Note: This report is adapted from "A Model of Reactor Kinetics,"
by A. Stanley Thompson and Bruce R. Thompson, Nuclear Science and
Engineering, The American Nuclear Society, September 1988
R E A C T O R K I N E T I C S
--------------------------------------
ABSTRACT
The analytical model of nuclear reactor transients, incorporating
both mechanical and nuclear effects, simulates reactor kinetics.
Linear analysis shows the stability borderline for small power
perturbations. In a stable system, initial power disturbances die
out with time. With an unstable combination of nuclear and
mechanical characteristics, initial disturbances persist, and may
increase with time. With large instability, oscillations of great
magnitude occur.
Stability requirements set limits on the power density at which
particular reactors can operate. The limiting power density
depends largely on the product of two terms: the fraction of
delayed neutrons and the frictional damping of vibratory motion in
reactor core components. As the fraction of delayed neutrons is
essentially fixed, mechanical damping largely determines the
maximum power density.
A computer program, based on the analytical model, calculates and
plots reactor power as a non-linear function of time in response
to assigned values of mechanical and nuclear characteristics.
A model, based on mathematical equations describing the nuclear
and mechanical characteristics of reactors, simulates the
transient response of reactor power. The nuclear and mechanical
equations are tied by terms describing the heating effect of power
transients, which distort the reactor structure, and in turn
affect the power level. The combined mechanical and nuclear
dynamics determine conditions for reactor stability.
THE MATHEMATICS OF REACTOR KINETICS
The analysis is based on two equations, one nuclear, the other
mechanical. The nuclear equation relates the fractional power
change to (1) the damping effect of delayed neutrons, and (2) the
restoring effect of reactivity coefficients for temperature and
density changes. The mechanical equation contains terms for (3)
inertial resistance to acceleration, (4) mechanical damping of
motion, (5) elastic resistance to distortion, and (6) thermal
expansion of material caused by energy absorption.
The Nuclear Equation
--------------------
We consider a "point" reactor[1],[2],[3], with time the only
independent variable. Parameters describing the point reactor must
be determined experimentally, or from equations dependent on
spatial variables and neutron energy levels as well as time.
We assume that the average delay of fission neutrons is long
compared to the period of power oscillations, so that the quantity
of delayed neutrons produced during a transient depends on the
neutron density accompanying the steady power level, , which
existed before the transient disturbance started.
With a negative temperature coefficient, a T, and a density
coefficient of reactivity, a y, where y is the fractional change
in density, the reactivity, r , is expressed,
r = r 0 + a T T + a y y (1)
The rate of change of power is then
dP/dT =[(aT T+ay y-b+r0 )P+P0 ]/ l (2)
where b is the fraction of delayed neutrons and l is the average
generation time of fission neutrons. A transient rise in the power
level heats reactor material in accord with the first law of
thermodynamics,
P-P0 = S dT/dt (3)
where S = cp M is the heat capacity per unit temperature rise, cp
is the specific heat at constant pressure and M is the heated mass
of reactor material.
Combining Equations (2) and (3) to eliminate the temperature terms
gives
(4)
where
cn = b/l , w2T = - aT P0 /(S l) (5)
and wT has the dimensions of inverse time, representing the
natural frequency of a small power oscillation associated with the
restoring effect of the negative temperature coefficient of
reactivity.
Equations (1) to (5) have been developed and studied by other
authors, including Thompson and Rodgers[1] and Weinberg and
Wigner[2] (without the density term).
The last term in Equation (4), containing the density term, y,
couples a rapid power transient to the mechanical dynamics of the
reactor. A similar coupling is noted by Hetrick[4], who reports
that the inertial effect increases the total energy produced
during a transient.
The Mechanical Equation[5]
--------------------------
Rapidly deformed reactor structures form a mechanical dynamic
system, characterized by one or more fundamental or harmonic
natural frequencies at which reactor parts may vibrate in response
to external stimuli or to internal instabilities.
Any transient disturbance in material density or in the
distribution of strains in certain parts or all of a heated
reactor structure may affect the reactivity. Such a part of a
particular reactor might be a vibrating fuel assembly, or
moderator, or coolant. Non-uniform heating bends fuel elements
away from a straight line. Heated moderators and reflectors
expand, increasing leakage of neutrons from the reactor. Vapor
cavities may form within reactor liquids, increasing or decreasing
reactivity.
Large vibratory changes can lead to "fatigue" failure, and
therefore are not generally acceptable in the design of permanent
structures. Small disturbances in density or strain are considered
in this analysis.
The partial differential equation for a mechanical vibration is
divided by "separation of variables"[5] into ordinary differential
equations, one dependent on time and one or more dependent on
space. For solving the time equation, it often suffices to know
the frequency of the system, without the details of the spatial
distribution. The equation describing the vibratory motion of a
heated material becomes,
(6)
where cm is the mechanical damping coefficient, and w m is the
mechanical natural frequency of a reactor component whose motion
changes reactivity. b is the coefficient of thermal expansion of
the heated material.
Of the four terms in Equation (6), the first three appear in the
conventional equation for the damped oscillation of a mechanical
system.[5],[6] The first represents the force needed to overcome
inertia, the second to overcome friction, and the third an
elastic, or spring, force. The fourth term in Equation (6) is a
forcing function. It represents the force accompanying the thermal
strain caused by rapid changes in temperature, T.
Equations (4), (5) and (6) are a system for transients in reactor
power, P, temperature, T, and density, y. Eliminating temperature
and density terms yields a fourth-order differential equation with
power as the dependent variable. The result is
(7)
where
(8)
and wy has the dimensions of inverse time. It represents a nuclear
frequency dependent on the density coefficient of reactivity.
An equation essentially the same as Equation (7) was developed by
Thompson[7]. An equation considering heat transfer delays, but
without mechanical effects, was developed by Thompson and
Rodgers[1].
Equation (7) is nonlinear because of the logarithm of power in the
first three terms and the inverted power ratio in the next three
terms. Its solution requires the use of finite difference
equations and successive approximation procedures.
STABILITY OF EQUATION (7)
-------------------------
The stability of a linear system, operating at or near equilibrium
is a necessary, but not a sufficient, condition for stability
under large ,non-linear departures from equilibrium. When only the
temperature coefficient of reactivity, , is considered, with no
density coefficient, , Equation (7) reduces to a second-order
differential equation, always stable against small oscillations if
its coefficients are positive.
The methods of linear transients pose two stability requirements
for the linearized form of Equation (7). First, all coefficients
must be positive. Second, the coefficients must satisfy Routh's
criterion[8], namely,
From Equation (7), the coefficients are
Replacing the 's with their values gives the stability
requirement for Equation (7),
(9)
For the combined nuclear and mechanical system, Equation (9)
determines linear stability. The last term involving has only a
secondary effect on stability.
Particular values of two ratios of the variables in Equation (9)
serve as reference points. For the damping for an independent
mechanical system is critical; above that value sustained
oscillations cannot exist. Similarly, for , where is defined
by
the independent nuclear system is critically damped, precluding
sustained power oscillations. For most reactors nuclear damping is
above the critical value, the right side of the inequality in
Equation (9) is close to unity, and the expression for stability
simplifies to
(10)
THE COMPUTER PROGRAM
--------------------
The computer program solves Equation (7) by finite difference
methods, using trial-and-error procedures to handle the highly
non-linear equations. The program, in ZBasic[9] for an
IBM-compatible personal computer, is available from the authors.
ESTIMATED VALUES OF THE COEFFICIENTS
------------------------------------
The coefficients in the kinetic equation depend not only on
materials, but also on specific reactor configuration. Let us
develop possible values, based only on the properties of
materials, and perhaps wide of the mark.
In Equation (7), the ratio, , measures the rate of temperature
rise in material heated by continuing steady-state power if the
cooling system failed. An extreme high value may be 100 oC per
sec, the lowest possible value being zero. The generation time,
, for fission neutrons may vary from 10-4 seconds for a thermal
reactor to 10-7 seconds for a fast reactor. The negative
temperature coefficient, , (or positive density coefficient,
) may vary from zero (or even reverse sign) to 10-3 per oC. These
figures would put the value of the nuclear frequency in the range,
0 < < 1000 radians per second.
The neutron damping ratio is defined, . The fraction of delayed
neutrons for uranium is about = .007. The neutron damping ratio is
considerably greater than the critical damping ratio, .
The mechanical frequencies and the mechanical damping of parts of
the reactor structure depend on the specific design.
CONCLUSIONS
-----------
For reactor conditions not satisfying the inequality of Equation
(9), self-excited oscillations of power, temperature and density
may occur. For stability both mechanical and neutron damping are
necessary. Equation (10) appears to prescribe an upper limit to
stable operating power density. Substituting the values for cn and
y gives the requirement,
(11)
Neutron generation time has disappeared from this expression.
Increasing reactor power above incipient instability results in
ever increasing severity of persistent oscillations.
DEFINITIONS
-----------
a0, a1, a2, a3 = coefficients in characteristic equation
b = volume coefficient of thermal expansion, /oC
cm = coefficient of mechanical damping, /sec
cn = coefficient of neutron damping, b / l, /sec
cp = specific heat at constant pressure, cal/gram-oC
l = average generation time of neutron production, sec
M = mass of heated materials, gram
n = dimensionless time, wmt
P = instantaneous power level, cal/sec
Po = average power level, cal/sec
S = heat capacity per unit temperature, cp M, cal/oC
t = time, sec
T = perturbation in temperature, oC
TF = temperature, after cooling failure for time,
t = p /w m
y = perturbation on average density, dimensionless
a y = density coefficient of reactivity, dimensionless
a T = (negative) temperature coefficient of reactivity,
/oC
b = fraction of delayed neutrons
r = reactivity, dimensionless
w m = mechanical frequency, /sec
w n = nuclear frequency =(w2y+ w2T)0.5
wT = nuclear frequency based upon temperature coefficient
of reactivity, w 2T = aTP0 /(S l), /sec
wy = nuclear frequency based upon density coefficient of
reactivity, w2y = ay bP0 /(S l), /sec
Definitions for Use in Examples:
A = time interval used in integration
B = time at end of calculation run
C = ratio, nuclear and mechanical frequency, wy/wm
D = ratio, nuclear and mechanical frequency, wT/wm
E = ratio, mechanical damping to mechanical frequency, cm /wm
F = ratio, neutron damping to nuclear frequency, cn/wn
G = initial excess reactivity, r0 /(wnl)
H = initial power displacement, P1 /P0
COMMENT
-------
Reactor proponents now promise us a new generation of inherently
safe reactors based primarily on built-in physical
characteristics, primarily a large negative temperature
coefficient of reactivity. Equation (11) for the requirement for
stability can be arranged,
-dr 0/dt < b cm (12)
where the term on the left is the rate of reduction in reactivity
which would occur if the cooling system suddenly failed. It
appears that the mechanical friction available in moving reactor
parts may set a different stability limit. A given required amount
of (positive) mechanical friction is notoriously difficult to
assure in the presence of rapidly flowing cooling fluids.
REFERENCES
----------
1. Thompson and Rodgers, Thermal Power from Nuclear Reactors,
John Wiley & Sons, Inc., 1956, p. 116.
2. Weinberg and Wigner, The Physical Theory of Neutron Chain
Reactors, The University of Chicago Press, 1958, pp 603-609.
3. Glasstone and Sesonske, Nuclear Reactor Engineering, Van
Nostrand Reinhold Company, Chicago, 1967.
4. David L. Hetrick, Dynamics of Nuclear Reactors, University of
Chicago Press, Chicago, 1971.
5. S. Timoshenko and D. H. Young, Vibration Problems in
Engineering, D. Van Nostrand Company, Inc., Princeton, NJ,
1955, pp. 299, 409, 416.
6. J. P. Den Hartog, Mechanical Vibrations, McGraw-Hill Book
Company, Inc., New York, 1940, Ch. 7, p. 300.
7. A. Stanley Thompson, "Study of Reactor Kinetics," American
Society of Mechanical Engineers, Paper No. 62-WA-218, 1962.
8. Chestnut and Mayer, Servomechanisms and Regulating System
Design, John Wiley & Sons, Inc., New York, 1951.
9. Andrew R. Gariepy, ZBasic Interactive BASIC Compiler, ZEDCOR,
INC., Tucson, Arizona, 1987.