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Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
1
)(),(),(),(),(,1
\\ rDrrrrv gsgaggsgfg
g
Multi-group Model
),()(),()(),()(
),()(),()(),(1
11 \
\\
\
\\\
trrDtrrtrr
Strrtrrtrtv
gggsggag
extg
G
gggsg
G
ggfgggg
g
Calculate group-averaged:
Or for,
we need group-averaged )(),( rr sggrg
),()(),()(),()(
),()(),()(),(1
11 \
\\
\
\\\
trrDtrrtrr
Strrtrrtrtv
gggsgggrg
extg
G
gggsg
G
ggfgggg
g
Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
2
Multi-group Model• Group-averaged parameters?• ENDF.
• Integrate term by term over groups and equate to equation of multi-group model.
),,(),(
),,(),(),,(),(
),,(),(
),,(),()()(),,(
)(
1
0
\\\
\
0
\\\
tErErD
tErErtErEr
SdEtErEEr
dEtErErEEt
tEr
Ev
sa
exts
f
�
Units!
Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
3
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 Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
4
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 Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
5
Multi-group Model
dEtErErtrrg
g
E
E
agag
1
),,(),(),()(
dEtEr
dEtErEr
rg
g
g
g
E
E
E
E
a
ag
1
1
),,(
),,(),(
)(
Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
6
Multi-group Model
dEtErErtrrg
g
E
E
sgsg
1
),,(),(),()(
dEtEr
dEtErEr
rg
g
g
g
E
E
E
E
s
sg
1
1
),,(
),,(),(
)(
Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
7
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
\\\
1
\
1 1\
\
1
\
1\
\
1
\
\\
),,(),(
),,(),(
),,(),(),()(
1 1\
\\
\\\\ ),,()(
),(
1)(
g
g
g
g
E
E
E
E
s
ggsg
dEdEtErEEtr
r
Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
8
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\
\
1
1
\
\\\
),,()()(
),,()()(
),,()()()(
),,()()()(),()(
1\
\\
\\\\\\ ),,()()(
),(
1)(
g
g
E
E
f
gfgg
dEtErEEtr
r
1
)(g
g
E
E
g dEE
Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
9
Multi-group Model
ENDF
High G, few mesh
points.
Small G, more mesh
points.
Poison, burnup (or better consumption), temperature, control rod position, etc…
Flux
Flux
Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
10
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 Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
11
Multi-group ModelCriticalityCriticality
G
ggfggg
g
ggg
ggsggrggg
trrK
trrtrrtrrD
1
1
1
\
\\\
\
\
\\
),()(1
),()(),()(),()(
Fk
M1
No upscatter
Not only sinks
Not all sources, only fission.
Iterations.Iterations.
),()(),()(
),()(),()(),(1
\
\
\\
\
\\\
11
trrDtrr
Strrtrrtrtv
gggrg
extg
G
ggg
ggsg
G
ggfgggg
g
Redundant when no upscatter.
Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
12
Multi-group Model
3
2
1
332313
2212
11
0
00
rss
rs
r
D
D
D
M
333223113
332222112
331221111
fff
fff
fff
F
No upscatter
Iterations.Iterations.
Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
13
Multi-group Model
3
2
1
3323
2212
11
0
0
00
rs
rs
r
D
D
D
M
333223113
332222112
331221111
fff
fff
fff
F
No upscatter
Directly coupled
Iterations.Iterations.
Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
14
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 Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
15
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 Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
16
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 Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
17
Multi-group Model
),()(),()(),()(
),()(),()(),(1
11 \
\\
\
\\\
trrDtrrtrr
Strrtrrtrtv
gggsggag
extg
G
gggsg
G
ggfgggg
g
),()(),()(
),()(),(1
trrDtrr
Strrtrtv
a
extf
Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
18
Multi-group Model
Project 3Project 3
Work out the multi-group to two-group multi-group to two-group collapsing and investigate criticality.
Write down the appropriate matrices.
Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
19
Poisoning
Saturates
135Xe106 b
Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
20
Poisoning
Continuously accumulates
149Sm105 b
Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
21
Poisoning• Not anticipated! Reactor shut down! Time scale:Time scale:
Hours and days.Hours and days.135Xe106 b
149Sm105 b
XeI
Ia
Xea
mXe
a
Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
22
HW 28HW 28
Poisoning
eratora
clada
fuela
poisona
poisona
eratora
clada
fuela
fuela
eratora
clada
fuela
fuela
f
f
kk
k
mod12
mod2
mod1
that Show
(critical)
reactor). (Infinite use uslet ,1
Reactivity
Negative reactivity due to poison buildup. It is proportional to the amount of poison.
Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
23
Poisoning
),(),(),(),(),(),(
),(),(),(),(),(
trtrXetrXetrItrt
trXe
trtrItrItrt
trI
XeaXeIfXe
IaIfI
Initial conditions?Initial conditions?• Clean Core Startup.Clean Core Startup.• Shutdown (later).
constant.)0()( assume uslet and
Fuel.Fresh 0)0()0(
t
XeI
small
Assume no spacial dependence.
Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
24
Poisoning
)(
)1()(
)(
)(
0
0
)(
0
0
0
0
ttXeaIXe
fI
tXeaXe
fXeI
IXeaXe
XeaXe
ee
etXe
)1()( 0 t
I
fI IetI
HW 29HW 29 Show that:
and
)(I)(Xe
Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
25
Poisoning
)(I)(Xe
eratora
clada
fuela
Xea
eratora
clada
fuela
poisona tXet
modmod
)()(
• Now, we know Xe(t)
Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
26
Poisoning• Shutdown. Shutdown. After the reactor has been operating for a “long” time.
.0)0()(
)()0(
)()0(
t
XeXe
II
),(),(),(
),(),(
trXetrIt
trXe
trIt
trI
XeI
I
),(),(),(),(),(),(
),(),(),(),(),(
trtrXetrXetrItrt
trXe
trtrItrItrt
trI
XeaXeIfXe
IaIfI
Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
27
PoisoningHW 30HW 30 Show that
)()(
)()(
)()(
tt
XeI
It
t
IXeXe
I
eeI
eXetXe
eItI
Height of the peak depends on I() and Xe(), i.e. depends on .
> 0 ?
Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
28
Poisoning
If, the available excess reactivity can compensate for less than 30 minutes of poison buildup, can’t startup again after ~30 minutes of shutdown, because you can’t achieve criticality. Need to wait some 40 hours (in this case) for Xe to decay down.
Shutdown Xe negative try to add positive reactivity move control rods out need to have enough reserve costly to do that.
Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
29
Poisoning
Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
30
PoisoningStrategiesStrategies• If you plan to shut down for “short maintenance”, think about stepback.• Examine different scenarios using this code from
http://www.nuceng.ca/ • Prepare your own report, code, calculations, graphs, comments, conclusions etc…..• Be creative and use whatever experience you gained during your study in this program.
Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
31
PoisoningXe OscillationsXe Oscillations• (r,t) (spacial dependence) flux locally Xe burnup (reactivity) flux further control rods globally in flux elsewhere Xe burnup ….. Xe oscillation but limited by opposite effect due to increase (decrease) of I in the high (low) flux region.• In large reactors (compared to neutron diffusion length) local flux, power and temperature could reach unacceptable values for certain materials safety issues. • Think of one sensor and one control rod feel average flux apparently OK more sensors and control rods to locate and deal with local changes.
Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
32
Poisoning
Permanent PoisonsPermanent Poisons• 149Sm has sizeable but lower cross section than 135Xe.• It does not decay.
• Accumulates with time.• Consequences?????????
....................).........,(),(),(
trtrt
trSmfSm
????