On the theory of the initial stage of slow nuclear burning
Scientific principles of linear nuclear power systems of slow burning are elaborated. Two concepts of a slow burning reactor, one using U-Pu cycle and the other Th-U cycle are proposed. In the first concept the reactor is approximated by a homogeneous media and operates with fast neutrons. In the se...
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| Опубліковано в: : | Вопросы атомной науки и техники |
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| Дата: | 2001 |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2001
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| Цитувати: | On the theory of the initial stage of slow nuclear burning / N. Khizhnyak // Вопросы атомной науки и техники. — 2001. — № 6. — С. 279-282. — Бібліогр.: 2 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859818409139634176 |
|---|---|
| author | Khizhnyak, N. |
| author_facet | Khizhnyak, N. |
| citation_txt | On the theory of the initial stage of slow nuclear burning / N. Khizhnyak // Вопросы атомной науки и техники. — 2001. — № 6. — С. 279-282. — Бібліогр.: 2 назв. — англ. |
| collection | DSpace DC |
| container_title | Вопросы атомной науки и техники |
| description | Scientific principles of linear nuclear power systems of slow burning are elaborated. Two concepts of a slow burning reactor, one using U-Pu cycle and the other Th-U cycle are proposed. In the first concept the reactor is approximated by a homogeneous media and operates with fast neutrons. In the second concept the reactor is a heterogeneous assembly using both fast and thermal neutrons. The fast neutron flux at the initial stage after the reactor was ignited was determined for both concepts.
|
| first_indexed | 2025-12-07T15:23:25Z |
| format | Article |
| fulltext |
ON THE THEORY
OF THE INITIAL STAGE OF SLOW NUCLEAR BURNING
N.A. Khizhnyak
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
Scientific principles of linear nuclear power systems of slow burning are elaborated. Two concepts of a slow
burning reactor, one using U-Pu cycle and the other Th-U cycle are proposed. In the first concept the reactor is
approximated by a homogeneous media and operates with fast neutrons. In the second concept the reactor is a
heterogeneous assembly using both fast and thermal neutrons. The fast neutron flux at the initial stage after the
reactor was ignited was determined for both concepts.
PACS: 28.41.Ak
In the present paper• the scientific fundamentals of
linear nuclear power systems of slow burning are
developed.
The nuclear disarmament has resulted in
accumulation of significant stocks of fissile materials
235U and 239Pu, which are expedient to burn in nuclear
power plants. On the other hand, the long-term
production of military fissile materials has resulted in
accumulation (on a scale of hundreds thousand tons) of
the fertile uranium (uranium-238 with density of the
uranium-235 isotope about 0.2... 0.3 %, instead of 0.7 %
in natural uranium). Feoktistov [1] has proposed to use
these stocks in power nuclear reactors of a new type, in
which the initial critical mass of 239Pu is enclosed by
fertile uranium. An active zone neutron exposure of this
uranium will convert uranium-238 to a plutonium-239
and the active zone sequentially displaces itself to a new
position, leaving behind itself nuclear "ashes" consisting
of fission fragments and a not burnt out fissile material.
Analogous linear reactor (TIW reactor — E. Teller,
M. Ishikawa, and L. Wood) was proposed and
calculated in USA [2]. Such nuclear burning process can
be realized on the basis of both 232Th and 233U fuel
cycles. In this paper an attempt of the systematic
analysis of reactors of a Feoktistov and TIW types is
undertaken.
Let us consider a reactor in a form of cylinder of
radius R and length L . Fig. 1 shows the burning
process in a such prolate reactor. Let us identify the
principal processes and the corresponding zones within
a reactor.
Fertile
zone
Breeding
zone
Burning
zone
Extinction
zone
Nuclear
ashes
zone
Fig. 1
Prepared for publication by D.P. Belozorov and L.N. Davydov
Zone of burning where the fissile 239Pu burns,
producing energy.
Breeding zone, where 238U catches a neutron,
diffusing from the burning zone, and transforms finally
into 239Pu.
Fertile material zone is a zone containing fertile
232Th or 238U which can be transformed under neutron
irradiation into fissile nuclear fuel. In the case of Th this
zone should be heterogeneous interlacing fertile 232Th
isotope with a moderator, because the capture cross
section is higher for thermal neutron than for fast ones.
Heterogeneity is, thus, another parameter which can be
used to control the rate of burning.
Zone of extinction of burning is not precisely
defined. Here, due to the burn-out of the fissile fuel, to
the poisoning of the zone with fission products and to
other factors, the condition of a self-sustaining nuclear
reaction is not fulfilled any more, but the heat release is
still high. This zone plays an important role in
determination of the criticality conditions of the active
or burning zone.
Nuclear ashes zone, where products of the
radioactive decay of the nuclear fuel are concentrated
together with the unburned material.
From the physical point of view the formulated
problem has much in common with a known problem of
the slow burning due to chemical reactions, therefore,
we shall speak about the reactor with slow nuclear
burning.
The following problems can be formulated:
1. To determine velocity of propagation of the zone of
slow nuclear burning depending on parameters of
fertile material and other parameters of the problem
(radius of assembly, structure and composition of the
fertile material and moderator).
2. To calculate the multiplication factor keff in the
burning zone and to find conditions of stationarity
and stability of the burning zone, elemental
composition and its modification lengthwise of
assembly from a beginning up to its extremity.
3. To evaluate total heat release in a burning zone and
the role of an additional neutron illumination from
the zone of extinction.
4. To examine by the numerical methods an elemental
composition in the zone of extinction as well as the
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2001, № 6, p. 279-282. 23
dynamics of accumulation and disintegration of
different elements in this zone.
5. To determine the elemental composition of nuclear
ashes.
6. To compare the results to obtain with the results of
Feoktistov[1] and Teller[2].
To begin with let us write down the balance equation
for scalar neutron flux nv=Φ ( n is the neutron
density, v is their velocity)
( ) ΦΣν+ΦΣ−Φ=
∂
Φ∂
α fD
tv effgraddiv1 (1)
Here D is the neutron diffusivity, αΣ and fΣ are the
macroscopic cross sections of neutron absorption and
nuclear fuel fission, effν is the effective number of
surplus neutrons. The number of nucleus of fissile
element fN , bred by neutrons, is governed by the burn-
out equations
ΦΣ= α f
f
dt
dN
, (2)
where fαΣ is a macroscopic cross-section of neutron
capture resulting in breeding of fissile element.
Eq. (1) is the principal equation describing the
neutron dynamics. It is a nonlinear equation respective
to Φ , because of the last term in the RHS of equation.
At the first phase of the burning process, while the
density of the fissile nuclei in a breeding zone is small,
this term can be neglected and we can consider, at first,
a linear stage of the process development.
Two scenarios of the breeding process differing in
initial and boundary conditions of the problem are
conceivable.
A) Uranium − Plutonium assembly. Fertile material
is uranium-238. Nuclear fuel in the active zone
consists of plutonium-239. The entire system
operates at fast neutrons. There is no necessity of
heterogeneous structure of the breeding zone;
therefore, hereinafter we will examine this
scenario in the supposition, that only fast
neutrons are of importance (one-group
approximation).
B) Thorium − uranium assembly. Fertile material
is thorium-232. Nuclear fuel in the active zone
consists of uranium-233. Essentially, that both
nuclear reactor and breeding zone operate with
thermal neutrons, while the fission process
generates fast neutrons. In this scenario it is
necessary either to employ a complete kinetic
equation of neutron dynamics, or as it is used in
the theory of reactors, to resort to a multigroup
approximation (in the simplest case, two systems
of connected equations, for fast and thermal
neutrons).
Let us consider first the problem (A). The truncated
Eq. (1) is linear
( ) ΦΣ−Φ=
∂
Φ∂
αgraddiv D
tv
1 (3)
with the following initial and boundary conditions:
( ) ( )krJz 00δψ=Φ for 0=t ,
Φ=
∂
Φ∂−
v
v
z
D z for 0=z , (4)
0=Φ for ,extrλ+= Rr
where 0ψ is a constant characterizing the strength of
the neutron flux produced in the active zone, 0J is a
zero-order Bessel function (see below), R is the radius
of the cylinder reactor, extrλ is the length of the
extrapolation of the neutron flux into the moderator
(Fig. 2).
Ф =
( Ф/ )
nv
j = nv =
= _ D z
z z
∂ ∂
Ф| =0t=0
A
O
Active zone Breeding
zone
z
Fig. 2
In Fig. 2. AO is an impermeable membrane. At the
moment 0=t the membrane is removed and the
breeding process starts. Here jz is a flux of the fast
neutron in the reactor directed to the breeding zone,
vvj zz Φ= , where zv is z -component of the velocity.
Apparently, ( ) 1≤vvz .
The equation for neutron flux takes the form
ΦΣ−
∂
Φ∂+
∂
Φ∂
∂
∂=
∂
Φ∂
α2
211
zr
r
rr
D
tv
.
Separating the variables
( ) ( ) ( ) ( )tTzZrRtzr =Φ ,, ,
after some calculation we obtain the solution for 0>z
and 0>t in the form
( ) ( ) α
−Φ=Φ ∫ α−
α dzk
Dvk
vzk
„
krJetzr zvt
3
3
30
2
sincos,, ,
where ( )22
zkkD ++Σ=α α is the separation constant,
zk is the Fourier transform variable for ( )zZ and
( ) 4052.extr =λ+Rk is the first root of the zero-order
Bessel function 0J . The above solution satisfies all
boundary conditions at 0=z and extrλ+= Rr .
However, path of integration C and function αΦ has to
be determined to satisfy the initial condition at 0=t .
Below, for convenience, we shall use zk as an
integration variable,
( ) [ ] ( )(∫ −Φ=Φ ++Σ−
C
z
kDkvt
k zkkrJetzr za
z
cos,, 0
22
zz
z
z dkzk
kD
)
ν
ν− . (5)
24
Here the integration path C is all real values of zk
in the interval ( )∞,0 . However, one can see that the
complex value χ= ik z , where
Dv
vz=χ , also satisfies
the equation and conditions of the problem. Then
( ) ( ) [ ]
.dsincos
,,
−Φ+
+
=Φ
∫
∞
−
−+Σν− α
0
0
2
2
2
zz
z
z
z
vtDk
k
z
Dv
v
Dv
vtDkt
kzk
Dvk
vzke
AeekrJtzr
z
z
zz
(6)
The constant A and function zkΦ should be
determined from the initial condition.
Let us analyze the physical meaning of each term in
Eq. (6). The factor [ ]2+Dkvte αΣ− testifies, that the
neutrons coming from zone 0<z into zone 0>z are
consumed in the processes of absorption and diffusion
into the reflector. Therefore, the main concern is
connected with the dynamics of neutrons in variables
tz, . The term
z
Dv
v
Dv
v
t zz
Ae
−
2
describes the neutron “pumping” through the AO section
(Fig. 2). The neutron density increases exponentially
with a characteristic rise time
2
zv
DvТ =
and exponentially goes down with length. In order to
displace the equilibrium to the right (Fig. 2) the
condition
( ) ε+Σ= α
2+Dkv
Dv
vz
2
,
should be, evidently, satisfied. Here ε is a certain small
quantity. The term
∫
∞
−Φ −
0
2
z
zz
z
z
z
vtD
k kzk
Dvk
vzke
z
dsincosk (7)
describes ordinary diffusion broadening of neutron
density. Indeed, the integral
( ) vDt
z
zz
vtD e
vtD
dkzke 4
0
2
2
z
4∫
∞ π=− cosk (8)
is responsible for diffusion spreading of an initially
given (at 0=z ) flat distribution of neutron density in
the form of ( )zδ , because
( ) ( ) ( )∫∫
∞∞
=δδ=−
∞→ 00 2
12
z
zzz
vtD
t
dkzzdkzke ,coslim k .
(9)
To satisfy the remaining initial condition at 0=t let us
investigate the behavior of the integral Eq. (7) for
0→t . Integrating Eq. (8) over z from 0 to z , one
finds
∫ ∫
∞
ξπ= ξ−−
0
4
0
22
z
vDt
z
z
z
zvtD dedk
k
zk
e
sink ,
therefore, in the limit of small 0→t
( )
Dv
vzdkzk
Dvk
vzke z
zz
z
z
z
vtD
t 200
2
3
π−δ=
−∫
∞
−
→
sincoslim k .
Assuming next that
zz kk g+ψ=Φ 0 (10)
we can write down the initial condition at 0=t for the
desired solution Φ in the form
0
20
0
=πψ−
−+ ∫
∞−
Dv
vdkzk
Dvk
vzkgAe z
zz
z
z
zk
z
Dv
v
z
z
sincos
(11)
for all z values. Using the well-known integral
representation of the exponential function
∫
∞
+
⋅
π
=
−
0
2
2
2
z
z
z
zz
z
Dv
v
k
Dv
vk
zk
Dv
ve
z
dcos
,
we find
−
+
π
=− ∫
∞−
zk
Dvk
vzk
Dv
vk
dk
Dv
ve z
z
z
z
z
z
zz
z
Dv
vz
sincos
0
2
2
212
Substituting this equation into Eq. (11), we see that all
initial and boundary conditions (4) would be satisfied if
.
,
2
2
2
0
0
1
+
ψ−=
πψ=
Dv
vk
Dv
vg
Dv
vA
z
z
z
k
z
z
(12)
Then the neutron flux in the region 0>z for 0>t
will be given by
( ) ( ) [ ]
.sincosd
,,
k
k 2
−
+
+
+
πψ=Φ
−
−Σ−
∫
∞
α
zk
Dvk
vzke
Dv
vk
kk
e
Dv
vekrJtzr
zvtD
z
z
Dv
v
Dv
v
t
zDvt
zz+
3
3
3
0
2
2
3
3
2
3
00
2
2
3
(13)
The asymptotics of Eq. (13) for large enough
argument 0>− ztvz takes the form
( ) ( ) ( )
( )
π+π×
×ψ∝Φ
−−
+Σ−
ztv
Dv
v
zvtD
z
Dkvt
z
z
a
e
Dv
ve
vtD
ekrJtzr
4
00
2
2
2
1
,,
(14)
One can see that the first exponential describes the
losses due to absorption and escape through the cylinder
reactor surface. The first term in the curly brackets relates
to the usual diffusion and the second one is the source due
to the neutron flux incoming from the reactor active zone.
25
Once the neutron flux is found, the newly produced
fuel can be determined as ΦΣ f .
Considering the problem (B), i.e. the Th-U
heterogeneous assembly (see Fig. 3), we shall use a two-
group approach. The fast neutrons of the first group
diffuse into the breeding zone and slow down to the
thermal velocities. Thus they pass to the second group
of thermal neutrons.
a b L
z
2R
Fig. 3
The dynamics of the process is described by the
balance equations
( ) I
s
III
I
graddiv ΦΣ−ΦΣ−Φ=
∂
Φ∂
aD
tv
1 , (15)
( ) IIIIIII
II
graddiv ΦΣ+ΦΣ−Φ=
∂
Φ∂
saD
tv
1 , (16)
where sΣ is the macroscopic cross section of the
neutron transition from the first group into the second
one. Evidently, -1
s~ DsΣ , where -1
sD is the
characteristic length of neutron slowing-down. The
problem should be solved with the following initial and
boundary conditions for 0>z
( ) ( )
( ) ( ) IIIII
II
ΦΣ+δψ=Φ
δψ=Φ
skrJz
krJz
00
00
for 0=t and for any 0≥z
( ) III,
III,
III,
0
0 j
z
D =
∂
Φ∂−
for 0=z , (17)
0=Φ I,II for ( )extrλ+= Rr .
We study, as before, the initial stage of the process when
the number of the surplus neutrons is insignificant. But,
contrary to the previous problem, the zone of preparation
to burning is heterogeneous. We suppose that it consists
of the periodically placed layers, or discs, of two different
materials (fertile fuel and moderator) with the
corresponding width a and b , diffusivities 1D and 2D ,
and the absorption cross sections 1aΣ and 2aΣ .
Separating the variables for the neutron flux of the
first group
( ) ( ) ( ) ( )tTzZrRtzr =Φ ,,I
and searching for the solution in the first period of the
assembly, one can find
( ) ( ) ( )
( ) ( )
−+=Ζ
ϕ±
zu
Lu
Luezuz
i
2
2
1
1 (18)
where ( )zu1 and ( )zu2 are the fundamental solutions
and ϕ is a parameter which determines the evolution of
the solution at the length of the period, baL += , i.e.,
for first layer
( ) ( )
( ) ( ) 10,00
,00,10
212
111
=′=
=′=
uDu
uDu
(19)
and
( ) ( )[ ]LuDLu 2112
1cos ′+=ϕ . (20)
At the second period of the assembly the solution will
differ from the solution of the first period by a factor
ϕ± ie , at the third period by a factor ϕ± ie 2 , and at ( )1+n
-period by a factor ϕ± ine . With the use of Eq. (18) it is
possible to construct the solutions satisfying the
boundary conditions (17).
Next, for simplicity, we shall consider a physically
interesting case of closely spaced disks, when 11 < <ap ,
12 < <bp , and, therefore, 1< <=ϕ Lk z . After somewhat
tedious calculations the problem can be reduced to that
of the homogeneous medium but with somewhat
different parameters,
( ) ( ) [ ]
.sincosd
,,
||
k
||
II
II
||
||||
−
+
+
+
πψ=Φ
−
−
+Σ−
∫
∞
⊥
zk
vkD
vzke
vD
vk
kk
e
vD
vekrJtzr
z
z
z
z
vtD
z
z
zz
z
vD
vlz
vD
v
t
zkDvt
z
a
2
z
0
2
2
2
00
2
2
(21)
Namely, the mean macroscopic absorption cross
section of the periodic assembly is equal to one
averaged over the period.
,
ba
ba aaa
+
ΣΣ=Σ + 11 (22)
The diffusivity becomes anisotropic. The transversal
diffusivity is averaged over the period, but as for the
longitudinal diffusivity, averaged are the reciprocal
quantities, or diffusional times,
ba
bDaD
D
+
+
=⊥
21 , (23)
( )
12
12
bDaD
DDba
D
+
+
= . (24)
The product ( )tzra ,,ΦΣ gives the thermal neutron
source strength (in cm–3s–1) due to the slowing-down of
fast neutrons in the volume with 0>z .
Author acknowledges the financial support of STCU
(project №1480).
REFERENCES
1.L.P. Feoktistov. Neutron-fissioning wave //
Doklady AS USSR, 1989, v. 309, p. 864-867 (in
Russian).
2.E. Teller, M. Ishikawa, L. Wood, R. Hyde, and
26
J. Nuckolls. Completely automated nuclear reactors
for long-term operation. Int. Conf. on Emerging
Nuclear Energy Systems, Obninsk, Russia, 24-28
June 1996.
27
REFERENCES
|
| id | nasplib_isofts_kiev_ua-123456789-80032 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T15:23:25Z |
| publishDate | 2001 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Khizhnyak, N. 2015-04-09T16:04:09Z 2015-04-09T16:04:09Z 2001 On the theory of the initial stage of slow nuclear burning / N. Khizhnyak // Вопросы атомной науки и техники. — 2001. — № 6. — С. 279-282. — Бібліогр.: 2 назв. — англ. 1562-6016 PACS: 28.41.Ak https://nasplib.isofts.kiev.ua/handle/123456789/80032 Scientific principles of linear nuclear power systems of slow burning are elaborated. Two concepts of a slow burning reactor, one using U-Pu cycle and the other Th-U cycle are proposed. In the first concept the reactor is approximated by a homogeneous media and operates with fast neutrons. In the second concept the reactor is a heterogeneous assembly using both fast and thermal neutrons. The fast neutron flux at the initial stage after the reactor was ignited was determined for both concepts. Author acknowledges the financial support of STCU (project №1480). en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Kinetic theory On the theory of the initial stage of slow nuclear burning К теории начальной стадии медленного ядерного горения Article published earlier |
| spellingShingle | On the theory of the initial stage of slow nuclear burning Khizhnyak, N. Kinetic theory |
| title | On the theory of the initial stage of slow nuclear burning |
| title_alt | К теории начальной стадии медленного ядерного горения |
| title_full | On the theory of the initial stage of slow nuclear burning |
| title_fullStr | On the theory of the initial stage of slow nuclear burning |
| title_full_unstemmed | On the theory of the initial stage of slow nuclear burning |
| title_short | On the theory of the initial stage of slow nuclear burning |
| title_sort | on the theory of the initial stage of slow nuclear burning |
| topic | Kinetic theory |
| topic_facet | Kinetic theory |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/80032 |
| work_keys_str_mv | AT khizhnyakn onthetheoryoftheinitialstageofslownuclearburning AT khizhnyakn kteoriinačalʹnoistadiimedlennogoâdernogogoreniâ |