neutron neutron life cycle reproduction factor...

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Neutron reproduction Neutron Life Cycle x x 0.9 factor k eff = 1.000 Thermal Thermal utilization utilization factor “f” factor “f” x η x x 0.9 Resonance Resonance escape escape ii ”” ii ”” x x 1.03 03 Fast fission Fast fission probability pprobability pWhat is: • Migration length? Fast fission Fast fission factor “ factor “ε ε• Critical size? How does the t ff t th geometry affect the reproduction factor? 1 Nuclear Reactor Theory, BAU, Second Semester, 2011-2012 (Saed Dababneh).

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Neutron reproduction

Neutron Life Cycle

x x 00..99

factork eff = 1.000

Thermal Thermal utilization utilization factor “f”factor “f”

x η

x x 00..99Resonance Resonance

escape escape i i ” ”i i ” ”

x x 11..0303Fast fissionFast fission

probability ”p”probability ”p”What is:• Migration length? Fast fission Fast fission

factor “factor “εε””

g g• Critical size?How does the

t ff t thgeometry affect the reproduction factor?

1Nuclear Reactor Theory, BAU, Second Semester, 2011-2012 (Saed Dababneh).

Neutron Life Cycley

Why should we b t th ?worry about these?

How?f

How?

2Nuclear Reactor Theory, BAU, Second Semester, 2011-2012 (Saed Dababneh).

Controlled Fission

k = fpεη(1-lf )(1-l h l) Not fixed !k fpεη(1 lfast)(1 lthermal)• Thermal utilization factor f can be changed, as an

Not fixed…!

g ,example, by adding absorber to coolant (PWR)(chemical shim, boric acid), or( , ),by inserting movable control rods in & out. Poison.• Reactors can also be controlled by altering neutron y gleakages using movable neutron reflectors.• f and p factors change as fuel is burned.p g• f, p, η change as fertile material is converted to fissilematerial.

Nuclear Reactor Theory, BAU, Second Semester, 2011-2012 (Saed Dababneh).

3

Controlled Fission• Attention should be paid also to the fact that reactor power changes occur due to changes in resonance escape probability p. If Fuel T↑, p↓ due to Doppler broadening ofresonance peaks.

U d d tiUnder-moderation and

over-moderationover moderation.

Nuclear Reactor Theory, BAU, Second Semester, 2011-2012 (Saed Dababneh).

4

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

Time Average number of thermal neutronsTime Average number of thermal neutronst n

t + τ knt + τ knt + 2τ k2n

nkndn• For a short time dtτ

nkndtdn −

=

•• Show thatShow that τtkentn )1(0)( −=

5

0)(Nuclear Reactor Theory, BAU, Second Semester, 2011-2012

(Saed Dababneh).

Controlled Fissionτtkentn )1()( −=

• k = 1 n is constant (Desired).• k < 1 n decays exponentially

entn 0)( =Reactivity.

• k < 1 n decays exponentially.• k > 1 n grows exponentially with time constant τ / (k-1).• k = 1.01 (slightly supercritical..!) e(0.01/0.001)t = e10 = 22026 in in 11s. s. ( g y p )• Design the reactor to be slightly subcritical for prompt neutrons.• The “few” “delayed” neutrons will be used to achieve criticality, allowing enough time toallowing enough time tomanipulate the controlrods (or use shim or …).

6

( )Cd control rodsCd control rods

Nuclear Reactor Theory, BAU, Second Semester, 2011-2012 (Saed Dababneh).

Fission ReactorsEssential elements:Essential elements:• Fuel [fissile (or fissionable) material].• Moderator (not in reactors using fast neutrons). Core

• Reflector (to reduce leakage and critical size).• Containment vessel (to prevent leakage of waste).• Core catcher.• Shielding (for neutrons and γ’s).• Coolant.• Control system.• Emergency systems (to prevent runaway during failure).

Chapter 4 in Lamarsh

7Nuclear Reactor Theory, BAU, Second Semester, 2011-2012 (Saed Dababneh).

p

Fission ReactorsTypes of reactors:Types of reactors:Used for what?Used for what?• Power reactors: extract kinetic energy of fragments as heat boil water steam drives turbine electricity.• Research reactors: low power (1-10 MW) to generate neutrons (~1013 n.cm-2.s-1 or higher) for research.• Converters and breeders: Convert non-thermally-fissionable material (non-fissile) to a thermally-fissionable material (fissile).• ADS.• Fusion. What are neutron generators?What are neutron generators?

8Nuclear Reactor Theory, BAU, Second Semester, 2011-2012 (Saed Dababneh).

What are neutron generators?What are neutron generators?

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

From converter or breeder reactorHow??? breeder reactor.How???

9Nuclear Reactor Theory, BAU, Second Semester, 2011-2012 (Saed Dababneh).

Fission ReactorsWhat assembly?What assembly?

H t d t d f l l d• Heterogeneous: moderator and fuel are lumped. • Homogeneous: moderator and fuel are mixed together.

I h t it i i t l l t d f• In homogeneous systems, it is easier to calculate p and ffor example, but a homogeneous natural uranium-graphite mixture (for example) can not go critical Why?graphite mixture (for example) can not go critical. Why?

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

10Nuclear Reactor Theory, BAU, Second Semester, 2011-2012 (Saed Dababneh).

More on ModeratorsWhat moderator?What moderator?1. Cheap and abundant.2. Chemically stable.3. Low mass (high ζ logarithmic energy decrement).4. High density.5. High Σs and very low Σa.• Graphite (1,2,4,5) increase amount to compensate 3.• Water (1,2,3,4) but n + p → d + γ enriched uranium.• D2O (heavy water) (1!) but has low capture cross

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

• ….. 11Nuclear Reactor Theory, BAU, Second Semester, 2011-2012

(Saed Dababneh).

More on Moderators

Moderating ratio ≡ s∑ζg

Calculate both a∑HW HW 1212 α+→→+ LiBnB 7*1110

moderating power and ratio for water, heavy

B-101010BBywater, graphite, polyethylene and boronboron.

1/v region

p y yTabulate your results and comment.

12Nuclear Reactor Theory, BAU, Second Semester, 2011-2012 (Saed Dababneh).

More on Moderators

HW HW 12 12 (continued)(continued)

Calculate the moderating power and ratio for pure D O ll f D O t i t d ith ) 0 25%D2O as well as for D2O contaminated with a) 0.25% and b) 1% H2O.C t th ltComment on the results.In CANDU systems there is a need for heavy water

dupgradors.

Nuclear Reactor Theory, BAU, Second Semester, 2011-2012 (Saed Dababneh).

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More on Moderators

ζnEE −= lnln \ )/ln( \nEEn =Recall ζnEEn −= lnln

ζn =Recall

After n collisionsAfter n collisions

)/ln( thf EE After one collision

11ln

2)1(1ln

2

\ +−−

+=⎥⎦⎤

⎢⎣⎡==∆

AA

AA

EEu

av

ζζ)( thfn =

ζ

Total mean free path = n λf

Total mean free path = n λsIs it random walk or there is a

f d di ti ???Nuclear Reactor Theory, BAU, Second Semester, 2011-2012

(Saed Dababneh).14

preferred direction??? th

More on Moderators

Nuclear Reactor Theory, BAU, Second Semester, 2011-2012 (Saed Dababneh).

15

More on Moderators

A ⎞⎛21 After one collision.

Recall (head-on). Then the maximum energy loss is (1-α)E, or αE ≤ E\ ≤ E.

EEAAE α≡⎟

⎠⎞

⎜⎝⎛

+−

=min\

11

For an ss--wavewave collision:EEPdEEEP

E 1)(1)( \\\ =→∴=→∫Assumptions:Assumptions:1.1. Elastic scattering.Elastic scattering. E↓Flat-top probability

EEEPdEEEP

E )1()(1)(

αα −→∴→∫

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

E↑3.3. Spherical symmetry in Spherical symmetry in

CMCMEE )1(2

1\ α+=CM.CM.

Obviously

2

⎪⎨⎧ ≤≤

−=≡→EEE

EE

dEdEE

ss

s )1()(

)(|

|\ α

ασσσ

Nuclear Reactor Theory, BAU, Second Semester, 2011-2012 (Saed Dababneh).

16

⎪⎩ otherwisedE 0

More on ModeratorsHW HW 13 13 (or (or 66\\)) •• Scattering Kernel.Scattering Kernel.

•• Slowing down density.Slowing down density.

[ ]2\ 1cos21++ AAE CMθ

(Re)-verifyg yg y

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

[ ]

[ ]22 cos)1()1(

21

)1(cos21

−++=+

++=

AAA

EE CMθααθ

[ ]2

222

)1(sincos

+−+

=AA θθ

)(For doing so, you need to verify and use

CMA θ1CM

CM

AAA

θθθcos21

cos1cos2 ++

+=

Nuclear Reactor Theory, BAU, Second Semester, 2011-2012 (Saed Dababneh).

17

More on ModeratorsHW HW 13 13 (or (or 66\\) ) continued…continued…

• Forward scattering is preferred for “practical”preferred for practical moderators (small A).• If isotropic neutronIf isotropic neutron scattering (spherically symmetric) in thesymmetric) in the laboratorylaboratory frame average cosine of the Show that

2)(cos == θµaverage cosine of the scattering angle is zero.

Show that A3

)(cos θµ

Nuclear Reactor Theory, BAU, Second Semester, 2011-2012 (Saed Dababneh).

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More on Moderators

1dHW HW 13 13 (or (or 66\\) ) continued…continued…

Spherically symmetric in CM )(41)( E

dd

sCM

sCMs σ

πθσσ

==Ω

CM 2312

Show thatCM

CMs

s AAAE

θθ

πσθσ

cos1)1cos2(

4)()( 1

2312

−−

+++

=

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

A l t di t ib tidi t ib tiNuclear Reactor Theory, BAU, Second Semester, 2011-2012

(Saed Dababneh).19

• Angular neutron distribution.distribution.

More on ModeratorsModeratorModerator--toto--fuel ratio fuel ratio ≡≡ Nm/Nu. Self regulation.• Ratio ↑ p ↑ Σa of the moderator ↑ f ↓ (leakage ↓).• Ratio ↓ p ↓ f ↑ (leakage ↑).• T ↑ ratio ↓ (why).• Other factors alsoOther factors also change.• Temperature coefficient of reactivity.• Moderator temperaturetemperature coefficient of reactivity.

Nuclear Reactor Theory, BAU, Second Semester, 2011-2012 (Saed Dababneh).

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