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HW 14HW 14
More on Moderators
Calculate the moderating power and ratio for pure D2O as well as for D2O contaminated with a) 0.25% and b) 1% H2O.Comment on the results.In CANDU systems there is a need for heavy water upgradors.
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u u
n n0 01 2 3 4 5 6 7 1 2 3
slowing down in large massnumber material
slowing down in hydrogeneousmaterial
continuous slowing-down model
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1
1ln
2
)1(1ln
2
\
A
A
A
A
E
Eu
av
ContinuousContinuous slowing down model or Fermi model. slowing down model or Fermi model.
• The scattering of neutrons is isotropic in the CM system, thus is independent on neutron energy. also represents the average increase in lethargy per collision, i.e. after n collisions the neutron lethargy will be increased by n units.
• Materials of low mass number is large Fermi model is inapplicable.
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Moderator-to-fuel ratio Moderator-to-fuel ratio Nm/Nu.• Ratio leakage a of the moderator f .• Ratio slowing down time p leakage .
• Water moderated reactors, for example, should be under moderated.• T ratio (why).
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One-Speed Interactions• Particular general.Recall:• Neutrons don’t have a chance to interact with each other (review test!) Simultaneous beams, different intensities, same energy:
Ft = t (IA + IB + IC + …) = t (nA + nB + nC + …)v• In a reactor, if neutrons are moving in all directions n = nA + nB + nC + …
Rt = t nv = t
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drn ),(
r
d
Neutrons per cm3 at
r whose velocity vector lies within d about .
4
),()( drnrn
• Same argument as before vdrnrdI ),(),(
)()(),(),()()(
),(),(
4
rrnvdrnvrdFrFrR
rdIrdF
ttt
t
One-Speed Interactions
drnvr ),()(4
where
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Multiple Energy Interactions
dEdErn ),,(
Neutrons per cm3 at r with energy interval (E, E+dE) whose velocity vector lies within d about .
• Generalize to include energy
4
),,(),( dEdErndEErn
0 4
),,()(
dEdErnrn
dEErEdEEvErnEdEErR tt ),()()(),()(),(
0
),()()( dEErErR t
Thus knowing the material properties t and the neutron flux as a function of space and energy, we can calculate the interaction rate throughout the reactor.
Scalar
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Neutron Current
• Similarly and so on …
• Redefine as
0
),()()( dEErErR SS
Scalar
vdrnrdI ),(),( dvrnrId
),(),(
drnvr ),()(4
drnvJ ),(
4
Neutron current densityNeutron current density
J• From larger flux to smaller flux!
• Neutrons are not pushed!• More scattering in one direction than in the other.
xJxJ ˆ
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4
cos),(ˆ dvrnJxJ xx
Net flow of neutrons per second per unit area normal to the x direction:
In general: nJnJ ˆ
Equation of ContinuityEquation of Continuity
A
a dAntrJdtrrdtrSdtrnt
ˆ),(),()(),(),(
Rate of change in neutron density
Production rate
Absorption rate
“Leakage in/out” rate
Volume Source distribution
function
Surface area
bounding
Normal to A
Equation of Continuity
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Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
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Using Gauss’ Divergence Theorem S V
rdBAdB 3
dtrJdAntrJA
),(ˆ),(
A
a dAntrJdtrrdtrSdtrnt
ˆ),(),()(),(),(
),(),()(),(),(1
trJtrrtrStrtv a
Equation of Continuity
Equation of ContinuityEquation of Continuity
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For steady state operation
0)()()()( rSrrrJ a
For non-spacial dependence
)()()( ttStnt a
Delayed sources?
Equation of Continuity
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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.
)(rnot
Restrictive! Applicability??
Restrictive! Applicability??
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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”.
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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(assuming no
buildup)
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Fick’s Law
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Fick’s Law
2
0
2/
0 0
sincos)(4 r
rzszz ddrder
dAdAJ t
HW 15HW 15
023
zJJJ
t
szzz
?
zz dAJ
and show that
and generalize23 t
sDDJ
Diffusion Diffusion coefficientcoefficient
Fick’s law
Fick’s law
The current density is proportional to the negative of the gradient of the neutron flux.
s
D
3
1