Steady State Diffusion Equation

One--speed neutron diffusion in a finite medium

One
• At the interface A B
φ A = φB
dφ A dφ B
J A = J B ⇒ − DA = − DB
dx dx
x
• What if A or B is a vacuum?
• Linear extrapolation distance.

Nuclear Reactors, BAU, 1st Semester, 2008-2009 1

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 More realistic multiplying medium
One-speed neutron diffusion in a multiplying medium
One-
The reactor core is a finite multiplying medium.
• Neutron flux?
• Reaction rates?
• Power distribution in the reactor core?
Recall:
• Critical (or steady-state):
steady state):
Number of neutrons produced by fission = number
of neutrons lost by: y
d i rate (S)
neutron production
absorption k∞ =
neutron absorption rate ( A)
and
neutron production rate ( S )
l k
leakage k =
neutron absorption rate ( A) + neutron leakage rate ( LE )
eff

Nuclear Reactors, BAU, 1st Semester, 2008-2009 2

(Saed Dababneh).
 More realistic multiplying medium
keff Things to be used later
later…!!
A
= = Pnon −leak non - leakage probability
k∞ A + LE
LE ∝ SA
Recall:
surface area
For a critical reactor:
S ∝V Volume Keff = 1
LE SA a 2 1 K∞ > 1
∝ ∝ 3 =
S V a a
Steady state homogeneous reactor
r r r
0 = ∑ a k∞φ (r ) − ∑ a φ (r ) + D∇ φ (r )
2

r r k∞ − 1
∇ φ (r ) + B φ (r ) = 0
2 2
B ≡ 2
2

L
Nuclear Reactors, BAU, 1st Semester, 2008-2009
(Saed Dababneh).
Material buckling 3
 More on One-Speed Diffusion
HW 20
Show that for a critical homogeneous reactor
1 ∑a φ ∑a φ
Pnon −leak = 2 2 = =
B L + 1 ∑ a φ − D∇ φ ∑ a φ + B Dφ
2 2

Infinite Bare Slab Reactor (one

(one--speed diffusion) z
• Vacuum beyond
beyond. φ
• Return current = 0.
Reactor x
d = linear extrapolation distance
= 0.71 λtr (for plane surfaces) a/2
= 2.13 D. a0/2 a
d d
Nuclear Reactors, BAU, 1st Semester, 2008-2009 4
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 More on One-Speed Diffusion
HW 21
d 2φ
For the infinite slab 2 + B 2φ = 0 . Show that the
dx
general solution
φ ( x) = A cos Bx + C sin Bx
with BC’s a0
φ (± )=0
2 Flux is symmetric about
dφ ( x) the origin
origin.
=0
dx x =0
T
φ ( x) = A cos Bx A = φ0
a0 a0 a0 π 3π 5π
φ (± ) = A cos B(± ) = 0 ⇒ B(± ) = , , ,...
2 2 2 2 2 2
Nuclear Reactors, BAU, 1st Semester, 2008-2009 5
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 More on One-Speed Diffusion
HW 21 (continued) a0 π 3π 5π
B (± ) = , , ,...
2 2 2 2
π 3π 5π
a0 = , , ,...
B B B
Fundamental mode, the only mode significant in
critical reactors
reactors.
π π
φ ( x) = φ0 cos x B= ≡ Geometrical Buckling
a0 a0
For a critical reactor, the geometrical buckling is equal
to the material buckling. 2
To achieve criticality ⎛π ⎞ k∞ − 1
⎜ ⎟ ⎜ a ⎟ = L2
Nuclear Reactors, BAU, 1st Semester, 2008-2009
(Saed Dababneh). ⎝ 0⎠ 6
 More on One-Speed Diffusion
φ0 ???
2
⎛π ⎞ k∞ − 1
• To achieve criticality ⎜⎜ ⎟ = 2
⎝ a0 ⎠ L
• But
B t criticality
iti lit att what
h t power llevel??
l??
• φ0 can not be determined by the geometry alone.

π
φ ( x) = φ0 ( P,..,..) cos x
a0

Nuclear Reactors, BAU, 1st Semester, 2008-2009 7

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 More realistic multiplying medium
r r
∇ φ (r ) + B φ (r ) = 0
2 2

r
∇ φ (r )
2
B =−
2
r
φ (r )
• The
Th buckling
b kli iis a measure off extent
t t to
t which
hi h th
the flflux
curves or “buckles.”
• For a slab reactor,
reactor the buckling goes to zero as “a”
a
goes to infinity. There would be no buckling or curvature
in a reactor of infinite width.
• Buckling can be used to infer leakage. The greater the
curvature, the more leakage would be expected.

Nuclear Reactors, BAU, 1st Semester, 2008-2009 8

(Saed Dababneh).
 More on One-Speed Diffusion
S h i l Bare
Spherical B Reactor
R (one
(one-
( -speed
d diffusion)
diff i )
6a 2
4πa 2
Cube > 4 3 Sphere
3 πa
3
a
Minimum leakage X minimum fuel to achieve criticality
criticality.
φ
HW 22 d φ + 2 dφ + B 2φ = 0
2

2
dr r dr
A C
φ = cos Br + sin Br
r r r
T
C πr π Reactor
φ = sin , r0 = Continue!
r r0 B r0
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 More on One-Speed Diffusion
HW 23
Infinite planer source in an infinite φ
medium.
d 2φ ( x) 1 Sδ ( x) SL − x / L
2
− 2φ =− X φ ( x) = e
dx L D 2D

HW 24
Infinite planer source in a finite
medium. x
SL sinh [(a0 − 2 x ) / 2 L ] a/2
φ= a
2 D cosh((a0 / 2 L) a0/2

Nuclear Reactors, BAU, 1st Semester, 2008-2009

(Saed Dababneh).
Source 10
 More on One-Speed Diffusion
Infinite planer source in a multi-
multi-region medium.
φ1 (± a / 2) = φ2 (± a / 2)
dφ1 dφ2
D1 = D2 Infinite Finite Infinite
dx dx
x=± a / 2 x=± a / 2
φ
+ more BC

Project 2

Nuclear Reactors, BAU, 1st Semester, 2008-2009 11

(Saed Dababneh).
 Back to Multiplication Factor
Things to be used later
later…!!
keff
k∞ = fpεη, k∞
= Pnon −leak X k
eff = f ρεη P non −leak
1
• Fast from thermal, η =
Σ
∑ ν (i )Σ f (i )
• Fast from fast, ε. a i

• Thermal from fast, p.

∑ afuel
• Thermal available for fission f = ∑ fuel + ∑ clad + ∑ mod erator + ∑ poison
a a a a
Recall:
Thinking QUIZ
• For each thermal neutron absorbed
absorbed, how many fast
neutrons are produced?

Nuclear Reactors, BAU, 1st Semester, 2008-2009 12

(Saed Dababneh).