Nuclear Reactor Theory 0302750Nuclear Reactor Theory 0302750
Course webCourse webhttp://nuclear bau edu jo/ju/ju-reactors/http://nuclear.bau.edu.jo/ju/ju reactors/
orhttp://nuclear dababneh com/ju/ju-reactors/http://nuclear.dababneh.com/ju/ju reactors/
Nuclear Reactor Theory, JU, First Semester, 2010-2011(Saed Dababneh).
1
GradingReview Test 10%Mid term Exam 30%Mid-term Exam 30%Projects, quizzes and HWs 20%Final Exam 40%
• Homeworks and small projects are due after one• Homeworks and small projects are due after one week unless otherwise announced.• Remarks or questions marked in red without being• Remarks or questions marked in red without being announced as homeworks should be also seriously considered!considered!• Some tasks can (or should) be sent by email:
saed@dababneh comNuclear Reactor Theory, JU, First Semester, 2010-2011
(Saed Dababneh).2
Review Test
R d
Will do the test
afterwardsRead LamarshChs 1 2
afterwards.
Review relevant
Chs. 1, 2 and 3.Readrelevant
material.Read
Krane Ch. 13.
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
3
Projects
Topics related to:Topics related to:• Heat removal.• Radiation protection.• Radiation shielding.• Reactor licensing and safety.
N l it• Nuclear security.• Uranium mining or other front-end fuel cycle elements.• Back-endBack end.• Other topics that you would like to suggest…..
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
4
Projects
• Please do your own thorough research on relevant• Please do your own thorough research on relevant topics you may find appropriate.• Provide your suggestion next week• Provide your suggestion next week.• Final decision on the subject of your project should be taken before mid Octoberbe taken before mid October.• Due date for written version is Monday, December 20th20 .• Presentation date will be decided later.
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
5
Nuclear Reaction Energetics (revisited)
Conservation Laws• Charge, Baryon number, total energy, linear momentum, angular g y gy gmomentum, parity, (isospin??) …….
bQTT22
θφ
ap X
pb QTTcmcm iffi =−=− 22
φpa XpY Y
+ve Q-value exoergic reaction.-ve Q-value endoergic reaction.
aYb TQTT +=++ve Q-value reaction possible if T 0+ve Q-value reaction possible if Ta 0.-ve Q-value reaction not possible if Ta 0. (Is Ta > |Q| sufficient?).
Conservation of momentum ……
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
6
Nuclear Reaction Energetics (revisited)
• Conservation of momentum.• We usually do not detect Y.
HW HW 11y
Show that:aaYYbYabaaba TmmQmmmTmmTmm
T−+++± ])()[(coscos 2θθ
• The threshold energy (for Ta): (the condition occurs for θ = 0º).bY
aaYYbYabaabab mm
T+
=
gy ( a)
abY
bYTh mmm
mmQT−+
+−=
• +ve Q-value reaction possible if Ta 0.• -ve Q-value reaction possible if Ta > TTh.
abY
a Th• Coulomb and other barriers…….!!!• Neutrons vs. charged particles.
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
7
Nuclear Reaction Energetics (revisited)
Th d bl l d it ti b t T d th
HW HW 11 (continued)(continued)
• The double valued situation occurs between TTh and the upper limit Ta
\.YmQT \
aY
Ya mm
QT−
−=\
• Double-valued in a forward cone.
aaYYbY TmmQmmm ])()[(cos max2 −++
−=θaba Tmmmax
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
8
Nuclear Reaction Energetics (revisited)
Di th hl th 7Li( ) ti
HW HW 11 (continued)(continued)
• Discuss thoroughly the 7Li(p,n) reaction.• During the discussion emphasize on the case
h th i id t t b i 30 k V bwhen the incident proton beam is 30 keV above the threshold.
U ti kill• Use your computing skills.
• Discuss the elasticelastic and inelastic inelastic scatteringscattering of neutronsneutrons using thesescatteringscattering of neutronsneutrons using these relations.
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
9
Nuclear Reaction Energetics
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
10
Nuclear Reaction Energetics (revisited)
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
11
Nuclear Reaction Energetics (revisited)
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
12
Nuclear Reaction Energetics (revisited)
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
13
Nuclear Reaction Energetics (revisited)
• If the reaction reaches excited states of Y
EQcmEcmcmcmQ =++= 2222 )(
58 61
exbexYaXex EQcmEcmcmcmQ −=−+−+= 0)(
58Ni(α,p)61Cu
l
less proton energy
even less ….
Highest proton energy
See Figures 11.4 i K
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
14
in Krane
Nuclear Reaction Energetics (revisited)
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
15
Neutron Interactions (revisited)
• Chadwick’s discovery.• Neutrons interact with nuclei, not with atoms. (Exceptions).
• Recall from basic Nuclear Physics:o Inelastic scattering (n,n\). Q = -E* Inelastic gammas.
Threshold?o Elastic scattering (n,n). Q = ?? (Potential and CN).
N t d ti ?Neutron moderation?o Radiative capture (n,γ). Q = ?? Capture gammas.
(n α) (n p) Q = ?? Absorption Reactionso (n,α), (n,p). Q = ?? Absorption Reactions.o (n,2n), (n,3n) Q = ?? Energetic neutrons on heavy water can easily eject the loosely bound neutroncan easily eject the loosely bound neutron.o Fission. (n,f).
HWHW 22 Examples of such exo- and endo-thermic reactions with Q
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
16
HW HW 22 Examples of such exo and endo thermic reactions with Q calculations.
Neutron Scattering (revisited)
• Elastic or inelastic.• Analogous to diffraction.• Alternating maxima and minima• Alternating maxima and minima.• First maximum at
h=λ R
λθ ≈
(3
1ARR
p
o=
λ
• Minimum not at zero (sharp edge of the nucleus??)• Clear for neutrons• Clear for neutrons.• Protons? High energy, large angles. Why? 222 11)(θσ ⎞
⎜⎛⎞
⎜⎛ zZedg y
• Inelastic Excited states, 2
4sin1
41
4)(
θπεθσ
⎟⎠
⎞⎜⎜⎝
⎛⎟⎠
⎞⎜⎜⎝
⎛=
Ω ao TzZe
dd
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
17
energy, X-section and spin-parity.
Reaction Cross Section (revisited)
• Probability.• Projectile a will more probably hit target X if area is larger.• Classically: σ = π(Ra + RX)2.Classical σ = ??? (in b) n + 1H, n + 238U, 238U + 238U
Q t h i ll D2• Quantum mechanically: σ = π D2.
CMXa
Emm
2hh
D =+
=
• Coulomb and centrifugal barriers energy dependence of σ.What about neutrons?What about neutrons?
CMaXaXaaX EEmm µ22
What about neutrons?What about neutrons?• Nature of force:
Strong: 15N(p,α)12C σ ~ 0.5 b at Ep = 2 MeV.Strong: N(p,α) C σ 0.5 b at Ep 2 MeV.Electromagnetic: 3He(α,γ)7Be σ ~ 10-6 b at Eα = 2 MeV.Weak: p(p,e+ν)D σ ~ 10-20 b at Ep = 2 MeV.
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
18
p• Experimental challenges to measure low X-sections..
Reaction Cross Section (Simple terms)A (A f h ??!!)
XA (Area of what??!!)|v|
Monoenergetic (and Target with N atoms.cm-3 or NAX atomsPosition of a neutron 1 s
b f i iMonoenergetic (and unidirectional) neutrons of speed v (cm.s-1) and
density n (cm-3)
gbefore arriving at target
Volume = vAcontaining nvA neutrons that hit the
NX??y ( ) containing nvA neutrons that hit the
“whole!!” target in 1 s.Beam Intensity I ≡ nvA/A = nv (cm-2s-1)
Number of neutrons interacting with target per secondNumber of neutrons interacting with target per second∝ I, A, X and N= σt I N A XTotal microscopic cross
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
19
σt I N A XTotal microscopic cross section
Reaction Cross Section (Simple terms)
Number of neutrons interacting with target per second= σt I N A Xt
Total microscopic Total number of
Number of interactions with a single nucleus per second
pcross section nuclei in the
target
Number of interactions with a single nucleus per second = σt I Interpretation and units of σ.
nvA = IA neutrons strike the target per second, of these σtI neutrons interact with any single nucleus. Thus,
measures the probability for a neutron to hit a nucleus (per unit area of target)AAI
I tt σσ=
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
20
(per unit area of target).AAIEffective cross-sectional area of the nucleus.
Reaction Cross Section (Simple terms)The probability for a neutron to hit a nucleus (per unit area of target):
AAII tt σσ=
AAI
Typical nucleus (R=6 fm): geometrical πR2 ≈ 1 b.Typical σ: <µb to >106 b
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
21
Typical σ: <µb to >106 b.
Reaction Cross Section (Simple terms)
Number of neutrons interacting with target per second= σt I N A X t
Volume of the Total
microscopic
Number of interactions per cm3 per second (Collision Density)
targetp
cross section
Number of interactions per cm3 per second (Collision Density)Ft = σt I N = I Σt
Σt = N σtt t
Macroscopic total cross
section
XteIXI = Σ−)( 0section.
Probability per unit path length. t Σ
=1λMean free path
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
22
g
tΣ
Neutron AttenuationX
I II0 IRecall Σt = N σt XteIXI Σ−= 0)(
Probability per unit path
mfp for scattering λs = 1/Σsmfp for absorption λ = 1/Σ
length.mfp for absorption λa 1/Σa
………….
total mfp λt = 1/Σt
XteXP Σ−=)(i iProbability
total mfp λt 1/Σt
Nuclear Reactor Theory, JU, First Semester,2010-2011 (Saed Dababneh).
23XteXP
eXPΣ−
−
−=1)(
)(
ninteractio
ninteractionoProbability
Reaction Cross Section (Simple terms)
Homogeneous Mixture
yyxxyx NN σσ +=Σ+Σ=Σ
Molecule xmyn Nx=mN, Ny=nN
yx nm σσσ +=given that events at x and y are independent.
yx
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
24
Reaction Cross Section
Detector for particle “b”dΩ
θ,φIa\
p
d“b” particles / s
2θ,φ
\\ NIdRd b=σ
cm2
NIa
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
25
Reaction Cross Section
Many different quantities are called“cross section”.
dΩAngular distribution
Krane Table 11.14
),( drdRb πφθ Ω
=Units … !
\\4),(
NIr
dd
πφθσ
=Ω
“Differential” cross sectionσ(θ,φ) or σ(θ )or “cross section” !! 4 NId aπΩor “cross section” …!!
=Ω ddd φθθ2
sin Doubly differential
∫ ∫ ∫ Ω=Ω
Ω=
ddddd
dd σφθθσσ
π π
0
2
0
sin d σ2
dEdσ
0 0 ΩdEddE
σ for all “b” particlesNuclear Reactor Theory, JU, First Semester, 2010-2011
(Saed Dababneh).26
σt for all b particles.
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
27
nn--TOFTOFCERNCERN
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
28
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
29
Neutron Cross Section (Different Features)
1/v Fast neutrons should be
d t dmoderated.
235U thermal cross sectionsσfission ≈ 584 b.σscattering ≈ 9 b.σ ≈ 97 bσradiative capture ≈ 97 b.
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
30Fission Barriers
Neutron Induced Reactions22 nXHCCHbY IIIn ++∝ DσX(n,b)Y
Γn(En)Γb(Q+En)2
11vE
∝∝
)(EP )( nln EPvn
∝Probability to ypenetrate the potential barrier
P (E ) 1For thermal neutronsQ >> En
Γb(Q) ≈ constant Po(Ethermal) = 1P>o(Ethermal) = 0
1v
Enn1)( ∝σNon-resonant
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
31
v
Neutron Induced Reactions
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
32
Statistical Factor (Introduction)
D
hh bbplL ===
D
Dlb =222
1max, )12( Dπππσ +=−= + lbb lll
)()(7.656)(2
keVEub CMµ
π =DHW HW 33)()( keVEuµ
122 J +Generalization
)1()12)(12(
122max aX
XaaX JJ
J δπσ +++
+= D
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
33
ω
Reaction Cross Section
ExcitedState
ExJπ a + X Y + b Q > 0b + Y X + a Q < 0
Entrance Channela + X
ExitChannel
b Y X a Q 0
Inverse Reactiona X Channelb + YCompound
Nucleus C*
Inverse Reaction
212J +More Generalization
22 )1()12)(12(
12 XaHCCHbYJJ
JIIIaX
XaaXaX +++
+++
= δπσ D
QM StatisticalFactor (ω)
Identicalparticles
• Nature of force(s).• Time-reversal invariance.
212J + 22 )1()12)(12(
12 YbHCCHXaJJ
JIIIbY
YbbYbY +++
+++
= δπσ D
??=aXσHWHW 44Nuclear Reactor Theory, JU, First Semester, 2010-2011
(Saed Dababneh).34
??=bYσ
HW HW 44
Resonance Reactions
Projectile Projectile
TargetQ l Q-value
TargetQ-value Q-value
Q + ER = Er
Direct ResonantEγ = E + Q - Eex
Direct Capture(all energies)
Resonant Capture(selected energies ( g ) ( gwith large X-section)
2XaHY +∝ γγσ
22XaHEEHE CNrrf +∝ γγσ
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
35
γγ CNrrf γγ
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
36
Resonance Reactions
fDamped OscillatorDamped Oscillator Oscillator strength
22
2 )()( δωω +−∝
o
fresponse
Dampingf t
1t
=δ
eigenfrequencyfactor0t
)( ΓΓ∝ baEσ 2
22 )()(
)(Γ+−
∝REE
Eσ
hΓtNuclear Reactor Theory, JU, First Semester, 2010-2011
(Saed Dababneh).37
h=Γ ot
Resonance Reactions
222 )1(12)(
Γ
ΓΓ+
+= ba
aXaXJE δπσ D 2
22 )()(
)()12)(12(
)(Γ+−++ R
aXXa
aX EEJJBreitBreit--Wigner formulaWigner formula Γ+ΓΓBreitBreit Wigner formulaWigner formula
• All quantities in CM systemO l f i l t d
ba Γ+Γ=Γ
• Only for isolated resonances.baR ΓΓ∝σ Reaction Usually Γa >> Γb.
bR
aae
ΓΓΓ∝
σσ Elastic scattering
HWHW 55 When does σ take its maximum value?
y a b.
a
b
e
R
ΓΓ
=σσ HW HW 55 When does σR take its maximum value?
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
38
Resonance Reactions
Ja + JX + l = JExitChannelExcited
ExJπ
(-1)l π(Ja) π(JX) = π(J) b + YExcitedState
Entrance Ch l(-1)l = π(J) Natural parity.
Compound
Channela + X
Nucleus C*
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
39
Resonance Reactions
What is the “Resonance Strength” …?What is its significance?What is its significance?In what units is it measured?
ΓΓ+
+ baJ )1(12 δωγΓ
+++
= baaX
Xa JJ)1(
)12)(12(δωγ
ectio
n
Charged particledi ti t ( )
Cro
ss se
E
radiative capture (a,γ)(What about neutrons?)(What about neutrons?)
C ECωγ ∝ Γa ωγ ∝ Γγ
Energy
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
40
Neutron Resonance Reactions
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
41
Neutron Activation Analysis
(Z,A) + n (Z, A+1)β-
γ (β-delayed γ-ray)
(Z+1, A+1)
ProjectProject 11Project Project 11NAA and UNAA and UNAA and UNAA and U
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
42
Neutron Flux and Reaction Rate
Recall Ft = n v σt N = I ΣtSimultaneous beams, different intensities, same energysame energy.Simultaneous beams, different intensities, same energysame energy.
Ft = Σt (IA + IB + IC + …) = Σt (nA + nB + nC + …)vIn a reactorreactor, if neutrons are moving in all directionsall directionsIn a reactorreactor, if neutrons are moving in all directionsall directions
n = nA + nB + nC + …F = Σ nvFt = Σt nv
Not talkingneutron flux φ = nv
Not talking about a beam
anymore.
Reaction Rate Rt ≡ Ft = Σt φ = φ /λt (=nvNσt)Nuclear Reactor Theory, JU, First Semester, 2010-2011
(Saed Dababneh).43same energysame energy
Neutron Flux and Reaction Rate
Different energiesDifferent energiesDensity of neutrons with energy between E and E+dEy gy
n(E)dEReaction rate for those “monoenergetic” neutronsg
dRt = Σt(E) n(E)dE v(E)∞ ∞∞
∫=0
)( dEEnn ∫∫∞∞
==00
)()()( dEEEndEE υφφ
∫∫∞∞
∑=∑= )()()()()( dEEEdEEEnER φυ ∫∫ ∑=∑=00
)()()()()( dEEEdEEEnER ttt φυ
∫∞
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
44∫∑=0
)()( dEEER ii φ
Neutron Flux and Reaction Rate
In general, neutron flux depends on:• Neutron energy, E.gy• Neutron spatial position, r.• Neutron angular direction, Ω.g• Time, t.
Various kinds of neutron fluxes (depending on the ( p gdegree of detail needed).Time-dependent and time-independent angularTime dependent and time independent angular neutron flux. ),,,( tEr Ωφ
),,( ΩErφ),,,(φ
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
45
),,(φ
Neutron Flux and Reaction RateIn Thermal ReactorsThermal Reactors, the absorptionabsorption rate in a “medium” of thermal (MaxwellianMaxwellian) neutrons
∫∑=Th l
aa dEEvEnER )()()(
Usually 1/v cross section, thus
Thermal
)( 0vEa =∑
∫
)()( 0 EvEa∑
then 000000 )()()()( φEnvEdEEnvER aaThermal
aa ∑=∑=∑= ∫Independent of Independent of n(E)n(E)..
The reference energy is chosen at 0.0253 eV. • Look for Thermal Cross Sections.• Actually, look for evaluated nuclear data.Nuclear Reactor Theory, JU, First Semester, 2010-2011
(Saed Dababneh).46ENDF
Neutron Moderation
Show that, after one elasticelastic scattering the ratio between the final neutron energy E\ and its initial
HW HW 66gy
energy E is given by:[ ]2222\ sincoscos21 −+++ AAAE CM θθθ 11[ ]
22 )1(sincos
)1(cos21
++
=+
++=
AA
AAA
EE θθθ
⎞⎛⎞⎛ 2\ 1AE
11H ?H ?
For a head-on collision: α≡⎟⎠⎞
⎜⎝⎛
+−
=⎟⎟⎠
⎞⎜⎜⎝
⎛
min 11
AA
EE
After n ss--wavewave collisions:where the average change in lethargy lethargy
ζnEEn −= lnln \
)ln( EEu M=g g gygyis
11ln
2)1(1ln
2
\ +−−
+=⎥⎦⎤
⎢⎣⎡==∆
AA
AA
EEu ζ
)( M
12 +⎦⎣ AAE av
47Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
Average decrease in ln(E) after one collision.
Neutron Moderation HW HW 6 6 (continued)(continued)
• Reproduce the plot.• Discuss the effect of the
On 12C.
thermal motion of the moderator atoms.
First collision.
MostMostMost Most probable probable
and average and average energies?energies?
Second collision.energies?energies?
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
48
Neutron Moderation HW HW 6 6 (continued)(continued)
Neutron scattering by light nuclei then the average energy loss
EE )1(21\ α+=
EEEE )1(1\ α==∆then the average energy loss and the average fractional energy loss
EEEE )1(2 α−=−=∆
)1(1∆E
• How many collisions are needed to thermalize a 2 MeV
)1(21 α−=
EHow many collisions are needed to thermalize a 2 MeV
neutron if the moderator was:1H 2H 4He graphite 238U ?H H He graphite U ?
• What is special about 1H?• Why we considered elastic scattering?Why we considered elastic scattering?• When does inelastic scattering become important?
49Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
Nuclear Fission
Coulomb effectSurface effect
~200 MeV
Coulomb effectSurface effect
50Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
Nuclear Fission• B.E. per nucleon for 238U (BEU) and 119Pd (BEPd) ?
2 119 BE 238 BE ?? K E f th• 2x119xBEPd – 238xBEU = ?? K.E. of the fragments ≈ 1011 J/g
B i l 105 J/• Burning coal 105 J/g• Why not spontaneous?
T 119Pd f t j t t hi• Two 119Pd fragments just touchingThe Coulomb “barrier” is:
)46( 2
C d ! Wh t if 79Z d 159S ? L t
MeVMeVfm
fmMeVV 2142502.12
)46(.44.12
>≈=
• Crude …! What if 79Zn and 159Sm? Large neutron excess, released neutrons, sharp potential edge,
h i l U !
f
spherical U…!51Nuclear Reactor Theory, JU, First Semester, 2010-2011
(Saed Dababneh).
Nuclear Fission
• 238U (t½ = 4.5x109 y) for α-decay.• 238U (t½ ≈ 1016 y) for spontaneous fissionU (t½ ≈ 10 y) for spontaneous fission.• Heavier nuclei??• Energy absorption from a neutron (for example) couldEnergy absorption from a neutron (for example) could form an intermediate state probably above barrier induced fissioninduced fission.• Height of barrier is called activation energy.
52Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
Nuclear Fission
Liquid Drop
MeV
)
Shell
Ene
rgy
(va
tion
EA
cti
53Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
Nuclear Fission34 Rπ ε+= )1(Ra23 abR =3
24 abπ=
=Rb
abR =
Surface Term B = a A⅔
3abπ ε+1
)1( 22 ++ ε
Volume Term (the same)
Surface Term Bs = - as A⅔
Coulomb Term BC = - aC Z(Z-1) / A⅓
...)1( 52 ++ ε
...)1( 251 +− ε
32
31
52
51 )1( AaAZZa SC >− − fission
47~2
>Z Crude: QM and original shape
could be different from sphericalA could be different from spherical.54Nuclear Reactor Theory, JU, First Semester, 2010-2011
(Saed Dababneh).
Nuclear Fission
)120( 2
48300
)120(=
300Consistent with activation energy curve for A = 300curve for A = 300.
Extrapolation to 47 10-20 sExtrapolation to 47 ≈ 10 20 s.55Nuclear Reactor Theory, JU, First Semester, 2010-2011
(Saed Dababneh).
Nuclear Fission
235U + n235U + n
93Rb + 141Cs + 2nN t iNot unique.
Low-energy fissionfission processes.
56Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
Nuclear FissionZ1 + Z2 = 92Z 37 Z 55Z1 ≈ 37, Z2 ≈ 55A1 ≈ 95, A2 ≈ 140L tLarge neutron excessMost stable:Z=45 Z=58Z=45 Z=58
Prompt neutronsPrompt neutrons within 10-16 sPrompt neutronsPrompt neutrons within 10 16 s.Number ν depends on nature of fragments and on incident neutron energyon incident neutron energy.The average number is characteristic of the processprocess.
57Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
Nuclear Fission
The average number of neutrons isneutrons is different, but the distribution is G iGaussian.
58Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
Why only left side of the
mass parabola?
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
59
Higher than S ?
Delayed neutronsDelayed neutrons
Higher than Sn?
Delayed neutronsDelayed neutrons~ 1 delayed neutron
100 fi i b tper 100 fissions, but essential for control of the reactor
In general, of the reactor.
• Waste
g ,β decay favors high • Waste.
• Poison.high
energy.
60Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
Nuclear Fission
61Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
Nuclear Fission
1/1/vFast neutrons should be
d t dmoderated.
235U thermal cross sectionsσfission ≈ 584 b.σscattering ≈ 9 b.σ ≈ 97 bσradiative capture ≈ 97 b.
Fission Barriers 62Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
Nuclear Fission
• Q for 235U + n 236U is 6.54478 MeV.• Table 13 1 in Krane: Activation energy E for 236U ≈ 6 2 MeV• Table 13.1 in Krane: Activation energy EA for 236U ≈ 6.2 MeV (Liquid drop + shell) 235U can be fissioned with zero-energy neutrons.
238 239• Q for 238U + n 239U is 4.??? MeV.• EA for 239U ≈ 6.6 MeV MeV neutrons are needed.• Pairing term: δ = ??? (Fig 13 11 in Krane)• Pairing term: δ = ??? (Fig. 13.11 in Krane).• What about 232Pa and 231Pa? (odd Z).• Odd-N nuclei have in general much larger thermal fission
63
g gcross sections than even-N nuclei (Table 13.1 in Krane).Nuclear Reactor Theory, JU, First Semester, 2010-2011
(Saed Dababneh).
Nuclear FissionWhy not use it?Why not use it?
σf,Th 584 2.7x10-6 700 0.019 b
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
64
Nuclear Fission• Thermal neutron fission of 235U forms compound nucleus that splits up in more than 40 different ways, yielding over 80 primary fission fragments (products).235
92U + 10n 9037Rb + 144
55Cs + 210n
23592U + 10n 87
35Br + 14657La + 31
0n
23592U + 10n 72
30Zn + 16062Sm + 41
0n • The fission yield is defined as the proportion• The fission yield is defined as the proportion (percentage) of the total nuclear fissions that form products of a given mass number.
65Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
p g
65Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
Nuclear Fission
• Remember neutron excess.(A Z) (A Z+1) or (A 1 Z)• (A,Z) ⇒ (A,Z+1) or (A-1,Z).
Only leftOnly left side of the
mass parabola
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
66
parabola.
Nuclear Fission• 235U + n 93Rb + 141Cs + 2n ≈ 165 MeV
average kinetic• Q = ????• What if other fragments?
Diff t b f t
average kinetic energy carried
by fission• Different number of neutrons.• Take 200 MeV as a representative value.
by fission fragments per
fission.
66 MeV 98 MeV
Heavy LightHeavyfragments
g tfragments
67
miscalibratedNuclear Reactor Theory, JU, First Semester, 2010-2011
(Saed Dababneh).
Nuclear Fission• ν neutrons
itt demitted per fission.• ν depends• ν depends on fissioning nuclide and
India?
on neutron energy i d iinducing fission.
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
68
Nuclear Fission• Mean neutron energy ≈ 2 MeVMeV.• ≈ 2.5 neutrons per fission (average) ≈ 5 MeV(average) 5 MeV average kinetic energy carried by prompt neutrons per fission.
• Show that the average momentum carried by a neutron is only ≈1.5 % that carried by a fragment. • Thus neglecting neutron momenta, show that the ratio between kinetic energies of the two fragments is the inverse of the ratio of their masses E
69
their masses.
1
2
2
1
mm
EE
≈14095
9866
≈Nuclear Reactor Theory, JU, First Semester, 2010-2011
(Saed Dababneh).
Nuclear Fission
EeE E 292sinh4530)( 036.1−=χHW HW 77 EeE 29.2sinh453.0)( =χ
The experimental pspectrum of prompt neutrons is fitted by ythe above equation. Calculate the mean and the most probable neutron penergies.
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
70
Nuclear Fission• The fission gamma radiation
Prompt with average energy of 0.9 MeV.β delayed gammas.
HW HW 88•• Investigate how prompt Investigate how prompt gammas interact with gammas interact with water, uranium and lead.water, uranium and lead.
71Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
71Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
Fission Products• β and γemissions fromemissions from radioactive fission products carry partproducts carry part of the fission energy, even after shut down. • On approaching end of the chain, the decay energy decreases and half-life increases Long-lived isotopes constitute the mainand half-life increases. Long-lived isotopes constitute the main hazard.• Can interfere with fission process in the fuel. Example?Example? (poisoning).(poisoning).p pp (p g)(p g)• Important for research.• β-decay favors high energy ~20 MeV compared to ~6 MeV for γ.
72Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
• Only ~ 8 MeV from β-decay appears as heat. Why?Why?
Nuclear FissionSegrè Distribution of fission energy
aLost … ! b
c
•• How much is recoverable?How much is recoverable?How much is recoverable?How much is recoverable?•• What about capture gammas? What about capture gammas? (produced by (produced by νν--1 1 neutronsneutrons))•• Note again that c < (Note again that c < (a+ba+b).).
73
Note again that c < (Note again that c < (a+ba+b).).Nuclear Reactors, BAU, 1st Semester, 2007-2008
(Saed Dababneh).Nuclear Reactor Theory, JU, First Semester, 2010-2011
(Saed Dababneh).
Nuclear Fission
Distribution of fission energyEnge Distribution of fission energyg
Krane sums
them up as β Lost … !
decays.
74Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
Fission Productsβ-f
A, Zi
A, Z-1k
A-1, Zj
(n,γ)j
β-(n,γ)
A, Z+1A+1, Z
dNi/dt = Formation Rate - Destruction rate - Decay Rate
i NNNNNdN λφλφφγ ++ iiiikkjjffii NNNNN
dtλφσλφσφσγ −−++=
Ni saturates and is higher with higher neutron flux, larger “fission yield” and
75Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
Ni saturates and is higher with higher neutron flux, larger fission yield and longer half-live.
75Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
Fission Products
HWHW 99HW HW 99Investigate the activity, decay and gamma
i f fi i d t f ti f tienergies of fission products as a function of time. Comment on consequences (e.g. rod cooling).
iikki NN
dtdN λλ −=• Shutdown
HWHW 1010Investigate both and
dt
iikk NN λλ > iikk NN λλ <HW HW 1010
giving full description for the buildup and decay of fission fragment i.
iikk
76Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
76Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
Fission Products[ ] sMeVTttxtP /)(101.4)( 2.02.011 −− +−= [ ])()(
per watt of original operating power.T = time of operation.
Fission product activity after
reactor shutdown?
77Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
77Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
Nuclear Fission
It is necessary to evaluate the potential hazards• It is necessary to evaluate the potential hazards associated with an accidental release of fission products into the environment.products into the environment.
• It is required to determine a proper cooling time of the spent fuel (before it becomes ready for reprocessing) p ( y p g)that depends on the decay times of fission products.
• It is necessary to estimate the rate at which the heat is released as a result of radioactive decay of the fission products after the shut down of a reactor.
• The poisoning is needed to be calculated (the parasitic capture of neutrons by fission products that accumulate during the reactor operation)during the reactor operation).
78Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
78Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
Nuclear Fission• Recoverable energy release ≈ 200 MeV per 235U fission.• Fission rate = 2 7x1021 P fissions per day P in MW• Fission rate = 2.7x1021 P fissions per day. P in MW.• 3.12x1016 fissions per second per MW, or 1.2x10-5 gram of 235U per second per MW (thermal).
B tB t 1 05 P /d P i MW• Burnup rateBurnup rate: 1.05 P g/day. P in MW. • The fissioning of 1.05 g of 235U yields 1 MWd of energy. • Specific Burnup Specific Burnup = 1 MWd / 1.05 g ≈ 950000 MWd/t (pure (pure 235235U !!!!!!!!!).U !!!!!!!!!).• Fractional Burnup Fractional Burnup = ???• Thermal reactor loaded with 98 metric tons of UO2, 3% enriched, operates at 3300 MWt for 750 days.
Actually much less (all heavy material).Actually much less (all heavy material).
y• ≈ 86.4 t U. Specific burnup ≈ 28650 MWd/t. • Fast fission of 238U.• 238U converted to plutonium ⇒ more fission Not all fissions from 235UU converted to plutonium ⇒ more fission. Not all fissions from U.
79Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
Nuclear Fission
)(EσγC t t fi i ti)()(
)(E
Efσ
σα γ=• Capture-to-fission ratio:
1 05(1 ) /d• Consumption rateConsumption rate: 1.05(1+α) P g/day.
•• Read all relevant material in Lamarsh Read all relevant material in Lamarsh ChCh 44 We will come back to this laterWe will come back to this later
•Two neutrinos are expected immediately from the
Ch. Ch. 44. We will come back to this later.. We will come back to this later.
•Two neutrinos are expected immediately from the decay of the two fission products, what is the minimumflux of neutrinos expected at 1 km from the reactor
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
80
flux of neutrinos expected at 1 km from the reactor.4.8x1012 m-2s-1
Nuclear Fission• 3.1x1010 fissions per second per W.
I th l t j it f fi i i• In thermal reactor, majority of fissions occur in thermal energy region, φ and Σ are maximum.
T t l fi i t i th l t f l• Total fission rate in a thermal reactor of volume V
φVΣ• Thermal reactor powerThermal reactor power (quick calculation)(quick calculation)
φfVΣ
VP fφΣ
10101.3 xP f
th =
81Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
81Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
Controlled Fission• 235U + n X + Y + ν n
Fast second generation neutrons
• Moderation of second generation neutrons Chain reaction.• Water, D2O or graphite moderator.
R ti f b f “ t ” (fi i ) i ti t
g
• Ratio of number of “neutrons” (fissions) in one generation to the preceding ≡ k∞ (neutron reproduction or multiplication factor)factor).
• k ≥ 1 Chain reaction.Infinite medium (ignoring leakage at the surface).
Chain reacting pileChain reacting pile• k < 1 subcritical.• k = 1 critical system.
k 1 iti l
Chain reacting pileChain reacting pile
• k > 1 supercritical.For steady release of energy (steady-state operation) we need k =1
82
state operation) we need k =1.
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
Controlled Fission• Average number of allall neutrons released per fission
ν (for thermal neutrons 0 0253 eV)ν (for thermal neutrons, 0.0253 eV).• 233U : 2.492• 235U : 2.418• 239Pu : 2.871• 241Pu : 2.927
• Reactor is critical (k = 1): rate of neutrons produced by fission = rate• Reactor is critical (keff = 1): rate of neutrons produced by fission = rate of neutrons absorbed absorbed + leaked.
Size and composition of the reactor. 83Nuclear Reactor Theory, JU, First Semester,2010-2011 (Saed Dababneh).
Controlled FissionProbability for a thermal neutron to
fi i 235 i235U thermal cross sections
cause fission on 235U isU t e a c oss sect o s
σfission ≈ 584 b.σscattering ≈ 9 b. σ
=≈1fCheck Check
numbers!numbers!σradiative capture ≈ 97 b. ασσ γ ++ 1f
If each fission produces an average of ν neutrons, then the mean number of fastfast fission neutrons produced per thermal neutron = ηnumber of fastfast fission neutrons produced per thermal neutron η
νσσ ff
αν
σσσ
νσσ
νηγ +=
+==
1f
f
a
f η <ν
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
84
Controlled Fission235U• Assume Assume natural natural uranium:uranium:
99 2745% 238U 0 7200% 235U99.2745% 238U, 0.7200% 235U.
Thermal σf = 0 b 584 bTh l 2 75 b 97 b
24 RπWhy?
Thermal σγ = 2.75 b 97 b
NN yyxxyx σσ +=Σ+Σ=Σ
4 Rπ
238UΣ / N (0 992745)(0)
Nyyxx )( σγσγ +=
• Σf / N = (0.992745)(0) + (0.0072)(584)
= 4 20 b24 Rπ
Doppler effect?Doppler effect?
4.20 b.• Σγ / N = (0.992745)(2.75) +
(0.0072)(97)
85
= 3.43 b.Nuclear Reactor Theory, JU, First Semester, 2010-
2011 (Saed Dababneh).
Using the experimental elastic scattering data the radius of the nucleus can be estimated.
Controlled Fission• Probability for a thermal neutron to cause fission in natural natural uraniumuranium
55.0433204
20.4=
+=
Compare to pure 235U and to 3% enriched fuel.
• If each fission produces an average of ν = 2.4 neutrons, then the mean number of fast fission neutrons produced per thermal
43.320.4 + to 3% e c ed ue
mean number of fast fission neutrons produced per thermal neutron = η = 2.4 x 0.55 ≈ 1.3• This is close to 1. If neutrons are still to be lost, there is a danger of losing criticality. (Heavy water?).• For enriched uraniumenriched uranium (235U = 3%) η = ????? (> 1.3). (Light water?)water?).• In this case η is further from 1 and allowing for more neutrons to be lost while maintaining criticality.
86
be lost while maintaining criticality.
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
Moderation (to compare x-section)
1H2H 1H(n,n)(n,n)
2H
(n,γ) (n,γ)(n,γ) (n,γ)
• Resonances?Resonances?• 3H production.
87Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
Controlled FissionHW HW 1111
∑ Σ= f ii )()(1 νη• Verify ∑Σ if
a
ii )()(νηy
• Comment on the calculation for thermal neutrons and a mixture of fissile and non-fissile materials, ,giving an example.• Comment for fast neutrons and a mixture of fissionable materials, giving an example.
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
88
Conversion and Breeding
Converters: Convert non thermally fissionable materialConverters: Convert non-thermally-fissionable material to a thermally-fissionable material.
_239min23239238 νβ ++⎯⎯ →⎯→+ −NpUnU
_2393.2 νβ ++⎯⎯→⎯ −Pud
σf,th = 742 bf,th
_233min22233232 νβ ++⎯⎯ →⎯→+ −PaThnTh
_23327 νβ
β
++⎯⎯→⎯ −UdIndia
89Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
βσf,th = 530 b
89Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
Conversion and Breeding
If 2 C i d fi i ibl• If η = 2 Conversion and fission possible.• If η > 2 Breeder reactor.• 239Pu: Thermal neutrons (η ~ 2 1) hard for breeding• Pu: Thermal neutrons (η ~ 2.1) hard for breeding.
Fast neutrons (η ~ 3) breeding fast breeder reactors.
• After sufficient time of breeding, fissile material can be easily (chemically) separated from fertile material(chemically) separated from fertile material.Compare to separating 235U from 238U.• Reprocessing.
90
p g
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
90Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
Controlled Fission• Note that η is greater than 2
t th l i dat thermal energies andalmost 3 at high energies.
Th “ t ” t
Variations in Variations in ηη• These “extra” neutrons are Used to convert fertile into fi il f lfissile fuel.• Plutonium economy.
I di d th i• India and thorium.• Efficiency of this process is d t i d b tdetermined by neutronenergy spectrum.
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
91
Controlled Fission
• Conversion 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 (fast breeder).• They are called “fast” because primary fissions inducing neutrons are fast not thermal, thus η > 2.5 but σf is only a few barns.• Moderator??
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
92
Controlled Fission• N thermalthermal neutrons in one generation have produced have produced so far so far ηηNN fastfast neutronsneutronsηηNN fastfast neutrons.neutrons.• Some of these fastfast neutrons can cause 238U fission more fast neutrons fast fission factorfast fission factor = ε (= 1.03 for natural uranium).neutrons fast fission factor fast fission factor ε ( 1.03 for natural uranium).• Now we have Now we have εηεηNN fastfast neutrons.neutrons.• We need to moderate these fast neutrons use graphite as an example for 2 MeV neutrons we need ??? collisions. How many for 1 MeV neutrons?• The neutron will pass through the 10 100 eV region during the• The neutron will pass through the 10 - 100 eV region during the moderation process. This energy region has many strongstrong 238Ucapture resonances (up to ????? b) Can not mix uranium and captu e eso a ces (up to b) Ca ot u a u a dmoderator.• In graphite, an average distance of 19 cm is needed for
93
thermalization the resonance escape probability resonance escape probability p (≈ 0.9).Nuclear Reactor Theory, JU, First Semester, 2010-2011
(Saed Dababneh).Reactor design.
Controlled Fission• Now we have Now we have ppεηεηN N thermalthermal neutrons.neutrons.• Moderator must not be too large to capture thermal neutrons;Moderator must not be too large to capture thermal neutrons; when thermalized, neutrons should have reached the fuel.• Graphite thermal cross section = 0.0034 b, but there is a lot of it present.• Capture can also occur in the material encapsulating the fuel l t ( l d)elements (clad).
• The thermal utilization factor thermal utilization factor f (≈ 0.9) gives the fraction of thermal neutrons that are actually available for the fuelthermal neutrons that are actually available for the fuel.• Now we have Now we have ffppεηεηNN thermalthermal neutronsneutrons, could be > or < Nthus determining the criticality of the reactor.
k fk∞ = fpεη The fourThe four--factor formula.factor formula.
k = k ff = fpεη(1-lf )(1-l h l)94
k keff fpεη(1 lfast)(1 lthermal)Fractions lost at surfaceNuclear Reactor Theory, JU, First Semester, 2010-2011
(Saed Dababneh).
Controlled Fission
k∞ = fpεη, leaknoneff Pfk −= ρεη
• Fast from thermal, as defined in HW 11.
• Fast from fast ε∑ Σ
Σ=
if
a
ii )()(1 νη• Fast from fast, ε.• Thermal from fast, p.• Thermal available for fuel
a
d
∑= idtl df l
fuelaf• Thermal available for fuel
Thinking QUIZThinking QUIZ
..mod +∑+∑+∑+∑+∑ poisona
rodsa
eratora
clada
fuela
f
Thinking QUIZThinking QUIZ• For each thermal neutron absorbed, how many fast neutrons are produced? Will need this when discuss two group diffusion
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
95
neutrons are produced? Will need this when discuss two-group diffusion.
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?
96Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
Neutron Life Cycley
Why should we b t th ?worry about these?
How?f
How?
97Nuclear Reactor Theory, JU, First Semester, 2010-2011 (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, JU, First Semester, 2010-2011 (Saed Dababneh).
98
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, JU, First Semester, 2010-2011 (Saed Dababneh).
99
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)( −=
100
0)(Nuclear Reactor Theory, JU, First Semester, 2010-2011
(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 …).
101
( )Cd control rodsCd control rods
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (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).• Shielding (for neutrons and γ’s).• Coolant.• Control system.• Emergency systems (to prevent runaway during failure).
Chapter 4 in Lamarsh
102Nuclear Reactor Theory, JU, First Semester, 2010-2011 (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?
103Nuclear Reactor Theory, JU, First Semester, 2010-2011 (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???
104Nuclear Reactor Theory, JU, First Semester, 2010-2011 (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?
105Nuclear Reactor Theory, JU, First Semester, 2010-2011 (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).
• ….. 106Nuclear Reactor Theory, JU, First Semester, 2010-2011
(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.
107Nuclear Reactor Theory, JU, First Semester, 2010-2011 (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, JU, First Semester, 2010-2011 (Saed Dababneh).
108
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, JU, First Semester, 2010-2011
(Saed Dababneh).109
preferred direction??? th
More on Moderators
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
110
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, JU, First Semester, 2010-2011 (Saed Dababneh).
111
⎪⎩ 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
+−+
=A
A θθ)(
For doing so, you need to verify and useCMA θ1
CM
CM
AAA
θθθcos21
cos1cos2 ++
+=
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
112
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, JU, First Semester, 2010-2011 (Saed Dababneh).
113
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, JU, First Semester, 2010-2011
(Saed Dababneh).114
• 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, JU, First Semester, 2010-2011 (Saed Dababneh).
115
One-Speed Interactions• Particular general.Recall:Recall:• Neutrons don’t have a chance to interact with each other (BAU 2007 review test!) Simultaneousother (BAU 2007 review test!) Simultaneous beams, different intensities, same energy:
Ft = Σt (IA + IB + IC + …) = Σt (nA + nB + nC + …)vFt Σt (IA IB IC …) Σt (nA nB nC …)v• In a reactor, if neutrons are moving in all directions
n = nA + nB + nC + …n nA nB nC …
Rt = Σt nv = Σt φRt Σt nv Σt φ
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
116
One-Speed Interactions
Ωdrn ),( ωrr
ωr≡ Neutrons per cm3 at )( ωr whose velocity
vector lies within dΩΩdabout ω.
∫ Ω= ω),()( drnrn rrr
rr∫π4
• Same argument as before ΩddI )()( rrrrΩ= vdrnrdI ),(),( ωω
rrrr
),()(),( rdIrrdF trrrrr ωω ∑=
)()()()(),()(),()()(
),()(),(
4
rrrvnrdrnvrrdFrFrR
rdIrrdF
ttt
trrrrrrrrrrr φωω
ωω
πω
∑=∑=Ω∑===
∑
∫∫Scalar
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
117
4πω
Ω= ∫ drnvrπ
ωφ4
),()( rrrwhere
Multiple Energy Interactions
ΩdEdErn )( ωrr
≡ Neutrons per cm3 at r with energy• Generalize to include energy
ΩdEdErn ),,( ω ≡ Neutrons per cm3 at r with energy interval (E, E+dE) whose velocity vector lies within dΩ about ωvector lies within dΩ about ω.
∫ Ω= ω),,(),( dEdErndEErnrrr ∫ ∫
∞
Ω= ),,()( ω dEdErnrnrrr
∫π4
∫ ∫0 4π
dEErErdEEvErnErdEErR ttt ),(),()(),(),(),( rrrrr φ∑=∑=
∫∞
∑= ),(),()( dEErErrR ttrrr φ
Scalar0Thus knowing the material properties Σt and the neutron flux φ, both as functions of space and energy, we can calculate the interaction rate
Scalar
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
118
p gy,throughout the reactor.
Neutron Current• Similarly and so on …∫
∞
∑=0
),(),()( dEErErrR SSrrr φ
• Redefine as
0Scalar
ΩddI )()( rrrr ΩdvrnrId rrrrrr )()( ωω• Redefine as Ω= vdrnrdI ),(),( ωω Ω= dvrnrId ),(),( ωω
Ω= ∫ drnvr )()( ωφrrr
Ω= ∫ drnvJ )( ωrrrrOne group!
Ω= ∫ drnvr ),()(4
ωφπ
Ω= ∫ drnvJ ),(4
ωπ
Neutron current densityNeutron current densityNeutron current densityNeutron current density
Jr• From larger flux to smaller flux!
Neutrons are not pushed! J• Neutrons are not pushed!• More scattering in one direction than in the other
119
than in the other.xJxJ =• ˆ
r
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
Equation of ContinuityNNetet flow of neutrons per second per unit area normal to the x direction:
∫ Ω==•π
θω4
cos),(ˆ dvrnJxJ xxrrr
π4In general: nJnJ =• ˆ
r
Equation of ContinuityEquation of ContinuityEquation of ContinuityEquation of Continuity
∫∫ ∫∫ •−∀∑−∀=∀∂∂
a dAntrJdtrrdtrSdtrn ˆ),(),()(),(),( rrrrrr φ±
∫∫ ∫∫∂ ∀ ∀∀ Aat
)()()()()( φ
R f h i P d i Ab i “L kRate of change in number of neutrons
Production rate
Absorption rate
“Leakage in/out” rate
So rce S rface Normal to A
Nuclear Reactor Theory, JU, First Semester,2010-2011 (Saed Dababneh).
120
Volume Source distribution
function
Surface area
bounding ∀
Normal to A (outwards)
Equation of ContinuityUsing Gauss’ Divergence Theorem ∫∫ •∇=•
VS
rdBAdB 3rrrr
Recall:
∫ Q
∀•∇=• ∫∫∀
dtrJdAntrJA
),(ˆ),( rrrrr
0
0
ερ
ε
=⋅∇
=⋅∫E
AdES
Q
∀A0ε
∫∫ ∫∫ •−∀∑−∀=∀∂∂
∀ ∀∀ Aa dAntrJdtrrdtrSdtrn
tˆ),(),()(),(),( rrrrrr φ
∂ ∀ ∀∀ AtBoth flux and current!!
Convert current to flux?One group!
),(),()(),(),(1 trJtrrtrStrtv a
rrrrrrr•∇−∑−=
∂∂ φφtv ∂
E ti f C ti itE ti f C ti itNuclear Reactor Theory, JU, First Semester, 2010-2011
(Saed Dababneh).121
Equation of ContinuityEquation of Continuity
Equation of Continuity
Steady statey
0)()()()( •∇∑ rJrrrS rrrrrr φ 0)()()()( =•∇−∑− rJrrrS a φ
Non-spacial dependence
)()()( ttStnt a φ∑−=∂∂t∂
Delayed sources? Will do it later.Nuclear Reactor Theory, JU, First Semester, 2010-2011
(Saed Dababneh).122
Delayed sources? Will do it later.
Fick’s Law
• The exact interpretation of neutron transport in heterogeneous domains is so complexheterogeneous domains is so complex.• Assumptions and approximations.• Simplified approachesSimplified approaches.• Simplified but accurate enough to give an estimateestimate of the average characteristicsaverage characteristics of neutron populationneutron populationthe average characteristics average characteristics of neutron populationneutron population.• Numerical solutions.• Monte Carlo techniquesMonte Carlo techniques.
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
123
Fick’s LawAssumptions:1. The medium is infinite.1. The medium is infinite.2. The medium is uniform 3. There are no neutron sources in the medium.
).(rnot r∑∑
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 of5. The neutron flux is a slowly varying function of
position.6. The neutron flux is not a function of time.6. The neutron flux is not a function of time.
htt // i /EJ/ ti l /0143 0807/26/5/023/ j 5 5 023 df
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
124
http://www.iop.org/EJ/article/0143-0807/26/5/023/ejp5_5_023.pdf
Fick’s Law
Lamarsh puts it more bluntly:Lamarsh puts it more bluntly:“Fick’s Law is invalid: a) in a medium that strongly absorbs neutrons;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; andsource or the surface of a material; and c) when neutron scattering is strongly anisotropic.”
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
125
Fick’s Law
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
126
Fick’s Lawφ(x) Negative Flux Gradient
Current Jx• Diffusion: random walk of an ensemble of particles
High flux
pfrom region of high “concentration” to region g f
More collisions
gof small “concentration”.• Flow is proportional to
Low flux
p pthe negative gradient of the “concentration”.
x
Less collisions
• From larger flux to smaller flux!Recall:
x
DJ x ∂∂
−=φ
From larger flux to smaller flux!• Neutrons are not pushed!• More scattering in one direction
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
127
xx ∂than in the other.
Fick’s Lawz Number of neutrons scatteredscattered per
second from d∀ at rr and going d∀
θ ∀∑ Σ− dedAr rz tcos)( θφ r
g gthrough dAz
d∀
rdAz
θ ∀∑ der
rs 24)(
πφ
R dy
z
φ ∑∑
Removed en route
(assuming no
xφ
Slowly varying)(rnot ssr
∑∑ (assuming no buildup)
Isotropic
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
128
Fick’s Law
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
129
Fick’s Law
∞Σ π π2 2/dAHW HW 1414
[ ]∫ ∫ ∫= = =
Σ−− Σ=
φ θ
φθθθφπ 0 0 0
sincos)(4 r
rzszz ddrderdAdAJ t
r
= = =φ θ0 0 0r
⎞⎛⎞⎜⎛ ∑ ∂φ
?=+zz dAJ
023
⎟⎠⎞
⎜⎝⎛⎟⎠
⎞⎜⎜⎝
⎛Σ∑
−=−= −+
zJJJ
t
szzz ∂
∂φand show that
D ≈1
and generalize2sDDJ ∑
=∇−= φr Diffusion Diffusion
coefficientcoefficient
sΣ3
g23 tΣ
φ coefficientcoefficient
The current density is proportional to the negative of the gradient Total removal
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
130
y p p g gof the neutron flux.
Fick’s LawValidity:1. The medium is infinite. Integration over all space.1. The medium is infinite. Integration over all space.
after few mean free paths 0 corrections at the surface are still required.
rte ∑−
corrections at the surface are still required.2. The medium is uniform.
φ and Σ are functions of space re-)(rnot ssr
∑∑)(rsr
∑ φ and Σ are functions of space rederivation of Fick’s law? locally larger Σs extra JJ cancelled by iff ???
)(s
rr sat ee )( ∑+∑−∑− = HWHW 1515JJ cancelled by iff ???Note: assumption 5 is also violated!
3. There are no neutron sources in the medium.
ee HW HW 1515
3. There are no neutron sources in the medium.Again, sources are few mean free paths away and corrections otherwise.
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
131
co ect o s ot e se
Fick’s Law4. Scattering is isotropic in the lab. coordinate system!
If reevaluate D.02)(cos ≠== θµ HWHW 1616If reevaluate D.3
)(A
µ
11 trD λ===
HW HW 1616
33)(3 trst
Dµ ∑∑−∑
λ
Weekly absorbing Σt = Σs.
For “practical” moderators:µ
λλ−
≅1
str
5. The flux is a slowly varying function of position.Σa ↑ variation in φ ↑.Σa ↑ variation in φ ↑.
)(2
2
rr
r
∂∂ φ ??
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
132
r∂
Fick’s Law
Estimate the diffusion coefficient of graphite at 1 eV.HW HW 1717
g pThe scattering cross section of carbon at 1 eV is 4.8 b.
Scattering OthScattering Other materials?
Absorption
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
133
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!!!!!! g y p
43 101101.011
/ −− ==∆
=∆ xxmsφφφφ
Very small fractional change during the time 1 1smsφ
y g gneeded for the neutron to travel this “significant” distance.
φ∇−≅r
DJNuclear Reactor Theory, JU, First Semester, 2010-2011
(Saed Dababneh).134
φ∇≅ DJ
Back to the Continuity Equation
1 ∂ ),(),()(),(),(1 trJtrrtrStrtv a
rrrrrrr•∇−∑−=
∂∂ φφ
),(),()(),(),(1 trDtrrtrStr rrrrrrr φφφ ∇•∇+∑−=∂ ),(),()(),(),( trDtrrtrStrtv a φφφ ∇∇+∑∂
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
135
The Diffusion Equation
),(),()(),(),(1 trDtrrtrStrtv a
rrrrrrr φφφ ∇•∇+∑−=∂∂tv ∂
If D is independent of r (uniform medium)Laplacian
)()()()()(1 2 trDtrrtrStr rrrrr φφφ ∇+∑∂
Laplacian
),(),()(),(),( trDtrrtrStrtv a φφφ ∇+∑−=∂
l H l h lt ti
)()()()(0 2 rDrrrS arrrr φφ ∇+∑−=
or scalar Helmholtz equation.
)()()()( a φφ
)()()(0 2 rDrr rrr φφ ∇+∑Nuclear Reactor Theory, JU, First Semester, 2010-2011
(Saed Dababneh).136
)()()(0 rDrra φφ ∇+∑−= Buckling equation.
Steady State Diffusion Equation)()()()(0 2 rDrrrS arrrr φφ ∇+∑−=
DDL∑
=2Define L ≡ Diffusion LengthL2 ≡ Diffusion Area a∑ us o eaModeration Length
S1DS
L−=−∇ φφ 2
2 1
B d C ditiB d C diti01
22 =−∇ φφ
L
Boundary ConditionsBoundary Conditions• Solve DE get φ.
Solution must satisfy BC’s2 φφL • Solution must satisfy BC’s.
• Solution should be real and non-negative.
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
137
non negative.
Steady State Diffusion EquationOneOne--speed neutron diffusion in infinite mediumspeed neutron diffusion in infinite medium
Point sourcePoint source
0)(1)( 22 =−∇ r
Lr rr φφ 2L
2HWHW 1818
0)(1)(2)( 22
2
=−+ rL
rdrd
rr
drd φφφ
HW HW 1818
LdrrdrLrLr //−General solution
reC
reA
LrLr //
+=φ
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
138
A, C determined from BC’s.
Steady State Diffusion EquationBC BC r → ∞ φ → 0 C = 0.
e Lr /−HW HW 18 18 (continued)(continued)
reA=φ
S eS Lr /−
Show that D
SAπ4
=r
eD
S4
=π
φa
DL∑
=2
neutrons per second absorbed in the ring.φπ adrr ∑24 neutrons per second absorbed in the ring.
rdr22
φa
Show that22 6Lr =
r
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
139
Steady State Diffusion Equation
HWHW 1919HW HW 1919Study example 5.3 and solve problem 5.8 in Lamarsh.Study example 5.3 and solve problem 5.8 in Lamarsh.
Multiple Point Sources?
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
140
Steady State Diffusion EquationOneOne--speed neutron diffusion in a finite mediumspeed neutron diffusion in a finite medium
At th i t f A B
BA φφ =• At the interface
BA φφdDdDJJ BA
AAφφ
−=−⇒=
What if A or B is a ac m?
dxD
dxDJJ BABA =⇒=
x• What if A or B is a vacuum?• Linear extrapolation distance.
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
141
More realistic multiplying mediumOneOne--speed neutron diffusion in a multiplying mediumspeed neutron diffusion in a multiplying mediumThe reactor core is a finite multiplying mediumThe reactor core is a finite multiplying medium.• Neutron flux?• Reaction rates?• Reaction rates?• Power distribution in the reactor core?Recall:Recall:• Critical (or steady-state):Number of neutrons produced by fission = numberNumber of neutrons produced by fission = number of neutrons lost by:absorption rate productionneutron (S)k =absorptionandleakage
)( rate absorptionneutron Ak =∞
)( rate productionneutron Sk =
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
142
leakage)( rate leakageneutron )( rate absorptionneutron LEA
keff +=
More realistic multiplying medium
yprobabilit leakage-nonleaknoneff P
LEAA
kk
−=+
=Things to be used later…!
LEAk∞ +SALE area surface∝ For a critical reactor:
Recall:
aSALEVolumeVS
12
∝For a critical reactor:
Keff = 1K > 1
aaa
VSA
SLE 1
3 =∝∝
St d t t h tSt d t t h t
K∞ > 1
)()()(0 2 rDrrk aarrr φφφ ∇+∑−∑= ∞
Steady state homogeneous reactorSteady state homogeneous reactor)()()( aa φφφ∞
2222 10)()( kBrBr −≡=+∇ ∞rr φφ
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
143
2)()(L
φφ
Material buckling
More realistic multiplying medium0)()( 22 =+∇ rBr rr φφ
)(2 rφ)(
)(22
rrB r
r
φφ∇
−=)(rφ
• The buckling is a measure of extent to which the flux curves or “buckles ”curves or buckles.• For a slab reactor, the buckling goes to zero as “a”goes to infinity There would be no buckling or curvaturegoes 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• Buckling can be used to infer leakage. The greater the curvature, the more leakage would be expected.
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
144
More on One-Speed DiffusionHW HW 2020
Show that for a critical homogeneous reactorcritical homogeneous reactorShow that for a critical homogeneous reactorcritical homogeneous reactorφφP aa1 ∑
=∑
==φφφφ DBDLB
Paa
leaknon 2222 1 +∑=
∇−∑=
+=−
Infinite Bare Slab ReactorInfinite Bare Slab Reactor (( d diff i )d diff i )Infinite Bare Slab Reactor Infinite Bare Slab Reactor (one(one--speed diffusion)speed diffusion)φ• Vacuum beyond.
R t t 0
z
xa/2
• Return current = 0.d = linear extrapolation distance
0 71 λ (for plane s rfaces)
Reactor
aa/2
a0/2= 0.71 λtr (for plane surfaces)= 2.13 D.
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
145
d d
More on One-Speed DiffusionHW HW 2121
022
=+ φφ BdFor the infinite slab Show that the02 =+ φBdx
For the infinite slab . Show that the general solution
BxCBxAx sincos)( +=φwith BC’s
BxCBxAx sincos)( +=φ0)
2( 0 =±
aφFlux is symmetric about
0)(2
0
=xdx
xdφFlux is symmetric about
the origin.0=xdx
0cos)( φφ == ABxAx 0)( φφ
,...5,3,)(0)(cos)( 000 πππφ =±⇒=±=±aBaBAa
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
146
,...2
,2
,2
)2
(0)2
(cos)2
(φ ±⇒±± BBA
More on One-Speed Diffusion
,...2
5,2
3,2
)2
( 0 πππ=±
aBHW HW 21 21 (continued)(continued)
222253a πππ
= ,...,,0 BBBa =
Fundamental mode the only mode significant inFundamental mode, the only mode significant in critical reactors. 2
⎞⎜⎛ ππ BucklinglGeometricacos)(
0
2
00 ≡⎟
⎠
⎞⎜⎜⎝
⎛==
aBx
ax ππφφ
For a critical reactor the geometrical buckling is equalFor a critical reactor, the geometrical buckling is equal to the material buckling.To achieve criticality
21k −⎞
⎜⎛ π
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
147
To achieve criticality 2
0
1L
ka
−=⎟
⎠
⎞⎜⎜⎝
⎛ ∞π
More on One-Speed Diffusion???0φ
2⎞⎛
• To achieve criticality 2
2
0
1L
ka
−=⎟⎟
⎠
⎞⎜⎜⎝
⎛ ∞π
• But criticality at what power level??• φ can not be determined by the geometry alone
0 La ⎠⎝
• φ0 can not be determined by the geometry alone.
π xa
Px0
0 cos,..,..)()( πφφ =
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
148
More on One-Speed DiffusionSpherical Bare Reactor (oneSpherical Bare Reactor (one--speed diffusion)speed diffusion)
22 46 aa π3
343
46aa
aa
ππ
>CubeCube SphereSphere
Minimum leakage minimum fuel to achieve criticality.φ2 2
2 φφ ddHWHW 2222 02 22 =++ φφφ B
drd
rdrdHW HW 2222
CA BrrCBr
rA sincos +=φ
r
r0Br
rr
rC ππφ == 0
0
,sin Continue!Reactor
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
149
r0Brr 0
More on One-Speed DiffusionHW HW 2323
Infinite planer source in an infinite Infinite planer source in an infinite didi
φφmedium.medium.
LxeSLx /)( −=φxSxd )(1)(2 δφφ
φφ
eD
x2
)( =φDLdx
)()(22 φφ
−=−
HW HW 2424Infinite planer source in a finiteInfinite planer source in a finite
xa/2( )[ ]2/2sinh LxaSL
Infinite planer source in a finite Infinite planer source in a finite medium.medium.
aa/2
a0/2( )[ ]
)2/cosh(2/2sinh
2 0
0
LaLxa
DSL −
=φ
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
150Source
More on One-Speed DiffusionInfinite planer source in a multiInfinite planer source in a multi--region medium.region medium.
aa ±=± )2/()2/( φφ
FiniteInfinite InfinitedDdD
aa
=
±=±
22
11
21 )2/()2/(φφ
φφ
φφFiniteInfinite Infinite
BCmoredxdx axax
+±=±= 2/
22/
1
P j tP j t 22Project Project 22
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
151
Back to Multiplication Factor
k∞ = fpεη, leaknoneff P
kk
−= leaknoneff Pfk −= ρεηThings to be used later…!Recall:
k∞leaknoneff
• Fast from thermal,F t f f t
∑ ΣΣ
=i
f ii )()(1 νη• Fast from fast, ε.• Thermal from fast, p.
Th l il bl f fi i
Σ ia
fuelaf ∑• Thermal available for fission
Thi ki QUIZThi ki QUIZ
poisona
eratora
clada
fuela
af∑+∑+∑+∑
∑= mod
Recall:Thinking QUIZThinking QUIZ• For each thermal neutron absorbed, how many fast
t d d?neutrons are produced?
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
152
Two-Group Neutron Diffusion•• Introductory to multiIntroductory to multi--group (Hence crude).group (Hence crude).• All neutrons are either in a fast or in a thermal energy group.• Boundary between two groups is set to ~1 eV.• Thermal neutrons diffuse in a medium and cause fission, are captured, or leak out from the system.• Source for thermal neutrons is provided by the slowing down of fast neutrons (born in fission).• Fast neutrons are lost by slowing down due to elastic scattering in the medium, or leak out from the system, or due to fission or capture.• Source for fast neutrons is thermal and fast neutron
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
153
fission.
Two-Group Neutron Diffusion
FastdErErMeV
∫=10
1 ),()( rr φφ
ThermaldErEreV
eV
∫
∫1
1
)()( rr φφ ThermaldErEr ∫=0
2 ),()( φφ
φνφν ∑+∑
221122
212
1
222111
φφφφφνφν ff
eff DDk
∑+∑+∇−∇−
∑+∑=
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
154
22112211 φφφφ aaDD ∑+∑+∇∇
Two-Group Neutron Diffusion)()()(0 1
21111 rDrrS a
rrr φφ ∇+∑−=
RemovalRemoval cross section Depends on th l d f t
Fast diffusion coefficient
= fission + capture + scattering to group 2
thermal and fast fluxes.
)()()()(0 12
1112211 rDrrr affrrrr φφφνφν ∇+∑−∑+∑=
koror
)()()(0 12
11122 rDrrkaa
rrr φφφρ
∇+∑−∑= ∞
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
155
ρ
Two-Group Neutron Diffusion)()()(0 2
22222 rDrrS a
rrr φφ ∇+∑−=
Thermal diffusionThermal absorption D d f t Thermal diffusion
coefficientcross section = fission + capture.
Depends on fast flux.
)()()(0 22
222121 rDrr asrrr φφφ ∇+∑−∑= →
)()()(0 2D rrr φφφ ∇∑∑oror
)()()(0 22
22211 rDrr aarrr φφφρ ∇+∑−∑=
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
156
Two-Group Neutron Diffusion
)()()(0 12
11122 rDrrkaa
rrr φφφρ
∇+∑−∑= ∞
ρ)()()(0 2
222211 rDrr aa
rrr φφφρ ∇+∑−∑= )()()( 222211 aa φφφρ ∑∑
• A coupled system of equations; both depend on p y q pboth fluxes.• Recall also, for a steady state system:y y
0)()( 12
12 =+∇ rBr rr φφ
0)()(
)()(
22
22
11
=+∇ rBr rr φφ
φφ
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
157
)()( 22 φφGeometrical
Two-Group Neutron Diffusion
2 ∑∑ ∞kBD
Homogeneous system Determinant of coefficients matrix = 0Review Cramer’s
02
221
22
11 =−∑−∑
∑−∑− ∞
BD
BD
aa
aa
ρρ
Review Cramer s rule!
Do we need it here?
0))(( 122
222
11 =∑∑−−∑−−∑− ∞kBDBD aaaa ρ
here?
0))(( 122
222
11
122211
=∑∑−+∑+∑ ∞kBDBD
aaaa ρ
011)1)(1(
0))((
222
22
2
122211
=−++
∑∑+∑+∑ ∞
kBB
kBDBD aaaa
0)1)(1(
0))((
2222
2222
=++
++ ∞
kLBLB
LLkB
LB
L ThermalFastThermalFast
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
158
0)1)(1( =−++ ∞kLBLB ThermalFast
Two-Group Neutron Diffusion0)1)(1( 2222 =−++ ∞ThermalFast
kkLBLB
1)1)(1( 2222 =
++∞
ThermalFast LBLBk
11k1
11
12222 ++
== −−∞ FastThermal
Thermalleaknon
Fastleaknon
eff
LBLBPP
kk
∞ FastThermal
1kkFor large reactors
22
222
11)(1 M
kBLLB
k
ThermalFast
−=⇒=
++∞∞
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
159
Two-Group Neutron Diffusion222FastThermal LLM += If any ↑
leakageFastThermal
D 1λ
leakage ↑.
traa
tr
aThermal
DL∑∑
=∑
=∑
=3
13
2 λ
FastnL∑∑
=≡3
ageFermi 2
trsFast ∑∑3
g
• Slowing down density.g y• Fermi model.
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
160
Reactor Model: One-Group• Before considering multi-group.• So far we did 1-D.So far we did 1 D.• Back to oneone--groupgroup but extend to 33--DD.
z
F h h i fi iHW HW 2121||
φ
ReactorFor the homogeneous infinite slab reactor, extend the
iti lit diti th t
xa/2Reactor
criticality condition that you found in HW 21.
ad d
a0/2
DLkB
aB af
mg
∑−∑=
−==⎟
⎠
⎞⎜⎜⎝
⎛= ∞
νπ2
22
2 1 11--DD
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
161
DLa ⎠⎝ 0
Reactor Model: One-Group
)( 22 xd φ
• In 3-D
02222 ∂∂∂ φφφφ B
Daf ∑−∑ν
0)()( 22 =+ xB
dxxd φφ 02
222 =+∂∂
+∂∂
+∂∂ φφφφ B
zyx
Bxcos0φφ = zByBxB zyx coscoscos0φφ =k f ∑−∑−⎞
⎜⎛ νπ 2
22 1
kBBBBB af ∑−∑−⎞⎜⎜⎛
+⎞
⎜⎜⎛
+⎞
⎜⎜⎛
++ ∞νπππ 2
2222222 1DL
kBa
B afmg
∑∑===⎟
⎠
⎞⎜⎜⎝
⎛= ∞
νπ2
2
0
2 1
DLB
cbaBBBB f
mzyxg ===⎠
⎜⎜⎝
+⎠
⎜⎜⎝
+⎠
⎜⎜⎝
=++= ∞2
2
000
2222
Critical dimensions (size), for the given material properties, predicted by the model.
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
162
Reactor Model: One-Group• Transient case.
t !
),()(),()(),(),(1 trrDtrrtrStrtv a
rrrrrrrr φφφ ∇•∇+∑−=∂∂tv ∂
Moderator structure
fuelfuelf
fuela γ∑+∑=∑t !
• Delayed neutrons!!
Moderator, structure, coolant, fuel, …
• Reflectors!!• For homogeneous 1-D:
2
),(),(),(),(12
2
txx
DtxtxStxtv a φφφ
∂∂
+∑−=∂∂
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
163
xtv ∂∂),( txf φν ∑
Reactor Model: One-Group
),(),(),(),(12
2
txDtxtxtx af φφφνφ∂∂
+∑−∑=∂∂ ),(),(),(),( 2xtv af φφφφ
∂∂HW HW 2525Separation of variables: )()(),( tTxtx ψφ =
21 T ∂∂ ψ2
1x
DTTTtT
v af ∂∂
+∑−∑=∂∂ ψψψνψ
constant)(12
2
=−≡⎥⎦
⎤⎢⎣
⎡∑−∑+
∂∂
=∂∂ λψνψ
afDvTT
)(2 ⎥⎦
⎢⎣ ∂∂
ψψ afxtT
)0()( teTtT λ−= )( 2DBv ∑+∑= νλShow that
= 0 for steady state.
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
164
,)0()( eTtT = )( fa DBv ∑−+∑= νλShow that
Reactor Model: One-GroupHW HW 25 25 (continued)(continued)
0)( 0±aφ xBx cos)( =ψ
22 ⎞
⎜⎛ nB πt0)
2( 0 =±φ xBx nn cos)( =ψ
0
2
⎠⎜⎜⎝
=a
Bntry
2 )( 2fnan DBv ∑−+∑= νλeigenvalues
⎞⎛ π?Solution ∑ ⎟⎟⎠
⎞⎜⎜⎝
⎛= −
oddn
tn a
xneAtx n
0
cos),( πφ λ?
?
Initial condition ∑ ⎟⎠
⎞⎜⎜⎝
⎛= n a
xnAx cos)0,( πφ
?
Show that ⎠
⎜⎝oddn a0
∫+
⎟⎠
⎞⎜⎜⎝
⎛= 2
0
0cos)0,(2 a
a dxxnxAnπφ
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
165
∫− ⎠⎜⎝2
000 aa
Reactor Model: One-Group2
2 ⎟⎠
⎞⎜⎜⎝
⎛=
nBnπ
...25
23
21 <<< BBB
0 ⎠⎜⎝ a
)( 2fnan DBv ∑−+∑= νλ
531
...25
23
21 <<< λλλ)( fnan 531
)( 2DBv ∑−+∑= νλ Slowest decaying eigenvalue)( 11 fa DBv ∑+∑= νλ Slowest decaying eigenvalue.
xBeAxeAtx tt coscos)( 11 λλ πφ −− =⎞
⎜⎜⎛
≅ xBeAa
eAtx 110
1 coscos),(φ =⎠
⎜⎜⎝
≅
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
166
Reactor Model: One-Group0)( 2
11 =∑−+∑= fa DBv νλFor steady state
222 af BBB ≡∑−∑ν
CriticalityCriticality 0=λ1 mf
g BD
BB ≡==CriticalityCriticality 01 =λ
22mg BB < 01 <λSuper criticalitySuper criticality LE ↓
22mg BB > 01 >λSub criticalitySub criticality LE ↑g 1
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
167
Reactor Model: One-Group• That was for the bare slab reactor.• What about more general bare reactor models?g
),()(),()(),(),(1 trrDtrrtrStrt a
rrrrrrrr φφφ ∇•∇+∑−=∂∂
• For steady state, homogeneous model:
tv ∂
y g
0),(1),(),(),( 222 =
−+∇=
∑−∑+∇ ∞ tr
Lktrtr
Dtr af rrrr φφφ
νφ
• BC: φ(extrapolated boundary) = 0.
2LDφ( p y)
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
168
Reactor Model: One-Group• R0, H0 are the extrapolated dimensions.
1 22∂⎞
⎜⎛ ∂∂ φφ 01 2
2 =+∂
+⎠⎞
⎜⎝⎛ ∂
∂∂ φφφ B
dzdrr
rrR
• BC’s:0),( 0 =zRφ H
z
0),(
),(
2
0
0 =± Hrφ
φ Hx
yrθ
• Let)()(),( zrzr Ζℜ=φ
B lReactor
θ
• Solve the problem and discuss criticality condition.Solve the problem and discuss criticality condition.cosBesselHW HW 2626
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
169
Reactor Model: One-Group
R• Briefly, we go through HW 26.
1 2∂⎞⎛ ∂∂ φφ 01 22 =+
∂+⎠⎞
⎜⎝⎛ ∂
∂∂ φφφ B
dzdrr
rr)()()( Ζℜφ
z
H)()(),( zrzr Ζℜ=φ2Ζ πzd
y
Reactor
coscos00
22
⎞⎛ ℜ
==Ζ⇒=Ζ+Ζ πλλ
ddH
zzdzd x rθ
Reactor02 =ℜ+⎟⎠⎞
⎜⎝⎛ ℜ α
drdr
drd
)()( 00 rCYrAJ αα +=ℜNuclear Reactor Theory, JU, First Semester, 2010-2011
(Saed Dababneh).170
Reactor Model: One-Group
0)(0 =⇒−∞→⇒→ CxYx n
00 4048.20)4048.2( RJ α=⇒=Nuclear Reactor Theory, JU, First Semester, 2010-2011
(Saed Dababneh).171
Reactor Model: One-GroupR
0 cos)4048.2(,...)(H
zR
rJPA πφ =
H
z00 HR
Hx
yrθ
φφ
Reactor
rθ
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
172
Reactor Model: One-Group
R0
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
173H0/2
Reflected Slab: One-GroupzReflected Slab ReactorReflected Slab Reactor
or or1 ∂
x
Core
Ref
lect
o
Ref
lect
o
),()(),()(
),()(),(1
trrDtrr
trrtrtv
a
f
rrrrrr
rrr
φφ
φνφ
∇•∇+∑−
∑=∂∂
xaa/2 bb
),()(),()(a φφ
For steady-state, homogeneous, 1-D)(2 xd Cφ
)(
0)()()(
2
2 =∑−∑+
xd
xdx
xdD
R
CCa
Cf
CC
φ
φνφ
C ≡ Core1
Recall:
0)()(2 =∑− x
dxxdD RR
aR φφ
R ≡ Reflector0)(1)( 2
2 =−∇ rL
r rr φφ
)2
()2
(),2
()2
(,0)2
(BCs aJaJaaba CRCRR ===+ φφφ
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
174
Reflected Slab: One-Group
C
Ca
CfC
mCm
CC
DBxBAx
∑−∑== 2)()cos()(ν
φ
RRRR DL
xba
A =⎥⎥⎤
⎢⎢⎡ −+
= 2)(2sinhφ Ra
R LL
A∑
=⎥⎥
⎦⎢⎢
⎣
= )(sinhφ
⎤⎡C bB
BC
⎤⎡
⎥⎦⎤
⎢⎣⎡=
RC
RR
CmC
bDBLbAaBA sinh)
2cos(
⎥⎦⎤
⎢⎣⎡= RR
RCmC
mC
Lb
LDaBBD coth)
2tan(
⎥⎦⎤
⎢⎣⎡= R
RR
RCmCC
mC
LbA
LDaBABD cosh)
2sin(
⎥⎦⎢⎣LL2
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
175
Reflected Slab: One-Group
⎥⎦⎤
⎢⎣⎡= RR
RCmC
mC
Lb
LDaBBD coth)
2tan(Criticality condition.Criticality condition.
⎦⎣LL2
For bare slab CC• For bare slab CC was π / 2.
Smaller core for• Smaller core for reflected reactor ( ith )(even with a0 > a).• Save fuel.
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
176
Criticality “Calculation”• Can we solve “real” reactor problems analytically?• The previous discussion provides understanding of the concepts b t l i di t th d f t ti l t h it ti l t h ibut also indicates the need for computational techniques.computational techniques.
),()(),()(),()(),(1 trrDtrrtrrtr frrrrrrrrr φφφνφ ∇•∇+∑−∑=
∂
• Assume:
),()(),()(),()(),( trrDtrrtrrtrtv af φφφνφ ∇∇+∑∑
∂
)(),( retr t rr ψφ λ= )(),( ψφ
)()()()()()()( rrDrrrrr rrrrrrrrr ψψψνψλ ∇•∇+∑∑=
• Adjust parameters so that λ = 0 (Steady-state)
)()()()()()()( rrDrrrrrv af ψψψνψ ∇•∇+∑−∑=
Adjust parameters so that λ 0 (Steady state).• What parameters and how to adjust them?
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
177
Criticality “Calculation”)()()()()()()( rrDrrrrr
v afrrrrrrrrr ψψψνψλ ∇•∇+∑−∑=
• Fixed design and geometry one free variable is krr ν
∑∑ )()()()()()( rrk
rrrrD ffudge
arrrrrr ψνψψ ∑=∑+∇•∇−
operators are ,1 FMFK
Mfudge
ψψ =
•• As we did earlier (be guided by HW As we did earlier (be guided by HW 2020):):∑∑∑ νψν
fudge
221 LBDk aff
fudge +
∑∑=
∑+∇•∇−
∑=
νψψ
ψνrr
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
178
1 LBD a +∑+∇•∇− ψψ
Criticality “Calculation”
22k affudge
∑∑=ν
ψψ FK
M 1=
221 LBfudge + K fudge
• Build an algorithm.g• “Guess” (reasonably) initial kfudge and ψ (or φ) for the zeroth iteration.• Calculate the initial source term.• Iterate: .andGuess 00 kφ
get 1 10
00
01
kSF
kM ⇒== φφφ
φ
converges.flux until .....on soand0
101
11
SSkk
FS
=
= φ
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
179
g0S
Criticality “Calculation”• Or:
==Fk sourcesfission φ
∫∫ ++ ii dVSdVF
Mk
sinks11φ
φ
∫
∫
∫ ++
+ == volume
i
volume
i
i
dVSdVFk
11
1
φ
∫∫ + ii
volume
idVS
kdVM
k 11φ
• If for example k > 1, take action to reduce source or
volumevolume k
increase absorption.•• How?How?
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
180
How to Adjust CriticalityReactor Kinetics Reactor Kinetics Reactor kinetics refers to the manipulation of parameters that affect k and to the subsequent direct response of the reactor system Examples are:the subsequent direct response of the reactor system. Examples are:
• Absorber rods or shim movements to compensate for fuel burnup. • Safety scram rods to rapidly shutdown the chain reaction• Safety scram rods to rapidly shutdown the chain reaction. • Control rods to provide real-time control to keep k = 1 or to maneuver up and down in power.• …..
Reactor Dynamics Reactor Dynamics Reactor dynamics refers to the more indirect feedback mechanisms due to power level effects and other overall system effects such as:
• Temperature feedback. • VoidVoid feedback. • Pump speed control (affects water density and temperature).
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
181
Pump speed control (affects water density and temperature). • …
How to Adjust Criticality
B f llB f llBefore all:Before all:
Core Design Core Design The transient response of the reactor to the above direct and indirect changes in basic parameters is highly dependent on the design details of the reactor. Sample issues are: p
• Where should the control rods be placed for maximum effectiveness? • Will the power go up or down if a void is introduced into the reactor? Will the power go up or down if a void is introduced into the reactor? • Will the power go up or down if core temperature goes up? • How often should the reactor be refueled? • and so on• and so on...
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
182
Multi-group Model• Wide neutron spectrum.• One-group, two-group? Should be generalized.g p g p g
GG
Fraction of an eV
),()(),()(),(1
11 \\\
\\\\ Strrtrrtr
tvextg
G
gggsg
G
ggfgggg
grr
rrrrr φφνχφ +∑+∑=∂∂ ∑∑
==
Flux averaged ),()(),()(),()( trrDtrrtrr gggsggagrrrrrrrr φφφ ∇•∇+∑−∑−
Identify the terms NOW∫−
≡1
)()(gE
dEtErtr rr φφ
Flux-averaged quantities.
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
183
Identify the terms, NOW.∫≡ ),,(),(gE
g dEtErtr φφ
Multi-group Model
Total fissionactio
n
Scattering in er s
ourc
es
)()()()()(1 S extGG rrrrr φφφ ∑∑
∂ ∑∑
Total fission
Fra Scattering in
Oth
e
)()()()()()(
),()(),()(),(1
11 \\\
\\\\
trrDtrrtrr
Strrtrrtrtv
extg
gggsg
ggfgggg
grrrrrrrr
rrrrr
φφφ
φφνχφ
∇•∇+∑−∑−
+∑+∑=∂∂ ∑∑
==
),()(),()(),()( trrDtrrtrr gggsggag φφφ ∇•∇+∑∑
LeakageScattering outAbsorption LeakagegAbsorption
Fraction of
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
184
an eV
Multi-group ModelMaxwellian
Fission1/E
ss o
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
185
55--group example.group example.
Multi-group Model
5
Total fission
[ ]
5
1\\\\ ),()( φνχ
ggfggg trr =∑∑
=
rr
[ ]111222333444555 φνφνφνφνφνχ fffffg ∑+∑+∑+∑+∑
Th l fi i F t fi iThermal fission (~ 97%)
Fast fission (~ 3%)
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
186
Multi-group Model
5
Scattering in
1\\\ ),()( φ
gggsg
trr =∑∑=
rr
5544332211 φφφφφ gsgsgsgsgs ∑+∑+∑+∑+∑
33=g
Upscattering!!??? Skipping!!???
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
187
Multi-group Model
tφ )()(∑rr
Scattering out
gsggsggsggsggsg
gsg trr
φφφφφ
φ
54321
),()(
∑+∑+∑+∑+∑
=∑
gggggggggg
3=g 3g
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
188
Multi-group Model5
)()()(1 φνχφ extStrrtr +∑=∂ ∑ rrr
Group 3
31
333
)()()()(
),()(),(\
\\\
φφ
φνχφg
gfgg
trrDtrr
Strrtrtv
∇•∇+∑−
+∑=∂ ∑
=
rrrrrr
[ ][ ]
335334333332331
3333 ),()(),()(
φφφφφφφφφφ
φφ
sssss
a trrDtrr
∑+∑+∑+∑+∑+∑+∑+∑+∑+∑−
∇•∇+∑
[ ]553443333223113 φφφφφ sssss ∑+∑+∑+∑+∑+
Removal cross section
∑+∑+∑+∑+∑=∑−∑+∑≡∑
353432313
33333
ssssa
ssar
∑=
∑+∑=5
133
353432313
\\
ggsa
ssssa
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
189
≠=
31
\gg
Multi-group Model
actio
n
er s
ourc
es
)()()()()(1 S extGG rrrrr φφφ ∑∑
∂ ∑∑
Total fissionFra Scattering in O
the
)()()()()()(
),()(),()(),(1
11 \\\
\\\\
trrDtrrtrr
Strrtrrtrtv
extg
gggsg
ggfgggg
grrrrrrrr
rrrrr
φφφ
φφνχφ
∇•∇+∑−∑−
+∑+∑=∂∂ ∑∑
==
),()(),()(),()( trrDtrrtrr gggsgggrg φφφ ∇•∇+∑∑
LeakageRemoval In-group Scattering
)()()()()(1 Sttt extGG rrrrr φφφ +∑+∑
∂ ∑∑ ),()(),()(),(\
\\\
\\\\
11
Strrtrrtrtv
extg
ggg
ggsgg
gfggggg
rr
φφνχφ +∑+∑=∂ ∑∑
≠== Net Scattering in
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
190
),()(),()( trrDtrr gggrgrrrrrr φφ ∇•∇+∑−
Multi-group Model),()(),()(),(1
11 \\\
\\\\ Strrtrrtr
tvextg
G
gggsg
G
ggfgggg
g
rrrrr φφνχφ +∑+∑=∂∂ ∑∑
==
),()(),()(),()( trrDtrrtrr gggsggagrrrrrrrr φφφ ∇•∇+∑−∑−
Calculate group-averaged:
)(),(),(),(),(,1\\ rDrrrr gsgagf
rrrrr∑∑∑∑
Calculate group averaged:
)(),(),(),(),(, \\v gsgaggsgfg
g
Or for,Or for,),()(),()(),(1
11 \\\
\\\\ Strrtrrtr
tvextg
G
gggsg
G
ggfgggg
g
rrrrr φφνχφ +∑+∑=∂∂ ∑∑
==
we need group-averaged )(),( rr rr∑∑
),()(),()(),()( trrDtrrtrr gggsgggrgrrrrrrrr φφφ ∇•∇+∑−∑−
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
191
we need group averaged )(),( rr sggrg ∑∑
Multi-group Model• Group-averaged parameters?• ENDF.
),,(),()()(),,()(
1 \\\\ dEtErErEEt
tErEv f
rsr
φνχφ∑=
∂∂
∫∞
)()(
)(
\\\
0
SdEtErEEr
tEv
extrr φ +→∑+
∂
∫∞ Units!
)()()()(
),,(),(0
tErErtErEr
SdEtErEErs
rrrr φφ
φ
∑−∑−
+→∑+ ∫
),,(),(
),,(),(),,(),(
tErErD
tErErtErEr sarrrr
φ
φφ
∇•∇+
∑∑
• Integrate term by term over groups and equate to equation of multi-group model.
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
192
Multi-group Model
∫−
≡1
),,(),(gE
Eg dEtErtr rr φφ• Define group flux
gE
∫−∂
=∂
=∂ 1
),,(1),(),(1 gE
gg dEtEr
trtr r
rr φ
φφ ∫∂∂∂
),,()(
),(gEg
gg Evtvttv
φφ
E
∫−1
),,()(
1
1
gE
E
dEtErEv
rφ
∫−
=1
),,(
)(1g
g
EE
g dEtErv rφ
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
193
∫gE
Multi-group Model
∫−
∇•∇=∇•∇1
)()()()(gE
dEtErErDtrrD rrrrrrrrφφ ∫ ∇•∇∇•∇ ),,(),(),()(
gEgg dEtErErDtrrD φφ
∫−
∇1
)()(gE
dEEED rrr φ∫ ∇
=
),,(),(
)( gE
dEtErErD
rD
rr
rφ
∫−
∇
=1
),,()(
gEg
dEtErrD
rrφ∫
gE
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
194
Multi-group Model
dEtErErtrrgE
∫−
∑=∑1
)()()()( rrrr φφ dEtErErtrrgE
agag ∫ ∑∑ ),,(),(),()( φφ
gE
∫−1
dEtErEr
r gEa∫ ∑
∑
),,(),(
)(
rr
rφ
dEtErr
g
g
Eag
∫−
=∑1
),,()(
rφgE∫ ),,(φ
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
195
Multi-group Model
dEtErErtrrgE
∫−
∑=∑1
)()()()( rrrr φφ dEtErErtrrgE
sgsg ∫ ∑∑ ),,(),(),()( φφ
gE
∫−1
dEtErEr
r gEs∫ ∑
∑
),,(),(
)(
rr
rφ
dEtErr
g
g
Esg
∫−
=∑1
),,()(
rφgE∫ ),,(φ
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
196
Multi-group Model
∫ ∫∑∞−
→∑=∑E
s
G
ggsg
g
dEdEtErEErtrr \\\
1
1
\\\ ),,(),(),()( rrrr φφ
∫ ∑ ∫
=
− −
→∑=E G
E
Eg
g g
g
dEdEtErEEr \\\
01
1 1\
\
)()( rr φ∫ ∑ ∫=
→∑=
E E
E g Es
g g
dEdEtErEEr1
\
\\
),,(),( φ
∑ ∫ ∫=
− −
→∑=G
g
E
E
E
Es
g
g
g
dEdEtErEEr1
\\\
\
1 1\
\
),,(),( rr φg g \
1 1\E E
∫ ∫− −
→∑=∑1 1\
\\
\\\\ ),,()(
),(1)(
g
g
g
g
E
E Es
ggsg dEdEtErEE
trr r
rr φ
φ
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
197
g g
Multi-group Model
∫ ∫∑∞−
∑=∑E
Ef
G
gfggg
g
dEdEtErEEEtrr \
0
\\\
1
1
\\\\ ),,()()()(),()( rrr φνχφνχ
∫∫∞
=
−
∑= f
E
Eg
g
g
dEtErEEdEE \\\\
01
1
\
),,()()()( rφνχ
∫
∫∫∞
∑=
fE g
dEtErEE \\\\
0
)()()(
)()()()(
rφνχ
φχ
∫−1
)(gE
dEEχχ
∑ ∫
∫=
∑=
GE
fg
g
dEtErEE
\\\\
0
1\
),,()()( φνχ∫= )(gE
g dEEχχ
∑ ∫=
∑=g E
fg
g
dEtErEE1
\\\\
\\
),,()()( rφνχ
∫=
∑=∑1\
\\\\\\ ),,()()(
)(1)(
gE
ffgg dEtErEEtr
r rr
r φνφ
ν
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
198
∫\
\ ),(g
Ef
gfgg trφ
Multi-group ModelENDF
High G, few meshmesh
points.Flux
Small G
Poison, burnup (or better consumption), Small G,
more mesh points.
)temperature, control rod position, etc…
Flux
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
199
Flux
Multi-group ModelWhat could we make life a little easier?!• No upscattering .for0)( \
\ ggr >=∑rNo upscattering
set group G to include neutrons up to ~1 eV..for 0)(\ ggr
gsg>∑
( ))()()()()()(1gG rrrrrr φφφ ∑∑∑ ∑∑
−
( )),()(),()(),()(11 \
\\\
\\ trrtrrtrr gsggg
ggsgg
ggsg
rrrrrr φφφ ∑+∑⇒∑ ∑∑==
Your choice of how to
• No group skipping when scattering down (directly
Your choice of how to tackle in-scattering.
No group skipping when scattering down (directly coupled).
( )G
( )),()(),()(),()( 1)1(1\
\\ trrtrrtrr gsgggggsg
ggsg
rrrrrr φφφ ∑+∑⇒∑ −−=∑HWHW 2727 H l d thi ? Wh t b t H?
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
200
HW HW 2727 How can we pledge this? What about H?
Multi-group ModelCriticalityCriticality
1Not all sources,
only fission.φφ F
kM 1
=Not only sinks
y
yIterations.Iterations.
),()(),()(),(1\\\\\ Strrtrrtr ext
g
G
ggsg
G
gfggggrrrrr φφνχφ +∑+∑=
∂∂ ∑∑
No upscatter
),()(),()(
),()(),()(),(\
\\ 11
trrDtrr
tv
gggrg
g
ggg
ggsgg
gfggggg
rrrrrr φφ
φφχφ
∇•∇+∑−
∂ ∑∑≠==
∑−
=
∑−∑+∇•∇−g
gggsggrggg trrtrrtrrD
1
1\\\ ),()(),()(),()( rrrrrrrr
φφφupscatter
∑
≠
∑=G
gfggg
ggg
trrK
\\\
\
),()(1 rr φνχRedundant when no upscatter.
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
201
∑=g
gfgggK 1\
)()( φχ
Multi-group Model
⎤⎡rr
No upscatter
⎥⎥⎥⎤
⎢⎢⎢⎡
∑+∇•∇−∑−∑+∇•∇−
= rrL
rrL
rr
2212
11
000
rs
r
DD
M
⎥⎥⎥
⎦⎢⎢⎢
⎣
∑+∇•∇−∑−∑−OMMM
L332313 rss D
⎥⎤
⎢⎡ 1φ⎥
⎤⎢⎡ ∑∑∑ L331221111 fff νχνχνχ
⎥⎥⎥
⎢⎢⎢
= 2
1
φφφ
φ⎥⎥⎥
⎢⎢⎢
∑∑∑∑∑∑
=L332222112 fffF
νχνχνχνχνχνχ
⎥⎥
⎦⎢⎢
⎣ M3φ
⎥⎥
⎦⎢⎢
⎣
∑∑∑OMMM
L333223113 fff νχνχνχ
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
202
⎦⎣Iterations.Iterations.
Multi-group Model
⎤⎡rr
No upscatter
⎥⎥⎥⎤
⎢⎢⎢⎡
∑+∇•∇−∑−∑+∇•∇−
= rrL
rrL
rr
2212
11
000
rs
r
DD
M
⎥⎥⎥
⎦⎢⎢⎢
⎣
∑+∇•∇−∑−OMMM
L33230 rs DDirectly coupled
⎥⎤
⎢⎡ 1φ⎥
⎤⎢⎡ ∑∑∑ L331221111 fff νχνχνχ
p
⎥⎥⎥
⎢⎢⎢
= 2
1
φφφ
φ⎥⎥⎥
⎢⎢⎢
∑∑∑∑∑∑
=L332222112 fffF
νχνχνχνχνχνχ
⎥⎥
⎦⎢⎢
⎣ M3φ
⎥⎥
⎦⎢⎢
⎣
∑∑∑OMMM
L333223113 fff νχνχνχ
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
203
⎦⎣Iterations.Iterations.
Multi-group ModelMultiMulti--group group ⇒⇒ oneone--groupgroup
E
∫∫∞
≡⇒≡−
0
),,(),(),,(),(1
dEtErtrdEtErtrgE
Eg
rrrr φφφφ0gE
E
∫∫∞−
0
),,()(
11
),,()(
1
1
1
dEtErEv
dEtErEv
g
g
E
E
rrφφ
∫∫∞=⇒=
−
0
),,(
)(
),,(1
dEtErv
dEtErv g
g
Eg rr φφ∫
0gE
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
204
Multi-group Model
∫∫∞
∇∇−
),,(),(),,(),(1
dEtErErDdEtErErDgE
rrrrrrφφ
∫
∫
∫
∫∞
∇=⇒
∇
=−
0
),,()(
),,()(
1
dEtErrD
dEtErrD
g
g
EE
grr
r
rr
r
φφ ∫∫0
),,(gE
φ
dEtErErdEtErErE g
∫∫∞
∑∑−
)()(),,(),(1
rrrrφφ
dEtEr
dEtErErr
dEtEr
dEtErEr
ra
aEE
a
ag g
g
∫
∫
∫
∫∞
∑=∑⇒
∑
=∑−
0
)(
),,(),()(
)(
),,(),(
)(1 r
r
r
r
φ
φ
φ
φ
dEtErdEtErE g
∫∫0
),,(),,( φφ
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
205
Multi-group Model
1 when 0),()(),()(\
\\ =⇒∑−∑∑ Gtrrtrr gsg
G
ggsg
rrrr φφ1\ =g
ggg
1
∫∫∞−gE
1)()(0
==⇒= ∫∫ dEEdEEgE
g χχχχ
1 when ),()(),()(1\
\\\ =∑⇒∑∑=
Gtrrtrr f
G
ggfgg
rrrr φνφνg
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
206
Multi-group ModelSubstituting all of the above into
)()()()()(1 Sttt extGG rrrrr φφφ +∑+∑
∂ ∑∑
),()(),()(),()(
),()(),()(),(11 \
\\\
\\\
trrDtrrtrr
Strrtrrtrtv
extg
gggsg
ggfgggg
grrrrrrrr φφφ
φφνχφ
∇•∇+∑−∑−
+∑+∑=∂ ∑∑
==
yields
),()(),()(),()( trrDtrrtrr gggsggag φφφ ∇∇+∑∑
yields
),()(),(1 Strrtr extf
rrr φνφ +∑=∂∂
),()(),()(
)()()(
trrDtrrtv
a
f
rrrrrr φφ
φφ
∇•∇+∑−∂
which is the one-group diffusion equation.
)()()()(a φφ
c s t e o e g oup d us o equat o
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
207
Multi-group Model
Project Project 33
Work out the multimulti--group to twogroup to two--groupgroup collapsingWork out the multimulti--group to twogroup to two--group group collapsing and investigate criticality.
Write down the appropriate matricesWrite down the appropriate matrices.
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
208
Poisoning135Xe106 b
S t tSaturates
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
209
Poisoning149Sm105 b
Continuously accumulates
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
210
accumulates
Poisoning• Not anticipated! Reactor shut down! Time scale:Time scale:
Hours and days.Hours and days.135Xe 149SmXe106 b
Sm105 b
φσmXe
a
XeγIγ≈ I
I φXe
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
211
φσ Ia φσ Xe
a
PoisoningHW HW 2828 k
kk reactor). (Infinite use uslet ,1Reactivity −
=≡ ∞ρ
eratorcladfuel
fuelaf
k
mod1 (critical) ∑+∑+∑
∑=
itl df l
fuela
aafa
f d2∑
=
∑+∑+∑
poisona
poisona
eratora
clada
fuela
f mod2
thatShow ∑==∆
∑+∑+∑+∑
ρρρ eratora
clada
fuela
mod12 that Show∑+∑+∑
−=−=∆ ρρρ
Negative reactivity due to poison buildup. It is proportional to the amount of poison.
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
212
p opo t o a to t e a ou t o po so
Poisoning),(),(),(),(),( trtrItrItr
ttrI I
aIfIrrrr
r
φσλφγ −−∑=∂
∂ small
),(),(),(),(),(),( trtrXetrXetrItrt
trXet
XeaXeIfXe
rrrrrr
φσλλφγ −−+∑=∂
∂∂
t∂Initial conditions?Initial conditions?• Clean Core StartupClean Core Startup Assume no spacialClean Core Startup.Clean Core Startup.• Shutdown (later).
Assume no spacial dependence.
)()(ldFuel.Fresh 0)0()0( ==φφ
XeIconstant.)0()( assumeuslet and ==φφ t
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
213
Poisoning
)1()( 0 tfI IetI λ
λφγ −−
∑=HW HW 2929 Show that:
Iλ
)(∞I )(∞I)(∞Xe
)1()(
)( )(0 0 tXe
fXeI XeaXeetXe φσλ
φλφγγ +−−
∑+=and )()(
)(0
0
ttfI
XeaXe
Xe λφλφγφσλ
+∑+
)( )(
0
0 0 ttXeaIXe
fI IXeaXe ee λφσλ
φσλλφγ −+− −
+−
∑+
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
214
Poisoning
)(∞I )(I)(∞Xe
• Now, we know Xe(t)
eratorcladfuel
Xea
eratorcladfuel
poisona tXet
modmod
)()(∑+∑+∑
−=∑+∑+∑
∑−=∆
σρ
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
215
aaaaaa ∑+∑+∑∑+∑+∑
Poisoning•• Shutdown. Shutdown. After the reactor has been operating for a “long” time.
)()()()(),( ttItIttrI I rrrrr
φλφ∑∂
)()()()()(),(
),(),(),(),(),(
ttXtXtIttrXe
trtrItrItrt
Xe
IaIfI
rrrrrr
φλλφ
φσλφγ
+∑∂
−−∑=∂
),(),(),(),(),(),( trtrXetrXetrItrt
XeaXeIfXe φσλλφγ −−+∑=
∂
)()0( ∞= II ),(),( trItrII
rr
λ−=∂
0)0()()()0(
==∞=
φφ tXeXe
)()(),(
),(
tXtItrXet I
rrr
λλ∂∂
.0)0()( ==φφ t ),(),(),( trXetrIt XeI λλ −=∂
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
216
PoisoningHW HW 3030 Show that
)()()()(
)()(
ttIt
tI
IXX
eItI
λλλ
λ
λ
−
∞∞=
)()()()( tt
XeI
It IXeXe eeIeXetXe λλλ
λλλ −−− −−
+∞=
> 0 ?
Height of the peak depends on I(∞)depends on I(∞)and Xe(∞), i.e. depends on φ
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
217
depends on φ.
PoisoningShutdown Xe negative ∆ρ try to add positive reactivity move control rods out need to have
If, the available excess
yenough reserve costly to do that.
reactivity can compensate for less than 30 minutes ofthan 30 minutes of poison buildup, can’t startup again after ~30startup again after 30 minutes of shutdown, because you can’t achieve criticality. Need to wait long hours for Xeto decay down
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
218
to decay down.
Poisoning
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
219
PoisoningStrategiesStrategies• If you plan to shut down for “short maintenance”, think y pabout stepback.• Examine different scenarios using a code from g
http://www.nuceng.ca/• Prepare your own report, code, calculations, graphs, p y p g pcomments, conclusions etc…..• Be creative.
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
220
PoisoningXeXe OscillationsOscillations• φ(r,t) (spacial dependence) flux locally Xeburnup ρ (reactivity) flux further control rods globally in flux elsewhere Xe burnup ρ ….. limited by opposite effect due to increase (decrease) of I in the high (low) flux region.• In large reactors (compared to neutron diffusion length) local flux, power and temperature could reach unacceptable values for certain materials safety issues. • Think of one sensor and one control rod feel average flux apparently OK more sensors and
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
221
control rods to locate and deal with local changes.
Poisoning
Permanent PoisonsPermanent PoisonsPermanent PoisonsPermanent Poisons• 149Sm has sizeable but lower cross section than 135Xe.• It does not decayIt does not decay.
)()(),( trtrtrSm rrr
φγ ∑≈∂ ????
• Accumulates with time
....................).........,(),( trtrt fSm φγ ∑≈∂
????
Accumulates with time.• Consequences?????????
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
222
Fuel DepletionTime scale:Time scale:
Days and months.Days and months.
1214322 10~,10~ −−− scmcmN φ• More depletion increase steady state flux by means of reducing absorbers.
)(N r∂• For a given fuel isotope ),(),(
),(trtrN
ttrN f
aff rrr
φσ−=∂
∂
• For constant flux constant flux φφ00 the solution is),()( )0()0()( 0 trtr f
af
a erNerNtrNrr rrr Φ−− == σφσ
Exponential burnup
• For time varying fluxtime varying flux
)0,()0,(),( 0fff
aa erNerNtrN ==
Neutron fluencet
∫ ),(),(
)0,()0,(),( 0
\\
trf
dttr
ff
fa
fa
erNerNtrNr
r
rrr Φ−−
=∫
= σφσ
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
223Solve numerically.
Fuel Depletion•• Constant power.Constant power.
)()0,(),(),(),( 0 rPrPtrtrwNtrP fff
rrrrr=== φσ
Energy released per
Fission rate )0,()0,(),(),(
)0,()0,(),(),(
rrtrtr
rrNtrtrN
ff
ffrrrr
rrrr
φφ
φφ
∑=∑
=
• Power ~ flux only over short time periods during which Nf is constant.
released per fission
N )( rr∂
ff
wrPtrtrN
ttrN f
aff )(),(),(
),( 0r
rrr
−≈−=∂
∂φσ Linear
depletion!• The solution is obviously
trPrNtrN ff)()0,(),( 0r
rr−≈
fa
ff σσ ≈
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
224
tw
NtN ff )0,(),(
Fuel Depletion
Do the
HW HW 3131
Do the calculations for differentfor different
flux and power levels.
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
225
Poisoning and Fuel DepletionInfinite, critical homogeneous reactor.Infinite, critical homogeneous reactor.
)(tf∑)()()()(
)(mod tttt
tfk controla
poisona
eratora
clada
fa
fa
∑+∑+∑+∑+∑∑
==∞ εηρρεη
thus
trPrNtrN ff)()0,(),( 0r
rr−≈Constant powerConstant power
ttrtrNrN
tw
rNtrN
faff
ff
),(),()0,(
)0,(),(rrr φσ−=
Constant powerConstant power
rrrN
tr f )0,()0()0,(
)(r
rr
r φφφ ==
[ ]trrNrN f
aff
aff
)0,()0,()0,(
),(),(),(rrr φσ
φ
−=tr
rtrN
tr faf )0,(1
)0,(),(
),( rr φσφφ
−
[ ]trrN faf )0,(1)0,( rr φσ−=
[ ]tt fff )0(1)0()( rrr φ∑∑Nuclear Reactor Theory, JU, First Semester, 2010-2011
(Saed Dababneh).226
[ ]trrtr fa
fa
fa )0,(1)0,(),( φσ−∑=∑
Poisoning and Fuel Depletion
)( )(0fXI Xeφλφγγ ∑+
∞XeConstant
)1()(
)( )(
0
0 0 tXeaXe
fXeI XeaXeetXe φσλ
φγφσλφγγ +−
∑
−+
∑+=
)( )(
0
0 0 ttXeaIXe
fI IXeaXe ee λφσλ
φσλλφγ −+− −
+−
∑+
Constant
)0,()0,()(),(),(
rrtrXetr fXeIXe
aXea
rrrr
λφγγ
σ∑+
==∑ ∞
),()()(
trXea
Xeaa rφ
σλ
+∞
trrtr fSmSma
Sma )0,()0,(),( rrr φγσ ∑≈∑ tr
rtr fa )0,(1
)0,(),( r
rr
φσφφ
−=
Oth fi i d t ( i ) ith l t tiNuclear Reactor Theory, JU, First Semester, 2010-2011
(Saed Dababneh).227
• Other fission products (poisons) with less capture cross sections.
Poisoning and Fuel Depletion• Now we know all macroscopic cross sections.
)()()()()(
mod tttttfk control
apoisona
eratora
clada
fa
fa
∑+∑+∑+∑+∑∑
==∞ εηρρεη)()()()( aaaaa
• When there are no absorbers left to Until = 0.
Solve for t to get remove, we need to refuel.• Absorbers are not only control rods.
upper limit for “core loading
lif ti ”
y• All fuel nuclei should be considered.• For each species, all sources and
lifetime”.Damaged
fuel !
sinks should be taken into account.• Online loading environmental.
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
228
fuel…!• 3H.
Poisoning and Fuel Depletion
dN )(tFNNNNdt
dNC
CBBAAAA
A +++−−= γφσλφσλ
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
229
dtFuel loading
Poisoning and Fuel Depletion• Some poisons are intentionally introduced into Some poisons are intentionally introduced into the reactor.the reactor.• Fixed burnable poisons.B, Gd. More uniform distribution than rods, more intentionally localized than shim.• Soluble poisons (chemical shim) with caution.Boric acid (soluble boron, solbor) in coolant.Boration and dilution.Scram emergency shutdown (sodium polyborate or gadolinium nitrate).• Non-burnable poisons.C f fNuclear Reactor Theory, JU, First Semester, 2010-2011
(Saed Dababneh).230
Chain of absorbers or self shielding.
Delayed Precursors
1 GG∂
)()()()()()(
),()(),()(),(1
11 \\\
\\\\
tDtt
Strrtrrtrtv
extg
gggsg
ggfgggg
grrrrrrrr
rrrrr
φφφ
φφνχφ
∇∇∑∑
+∑+∑=∂∂ ∑∑
==
),()(),()(),()( trrDtrrtrr gggsggag φφφ ∇•∇+∑−∑−
1 ∂• For one-group
),()(),(1 Strrtrtv
extf
rrr φνφ +∑=∂∂
),()(),()( trrDtrrarrrrrr φφ ∇•∇+∑−
Wh t b t d l d t ?• What about delayed neutrons?
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
231
Delayed PrecursorsDelayed
neutron emitter
+One of 66 delayed
t
dp ννν +=
Delayed neutron fractionνβ d=neutron
precursors known so far.
Delayed neutron fractionν
β =
Data for all precursors are not accurately
kknown.
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
232
Delayed PrecursorsFissile nucleus Delayed neutron / 100 fissions
233U 0 667233U235U
238U*
0.6671.6214 39
Increases with N.238U* 4.39
239Pu240P *
0.6280 95
t
240Pu*241Pu
242P *
0.951.522 21242Pu* 2.21
Data for thermal neutron induced fission, except for , p* fast neutron induced fission.
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
233
Delayed Precursors
(s)
β < 0.7% ≅ 0.016 / ν235U
pν
),()()1(),(1 6
1SCtrrtr
tvext
iiif
rrr λφνβφ ++∑−=∂∂ ∑
),()(),()(1
trrDtrrtv
a
irrrrrr φφ ∇•∇+∑−
∂ =
),()(),(),( trrtrCt
trCfiii
i rrrr
φνβλ ∑+−=∂
∂
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
234
t∂
Delayed Precursors• The multi-group equation now becomes
Different energy spectra
),(),()()1(),(1 6
11\\\\ trCtrrtr
tv iii
Cg
G
ggfgg
pgg
g
rrrr λχφνβχφ +∑−=∂∂ ∑∑
==
),()(1\
\\ Strr extg
G
gggsg
rr φ +∑+ ∑=
),()(),()(),()( trrDtrrtrr gggsggag
grrrrrrrr φφφ ∇•∇+∑−∑−
∂ G)( r ∑=
∑+−=∂
∂ G
ggfggiii
i trrtrCt
trC
1\\\\ ),()(),(),( rrr
r
φνβλ
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
235
=g 1
Full Blown Diffusion Equation• In steady statesteady state
∑G
∑=
∑=g
gfggiii trrtrC1\
\\\ ),()(),( rrr φνβλ
)()()()()1(011 \
\\\\
\\\ rrrr
G
G
ggfgg
Cg
G
ggfgg
pg
rrrr φνβχφνβχ ∑+∑−= ∑∑==
)()()()()()()()(1\
\\ rrDrrrrSrr gggsggagextg
G
gggsg
rrrrrrrrrr φφφφ ∇•∇+∑−∑−+∑+ ∑=
GC [ ] )()()(01\
\\\ rr
G
G
ggfgg
pg
Cg
pg
rr φνβχχχ ∑−+= ∑=
Significance of ggg depends on whether
Cgχ
)()()()(1\
\\ rrSrr gagextg
G
gggsg
rr
rrrr φφ ∑−+∑+ ∑=
pwe have fine or
course energy groups.
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
236
)()()()( rrDrr gggsgrrrrrr φφ ∇•∇+∑−
gy g p
Top Related