nuclear reactors, bau, 1st semester, 2007-2008 (saed dababneh). 1 hw 14 more on moderators calculate...
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Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
1
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.
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
2
More on Moderators
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
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
3
More on Moderators
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.
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
4
More on Moderators
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).
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
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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
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
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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
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
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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
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
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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
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
9
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
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
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
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For steady state operation
0)()()()( rSrrrJ a
For non-spacial dependence
)()()( ttStnt a
Delayed sources?
Equation of Continuity
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
12
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??
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
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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”.
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (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(assuming no
buildup)
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
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Fick’s Law
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
16
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