Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
1
Steady State Diffusion Equation
HW 20HW 20
Study example 5.3 and solve problem 5.8 in Lamarsh.
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
2
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.• 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!!
dx
dD
dx
dDJJ B
BA
ABA
x
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
3
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:(1)absorption
(1)leakage
)( rate absorptionneutron
rate productionneutron
A
(S)k
)( rate leakageneutron )( rate absorptionneutron
)( rate productionneutron
LEA
Skeff
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
4
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
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
5
More on One-Speed DiffusionHW 21HW 21
Show that for a critical homogeneous reactorcritical homogeneous reactor
DBDLBP
a
a
a
aleaknon 2222 1
1
Infinite Slab Reactor (one-speed diffusion)Infinite Slab Reactor (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, 2007-2008 (Saed Dababneh).
6
HW 22HW 22
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, 2007-2008 (Saed Dababneh).
7
,...2
5,
2
3,
2)
2( 0
a
BHW 22 HW 22 (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, 2007-2008 (Saed Dababneh).
8
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.
xr
r0
0
2 22
2
B
dr
d
rdr
dHW 23HW 23
Brr
CBr
r
Asincos
Br
r
r
r
C 00
,sin
Continue!
Reactor
More on One-Speed Diffusion
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
9
HW 24HW 24Infinite 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 25HW 25
More on One-Speed Diffusion
Infinite planer source in a finite Infinite planer source in a finite medium.medium.
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
10
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, 2007-2008 (Saed Dababneh).
11
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
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
12
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).
13
Two-Group Neutron Diffusion
ThermaldErEr
FastdErEr
eV
MeV
eV
1
0
2
10
1
1
),()(
),()(
221122
212
1
222111
aa
ffeff DD
k
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
14
)()()(0 12
1111 rDrrS a
Two-Group Neutron Diffusion
Removal cross section = fission + capture + scattering to group 2
Depends on thermal flux.
Fast diffusion coefficient
)()()()(0 12
1112211 rDrrr aff
)()()(0 12
11122 rDrrk
aa
oror
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
15
Two-Group Neutron Diffusion
)()()(0 22
2222 rDrrS a
Thermal diffusion coefficient
Thermal absorption cross section = fission
+ capture.
)()()(0 22
222121 rDrr as
Depends on fast flux.
)()()(0 22
22211 rDrr aa
oror
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
16
Two-Group Neutron Diffusion
)()()(0 12
11122 rDrrk
aa
)()()(0 22
22211 rDrr aa
• A coupled system of equations; both depend on both fluxes.• For a critical, steady state system:
0)()(
0)()(
22
22
12
12
rBr
rBr
Geometrical
Review Cramer’s
rule!