Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh). 1 k = fp, Fast from thermal,...

Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).

1

k = fp, leaknoneff Pfk

• Fast from thermal, as defined in HW 11.

• Fast from fast, .• Thermal from fast, p.• Thermal available for fuel

Thinking QUIZThinking QUIZ• For each thermal neutron absorbed, how many fast neutrons are produced? Will need this when discuss two-group diffusion.

i

fa

ii )()(1

..mod

poisona

rodsa

eratora

clada

fuela

fuelaf

Controlled Fission

2

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

x 0.9x 0.9Resonance Resonance

escape escape probability ”p”probability ”p”

x 0.9x 0.9Thermal Thermal

utilization utilization factor “f”factor “f”

x

What is:• Migration length?• Critical size?How does the geometry affect the reproduction factor?

Neutron reproduction

factork eff = 1.000


Neutron Life Cycle

3Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).

Neutron Life Cycle

Why should we worry about these?

How?f


4

k = fp(1-lfast)(1-lthermal)

Controlled Fission

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

Not fixed…!


5

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.

Under-moderation and

over-moderation.

6

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)(


7

• 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.

8

Fission Reactors

Essential elements:Essential elements:• Fuel [fissile (or fissionable) 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

9

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?

10

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???


11

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 (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?


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).

• ….. 12

More on Moderators


13

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


14

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.


15

More on Moderators

1

1ln

2

)1(1ln

2

\

A

A

A

A

E

Eu

av

nEEn lnln \

)/ln( \

nEEn Recall

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


16

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:

Flat-top probability

Obviously


17

More on Moderators

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.


18

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\\))• 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.


19

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


20

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.


21

More on Moderators

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

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

Self regulation.


22

One-Speed Interactions• Particular general.Recall:• Neutrons don’t have a chance to interact with each other (BAU 2007 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

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Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh). 1 k = fp, Fast from thermal,...

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