lecture 2008 4 - nuclear.dababneh.comnuclear.dababneh.com/reactors/notes/2008/lecture_2008_4.pdf ·...

Review Test

• We have done today the review test.http://nuclear.bau.edu.jo/Reactors/Notes/2008/ReviewTest.ppt

• We will discuss it next time.• Please study the review test of last year (2007-2008).http://nuclear.bau.edu.jo/Reactors/Notes/2007/ReviewTest.ppt

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

1

Controlled Fission• 235U + n X + Y + (~2.4)n• Moderation of second generation neutrons Chain reaction

Fast second generation neutrons

ν

Moderation of second generation neutrons Chain reaction.• Water, D2O or graphite moderator.• Ratio of number of “neutrons” (fissions) in one generation to th di k ( t d ti lti li tithe preceding ≡ k∞ (neutron reproduction or multiplication factor).

Infinite medium (ignoring leakage at the surface).

• k ≥ 1 Chain reaction.• k < 1 subcritical.• k = 1 critical system

Chain reacting pileChain reacting pile

• k = 1 critical system.• k > 1 supercritical.For steady release of energy (steady-t t ti ) d k 1

2

state operation) we need k =1.


Controlled Fission• Average number of allall neutrons released per fission

ν (for thermal neutrons, 0.0253 eV).233U 2 492• 233U : 2.492

• 235U : 2.418• 239Pu : 2.871• 241Pu : 2.927

• Reactor is critical (k = 1): rate of neutrons produced by fission = rate of neutrons absorbed absorbed + leaked.

Size and composition of the reactor. 3Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).

Controlled FissionProbability for a thermal neutron to cause fission on 235U is

235U thermal cross sectionsσ ≈ 584 bσfission ≈ 584 b.σscattering ≈ 9 b.σradiative capture ≈ 97 b. ασσ

σ

γ +=

+≈

11

f

fCheck Check numbers!numbers!

ασσ γ ++ 1f

If each fission produces an average of ν neutrons then the meanIf each fission produces an average of ν neutrons, then the mean number of fastfast fission neutrons produced per thermal neutron = η

αν

σσσ

νσσ

νηγ +=

+==

1f

f

a

f η <ν


4

γf

Controlled Fission235U• Assume Assume natural natural uranium:uranium:

99.2745% 238U, 0.7200% 235U.

Thermal σf = 0 b 584 bThermal σγ = 2.75 b 97 b

24 RπWhy?

N

NN

yyxx

yyxxyx

)( σγσγ

σσ

+=

+=Σ+Σ=Σ

238U• Σf / N = (0.992745)(0) +

(0.0072)(584)

yyxx

24 RπDoppler effect?Doppler effect?

( )( )= 4.20 b.

• Σγ / N = (0.992745)(2.75) + (0 0072)(97)

4 Rπ

5

(0.0072)(97)= 3.43 b.


Using the experimental elastic scattering data the radius of the nucleus can be estimated.

Moderation (to compare x-section)

1H( )(n n)

2H(n,n)(n,n)

(n,γ) (n,γ)

• Resonances?6Nuclear Reactors, BAU, 1st Semester, 2008-2009

(Saed Dababneh).

Controlled Fission• Probability for a thermal neutron to cause fission in natural natural uraniumuranium 204 Compare touraniumuranium

If h fi i d f 2 4 t th th

55.043.320.4

20.4=

+=

Compare to pure 235U.

• If each fission produces an average of ν = 2.4 neutrons, then the 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) (LightFor enriched uraniumenriched uranium ( U 3%) η ????? ( 1.3). (Light water?).• In this case η is further from 1 and allowing for more neutrons to be lost while maintaining criticality

7

be lost while maintaining criticality.


Controlled Fission

1HW 11HW 11

∑ ΣΣ

=i

f ii )()(1 νη• VerifyΣ ia

• Comment on the calculation for thermal neutrons and a mixture of fissile and non-fissile materials, giving an example.

C t f f t t d i t f• Comment for fast neutrons and a mixture of fissionable materials, giving an example.


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Conversion and Breeding

Converters: Convert non-thermally-fissionable material to a thermally-fissionable material.

_239min23239238 νβ ++⎯⎯ →⎯→+ −NpUnU

_2393.2 νβ

νβ

++⎯⎯→⎯

++→→+

−Pu

NpUnU

d βσf,th = 742 b

_233min22233232 β −PThTh

_23327

233min22233232

νβ

νβ

++⎯⎯→⎯

++⎯⎯ →⎯→+

−U

PaThnTh

dIndia

9Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).

νβ ++→ Uσf,th = 530 b



• If η = 2 Conversion and fission.If 2 B d t• If η > 2 Breeder reactor.

• 239Pu: Thermal neutrons (η ~ 2.1) hard for breeding.Fast neutrons (η ~ 3) possible breeding fast (η ) p gbreeder reactors.

• After sufficient time of breeding, fissile material can be easily g, y(chemically) separated from fertile material.Compare to separating 235U from 238U.• Reprocessing

10

• Reprocessing.





11

Controlled Fission• N thermalthermal neutrons in one generation have produced have produced so far so far ηηNNfastfast neutrons.neutrons.• Some of these fastfast neutrons can cause 238U fission more fast neutrons fast fission factor fast fission factor = ε (= 1.03 for natural uranium).• Now we haveNow we have εηεηNN fastfast neutrons.neutrons.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?for 1 MeV neutrons?• 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 moderator.• In graphite, an average distance of 19 cm is needed for

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g ap te, a a e age d sta ce o 9 c s eeded othermalization the resonance escape probability resonance escape probability p (≈ 0.9).


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;

h th li d t h ld h h d th f lwhen 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 elements.• The thermal utilization factorthermal utilization factor f (≈ 0 9) gives the fraction ofThe thermal utilization factor thermal utilization factor f (≈ 0.9) gives the fraction of thermal neutrons that are actually available for the fuel.• Now we have Now we have ffppεηεηNN thermalthermal neutronsneutrons, could be > or < Nth d t i i th iti lit f th tthus determining the criticality of the reactor.

k∞ = fpεη The fourThe four--factor formula.factor formula.

k = k = f εη(1 l )(1 l )13

k = keff = fpεη(1-lfast)(1-lthermal)Fractions lost at surfaceNuclear Reactors, BAU, 1st Semester, 2008-2009

(Saed Dababneh).

Controlled Fission

k = fpεη Pfk = ρεηk∞ = fpεη, leaknoneff Pfk −= ρεη

• Fast from thermal ∑ Σ= ii )()(1 νη• Fast from thermal, as defined in HW 11.

• Fast from fast, ε.• Thermal from fast, p

∑ ΣΣ

=i

fa

ii )()(νη

Thermal from fast, p.• Thermal available for fission poison

aerator

aclada

fuela

fuelaf

∑+∑+∑+∑∑

= mod

Thinking QUIZThinking QUIZ• For each thermal neutron absorbed, how many fast

t d d?Nuclear Reactors, BAU, 1st Semester, 2008-2009

(Saed Dababneh).14

neutrons are produced? Will need this when discuss two-group diffusion.

Neutron reproduction

factor

x 0.9x 0.9Thermal Thermal

x η

factork = 1.000

x 0.9x 0.9

utilization utilization factor “f”factor “f”

x η

Resonance Resonance escape escape

probability ”p”probability ”p”What is:

x 1.03x 1.03Fast fission Fast fission factor “factor “εε””

• Migration length?• Critical size?How does theHow does the geometry affect the reproduction factor?


Neutron Life Cycle

Why should we worry about these?

How?


Controlled Fission

k = fpεη(1-lfast)(1-lthermal) Not fixed…!

• Thermal utilization factor f can be changed, as an example, by adding absorber to coolant (PWR)example, by adding absorber to coolant (PWR)(chemical shim, boric acid), orby inserting movable control rods in & out.• Reactors can also be controlled by altering neutron leakages using movable neutron reflectors.

f and factors change as f el is b rned• f and p factors change as fuel is burned.• f, p, η change as fertile material is converted to fissilematerial


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material.

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

Under-moderation and

over-moderation.


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

Variations in Variations in ηη

Used to convert fertile into fissile fuel.

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


19

gy

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