Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh). 1 Controlled Fission Note that is greater than 2 at thermal energies and almost 3 at

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

1

Controlled Fission• Note that is greater than 2 at thermal energies andalmost 3 at high energies.• These “extra” neutrons are Used to convert fertile into fissile fuel.• Plutonium economy.• India and thorium.• Efficiency of this process is determined by neutronenergy spectrum.

Variations in Variations in


2

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 reactors have BR > 1.• 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??

3

Controlled Fission

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

Time Average number of thermal neutrons t nt + knt + 2 k2n

• For a short time dt

• Show thatShow that

nkn

dt

dn

tkentn )1(0)(


4

• k = 1 n is constant (Desired).• k < 1 n decays exponentially.• k > 1 n grows exponentially with time constant / (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, allowing enough time tomanipulate the controlrods (or use shim or …).

Controlled Fission

Dan

gero

us

Dan

gero

us

Cd control rodsCd control rods


tkentn )1(0)(

Reactivity.

5

Fission Reactors

Essential elements:Essential elements:• Fuel (fissile material).• Moderator (not in reactors using fast neutrons).• Reflector (to reduce leakage and critical size).• Containment vessel (to prevent leakage of waste).• Shielding (for neutrons and ’s).• Coolant.• Control system.• Emergency systems (to prevent runaway during failure).

Core


Chapter 4 in Lamarsh

6

Fission Reactors

Types 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?

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Fission Reactors

What 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 reactor.How???


8

Fission ReactorsWhat assembly?What assembly?• 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?

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


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

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

• ….. 9

More on Moderators


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More on Moderators


Moderating ratio

Calculate both moderating power and ratio for water, heavy water, graphite, polyethylene and boronboron.Tabulate your results and comment.

a

s

Moderating power

HW 12HW 12 LiBnB 7*1110

B-10

1/v region

Good absorber,

Good absorber, bad moderator.

bad moderator.

Never consider this only!

For a compound?

1010BB


11

HW 12 HW 12 (continued)(continued)

More on Moderators

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


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More on Moderators

1

1ln

2

)1(1ln

2

\

A

A

A

A

E

Eu

av

nEEn lnln \

)/ln( \

nEEnRecall

After n collisions

)/ln( thf EE

n

Total mean free path = n s

Is it random walk or there is a preferred direction???

creation

absorption

After one collision

f

th


13

More on Moderators

Assumptions:Assumptions:1.1.Elastic scattering. Elastic scattering. E2.2.Target nucleus at rest. Target nucleus at rest. E3.3.Spherical symmetry in Spherical symmetry in CM.CM.

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

show that

Obviously


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

EEA

AE

2

min\

1

1

EEEPdEEEP

E

E )1(

1)(1)( \\\

EE )1(21\

otherwise

EEEE

E

dE

dEE

ss

s

0)1(

)()(

|

|\

After one collision.


15

More on Moderators

2

222

2

2\

)1(

sincos

cos)1()1(2

1

)1(

cos21

A

A

A

AA

E

E CMCM

(Re)-verify

For doing so, you need to verify and use

CM

CM

AA

A

cos21

cos1cos

2

HW 13 (or 6HW 13 (or 6\\) ) continued…continued…• Scattering Kernel.Scattering Kernel.• Slowing down density.Slowing down density.• Migration length.Migration length.• Fermi age and continuous fermi model.Fermi age and continuous fermi model.


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More on Moderators

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

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

Show that A3

2)(cos


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More on Moderators

Spherically symmetric in CM

Show that

)(4

1)( E

d

ds

CMsCM

s

CM

CMs

s A

AAE

cos1

)1cos2(

4

)()(

1

2312

Try to sketch.

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

• 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

• Angular neutron distribution.distribution.


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More on Moderators

Moderator-to-fuel ratio Moderator-to-fuel ratio Nm/Nu.• Ratio p a of the moderator f (leakage ).• Ratio slowing down time p f (leakage ).

• T ratio (why).• Other factors also change.• Temperature coefficient of reactivity.• Moderator temperature coefficient of reactivity.

Self regulation.


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One-Speed Interactions• Particular general.Recall:• Neutrons don’t have a chance to interact with each other (review test!) Simultaneous beams, different intensities, same energy:

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

Rt = t nv = t


20

drn ),(

r

d

Neutrons per cm3

at r whose velocity vector lies within d about .

4

),()( drnrn

• Same argument as before vdrnrdI ),(),(

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

),()(),(

4

rrrnrvdrnvrrdFrFrR

rdIrrdF

ttt

t

One-Speed Interactions

drnvr ),()(4

where


21

Multiple Energy Interactions

dEdErn ),,(

Neutrons per cm3 at r with energy interval (E, E+dE) whose velocity vector lies within d about .

• Generalize to include energy

4

),,(),( dEdErndEErn

0 4

),,()(

dEdErnrn

dEErErdEEvErnErdEErR tt ),(),()(),(),(),(

0

),(),()( dEErErrR t

Thus knowing the material properties t and the neutron flux as a function of space and energy, we can calculate the interaction rate throughout the reactor.

Scalar


22

Neutron Current

• Similarly and so on …

• Redefine as

0

),()()( dEErErR SS

Scalar

vdrnrdI ),(),( dvrnrId

),(),(

drnvr ),()(4

drnvJ ),(

4

Neutron current densityNeutron current density

J• From larger flux to smaller flux!

• Neutrons are not pushed!• More scattering in one direction than in the other.

xJxJ ˆ


23

4

cos),(ˆ dvrnJxJ xx

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

In general: nJnJ ˆ

Equation of ContinuityEquation of Continuity

A

a dAntrJdtrrdtrSdtrnt

ˆ),(),()(),(),(

Rate of change in neutron density

Production rate

Absorption rate

“Leakage in/out” rate

Volume Source distribution

function

Surface area

bounding

Normal to A

Equation of Continuity

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Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh). 1 Controlled Fission Note that is greater than 2 at thermal energies and almost 3 at