Lecture 2008 5 - Saed Dababneh Home...

Controlled Fission• Note that η is greater than 2 at thermal energies andat thermal energies andalmost 3 at high energies.• These “extra” neutrons are

Variations in Variations in ηη

Used to convert fertile into fissile fuel.

Plutonium economy• Plutonium economy.• India and thorium.• Efficiency of this process is Efficiency of this process is determined by neutronenergy spectrum.

Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).

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gy

Controlled Fission

• Conversion ratioConversion ratio CR is defined as the average rate ofConversion ratio Conversion ratio CR is defined as the average rate of fissile atom production to the average rate of fissile atom consumption.• For LWR's CR ≅ 0.6.• CR is called BR for values > 1.

F t b d t h BR > 1• Fast breeder reactors have BR > 1.• They are called “fast” because primary fissions inducing neutrons are fast not thermal thus η > 2 5 butinducing neutrons are fast not thermal, thus η > 2.5 but σf is only a few barns.• Moderator??


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Controlled FissionTime scale for neutron multiplicationTime scale for neutron multiplication• Time constant τ includes moderation time (~10-6 s) and diffusionTime constant τ includes moderation time ( 10 s) and diffusion time of thermal neutrons (~10-3 s).

Time Average number of thermal neutronstt n

t + τ knt + 2τ k2n

• For a short time dtτ

nkndtdn −

=

•• Show thatShow that

τdtτtkentn )1(

0)( −=3

0)(Nuclear Reactors, BAU, 1st Semester, 2008-2009

(Saed Dababneh).

Controlled Fissionτtkentn )1(

0)( −=• k = 1 n is constant (Desired).• k < 1 n decays exponentially.• k > 1 n grows exponentially with time constant τ / (k 1)

0)(Reactivity.

• k > 1 n grows exponentially with time constant τ / (k-1).• k = 1.01 (slightly supercritical..!) e(0.01/0.001)t = e10 = 22026 in 1s. in 1s. • Design the reactor to be slightly subcritical for prompt neutrons.• The “few” “delayed” neutrons will be used to achieve criticality, y,allowing enough time tomanipulate the controlrods (or use shim or )

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rods (or use shim or …).Cd control rodsCd control rods


Fission ReactorsEssential elements:Essential elements:• Fuel (fissile material)Fuel (fissile material).• Moderator (not in reactors using fast neutrons).• Reflector (to reduce leakage and critical size).

Core

( g )• Containment vessel (to prevent leakage of waste).• Shielding (for neutrons and γ’s).

C l• Coolant.• Control system.• Emergency systems (to prevent runaway during failure)• Emergency systems (to prevent runaway during failure).

Chapter 4 in Lamarsh

5Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).

Chapter 4 in Lamarsh

Fission ReactorsTypes of reactors:Types of reactors:Used for what?Used for what?Used for what?Used for what?• Power reactors: extract kinetic energy of fragments as heat boil water steam drives turbine electricity.y• Research reactors: low power (1-10 MW) to generate neutrons (~1013 n.cm-2.s-1 or higher) for research.

C d b d C h ll• Converters and breeders: Convert non-thermally-fissionable material (non-fissile) to a thermally-fissionable material (fissile)fissionable material (fissile).• ADS.• Fusion. Wh t t t ?Wh t t t ?

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Fusion.


What are neutron generators?What are neutron generators?

Fission ReactorsWhat neutron energy?What neutron energy?• Thermal fast reactorsThermal, fast reactors.• Large, smaller but more fuel.What fuel?What fuel?What fuel?What fuel?• Natural uranium, enriched uranium, 233U, 239Pu,Mixtures.

From converter or breeder reactor.How???


Fission ReactorsWhat assembly?What assembly?• Heterogeneous: moderator and fuel are lumped.Heterogeneous: moderator and fuel are lumped. • Homogeneous: moderator and fuel are mixed together.• In homogeneous systems, it is easier to calculate p and

f for example, but a homogeneous natural uranium-graphite mixture can not go critical. Why?

Wh t l t?Wh t l t?What coolant?What coolant?• Coolant prevents meltdown of the core.• It transfers heat in power reactors• It transfers heat in power reactors.• Why pressurized-water reactors.• Why liquid sodium?

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y q


More on ModeratorsWhat moderator?What moderator?1 Cheap and abundant1. Cheap and abundant.2. Chemically stable.3. Low mass (high ζ logarithmic energy decrement).( g ζ )4. High density.5. High Σs and very low Σa.

G hi (1 2 4 ) i 3• Graphite (1,2,4,5) increase amount to compensate 3.• Water (1,2,3,4) but n + p → d + γ enriched uranium.• D O (heavy water) (1!) but has low capture cross• D2O (heavy water) (1!) but has low capture cross

section natural uranium, but if capture occurs, produces tritium (more than a LWR).produces tritium (more than a LWR).

• ….. 9Nuclear Reactors, BAU, 1st Semester, 2008-2009

(Saed Dababneh).

More on Moderators

∑ζModerating ratio ≡ s

∑∑ζ

HW 12HW 12 α+→→+ LiBnB 7*1110

Calculate both moderating power and

a∑HW 12HW 12 α+→→+ LiBnB

B-101010BBratio for water, heavy water, graphite,

l th l d bb1/v region

1010BB

polyethylene and boronboron.Tabulate your results and comment


comment.

More on Moderators

HW 12 HW 12 (continued)(continued)

Calculate the moderating power and ratio for pure D2O as well as for D2O contaminated with a) 0.25% and b) 1% H Oand b) 1% H2O.Comment on the results.In CANDU systems there is a need for heavy waterIn CANDU systems there is a need for heavy water upgradors.


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More on Moderators

ζnEEn −= lnln \

ζ)/ln( \

nEEn =Recall ζ

After n collisions

)/l ( EE Aft lli i

1ln)1(1ln2

\

−−+=⎥⎦

⎤⎢⎣⎡==∆

AAEu ζζ)/ln( thf EE

n =After one collision

12\ +⎥⎦⎢⎣ AAE av

ζζf

Total mean free path = n λs

Is it random walk or there is a


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preferred direction??? th

More on Moderators


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More on Moderators

Recall (head on) Then the

HW 13 (or 6HW 13 (or 6\\))EEAE α≡⎞

⎜⎛ −

=2

min\ 1 After one collision.

Recall (head-on). Then the maximum energy loss is (1-α)E, or αE ≤ E\ ≤ E.For an ss--wavewave collision:

EEA

E α≡⎠

⎜⎝ +

=min1

A tiA tiE

EEPdEEEPE

E )1(1)(1)( \\\

αα −=→∴=→∫

Assumptions:Assumptions:1.1. Elastic scattering. Elastic scattering. E↓2.2. Target nucleus at rest. Target nucleus at rest.

E↑33 S h i l t iS h i l t i

show that

EE )1(1\ α+= 3.3. Spherical symmetry in Spherical symmetry in CM.CM.

Obviously

EE )1(2 α+=⎪⎨⎧ ≤≤

−=≡→EEE

EEdEE

ss

s )1()(

)(|

|\ α

ασσσObviously


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⎪⎩⎨

otherwiseEdEs

0)1()( | α

More on Moderators

(Re) erifHW 13 (or 6HW 13 (or 6\\) ) continued…continued…

•• Scattering Kernel.Scattering Kernel.•• Slowing down density.Slowing down density.•• Migration length.Migration length.

[ ]2

2\

cos)1()1(21

)1(cos21

−++=++

=A

AAEE CM

CM

θααθ(Re)-verify Migration length.Migration length.

•• Fermi age and continuous fermi model.Fermi age and continuous fermi model.

[ ]

[ ]222

2

sincos

2)1(

−+=

+

A

AE

θθ[ ]2)1( +

=A

For doing so, you need to verify and use

CM

CM

AAA

θθθcos21

cos1cos2 ++

+=


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AA θcos21++

More on ModeratorsHW 13 (or 6HW 13 (or 6\\) ) continued…continued…

• Forward scattering is preferred for “practical” moderators (small A)moderators (small A).• If isotropic neutron scattering (sphericallyscattering (spherically symmetric) in the laboratorylaboratory frame 2average cosine of the scattering angle is zero.

Show that A32)(cos == θµ


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More on Moderators

Sphericall s mmetric in CM )(1)( Ed CMs σθσσ==

HW 13 (or 6HW 13 (or 6\\) ) continued…continued…

Spherically symmetric in CM

Show that

)(4

)( Ed ssCM σ

πθσ ==

ΩCM

s AAE θσθσ )1cos2()()(2312 −− ++

=Show thatCMs A θπ

θσcos14

)( 1−+=

• Neutron scatteringscattering is isotropic in the laboratory• Neutron scatteringscattering is isotropic in the laboratory system?! valid for neutron scattering with heavy nuclei, which is not true for usual thermal reactor moderators (corrections are applied).Distinguish fromDistinguish from


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• Angular neutron distribution.distribution.

More on ModeratorsModeratorModerator--toto--fuel ratio fuel ratio ≡≡ Nm/Nu.• Ratio ↑ p ↑ Σ of the moderator ↑ f ↓ (leakage ↓)

Self regulation.• Ratio ↑ p ↑ Σa of the moderator ↑ f ↓ (leakage ↓).• Ratio ↓ slowing down time ↑ p ↓ f ↑ (leakage ↑).• T ↑ ratio ↓ (why)• T ↑ ratio ↓ (why).• Other factors also change.• Temperature coefficient of reactivity.• ModeratorModerator temperature coefficient of reactivity.


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One-Speed Interactions• Particular general.Recall:

N t d ’t h h t i t t ith h• Neutrons don’t have a chance to interact with each other (review test!) Simultaneous beams, different intensities same energy:intensities, same energy:

Ft = Σt (IA + IB + IC + …) = Σt (nA + nB + nC + …)v• In a reactor, if neutrons are moving in all directions, g

n = nA + nB + nC + …

Rt = Σt nv = Σt φ


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One-Speed Interactions

Ωdrn ),( ωrr

ωr≡ Neutrons per cm3 at

r whose velocity

Ωd

r whose velocity vector lies within dΩabout ω.

∫rr

d∫ Ω=π

ω4

),()( drnrn rrr

• Same argument as before r• Same argument as before Ω= vdrnrdI ),(),( ωω

rrrr

)()()()(),()(),()()(

),()(),(

rrrnrvdrnvrrdFrFrR

rdIrrdF

ttt

trrrrrrrrrrr

rrrrr

φωω

ωω

∑=∑=Ω∑===

∑=

∫∫


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4πω

Ω= ∫ drnvr ),()(4

ωφπ

rrrwhere

Multiple Energy Interactions

ΩdEdErn ),,( ωrr

≡ Neutrons per cm3 at r with energy • Generalize to include energy

interval (E, E+dE) whose velocity vector lies within dΩ about ω.

∫ ∫∞

∫ Ω=π

ω4

),,(),( dEdErndEErnrrr ∫ ∫ Ω=

0 4

),,()(π

ω dEdErnrnrrr

dEEEdEEEEdEER )()()()()()( rrrrr φ∑∑ dEErErdEEvErnErdEErR tt ),(),()(),(),(),( φ∑=∑=

∫∞

∑= ),(),()( dEErErrR trrr φ∫∑

0

),(),()( t φ

Thus knowing the material properties Σt and the neutron flux φ as a function of space and energy we can calculate the interaction rate

Scalar


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function of space and energy, we can calculate the interaction rate throughout the reactor.

Neutron Current∞

• Similarly and so on …∫∞

∑=0

),()()( dEErErR SSrr φ

Scalar

• Redefine as

Scalar

Ω= vdrnrdI ),(),( ωωrrrr Ω= dvrnrId rrrrrr ),(),( ωω

Ω= ∫ drnvr ),()(4

ωφπ

rrrΩ= ∫ drnvJ ),(

4

ωπ

rrrr

4π 4π

Neutron current densityNeutron current density• From larger flux to smaller flux!

Jrg

• Neutrons are not pushed!• More scattering in one direction


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than in the other.xJxJ =• ˆ

r

NN fl f d i l

Equation of Continuity

∫ Ωθ)(ˆ dJJrrr

NNetet flow of neutrons per second per unit area normal to the x direction:

∫ Ω==•π

θω4

cos),(ˆ dvrnJxJ xxrr

In general: JnJ =• ˆr

ge e a nJnJEquation of ContinuityEquation of Continuity

∫∫ ∫∫∂ r

∫∫ ∫∫ •−∀∑−∀=∀∂∂

∀ ∀∀ Aa dAntrJdtrrdtrSdtrn

tˆ),(),()(),(),( rrrrrr φ

Rate of change in neutron density

Production rate

Absorption rate

“Leakage in/out” rate

Normal


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Volume Source distribution

function

Surface area

bounding ∀

Normal to A

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