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Controlled Fission• Note that η is greater than 2 at thermal energies andat thermal energies andalmost 3 at high energies.• These “extra” neutrons are
Variations in Variations in ηη
Used to convert fertile into fissile fuel.
Plutonium economy• Plutonium economy.• India and thorium.• Efficiency of this process is Efficiency of this process is determined by neutronenergy spectrum.
Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).
1
gy
Controlled Fission
• Conversion ratioConversion ratio CR is defined as the average rate ofConversion ratio Conversion ratio CR is defined as the average rate of fissile atom production to the average rate of fissile atom consumption.• For LWR's CR ≅ 0.6.• CR is called BR for values > 1.
F t b d t h BR > 1• Fast breeder reactors have BR > 1.• They are called “fast” because primary fissions inducing neutrons are fast not thermal thus η > 2 5 butinducing neutrons are fast not thermal, thus η > 2.5 but σf is only a few barns.• Moderator??
Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).
2
Controlled FissionTime scale for neutron multiplicationTime scale for neutron multiplication• Time constant τ includes moderation time (~10-6 s) and diffusionTime constant τ includes moderation time ( 10 s) and diffusion time of thermal neutrons (~10-3 s).
Time Average number of thermal neutronstt n
t + τ knt + 2τ k2n
• For a short time dtτ
nkndtdn −
=
•• Show thatShow that
τdtτtkentn )1(
0)( −=3
0)(Nuclear Reactors, BAU, 1st Semester, 2008-2009
(Saed Dababneh).
Controlled Fissionτtkentn )1(
0)( −=• k = 1 n is constant (Desired).• k < 1 n decays exponentially.• k > 1 n grows exponentially with time constant τ / (k 1)
0)(Reactivity.
• 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 1s. in 1s. • Design the reactor to be slightly subcritical for prompt neutrons.• The “few” “delayed” neutrons will be used to achieve criticality, y,allowing enough time tomanipulate the controlrods (or use shim or )
4
rods (or use shim or …).Cd control rodsCd control rods
Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).
Fission ReactorsEssential elements:Essential elements:• Fuel (fissile material)Fuel (fissile material).• Moderator (not in reactors using fast neutrons).• Reflector (to reduce leakage and critical size).
Core
( g )• Containment vessel (to prevent leakage of waste).• Shielding (for neutrons and γ’s).
C l• Coolant.• Control system.• Emergency systems (to prevent runaway during failure)• Emergency systems (to prevent runaway during failure).
Chapter 4 in Lamarsh
5Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).
Chapter 4 in Lamarsh
Fission ReactorsTypes of reactors:Types of reactors:Used for what?Used for what?Used for what?Used for what?• Power reactors: extract kinetic energy of fragments as heat boil water steam drives turbine electricity.y• Research reactors: low power (1-10 MW) to generate neutrons (~1013 n.cm-2.s-1 or higher) for research.
C d b d C h ll• Converters and breeders: Convert non-thermally-fissionable material (non-fissile) to a thermally-fissionable material (fissile)fissionable material (fissile).• ADS.• Fusion. Wh t t t ?Wh t t t ?
6
Fusion.
Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).
What are neutron generators?What are neutron generators?
Fission ReactorsWhat neutron energy?What neutron energy?• Thermal fast reactorsThermal, fast reactors.• Large, smaller but more fuel.What fuel?What fuel?What fuel?What fuel?• Natural uranium, enriched uranium, 233U, 239Pu,Mixtures.
From converter or breeder reactor.How???
7Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).
Fission ReactorsWhat assembly?What assembly?• Heterogeneous: moderator and fuel are lumped.Heterogeneous: moderator and fuel are lumped. • Homogeneous: moderator and fuel are mixed together.• In homogeneous systems, it is easier to calculate p and
f for example, but a homogeneous natural uranium-graphite mixture can not go critical. Why?
Wh t l t?Wh t l t?What coolant?What coolant?• Coolant prevents meltdown of the core.• It transfers heat in power reactors• It transfers heat in power reactors.• Why pressurized-water reactors.• Why liquid sodium?
8
y q
Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).
More on ModeratorsWhat moderator?What moderator?1 Cheap and abundant1. Cheap and abundant.2. Chemically stable.3. Low mass (high ζ logarithmic energy decrement).( g ζ )4. High density.5. High Σs and very low Σa.
G hi (1 2 4 ) i 3• Graphite (1,2,4,5) increase amount to compensate 3.• Water (1,2,3,4) but n + p → d + γ enriched uranium.• D O (heavy water) (1!) but has low capture cross• D2O (heavy water) (1!) but has low capture cross
section natural uranium, but if capture occurs, produces tritium (more than a LWR).produces tritium (more than a LWR).
• ….. 9Nuclear Reactors, BAU, 1st Semester, 2008-2009
(Saed Dababneh).
More on Moderators
∑ζModerating ratio ≡ s
∑∑ζ
HW 12HW 12 α+→→+ LiBnB 7*1110
Calculate both moderating power and
a∑HW 12HW 12 α+→→+ LiBnB
B-101010BBratio for water, heavy water, graphite,
l th l d bb1/v region
1010BB
polyethylene and boronboron.Tabulate your results and comment
10Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).
comment.
More on Moderators
HW 12 HW 12 (continued)(continued)
Calculate the moderating power and ratio for pure D2O as well as for D2O contaminated with a) 0.25% and b) 1% H Oand b) 1% H2O.Comment on the results.In CANDU systems there is a need for heavy waterIn CANDU systems there is a need for heavy water upgradors.
Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).
11
More on Moderators
ζnEEn −= lnln \
ζ)/ln( \
nEEn =Recall ζ
After n collisions
)/l ( EE Aft lli i
1ln)1(1ln2
\
−−+=⎥⎦
⎤⎢⎣⎡==∆
AAEu ζζ)/ln( thf EE
n =After one collision
12\ +⎥⎦⎢⎣ AAE av
ζζf
Total mean free path = n λs
Is it random walk or there is a
Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).
12
preferred direction??? th
More on Moderators
Recall (head on) Then the
HW 13 (or 6HW 13 (or 6\\))EEAE α≡⎞
⎜⎛ −
=2
min\ 1 After one collision.
Recall (head-on). Then the maximum energy loss is (1-α)E, or αE ≤ E\ ≤ E.For an ss--wavewave collision:
EEA
E α≡⎠
⎜⎝ +
=min1
A tiA tiE
EEPdEEEPE
E )1(1)(1)( \\\
αα −=→∴=→∫
Assumptions:Assumptions:1.1. Elastic scattering. Elastic scattering. E↓2.2. Target nucleus at rest. Target nucleus at rest.
E↑33 S h i l t iS h i l t i
show that
EE )1(1\ α+= 3.3. Spherical symmetry in Spherical symmetry in CM.CM.
Obviously
EE )1(2 α+=⎪⎨⎧ ≤≤
−=≡→EEE
EEdEE
ss
s )1()(
)(|
|\ α
ασσσObviously
Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).
14
⎪⎩⎨
otherwiseEdEs
0)1()( | α
More on Moderators
(Re) erifHW 13 (or 6HW 13 (or 6\\) ) continued…continued…
•• Scattering Kernel.Scattering Kernel.•• Slowing down density.Slowing down density.•• Migration length.Migration length.
[ ]2
2\
cos)1()1(21
)1(cos21
−++=++
=A
AAEE CM
CM
θααθ(Re)-verify Migration length.Migration length.
•• Fermi age and continuous fermi model.Fermi age and continuous fermi model.
[ ]
[ ]222
2
sincos
2)1(
−+=
+
A
AE
θθ[ ]2)1( +
=A
For doing so, you need to verify and use
CM
CM
AAA
θθθcos21
cos1cos2 ++
+=
Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).
15
AA θcos21++
More on ModeratorsHW 13 (or 6HW 13 (or 6\\) ) continued…continued…
• Forward scattering is preferred for “practical” moderators (small A)moderators (small A).• If isotropic neutron scattering (sphericallyscattering (spherically symmetric) in the laboratorylaboratory frame 2average cosine of the scattering angle is zero.
Show that A32)(cos == θµ
Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).
16
More on Moderators
Sphericall s mmetric in CM )(1)( Ed CMs σθσσ==
HW 13 (or 6HW 13 (or 6\\) ) continued…continued…
Spherically symmetric in CM
Show that
)(4
)( Ed ssCM σ
πθσ ==
ΩCM
s AAE θσθσ )1cos2()()(2312 −− ++
=Show thatCMs A θπ
θσcos14
)( 1−+=
• Neutron scatteringscattering is isotropic in the laboratory• Neutron scatteringscattering is isotropic in the laboratory system?! valid for neutron scattering with heavy nuclei, which is not true for usual thermal reactor moderators (corrections are applied).Distinguish fromDistinguish from
Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).
17
• Angular neutron distribution.distribution.
More on ModeratorsModeratorModerator--toto--fuel ratio fuel ratio ≡≡ Nm/Nu.• Ratio ↑ p ↑ Σ of the moderator ↑ f ↓ (leakage ↓)
Self regulation.• Ratio ↑ p ↑ Σa of the moderator ↑ f ↓ (leakage ↓).• Ratio ↓ slowing down time ↑ p ↓ f ↑ (leakage ↑).• T ↑ ratio ↓ (why)• T ↑ ratio ↓ (why).• Other factors also change.• Temperature coefficient of reactivity.• ModeratorModerator temperature coefficient of reactivity.
Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).
18
One-Speed Interactions• Particular general.Recall:
N t d ’t h h t i t t ith h• Neutrons don’t have a chance to interact with each other (review test!) Simultaneous beams, different intensities same energy:intensities, same energy:
Ft = Σt (IA + IB + IC + …) = Σt (nA + nB + nC + …)v• In a reactor, if neutrons are moving in all directions, g
n = nA + nB + nC + …
Rt = Σt nv = Σt φ
Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).
19
One-Speed Interactions
Ωdrn ),( ωrr
ωr≡ Neutrons per cm3 at
r whose velocity
Ωd
r whose velocity vector lies within dΩabout ω.
∫rr
d∫ Ω=π
ω4
),()( drnrn rrr
• Same argument as before r• Same argument as before Ω= vdrnrdI ),(),( ωω
rrrr
)()()()(),()(),()()(
),()(),(
rrrnrvdrnvrrdFrFrR
rdIrrdF
ttt
trrrrrrrrrrr
rrrrr
φωω
ωω
∑=∑=Ω∑===
∑=
∫∫
Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).
20
4πω
Ω= ∫ drnvr ),()(4
ωφπ
rrrwhere
Multiple Energy Interactions
ΩdEdErn ),,( ωrr
≡ Neutrons per cm3 at r with energy • Generalize to include energy
interval (E, E+dE) whose velocity vector lies within dΩ about ω.
∫ ∫∞
∫ Ω=π
ω4
),,(),( dEdErndEErnrrr ∫ ∫ Ω=
0 4
),,()(π
ω dEdErnrnrrr
dEEEdEEEEdEER )()()()()()( rrrrr φ∑∑ dEErErdEEvErnErdEErR tt ),(),()(),(),(),( φ∑=∑=
∫∞
∑= ),(),()( dEErErrR trrr φ∫∑
0
),(),()( t φ
Thus knowing the material properties Σt and the neutron flux φ as a function of space and energy we can calculate the interaction rate
Scalar
Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).
21
function of space and energy, we can calculate the interaction rate throughout the reactor.
Neutron Current∞
• Similarly and so on …∫∞
∑=0
),()()( dEErErR SSrr φ
Scalar
• Redefine as
Scalar
Ω= vdrnrdI ),(),( ωωrrrr Ω= dvrnrId rrrrrr ),(),( ωω
Ω= ∫ drnvr ),()(4
ωφπ
rrrΩ= ∫ drnvJ ),(
4
ωπ
rrrr
4π 4π
Neutron current densityNeutron current density• From larger flux to smaller flux!
Jrg
• Neutrons are not pushed!• More scattering in one direction
Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).
22
than in the other.xJxJ =• ˆ
r
NN fl f d i l
Equation of Continuity
∫ Ωθ)(ˆ dJJrrr
NNetet flow of neutrons per second per unit area normal to the x direction:
∫ Ω==•π
θω4
cos),(ˆ dvrnJxJ xxrr
In general: JnJ =• ˆr
ge e a nJnJEquation of ContinuityEquation of Continuity
∫∫ ∫∫∂ r
∫∫ ∫∫ •−∀∑−∀=∀∂∂
∀ ∀∀ Aa dAntrJdtrrdtrSdtrn
tˆ),(),()(),(),( rrrrrr φ
Rate of change in neutron density
Production rate
Absorption rate
“Leakage in/out” rate
Normal
Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).
23
Volume Source distribution
function
Surface area
bounding ∀
Normal to A