in d v at v at time t.
The origin of the time dependence of the macroscopic cross sections
will be discussed later.
Depending on the type of the nuclear reaction taking place in a
collision, a neutron may be captured, scattered, or may cause fission.
Each of these events is characterized by a partial macroscopic cross
section, which we denote by E , E, and E, respectively. Since the
0 8 f
reaction rates associated with these events are additive, we have
E = E + E + 27 (3)
c a f
The mathematical description of a scattering event requires the
introduction of the differential scattering cross section E (v —• v', r, i),
a
which is defined as
v v r 3 3 3
^s( ~+ '> > i) vn(r, v, i) dr dv dv
3 3
i.e., the expected number of neutrons in d r at r scattered into d v at
v's per second at time t due to collisions of neutrons with, velocities in
dv at v in the same volume element at r.
It follows from this definition that
3
j E(v v', r, t) dv' = Z(r, v, t) (4)
s a
In a fission event, on the average, more than one neutron is produced.
These neutrons are emitted at the point in space where the fission
4 1 * Kinetic Equations
occurs. Some of -t8hem are emitted instantaneously (in time intervals of
the order of 10 sec or shorter following the fission event), and are
termed prompt neutrons [2]. The others are emitted with long time
delays (of the order of seconds), and are called delayed neutrons.
Figure 1.1.1 shows the origin of delayed neutron8s. 7Their emission
follows the deexcitation of certain fission fragments ( Br in Figure 1.1.1)
STABLE
FIGURE 1.1.1. Decay scheme of a typical delayed neutron precursor.
by beta decay. These fission fragments are referred to as the delayed
neutron precursors, or just precur8so7r for brevity. A delayed neutron is
emitted by the daughter nucleus ( Kr in Figure 1.1.1) which is produced
by the beta decay of the precursor. The long time delays distinguishing
the delayed neutrons from the prompt neutrons are associated with the
rather slow nuclear process of beta emission [2].
The migration of the precursors before the emission of a delayed
neutron can be neglected in solid-fuel reactors, because they lose their
kinetic energy very rapidly as a consequence of their large electric
charge. They are stopped in a short distance from the point of their