Transcript Document
Steady State Diffusion Equation
HW 20
Study example 5.3 and solve problem 5.8 in Lamarsh.
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed
Dababneh).
1
Steady State Diffusion Equation
One-speed neutron diffusion in a finite medium
• At the interface
A B
A
B
d A
dB
J A J B DA
DB
dx
dx
x
• What if A or B is a vacuum?
• Linear extrapolation distance.
• Bare slab with central infinite planar source (Lamarsh).
• Same but with medium surrounding the slab.
• Maybe we will be back to this after you try it!!
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed
Dababneh).
2
More realistic multiplying medium
One-speed neutron diffusion in a multiplying medium
The reactor core is a finite multiplying medium.
• Neutron flux?
• Reaction rates?
• Power distribution in the reactor core?
Recall:
• Critical (or steady-state):
Number of neutrons produced by fission = number
of neutrons lost by:
neutronproductionrate(S)
k
(1) absorption
neutronabsorptionrate( A)
(1) leakage
keff
neutronproductionrate( S )
neutronabsorptionrate( A) neutronleakage rate( LE )
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed
Dababneh).
3
More realistic multiplying medium
keff
A
Pnon leak
k A LE
LE SA
surface area
S V
Volum e
non- leakageprobability
For a critical reactor:
Keff = 1
K > 1
LE SA a 2 1
3
S
V
a
a
Steady state homogeneous reactor
2
0 a k (r ) a (r ) D (r )
k 1
2
2
2
(r ) B (r ) 0
B 2
L
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed
Dababneh).
Material buckling
4
More on One-Speed Diffusion
HW 21
Show that for a critical homogeneous reactor
Pnon leak
a
a
1
2 2
2
2
B L 1 a D a B D
Infinite Slab Reactor (one-speed diffusion) z
• Vacuum beyond.
• Return current = 0.
Reactor
x
d = linear extrapolation distance
a/2
= 0.71 tr (for plane surfaces)
a
= 2.13 D.
a0/2
d
d
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed
Dababneh).
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More on One-Speed Diffusion
HW 22
d 2
For the infinite slab 2 B 2 0 . Show that the
dx
general solution
( x) A cos Bx C sin Bx
With BC’s
a0
)0
2
d ( x)
0
dx x 0
(
Flux is symmetric about
the origin.
( x) A cos Bx
A 0
a0
a0
a0
3 5
( ) A cos B( ) 0 B( ) , , ,...
2
2
2
2 2 2
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed
Dababneh).
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More on One-Speed Diffusion
HW 22 (continued)
a0
3 5
B( ) , ,
,...
2
2 2 2
3 5
a0 , ,
,...
B B B
Fundamental mode, the only mode significant in
critical reactors.
( x) 0 cos
a0
x
B
a0
Geometrical Buckling
For a critical reactor, the geometrical buckling is equal
to the material buckling.
2
k 1
To achieve criticality
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed
Dababneh).
a L2
0
7
More on One-Speed Diffusion
Spherical Bare Reactor (one-speed diffusion)
6a
4a
4 3
3
a
3 a
2
2
Minimum leakage minimum fuel to achieve criticality.
2
d
2 d
HW 23
2
B
0
2
dr
r dr
A
C
cos Br sin Br
r
r
Reactor
r
C
r
sin , r0
r
r0
B
Continue!
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed
Dababneh).
x
r0
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More on One-Speed Diffusion
HW 24
Infinite planer source in an infinite
medium.
SL x / L
d 2 ( x) 1
S ( x)
e
2
( x)
2
dx
L
D
2D
HW 25
Infinite planer source in a finite
medium.
SL sinha0 2 x / 2L
2D cosh(a0 / 2L)
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed
Dababneh).
x
a/2
a
a0/2
Source
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More on One-Speed Diffusion
Infinite planer source in a multi-region medium.
1 ( a / 2) 2 ( a / 2)
d1
d2
D1
D2
Infinite
Finite
Infinite
dx
dx
x a / 2
m ore
x a / 2
BC
Project 2
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed
Dababneh).
10
Back to Multiplication Factor
k
k = fp,
P
keff fPnon leak
eff
k
non leak
1
• Fast from thermal,
a
• Fast from fast, .
(i)
f
(i)
i
• Thermal from fast, p.
• Thermal available for fission
afuel
f fuel
mod erator
poison
a clad
a
a
a
Thinking QUIZ
• For each thermal neutron absorbed, how many fast
neutrons are produced?
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed
Dababneh).
11
Two-Group Neutron Diffusion
• Introductory to multi-group.
• All neutrons are either in a fast or in a thermal energy
group.
• Boundary between two groups is set to 1 eV.
• Thermal neutrons diffuse in a medium and cause
fission (or are captured) or leak out from the system.
• Source for thermal neutrons is provided by the slowing
down of fast neutrons (born in fission).
• Fast neutrons are lost by slowing down due to elastic
scattering in the medium or leak out from the system (or
fission or capture).
• Source for fast neutrons is thermal neutron fission.
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed
Dababneh).
12
Two-Group Neutron Diffusion
1 (r )
10 MeV
( E, r )dE
Fast
1eV
1eV
2 (r ) ( E , r )dE
Therm al
0
1 f 1 1 2 f 2 2
keff
2
2
D1 1 D2 2 a1 1 a 2 2
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed
Dababneh).
13
Two-Group Neutron Diffusion
2
0 S1 (r ) a1 1 (r ) D1 1 (r )
Depends on
thermal flux.
Fast diffusion
Removal cross section coefficient
= fission + capture +
scattering to group 2
2
0 f 1 1 (r ) f 2 2 (r ) a1 1 (r ) D1 1 (r )
or
0
k
2
a 2 2 (r ) a1 1 (r ) D1 1 (r )
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed
Dababneh).
14
Two-Group Neutron Diffusion
2
0 S2 (r ) a 2 2 (r ) D2 2 (r )
Depends on fast
flux.
Thermal absorption
cross section = fission
+ capture.
Thermal diffusion
coefficient
2
0 s12 1 (r ) a 2 2 (r ) D2 2 (r )
or
2
0 a1 1 (r ) a 2 2 (r ) D2 2 (r )
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed
Dababneh).
15
Two-Group Neutron Diffusion
0
k
2
a 2 2 (r ) a1 1 (r ) D1 1 (r )
2
0 a1 1 (r ) a 2 2 (r ) D2 2 (r )
• A coupled system of equations; both depend on
both fluxes.
• For a critical, steady state system:
2
1 (r ) B 1 (r ) 0
2
2
2 ( r ) B 2 ( r ) 0
2
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed
Dababneh).
Review
Cramer’s
rule!
Geometrical
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