3/7/09
EP 471  Engineering Problem Solving II
Exercise 10
Eigenvalue Problems: Calculation of Critical, OneSpeed Flux Shapes
One of the applications of the power iteration method comes in finding critical flux
shapes in nuclear reactor cores.
Here we have an eigenvalue problem where our interest
is confined to the fundamental mode; we can’t have any negative neutron populations,
and the first (fundamental) mode shape is the solution that’s positive everywhere.
This is
also a problem that doesn’t lend itself to the
bvp4c
utility because the composition of
the reactor changes discontinuously between the core and the reflector.
A qualitative flux
shape is shown below:
ϕ
x
R
C
The core region (C) contains the nuclear fuel.
The reflector region (R) may simply be
water; its purpose is to increase the neutron population at the edge of the core.
The
governing equation in the core region is:
C
fC
C
aC
C
C
k
dx
d
D
ϕ
ν
ϕ
ϕ
Σ
=
Σ
+
−
1
2
2
and in the reflector region:
0
2
2
=
Σ
+
−
R
aR
R
R
dx
d
D
ϕ
ϕ
The coefficients
D
C
,
Σ
aC
,
νΣ
fC
,
D
R
and
Σ
aR
are constant coefficients within the core and
reflector regions respectively, but they change discontinuously at the boundary between
core and reflector.
The eigenvalue is
k
, the neutron multiplication factor.
The reactor is
critical when
k
= 1 (as many neutrons exist in the next generation as in the previous
generation, or the power is constant).
These equations follow from something analogous to a continuity equation in fluid
mechanics: they are “conservation of neutron” equations instead of “conservation of
mass” equations.
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 Spring '08
 Witt
 Flux, σ, Normal mode, 0.1 cm, 0.02 cm, 1.0977xxx

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