Study of self-organizing regime of nuclear burning wave in fast reactor
An approach for description of the space-time evolution of self-organizing nuclear burning wave regime in a critical fast neutron reactor has been developed in the effective multigroup approximation. It is based on solving the non-stationary neutron diffusion equation together with the fuel burn-u...
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| Date: | 2005 |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2005
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| Cite this: | Study of self-organizing regime of nuclear burning wave in fast reactor / S.P. Fomin, Yu.P. Mel’nik, V.V. Pilipenko, N.F. Shul’ga // Вопросы атомной науки и техники. — 2005. — № 6. — С. 106-113. — Бібліогр.: 13 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859810734348697600 |
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| author | Fomin, S.P. Mel’nik, Yu.P. Pilipenko, V.V. Shul’ga, N.F. |
| author_facet | Fomin, S.P. Mel’nik, Yu.P. Pilipenko, V.V. Shul’ga, N.F. |
| citation_txt | Study of self-organizing regime of nuclear burning wave in fast reactor / S.P. Fomin, Yu.P. Mel’nik, V.V. Pilipenko, N.F. Shul’ga // Вопросы атомной науки и техники. — 2005. — № 6. — С. 106-113. — Бібліогр.: 13 назв. — англ. |
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| container_title | Вопросы атомной науки и техники |
| description | An approach for description of the space-time evolution of self-organizing nuclear burning wave regime in a critical
fast neutron reactor has been developed in the effective multigroup approximation. It is based on solving the
non-stationary neutron diffusion equation together with the fuel burn-up equations and the equations of nuclear
kinetics for delayed neutron precursor nuclei. The calculations have been carried out in the plane one-dimensional
model for a two-zone homogeneous reactor with the metal U-Pu fuel, the Na coolant and constructional material Fe.
The temperature effects and heat sink were not considered.
В ефективному багатогруповому наближенні розвинуто підхід для опису просторово-часової еволюції
хвильового режиму ядерного горіння, що самоорганізується у критичному реакторі на швидких нейтронах.
Він заснований на розв’язанні нестаціонарного дифузійного рівняння переносу нейтронів разом з
рівняннями вигоряння палива і кінетики попередників запізнілих нейтронів. Розрахунки проводилися у
плоскій одновимірній моделі двохзонного гомогенного реактора з металевим U-Pu паливом, Na-теплоносієм
та конструктційним матеріалом Fe. Температурні ефекти і відвід тепла не розглядались. Доведено, що за
певних умов у реакторі можна створити хвильовий режим ядерного горіння, у якому реактор без керування
протягом тривалого часу може підтримуватися у стані, близькому до критичного.
В эффективном многогрупповом приближении развит подход для описания пространственно-временной
эволюции самоорганизующегося волнового режима ядерного горения в критическом реакторе на быстрых
нейтронах. Он основан на решении нестационарного диффузионного уравнения переноса нейтронов сов-
местно с уравнениями выгорания топлива и кинетики предшественников запаздывающих нейтронов. Расче-
ты проводились в плоской одномерной модели двухзонного гомогенного реактора с металлическим U-Pu
топливом, Na-теплоносителем и конструкционным материалом Fe. Температурные эффекты и отвод тепла не
рассматривались. Показано, что при определенных условиях в реакторе можно создать волновой режим
ядерного горения, в котором реактор без управления в течение длительного времени может поддерживаться
в состоянии, близком к критическому.
|
| first_indexed | 2025-12-07T15:18:57Z |
| format | Article |
| fulltext |
A P P L I C A T I O N O F N U C L E A R M E T H O D S
STUDY OF SELF-ORGANIZING REGIME
OF NUCLEAR BURNING WAVE IN FAST REACTOR
S.P. Fomin, Yu.P. Mel’nik, V.V. Pilipenko, N.F. Shul’ga
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
e-mail: sfomin@kipt.kharkov.ua
An approach for description of the space-time evolution of self-organizing nuclear burning wave regime in a cri-
tical fast neutron reactor has been developed in the effective multigroup approximation. It is based on solving the
non-stationary neutron diffusion equation together with the fuel burn-up equations and the equations of nuclear
kinetics for delayed neutron precursor nuclei. The calculations have been carried out in the plane one-dimensional
model for a two-zone homogeneous reactor with the metal U-Pu fuel, the Na coolant and constructional material Fe.
The temperature effects and heat sink were not considered.
PACS: 28.41 T, 28.52 N
1. INTRODUCTION
The work is devoted to developing the physical
foundations of the new perspective conception of the
safe fast reactor (FR). This FR is a new type of long-life
operation reactor with the so-called inner safety. This
type of inherent safety prevents the appearance of
reactivity-initiated accidents in FR by virtue of the
physical principles that underlie its design. The FR in
distinction from conventional fast reactors has no initial
excess reactivity. Therefore, it does not need any
reactivity control. Another important peculiarity of FR
under consideration is that it operates till the end of its
life without any refueling or fuel shuffling. In FR of this
type natural or depleted uranium can be used as its fuel,
except for the active zone of the initial critical assembly
that serves as an ignition region. Thorium can also be
useable instead of uranium. The level of fuel burn-up
can be essentially high. The FR also has an important
merit from the point of view of nuclear proliferation. It
does not need a human access during its operation time
and so can be placed underground.
The FR operation is based on the non-linear self-
organizing regime of the nuclear burning wave (NBW)
that arises owing to a high conversion ratio from fertile
to fissile materials in the FR. Feoktistov [1,2] was the
first to show up this regime in the framework of a
schematic one-dimensional model of FR with the U-Pu
fuel cycle. In a self-similar approach he proved the
existence of the NBW and estimated the velocity of its
propagation. However, Feoktistov’s model of FR has
essential faults. In this model, the equilibrium and
critical plutonium densities were considered as certain
phenomenological parameters whereas in reality they
depend on both the neutron cross-sections and
characteristic neutron spectrum. This scheme was
restricted to a maximally simplified set of burn-up
equations that involved only four components of the
nuclear transformation chain and did not take into
account the fission products, constructional materials
and coolant.
The concept of NBW was further developed in the
framework of multigroup diffusion approximation in
[3,4]. A stationary equilibrium regime of nuclear burn-
up (named CANDLE) was studied in the cylindrical mo-
del of FR using the self-similar solution approach.
Calculations were carried out for a lead-bismuth-eutectic
cooled FR with metallic uranium fuel employing a long
nuclide chain in the burn-up equations. It was shown
that in this steady-state regime the NBW moved with a
constant velocity along the reactor without changing its
shape.
Teller et al. [5,6] proposed a long-life FR of a
prolate form with fuel of the U-To cycle. They sought
for a numerical solution of the corresponding non-
stationary problem and used a large amount of specially
composed software. The burning process had the form
of a running NBW that started in the ignition zone. Then
this wave propagated beyond the ignition zone in an
axial direction during the life of reactor.
The mechanism of self-organizing regime in the FR,
was also explored in the framework of the quasi-
diffusion approach with the help of mathematical
modeling in [7,8]. In the one-dimensional model of FR
it was shown that this regime (called as self-adjusting
neutron-nuclide regime) could be initiated inside its core
at certain conditions. However, as it follows from the
results [8], this regime does not go over into the NBW
regime in the non-stationary scheme considered. The
reactor starts to damp earlier than the wave is created.
The possibility of creating the NBW regime in a
linear FR was also confirmed in Ref. [9] by calculations
carried out for the simple model [1] both in the self-
106 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2005, № 6.
Series: Nuclear Physics Investigations (45), p. 106-113.
similar solution approach and in the corresponding non-
stationary one.
In our previous work [10], the possibility of creation
of a self-organizing regime in the form of a running
NBW was proved for a model of critical FR close to a
real situation. The requirements of the wave initiation
and evolution in space and time were studied. To
describe the space-time evolution of NBW, an approach
has been developed in the framework of one-group
approximation. It includes the non-stationary diffusion
equation for neutron transport, the burn-up equations for
fuel components and the equations of nuclear kinetics
for precursor nuclei of delayed neutrons. Certain
properties of the NBW regime were studied, such as the
mechanism of reactivity feedback and stability of the
NBW regime relative to distortions of the neutron flux.
The calculations have shown that the regime of
running NBW can be observed when one uses the
effective one-group cross sections obtained by averaging
the group cross sections over the group neutron
spectrum integrated over the length of the initial critical
assembly of FR. It should be noted that the NBW
regime did not arise if for this purpose the group neutron
fluxes at each space point of the initial critical assembly
were used (cf. [8]).
In both cases the effective one-group microscopic
cross sections remained unchanged during the lifetime
of the FR. As a matter of fact, the group neutron fluxes
essentially change inside the FR during its operation
time. Therefore, the effective one-group microscopic
cross sections must also be changed according with this
space-time alteration of the neutron spectrum with time.
The purpose of the present paper is to study the
influence of the space-time alteration of the group
neutron fluxes on the process of NBW initiation and
propagation inside the FR under consideration. This can
be done considering the multigroup criticality problem
at every moment of the FR operation time. The corres-
ponding calculations will be performed in the frame-
work of effective multigroup approximation, which is a
modified version of the approach developed in [10].
2. THE CALCULATION SCHEME
For the description of space-time evolution of the
self-organizing NBW in a critical FR it is necessary to
solve the set of equations in partial derivatives that
includes the non-stationary diffusion equation, the
equations of fuel burn-up and the equations of nuclear
kinetics.
Hereafter, these equations are written in the one-
group approximation for the case of the plane one-
dimensional model of FR under consideration.
The non-stationary diffusion equation with taking
into account delayed neutrons can be written as
1 ( ) (1 )( )∂ Φ ∂ ∂ Φ− Φ − − Σ Φ
∂ ∂ ∂
+ Σ a f fD
v t x x
β ν
= ∑ ∑ i i
l ll i
Cλ , (1)
where Φ(x,t) is the scalar neutron flux, Σα(x)=Σjσα
jNj(x)
is the macroscopic cross section of the neutron reaction
of the α-type (the index α corresponds to the reactions
of neutron absorption (a) and fission (f)), Nj(x) is the
concentration of j’th nuclide at the point x; σα
j is the cor-
responding effective one-group microscopic cross sec-
tion for the j’th nuclide; νfΣf=Σjνf
jσf
jNj(x), νf
j is the mean
number of neutrons produced at a single nuclear fission
event for the j’th fissile nuclide; β=Σjβj(νfΣf )j/νfΣf is
the effective fraction of delayed neutrons, βj=Σiβj
i, and
βj
i, Cj
i and λj
i are the portion of delayed neutrons, the
concentration and decay constant of the precursor nuclei
in the i’th group of the j’th fissile nuclide, correspond-
ingly; D(x)=1/(3Σtr(x)) is the diffusion coefficient, Σ
tr(x)) is the macroscopic transport cross-section, v is the
one-group neutron velocity.
We use the boundary conditions of the third kind for
the flux Φ(x,t) that take into account the presence of an
external neutron flux jex falling onto the left boundary of
FR while the right boundary is free:
0
( , )(0, ) 2 (0, ) 2
=
∂ ΦΦ − =
∂ x
x tt D t jexx , (2)
( , )( , ) 2 ( , ) 0
=
∂ ΦΦ + =
∂ x L
x tL t D L t
x . (3)
These conditions are valid for any moment of time
within the considered time interval 0 ≤ t ≤ T. Besides,
the scalar neutron flux in the corresponding critical
assembly Φ0(x) is chosen as an initial condition for Φ
(x,t) at the moment t = 0 for all values x from the space
interval 0 ≤ x ≤ L.
The burn-up equations describe changing the fuel
components with time according to the chain of nuclear
transformations. In the case of FR with the U-Pu fuel
cycle we consider the chain including only 10 nuclides,
whose numeration is presented in the table, to facilitate
writing down corresponding equations:
( )
( )( 1) ( 1) , 1 8,
l
al l l
cl l l
N N
t
N l
σ
σ − −
∂ = − Φ + Λ
∂
+ Φ + Λ = K
(4)
9
6 6
∂ = Λ
∂
N N
t
, (5)
10
1,4,5,6,7=
∂ = Φ
∂ ∑ fl l
l
N N
t
σ , (6)
where σal=σcl+σfl, σcl is the microscopic cross section of
radiation neutron capture by the l’th nuclide, Λl=ln2/Tl
1/2
and Tl
1/2 are the decay constant and half-life of the l’th
nuclide.
The numeration of nuclei in the 238U – 239Pu
transformation chain
№ 1 2 3 4 5
Nucleus 238U 239U 239Np 239Pu 240Pu
№ 6 7 8 9 10
Nucleus 241Pu 242Pu 243Am 241Am FP
The pair of fission fragments produced at every
fission event is considered to be one nuclide that we de-
note by the symbol FP (fission products).
107
At the initial moment of time the values of nuclide
concentrations are
0( , 0) ( )= =l lN x t N x . (7)
In the scheme studied we neglect the burn-up of nuc-
lei 239U, 239Np, 241Am, 243Am (σa2=σa3=σa8=σa9=0) be-
cause the decrease of their concentrations due to the
absorption reactions is small as compared with the
processes that have been considered. The changes of the
fission fragments as a result of neutron absorption also
were not considered.
The NBW regime is a slow process in which the
scalar neutron flux Φ varies very weakly during the cha-
racteristic decay time of the precursor nuclei that emit
delayed neutrons (see, for example, [10]). In this case,
for the nuclear kinetic equations can be used the appro-
ximation of one equivalent group of delayed neutrons
( )∂ = − + Σ Φ
∂
l
l l l f f l
C C
t
λ β ν , (8)
0( , 0) ( )= =l lC x t C x , (9)
where / /= ∑ i i
l l l li
λ β β λ .
The complete statement of the non-stationary prob-
lem considered above includes the set of 16 non-linear
partial differential equations and the corresponding ini-
tial and boundary conditions for them as well. We have
solved this set of the equations using the finite-diffe-
rence method. To apply the finite-difference technique,
we introduce a rectangular mesh with steps h and τ (uni-
form for x and variable for t) in the range of the variab-
les x and t. The FR length L was divided into M intervals
of a spatial calculation mesh. We have found the solu-
tions of the set of algebraic equations obtained from
Eq. (1) in this way using, like in Ref. [10], the implicit
Crank-Nicolson difference scheme [11]. This symmet-
ric-in-time scheme shows an unconditional stability at
any relation between space and time steps. It has the ap-
proximation of the second order of accuracy in h and τ .
The solution of the burn-up equations (4)-(6) and
equations of nuclear kinetics (8) has been simplified
assuming that the effective one-group cross sections and
the neutron flux Φ are constant during the time intervals
τ. As for the constancy of cross-sections, this assump-
tion is well fulfilled for the FR conditions because of a
weak sensitivity of the effective cross-sections to chan-
ges in the fast neutron energy spectrum. The assumption
for Φ can be easily satisfied by choosing sufficiently
small time intervals τ, within which the flux value
should be taken as Φ = (Φn + Φn+1)/2 (Φn is the flux
value at the time layer n). This fact allowed us to obtain
an approximate analytical solution of Eqs. (4)-(6) and
(8) for the concentrations of the corresponding nuclides
at every node of the space mesh for the new time layer
n+1 via the solution for layer n (see for details [10]).
It should be noted that the implicit finite-difference
scheme that was used for solving the set of algebraic
equations under consideration is non-linear. The neutron
flux has been found using the method of successive
approximations, in which its values at a new time layer
were determined by an iteration procedure (see [10]).
The expression for the equilibrium concentration of
plutonium in a stationary state is written as
1
1
4 4 4
( ,2 )
c
eq
f c n n
NN σ
σ σ σ
=
+ + , (10)
where σ4
(n,2n) is the cross section of the (n,2n) reaction
for 239Pu.
At the initial stage of the nuclear burning inside the
FR the following expression for Neq is more correct
3 3
4 4 4
( ,2 )( )eq
f c n n
NN
σ σ σ
Λ=
+ + Φ . (11)
For calculations of the effective one-group micros-
copic cross sections we used the group neutron fluxes Φ
g (g is the number of neutron energy group) for the
initial critical assembly, found from solving the stationa-
ry multigroup problem. The calculations were perform-
ed in the 26-group approximation using the group
neutron constants from Ref. [12]. The method of averag-
ing the group cross-sections when passing from a greater
number of energy groups to a smaller one is well known
(see, for example, [13]). We have used the approach that
takes into account the requirement of conservation of
rates of corresponding reactions during this procedure.
The scheme of passing from the group microscopic
cross sections to the effective one-group cross sections
is defined by the relations
26
1
gl g
l
g S
α
α
σσ
=
Φ=
Φ∑ ,
26
1
g
S
g =
Φ = Φ∑ , (12)
where ΦS is the neutron flux summed over 26 groups,
the index α corresponds to the reactions of neutron cap-
ture (c), fission (f) and scattering (s).
The one-group neutron velocity is given by
26
1
1 1 g
g
gSv v=
Φ=
Φ ∑ , (13)
where vg is the neutron velocity for the group g.
When averaging the transport cross section σtr, the
requirement of conservation of the neutron leakage
value was taken into account, which leads to the
following expression
26 26
1 1
/l gl g g g g
tr tr
g g
D Dσ σ
= =
= Φ Φ∑ ∑ . (14)
3. RESULTS AND DISCUSSION
We have carried out a series of calculations of the
NBW regime in the critical FR using the developed pro-
cedure of numerical solution of the set of non-linear
algebraic equations under consideration. The FR core
length, 0≤x≤ L, is divided into M=200 intervals of the
spatial calculation mesh. We impose boundary condi-
tions (3), (4) on the scalar neutron flux Φ(x,t). To create
a neutron field in the system, that has to initiate the pro-
cess of nuclear burning, we assume that the left boun-
dary of the system is exposed to an external neutron flux
coming from a source of certain intensity jex.
We consider a two-zone homogeneous FR with a
metal U–Pu fuel of porosity p=1 (see Fig. 1). In the first
zone (near the left edge of the reactor) the fuel consists
of uranium enriched with plutonium (the ignition regi-
108
on). The second zone (the breeding zone), adjacent to
the ignition one, is filled with 238U (the presence of 235U
was neglected in the present FR model). Both zones also
include the constructional material Fe and the Na cool-
ant. The plutonium concentration in the enriched zone
was chosen so as to be less than the equilibrium pluto-
nium concentration and was equal to 10% of the concen-
tration of fuel nuclei. The isotope composition of pluto-
nium in the fuel was chosen to be as follows:
239Pu : 240Pu : 241Pu : 242Pu = 0.70 : 0.22 : 0.05 : 0.03.
X 0 XB L
Composition: U-Pu fuel - 40 %, Na – 25 % Fe – 35 %
Zone 1 Zone 2
jex
Fuel: Fuel:
Pu – 10 % 238U – 100 %
238U – 90 %
Fig. 1. The initial critical assembly of FR
The sharp boundary between the zones that exists at
the initial moment of time would be quickly diffused
with time. Therefore, to simplify the calculations, the U–
Pu fuel distribution is assumed in the form of the Fermi
step function with the parameters xB and ∆ defining the
distribution width and diffuseness, respectively (see
[10]).
The initial space distribution of the neutron field cor-
responding to the neutron flux of the critical FR was
normalized so that the averaged energy production den-
sity in the enriched zone was equal to 10-8 kW cm–3. The
intensity of the external neutron flux falling onto the left
boundary of FR, which initiates the burning process, has
been chosen to be jex = 6⋅1011 cm-2 s-1 for all variants of
calculations.
We have chosen the following volume fractions of
components in each zone: for the nuclear fuel
Ffu = 40 %, the constructional material FFe = 35 % and
the coolant FNa = 25 % (see Fig.1). These values of
volume fractions were taken to correspond to the
composition of actual reactors.
In one of the variants that we have calculated in our
paper [10] the corresponding effective one-group mic-
roscopic cross sections σα (x) were calculated (see (12))
using the multigroup neutron fluxes Φ0
g(x) obtained for
the initial critical assembly under study. During the re-
actor operation period considered in our non-stationary
calculation, the values of effective microscopic cross
sections σα
l(x) (dependent on the space variable)
considered to be unchanged, as it was done in [7]. In
this case the variation of the effective one-group cross
sections according to the change of neutron spectrum
with time was not taken into account. As result the
running NBW does not arise, although a little shift of
maxima in the space distributions of neutron flux and
power is observed. In the reactor, the self-organized
regime of nuclear burning is established for a certain
time period, being similar to the so-called self-adjusting
neutron-nuclide regime that was described in [7].
0 50 100 150 200
0.0
0.5
1.0
1.5
2.0
2.5 a)
Φ
x
0 50 100 150 200
0.0
0.1
0.2
0.3
0.4
0.5
b)
Φ g
x
0 2 4 6 8 10 12 14
0.00
0.05
0.10
0.15
0.20
c)
u
χ g
Fig. 2. Calculations for critical assembly of the FR:
a) the neutron flux Φ (in an arbitrary units) both
summed over 26 groups (ΦS in (12), solid curve) and
obtained in the one-group approach (dashed curve)
versus x (in cm); b) the spatial distribution of group
neutron fluxes Φg: short dashes are for g = 5
(0.8 <En< 1.4), solid curve is for g = 7 (0.2 <En< 0.4),
long dashes are for g = 8 (0.1 <En< 0.2), dotted curve
is for g = 10 (0.0215 <En< 0.0465), the bounds of
energy groups are presented in parentheses; c) the
group neutron spectrum χg averaged over the FR
volume via lethargy u = ln(10.5/En), En is the neutron
energy in MeV
In the present paper we take into consideration the
variation of neutron flux and the effective one-group
cross sections in the FR with time. Therefore, at each
time layer we solved the multigroup problem for fluxes
in the critical FR assembly whose composition changes
according to the burn-up equations. In the calculations
that are described below, the group neutron fluxes Φg(x)
109
found for the corresponding critical assemblies were
used to obtain the effective one group cross sections.
Thus, during the whole lifetime of FR the cross sections
were corrected in the correspondence to the neutron
spectrum alteration that occurred.
Fig. 2 presents the results of calculations of the
scalar neutron fluxes in a critical assembly of FR that
were carried out in the 26-group approximation with the
parameters xB = 64 см, ∆ = 4.3786 cm and L = 200 cm.
These parameter values correspond to keff = 1 for the
given variant of FR.
A comparison of the calculation results for the sum-
med neutron flux ΦS(x) with the corresponding flux Φ(x)
calculated in the one-group approximation with the
effective cross sections σα
l(x) obtained using the 26-
group fluxes Φg(x) is made in Fig. 2a. As can be seen,
the results of calculations in the multigroup approach
and in the considered variant of the one-group calcula-
tions are close to each other. Therefore, this one-group
calculation reproduces the result obtained for the
neutron flux in the multigroup approximation in a
sufficiently accurate way.
Fig. 2b presents, as an example, the space distribu-
tions for the group neutron fluxes calculated for four
energy groups.
As can be seen from Fig. 2c, the maximum of the
energy spectrum of the considered FR with the metal
fuel lies in the region of 200 keV (on the axis of ordina-
tes we plot the integral neutron flux normalized to uni-
ty). It should be noted that, depending on the reactor
composition, essential distinctions in spectra can be
observed, especially in the low energy part. For examp-
le, the calculations show that when passing to the oxide
fuel a considerable softening of the spectrum occurs in
the low energy region. It can also be seen that at neutron
energies lower than 10 keV that corresponds to the
energy groups with numbers less than 12 the flux rapidly
drops. Therefore, the neutrons in FR do not practically
get to the thermal energy region. At the energies higher
than 1 MeV the form of the calculated multigroup
spectrum approaches to that of the fission neutron
spectrum.
We have found approximate solutions of the non-
stationary problem under consideration in the effective
multigroup approach. This approach is based on the
above-obtained result that the neutron flux in a critical
FR summed over all groups does not practically differ
from the one-group flux calculated with the effective
cross sections obtained as described above. In this case,
the values of effective one-group cross sections at each
space point are corrected at each time layer according to
the fuel composition that changes with time. The
effective cross sections define the coefficients of the
implicit Crank-Nicolson difference scheme for the non-
stationary one-group diffusion equation. Thus, the scalar
one-group neutron flux was found using the procedure
developed in [10] for the numerical solution of the
corresponding set of non-linear algebraic equations.
Fig. 3 presents the results of calculations of the main
characteristics that describe the burning process in FR.
It can be seen that the neutron flux Φ and the density of
energy production P change with time. By the 110-th
day they become approximately 400 times higher as
compared with their values initiated by the external flux
at the very beginning (cf. the curves for t = 1 day and
t = 100 days).
0 50 100 150 200
0
20
40
60
80
100 a)
Φ
x
t5, x5
t1,x100
t2, x2
t3 t4
0 50 100 150 200
0
50
100 b)
P
x
t1, x100
t2, x2
t3 t4
t5
0 50 100 150 200
0.0
0.5
1.0
1.5
c)
NPu
x
t1
t2
t3
t4
t5
0 50 100 150 200
0
10
20
30
40
d)
B
x
t5
t4
t3
t2, x5
Fig. 3. The NBW regime initiated by the external
neutron flux jex= 6⋅1011 cm- 2 s- 1 falling onto the left
boundary of FR via the reactor length x (cm): a) the
scalar neutron flux Φ(x) (×1016 cm- 2 s- 1); b) the power
density P(x) (kW cm- 3); c) the 239Pu concentration
NPu(x) (×1021 cm- 3); d) the fuel burn-up depth B(x) (%)
for t1=1, t2=70, t3=109, t4=130 and t5=180 days
At the initial stage during about 70 days, we observe
slow changes of the initial distribution of 239Pu and its
110
accumulation by means of the transformation of 238U
nuclei that capture neutrons and then suffer two
successive β-decays 239U → 239Np → 239Pu with the
characteristic time t ≈ 2.5 days. Owing to this process
that occurs mainly near the right boundary of the
ignition zone, the front of the future NBW is gradually
formed. In this case, as can be seen from Fig. 3, the
allowance for the neutron spectrum alteration effects
leads to the space-time behavior of FR qualitatively
different from the results of Ref. [7] obtained with the
effective one-group cross sections σα
l(x) for the initial
critical assembly. So after the moment t ≈ 100 days the
space profiles Φ(x) and P(x) do not practically change
during 50…60 days, while the maxima of the
corresponding curves move along the x-axis.
The NBW velocity V defined as the velocity of
shifting the scalar neutron flux maximum is shown in
Fig. 4a. The function ΦI(t) presented in Fig. 4b is the
scalar neutron flux integrated over the reactor length.
One can observe a jump of V at the beginning of the
NBW propagation, which is caused by the influence of
the near ignition zone. After this initial velocity jump a
stable propagation of the NBW occurs with an almost
constant velocity V~ 1.7 cm/d during about thirty days.
This stage of NBW corresponds to the wave profiles for
the moment t = 130 days in Fig. 3. The space width of
NBW for this case is large enough so that the chosen
length of reactor core is relatively small. Owing to this
fact, the formed NBW quickly reaches the right
boundary of the reactor and the stage of stable motion of
the wave is rather short. Then the wave velocity starts to
decrease and the stage of slow extinction of the process
in the vicinity of the right boundary of the reactor
begins. Duration of the stage of extinction is about 60…
70 days. Characteristic NBW profiles for this stage are
presented for the moment t = 190 days, which are
featured by considerably lower values of the neutron
flux and energy production. Thus, in contrast to Ref. [7],
the NBW arises and moves with a constant velocity
towards the region with low plutonium concentration.
After the extinction of the reactor, plutonium and the
nuclear fission products are distributed with a practically
uniform concentration over the whole volume of the
reactor core, except for the regions close to its
boundaries (see Fig. 3). By the extinction moment the
fuel burn-up depth reaches a high level of 30…40 %
practically in the whole volume of the reactor. It can be
seen from Fig. 3 that in the breeding zone an intense
accumulation of plutonium occurs and the isotope that
burns up is practically 238U.
It follows from Fig. 4 that the velocity V and integral
neutron flux ΦI calculated without turning the external
flux jex off increase more rapidly than in the case when
the source of this flux is turned off after 10 days from
the moment of its turning on. The cause of this effect is
demonstrated in Fig. 5.
At the moment of turning the external neutron flux
off the magnitude of the integral flux in the reactor first
decreases. This decrease is followed by an increase of
the flux magnitude and then by oscillations that are
gradually damped. Therefore, the reactor itself quenches
the arising perturbation in approximately 30 days. Thus,
the self-organized burning regime that arose before the
moment of turning jex off is stable.
If we increase the external flux intensity by an order
of magnitude, then the curves for the wave velocity and
integral neutron flux in the FR are similar to those shown
0 40 80 120 160 200 240
0.0
0.5
1.0
1.5
2.0 a)
V
t
0 40 80 120 160 200 240
0
20
40
60
80
t
Φ I b)
Fig. 4. The NBW velocity V (cm d- 1) (a) and the
integral neutron flux ΦI (×1018 cm- 1 s- 1) (b) versus time
(days). Solid curves were calculated without turning jex
off, the dashed ones are for jex turned off at
toff = 10 days. The conditions correspond to Fig. 3
0 5 10 15 20 25 30
0.0
0.1
0.2
0.3
t
Φ
I
Fig. 5. The integral neutron flux ΦI (×1018 cm- 1 s- 1)
at the initial stage of burning versus time (days). The
solid curve was calculated without turning jex off and
the dashed one is for jex turned off at toff = 10 days. The
conditions correspond to Fig. 3
in Fig. 4 but are shifted down in time. Although the
perturbation of the integral neutron flux that arises when
the more intense external flux is turned off is much
greater, the picture of the damped oscillations of this
111
perturbation is similar to the one shown in Fig. 5. The
amplitude and duration of these oscillations are
essentially greater than in the previous case,
nevertheless the stability of the regime of our interest is
not violated. Turning the external flux source off at a
much later time moment (after 30 days) does not
practically affect the integral neutron flux behavior.
0 50 100 150 200
0
1
2
3
a)
N
eq
x
1 2
3
4
0 50 100 150 200
0
1
2
3
b)
Neq
x
1
2
3 4
Fig. 6. The spatial distribution of equilibrium 239Pu
concentration Neq (×1021 cm- 3), calculated by formulae
(10) (dashed curves) and (11) (solid curves), for
t = 20 days (1 and 2) and t = 200 days (3 and 4): (a)
without and (b) with allowance for the spectrum
alteration
Fig. 6 presents the space distributions of the
equilibrium 239Pu concentration calculated according to
formulae (10) and (11). It can be seen that these
quantities have a considerable time and space variation.
It should be noted that in Fig. 6b the changes of the
equilibrium concentration with time occur owing to the
decrease of the uranium concentration and to the change
of the group neutron spectrum. The results for Neq
obtained with the allowance for the spectrum alteration
and without it essentially differ at large t values (cf.
Figs. 6a and 6b). The values of the equilibrium
concentrations calculated by formulae (10) and (11)
significantly differ from each other due to the violation
of the condition of constancy of the 239U and 239Np
concentrations [10].
In the NBW regime the FR is automatically
sustained in a state close to the critical one during a long
time despite forming large amount of fission products.
The self-organizing NBW regime is determined by the
nuclear processes, in which the burn-up of the produced
239Pu excess occurs instantaneously, whereas the
increase of its concentration by means of two successive
β-decays of 239U and 239Np nuclei is a slow process
developing during a long time period. Therefore, the
described processes implement an intrinsic reactivity
feedback that ensures the reactor operation stability (see
the discussion in [10]).
4. CONCLUSIONS
The feasibility of creating a self-organizing non-
linear regime in the form of a running wave of nuclear
burning in a critical FR is studied.
The solution of the corresponding non-stationary
problem was found in the framework of effective
multigroup diffusion approach. The approach takes into
account the variation of the effective one-group cross
sections according to the alteration of the group neutron
spectrum with time that corresponds to the real situation.
This allowed us to describe the peculiarities of the NBW
regime in the FR.
The allowance for the neutron spectrum alteration
effects leads to qualitative changes of the space-time
behavior of the scalar neutron flux, as compared with
the results obtained in Ref. [7]. In contrast to [7], the
NBW arises in the FR. Its front moves with an
approximately constant velocity from the ignition zone
boundary towards the region with low plutonium
concentration.
In the FR of length 2 m under consideration the self-
organizing nuclear burning regime is set lasting for a
time period of about 200 days. In the NBW regime FR
sustains in a state close to the critical one without any
control during a fairly long time. An intrinsic reactivity
feedback is governed by the non-linearity of the NBW
regime. This feedback prevents the reactor from the
runaway regime and ensures the stable evolution of the
self-organizing NBW process.
The average fuel burn-up of about 40 % can be
attained inside the FR.
The present results show a notable stability of the
NBW regime to distortions of the neutron flux.
It should be noted that the values of the neutron flux
and density of energy production in the hypothetical FR
under consideration are rather high. To obtain more
realistic parameters of the FR that would be applicable
from the practical point of view, it is necessary to use
more complete mathematical models for the description
of the NBW regime. Of great interest would also be
similar investigations for the FR with the thorium-
uranium fuel.
ACKNOWLEDGEMENT
We express our gratitude to Drs. V.A. Apse,
E.F. Kryuchkov and their colleagues from MEPhI
(Moscow, Russia) for providing us with the package of
codes ”TIME-26”, which was very useful at solving the
multigroup criticality problem.
REFERENCES
1. L.P. Feoktistov. An analysis of a concept of a
physically safe reactor: Preprint IAE-4605/4,
Moscow: IAE, 1988, 4 p. (in Russian).
112
2. L.P. Feoktistov. Neutron-fissioning wave // Dokl.
Akad. Nauk SSSR. 1989, v. 309, p. 864-867 (in
Russian).
3. H. Sekimoto, K. Ryu, Y. Yoshimura. CANDLE: The
New Burnup Strategy // Nucl. Sci. Engin. 2001,
v. 139, p. 306-317.
4. H. Sekimoto, K. Tanaka. Application of CANDLE
Burnup Strategy to Small Reactors// Trans. Am.
Nucl. Soc. 2002, v. 87, p. 399-400.
5. E. Teller. Nuclear Energy for the Third Millennium:
Preprint UCRL-JC-129547, LLNL, 1997, 21 p.
6. E. Teller, M. Ishikawa, L. Wood et al. Completely
automated nuclear reactors for long-term operation.
Int. Conf. on Emerging Nuclear Energy Systems,
1996, p. 1-25.
7. V.Ya. Goldin, D.Yu. Anistratov. Fast neutron
reactor in a self-adjusting neutron-nuclide regime //
Mathematical Modelling. 1995, v. 7, p. 12-32. (in
Russian).
8. V.Ya. Goldin, N.V. Sosnin, Yu.V. Troshchiev. Fast
neutron reactor in a self-controlled regime of 2d
type// Dokl. Ros. Acad. Nauk. 1998, v. 358, p. 747-
748 (in Russian).
9. V.V. Pilipenko, D.P. Belozorov, L.N. Davydov, N.F
Shul’ga. Some Aspects of Slow Nuclear Burning.
Proc of ICAPP’03, 2003, Pap. 3169.
10. S.P. Fomin, Yu.P. Mel’nik, V.V. Pilipenko,
N.F. Shul’ga. Investigation of Self-Organization of
the Non-Linear Nuclear Burning Regime in Fast
Neutron Reactors // Annals of Nuclear Energy. 2005
(in press).
11. J. Crank, P. Nicolson. A practical method for
numerical evaluation of solutions of partial
differential equations of the heat-conduction type //
Proc. Camb. Phil. Soc. 1947, v. 43, p. 50-67.
12. L.P. Abagyan et al. Group Constants for
Calculations of Reactor and Shielding. Moscow: ″
Energoizdat″, 1981, 231 p. (in Russian).
13. A.E. Walter, A.B. Reynolds. Fast Breeder Reactors.
New York: ″Pergamon Press″, 1981, 605 p.
ИССЛЕДОВАНИЕ САМООРГАНИЗУЮЩЕГОСЯ ВОЛНОВОГО РЕЖИМА
ЯДЕРНОГО ГОРЕНИЯ В РЕАКТОРЕ НА БЫСТРЫХ НЕЙТРОНАХ
С.П. Фомин, Ю.П. Мельник, В.В. Пилипенко, Н.Ф. Шульга
В эффективном многогрупповом приближении развит подход для описания пространственно-временной
эволюции самоорганизующегося волнового режима ядерного горения в критическом реакторе на быстрых
нейтронах. Он основан на решении нестационарного диффузионного уравнения переноса нейтронов сов-
местно с уравнениями выгорания топлива и кинетики предшественников запаздывающих нейтронов. Расче-
ты проводились в плоской одномерной модели двухзонного гомогенного реактора с металлическим U-Pu
топливом, Na-теплоносителем и конструкционным материалом Fe. Температурные эффекты и отвод тепла не
рассматривались. Показано, что при определенных условиях в реакторе можно создать волновой режим
ядерного горения, в котором реактор без управления в течение длительного времени может поддерживаться
в состоянии, близком к критическому.
ДОСЛІДЖЕННЯ ХВИЛЬОВОГО РЕЖИМУ ЯДЕРНОГО ГОРІННЯ,
ЩО САМООРГАНІЗУЄТЬСЯ У РЕАКТОРІ НА ШВИДКИХ НЕЙТРОНАХ
С.П. Фомін, Ю.П. Мельник, В.В. Пилипенко, , М.Ф. Шульга
В ефективному багатогруповому наближенні розвинуто підхід для опису просторово-часової еволюції
хвильового режиму ядерного горіння, що самоорганізується у критичному реакторі на швидких нейтронах.
Він заснований на розв’язанні нестаціонарного дифузійного рівняння переносу нейтронів разом з
рівняннями вигоряння палива і кінетики попередників запізнілих нейтронів. Розрахунки проводилися у
плоскій одновимірній моделі двохзонного гомогенного реактора з металевим U-Pu паливом, Na-теплоносієм
та конструктційним матеріалом Fe. Температурні ефекти і відвід тепла не розглядались. Доведено, що за
певних умов у реакторі можна створити хвильовий режим ядерного горіння, у якому реактор без керування
протягом тривалого часу може підтримуватися у стані, близькому до критичного.
113
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
An approach for description of the space-time evolution of self-organizing nuclear burning wave regime in a critical fast neutron reactor has been developed in the effective multigroup approximation. It is based on solving the non-stationary neutron diffusion equation together with the fuel burn-up equations and the equations of nuclear kinetics for delayed neutron precursor nuclei. The calculations have been carried out in the plane one-dimensional model for a two-zone homogeneous reactor with the metal U-Pu fuel, the Na coolant and constructional material Fe. The temperature effects and heat sink were not considered.
PACS: 28.41 T, 28.52 N
ACKNOWLEDGEMENT
REFERENCES
С.П. Фомин, Ю.П. Мельник, В.В. Пилипенко, Н.Ф. Шульга
С.П. Фомін, Ю.П. Мельник, В.В. Пилипенко, , М.Ф. Шульга
|
| id | nasplib_isofts_kiev_ua-123456789-81239 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T15:18:57Z |
| publishDate | 2005 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Fomin, S.P. Mel’nik, Yu.P. Pilipenko, V.V. Shul’ga, N.F. 2015-05-13T18:09:15Z 2015-05-13T18:09:15Z 2005 Study of self-organizing regime of nuclear burning wave in fast reactor / S.P. Fomin, Yu.P. Mel’nik, V.V. Pilipenko, N.F. Shul’ga // Вопросы атомной науки и техники. — 2005. — № 6. — С. 106-113. — Бібліогр.: 13 назв. — англ. 1562-6016 PACS: 28.41 T, 28.52 N https://nasplib.isofts.kiev.ua/handle/123456789/81239 An approach for description of the space-time evolution of self-organizing nuclear burning wave regime in a critical fast neutron reactor has been developed in the effective multigroup approximation. It is based on solving the non-stationary neutron diffusion equation together with the fuel burn-up equations and the equations of nuclear kinetics for delayed neutron precursor nuclei. The calculations have been carried out in the plane one-dimensional model for a two-zone homogeneous reactor with the metal U-Pu fuel, the Na coolant and constructional material Fe. The temperature effects and heat sink were not considered. В ефективному багатогруповому наближенні розвинуто підхід для опису просторово-часової еволюції хвильового режиму ядерного горіння, що самоорганізується у критичному реакторі на швидких нейтронах. Він заснований на розв’язанні нестаціонарного дифузійного рівняння переносу нейтронів разом з рівняннями вигоряння палива і кінетики попередників запізнілих нейтронів. Розрахунки проводилися у плоскій одновимірній моделі двохзонного гомогенного реактора з металевим U-Pu паливом, Na-теплоносієм та конструктційним матеріалом Fe. Температурні ефекти і відвід тепла не розглядались. Доведено, що за певних умов у реакторі можна створити хвильовий режим ядерного горіння, у якому реактор без керування протягом тривалого часу може підтримуватися у стані, близькому до критичного. В эффективном многогрупповом приближении развит подход для описания пространственно-временной эволюции самоорганизующегося волнового режима ядерного горения в критическом реакторе на быстрых нейтронах. Он основан на решении нестационарного диффузионного уравнения переноса нейтронов сов- местно с уравнениями выгорания топлива и кинетики предшественников запаздывающих нейтронов. Расче- ты проводились в плоской одномерной модели двухзонного гомогенного реактора с металлическим U-Pu топливом, Na-теплоносителем и конструкционным материалом Fe. Температурные эффекты и отвод тепла не рассматривались. Показано, что при определенных условиях в реакторе можно создать волновой режим ядерного горения, в котором реактор без управления в течение длительного времени может поддерживаться в состоянии, близком к критическому. We express our gratitude to Drs. V.A. Apse, E.F. Kryuchkov and their colleagues from MEPhI (Moscow, Russia) for providing us with the package of codes ”TIME-26”, which was very useful at solving the multigroup criticality problem. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Применение ядерных методов Study of self-organizing regime of nuclear burning wave in fast reactor Дослідження хвильового режиму ядерного горіння, що самоорганізується у реакторі на швидких нейтронах Исследование самоорганизующегося волнового режима ядерного горения в реакторе на быстрых нейтронах Article published earlier |
| spellingShingle | Study of self-organizing regime of nuclear burning wave in fast reactor Fomin, S.P. Mel’nik, Yu.P. Pilipenko, V.V. Shul’ga, N.F. Применение ядерных методов |
| title | Study of self-organizing regime of nuclear burning wave in fast reactor |
| title_alt | Дослідження хвильового режиму ядерного горіння, що самоорганізується у реакторі на швидких нейтронах Исследование самоорганизующегося волнового режима ядерного горения в реакторе на быстрых нейтронах |
| title_full | Study of self-organizing regime of nuclear burning wave in fast reactor |
| title_fullStr | Study of self-organizing regime of nuclear burning wave in fast reactor |
| title_full_unstemmed | Study of self-organizing regime of nuclear burning wave in fast reactor |
| title_short | Study of self-organizing regime of nuclear burning wave in fast reactor |
| title_sort | study of self-organizing regime of nuclear burning wave in fast reactor |
| topic | Применение ядерных методов |
| topic_facet | Применение ядерных методов |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/81239 |
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