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Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
1
Multi-group Model• Wide neutron spectrum.• One-group, two-group? Should be generalized.
),()(),()(),()(
),()(),()(),(1
11 \
\\
\
\\\
trrDtrrtrr
Strrtrrtrtv
gggsggag
extg
G
gggsg
G
ggfgggg
g
Identify the terms, NOW.
Fraction of an eV
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
2
Multi-group Model
),()(),()(),()(
),()(),()(),(1
11 \
\\
\
\\\
trrDtrrtrr
Strrtrrtrtv
gggsggag
extg
G
gggsg
G
ggfgggg
g
Total fission
Fra
ctio
n
Scattering in
Oth
er s
ourc
es
LeakageScattering outAbsorption
Fraction of an eV
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
3
Multi-group Model
5-group example.5-group example.
Maxwellian
1/EFission
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
4
Multi-group Model
111222333444555
5
1\
\\\ ),()(
fffffg
ggfggg trr
Total fission
Thermal fission (~ 97%)
Fast fission (~ 3%)
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
5
Multi-group Model
5544332211
5
1\
\\ ),()(
gsgsgsgsgs
gggsg
trr
Scattering in
Upscattering!!???
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
6
Multi-group Model
gsggsggsggsggsg
gsg trr
54321
),()(
Scattering out
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
7
Multi-group Model
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5
133
3
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),()(),(1
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ext
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Group 3
Removal cross section
5
31
33
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\
\
gg
gsa
ssssa
ssar
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
8
),()(),()(),()(
),()(),()(),(1
11 \
\\
\
\\\
trrDtrrtrr
Strrtrrtrtv
gggsgggrg
extg
G
gggsg
G
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g
Multi-group Model
Total fissionFra
ctio
nScattering in O
ther
sou
rces
LeakageRemoval In-group Scattering
),()(),()(
),()(),()(),(1
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11
trrDtrr
Strrtrrtrtv
gggrg
extg
G
ggg
ggsg
G
ggfgggg
g
Net Scattering in
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
9
)(),(),(),(),(,,1
\\ rDrrrrv gsgaggsgfggg
Multi-group Model
),()(),()(),()(
),()(),()(),(1
11 \
\\
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\\\
trrDtrrtrr
Strrtrrtrtv
gggsggag
extg
G
gggsg
G
ggfgggg
g
Calculate group-averaged:
Or for,
we need group-averaged )(),( rr sggrg
),()(),()(),()(
),()(),()(),(1
11 \
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\\\
trrDtrrtrr
Strrtrrtrtv
gggsgggrg
extg
G
gggsg
G
ggfgggg
g
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
10
Multi-group Model• Group-averaged parameters?• ENDF.
• Integrate term by term over groups and equate to equation of multi-group model.
),,(),(
),,(),(),,(),(
),,(),(
),,(),()()(),,(
)(
1
0
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tErErD
tErErtErEr
SdEtErEEr
dEtErErEEt
tEr
Ev
sa
exts
f
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Units!
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
11
Multi-group Model
1
),,()(
1),(),(
1 g
g
E
Eg
gg
g
dEtErEvtv
tr
ttr
tv
1
),,(),(g
g
E
E
g dEtErtr • Define group flux
1
1
),,(
),,()(
1
1g
g
g
g
E
E
E
E
gdEtEr
dEtErEv
v
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
12
Multi-group Model
1
),,(),(),()(g
g
E
E
gg dEtErErDtrrD
1
1
),,(
),,(),(
)(g
g
g
g
E
E
E
E
g
dEtEr
dEtErErD
rD
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
13
Multi-group Model
dEtErErtrrg
g
E
E
agag
1
),,(),(),()(
dEtEr
dEtErEr
rg
g
g
g
E
E
E
E
a
ag
1
1
),,(
),,(),(
)(
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
14
Multi-group Model
dEtErErtrrg
g
E
E
sgsg
1
),,(),(),()(
dEtEr
dEtErEr
rg
g
g
g
E
E
E
E
s
sg
1
1
),,(
),,(),(
)(
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
15
Multi-group Model
G
g
E
E
E
E
s
E
E
G
g
E
E
s
E
E
s
G
gggsg
g
g
g
g
g
g
g
g
g
g
dEdEtErEEr
dEdEtErEEr
dEdEtErEErtrr
1
\\\
1
\\\
0
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1
\
1 1\
\
1
\
1\
\
1
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),,(),(
),,(),(
),,(),(),()(
1 1\
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),(
1)(
g
g
g
g
E
E
E
E
s
ggsg
dEdEtErEEtr
r
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
16
Multi-group Model
G
g
E
E
fg
fg
f
E
E
E
E
f
G
ggfggg
g
g
g
g
g
g
dEtErEE
dEtErEE
dEtErEEdEE
dEdEtErEEEtrr
1
\\\\
\
0
\\\
\
0
\\\
\
0
\\\
1
\
1\
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1
1
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\\\
),,()()(
),,()()(
),,()()()(
),,()()()(),()(
1\
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),(
1)(
g
g
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E
f
gfgg
dEtErEEtr
r
1
)(g
g
E
E
g dEE
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
17
Multi-group Model
ENDF
High G, few mesh
points.
Small G, more mesh
points.
Poison, burnup, temperature, control rod position, etc… Flux
Flux
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
18
Multi-group ModelWhat could make life a little easier?!• No upscattering
set group G to include neutrons up to ~1 eV.
• No group skipping when scattering down (directly coupled).
.for 0)( \\ ggrgsg
),()(),()(),()(1
11 \
\\
\
\\ trrtrrtrr gsgg
g
gggsg
G
gggsg
Your choice of how to tackle in-scattering.
),()(),()(),()( 1)1(1\
\\ trrtrrtrr gsgggggs
G
gggsg
HW 27HW 27 How can we pledge this? What about H?
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
19
Multi-group ModelCriticalityCriticality
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ggfggg
g
ggg
ggsggrggg
trrK
trrtrrtrrD
1
1
1
\
\\\
\
\
\\
),()(1
),()(),()(),()(
Fk
M1
No upscatterNot only sinks
Not all sources, only fission.
Iterations.Iterations.
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
20
Multi-group Model
3
2
1
332313
2212
11
0
00
rss
rs
r
D
D
D
M
333223113
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fff
fff
fff
F
No upscatter
Iterations.Iterations.
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
21
Multi-group Model
3
2
1
3323
2212
11
0
0
00
rs
rs
r
D
D
D
M
333223113
332222112
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fff
fff
fff
F
No upscatter
Directly coupled
Iterations.Iterations.
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
22
Multi-group ModelMulti-group Multi-group one-group one-group
0
),,(),(),,(),(1
dEtErtrdEtErtrg
g
E
E
g
0
0
),,(
),,()(
1
1
),,(
),,()(
1
11
1
dEtEr
dEtErEv
vdEtEr
dEtErEv
v g
g
g
g
E
E
E
E
g
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
23
Multi-group Model
0
0
),,(
),,(),(
)(
),,(
),,(),(
)(1
1
dEtEr
dEtErErD
rD
dEtEr
dEtErErD
rDg
g
g
g
E
E
E
E
g
dEtEr
dEtErEr
r
dEtEr
dEtErEr
ra
aE
E
E
E
a
ag g
g
g
g
0
0
),,(
),,(),(
)(
),,(
),,(),(
)(1
1
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
24
Multi-group Model
1 when 0),()(),()(1\
\\
Gtrrtrr gsg
G
gggsg
1)()(0
1
dEEdEEg
g
E
E
g
1 when ),()(),()(1\
\\\
Gtrrtrr f
G
ggfgg
Substituting all of the above into
yields
which is the one-group diffusion equation.
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
25
Multi-group Model
),()(),()(),()(
),()(),()(),(1
11 \
\\
\
\\\
trrDtrrtrr
Strrtrrtrtv
gggsggag
extg
G
gggsg
G
ggfgggg
g
),()(),()(
),()(),(1
trrDtrr
Strrtrtv
a
extf
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
26
Multi-group Model
Project 4Project 4Work out the multi-group to two-group multi-group to two-group collapsing
and investigate criticality.
Very important!!!
Happy Eid.
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