nuclear reactors, bau, 1st semester, 2008-2009 (saed dababneh). 1 one-speed neutron diffusion in a...
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Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).
1
One-speed neutron diffusion in a finite mediumOne-speed neutron diffusion in a finite medium
Steady State Diffusion Equation
A B
BA • At the interface
• What if A or B is a vacuum?• Linear extrapolation distance.
dx
dD
dx
dDJJ B
BA
ABA
x
Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).
2
One-speed neutron diffusion in a multiplying mediumOne-speed neutron diffusion in a multiplying medium
More realistic 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:absorptionandleakage
)( rate absorptionneutron
rate productionneutron
A
(S)k
)( rate leakageneutron )( rate absorptionneutron
)( rate productionneutron
LEA
Skeff
Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).
3
More realistic multiplying medium
yprobabilit leakage-nonleaknoneff P
LEA
A
k
k
aa
a
V
SA
S
LE
VolumeVS
SALE
1
area surface
3
2
)()()(0 2 rDrrk aa
Steady state homogeneous reactorSteady state homogeneous reactor
2222 1
0)()(L
kBrBr
Material buckling
For a critical reactor:Keff = 1K > 1
Recall:
multiplying m
edium
multiplying m
edium
Things to be used later…!
Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).
4
More on One-Speed DiffusionHW 20HW 20
Show that for a critical homogeneous reactorcritical homogeneous reactor
DBDLBP
a
a
a
aleaknon 2222 1
1
Infinite Bare Slab Reactor Infinite Bare Slab Reactor (one-speed diffusion)(one-speed diffusion)
x
aa/2
d da0/2
• Vacuum beyond.• Return current = 0.d = linear extrapolation distance = 0.71 tr (for plane surfaces) = 2.13 D.
z
Reactor
Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).
5
HW 21HW 21
022
2
B
dx
dFor the infinite slab . Show that the general solution
with BC’sBxCBxAx sincos)(
0)(
0)2
(
0
0
xdx
xd
a
Flux is symmetric about
the origin.
0cos)( ABxAx
,...2
5,
2
3,
2)
2(0)
2(cos)
2( 000
aB
aBA
a
More on One-Speed Diffusion
Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).
6
,...2
5,
2
3,
2)
2( 0
a
BHW 21 HW 21 (continued)(continued)
,...5
,3
,0 BBBa
Fundamental mode, the only mode significant in critical reactors.
Buckling lGeometricacos)(00
0 a
Bxa
x
For a critical reactor, the geometrical buckling is equal to the material buckling.To achieve criticality
2
2
0
1
L
k
a
More on One-Speed Diffusion
Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).
7
More on One-Speed Diffusion
???0
• To achieve criticality
• But criticality at what power level??• 0 can not be determined by the geometry alone.
2
2
0
1
L
k
a
xa
Px0
0 cos,..,..)()(
Do it.
Do it.
Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).
8
More realistic multiplying medium0)()( 22 rBr
)(
)(22
r
rB
• The buckling is a measure of extent to which the flux curves or “buckles.”• For a slab reactor, the buckling goes to zero as “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 (Saed Dababneh).
9
Spherical Bare Reactor (one-speed diffusion)Spherical Bare Reactor (one-speed diffusion)
334
2
3
2 46
a
a
a
a
Minimum leakage minimum fuel to achieve criticality.
r0
0
2 22
2
B
dr
d
rdr
dHW 22HW 22
Brr
CBr
r
Asincos
Br
r
r
r
C 00
,sin
Continue!Reactor
More on One-Speed Diffusion
CubeCube SphereSphere
r
Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).
10
HW 23HW 23Infinite planer source in an infinite Infinite planer source in an infinite medium.medium.
LxeD
SLx /
2)(
D
xS
Ldx
xd )(1)(22
2
x
aa/2
a0/2
Source
)2/cosh(
2/2sinh
2 0
0
La
Lxa
D
SL
HW 24HW 24
More on One-Speed Diffusion
Infinite planer source in a finite Infinite planer source in a finite medium.medium.
Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).
11
More on One-Speed Diffusion
Infinite planer source in a multi-region medium.Infinite planer source in a multi-region medium.
FiniteInfinite Infinite
BCmore
dx
dD
dx
dD
aa
axax
2/
22
2/
11
21 )2/()2/(
Project 2Project 2
Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).
12
Back to Multiplication Factor
k = fp, leaknoneff P
k
k
leaknoneff Pfk
• Fast from thermal,• Fast from fast, .• Thermal from fast, p.• Thermal available for fission
Thinking QUIZThinking QUIZ• For each thermal neutron absorbed, how many fast neutrons are produced?
i
fa
ii )()(1
poisona
eratora
clada
fuela
fuelaf
mod
Recall:
Things to be used later…!