Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh). 1 HW 14 More on Moderators Calculate...

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

1

HW 14HW 14

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.

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

2

More on Moderators

u u

n n0 01 2 3 4 5 6 7 1 2 3

slowing down in large massnumber material

slowing down in hydrogeneousmaterial

continuous slowing-down model

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

3

More on Moderators

1

1ln

2

)1(1ln

2

\

A

A

A

A

E

Eu

av

ContinuousContinuous slowing down model or Fermi model. slowing down model or Fermi model.

• The scattering of neutrons is isotropic in the CM system, thus is independent on neutron energy. also represents the average increase in lethargy per collision, i.e. after n collisions the neutron lethargy will be increased by n units.

• Materials of low mass number is large Fermi model is inapplicable.

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

4

More on Moderators

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

• Water moderated reactors, for example, should be under moderated.• T ratio (why).

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

5

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

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

6

drn ),(

r

d

Neutrons per cm3 at

r whose velocity vector lies within d about .

4

),()( drnrn

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

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

),(),(

4

rrnvdrnvrdFrFrR

rdIrdF

ttt

t

One-Speed Interactions

drnvr ),()(4

where

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

7

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

dEErEdEEvErnEdEErR tt ),()()(),()(),(

0

),()()( dEErErR 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

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

8

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

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

9

4

cos),(ˆ dvrnJxJ xx

Net 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

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

10

Using Gauss’ Divergence Theorem S V

rdBAdB 3

dtrJdAntrJA

),(ˆ),(

A

a dAntrJdtrrdtrSdtrnt

ˆ),(),()(),(),(

),(),()(),(),(1

trJtrrtrStrtv a

Equation of Continuity

Equation of ContinuityEquation of Continuity

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

11

For steady state operation

0)()()()( rSrrrJ a

For non-spacial dependence

)()()( ttStnt a

Delayed sources?

Equation of Continuity

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

12

Fick’s LawAssumptions:1.The medium is infinite.2.The medium is uniform 3.There are no neutron sources in the medium.4.Scattering is isotropic in the lab. coordinate system.5.The neutron flux is a slowly varying function of position.6.The neutron flux is not a function of time.

)(rnot

Restrictive! Applicability??

Restrictive! Applicability??

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

13

Fick’s LawCurrent Jx

x

Con

cent

ratio

n C

dC/dx

x

(x)

High flux

More collisions

Low flux

Less collisions

Negative Flux GradientCurrent Jx

• Diffusion: random walk of an ensemble of particles from region of high “concentration” to region of small “concentration”.• Flow is proportional to the negative gradient of the “concentration”.

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

14

x

y

z

rdAz

Fick’s Law

der

dAr rz

st

24

cos)(

Number of neutrons scatteredscattered per second from d at rr and going through dAz

Slowly varying)(rnot ss

Isotropic

Removed(assuming no

buildup)

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

15

Fick’s Law

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

16

Fick’s Law

2

0

2/

0 0

sincos)(4 r

rzszz ddrder

dAdAJ t

HW 15HW 15

023

zJJJ

t

szzz

?

zz dAJ

and show that

and generalize23 t

sDDJ

Diffusion Diffusion coefficientcoefficient

Fick’s law

Fick’s law

The current density is proportional to the negative of the gradient of the neutron flux.

s

D

3

1