Self-sustained regime of nuclear burning wave in U–Pu fast reactor with Pb–Bi coolant
The regime of the nuclear burning wave in a fast reactor is described using the non-stationary diffusion equation for neutron transport and equations of the fuel component burn-up and of nuclear kinetics for precursor nuclei of delayed neutrons. A critical two-zone fast reactor of cylindrical form w...
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2007
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| Cite this: | Self-sustained regime of nuclear burning wave in U–Pu fast reactor with Pb–Bi coolant / S.P. Fomin, Yu.P. Mel’nik, V.V. Pilipenko, and N.F. Shul’ga // Вопросы атомной науки и техники. — 2007. — № 3. — С. 156-163. — Бібліогр.: 18 назв. — англ. |
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Fomin, S.P. Mel’nik, Yu.P. Pilipenko, V.V. Shul’ga, N.F. 2017-01-07T11:35:24Z 2017-01-07T11:35:24Z 2007 Self-sustained regime of nuclear burning wave in U–Pu fast reactor with Pb–Bi coolant / S.P. Fomin, Yu.P. Mel’nik, V.V. Pilipenko, and N.F. Shul’ga // Вопросы атомной науки и техники. — 2007. — № 3. — С. 156-163. — Бібліогр.: 18 назв. — англ. 1562-6016 PACS: 28.41 T, 28.52 N https://nasplib.isofts.kiev.ua/handle/123456789/110950 The regime of the nuclear burning wave in a fast reactor is described using the non-stationary diffusion equation for neutron transport and equations of the fuel component burn-up and of nuclear kinetics for precursor nuclei of delayed neutrons. A critical two-zone fast reactor of cylindrical form with metal fuel of the U–Pu cycle with Pb–Bi coolant is considered. The neutron leakage in transverse direction was taken into account using the concept of radial buckling. The calculation results of the space-time evolution of neutron flux in this system using the effective multi-group approximation are presented. На основі рішення нестаціонарного дифузійного рівняння переносу нейтронів сумісно з рівняннями вигоряння палива та кінетики ядер – попередників запізнілих нейтронів описано режим хвилі ядерного горіння у реакторі на швидких нейтронах. Розглянуто критичний двозонний швидкий реактор циліндричної форми з металевим U-Pu-паливом та Pb–Bi-теплоносієм. Для урахування поперечної втрати нейтронів використовувалася концепція радіального баклингу. Представлено результати розрахунків просторово-часової еволюції нейтронного потоку в цій системі, що проводилися в ефективному багатогруповому наближенні. На основе решения нестационарного диффузионного уравнения переноса нейтронов совместно с уравнениями выгорания топлива и кинетики ядер – предшественников запаздывающих нейтронов описан режим волны ядерного горения в реакторе на быстрых нейтронах. Рассмотрен критический двухзонный быстрый реактор цилиндрической формы с металлическим U-Pu-топливом и Pb–Bi-теплоносителем. Для учета поперечной утечки нейтронов использовалась концепция радиального баклинга. Представлены результаты расчетов пространственно-временной эволюции нейтронного потока в этой системе, выполненные в эффективном многогрупповом приближении. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники High-energy electrodynamics in matter Self-sustained regime of nuclear burning wave in U–Pu fast reactor with Pb–Bi coolant Режим хвилі ядерного горіння, що самопідтримується у U–Pu реакторі на швидких нейтронах з Pb–Bi теплоносієм Самоподдерживающийся режим волны ядерного горения в U–Pu реакторе на быстрых нейтронах c Pb–Bi теплоносителем Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Self-sustained regime of nuclear burning wave in U–Pu fast reactor with Pb–Bi coolant |
| spellingShingle |
Self-sustained regime of nuclear burning wave in U–Pu fast reactor with Pb–Bi coolant Fomin, S.P. Mel’nik, Yu.P. Pilipenko, V.V. Shul’ga, N.F. High-energy electrodynamics in matter |
| title_short |
Self-sustained regime of nuclear burning wave in U–Pu fast reactor with Pb–Bi coolant |
| title_full |
Self-sustained regime of nuclear burning wave in U–Pu fast reactor with Pb–Bi coolant |
| title_fullStr |
Self-sustained regime of nuclear burning wave in U–Pu fast reactor with Pb–Bi coolant |
| title_full_unstemmed |
Self-sustained regime of nuclear burning wave in U–Pu fast reactor with Pb–Bi coolant |
| title_sort |
self-sustained regime of nuclear burning wave in u–pu fast reactor with pb–bi coolant |
| 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. |
| topic |
High-energy electrodynamics in matter |
| topic_facet |
High-energy electrodynamics in matter |
| publishDate |
2007 |
| language |
English |
| container_title |
Вопросы атомной науки и техники |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| format |
Article |
| title_alt |
Режим хвилі ядерного горіння, що самопідтримується у U–Pu реакторі на швидких нейтронах з Pb–Bi теплоносієм Самоподдерживающийся режим волны ядерного горения в U–Pu реакторе на быстрых нейтронах c Pb–Bi теплоносителем |
| description |
The regime of the nuclear burning wave in a fast reactor is described using the non-stationary diffusion equation for neutron transport and equations of the fuel component burn-up and of nuclear kinetics for precursor nuclei of delayed neutrons. A critical two-zone fast reactor of cylindrical form with metal fuel of the U–Pu cycle with Pb–Bi coolant is considered. The neutron leakage in transverse direction was taken into account using the concept of radial buckling. The calculation results of the space-time evolution of neutron flux in this system using the effective multi-group approximation are presented.
На основі рішення нестаціонарного дифузійного рівняння переносу нейтронів сумісно з рівняннями вигоряння палива та кінетики ядер – попередників запізнілих нейтронів описано режим хвилі ядерного горіння у реакторі на швидких нейтронах. Розглянуто критичний двозонний швидкий реактор циліндричної форми з металевим U-Pu-паливом та Pb–Bi-теплоносієм. Для урахування поперечної втрати нейтронів використовувалася концепція радіального баклингу. Представлено результати розрахунків просторово-часової еволюції нейтронного потоку в цій системі, що проводилися в ефективному багатогруповому наближенні.
На основе решения нестационарного диффузионного уравнения переноса нейтронов совместно с уравнениями выгорания топлива и кинетики ядер – предшественников запаздывающих нейтронов описан режим волны ядерного горения в реакторе на быстрых нейтронах. Рассмотрен критический двухзонный быстрый реактор цилиндрической формы с металлическим U-Pu-топливом и Pb–Bi-теплоносителем. Для учета поперечной утечки нейтронов использовалась концепция радиального баклинга. Представлены результаты расчетов пространственно-временной эволюции нейтронного потока в этой системе, выполненные в эффективном многогрупповом приближении.
|
| issn |
1562-6016 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/110950 |
| citation_txt |
Self-sustained regime of nuclear burning wave in U–Pu fast reactor with Pb–Bi coolant / S.P. Fomin, Yu.P. Mel’nik, V.V. Pilipenko, and N.F. Shul’ga // Вопросы атомной науки и техники. — 2007. — № 3. — С. 156-163. — Бібліогр.: 18 назв. — англ. |
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| fulltext |
SELF-SUSTAINED REGIME OF NUCLEAR BURNING WAVE
IN U–Pu FAST REACTOR WITH Pb–Bi COOLANT
S.P. Fomin, Yu.P. Mel’nik, V.V. Pilipenko, and N.F. Shul’ga
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine;
e-mail: sfomin@kipt.kharkov.ua
The regime of the nuclear burning wave in a fast reactor is described using the non-stationary diffusion equation
for neutron transport and equations of the fuel component burn-up and of nuclear kinetics for precursor nuclei of
delayed neutrons. A critical two-zone fast reactor of cylindrical form with metal fuel of the U–Pu cycle with Pb–Bi
coolant is considered. The neutron leakage in transverse direction was taken into account using the concept of radial
buckling. The calculation results of the space-time evolution of neutron flux in this system using the effective multi-
group approximation are presented.
PACS: 28.41 T, 28.52 N
1. INTRODUCTION
The new conception of the safe fast reactor (FR)
with the inner safety [1–13] has got further development
in the present work. This FR is a new type of long-life
operation critical reactor which can solve the problems
of radioactive waste transmutation and nuclear prolif-
eration constraints. The main advantage of this FR is
that it does not require a reactivity control and therefore
the initial fuel composition of the reactor will evolve
according to nuclear processes without external inter-
vention during the full FR lifetime without any refuel-
ing or fuel shuffling. In this regime the FR is automati-
cally sustained in a state close to the critical one due to
the mechanism of reactivity feedback (see [10]). The
lifetime of the reactor could be very long without hu-
man access inside the FR core during its operation time
and so can be placed underground.
The operation of FR 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 a schematic model of FR
with the U–Pu fuel cycle. Later Teller et al. [3, 4] pro-
posed a concept of long-life FR with fuel of the Th–U
cycle. The corresponding complicated non-stationary
problem was solved with help of the Monte-Carlo simu-
lation. Further the concept of NBW (named CANDLE)
was developed in the framework of multigroup diffu-
sion approximation [5, 6] using the self-similar solution
approach. In [7, 8] authors showed by means of mathe-
matical modeling the possibility of formation of non-
linear self-organizing regime in the FR (called as self-
adjusting neutron-nuclide regime) that however did not
go over into the NBW regime in the non-stationary
scheme considered.
The possibility of creating the NBW regime in a linear
FR was also confirmed in [9] by calculations carried out
for the simple model [1] both in the self-similar solution
approach and in the corresponding non-stationary one.
In our previous works [10–12], the space-time evo-
lution of a self-organizing regime in the form of NBW
was studied for a plane one-dimensional model of criti-
cal FR on the basis of one-group and effective multi-
group diffusion approaches. The mathematical formula-
tion of the problem included the solution of the non-
stationary diffusion equation for neutron transport to-
gether with the burn-up equations for fuel components
and the equations of nuclear kinetics for precursor nu-
clei of delayed neutrons. The questions of stability of
the NBW regime relative to distortions of the neutron
flux and of the mechanism of reactivity feedback were
investigated. The calculations performed in [11, 12]
showed that the allowance for the space-time alteration
of the neutron spectrum during burn-up process is very
important for the numerical simulation of the regime of
the propagating NBW. In [13] the approach [11, 12]
was further developed for simulating the initiation and
propagation of the NBW in a cylindrical FR with U–Pu
fuel and Na coolant taking into account the neutron
leakage in the transversal direction by using the concept
of radial buckling. This made it possible to study the
influence of finite transversal size of FR upon rise and
evolution of the NBW regime and to find certain vari-
ants of parameters of the FR under consideration in
which the nuclear burning rate, the energy production
intensity and the neutron flux in NBW take values ac-
ceptable from the practical point of view.
Since the commonly used in fast neutrons reactors
Na coolant seems to lead to rather strong safety con-
straints on the construction and operation of the reac-
tors, in this work we present the results of calculations
in which the Pb-Bi eutectic has been considered as
coolant. These investigation is the next step of devel-
opment of the deterministic approach [13] for descrip-
tion of the space-time evolution of the burn-up process
in cylindrical FR with taking into account of the trans-
versal neutron leakage using the radial buckling con-
cept.
2. THE DETAILS OF CALCULATION
We study the space-time evolution of the nuclear
burning process in a critical FR by solving the set of
partial differential equations that includes the non-
stationary neutron diffusion equation, equations of fuel
burn-up and equations of nuclear kinetics for precursors
of delayed neutrons. The calculations are carried out in
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2007, N3 (1), p. 156-163. 156
the effective multi-group approach [11, 12] in which the
effective one-group cross sections are determined with
the help of averaging the group cross-sections over the
local neutron spectra found from solving the corre-
sponding critical multi-group problem for the given
time moment. These effective cross section are used for
solving the non-stationary problem under consideration
in the one-group approximation.
The one-group non-stationary neutron diffusion
equation with taking into account delayed neutrons can
be written for the cylindrical model of FR in the form
1 1 (1 )( )a fD rD
v t z z r r r
β ν
∂Φ ∂ ∂Φ ∂ ∂Φ
− − Φ − − Σ
∂ ∂ ∂ ∂ ∂
+Σ f Φ
i
l ll i
Cλ= ∑ ∑ i . (1) + Φ
Here Φ is the scalar neutron flux, Σα=Σjσα
jNj is the mac-
roscopic cross section of the neutron reaction (the index
α corresponds to the reactions of neutron absorption (a)
and fission (f)), Nj is the concentration of j’th nuclide;
σα
j is the corresponding effective one-group micro-
scopic cross section; νfΣf=Σjνf
jσf
jNj, νf
j is the mean num-
ber 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-
dingly; D=1/(3Σtr) is the diffusion coefficient, Σtr is the
macroscopic transport cross-section, v is the one-group
neutron velocity.
The boundary conditions for the neutron flux Φ are
written in the form
0
2 2 ex
z
D j
z =
∂Φ Φ − = ∂
, 2 0
z L
D
z =
∂Φ Φ + = ∂
. (2)
In (2) jex is the external neutron flux falling onto the left
boundary of FR ( ) while the right boundary
( ) is free.
0z =
z L=
The conditions at the central axis and at the lateral
surface of FR are:
0
r
∂Φ
=
∂
at ; at , (3) 0r = 0Φ = extrr R δ= +
where R is the FR radius and 0.71extr trΣδ = is the ex-
trapolation length.
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 as-
sembly Φ0 is chosen as an initial condition for Φ at the
moment t = 0.
We consider the nuclear burning wave propagation
in the axial direction and reduce the problem to a one-
dimensional case using the concept of radial buckling
that takes an approximate account of the neutron leak-
age in the transversal direction. Then the equation (1)
takes the form
21 (1 )( )r a f f
V DB
v t z
β ν
∂Φ ∂
+ + Φ Φ− − Σ
∂ ∂
+Σ Φ
i
l ll i
Cλ= ∑ ∑ i , V D , (4)
z
∂Φ
= −
∂
where ( )2.405rB R δ= + extr is the radial buckling pa-
rameter.
The boundary conditions (2) can be rewritten as
( )
0
2 2 exz
V j
=
Φ + = , ( )2 2 exz L
V j
=
Φ − = . (5)
The burn-up equations describe the time evolution
of the fuel composition according to the chain of nu-
clear transformations. In the case of FR with the U–Pu
fuel cycle we consider the chain including only 10 nu-
clides, whose numeration is presented in Table 1:
( )
( )( 1) ( 1) , 1
l
al l l
cl l l
N N
t
N l
σ
σ − −
∂
= − Φ +Λ
∂
+ Λ = …
(6)
8,
9
6 6
∂
= Λ
∂
N N
t
, (7)
10
1,4,5,6,7=
∂
=
∂ ∑ fl l
l
N N
t
σ Φ . (8)
Here 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.
It should be noted that using the radial buckling con-
ception we neglect the effects of the fuel nuclide burn-
up non-uniformity in the radial direction.
Table 1. 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
In Table 1 FP denotes the fission product nuclide that
effectively represents any pair of fission fragments whose
properties are averaged for all possible fission events.
The initial values of nuclide concentrations are
0( , 0) ( )lN z t N z= = l . (9)
The burn-up of nuclei 239U, 239Np, 241Am, 243Am is
neglected (σa2=σa3=σa8=σa9=0) since it is small as. The
changes of the fission fragments due to the neutron ab-
sorption also were not considered.
An important characteristics of nuclear processes in
FR is equilibrium concentration of plutonium (see, for
example, [7]) which in a stationary state can be 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)
Since the decay time of the precursor nuclei emitting
delayed neutrons is much less than the characteristic
157
time of variation of the scalar neutron flux Φ in the
NBW regime (see, for example, [10]), we can use the
approximation of one equivalent group of delayed neu-
trons for the nuclear kinetic equations
( )∂
= − + Σ Φ
∂
l
l l l f f l
C C
t
λ β ν , (12)
0( , 0) ( )l lC z t C z= = , (13)
where / /= ∑ i i
l l li
λ β β λl .
If the reactivity is less than the effective fraction of
delayed neutrons β , the behaviour of FR is completely
controlled by delayed neutrons [14]. For this reason, in
the considered case of small deviations from the critical
state delayed neutrons play an important role in the ini-
tiation and stability of the nuclear burning process.
The complete formulation of the non-stationary
problem under consideration includes the set of 16 non-
linear partial differential equations and the correspond-
ing initial and boundary conditions for them. For nu-
merical solution of the diffusion equation (4) we use the
conservative finite-difference method (see, for example,
[15]) and the implicit Crank-Nicolson difference
scheme [16] with variable time step. This symmetric-in-
time scheme has the approximation of the second order
of accuracy in space and time steps and shows an un-
conditional stability at any relation between them.
The solution of the burn-up equations (4)-(6) and
equations of nuclear kinetics (10) has been simplified
assuming that the effective one-group cross sections and
the neutron flux Φ are constant during the time step τ.
The assumption about the constancy of cross-sections is
well fulfilled for the FR conditions because of the weak
sensitivity of the effective cross-sections to changes in the
fast neutron spectrum. The assumption for Φ can be eas-
ily satisfied by choosing sufficiently small time intervalsτ.
This fact allowed us to obtain an approximate analytical
solution of Eqs. (5)-(7) and (10) for the concentrations of
the corresponding nuclides (see for details [10]).
Because of non-linearity of the used implicit finite-
difference scheme, the neutron flux at a new time layer
has been found by means of successive iterations [10].
To calculate the effective one-group microscopic
cross sections we use the group neutron fluxes Φg (g is
the number of neutron energy group) found from solv-
ing the stationary multigroup problem. The calculations
were performed in the 26-group approximation using
the group neutron constants from [17] and [18]. We
have used the method of averaging the group cross-
sections that takes into account the requirement of con-
servation of rates of corresponding reactions during this
procedure (see, for example, [14]). The scheme of pass-
ing from the group microscopic cross sections to the
effective one-group cross sections is described in
[11,12].
3. RESULTS AND DISCUSSION
The approach described above has been employed to
carry out a series of calculations that simulate the initia-
tion and propagation of the NBW in the critical FR un-
der consideration. According to the boundary condi-
tions (5) the process of nuclear burning in the FR is
initiated by the external neutron flux jex coming onto the
left end of the system.
Fig.1 presents a schematic layout of the two-zone
homogeneous cylindrical FR under consideration. In the
reactor a metal U–Pu fuel of porosity p=0.8 is used.
Near the left end of the reactor there is an ignition zone
in which the uranium fuel is enriched with plutonium
with concentration 10%. This value has been chosen so
as to be less than the equilibrium plutonium concentra-
tion Neq (10). The plutonium in the fuel has the standard
isotope composition: 239Pu : 240Pu : 241Pu : 242Pu =
0.70 : 0.22 : 0.05 : 0.03. In the breeding zone that is
adjacent to the ignition one and occupies the main part
of the FR volume the fuel consists of the fertile 238U (we
neglect the presence of 235U in the present FR model).
Both zones also include the constructional material Fe
and the Pb-Bi eutectic coolant.
r
R
jex
0 Lig L z
Ignition zone Breeding zone
Fig. 1. The initial critical assembly of FR
As the neutron flux Φ0 we choose the eigenfunction
for the neutron flux in the initial critical FR normalized
to a small magnitude corresponding to the average en-
ergy production density in the ignition zone equal to 10-
8 kW cm–3. The intensity of the external neutron flux
that initiates the burning process has been chosen to be
jex=6⋅1011 cm-2 s-1.. This external neutron flux does not
irradiate the ignition zone during all the time of the re-
actor campaign but it was turned off at an early stage of
the FR operation. The time moment toff of the turning
off was chosen empirically so that the neutron flux in
the system had reached a level high enough for the fur-
ther development of the nuclear burning process. In the
calculations that are presented below the value toff =
10 days has been used.
During the calculations the volume fractions of the
FR components in each zone had the following values
corresponding to composition of actual reactors: for the
nuclear fuel Ffu = 44 %, the constructional material
FFe = 20 % and the Pb–Bi coolant Fcool = 36 %.
In the calculations 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 solve the multi-group problem for fluxes
in the critical FR assembly whose composition changes
according to the burn-up equations.
The group neutron fluxes Φg(z) found for the corre-
sponding critical assemblies are used to obtain the ef-
fective one group cross sections. Thus, during the whole
158
lifetime of FR the cross sections are averaged in the
correspondence to the neutron spectrum alteration that
occurs. 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 changing
with time.
We have performed a set of variants of calculations
of the nuclear burning evolution in reactors with differ-
ent geometrical sizes. In these calculations we have
studied the possibility of initiation and character of
propagation of the NBW in the FR under consideration
depending on the reactor radius and length. As an ex-
ample, the results of calculations for the reactor length
L = 500 cm, its radius R = 100.5 cm and the ignition
zone width Lig = 72.12 cm are shown in Figs. 2 and 3.
0 100 200 300 400 500
50
100
150
200
250 a)
Φ
z
0 100 200 300 400 500
10
20
30
40
50
60 b)
Φg
z
0 2 4 6 8 10 12
0.05
0.10
0.15
0.20
14
c)
u
χg
Fig. 2. Calculations of neutron field inside initial
critical FR: a) the neutron flux Φ (×1012 neutron/(cm2s))
normalized to the average energy production
0.02 kW/cm3 both summed over 26 groups (solid curve)
and obtained in the one-group approach (dashed curve)
vs z (in cm); b) the group neutron fluxes Φg: g = 5
(0.8 <En< 1.4, short dashes), g = 7 (0.2 <En< 0.4, solid
curve), g = 8 (0.1 <En< 0.2, long dashes), g = 10
(0.0215 <En< 0.0465, dots), the bounds of energy
groups (MeV) are presented in parentheses; c) the neu-
tron zone spectrum χg (ignition zone, solid curve) and
(breeding zone, dashes) vs lethargy u = ln(10.5/En), En
is the neutron energy in MeV
Fig. 2 presents the results of calculations of the sca-
lar neutron fluxes in a initial critical assembly of FR
that were carried out in the 26-group approximation.
The parameter values presented above correspond to
keff = 1 for the given variant of FR.
0 100 200 300 400 500
0.05
0.10
0.15
0.20
0.25
a)
t5
t4
t
3
t2,*0.5
t1
z
Φ
0 100 200 300 400 500
0.5
1.0
1.5
2.0
2.5
b)
t
5
t
4t
3
t
2
,*0.5
t
1
z
P
0 100 200 300 400 500
0.5
1.0
1.5
c)
t
5t
4t
3
t
2
t
1
z
N
Pu
0 100 200 300 400 500
5
10
15
20
25
30
35
d)
t
5t
4t
3
t
2
z
B
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 z (cm): a) the
scalar neutron flux Φ(z) (×1017 cm-2 s-1); b) the power
density P(z) (kW cm-3); c) the 239Pu concentration
NPu(z) (×1021 cm-3); d) the fuel burn-up depth B(z) (%)
for t1=5, t2=100, t3=2500, t4=4000 and t5=5500 days
159
In Fig. 2,a we compare the neutron flux ΦS(z) sum-
med over 26 groups to the corresponding flux Φ(z) cal-
culated in the one-group approximation using the ef-
fective cross sections σα
l(z) averaged with 26-group
fluxes Φg(z) (the space distributions for some of them
are shown in Fig. 2,b ). The results of calculations in the
multigroup approach and in the considered one-group
calculations are close to each other. This fact can war-
rant the usage of one-group calculations with taking
account of the time alteration of the effective one-group
cross sections (the effective multigroup approximation),
as in the case of the plane one-dimensional model
[11, 12], for the simulation of NBW evolution in the
cylindrical FR.
The neutron energy spectra in the ignition zone and
breeding one represented in Fig. 2,c, are rather hard, the
thermal neutrons are practically missing there.
Fig. 3 presents the results of calculations of the main
characteristics that describe the burning process in FR.
It can be seen that at initial stage of the process the neu-
tron flux Φ and the density of energy production P es-
sentially increase (cf. the corresponding curves for the
time moments t1 = 1 day and t2 = 100 days). The neu-
tron flux has a rather sharp maximum about the 170-th
day and then quickly decreases during next 250 days.
This evolution is clearly seen in Fig. 4 (see the
curves for the reactor with R=100.5 cm) which presents
the integral neutron flux ΦI being the scalar neutron
flux at the reactor axis integrated over its length.
0 1000 2000 3000 4000 5000
10
20
30
40
a)
Φ
I
t
0 200 400 600 800 1000
50
100
150
200
250
300
b)
Φ
I
t
Fig. 4. The integral (over the reactor axis) neutron
flux ΦI (×1017 cm-1 s-1) versus time (in days): a) for the full
lifetime of the reactor with radius R = 100.5 cm; b) com-
parison for the reactors with R=100.5 cm (solid curve);,
R=110 cm (dashed curve) and R=130 cm (dotted curve)
Further, the integral neutron flux value does not
practically change during the reactor campaign that lasts
about 5500 days for the considered reactor (L=500 cm,
R=100.5 cm). At the same time the maxima of the spa-
tial distributions for Φ and P move with a constant ve-
locity V along the z-axis. We define the NBW velocity
as the velocity of shifting the scalar neutron flux maxi-
mum whose dependence on time is shown in Fig. 5.
0 1000 2000 3000 4000 5000
0.05
0.10
0.15
0.20
0.25 a)
V
t
0 200 400 600 800 1000
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6 b)
V
t
Fig. 5. The NBW velocity V (cm d-1) versus time (in
days): a) for the full lifetime of the reactor with radius
R = 100.5 cm; b) comparison for the reactors with
R = 100.5 cm (solid curve); R = 110 cm (dashed curve)
and R = 130 cm (dotted curve)
In this figure one can observe a jump of V at the ini-
tial stage which corresponds to the sharp maximum of
the neutron flux ΦI and is caused by the quick forma-
tion of the NBW front. After this velocity jump a stable
propagation of the NBW occurs with an almost constant
velocity V~ 0.06 cm/d till the reactor extinction. This
stage of NBW corresponds to the wave profiles for the
moments t3 = 2500 days and t4 = 4000 days in Fig. 3.
When the wave approaches the right reactor edge, the
NBW velocity decreases and the stage of slow extinc-
tion of the burning process begins. The stage of the ex-
tinction lasts several hundreds days. A characteristic
NBW profile for this stage is presented for the moment
t5 = 5500 days. This profile is featured by lower values
of the neutron flux and energy production.
At the initial stage of the evolution of burning proc-
ess (t~200 days), there are slow changes of the initial
axial distribution of 239Pu (see Fig. 3,c). It is gradually
160
accumulated mainly near the right boundary of the igni-
tion zone due to the transformation of the 238U nuclei
through the radiative neutron capture and two succes-
sive β-decays. This leads to the formation of the front of
the future NBW.
The quickest changes that create the NBW front oc-
cur during the period between 150 and 300 days. Fur-
ther, when the NBW starts to move, we observe a char-
acteristic bump in the 239Pu distribution that is situated
just in the zone of the most intense burning. This bump
first arises near the ignition zone boundary and then
gradually shifts along the reactor axis leaving a certain
residual level of plutonium concentration behind it.
When the reactor campaign is over, plutonium and
the nuclear fission products are distributed with a prac-
tically uniform concentration over the whole length of
the reactor, except for the regions close to its bounda-
ries. By the extinction moment the fuel burn-up depth
reaches a high level over 30 % in the whole volume of
the reactor, except for the parts near the reactor bounda-
ries (see Fig. 3,d). In the breeding zone an intense ac-
cumulation of plutonium occurs and the isotope that
burns up is practically 238U. This is clearly demonstrated
by Fig. 6 that presents the axial distributions of the ini-
tial (t = 0) and final (t = 5500 days) fuel composition in
FR.
0 100 200 300 400 500
0.0
0.2
0.4
0.6
0.8
1.0
a)
Pu 238U
z
N
fuel
0 100 200 300 400 500
0.2
0.4
0.6
0.8
1.0
b)
Fission products
Pu 238U
z
N
fuel
Fig. 6. The initial (a, t = 0) and final (b, t = 5500
days) distributions of the relative fuel component con-
centrations in FR via the reactor length z (cm)
The course of the evolution of nuclear burning proc-
ess in the FR under consideration strongly depends on
the reactor transverse size (the cylinder radius R).
Increasing R leads to the increase of the neutron flux
value in the NBW and as a result to a higher NBW ve-
locity. These effects are demonstrated in Figs. 4,b and
5,b, where are presented time dependencies of ΦI and V
for the reactors with R = 110 cm (ignition zone width of
70.1 cm) and R = 130 cm (ignition zone width of
67.2 cm) together with results for variant with
R = 100.5 cm (the reactor length in all the cases is
500 cm).
These results show that the reactor with R = 130 cm
is characterized by a very high neutron flux level in the
NBW regime, high NBW velocity of 1.4 cm/day and a
very short duration of the reactor campaign.
100 105 110 115 120 125 130
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
V
R
Fig. 7. Dependence of the NBW velocity V (cm d-1)
on the reactor radius R (cm) at the reactor length
500 cm
Fig. 7 presents the dependence of NBW velocity V
on the reactor radius R. From this figure one can see
that with R decreasing somewhat below 100.5 cm the
velocity V goes to zero. This means that at such small
reactor radius values the NBW cannot propagate. Our
calculation for a reactor with R = 100 cm confirms
this conclusion. In this case the neutron flux gradu-
ally decreases after reaching its maximum. At that,
there is no spatial shift of the burning process along
the reactor axis and NBW does not arise. Note that
we have not considered the influence of the radial
reflector which would lead to decrease of the trans-
versal active zone size necessary for the NBW exis-
tence.
Of special interest is the question about the stabil-
ity of the self-sustaining nuclear burning regime at its
initial stage. In Fig. 8 we present the integral neutron
flux ΦI variation during the first 45 days of FR opera-
tion. The curve shown in Fig. 8 demonstrates the ef-
fect of turning the external neutron flux jex off at
days. It is seen that at the moment t the
magnitude of the integral flux in the reactor first
sharply decreases. This decrease is followed by an
increase of the flux magnitude and then by the flux
oscillations being gradually damped. Therefore, the
arising perturbation is damped by the reactor itself
during approximately 30 days. Turning the external
flux off at later time moments t causes much less
perturbation of the neutron flux behavior. Thus, the
10offt = off
off
161
self-organized burning regime that arose before the
moment of turning jex off is fairly stable.
0 10 20 30 40
0.5
1.0
1.5
Φ
I
t
Fig. 8. The integral (over the reactor axis) neutron
flux ΦI (×1017 cm-1 s-1) at the initial stage of nuclear
burning process versus time (days) for the reactor with
radius R = 100.5 cm. The external neutron flux jex is
turned off at the time moment toff = 10 days. The condi-
tions correspond to Fig. 3
It is also interesting to consider the time evolution of
such an important quantity as the equilibrium 239Pu con-
centration Neq. Fig. 9 presents the its space distributions
calculated according to formulae (10) and (11) at the
beginning of the nuclear burning process and at the
stage of NBW propagation. It can be seen that these
quantities have a considerable time and space variation.
At the initial stage of the nuclear burning the values Neq
calculated by formulae (10) and (11) significantly differ
from each other due to the essential non-stationarity of
the process. When the process passes to the stationary
NBW regime, the results given by formulae (10) and
(11) practically coincide.
0 100 200 300 400 500
1
2
3
Neq
z
1
2
3
4
Fig. 9. The spatial distribution of equilibrium 239Pu
concentration Neq (×1021 cm-3), calculated by formulae
(10) (dashed curves) and (11) (solid curves), for t = 5
days (1 and 2) and t = 4000 days (3 and 4)
The results of calculations carried out in this work
for the cylindrical model of FR with taking account of
the transversal neutron leakage using the radial buckling
concept show the essential decrease of the NBW veloc-
ity as compared to the calculation in the one-
dimensional plane model [11, 12]. These results con-
firm the conclusion that FR in the NBW regime is
automatically sustained during a long time (many years)
in a state close to the critical one despite the creation of
large amount of fission products.
4. CONCLUSIONS
Initiation and evolution of a self-organizing non-
linear regime in the form of a nuclear burning wave in a
critical FR with metal fuel of U–Pu cycle and Pb–Bi
eutectic alloy coolant has been studied on the basis of
solving of the non-stationary neutron diffusion equation
together with the set of fuel component burn-up equa-
tions and nuclear kinetics equations for the precursors
of delayed neutrons. The calculations have been per-
formed for a cylindrical reactor consisting of two axial
homogeneous zones: the ignition zone enriched with
plutonium and the breeding one. The process of nuclear
burning is initiated by an external neutron flux coming
into the ignition zone and then propagates along the
breeding zone in the axial direction. The problem was
reduced to the one-dimensional case by using the radial
buckling concept that takes approximate account of the
neutron leakage in transversal direction. The calcula-
tions were carried out in the framework of effective
multigroup diffusion approach that allows for the varia-
tion of effective one-group cross sections according to
alteration of the group neutron spectrum at each spatial
point of FR with time.
The space-time evolution of the nuclear burning
process has been studied for different values of radius
and length of the FR under consideraion. The calcula-
tion results have demonstrated the possibility of arising
of a self-organizing regime of a running NBW in the
breeding zone along the reactor axis after turning the
external neutron flux off at an early stage. The reactor is
automatically sustained in the state close to the critical
one during a long time. In a particular case of the reac-
tor with the 5 m length and 1.05 m radius, the simulated
reactor campaign lasts about 15 years. In this case at the
stationary NBW stage the neutron flux reaches about
2.5·1016 n/cm2 s, the NBW velocity is approximately
22 cm per year, the burn-up depth takes the value more
than 30 %. The increasing of the reactor radius leads to
an essential increase of the neutron flux and the NBW
velocity. Decreasing the reactor radius can make forma-
tion of the NBW regime impossible. For example, at the
reactor radius of 1.0 m the creation of the running burn-
ing wave was not observed in our calculations.
The results obtained in this work for FR with the
heavy Pb-Bi coolant are, in many respects, analogous to
those obtained when using the Na coolant [13], how-
ever, the same NBW velocity values are reached at sig-
nificantly lower transversal sizes of FR Besides, when
Pb–Bi is used, the initial ignition zone has a smaller
width than in the case of Na coolant. These facts are
caused by the better neutron economy and greater con-
version ratio from the fertile 238U into the fissile isotope
239Pu when using the heavy coolant.
The present results show that the distortion of the
neutron flux caused by turning the external neutron
source off at a very early stage of the process (e.g. 10
days), when the NBW is not formed yet, is damped
quickly enough by the reactor itself. This notable stabil-
ity of the nuclear burning regime in the FR under con-
sideration is due to an intrinsic reactivity feedback gov-
erned by the nonlinearity of NBW regime. This feed-
162
back prevents the reactor from the runaway regime and
ensures the stable evolution of the self-organizing NBW
regime.
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САМОПОДДЕРЖИВАЮЩИЙСЯ РЕЖИМ ВОЛНЫ ЯДЕРНОГО ГОРЕНИЯ В U–Pu-РЕАКТОРЕ
НА БЫСТРЫХ НЕЙТРОНАХ C Pb–Bi-ТЕПЛОНОСИТЕЛЕМ
С.П. Фомин, Ю.П. Мельник, В.В. Пилипенко, Н.Ф. Шульга
На основе решения нестационарного диффузионного уравнения переноса нейтронов совместно с урав-
нениями выгорания топлива и кинетики ядер – предшественников запаздывающих нейтронов описан режим
волны ядерного горения в реакторе на быстрых нейтронах. Рассмотрен критический двухзонный быстрый
реактор цилиндрической формы с металлическим U-Pu-топливом и Pb–Bi-теплоносителем. Для учета попе-
речной утечки нейтронов использовалась концепция радиального баклинга. Представлены результаты рас-
четов пространственно-временной эволюции нейтронного потока в этой системе, выполненные в эффектив-
ном многогрупповом приближении.
РЕЖИМ ХВИЛІ ЯДЕРНОГО ГОРІННЯ, ЩО САМОПІДТРИМУЄТЬСЯ У U–Pu-РЕАКТОРІ
НА ШВИДКИХ НЕЙТРОНАХ З Pb–Bi-ТЕПЛОНОСІЄМ
С.П. Фомін, Ю.П. Мельник, В.В. Пилипенко, М.Ф. Шульга
На основі рішення нестаціонарного дифузійного рівняння переносу нейтронів сумісно з рівняннями ви-
горяння палива та кінетики ядер – попередників запізнілих нейтронів описано режим хвилі ядерного горіння
у реакторі на швидких нейтронах. Розглянуто критичний двозонний швидкий реактор циліндричної форми з
металевим U-Pu-паливом та Pb–Bi-теплоносієм. Для урахування поперечної втрати нейтронів використову-
валася концепція радіального баклингу. Представлено результати розрахунків просторово-часової еволюції
нейтронного потоку в цій системі, що проводилися в ефективному багатогруповому наближенні.
163
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