Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh). 1 • The exact interpretation of neutron transport in heterogeneous domains is so complex. • Assumptions and approximations. • Simplified approaches. • Simplified but accurate enough to give an estimate estimate of the average characteristics average characteristics of neutron population neutron population. • Numerical solutions. • Monte Carlo techniques. Fick’s Law MCNP MCNP
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The exact interpretation of neutron transport in heterogeneous domains is so complex.
Fick’s Law. The exact interpretation of neutron transport in heterogeneous domains is so complex. Assumptions and approximations. Simplified approaches. Simplified but accurate enough to give an estimate of the average characteristics of neutron population . Numerical solutions. - PowerPoint PPT Presentation
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Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
1
• The exact interpretation of neutron transport in heterogeneous domains is so complex.• Assumptions and approximations.• Simplified approaches.• Simplified but accurate enough to give an estimateestimate of the average characteristics average characteristics of neutron populationneutron population.• Numerical solutions.• Monte Carlo techniques.
Fick’s Law
MCNPMCNP
Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
2
Fick’s LawAssumptions:1.The medium is infinite.2.The medium is uniform 3.There are no neutron sources in the medium.4.Scattering is isotropic in the lab coordinate system.5.The neutron flux is a slowly varying function of position.6.The neutron flux is not a function of time.
Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
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Lamarsh puts it more bluntly:“Fick’s Law is invalid: a) in a medium that strongly absorbs neutrons; b) within three mean free paths of either a neutron source or the surface of a material; and c) when neutron scattering is strongly anisotropic.”
Fick’s Law
Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
4
Fick’s Law
Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
5
Fick’s LawCurrent Jx
x
Con
cent
ratio
n C
dC/dx
x
(x)
High flux
More collisions
Low flux
Less collisions
Negative Flux GradientCurrent Jx
• Diffusion: random walk of an ensemble of particles from region of high “concentration” to region of small “concentration”.• Flow is proportional to the negative gradient of the “concentration”.
xDJ x
• From larger flux to smaller flux!• Neutrons are not pushed!• More scattering in one direction than in the other.
Recall:
Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
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x
y
z
rdAz
Fick’s Law
der
dAr rz
st
24
cos)(
Number of neutrons scatteredscattered per second from d at rr and going through dAz
Slowly varying)(rnot ss
Isotropic
Removed en route
(assuming no buildup)
d
Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
7
Fick’s Law
Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
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Fick’s Law
2
0
2/
0 0
sincos)(4 r
rzszz ddrder
dAdAJ t
HW 14HW 14
023
zJJJ
t
szzz
?
zz dAJ
and show that
and generalize23 t
sDDJ
Diffusion Diffusion coefficientcoefficient
Fick’s lawFick’s law
The current density is proportional to the negative of the gradient of the neutron flux.
s
D
3
1
Total removal
Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
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Fick’s LawValidity:1. The medium is infinite. Integration over all space. after few mean free paths 0 corrections at the surface are still required. 2. The medium is uniform.
and are functions of space re-derivation of Fick’s law? locally larger s extra JJ cancelled by iff ???
Note: assumption 5 is also violated!3. There are no neutron sources in the medium.
Again, sources are few mean free paths away and corrections otherwise.
rte
)(rnot ss
)(rs
rr sat ee )( HW 15HW 15
Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
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Fick’s Law4. Scattering is isotropic in the lab. coordinate system.
If reevaluate D.
For “practical” moderators:
5. The flux is a slowly varying function of position.a variation in .
03
2)(cos
A
33
1
)(3
1 tr
trst
D
1
str
)(2
2
rr
HW 16HW 16
Weekly absorbing t = s.
Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
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Estimate the diffusion coefficient of graphite at 1 eV.
The scattering cross section of carbon at 1 eV is 4.8 b.
Fick’s LawHW 17HW 17
Scattering
Absorption
Other materials?
Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
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Fick’s Law6. The neutron flux is not a function of time.
Time needed for a thermal neutron to traverse 3 mean free paths 1 x 10-3 s (How?).If flux changes by 10% per second!!!!!!
Very small fractional change during the time needed for the neutron to travel this “significant” distance.
43
1
101101.011
/
xxmss
ms
DJ
Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
13
Back to the Continuity Equation
),(),()(),(),(1
trJtrrtrStrtv a
),(),()(),(),(1
trDtrrtrStrtv a
Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
14
The Diffusion Equation
),(),()(),(),(1
trDtrrtrStrtv a
If D is independent of r (uniform medium)
),(),()(),(),(1 2 trDtrrtrStr
tv a
)()()()(0 2 rDrrrS a
or scalar Helmholtz equation.
)()()(0 2 rDrra
Buckling equation.
Laplacian
Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
15
)()()()(0 2 rDrrrS a
a
DL
2Define L Diffusion Length
L2 Diffusion Area Moderation Length
D
S
L
22 1
Steady State Diffusion Equation
01
22
L
Boundary ConditionsBoundary Conditions• Solve DE get .• Solution must satisfy BC’s.• Solution should be real and non-negative.
Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
16
Steady State Diffusion EquationOne-speed neutron diffusion in infinite mediumOne-speed neutron diffusion in infinite medium
Point source
0)(1
)(2
2 rL
r
0)(1
)(2
)(22
2
rL
rdr
d
rr
dr
d HW 18HW 18
r
eC
r
eA
LrLr //
General solution
A, C determined from BC’s.
BC BC r 0 C = 0.
Show that
neutrons per second absorbed in the ring.
Show that
Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
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Steady State Diffusion Equation
r
eA
Lr /
D
SA
4
r
e
D
S Lr /
4
HW 18 HW 18 (continued)(continued)
a
DL
2
rdr
22 6Lr r
adrr 24
Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
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Steady State Diffusion Equation
HW 19HW 19
Study example 5.3 and solve problem 5.8 in Lamarsh.
Multiple Point Sources?
Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
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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.
dx
dD
dx
dDJJ B
BA
ABA
x
Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
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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:absorptionandleakage
)( rate absorptionneutron
rate productionneutron
A
(S)k
)( rate leakageneutron )( rate absorptionneutron
)( rate productionneutron
LEA
Skeff
Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
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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
Recall:
multiplying m
edium
multiplying m
edium
Things to be used later…!
Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
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More realistic multiplying medium0)()( 22 rBr
)(
)(22
r
rB
• The buckling is a measure of extent to which the flux curves or “buckles.”• For a slab reactor, the buckling goes to zero as “a” goes to infinity. There would be no buckling or curvature in a reactor of infinite width. • Buckling can be used to infer leakage. The greater the curvature, the more leakage would be expected.
Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
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More on One-Speed DiffusionHW 20HW 20
Show that for a critical homogeneous reactorcritical homogeneous reactor
DBDLBP
a
a
a
aleaknon 2222 1
1
Infinite Bare Slab Reactor Infinite Bare Slab Reactor (one-speed diffusion)(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.