They tell us "a nuclear reactor can't blow up like a bomb."
Too bad!
We're going to miss that beautiful mushroom cloud.
It can, however, blow up like Chernobyl,
by chemical, mechanical, and nuclear means,
polluting the whole northern hemisphere.
Perhaps that's good enough.
U N S T A B L E N U C L E A R P O W E R
---------------------------------------------------
Stable power processes are never guaranteed. An assortment of
unstable behaviors wrecks power apparatus, including mechanical
vibration, malfunctioning control apparatus, unstable fluid flow,
unstable boiling of liquids, or combinations thereof.
"INHERENTLY SAFE" REACTORS AREN'T SAFE
--------------------------------------
The experts tell us that they will build "safe" reactors, using
inherent reactor characteristics to limit any unwanted rise in
reactor power. A rise in power increases the temperature of
reactor parts, causing them to swell, or bend, or otherwise
distort. A necessary condition for stable reactor power stable is
that the reactor distortion accompanying a temperature rise
decrease the reactivity, thus slowing any further rise in power.
For instance, the reactor distortion caused by a temperature rise
could allow more neutrons to be wasted by leakage, thus decreasing
the reactivity. Overly simple equations for reactor power dynamics
indicate that this idea should work, and sometimes it does.
The supposition that satisfying the above condition guarantees
power stability is based on a false assumption that the reactor
distortion occurs simultaneously with the temperature rise which
caused it. The distortion lags behind the temperature change by a
time determined by the mechanical dynamics of the reactor system.
Therefore the change in reactivity also lags behind the
temperature change, raising the possibility of unstable power
oscillations. This conclusion is supported by actual experience
with seriously unstable reactors such as the inoperable Ft. St.
Vrain power reactor discussed later.
It is shown in the last section of this pamphlet that stability
against power oscillations depends on having sufficient mechanical
friction in the distorting reactor parts. Guaranteeing this
requirement may be outside the power of a designer.
EVIDENCE OF REACTOR INSTABILITY
-------------------------------
My interest in the subject of reactor instability was aroused by
two sets of experiments.
I heard of the first set of experiments, called "teasing the
dragon," from a physicist who had worked on the atomic bomb
project. In a low-level simulation of a bomb test, a sudden pulse
of nuclear energy was initiated, sufficient to cause parts of the
test assembly to jump apart momentarily (and occasionally to
destroy the experimenter, but not quite enough to destroy the
apparatus and its surroundings). I wanted to understand how the
dynamic mechanism which controlled the energy pulse might be
related to the control of reactor power.
The second set of experiments, under the acronym, SPERT, was
carried out at the National Reactor Test Station to study
transients of reactor power. An assembly of thin sheets of
aluminum bearing the nuclear fuel was immersed in a pool of water,
the surface of the water being open to the atmosphere. Reactor
power was slowly increased by gradually pulling out control rods.
At a certain low power a spontaneous unstable oscillation suddenly
appeared and built rapidly to violent proportions, resulting in a
flash of blue light from Cerenkov radiation and the explosive
expulsion of water from the reactor pool. The peak of power was
greater by a factor of thousands than the level from which the
oscillations had started. It was concluded that the unexpected
instability was related to coolant boiling.
In the early 1960's I developed an analysis of a possible unstable
mode of power oscillation which I thought could lead to reactor
explosions. I developed mathematical criteria to try to predict
the threshold of reactor power at which such unstable oscillations
would occur. Based on my analyses, I concluded that coolant
boiling in the SPERT reactor set up a dynamic configuration in
which a heavy mass of water bounced at a low natural frequency on
the soft spring provided by vapor bubbles. The mechanical
oscillation was coupled with the nuclear power oscillation in an
unstable combination. Water-moderated reactors, the type most
commonly used for the generation of power, might be susceptible to
catastrophic nuclear power oscillations if the pressure of the
water in the reactor suddenly decreased, creating vapor bubbles in
the core.
I wrote several reports predicting that various power reactors
could be susceptible to power oscillations of the type observed in
the SPERT experiments. One preliminary report entitled "Study of
Reactor Kinetics" was published by the American Society of
Mechanical Engineers as Paper Number 62-WA-218 late in 1961. I
presented the report at the annual meeting of the Society in New
York in November 1962. I received encouragement from engineering
friends, but was unable to get any attention from the nuclear
establishment. I was denied publication in the nuclear journals.
The internal study and public scrutiny of potential instabilities
of reactor power was (and is) not popular with reactor proponents.
Even when instabilities cause accidents, they are not acknowledged
as such.
UNSTABLE REACTORS
-----------------
The SPERT experiments seemed to us pertinent to our claim that
mechanical effects contribute to reactor instability. Subsequent
reactors have demonstrated instabilities similar to our
predictions.
A non-nuclear model of a nuclear reactor, for a space propulsion
project named ROVER, was composed of large graphite blocks through
which a gas was circulated in a flow test. The blocks were
restrained against the radial pressure of the gas by springs at
their outer periphery. A motion picture taken during a flow test
showed the blocks banging to pieces as the pressure of the
circulating gas bounced them against the restraining springs in a
self-excited oscillation, equivalent to a negative mechanical
damping in the system. No nuclear catastrophe occurred only
because there was no nuclear power to couple with the mechanical
oscillation of the graphite blocks.
A high-temperature, gas-cooled reactor for electrical power was
built for Colorado Public Service at Ft. St. Vrain. On completion
it could not be brought to a useful level of operating power
because persistent power oscillations worsened as power was
increased. Because of the oscillation and other problems the
reactor was abandoned. A successful ratepayers' suit removed the
useless power plant from the rate base.
Other examples of unstable reactor behavior exist.
REACTOR KINETICS
----------------
Twenty-five years after my failure to achieve publication in
nuclear journals, my son, Bruce Thompson, volunteered to help me
write a straight technical article about stability, based on a
computer program which we had developed together. Our program
solves by finite difference mathematics a non-linear, fourth-order
differential equation involving nuclear, mechanical and thermal
characteristics of reactors. Our paper, "A Model of Reactor
Kinetics," was finally published in Nuclear Science and
Engineering, the technical journal of the American Nuclear
Society, in September 1988[1]. It demonstrates a mechanism for
catastrophic instability.
Our paper demonstrates that mechanical friction in a reactor core
structure, like the shock absorber in an automobile, is necessary
to limit oscillations of reactor power. Without adequate internal
friction, a nuclear power driven mechanical oscillation increases
toward destruction of the reactor core. Design engineers in many
fields have found to their sorrow that any given level of
mechanical friction is difficult to guarantee. Some of the
computer-generated examples later in this section show the
changing core temperature and reactor power for a reactor without
adequate friction to provide stability. A small perturbation in
power causes an initially small oscillation which builds rapidly
to destruction, either blowup or melt down.
In the design, construction and operation of nuclear reactors an
attempt is made to maintain steady operation at any desired power
level from fission of the nuclear fuels, uranium or plutonium.
This attempt can fail, sometimes catastrophically. A nuclear power
plant is a nuclear system and a mechanical system. It is also a
heat-transfer system, tied in with controls, boilers, turbines,
human operators and a multiplicity of other complicating factors.
The possibilities for instability are myriad. This fiercely
complicated set of systems is such as to preclude any possibility
of the formation of adequate analytical equations and their
solution to guarantee the stability of power. An experimental
program sufficient to eliminate the possibilities of power
instability in reactors would be expected to be ruinous of both
the economy and the environment. Our program covers only a small
part of the complicated possibilities, and demonstrates only one
type of power instability.
PERTURBATIONS TO REACTOR POWER
------------------------------
We outline here the calculation of the power dynamics of a
reactor, based on Reference 1. We present some necessary
conditions for power stability provided by nuclear physics, as
analyzed by Weinberg and Wigner in Reference 3. Their analysis
generated a second-order differential equation whose solution is
always stable if the coefficients in the equation are all
positive. Then we consider mechanical characteristics of reactors
which render the nuclear analysis insufficient to assure
stability. This results in a fourth-order differential equation
whose stability requires not only that all coefficients be
positive, but also that a relationship among the coefficients be
satisfied (Routh's criterion). It turns out that, for power
stability, motion of any reactor parts which affect reactivity
must be sufficiently damped by mechanical friction in the moving
parts.
Reactivity and stability are unavoidably affected by depletion of
reactor fuels and the accumulation of "poisons" during operation.
Other destabilizing effects not considered here, or anywhere else,
are so numerous and so complicated in their interrelationships as
to thwart human efforts to guarantee reactor stability.
EXAMPLES
--------
We attempt with twelve examples to demonstrate some physical
properties of reactors on which stability depends. The examples
show the influence on power stability of: reactivity, delayed
neutrons, coefficients of reactivity, mechanical inertia,
friction, heat transfer.
The defining parameters for these examples are:
A: Time interval
B: Final time
C: Frequency ratio, density coefficient
D: Frequency ratio, temperature coefficient
E: Mechanical damping
F: Nuclear damping
G: Initial reactivity
H: Initial power ratio
I: Mechanical effects on (off)
J: Heat transfer on (off)
K Coefficient of Heat Transfer
These parameters are more precisely defined in Reference 1.
The values used for A through K, are shown with each of the
computer-generated solutions.
Nuclear and Mechanical Characteristics
--------------------------------------
The first examples show the effects of the "in-built" nuclear
characteristics, delayed neutrons and temperature coefficients of
reactivity, which reactor proponents say will "guarantee" the
stability of their new generation of "safe" reactors. The later
examples show some of the destabilizing mechanical effects which
may defeat both the proponents of reactors and the rest of us.
NUCLEAR PHYSICS
---------------
Reactivity
----------
Reactivity is the name of the quantity which determines the rate
of change of reactor power. For reactor power to be steady the
reactivity must be zero. Maintaining zero reactivity requires
moving "control rods" to maintain a balance among the rates of
leakage and absorption of neutrons in nuclear fuel and in other
reactor materials. Because the absorption and leakage of neutrons
have been extensively studied, these balancing dependencies are
relatively well understood.
Positive reactivity causes power to rise exponentially at a rate
proportional to the reactivity. Negative reactivity causes power
to decrease. To change power in a planned manner, reactivity is
adjusted by moving the control rods, either manually or by means
of automatic controls. Partially removing a control rod is
expected to increase the reactivity, causing the power to rise to
a new level. Inserting a properly designed control rod farther
into the reactor should, but doesn't always, lower the power
level.
[Example 1]
Example 1
Example 1 shows the effect of a relatively large initial
reactivity, ignoring several important limiting effects, leading
to a rapid rise to a "blowup." The unacceptable behavior which
would result from a continuation of this exponential response is
modified in real reactors by two important effects.
Delayed Neutrons
----------------
The first nuclear effect is that of "delayed neutrons." Of all
neutrons formed in fission a small fraction (.007) are delayed by
significant time periods after absorption of the fission neutrons.
The delayed neutrons, not being available immediately to support a
transient rise in power, act as a damping factor on transients,
providing the strong beneficial effect of limiting sudden changes
in power.
Examples 2, 3, and 4 show the classical "damping effect" of
delayed neutrons on the rise of power, ignoring the temperature
coefficient of reactivity and mechanical effects. Example 3 shows
the reaction to "prompt critical" reactivity, in which by
definition the initial reactivity is numerically equal to the
fraction of delayed neutrons (G=F=15). The initial rate of power
rise continues indefinitely on a straight line. In Example 2 the
addition of reactivity greater than prompt critical (G/F = 15/14)
causes an exponential rise of power faster than in Example 3.
Example 4 shows a "sub-critical" reaction, in which a value of
reactivity less than the fraction of delayed neutrons (G/F =
15/16) leads to the leveling-off of power at a new value,
P/P0=F/(G-F) = 16.
[Example 2]
Example 2
[Example 3]
Example 3
[Example 4]
Example 4
Temperature Coefficient of Reactivity
-------------------------------------
The rise in temperature which accompanies a rapid rise in power
generally changes the reactivity. The "temperature coefficient of
reactivity" measures the effect of changing temperature on
reactivity. Because power stability requires a rise in temperature
to decrease the reactivity, a "negative temperature coefficient"
is necessary for stability. Examples 5, 6, and 9 show the effects
on power stability of including delayed neutron damping and
temperature coefficients of reactivity, but neglecting all other
effects.
[Example 5]
Example 5
Example 5 shows the destabilizing effect on reactor power of a
positive temperature coefficient (D= -2.0), leading to "blowup."
[Example 6]
Example 6
Example 6 shows an oscillation of power at nuclear frequency, with
a negative temperature coefficient (D=2.0) and with delayed
neutron damping (F=0.20). Because of neutron damping the
oscillations decrease in magnitude with time.
The effects of the nuclear factors, reactivity, delayed neutrons
and temperature coefficients, are described by a non-linear
second-order differential equation. This relatively simple
equation predicts stable reactor power if the temperature
coefficient of reactivity is negative. Neutron damping in reactors
is generally relatively large, providing a strong tendency toward
stability for a simple nuclear system. The real situation is more
complicated, and more difficult than can be described in terms
only of nuclear factors.
MECHANICAL CHARACTERISTICS
--------------------------
Mechanical Inertia and Mechanical Friction
------------------------------------------
Changes in reactor temperature also affect the density. A decrease
in the density of reactor parts tends to increase the leakage of
neutrons which in turn decreases the reactivity. This adds the
complications of mechanical dynamics to the analysis of reactor
stability. Fast density changes in a reactor structure may cause
it to vibrate, and reactor power may oscillate as a result.
Accounting for this effect requires adding to the temperature
coefficient described above a "density coefficient of reactivity."
The density coefficient couples mechanical and nuclear power
oscillations. As previously mentioned, inclusion of mechanical
effects requires solution of a fourth order differential equation
of reactor kinetics, as opposed to the second order nuclear
equation. Prediction of power stability becomes more precarious.
Examples 7 through 9 and A through C demonstrate the added
mechanical effects on the nuclear requirements for power
stability. Any amount of mechanical friction is sufficient to damp
oscillations of simple mechanical systems. But for stability with
the combined mechanical and nuclear systems there is an additional
requirement. Reference 1 shows that the nuclear frequency ratio,
C, must be less than the product of mechanical damping, E and
neutron damping, F, or in equation form, CC, the configuration is
stable. After its initial burst power finally levels off near its
steady state value, P/P0=1.
[Example 8]
Example 8
Example 8 shows the destabilizing effect of insufficient
mechanical friction on the pulsed reactor of Example 7. Because
mechanical friction is zero (E=0), the value of C exceeds that
allowed by the stability criterion, (C>E*F). In contrast to the
leveling off shown in Example 7, bursts of energy of increasing
magnitude occur at intervals of approximately 2p , in other words
at the mechanical natural frequency of whatever moving reactor
part is responsible for the positive density coefficient. In the
intervals between peaks, energy generation becomes very small, a
period in which one might mistakenly and disastrously assume that
the reactor had shut down. In this example a "blowup" occurs just
past the end of the first mechanical cycle.
In Examples 6 and 9 through C the scale has been expanded by a
large factor to accommodate smaller transients appropriate to the
operation of power reactors.
[Example 9]
Example 9
In Example 9, with mechanical effects included, two frequencies
occur simultaneously, one mechanical and the other nuclear. The
nuclear frequency is twice the mechanical (D=2). Both are damped
by the nuclear damping (F=.10) and mechanical damping (E=.10).
[Example A]
Example A
The transient in Example A shows the destabilizing effect of a
positive density coefficient (C=20), with neutron damping (F=5)
and a small initial reactivity (G=.05), but with inadequate
mechanical damping (E=.05). This, as expected from stability
analysis (C>E*F), causes power bursts at approximately mechanical
frequency of ever increasing magnitude.
[Example B]
Example B
In Example B addition of sufficient mechanical damping (E=10)
returns the system to stability (C prompt on my computer I type DEL *.* and press
RETURN, there appears on the screen the question, Are you sure
(Y/N). Because * is a wild-card, I am warned that all the
information on the disk is at risk. We believe mechanical friction
is a wild-card of nuclear reactors whose efficacy cannot be
guaranteed by reactor designers.
We cannot afford one hundred years of automobile type
make-and-break experimental development of nuclear reactors.
References:
1. "A Model of Reactor Kinetics," by A. Stanley Thompson and
Bruce R. Thompson, Nuclear Science and Engineering, September
1988.
2. Thermal Power from Nuclear Reactors, by A. Stanley Thompson
and Oliver E. Rodgers, John Wiley & Sons, Inc., New York,
1956.
3. The Physical Theory of Neutron Chain Reactors, by Weinberg
and Wigner, The University of Chicago Press, 1958, pp
603-609.
Note:
The "Reactor Kinetics" program for solving power stability
problems is available for IBM clone computers, on either a 5.25
inch or a 3.5 inch floppy disk, from:
A. Stanley Thompson
1910 Monroe Street
Eugene, OR 97405