Transcript Document
Two-Group Neutron Diffusion
Homogeneous system Determinant of coefficients matrix = 0
a
1
D
1
B
2
a
1
k
a
2
a
2
D
2
B
2 0 (
a
1
D
1
B
2 )(
a
2
D
2
B
2 )
k
a
2
a
1 0 (
a
1
D
1
B
2 )(
a
2
D
2
B
2 )
k
a
2
a
1 0 1 (
L
2
Fast
B
2 )( 1
L
2
Thermal
B
2 )
k
L
2 1
Fast
( 1
B
2
L
2
Fast
)( 1
B
2
L
2
Thermal
)
k
1
L
2
Thermal
0 0 Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
1
Two-Group Neutron Diffusion
( 1
B
2
L
2
Fast
)( 1
B
2
L
2
Thermal
)
k
0 ( 1
B
2
L
2
Fast k
)( 1
B
2
L
2
Thermal
) 1
k eff
P Fast non
leak P Thermal non
leak k
For large reactors 1
B
2
L
2
Thermal
1 1
B
2 (
L
2
k
Fast L
2
Thermal
) 1
B
2
k
M
2 1 1
B
2
L
2
Fast
1 Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
2
M
2
Two-Group Neutron Diffusion
L
2
Thermal
L
2
Fast
If any leakage .
2
L Thermal
D
a
3
tr
a
3
a
1
tr
Fermi age 2
L Fast
3
s n
tr
Slowing down density.
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
3
Reactor Model: One-Group
• Before considering multi-group.
• So far we did 1-D.
• Let us consider one-group but extend to 3-D.
HW 22 |
For the homogeneous infinite slab reactor, extend the criticality condition that you found in HW 22.
B g
2
a
0 2
B m
2
k
L
2 1
f D
a d
z
Reactor
a/2 a a 0 /2
1-D
d
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
4
x
Reactor Model: One-Group
• In 3-D
d
2 (
x
)
B
2 (
x
) 0
dx
2 2
x
2 2
y
2 2
z
2
B
2 0
f
a D
0 cos
Bx
0 cos
B x x
cos
B y y
cos
B z z B g
2
a
0 2
B m
2
B g
2
k
L
2 1
B x
2
B y
2
f
a
B z
2
D
a
0 2
b
0 2
c
0 2
B m
2
k
L
2 1
f D
a
Critical dimensions (size), for the given material properties, predicted by the model.
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
5
Reactor Model: One-Group
1
v
• Transient case.
t
(
r
,
t
)
t
!
S
(
r
,
t
)
a
(
r
) (
r
,
t
)
D
(
r
) (
r
,
t
)
t
!
a fuel
Moderator, structure, coolant, fuel, …
f fuel
fuel
• Delayed neutrons!!
• For homogeneous 1-D: 1
v
t
(
x
,
t
)
S
(
x
,
t
)
a
(
x
,
t
)
D
2
x
2 (
x
,
t
) Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
f
(
x
,
t
) 6
Reactor Model: One-Group
1
v
t
(
x
,
t
)
f
(
x
,
t
)
a
(
x
,
t
)
D
2
x
2 (
x
,
t
)
HW 26
1
T
Separation of variables: 1
v
T
t
f
(
x
,
t
)
T
a
(
x
)
T
(
t
)
T
DT
T
t
v
D
2
x
2 (
f
a
) 2
x
2 constant = 0 for steady state.
Show that
T
(
t
)
T
( 0 )
e
t
,
d
2
dx
2 Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
B
2 0 ,
v
(
a
DB
2
f
) 7
Reactor Model: One-Group
HW 26 (continued)
(
a
0 ) 2 0 try eigenvalues
n
(
x
)
n
cos
B n x
v
(
a
DB n
2
B n
2
f
)
n
a
0 2 Solution (
x
,
t
)
n
odd
?
A n e
n t
cos
n
x a
0 ?
Initial condition Show that
A n
( 2
a
0
x
, 0 )
a
2 0
a
2 0 (
n
odd x
, 0 )
A n
cos cos
n
x a
0
n
x
a
0
dx
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
8
Reactor Model: One-Group
B n
2
n
v
(
n
a
0
a
2
DB n
2
B
1 2
B
2 2
f
)
B
3 2 ...
2 1 2 2 2 3 ...
1
v
(
a
DB
1 2
f
) Slowest decaying eigenvalue .
(
x
,
t
)
A
1
e
1
t
cos
x a
0
A
1
e
1
t
cos
B
1
x
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
9
Reactor Model: One-Group
For steady state
Criticality
B
1 2 1
B g
2
v
(
a
DB
1 2
f D
a
f B m
2 ) 0 1 0
Super criticality
B g
2
B m
2
LE
1 0
Sub criticality
B g
2
B m
2
LE
1 0 Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
10
Reactor Model: One-Group
1
v
• That was for the bare slab reactor.
• What about more general bare reactor models?
t
(
r
,
t
)
S
(
r
,
t
)
a
(
r
) (
r
,
t
)
D
(
r
) (
r
,
t
) • For steady state, homogeneous model: 2 (
r
,
t
)
f D
a
(
r
,
t
) 2 (
r
,
t
)
k
L
2 1 (
r
,
t
) 0 • BC:
(extrapolated boundary) = 0.
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
11
Reactor Model: One-Group
•
R 0 , H 0
1
r
r
are the extrapolated dimensions.
r
dr
dz
2 2
B
2 0
R
• BC’s: (
R
0 ,
z
) 0 (
r
,
H
0 2 ) 0 Reactor
H
• Let (
r
,
z
) (
r Bessel
) (
z cos
) •
Solve the problem and discuss criticality condition.
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
Project 3
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