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Reactor Theory
1. One Group Reactor Equation
a) Assumptions {bare reactor}
(1) Rx is a homogeneous fuel & moderator
(2) Rx consists of one region and has neither a
blanket or reflector.
b) Diffusion Equation
Nuclear Reactor Theory
Chapter 6
∅ − Σ ∅ + =
1
Eq. 1
D – Diffusion coefficient
Σ - x section for absorption
v- n yield
S- source density (n/cm3 –s)
= νΣ ϕ
Reactor Theory
2
Reactor Theory
So we have
1
∅ − Σ ∅ + Σ ∅ = 0
If the fission source neutrons do not balance the
leakage and absorption terms, then the RHS of Eq. 1 is
non-zero.
Let
= { Σ - Σ }
then ∅ = − ∅
substituting − ∅ for ∅
In this case we could balance the equation by
multiplying the source term by a constant say “1/k ”.
We get − ∅ − Σ ∅ + Σ ∅ = 0 Eq.2
If the source is too small then k <1.
If the source is too large then k >1.
∅
Solving for k : = ∅
! ∅
= !
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Reactor Theory
k = the multiplication factor
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Reactor Theory
Notice that = νΣ ϕ = ηΣ# ϕ = η !$ Σ ϕ
!
∅
= ∅
! ∅
= #n/fiss
!
Numerator = neutrons born via fission
Denominator = neutrons lost to leakage and
absorption.
f=
!$
!
Fuel
f
n per abs
= fuel utilization: the fraction of all
neutrons absorbed in the reactor that are
absorbed in the fuel.
= η%Σ ϕ
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Eq. 3.
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Reactor Theory
C. Multiplication factor in the infinite reactor.
Assumptions:
1. All neutrons are absorbed. No neutrons leak.
2. The neutron flux φ is constant everywhere. Is
independent of position.
Concept:
1. Since all neutrons born are eventually
absorbed, then Σ ϕ is the total number of
neutrons.
2. Of these %Σ ϕ are absorb in the fuel, and
release η%Σ ϕ in the next generation
Reactor Theory
Multiplication factor in the infinite reactor.
Reactor Theory
D. Buckling B2 in the Critical bare reactor.
Reactor Theory
D. Buckling B2 in the Critical bare reactor.
Concept:
3. Dividing the number of neutrons in one generation
by those in the next gives
effective yield/absorp
& =
'! (
! (
= η%
Fuel utilization
4. & is k for and infinite reactor.
5. Since η and % are constants that depend on the
material properties of the reactor, & is the same for a
bare reactor as for an infinite reactor of the same
composition.
6. Thus & refers to a Rx in which no neutrons leak.
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1. Since & =
'! (
! (
= η%
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4. If Rx is critical, k = 1, and
=
0,
− ∅ + (& −1)Σ ∅ =0
5. Let / = 0! = diffusion area, then
(& − 1)∅
− ∅ + = 0
/
2
6. Solving for B
(& − 1)
= /
2. And the source term = η%Σ ϕ,
then = & Σ ϕ
3. Using this in the one-group Rx eqn. {use Eqn. 2 & 3}
1
*+
− ∅ − Σ ∅ + =
*,
1
*+
− ∅ − Σ ∅ + (& Σ ∅) =
*,
4. If Rx is critical, k = 1, and
=
0,
− ∅ + (& −1)Σ ∅ =0
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10
Reactor Theory
Reactor Theory
Ex 6.1 … Find f and k∞for a mixture of U-235 and sodium
(Na) in which the U is present to 1 w/0.
Solution:
%=
=
!$
!
=
!$
!2
!$
=
3!2
03
!$
$ !$
67 &6# =atomic concentrations of Na and U
So
A2
A$
=
B2 C$
C2 B$
where
But :
B$
B$ B2
B C$ V!2
$ C2 V!$
= 1 + B2
in 9:;<=/?9@
H
DE=F<+=E,G{IJ@}
So:
B
= 0.01 ⇒ B 2 = 99
Thus: % U = 1 + 99
4 5
4 25!2
Now:
% U
$
@Z
@
[.[[[\
.]Z
= 1.48
% = 0.671
Next: & = +% = 2.2 0.671 = 1.48
and LE=MNO9O,:9E?PM,{H⁄JQRS}
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Thus an infinite reactor of this composition would
be super critical.
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Reactor Theory
Reactor Theory
c ∅
2. Rx Eqn. is cd + ∅ = 0
With the general solution ∅ e = f cos e + f sin e
Where C1 and C2 are to be determined using Ficks law.
II. Solutions to the one group reactor equation.
The Slab Reactor
3. l = −
Assume:
1. Infinite bare reactor of thickness ‘a’.
2. Is critical (flux in steady-state)
a
c∅
=
cd
0 at x = 0 due to symmetry of slab, so
0
a/2
c∅
Since: ∅
x
Finally:
and:
= f cos
∅ e = f cos
13
Reactor Theory
∅ e = f f:=(e)
= 0 and = n
e
n
+
= 1,3,5, …
+ = 1,3,5, …
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Reactor Theory
The Slab Reactor
∅ e = f cos
= 0.
Thus: cd = −f sin 0 + f cos 0 = 0O,e = 0
So: f = 0!
-a/2
c∅
cd
The Slab Reactor
The value of C1 depends on the reactor power level (P).
From the text page 274,
+r
e + = 1,3,5, …
O
For a critical reactor only the n = 1 eigenvalue will exist.
n
f = s
t !
+r
∅ e = f cos
e
O
u
P = power per unit area {v7⁄IJw }
xy =constant conversion factor of 3.2 x 10U
n=1
n
∅ e = s
n=3
t !
-a/2
0
a/2
n
u?:=( )
x
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Reactor Theory
Reactor Theory
C. Buckling
Recall the solution for the slab reactor
∅ e = f cos( e)
The square of the first eigenvalue is called the
buckling of the reactor.
B. Other Reactor Shapes
1. Sphere of radius R
∅(N) =
sin(rN0z )
u
4xy Σ z
z
2. Infinite Cylinder of radius R
∅ N =
Since
[.{@\|
.}[Z~
l (
)
st y Q
y
Then
3. Solutions for other shapes listed in table 6.2.
c ∅
cd + ∅ = 0
=
U c ∅
∅ cd The RHS is an expression that is proportional to the
curvature of the flux, which in turn is a measure of
how much the flux curves or “buckles”.
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Reactor Theory
Reactor Theory
C. Buckling
D. Max. to Avg. Flux Ratio Ω (table 6.2)
n
Notice that = ( ) for a slab, and that decreases as ‘a’ increases. In the limit as ⇒ ∞,
=0, φ = constant and thus has no buckle.
For the reasons listed below, it is important that
the flux distribution in an actual reactor be as
flat as possible..
Ω=
is a measure of the variation of the flux w/ in a reactor,
and the extent to which the max. power density at the
center exceeds the avg power density.
e.g. For a bare infinite slab: ∅ e = f cos
so
∅Jd = f
– Reduced power spikes
– Uniform fuel burn-up
– Prevent ‘hot channel’ effects
And
/
∅H = U/ f cos(

Thus Ω = 0 =

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Reactor Theory
E. Ex 6.2. A bare spherical rx radius R = 50 cm,
operates at 100Mw=10\ J/s and Σ =
0.0047?9U .
What are the max and avg. flux values?
∅Jd = lim }s
~→[
|
t y

[ n

( t )
~
=
=}
@.d[
=
[.[[}{(Z[)w
∅H =
}.\d[
@.
= 1.27e10Z IJ
n
nd
)Fe
=
n
e
,

n
= 1.57
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Reactor Theory
III. k for a non- infinite reactor
Recall a critical reactor has:
=
|
n
}st y y
4.18e10Z
∅Jd
∅H
IJ U7
U

rearranging : which was found for & =1

= 1 Eqn. 1
Since n’s either leak or are absorbed, the
relative probability that a n will be absorbed is
called the non-leakage probability u .
U7
Table 6.2
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Reactor Theory
Reactor Theory
III. k for a non- infinite reactor
For other then critical conditions we can write:
III. k for a non- infinite reactor
u =
=
u =
#:%+ =O=:N<F
Σ ∅
=
#:%+ =O=:N<F:N;<O<F Σ ∅ + ∅

0
!
3!
! 03!
=
22
= & u
Eqn. 2
Comparing Eqn. 1 and 2
& u = 1%:NO?NE,E?O;N<O?,:N.
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4/14/2015
Reactor Theory
III. k for a non- infinite reactor
Ex…Given a reactor bare sphere of radius R = 48.5
cm, with / = 384?9 .
What is the probability that a fission neutron
will be absorbed?
1
1
u =
=
= 0.38
1 + / 1 + 384(r0
48.5)
table 6.2
38% chance of absorption, 62% chance of leakage.
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Reactor Theory
IV. Thermal Reactors
A. Recall that thermal rx’s contain a moderator
to slow down fission neutrons to thermal
energies. For convenience all materials in the
reactor other than fuel are considered
moderator.
B. Thermal Utilization (f) {formerly the fuel
utilization} is the fraction of all neutrons
absorbed that are absorbed in the fuel in a
thermal reactor.
Σ
%=
Σ
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Reactor Theory
IV. Thermal Reactors
D. The resonance escape probability (p) is the
probability that a neutron is not absorbed while
slowing down by nuclides having absorption
resonances above thermal energies.
E. Based on previous discussion, the absorption
of Σ ∅ thermal neutrons leads to the
production of η %Σ ∅ new neutrons in an
infinite thermal reactor.
The infinite rx multiplication factor is;
η %Σ ∅
& =
= η %ε
Σ ∅
Reactor Theory
IV. Thermal Reactors
C. Fast fission factor (ε) is defined as the ratio of
the total number of fission neutrons produced
by both fast and thermal fission to the number
produced by thermal fission alone. The value of
ε (fast fission factor) ~ 1.02- 1.08. Thus about 2 –
8 % of fission are from fast neutron reactions.
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Reactor Theory
IV. Thermal Reactors
F. Although in a thermal reactor most fissions
occur with neutrons at thermal energies, all
fission neutrons are born at fast energies.
Therefore it is customary to describe a thermal
reactor by two groups of neutrons (fast and
thermal). Thus there are to groups of neutron
flux terms:
Let ∅ = ∅,<%O=,+<,N:+%;ex+1
And
∅ = ∅,<,<N9O;+<,N:+%;ex+2
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Reactor Theory
IV. Thermal Reactors
Substituting and rearranging the diffusion
equation leads to (see page 289 of text)
The Two Group Critical Equation

( )( )
P<N</ =
O+F =
29
=1
= ,<N9O;FE%%=E:+ON<O
Σ
= +<,N:+OM<
Σ
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Reactor Theory
Reactor Theory
IV. Thermal Reactors
Neutron life cycle
= η %εu u#
IV. Thermal Reactors
/<,,E+Mu =
(
probability
O+Fu# =
=
)
( )
thermal non-leakage
= fast non-leakage probability
Σ# ∅ → η → ¡ → u# → → u → % →
n’s
absorbed
in fuel
Which leads to & u u# = 1for a critical reactor.
Or
= & u u# = η %εu u#
Which is the six factor formula for a thermal reactor!
n’s via
thermal
fission
Fast n’s
that don’t
leak
n’s from
thermal + fast
fission
Slow n’s
that don’t
leak
Slow n’s
Slow n’s
absorb in
fuel
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35
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Reactor Theory
IV. Thermal Reactors
Reactor Theory
IV. Thermal Reactors
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Reactor Theory
IV. Thermal Reactors
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Reactor Theory
IV.
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Reactor Theory
Reactor Theory
IV.
IV.
41
42
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