The velocity of slow nuclear burning in the two-group approximation
The velocity of slow nuclear burning was obtained in the two-group approximation. Two groups of neutrons were considered: the group of thermal (slow) neutrons and the group of fast neutrons; each group being described with its diffusion equation. It was shown that in the case of heavy moderators the...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2001
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| Назва видання: | Вопросы атомной науки и техники |
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| Цитувати: | The velocity of slow nuclear burning in the two-group approximation / A.I. Akhiezer, D.P. Belozorov, F.S. Rofe-Beketov, L.N. Davydov, Z.A. Spolnik // Вопросы атомной науки и техники. — 2001. — № 6. — С. 276-278. — Бібліогр.: 5 назв. — англ. |
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irk-123456789-800312015-04-10T03:02:55Z The velocity of slow nuclear burning in the two-group approximation Akhiezer, A.I. Belozorov, D.P. Rofe-Beketov, F.S. Davydov, L.N. Spolnik, Z.A. Kinetic theory The velocity of slow nuclear burning was obtained in the two-group approximation. Two groups of neutrons were considered: the group of thermal (slow) neutrons and the group of fast neutrons; each group being described with its diffusion equation. It was shown that in the case of heavy moderators the obtained expression for the two-group velocity had the same structure as the one-group velocity studied by authors before if new effective diffusion and multiplication coefficients were introduced. The expressions for corresponding effective coefficients are presented. 2001 Article The velocity of slow nuclear burning in the two-group approximation / A.I. Akhiezer, D.P. Belozorov, F.S. Rofe-Beketov, L.N. Davydov, Z.A. Spolnik // Вопросы атомной науки и техники. — 2001. — № 6. — С. 276-278. — Бібліогр.: 5 назв. — англ. 1562-6016 PACS: 28.20.Ld http://dspace.nbuv.gov.ua/handle/123456789/80031 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
| topic |
Kinetic theory Kinetic theory |
| spellingShingle |
Kinetic theory Kinetic theory Akhiezer, A.I. Belozorov, D.P. Rofe-Beketov, F.S. Davydov, L.N. Spolnik, Z.A. The velocity of slow nuclear burning in the two-group approximation Вопросы атомной науки и техники |
| description |
The velocity of slow nuclear burning was obtained in the two-group approximation. Two groups of neutrons were considered: the group of thermal (slow) neutrons and the group of fast neutrons; each group being described with its diffusion equation. It was shown that in the case of heavy moderators the obtained expression for the two-group velocity had the same structure as the one-group velocity studied by authors before if new effective diffusion and multiplication coefficients were introduced. The expressions for corresponding effective coefficients are presented. |
| format |
Article |
| author |
Akhiezer, A.I. Belozorov, D.P. Rofe-Beketov, F.S. Davydov, L.N. Spolnik, Z.A. |
| author_facet |
Akhiezer, A.I. Belozorov, D.P. Rofe-Beketov, F.S. Davydov, L.N. Spolnik, Z.A. |
| author_sort |
Akhiezer, A.I. |
| title |
The velocity of slow nuclear burning in the two-group approximation |
| title_short |
The velocity of slow nuclear burning in the two-group approximation |
| title_full |
The velocity of slow nuclear burning in the two-group approximation |
| title_fullStr |
The velocity of slow nuclear burning in the two-group approximation |
| title_full_unstemmed |
The velocity of slow nuclear burning in the two-group approximation |
| title_sort |
velocity of slow nuclear burning in the two-group approximation |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2001 |
| topic_facet |
Kinetic theory |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/80031 |
| citation_txt |
The velocity of slow nuclear burning in the two-group approximation / A.I. Akhiezer, D.P. Belozorov, F.S. Rofe-Beketov, L.N. Davydov, Z.A. Spolnik // Вопросы атомной науки и техники. — 2001. — № 6. — С. 276-278. — Бібліогр.: 5 назв. — англ. |
| series |
Вопросы атомной науки и техники |
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2025-07-06T03:58:24Z |
| last_indexed |
2025-07-06T03:58:24Z |
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1836868493008437248 |
| fulltext |
THE VELOCITY OF SLOW NUCLEAR BURNING
IN THE TWO-GROUP APPROXIMATION
A.I. Akhiezer, D.P. Belozorov, F.S. Rofe-Beketov*, L.N. Davydov, Z.A. Spolnik
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
*Verkin Institute for Low Temperature Physics & Engineering, Kharkov, Ukraine
The velocity of slow nuclear burning was obtained in the two-group approximation. Two groups of neutrons
were considered: the group of thermal (slow) neutrons and the group of fast neutrons; each group being described
with its diffusion equation. It was shown that in the case of heavy moderators the obtained expression for the two-
group velocity had the same structure as the one-group velocity studied by authors before if new effective diffusion
and multiplication coefficients were introduced. The expressions for corresponding effective coefficients are
presented.
PACS: 28.20.Ld
The simplest variant of many-group approximation
is so named two-group approximation, which considers
only two mutually connected neutron groups; namely
the groups of thermal and fast neutrons. The neutrons
with thermal energy belong to the thermal group and all
the neutrons with the energy exceeding the thermal one
belong to the fast neutron group. It worth noting, that
such division is justified also from the physical
consideration because cross-sections of reactions with
thermal neutrons differ substantially from the cross-
sections of fast neutrons. When we consider diffusion
processes each of these groups is taken to be a
“monoenergetic” one described by its own diffusion
equation with constant coefficients not depending on
energy. In other words it is supposed that the neutrons
diffuse without energy loss within each group until
some of the fast neutrons undergo a number of
collisions necessary to decrease their energy down to
the level of the lower thermal group; at that moment
these neutrons immediately jump to the thermal group
[1-3].
According to this picture, the diffusion equation that
describes the neutron of the fast group takes the form
1
1
2
2
22
2
τ
ν+
τ
−∆=
∂
∂ nn
nD
t
n
, (1)
where ),(2 tn r is the neutron density of the fast group;
2D is the neutron diffusion coefficient for the fast
group; 2τ is the fast neutron life-time during which it
undergoes slowing down to the thermal energy range;
ν is the average number of fast neutrons born at
thermal fission; ),(1 tn r is the density of thermal
neutrons; and τ1 is the thermal fission capture life-time.
The slow neutron density ),(1 tn r satisfies the
following diffusion equation
2
2
1
1
11
1
τ
+
τ
−∆=
∂
∂ nn
nD
t
n
, (2)
where 1D is the thermal diffusion coefficient.
The diffusion coefficient can be estimated, if we use
the relation 22
2
2 τ= DL , where 2
L is the diffusion
length for the fast neutron group. In the case of graphite
moderator 1.02
1
2
2 ≈−LL [2], where 1L is the thermal
diffusion length and the ratio of fast and thermal
neutron lifetimes 1
12
−ττ has the order of magnitude
21
12 10 −− ≈ττ (see, e.g., [2,3]). We obtain for the
graphite moderator that 101
12 ≈−DD .
We shall solve the system of equations (1), (2) for
the case of δ–shaped thermal source having the output
02 nπ and placed at the initial moment in the 0=z
plane. Therefore the initial conditions for our problem
are
.),(
,)(sinsin),(
00
20
2
01
=
δπππ=
zn
zy
a
x
a
nzn
(3)
As for the physical picture, we can speak here about
the propagation of an initial fluctuation of thermal
neutron density.
The boundary conditions on the cylinder surface are
assumed to be zero so we seek the solution of our
system in the form ( ) =tzyxn j ,,,
),(,sinsin),( 21=ππ= jy
a
x
a
tzn j . As is easy to see the
existence of boundaries along x and y axes results in
renormalization of quantities ),( 211 =τ − jj ,
2,1
2
2
2,1
2,1
121
τ
+π=
τ ′ a
D . (4)
Finally the system of equations which describes the
neutron diffusion along the reactor axis has the form
.),(),(),(),(
,),(),(),(),(
1
1
2
2
2
1
2
2
2
2
2
1
1
2
1
2
1
1
τ
ν+
τ ′
−
∂
∂=
∂
∂
τ
+
τ ′
−
∂
∂=
∂
∂
tzntzn
z
tznD
t
tzn
tzntzn
z
tznD
t
tzn
(5)
The set of equations (5) can be solved by the Fourier
transform. Taking the Fourier transform with respect to z,
20 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2001, № 6, p. 276-278.
( )2,1,),(
2
1),( =
π
= ∫
+ ∞
∞−
jdztznetkN j
ikz
j (6)
we obtain the system of ordinary differential equations
for ),( tkN j :
.)(
,)(
1
1
2
2
2
2
2
2
2
1
1
2
1
1
1
1
τ
ν+
τ ′
+−=
τ
+
τ ′
+−=
N
NkD
dt
dN
N
NkD
dt
dN
(7)
We seek the solution of equations (7) in the form
( )2,1== λ jeCN t
jj , and after equating the
determinant to zero we obtain the characteristic
equation of the system
011
212
2
2
1
2
1 =
ττ
ν−
τ ′
++λ
τ ′
++λ kDkD . (8)
We are interested in multiplicative solutions, namely
the solutions for which the neutron density increase
exponentially with time at any point. As it was shown
earlier [4,5] only such solutions describe a slow nuclear
burning wave. Thus we can restrict ourselves only to
one (positive) root of characteristic equation.
ττ
ν+
τ ′
−
τ ′
+−+
+
τ ′
+
τ ′
−+−=λ +
21
2
21
2
21
21
2
21
411
112
)()(
)()(
kDD
kDD
, (9)
though the coefficient )(1 kC + is calculated accounting
for both roots.
The problem of determination of slow nuclear
burning velocity 0v reduces itself to finding such
velocity value 0vv = that the asymptotic behavior of
the thermal neutron density ),( tzn1 for the case vtz =
and ∞→t undergoes a change when passing through
this velocity value. This asymptotics can be obtained by
the saddle-point method from the integral expression
for ),(1 tvtn
∫
+ ∞
∞−
+
−λ +
π
= )(),( ))(( kCdketvtn tikvk
11
2
1
, (10)
where )(k+λ is given by Eq.(9).
Asymptotic velocity 0v in this case will be
determined from the solution of two algebraic equations
dk
kd
iv
)(+λ
= , (11)
[ ] 0=λ− + )(Re kikv .
The first of these equations determines the saddle-point
)(0 vk , whereas the second represents the condition for
the change in the asymptotics character of ( )tvtn ,1
(from the exponential increase at 0vv < to damped
exponential at 0vv > ).
Using the expression (9) for the )(k+λ we can
write the equations (11) in explicit form
,
4)11()(
)()11()(
)(
21
2
21
2
21
21
21
2
21
21
k
kDD
DDkDD
kDDiv
ττ
ν+
τ ′
−
τ ′
+−
−
τ ′
−
τ ′
+−
+
++−=
(12)
ττ
ν+
τ ′
−
τ ′
+−+
+
τ ′
+
τ ′
−+−=
21
2
21
2
21
21
2
21
4)11()(
)11()(2
kDD
kDDikv
.
The solution of system (12) can be easily obtained in the
approximation of small multiplication 1< <γ . As can
be shown in this case 4
0
24
0
2
21 )( kDkDD ≈− has the
order of magnitude 2γ and we can expand the square
root in (9) up to terms of the order of γ. Indeed,
according to (11) characteristic 0k have the order of
1
0
−≈ vDk and the velocity of slow nuclear burning is
the quantity of the order of RDv ≈0 [4,5], where R
is the characteristic coefficient of multiplication and D
is the characteristic diffusion coefficient. In the case of
small γ under consideration here, 1−γ τ≈R . Therefore,
11
00 )( −− τγ≈≈ DDvk
. Expanding the square root in
Eq.(9) we obtain
.
))((
)(
12
2
12
1221
212
1
τ ′+τ ′
γ+
+
τ ′+τ ′
τ ′−τ ′−
−+−≈λ + k
DD
DD
(13)
Using Eq.(13) we shall put the system (11) into the
form
,
))((
)( k
DD
DDiv
τ ′+τ ′
τ ′−τ ′−
−+−=
12
1221
21
(14)
.
))((
)(
12
2
12
1221
21
2
2
τ ′+τ ′
γ+
+
τ ′+τ ′
τ ′−τ ′−
−+−= k
DD
DDikv
Solving these equations with respect to v one
obtains
12
effeff
0 2
τ ′+τ ′
=
AD
v , (15)
where Deff is the effective diffusion coefficient
τ+τ ′
τ ′−τ ′−
−+=
12
1221
212
1 ))((
)(eff
DD
DDD (16)
and Aeff is the effective coefficient of multiplication
1
21
21
eff −
ττ
τ ′τ ′ν
=γ=A . (17)
The role of the effective lifetime plays the quantity
)( 21 τ ′+τ ′ Therefore the concept of slow nuclear burning
velocity allows generalization for the case of two
neutron groups. In the case of heavy moderator, as we
saw before, there exists the relation 11
12 < <τ ′τ ′ −)(
between lifetimes 1τ ′ and 2τ ′ . In this case it follows
from Eq.(16)
1eff DD ≅ . (18)
Therefore, up to the terms of the order of γ Eq.(15)
obtained in the two-group approximation turns into the
expression for the velocity which was obtained earlier
[4,5] in the one-group approximation
1
*
1
0
~
2
τ
=
AD
v (19)
and the quantity ∗A~ equals to
1
1
2
2*~
τ ′
τ
−
τ
τ ′ν
=A . (20)
We consider now the solution of the system (12) in
the general case of heavy moderators for which we
assume the validity of the following conditions
21 DD /=ζ , 112 < <ζ< <ττ / , 11122 < <ττ DD / ,
(note that for graphite 1021 ./ ∝DD , 01012 ./ ∝ττ ).
As for quantity 12 τν τ / we have ζ≤τν τ 12 / (for U235
52.=ν ).
Using these relations we consider now the radicand
in (9) which after taking the quantity 1
2 )( −τ ′ out of
radical we can write in the form
( )
21
21
2
2
2
1
22
21
2
111
/
'
'
''
'
ττ
ν τ+
τ
τ−+ζ−τ
τ
kD . (21)
As it is easy to see two summands under the radical are
comparable by magnitude only in the case when the
dimensionless parameter 2
22 kD τ ′ is close to –1.
Indeed, assuming ykD +−=τ ′ 12
22 we obtain that the
condition ( ) 21
12
// τν τ∝ζ+y must be fulfilled .For all
other values of 2k the first term exceeds considerably
the second one and the quantity )(k+λ has
correspondingly the form
1
2
21
2
2
21)(
τ
γ+
ττ
τ ′ν
−−=λ + kDDk ; (22)
here γ is given by Eq.(17).
Substituting the obtained expression for the )(k+λ
into the system (11) we find for the velocity of slow
nuclear burning the expression
1
0 τ
=
*~
eff AD
v , (23)
where
12
12
11
22
1eff ),1(
ττ
τ ′τ ′ν
τ ′
τ ′
=∆∆−=
D
D
DD (24)
(for graphite %25≈∆ ).The quantity ∗A~ is given by
formula (20).
Therefore, according to Eq. (23) in this case we can
also speak about the slow nuclear burning velocity the
expression for which has a characteristic structure
similar to the structure of the slow nuclear burning
velocity in the one-group approximation [4,5].
According to the above discussion we proved that
both in the one-group and the two-group
approximations we could say about the slow nuclear
burning velocity. This velocity is proportional to the
square root from the product of diffusion and
multiplication coefficients.
Finally, the account of the neutrons with the energy
above the thermal one as a separate neutron group in the
framework of two-group approximation doesn’t change
the result qualitatively though it leads to not so large
quantitative renormalization of the thermal diffusion
coefficient in the case of heavy moderators.
Authors gratefully acknowledge the financial
support of STCU (project №1480).
REFERENCES
1. A.I. Akhiezer, I.Ya. Pomeranchuk. Nekotorye
voprosy teorii yadra (Some Problems in the Theory
of the Nucleus. Moscow: “Gostekhizdat”, 1950,
416 p. (in Russian).
2. A.D. Galanin. Thermal Reactor Theory. Oxford:
“Pergamon”, 1960, 383 p.
3. S. Glasstone and M. Edlund. The Elements of
Nuclear reactor Theory. Toronto, 1952, 458 p.
4. A.I. Akhiezer, D.P. Belozorov, F.S. Rofe-Beketov,
L.N. Davydov, Z.A. Spolnik. Propagating of a
Nuclear Chain Reaction in the Diffusion
Approximation // Yad. Fiz. 1999, v. 62, №9, p. 1-9.
5. A.I. Akhiezer, D.P. Belozorov, F.S. Rofe-Beketov,
L.N. Davydov, Z.A. Spolnik On the theory of
propagation of chain nuclear reaction // Physica.
1999, v. A273, p. 272-285.
22
A.I. Akhiezer, D.P. Belozorov, F.S. Rofe-Beketov*, L.N. Davydov, Z.A. Spolnik
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
The velocity of slow nuclear burning was obtained in the two-group approximation. Two groups of neutrons were considered: the group of thermal (slow) neutrons and the group of fast neutrons; each group being described with its diffusion equation. It was shown that in the case of heavy moderators the obtained expression for the two-group velocity had the same structure as the one-group velocity studied by authors before if new effective diffusion and multiplication coefficients were introduced. The expressions for corresponding effective coefficients are presented.
PACS: 28.20.Ld
REFERENCES
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