Neutron reproduction
Neutron Life Cycle
x x 00..99
factork eff = 1.000
Thermal Thermal utilization utilization factor “f”factor “f”
x η
x x 00..99Resonance Resonance
escape escape i i ” ”i i ” ”
x x 11..0303Fast fissionFast fission
probability ”p”probability ”p”What is:• Migration length? Fast fission Fast fission
factor “factor “εε””
g g• Critical size?How does the
t ff t thgeometry affect the reproduction factor?
1Nuclear Reactor Theory, BAU, Second Semester, 2011-2012 (Saed Dababneh).
Neutron Life Cycley
Why should we b t th ?worry about these?
How?f
How?
2Nuclear Reactor Theory, BAU, Second Semester, 2011-2012 (Saed Dababneh).
Controlled Fission
k = fpεη(1-lf )(1-l h l) Not fixed !k fpεη(1 lfast)(1 lthermal)• Thermal utilization factor f can be changed, as an
Not fixed…!
g ,example, by adding absorber to coolant (PWR)(chemical shim, boric acid), or( , ),by inserting movable control rods in & out. Poison.• Reactors can also be controlled by altering neutron y gleakages using movable neutron reflectors.• f and p factors change as fuel is burned.p g• f, p, η change as fertile material is converted to fissilematerial.
Nuclear Reactor Theory, BAU, Second Semester, 2011-2012 (Saed Dababneh).
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Controlled Fission• Attention should be paid also to the fact that reactor power changes occur due to changes in resonance escape probability p. If Fuel T↑, p↓ due to Doppler broadening ofresonance peaks.
U d d tiUnder-moderation and
over-moderationover moderation.
Nuclear Reactor Theory, BAU, Second Semester, 2011-2012 (Saed Dababneh).
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Controlled FissionTime scale for neutron multiplicationTime scale for neutron multiplication• Time constant τ includes moderation time (~10-6 s) and diffusion time of thermal neutrons (~10-3 s).
Time Average number of thermal neutronsTime Average number of thermal neutronst n
t + τ knt + τ knt + 2τ k2n
nkndn• For a short time dtτ
nkndtdn −
=
•• Show thatShow that τtkentn )1(0)( −=
5
0)(Nuclear Reactor Theory, BAU, Second Semester, 2011-2012
(Saed Dababneh).
Controlled Fissionτtkentn )1()( −=
• k = 1 n is constant (Desired).• k < 1 n decays exponentially
entn 0)( =Reactivity.
• k < 1 n decays exponentially.• k > 1 n grows exponentially with time constant τ / (k-1).• k = 1.01 (slightly supercritical..!) e(0.01/0.001)t = e10 = 22026 in in 11s. s. ( g y p )• Design the reactor to be slightly subcritical for prompt neutrons.• The “few” “delayed” neutrons will be used to achieve criticality, allowing enough time toallowing enough time tomanipulate the controlrods (or use shim or …).
6
( )Cd control rodsCd control rods
Nuclear Reactor Theory, BAU, Second Semester, 2011-2012 (Saed Dababneh).
Fission ReactorsEssential elements:Essential elements:• Fuel [fissile (or fissionable) material].• Moderator (not in reactors using fast neutrons). Core
• Reflector (to reduce leakage and critical size).• Containment vessel (to prevent leakage of waste).• Core catcher.• Shielding (for neutrons and γ’s).• Coolant.• Control system.• Emergency systems (to prevent runaway during failure).
Chapter 4 in Lamarsh
7Nuclear Reactor Theory, BAU, Second Semester, 2011-2012 (Saed Dababneh).
p
Fission ReactorsTypes of reactors:Types of reactors:Used for what?Used for what?• Power reactors: extract kinetic energy of fragments as heat boil water steam drives turbine electricity.• Research reactors: low power (1-10 MW) to generate neutrons (~1013 n.cm-2.s-1 or higher) for research.• Converters and breeders: Convert non-thermally-fissionable material (non-fissile) to a thermally-fissionable material (fissile).• ADS.• Fusion. What are neutron generators?What are neutron generators?
8Nuclear Reactor Theory, BAU, Second Semester, 2011-2012 (Saed Dababneh).
What are neutron generators?What are neutron generators?
Fission ReactorsWhat neutron energy?What neutron energy?• Thermal, fast reactors.• Large, smaller but more fuel.What fuel?What fuel?• Natural uranium, enriched uranium, 233U, 239Pu,, , , ,Mixtures.
From converter or breeder reactorHow??? breeder reactor.How???
9Nuclear Reactor Theory, BAU, Second Semester, 2011-2012 (Saed Dababneh).
Fission ReactorsWhat assembly?What assembly?
H t d t d f l l d• Heterogeneous: moderator and fuel are lumped. • Homogeneous: moderator and fuel are mixed together.
I h t it i i t l l t d f• In homogeneous systems, it is easier to calculate p and ffor example, but a homogeneous natural uranium-graphite mixture (for example) can not go critical Why?graphite mixture (for example) can not go critical. Why?
What coolant?What coolant?• Coolant prevents meltdown of the core.• It transfers heat in power reactors.• Why pressurized-water reactors.• Why liquid sodium?
10Nuclear Reactor Theory, BAU, Second Semester, 2011-2012 (Saed Dababneh).
More on ModeratorsWhat moderator?What moderator?1. Cheap and abundant.2. Chemically stable.3. Low mass (high ζ logarithmic energy decrement).4. High density.5. High Σs and very low Σa.• Graphite (1,2,4,5) increase amount to compensate 3.• Water (1,2,3,4) but n + p → d + γ enriched uranium.• D2O (heavy water) (1!) but has low capture cross
section natural uranium, but if capture occurs, produces tritium (more than a LWR).
• ….. 11Nuclear Reactor Theory, BAU, Second Semester, 2011-2012
(Saed Dababneh).
More on Moderators
Moderating ratio ≡ s∑ζg
Calculate both a∑HW HW 1212 α+→→+ LiBnB 7*1110
moderating power and ratio for water, heavy
B-101010BBywater, graphite, polyethylene and boronboron.
1/v region
p y yTabulate your results and comment.
12Nuclear Reactor Theory, BAU, Second Semester, 2011-2012 (Saed Dababneh).
More on Moderators
HW HW 12 12 (continued)(continued)
Calculate the moderating power and ratio for pure D O ll f D O t i t d ith ) 0 25%D2O as well as for D2O contaminated with a) 0.25% and b) 1% H2O.C t th ltComment on the results.In CANDU systems there is a need for heavy water
dupgradors.
Nuclear Reactor Theory, BAU, Second Semester, 2011-2012 (Saed Dababneh).
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More on Moderators
ζnEE −= lnln \ )/ln( \nEEn =Recall ζnEEn −= lnln
ζn =Recall
After n collisionsAfter n collisions
)/ln( thf EE After one collision
11ln
2)1(1ln
2
\ +−−
+=⎥⎦⎤
⎢⎣⎡==∆
AA
AA
EEu
av
ζζ)( thfn =
ζ
Total mean free path = n λf
Total mean free path = n λsIs it random walk or there is a
f d di ti ???Nuclear Reactor Theory, BAU, Second Semester, 2011-2012
(Saed Dababneh).14
preferred direction??? th
More on Moderators
A ⎞⎛21 After one collision.
Recall (head-on). Then the maximum energy loss is (1-α)E, or αE ≤ E\ ≤ E.
EEAAE α≡⎟
⎠⎞
⎜⎝⎛
+−
=min\
11
For an ss--wavewave collision:EEPdEEEP
E 1)(1)( \\\ =→∴=→∫Assumptions:Assumptions:1.1. Elastic scattering.Elastic scattering. E↓Flat-top probability
EEEPdEEEP
E )1()(1)(
αα −→∴→∫
1.1. Elastic scattering. Elastic scattering. E↓2.2. Target nucleus at rest. Target nucleus at rest.
E↑3.3. Spherical symmetry in Spherical symmetry in
CMCMEE )1(2
1\ α+=CM.CM.
Obviously
2
⎪
⎪⎨⎧ ≤≤
−=≡→EEE
EE
dEdEE
ss
s )1()(
)(|
|\ α
ασσσ
Nuclear Reactor Theory, BAU, Second Semester, 2011-2012 (Saed Dababneh).
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⎪⎩ otherwisedE 0
More on ModeratorsHW HW 13 13 (or (or 66\\)) •• Scattering Kernel.Scattering Kernel.
•• Slowing down density.Slowing down density.
[ ]2\ 1cos21++ AAE CMθ
(Re)-verifyg yg y
•• Migration length.Migration length.•• Fermi age and continuous fermi model.Fermi age and continuous fermi model.
[ ]
[ ]22 cos)1()1(
21
)1(cos21
−++=+
++=
AAA
EE CMθααθ
[ ]2
222
)1(sincos
+−+
=AA θθ
)(For doing so, you need to verify and use
CMA θ1CM
CM
AAA
θθθcos21
cos1cos2 ++
+=
Nuclear Reactor Theory, BAU, Second Semester, 2011-2012 (Saed Dababneh).
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More on ModeratorsHW HW 13 13 (or (or 66\\) ) continued…continued…
• Forward scattering is preferred for “practical”preferred for practical moderators (small A).• If isotropic neutronIf isotropic neutron scattering (spherically symmetric) in thesymmetric) in the laboratorylaboratory frame average cosine of the Show that
2)(cos == θµaverage cosine of the scattering angle is zero.
Show that A3
)(cos θµ
Nuclear Reactor Theory, BAU, Second Semester, 2011-2012 (Saed Dababneh).
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More on Moderators
1dHW HW 13 13 (or (or 66\\) ) continued…continued…
Spherically symmetric in CM )(41)( E
dd
sCM
sCMs σ
πθσσ
==Ω
CM 2312
Show thatCM
CMs
s AAAE
θθ
πσθσ
cos1)1cos2(
4)()( 1
2312
−
−−
+++
=
• Neutron scatteringscattering is isotropic in the laboratory system?! valid for neutron scattering with heavysystem?! valid for neutron scattering with heavy nuclei, which is not true for usual thermal reactor moderators (corrections are applied).moderators (corrections are applied).Distinguish fromDistinguish from
A l t di t ib tidi t ib tiNuclear Reactor Theory, BAU, Second Semester, 2011-2012
(Saed Dababneh).19
• Angular neutron distribution.distribution.
More on ModeratorsModeratorModerator--toto--fuel ratio fuel ratio ≡≡ Nm/Nu. Self regulation.• Ratio ↑ p ↑ Σa of the moderator ↑ f ↓ (leakage ↓).• Ratio ↓ p ↓ f ↑ (leakage ↑).• T ↑ ratio ↓ (why).• Other factors alsoOther factors also change.• Temperature coefficient of reactivity.• Moderator temperaturetemperature coefficient of reactivity.
Nuclear Reactor Theory, BAU, Second Semester, 2011-2012 (Saed Dababneh).
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