nuclear reactor theory, ju, first semester, 2010-2011 (saed dababneh). 1 reactor model: one-group...
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Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
1
Reactor Model: One-Group• That was for the bare slab reactor.• What about more general bare reactor models?
• For steady state, homogeneous model:
• BC: (extrapolated boundary) = 0.
),()(),()(),(),(1
trrDtrrtrStrtv a
0),(1
),(),(),(2
22
trL
ktrtr
Dtr af
• R0, H0 are the extrapolated dimensions.
• BC’s:
• Let
• Solve the problem and discuss criticality condition.Solve the problem and discuss criticality condition.
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
2
Reactor Model: One-Group
01 2
2
2
Bdzdr
rrr
0),(
0),(
2
0
0
Hr
zR
)()(),( zrzr cosBesselHW 26HW 26
Reactor
R
H
z
xy
r
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
3
Reactor Model: One-Group
Reactor
R
H
• Briefly, we go through HW 26.
01 2
2
2
Bdzdr
rrr
)()(),( zrzr
0
coscos0
2
0
22
2
dr
dr
dr
d
H
zz
dz
d
z
xy
r
)()( 00 rCYrAJ
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
4
Reactor Model: One-Group
0)(0 CxYx n
00 4048.20)4048.2( RJ
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
5
Reactor Model: One-Group
Reactor
R
H
z
xy
r
000 cos)
4048.2(,...)(
H
z
R
rJPA
Criticality condition?
Criticality condition?
Do it.Do it.
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
6
Reactor Model: One-Group
R0
H0/2
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
7
Reflected Slab: One-Group
x
aa/2
Core
zReflected Slab ReactorReflected Slab Reactor
bb
Re
flect
or
Re
flect
or
),()(),()(
),()(),(1
trrDtrr
trrtrtv
a
f
For steady-state, homogeneous, 1-D
0)()(
0)()()(
2
2
2
2
xdx
xdD
xdx
xdD
RRa
RR
CCa
Cf
CC
C
C Core
R Reflector
)2
()2
(),2
()2
(,0)2
(BCsa
Ja
Jaa
ba CRCRR
0)(1
)(2
2 rL
r
Recall:
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
8
Ra
RR
RRR
C
Ca
CfC
mCm
CC
DL
L
xba
A
DBxBAx
2
2
)(2sinh
)()cos()(
Verify.
RR
R
RCmCC
mC
RR
CmC
L
bA
L
DaBABD
L
bA
aBA
cosh)2
sin(
sinh)2
cos(
BC
RR
RCmC
mC
L
b
L
DaBBD coth)
2tan(
Reflected Slab: One-Group
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
9
RR
RCmC
mC
L
b
L
DaBBD coth)
2tan(Criticality condition.Criticality condition.
• For bare slab CC was / 2.• Smaller core for reflected reactor (even with a0 > a).• Save fuel.
Reflected Slab: One-Group
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
10
Criticality “Calculation”• Can we solve “real” reactor problems analytically?• The previous discussion provides understanding of the concepts but also indicates the need for computational techniques.computational techniques.
• Assume:
• Adjust parameters so that = 0 (Steady-state).
• What parameters and how to adjust them?
),()(),()(),()(),(1
trrDtrrtrrtrtv af
)(),( retr t
)()()()()()()( rrDrrrrrv af
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
11
)()()()()()()( rrDrrrrrv af
Criticality “Calculation”
• Fixed design and geometry one free variable is k
• As we did earlier (be guided by HW 20):As we did earlier (be guided by HW 20):
)()()()()()( rrk
rrrrD ffudge
a
221 LBDk af
a
ffudge
operators are ,1
FMFK
Mfudge
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
12
Criticality “Calculation”
221 LBk af
fudge
FK
Mfudge
1
• Build an algorithm.• “Guess” (reasonably) initial kfudge and (or ) for the zeroth iteration.• Calculate the initial source term.• Iterate:
converges.flux until .....on so and
get 1
. and Guess
0
101
11
10
00
01
00
S
Skk
FS
k
SF
kM
k
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
13
Criticality “Calculation”• Or:
• If for example k > 1, take action to reduce source or increase absorption.• How?How?
volume
ii
volume
i
volume
i
volume
i
i
dVSk
dVS
dVM
dVF
k
M
Fk
1
sinks
sourcesfission
1
1
1
1
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
14
Reactor Kinetics Reactor Kinetics Reactor kinetics refers to the manipulation of parameters that affect k and to the subsequent direct response of the reactor system. Examples are:
• Absorber rods or shim movements to compensate for fuel burnup. • Safety scram rods to rapidly shutdown the chain reaction. • Control rods to provide real-time control to keep k = 1 or to maneuver up and down in power.• …..
Reactor Dynamics Reactor Dynamics Reactor dynamics refers to the more indirect feedback mechanisms due to power level effects and other overall system effects such as:
• Temperature feedback. • VoidVoid feedback. • Pump speed control (affects water density and temperature). • …
How to Adjust Criticality
Negative or positive reactivity.reactivity.
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
15
How to Adjust Criticality
Before all:Before all:
Core Design Core Design The transient response of the reactor to the above direct and indirect changes in basic parameters is highly dependent on the design details of the reactor. Sample issues are:
• Where should the control rods be placed for maximum effectiveness? • Will the power go up or down if a void is introduced into the reactor? • Will the power go up or down if core temperature goes up? • How often should the reactor be refueled? • and so on...
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
16
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
1
),,(),(g
g
E
E
g dEtErtr
Flux-averaged quantities.
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
17
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 Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
18
Multi-group Model
5-group example.5-group example.
Maxwellian
1/EFission
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
19
Multi-group Model
111222333444555
5
1\
\\\ ),()(
fffffg
ggfggg trr
Total fission
Thermal fission (~ 97%)
Fast fission (~ 3%)
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
20
Multi-group Model
5544332211
5
1\
\\ ),()(
gsgsgsgsgs
gggsg
trr
Scattering in
Upscattering!!???
3g
Skipping!!???
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
21
Multi-group Model
gsggsggsggsggsg
gsg trr
54321
),()(
Scattering out
3g
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
22
Multi-group Model
553443333223113
335334333332331
3333
3
5
133
3
),()(),()(
),()(),(1
\
\\\
sssss
sssss
a
ext
ggfgg
trrDtrr
Strrtrtv
Group 3
Removal cross section
5
31
33
353432313
33333
\
\
\
gg
gsa
ssssa
ssar
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
23
),()(),()(),()(
),()(),()(),(1
11 \
\\
\
\\\
trrDtrrtrr
Strrtrrtrtv
gggsgggrg
extg
G
gggsg
G
ggfgggg
g
Multi-group Model
Total fissionFra
ctio
nScattering in O
ther
sou
rces
LeakageRemoval In-group Scattering
),()(),()(
),()(),()(),(1
\
\
\\
\
\\\
11
trrDtrr
Strrtrrtrtv
gggrg
extg
G
ggg
ggsg
G
ggfgggg
g
Net Scattering in