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MICROCOPY RESOLUTION TEST CHART
NATIONAL BUREAU OF STANDARDS - 1963
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LEGAL NOTICE
This report was prepared as an account of
Government sponsored work. Neither the
United States, nor the Commission, nor any
person acting on behalf of the Commission:
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A. Makes any warranty or representa-
tion, expressed or implied, with respect to
the accuracy, completeness, or usefulness
of the information contained in this report,
or that the use of any information, appa-
ratus, method, or process disclosed in this
report may not infringe privately owned
rights; or
B. Assumes any liabilities with respect
to the use of, or for damages resulting from
the use of any information, apparatus,
method, or process disclosed in this report.
As used in the above, "person acting on
behalf of the Commission" includes any em-
ployee or contractor of the Commission, or
employee of such contractor, to the extent
that such employee or contractor of the
Commission, or employee of such contractor
prepares, disseminates, or provides access
to, any information pursuant to his employ-
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ORN-P-1235
CONF-650501-10
COMPUTATIONAL PROBLEMS OF REACTOR SAFETY*
MAY 4 7 1968
MASTER
W. K. Ergen, Oak Ridge National Laboratory
Prepared for presentation at the Conference on Application of ,
Conputing Methods to Reactor Problems, Argonne National Laboratory,
May 17-19, 1965
Introduction
I am probably the last person who should give this talk, as I have never
designed a computer code in my life, My work is directed toward a qualitative,
or maybe semi-quantitative, understanding of the complicated and interlocking
phenomena affecting reactor safety. Pencil-and-paper mathematics is all that
can usefully be applied to this end of the business; and if a problem ever
gets to the stage where more sophisticated methods pay off, the work is trans-
ferred to mathematical experts.
You may, however, enjoy hearing a broad survey
of the problems of reactor safety in order to see where modern computational
techniques can rsefully be employed. I will try to present such a survey.
Nuclear-reactor hazards stem mainly from the possibility of dispersing
fission fragments. The other hazards - such as that resulting from flying
objects - are relatively minor compared to the fission-fragment hazard, or
compared to similar hazards in non-nuclear plants. The largest reactor explosion
actually attained so far was a deliberate destructive test of a space-propulsion
reactor, and it yielded the equivalent of 100 lbs of TNT, rather a small amount
compared with the yield of some chemical accidents.
In almost all reactors the fissions occur in solid fuel elements, and the
fission fragments are therefore normally trapped inside the solid matrix of
these elements, or at least inside the cladding of the fuel.
To release the
fission fragments, some sort of overheating is required resulting in a phase
Research sponsored by the United States Atomic Energy Commission under
contract with the Union Carbide Corporation.
PATENT CLEARS APPROVED. PROSECTION.
PATENT CLEARANCE OBTAINED. RELEASE TO
THE PUBLIC IS APPROVED. PROCEDURES
ARE ON EILE IN THE RECEIVING SECTION,
-2-
change in the solid fuel, or in its melting, or vaporization, or in damage to
the cladding. Overheating occurs when the ratio of heat production to cooling
becomes unduly large. I would like to discuss first the case where this results
from excessive heating during a power excursion of the reactor.
Reactor Dynamics - Doppler Effect
A power excursion occurs when the population n of each neutron generation
exceeds the population of the preceding generation by a fraction, called the
excess reactivity key. If t is the lifetime of one generation,
du/dt = (kex/t) n
(1)
ror
2
and the neutron population and reactor power increases exponentially as long
as kex and t remain constant. Actually, things are very much more complicated.
re
are ver
mo
In the first place, a few tenths of a percent of the neutrons are "delayed";
that is, they have lifetimes orders of magnitude greater than the majority of
the neutrons. Eq. 1 then is modified, inasmuch as we have to subtract from
kov a fraction corresponding to the neutrons which are delayed; however, on
the right side of Eq.1 new terms are added corresponding to the delayed neutrons
as they are formed. These terms are obtained from a set of first order differ-
ential equations. Eq. 1, and these differential equations form a system which
is frequently solved by analog computers, sometimes simply involving electrical
networks.
Much more fundamental modifications of Eq. I are caused by feedback, that
is the fact that the excess reactivity is influenced by the neutron population n.
A relatively simple example of this is produced by the Doppler effect. For
reasons known to many of you, the reactivity changes if the fuel temperature
-3-
varies. In practical reactors the reactivity usually goes down as the fuel
nas
gets hotter, that is we have a negative feedback. Now the temperature depends
on the energy density deposited by the nuclear heating, and the energy density
is the time integral of the power density, which, in turn, is essentially pro-
portional to the neutron population n. Thus, if k. is the originally inserted
excess reactivity,
kex = k. - function of ſ
dt' n(t') .
(2)
If we have indeed a negative feedback, this function is positive, and after a
while kex goes to zero and n reaches a maximum. After this, key becomes
more and more negative and n decreases assymptotically to zero.
In case the above function is linear, an analytical solution in terms of
hyperbolic functions is easily obtained, at least if Eq. 1.can be used and the
delayed neutrons appear. Note, that the feedback makes Eq. l non-linear, even
if the function in Eq. 2 is linear, because n is multiplied by its own time
integral.
In practice, however, the function in Eq. 2 is not linear.
In the first
place, the reactivity decrease per degree decreases with increasing temperature,
On the other hand, the heat capacity decreases usually with increasing tempera-
ture so that usually at higher temperature more degrees are obtained for a given
increase in energy density. The two non-linearities thus compensate each other,
but only partially. Furthermore, if the temperature increases are such as to
constitute a safety problem, they must involve some sort of a phase change, as
mentioned above; and then a violent non-linearity is introduced by the latent
heat of the phase change, which prevents a temperature increase for a finite
increase in energy density. Digital methods are used overwhelmingly in the
non-linear case though I am not absolutely sure that this is necessary,
-4-
The Doppler effect is usually regarded as "instantaneous", that is, the
reactivity is usually assumed to follow the temperature or energy density with-
out time delay. However, in some important cases even this simplication is not
2
quite correct.? For various reasons, the fissionable material, such as plutonium,
is sometimes concentrated in small grains, embedded in fertile inaterial, such
as wºu. The decrease in reactivity with increasing temperature comes from the
fertile material. The Doppler effect of the fissionable material is small,
and probably goes in the other direction.
In a transient, the grains of fission-
able material heat up first, giving a small and probably positive reactivity
change. Only after a time delay will the fertile material heat up and give
the strong negative Doppler effect. This time delay is described by the heat-
conduction equation, a partial differential equation of second order in the
space coordinates and of first order in time. As a practical matter, the
delay allows the reactor power to increase longer than would otherwise be the
case, and consequently the transient becomes more violent.
It has recently been pointed out, however, that the above computational
model might be misleading. If the small grains of fissionable material are to
absorb, initially, almost all the energy generated, they must get very hot
indeed. They will soon vaporize and fissionable material will condense in small
cracks in the fertile material which never has exactly the maximum theoretical
density. From then on, the heat transfer will be very rapid and the negative
Doppler effect will terminate the transient sooner than the previous model
indicated. I just mentioned this somewhat detailed argument in order to emphasize
the difficulty of arriving at a suitable model, a difficulty which is superimposed
on your problem of computationally describing a given model.
-5 -
Other Reactivity Effects
The Doppler effect is of course not the only reactivity effect to be
COV
considered. Some reactivity change will result from the thermal expansion of
the heated fuel. As long as it is a question of a metallic fuel rod that
expands longitudinally as a function of its temperature, the problem is fairiy
straightforward, complicated only moderately by the temperature dependence of
the expansion coefficient. Sometimes it is even possible to neglect the time
delay connected with this expansion, since it is small compared to the time
constant of some transients. However, if it is a question of a fuel rod that
is constrained at the ends or elsewhere, that is "fluffed up" by the effect of
many fissions, or that has many voids or cracks, the matter is usually not
accessible to analytical solution. Other thermal-mechanical effects will be
discussed later.
Reactivity effects are also caused by changes in coolant density, which
may result from the transfer of heat from the fuel into the coolant. Here it
is a question of coupling the reactor dynamics equations with the heat-transfer
equations, which involve the heat conduction equation in several regions, coupled
by boundary conditions. The great difficulty stems here from the non-linearity,
and above all, from the threshold effects such as boiling. In water-cooled
reactors, the nhenomena governing the heat transfer are quite different in the
subcooled region and in the region where boiling is taking place. The difference
also applies to the effect one unit of heat has on reactivity: 18 the unit is
used to generate steam, it changes the reactivity much more than if it heats
subcooled water.
Further reactivity effects are caused by the heating of the moderator,
which again is connected to heat generation in the fuel by heat transfer
-6-
equations, involving sometimes variable contact resistances at the boundaries
between regions. The moderator heating acts through a change in the low-
energy spectrum of the neutrons which interacts with the energy dependence of
the cross sections in a manner that is hard to describe by analytical functions
and hence particularly suitable for computational solutions. Furthermore, some
of the nuclides with important cross-section variations in the low-energy
region vary in concentration throughout the life of the reactor. For instance
4°xe, the fissionable nuclides and some fission fragments belong into this
category. For this reason, very many calculations are required again calling
for computational methods.
Advanced Topics of Reactor Dynamics
In Eq. 2 I have tacitly assumed that the reactivity is inserted as one
step at low, but not extremely low, power. This assumption is frequently made
because it is simple and gives pessimistic results, something required in most
safety analyses. If the reactivity is inserted more gradually, the negative
reactivity effects mentioned above keep up with the reactivity insertion and
the reactor never gets very super-critical. Also, slower but larger reactivity
effects get a chance to act. Thus consideration of the time dependence of the
reactivity addition results frequently in reduced severity of the computed
transient. However, even where analytical solutions are available for the
inotantaneous reactivity addition, computational methods might be required for
the time-dependent case of reactivity insertion. Furthermore, the postulated
reactivity additions, such as rod withdrawal at maximum speed etc., are fairly
well standardized, resulting in a large number of calculations of the same kind,
which favors machine calculations,
-7-
Equation 1 breaks down if the start-up occurs at extremely low power or
SOUTCO
source strength. In that case, the reactor power may fail to rise, even if
er
even
the reactor is supercritical, simply because there is no neutron present to
get the reaction started, or because a chain reaction, which did start, breaks
off in accordance with the law of statistics. Thus even a gradual reactivity
addition may result in considerable excess reactivity and hence a large excur-
sion, because there is no immediate power increase and no immediate negative
reactivity effent. Codes nave been written for this low-source startup.
The capability of even the largest computing machines is challenged
when the space dependence of the reactor excursions is taken into account.
In the case of large reactors that is necessary. The point is that the power
density is not the same everywhere in the reactor. Hence the feedback effects
will vary from point to point, and the reactivity distribution will change.
This in turn distorts the power distribution. Considering that the computa -
tions usually require multigroup calculations, and that the space dependence
might have to be worked uut in several regions and in two or three dimensions ,
it is clear that one cannot be very generous with the number of groups, the
number of space points and the length of the time intervals. A procedure
usually adopted is the following: First the space-dependent problem is solved
for one moment in time.
This gives a power distribution, and an assymtotic
reactor period. Next, for a certain time interval, the power, distributed in
space as just computed, is allowed to increase exponentially with the above
period. The energy density deposited during the time interval causes a space-
dependent change in the local reactivity. The new reactivity distribution is
used in a new space-dependent computation to obtain a power distribution and
reactor period, and so.
-8-
Quite apart from the difficulties with excessive coarseness of the space
EXCE
coal
and time grid, and the popsibility of computational instabilities, it is diffi-
cult to prove that the above method approximates the correct solution.
In
reality, of course, the power does not increase with the same period all over
the reactor, something that is a basic assumption in the above method.
In some simplified cases, I have been able to obtain analytical solutions
to the space dependent problem with feedback. The solutions are based on the
Weierstrass P-function, and they may be used to check the validity of some of
the above computational methods. If the analytical method is applied to the
multi-region case, it becomes a cumbersome system of coupled equations, involv-
ing P-functions just as the no-feedback case becomes a system of coupled equa-
tions involving sine, cosine, and exponential functions. The system with the
P-functions could presumably be coded and that might be corisiderably simpler
than the coding of the problem based on a grid of discrete space and time points.
Hydraulics
Overheating of the fuel elements and the resulting fission fragment release
may also result from too low a coolant flow, or loss of the coolant altogether.
In fact the assumed "maximum credible accident" usually involves a break of one
of the main coolant pipes, followed by the loss of the coolant through the result-
ing openings. Sometimes it is assumed that a coolant pump loses power, and the
flow then coasts down ultimately leaving insufficient flow for heat removal.
The events immediately after the onset of the disturbance are the most important
because the heat generation is largest at that time:
The nuclear chain reaction
has not yet quite decayed, and the afterheat from fission fragments is greatest.
-9-
Basically, the fluid flow problem involves nothing but the conservation of mass,
momentum and energy. However, the complicated nature of the problem may be
understood, when it is remembered that the mass flow distributes itself over
several channels in series or in parallel; that the momentum equation involves
friction factors in complicated channel shapes; and that the energy equations
involved the pumps and frequently also gravity (such as in the case of natural
convection). The matter is particularly involved when boiling is present.
Here, large density changes, large accelerations, and two-phase flow enter in.
Coupling with the heat transfer equations influence the hydrodynamics, and more
importantly, the flow pattern influences the heat transfer. For transient
phenomena of the order of milliseconds' duration, the interaction between heat
transfer, boiling, and flow is not well understood. The situation is bad for
water, and worse for other coolants. It is frequently not a question of solving
the equations applicable to a specific model, but rather of finding a reasonahie
model.
Thermal-Mechanical Effects
I mentioned already, that thermal-mechanical effects, such as fuel-element
expansion or bowing, influence reactivity.
They also influence fluid flow to
some extent, and are of particular interest with respect to possible interfer-
ence with control rod motion etc. The interaction of time-dependent heat genera-
tion, heat transfer, and mechanics prevents solution in ail cases except the
simplest cnes. The time-dependent equation describing the bowing of a single
rod is a partial differential equation, of the second order in time and the
fourth order in space. Constraints in the mechanical motion, and possible creep,
further complicate matters.
-10-
For really large excursions, and those are the most important ones for
safety purposes, the pressure waves caused by thermal expansion or vaporization
of the fuel are to be computed. The results of these pressure waves are fuel
element destruction, elastic or plastic deformation of the pressure vessel,
water hammer effects and so on. Frequently, the fuel element destruction and
dissassembly of the core are the shut-down mechanisms which ultimately termi-
nates the nuclear excursion. The interaction between the nuclear equations
and the mechanical equations is the subject of the famous AX-1 code for very
large destructive transients. This code has been subject to many refinements,
to take into account the Doppler effect, and two dimensional geometry. The
latter is of importance for instance in the case of a reactor of a shape
approximating a flat cylinder.
Fission Fragment Transport
Once the above effects are assumed to conspire to release fission frag-
ments from the reactor, or even from the fuel elements, rather simple assump-
tions have been made in the past as to the further fate of the fission fragments.
For instance it was postulated that 100% of the rare gases and 25% of the iodine
leaks out to the atmosphere. Under this assumption, the iodine is' by far the
dominan't hazard. Practically that is unrealistic because the iodine will, to
a large extent, be retained in the coolant, plate out on the containment walls
or be trapped in purposely supplied filters. Mathematical models are used to
describe these effects. However, here again, we are only just on the verge of
being able to understand the phenomena which have to be considered in these
len
Se
models. For instance it is quite important to know what chemical form the iodine
is in, and whether it is adscrbed on particulate matter or not. There is thus
not much point in discussing these models in detail.
Wačer sprays may be used to reduce the pressure inside the containment
and this leads, in a computable manner, to a reduction and final termination
of any release from the containment. On the other hand, some of the postulated
accidents responsible for the fission fragment release from the fuel do not
affect all the fuel simultaneously. The fuel elements in the highest flux
release the fragments first, the other elements do so successively later. This
convoluted with the time function describing the release from the containment
gives an overall release more complicated, but more favorable, than the above
mentioned release fractions.
Meterology
The leakage from the containment then represents a source term for
atmospheric dispersion. Previously this dispersion had been assumed to take
place according to the Sutton diffusion equation, an equation quite similar
to the diffusion equation describing the neutron flux, and also to the heat
conduction equation. The meteorological equation has some wrinkles of its
own, however, for instance it is anisotropic. This relatively simple treat-
ment neglects a few factors that reduce the computed radiation dose to an
individual affected. Wind variability is one of them: is has been customary
to assume that the wind has a constant direction after the incident and hence
exposes a certain unfortunate group of individuals for the duration of the
release. Actually, the wind rarely behaves like this, but sweeps back and forth
over a larger arc. Sometimes the wind variability is neglected because of con-
cern over the rare cases in which it does not apply, but frequently it is
disregarded because of mathematical complexities which it entails. In these
latter cases the convenience of computing machines may make it practical to
take credit for the wind variability and similar mitigating factors.
-12-
Another example is the whole-hody radiation exposure resulting from a
cloud of rare gases. A simple method of computing this exposure is based on
the assumption that the cloud is infinite in extent.
This is far from true
close to the source, and a more sophisticated calculation facilitated by
computing machines“ gives more favorable, and still correct results.
To compute from given concentrations of the various fission fragments the
doses to the whole body and to various organs, and to find ground and crop con-
tamination, and other economic losses requires in any given case the performance
of a large number of fairly elementary operations, something computers are well
suited for.
· Concluding Remarks
I would like to mention an example of a somewhat peripheral use of computers
in nuclear safety:
The Nuclear Safety Information Center 'uses data processing
machines for the storage and retrieval of information.
In conclusion, I must apologize again for having given a talk at this
meeting without understanding much about computers myself. What I tried to
do was to give a survey of the problems that occur in nuciear safety, and that
are suitable - to a varying degree - for solution by computational methods.
This might help in establishing a background for the excellent safety-connected
papers of this session. I am particularly referring to the papers by Galligani,
by Blaine, and Berland, by Perks, by Brickstock, Davies and Smith, and by Russell.
The emphasis on reactor physics was intentional, and based on the relatively
advanced state of the discipline, Much of the information I presented can be
found in the SIFTOR book. I have tried, to include a few examples of work
ey
done since these books were written.
Thank you.
-13-
References
1. T. J. Thompson, and J. G. Beckerley, "The Technology of Nuclear Reactor
Safety," Volume 1, the M.1.1, Press, Cambridge, Massachusetts
2.. Axel Fraude, "Institute of Neutron Physics and Reactor Technology," INR-NR 59/63,
July 21, 1964,
R. E. Peterson, "Neutron Kinetics Problems Associated with Mixed Oxide Fuels,"
Hanford Atomic Products Operation, Richland, Washington, HW-81259, July 1964.
V. W. Fustafson and R. E. Peterson, "A Computer Investigation of Doppler
Delay in Mixed Oxide Fuels," Trans. Am. Nuclear Soc., vol. 7, no. 2,
November, 1964.
H. Hurwitz, D, B. MacMillan, J. H, Smith, and M. L. Storm, 'Kinetics of Low
Source Reactor Startups, Part II," General Electric Research Laboratory
and and Knolls Atomic Power Laboratory, Schenectady, New York, July 6, 1962,
Nuclear Science and Engineering, 15, 187–196 (1963) ; KĄPL-P-2245.
D. B, MacMillan, and M. L. Storm, "Kinetics of Low Neutron Source Reactor
Startups - Part III, Nuclear Science and Engineering 16, 369-380 (1963).
H. Hurwitz, Jr., "Kinetics of Low Source Reactor Startups, Part I,"
Nuclear Science and Engineering, 15(2): 166-186, February 1963,
II
D. S. Duncan, "CLOUD -An IBM 709 Program for Computing Gamma-Ray Dose Rate
from a Radioactive Cloud," Atomics International, Canoga Park, California,
NAA-SR-Memo 4822, 1959.
END
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